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TECHNICAL GUIDE OPTICAL COATINGS & MATERIALS marketplace.idexop.com A5 MATERIAL PROPERTIES A57 OPTICAL SPECIFICATIONS A83 FUNDAMENTAL OPTICS A93 GAUSSIAN BEAM OPTICS A157 MACHINE VISION GUIDE A173 LASER GUIDE A197 A3 Technical Guide A4 1-505-298-2550 Optical Coatings & Materials OPTICAL COATINGS & MATERIALS A6 THE REFLECTION OF LIGHT A7 SINGLE-LAYER ANTIREFLECTION COATINGS Material Properties OPTICAL COATINGS A12 MULTILAYER ANTIREFLECTION COATINGS A17 THIN-FILM PRODUCTION A25 CVI LASER OPTICS ANTIREFLECTION COATINGS A29 SINGLE-LAYER MGF2 COATINGS A37 Fundamental Optics A20 Optical Specifications HIGH-REFLECTION COATINGS METALLIC HIGH-REFLECTION COATINGS A38 A43 LASER-LINE MAX-R™ COATINGS A45 ULTRAFAST COATING (TLMB) A46 OPTICAL FILTER COATINGS A47 NEUTRAL DENSITY FILTERS A53 LASER-INDUCED DAMAGE A55 OEM AND SPECIAL COATINGS A56 Gaussian Beam Optics MAXBRITE™ COATINGS (MAXB) Machine Vision Guide Laser Guide marketplace.idexop.com A5 OPTICAL COATINGS OPTICAL COATINGS Optical Coatings and Materials The vast majority of optical components are made of various types of glass, and most are coated with thin layers of special materials. The purpose of these coatings is to modify the reflection and transmission properties of the components’ surfaces. Whenever light passes from one medium into a medium with different optical properties (most notably refractive index), part of the light is reflected and part of the light is transmitted. The intensity ratio of the reflected and transmitted light is primarily a function of the change in refractive index between the two media, and the angle of incidence of the light at the interface. For most uncoated optical glasses, 4-5% of incident light is reflected at each surface. Consequently, for designs using more than a few components, transmitted light losses can be significant. More important are the corresponding losses in image contrast and lens resolution caused by reflected ghost images (usually defocused) superimposed on the desired image. Applications generally require that the reflected portion of incident light approach zero for transmitting optics (lenses), 100% for reflective optics (mirrors), or some fixed intermediate value for partial reflectors (beamsplitters). The only suitable applications for uncoated optics are those where only a few optical components are in the optical path, and significant transmission inefficiencies can be tolerated. In principle, the surface of any optical element can be coated with thin layers of various materials (called thin films) in order to achieve the desired reflection/ transmission ratio. With the exception of simple metallic coatings, this ratio depends on the nature of the material from which the optic is fabricated, the wavelength of the incident light, and the angle of incidence of the light (measured from the normal). There is also polarization dependence to the reflection/transmission ratio when the angle of incidence is not normal to the surface. A multilayer coating, sometimes made up of more than 100 individual fractional-wavelength layers, may be used to optimize the reflection/transmission ratio for a specific wavelength and angle of incidence or to optimize it over a specific range of conditions. can have a long life. In fact, the surfaces of many highindex glasses that are soft or prone to staining can be protected with a durable antireflection coating. Several factors influence coating durability. Coating designs should be optimized to minimize thickness and reduce mechanical stresses that may distort the optical surfaces or cause detrimental polarization effects. Resilient material must used. Great care must be taken in coating fabrication to produce high-quality, nongranular, even layers. CVI Laser Optics is a leading supplier of precision optical components and multielement optical systems. We have achieved our market-leading position by having an extensive knowledge of the physics of thin-film coatings and without the advanced production systems and methods required to apply such coatings in production. With state-of-the-art coating facilities CVI Laser Optics not only is able to coat large volumes of standard catalog and custom optical components, but also is able to develop and evaluate advanced new coatings for customers’ special requirements. Although our optical-coating engineers and technicians have many years of experience in designing and fabricating various types of dielectric and metallic coatings, the science of thin films continues to evolve. CVI Laser Optics continually monitors and incorporates new technology and equipment to be able to offer our customers the most advanced coatings available. The CVI Laser Optics range of coatings currently includes antireflection coatings, metallic reflectors, alldielectric reflectors, hybrid reflectors, partial reflectors (beamsplitters), and filters for monochromatic, dichroic, and broadband applications.With new and expanded coating capabilities, including the new deep-UVoptimized Leybold SYRUSpro 1100™, CVI Laser Optics offers the same high-quality coatings to customers who wish to supply their own substrates. As with any special or OEM order, please contact CVI Laser Optics to discuss your requirements with one of our qualified applications engineers. Today’s multilayer dielectric coatings are remarkably hard and durable. With proper care and handling, they A6 Optical Coatings 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials THE REFLECTION OF LIGHT REFLECTIONS AT UNCOATED SURFACES light, as a proportion of the incident light, is given by INTENSITY When a beam of light is incident on a plane surface at normal incidence, the relative amplitude of the reflected UHIOHFWHG UD\ L L DLU U U n W UHIUDFWHG UD\ VLQ VLQ W L n DLU n JODVV Figure 1.1 Reflection and refraction at a simple air to glass interface INCIDENCE ANGLE The intensity of reflected and transmitted beams at a surface is also a function of the angle of incidence. Because of refraction effects, it is necessary to differentiate between external reflections, where the incident beam originates in the medium with a lower refractive index (e.g., air in the case of an air to glass or air to water interface), and external reflection, where the beam originates in the medium with a higher refractive index (e.g., glass in the case of a glass to air interface, or flint glass in the case of a flint to crown-glass interface), and to consider them separately. EXTERNAL REFLECTION AT A DIELECTRIC BOUNDARY Fresnel’s laws of reflection precisely describe amplitude and phase relationships between reflected and incident light at a boundary between two dielectric media. It is convenient to think of the incident radiation as the superposition of two plane-polarized beams, one with its electric field parallel to the plane of incidence (p-polarized), and the other with its electric field perpendicular to the plane of incidence (s-polarized). Fresnel’s laws can be summarized in the following two equations, which give the reflectance of the s- and p-polarized components: Machine Vision Guide JODVVn The greater the disparity between the two refractive indexes, the greater the reflection. For an air to glass interface, with glass having a refractive index of 1.5, the intensity of the reflected light will be 4% of the incident light. For an optical system containing ten such surfaces, the transmitted beam will be attenuated to approximately 66% of the incident beam due to reflection losses alone, emphasizing the importance of antireflection coatings to system performance. Gaussian Beam Optics LQFLGHQW UD\ where p is the ratio of the refractive indexes of the two materials (n1/n2). Intensity is the square of this expression. Fundamental Optics At a simple interface between two dielectric materials, the amplitude of reflected light is a function of the ratio of the refractive index of the two materials, the polarization of the incident light, and the angle of incidence. (1.1) Optical Specifications The law of reflection states that the angle of incidence (θ1) equals the angle of reflection (θr). This is illustrated in figure 1.1, which shows reflection of a light ray at a simple air to glass interface. The incident and reflected rays make an equal angle with respect to the axis perpendicular to the interface between the two media. (1 − p ) (1 + p ) Material Properties Whenever light is incident on the boundary between two media, some light is reflected and some is transmitted into the second medium, undergoing refraction. Several physical laws govern the direction, phase, and relative amplitude of the reflected light. For our purposes, it is necessary to consider only polished optical surfaces. Diffuse reflections from rough surfaces are not considered in this discussion. Laser Guide marketplace.idexop.com The Reflection of Light A7 OPTICAL COATINGS Optical Coatings and Materials This angle, called Brewster’s angle, is the angle at which the reflected light is completely polarized. This situation occurs when the reflected and refracted rays are perpendicular to each other (θ1=θ2 = 90º), as shown in figure 1.3. 3HUFHQW5HIOHFWDQFH This leads to the expression for Brewster’s angle, θB: θ1 = θB= arctan (n2 / n1 ) (1.5) S $QGOHRI,QFLGHQFHLQ'HJUHHV Figure 1.2 External reflection at a glass surface (n = 1.52) showing s- and p-polarized components Under these conditions, electric dipole oscillations of the p-component will be along the direction of propagation and therefore cannot contribute to the reflected ray. At Brewster’s angle, reflectance of the s-component is about 15%. 2 sin (v1 − v 2 ) rs = 2 (1.2) sin − vv 22 )) sin ((vv11 + rs = (v + v ) 2 sin tan (v11 − v22 ) rp = . tan ((vv1 + v ) 2 tan 1 − v 22 ) rp = . (1.3) tan (v1 + v 2 ) In the limit of normal incidence in air, Fresnel’s laws reduce to the following simple equation: SSRODUL]HG DEVHQWSSRODUL]HG UHIOHFWHGUD\ QRUPDO LQFLGHQWUD\ v1 v1 DLURUYDFXXP LQGH[ Q 1 LVRWURSLFGLHOHFWULFVROLG LQGH[ Q 2 LV D[ ROH RQ S L WL G HF GLU UHIUDFWHGUD\ GLSROHUDGLDWLRQ SDWWHUQVLQ2v v2 SSRODUL]HG 2 n − 1 r= . n + 1 UHIUDFWHGUD\ (1.4) It can easily be seen that, for a refractive index of 1.52 (crown glass), this gives a reflectance of 4%. This important result reaffirms that, in general, 4% of all illumination incident normal to an air-glass surface will be reflected. The variation of reflectance with angle of incidence for both the s- and p-polarized components, plotted using the formulas above, is shown in figure 1.2. It can be seen that the reflectance remains close to 4% up to about 25º angle of incidence, and that it rises rapidly to nearly 100% at grazing incidence. In addition, note that the p-component vanishes at 56°. Figure 1.3 Brewster’s angle: at this angle, the p-polarized component is completely absent in the reflected ray D vF FULWLFDODQJOH E Q DLU F Q JODVV vF G G F F E D D E Figure 1.4 Internal reflection at a glass surface (n = 1.52) showing s- and p-polarized components A8 The Reflection of Light 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials &ULWLFDODQJOH US Quantum theory shows us that light has wave/particle duality. In most classical optics experiments, the wave properties generally are most important. With the exception of certain laser systems and electro-optic devices, the transmission properties of light through an optical system can be well predicted and rationalized by wave theory. 7RWDOUHIOHFWLRQ ,QWHUQDO$QJOHRI,QFLGHQFHLQ'HJUHHV INTERNAL REFLECTION AT DIELECTRIC BOUNDARY Light waves that are exactly out of phase with one another (by 180º or π radians) undergo destructive interference, and, as shown in the figure, their amplitudes cancel. In intermediate cases, total amplitude is given by the vector resultant, and intensity is given by the square of amplitude. Fundamental Optics For light incident from a higher to a lower refractive index medium, we can apply the results of Fresnel’s laws in exactly the same way. The angle in the high-index material at which polarization occurs is smaller by the ratio of the refractive indices in accordance with Snell’s law. The internal polarizing angle is 33° 21' for a refractive index of 1.52, corresponding to the Brewster angle (56° 39') in the external medium, as shown in figure 1.4. One consequence of the wave properties of light is that waves exhibit interference effects. Light waves that are in phase with one another undergo constructive interference, as shown in figure 1.6. Optical Specifications Figure 1.5 Critical angle: at this angle, the emerging ray is at grazing incidence Material Properties 3HUFHQW5HIOHFWDQFH INTERFERENCE %UHZVWHU V DQJOH UV &RQVWUXFWLYHLQWHUIHUHQFH ZDYH , $PSOLWXGH ZDYH,, Gaussian Beam Optics The angle at which the emerging refracted ray is at grazing incidence is called the critical angle (see figure 1.5). For an external medium of air or vacuum (n = 1), the critical angle is given by UHVXOWDQW ZDYH (1.6) 7LPH GHVWUXFWLYHLQWHUIHUHQFH and depends on the refractive index nλ, which is a function of wavelength. For all angles of incidence higher than the critical angle, total internal reflection occurs. ZDYH , Machine Vision Guide PHASE CHANGES ON REFLECTION There is another, more subtle difference between internal and external reflections. During external reflection, light waves undergo a 180º phase shift. No such phase shift occurs for internal reflection (except in total internal reflection). This is one of the important principles on which multilayer films operate. $PSOLWXGH ZDYH,, ]HURDPSOLWXGH UHVXOWDQW ZDYH 7LPH Figure 1.6 A simple representation of constructive and destructive wave interference Laser Guide marketplace.idexop.com The Reflection of Ligh A9 OPTICAL COATINGS Optical Coatings and Materials Various experiments and instruments demonstrate light interference phenomena. Some interference effects are possible only with coherent sources (i.e., lasers), but many are produced by incoherent light. Three of the best-known demonstrations of visible light interference are Young’s slits experiment, Newton’s rings, and the Fabry-Perot interferometer. These are described in most elementary optics and physics texts. In all of these demonstrations, light from a source is split in some way to produce two sets of wavefronts. These wavefronts are recombined with a variable path difference between them. Whenever the path difference is an integral number of half wavelengths, and the wavefronts are of equal intensity, the wavefronts cancel by destructive interference (i.e., an intensity minimum is produced). An intensity minimum is still produced if the interfering wavefronts are of differing amplitude, the result is just non-zero. When the path difference is an integral number of wavelengths, the wavefront intensities sum by constructive interference, and an intensity maximum is produced. THIN-FILM INTERFERENCE Thin-film coatings may also rely on the principles of interference. Thin films are dielectric or metallic materials whose thickness is comparable to, or less than, the wavelength of light. CVI Laser Optics offers a variety of single- and multilayer antireflection and high-reflection coatings When a beam of light is incident on a thin film, some of the light will be reflected at the front surface, and some light will be reflected at the rear surface, as shown in figure 1.7. The remainder will be transmitted. At this stage, we shall ignore multiple reflections and material absorption effects. element of interest). In other words, the optical thickness of a piece of material is the thickness of that material corrected for the apparent change of wavelength passing through it. The optical thickness is given by top = tn, where t is the physical thickness, and n is the ratio of the speed of light in the material to the speed of light in vacuum: n= cvacuum . cmedium (1.7) To a very good approximation, n is the refractive index of the material. Returning to the thin film at normal incidence, the phase difference between the external and internal reflected wavefronts is given by (top/λ)x2π, where λ is the wavelength of light. Clearly, if the wavelength of the incident light and the thickness of the film are such that a phase difference of π exists between reflections, the reflected wavefronts interfere destructively and overall reflected intensity is a minimum. If the two interfering reflections are of equal amplitude, the amplitude (and hence intensity) minimum will be zero. In the absence of absorption or scatter, the principle of conservation of energy indicates that all “lost” reflected intensity will appear as enhanced intensity in the transmitted beam. The sum of the reflected and transmitted beam intensities is always equal to the incident intensity. Conversely, when the total phase shift between two reflected wavefronts is equal to zero (or multiples of 2π), then the reflected intensity will be a maximum, and the transmitted beam will be reduced accordingly. The two reflected wavefronts can interfere with each other. The degree of interference will depend on the optical thickness of the material and the wavelength of the incident light (see figure 1.8). The optical thickness of an element is defined as the equivalent vacuum thickness (i.e., the distance that light would travel in vacuum in the same amount of time as it takes to traverse the optical A10 The Reflection of Light 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials Ȝ n GHQVH PHGLXP n Ȝ Ȝ KRPRJHQHRXV WKLQ ILOP IURQWDQGEDFN VXUIDFHUHIOHFWLRQV n t Ȝn Ȝ tRS tn Ȝ DLU Q n Material Properties DLU n a t WUDQVPLWWHGOLJKW t RS RSWLFDOWKLFNQHVV UHIUDFWLYH LQGH[ Q RSWLFDOWKLFNQHVV RIILOPW RS QW Optical Specifications W SK\VLFDO WKLFNQHVV Figure 1.7 Front and back surface reflections for a thin film at near-normal incidence Figure 1.8 A schematic diagram showing the effects of lower light velocity in a dense medium (in this example, the velocity of light is halved in the dense medium n = n/n0, and the optical thickness of the medium is 2 x the real thickness) Fundamental Optics Gaussian Beam Optics Machine Vision Guide Laser Guide marketplace.idexop.com The Reflection of Light A11 OPTICAL COATINGS SINGLE-LAYER ANTIREFLECTION COATINGS Optical Coatings and Materials The basic principles of single-layer antireflection coatings should now be clear. Ignoring scattering and absorption,transmitted energy = incident energy– reflected energy. If the substrate (glass, quartz, etc.) is coated with a thin layer (film) of material, and if the reflections from the air/film interface and from the film/substrate interface are of equal magnitude and 180º (π radians) out of phase, then the reflected waves will cancel each other out by destructive interference, and the intensity of the transmitted beam will approach the intensity of the incident beam. FILM THICKNESS To eliminate reflections at a specific wavelength, the optical thickness of a single-layer antireflection film must be an odd number of quarter wavelengths. This requirement is illustrated in figure 1.9. The reflections at both the air/film and film/substrate interfaces are “internal” (low index to high index) and the phase changes caused by the reflections themselves cancel out. Consequently, the net phase difference between the two reflected beams is determined solely by their optical path difference 2tnc, where t is the physical thickness and nc is the refractive index of the coating layer. For a 180º phase shift, 2tnc =Nλ/2 and tnc =Nλ/4 where N=1, 3, 5 . . . DLU n WKLQ ILOP n JODVV n ,I tRS WKHRSWLFDO WKLFNQHVVnt l WKHQUHIOHFWLRQV LQWHUIHUHGHVWUXFWLYHO \ ZDYHOHQJWK l UHVXOWDQWUHIOHFWHG LQWHQVLW\ ]HUR t SK\VLFDO WKLFNQHVV Figure 1.9 Schematic representation of a single-layer antireflection coating A12 Single-Layer Antireflection Coatings Single-layer antireflection coatings are generally deposited with a thickness of λ/4, where λ is the desired wavelength for peak performance. The phase shift is 180º (π radians), and the reflections are in a condition of exact destructive interference. REFRACTIVE INDEX The intensity of the reflected beam from a single surface, at normal incidence, is given by 2 1 − p 1 + p × the incident intensity (1.8) where p is the ratio of the refractive indexes of the two materials at the interface. For the two reflected beams to be equal in intensity, it is necessary that p, the refractive index ratio, be the same at both the interfaces nair nfilm . = nfilm nsubstrate (1.9) Since the refractive index of air is 1.0, the thin antireflection film ideally should have a refractive index of nfilm = nsubstrate (1.10) . Optical glasses typically have refractive indexes between 1.5 and 1.75. Unfortunately, there is no ideal material that can be deposited in durable thin layers with a low enough refractive index to satisfy this requirement exactly (n = 1.23 for the optimal antireflection coating on crown glass). However, magnesium fluoride (MgF2) is a good compromise because it forms high quality, stable films and has a reasonably low refractive index (1.38) and low absorbance at a wavelength of 550 nm. Magnesium fluoride is probably the most widely used thin-film material for optical coatings. Although its performance is not outstanding for all applications, it represents a significant improvement over an uncoated surface. At normal incidence, typical crown glass surfaces 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials v DQJOHRILQFLGHQFH v 0J ) ZDYHOHQJWKRSWLFDOWKLFNQHVV DWQPn VLQJOHOD\HU 0J ) $QJOHRI,QFLGHQFHLQ$LU,Q'HJUHHV Fundamental Optics XQFRDWHGJODVV Optical Specifications JODVV Material Properties 3HUFHQW5HIOHFWDQFHDWQP VXEVFULSWVR V UHIOHFWDQFHIRUsSRODUL]DWLRQ R DY UHIOHFWDQFHIRUDYHUDJHSRODUL]DWLRQ R S UHIOHFWDQFHIRUpSRODUL]DWLRQ R V ∞LQFLGHQFHFRDWLQJ Gaussian Beam Optics 3HUFHQW5HIOHFWDQFH DW ∞LQFLGHQFH R V QRUPDOLQFLGHQFHFRDWLQJDW ∞ R DY QRUPDOLQFLGHQFHFRDWLQJDW ∞ R DY ∞LQFLGHQFHFRDWLQJ R S QRUPDOLQFLGHQFHFRDWLQJDW ∞ Machine Vision Guide R S ∞LQFLGHQFHFRDWLQJ :DYHOHQJWKQP Figure 1.10 MgF2 performance at 45° incidence on BK7 for a normal-incidence coating design and for a coating designed for 45° incidence (design wavelength: 550 nm) Laser Guide marketplace.idexop.com Single-Layer Antireflection Coatings A13 OPTICAL COATINGS Optical Coatings and Materials reflect from 4 to 5% of visible light. A high-quality MgF2 coating can reduce this value to 1.5%. For many applications this improvement is sufficient, and higher performance multilayer coatings are not necessary. Single-layer quarter-wavelength coatings work extremely well over a wide range of wavelengths and angles of incidence even though the theoretical target of zeropercent reflectance applies only at normal incidence, and then only if the refractive index of the coating material is exactly the geometric mean of the indexes of the substrate and of air. In actual practice, the single layer quarter-wave MgF2 coating makes its most significant contribution by improving the transmission of optical elements with steep surfaces where most rays are incident at large angles (see figure 1.10). ANGLE OF INCIDENCE The optical path difference between the front and rear surface reflections of any thin-film layer is a function of angle. As the angle of incidence increases from zero (normal incidence), the optical path difference is increased. This change in optical path difference results in a change of phase difference between the two interfering reflections, which, in turn, causes a change in reflection. WAVELENGTH DEPENDENCE With any thin film, reflectance and transmission depend on the wavelength of the incident light for two reasons. First, since each thin-film layer is carefully formed at a thickness of a quarter of the design wavelength for optimal single-wavelength performance, the coating is suboptimal at any other wavelength. Second, the indexes of refraction of the coating and substrate change as a function of wavelength (i.e., dispersion). Most up-to-date thin-film coating design optimization programs, such as those used by CVI Laser Optics, include the capability to account for material dispersion when calculating thin-film performance and monitoring the thin film deposition process. A14 Single-Layer Antireflection Coatings COATING FORMULAS Because of the practical importance and wide usage of single-layer coatings, especially at oblique (non-normal) incidence angles, it is valuable to have formulas from which coating reflectance curves can be calculated as functions of wavelength, angle of incidence, and polarization. COATING DISPERSION FORMULA The first step in evaluating the performance of a singlelayer antireflection coating is to calculate (or measure) the refractive index of the film and substrate at the primary or center wavelength of interest. In our example, we will assume that the thin film may be considered to be homogeneous. The refractive index of crystalline MgF2 is related to wavelength by the Lorentz-Lorenz formulas (3.5821) (10 −3 ) (1.11) (l − 0.14925) (3.5821) (10 −3 ) no = 1.36957 + (3.7415 0 −3 ) ) (l −)0(.114925 ne = 1.381 + (l − 0.14947 ) (3.7415) (10 −3 ) ne = 1.381 + (1.12) (l − 0.14947 ) no = 1.36957 + for the ordinary and extraordinary rays, respectively, where λ is the wavelength in micrometers. The index for the amorphous phase is the average of the crystalline indexes: n = n(l ) = 1 ( no + ne ). 2 (1.13) The value 1.38 is the universally accepted amorphous film index for MgF2 at a wavelength of 550 nm, assuming a thin-film packing density of 100%. Real films tend to be slightly porous, reducing the net or actual refractive index from the theoretical value. Because it is a complex function of the manufacturing process, packing density itself varies slightly from batch to batch. Air and water vapor can also settle into the film and affect its refractive index. For CVI Laser Optics MgF2 coatings, our tightly controlled procedures result in packing densities that yield refractive indexes that are within three percent of the theoretical value. 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials COATED SURFACE REFLECTANCE AT NORMAL INCIDENCE For a thin-film coating having an optical thickness of one-quarter wavelength for wavelength λ, let na denote the refractive index of the external medium at that wavelength (1.0 for air or vacuum) and let nf and ns, respectively, denote the film and substrate indexes, as shown in figure 1.11. For normal incidence at wavelength λ, the single-pass reflectance of the coated surface can be shown to be 1%. /D6)1 6) (1.14) 5HIUDFWLYH,QGH[ nJ Figure 1.12 Reflectance at surface of substrate with index ng when coated with a quarter wavelength of magnesium fluoride (index n=1.38) The extremum is a minimum if n2 is less than n3 and a maximum if n2 exceeds n3. The same formulas apply in either case. Corresponding to the angle of incidence in the external media θ1d is the angle of refraction within the thin film: n (l ) sin v1d . (1.15) v 2d = arcsin 1 d n2 (l d ) Assume that the coating exhibits a reflectance extremum of the first order for some wavelength λd and angle of incidence θ1d in the external medium. The coating is completely specified when θ1d and λd are known. As θ1 is reduced from θ1d to zero, the reflectance extremum shifts in wavelength from λd to λn, where the subscript n denotes normal incidence. The wavelength is given by the equation ln = 0J ) DQWLUHIOHFWLRQ FRDWLQJ LQGH[ n I ZDYHOHQJWK l VXEVWUDWH LQGH[ n V n2 (l d ) ld cos v 2d . (1.16) Machine Vision Guide DLURUYDFXXP LQGH[ n D n2 (l n ) Gaussian Beam Optics At oblique incidence, the situation is more complex. Let n1, n2, and n3, respectively, represent the wavelengthdependent refractive indexes of the external medium (air or vacuum), coating film, and substrate as shown in figure 1.13. Fundamental Optics COATED SURFACE REFLECTANCE AT OBLIQUE INCIDENCE 2 regardless of the state of polarization of the incident radiation. The reflectance is plotted in figure 1.12 for various substrate types (various indexes of refraction). IXVHGVLOLFD Optical Specifications n n − nf 2 R= a s na ns + nf 2 Material Properties 3HUFHQW5HIOHFWDQFH3HU6XUIDFH Corresponding to the arbitrary angle of incidence θ1 and arbitrary wavelength λ are angles of refraction in the coating and substrate, given by n (l ) sin v1 v 2 = arcsin 1 (1.17) n2 (l ) and Figure 1.11 Reflectance at normal incidence Single-Layer Antireflection Coatings Laser Guide marketplace.idexop.com n (l ) sin v1 . v3 = arcsin 1 n3 (l ) A15 OPTICAL COATINGS Optical Coatings and Materials DLURUYDFXXPLQGH[ n ZD YHO HQ JWK l RSWLFDOSDWKGLIIHUHQFH v 0J )DQWLUHIOHFWLRQ FRDWLQJLQGH[ n JODVVRUVLOLFDVXEVWUDWH LQGH[ n nb±na The corresponding reflectance for the coated surface, accounting for both interfaces and the phase differences between the reflected waves, are given by a b b v Rp = h Rp = Rs = v Rs = n (l ) sin v1 v 2 = arcsin 1 n2 (l ) and 1 2+ r122pr2232p + 2r12pr23p cos( 2β ) r12s + r23s + 2r12s r23s cos( 2β ) 22 1r12+2 sr12 co 2β ) + 22rr12 +2 srr23 23ss + 12ssrr23 23ss cos( 2 2 os( 2β ) 1 + r12s r23s + 2r12s r23s co (1.23) (1.24) Where β (in radians) is the phase difference in the external medium between waves reflected from the first and second surfaces of the coating and (1.18) The following formulas depict the single-interface amplitude reflectance for both the p- and s-polarizations: n2 cos v1 − n1 cos v 2 n2 cos v1 + n1 cos v 2 (1.19) n cos v1 − n1 cos v 2 r12p = 2 cosvvv223 nn223 cos −− nnn112cos cos vv112 + cos r12 23p = cos v12 ++ nn12cos cosvv n23 cos n23 cos v12 −− nn12cos cosvv2233 r12 = (1.20) 23p cos cos ccos osvvv1122−++ cos nnn1233cos −nnnn2122cos cosvvvv2233 r12 = 23sp = nn13cos cosvv ++nn cos cosvv cosvv1122−−nn2222cos cosvv2233 nn cos r12 = 13 23sp = +nn22cos osvvvv12+ cosvv3 cos cos ccos nnn1123cos 23 cosvv223 . 12−−nn r12 23ss = cosvv12++nn23cos cosvv23 (1.21) n cos n1122cos cosvv12−−nn23cos cosvv23 r12 = . 23ss nn12cos + n cos v cos v122 + − nn233 cosvv233 2c r23s = n +n n2 ccos vv 2 − n33 cos vv33 . r23s = 2 cos v2 cos vv . (1.22) n2 cos v 2 + n3 cos 3 r12p = 2 22 + 2r12 r23pp cos( cos(22bβ) 1r12+pr12+2 prr23 23pp + 2r 12pp r23 os( 2β ) Figure 1.13 Reflectance at oblique incidence n (l ) sin v1 . v3 = arcsin 1 n3 (l ) r122p + r232p + 2r12pr23p cos( 2b ) b= 2p n2 (l ) h cos v 2 . (1.25) l The average reflectance is given by R= 1 (1.26) ( Rp + Rs ) . 2 By applying these formulas, reflectance curves can be calculated as functions of either wavelength λ or angle of incidence θ1. The subscript “12p,” for example, means that the formula gives the amplitude reflectance for the p-polarization at the interface between the first and second media. A16 Single-Layer Antireflection Coatings 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials MULTILAYER ANTIREFLECTION COATINGS THE QUARTER/QUARTER COATING where n0 is the refractive index of air (approximated as 1.0), n3 is the refractive index of the substrate material, and n1 and n2 are the refractive indices of the two film materials, as indicated in figure 1.14. If the substrate is crown glass with a refractive index of 1.52 and if the first layer is the lowest possible refractive index, 1.38 (MgF2), the refractive index of the highindex layer needs to be 1.70. Either beryllium oxide or magnesium oxide could be used for the inner layer, but both are soft materials and will not produce very durable coatings. Although it allows some freedom in the choice of coating materials and can give very low reflectance, the quarter/quarter coating is constrained in its design owing to the lack of materials with suitable refractive index and physical or durability properties. In principle, it is possible to deposit two materials simultaneously to achieve layers of almost any required refractive index, but such coatings are not very practical. As a consequence, thin-film engineers have developed multilayer and special two-layer antireflection coatings that allow the refractive index of each layer and, therefore, coating performance to be optimized. Gaussian Beam Optics Machine Vision Guide Multilayer coating performance is calculated in terms of relative amplitudes and phases, which are summed to give the overall (net) amplitude of the reflected beam. The overall amplitude is then squared to give the intensity. If one knows the reflected light intensity goal, how does one calculate the required refractive index of the inner layer? Several methodologies have been developed over the last 40 to 50 years to calculate thin-film coating properties and converge on optimum n12 n3 = n0 (1.27) n22 Fundamental Optics This coating is used as an alternative to the single-layer antireflection coating. It was developed because of the lack of available materials with the indexes of refraction needed to improve the performance of single-layer coatings. The basic problem associated with single-layer antireflection coatings is that the refractive index of the coating material is generally too high, resulting in too strong a reflection from the first surface which cannot be completely canceled through destructive interference with the weaker reflection from the substrate’s top or first surface. In a two-layer coating, the first reflection is canceled through destructive interference with two weaker out-of-phase reflections from underlying surfaces. A quarter/quarter coating consists of two layers, both of which have an optical thickness of a quarter wave at the wavelength of interest. The outer layer is made of a low-refractive-index material, and the inner layer is made of a high-refractive-index material (compared to the substrate). As illustrated in figure 1.14, the second and third reflections are both exactly 180º out of phase with the first reflection. When considering a two-layer quarter/quarter coating optimized for one wavelength at normal incidence, the required refractive indexes for minimum reflectivity can be calculated easily by using the following equation: Optical Specifications Two basic types of antireflection coating are worth examining in detail: the quarter/quarter coating and the multilayer broadband coating. designs. The field has been revolutionized in recent years through the availability of powerful PC’s and efficient application-specific thin-film-design software programs. Material Properties Previously, we discussed the basic equations of thin-film design and their application to a simple magnesium fluoride antireflection coating. It is also useful to understand the operation of multilayer coatings. While it is beyond the scope of this chapter to cover all aspects of modern multilayer thin-film design, it is hoped that this section will provide the reader with insight into thin films that will be useful when considering system designs and specifying cost-effective real-world optical coatings. TWO-LAYER COATINGS OF ARBITRARY THICKNESS Optical interference effects can be characterized as either constructive or destructive interference, where the phase shift between interfering wavefronts is 0º or 180º respectively. For two wavefronts to completely cancel each other, as in a single-layer antireflection coating, a phase shift of exactly 180º is required. Where three or more reflecting surfaces are involved, complete Laser Guide marketplace.idexop.com Multilayer Antireflection Coatings A17 OPTICAL COATINGS Optical Coatings and Materials a skewed V shape with a reflectance minimum at the design wavelength. TXDUWHUTXDUWHUDQWLUHIOHFWLRQFRDWLQJ $ % & DLUQ 0 ORZLQGH[OD\HUQ 1 KLJKLQGH[OD\HUQ VXEVWUDWHQ 3 V-coatings are very popular, economical coatings for near monochromatic applications, such as optical systems using nontunable laser radiation (e.g., helium neon lasers at 632.8 nm). BROADBAND ANTIREFLECTION COATINGS $PSOLWXGH ZDYHIURQW A ZDYHIURQW B ZDYHIURQW C UHVXOWDQW ZDYH 7LPH Figure 1.14 Interference in a typical quarter/quarter coating cancellation can be achieved by carefully choosing the relative phase and intensity of the interfering beams (i.e., optimizing the relative optical thicknesses). This is the basis of a two-layer antireflection coating, where the layers are adjusted to suit the refractive index of available materials, instead of vice versa. For a given combination of materials, there are usually two combinations of layer thicknesses that will give zero reflectance at the design wavelength. These two combinations are of different overall thickness. For any type of thin-film coating, the thinnest possible overall coating is used because it will have better mechanical properties (less stress). A thinner combination is also less wavelength sensitive. Two-layer antireflection coatings are the simplest of the so-called V-coatings. The term V-coating arises from the shape of the reflectance curve as a function of wavelength, as shown in figure 1.15, which resembles A18 Multilayer Antireflection Coatings Many optical systems (particularly imaging systems) use polychromatic (more than one wavelength) light. In order for the system to have a flat response over an extended spectral region, transmitting optics are coated with a dichroic broadband antireflection coating. The main technique used in designing antireflection coatings that are highly efficient at more than one wavelength is to use “absentee” layers within the coating. Additional techniques can be used for shaping the performance curves of high reflectance coatings and wavelengthselective filters, but these are not applicable to antireflection coatings. ABSENTEE LAYERS An absentee layer is a film of dielectric material that does not change the performance of the overall coating at one particular wavelength. Usually that particular wavelength is the wavelength for which the coating is being optimized. The absentee layer is designed to have an optical thickness of a half wave at that specific wavelength. The “extra” reflections cancel out at the two interfaces because no additional phase shifts are introduced. In theory, the performance of the coating is the same at that specific design wavelength whether or not the absentee layer is present. At other wavelengths, the absentee layer starts to have an effect for two reasons: the ratio between physical thickness of the layer and the wavelength of light changes with wavelength, and the dispersion of the coating material causes optical thickness to change with wavelength. These effects give the designer extra degrees of freedom not offered by simpler designs. The complex, computerized, multilayer antireflection coating design techniques used by CVI Laser Optics are 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials 5HIOHFWDQFH Material Properties Optical Specifications l 0 :DYHOHQJWK Figure 1.15 Characteristic performance curve of a V-coating Fundamental Optics based on the simple principles of interference and phase shifts described in the preceding text. Because of the properties of coherent interference, it is meaningless to consider individual layers in a multilayer coating. Each layer is influenced by the optical properties of the other layers in the multilayer stack. A complex series of matrix multiplications, in which each matrix corresponds to a single layer, is used to mathematically model the performance of multilayer thin-film coatings Gaussian Beam Optics There also are multiple reflections within each layer of a coating. In the previous discussions, only first-order or primary reflections were considered. This oversimplified approach is unable to predict accurately the true behavior of multilayer coatings. Second-, third-, and higher-order terms must be considered if real coating behavior is to be modeled accurately. Machine Vision Guide Laser Guide marketplace.idexop.com Multilayer Antireflection Coatings A19 OPTICAL COATINGS HIGH-REFLECTION COATINGS Optical Coatings and Materials High-reflection coatings can be applied to the outside of a component, such as a flat piece of glass, to produce a first-surface mirror. Alternately, they can be applied to an internal surface to produce a second-surface mirror, which is used to construct certain prisms. the coatings are effective for both s- and p-polarization components, and can be designed for a wide angle of incidence range. However, at angles that are significantly distant from the design angle, reflectance is markedly reduced. High-reflection coatings can be classified as either dielectric or metallic coatings. PERFORMANCE CURVE DIELECTRIC COATINGS High-reflectance dielectric coatings are based upon the same principles as dielectric antireflection coatings. Quarter-wave thicknesses of alternately high- and lowrefractive-index materials are applied to the substrate to form a dielectric multilayer stack, as shown in figure 1.16. By choosing materials of appropriate refractive indexes, the various reflected wavefronts can be made to interfere constructively to produce a highly efficient reflector. The peak reflectance value is dependent upon the ratio of the refractive indices of the two materials, as well as the number of layer pairs. Increasing either increases the reflectance. The width of the reflectance curve (as a function of wavelength) is also determined by the films’ refractive index ratio. The larger the ratio is, the wider the high-reflectance region will be. Over limited wavelength intervals, the reflectance of a dielectric coating easily can be made to exceed the highest reflectance of a metallic coating. Furthermore, The reflection versus wavelength performance curve of a single dielectric stack has the characteristic flat-topped, inverted-V shape shown in figure 1.17. Clearly, reflectance is a maximum at the wavelength for which both the high- and low-index layers of the multilayer are exactly one-quarter-wave thick. Outside the fairly narrow region of high reflectance, the reflectance slowly reduces toward zero in an oscillatory fashion. The width and height (i.e., peak reflectance) of the high-reflectance region are functions of the refractive-index ratio of the two materials used and the number of layers actually included in the stack. The peak reflectance can be increased by adding more layers, or by using materials with a higher refractive index ratio. Amplitude reflectivity at a single interface is given by (1 − p ) (1 + p ) (1 − p ) (1 + p ) where(1.28) where n p = H nL N −1 × nH2 , nS DLU where nS is the index of the substrate and nH and nL are the indices of the high- and low-index layers. N is the total number of layers in the stack. The width of the highreflectance part of the curve (versus wavelength) is also determined by the film index ratio. The higher the ratio is, the wider the high-reflectance region will be. VXEVWUDWH TXDUWHU ZDYHWKLFNQHVVRIKLJKLQGH[PDWHULDO TXDUWHU ZDYHWKLFNQHVVRIORZLQGH[PDWHULDO Figure 1.16 A simple quarter-wave stack A20 High-Reflection Coatings SCATTERING The main parameters used to describe the performance of a thin film are reflectance and transmittance plus absorptance, where applicable. Another less well-defined 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials BROADBAND COATINGS HIIHFWLYHEURDGEDQGKLJKUHIOHFWLRQFRDWLQJ LQFLGHQW ZDYHOHQJWK l Fundamental Optics 127(,IDWOHDVWRQHFRPSRQHQWLVWRWDOO\ UHIOHFWLYHWKHFRDWLQJZLOOQRWWUDQVPLW OLJKWDWWKDWZDYHOHQJWK Gaussian Beam Optics There is a subtle difference between multilayer antireflection coatings and multilayer high-reflection coatings, which allows the performance curves of the latter to be modified by using layer thicknesses designed for different wavelengths within a single coating. Consider a multilayer coating consisting of pairs, or stacks of layers, that are optimized for different wavelengths. At any given wavelength, providing at least one of the layers is highly reflective for that wavelength, the overall coating will be highly reflective at that wavelength. Whether the other components transmit or are partially reflective at that wavelength is immaterial. Transmission of light of that wavelength will be blocked by reflection of one of the layers. This can be summarized by an empirical rule. At any wavelength, the reflection of a multilayer coating consisting of several discrete components will be at least that of the most reflective component. Exceptions to this rule are coatings that have been designed to produce interference effects involving not just the surfaces within the two-layer or multilayer component stack, but also between the stacks themselves. Obvious examples are narrowband interference filters. Optical Specifications In contrast to antireflection coatings, the inherent shape of a high-reflectance coating can be modified in several different ways. The two most effective ways of modifying a performance curve are to use two or more stacks centered at slightly shifted design wavelengths or to finetune the layer thicknesses within a stack. On the other hand, in an antireflection coating, even if one of the stacks is exactly antireflective at a certain wavelength, the overall coating may still be quite reflective because of reflections by the other components (see figure 1.18). Material Properties parameter is scattering. This is hard to define because of the inherently granular properties of the materials used in the films. Granularity causes some of the incident light to be lost by diffraction effects. Often it is scattering, not mechanical stress and weakness in the coating, that limits the maximum practical thickness of an optical coating. QRQHI IHFWLYHEURDGEDQGDQWLUHIOHFWLRQFRDWLQJ LQFLGHQW ZDYHOHQJWK l 127(8QOHVVHYHU\FRPSRQHQWLVWRWDOO\ QRQUHIOHFWLYHVRPHUHIOHFWLRQORVVHVZLOORFFXU WRWDOO\UHIOHFWLYHFRPSRQHQWIRU l SDUWLDOO\UHIOHFWLYHFRPSRQHQWIRU WRWDOO\QRQUHIOHFWLYHFRPSRQHQWIRU 5HODWLYH:DYHOHQJWK Figure 1.17 Typical reflectance curve of an unmodified quarter-wave stack Machine Vision Guide 3HUFHQW5HIOHFWDQFH l l Figure 1.18 Schematic multicomponent coatings with only one component exactly matched to the incident wavelength, λ. The high-reflection coating is successful; the antireflection coating is not. Laser Guide marketplace.idexop.com High-Reflection Coatings A21 OPTICAL COATINGS Optical Coatings and Materials BROADBAND REFLECTION COATINGS POLARIZATION EFFECTS The design procedure for a broadband reflection coating should now be apparent. Two design techniques are used. The most obvious approach is to use two quarterwave stacks with their maximum reflectance wavelengths separated on either side of the design wavelength. This type of coating, however, tends to be too thick and often has poor scattering characteristics. This basic design is very useful for dichroic high reflectors, where the peak reflectances of two stacks are at different wavelengths. When light is incident on any optical surface at angles other than normal incidence, there is always a difference in the reflection/transmission behavior of s- and p-polarization components. In some instances, this difference can be made extremely small. On the other hand, it is sometimes advantageous to design a thin-film coating that maximizes this effect (e.g., thinfilm polarizers). Polarization effects are not normally considered for antireflection coatings because they are nearly always used at normal incidence where the two polarization components are equivalent. A more elegant approach to broadband dielectric coatings involves using a single modified quarter-wave stack in which the layers are not all the same optical thickness. Instead, they are graded between the quarterwave thickness for two wavelengths at either end of the intended broadband performance region. The optical thicknesses of the individual layers are usually chosen to follow a simple arithmetic or geometric progression. By using designs of this type, multilayer, broadband coatings with reflectance in excess of 99% over several hundred nanometers are possible. In many scanning dye laser systems, high reflectance over a large wavelength region is absolutely essential. In many non-laser instruments, all-dielectric coatings are favored over metallic coatings because of their high reflectance. Multilayer broadband coatings are available with highreflectance regions spanning almost the entire visible spectrum. sSODQH pSODQH 3HUFHQW5HIOHFWDQFH 5HODWLYH:DYHOHQJWK Figure 1.19 The s-polarization reflectance curve is always broader and higher than the p-polarization reflectance curve A22 High-Reflection Coatings High-reflectance or partially reflecting coatings are frequently used at oblique angles, particularly at 45º, for beam steering or beam splitting. Polarization effects can therefore be quite important with these types of coating. At certain wavelengths, a multilayer dielectric coating shows a remarkable difference in its reflectance of the s- and p-polarization components (see figure 1.19). The basis for the effect is the difference in effective refractive index of the layers of film for s- and p-components of the incident beam, as the angle of incidence is increased from the normal. This effect should not be confused with the phenomenon of birefringence in certain crystalline materials, most notably calcite. Unlike birefringence, it does not require the symmetric properties of a crystalline phase. It arises from the difference in magnitude of magnetic and electric field vectors for s- and p-components of an electromagnetic wave upon reflection at oblique incidence. Maximum s-polarization reflectance is always greater than the maximum p-polarization reflectance at oblique incidence. If the reflectance is plotted as a function of wavelength for some arbitrary incidence angle, the s-polarization high reflectance peak always extends over a broader wavelength region than the p-polarization peak. Many dielectric coatings are used at peak reflectance wavelengths where polarization differences can be made negligible. In some cases, the polarization differences can be put to good use. The “edge” region of the reflectance curve is a wavelength region in which the s-polarization reflectance is much higher than the p-polarization reflectance. This can be maximized in a design to produce a very efficient thin-film polarizer. 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials INTERFERENCE FILTERS An interference filter is produced by applying a complex multilayer coating to a glass blank. The complex Figure 1.20 Spectral performance of an interference filter PARTIALLY TRANSMITTING COATINGS In many applications, it is desirable to split a beam of light into two components with a selectable intensity ratio. This is performed by inserting an optical surface at an oblique angle (usually 45º) to separate reflected and transmitted components. In most cases, a multilayer coating is applied to the surface in order to modify intensity and polarization characteristics of the two beams. An alternative to the outdated metallic beamsplitter is a broadband (or narrowband) multilayer dielectric stack with a limited number of pairs of layers, which transmits a fixed amount of the incident light. Just as in the case of metallic beamsplitter coatings, the ratio of reflected and transmitted beams depends on the angle of incidence. Unlike a metallic coating, a highquality film will introduce negligible losses by either absorption or scattering. There are, however, two drawbacks to dielectric beamsplitters. The performance of these coatings is more wavelength sensitive than that of metallic coatings, and the ratio of transmitted and reflected intensities may be quite different for Machine Vision Guide In many applications, particularly those in the field of resonance atomic or molecular spectroscopy, a filtering system is required that transmits only a very narrow range of wavelengths of incident light. For particularly highresolution applications, monochromators may be used, but these have very poor throughputs. In instances where moderate resolution is required and where the desired region(s) is (are) fixed, interference filters should be used. :DYHOHQJWKQP Gaussian Beam Optics This type of thin-film filter is used in high-power image-projection systems in which the light source often generates intense amounts of heat (infrared and nearinfrared radiation). Thin-film filters designed to separate visible and infrared radiation are known as hot or cold mirrors, depending on which wavelength region is rejected. CVI Laser Optics offers both hot and cold mirrors. 7\SLFDOUHIOHFWDQFHFXUYH Fundamental Optics Thin films acting as edge filters are now routinely manufactured using a modified quarter-wave stack as the basic building block. CVI Laser Optics produces many custom edge filters specially designed to meet customers’ specifications. A selection suitable for various laser applications is offered as standard catalog items. Optical Specifications Traditionally, such absorption filters have been made from colored glasses. CVI Laser Optics offers a range of these economical and useful filters. Although they are adequate for many applications, they have two drawbacks: they function by absorbing unwanted wavelengths, which may cause reliability problems in such high-power situations as projection optics; also the edge of the transmission curve may not be as sharp as necessary for many applications. Material Properties In many optical systems, it is necessary to have a wavelength filtering system that transmits all light of wavelengths longer than a reference wavelength or transmits light at wavelengths shorter than a reference wavelength. These types of filters are often called shortwavelength or long-wavelength cutoff filters. coating consists of a series of broadband quarter-wave stacks, which act as a very thin, multiple-cavity FabryPerot interferometer. Colored-glass substrates can be used to absorb unwanted light. Figure 1.20 shows the transmission curve of a typical CVI Laser Optics interference filter, the 550nm filter from the visible-40 filter set. Notice the notch shape of the transmission curve, which dies away very quickly outside the hightransmission (low-reflectance) region. 3HUFHQW5HIOHFWDQFH EDGE FILTERS AND HOT OR COLD MIRRORS Laser Guide marketplace.idexop.com High-Reflection Coatings A23 OPTICAL COATINGS Optical Coatings and Materials the s- and p-polarization components of the incident beam. In polarizers, this can be used to advantage. The difference in partial polarization of the reflected and transmitted beams is not important, particularly when polarized lasers are used. In beamsplitters, this is usually a drawback. A hybrid metal-dielectric coating is often the best compromise. CVI Laser Optics produces coated beamsplitters with designs ranging from broadband performance without polarization compensation, to broadband with some compensation for polarization, to a range of cube beamsplitters that are virtually nonpolarizing at certain laser wavelengths. These nonpolarizing beamsplitters offer unparalleled performance with the reflected s- and p-components matched to better than 5%. METALLIC COATINGS Metallic coatings are used primarily for mirrors and are not classified as thin films in the strictest sense. They do not rely on the principles of optical interference, but rather on the physical and optical properties of the coating material. However, metallic coatings are often overcoated with thin dielectric films to increase the reflectance over a desired range of wavelengths or range of incidence angles. In these cases, the metallic coating is said to be “enhanced.” Overcoating metallic coatings with a hard, single, dielectric layer of halfwave optical thickness improves abrasion and tarnish resistance but only marginally affects optical properties. Depending on the dielectric used, such overcoated metals are referred to as durable, protected, or hardcoated metallic reflectors. The main advantages of metallic coatings are broadband spectral performance, insensitivity to angle of incidence and polarization, and low cost. Their primary disadvantages include lower durability, lower reflectance, and lower damage threshold. A24 High-Reflection Coatings 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials THIN-FILM PRODUCTION VACUUM DEPOSITION The evaporation source is usually one of two types. The simpler, older type relies on resistive heating of a thin folded strip (boat) of tungsten, tantalum, or molybdenum which holds a small amount of the coating material. During the coating process, a high current (10 – 100 A) SOFT FILMS PRQLWRULQJ SODWH VXEVWUDWHV VXEVWUDWHV YDFXXP V\VWHP TXDUW]ODP S VKXWWHU Until the advent of electron bombardment vaporization, only materials that melted at moderate temperatures (2000ºC) could be incorporated into thin film coatings. Unfortunately, the more volatile low-temperature materials also happen to be materials that produce softer, less durable coatings. Consequently, early multilayer coatings deteriorated fairly quickly and required undue amounts of care during cleaning. More importantly, higher performance designs, with performance specifications at several wavelengths, could not be produced easily owing to the weak physical properties and lack of durability of such materials. Gaussian Beam Optics WKHUPRFRXSOH TXDUW]ODPS KHDWLQJ Fundamental Optics URWDWLRQPRWRU Several problems are associated with thermal evaporation. Some useful substances can react with the hot boat, which can cause impurities to be deposited with the layers, changing the optical properties of the resulting thin-film stack. In addition, many materials, particularly metal oxides, cannot be vaporized this way because the material of the boat (tungsten, tantalum, or molybdenum) melts at a lower temperature than the material to be vaporized. Instead of a layer of zirconium oxide, a layer of tungsten would be deposited on the substrate. Optical Specifications THERMAL EVAPORATION Material Properties CVI Laser Optics manufactures thin films by a process known as vacuum deposition. Uncoated substrates are placed in a large vacuum chamber capable of achieving a vacuum of at least 10–6 torr. At the bottom of the chamber is the source of the film material to be vaporized, as shown in figure 1.21. The substrates are mounted on a series of rotating carousels, arranged so that each substrate sweeps in planetary style through the same time-averaged volume in the chamber. is passed through the boat, thermally vaporizing the coating material. Because the chamber is at a greatly reduced pressure, there is a very long, mean-free-path for the free atoms or molecules, and the heavy vapor is able to reach the moving substrates at the top of the chamber. Here it condenses back to the solid state, forming a thin uniform film. YDSRU (EHDPJXQ EDVHSODWH SRZHU VXSSO\ GHWHFWRU FKRSSHU OLJKWVRXUFH ZDWHU FRROLQJ UHIOHFWLRQVLJQDO SRZHU VXSSO\ RSWLFDOPRQLWRU Figure 1.21 Schematic view of a typical vacuum desposition chamber Electron bombardment has become the accepted method of choice for advanced optical-thin-film fabrication. This method is capable of vaporizing even difficult-to-vaporize materials such as titanium oxide and zirconium oxide. Using large cooled crucibles precludes or eliminates the chance of reaction between the heated coating material and the metal of the boat or crucible. Machine Vision Guide ILOWHU ELECTRON BOMBARDMENT A high-flux electron gun (1 A at 10 kV) is aimed at the film material contained in a large, water-cooled, copper Laser Guide marketplace.idexop.com Thin-Film Production A25 OPTICAL COATINGS Optical Coatings and Materials crucible. Intense local heating melts and vaporizes some of the coating material in the center of the crucible without causing undue heating of the crucible itself. For particularly involatile materials, the electron gun can be focused to intensify its effects. Careful control of the temperature and vacuum conditions ensures that most of the vapor will be in the form of individual atoms or molecules, as opposed to clusters of atoms. This produces a more uniform coating with better optical characteristics and improved longevity. PLASMA ION-ASSISTED BOMBARDMENT Plasma ion-assisted deposition (PIAD) is a coating technique, often applied at low temperatures, which offers unique benefits in certain circumstances. Ion assist during the coating process leads to a higher atomic or molecular packing density in the thin-film layers (increasing index of refraction), minimizes wavelength shift, and achieves the highest adhesion levels and the lowest absorption available. This performance level is particularly crucial in many semiconductor, microelectronics, and telecommunications applications. The lack of voids in the more efficiently packed film means that it is far less susceptible to water-vapor absorption. Water absorption by an optical coating can change the index of refraction of layers and, hence, the optical properties. Water absorption can also cause mechanical changes that can ultimately lead to coating failure. Ion-assisted coating can also be used for cold or low-temperature processing. Eliminating the need to heat parts during coating allows cemented parts, such as cemented achromats, to be safely coated. From a materials standpoint, PIAD is often used when depositing metal oxides, metal nitrides, pure metals, and nonmetal oxides. Therefore, PIAD can significantly improve the performance of antireflection coatings, narrow- and wide-passband filters, edge filters, dielectric mirrors, abrasion-resistant transparent films, gain-flattening filters, and Rugate (gradient) filters. A26 Thin-Film Production ION-BEAM SPUTTERING (IBS) Ion-beam sputtering is a deposition method using a very high kinetic energy ion beam. The target is external to the ion source which allows for independent or automated control of the ion energy and flux. The energy and flux of ions is composed of neutral atoms which allow either insulating or conducting targets to be sputtered directly onto the substrate; this allows for a wide range of coating options. The high energy flux impacts the target source and ejects atoms directly towards the intended substrate. Direct sputtering provides a high level of accuracy and repeatability over numerous coating runs. IBS deposition produces dense coating layers with almost no scatter or absorption which minimizes or eliminates spectral shift due to moisture absorption. In addition, the coating density and durability allows for high damage threshold coating designs. MAGNETRON SPUTTERING Magnetron sputtering is a thin film deposition process that utilizes a magnet behind a cathode to trap free electrons in a circuitous magnetic field close to the target surface. A metered gaseous plasma of ions or neutral particles is introduced and the accelerated electrons collide with the neutral gas atoms in their path. These interactions cause ionizing collisions and drive electrons off the gas atoms. The gas atom becomes unbalanced and will have more positively charged protons than negatively charged electrons. The positively charged ions are accelerated towards the negatively charged electrode and impact the target material. The energy transfer is greater than the binding energy of the target material, causing the release of free electrons, erosion of the target material, and ultimately the sputtering process. The ejected source material particles are neutrally charged and therefore unaffected by the negative magnetic field. The ejected atoms are transferred to a substrate into densely packed coating layers resulting in little or no spectral shift caused by moisture absorption. The release of free electrons feed the formation of ions and the propagation of the plasma. 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials prescribed value. Highly accurate optical monitoring is essential for the production and optimization of specific optical effects, such as setting the exact edge position of an interference filter or sharp-cut off reflector. Material Properties Due to close proximity the percentage of confined electrons that cause ionizing collisions dramatically increases. This allows for very high deposition rates at which the target material is eroded and subsequently deposited onto the substrate. SCATTERING Magnetron sputtering has the advantages of exceptional uniformity, high deposition rates, low deposition pressure, and low substrate temperature allowing a wide variation of industrial production. LQFLGHQWOLJKW Machine Vision Guide As each layer is deposited onto the witness sample, the intensity of reflected and/or transmitted light oscillates in a sinusoidal manner due to optical interference effects. The turning points represent quarter- and half-wave thicknesses at the monitoring wavelength. Deposition is automatically stopped when the reflectance and/ or transmittance of the reference surface achieves a The most notable example of applications in which scattering is critical are intracavity mirrors for lowgain lasers, such as certain helium neon lasers, and continuous-wave dye lasers. Gaussian Beam Optics Optical monitoring is the most common method of observing the deposition process. A double-beam monochromator-photometer monitors, at applicationspecific wavelengths, the optical characteristics of a witness sample located within the vacuum chamber. In certain cases, the detection system can directly monitor the changing optical characteristics of the actual substrate being coated. During operation, a beam of light passes through the chamber and is incident on the witness sample or the substrate to be coated. Reflected and/or transmitted light is detected using photomultiplier detectors and phase-sensitive detection techniques to maximize signal-tonoise ratio. Fundamental Optics A chamber set up for multilayer deposition has several sources that are preloaded with various coating materials. The entire multilayer coating is deposited without opening the chamber. A source is heated, or the electron gun is turned on, until the source is at the proper molten temperature. The shutter above the source is opened to expose the chamber to the vaporized material. When a particular layer is deposited to the correct thickness, the shutter is closed and the source is turned off. This process is repeated for the other sources. Optical Specifications MONITORING AND CONTROLLING LAYER THICKNESS Reflectance and transmittance are usually the most important optical properties specified for a thin film, closely followed by absorption. However, the degree of scattering caused by a coating is often the limiting factor in the ability of coated optics to perform in certain applications. Scattering is quite complex. The overall degree of scattering is determined by imperfections in layer interfaces, bulk substrate material characteristics, and interference effects between the photons of light scattered by these imperfections, as shown in figure 1.22. Scattering is also a function of the granularity of the layers. Granularity is difficult to control as it is often an inherent characteristic of the materials used. Careful modification of deposition conditions can make a considerable difference in this effect. Figure 1.22 Interface imperfections scattering light in a multilayer coating Laser Guide marketplace.idexop.com Thin-Film Production A27 OPTICAL COATINGS Optical Coatings and Materials TEMPERATURE AND STRESS PRODUCTION CONTROL Mechanical stress within the thin-film coating can be a major problem. Even with optimized positioning of the optics being coated and careful control of the source temperature and vacuum, many thin-film materials do not deposit well on cold substrates causing stresses within the layers. This is particularly true of involatile materials. Raising the substrate, temperature a few hundred degrees improves the quality of these films, often making the difference between a usable and a useless film. The elevated temperature seems to allow freshly condensed atoms (or molecules) to undergo a beneficial but limited amount of surface diffusion. Two major factors are involved in producing a coating that performs to a particular set of specifications. First, sound design techniques must be used. If design procedures cannot accurately predict the behavior of a coating, there is little chance that satisfactory coatings will be produced. Second, if the manufacturing phase is not carefully controlled, the thin-film coatings produced may perform quite differently from the computer simulation. Optics that have been coated at an elevated temperature require very slow cooling to room temperature. The thermal expansion coefficients of the substrate and the film materials are likely to be somewhat different. As cooling occurs, the coating layer or layers contract at different rates which produces stress. Many pairs of coating materials also do not adhere particularly well to each other owing to different chemical properties and bulk packing characteristics. Temperature-induced stress and poor interlayer adhesion are the most common thickness-related limitations in optical thin-film production. Ignoring such techniques as ion-assisted deposition, stress must be reduced by minimizing overall coating thickness and by carefully controlling the production process. INTRINSIC STRESS Even in the absence of thermal-contraction-induced stress, the layers often are not mechanically stable because of intrinsic stress from interatomic forces. The homogeneous thin film is not the preferred phase for most coating materials. In the lowest energy state, molecules are aligned in a crystalline symmetric fashion. This is the natural form in which intermolecular forces are more nearly in equilibrium. At CVI Laser Optics, great care is taken in coating production at every level. Not only are all obvious precautions taken, such as thorough precleaning and controlled substrate cool down, but even the smallest details of the manufacturing process are carefully controlled. Our thoroughness and attention to detail ensure that the customer will always be supplied with the best design, manufactured to the highest standards. QUALITY CONTROL All batches of CVI Laser Optics coatings are rigorously and thoroughly tested for quality. Even with the most careful production control, this is necessary to ensure that only the highest quality parts are shipped. Our inspection system meets the stringent demands of MIL-I-45208A, and our spectrophotometers are calibrated to standards traceable to the National Institute of Standards and Technology (NIST). Upon request, we can provide complete environmental and photometric testing to MIL-C-675 and MIL-M-13508. All are firm assurances of dependability and accuracy. In addition to intrinsic molecular forces, intrinsic stress results from poor packing. If packing density is considerably less than percent, the intermolecular binding may be sufficiently weak that it makes the multilayer stack unstable. A28 Thin-Film Production 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials CVI LASER OPTICS ANTIREFLECTION COATINGS BROADBAND MULTILAYER ANTIREFLECTION COATINGS 3HUFHQW5HIOHFWDQFH There are two families of broadband antireflection coatings from CVI Laser Optics: HEBBAR™ and BBAR. 1RUPDOLQFLGHQFH :DYHOHQJWKQP X HEBBAR™ Optical Specifications coating for 245 – 440 nm X R < 0.5%, Rabs < 1.0% avg X Damage threshold: 3.5 J/cm2, 10 nsec pulse at 355 nm typical 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH ÛLQFLGHQFH :DYHOHQJWKQP X HEBBAR™ Gaussian Beam Optics 3HUFHQW5HIOHFWDQFH Fundamental Optics coating for 415 – 700 nm X R < 0.4%, Rabs < 1.0% avg X Damage threshold: 3.8 J/cm2, 10 nsec pulse at 532 nm typical Machine Vision Guide The typical reflectance curves shown below are for N-BK7 substrates, except for the ultraviolet 245 – 440 nm and 300-500 nm coatings which are applied to fused silica substrates or components. The reflectance values given below apply only to substrates with refractive indices ranging from 1.47 to 1.55. Other indices, while having their own optimized designs, will exhibit reflectance values approximately 20% higher for incidence angles from 0 to 15º and 25% higher for incidence angles of 30+º. Material Properties Broadband antireflection coatings provide a very low reflectance over a broad spectral bandwidth. These advanced multilayer films are optimized to reduce overall reflectance to an extremely low level over a broad spectral range. HEBBAR™ COATINGS HEBBAR coatings exhibit a characteristic doubleminimum reflectance curve covering a spectral range of some 250 nm or more. The reflectance does not exceed 1.0%, and is typically below 0.6%, over this entire range. Within a more limited spectral range on either side of the central peak, reflectance can be held to well below 0.4%. HEBBAR coatings are relatively insensitive to angle of incidence. The effect of increasing the angle of incidence (with respect to the normal to the surface) is to shift the curve to slightly shorter wavelengths and to increase the long wavelength reflectance slightly. These coatings are extremely useful for high numerical-aperture (low f-number) lenses and steeply curved surfaces. In these cases, incidence angles vary significantly over the aperture. 7\SLFDOUHIOHFWDQFHFXUYH To order a HEBBAR coating, append the coating suffix given in the table below to the product number. In some instances it will be necessary to specify which surfaces are to be coated. Laser Guide marketplace.idexop.com CVI Laser Optics Antireflection Coatings A29 OPTICAL COATINGS Optical Coatings and Materials 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH ÛLQFLGHQFH 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH ÛLQFLGHQFH :DYHOHQJWKQP X HEBBAR™ coating for 780 – 850 nm diode lasers < 0.25%, Rabs < 0.4% avg X R X R X Damage X Damage threshold: 6.5 J/cm2, 20 nsec pulse at 1064 nm typical 3HUFHQW5HIOHFWDQFH X HEBBAR™ HEBBAR™ coating for 300 – 500 nm threshold: 3.2 J/cm2, 10 nsec pulse at 355 nm typical :DYHOHQJWKQP coating for 750 – 1100 nm X R < 0.4%, Rabs < 0.6% avg X Damage threshold: 6.5 J/cm2, 20 nsec pulse at 1064 nm typical CVI Laser Optics Antireflection Coatings 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH ÛLQFLGHQFH 7\SLFDOUHIOHFWDQFHFXUYH < 1.0% abs 3HUFHQW5HIOHFWDQFH A30 X Specialty :DYHOHQJWKQP ÛLQFLGHQFH :DYHOHQJWKQP X Specialty HEBBAR™ coating for 425 – 670 nm optimized for 45° X R avg < 0.6%, Rabs < 1.0% X Damage threshold: 3.8 J/cm2, 10 nsec pulse at 532 nm typical 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH ÛLQFLGHQFH lasers X R avg :DYHOHQJWKQP HEBBAR™ coating for 660 – 835 nm diode < 0.5%, Rabs < 1.0% X Damage abs < 0.5%@ 780-830 nm and 1300 nm X Damage threshold: 5.4 J/cm2, 20 nsec pulse at 1064 nm typical < 1.25% @ 450-700 nm, Rabs <0.25 @ 1064 nm 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH BandHEBBAR™ coating for 780 – 830 nm and 1300 nm X R Band HEBBAR™ coating for 450 – 700 nm and 1064 nm 1RUPDOLQFLGHQFH :DYHOHQJWKQP X Extended X R HEBBAR™ coating for 420 – 1100 nm Machine Vision Guide Gaussian Beam Optics X Dual Fundamental Optics threshold: 1.3 J/cm2, 10 nsec pulse at 532 nm typical, 5.4 J/cm2, 20 nsec pulse at 1064 nm typical QRUPDOLQFLGHQFH :DYHOHQJWKQP :DYHOHQJWKQP X Damage X Dual 7\SLFDOUHIOHFWDQFHFXUYH avg threshold: 3.8 J/cm , 10 nsec pulse at 532 nm typical QRUPDOLQFLGHQFH X R 2 Optical Specifications X Specialty 3HUFHQW5HIOHFWDQFH 7\SLFDOUHIOHFWDQFHFXUYH Material Properties 3HUFHQW5HIOHFWDQFH < 1.0% , Rabs <1.75 @ 1064 nm avg X Damage threshold: 4.5 J/cm2, 10 nsec pulse at 532 nm typical 6.4 J/cm2, 20 nsec pulse at 1064 nm typical Laser Guide marketplace.idexop.com CVI Laser Optics Antireflection Coatings A31 OPTICAL COATINGS Optical Coatings and Materials Standard HEBBAR™ Coatings Description Wavelength Range (nm) Reflectance (%) Optimized for Angle of Incidence (degrees) HEBBAR™ 245 – 440 nm 245 – 440 Ravg < 0.50 HEBBAR™ 415 – 700 nm 415 – 700 HEBBAR™ 780 – 850 nm HEBBAR™ 750 – 1100 nm COATING SUFFIX FORMER‡ REPLACED BY 0 /072 HE-245-440 Ravg < 0.40 0 /078 HE-415-700 780 – 850 Ravg < 0.25 0 /076 HE-780-850 750 – 1100 Ravg < 0.40 0 /077 HE-750-1100 Specialty HEBBAR™ Coatings, optional designs for OEM and Prototype applications Description Wavelength Range (nm) Reflectance (%) Optimized for Angle of Incidence (degrees) HEBBAR™ 300 – 500 nm 300 – 500 Rabs < 1.0 HEBBAR™ 425 – 670 nm 425 – 670 HEBBAR™ 660 – 835 nm 660 – 835 COATING SUFFIX FORMER‡ REPLACED BY 0 /074 HE-300-500 Ravg < 0.60 45 UNP /079 HE-425-675-45UNP Ravg < 0.50 0 /075 HE-660-835 Dual Band HEBBAR™ Coatings COATING SUFFIX Description Wavelength Range (nm) Reflectance (%) Optimized for Angle of Incidence (degrees) FORMER‡ REPLACED BY HEBBAR™ 450 – 700 nm and 1064 nm 450 – 700 and 1064 Ravg < 0.60 0 /083 HE-450-700/1064 HEBBAR™ 780 – 830 nm and 1300 nm 780 – 830 and 1300 Ravg < 0.40 0 /084 HE-780-830/1300 Extended-Range HEBBAR™ Coating Description Wavelength Range (nm) Reflectance (%) Optimized for Angle of Incidence (degrees) FORMER‡ REPLACED BY HEBBAR™ 420 – 1100 nm 420 – 1100 Ravg < 0.50 0 /073 HE-420-1100 ‡ Former Melles Griot part number is replaced by new CVI Laser Optics part number A32 CVI Laser Optics Antireflection Coatings 1-505-298-2550 OPTICAL COATINGS OPTICAL COATINGS Optical Coatings & Materials 3HUFHQW5HIOHFWDQFH LQFLGHQFH VSODQH SSODQH :DYHOHQJWKQP Optical Specifications BBAR/45 425 – 675 coating for the visible region (45° incidence) 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH LQFLGHQFH sSODQH pSODQH Fundamental Optics CVI Laser Optics also provides three mid infrared and far infrared broad band antireflection coatings from 2.0 µm to 12.0 µm. These coatings are available on a wide range of materials including Si, Ge, ZnS, ZnSe, or CaF2. Our standard coatings cover 2 - 2.5 µm, 3 - 5 µm and the 8 - 12 µm region. Custom coatings are also available for mid and far infrared applications 7\SLFDOUHIOHFWDQFHFXUYH Material Properties BBAR-SERIES COATINGS CVI Laser Optics offers six overlapping broad band antireflection (BBAR) coating designs covering the entire range from 193 nm to 1600 nm. This includes very broad coverage of the entire Ti:Sapphire region. The BBAR coatings are unique in the photonics industry by providing both a low average reflection of ≤ 0.5% over a very broad range and also providing the highest damage threshold for pulsed and continuous wave laser sources (10J/cm2, 20 ns, 20 Hz at 1064 nm and 1MW/cm2 cw at 1064 nm respectively). Typical performance curves are shown in the graphs for each of the standard range offerings. If your application cannot be covered by a standard design, CVI Laser Optics can provide a special broad band antireflection coating for your application. :DYHOHQJWKQP BBAR/45 1050 – 1600 coating for the NIR region (45° incidence) QRUPDOLQFLGHQFH :DYHOHQJWKQP BBAR 193 – 248 coating for the UV region (0° incidence) QRUPDOLQFLGHQFH :DYHOHQJWKQP Machine Vision Guide 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH Gaussian Beam Optics 7\SLFDOUHIOHFWDQFHFXUYH BBAR 248 – 355 coating for the UV region (0° incidence) Laser Guide marketplace.idexop.com CVI Laser Optics Antireflection Coatings A33 OPTICAL COATINGS Optical Coatings and Materials 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH :DYHOHQJWKQP BBAR 355 – 532 coating for the UV region (0° incidence) QRUPDOLQFLGHQFH QRUPDOLQFLGHQFH :DYHOHQJWKLQ0LFURPHWHUV :DYHOHQJWKQP BBAR 425 – 675 coating for VIS and NIR regions (0° incidence) :DYHOHQJWKQP BBAR 670 – 1064 coating for VIS and NIR regions (0° incidence) A34 CVI Laser Optics Antireflection Coatings 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH BBAR 3500 – 5000 coating for the IR region (0° incidence) 7\SLFDOUHIOHFWDQFHFXUYH 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH BBAR 1050 – 1600 coating for VIS and NIR regions (0° incidence) 7\SLFDOUHIOHFWDQFHFXUYH :DYHOHQJWKQP *H =Q6H QRUPDOLQFLGHQFH :DYHOHQJWKLQ0LFURPHWHUV BBAR 8000 – 12000 coating for the IR region (0° incidence) 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials 7\SLFDOUHIOHFWDQFHFXUYH X Standard reflectance of less than 0.1% coatings available for most laser lines X Custom LQFLGHQFH sSODQH pSODQH X Maximum center wavelengths at specific angles of incidence available per request Material Properties 3HUFHQW5HIOHFWDQFH V-Coating Center Wavelengths :DYHOHQJWKQP CVI Laser Optics will manufacture V-Type AR coatings for wavelengths from 193 nm to 10.6 µm. Damage thresholds for AR coatings on SF11 and similar glasses are limited not by the coating, but by the bulk material properties. Our damage testing has shown a damage threshold for SF11 and similar glasses to be 4 J/cm2. X Wavelength X Substrate X Angle material of incidence REPLACED BY 193 248 ArF 0.5 /101 193-0 ArF 0.25 /102 248-0 Nd 3rd harmonic 266 0.25 /103 266-0 308 XeCl 0.25 /104 308-0 351 Ar ion 0.25 /105 351-0 364 Ar ion 0.25 /107 364-0 442 HeCd 0.25 /111 442-0 458 Ar ion 0.25 /112 458-0 466 Ar ion 0.25 /113 466-0 473 Ar ion 0.25 /114 473-0 476 Ar ion 0.25 /115 476-0 488 Ar ion 0.25 /116 488-0 496 Ar ion 0.25 /117 496-0 502 Ar ion 0.25 /118 502-0 514 Ar ion 0.25 /119 514-0 532 Nd 2nd harmonic 0.25 /122 532-0 543 HeNe 0.25 /121 543-0 633 HeNe 0.25 /123 633-0 670 GaAlAs 0.25 /128 670-0 694 Ruby 0.25 /124 694-0 780 GaAlAs 0.25 /163 780-0 830 GaAlAs 0.25 /166 830-0 850 GaAlAs 0.25 /167 850-0 904 GaAs 0.25 /125 904-0 1064 Nd 0.25 /126 1064-0 1300 InGaAsP 0.25 /168 1300-0 1523 HeNe 0.25 /169 1523-0 1550 InGaAsP 0.25 /169 1550-0 Machine Vision Guide When ordering, be sure to specify the following: FORMER‡ Gaussian Beam Optics V-type AR coatings on Fused Silica, Crystal Quartz, Suprasil, and N-BK7 have damage threshold of 15 J/cm2 at 1064 nm, 20 ns, 20 Hz. Typical performance can often exceed 20 J/cm2. Maximum Reflectance (%) Fundamental Optics V-COATINGS CVI Laser Optics V-type AR Coatings are the best choice for a single laser wavelength or multiple, closely-spaced wavelengths. Examples are the principle argon laser lines at 488 nm and 514 nm, the neodymium transitions in a variety of host materials at 1047 – 1064 nm, and the individual excimer laser lines. Laser Type Optical Specifications BBAR/45 425 – 675 coating for the visible region (45° incidence) Wavelength (nm) ‡ Former Melles Griot part number is replaced by new CVI Laser Optics part number X Polarization X Fluence in J/cm2 X Near-zero reflectance at one specific wavelength and incidence angle Laser Guide marketplace.idexop.com CVI Laser Optics Antireflection Coatings A35 OPTICAL COATINGS 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH :DYHOHQJWKQP The reflectance curve for a typical V-coating, on N-BK7 glass, designed for operation at 632.8 nm is shown below. DOUBLE-V AND TRIPLE-V COATINGS CVI Laser Optics offers Double-V and Triple-V multilayer antireflection coatings for use in Nd:YAG laser systems at normal incidence. Highly damage resistant, electron beam deposited dielectrics are used exclusively as coating materials. As shown in the curves, the antireflection peaks at the harmonics are quite narrow. Also, due to the coating design and dispersion, they do not fall exactly at a wavelength ratio of 1 : 1/2 : 1/3. Consequently, the reflectivity specifications of these AR coatings are not as good as V-coatings for any one wavelength. CVI Laser Optics offers these Double-V coatings on W2 windows, in all standard sizes. Contact CVI Laser Optics for the performance of 45° Double-V and Triple-V AR coatings or for other harmonic combinations. A36 CVI Laser Optics Antireflection Coatings QRUPDOLQFLGHQFH Example of a V-coating for 632.8 nm 7\SLFDOUHIOHFWDQFHFXUYH :DYHOHQJWKQP Double-V antireflection coating for 532 nm and 1064 nm X Designed for normal incidence X R < 0.3% 1064 nm X R < 0.6% at 532 nm X Damage threshold 5 J/cm2 at 532 nm X Damage threshold 10 J/cm2 at 1064 nm 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH Optical Coatings and Materials 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH :DYHOHQJWKQP Triple-V antireflection coating for 355 nm, 532 nm, and 1064 nm X Designed for normal incidence X R < 0.3% 1064 nm X R < 0.6% at 532 nm X R < 1.5% at 355 nm 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials SINGLE-LAYER MgF2 COATINGS Percent Reflectance normal incidence 45˚ incidence 3 2 1 400 500 600 700 Wavelength (nm) Optical Specifications Single-layer antireflection coatings for use on very steeply curved or short-radius surfaces should be specified for an angle of incidence approximately half as large as the largest angle of incidence encountered by the surface. Typical reflectance curve 4 Material Properties Magnesium fluoride (MgF2) is commonly used for singlelayer antireflection coatings because of its almost ideal refractive index (1.38 at 550 nm) and high durability. These coatings can be optimized for 550 nm for normal incidence, but as can be seen from the reflectance curves, they are extremely insensitive to wavelength and incidence angle. Single-layer MgF2 400 – 700 nm coating X Popular Single-Layer MgF2 Antireflection Coating Normal Incidence Maximum Reflectance on N-BK7 (%) Maximum Reflectance on Fused Silica (%) COATING SUFFIX 400 – 700 2.0 2.25 SLMF-400-700 520 – 820 2.0 2.25 SLMF-520-820 durable and most economical X Optimized X Relatively for 550 nm, normal incidence insensitive to changes in incidence angle X Damage threshold: 13.2 J/cm2, 10 nsec pulse at 532 nm typical 7\SLFDOUHIOHFWDQFHFXUYH 1RUPDOLQFLGHQFH Gaussian Beam Optics 3HUFHQW5HIOHFWDQFH X Highly Fundamental Optics Wavelength Range (nm) and versatile antireflection coating for visible wavelengths :DYHOHQJWKQP X Optimized Machine Vision Guide Single-layer MgF2 520 – 820 nm coating for 670 nm, normal incidence X Useful for most visible and near-infrared diode wavelengths X Highly durable and insensitive to angle X Damage threshold: 13.2 J/cm2, 10 nsec pulse at 532 nm typical Laser Guide marketplace.idexop.com Single-Layer MgF2 Coatings A37 OPTICAL COATINGS METALLIC HIGH-REFLECTION COATINGS Optical Coatings and Materials METALLIC HIGH-REFLECTION COATINGS CVI Laser Optics offers eight standard metallic highreflection coatings formed by vacuum deposition. These coatings can be used at any angle of incidence and can be applied to most optical components. To specify this coating, simply append the coating suffix number to the component product number. CVI Laser Optics Coating Chambers CVI Laser Optics thin-film coating chambers have X Multiple X Optical e-beam sources and crystal controls X Residual-gas Metallic reflective coatings are delicate and require care during cleaning. Dielectric overcoats substantially improve abrasion resistance, but they are not impervious to abrasive cleaning techniques. Clean, dry, pressurized gas can be used to blow off loose particles. This can be followed by a very gentle wipe using deionized water, a mild detergent, or alcohol. Gentle cleaning with an X Mass-flow X Quartz analyzers controls substrate heaters X Compound planetary rotation capabilities appropriate swab can be effective. Metallic High-Reflection Coatings Coating Type Wavelength Range (nm) Average Reflectance (%) Vacuum UV Aluminum 157 Deep UV Aluminum Damage Threshold Former Coating Suffix ‡ PRODUCT CODE Pulsed (J / cm2) cw (MW / cm2) > 80 not tested not tested VUVA 193 > 90 not tested not tested DUVA UV Enhanced Aluminum 250 – 600 85 0.3 22.0 /028 PAUV Protected Aluminum 400 – 10,000 90 0.5 22 /011 PAV Enhanced Aluminum 450 – 650 92 0.3 12.0 /023 EAV Protected Silver 400 – 20,000 95 1.6 73.0 /038 PS Protected Gold 650 – 10,000 95 0.4 17.0 /055 PG Bare Gold 700 – 20,000 99 1.1 48.0 /045 PG BARE ‡ Former Melles Griot part number is replaced by new CVI Laser Optics part number A38 Metallic High-Reflection Coatings 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH :DYHOHQJWKQP 7\SLFDOUHIOHFWDQFHFXUYH Material Properties 3HUFHQW5HIOHFWDQFH :DYHOHQJWKQP DEEP UV ALUMINUM (DUVA) X Enhanced X Enhanced performance for 157 nm X Provides X Provides X Dielectric X Dielectric X R X R consistently high reflectance throughout the vacuum ultraviolet, visible, and near-infrared regions overcoat minimizes oxidation and increases abrasion resistance Based on CVI Laser Optics high density aluminum coating technology, VUVA mirrors are designed for optimized performance at 157 nm. Certification of performance at wavelength is available for an additional charge. Call CVI Laser Optics for details. performance for 193 nm consistently high reflectance throughout the vacuum ultraviolet, visible, and near-infrared regions overcoat minimizes oxidation and increases abrasion resistance > 90% @ 193 nm, Rave ≥85% @ 400 – 1200 nm Fundamental Optics > 80% @ 157 nm Optical Specifications VACUUM UV ALUMINUM (VUVA) Gaussian Beam Optics Based on CVI Laser Optics high-density Al coating technology, broadband DUVA mirrors provide significantly higher 193 nm reflectance and durability than standard UV-protected Al mirrors. Choose buildto-print or off-the-shelf optics for your ellipsometry, spectroscopy, and semiconductor lithography or metrology applications. Machine Vision Guide Laser Guide marketplace.idexop.com Metallic High-Reflection Coatings A39 OPTICAL COATINGS Percent Reflectance 90 80 70 normal incidence 60 200 250 300 350 400 Wavelength (nm) ULTRAVIOLET PROTECTED ALUMINUM (PAUV) X Maintains reflectance in the ultraviolet region X Dielectric overcoat prevents oxidation and increases abrasion resistance X R > 86% from 250 to 400 nm avg X R avg > 85% from 400 to 700 nm X Damage threshold: 0.07J/cm2, 10 nsec pulse (5.7 MW/cm2) at 355 nm typical The protective dielectric layer prevents oxidation and improves abrasion resistance. While the resulting surface is not as abrasion resistant as our protected aluminum it can be cleaned with care. A40 Metallic High-Reflection Coatings 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH Optical Coatings and Materials Typical reflectance curve 100 QRUPDOLQFLGHQFH LQFLGHQFH VSODQH SSODQH :DYHOHQJWKQP PROTECTED ALUMINUM (PAV) X The best general-purpose metallic reflector for visible to near-infrared X Protective overcoat extends life of mirror and protects surface X R avg > 90% from 400 to 10.0 µm X Damage threshold: 0.3 J/cm2, 10 nsec pulse (21 MW/cm2) at 532 nm typical; 0.5 J/cm2, 20 nsec pulse (22 MW/cm2) at 1064 nm typical Protected aluminum is the very best general-purpose metallic coating for use as an external reflector in the visible and nearinfrared spectra. The protective film arrests oxidation and helps maintain a high reflectance. It is also durable enough to protect the aluminum coating from minor abrasions. 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials Typical reflectance curve 3HUFHQW5HIOHFWDQFH 95 90 normal incidence 45° incidence s-plane p-plane 85 80 400 450 500 550 600 650 700 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH 750 PROTECTED SILVER (PS) X Durability X Extremely of protected aluminum X Damage threshold: 0.4 J/cm2, 10 nsec pulse (33 MW/cm2) at 532 nm typical; 0.3 J/cm2, 20 nsec pulse (12 MW/cm2) at 1064 nm typical X Excellent region X Ravg Optical Specifications ENHANCED ALUMINUM (EAV) > 92% from 450 to 650 nm :DYHOHQJWKP Wavelength (nm) X Ravg Material Properties Percent Reflectance 100 versatile mirror coating performance for the visible to infrared > 95% from 400 nm to 20 µm X Can Fundamental Optics be used for ultrafast Ti:Sapphire laser applications X Damage By coating the aluminum with a multilayer dielectric film, reflectance is increased over a wide range of wavelengths. This coating is well suited for applications requiring the durability and reliability of protected aluminum, but with higher reflectance in the mid-visible regions. threshold: 0.9 J/cm2, 10 nsec pulse (75 MW/cm2) at 532 nm typical; 1.6 J/cm2, 20 nsec pulse (73 MW/cm2) at 1064 nm typical Gaussian Beam Optics CVI Laser Optics uses a proprietary coating and edgesealing technology to offer a first-surface external protected silver coating. In recent tests, the protected silver coating has shown no broadening effect on a 52 femtosecond pulse. This information is presented as an example of performance for femtosecond applications, but no warranty is implied. Machine Vision Guide Laser Guide marketplace.idexop.com Metallic High-Reflection Coatings A41 OPTICAL COATINGS 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH :DYHOHQJWKLQ0LFURPHWHUV QRUPDOLQFLGHQFH :DYHOHQJWKP PROTECTED GOLD (PG) BARE GOLD (PG-BARE) X Protective X Widely X Ravg overcoat extends coating life ≥ 95.0% from 650 nm to 10 µm X Damage threshold: 0.4 J/cm , 20 nsec pulse (17 MW/cm2) at 1064 nm typical 2 X R avg The CVI Laser Optics proprietary protected gold mirror coating combines the natural spectral performance of gold with the durability of hard dielectrics. Protected gold provides over 95% average reflectance from 650 nm to 10 µm. At a wavelength of 3 µm, the PG coating was tested for laser-induced damage and was found to withstand up to 18 2 J/cm2 with a 260 ms pulse (0.4 MW/cm2). These mirrors can be cleaned regularly using standard organic solvents, such as alcohol or acetone. Metallic High-Reflection Coatings used in the near, middle, and far infrared X Effectively controls thermal radiation > 99% from 700 nm to 20 µm X Damage A42 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH Optical Coatings and Materials 7\SLFDOUHIOHFWDQFHFXUYH threshold: 1.1 J/cm2, 20 nsec pulse (48 MW/cm2) at 1064 nm typical Bare gold combines good tarnish resistance with consistently high reflectance throughout the near, mid-, and far-infrared regions. Because bare gold is soft and scratches easily, CVI Laser Optics recommends using flow-washing with solvents and clean water or blowing the surface clean with a low-pressure stream of dry air for cleaning the coated mirror surface. 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials MAXBRITE™ COATINGS (MAXB) The extended MAXB-420-700 coating offers superior response below 500 nm, and it is particularly useful for helium cadmium lasers at 442 nm, or the blue lines of argon-ion lasers. Like MAXB-245-390, mechanical stresses in this complex coating limit its use to substrates with a surface figure accuracy specification of no greater than λ/4. QRUPDOLQFLGHQFH LQFLGHQFH :DYHOHQJWKQP MAXB-245-390 COATING X R avg Gaussian Beam Optics > 98% from 245 to 390 nm X Damage threshold: 0.92 J/cm2, 10 nsec pulse at 532 nm typical Machine Vision Guide The MAXB-630-850 MAXBRIte coating covers all the important visible and near-infrared diode laser wavelengths from 630 to 850 nm. This broadband coating is ideal for applications employing non-temperature stabilized diode lasers where wavelength drift is likely to occur. The MAXB-630-850 also makes it possible to use a HeNe laser to align diode systems. 3HUFHQW5HIOHFWDQFH The MAXB-420-700 MAXBRIte coating is particularly useful for helium cadmium lasers at 442 nm, or the blue lines of argon-ion lasers. The MAXB-480-700 MAXBRIte coating is suitable for instrumental and external laser-beam manipulation tasks. It is the ideal choice for use with tunable dye and parametric oscillator systems. 7\SLFDOUHIOHFWDQFHFXUYH Fundamental Optics The MAXB-248-390 ultraviolet MAXBRIte coating provides superior performance for a broad range of ultraviolet applications using some of the excimer lasers, third and fourth harmonics of most solid-state lasers, and mercury and xenon lamps. Optical Specifications These coatings exhibit exceptionally high reflectances for both s- and p-polarizations. In each case, at the most important laser wavelengths and for angles of incidence as high as 45º, the average of s- and p-reflectances exceeds 99%. For most applications, they are superior to metallic or enhanced metallic coatings. limited to substrates having a surface figure accuracy specification of no greater than λ/4 (versus an absolute standard). Material Properties MAXBRIte™ (multilayer all-dielectric xerophilous broadband reflecting interference) coatings are the best broadband mirror coatings commercially available. The MAXBRIte™ coatings are available for four broad regions. 245 nm – 390 nm, 420 nm – 700 nm, 480 nm – 700 nm, and 630 nm – 850 nm. They all reflect over 98% of incident laser radiation within their respective wavelength ranges. The ultraviolet MAXB-245-390 coating provides superior performance for ultraviolet applications. It is ideal for use with many of the excimer lasers, as well as third and fourth harmonics of most solid-state lasers. It is also particularly useful with broadband ultraviolet light sources, such as mercury and xenon lamps. Due to mechanical stresses within this advanced coating, it is Laser Guide marketplace.idexop.com MAXBRIte™ Coatings (MAXB) A43 OPTICAL COATINGS Optical Coatings and Materials 7\SLFDOUHIOHFWDQFHFXUYH QRUPDOLQFLGHQFH LQFLGHQFH :DYHOHQJWKQP MAXB-420-700 COATING X R avg X R avg threshold: 0.4 J/cm , 10 nsec pulse at 532 nm typical :DYHOHQJWKQP > 98% from 630 – 850 nm X Damage 2 threshold: 0.92 J/cm2, 10 nsec pulse at 532 nm typical 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH LQFLGHQFH MAXB-630-850 COATING > 98% from 420 – 700 nm X Damage 7\SLFDOUHIOHFWDQFHFXUYH 3HUFHQW5HIOHFWDQFH 3HUFHQW5HIOHFWDQFH QRUPDOLQFLGHQFH LQFLGHQFH :DYHOHQJWKQP MAXB-480-700 COATING X R avg > 98% from 480 – 700 nm X Damage A44 threshold: 0.92 J/cm2, 10 nsec pulse at 532 nm typical MAXBRIte™ Coatings (MAXB) 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials LASER-LINE MAX-R™ COATINGS The Laser-Line MAX-R™ mirrors have been upgraded to fit within our broader offering of laser-line mirrors. A full list of the former Melles Griot part numbers and the new CVI Laser Optics part numbers has been included in the Mirror section of this catalog under Laser-Line MAX-R™. MAX-R™ coatings available for popular laser wavelengths, at both 0º and 45º angle of incidence Laser-Line MAX-R™ Coatings, 45º Incidence X Standard X Custom coatings available from 193 to 1550 nm Laser-Line MAX-R™ Coatings, Normal Incidence Laser Type Minimum Reflectance Rp (%) 45º Incidence 45º ± 15º Incidence Former Coating Suffix ‡ PRODUCT CODE 193 ArF 97.0 94.0 /251 ARF 248 KrF 98.0 95.0 /252 KRF 266 Nd 4th harmonic 98.0 95.0 /253 Y4 0º ± 15º ArF 97.0 94.0 /201 ARF 248 KrF 98.0 95.0 /202 KRF 266 Nd 4th harmonic 98.0 95.0 /203 Y4 308 XeCl 99.0 96.0 /204 XECL 351 Ar Ion 99.0 96.0 /205 AR3 364 Ar ion 99.0 96.0 /207 AR3 442 HeCd 99.3 99.0 /209 HC1 476 Ar ion 99.3 98.5 /267 AR2 488 Ar ion 99.3 98.5 /269 AR1 193 PRODUCT CODE 308 XeCl 98.0 95.0 /254 XECL 351 Ar ion 98.0 96.0 /255 AR3 364 Ar ion 98.0 96.0 /257 AR3 442 HeCd 99.0 98.0 /259 HC1 458 Ar ion 99.3 98.0 /261 AR2 466 Ar ion 99.3 98.5 /263 AR2 473 Ar ion 99.3 98.5 /265 AR2 Ar ion 99.5 99.3 /211 AR2 466 Ar ion 99.5 99.3 /213 AR2 496 Ar ion 99.5 98.5 /271 AR1 473 Ar ion 99.5 99.3 /215 AR2 502 Ar ion 99.5 98.5 /272 AR1 476 Ar ion 99.5 99.3 /217 AR2 514 Ar ion 99.5 98.5 /273 AR1 488 Ar ion 99.5 99.3 /219 AR1 496 Ar ion 99.5 99.3 /221 AR1 532 Nd 2 harmonic 99.5 98.5 /275 Y2 502 Ar ion 99.5 99.3 /222 AR1 543 HeNe 99.5 98.5 /276 CV 514 Ar ion 99.5 99.3 /223 AR1 Nd 2nd harmonic 633 HeNe 99.5 98.5 /279 HN 532 99.5 99.3 /225 Y2 543 HeNe 99.5 99.3 /226 CV 633 HeNe 99.5 99.3 /229 HN 670 GaAlAs 99.5 99.3 /228 LDM 780 GaAlAs 99.3 99.0 /233 LDM 830 GaAlAs 99.3 99.0 /237 LDM 1064 Nd 99.2 99.0 /241 Y1 1300 InGaAsP 99.2 99.0 /245 LDM 1523, 1550 HeNe, InGaAsP 99.2 99.0 /247 LDM nd 670 GaAlAs 99.0 98.5 /278 LDM 780 GaAlAs 99.0 98.5 /283 LDM 830 GaAlAs 99.0 98.5 /287 LDM 1064 Nd 99.0 98.0 /291 Y1 1300 InGaAsP 99.0 98.5 /295 LDM 1523, 1550 HeNe, InGaAsP 99.0 98.5 /297 LDM Machine Vision Guide 458 Gaussian Beam Optics 0º Laser Type Fundamental Optics Minimum Reflectance Rp (%) Wavelength (nm) Former Coating Suffix ‡ Wavelength (nm) Optical Specifications X Highest possible reflectance achieved at specific laser wavelengths and typical angles of incidence Product Upgrade Material Properties Laser-line MAX-R™ coatings have been upgraded to higher damage threshold designs. While maintaining the high reflectivity and same optimized coating for angles of incidence at 0º or 45º, the damage thresholds have been significantly improved. The table below has been created to identify the new product codes. Please refer to the product code index for the additional specifications for these mirrors. If you have any additional questions please contact our customer service representatives for assistance. ‡ Former Melles Griot part number is replaced by new CVI Laser Optics part number Laser Guide marketplace.idexop.com Laser-Line MAX-R™ Coatings A45 OPTICAL COATINGS ULTRAFAST COATING (TLMB) Optical Coatings and Materials CVI Laser Optics has developed a new coating for ultrafast laser systems operating in the near-infrared spectral region. This all-dielectric coating, centered at 800 nm, minimizes pulse broadening for ultrafast applications. The coating also offers exceptionally high reflectance for both s- and p-polarizations in the 750 – 870 nm spectral region. Ultrafast Coating (TLMB) Wavelength Range (nm) Minimum Reflectance Rp (%) Angle of Incidence (degrees) Pulse Broadending (%) Former Coating Suffix ‡ Product Code 770 – 830 99.0 45 <18.0 /091 TLMB ‡ Former Melles Griot part number is replaced by new CVI Laser Optics part number The ultrafast coating is ideal for high-power Ti:sapphire laser applications. This coating is superior to protected and enhanced metallic coatings because of its ability to handle higher powers. 3HUFHQW5HIOHFWDQFH 813 :DYHOHQJWKQP TLMB Ultrafast coating A46 5HIOHFWDQFH*URXS 'HOD\'LVSHUVLRQIV 7\SLFDOUHIOHFWDQFHFXUYH Ultrafast Coating (TLMB) 7/0% WUDGLWLRQDOEURDGEDQG WUDGLWLRQDOKLJK/'7 :DYHOHQJWKQP A comparison of Reflectance Group Delay Dispersion vs. Wavelength of traditional broadband, traditional high LDT, and the CVI Laser Optics TLMB Ultrafast mirror 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials OPTICAL FILTER COATINGS ABSORPTION, TRANSMITTANCE, AND OPTICAL DENSITY The transmittance of a series of filters is the product of their individual external transmittance, T1xT2xT3, etc. Because transmittance (and hence opacity) is multiplicative, and since transmittance may extend over many orders of magnitude, it is often more convenient to use a logarithmic expression to define transmittance. OPTICAL DENSITY Optical density, or “density,” is the base 10 logarithm of opacity: D = log(1/T) As optical density increases, the amount of light blocked by the filter (by reflection and/or absorption) increases. The most important point to note is that optical density is additive. If several filters are stacked in series, their combined optical density is the sum of the individual optical densities. Machine Vision Guide TRANSMITTANCE As a beam of light passes through an absorbing medium, the amount of light absorbed is proportional to the intensity of incident light times the absorption coefficient. Consequently, the intensity of an incident Alternatively, a filter is defined by the amount of light it blocks, as opposed to the amount of light it transmits. This parameter is opacity, which is simply the reciprocal of the transmittance, 1/T. Gaussian Beam Optics Absorption occurs when the electric field of a light wave interacts with absorbing atoms or molecules in an oscillating dipole interaction. The photon is absorbed and the atom or molecule is placed in an excited state. This process occurs only at resonant wavelengths. In a solid or liquid absorber, excitation energy is dissipated as heat (vibrations of particles). Therefore, filters that rely mainly on absorption are not ideal for high-power laser applications. The intense local heating can lead to structural damage. Internal transmittance is the transmittance of an optical element when surface (coated or uncoated) losses are ignored. The measured transmittance of the element (including surface effects), transmittance, is called external transmittance, T. Fundamental Optics ABSORPTION All materials will absorb radiation in some parts of the electromagnetic spectrum. The amount of absorption depends on the wavelength, the amount of absorbing material in the radiation path, and the absorption of that material at that wavelength. Materials that absorb some visible wavelengths appear colored. For purposes of this catalog, colored glass refers to glass that is a wavelength-selective absorber in the near-ultraviolet to the near-infrared region. where Ti is internal transmittance, a is the absorption coefficient, c is the concentration of absorbers, and x is the overall thickness of the absorbing medium. Clearly α, and hence Ti, are wavelength dependent. For solid absorbing mediums, c = 1. Optical Specifications Metallic films, colored glasses, and thin dielectric films (sometimes all in the same unit) are used in CVI Laser Optics filters. These filters include wavelength-invariant varieties (neutral-density filters) and various wavelengthselective filters (colored-glass, high-pass and low-pass filters, edge filters, dichroics, and interference filters). Ti = 10–αcx Material Properties Absorption, particularly wavelength-selective absorption, is an important factor in the function of many of the filters described in the catalog. The two most commonly used absorbers are thin metallic films and “colored” glass. Some metallic films, such as Inconel®, chromium, and nickel, are particularly insensitive to wavelength for absorption. On the other hand, the amount of absorption by colored glass can vary by as much as several orders of magnitude in only tens of nanometers. beam drops exponentially as it passes through the absorber. This is often expressed as Beer’s law: Optical density is particularly useful for neutraldensity (ND) filters. These filters, which have a very flat wavelength response, are used to attenuate light in a calibrated, chromatically invariant fashion. ND filters Laser Guide marketplace.idexop.com Optical Filter Coatings A47 OPTICAL COATINGS Optical Coatings and Materials are supplied in sets of various calibrated densities. Combinations of these filters can be used to produce many different calibrated optical densities. INTERFERENCE FILTERS Interference filter applications are extremely diverse, including disease diagnosis, spectral radiometry, calorimetry, and color separation in television cameras. Used with even the least expensive broadband photometers or radiometers, CVI Laser Optics interference filters enable rapid and accurate measurement of the amplitude of specific spectral lines. This combination has an enormous throughput advantage since the collecting area of filters is very large compared to instrumental slits. Additionally, interference filters enable the viewing and near-instantaneous recording of very spectrally selective images. Spatial and spectral scanning instruments can provide similar images but take much longer. Narrowband interference filters permit isolation of wavelength intervals a few nanometers or less in width, without dispersion elements such as prisms or gratings. For example, a single line in the emission spectrum of a flame can be monitored without confusion from other nearby lines, or the signal from a laser communications transmitter can be received without interference from a brightly sunlit landscape. Colored-glass and gelatin filters are incapable of such discrimination. Interference filters are multilayer thin-film devices. While many interference filters may be correctly described as “all dielectric” in construction, metallic layers are often present in auxiliary blocking structures. Broadband interference filters almost always contain a metallic layer (in their spacers, not in their stacks). Interference filters come in two basic types, which transmit a desired wavelength interval while simultaneously rejecting both longer and shorter wavelengths, and edge filters. FABRY-PEROT INTERFEROMETER Narrowband interference filters (bandpass filters) operate with the same principles as the Fabry-Perot interferometer. In fact, they can be considered FabryPerot interferometers since they usually operate in the first order. A48 Optical Filter Coatings The Fabry-Perot is a simple interferometer, which relies on the interference of multiple reflected beams. The accompanying figure shows a schematic Fabry-Perot cavity. Incident light undergoes multiple reflections between coated surfaces which define the cavity. Each transmitted wavefront has undergone an even number of reflections (0, 2, 4, . . . ). Whenever there is no phase difference between emerging wavefronts, interference between these wavefronts produces a transmission maximum. This occurs when the optical path difference is an integral number of whole wavelengths, i.e., when mλ = 2topcosθ where m is an integer, often termed the order, top is the optical thickness, and θ is the angle of incidence. At other wavelengths, destructive interference of transmitted wavefronts reduces transmitted intensity toward zero (i.e., most, or all, of the light is reflected back toward the source). Transmission peaks can be made very sharp by increasing the reflectivity of the mirror surfaces. In a simple FabryPerot interferometer transmission curve (see figure), the ratio of successive peak separation to full width at half-maximum (FWHM) transmission peak is termed finesse. High reflectance results in high finesse (i.e., high resolution). In most Fabry-Perot interferometers, air is the medium between high reflectors; therefore, the optical thickness, top, is essentially equal to d, the physical thickness. The PXOWLSOHUHIOHFWLRQVLQDLUVSDFH v W RS RSWLFDOWKLFNQHVV KLJK UHIOHFWDQFH ! ORZDEVRUEDQFH PLUURUV Schematic of a Fabry-Perot interferometer 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials ):+0 ILQHVVH )65 ):+0 )65 P P P Transmission pattern showing the free spectral range (FSR) of a simple Fabry-Perot interferometer FWHM 50 10 1 cavity 5 1 cavity 1 .5 2 cavities 2 cavities .1 3 cavities .05 4 cavities .01 -7 -5 -3 -1 1 3 5 Machine Vision Guide The entire assembly of two quarter-wave stacks, separated by a half-wave spacer, is applied to a single surface in one continuous vacuum deposition run. By analogy with interferometers, the simplest bandpass interference filters are sometimes called cavities. Two or more such filters can be deposited one on top of the other, separated by an absentee layer, to form a multiple-cavity filter. Increasing the number of cavities has a significant effect on the shape of the passband (see figure). The resulting overall passband transmittance is given approximately by the product of the passbands of individual cavities. The advantages of multiple-cavity filters are steeper band slopes, improved near-band 100 Gaussian Beam Optics BANDPASS FILTER DESIGN The simplest bandpass filter is a very thin Fabry-Perot interferometer. The air gap is replaced by a thin layer of dielectric material with a half-wave optical thickness (optimized at the wavelength of the desired transmission peak). The high reflectors are normal quarter-wave stacks with a broadband reflectance peaking at the design wavelength. Fundamental Optics )ULQJH2UGHU Optical Specifications 3HUFHQW7UDQVPLWWDQFH Percent Normalized Transmitance rejection, and “square” (not Gaussian or Lorentzian) passband peaks. This last result, especially desirable in intermediate-bandwidth filters, is achieved in part by reducing stack reflectance, which broadens individual cavity passbands.The construction of a typical two-cavity interference filter, along with an exploded view showing the detailed structure of the all-dielectric multilayer bandpass filter film, is shown in the accompanying figure. H symbolizes a precisely quarter-wavelength optical thickness layer of a high-index material (typically zinc sulfide, ZnS), while L symbolizes a precisely quarterwavelength optical thickness layer of a low-index material (typically cryolite, Na3AIF6). The spacer is a layer of high-index material of half-wavelength thickness, and the absentee, or coupling, layer is a layer of low-index material of half-wavelength thickness. Here, wavelength refers to the wavelength of peak transmittance. Layers are formed by vacuum deposition. The aluminum rings protect the edges, and epoxy cement protects the films from moisture and laminates the bandpass and blocker sections together. Material Properties air gap may vary from a fraction of a millimeter to several centimeters.The Fabry-Perot is a useful spectroscopic tool. It provided much of the early motivation to develop quality thin films for the high-reflectance mirrors needed for high finesse. Fabry-Perot interferometers can be constructed from purely metallic coatings, but high absorption losses limit performance. 7 Deviation from Center Wavelength in FWHM Units x= (FWHM ) l-lmax Note: The actual FWHM will be different in each case. Effect of number of cavities on passband shape for typical interference filters with 10 nm FWHM Laser Guide marketplace.idexop.com Optical Filter Coatings A49 OPTICAL COATINGS WAVELENGTH DEPENDENCE ON ANGLE OF INCIDENCE A common characteristic of single and multilayer dielectric coatings and interference filters is that transmittance and reflectance spectra shift to shorter wavelengths as they are tilted from normal to oblique incidence. This applies to both edge and bandpass filters. As tilt is increased in filters constructed with metallic layers, the transmittance peak splits into two orthogonally polarized peaks which shift to shorter wavelengths at different rates. CVI Laser Optics narrowband filters are made with all-dielectric multilayers to prevent this transmittance split from occurring. Optical Coatings and Materials ADDITIONAL BLOCKING Close to the passband, and on the long wavelength side, multilayer blocking structures (usually metal dielectric hybrid filters) are used in CVI Laser Optics passband filters to limit transmittance to 0.01%. More stringent blocking is possible, but this increases filter cost and compromises maximum transmission. Colored glass is often used to suppress transmission on the short wavelength side of the passband. TABLE OF NORMALIZED PASSBAND SHAPE The graph showing change in filter performance as a function of the number of cavities is qualitatively useful, but the following bandwidth table gives quantitative data. This table applies to zinc sulfide (ZnS)/cryolite (Na3AlF6) interference filters of any FWHM. The shift to shorter wavelengths at oblique incidence is very useful in tuning bandpass filters from one wavelength to another, or adjusting the half-power point wavelengths of edge filters in collimated light. If the to shift wavelength enhances the usefulness to interference filter sets. Each filter in variable bandpass sets can be angle tuned down to the normal incidence transmission wavelength of the next filter in the set. Wavelengths of transmittance peaks or cavity resonances for Fabry-Perot interferometers and bandpass interference filters are approximately governed, for observers within the cavity or spacer, by the equation Although the table is strictly applicable from 400 nm to 1100 nm, CVI Laser Optics ultraviolet filters, which are of different composition, have very similar characteristics. The table shows the functional dependence of normalized passband shape on the number of cavities used in filter construction, with FWHM arbitrary but held fixed. Because transmittance is normalized to peak value, the table is applicable to blocked and unblocked filters. To apply the table to a specific filter, simply multiply by peak transmittance. Both minimum and maximum full bandwidths are shown at various normalized transmittance levels. The difference between minimum and maximum full bandwidths allows for spacer material choice and filter-to-filter variation. Normal incidence is assumed. Beyond the spectral range displayed here, our filters of two-, three-, and four-cavity construction are supplied with blocking structures tthat limit absolute transmittance out of band to less than 10–4. 2netcosθ = mλ where ne is the spacer refractive index, t is the spacer thickness, θ is the internal angle of incidence (measured within the cavity or spacer), m is the order number of interference (a positive integer), and λ is the wavelength VXEVWUDWH VWDFN XQILOWHUHGOLJKWLQ VLPSOHVWSHULRG VSDFHU EDQGSDVV VHFWLRQ VWDFN PXOWLOD\HU GLHOHFWULFEDQGSDVVILOWHU HSR[\ PHWDOGLHOHFWULFPXOWLOD\HUEORFNLQJILOWHU EORFNHU VHFWLRQ VLPSOHVWSHULRG VSDFHU RSWLRQDOFRORUHGJODVV VWDFN ILOWHUHGOLJKWRX W VLPSOHVWSHULRG DEVHQWHH VWDFN VXEVWUDWH DOXPLQXP ULQ J DOOGLHOHFWUL F FDYLW\ + / DOOGLHOHFWUL F FDYLW\ VLPSOHVWSHULRG HSR[\ Cross section of a typical two-cavity interference filter A50 Optical Filter Coatings 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials Bandwidth at Various Normalized Transmittances Normalized Transmittance Level (% of peak) Minimum Maximum 1 90 0.30 0.35 10 2.50 3.00 1 8.00 10.00 90 0.50 0.60 3.50 2 4 2.80 5.50 6.30 0.01 10.00 15.00 90 0.70 0.80 1 1.90 2.20 0.1 2.90 3.20 0.01 4.90 5.40 90 0.85 0.90 1 1.50 1.65 0.1 2.00 2.25 0.01 3.50 4.25 2 n l = l max 1 4 0 sin 2 f ne CVI Laser Optics can supply filters listed in this section in volume to OEM users. Volume users frequently do not require an individual spectrophotometer curve for each filter. CVI Laser Optics can also supply custom interference filters. When specifying a custom filter, please give us the center wavelength, FWHM, blocking, minimum peak transmission, and size. Each of these factors has a significant impact on cost and therefore should not be specified more tightly than required by the application. By curve-fitting the second formula above (from which t is absent) to measured angle shifts at small angles, the effective index and angle at which blocker displacement of the peak becomes significant can, in principle, be found. In the absence of actual measurements, the formula probably should not be trusted much beyond five or ten degrees. With suitable interpretation, the formula can be applied to prominent landmarks in transmittance and reflectance spectra of edge filters, multi- and single-layer coatings, and all interference filters. Machine Vision Guide In terms of the external angle of incidence, Ø, it can be shown that the wavelength of peak transmittance at small angles from normal incidence is given by High-Volume or Special Filters for OEMs Gaussian Beam Optics there are, between cavity resonance transmittance peaks, additional broader peaks that correspond to the wavelengths at which the dielectric stacks are ineffective as resonant reflectors. Only a single resonance transmittance peak is selected for use and allowed to appear in the output spectrum of a complete (blocked) interference filter. Blocking techniques are highly effective. A three-cavity filter at the 1% normalized transmittance level (1% of peak) would have a nominal full bandwidth (full width at 1% of maximum) between the limits of 1.9 and 2.2. If the FWHM were 5.0 nm, the full width at 1% of maximum would be between 9.5 and 11.0 nm. Fundamental Optics 3 1 0.1 FWHM Example Optical Specifications Number of Cavities where n0 is the external medium refractive index (n0 = 1.0 in air) and ne is the spacer effective refractive index. The difference λmax –λ is the angle shift. The spacer effective index is dependent on wavelength, film material, and order number because of multilayer effects. The effective index and actual refractive index of spacer material is not equivalent, although the same symbol ne is used for both. Material Properties of a particular resonance transmittance peak. This equation is often called the monolayer approximation. The formula can be satisfied simultaneously for many different order number and wavelength combinations. Corresponding to each such combination there is, in principle, a different resonance transmittance peak for an unblocked filter. For an all-dielectric filter Laser Guide marketplace.idexop.com Optical Filter Coatings A51 OPTICAL COATINGS Optical Coatings and Materials In many applications, angle shifts can be safely ignored. Advanced radiometer designs are necessary only when wide fields and narrow bandwidths are simultaneously required. For example, if the desired monochromatic signal is to be at least 90 % of Tpeak throughout the field and the filter has a narrow 1.0 nm FWHM, the angular radius is only about 2.5º. Most CVI Laser Optics filters use a high-index spacer (usually zinc sulfide) to minimize angle shift. Some use low-index spacers (usually cryolite) to achieve higher transmittance or narrower bandwidths. CORRECT INTERFERENCE FILTER ORIENTATION A good rule of thumb, especially important if there is risk of overheating or solarization, is that interference filters should always be oriented with the shiniest (metallic) and most nearly colorless side toward the source in the radiant flux. This orientation will minimize thermal load on the absorbing-glass blocking components. Reversing filter orientation will have no effect on filter transmittance near or within the passband. TEMPERATURE EFFECTS, LIMITS, AND THERMAL SHOCK The transmittance spectrum of an interference filter is slightly temperature dependent. As temperature increases, all layer thicknesses increase. At the same time, all layer indices change. These effects combine in such a way that the transmittance spectrum shifts slightly to longer wavelengths with increasing temperature. The thermal coefficient is a function of wavelength, as shown in the following table. CVI Laser Optics interference filters are designed for use at 20°C. Unless bandpass filters with extremely narrow FWHMs are used at very different temperatures, the transmittance shifts indicated in the table are negligible. Our standard interference filters can be used at temperatures down to –50°C. Thermal contraction will result in permanent filter damage below this temperature. High-temperature limits depend on filter design: 70°C is a safe and conservative limit for all filters. Some of our standard filters can accommodate temperatures up to 125°C. As a general rule, it is unwise to subject interference filters to thermal shock, especially as the lower limit of –50°C is approached. Temperature change rates should not exceed 5°C per minute. A52 Optical Filter Coatings Temperature Dependence of Peak Transmittance Wavelength (nm) Temperature Coefficient of Shift (nm per ºC) 400 0.016 476 0.019 508 0.020 530 0.021 557 0.021 608 0.023 630 0.023 643 0.024 710 0.026 820 0.027 APPLICATION NOTE Interference Filter Usage Narrowband interference filters are extremely angle sensitive. The transmittance of a filter with a FWHM of 1.0 nm will decrease by 10%, at the transmission wavelength, for field angles of only 2.5º. For field angles of 5º, the transmittance decreases collimated portions of optical paths by over 90%. It is important, therefore, to use narrowband interference filters. The illustration shows the design of a narrow-field spectral radiometer for infinite conjugate ratio use, and it indicates the proper interference filter location. The radiometer consists of an interference filter, objective lens, field lens, field stop, and detector. The field lens, which images the objective lens onto the detector’s sensitive area, ensures uniform illumination of the detector. The field of view is limited by a field stop placed close to the field lens. LQWHUIHUHQFHILOWHU REMHFWLYHOHQV GHWHFWRU ILHOGVWRS ILHOGOHQV 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials NEUTRAL DENSITY FILTERS removal or perforation of the substrate to achieve D=0.0 accurately. In such instruments, performance is referenced to a blank, and density differences, not the densities themselves, are important. All CVI Laser Optics ND filters pass stringent optical and mechanical tests. Individual ND filters and ND filter sets are in stock and ready to ship. Our applications engineers will be pleased to assist you in the selection and application of standard or custom filters. Transmittance and density values may, like reflectance values, refer to either small angular fields (specular or undeviated values) or very large angular fields (diffuse or hemispherical values). The measurements that determine hemispherical values include both specular and scattered contributions. Density and relative density values for CVI Laser Optics ND filters are specular values based on external transmittance. D= log(1/T), or T = 10–D . Because of Beer’s and Fechner’s laws (sensation proportional to logarithm of stimulus, applicable to vision as a special case), it has been historically convenient to use the logarithmic density scale, instead of a transmittance scale. While optical density is dimensionless, the notation 0.50 D is sometimes used to mean 0.50 density units, or simply a density of 0.50. Two or more ND filters can be used to achieve values of transmittance or density not otherwise available.If they are arranged so that multiple reflections between them do not occur in the direction of interest, transmittance values are multiplicative, whereas optical densities are additive. By combining various filters, many separate density values may be achieved. Gaussian Beam Optics Dr = D – D 0 4n Fundamental Optics Optical density is analogous to the definition of decibel as used in electronics. ND filters used in combinations are additive if multiple reflections between filters do not occur in the direction of interest. The reciprocal of transmittance, 1/T, is called opacity. Also in widespread use is relative optical density, Dr, the difference between density D of a coated substrate and density D0 of an uncoated region of the same substrate: (n + 1) 2 Optical Specifications Optical density (D) is defined as the base 10 logarithm of the reciprocal of transmittance (T): D o = 2 log Material Properties Neutral-density (ND) filters attenuate, split, or combine beams in a wide range of irradiance ratios with no significant dependence on wavelength. These carefully prepared filters find wide application for precise attenuation or control of light. For example, beams can be attenuated to levels where photometers or radiometers are most accurate and linear, thereby extending their useful range. or CVI Laser Optics provides two types of ND filters: metallic (reflective) and glass (absorptive). D = D r = D0 METALLIC NEUTRAL-DENSITY FILTERS At 550 nm, D0 is typically about 0.0376 for N-BK7, and about 0.0309 for synthetic fused silica. Relative density Dr, not absolute density D, is the quantity that appears on individual microdensitometer traces supplied with CVI Laser Optics circular variable filters, because many variable filter applications require focal plane position or plate aberration constancy. This requirement prohibits All CVI Laser Optics metallic ND filters are made with N-BK7-fine annealed glass, or optical-quality synthetic fused silica. Vacuum deposition is used to apply a thin film of several special metallic alloys to the substrate. These alloys have been chosen to create a spectraldensity curve that is flatter over a wider range than the curves of most pure metals. Substrate materials are chosen for homogeneity, transmittance uniformity, Machine Vision Guide In terms of refractive index n, Laser Guide marketplace.idexop.com Neutral Density Filters A53 OPTICAL COATINGS Optical Coatings and Materials finishing characteristics, and (in the case of synthetic fused silica) ultraviolet transmittance. Substrates are polished to minimize light scattering. Metallic ND filters can be used at any wavelength between 200 and 2500 nm (fused silica), or between 350 and 2500 nm (N-BK7). Their operation depends on absorption in, and reflection from, the thin metallic film. When used in high-intensity beams, ND filters should be oriented with the metallic film facing toward the source to minimize substrate absorption and heating. Alloy films are corrosion resistant and do not age at normal temperatures. Adhesion of alloy films to their substrates is tenacious and unaffected by moisture and most solvents from –73°C (–100°F) to +150°C (302°F). Exposure to higher temperatures should be avoided because it causes film oxidation and increased transmittance. These filters are not suitable for use with high-power pulsed lasers. ABSORPTIVE NEUTRAL-DENSITY FILTERS Absorptive ND filters provide an alternative to metallic ND filters. The neutrality of the filter is a function of material and thickness. Since there can be large variations between glass melts, actual thickness and glass material may vary in order to guarantee optical density. These filters are recommended for low-power applications only, because of their absorbing properties. FILTER SET CONTENTS Each individual filter is checked, and an optical density spectrophotometer curve from the coating run is included with each filter. Measured ranges are from 200 to 700 nm for sets on synthetic fused silica, and from 350 to 700 nm for sets on N-BK7 substrates. Individual spectrophotometer curves are available on special request. Some sets include a blank (uncoated) substrate of the same material and thickness used for the filters. This blank is often very helpful for aligning and focusing optical systems before inserting the ND filter. Each ND filter set is packaged in a wooden case. A54 Neutral Density Filters 1-505-298-2550 OPTICAL COATINGS Optical Coatings & Materials LASER-INDUCED DAMAGE Optical Specifications Fundamental Optics For each damage-threshold specification, the information given is the peak fluence (energy per square centimeter), pulse width, peak irradiance (power per square centimeter), and test wavelength. The peak fluence is the total energy per pulse, the pulse width is the full width at half maximum (FWHM), and the test wavelength is the wavelength of the laser used to incur the damage. The peak irradiance is the energy of each pulse divided by the effective pulse length, which is from 12.5 to 25 percent longer than the pulse FWHM. All tests are performed at a repetition rate of 20 Hz for 10 seconds at each test point. This is important because longer durations can cause damage at lower fluence levels, even at the same repetition rate. When choosing a coating for its power-handling capabilities, some simple guidelines can be followed to make the decision process easier. First, the substrate material is very important. Higher damage thresholds can be achieved using fused silica instead of N-BK7. Second, consider the coating. Metal coatings have the lowest damage thresholds. Broadband dielectric coatings, such as the HEBBAR™ and MAXBRIte™ are better, but singlewavelength or laser-line coatings, such as the V coatings and the MAX-R™ coatings, are better still. If even higher thresholds are needed, then high energy laser (HEL) coatings are required. If you have any questions or concerns regarding the damage levels involved in your applications, please contact a CVI Laser Optics applications engineer. Material Properties CVI Laser Optics conducts laser-induced damage testing of our optics. Although our damage thresholds do not constitute a performance guarantee, they are representative of the damage resistance of our coatings. Occasionally, in the damage-threshold specifications, a reference is made to another coating because a suitable high-power laser is not available to test the coating within its design wavelength range. The damage threshold of the referenced coating should be an accurate representation of the coating in question. Gaussian Beam Optics The damage resistance of any coating depends on substrate, wavelength, and pulse duration. Improper handling and cleaning can also reduce the damage resistance of a coating, as can the environment in which the optic is used. These damage threshold values are presented as guidelines and no warranty is implied. Machine Vision Guide Laser Guide marketplace.idexop.com Laser-Induced Damage A55 OPTICAL COATINGS OEM AND SPECIAL COATINGS Optical Coatings and Materials CVI Laser Optics maintains advanced coating capabilities. In the last few years, CVI Laser Optics has expanded and improved these coating facilities to take advantage of the latest developments in thinfilm technology. The resulting operations can provide high-volume coatings at competitive prices to OEM customers, as well as specialized, high-performance coatings for the most demanding user. The most important aspect of our coating capabilities is our expert design and manufacturing staff. This group blends years of practical experience with recent academic research knowledge. With a thorough understanding of both design and production issues, CVI Laser Optics excels at producing repeatable, high-quality coatings at competitive prices. TECHNICAL SUPPORT Expert CVI Laser Optics applications engineers are available to discuss your system requirements. Often a simple modification to a system design can enable catalog components or coatings to be substituted for special designs at a reduced cost, without affecting performance. USER-SUPPLIED SUBSTRATES CVI Laser Optics not only coats catalog and custom optics with standard and special coatings but also applies these coatings to user-supplied substrates. A significant portion of our coating business involves applying standard or slightly modified catalog coatings to special substrates. HIGH VOLUME The high-volume output capabilities of the CVI Laser Optics coating departments result in very competitive pricing for large-volume special orders. Even the smallorder customer benefits from this large volume. Small quantities of special substrates can be cost-effectively coated with popular catalog coatings during routine production runs. CUSTOM DESIGNS A large portion of the work done at the CVI Laser Optics coating facilities involves special coatings designed and manufactured to customer specifications. These designs cover a wide range of wavelengths, from the infrared to deep ultraviolet, and applications ranging from basic research through the design and manufacture of industrial and medical products. The most common special coating requests are for modified catalog coatings, which usually involve a simple shift in the design wavelength. A56 OEM and Special Coatings 1-505-298-2550 Optical Coatings & Materials MATERIAL PROPERTIES OPTICAL PROPERTIES Material Properties INTRODUCTIONA58 A59 MECHANICAL AND CHEMICAL PROPERTIESA61 MAGNESIUM FLUORIDE A64 CALCIUM FLUORIDE A65 SUPRASIL 1 A66 UV-GRADE SYNTHETIC FUSED SILICA A67 CRYSTAL QUARTZ A69 Fundamental Optics A63 Optical Specifications LENS MATERIALS CALCITEA70 A71 OPTICAL CROWN GLASS A73 LOW-EXPANSION BOROSILICATE GLASS A74 Gaussian Beam Optics SCHOTT GLASS ZERODUR®A75 INFRASIL 302 A76 SAPPHIREA77 ZINC SELENIDE A78 Machine Vision Guide SILICONA79 GERMANIUMA80 MATERIAL PROPERTIES OVERVIEW A81 Laser Guide marketplace.idexop.com A57 MATERIAL PROPERTIES INTRODUCTION Material Properties Glass manufacturers provide hundreds of different glass types with differing optical transmission and mechanical strengths. CVI Laser Optics has simplified the task of selecting the right material for an optical component by offering each of our standard components in a single material, or in a small range of materials best suited to typical applications. There are, however, two instances in which one might need to know more about optical materials: one might need to determine the performance of a catalog component in a particular application, or one might need specific information to select a material for a custom component. The information given in this chapter is intended to assist in that process. The most important material properties to consider in regard to an optical element are as follows: X Transmission X Index characteristics X Mechanical X Chemical The thermal expansion coefficient can be particularly important in applications in which the part is subjected to high temperatures, such as high-intensity projection systems. This is also of concern when components must undergo large temperature cycles, such as in optical systems used outdoors. MECHANICAL CHARACTERISTICS The mechanical characteristics of a material are significant in many areas. They can affect how easy it is to fabricate the material into shape, which affects product cost. Scratch resistance is important if the component will require frequent cleaning. Shock and vibration resistance are important for military, aerospace, or certain industrial applications. Ability to withstand high pressure differentials is important for windows used in vacuum chambers. versus wavelength of refraction X Thermal THERMAL CHARACTERISTICS characteristics characteristics X Cost CHEMICAL CHARACTERISTICS The chemical characteristics of a material, such as acid or stain resistance, can also affect fabrication and durability. As with mechanical characteristics, chemical characteristics should be taken into account for optics used outdoors or in harsh conditions. TRANSMISSION VERSUS WAVELENGTH COST A material must transmit efficiently at the wavelength of interest if it is to be used for a transmissive component. A transmission curve allows the optical designer to estimate the attenuation of light as a function of wavelength caused by internal material properties. For mirror substrates, the attenuation may be of no consequence. Cost is almost always a factor to consider when specifying materials. Furthermore, the cost of some materials, such as UV-grade synthetic fused silica, increases sharply with larger diameters because of the difficulty in obtaining large pieces of the material. INDEX OF REFRACTION The index of refraction, as well as the rate of change of index with wavelength (dispersion), might require consideration. High-index materials allow the designer to achieve a given power with less surface curvature, typically resulting in lower aberrations. On the other hand, most high-index flint glasses have higher dispersion, resulting in more chromatic aberration in polychromatic applications. They also typically have poorer chemical characteristics than lower-index crown glasses. A58 Introduction 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials OPTICAL PROPERTIES TRANSMISSION t1 t2 = 1 − 2r + r 2 (2.2) m=− 1 ln Ti .(2.5) tc When it is necessary to find transmittance at wavelengths other than those for which Ti is tabulated, use linear interpolation. The on-axis Te value is normally the most useful, but some applications require that transmittance be known along other ray paths, or that it be averaged over the entire lens surface. The method outlined above is easily extended to encompass such cases. Values of t1 and t2 must be found from complete Fresnel formulas for arbitrary angles of incidence. The angles of incidence will be different at the two surfaces; therefore, t1 and t2 will generally be unequal. Distance tc, which becomes the surface-to-surface distance along a particular ray, must be determined by ray tracing. It is necessary to account separately for the s- and p-planes of polarization, and it is usually sufficient to average results for both planes at the end of the calculation. Gaussian Beam Optics where e is the base of the natural system of logarithms, µ is the absorption coefficient of the lens material, and tc is the lens center thickness. This allows for the possibility that the lens surfaces might have unequal transmittances (for example, one is coated and the other is not). Assuming that both surfaces are uncoated, Thus where the bar denotes averaging. In portions of the spectrum where absorption is strong, a value for Ti is typically given only for the lesser thickness. Then Fundamental Optics (2.1) Te = t1 t2Ti = t1 t2 e − mtc To calculate either Ti or the Te for a lens at any wavelength of interest, first find the value of absorption coefficient µ. Typically, internal transmittance Ti is tabulated as a function of wavelength for two distinct thicknesses tc1 and tc2, and m must be found from these. Optical Specifications External transmittance is the single-pass irradiance transmittance of an optical element. Internal transmittance is the single-pass irradiance transmittance in the absence of any surface reflection losses (i.e., transmittance of the material itself). External transmittance is of paramount importance when selecting optics for an image-forming lens system because external transmittance neglects multiple reflections between lens surfaces. Transmittance measured with an integrating sphere will be slightly higher. If Te denotes the desired external irradiance transmittance, Ti the corresponding internal transmittance, t1 the single-pass transmittance of the first surface, and t2 the single-pass transmittance of the second surface, then material dispersion formula found in the next section. These results are applicable to monochromatic. Both µ and n are functions of wavelength. Material Properties The most important optical properties of a material are its internal and external transmittances, surface reflectance, and refractive indexe. The formulas that connect these variables in the on-axis case are presented below. where n − 1 r= n + 1 Machine Vision Guide t1t2n =− 11−22r + r 2 where r= n + 1 where REFRACTIVE INDEX AND DISPERSION 2 (2.3) is the single-surface single-pass irradiance reflectance at normal incidence as given by the Fresnel formula. The refractive index n must be known or calculated from the The Schott Optical Glass catalog offers nearly 300 different optical glasses. For lens designers, the most important difference among these glasses is the index of refraction and dispersion (rate of change of index with wavelength). Typically, an optical glass is specified by its index of refraction at a wavelength in the middle of the visible spectrum, usually 587.56 nm (the helium d-line), and by the Abbé v-value, defined to be Laser Guide marketplace.idexop.com Optical Properties A59 MATERIAL PROPERTIES Material Properties vd = (nd–1)/ (nF-nC). The designations F and C stand for 486.1 nm and 656.3 nm, respectively. Here, vd shows how the index of refraction varies with wavelength. The smaller vd is, the faster the rate of change is. Glasses are roughly divided into two categories: crowns and flints. Crown glasses are those with nd < 1.60 and vd > 55, or nd > 1.60 and vd > 50. The others are flint glasses. The refractive index of glass from 365 to 2300 nm can be calculated by using the formula n2 − 1 = B1l 2 B l2 B l2 + 22 + 23 2 l − C1 l − C2 l − C3 (2.6) glass is annealed (heated and cooled) to remove any residual stress left over from the original manufacturing process. Schott Glass defines fine annealed glass to have a stress birefringence of less than or equal to 10 nm/ cm for diameters less than 300 mm and for thicknesses less than or equal to 60 mm. For diameters between 300 and 600 mm and for thicknesses between 60 and 80 mm, stress birefringence would be less than or equal to 12 nm/cm. APPLICATION NOTE Fused-Silica Optics Here λ, the wavelength, must be in micrometers, and the constants B1 through C3 are given by the glass manufacturer. Values for other glasses can be obtained from the manufacturer’s literature. This equation yields an index value that is accurate to better than 1x10–5 over the entire transmission range, and even better in the visible spectrum. Synthetic fused silica is an ideal optical material for many laser applications. It is transparent from as low as 180 nm to over 2.0 µm, has low coefficient of thermal expansion, and is resistant to scratching and thermal shock. OTHER OPTICAL CHARACTERISTICS REFRACTIVE INDEX HOMOGENEITY The tolerance for the refractive index within melt for all Schott fine annealed glass used in CVI Laser Optics catalog components is ±1x10–4. Furthermore, the refractive index homogeneity, a measure of deviation within a single piece of glass, is better than ±2x10–5. STRIAE GRADE Striae are thread-like structures representing subtle but visible differences in refractive index within an optical glass. Striae classes are specified in ISO 10110. All CVI Laser Optics catalog components that utilize Schott optical glass are specified to have striae that conform to ISO 10110 class 5 indicating that no visible striae, streaks, or cords are present in the glass. STRESS BIREFRINGENCE Mechanical stress in optical glass leads to birefringence (anisotropy in index of refraction) which can impair the optical performance of a finished component. Optical A60 Optical Properties 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials MECHANICAL AND CHEMICAL PROPERTIES MICROHARDNESS The most important mechanical property of glass is microhardness. A precisely specified diamond scribe is placed on the glass surface under a known force. The indentation is then measured. The Knoop and the Vickers microhardness tests are used to measure the hardness of a polished surface and a freshly fractured surface, respectively. Fundamental Optics CLIMATIC RESISTANCE Humidity can cause a cloudy film to appear on the surface of some optical glass. Climatic resistance expresses the susceptibility of a glass to high humidity and high temperatures. In this test, glass is placed in a water vapor-saturated environment and subjected to a temperature cycle which alternately causes condensation and evaporation. The glass is given a rating from 1 to 4 depending on the amount of surface scattering induced by the test. A rating of 1 indicates little or no change after 30 hours of climatic change; a rating of 4 means a significant change occurred in less than 30 hours. ALKALI AND PHOSPHATE RESISTANCE Alkali resistance is also important to the lens manufacturer since the polishing process usually takes place in an alkaline solution. Phosphate resistance is becoming more significant as users move away from cleaning methods that involve chlorofluorocarbons (CFCs) to those that may be based on traditional phosphate-containing detergents. In each case, a twodigit number is used to designate alkali or phosphate resistance. The first number, from 1 to 4, indicates the length of time that elapses before any surface change occurs in the glass, and the second digit reveals the extent of the change. Optical Specifications To quantify the chemical properties of glasses, glass manufacturers rate each glass according to four categories: climatic resistance, stain resistance, acid resistance, and alkali and phosphate resistance. 0.3 acid solution, and values from 51 to 53 are used for glass with too little resistance to be tested with such a strong solution. Material Properties Mechanical and chemical properties of glass are important to lens manufacturers. These properties can also be significant to the user, especially when the component will be used in a harsh environment. Different polishing techniques and special handling may be needed depending on whether the glass is hard or soft, or whether it is extremely sensitive to acid or alkali. Gaussian Beam Optics STAIN RESISTANCE Stain resistance expresses resistance to mildly acidic water solutions, such as fingerprints or perspiration. In this test, a few drops of a mild acid are placed on the glass. A colored stain, caused by interference, will appear if the glass starts to decompose. A rating from 0 to 5 is given to each glass, depending on how much time elapses before stains occur. A rating of 0 indicates no observed stain in 100 hours of exposure; a rating of 5 means that staining occurred in less than 0.2 hours. Machine Vision Guide ACID RESISTANCE Acid resistance quantifies the resistance of a glass to stronger acidic solutions. Acid resistance can be particularly important to lens manufacturers because acidic solutions are typically used to strip coatings from glass or to separate cemented elements. A rating from 1 to 4 indicates progressively less resistance to a pH Laser Guide marketplace.idexop.com Mechanical and Chemical Properties A61 MATERIAL PROPERTIES Material Properties Knoop Hardness Values for Standard Optical Materials Material Knoop Hardness Magnesium Fluoride 415 Calcium Fluoride 158 Fused Silica 522 BK7 (N-BK7) 610 Optical Crown Glass 542 Borosilicate Glass 480 Zerodur 620 Zinc Selenide 112 Silicon 1100 Germanium 780 APPLICATION NOTE Glass Manufacturers The catalogs of optical glass manufacturers contain products covering a very wide range of optical characteristics. However, it should be kept in mind that the glass types that exhibit the most desirable properties in terms of index of refraction and dispersion often have the least practical chemical and mechanical characteristics. Furthermore, poor chemical and mechanical attributes translate directly into increased component costs because working these sensitive materials increases fabrication time and lowers yield. Please contact us before specifying an exotic glass in an optical design so that we can advise you of the impact that that choice will have on part fabrication. A62 Mechanical and Chemical Properties 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials LENS MATERIALS Lens Material Table Material Synthetic Fused Silica UV Grade N-BK7 Grade A Fine Annealed LUK-UV PLCC-UV LUD-UV LUB-UV BICX-UV BICC-UV RCX-UV RCC-UV SCX-UV SCC-UV CLCX-UV CLCC-UV PLCX-EUV LPX-C LPK-C PLCX-C PLCC-C LDX-C LDK-C BICX-C BICC-C LCP-C LCN-C RCX-C RCC-C SCX-C SCC-C CLCC-C CLCX-C MENP-C MENN-C Gaussian Beam Optics BFPL-C LaSFN9 Grade A Fine Annealed SK11 and SF5 Grade A Fine Annealed BaK1 Grade A Fine Annealed SF11 Grade-A Fine Annealed LMS and selected LPX series LAI selected LPX series PLCX-SF11 PLCC-SF11 LAP LAN Optical Crown Glass LAG Low-Expansion Borosilicate Glass (LEBG) selected CMP series Sapphire Calcium Fluoride Magnesium Fluoride Zinc Selenide Various Glass Combinations (including lead- and arsenic-free glasses) PXS PLCX-CFUV PLCX-CFIR RCX-CFUV RCC-CFUV BICX-MF PLCX-MF PLCX-ZnSe MENP-ZnSe LAO LAL AAP FAP HAP HAN YAP YAN LBM LSL GLC OAS Machine Vision Guide A borosilicate crown glass, N-BK7, is the material used in many CVI Laser Optics products. N-BK7 performs well in chemical tests so that special treatment during polishing is not necessary. N-BK7, a relatively hard glass, does not scratch easily and can be handled without special precautions. The bubble and inclusion content of N-BK7 is very low, with a cross-section total less than 0.029 mm2 per 100 cm3. Another important characteristic of N-BK7 is its excellent transmittance, at wavelengths as low as 350 nm. Because of these properties, N-BK7 is used widely throughout the optics industry. A variant of N-BK7, designated UBK7, has transmission at wavelengths as low as 300 nm. This special glass is useful in applications requiring a high index of refraction, the desirable chemical properties of N-BK7, and transmission deeper LUP-UV PLCX-UV BFPL-UV Synthetic Fused Silica Excimer Grade The following physical constant values are reasonable averages based on historical experience. Individual material specimens may deviate from these means. Materials having tolerances more restrictive than those published in the rest of this chapter, or materials traceable to specific manufacturers, are available only on special request. N-BK7 OPTICAL GLASS Product Code Fundamental Optics The performance of optical lenses and prisms depends on the quality of the material used. No amount of skill during manufacture can eradicate striae, bubbles, inclusions, or variations in index. CVI Laser Optics takes considerable care in its material selection, using only first-class optical materials from reputable glass manufacturers. The result is reliable, repeatable, consistent performance. CVI Laser Optics reserves the right, without prior notice, to make material changes or substitution on any optical component. Optical Specifications Glass type designations and physical constants are the same as those published by Schott Glass. CVI Laser Optics occasionally uses corresponding glasses made by other glass manufacturers but only when this does not result in a significant change in optical properties. into the ultraviolet range. N-BK7 refers to the lead and arsenic-free version of BK7, with most optical properties identical between the two. Material Properties CVI Laser Optics lenses are made of synthetic fused silica, N-BK7 grade A fine annealed glass, and several other materials. The following table identifies the materials used in CVI Laser Optics lenses. Some of these materials are also used in prisms, mirror substrates, and other products. Laser Guide marketplace.idexop.com Lens Materials A63 MATERIAL PROPERTIES MAGNESIUM FLUORIDE Material Properties Magnesium Fluoride (MgF2) is a tetragonal positive birefringent crystal grown using the vacuum Stockbarger technique. MgF2 is a rugged material resistant to chemical etching as well as mechanical and thermal shock. High-vacuum UV transmission and resistance to laser damage make MgF2 a popular choice for VUV and excimer laser windows, polarizers, and lenses. Refractive Index of Magnesium Fluoride Wavelength (nm) Index of Refraction Ordinary Ray (nO) Index of Refraction Extraordinary Ray (nE) 193 1.42767 1.44127 213 1.41606 1.42933 222 1.41208 1.42522 226 1.41049 1.42358 244 1.40447 1.41735 248 1.40334 1.41618 257 1.40102 1.41377 Knoop Hardness: 415 266 1.39896 1.41164 Coefficient of Thermal Expansion: 8.48x10–6/°C (perpendicular to c axis) 13.70x10–6/°C (parallel to c axis) 280 1.39620 1.40877 308 1.39188 1.40429 Melting Point: 1585°C 325 1.38983 1.40216 337 1.38859 1.40086 351 1.38730 1.39952 355 1.38696 1.39917 Specifications Density: 3.177 g/cm3 Young’s Modulus: 138.5 GPa Poisson’s Ratio: 0.271 Dispersion Constants (Ordinary Ray): B1=4.87551080x10–1 B2=3.98750310x10–1 B3=2.31203530 C1=1.88217800x10–3 C2=8.95188847x10–3 C3=5.66135591x102 Dispersion Constants (Extraordinary Ray): B1=4.13440230x10–1 B2=5.04974990x10–1 B3=2.49048620 C1=1.35737865x10–3 C2=8.23767167x10–3 C3=5.65107755x102 A64 Magnesium Fluoride 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials CALCIUM FLUORIDE Specifications Density: 3.18 g/cm3 @ 25°C Young’s Modulus: 1.75x107 psi Material Properties Calcium fluoride (CaF2), a cubic single-crystal material, has widespread applications in the ultraviolet and infrared spectrum. CaF2 is an ideal material for use with excimer lasers. It can be manufactured into windows, lenses, prisms, and mirror substrates. Poisson’s Ratio: 0.26 Knoop Hardness: 158 3HUFHQW([WHUQDO 7UDQVPLWWDQFH RUGLQDU\UD\ H[WUDRUGLQDU\UD\ :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 5 mm thick uncoated calcium fluoride Dispersion Constants: B1 = 0.5675888 B2 = 0.4710914 B3 = 3.8484723 C1 = 0.00252643 C2 = 0.01007833 C3 = 1200.5560 Refractive Index of Calcium Fluoride Wavelength (µm) Index of Refraction 0.193 1.501 0.248 1.468 0.257 1.465 0.266 1.462 0.308 1.453 0.355 1.446 0.486 1.437 0.587 1.433 0.65 1.432 0.7 1.431 1.0 1.428 1.5 1.426 2.0 1.423 2.5 1.421 3.0 1.417 4.0 1.409 5.0 1.398 6.0 1.385 7.0 1.369 8.0 1.349 Machine Vision Guide Melting Point: 1360°C Gaussian Beam Optics Excimer-grade CaF2 provides the combination of deep ultraviolet transmission (down to 157 nm), high damage threshold, resistance to color-center formation, low fluorescence, high homogeneity, and low stressbirefringence characteristics required for the most demanding deep ultraviolet applications. Coefficient of Thermal Expansion: 18.9x10–6/°C (20°–60°C) Fundamental Optics To meet the need for improved component lifetime and transmission at 193 nm and below, manufacturers have introduced a variety of inspection and processing methods to identify and remove various impurities at all stages of the production process. The needs for improved material homogeneity and stress birefringence have also caused producers to make alterations to the traditional Stockbarger approach. These changes allow tighter temperature control during crystal growth, as well as better regulation of vacuum and annealing process parameters. Thermal Coefficient of Refraction: dn/dT =–10.6x10–6/°C Optical Specifications CaF2 transmits over the spectral range of about 130 nm to 10 mm. Traditionally, it has been used primarily in the infrared, rather than in the ultraviolet. CaF2 occurs naturally and can be mined. It is also produced synthetically using the time- and energy-consuming Stockbarger method. Unfortunately, achieving acceptable deep ultraviolet transmission and damage resistance in CaF2 requires much greater material purity than in the infrared, and it completely eliminates the possibility of using mined material. Laser Guide marketplace.idexop.com Calcium Fluoride A65 MATERIAL PROPERTIES SUPRASIL 1 Material Properties Suprasil 1 is a type of fused silica with high chemical purity and excellent multiple axis homogeneity. With a metallic content less than 8 ppm, Suprasil 1 has superior UV transmission and minimal fluorescence. Suprasil 1 is primarily used for low fluorescence UV windows, lenses and prisms where multiple axis homogeneity is required. Refractive Index of Suprasil 1* Wavelength (nm) Index of Refraction 193.4 1.56013 248.4 1.50833 266.0 1.49968 308.0 1.48564 325.0 1.48164 337.0 1.47921 365.5 1.47447 404.7 1.46962 435.8 1.46669 441.6 1.46622 Continuous Operating Temperature: 900°C maximum 447.1 1.46578 Coefficient of Thermal Expansion: 5.5x10–7/ºC 486.1 1.46313 Specific Heat: 0.177 cal/g/ºC @ 25°C 488.0 1.46301 Dispersion Constants: B1 = 0.6961663 B2 = 0.4079426 B3 = 0.8974794 C1 = 0.0046791 C2 = 0.0135121 C3 = 97.9340025 514.5 1.46156 532.0 1.46071 546.1 1.46008 587.6 1.45846 632.8 1.45702 656.3 1.45637 694.3 1.45542 752.5 1.45419 905.0 1.45168 1064.0 1.44963 1153.0 1.44859 1319.0 1.44670 Specifications Abbé Constant: 67.8±0.5 Change of Refractive Index with Temperature (0° to 700°C): 1.28x10–5/ºC Homogeneity (maximum index variation over 10-cm aperture): 2x10–5 Knoop Hardness: 590 3HUFHQW([WHUQDO 7UDQVPLWWDQFH Density: 2.20 g/cm3 @ 25°C * Accuracy ±3 x 10-5 :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 5 mm thick uncoated calcium fluoride A66 Suprasil 1 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials UV-GRADE SYNTHETIC FUSED SILICA Synthetic fused silica (amorphous silicon dioxide), by chemical combination of silicon and oxygen, is an ideal optical material for many applications. It is transparent over a wide spectral range, has a low coefficient of thermal expansion, and is resistant to scratching and thermal shock. Specifications The synthetic fused silica materials used by CVI Laser Optics are manufactured by flame hydrolysis to extremely high standards. The resultant material is colorless and non-crystalline, and it has an impurity content of only about one part per million. Density: 2.20 g/cm3 @ 25°C ultraviolet and infrared transmission X Low coefficient of thermal expansion, which provides stability and resistance to thermal shock over large temperature excursions Continuous Operating Temperature: 900°C maximum Coefficient of Thermal Expansion: 5.5x10-7/ºC Specific Heat: 0.177 cal/g/ºC @ 25°C Dispersion Constants: B1 = 0.6961663 B2 = 0.4079426 B3 = 0.8974794 C1 = 0.0046791 C2 = 0.0135121 C3 = 97.9340025 thermal operating range X Increased hardness and resistance to scratching X Much Knoop Hardness: 522 higher resistance to radiation darkening from ultraviolet, x-rays, gamma rays, and neutrons. Wavelength (nm) Index of Refraction 180.0 1.58529 190.0 1.56572 200.0 1.55051 213.9 1.53431 226.7 1.52275 230.2 1.52008 239.9 1.51337 248.3 1.50840 265.2 1.50003 275.3 1.49591 280.3 1.49404 289.4 1.49099 296.7 1.48873 302.2 1.48719 330.3 1.48054 340.4 1.47858 351.1 1.47671 361.1 1.47513 365.0 1.47454 404.7 1.46962 435.8 Machine Vision Guide The batch-to-batch internal transmittance of synthetic fused silica may fluctuate significantly in the near infrared between 900 nm and 2.5 µm due to resonance absorption by OH chemical bonds. If the optic is to be used in this region, Infrasil 302 may be a better choice. Refractive Index of UV-Grade Synthetic Fused Silica* Gaussian Beam Optics UV-grade synthetic fused silica (UVGSFS or Suprasil 1) is selected to provide the highest transmission (especially in the deep ultraviolet) and very low fluorescence levels (approximately 0.1% that of fused natural quartz excited at 254 nm). UV-grade synthetic fused silica does not fluoresce in response to wavelengths longer than 290 nm. In deep ultraviolet applications, UV-grade synthetic fused silica is an ideal choice. Its tight index tolerance ensures highly predictable lens specifications. Fundamental Optics X Wider Homogeneity (maximum index variation over 10-cm aperture): 2x10–5 Optical Specifications X Greater Change of Refractive Index with Temperature (0° to 700°C): 1.28x10-5/ºC Material Properties Synthetic fused silica lenses offer a number of advantages over glass or fused quartz: Abbé Constant: 67.8±0.5 1.46669 * Accuracy ±3 x 10 -5 Laser Guide marketplace.idexop.com UV-Grade Synthetic Fused Silica A67 MATERIAL PROPERTIES Material Properties Refractive Index of UV-Grade Synthetic Fused Silica* Refractive Index of UV-Grade Synthetic Fused Silica* Wavelength (nm) Index of Refraction Wavelength (nm) Index of Refraction 441.6 1.46622 830.0 1.45282 457.9 1.46498 852.1 1.45247 476.5 1.46372 904.0 1.45170 486.1 1.46313 1014.0 1.45024 488.0 1.46301 1064.0 1.44963 496.5 1.46252 1100.0 1.44920 514.5 1.46156 1200.0 1.44805 532.0 1.46071 1300.0 1.44692 546.1 1.46008 1400.0 1.44578 587.6 1.45846 1500.0 1.44462 589.3 1.45840 1550.0 1.44402 632.8 1.45702 1660.0 1.44267 643.8 1.45670 1700.0 1.44217 656.3 1.45637 1800.0 1.44087 694.3 1.45542 1900.0 1.43951 706.5 1.45515 2000.0 1.43809 786.0 1.45356 2100.0 820.0 1.45298 %. 1 :DYHOHQJWKLQ1DQRPHWHUV :DYHOHQJWKLQ0LFURPHWHUV E8SSHU/LPLWV 1%. 89 3HUFHQW([WHUQDO 7UDQVPLWWDQFH *6 )6 3HUFHQW([WHUQDO 7UDQVPLWWDQFH D/RZHU/LPLWV 89*6)6 * Accuracy ±3 x 10 1.43659 -5 Comparison of uncoated external transmittances for UVGSFS and N-BK7, all 10 mm in thickness A68 UV-Grade Synthetic Fused Silica 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials CRYSTAL QUARTZ 3HUFHQW([WHUQDO 7UDQVPLWWDQFH The dispersion for the index of refraction is given by the Laurent series shown below. l2 = A3 l4 = A4 l6 = :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 10 mm thick uncoated crystal quartz A5 Optical Specifications h2 = A0 = A1l2 = A2 Material Properties Crystal quartz is a positive uniaxial birefringent single crystal grown using a hydrothermal process. Crystal quartz from CVI Laser Optics is selected to minimize inclusions and refractive index variation. Crystal quartz is most commonly used for high-damage-threshold waveplates and solarization-resistant Brewster windows for argon lasers. l8 Refractive Index of Crystal Quartz 193 1.66091 1.67455 Transmission Range: 0.170–2.8 µm 213 1.63224 1.64452 222 1.62238 1.63427 226 1.61859 1.63033 244 1.60439 1.61562 Density: 2.64 g/cm3 248 1.60175 1.61289 Young’s Modulus, Perpendicular: 76.5 GPa 257 1.59620 1.60714 Young’s Modulus, Parallel: 97.2 GPa 266 1.59164 1.60242 280 1.58533 1.59589 308 1.57556 1.58577 325 1.57097 1.58102 337 1.56817 1.57812 351 1.56533 1.57518 355 1.56463 1.57446 400 1.55772 1.56730 442 1.55324 1.56266 458 1.55181 1.56119 488 1.54955 1.55885 515 1.54787 1.55711 532 1.54690 1.55610 590 1.54421 1.55333 633 1.54264 1.55171 670 1.54148 1.55051 694 1.54080 1.54981 755 1.53932 1.54827 780 1.53878 1.54771 800 1.53837 1.54729 820 1.53798 1.54688 860 1.53724 1.54612 980 1.53531 1.54409 Melting Point: 1463°C Knoop Hardness: 741 Thermal Expansion Coefficient, Perpendicular: 13.2x10–6/°C Thermal Expansion Coefficient, Parallel: 7.1x10 /°C –6 Dispersion Constants (Ordinary Ray): A0=2.35728 A1=41.17000x10–2 A2=1.05400x10–2 A3=1.34143x10–4 A4=44.45368x10–7 A5=5.92362x10–8 Dispersion Constants (Extraordinary Ray): A0=2.38490 A1=41.25900x10–2 A2=1.07900x10–2 A3=1.65180x10–4 A4=41.94741x10–7 A5=9.36476x10–8 Machine Vision Guide Index of Refraction Extraordinary Ray (nE) Gaussian Beam Optics Specifications Index of Refraction Ordinary Ray (nO) Fundamental Optics Wavelength (nm) Laser Guide marketplace.idexop.com Crystal Quartz A69 MATERIAL PROPERTIES CALCITE Since calcite is a natural crystal, the transmission will vary from piece to piece. In general, a 10 mm thick sample will fall within the following ranges: 350 nm, 40 – 45%; 400 nm, 70 – 75%; 500 – 2300 nm, 86 – 88%. RUGLQDU\UD\ H[WUDRUGLQDU\UD\ :DYHOHQJWKLQ0LFURPHWHUV Typical external transmittance for 10 mm thick calcite Refractive Index of Calcite Wavelength (nm) Index of Refraction Ordinary Ray (nO) Index of Refraction Extraordinary Ray (nE) 250 1.76906 1.53336 350 1.69695 1.50392 Specifications 450 1.67276 1.49307 Density: 2.71 g/cm 550 1.66132 1.48775 Mohs Hardness: 3 650 1.65467 1.48473 Melting Point: 1339 °C 750 1.65041 1.48289 850 1.64724 1.48157 950 1.64470 1.48056 1050 1.64260 1.47980 1150 1.64068 1.47916 1250 1.63889 1.47792 1350 1.63715 1.47803 1450 1.63541 1.47765 1550 1.63365 1.47722 1650 1.63186 1.47681 1750 1.62995 1.47638 1850 1.62798 1.47597 1950 1.62594 1.47555 2050 1.62379 1.47513 2150 1.62149 1.47486 2250 1.61921 1.47482 2350 1.61698 1.47528 3 Dispersion Constants (Ordinary Ray): B1 = 1.56630 B2 = 1.41096 B3 = 0.28624 C1 = 105.58893 C2 = 0.01583669 C3 =40.01182893 Dispersion Constants (Extraordinary Ray): B1 = 8.418192x10–5 B2 = 1.183488 B3 = 0.03413054 C1 = 0.3468576 C2 = 7.741535x10–3 C3 = 12.185616 A70 3HUFHQW([WHUQDO 7UDQVPLWWDQFH Material Properties Calcite (CaCO3) is a naturally occurring negative uniaxial crystal which exhibits pronounced birefringence. Strong birefringence and a wide transmission range have made this mineral popular for making polarizing prisms for over 100 years. Although it can now be grown artificially in small quantities, most optical calcite is mined in Mexico, Africa, and Siberia. Finding optical grade crystals remains a time-consuming task requiring special skills. Cutting and polishing calcite is also challenging due to the softness of the mineral and its tendency to cleave easily. These factors explain why, even many years after the techniques were developed, calcite prisms remain expensive when compared to other types of polarizers. Calcite 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials SCHOTT GLASS Material Properties The following tables list the most important optical and physical constants for Schott optical glass types BK7, SF11, LaSFN9, BaK1, and F2, with N-BK7 and N-BaK1 denoting the lead and arsenic-free versions of BK7 and BaK1. These types are used in most CVI Laser Optics simple lens products and prisms. Index of refraction as well as the most commonly required chemical characteristics and mechanical constants, are listed. Further numerical data and a more detailed discussion of the various testing processes can be found in the Schott Optical Glass catalog. Optical Specifications Physical Constants of Schott Glasses Glass Type Melt-to-Melt Mean Index Tolerance Stress Birefringence nm/cm Yellow Light SF11 LaSFN9 BaK1 (N-BaK1) F2 ±0.0005 ±0.0005 ±0.0005 ±0.0005 ±0.0005 10 10 10 10 10 64.17 25.76 32.17 57.55 36.37 Fundamental Optics Abbé Factor (vd) BK7 (N-BK7) Constants of Dispersion Formula: B1 1.03961212 B2 2.31792344x10 1.73848403 –1 1.97888194 3.11168974x10 –1 1.12365662 3.20435298x10 –1 3.09276848x10 1.34533359 –1 2.09073176x10–1 1.01046945 1.17490871 1.92900751 8.81511957x10–1 9.37357162x10–1 C1 6.00069867x10–3 1.36068604x10-4 1.18537266x10–2 6.44742752x10–3 9.97743871x10–3 C2 2.00179144x10–2 6.15960463x10-2 5.27381770x10–2 2.22284402x10–2 4.70450767x10–2 1.03560653x102 1.21922711x102 1.66256540x102 1.07297751x102 1.11886764x102 2.51 4.74 4.44 3.19 3.61 C3 Density (g/cm ) 3 Gaussian Beam Optics B3 Coefficient of Linear Thermal Expansion (cm/°C): 7.1x1046 6.1x1046 7.4x1046 7.6x1046 8.2x1046 +20º to +300ºC 8.3x1046 6.8x1046 8.4x1046 8.6x1046 9.2x1046 557ºC 505ºC 703ºC 592ºC 438ºC 8.20x109 6.60x109 1.09x1010 7.30x109 5.70x109 Climate Resistance 2 1 2 2 1 Stain Resistance 0 0 0 1 0 Acid Resistance 1.0 1.0 2.0 3.3 1.0 Alkali Resistance 2.0 1.2 1.0 1.2 2.3 Phosphate Resistance 2.3 1.0 1.0 2.0 1.3 Knoop Hardness 610 450 630 530 420 0.206 0.235 0.286 0.252 0.220 Transition Temperature Young’s Modulus (dynes/mm2) Poisson’s Ratio Machine Vision Guide –30º to +70ºC Laser Guide marketplace.idexop.com Schott Glass A71 MATERIAL PROPERTIES Material Properties Refractive Index of Schott Glass A72 Refractive Index, n Wavelength λ (nm) BK7 (N-BK7) SF11 LaSFN9 BaK1 (N-BaK1) F2 351.1 1.53894 — — 1.60062 363.8 1.53649 — — 404.7 1.53024 1.84208 435.8 1.52668 441.6 1.52611 457.9 Fraunhofer Designation Source Spectral Region 1.67359 Ar laser UV 1.59744 1.66682 Ar laser UV 1.89844 1.58941 1.65064 h Hg arc Violet 1.82518 1.88467 1.58488 1.64202 g Hg arc Blue 1.82259 1.88253 1.58415 1.64067 HeCd laser Blue 1.52461 1.81596 1.87700 1.58226 1.63718 Ar laser Blue 465.8 1.52395 1.81307 1.87458 1.58141 1.63564 Ar laser Blue 472.7 1.52339 1.81070 1.87259 1.58071 1.63437 Ar laser Blue 476.5 1.52309 1.80946 1.87153 1.58034 1.63370 Ar laser Blue 480.0 1.52283 1.80834 1.87059 1.58000 1.63310 F’ Cd arc Blue F 486.1 1.52238 1.80645 1.86899 1.57943 1.63208 H2 arc Blue 488.0 1.52224 1.80590 1.86852 1.57927 1.63178 Ar laser Blue 496.5 1.52165 1.80347 1.86645 1.57852 1.63046 Ar laser Green 501.7 1.52130 1.80205 1.86524 1.57809 1.62969 Ar laser Green 514.5 1.52049 1.79880 1.86245 1.57707 1.62790 Ar laser Green 532.0 1.51947 1.79479 1.85901 1.57580 1.62569 Nd laser Green 546.1 1.51872 1.79190 1.85651 1.57487 1.62408 e Hg arc Green 587.6 1.51680 1.78472 1.85025 1.57250 1.62004 d He arc Yellow 589.3 1.51673 1.78446 1.85002 1.57241 1.61989 D 632.8 1.51509 1.77862 1.84489 1.57041 1.61656 643.8 1.51472 1.77734 1.84376 1.56997 1.61582 656.3 1.51432 1.77599 1.84256 1.56949 1.61503 694.3 1.51322 1.77231 1.83928 1.56816 1.61288 786.0 1.51106 1.76558 1.83323 1.56564 1.60889 IR 821.0 1.51037 1.76359 1.83142 1.56485 1.60768 IR 830.0 1.51020 1.76311 1.83098 1.56466 1.60739 852.1 1.50980 1.76200 1.82997 1.56421 1.60671 904.0 1.50893 1.75970 1.82785 1.56325 1.60528 1014.0 1.50731 1.75579 1.82420 1.56152 1.60279 1060.0 1.50669 1.75445 1.82293 1.56088 Na arc Yellow HeNe laser Red C’ Cd arc Red C H2 arc Red Ruby laser Red s GaAlAs laser IR Ce arc IR GaAs laser IR Hg arc IR 1.60190 Nd laser IR InGaAsP laser IR t 1300.0 1.50370 1.74901 1.81764 1.55796 1.59813 1500.0 1.50127 1.74554 1.81412 1.55575 1.59550 IR 1550.0 1.50065 1.74474 1.81329 1.55520 1.59487 IR 1970.1 1.49495 1.73843 1.80657 1.55032 1.58958 Hg arc IR 2325.4 1.48921 1.73294 1.80055 1.54556 1.58465 Hg arc IR Schott Glass 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials OPTICAL CROWN GLASS Specifications Glass Type Designation: B270 Dispersion: (nF–nC) = 0.0089 Knoop Hardness: 542 Wavelength (nm) Index of Refraction Fraunhofer Designation Source Spectral Region 435.8 1.53394 g Hg arc Blue 480.0 1.52960 F’ Cd arc Blue 486.1 1.52908 F H2 arc Blue 546.1 1.52501 e Hg arc Green 587.6 1.52288 d He arc Yellow 589.0 1.52280 D Na arc Yellow 643.8 1.52059 C’ Cd arc Red 656.3 1.52015 C H2 arc Red Optical Specifications Abbé Constant: 58.5 Refractive Index of Optical Crown Glass Material Properties In optical crown glass, a low-index commercial-grade glass, the index of refraction, transmittance, and homogeneity are not controlled as carefully as they are in optical-grade glasses such as N-BK7. Optical crown glass is suitable for applications in which component tolerances are fairly loose, and as a substrate material for mirrors. Density: 2.55 g/cm3 @ 23°C Young’s Modulus: 71.5 kN/mm2 Transmission Values for 6 mm thick Sample Specific Heat: 0.184 cal/g/°C (20°C to 100°C) 300.0 0.3 310.0 7.5 320.0 30.7 330.0 56.6 340.0 73.6 350.0 83.1 360.0 87.2 380.0 88.8 400.0 90.6 450.0 90.9 500.0 91.4 600.0 91.5 Transformation Temperature: 521°C 3HUFHQW([WHUQDO 7UDQVPLWWDQFH Softening Point: 708°C :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 10 mm thick uncoated optical crown glass Gaussian Beam Optics Transmission (%) Fundamental Optics Wavelength (nm) Coefficient of Thermal Expansion: 93.3x10–7/°C (20°C to 300°C) Machine Vision Guide Laser Guide marketplace.idexop.com Optical Crown Glass A73 MATERIAL PROPERTIES LOW-EXPANSION BOROSILICATE GLASS Specifications Abbé Constant: 66 Knoop Hardness: 480 Density: 2.23 g/cm3 @ 25°C Young’s Modulus: 5.98x109 dynes/mm2 Poisson’s Ratio: 0.20 Specific Heat: 0.17 cal/g/°C @ 25°C Coefficient of Thermal Expansion: 3.25x10–6/°C (0°–300°C) Softening Point: 820°C Melting Point: 1250°C Low-Expansion Borosilicate Glass Wavelength (nm) Index of Refraction Fraunhofer Designation Source Spectral Region 486.1 1.479 F H2 arc Blue 514.5 1.477 Ar laser Green 546.1 1.476 e Hg arc Green 587.6 1.474 d Na arc Yellow 643.8 1.472 C’ Cd arc Red 3HUFHQW([WHUQDO 7UDQVPLWWDQFH Material Properties The most well-known low-expansion borosilicate glass (LEBG) is Pyrex® made by Corning. It is well suited for applications in which high temperature, thermal shock, or resistance to chemical attack are primary considerations. On the other hand, LEBG is typically less homogeneous and contains more striae and bubbles than optical glasses such as N-BK7. This material is ideally suited to such tasks as mirror substrates, condenser lenses for high-power illumination systems, or windows in hightemperature environments. Because of its low cost and excellent thermal stability, it is the standard material used in test plates and optical flats. The transmission of LEBG extends into the ultraviolet and well into the infrared. The index of refraction in this material varies considerably from batch to batch. Typical values are shown in the accompanying table. :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 8 mm thick uncoated low-expansion borosilicate glass A74 Low-Expansion Borosilicate Glass 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials ZERODUR® Specifications Abbé Constant: 66 Dispersion: (nF–nC) = 0.00967 Material Properties Many optical applications require a substrate material with a near-zero coefficient of thermal expansion and/ or excellent thermal shock resistance. ZERODUR® with its very small coefficient of thermal expansion at room temperature is such a material. Knoop Hardness: 620 Density: 2.53 g/cm3 @ 25ºC Specific Heat: 0.196 cal/g/°C Coefficient of Thermal Expansion: 0.0580.10x10–6/° (20°–300°C) Maximum Temperature: 600°C Refractive Index of ZERODUR® Wavelength (nm) Index of Refraction Fraunhofer Designation 435.8 1.5544 g 480.0 1.5497 F’ 486.1 1.5491 F 546.1 1.5447 e 587.6 1.5424 d 643.8 1.5399 C’ 656.3 1.5394 C Gaussian Beam Optics Machine Vision Guide The figure below shows the variation of expansion coefficient with temperature for a typical sample. The actual performance varies very slightly, batch to batch, with the room temperature expansion coefficient in the range of ±0.15x10-6/°C. By design, this material exhibits a change in the sign of the coefficient near room temperature. A comparison of the thermal expansion coefficients of ZERODUR and fused silica is shown in the figure. ZERODUR is markedly superior over a large temperature range, and consequently, makes ideal mirror substrates for such stringent applications as multiple-exposure holography, holographic and general interferometry, manipulation of moderately powerful laser beams, and space-borne imaging systems. Poisson’s Ratio: 0.24 Fundamental Optics Typical of amorphous substances, the vitreous phase has a positive coefficient of thermal expansion. The crystalline phase has a negative coefficient of expansion at room temperature. The overall linear thermal expansion coefficient of the combination is almost zero at useful temperatures. Young’s Modulus: 9.1x109 dynes/mm2 Optical Specifications ZERODUR, which belongs to the glass-ceramic composite class of materials, has both an amorphous (vitreous) component and a crystalline component. This Schott glass is subjected to special thermal cycling during manufacture so that approximately 75% of the vitreous material is converted to the crystalline quartz form. The crystals are typically only 50 nm in diameter, and ZERODUR appears reasonably transparent to the eye because the refractive indices of the two phases are almost identical. Laser Guide marketplace.idexop.com ZERODUR® A75 MATERIAL PROPERTIES INFRASIL 302 Material Properties Infrasil 302 is an optical-quality quartz glass made by fusing natural quartz crystals in an electric oven. It combines excellent physical properties with excellent optical characteristics, especially in the near infrared region (1 to 3 µm) because it does not exhibit the strong OH absorption bands typical of synthetic fused silica. Specifications OH content: <8 ppm Knoop Hardness: 590 Thermal Expansion Coefficient: 0.58x10–6/°C (0°C to 200°C) Bubble class: 0 Maximum bubble diameter: ≤0.15 mm typical Infrasil 302 is homogeneous in the primary functional direction. Weak striations, if any, are parallel to the major faces and do not affect optical performance. Optical Homogeneity: Granular Structure: None Striations: In all three dimensions free of striations Index Homogeneity: In all three dimensions guaranteed total Δn ≤5x10–6 Spectral Transmittance: Very weak absorption band occur at wavelengths arouns 1.39 mm, 2.2 mm, and 2.72 µm according to an OH content of ≤8 ppm (weight). 100 Percent External Transmittance 80 60 Refractive Index of Infrasil 302 40 20 0.20 0.25 0.30 0.35 1.0 2.0 3.0 Wavelength in Micrometers Infrasil 302 Index of Refraction 435.8 1.46681 486.1 1.46324 587.6 1.45856 656.3 1.45646 4.0 External transmittance for 10 mm thick uncoated Infrasil 302 A76 Wavelength (nm) 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials SAPPHIRE Young’s Modulus: 3.7x1010 dynes/mm2 Poisson’s Ratio: –0.02 Mohs Hardness: 9 Specific Heat: 0.18 cal/g/ºC @ 25°C Coefficient of Thermal Expansion: 7.7x10 /ºC (0°–500°C) Softening Point: 1800°C Dispersion Constants (Ordinary Ray): B1 = 1.4313493 B2 = 0.65054713 B3 = 5.3414021 C1 = 0.00527993 C2 = 0.01423827 C3 = 325.0178 Dispersion Constants (Extraordinary Ray): B1 = 1.5039759 B2 = 0.55069141 B3 = 6.5927379 C1 = 0.00548026 C2 = 0.01479943 C3 = 402.8951 Fundamental Optics Refractive Index of Sapphire Wavelength (nm) Index of Refraction Ordinary Ray (nO) Index of Refraction Extraordinary Ray (nE) 265.2 1.83359 1.82411 351.1 1.79691 1.78823 404.7 1.78573 1.77729 488.0 1.77533 1.76711 514.5 1.77304 1.76486 532.0 1.77170 1.76355 546.1 1.77071 1.76258 632.8 1.76590 1.75787 1550.0 1.74618 1.73838 2000.0 1.73769 1.72993 Gaussian Beam Optics Machine Vision Guide The transmission of sapphire is limited primarily by losses caused by surface reflections. The high index of sapphire makes magnesium fluoride almost an ideal single-layer antireflection coating. When a single layer of magnesium fluoride is deposited on sapphire and optimized for 550 nm, total transmission of a sapphire component can be kept above 98% throughout the entire visible spectrum. Density: 3.98 g/cm3 @ 25ºC Optical Specifications Sapphire is single-crystal aluminum oxide (Al2O3). Because of its hexagonal crystalline structure, sapphire exhibits anisotropy in many optical and physical properties. The exact characteristics of an optical component made from sapphire depend on the orientation of the optic axis or c-axis relative to the element surface. Sapphire exhibits birefringence, a difference in index of refraction in orthogonal directions. The difference in index is 0.008 between light traveling along the optic axis and light traveling perpendicular to it. Specifications Material Properties Sapphire is a superior window material in many ways. Because of its extreme surface hardness, sapphire can be scratched by only a few substances other than itself (such as diamond or boron nitride). Chemically inert and insoluble in almost everything except at highly elevated temperatures, sapphire can be cleaned with impunity. For example, even hydrogen fluoride fails to attack sapphire at temperatures below 300ºC. Sapphire exhibits high internal transmittance all the way from 150 nm (vacuum ultraviolet) to 6 µm (mid-infrared). Because of its great strength, sapphire windows can safely be made much thinner than windows of other glass types, and therefore are useful even at wavelengths that are very close to their transmission limits. Because of the exceptionally high thermal conductivity of sapphire, thin windows can be very effectively cooled by forced air or other methods. Conversely, sapphire windows can easily be heated to prevent condensation. 3HUFHQW([WHUQDO 7UDQVPLWWDQFH :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 1 mm thick uncoated sapphire Laser Guide marketplace.idexop.com Sapphire A77 MATERIAL PROPERTIES ZINC SELENIDE Material Properties ZnSe is produced as microcrystalline sheets by synthesis from H2Se gas and zinc vapor. It has a remarkably wide transmission range and is used extensively in CO2 laser optics. Wavelength (µm) Index of Refraction 0.63 2.590 1.40 2.461 Specifications 1.50 2.458 Transmission Range: 0.5–22 µm 1.66 2.454 Refractive Index Inhomogeneity @ 633 nm: <3x10–6 1.82 2.449 Temperature Coefficient of Refractive Index @ 10.6 µm: 61x10–6/°C 2.05 2.446 2.06 2.446 2.15 2.444 2.44 2.442 2.50 2.441 2.58 2.440 2.75 2.439 3.00 2.438 3.42 2.436 3.50 2.435 Bulk Absorption Coefficient @ 10 µm: 0.0004/cm Melting Point: 1520°C Knoop Hardness: 112 Density: 5.27 g/cm3 Rupture Modulus: 55.2 MPa Young’s Modulus: 67.2 GPa Fracture Toughness: 0.5 MPa/m Thermal Expansion: Coefficient 7.6x10–6/°C 2.432 5.00 2.430 6.00 2.426 6.24 2.425 7.50 2.420 8.66 2.414 9.50 2.410 9.72 2.409 10.60 2.400 11.00 2.400 11.04 2.400 12.50 2.390 13.02 2.385 13.50 2.380 15.00 2.370 16.00 2.360 16.90 2.350 17.80 2.340 18.60 2.330 19.30 2.320 20.00 2.310 3HUFHQW([WHUQDO 7UDQVPLWWDQFH 4.36 :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 10 mm thick uncoated zinc selenide A78 Refractive Index of Zinc Selenide Zinc Selenide 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials SILICON Silicon is commonly used as substrate material for infrared reflectors and windows in the 1.5–8 µm region. The strong absorption band at 9 µm makes it unsuitable for CO2 laser transmission applications, but it is frequently used for laser mirrors because of its high thermal conductivity and low density. Refractive Index of Silicon Index of Refraction 0.63 3.920 1.40 3.490 1.50 3.480 1.66 3.470 1.82 3.460 2.05 3.450 2.06 3.490 Melting Point: 1417°C 2.15 3.470 Knoop Hardness: 1100 2.44 3.470 Density: 2.33 g/cm3 2.50 3.440 Young’s Modulus: 131 GPa 2.58 3.436 2.75 3.434 3.00 3.431 3.42 3.428 3.50 3.427 Specifications Transmission Range: 1.5–7 µm Thermal Expansion Coefficient: 4.50x10 /°C –6 3HUFHQW([WHUQDO 7UDQVPLWWDQFH 80 5.00 3.420 40 6.00 3.419 6.24 3.419 7.50 3.417 8.66 3.416 9.50 3.416 9.72 3.416 10.60 3.416 11.00 3.416 11.04 3.416 12.50 3.416 13.02 3.416 13.50 3.416 15.00 3.416 16.00 3.416 16.90 3.416 17.80 3.416 18.60 3.416 19.30 3.416 20.00 3.416 20 .7 1 2 3 5 7 10 20 30 :DYHOHQJWKLQ0LFURPHWHUV 50 70 External transmittance for 5 mm thick uncoated silicon Machine Vision Guide 3.422 Gaussian Beam Optics 4.36 60 Fundamental Optics 100 Optical Specifications Temperature Coefficient of Refractive Index @ 10.6 µm: 160x10–6/°C Material Properties Wavelength (µm) Laser Guide marketplace.idexop.com Silicon A79 MATERIAL PROPERTIES GERMANIUM Material Properties Germanium is commonly used in imaging systems working in the 2 to 12 µm wavelength region. It is an ideal substrate material for lenses, windows and mirrors in low-power cw and CO2 laser applications. Wavelength (µm) Index of Refraction 0.63 5.390 1.40 4.340 Specifications 1.50 4.350 Transmission Range: 2–23 µm 1.66 4.330 Temperature Coefficient of Refractive Index @ 10.6 µm: 277x10–6/°C 1.82 4.290 2.05 4.250 Bulk Absorption Coefficient @ 10 µm: 0.035/cm 2.06 4.240 Melting Point: 973°C 2.15 4.240 2.44 4.070 2.50 4.220 2.58 4.060 2.75 4.053 3.00 4.054 3.42 4.037 3.50 4.036 Knoop Hardness: 692 Density: 5.323 g/cm3 Young’s Modulus 102.6: GPa Thermal Expansion Coefficient: 5.7x10–6/°C 3HUFHQW([WHUQDO 7UDQVPLWWDQFH 4.36 4.023 5.00 4.018 6.00 4.014 6.24 4.010 7.50 4.010 8.66 4.007 9.50 4.006 9.72 4.006 10.60 4.006 11.00 4.006 11.04 4.006 12.50 4.000 13.02 4.000 13.50 4.000 15.00 4.000 16.00 4.000 16.90 4.000 17.80 4.000 18.60 4.000 19.30 4.000 20.00 4.000 :DYHOHQJWKLQ0LFURPHWHUV External transmittance for 10 mm thick uncoated germanium A80 Refractive Index of Germanium Germanium 1-505-298-2550 MATERIAL PROPERTIES Optical Coatings & Materials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achine Vision Guide Gaussian Beam Optics =HURGXU Fundamental Optics &DOFLWH Optical Specifications 89)XVHG6LOLFD Material Properties 0J) Laser Guide marketplace.idexop.com Material Properties Overview A81 MATERIAL PROPERTIES Material Properties A82 1-505-298-2550 Optical Coatings & Materials OPTICAL SPECIFICATIONS A84 Material Properties WAVEFRONT DISTORTION CENTRATIONA85 A86 COSMETIC SURFACE QUALITY – U.S. MILITARY SPECIFICATIONS A89 SURFACE ACCURACY A91 Optical Specifications MODULATION TRANSFER FUNCTION Fundamental Optics Gaussian Beam Optics Machine Vision Guide Laser Guide marketplace.idexop.com A83 OPTICAL SPECIFICATIONS WAVEFRONT DISTORTION Optical Specifications Sometimes the best specification for an optical component is its effect on the emergent wavefront. This is particularly true for optical flats, collimation lenses, mirrors, and retroreflectors where the presumed effect of the element is to transmit or reflect the wavefront without changing its shape. Wavefront distortion is often characterized by the peak-to-valley deformation of the emergent wavefront from its intended shape. Specifications are normally quoted in fractions of a wavelength. Consider a perfectly plane, monochromatic wavefront, incident at an angle normal to the face of a window. Deviation from perfect surface flatness, as well as inhomogeneity of the bulk material refractive index of the window, will cause a deformation of the transmitted wavefront away from the ideal plane wave. In a retroreflector, each of the faces plus the material will affect the emergent wavefront. Consequently, any reflecting or refracting element can be characterized by the distortions imparted to a perfect incident wavefront. INTERFEROMETER MEASUREMENTS CVI Laser Optics measures wavefront distortion with a laser interferometer. The wavefront from a helium neon laser (λ=632.8 nm) is expanded and then divided into a reference wavefront and test wavefronts by using a partially transmitting reference surface. The reference wavefront is reflected back to the interferometer, and the test wavefront is transmitted through the surfaces to the test element. The reference surface is a known flat or spherical surface whose surface error is on the order of λ/20. When the test wavefront is reflected back to the interferometer, either from the surface being tested or from another λ/20 reference surface, the reference and test wavefronts recombine at the interferometer. Constructive and destructive interference occurs between the two wavefronts. difference in the optical path traveled by the two wavefronts. In surface and transmitted wavefront testing, the test wavefront travels through an error in the test piece twice. Therefore, one fringe spacing represents one-half wavelength of surface error or transmission error of the test element. A determination of the convexity or concavity of the error in the test element can be made if the zero-order direction of the interference cavity (the space between the reference and test surfaces) is known. The zero-order direction is the direction of the center of tilt between the reference and test wavefronts. Fringes that curve around the center of tilt (zero-order) are convex as a result of a high area on the test surface. Conversely, fringes that curve away from the center of tilt are concave as a result of a low area on the test surface. By using a known tilt and zero-order direction, the amount and direction (convex or concave) of the error in the test element can be determined from the fringe pattern. Six fringes of tilt are introduced for typical examinations. CVI Laser Optics uses wavefront distortion measurements to characterize achromats, windows, filters, beamsplitters, prisms, and many other optical elements. This testing method is consistent with the way in which these components are normally used. INTERFEROGRAM INTERPRETATION CVI Laser Optics tests lenses with a noncontact phasemeasuring interferometer. The interferometer has a zoom feature to increase resolution of the optic under test. The interferometric cavity length is modulated, and a computerized data analysis program is used to interpret the interferogram. This computerized analysis increases the accuracy and repeatability of each measurement and eliminates subjective operator interpretation. A slight tilt of the test wavefront to the reference wavefront produces a set of fringes whose parallelism and straightness depend on the element under test. The distance between successive fringes (usually measured from dark band to dark band) represents one wavelength A84 Wavefront Distortion 1-505-298-2550 OPTICAL SPECIFICATIONS Optical Coatings & Materials CENTRATION The mechanical axis and optical axis exactly coincide in a perfectly centered lens. For a simple lens, the optical axis is defined as a straight line that joins the centers of lens curvature. For a planoconvex or plano-concave lens, the optical axis is the line through the center of curvature and perpendicular to the plano surface. Centration error is measured by rotating the lens on its mechanical axis and observing the orbit of the focal CYLINDRICAL OPTICS Cylindrical optics can be evaluated for centering error in a manner similar to that for simple lenses. The major difference is that cylindrical optics have mechanical and optical planes rather than axes. The mechanical plane is established by the expected mounting, which can be edge only or the surface-edge combination described above. The radial separation between the focal line and the established mechanical plane is the centering error and can be converted into an angular deviation in the same manner as for simple lenses. The centering error is measured by first noting the focal line displacement in one orientation, then rotating the lens 180 degrees and noting the new displacement. The centering error angle is the inverse tangent of the total separation divided by twice the focal length. Gaussian Beam Optics MEASURING CENTRATION ERROR It is more difficult to achieve a given centration specification for a doublet than it is for a singlet because each element must be individually centered to a tighter specification, and the two optical axes must be carefully aligned during the cementing process. Centration is even more complex for triplets because three optical axes must be aligned. The centration error of doublets and triplets is measured in the same manner as that of simple lenses. One method used to obtain precise centration in compound lenses is to align the elements optically and edge the combination. Fundamental Optics Ideally, the optical and mechanical axes coincide. The tolerance on centration is the allowable amount of radial separation of these two axes, measured at the focal point of the lens. The centration angle is equal to the inverse tangent of the allowable radial separation divided by the focal length. DOUBLETS AND TRIPLETS Optical Specifications The mechanical axis is determined by the way in which the lens will be mounted during use. There are typically two types of mounting configurations: edge mounting and surface mounting. With edge mounting, the mechanical axis is the centerline of the lens mechanical edge. Surface mounting uses one surface of the lens as the primary stability reference for the lens tip and then encompasses the lens diameter for centering. The mechanical axis for this type of mounting is a line perpendicular to the mounting surface and centered on the entrapment diameter. Material Properties OPTICAL AND MECHANICAL AXES point, as shown in figure 3.1. To determine the centration error, the radius of this orbit is divided by the lens focal length and then converted to an angle. RUELWRI DSSDUHQWIRFXV FDOD[L V PHFKDQL v + +d WUXHIRFXV Machine Vision Guide +d¢ & RSWLFDOD[LV & )d F HGJHJULQGLQJUHPRYHV PDWHULDORXWVLGHLPDJLQDU\F\OLQGHU Figure 3.1 Centration and orbit of apparent focus Laser Guide marketplace.idexop.com Centration A85 OPTICAL SPECIFICATIONS MODULATION TRANSFER FUNCTION Optical Specifications The modulation transfer function (MTF), a quantitative measure of image quality, is far superior to any classic resolution criteria. MTF describes the ability of a lens or system to transfer object contrast to the image. MTF plots can be associated with the subsystems that make up a complete electro-optical or photographic system. MTF data can be used to determine the feasibility of overall system expectations. Bar-chart resolution testing of lens systems is deceptive because almost 20 percent of the energy arriving at a lens system from a bar chart is modulated at the third harmonic and higher frequencies. Consider instead a sine-wave chart in the form of a positive transparency in which transmittance varies in one dimension. Assume that the transparency is viewed against a uniformly illuminated background. The maximum and minimum transmittances are Tmax and Tmin, respectively. A lens system under test forms a real image of the sine-wave chart, and the spatial frequency (u) of the image is measured in cycles per millimeter. Corresponding to the transmittances Tmax and Tmin are the image irradiances Imax and Imin. By analogy with Michelson’s definition of visibility of interference fringes, the contrast or modulation of the chart and image are defined, respectively, as Mc = Tmax − Tmin Tmax + Tmin (3.1) I max − I min I max + I min (3.2) where Mc is the modulation of the chart and Mi is the modulation of the image. The modulation transfer function (MTF) of the optical system at spatial frequency u is then defined to be MTF = MTF( u ) = M i / M c . A86 It is often convenient to plot the magnitude of MTF(u) versus u. Changes in MTF curves are easily seen by graphical comparison. For example, for lenses, the MTF curves change with field angle positions and conjugate ratios. In a system with astigmatism or coma, different MTF curves are obtained that correspond to various azimuths in the image plane through a single image point. For cylindrical lenses, only one azimuth is meaningful. MTF curves can be either polychromatic or monochromatic. Polychromatic curves show the effect of any chromatic aberration that may be present. For a well-corrected achromatic system, polychromatic MTF can be computed by weighted averaging of monochromatic MTFs at a single image surface. MTF can also be measured by a variety of commercially available instruments. Most instruments measure polychromatic MTF directly. PERFECT CIRCULAR LENS The monochromatic, diffraction-limited MTF (or MDMTF) of a circular aperture (perfect aberration-free spherical lens) at an arbitrary conjugate ratio is given by the formula MDMTF( x ) = and Mi = The graph of MTF versus u is a modulation transfer function curve and is defined only for lenses or systems with positive focal length that form real images. 2 arccos( x ) − x 1 − x 2 p (3.4) where the arc cosine function is in radians and x is the normalized spatial frequency defined by x= u (3.5) uic where u is the absolute spatial frequency and uic is the incoherent diffraction cutoff spatial frequency. There are several formulas for uic including (3.3) Modulation Transfer Function 1-505-298-2550 OPTICAL SPECIFICATIONS Optical Coatings & Materials uic = = MDMTF( x ) = (1 − x ) 1.22l n′′D 1 − n′′ D 2 where x is again the normalized spatial frequency u/uic, where, in the present cylindrical case, l s ′′ 1.22l 2n′′ sin( u ′′ ) 1 − n " D 2 uic = l n′′D l s ′′ (3.6) D (3.7) 2s ′′ implies that the secondary principal surface is a sphere centered upon the secondary conjugate point. This means that the lens is completely free of spherical aberration and coma, and, in the special case of infinite conjugate ratio (s" = f"), D .(3.8) lf PERFECT RECTANGULAR LENS The MDMTF of a rectangular aperture (perfect aberration-free cylindrical lens) at arbitrary conjugate ratio is given by the formula 2n′′ sin (u ′′ ) l (3.11) The remaining three expressions for uic in the circular aperture case can be applied to the present rectangular aperture case provided that two substitutions are made. Everywhere the constant 1.22 formerly appeared, it must be replaced by 1.00. Also, the aperture diameter D must now be replaced by the aperture width w. The relationship sin(u") = w/2s" means that the secondary principal surface is a circular cylinder centered upon the secondary conjugate line. In the special case of infinite conjugate ratio, the incoherent cutoff frequency for cylindrical lenses is uic = n′′ w (3.12) lf Machine Vision Guide uic = n′′ uic = Gaussian Beam Optics sin( u ′′ ) = and rd is one-half the full width of the central stripe of the diffraction pattern measured from first maximum to first minimum. This formula differs by a factor of 1.22 from the corresponding formula in the circular aperture case. The following applies to both circular and rectangular apertures: Fundamental Optics where rd is the linear spot radius in the case of pure diffraction (Airy disc radius), D is the diameter of the lens clear aperture (or of a stop in near-contact), λ is the wavelength, s is the secondary conjugate distance, u" is the largest angle between any ray and the optical axis at the secondary conjugate point, the product n" sin(u") is by definition the image space numerical aperture, and n" is the image space refractive index. It is essential that D, λ, and s" have consistent units (usually millimeters, in which case u and uic will be in cycles per millimeter). The relationship 1 (3.10) rd Optical Specifications 2n′′ sin( u ′′ ) = l = (3.9) Material Properties = 1.22 rd IDEAL PERFORMANCE & REAL LENSES In an ideal lens, the x-intercept and the MDMTFintercept are at unity (1.0). MDMTF(x) for the rectangular case is a straight line between these intercepts. For the circular case, MDMTF(x) is a curve that dips slightly Laser Guide marketplace.idexop.com Modulation Transfer Function A87 OPTICAL SPECIFICATIONS All real cylindrical, monochromatic MTF curves fall on or below the straight MDMTF(x) line. Similarly, all real spherical and monochromatic MTF curves fall on or below the circular MDMTF(x) curve. Thus the two ideal MDMTF(x) curves represent the perfect (ideal) optical performance. Optical element or system quality is measured by how closely the real MTF curve approaches the corresponding ideal MDMTF(x) curve (see figure 3.3). MTF is an extremely sensitive measure of image degradation. To illustrate this, consider a lens having a quarter wavelength of spherical aberration. This aberration, barely discernible by eye, would reduce the MTF by as much as 0.2 at the midpoint of the spatial frequency range. 0'07) Optical Specifications below the straight line. These curves are shown in figure 3.2. Maximum contrast (unity) is apparent when spatial frequencies are low (i.e., for large features). Poor contrast is apparent when spatial frequencies are high (i.e., small features). All examples are limited at high frequencies by diffraction effects. A normalized spatial frequency of unity corresponds to the diffraction limit. UHFWDQJXODUDSHUWXUH FLUFXODUDSHUWXUH 1RUPDOL]HG6SDWLDO)UHTXHQF\x Figure 3.2 MDMTF as a function of normalized spatial frequency GLIIUDFWLRQOLPLWHGOHQV 07) OHQVZLWK ZDYHOHQJWK DEHUUDWLRQ 1RUPDOL]HG6SDWLDO)UHTXHQF\x Figure 3.3 MTF as a function of normalized spatial frequency A88 Modulation Transfer Function 1-505-298-2550 OPTICAL SPECIFICATIONS Optical Coatings & Materials COSMETIC SURFACE QUALITY – U.S. MILITARY SPECIFICATIONS Material Properties and dig standards according to U.S. military drawing C7641866 Rev L. Additionally, our inspection areas are equipped with lighting that meets the specific requirements of MIL-PRF-13830B. Cosmetic surface quality describes the level of defects that can be visually noted on the surface of an optical component. Specifically, it defines state of polish, freedom from scratches and digs, and edge treatment of components. These factors are important, not only because they affect the appearance of the component, but also because they scatter light, which adversely affects performance. Scattering can be particularly important in laser applications because of the intensity of the incident illumination. Unwanted diffraction patterns caused by scratches can lead to degraded system performance, and scattering of high-energy laser radiation can cause component damage. Overspecifying cosmetic surface quality, on the other hand, can be costly. CVI Laser Optics components are tested at appropriate levels of cosmetic surface quality according to their intended application. The scratch-and-dig designation for a component or assembly is specified by two numbers. The first defines allowable maximum scratch visibility, and the second refers to allowable maximum dig diameter, separated by a hyphen; for example, represents a commonly acceptable cosmetic standard. Optical Specifications X 80-50 X 60-40 represents an acceptable standard for most low power scientific research applications. X 40-20 represents an acceptable standard for low to moderate power laser or imaging applications that tolerate low light scatter. X 20-10 Because of the subjective nature of this examination, it is critical to use trained inspectors who operate under standardized conditions in order to achieve consistent results. CVI Laser Optics optics are compared by experienced quality assurance personnel using scratch X 10-5 represents a precise standard for very demanding high power laser applications. SCRATCHES A scratch is defined as any marking or tearing of a polished optical surface. The numeric designations for scratches are not related in any way to the width of a scratch, as the appearance of a scratch can depend upon the shape of the scratch, or how it scatters the light, as well as the component material and the presence of any coatings. Therefore, a scratch on the test optic that appears equivalent to the 80 standard scratch is not necessarily 8 mm wide. The combined length of the largest scratches on each surface cannot exceed one-quarter of the diameter of the element. If maximum visibility scratches are present (e.g., several 60 scratches on a 60-40 lens), the sum of the products of the scratch numbers times the ratio of their length to the diameter of the element cannot exceed half the maximum scratch number. Even with some maximum visibility scratches present, MIL-PRF-13830B still allows many combinations of smaller scratch sizes and lengths on the polished surface. Machine Vision Guide As stated above, all optics in this catalog are referenced to MIL-PRF-13830B standards. These standards include scratches, digs, grayness, edge chips, and cemented interfaces. It is important to note that inspection of polished optical surfaces for scratches is accomplished by visual comparison to scratch standards. Thus, it is not the actual width of the scratch that is ascertained, but the appearance of the scratch as compared to these standards. A part is rejected if any scratches exceed the maximum size allowed. Digs, on the other hand, specified by actual defect size, can be measured quantitatively. represents a minimum standard for laser mirrors or extra-cavity optics used in moderate power laser and imaging applications. Gaussian Beam Optics SPECIFICATION STANDARDS Fundamental Optics The most common and widely accepted convention for specifying surface quality is the U.S. Military Surface Quality Specification, MIL-PRF-13830B. The surface quality of all CVI Laser Optics optics is tested in accordance with this specification. In Europe, an alternative specification, the DIN (Deutsche Industrie Norm) specification, DIN 3140, Sheet 7, is used. CVI Laser Optics can also work to ISO-10110 requirements. Laser Guide marketplace.idexop.com Cosmetic Surface Quality – U.S. Military Specifications A89 OPTICAL SPECIFICATIONS Optical Specifications DIGS A dig is a pit or small crater on the polished optical surface. Digs are defined by their diameters, which are the actual sizes of the digs in hundredths of a millimeter. The diameter of an irregularly shaped dig is ½ (L + W), where L and W are, respectively, the length and width of the dig: X 50 dig = 0.5 mm in diameter X 40 dig = 0.4 mm in diameter X 30 dig = 0.3 mm in diameter X 20 dig = 0.2 mm in diameter X 10 dig = 0.1 mm in diameter The permissible number of maximum-size digs shall be one per each 20 mm of diameter (or fraction thereof) on any single surface. The sum of the diameters of all digs, as estimated by the inspector, shall not exceed twice the diameter of the maximum size specified per any 20 mm diameter. Digs less than 2.5 mm are ignored. clear aperture up to a maximum of 1.0 mm. The sum of edge separations deeper than 0.5 mm cannot exceed 10 percent of the element perimeter. BEVELS Although bevels are not specified in MIL-PRF-13830B, our standard shop practice specifies that element edges are beveled to a face width of 0.25 to 0.5 mm at an angle of 45°±15°. Edges meeting at angles of 135° or larger are not beveled. COATING DEFECTS Defects caused by an optical element coating, such as scratches, voids, pinholes, dust, or stains, are considered with the scratch-and-dig specification for that element. Coating defects are allowed if their size is within the stated scratch-and-dig tolerance. Coating defects are counted separately from substrate defects. EDGE CHIPS Lens edge chips are allowed only outside the clear aperture of the lens. The clear aperture is 90 percent of the lens diameter unless otherwise specified. Chips smaller than 0.5 mm are ignored, and those larger than 0.5 mm are ground so that there is no shine to the chip. The sum of the widths of chips larger than 0.5 mm cannot exceed 30 percent of the lens perimeter. Prism edge chips outside the clear aperture are allowed. If the prism leg dimension is 25.4 mm or less, chips may extend inward 1.0 mm from the edge. If the leg dimension is larger than 25.4 mm, chips may extend inward 2.0 mm from the edge. Chips smaller than 0.5 mm are ignored, and those larger than 0.5 mm must be stoned or ground, leaving no shine to the chip. The sum of the widths of chips larger than 0.5 mm cannot exceed 30 percent of the length of the edge on which they occur. CEMENTED INTERFACES Because a cemented interface is considered a lens surface, specified surface quality standards apply. Edge separation at a cemented interface cannot extend into the element more than half the distance to the element A90 Cosmetic Surface Quality – U.S. Military Specifications 1-505-298-2550 OPTICAL SPECIFICATIONS Optical Coatings & Materials SURFACE ACCURACY During manufacture, a precision component is frequently compared with a test plate that has an accurate polished surface that is the inverse of the surface under test. When the two surfaces are brought together and viewed in nearly monochromatic light, Newton’s rings (interference fringes caused by the near-surface contact) appear. The number of rings indicates the difference in radius between the surfaces. This is known as power or sometimes as figure. It is measured in rings that are equivalent to half wavelengths. Optical Specifications Beyond their number, the rings may exhibit distortion that indicates nonuniform shape differences. The distortion may be local to one small area, or it may be in the form of noncircular fringes over the whole aperture. All such nonuniformities are known collectively as irregularity. Fundamental Optics Modern techniques for measuring surface accuracy utilize phase-measuring interferometry with advanced computer data analysis software. Removing operator subjectivity has made this approach considerably more accurate and repeatable. A zoom function can increase the resolution across the entire surface or a specific region to enhance the accuracy of the measurement. POWER AND IRREGULARITY Material Properties When attempting to specify how closely an optical surface conforms to its intended shape, a measure of surface accuracy is needed. Surface accuracy can be determined by interferometry techniques. Traditional techniques involve comparing the actual surface to a test plate gauge. In this approach, surface accuracy is measured by counting the number of rings or fringes and examining the regularity of the fringe. The accuracy of the fit between the lens and the test gauge (as shown in figure 3.4) is described by the number of fringes seen when the gauge is in contact with the lens. Test plates are made flat or spherical to within small fractions of a fringe. The accuracy of a test plate is only as good as the means used to measure its radii. Extreme care must be used when placing a test plate in contact with the actual surface to prevent damage to the surface. SURFACE FLATNESS VXUIDFHLQFRQWDFW DLUJDSEHWZHHQVXUIDFHV WHVWVXUIDFH UHIHUHQFHVXUIDFH Machine Vision Guide PD[LPXPDOORZDEOH GHYLDWLRQ Gaussian Beam Optics Surface flatness is simply surface accuracy with respect to a plane reference surface. It is used extensively in mirror and optical-flat specifications. VXUIDFHDFFXUDF\ Figure 3.4. Surface accuracy Laser Guide marketplace.idexop.com Surface Accuracy A91 OPTICAL SPECIFICATIONS Optical Specifications A92 1-505-298-2550 Optical Coatings & Materials FUNDAMENTAL OPTICS PARAXIAL FORMULAS Material Properties INTRODUCTIONA94 A95 IMAGING PROPERTIES OF LENS SYSTEMS A99 A105 LENS SHAPE A111 LENS COMBINATIONS A112 DIFFRACTION EFFECTS A114 LENS SELECTION A118 SPOT SIZE A121 ABERRATION BALANCING A122 DEFINITION OF TERMS A124 PARAXIAL LENS FORMULAS A127 PRINCIPAL-POINT LOCATIONS A132 Gaussian Beam Optics PERFORMANCE FACTORS Fundamental Optics A101 Optical Specifications LENS COMBINATION FORMULAS PRISMSA133 POLARIZATIONA137 WAVEPLATESA143 ETALONSA147 Machine Vision Guide ULTRAFAST THEORY A150 Laser Guide marketplace.idexop.com A93 FUNDAMENTAL OPTICS INTRODUCTION Fundamental Optics The process of solving virtually any optical engineering problem can be broken down into two main steps. First, paraxial calculations (first order) are made to determine critical parameters such as magnification, focal length(s), clear aperture (diameter), and object and image position. These paraxial calculations are covered in the next section of this chapter. Second, actual components are chosen based on these paraxial values, and their actual performance is evaluated with special attention paid to the effects of aberrations. A truly rigorous performance analysis for all but the simplest optical systems generally requires computer ray tracing, but simple generalizations can be used, especially when the lens selection process is confined to a limited range of component shapes. In practice, the second step may reveal conflicts with design constraints, such as component size, cost, or product availability. System parameters may therefore require modification. Because some of the terms used in this chapter may not be familiar to all readers, a glossary of terms is provided in Definition of Terms. Finally, it should be noted that the discussion in this chapter relates only to systems with uniform illumination; optical systems for Gaussian beams are covered in Gaussian Beam Optics. Engineering Support CVI Laser Optics maintains a staff of knowledgeable, experienced applications engineers at each of our facilities worldwide. The information given in this chapter is sufficient to enable the user to select the most appropriate catalog lenses for the most commonly encountered applications. However, when additional optical engineering support is required, our applications engineers are available to provide assistance. Do not hesitate to contact us for help in product selection or to obtain more detailed specifications on CVI Laser Optics products. A94 Introduction 7+( 237,&$/ (1*,1((5,1* 352&(66 'HWHUPLQHEDVLFV\VWHP SDUDPHWHUVVXFKDV PDJQLILFDWLRQDQG REMHFWLPDJHGLVWDQFHV 8VLQJSDUD[LDOIRUPXODV DQGNQRZQSDUDPHWHUV VROYHIRUUHPDLQLQJYDOXHV 3LFNOHQVFRPSRQHQWV EDVHGRQSDUD[LDOO\ GHULYHGYDOXHV 'HWHUPLQHLIFKRVHQ FRPSRQHQWYDOXHVFRQIOLFW ZLWKDQ\EDVLF V\VWHPFRQVWUDLQWV (VWLPDWHSHUIRUPDQFH FKDUDFWHULVWLFVRIV\VWHP 'HWHUPLQHLISHUIRUPDQFH FKDUDFWHULVWLFVPHHW RULJLQDOGHVLJQJRDOV 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials PARAXIAL FORMULAS SIGN CONVENTIONS FOR LENSES: (refer to figure 4.1) FOR MIRRORS: s is + for object to left of H (the first principal point) ƒ is + for convex (diverging) mirrors s" is + for image to right of H" (the second principal point) s is + for object to left of H s is – for object to right of H s" is – for image to left of H" m is + for an inverted image Material Properties The validity of the paraxial lens formulas is dependent on adherence to the following sign conventions: ƒ is – for concave (converging) mirrors s is – for object to right of H s" is – for image to right of H" m is – for an upright image s" is + for image to left of H" Optical Specifications m is + for an inverted image m is – for an upright image Fundamental Optics When using the thin-lens approximation, simply refer to the left and right of the lens. UHDUIRFDOSRLQW IURQWIRFDOSRLQW h REMHFW f v + +s &$ )s LPDJH f Gaussian Beam Optics ) hs f s ss SULQFLSDOSRLQWV 1RWHORFDWLRQRIREMHFWDQGLPDJHUHODWLYHWRIURQWDQGUHDUIRFDOSRLQWV f OHQVGLDPHWHU REMHFWGLVWDQFHSRVLWLYHIRUREMHFWZKHWKHUUHDO RUYLUWXDOWRWKHOHIWRISULQFLSDOSRLQW+ f HIIHFWLYHIRFDOOHQJWK()/ZKLFKPD\EHSRVLWLYH DVVKRZQRUQHJDWLYHfUHSUHVHQWVERWK)+DQG +s)sDVVXPLQJOHQVLVVXUURXQGHGE\PHGLXP RILQGH[ ss LPDJHGLVWDQFHsDQGs sDUHFROOHFWLYHO\FDOOHG FRQMXJDWHGLVWDQFHVZLWKREMHFWDQGLPDJHLQ FRQMXJDWHSODQHVSRVLWLYHIRULPDJHZKHWKHUUHDO RUYLUWXDOWRWKHULJKWRISULQFLSDOSRLQW+s m s s/s hs h PDJQLILFDWLRQRU FRQMXJDWHUDWLRVDLGWREHLQILQLWHLI HLWKHUs s RUsLVLQILQLWH h REMHFWKHLJKW hs LPDJHKHLJKW v Machine Vision Guide s &$ FOHDUDSHUWXUHW\SLFDOO\RIf DUFVLQ&$s Figure 4.1 Sign conventions Laser Guide marketplace.idexop.com Paraxial Formulas A95 FUNDAMENTAL OPTICS Fundamental Optics Typically, the first step in optical problem solving is to select a system focal length based on constraints such as magnification or conjugate distances (object and image distance). The relationship among focal length, object position, and image position is given by = + (4.1) f s s″ This formula is referenced to figure 4.1 and the sign conventions given in Sign Conventions. By definition, magnification is the ratio of image size to object size or m= s″ h″ = .(4.2) s h This relationship can be used to recast the first formula into the following forms: (s + s ″) (4.3) ( m + ) (s + s ″) f = msm f = ((m s ++s″)) f = m + ( m + ) sm s″ f = s + (4.4) f = msm + s + ( = m + +s ″ ) ff = s + s ″ ) f = m (+m + m s) += ss″+ s ″ s( m +msm f = ++ m f = (4.5) m + + s( m + ) = s +ms ″ s + s″ = + ) = s + s ″ sf( m m++ m f =m s( m + ) = s + s ″ (4.6) where (s+s”) is the approximate object-to-image distance. With a real lens of finite thickness, the image distance, object distance, and focal length are all referenced to the principal points, not to the physical center of the lens. By neglecting the distance between the lens’ principal A96 Paraxial Formulas points, known as the hiatus, s+s” becomes the object-toimage distance. This simplification, called the thin-lens approximation, can speed up calculation when dealing with simple optical systems. EXAMPLE 1: OBJECT OUTSIDE FOCAL POINT A 1 mm high object is placed on the optical axis, 200 mm left of the left principal point of a LDX-25.0-51.0-C (f = 50 mm). Where is the image formed, and what is the magnification? (See figure 4.2.) = − s″ f s = − s ″ s ″ = . mm m= s ″ . = . = s or real image is 0.33 mm high and inverted. object F2 image F1 200 66.7 Figure 4.2 Example 1 (f = 50 mm, s = 200 mm, s” = 66.7 mm) EXAMPLE 2: OBJECT INSIDE FOCAL POINT The same object is placed 30 mm left of the left principal point of the same lens. Where is the image formed, and what is the magnification? (See figure 4.3.) = − s ″ s ″ = − mm s ″ − m= = = −. s or virtual image is 2.5 mm high and upright. 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials object F2 Material Properties F1 exit angle with the optical axis is the same as its entrance angle). This method has been applied to the three previous examples illustrated in figures 4.2 through 4.4. Note that by using the thin-lens approximation, this second property reduces to the statement that a ray passing through the center of the lens is undeviated. image Figure 4.3 Example 2 (f = 50 mm, s = 30 mm, s”= 475 mm) image Optical Specifications The paraxial calculations used to determine the necessary element diameter are based on the concepts of focal ratio (f-number or f/#) and numerical aperture (NA). The f-number is the ratio of the focal length of the lens to its “effective” diameter, the clear aperture (CA). object F2 F-NUMBER AND NUMERICAL APERTURE F1 f-number = Figure 4.4 Example 3 (f = 450 mm, s = 50 mm, s”= 425 mm) EXAMPLE 3: OBJECT AT FOCAL POINT A 1 mm high object is placed on the optical axis, 50 mm left of the first principal point of an LDK-50.0-52.2-C (f =450 mm). Where is the image formed, and what is the magnification? (See figure 4.4.) NA = sinv = CA f (4.8) Machine Vision Guide or virtual image is 0.5 mm high and upright. A simple graphical method can also be used to determine paraxial image location and magnification. This graphical approach relies on two simple properties of an optical system. First, a ray that enters the system parallel to the optical axis crosses the optical axis at the focal point. Second, a ray that enters the first principal point of the system exits the system from the second principal point parallel to its original direction (i.e., its The other term used commonly in defining this cone angle is numerical aperture. The NA is the sine of the angle made by the marginal ray with the optical axis. By referring to figure 4.5 and using simple trigonometry, it can be seen that Gaussian Beam Optics = − s ″ − s ″ = − mm s ″ − m= = = −. s To visualize the f-number, consider a lens with a positive focal length illuminated uniformly with collimated light. The f-number defines the angle of the cone of light leaving the lens which ultimately forms the image. This is an important concept when the throughput or lightgathering power of an optical system is critical, such as when focusing light into a monochromator or projecting a high-power image. Fundamental Optics In this case, the lens is being used as a magnifier, and the image can be viewed only back through the lens. f . (4.7) CA and NA = . ( f-number ) (4.9) Laser Guide marketplace.idexop.com Paraxial Formulas A97 FUNDAMENTAL OPTICS Fundamental Optics Ray f-numbers can also be defined for any arbitrary ray if its conjugate distance and the diameter at which it intersects the principal surface of the optical system are known. f CA 2 v principal surface Figure 4.5 F-number and numerical aperture NOTE Because the sign convention given previously is not used universally in all optics texts, the reader may notice differences in the paraxial formulas. However, results will be correct as long as a consistent set of formulas and sign conventions is used. A98 Paraxial Formulas 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials IMAGING PROPERTIES OF LENS SYSTEMS THE OPTICAL INVARIANT To understand the importance of the NA, consider its relation to magnification. Referring to figure 4.6, sin v = NA . When a lens or optical system is used to create an image of a source, it is natural to assume that, by increasing the diameter (Ø) of the lens, thereby increasing its CA, we will be able to collect more light and thereby produce a brighter image. However, because of the relationship EXAMPLE: SYSTEM WITH FIXED INPUT NA Two very common applications of simple optics involve coupling light into an optical fiber or into the entrance slit of a monochromator. Although these problems appear to be quite different, they both have the same limitation – they have a fixed NA. For monochromators, this limit is usually expressed in terms of the f-number. In addition to the fixed NA, they both have a fixed entrance pupil (image) size. Suppose it is necessary, using a singlet lens, to couple the output of an incandescent bulb with a filament 1 mm in diameter into an optical fiber as shown in figure 4.7. Assume that the fiber has a core diameter of 100 mm and an NA of 0.25, and that the design requires that the total distance from the source to the fiber be 110 mm. Which lenses are appropriate? By definition, the magnification must be 0.1. Letting s+s” total 110 mm (using the thin-lens approximation), we can use equation 4.3, f =m (s + s ″) , (m + ) Machine Vision Guide The magnification of the system is therefore equal to the ratio of the NAs on the object and image sides of the system. This powerful and useful result is completely independent of the specifics of the optical system, and it can often be used to determine the optimum lens diameter in situations involving aperture constraints. To understand how to use this relationship between magnification and NA, consider the following example. Gaussian Beam Optics NA m =NA . NA. ″ m = NA ″ (4.15) SAMPLE CALCULATION Fundamental Optics = Since the NA of a ray is given by CA/2s, once a focal length and magnification have been selected, the value of NA sets the value of CA. Thus, if one is dealing with a system in which the NA is constrained on either the object or image side, increasing the lens diameter beyond this value will increase system size and cost but will not improve performance (i.e., throughput or image brightness). This concept is sometimes referred to as the optical invariant. Optical Specifications s″ NA sarrive v v ″ NA ″ s s″ sin NA ″ sin we m at Since simply == is =. th . s sin NA s s ″v ″″ NA ″ simply magnification of the system, Since ofthe thesystem, system, Since NA isissimply tehthe emagnification magnification of hhe m sarrive =″ s at . we NA ″ Sincewe arrive is simply t we arrive s at at Material Properties CA NA (object side) = sin v = CA (4.10) NA (object side) = sin v = s sCA NA″ (image side) = sin v ″ = CA NA″ (image side) = sin v ″ = s ″ (4.11) s ″ w NA (object side) = sin v = CA s w which can be rearranged to show CA NAcan (object side) = sinto v =show which be rearranged CA which bevside) arranged CA can s sin NA″=(image = sin vto″ show =s s ″ CA = s sin v andNA″ (image vside) = sin v ″ = CA w CA andNA (object side) v ″ = sin v = s ″ CA can =tosbe ″ sin leading which rearranged to show s (4.12) = s sin v CA w CA ″ = sin v = NA =(object leading tos ″ sinside) sCA sNA″ v rearranged NA = sintov ″show sinbe ″ can CA and which =CA ==(image = vside) . s vsin sand NA sin ″ s ″ side)″ = sin v = CA s =NA v ″ = NA sin(object vNA CA =sin sv″″sin ″ ″=. sin v ″ =s NA″ (image side) s w CA s″ s″ Since simply the magnification of the system, leading toissbe NA″ (image side) = sin v show ″= and s ″ (4.13) v = sin ″ ″ which can rearranged to s is simply the magnification w s ″ of the system, Since s sin v rearranged sw″ can CA we arrive which leading CA s sin=v NA . to show ==toatbe swhichsin v ″be rearranged NA ″ we arrive at to show NA can s vsin vv NA s ″ = sin andCA leading to m = . = . sCA ″ NA = s sin v NA s ==sin ″″ NA Since is vsimply th″ m and CA leading tos ″ sin. ″ andss″ NA ″ v he magnification of the system, CA = sin iss ″simply ″ th Since sleading vsin v NA ″ CA =s =tos ″=sin ″ (4.14) . we arrive atv ″ NAhe leading s to sin ″ magnification of the system, between magnification and NA, there can be a theoretical limit beyond which increasing the diameter has no effect on light-collection efficiency or image brightness. (see eq. 4.3) to determine that the focal length is 9.1 mm. To determine the conjugate distances, s and s”, we utilize Laser Guide marketplace.idexop.com Imaging Properties of Lens Systems A99 FUNDAMENTAL OPTICS and Fundamental Optics equation 4.6, s ( m + ) = s + s ″, (see eq. 4.6) and find that s = 100 mm and s” = 10 mm. We can now use the relationship NA = CA/2s or NA”= CA/2s” to derive CA, the optimum clear aperture (effective diameter) of the lens. With an image NA of 0.25 and an image distance (s”) of 10 mm, . = CA = mm. Accomplishing this imaging task with a single lens therefore requires an optic with a 9.1 mm focal length and a 5 mm diameter. Using a larger diameter lens will not result in any greater system throughput because of the limited input NA of the optical fiber. The singlet lenses in this catalog that meet these criteria are LPX5.0-5.2-C, which is plano-convex, and LDX-6.0-7.7-C and LDX-5.0-9.9-C, which are biconvex. Making some simple calculations has reduced our choice of lenses to just three. The following chapter, Gaussian Beam Optics, discusses how to make a final choice of lenses based on various performance criteria. CA s″ s CA 2 v″ v CA image side object side Figure 4.6 Numerical aperture and magnification magnification = h″ = 0.1 = 0.1! h 1.0 filament h = 1 mm NA = optical system f = 9.1 mm CA = 0.025 2s NA″ = CA = 0.25 2s ″ CA = 5 mm fiber core h″ = 0.1 mm s = 100 mm s″ = 10 mm s + s″ = 110 mm Figure 4.7 Optical system geometry for focusing the output of an incandescent bulb into an optical fiber A100 Imaging Properties of Lens Systems 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials LENS COMBINATION FORMULAS For all values of f1, f2, and d, the location of the focal point of the combined system (s2"), measured from the secondary principal point of the second lens (H2"), is given by s ″ = f ( f − d ) . f + f − d (4.18) This can be shown by setting s1=d–f1 (see figure 4.8a), and solving Optical Specifications It is much simpler to calculate the effective (combined) focal length and principal-point locations and then use these results in any subsequent paraxial calculations (see figure 4.8). They can even be used in the optical invariant calculations described in the preceding section. COMBINATION FOCAL-POINT LOCATION Material Properties Many optical tasks require several lenses in order to achieve an acceptable level of performance. One possible approach to lens combinations is to consider each image formed by each lens as the object for the next lens and so on. This is a valid approach, but it is time consuming and unnecessary. EFFECTIVE FOCAL LENGTH The expression for the combination focal length is the same whether lens separation distances are large or small and whether f1 and f2 are positive or negative: f f . (4.16) f + f − d for s2". COMBINATION SECONDARY PRINCIPAL-POINT LOCATION Because the thin-lens approximation is obviously highly invalid for most combinations, the ability to determine the location of the secondary principal point is vital for accurate determination of d when another element is added. The simplest formula for this calculates the distance from the secondary principal point of the final (second) element to the secondary principal point of the combination (see figure 4.8b): Gaussian Beam Optics f = = + f s s″ Fundamental Optics The following formulas show how to calculate the effective focal length and principal-point locations for a combination of any two arbitrary components. The approach for more than two lenses is very simple: Calculate the values for the first two elements, then perform the same calculation for this combination with the next lens. This is continued until all lenses in the system are accounted for. This may be more familiar in the form (4.19) z = s ″ − f . Notice that the formula is symmetric with respect to the interchange of the lenses (end-for-end rotation of the combination) at constant d. The next two formulas are not. COMBINATION EXAMPLES Machine Vision Guide d = + − . (4.17) f f f f f It is possible for a lens combination or system to exhibit principal planes that are far removed from the system. When such systems are themselves combined, negative values of d may occur. Probably the simplest example of a negative d-value situation is shown in figure 4.9. Meniscus lenses with steep surfaces have external principal planes. When two of these lenses are brought Laser Guide marketplace.idexop.com Lens Combination Formulas A101 FUNDAMENTAL OPTICS Fundamental Optics into contact, a negative value of d can occur. Other combined-lens examples are shown in figures 4.10 through 4.13. Symbols fc = combination focal length (EFL), positive if combination final focal point falls to the right of the combination secondary principal point, negative otherwise (see figure 4.8c). Note: These paraxial formulas apply to coaxial combinations of both thick and thin lenses immersed in air or any other fluid with refractive index independent of position. They assume that light propagates from left to right through an optical system. f1 = focal length of the first element (see figure 4.8a). f2 = focal length of the second element. d = distance from the secondary principal point of the first element to the primary principal point of the second element, positive if the primary principal point is to the right of the secondary principal point, negative otherwise (see figure 4.8b). s1" = distance from the primary principal point of the first element to the final combination focal point (location of the final image for an object at infinity to the right of both lenses), positive if the focal point is to left of the first element’s primary principal point (see figure 4.8d). s2" = distance from the secondary principal point of the second element to the final combination focal point (location of the final image for an object at infinity to the left of both lenses), positive if the focal point is to the right of the second element’s secondary principal point (see figure 4.8b). zH = distance to the combination primary principal point measured from the primary principal point of the first element, positive if the combination secondary principal point is to the right of secondary principal point of second element (see figure 4.8d). 1 2 3 4 d>0 1 3 4 d<0 Figure 4.9 “Extreme” meniscus-form lenses with external principal planes (drawing not to scale) zH" = distance to the combination secondary principal point measured from the secondary principal point of the second element, positive if the combination secondary principal point is to the right of the secondary principal point of the second element (see figure 4.8c). A102 Lens Combination Formulas 2 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials H1″ H1 Hc″ Material Properties lens combination lens 1 zH″ f1 (a) (c) fc s1 = d4f1 Hc H2 H2” lens 1 and lens 2 Optical Specifications H1″ H1 lens combination zH d (b) (d) s2″ fc Fundamental Optics Figure 4.8 Lens combination focal length and principal planes z f1 f1 s2″ d f2 f<0 focal plane combination secondary principal plane f2 Figure 4.11 Achromatic combinations: Air-spaced lens combinations can be made nearly achromatic, even though both elements are made from the same material. Achieving achromatism requires that, in the thin-lens approximation, d= ( f + f ) Machine Vision Guide Figure 4.10 Positive lenses separated by distance greater than f1 = f2: f is negative and both s2" and z are positive. Lens symmetry is not required. d H2″ Gaussian Beam Optics H2 H1″ . This is the basis for Huygens and Ramsden eyepieces. This approximation is adequate for most thick-lens situations. The signs of f1, f2, and d are unrestricted, but d must have a value that guarantees the existence of an air space. Element shapes are unrestricted and can be chosen to compensate for other aberrations. Laser Guide marketplace.idexop.com Lens Combination Formulas A103 FUNDAMENTAL OPTICS Fundamental Optics f z<0 tc n d tc n s2 ″ H H″ combination secondary principal plane combination focus Figure 4.12 Telephoto combination: The most important characteristic of the telephoto lens is that the EFL, and hence the image size, can be made much larger than the distance from the first lens surface to the image would suggest by using a positive lens followed by a negative lens (but not necessarily the lens shapes shown in the figure). For example, f1 is positive and f2 = –f1/2. Then f is negative for d < f1/2, infinite for d = f1/2 (Galilean telescope or beam expander), and positive for d > f1/2. To make the example even more specific, catalog lenses LDX-50.8-130.4-C and LDK-42.0-52.2-C, with d = 78.2 mm, will yield s2" = 2.0 m, f = 5.2 m, and z = 43.2 m. A104 Lens Combination Formulas s s″ Figure 4.13 Condenser configuration: The convex vertices of a pair of identical plano-convex lenses are on contact. (The lenses could also be plano aspheres.) Because d = 0, f = f1/2 = f2/2, f1/2 = s2", and z = 0. The secondary principal point of the second element and the secondary principal point of the combination coincide at H", at depth tc/n beneath the vertex of the plano surface of the second element, where tc is the element center thickness and n is the refractive index of the element. By symmetry, the primary principal point of the combination is similarly located in the first element. Combination conjugate distances must be measured from these points. 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials PERFORMANCE FACTORS The performance of real optical systems is limited by several factors, including lens aberrations and light diffraction. The magnitude of these effects can be calculated with relative ease. To determine the precise performance of a lens system, we can trace the path of light rays through it, using Snell’s law at each optical interface to determine the subsequent ray direction. This process, called ray tracing, is usually accomplished on a computer. When this process is completed, it is typically found that not all the rays pass through the points or positions predicted by paraxial theory. These deviations from ideal imaging are called lens aberrations. The direction of a light ray after refraction at the interface between two homogeneous, isotropic media of differing index of refraction is given by Snell’s law: Z HQ JW K l v DIFFRACTION PDWHULDO LQGH[ n v Gaussian Beam Optics Diffraction, a natural property of light arising from its wave nature, poses a fundamental limitation on any optical system. Diffraction is always present, although its effects may be masked if the system has significant aberrations. When an optical system is essentially free from aberrations, its performance is limited solely by diffraction, and it is referred to as diffraction limited. HO Fundamental Optics PDWHULDO LQGH[ n DY Optical Specifications Numerous other factors, such as lens manufacturing tolerances and component alignment, impact the performance of an optical system. Although these are not considered explicitly in the following discussion, it should be kept in mind that if calculations indicate that a lens system only just meets the desired performance criteria, in practice it may fall short of this performance as a result of other factors. In critical applications, it is generally better to select a lens whose calculated performance is significantly better than needed. ABERRATIONS Material Properties After paraxial formulas have been used to select values for component focal length(s) and diameter(s), the final step is to select actual lenses. As in any engineering problem, this selection process involves a number of tradeoffs, including performance, cost, weight, and environmental factors. Figure 5.14 Refraction of light at a dielectric boundary In calculating diffraction, we simply need to know the focal length(s) and aperture diameter(s); we do not consider other lens-related factors such as shape or index of refraction. Machine Vision Guide Since diffraction increases with increasing f-number, and aberrations decrease with increasing f-number, determining optimum system performance often involves finding a point where the combination of these factors has a minimum effect. Laser Guide marketplace.idexop.com Performance Factors A105 FUNDAMENTAL OPTICS Fundamental Optics n1sinθ1 = n2sinθ2 (4.20) where θ1 is the angle of incidence, θ2 is the angle of refraction, and both angles are measured from the surface normal as shown in figure 4.14. Even though tools for the precise analysis of an optical system are becoming easier to use and are readily available, it is still quite useful to have a method for quickly estimating lens performance. This not only saves time in the initial stages of system specification, but can also help achieve a better starting point for any further computer optimization. The first step in developing these rough guidelines is to realize that the sine functions in Snell’s law can be expanded in an infinite Taylor series: sin v = v − v / ! + v / ! − v / ! + v / ! − . . . (4.21) The first approximation we can make is to replace all the sine functions with their arguments (i.e., replace sinθ1 with θ1 itself and so on). This is called first-order or paraxial theory because only the first terms of the sine expansions are used. Design of any optical system generally starts with this approximation using the paraxial formulas. The assumption that sinθ = θ is reasonably valid for θ close to zero (i.e., high f-number lenses). With more highly curved surfaces (and particularly marginal rays), paraxial theory yields increasingly large deviations from real performance because sinθ ≠ θ. These deviations are known as aberrations. Because a perfect optical system (one without any aberrations) would form its image at the point and to the size indicated by paraxial theory, aberrations are really a measure of how the image differs from the paraxial prediction. As already stated, exact ray tracing is the only rigorous way to analyze real lens surfaces. Before the advent of electronic computers, this was excessively tedious and time consuming. Seidel* addressed this issue by developing a method of calculating aberrations resulting A106 Performance Factors from the θ13/3! term. The resultant third-order lens aberrations are therefore called Seidel aberrations. To simplify these calculations, Seidel put the aberrations of an optical system into several different classifications. In monochromatic light they are spherical aberration, astigmatism, field curvature, coma, and distortion. In polychromatic light there are also chromatic aberration and lateral color. Seidel developed methods to approximate each of these aberrations without actually tracing large numbers of rays using all the terms in the sine expansions. In actual practice, aberrations occur in combinations rather than alone. This system of classifying them, which makes analysis much simpler, gives a good description of optical system image quality. In fact, even in the era of powerful ray-tracing software, Seidel’s formula for spherical aberration is still widely used. SPHERICAL ABERRATION Figure 4.15 illustrates how an aberration-free lens focuses incoming collimated light. All rays pass through the focal point F". The lower figure shows the situation more typically encountered in single lenses. The farther from the optical axis the ray enters the lens, the nearer to the lens it focuses (crosses the optical axis). The distance along the optical axis between the intercept of the rays that are nearly on the optical axis (paraxial rays) and the rays that go through the edge of the lens (marginal rays) is called longitudinal spherical aberration (LSA). The height at which these rays intercept the paraxial focal plane is called transverse spherical aberration (TSA). These quantities are related by TSA = LSA x tan(u").(4.22) Spherical aberration is dependent on lens shape, orientation, and conjugate ratio, as well as on the index of refraction of the materials present. Parameters for choosing the best lens shape and orientation for a given task are presented later in this chapter. However, the third-order, monochromatic, spherical aberration of a plano-convex lens used at infinite conjugate ratio can be estimated by 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials spot size due to spherical aberration = (4.23) aberration-free lens paraxial focal plane As shown in figure 4.16, the plane containing both optical axis and object point is called the tangential plane. Rays that lie in this plane are called tangential, or meridional, rays. Rays not in this plane are referred to as skew rays. The chief, or principal, ray goes from the object point through the center of the aperture of the lens system. The plane perpendicular to the tangential plane that contains the principal ray is called the sagittal or radial plane. The figure illustrates that tangential rays from the object come to a focus closer to the lens than do rays in the sagittal plane. When the image is evaluated at the tangential conjugate, we see a line in the sagittal direction. A line in the tangential direction is formed at the sagittal conjugate. Between these conjugates, the image is either an elliptical or a circular blur. Astigmatism is defined as the separation of these conjugates. Gaussian Beam Optics F″ When an off-axis object is focused by a spherical lens, the natural asymmetry leads to astigmatism. The system appears to have two different focal lengths. Fundamental Optics In general, simple positive lenses have undercorrected spherical aberration, and negative lenses usually have overcorrected spherical aberration. By combining a positive lens made from low-index glass with a negative lens made from high-index glass, it is possible to ASTIGMATISM Optical Specifications Theoretically, the simplest way to eliminate or reduce spherical aberration is to make the lens surface(s) with a varying radius of curvature (i.e., an aspheric surface) designed to exactly compensate for the fact that sin θ ≠ θ at larger angles. In practice, however, most lenses with high surface accuracy are manufactured by grinding and polishing techniques that naturally produce spherical or cylindrical surfaces. The manufacture of aspheric surfaces is more complex, and it is difficult to produce a lens of sufficient surface accuracy to eliminate spherical aberration completely. Fortunately, these aberrations can be virtually eliminated, for a chosen set of conditions, by combining the effects of two or more spherical (or cylindrical) surfaces. Material Properties . f . f/# produce a combination in which the spherical aberrations cancel but the focusing powers do not. The simplest examples of this are cemented doublets, such as the LAO series which produce minimal spherical aberration when properly used. The amount of astigmatism in a lens depends on lens shape only when there is an aperture in the system that is not in contact with the lens itself. (In all optical systems there is an aperture or stop, although in many cases it is simply the clear aperture of the lens element itself.) Astigmatism strongly depends on the conjugate ratio. F″ TSA LSA longitudinal spherical aberration transverse spherical aberration Figure 4.15 Spherical aberration of a plano-convex lens * Ludwig von Seidel, 1857. Machine Vision Guide u″ COMA In spherical lenses, different parts of the lens surface exhibit different degrees of magnification. This gives rise to an aberration known as coma. As shown in figure 4.17, each concentric zone of a lens forms a ring-shaped image called a comatic circle. This causes blurring in the image plane (surface) of off-axis object points. An off-axis object point is not a sharp image point, but it appears Laser Guide marketplace.idexop.com Performance Factors A107 FUNDAMENTAL OPTICS Fundamental Optics tangential image (focal line) tangential plane xis cal a opti sagittal image (focal line) principal ray sagittal plane optical system object point paraxial focal plane Figure 4.16 Astigmatism represented by sectional views as a characteristic comet-like flare. Even if spherical aberration is corrected and the lens brings all rays to a sharp focus on axis, a lens may still exhibit coma off axis. See figure 4.18. As with spherical aberration, correction can be achieved by using multiple surfaces. Alternatively, a sharper image may be produced by judiciously placing an aperture, or stop, in an optical system to eliminate the more marginal rays. FIELD CURVATURE Even in the absence of astigmatism, there is a tendency of optical systems to image better on curved surfaces than on flat planes. This effect is called field curvature (see figure 4.19). In the presence of astigmatism, this problem is compounded because two separate astigmatic focal surfaces correspond to the tangential and sagittal conjugates. Field curvature varies with the square of field angle or the square of image height. Therefore, by reducing the field angle by one-half, it is possible to reduce the blur from field curvature to a value of 0.25 of its original size. Positive lens elements usually have inward curving fields, and negative lenses have outward curving fields. Field curvature can thus be corrected to some extent by combining positive and negative lens elements. A108 Performance Factors DISTORTION The image field may not only have curvature but may also be distorted. The image of an off-axis point may be formed at a location on this surface other than that predicted by the simple paraxial equations. This distortion is different from coma (where rays from an off-axis point fail to meet perfectly in the image plane). Distortion means that even if a perfect off-axis point image is formed, its location on the image plane is not correct. Furthermore, the amount of distortion usually increases with increasing image height. The effect of this can be seen as two different kinds of distortion: pincushion and barrel (see figure 4.20). Distortion does not lower system resolution; it simply means that the image shape does not correspond exactly to the shape of the object. Distortion is a separation of the actual image point from the paraxially predicted location on the image plane and can be expressed either as an absolute value or as a percentage of the paraxial image height. It should be apparent that a lens or lens system has opposite types of distortion depending on whether it is used forward or backward. This means that if a lens were used to make a photograph, and then used in reverse to project it, there would be no distortion in the final screen image. Also, perfectly symmetrical optical systems at 1:1 magnification have no distortion or coma. 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials SRLQWVRQOHQV 6 c c 32 c c c c c c c c FRUUHVSRQGLQJ SRLQWVRQ S c c c Material Properties c c f S Optical Specifications Figure 4.17 Imaging an off-axis point source by a lens with positive transverse coma positive transverse coma Figure 4.18 Positive transverse coma Gaussian Beam Optics spherical focal surface Fundamental Optics focal plane The index of refraction of a material is a function of wavelength. Known as dispersion, this is discussed in Material Properties. From Snell’s law (see equation 4.20), it can be seen that light rays of different wavelengths or colors will be refracted at different angles since the index is not a constant. Figure 4.21 shows the result when polychromatic collimated light is incident on a positive lens element. Because the index of refraction is higher for shorter wavelengths, these are focused closer to the lens than the longer wavelengths. Longitudinal chromatic aberration is defined as the axial distance from the nearest to the farthest focal point. As in the case of spherical aberration, positive and negative elements have opposite signs of chromatic aberration. Once again, by combining elements of nearly opposite aberration to form a doublet, chromatic aberration can be partially corrected. It is necessary to use two glasses with different dispersion characteristics, so that the weaker negative element can balance the aberration of the stronger, positive element. LATERAL COLOR CHROMATIC ABERRATION The aberrations previously described are purely a function of the shape of the lens surfaces, and they can be observed with monochromatic light. Other aberrations, however, arise when these optics are used to transform light containing multiple wavelengths. Machine Vision Guide Figure 4.19 Field curvature Lateral color is the difference in image height between blue and red rays. Figure 4.22 shows the chief ray of an optical system consisting of a simple positive lens and a separate aperture. Because of the change in index with wavelength, blue light is refracted more strongly than red light, which is why rays intercept the image plane at different heights. Stated simply, magnification depends on color. Lateral color is very dependent on system stop location. Laser Guide marketplace.idexop.com Performance Factors A109 FUNDAMENTAL OPTICS Fundamental Optics For many optical systems, the third-order term is all that may be needed to quantify aberrations. However, in highly corrected systems or in those having large apertures or a large angular field of view, third-order theory is inadequate. In these cases, exact ray tracing is absolutely essential. OBJECT PINCUSHION DISTORTION BARREL DISTORTION Variations of Aberrations with Aperture Field Angle and Image Height Aberration Aperture (Ø) Field Angle (θ) Image Height (y) Lateral Spherical Ø3 — — Longitudinal Spherical Ø2 — — Coma Ø2 θ y Astigmatism Ø θ2 y2 Field Curvature Ø θ2 y2 Distortion — θ3 y3 Chromatic — — — Figure 4.20 Pinchusion and barrel distortion APPLICATION NOTE red focal point white light ray blue focal point blue light ray red light ray longitudinal chromatic aberration Figure 4.21 Longitudinal chromatic abberation red light ray lateral color Achromatic Doublets Are Superior to Simple Lenses Because achromatic doublets correct for spherical as well as chromatic aberration, they are often superior to simple lenses for focusing collimated light or collimating point sources, even in purely monochromatic light. Although there is no simple formula that can be used to estimate the spot size of a doublet, the tables in Spot Size give sample values that can be used to estimate the performance of catalog achromatic doublets. blue light ray aperture focal plane Figure 4.22 Lateral Color A110 Performance Factors 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials LENS SHAPE q= ( r + r ) . ( r − r ) (4.24) Fundamental Optics For wide-field applications, the best-form shape is definitely not the optimum singlet shape, especially at the infinite conjugate ratio, since it yields maximum field curvature. The ideal shape is determined by the situation and may require rigorous ray-tracing analysis. It is possible to achieve much better correction in an optical system by using more than one element. The cases of an infinite conjugate ratio system and a unit conjugate ratio system are discussed in the following section. Gaussian Beam Optics Figure 4.23 shows the transverse and longitudinal spherical aberrations of a singlet lens as a function of the shape factor, q. In this particular instance, the lens has a focal length of 100 mm, operates at f/5, has an index of refraction of 1.518722 (BK7 at the mercury green line, 546.1 nm), and is being operated at the infinite conjugate ratio. It is also assumed that the lens itself is the aperture stop. An asymmetric shape that corresponds to a q-value of about 0.7426 for this material and wavelength is the best singlet shape for on-axis imaging. It is important to note that the best-form shape is dependent on refractive index. For example, with a high-index material, such as silicon, the best-form lens for the infinite conjugate ratio is a meniscus shape. For imaging at unit magnification (s = s" = 2f), a similar analysis would show that a symmetric biconvex lens is the best shape. Not only is spherical aberration minimized, but coma, distortion, and lateral chromatic aberration exactly cancel each other out. These results are true regardless of material index or wavelength, which explains the utility of symmetric convex lenses, as well as symmetrical optical systems in general. However, if a remote stop is present, these aberrations may not cancel each other quite as well. Optical Specifications To further explore the dependence of aberrations on lens shape, it is helpful to make use of the Coddington shape factor, q, defined as At infinite conjugate with a typical glass singlet, the plano-convex shape (q = 1), with convex side toward the infinite conjugate, performs nearly as well as the bestform lens. Because a plano-convex lens costs much less to manufacture than an asymmetric biconvex singlet, these lenses are quite popular. Furthermore, this lens shape exhibits near-minimum total transverse aberration and near-zero coma when used off axis, thus enhancing its utility. Material Properties Aberrations described in the preceding section are highly dependent on application, lens shape, and material of the lens (or, more exactly, its index of refraction). The singlet shape that minimizes spherical aberration at a given conjugate ratio is called best-form. The criterion for best-form at any conjugate ratio is that the marginal rays are equally refracted at each of the lens/air interfaces. This minimizes the effect of sinθ ≠ θ. It is also the criterion for minimum surface-reflectance loss. Another benefit is that absolute coma is nearly minimized for best-form shape, at both infinite and unit conjugate ratios. $EHUUDWLRQVLQ0LOOLPHWHUV H[DFWORQJLWXGLQDOVSKHULFDODEHUUDWLRQ/6$ Machine Vision Guide H[DFWWUDQVYHUVHVSKHULFDO DEHUUDWLRQ76$ 4 4 4 4 6KDSH )DFWRU q Figure 4.23 Aberrations of positive singlets at infinite conjugate ratio as a function of shape Laser Guide marketplace.idexop.com Lens Shape A111 FUNDAMENTAL OPTICS LENS COMBINATIONS Fundamental Optics INFINITE CONJUGATE RATIO As shown in the previous discussion, the best-form singlet lens for use at infinite conjugate ratios is generally nearly plano-convex. Figure 4.24 shows a plano-convex lens (LPX-15.0-10.9-C) with incoming collimated light at a wavelength of 546.1 nm. This drawing, including the rays traced through it, is shown to exact scale. The marginal ray (ray f-number 4.5) strikes the paraxial focal plane significantly off the optical axis. This situation can be improved by using a two-element system. The second part of the figure shows a precision achromat (LAO-21.0-14.0), which consists of a positive low-index (crown glass) element cemented to a negative meniscus high-index (flint glass) element. This is drawn to the same scale as the plano-convex lens. No spherical aberration can be discerned in the lens. Of course, not all of the rays pass exactly through the paraxial focal point; however, in this case, the departure is measured in micrometers, rather than in millimeters, as in the case of the plano-convex lens. Additionally, chromatic aberration (not shown) is much better corrected in the doublet. Even though these lenses are known as achromatic doublets, it is important to remember that even with monochromatic light the doublet’s performance is superior. Figure 4.24 also shows the f-number at which singlet performance becomes unacceptable. The ray with f-number 7.5 practically intercepts the paraxial focal point, and the f/3.8 ray is fairly close. This useful drawing, which can be scaled to fit a plano-convex lens of any focal length, can be used to estimate the magnitude of its spherical aberration, although lens thickness affects results slightly. A dramatic improvement in performance is gained by using two identical plano-convex lenses with convex surfaces facing and nearly in contact. Those shown in figure 4.25 are both LPX-20.0-20.7-C. The combination of these two lenses yields almost exactly the same focal length as the biconvex lens. To understand why this configuration improves performance so dramatically, consider that if the biconvex lens were split down the middle, we would have two identical plano-convex lenses, each working at an infinite conjugate ratio, but with the convex surface toward the focus. This orientation is opposite to that shown to be optimum for this shape lens. On the other hand, if these lenses are reversed, we have the system just described but with a better correction of the spherical aberration. Previous examples indicate that an achromat is superior in performance to a singlet when used at the infinite conjugate ratio and at low f-numbers. Since the unit conjugate case can be thought of as two lenses, each working at the infinite conjugate ratio, the next step is to replace the plano-convex singlets with achromats, yielding a four-element system. The third part of figure 4.25 shows a system composed of two LAO-40.0-18.0 lenses. Once again, spherical aberration is not evident, even in the f/2.7 ray. PLANO-CONVEX LENS ray f-numbers 1.5 1.9 2.5 3.8 7.5 LPX-15.0-10.9-C UNIT CONJUGATE RATIO Figure 4.25 shows three possible systems for use at the unit conjugate ratio. All are shown to the same scale and using the same ray f-numbers with a light wavelength of 546.1 nm. The first system is a symmetric biconvex lens (LDX-21.0-19.2-C), the best-form singlet in this application. Clearly, significant spherical aberration is present in this lens at f/2.7. Not until f/13.3 does the ray closely approach the paraxial focus. paraxial image plane ACHROMAT 1.5 1.9 2.5 3.8 7.5 LAO-21.0-14.0 Figure 4.24 Single‑element plano‑convex lens compared with a two‑element achromat A112 Lens Combinations 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials Material Properties SYMMETRIC BICONVEX LENS ray f-numbers 2.7 3.3 4.4 6.7 13.3 paraxial image plane Optical Specifications LDX-21.0-19.2-C IDENTICAL PLANO-CONVEX LENSES Fundamental Optics 2.7 3.3 4.4 6.7 13.3 LPX-20.0-20.7-C Gaussian Beam Optics IDENTICAL ACHROMATS 2.7 3.3 4.4 6.7 13.3 Machine Vision Guide LAO-40.0-18.0 Figure 4.25 Three possible systems for use at the unit conjugate ratio Laser Guide marketplace.idexop.com Lens Combinations A113 FUNDAMENTAL OPTICS DIFFRACTION EFFECTS Fundamental Optics In all light beams, some energy is spread outside the region predicted by geometric propagation. This effect, known as diffraction, is a fundamental and inescapable physical phenomenon. Diffraction can be understood by considering the wave nature of light. Huygens’ principle (figure 4.26) states that each point on a propagating wavefront is an emitter of secondary wavelets. The propagating wave is then the envelope of these expanding wavelets. Interference between the secondary wavelets gives rise to a fringe pattern that rapidly decreases in intensity with increasing angle from the initial direction of propagation. Huygens’ principle nicely describes diffraction, but rigorous explanation demands a detailed study of wave theory. Diffraction effects are traditionally classified into either Fresnel or Fraunhofer types. Fresnel diffraction is primarily concerned with what happens to light in the immediate neighborhood of a diffracting object or aperture. It is thus only of concern when the illumination source is close to this aperture or object, known as the near field. Consequently, Fresnel diffraction is rarely important in most classical optical setups, but it becomes very important in such applications as digital optics, fiber optics, and near-field microscopy. Fraunhofer diffraction, however, is often important even in simple optical systems. This is the light-spreading effect of an aperture when the aperture (or object) is illuminated with an infinite source (plane-wave illumination) and the light is sensed at an infinite distance (far-field) from this aperture. From these overly simple definitions, one might assume that Fraunhofer diffraction is important only in optical systems with infinite conjugate, whereas Fresnel diffraction equations should be considered at finite conjugate ratios. Not so. A lens or lens system of finite positive focal length with plane-wave input maps the far-field diffraction pattern of its aperture onto the focal plane; therefore, it is Fraunhofer diffraction that determines the limiting performance of optical systems. More generally, at any conjugate ratio, far-field angles are transformed into spatial displacements in the image plane. A114 Diffraction Effects some light diffracted into this region secondary wavelets wavefront wavefront aperture Figure 4.26 Huygens’ principle CIRCULAR APERTURE Fraunhofer diffraction at a circular aperture dictates the fundamental limits of performance for circular lenses. It is important to remember that the spot size, caused by diffraction, of a circular lens is d = 2.44λ(f/#) (4.25) where d is the diameter of the focused spot produced from plane-wave illumination and λ is the wavelength of light being focused. Notice that it is the f-number of the lens, not its absolute diameter, that determines this limiting spot size. The diffraction pattern resulting from a uniformly illuminated circular aperture actually consists of a central bright region, known as the Airy disc (see figure 4.27), which is surrounded by a number of much fainter rings. Each ring is separated by a circle of zero intensity. The irradiance distribution in this pattern can be described by J ( x ) I x = I (4.26) x where 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials I = peak irradiance in the image where I = peak irradiance in image x= where l = wavelength w = slit wi idth v = angular deviation from pattern maximum. and x= pw sin v l Material Properties J ( x ) = Bessel function of the e first kind of order unity ∞ n + xn − x ∑ ( ) ( n − )! n ! n − = n = − ENERGY DISTRIBUTION TABLE where λ = wavelength D = aperture diameter θ = angular radius from the pattern maximum. The accompanying table shows the major features of pure (unaberrated) Fraunhofer diffraction patterns of circular and slit apertures. The table shows the position, relative intensity, and percentage of total pattern energy corresponding to each ring or band. It is especially convenient to characterize positions in either pattern with the same variable x. This variable is related to field angle in the circular aperture case by Fundamental Optics sinv = Optical Specifications pD sin v l lx (4.28) pD where D is the aperture diameter. For a slit aperture, this relationship is given by AIRY DISC DIAMETER = 2.44 l f/# This useful formula shows the far-field irradiance distribution from a uniformly illuminated circular aperture of diameter D. SLIT APERTURE sin x Ix = I (4.27) x lx (4.29) pw where w is the slit width, π has its usual meaning, and D, w, and λ are all in the same units (preferably millimeters). Linear instead of angular field positions are simply found from r=s"tanθ where s" is the secondary conjugate distance. This last result is often seen in a different form, namely the diffraction-limited spot-size equation, which, for a circular lens is d = . l (f /#) Machine Vision Guide A slit aperture, which is mathematically simpler, is useful in relation to cylindrical optical elements. The irradiance distribution in the diffraction pattern of a uniformly illuminated slit aperture is described by sinv = Gaussian Beam Optics Figure 4.27 Center of a typical diffraction pattern for a circular aperture (see eq. 4.25) This value represents the smallest spot size that can be achieved by an optical system with a circular aperture of a given f-number, and it is the diameter of the first dark Laser Guide marketplace.idexop.com Diffraction Effects A115 FUNDAMENTAL OPTICS APPLICATION NOTE Fundamental Optics ring, where the intensity has dropped to zero. The graph in figure 4.28 shows the form of both circular and slit aperture diffraction patterns when plotted on the same normalized scale. Aperture diameter is equal to slit width so that patterns between x-values and angular deviations in the far-field are the same. Rayleigh Criterion In imaging applications, spatial resolution is ultimately limited by diffraction. Calculating the maximum possible spatial resolution of an optical system requires an arbitrary definition of what is meant by resolving two features. In the Rayleigh criterion, it is assumed that two separate point sources can be resolved when the center of the Airy disc from one overlaps the first dark ring in the diffraction pattern of the second. In this case, the smallest resolvable distance, d, is GAUSSIAN BEAMS Apodization, or nonuniformity of aperture irradiance, alters diffraction patterns. If pupil irradiance is nonuniform, the formulas and results given previously do not apply. This is important to remember because most laser-based optical systems do not have uniform pupil irradiance. The output beam of a laser operating in the TEM00 mode has a smooth Gaussian irradiance profile. Formulas used to determine the focused spot size from such a beam are discussed in Gaussian Beam Optics. Furthermore, when dealing with Gaussian beams, the location of the focused spot also departs from that predicted by the paraxial equations given in this chapter. This is also detailed in Gaussian Beam Optics. 1$ I Energy Distribution in the Diffraction Pattern of a Circular or Slit Aperture Circular Aperture Ring or Band A116 Slit Aperture Position (x) Relative Intensity (Ix/I0) Energy in Ring (%) Position (x) Relative Intensity (Ix/I0) Energy in Band (%) Central Maximum 0.0 1.0 83.8 0.0 1.0 90.3 First Dark 1.22π 0.0 1.00π 0.0 First Bright 1.64π 0.0175 1.43π 0.0472 Second Dark 2.23π 0.0 2.00π 0.0 Second Bright 2.68π 0.0042 2.46π 0.0165 Third Dark 3.24π 0.0 3.00π 0.0 Third Bright 3.70π 0.0016 3.47π 0.0083 Fourth Dark 4.24π 0.0 4.00π 0.0 Fourth Bright 4.71π 0.0008 4.48π 0.0050 Fifth Dark 5.24π 0.0 5.00π 0.0 Diffraction Effects 7.2 2.8 1.5 1.0 1-505-298-2550 4.7 1.7 0.8 0.5 FUNDAMENTAL OPTICS Optical Coatings & Materials &LUFXODU$SHUWXUH Material Properties ZLWKLQILUVWEULJKWULQJ VOLW DSHUWXUH Optical Specifications 1RUPDOL]HG3DWWHUQ,UUDGLDQFH\ LQ$LU\GLVF FLUFXODU DSHUWXUH 4 4 4 4 4 4 4 4 3RVLWLRQLQ,PDJH3ODQHx Fundamental Optics LQ FHQWUDOPD[LPXP ZLWKLQWKHWZR DGMRLQLQJVXEVLGLDU\PD[LPD 6/,7$3(5785( Gaussian Beam Optics Figure 4.28 Fraunhofer diffraction pattern of a singlet slit superimposed on the Fraunhofer diffraction pattern of a circular aperture Machine Vision Guide Laser Guide marketplace.idexop.com Diffraction Effects A117 FUNDAMENTAL OPTICS LENS SELECTION Fundamental Optics Having discussed the most important factors that affect the performance of a lens or a lens system, we will now address the practical matter of selecting the optimum catalog components for a particular task. The following useful relationships are important to keep in mind throughout the selection process: X Diffraction-limited spot size = 2.44λ f/# X Approximate on-axis spot size of a plano-convex lens at the infinite conjugate resulting from spherical aberration = . f f /# X Optical invariant = m = NA . NA ″ EXAMPLE 1: COLLIMATING AN INCANDESCENT SOURCE Produce a collimated beam from a quartz halogen bulb having a 1 mm square filament. Collect the maximum amount of light possible and produce a beam with the lowest possible divergence angle. This problem, illustrated in figure 4.29, involves the typical tradeoff between light-collection efficiency and resolution (where a beam is being collimated rather than focused, resolution is defined by beam divergence). To collect more light, it is necessary to work at a low f-number, but because of aberrations, higher resolution (lower divergence angle) will be achieved by working at a higher f-number. In terms of resolution, the first thing to realize is that the minimum divergence angle (in radians) that can be achieved using any lens system is the source size divided by system focal length. An off-axis ray (from the edge of the source) entering the first principal point of the system exits the second principal point at the same angle. Therefore, increasing the system focal length improves this limiting divergence because the source appears smaller. An optic that can produce a spot size of 1 mm when focusing a perfectly collimated beam is therefore required. Since source size is inherently limited, it is pointless to strive for better resolution. This level of resolution can be achieved easily with a plano-convex lens. While angular divergence decreases with increasing focal length, spherical aberration of a plano-convex lens increases with increasing focal length. To determine the appropriate focal length, set the spherical aberration formula for a plano-convex lens equal to the source (spot) size: . f = mm. f /# This ensures a lens that meets the minimum performance needed. To select a focal length, make an arbitrary f-number choice. As can be seen from the relationship, as we lower the f-number (increase collection efficiency), we decrease the focal length, which will worsen the resultant divergence angle (minimum divergence = 1 mm/f ). In this example, we will accept f/2 collection efficiency, which gives us a focal length of about 120 mm. For f/2 operation we would need a minimum diameter of 60 mm. The LPX-60.0-62.2-C fits this specification exactly. Beam divergence would be about 8 mrad. Finally, we need to verify that we are not operating below the theoretical diffraction limit. In this example, the vPLQ f vPLQ VRXUFHVL]H f Figure 4.29 Collimating an incandescent source A118 Lens Selection 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials numbers (1 mm spot size) indicate that we are not, since diffraction-limited spot size = 2.44 x 0.5 µm x 2 = 2.44 µm. . ( ) = mm. This is slightly smaller than the 100 µm spot size we are trying to achieve. However, since we are not working at infinite conjugate, the spot size will be larger than that given by our simple calculation. This lens is therefore likely to be marginal in this situation, especially if we consider chromatic aberration. A better choice is the . f = . mm. . This formula yields a focal length of 4.3 mm and a minimum diameter of 1.3 mm. The LPX-4.2-2.3-BAK1 meets these criteria. The biggest problem with utilizing these tiny, short focal length lenses is the practical considerations of handling, mounting, and positioning them. Because using a pair of longer focal length singlets would result in unacceptable performance, the next step might be to use a pair of the slightly longer focal length, larger achromats, such as the LAO-10.0-6.0. The performance data, given in Spot Size, show that this combination does provide the required 8 mm spot diameter. Gaussian Beam Optics We will ignore, for the moment, that we are not working at the infinite conjugate. This problem, illustrated in figure 4.30, is essentially a 1:1 imaging situation. We want to collect and focus at a numerical aperture of 0.15 or f/3.3, and we need a lens with an 8 µm spot size at this f-number. Based on the lens combination discussion in Lens Combination Formulas, our most likely setup is either a pair of identical planoconvex lenses or achromats, faced front to front. To determine the necessary focal length for a plano-convex lens, we again use the spherical aberration estimate formula: Fundamental Optics spot size = EXAMPLE 3: SYMMETRIC FIBER-TO-FIBER COUPLING Couple an optical fiber with an 8 µm core and a 0.15 numerical aperture into another fiber with the same characteristics. Assume a wavelength of 0.5 µm. Optical Specifications We can immediately reject the biconvex lenses because of spherical aberration. We can estimate the performance of the LPX-5.0-5.2-C on the focusing side by using our spherical aberration formula: Material Properties EXAMPLE 2: COUPLING AN INCANDESCENT SOURCE INTO A FIBER In Imaging Properties of Lens Systems we considered a system in which the output of an incandescent bulb with a filament of 1 mm in diameter was to be coupled into an optical fiber with a core diameter of 100 µm and a numerical aperture of 0.25. From the optical invariant and other constraints given in the problem, we determined that f = 9.1 mm, CA = 5 mm, s = 100 mm, s" = 10 mm, NA" = 0.25, and NA = 0.025 (or f/2 and f/20). The singlet lenses that match these specifications are the planoconvex LPX-5.0-5.2-C or biconvex lenses LDX-6.0-7.7-C and LDX-5.0-9.9-C. The closest achromat would be the LAO-10.0-6.0. achromat. Although a computer ray trace would be required to determine its exact performance, it is virtually certain to provide adequate performance. Machine Vision Guide s=f s″= f Figure 4.30 Symmetric fiber-to-fiber coupling Laser Guide marketplace.idexop.com Lens Selection A119 FUNDAMENTAL OPTICS Fundamental Optics Because fairly small spot sizes are being considered here, it is important to make sure that the system is not being asked to work below the diffraction limit: . × . mm × . = mm . APPLICATION NOTE Spherical Ball Lenses for Fiber Coupling /06/6)1 FRXSOLQJVSKHUH Since this is half the spot size caused by aberrations, it can be safely assumed that diffraction will not play a significant role here. An entirely different approach to a fiber-coupling task such as this would be to use a pair of spherical ball lenses (LMS-LSFN series) or one of the gradient-index lenses (LGT series). EXAMPLE 4: DIFFRACTION-LIMITED PERFORMANCE Determine at what f-number a plano-convex lens being used at an infinite conjugate ratio with 0.5 mm wavelength light becomes diffraction limited (i.e., the effects of diffraction exceed those caused by aberration). To solve this problem, set the equations for diffractionlimited spot size and third-order spherical aberration equal to each other. The result depends upon focal length, since aberrations scale with focal length, while diffraction is solely dependent upon f-number. By substituting some common focal lengths into this formula, we get f/8.6 at f = 100 mm, f/7.2 at f = 50 mm, and f/4.8 at f = 10 mm. . × . mm × f /# = FROOLPDWHG OLJKWVHFWLRQ /06/6)1 FRXSOLQJVSKHUH RSWLFDO ILEHU fE RSWLFDO ILEHU XQFRDWHG QDUURZEDQG fE Spheres are arranged so that the fiber end is located at the focal point. The output from the first sphere is then collimated. If two spheres are aligned axially to each other, the beam will be transferred from one focal point to the other. Translational alignment sensitivity can be reduced by enlarging the beam. Slight negative defocusing of the ball can reduce the spherical aberration third-order contribution common to all coupling systems. Additional information can be found in “Lens Coupling in Fiber Optic Devices: Efficiency Limits,” by A. Nicia, Applied Optics, vol. 20, no. 18, pp 3136—45, 1981. Off-axis aberrations are absent since the fiber diameters are so much smaller than the coupler focal length. . × f f /# or f /# = (. × f ) / . When working with these focal lengths (and under the conditions previously stated), we can assume essentially diffraction-limited performance above these f-numbers. Keep in mind, however, that this treatment does not take into account manufacturing tolerances or chromatic aberration, which will be present in polychromatic applications. A120 Lens Selection 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials SPOT SIZE Material Properties are for on-axis, uniformly illuminated, collimated input light at 546.1 nm. They assume that the lens is facing in the direction that produces a minimum spot size. When the spot size caused by aberrations is smaller or equal to the diffraction-limited spot size, the notation “DL” appears next to the entry. The shorter focal length lenses produce smaller spot sizes because aberrations increase linearly as a lens is scaled up. In general, the performance of a lens or lens system in a specific circumstance should be determined by an exact trigonometric ray trace. CVI Laser Optics applications engineers can supply ray-tracing data for particular lenses and systems of catalog components on request. In certain situations, however, some simple guidelines can be used for lens selection. The optimum working conditions for some of the lenses in this catalog have already been presented. The following tables give some quantitative results for a variety of simple and compound lens systems, which can be constructed from standard catalog optics. Fundamental Optics In interpreting these tables, remember that these theoretical values obtained from computer ray tracing consider only the effects of ideal geometric optics. Effects of manufacturing tolerances have not been considered. Furthermore, remember that using more than one element provides a higher degree of correction but makes alignment more difficult. When actually choosing a lens or a lens system, it is important to note the tolerances and specifications clearly described for each CVI Laser Optics lens in the product listings. Optical Specifications The effect on spot size caused by spherical aberration is strongly dependent on f-number. For a plano-convex singlet, spherical aberration is inversely dependent on the cube of the f-number. For doublets, this relationship can be even higher. On the other hand, the spot size caused by diffraction increases linearly with f-number. Thus, for some lens types, spot size at first decreases and then increases with f-number, meaning that there is some optimum performance point at which both aberrations and diffraction combine to form a minimum. Unfortunately, these results cannot be generalized to situations in which the lenses are used off axis. This is particularly true of the achromat/aplanatic meniscus lens combinations because their performance degrades rapidly off axis. The tables give the diameter of the spot for a variety of lenses used at several different f-numbers. All the tables f/# Gaussian Beam Optics Focal Length = 10 mm Spot Size (µm) LDX-5.0-9.9-C LPX-8.0-5.2-C LAO-10.0-6.0 f/2 — 94 11 f/3 36 25 7 f/5 8 6.7 (DL) 6.7 (DL) f/10 13.3 (DL) 13.3 (DL) 13.3 (DL) LPX-18.5-15.6-C LPX-18.5-15.6-C f/2 295 — 3 f/3 79 7 4 (DL) f/5 17 6.7 (DL) 6.9 (DL) Focal Length = 30 mm Spot Size (µm) f/# Machine Vision Guide LAO-50.0-18.0 & MENP-18.0-4.0-73.5-NSF8 Focal Length = 60 mm Spot Size (µm) f/# LDX-50.0-60.0-C LPX-30.0-31.1-C LAO-60.0-30.0 LAO-100.0-31.5 & MENP-31.5-6.0-146.4-NSF8 f/2 816 600 — — f/3 217 160 34 4 (DL) f/5 45 33 10 6.7 (DL) f/10 13.3 (DL) 13.3 (DL) 13.3 (DL) 13.3 (DL) Laser Guide marketplace.idexop.com Spot Size A121 FUNDAMENTAL OPTICS ABERRATION BALANCING Fundamental Optics To improve system performance, optical designers make sure that the total aberration contribution from all surfaces taken together sums to nearly zero. Normally, such a process requires computerized analysis and optimization. However, there are some simple guidelines that can be used to achieve this with lenses available in this catalog. This approach can yield systems that operate at a much lower f-number than can usually be achieved with simple lenses. Specifically, we will examine how to null the spherical aberration from two or more lenses in collimated, monochromatic light. This technique will thus be most useful for laser beam focusing and expanding. Figure 4.31 shows the third-order longitudinal spherical aberration coefficients for six of the most common positive and negative lens shapes when used with parallel, monochromatic incident light. The plano-convex and plano-concave lenses both show minimum spherical aberration when oriented with their curved surface facing the incident parallel beam. All other configurations exhibit larger amounts of spherical aberration. With these lens types, it is now possible to show how various systems can be corrected for spherical aberration. A two-element laser beam expander is a good starting example. In this case, two lenses are separated by a distance that is the sum of their focal lengths, so that the overall system focal length is infinite. This system will not focus incoming collimated light, but it will change the beam diameter. By definition, each of the lenses is operating at the same f-number. The equation for longitudinal spherical aberration shows that, for two lenses with the same f-number, aberration varies directly with the focal lengths of the lenses. The sign of the aberration is the same as focal length. Thus, it should be possible to correct the spherical aberration of this Galilean-type beam expander, which consists of a positive focal length objective and a negative diverging lens. If a plano-convex lens of focal length f1 oriented in the normal direction is combined with a plano-concave lens of focal length f2 oriented in its reverse direction, the total spherical aberration of the system is LSA = . f . f + . f/# f/# After setting this equation to zero, we obtain f . =− = −.. f . To make the magnitude of aberration contributions of the two elements equal so they will cancel out, and thus positive lenses plano-convex (reversed) symmetric-convex plano-convex (normal) symmetric-concave plano-concave (normal) negative lenses plano-concave (reversed) aberration coefficient (k) 1.069 0.403 longitudinal spherical aberration (3rd order) = 0.272 kf f/#2 Figure 4.31 Third-order longitudinal spherical aberration of typical lens shapes A122 Aberration Balancing 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials correct the system, select the focal length of the positive element to be 3.93 times that of the negative element. or f = −.. f E&255(&7('!%($0(;3$1'(5 f 4PP PPGLDPHWHU SODQRFRQFDYH f PP PPGLDPHWHU V\PPHWULFFRQYH[ F63+(5,&$//<&255(&7('PP()/I2%-(&7,9( f 4PP PPGLDPHWHU SODQRFRQFDYH Machine Vision Guide . f + . f + . f = f PP PPGLDPHWHU SODQRFRQYH[ Gaussian Beam Optics These same principles can be utilized to create high numerical aperture objectives that might be used as laser focusing lenses. Figure 4.32c shows an objective consisting of an initial negative element, followed by two identical plano-convex positive elements. Again, all of the elements operate at the same f-number, so that their aberration contributions are proportional to their focal lengths. To obtain zero total spherical aberration from this configuration, we must satisfy f 4PP PPGLDPHWHU SODQRFRQFDYH Fundamental Optics The relatively low f numbers of these objectives is a great advantage in minimizing the length of these beam expanders. They would be particularly useful with Nd:YAG and argon-ion lasers, which tend to have large output beam diameters. D&255(&7('[%($0(;3$1'(5 Optical Specifications A beam expander of lower magnification can also be derived from this information. If a symmetric-convex objective is used together with a reversed plano-concave diverging lens, the aberration coefficients are in the ratio of 1.069/0.403 = 2.65. Figure 4.32b shows a system of catalog lenses that provides a magnification of 2.7 (the closest possible given the available focal lengths). The maximum wavefront error in this case is only a quarterwave, even though the objective is working at f/3.3. Material Properties Figure 4.32a shows a beam-expander system made up of catalog elements, in which the focal length ratio is 4:1. This simple system is corrected to about 1/6 wavelength at 632.8 nm, even though the objective is operating at f/4 with a 20 mm aperture diameter. This is remarkably good wavefront correction for such a simple system; one would normally assume that a doublet objective would be needed and a complex diverging lens as well. This analysis does not take into account manufacturing tolerances. Therefore, a corrected system should result if the focal length of the negative element is just about half that of each of the positive lenses. In this case, f1 = 425 mm and f2 = 50 mm yield a total system focal length of about 25 mm and an f-number of approximately f/2. This objective, corrected to 1/6 wave, has the additional advantage of a very long working distance. f PP PPGLDPHWHU SODQRFRQYH[ Figure 4.32 Combining catalog lenses for aberration balancing Laser Guide marketplace.idexop.com Aberration Balancing A123 FUNDAMENTAL OPTICS DEFINITION OF TERMS Fundamental Optics FOCAL LENGTH (f) FOCAL POINT (F OR F") Two distinct terms describe the focal lengths associated with every lens or lens system. The effective focal length (EFL) or equivalent focal length (denoted f in figure 4.33) determines magnification and hence the image size. The term f appears frequently in lens formulas and in the tables of standard lenses. Unfortunately, because ƒ is measured with reference to principal points which are usually inside the lens, the meaning of f is not immediately apparent when a lens is visually inspected. Rays that pass through or originate at either focal point must be, on the opposite side of the lens, parallel to the optical axis. This fact is the basis for locating both focal points. PRIMARY PRINCIPAL SURFACE Let us imagine that rays originating at the front focal point F (and therefore parallel to the optical axis after emergence from the opposite side of the lens) are singly refracted at some imaginary surface, instead of twice refracted (once at each lens surface) as actually happens. There is a unique imaginary surface, called the principal surface, at which this can happen. The second type of focal length relates the focal plane positions directly to landmarks on the lens surfaces (namely the vertices) which are immediately recognizable. It is not simply related to image size but is especially convenient for use when one is concerned about correct lens positioning or mechanical clearances. Examples of this second type of focal length are the front focal length (FFL, denoted ƒf in figure 4.33) and the back focal length (BFL, denoted fb). To locate this unique surface, consider a single ray traced from the air on one side of the lens, through the lens and into the air on the other side. The ray is broken into three segments by the lens. Two of these are external (in the air), and the third is internal (in the glass). The external segments can be extended to a common point of intersection (certainly near, and usually within, the lens). The principal surface is the locus of all such points of intersection of extended external ray segments. The The convention in all of the figures (with the exception of a single deliberately reversed ray) is that light travels from left to right. tF SULPDU\SULQFLSDOSRLQW VHFRQGDU\SULQFLSDOVXUIDFH VHFRQGDU\SULQFLSDOSRLQW tH SULPDU\SULQFLSDOVXUIDFH UD\IURPREMHFWDWLQILQLW\ UD\IURPREMHFWDWLQILQLW\ RSWLFDOD[LV r SULPDU\YHUWH[ $ ) + IURQWIRFDO SRLQW EDFNIRFDO SRLQW +s r UHYHUVHGUD\ORFDWHVIURQWIRFDO SRLQWRUSULPDU\SULQFLSDOVXUIDFH fI fE B f A f A B IURQWIRFXVWRIURQW HGJHGLVWDQFH UHDUHGJHWRUHDU IRFXVGLVWDQFH f HIIHFWLYHIRFDOOHQJWK PD\EHSRVLWLYHDVVKRZQ RUQHJDWLYH fI IURQWIRFDOOHQJWK fE EDFNIRFDOOHQJWK )s $ VHFRQGDU\YHUWH[ tH HGJHWKLFNQHVV tF FHQWHUWKLFNQHVV r UDGLXVRIFXUYDWXUHRIILUVW VXUIDFHSRVLWLYHLIFHQWHURI FXUYDWXUHLVWRULJKW r UDGLXVRIFXUYDWXUHRIVHFRQG VXUIDFHQHJDWLYHLIFHQWHURI FXUYDWXUHLVWROHIW Figure 4.33 Focal length and focal points A124 Definition of Terms 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials principal surface of a perfectly corrected optical system is a sphere centered on the focal point. Assuming that the lens is surrounded by air or vacuum (refractive index 1.0), this is both the distance from the front focal point (F) to the primary principal point (H) and the distance from the secondary principal point (H") to the rear focal point (F"). Later we use ƒ to designate the paraxial EFL for the design wavelength (λ0). Material Properties Near the optical axis, the principal surface is nearly flat, and for this reason, it is sometimes referred to as the principal plane. EFFECTIVE FOCAL LENGTH (EFL, f) SECONDARY PRINCIPAL SURFACE PRIMARY PRINCIPAL POINT (H) OR FIRST NODAL POINT SECONDARY PRINCIPAL POINT (H”) OR SECONDARY NODAL POINT This point is the intersection of the secondary principal surface with the optical axis. The conjugate distances are the object distance, s, and image distance, s". Specifically, s is the distance from the object to H, and s" is the distance from H" to the image location. The term infinite conjugate ratio refers to the situation in which a lens is either focusing incoming collimated light or is being used to collimate a source (therefore, either s or s" is infinity). The primary vertex is the intersection of the primary lens surface with the optical axis. SECONDARY VERTEX (A2) The secondary vertex is the intersection of the secondary lens surface with the optical axis. This length is the distance from the secondary vertex (A2) to the rear focal point (F"). EDGE-TO-FOCUS DISTANCES (A AND B) A is the distance from the front focal point to the primary vertex of the lens. B is the distance from the secondary vertex of the lens to the rear focal point. Both distances are presumed always to be positive. REAL IMAGE A real image is one in which the light rays actually converge; if a screen were placed at the point of focus, an image would be formed on it. VIRTUAL IMAGE A virtual image does not represent an actual convergence of light rays. A virtual image can be viewed only by looking back through the optical system, such as in the case of a magnifying glass. F-NUMBER (F/#) The f-number (also known as the focal ratio, relative aperture, or speed) of a lens system is defined to be the effective focal length divided by system clear aperture. Ray f-number is the conjugate distance for that ray divided by the height at which it intercepts the principal surface. f /# = f . CA Machine Vision Guide PRIMARY VERTEX (A1) BACK FOCAL LENGTH (FB) Gaussian Beam Optics CONJUGATE DISTANCES (S AND S”) This length is the distance from the front focal point (F) to the primary vertex (A1). Fundamental Optics This point is the intersection of the primary principal surface with the optical axis. FRONT FOCAL LENGTH (ff) Optical Specifications This term is defined analogously to the primary principal surface, but it is used for a collimated beam incident from the left and focused to the back focal point F" on the right. Rays in that part of the beam nearest the axis can be thought of as once refracted at the secondary principal surface, instead of being refracted by both lens surfaces. (see eq. 4.7) Laser Guide marketplace.idexop.com Definition of Terms A125 FUNDAMENTAL OPTICS Fundamental Optics NUMERICAL APERTURE (NA) The NA of a lens system is defined to be the sine of the angle, θ1, that the marginal ray (the ray that exits the lens system at its outer edge) makes with the optical axis multiplied by the index of refraction (n) of the medium. The NA can be defined for any ray as the sine of the angle made by that ray with the optical axis multiplied by the index of refraction: NA = nsinθ.(4.30) MAGNIFICATION POWER Often, positive lenses intended for use as simple magnifiers are rated with a single magnification, such as 4 x. To create a virtual image for viewing with the human eye, in principle, any positive lens can be used at an infinite number of possible magnifications. However, there is usually a narrow range of magnifications that will be comfortable for the viewer. Typically, when the viewer adjusts the object distance so that the image appears to be essentially at infinity (which is a comfortable viewing distance for most individuals), magnification is given by the relationship magnification = mm f ( f in mm ). (4.31) DEPTH OF FIELD AND DEPTH OF FOCUS In an imaging system, depth of field refers to the distance in object space over which the system delivers an acceptably sharp image. The criteria for what is acceptably sharp is arbitrarily chosen by the user; depth of field increases with increasing f-number. For an imaging system, depth of focus is the range in image space over which the system delivers an acceptably sharp image. In other words, this is the amount that the image surface (such as a screen or piece of photographic film) could be moved while maintaining acceptable focus. Again, criteria for acceptability are defined arbitrarily. In nonimaging applications, such as laser focusing, depth of focus refers to the range in image space over which the focused spot diameter remains below an arbitrary limit. APPLICATION NOTE Technical Reference For further reading about the definitions and formulas presented here, refer to the following publications: X Rudolph Thus, a 25.4 mm focal length positive lens would be a 10x magnifier. X Rudolph Smith, Modern Optical Engineering (McGraw Hill). X Donald The term diopter is used to define the reciprocal of the focal length, which is commonly used for ophthalmic lenses. The inverse focal length of a lens expressed in diopters is diopters = f Kingslake, System Design (Academic Press) X Warren DIOPTERS Kingslake, Lens Design Fundamentals (Academic Press) ( f in mm ). C. O’Shea, Elements of Modern Optical Design (John Wiley & Sons) X Eugene Hecht, Optics (Addison Wesley) X Max Born, Emil Wolf, Principles of Optics (Cambridge University Press) If you need help with the use of definitions and formulas presented in this guide, our applications engineers will be pleased to assist you. Thus, the smaller the focal length is, the larger the power in diopters will be. A126 Definiton of Terms 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials PARAXIAL LENS FORMULAS PARAXIAL FORMULAS FOR LENSES IN AIR s> 0 0 d 2 (r4s ) Optical Specifications Figure 4.34 Surface sagitta and radius of curvature d d r = (r − s ) + d s=r− r − > (4.36) d > s = rs − dr − r= + . s s d . r= + (4.37) s r = ( r − s ) + Fundamental Optics The paraxial formulas do not include effects of spherical aberration experienced by a marginal ray – a ray passing through the lens near its edge or margin. All EFL values (f) tabulated in this catalog are paraxial values which correspond to the paraxial formulas. The following paraxial formulas are valid for both thick and thin lenses unless otherwise noted. The refractive index of the lens glass, n, is the ratio of the speed of light in vacuum to the speed of light in the lens glass. All other variables are defined in figure 4.33. r> Material Properties The following formulas are based on the behavior of paraxial rays, which are always very close and nearly parallel to the optical axis. In this region, lens surfaces are always very nearly normal to the optical axis, and hence all angles of incidence and refraction are small. As a result, the sines of the angles of incidence and refraction are small (as used in Snell’s law) and can be approximated by the angles themselves (measured in radians). FOCAL LENGTH An often useful approximation is to neglect s/2. where n is the refractive index, tc is the center thickness, and the sign convention previously given for the radii r1 and r2 applies. For thin lenses, tc ≅ 0, and for plano lenses either r1 or r2 is infinite. In either case the second term of the above equation vanishes, and we are left with the familiar lens maker’s formula: SURFACE SAGITTA AND RADIUS OF CURVATURE (refer to figure 4.34) r = ( r − s ) + r= s d . + s f tc n t = ( n − ) f + − c nf (4.8) where, in the first form, the + sign is chosen for the square root if f is positive, but the – sign must be used if f is negative. In the second form, the + sign must be used regardless of the sign of f. d d f− (4.35) Laser Guide s = r − r − r = ( n − ) f ± Machine Vision Guide = ( n − ) − . f r r (4.34) SYMMETRIC LENS RADII (r2 = –r1) With center thickness constrained, Gaussian Beam Optics ( n − ) tc = ( n − ) − + (4.33) f r r n r r > marketplace.idexop.com Paraxial Lens Formulas A127 FUNDAMENTAL OPTICS Fundamental Optics PLANO LENS RADIUS Since r2 is infinite, HIATUS OR INTERSTITIUM (principal-point separation) r = ( n − ) f . (4.39) f ( n − ) tc HH″ = tc − − n r r (4.44) n f PRINCIPAL-POINT LOCATIONS (signed distances from vertices) which, in the thin-lens approximation (exact for plano lenses), becomes − r tc n ( r − r ) + tc ( n − ) (4.40) HH″ = tc − A 2 H″ = A1H = − r tc n ( r − r ) + tc ( n − ) (4.41) where the above sign convention applies. For symmetric lenses (r2 = –r1), = p sin r tc . nr − tc ( n − ) (4.42) If either r1 or r2 is infinite, l’Hôpital’s rule from calculus must be used. Thus, referring to Aberration Balancing, for plano-convex lenses in the correct orientation, A1H = and (4.43) t A 2 H″ = − c . n For flat plates, by letting r1 → ∞ in a symmetric lens, we obtain A1H = A2H"= tc/2n. These results are useful in connection with the following paraxial lens combination formulas. A128 SOLID ANGLE The solid angle subtended by a lens, for an observer situated at an on-axis image point, is Q = p ( − cos v ) A1H = − A 2H″ = Paraxial Lens Formulas . (4.45) n v (4.46) where this result is in steradians, and where v = arctan CA (4.47) s is the apparent angular radius of the lens clear aperture. For an observer at an on-axis object point, use s instead of s". To convert from steradians to the more intuitive sphere units, simply divide Ω by 4π. If the Abbé sine condition is known to apply, θ may be calculated using the arc sine function instead of the arctangent. BACK FOCAL LENGTH f b = f ″ + A 2 H″ = f″− r tc n ( r − r ) + tc ( n − ) (4.48) 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials where the sign convention presented above applies to A2H" and to the radii. If r2 is infinite, l’Hôpital’s rule from calculus must be used, whereby tc .(4.49) n FRONT FOCAL LENGTH = f + r tc n ( r − r ) + tc ( n − ) (4.50) where the sign convention presented above applies to A1H and to the radii. If r1 is infinite, l’Hôpital’s rule from calculus must be used, whereby tc .(4.51) n EDGE-TO-FOCUS DISTANCES For positive lenses, (4.52) LENS CONSTANT (k) This number appears frequently in the following formulas. It is an explicit function of the complete lens prescription (both radii, tc and n') and both media indices (n and n"). This dependence is implicit anywhere that k appears. k= and B = fb + s2 (4.53) where s1 and s2 are the sagittas of the first and second surfaces. Bevel is neglected. m= s″ s f = s − f s″− f = . f (4.55) EFFECTIVE FOCAL LENGTHS LENS f = n k f ″= n″ .(4.56) k Machine Vision Guide MAGNIFICATION OR CONJUGATE RATIO n ′ − n n ″ − n ′ tc ( n ′ − n )( n ″ − n ′) . + − r r n ′r r Gaussian Beam Optics A = ff + s The situation of a lens immersed in a homogenous fluid (figure 4.35) is included as a special case (n = n"). This case is of considerable practical importance. The two values f and f" are again equal, so that the lens combination formulas are applicable to systems immersed in a common fluid. The general case (two different fluids) is more difficult, and it must be approached by ray tracing on a surface-by-surface basis. Fundamental Optics ff = f − Optical Specifications ff = f − A1H These formulas allow for the possibility of distinct and completely arbitrary refractive indices for the object space medium (refractive index n'), lens (refractive index n"), and image space medium (refractive index n). In this situation, the EFL assumes two distinct values, namely f in object space and f" in image space. It is also necessary to distinguish the principal points from the nodal points. The lens serves both as a lens and as a window separating the object space and image space media. Material Properties fb = f ″ − PARAXIAL FORMULAS FOR LENSES IN ARBITRARY MEDIA FORMULA (Gaussian form) n n″ + = k. (4.57) s s″ (4.54) Laser Guide marketplace.idexop.com Paraxial Lens Formulas A129 FUNDAMENTAL OPTICS MAGNIFICATION Fundamental Optics LENS FORMULA (Newtonian form) xx ″ = f ″ f = nn ″ . k (4.58) m= ns ″ .(4.63) n ″s PRINCIPAL-POINT LOCATIONS LENS MAKER’S FORMULA ntc n ″− n ′ (4.59) k n′r ntc n ″− n ′ A1H = k− n ″tcn′rn ′ − n A 2H ″ = . k n ′r − n ″tc n ′ − n A 2H ″ = . (4.60) k n ′r n n″ = = k .(4.64) f f″ A1H = NODAL-POINT LOCATIONS (4.65) A1N = A1H + HN OBJECT-TO-FIRST-PRINCIPAL-POINT DISTANCE A N ″= =AAH2H ″ + H″ N ″. A12N 1 + HN ns ″ s= .(4.61) ks ″ − n ″ SECOND PRINCIPAL-POINT-TO-IMAGE DISTANCE s″= n ″s .(4.62) ks − n A 2 N ″ = A 2H″ + H″ N ″.(4.66) SEPARATION OF NODAL POINT FROM CORRESPONDING PRINCIPAL POINT HN = H"N" = (n"– n)/k, positive for N to right of H and N" to right of H". LQGH[ns ZDWHU LQGH[n DLURUYDFXXP fs f fI fE $ ) $ 1 1s + +s )s LQGH[ n %. Figure 4.35 Symmetric lens with disparate object and image space indices A130 Paraxial Lens Formulas 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials APPLICATION NOTE BACK FOCAL LENGTH (see eq. 4.48) FRONT FOCAL LENGTH ff = f − A1H. (see eq. 4.50) NUMERICAL APERTURES APPLICATION NOTE n sin v and n ″ sin v ″ where v ″ = arcsin CA . s ″ = p sin v where v = arctan Q = p 4 2 cos v ) v wherep sin ″ = arctan 2 v (see eq. 4.46) ″ 2s ″ CA Machine Vision Guide = p(1 CA 2s A ray directed at the primary nodal point N of a lens appears to emerge from the secondary nodal point N" without change of direction. Conversely, a ray directed at N" appears to emerge from N without change of direction. At the infinite conjugate ratio, if a lens is rotated about a rotational axis orthogonal to the optical axis at the secondary nodal point (i.e., if N" is the center of rotation), the image remains stationary during the rotation. This fact is the basis for the nodal slide method for measuring nodal‑point location and the EFL of a lens. The nodal points coincide with their corresponding principal points when the image space and object space refractive indices are equal (n = n"). This makes the nodal slide method the most precise method of principal‑point location. Gaussian Beam Optics Q = p ( − cos v ) Fundamental Optics Physical Significance of the Nodal Points CA s where v = arcsin Much time and effort can be saved by ignoring the differences among f, fb, and ff in these formulas by assuming that f = fb = ff. Then s becomes the lens‑to‑object distance; s" becomes the lens‑to‑image distance; and the sum of conjugate distances s+s" becomes the object‑to‑image distance. This is known as the thin‑lens approximation. Optical Specifications FOCAL RATIOS The focal ratios are f/CA and f "/CA, where CA is the diameter of the clear aperture of the lens. For Quick Approximations Material Properties f b = f ″ + A 2 H″ . . SOLID ANGLES (IN STERADIANS) To convert from steradians to spheres, simply divide by 4π. Laser Guide marketplace.idexop.com Paraxial Lens Formulas A131 FUNDAMENTAL OPTICS PRINCIPAL-POINT LOCATIONS Fundamental Optics Figure 4.36 indicates approximately where the principal points fall in relation to the lens surfaces for various standard lens shapes. The exact positions depend on the index of refraction of the lens material, and on the lens radii, and can be found by formula. In extreme meniscus lens shapes (short radii or steep curves), it is possible that both principal points will fall outside the lens boundaries. For symmetric lenses, the principal points divide that part of the optical axis between the vertices into three approximately equal segments. For plano lenses, one principal point is at the curved vertex, and the other is approximately one-third of the way to the plane vertex. F″ H″ F″ H″ F″ H″ F″ H″ F″ H″ H″ F″ F″ H″ F″ H″ F″ H″ F″ H″ Figure 4.36 Principal points of common lenses A132 Principal-Point Locations 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials PRISMS PRISM ORIENTATION The orientation of a prism determines its effect on a beam of light or an image. virtual image of object: visible only to observer entrance face right-angle prism hypotenuse face exit face Fundamental Optics A real image (see Figure 4.38) can be formed only if imaging optics are present in the system. Without imaging optics, the image is virtual. A virtual image has the same orientation as the real image shown, but it can be viewed by the observer only by looking back through the prism system. object Optical Specifications A viewer looks through a prism at an object and sees a virtual image (see Figure 4.37). This image may be displaced from the original object, or, if a dove prism is used, it may coincide with the object. Furthermore, image orientation may differ from the object; in the case of a right-angle prism, the image is reversed. the visible and near-infrared region. The possibility of significant TIR failure with convergent or divergent beams should be kept in mind if polarization is important. TIR can also fail if the hypotenuse face is not kept extremely clean. Even an almost invisible fingerprint can lead to TIR failure. An aluminum- or silver-coated hypotenuse is recommended for applications where the right-angle prism is frequently handled, or where convergent or divergent beams are used. There is a slight loss of reflectance at all internal angles with the coating, and no critical angle exists. Material Properties Prisms are blocks of optical material whose flat, polished sides are arranged at precisely controlled angles to each other. Prisms may be used in an optical system to deflect or deviate a beam of light. They can invert or rotate an image, disperse light into its component wavelengths, and be used to separate states of polarization. Figure 4.37 Virtual imaging using a prism object TOTAL INTERNAL REFLECTION vc (l ) = arcsin nl entrance face right-angle prism exit face Gaussian Beam Optics Rays incident upon a glass/air boundary (i.e., an internal reflection) at angles that exceed the critical angle are reflected with 100% efficiency regardless of their initial polarization state. The critical angle is given by hypotenuse face (4.67) The index of N-BK7 is sufficiently high to guarantee the TIR of a collimated beam at 45º internal incidence over ABERRATIONS FOR PRISMS Machine Vision Guide and depends on the refractive index, which is a function of wavelength. If, at some wavelength, the refractive index should fall to less than √2 = 1.414, the critical angle will exceed 45º, and total internal reflection (TIR) will fail for a collimated beam internally incident at 45º on the hypotenuse face of a right-angle prism. Reflectance decreases rapidly at angles of incidence smaller than the critical angle. Figure 4.38 Real imaging using a prism Prisms will introduce aberrations when they are used with convergent or divergent beams of light. Using prisms with collimated or nearly collimated light will help minimize aberrations. Conjugate distances that include prisms should be long. DISPERSING PRISMS Dispersing prisms are used to separate a beam of white light into its component colors. Generally, the light Laser Guide marketplace.idexop.com Prisms A133 FUNDAMENTAL OPTICS Fundamental Optics is first collimated and then dispersed by the prism. A spectrum is then formed at the focal plane of a lens or curved mirror. In laser work, dispersing prisms are used to separate two wavelengths following the same beam path. Typically, the dispersed beams are permitted to travel far enough so the beams separate spatially. A prism exhibits magnification in the plane of dispersion if the entrance and exit angles for a beam differ. This is useful in anamorphic (one-dimensional) beam expansion or compression, and may be used to correct or create asymmetric beam profiles. As shown in Figure 4.39, a beam of width W1 is incident at an angle α on the surface of a dispersing prism of apex angle A. The angle of refraction at the first surface, β, the angle of incidence at the second surface, γ, and the angle of refraction exiting the prism, δ, are easily calculated: β = sin–1((sin α) / η) γ = A– β δ = sin–1(ηsinγ) The magnification W2/W1 is given by: M= cos d cos b cos a cos g (4.69) The resolving power of a prism spectrometer angle α, the angular dispersion of the prism is given by: dd sinA d h = d l cos d cos b d l (4.70) If the spectrum is formed by a diffraction limited focal system of focal length f, the minimum spot size is dx ~ fλ/ W. This corresponds to a minimum angular resolution dδ ~ λ/w for a beam of diameter w. The diffraction limited angular resolution at a given beam diameter sets the limit on the spectral resolving power of a prism. Setting the expression for dδ equal to the minimum angular resolution, we obtain: (4.71) The beam deviation, ε, is of greatest importance. It is the angle the exit beam makes with its original direction. ε = α + δ – A wherelRP is the of the prism. Z resolving VLQ $ Gpower h 53 = = G l wavelength, FRV d FRV b the G l beam deviation ε is a At a given minimum at an angle of incidence: (4.68) αmin dev = sin–1[ηsin( A/2)] where η is the prism index of refraction at that wavelength. At this angle, the incident and exit angles are equal, the prism magnification is one, and the internal rays are perpendicular to the bisector of the apex angle. A W1 e a b g d W2 Figure 4.39 Diagram of dispersing prism A134 Prisms (4.72) By measuring the angle of incidence for minimum deviation, the index of refraction of a prism can be determined. Also, by proper choice of apex angle, the equal incident and exit angles may be made Brewster’s angle, eliminating losses for p-polarized beams. The apex angle to choose is: 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials A = p4 2 vB(4.73) where the relevant quantities are defined in Figure 4.41. In a Pellin Broca prism, an ordinary dispersing prism is split in half along the bisector of the apex angle. Using a right angle prism, the two halves are joined to create a dispersing prism with an internal right angle bend obtained by total internal reflection, as shown in Figure 4.42. In principle, one can split any type of dispersing prism to create a Pellin Broca prism. Typically the Pellin Broca prism is based on an Isosceles Brewster prism. Provided the light is p-polarized, the prism will be essentially lossless. Suppose wavelengths λ1 and λ2 are superimposed in a collimated beam, as at the output of a harmonic generating crystal, the diagram in Figure 4.42 suggests that it is always possible to find a rotation of the prism in its plane that ensures that one of the two wavelengths will operate at minimum deviation when refracting at the input face of the first of the half-dispersing prisms. This means that it will enter the right angle prism normal to one of its faces, be turned exactly 90°, be presented to Gaussian Beam Optics Figure 4.40 Translation of a prism at minimum deviation PELLIN BROCA PRISMS Fundamental Optics (4.74) As an example, consider CVI Laser Optics EDP-25-F2 prism, operating in minimum deviation at 590 nm. The angle of incidence and emergence are both then 54.09° and dη/dλ is –0.0854 µm–1 for F2 glass at 590 nm. If the 25 mm prism is completely filled, the resolving power, λ/dλ , is 2135. This is sufficient to resolve the Sodium D lines. Optical Specifications At minimum deviation, translating a prism along the bisector of the apex angle does not disturb the direction of the output rays. See Figure 4.40. This is important in femtosecond laser design where intracavity prisms are used to compensate for group velocity dispersion. By aligning a prism for minimum deviation and translating it along its apex bisector, the optical path length in material may be varied with no misalignment, thus varying the contribution of the material to overall group velocity dispersion. Finally, it is possible to show that at minimum deviation Material Properties If, in addition, the base angles of the prism are chosen as Brewster’s angle, an isosceles Brewster prism results. Another use is illustrated next. If the beam is made to fill the prism completely, b1=0, and b2 = b, the base of the prism. So, we have the classical result that the resolving power of a prism spectrometer is equal to the base of the prism times the dispersion of the prism material. Machine Vision Guide b1 b2 W b Figure 4.41 Ray path lengths of a prism at minimum deviation Figure 4.42 One of the wavelengths deviates at exactly 90º to its intitial direction Laser Guide marketplace.idexop.com Prisms A135 FUNDAMENTAL OPTICS Fundamental Optics the second half-dispersing prism in minimum deviation, and hence exit the Pellin Broca prism deviated at exactly 90° to its initial direction. mJ/cm2. Fused silica prisms track (i.e., suffer internal catastrophic damage) above this fluence, probably due to self-focusing. A simple dispersing prism always deviates the longer wavelength less than the shorter wavelength. In a Pellin Broca prism, whether the longer wavelength is deviated more or less depends on the orientation of the prism. This is an important consideration when designing a high power Pellin Broca beam separator, as shown in Figures 4.43 and 4.44. PORRO PRISMS CVI Laser Optics offers Brewster angle Pellin Brocca prisms in a number of sizes and materials. N-BK7 prisms are used in the visible and near IR, and are the least expensive. UV-grade fused silica Pellin Broca prisms are used from 240 nm to 2000 nm. Excimer-grade prisms are used in the 180 nm to 240 nm region. Crystal-quartz Pellin Broca prisms are specifically designed for high-power Q-switched 266 nm laser pulses at fluence levels of 50 l1, l2 l1 > l2 A Porro prism, named for its inventor Ignazio Porro, is a type of reflection prism used to alter the orientation of an image. In operation, light enters the large face of the prism, undergoes total internal reflection twice from the 45° sloped faces, and exits again through the large face. An image traveling through a Porro prism is rotated by 180° and exits in the opposite direction offset from its entrance point, as shown in Figure 4.45. Since the image is reflected twice, the handedness of the image is unchanged. Porro prisms have rounded edges to minimize breakage and facilitate assembly. Porro prisms are most often used in pairs, forming a double Porro prism, as shown in Figure 4.46. A second prism, rotated 90° with respect to the first, is placed such that the beam will traverse both prisms. The net effect of the prism system is a beam parallel to but displaced from its original direction, with the image rotated 180°. As before, the handedness of the image is unchanged. Double Porro prism systems are used in small optical telescopes to reorient an inverted image and in many binoculars to both re-orient the image and provide a longer, folded distance between the objective lenses and the eyepieces. l1 l2 Figure 4.43 Longer wavelength is deviated more than the shorter wavelength Figure 4.45 Porro prisms retroreflect and invert the image l1, l2 l1 > l2 l2 l1 Figure 4.44 Longer wavelength is deviated less than the shorter wavelength A136 Prisms Figure 4.46 Double Porro prisms results in a beam parallel to but displaced from its original direction, with the image rotated 180º 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials POLARIZATION POLARIZATION STATES 1. CARTESIAN REPRESENTATION In Cartesian coordinates, the propagation equation for an electric field is given by the formula where Ex, Ey, Øx, and Øy are real numbers defining the magnitude and the phase of the field components in the orthogonal unit vectors x and y. If the origin of time is irrelevant, only the relative phase shift As in the case of the Cartesian representation, we write E = (e+E+eiØ++ e–E–eiØ–) ei(kz-ωt)(4.79) Optical Specifications 4.75) Material Properties Four numbers are required to describe a single plane wave Fourier component traveling in the + z direction. These can be thought of as the amplitude and phase shift of the field along two orthogonal directions. e– is the unit vector for right circularly polarized light; for negative helicity light; for light that rotates clockwise in a fixed plane as viewed facing into the light wave; and for light whose electric field rotation disobeys the right hand rule with thumb pointing in the direction of propagation. where E+ , E–, Ø+, and Ø– are four real numbers describing the magnitudes and phases of the field components of the left and right circularly polarized components. y Fundamental Optics Ø= Øx– Øy(4.76) x z need be specified. 2. CIRCULAR REPRESENTATION Ey In the circular representation, we resolve the field into circularly polarized components. The basic states are represented by the complex unit vectors Gaussian Beam Optics ( ) = = ((11 // 22 )) (( xx + − iy iy )) and = (1 / 2 ) ( x − iy ) Ex E Figure 4.47 Linearly polarized light. Ex and Ey are in phase e+ = 1 / 2 ( x + iy ) and (4.77) ee+ − e− y x (4.78) Machine Vision Guide where e+ is the unit vector for left circularly polarized light; for positive helicity light, for light that rotates counterclockwise in a fixed plane as viewed facing into the light wave; and for light whose electric field rotation obeys the right hand rule with thumb pointing in the direction of propagation. z Ey E Ex Figure 4.48 Circularly polarized light. Ex and Ey are out of phase by angular frequency ω Laser Guide marketplace.idexop.com Polarization A137 FUNDAMENTAL OPTICS CONVERSIONS BETWEEN REPRESENTATIONS Fundamental Optics Note that B •( (= H (4.80) • B (= H ( B •( (= H • B (= H ( For brevity, we will provide only the Cartesian to circular and Cartesian to elliptical transformations. The inverse transformations are straightforward. We define the following quantities: (4.81) J = ( [[ FRV φ − (\\ VLQ φ 3. ELLIPTICAL REPRESENTATION An arbitrary polarization state is generally elliptically polarized. This means that the tip of the electric field vector will describe an ellipse, rotating once per optical cycle. Let a be the semimajor and b be the semiminor axis of the polarization ellipse. Let ψ be the angle that the semimajor axis makes with the x axis. Let ξ and η be the axes of a right-handed coordinate system rotated by an angle + ψ with respect to the x axis and aligned with the polarization ellipse as shown in Figure 4.49. The elliptical representation is: ( ) A ( = Dξ + EηA HLδ HL N] −ωW (4.82) JJ == (( [[ VLQ φ FRV \\ φ + FRV VLQ φφ −( (\\ [[ φ +( JJ = ( [[ FRV (\\ VLQ φφ VLQ φ FRV [[ \\ φ + φ FRV VLQ J == (([[ − (\\ φ =( ( [[ VLQ φφ −+ ( FRV \\ JJJ = FRV ( VLQ [[ \\ VLQ φ FRVφφφ [[ \\ φ + FRV VLQ J == (([[ −( (\\ = J( ([[ VLQJ φφ −+ ( FRV XJJJ=== +FRV \\ (\\ VLQφφφ [[ FRV VLQ −( φ VLQ φ + FRV ([[ = ( \\ [[ \\ φ φ FRV VLQ JJ == (([[ − ( \\ φ − ( \\ VLQ FRV XYJ=== JJ([[ [[ φ VLQ FRV VLQ φφ \\ JJφ + ( \\ ++FRV [[ φ φ VLQ FRV JJ == (([[ + ( \\ φ − FRV VLQ −+ ( (\\ VLQφφφ ([[ == J VLQ FRV +FRV J Jφ [[ \\ = =JDWDQ [[ \\ φXYJ J J + FRV VLQ φφ JJ = ( + (\\ [[ φJφ + VLQ FRV (DWDQ J ===( −J ( (\\ VLQφ FRV \\ FRV VLQ φφ [[ φ [[[ [ \\ X = J + J φY= =JDWDQ JJφ −J ( FRV φ +VLQ JJ = = ([[ \\ φ + ( FRV VLQ φφ (\\ VLQJ φ FRV J===J( (DWDQ [[ [[ J \\ + X φ J + = DWDQ J φY J J J = + = J( ([[ FRV XJJ== +VLQ Jφφ −+ ((\\ FRV VLQ φφ [[ \\ φY J J =JDWDQ = + J φ = DWDQ J J FRV φ XYJ = =J([[ +VLQ J φ − ( \\ J φ = DWDQ JJ φYX== =JJDWDQ J J +J φ = DWDQ + JJ J φ = DWDQ J J φY J J = =JDWDQ + J ( ( ( ( ((( ( ( ( ( ) ) ) ) ))) ) ) ) ) Y h y E (4.84) (4.85) (4.86) (4.87) (4.88) (4.89) φφ = = DWDQ DWDQ JJ JJ φ = DWDQ J J Note that the phase shift δo above is required to adjust the time origin, and the parameter ψ is implicit in the rotation of the ξ and η axes with respect to the x and y axes. (4.83) (4.90) In the above, atan(x,y) is the four quadrant arc tangent function. This means that atan(x,y) = atan(y/x) with the provision that the quadrant of the angle returned by the function is controlled by the signs of both x and y, not just the sign of their quotient; for example, if g2 = g1 = –1, then Ø12 above is 5π/4 or –3π/4, not π/4. a b W Figure 4.49 The polarization ellipse X A. CARTESIAN TO CIRCULAR TRANSFORMATION (+ = Y (4.91) (− = X φ+ = φ φ − = φ A138 Polarization 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials (+ = Y (4.92) (4.93) φ − = φ (4.94) where the phase shift of the transmitted field has been ignored. A real polarizer has a pass transmission, Tll, less than 1. The transmission of the rejected beam, T⊥, may not be 0. If r is a unit vector along the rejected direction, then ( ) E2 = T|| D = (Y + X ) (4.95) ED = = ((YY − + XX)) 7 = 7__ FRV v + 7⊥ VLQ v (4.97) (4.98) LINEAR POLARIZERS E2 = p ( p • E1 ) (4.99) 1/2 ⊥ (4.100) (4.101) The above equation shows that, when the polarizer is aligned so that θ = 0, T = Tll. When it is “crossed”, θ = π/2, and T = T⊥. The extinction ratio is ε = Tll / T⊥. A polarizer with perfect extinction has T⊥ = 0, and thus T = Tllcos2θ is a familiar result. Because cos2θ has a broad maximum as a function of orientation angle, setting a polarizer at a maximum of transmission is generally not very accurate. One has to either map the cos2θ with sufficient accuracy to find the θ = 0 point, or do a null measurement at θ = ±π/2. UHIOHFWHGEHDP VSRODUL]DWLRQ Machine Vision Guide Suppose the pass direction of the polarizer is determined by unit vector p. Then the transmitted field E2, in terms of the incident field E1, is given by + (T⊥ ) r ( r • E1 )ei Gaussian Beam Optics A linear polarizer is a device that creates a linear polarization state from an arbitrary input. It does this by removing the component orthogonal to the selected state. Unlike plastic sheet polarizers which absorb the rejected beam (which turns into heat), cube polarizers and thin-film plate polarizers reflect the rejected beam, creating two usable beams. Still others may refract the two polarized beams at different angles, thereby separating them. Examples are Wollaston and Rochon prism polarizers. || In the above, the phase shifts along the two directions must be retained. Similar expressions could be arrived at for the rejected beam. If θ is the angle between the field E1 and the polarizer pass direction p, the above equation predicts that (4.96) δ = φ + φ i Fundamental Optics ψ E ==(Yφ−X−) φ D = (Y + X ) δ == φφ −+φφ ψ ED = = (YY − + XX) δ = φ + φ ψ E ==(Yφ−X−) φ δ == φφ −+φφ ψ p ( p • E1 )e Optical Specifications B. CARTESIAN TO ELLIPTICAL TRANSFORMATION 1/2 Material Properties ( (+− = = XY φ+ = φ (− = = XY ( + φ − = φ φ+ = φ (− = X φ − = φ φ+ = φ WUDQVPLWWHG EHDP SSRODUL]HG LQFLGHQW EHDP Figure 4.50 At CVI Laser Optics, the DOT marks preferred input face. This is the tested direction for transmitted wavefront. Damage threshold is also higher for this orientation as well. Laser Guide marketplace.idexop.com Polarization A139 FUNDAMENTAL OPTICS POLARIZATION DEFINITIONS Fundamental Optics BIREFRINGENCE A birefringent crystal, such as calcite, will divide an entering beam of monochromatic light into two beams having opposite polarization. The beams usually propagate in different directions and will have different speeds. There will be only one or two optical axis directions within the crystal in which the beam will remain collinear and continue at the same speed, depending on whether the birefringent crystal is uniaxial or biaxial. If the crystal is a plane-parallel plate, and the optical axis directions are not collinear with the beam, radiation will emerge as two separate, orthogonally polarized beams (see Figure 4.51). The beam will be unpolarized where the beams overlap upon emergence. The two new beams within the material are distinguished from each other by more than just polarization and velocity. The rays are referred to as extraordinary (E) and ordinary (O). These rays need not be confined to the plane of incidence. Furthermore, the velocity of these rays changes with direction. Thus, the index of refraction for extraordinary rays is also a continuous function of direction. The index of refraction for the ordinary ray is constant and is independent of direction. XQSRODUL]HG LQSXWEHDP RUGLQDU\ UD\ OLQHDUO\SRODUL]HG RXWSXWEHDP $ ELUHIULQJHQW PDWHULDO H[WUDRUGLQDU\ UD\ OLQHDUO\SRODUL]HG RXWSXWEHDP % XQSRODUL]HGRXWSXWEHDP Figure 4.51 Double refraction in a birefringent crystal A140 Polarization Definitions The two indexes of refraction are equal only in the direction of an optical axis within the crystal. The dispersion curve for ordinary rays is a single, unique curve when the index of refraction is plotted against wavelength. The dispersion curve for the extraordinary ray is a family of curves with different curves for different directions. Unless it is in a particular polarization state, or the crystalline surface is perpendicular to an optical axis, a ray normally incident on a birefringent surface will be divided in two at the boundary. The extraordinary ray will be deviated; the ordinary ray will not. The ordinary ray index n, and the most extreme (whether greater or smaller) extraordinary ray index ne, are together known as the principal indices of refraction of the material. If a beam of linearly polarized monochromatic light enters a birefringent crystal along a direction not parallel to the optical axis of the crystal, the beam will be divided into two separate beams. Each will be polarized at right angles to the other and will travel in different directions. The original beam energy, which will be divided between the new beams, depends on the original orientation of the vector to the crystal. The energy ratio between the two orthogonally polarized beams can be any value. It is also possible that all energy will go into one of the new beams. If the crystal is cut as a plane-parallel plate, these beams will recombine upon emergence to form an elliptically polarized beam. The difference between the ordinary and extraordinary ray may be used to create birefringent crystal polarization devices. In some cases, the difference in refractive index is used primarily to separate rays and eliminate one of the polarization planes, for example, in Glantype polarizers. In other cases, such as Wollaston and Thompson beamsplitting prisms, changes in propagation direction are optimized to separate an incoming beam into two orthogonally polarized beams. DICHROISM Dichroism is selective absorption of one polarization plane over the other during transmission through a material. Sheet-type polarizers are manufactured with organic materials embedded into a plastic sheet. The sheet is stretched, aligning molecules and causing them 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials POLARIZATION BY REFLECTION Polarizing thin films are formed by using the patented Slocum process to deposit multiple layers of microscopic silver prolate spheroids onto a polished glass substrate. SPECTRAL PROPERTIES Trace amounts of chemical impurities, as well as lattice defects, can cause calcite to be colored, which changes absorption. For visible light applications, it is essential to use colorless calcite. For near-infrared applications, material with a trace of yellow is acceptable. This yellow coloration results in a 15 - 20% decrease in transmission below 420 nm. WAVEFRONT DISTORTION (STRIAE) Striae, or streaked fluctuations in the refractive index of calcite, are caused by dislocations in the crystal lattice. They can cause distortion of a light wavefront passing through the crystal. This is particularly troublesome for interferometric applications. Machine Vision Guide Optical radiation incident on small, elongated metal particles will be preferentially absorbed when the polarization vector is aligned with the long axis of the particle. CVI Laser Optics infrared polarizers utilize this effect to make polarizers for the near-infrared. These polarizers are considerably more effective than dichroic polarizers. There are three main areas of importance in defining calcite quality. Gaussian Beam Optics THIN METAL FILM POLARIZERS Calcite, a rhombohedral crystalline form of calcium carbonate, is found in various forms such as limestone and marble. Since calcite is a naturally occurring material, imperfections are not unusual. The highest quality materials, those that exhibit no optical defects, are difficult to find and are more expensive than those with some defects. Applications for calcite components typically fall into laser applications or optical research. CVI Laser Optics offers calcite components in two quality grades to meet those various needs. Fundamental Optics If a number of plates are stacked parallel and oriented at the polarizing angle, some vibrations perpendicular to the plane of incidence will be reflected at each surface, and all those parallel to it will be refracted. By making the number of plates within the stack large (more than 25), high degrees of linear polarization may be achieved. This polarization method is utilized in CVI Laser Optics polarizing beamsplitter cubes which are coated with many layers of quarter-wave dielectric thin films on the interior prism angle. This beamsplitter separates an incident laser beam into two perpendicular and orthogonally polarized beams. CALCITE Optical Specifications When a beam of ordinary light is incident at the polarizing angle on a transmissive dielectric such as glass, the emerging refracted ray is partially linearly polarized. For a single surface (with n=1.50) at Brewster’s angle, 100% of the light whose electric vector oscillates parallel to the plane of incidence is transmitted. Only 85% of the perpendicular light is transmitted (the other 15% is reflected). The degree of polarization from a single-surface reflection is small. The exact dimensions of these spheroids determine the optical properties of the film. Peak absorption can be selected for any wavelength from 400 to 3000 nm by controlling the deposition process. Contrast ratios up to 10,000:1 can be achieved with this method. Other CVI Laser Optics high-contrast polarizers exhibit contrasts as high as 100,000:1. Material Properties to be birefringent, and then dyed. The dye molecules selectively attach themselves to aligned polymer molecules, so that absorption is high in one plane and weak in the other. The transmitted beam is linearly polarized. Polarizers made of such material are very useful for low-power and visual applications. The usable field of view is large (up to grazing incidence), and diameters in excess of 100 mm are available. SCATTER Small inclusions within the calcite crystal account for the main source of scatter. They may appear as small cracks or bubbles. In general, scatter presents a significant problem only when the polarizer is being used with a laser. The amount of scatter centers that can be tolerated is partially determined by beam size and power. Laser Guide marketplace.idexop.com Polarization Definitions A141 FUNDAMENTAL OPTICS Fundamental Optics CVI LASER OPTICS CALCITE GRADES CVI Laser Optics has selected the most applicable calcite qualities, grouped into two grades: LASER GRADE Calcite with a wavefront deformation of λ/4 at 633 nm or better due to striae only. OPTICAL GRADE Calcite with a wavefront deformation of 1λ to λ/4 at 633 nm due to striae only. A142 Polarization Definitions 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials WAVEPLATES STANDARD WAVEPLATES: LINEAR BIREFRINGENCE E2 = s (s • E1 )ei s + (ff • E1 )ei f (4.102) The slow and fast axis phase shifts are given by: (2 = s (s • (1) HLf + f ( f • () (4.105) f Lpf hs (l ) − hf (l ) W l (2==fss (−s •ff(= 1) H + f ( f • () f 2 p Dh l) )HWLf l+ f ( f • () (2= f ==fss (−s •ff((= 1 p hs l ) − hf (l ) W l f= =2 fsp−Dh ff (=l )Wp l f hs (l ) − hf (l ) W l f = 2 p Dh (l ) W l (4.106) (4.107) In the above, Δη(λ) is the birefringence ηs(λ) - ηf(λ). The dispersion of the birefringence is very important in waveplate design; a quarter waveplate at a given wavelength is never exactly a half waveplate at half that wavelength. Let E1 be initially polarized along X, and let the waveplate slow axis make an angle θ with the x axis. This orientation is shown in Figure 4.52. When the waveplate is placed between parallel and perpendicular polarizers the transmissions are given by: Y f s Machine Vision Guide where s and f are unit vectors along the slow and fast axes. This equation shows explicity how the waveplate acts on the field. Reading from left to right, the waveplate takes the component of the input field along its slow axis and appends the slow axis phase shift to it. It does a similar operation to the fast component. To further analyze the effect of a waveplate, we throw away a phase factor lost in measuring intensity, and assign the entire phase delay to the slow axis: Gaussian Beam Optics The equation for the transmitted field E2, in terms of the incident field E1 is: where ηs and ηf are, respectively, the indices of refraction along the slow and fast axes, and t is the thickness of the waveplate. Fundamental Optics Suppose a waveplate made from a uniaxial material has light propagating perpendicular to the optic axis. This makes the field component parallel to the optic axis an extraordinary wave and the component perpendicular to the optic axis an ordinary wave. If the crystal is positive uniaxial, ηe > ηo, then the optic axis is called the slow axis, which is the case for crystal quartz. For negative uniaxial crystals ηe < ηo, the optic axis is called the fast axis. (4.104) Optical Specifications There are two types of birefringence. With linear birefringence, the index of refraction (and hence the phase shift) differs for two orthogonally polarized linear polarization states. This is the operation mode of standard waveplates. With circular birefringence, the index of refraction and hence phase shift differs for left and right circularly polarized components. This is the operation mode of polarization rotators. Øf = ηf(ω)ωt/c = 2πηf(λ)t/λ Material Properties Waveplates use birefringence to impart unequal phase shifts to the orthogonally polarized field components of an incident wave, causing the conversion of one polarization state into another. v E1 X Øs = ηs(ω)ωt/c = 2πηs(λ)t/λ(4.103) Figure 4.52. Orientation of the slow and fast axes of a waveplate with respect to an x-polarized input field Laser Guide marketplace.idexop.com Waveplates A143 FUNDAMENTAL OPTICS Fundamental Optics 7|| ∝| (2 [ |2 = 1 − sin2 2v sin2 f / (4.108) 2 2 2 2 7||⊥ ∝∝||((22[\|2|2==1sin − sin v sin 2v2 sin f /f / 7⊥ ∝| (2 \ |2 = sin2 2v sin2 f / (4.109) Note that θ is only a function of the waveplate orientation, and Øis only a function of the wavelength; the birefringence is a function of wavelength and the plate thickness. For a full-wave waveplate: Ø= 2mπ, T|| = 1, and T┴ = 0, regardless of waveplate orientation. For a half-wave waveplate: Ø= (2m + 1)π, T|| = cos22θ, and T┴= sin22θ. (4.110) This transmission result is the same as if an initial linearly polarized wave were rotated through an angle of 2θ. Thus, a half-wave waveplate finds use as a polarization rotator. For a quarter waveplate, Sometimes, waveplates described by the second line above are called 3/4 waveplates. For multiple order waveplates, CVI Laser Optics permits the use of either of the above classes of waveplates to satisfy the requirements of a quarter-wave waveplate. MULTIPLE-ORDER WAVEPLATES For the full-, half-, and quarter-wave waveplate examples given in standard waveplates, the order of the waveplate is given by the integer m. For m > 0, the waveplate is termed a multiple-order waveplate. For m = 0, we have a zero order waveplate. The birefringence of crystal quartz near 500 nm is approximately 0.00925. Consider a 0.5 mm thick crystal quartz waveplate. A simple calculation shows that this is useful as a quarter waveplate for 500 nm; in fact, it is a 37λ/ 4 waveplate at 500 nm with m = 18. Multiple-order waveplates are inexpensive, high-damage-threshold retarders. Further analysis shows that this same 0.5 mm plate is a 19λ/2 half waveplate at 488.2 nm and a 10λ full-wave waveplate at 466.5 nm. The transmission of this plate between parallel polarizers is shown in Figure 4.53 as a function of wavelength. The retardance of the plate at various key points is also shown. Note how quickly the retardance changes with wavelength. Because of this, multiple-order waveplates are generally useful only at their design wavelength. ZERO-ORDER WAVEPLATES (4.111) To analyze this, we have to go back to the field equation. Assume that the slow and fast axis unit vectors s and f form a right handed coordinate system such that s x f = +z, the direction of propagation. To obtain circularly polarized light, linearly polarized light must be aligned midway between the slow and fast axes. There are four possibilities listed in the table below. A144 Phase Shift Input Field Along (s + f)/√2 Input Field Along (s - f)/√2 Ø = π/2 + 2mπ RCP LCP Ø= 3π/2 + 2mπ LCP RCP Waveplates As discussed above, multiple-order waveplates are not useful with tunable or broad bandwidth sources (e.g., l l l 7SDUDOOHO Ø= (2m + 1)π/2 (i.e., an odd multiple of π/2) l l l l l l l l :DYHOHQJWKQP Figure 4.53 Transmission of a 0.5 mm-thick crystal quartz waveplate between parallel polarizers 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials femtosecond lasers). A zero-order waveplate can greatly improve the useful bandwidth in a compact, highdamage-threshold device. Figure 4.54 Zero-order crystal quartz half-wave waveplate for 800 nm Mica waveplates are an inexpensive zero-order waveplate solution for low-power applications and in detection schemes. WLOWDURXQG IDVWD[LV ,QFLGHQFH$QJOHGHJUHHV Figure 4.55 Retardance vs incidence angle for quartz and polymer waveplates 0.300 zero order achromatic multiple order 0.275 Machine Vision Guide CVI Laser Optics produces multiple-order and zero-order crystal quartz waveplates at any wavelength between 193 nm and 2100 nm. Virtually all popular laser wavelengths are kept in stock, and custom wavelength parts are available with short delivery time. TXDUW] SRO\PHU Gaussian Beam Optics :DYHOHQJWKQP 5HWDUGDQFHZDYHV Retardance (waves) 7SDUDOOHO WLOWDURXQG VORZD[LV Fundamental Optics 4:32;; Optical Specifications Retardance accuracy with wavelength change is often of key concern. For example, an off-the-shelf diode laser has a center wavelength tolerance of ±10 nm. Changes with temperature and drive conditions cause wavelength shifts which may alter performance. These polymer waveplates maintain excellent waveplate performance even with minor shifts in the source wavelength. The temperature sensitivity of laminated polymer waveplates is about 0.15 nm/°C, allowing operation over moderate temperature ranges without significantly degrading retardance accuracy. A comparison of different waveplate types and their dependence on wavelength is shown in figure 4.56. Material Properties As an example, consider the design of a broadband halfwave waveplate centered at 800 nm. Maximum tuning range is obtained if the plate has a single π phase shift at 800 nm. If made from a single plate of crystal quartz, the waveplate would be about 45 µm thick, which is too thin for easy fabrication and handling. The solution is to take two crystal quartz plates differing in thickness by 45 µm and align them with the slow axis of one against the fast axis of the other. The net phase shift of this zero-order waveplate is π. The two plates may be either air-spaced or optically contacted. The transmission of an 800 nm zero-order half-wave waveplate between parallel polarizers is shown in Figure 4.54 using a 0 - 10% scale. Its extinction is better than 100:1 over a bandwidth of about 95 nm centered at 800 nm. 4.55 compares the change in retardance as function of incidence angle for polymer and quartz waveplates. A polymer waveplate changes by less than 1% over a ±10° incidence angle. 0.250 0.225 0.200 0.80 0.90 1.00 1.10 1.20 Relative Wavelength (l/lc) POLYMER WAVEPLATES Polymer waveplates offer excellent angular field of view since they are true zero-order waveplates. Figure Figure 4.56 Wavelength performance of common quarter wave retarders Laser Guide marketplace.idexop.com Waveplates A145 FUNDAMENTAL OPTICS Fundamental Optics ACHROMATIC WAVEPLATES At 500 nm, a crystal quartz zero-order half-wave waveplate has a retardation tolerance of λ/50 over a bandwidth of about 50 nm. This increases to about 100 nm at a center wavelength of 800 nm. A different design which corrects for dispersion differences over the wavelength range is required for bandwidths up to 300 nm. If two different materials are used to create a zeroorder or low-order waveplate, cancellation can occur between the dispersions of the two materials. Thus, the net birefringent phase shift can be held constant over a much wider range than in waveplates made from one material. ACWP-series achromatic waveplates from CVI Laser Optics (see Figure 4.57) are comprised of crystal quartz and magnesium fluoride to achieve achromatic performance. Three wavelength ranges are available in both quarter and half wave retardances. Retardation tolerance is better than λ/100 over the entire wavelength range. :DYHOHQJWKQP DUAL-WAVELENGTH WAVEPLATES Dual-wavelength waveplates are used in a number of applications. One common application is separation of different wavelengths with a polarizing beamsplitter by rotating the polarization of one wavelength by 90°, and leaving the other unchanged. This frequently occurs in nonlinear doubling or tripling laser sources such as Nd:YAG (1064/532/355/266). One way to achieve the multiple retardation specifications is through careful selection of multipleorder waveplates which meet both wavelength and retardation conditions. This often results in the selection of a relatively high order waveplate. Therefore, these dual-wavelength waveplates operate best over a narrow bandwidth and temperature range. Another approach is to combine two quartz waveplates with their optical axes orthogonal to one another, effectively creating a zero-order waveplate. In this configuration, the temperature dependence is a function of the thickness difference between the waveplates, resulting in excellent temperature stability. The retardation of the compound waveplate is also a function of the thickness difference enabling wide bandwidth performance. 5HWDUGDWLRQLQZDYH 7UDQVPLVVLRQ For quarter-wave waveplates, perfect retardance is a multiple of 0.25 waves (or λ/4), and transmission through a linear polarizer must be between 33% and 67%. (In all but the shortest wavelength design, quarter-wave retardation tolerance is better than λ/100.) For half-wave waveplates, perfect retardance is 0.5 waves (or λ/2), while perfect transmission through a linear polarizer parallel to the initial polarization state should be zero. A high degree of achromatization is achievable by the dual material design. In addition, we manage low group velocity dispersion for ultrashort pulse applications through the use of thin plates. Figure 4.57 ACWP-400-700-10-2 A146 Waveplates 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials ETALONS The etalons described in this section are all of the planar Fabry-Perot type. Typical transmission characteristics for this type of etalon are shown in Figure 4.58. Solid Etalons are made from a single plate with parallel sides. Partially reflecting coatings are then deposited on both sides. The cavity is formed by the plate thickness between the coatings. I trans = Iinc 1+ 1 4R (1 − R )2 sin 2 ( / 2 ) (4.112) Here, R is the reflectance of each surface; δ is the phase shift (4.113) where, η is the refractive index (e.g., 1 for air-spaced etalons) δ is the etalon spacing or thickness Fundamental Optics θ is the angle of incidence The free spectral range (FSR) of the etalon is given by Gaussian Beam Optics Deposited Solid Etalons are a special type of solid etalon in which the cavity is formed by a deposited layer of coating material. The thickness of this deposited layer depends on the free spectral range required and can range from a few nanometers up to 15 micrometers. The cavity is sandwiched between the etalon reflector coatings and the whole assembly is supported on a fused-silica base plate. T= Optical Specifications Air-Spaced Etalons consist of pairs of very flat planoplano plates separated by optically contacted spacers. The inner surfaces of the plates are coated with partially reflecting coatings, the outer surfaces are coated with antireflection coatings. For a plane wave incident on the etalon, the transmission of the etalon is given by: Material Properties Etalons are most commonly used as line-narrowing elements in narrowband laser cavities or as bandwidthlimiting and coarse-tuning elements in broadband and picosecond lasers. Further applications are laser line profile monitoring and diagnosis. (4.114) )65 F QG ):+0 )65 The reflectivity finesse, FR is given by FR = Machine Vision Guide 7UDQVPLVVLRQ Etalon plates need excellent surface flatness and plate parallelism. To avoid peak transmission losses due to scatter or absorption, the optical coatings also have to meet the highest standards. p 5 (4.115) − 5 F5 Figure 4.59 shows the reflectivity finesse as a function of the coating reflectivity. )UHTXHQF\ Figure 4.58 Transmission characteristics of a Fabry-Perot etalon Laser Guide marketplace.idexop.com Etalons A147 FUNDAMENTAL OPTICS Fundamental Optics The defects that contribute to this reduction are as shown in Figure 4.60 (graphical representations are exaggerated for clarification). 5HIOHFWLYH)LQHVVH All three types of defects contribute to the total defect finesse Fd: 5HIOHFWLYLW\ Figure 4.59 Reflectivity finesse vs. coating reflectance of each surface The bandwidth (FWHM) is given by ):+0 )65 5 5 (4.117) − 5 5 =0 5 6 = − 5 QP 0 5 5 (4.118) 6 = = − QP 5 G WDQ 5 0 5 = &$ 6 = − 5 = QP ' GGWDQ 0 = &$ 6 = QP(4.119) ' = G WDQ G = &$ ' = G WDQ G &$ (4.120) ' = G = where FR is the reflectivity finesse, FS is the plate spherical deviation finesse coefficient, Fθ is the incident beam divergence finesse coefficient, and Fd is the diffractionlimited finesse coefficient. A148 Etalons (4.121) The beam divergence also influences the actual finesse of an etalon. Taking into account all these contributions, the effective finesse Fe) of an etalon (with FR being the reflectivity finesse and Fθ the divergence finesse) is: (4.116) However, the above applies to theoretical etalons which are assumed to be perfect. In reality, even the best etalon will show defects that limit theoretically expected performance. Therefore, in a real etalon, the actual finesse will usually be lower than the reflectivity finesse. 5 = 2+ + 2 2 Fdg Fdp2 Fd FS = + + + Fe F R2 FD2 F v2 FS2 (4.122) a. Spherical Defects (Fs) b. Spherical Irregularities (Fdg) c. Parallelism Defects (Fdp) Figure 4.60 Three types of defects contributing to the total defect finesse 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials The examples below show how the effective finesse varies with plate flatness and clear aperture. Example 1: Air-spaced etalon, X R = 95% (±1%) at 633 nm = 25 mm X Used (air gap) = 1 mm X Spherical / parallelism defects = <λ/20 (4.123) rms = 0.80 nm X Beam X F Temperature tuning: Primarily used for solid etalons, temperature-tuning changes both the actual spacing of the reflective surfaces via expansion and the index of refraction of the material, which changes the optical spacing. The tuning result can be given by aperture = 20 mm X Spacer X Plate Etalons can be tuned over a limited range to alter their peak transmission wavelengths. These techniques are: Angle tuning or tilting the etalon: As the angle of incidence is increased, the center wavelength of the etalon can be tuned down the spectrum. Optical Specifications X CA TUNING AN ETALON Material Properties The effective finesse a user sees when using the etalon depends not only on the absolute clear aperture, but also on the used aperture of the etalon, especially when a high finesse is required. divergence = 0.1 mRad = 61, Fe = 10 R Fundamental Optics Pressure tuning: Air-spaced etalons can be tuned by increasing the pressure in the cavity between the optics, thereby increasing the effective index of refraction, and thus the effective spacing. Example 2: Air-spaced etalon, X Same parameters as example 1 except: X Used aperture = 5 mm X F = 61, Fe = 40 (±4) R X Same parameters as example 1 except: X Spherical X Plate / parallelism defects = λ/100 rms = 0.40 nm X Beam X F Gaussian Beam Optics Example 3: Air-spaced etalon, The above examples illustrate how critical the optical surface flatness, plate parallelism and surface quality are to the overall performance of an etalon. At CVI Laser Optics we have developed sophisticated software that allows us to simulate all effects that influence the performance of an etalon. To order an etalon, FSR, finesse and used aperture are required. divergence: 0.1 mrad = 61, Fe = 40 (±8) R Machine Vision Guide These examples illustrate that, for large-aperture applications, it is important to use very high-quality plates to ensure a high finesse and good transmission values. Laser Guide marketplace.idexop.com Etalons A149 FUNDAMENTAL OPTICS ULTRAFAST THEORY Assume that the power, reflectivity, and polarization characteristics of a laser mirror are acceptable over the bandwidth of a femtosecond pulse. This means that, over the entire pulse bandwidth, a cavity mirror may have a reflectivity greater than 99.8%; a 50% beamsplitter may have a fairly constant reflection; a polarizer may maintain its rejection of one polarization with an acceptable transmission of the other. It is not enough, however, to simply preserve the power spectrum S(ω) = |E(ω)|2 when dealing with femtosecond pulses. The phase relationship among the Fourier components of the pulse must also be preserved in order that the pulse not be broadened or distorted. What constraint on the performance of a mirror or transmissive optic does this imply? linearly proportional to frequency with proportionality constant td. The reflected pulse is then: (r (W ) = U ∫ ( (q ) − Lq(W −Wd ) Gq = U( (W − Wd ) (4.126) Thus, provided the phase shift is linear in frequency over the pulse bandwidth, the reflected pulse is scaled by the amplitude reflectance r, and delayed in time by the constant group delay td. It is, otherwise, an undistorted replica of the original pulse. Examined over a large enough bandwidth, no optical system will exhibit the constant group delay over frequency needed for perfect fidelity. In general, the phase shift near some center frequency ω0 may be 2XWSXW3XOVH:LGWKIV Fundamental Optics The distinguishing aspect of femtosecond laser optics design is the need to control the phase characteristic of the optical system over the requisite wide pulse bandwidth. CVI Laser Optics has made an intensive theoretical study of these effects. Certain coating designs have been modified with control of the phase characteristics in mind. New proprietary designs have been created with desirable characteristics for femtosecond researchers. All optics in this section have been tested by researchers in the field and we are constantly fielding new requests. IV IV Consider a general initial pulse shape E0(t). As a function of its Fourier components, it may be expressed as: IV IV *''IVð *''IVð *''IVð (0 (W ) = ∫ ( ( q ) − LqW Gq (4.124) Suppose this pulse reflects off of a mirror. For this example, we assume the mirror is “ideal”, and use the Fourier transform of its complex amplitude reflectance: 2XWSXW3XOVH:LGWKIV IV IV IV (4.125) For this “ideal” mirror, r is a real constant equal to the amplitude reflectivity that is assumed constant over the pulse bandwidth. All phase effects have been assumed to be describable by a single phase shift Ø(ω) that is A150 Ultrafast Theory 2XWSXW3XOVH:LGWKIV IV IV IV Figure 4.61 Output pulse width vs. GDD 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials expanded in a Taylor series for frequencies near ω0: CVI Laser Optics uses three basic designs; TLM1 mirrors for energy fluence greater than 100 mJ/cm2, TLM2 mirrors for cw oscillators and low-fluence pulses, and TLMB mirrors which are a hybrid of the two. The reflectivity, GDD parameter, and cubic dispersion parameter for TLM2 high reflectors are shown in Figure 4.64. In these examples, the mirrors are centered at 800 nm and are designed for use at normal incidence and at 45º. Note that, at the design wavelength, (a) GDD is zero, (b) the cubic term is minimized, and (c) at 45° incidence, the GDD of the p-polarization component is very sensitive to wavelength, while the GDD for s-polarization component is nearly zero over a broad wavelength range. Thus one should avoid using mirrors at 45° incidence with the p-polarization. On the other hand, at 45° incidence, s-polarization provides very broad bandwidth and minimizes pulse distortion problems and should be used whenever possible. Machine Vision Guide where τ0 is the initial pulse duration (FWHM of the pulse intensity). Let the pulse enter a medium or reflect off of a mirror with non-zero Ø"(ω), measured in fsec2 radians. (For a continuous medium-like glass, Ø"(ω)= β"(ω) x z where β"(ω) is the group velocity dispersion (GVD) per centimeter of material, and z is the physical path length, in centimeters, traveled through the material.) The Gaussian pulse will be both chirped and temporally broadened by its encounter with group velocity dispersion. The power envelope will remain Gaussian; the result for the broadened FWHM is: DISPERSIVE PROPERTIES OF MIRRORS Gaussian Beam Optics (4.128) Figures 4.62 and 4.63 show the GVD and cubic dispersion respectively for some common used glasses. Some of the glasses can be used in the UV region. They should be useful in estimating material dispersion and pulse distortion effects. Please check these calculations independently before using them in a final design. Fundamental Optics To illustrate pulse distortion due to the dependence of the group delay on frequency, consider what happens when an unchirped, transform-limited Gaussian pulse passes through a medium, or is incident on a mirror whose dominating contribution to phase distortion is non-zero group velocity dispersion. The field envelope of the pulse is assumed to be of the form: GROUP-VELOCITY & CUBIC DISPERSION FOR VARIOUS OPTICAL MATERIALS Optical Specifications These derivatives are, respectively, the group delay Ø'(ω0), the group velocity dispersion Ø"(ω0), and the cubic term Ø'"(ω0), evaluated at a center frequency ω0. This expansion is heuristically useful, in an exactly soluble model, for the propagation of a transform-limited Gaussian pulse. Note, however, that for extremely short pulses the expansion above may be insufficient. A full numerical calculation may have to be performed using the actual phase shift function Ø(ω). CVI Laser Optics will be happy to assist those interested in the modeling of real optical elements. Material Properties (4.127) This result, valid only for initially unchirped, transformlimited Gaussian pulses, is nevertheless an excellent model to study the effects of dispersion on pulse propagation. The graphs shown in Figure 4.61 represent the theoretical broadening from dispersion for initial pulse widths ranging from 10 to 100 femtoseconds. Ti:Sapphire and other femtosecond laser systems need prismless compensation of the built-in positive chirp encountered in the laser optical circuit. This becomes mandatory in industrial and biomedical applications where the laser must provide a compact, stable, and reliable solution. (4.129) Laser Guide marketplace.idexop.com Ultrafast Theory A151 FUNDAMENTAL OPTICS Fundamental Optics 6) 6) 6) 6) 6) 6) /D)1 &4QH & )XVHG6LOLFD &D) &D) 72'IVñFP *9'IVðFP %. &4QH & )XVHG6LOLFD /D)1 %. :DYHOHQJWKmP Figure 4.62 GVD for common glasses In experiments using CVI Laser Optics TNM2 negative group velocity dispersion mirrors, 200 mW, 80 fsec pulses centered at 785 nm were achieved in a simple, prismless, Ti:Sapphire oscillator. The configuration is shown in Figure 4.65. OUTPUT COUPLERS & BEAMSPLITTERS Output-coupler partial reflectors and beamsplitters behave similarly; however, here is an additional consideration in their analysis. The behavior of the transmitted phase of the coating and the effect of material dispersion within the substrate on the transmitted beam have to be taken into account in a detailed analysis. In general, the coating transmitted phase has similar properties and magnitudes of GVD and cubic to the reflected phase. As usual, centering is important. As a beamsplitter, we recommend the 1.5 mm thick fused silica substrate PW-1006-UV. As an output A152 Ultrafast Theory :DYHOHQJWKmP Figure 4.63 Cubic dispersion for common glasses coupler substrate, we recommend the 3.0 mm thick, 30 minute wedge fused silica substrate IF-1012-UV. CVI Laser Optics has developed the TFPK Series Broadband Low Dispersion Polarizing Beamsplitters to satisfy requirements for very-high-power, short-pulse lasers. These optics are ideal for intracavity use in femtosecond regenerative amplifiers. The main emphasis is on linear phase characteristics. See Chapter 9 of Lasers, A. E. Siegman (University Science Books, Mill Valley, California, 1986), for a good discussion of linear pulse propagation. In chirped pulse regenerative amplification, the pulse may have to pass through one or two polarizers twice per round trip. There can be 10 to 20 round trips before the gain is saturated and the pulse is ejected. At this stage the pulse is long (100 - 1000 psec); however the phase shift at each frequency must still be maintained 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials 6 3 3mm crystal negative GVD mirror pump 7/0 output coupler :DYHOHQJWKQP 3 negative GVD mirror 7/0 Figure 4.65 Typical optical setup incorporating low GVD and Negative GVD mirrors in an ultrafast application 6 200 mW 80 fs pulses centered at 785 nm Optical Specifications *'' F´qIVHF low GVD mirror Material Properties 5HIOHFWLYLW\ :DYHOHQJWKQP 6 3 Fundamental Optics &XELF7HUP F´¶qIVHF 7/0 :DYHOHQJWKQP Figure 4.64 Dispersion and reflectivity for mirrors TLM2800-0 and TLM2-800-45 There are some subtleties associated with the TFPK. The near 72° angle has to be set properly and optimized. Some thought has to be given to mechanical clearances of the laser beam at such a steep incidence angle. The reflectivity for s-polarization is limited to 75%. Variant designs can increase this at a slight loss in bandwidth, increase in incidence angle, and increase in insertion loss for the transmitted p-polarized component. Machine Vision Guide Figure 4.67 shows the power transmission curves for both s- and p-polarization and the transmitted phase characteristics of the p component for a TFPK optimized at 800 nm. (Users may specify any wavelength from 250 nm to 1550 nm.) The phase characteristics shown are the GDD and the cubic phase term. Not shown are the reflected phase characteristics for s-polarization; they are similar to the p-polarization transmission curves, and have the same low nonlinearity and broad bandwidth. Note that both sides of the optic have the coating whose properties are described in Figure 4.67. Therefore, the s- and p-polarization transmissions per surface should be squared in determining the specifications. The phase characteristics show that in all modes of operation, the TFPK polarizer performance is dominated by the substrate. Gaussian Beam Optics to minimize the recompressed pulse width. The many round trips of the pulse in the regenerative amplifier put stringent requirements on the phase characteristics of the coatings. Figure 4.66 Typical optical set-up of negative GVD mirrors The FABS autocorrelator beamsplitters from CVI Laser Optics are broadband, 50% all-dielectric beamsplitters. They are useful in many types of pump-probe experiments and in the construction of antiresonant ring configurations. They are essentially lossless and extremely durable. Both have advantages over partially reflecting metal coatings. Laser Guide marketplace.idexop.com Ultrafast Theory A153 FUNDAMENTAL OPTICS ANTIREFLECTION COATINGS All CVI Laser Optics antireflection coating designs work well in femtosecond operation as the forward-going phasor is the dominant contribution to the phase shift; the AR coating is very thin and simply “fixes” the small Fresnel reflection of the substrate. 6 7UDQVPLWWHG3*''IVHFð 7UDQVPLWWHG3&XELFIVHFñ 7)3. 7)3. :DYHOHQJWKQP Figure 4.67 Properties for one coated side of a TFPK polarizing beamsplitter optimized for 800 nm. Both sides are coated for these properties. Ultrafast Theory GȘ GȜ O ȜO GȘ GȜ (4.130) ȜO l = tip to tip distance (AB) )$%63 F / η = refractive index of the prisms (assuming the same material) 7)3. ȦȦO ȜO where 7UDQVPLVVLRQ § A154 G ȥ GȦ *9' 3 Very-high-quality isosceles Brewster’s angle prisms for intra and extracavity use are available from CVI Laser Optics. The design of these prisms satisfies the condition of minimum loss due to entrance and exit at Brewster’s angle. To calculate GVD at Brewsters angle, refer to Figure 4.70 and use the following equation: 5HIOHFWHG%HDP*''IVHFð 7UDQVPLVVLRQ PRISMS 5HIOHFWHG%HDP&8%,&IVHFñ Fundamental Optics Power transmission curves for the s- and p-polarized versions of the FABS, along with the corresponding reflected phase characteristics for beamsplitters optimized at 800 nm, are shown in Figures 4.68 and 4.69. The linear pulse propagation properties of these beamsplitters are dominated by the substrate material dispersion. As with virtually all dielectric coated optics, the s-polarized version is broader than p-polarized version. CVI Laser Optics can produce FABS in other than 50:50 with excellent phase characteristics. )$%63 )$%63 :DYHOHQJWKQP Figure 4.68 Transmission characteristics for FABS series polarizers with p-polarized light 1-505-298-2550 FUNDAMENTAL OPTICS Optical Coatings & Materials L = total average glass path $ EURDGEDQG OLJKW ωl λl = 2πc (assumes Brewster prism at minimum deviation). Material Properties ψ = spectral phase of the electric field UHG EOXH EOXH UHG % For more on the Ultrafast phenomena, see J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic Press, 1996. Figure 4.70 Paired Brewster prisms Optical Specifications )$%66 Gaussian Beam Optics 5HIOHFWHG%HDP*''IVHFð )$%66 )$%66 Machine Vision Guide 5HIOHFWHG%HDP&XELFIVHFñ Fundamental Optics 7UDQVPLVVLRQ :DYHOHQJWKQP Figure 4.69 Transmission characteristics for FABS series polarizers with s-polarized light Laser Guide marketplace.idexop.com Ultrafast Theory A155 FUNDAMENTAL OPTICS Fundamental Optics A156 1-505-298-2550 Optical Coatings & Materials GAUSSIAN BEAM OPTICS A158 Material Properties GAUSSIAN BEAM PROPAGATION TRANSFORMATION AND MAGNIFICATION BY SIMPLE LENSES A163 REAL BEAM PROPAGATION A167 LENS SELECTION A170 Optical Specifications Fundamental Optics Gaussian Beam Optics Machine Vision Guide Laser Guide marketplace.idexop.com A157 GAUSSIAN BEAM OPTICS GAUSSIAN BEAM PROPAGATION Gaussian Beam Optics In most laser applications it is necessary to focus, modify, or shape the laser beam by using lenses and other optical elements. In general, laser-beam propagation can be approximated by assuming that the laser beam has an ideal Gaussian intensity profile, which corresponds to the theoretical TEM00 mode. Coherent Gaussian beams have peculiar transformation properties which require special consideration. In order to select the best optics for a particular laser application, it is important to understand the basic properties of Gaussian beams. Unfortunately, the output from real-life lasers is not truly Gaussian (although the output of a single mode fiber is a very close approximation). To accommodate this variance, a quality factor, M2 (called the “M-squared” factor), has been defined to describe the deviation of the laser beam from a theoretical Gaussian. For a theoretical Gaussian, M2 = 1; for a real laser beam, M2>1. The M2 factor for helium neon lasers is typically less than 1.1; for ion lasers, the M2 factor typically is between 1.1 and 1.3. Collimated TEM00 diode laser beams usually have an M2 ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 25 or 30. In all cases, the M2 factor affects the characteristics of a laser beam and cannot be neglected in optical designs. in most single-wavelength applications is primary (thirdorder) spherical aberration. Scatter from surface defects, inclusions, dust, or damaged coatings is of greater concern in laser-based systems than in incoherent systems. Speckle content arising from surface texture and beam coherence can limit system performance. Because laser light is generated coherently, it is not subject to some of the limitations normally associated with incoherent sources. All parts of the wavefront act as if they originate from the same point; consequently, the emergent wavefront can be precisely defined. Starting out with a well-defined wavefront permits more precise focusing and control of the beam than otherwise would be possible. For virtually all laser cavities, the propagation of an electromagnetic field, E(0), through one round trip in an optical resonator can be described mathematically by a propagation integral, which has the general form E (1) ( x, y ) = e − jkp ∫∫ InputPlane In the following section, Gaussian Beam Propagation, we will treat the characteristics of a theoretical Gaussian beam (M2=1); then, in the section Real Beam Propagation we will show how these characteristics change as the beam deviates from the theoretical. In all cases, a circularly symmetric wavefront is assumed, as would be the case for a helium neon laser or an argon-ion laser. Diode laser beams are asymmetric and often astigmatic, which causes their transformation to be more complex. Although in some respects component design and tolerancing for lasers is more critical than for conventional optical components, the designs often tend to be simpler since many of the constraints associated with imaging systems are not present. For instance, laser beams are nearly always used on axis, which eliminates the need to correct asymmetric aberration. Chromatic aberrations are of no concern in single-wavelength lasers, although they are critical for some tunable and multiline laser applications. In fact, the only significant aberration A158 Gaussian Beam Propagation ( ) K ( x, y, x0 , y0 ) E (0) x0, y0 dx0dy0 (5.1) where K is the propagation constant at the carrier frequency of the optical signal, p is the length of one period or round trip, and the integral is over the transverse coordinates at the reference or input plane. The function K is commonly called the propagation kernel since the field E(1)(x, y), after one propagation step, can be obtained from the initial field E(0)(x0, y0) through the operation of the linear kernel or “propagator” K(x, y, x0, y0). By setting the condition that the field, after one period, will have exactly the same transverse form, both in phase and profile (amplitude variation across the field), we get the equation g nm E nm ( x, y ) ≡ ∫∫ InputPlane ( ) K ( x, y, x0 , y0 ) E nm x0, y0 dx0dy0 1-505-298-2550 (5.2) GAUSSIAN BEAM OPTICS Optical Coatings & Materials where Enm represents a set of mathematical eigenmodes, and γnm a corresponding set of eigenvalues. The eigenmodes are referred to as transverse cavity modes, and, for stable resonators, are closely approximated by Hermite-Gaussian functions, denoted by TEMnm. (Anthony Siegman, Lasers) The lowest order, or “fundamental” transverse mode, TEM00 has a Gaussian intensity profile, shown in figure 5.1, which has the form Material Properties 3HUFHQW,UUDGLDQFH 4 Z 4 Z ( − k x2 + y 2 ) (5.3) Z Optical Specifications I ( x, y ) ∝ e &RQWRXU5DGLXV Figure 5.1 Irradiance profile of a Gaussian TEM00 mode In this section we will identify the propagation characteristics of this lowest-order solution to the propagation equation. In the next section, Real Beam Propagation, we will discuss the propagation characteristics of higher-order modes, as well as beams that have been distorted by diffraction or various anisotropic phenomena. HGLDPHWHURISHDN ):+0GLDPHWHURISHDN Fundamental Optics GLUHFWLRQ RISURSDJDWLRQ BEAM WAIST AND DIVERGENCE a perfectly collimated beam. The spreading of a laser beam is in precise accord with the predictions of pure diffraction theory; aberration is totally insignificant in the present context. Under quite ordinary circumstances, the beam spreading can be so small it can go unnoticed. The following formulas accurately describe beam spreading, making it easy to see the capabilities and limitations of laser beams. Even if a Gaussian TEM00 laser-beam wavefront were made perfectly flat at some plane, it would quickly acquire curvature and begin spreading in accordance with pw2 2 R ( z ) = z 1 + 0 lz (5.4) and 1/ 2 Gaussian Beam Propagation Laser Guide lz 2 w ( z ) = w0 1 + p w02 marketplace.idexop.com Machine Vision Guide Diffraction causes light waves to spread transversely as they propagate, and it is therefore impossible to have Figure 5.2 Diameter of a Gaussian beam Gaussian Beam Optics In order to gain an appreciation of the principles and limitations of Gaussian beam optics, it is necessary to understand the nature of the laser output beam. In TEM00 mode, the beam emitted from a laser begins as a perfect plane wave with a Gaussian transverse irradiance profile as shown in figure 5.1. The Gaussian shape is truncated at some diameter either by the internal dimensions of the laser or by some limiting aperture in the optical train. To specify and discuss the propagation characteristics of a laser beam, we must define its diameter in some way. There are two commonly accepted definitions. One definition is the diameter at which the beam irradiance (intensity) has fallen to 1/e2 (13.5%) of its peak, or axial value and the other is the diameter at which the beam irradiance (intensity) has fallen to 50% of its peak, or axial value, as shown in figure 5.2. This second definition is also referred to as FWHM, or full width at half maximum. For the remainder of this guide, we will be using the 1/e2 definition. A159 GAUSSIAN BEAM OPTICS pw2 2 R ( z ) = z 1 + 0 lz Gaussian Beam Optics and and lz 2 w ( z ) = w0 1 + p w02 v= 1/ 2 w (z) l = . z p w0 (5.8) (5.5) where z is the distance propagated from the plane where the wavefront is flat, λ is the wavelength of light, w0 is the radius of the 1/e2 irradiance contour at the plane where the wavefront is flat, w(z) is the radius of the 1/e2 contour after the wave has propagated a distance z, and R(z) is the wavefront radius of curvature after propagating a distance z. R(z) is infinite at z = 0, passes through a minimum at some finite z, and rises again toward infinity as z is further increased, asymptotically approaching the value of z itself. The plane z = 0 marks the location of a Gaussian waist, or a place where the wavefront is flat, and w0 is called the beam waist radius. The irradiance distribution of the Gaussian TEM00 beam, namely, This value is the far-field angular radius (half-angle divergence) of the Gaussian TEM00 beam. The vertex of the cone lies at the center of the waist, as shown in figure 5.3. It is important to note that, for a given value of λ, variations of beam diameter and divergence with distance z are functions of a single parameter, w0, the beam waist radius. NEAR-FIELD VS FAR-FIELD DIVERGENCE Unlike conventional light beams, Gaussian beams do not diverge linearly. Near the beam waist, which is typically close to the output of the laser, the divergence angle is extremely small; far from the waist, the divergence angle approaches the asymptotic limit described above. The Raleigh range (zR), defined as the distance over which the beam radius spreads by a factor of √2, is given by (5.6) zR = where w = w(z) and P is the total power in the beam, is the same at all cross sections of the beam. The invariance of the form of the distribution is a special consequence of the presumed Gaussian distribution at z = 0. If a uniform irradiance distribution had been presumed at z = 0, the pattern at z = ∞ would have been the familiar Airy disc pattern given by a Bessel function, whereas the pattern at intermediate z values would have been enormously complicated. pw (5.9) l At the beam waist (z = 0), the wavefront is planar [R(0) = ∞]. Likewise, at z = ∞, the wavefront is planar [R(∞) = ∞]. As the beam propagates from the waist, the wavefront curvature, therefore, must increase to a maximum and then begin to decrease, as shown in figure 5.4. The Raleigh range, considered to be the dividing line between near-field divergence and mid-range Simultaneously, as R(z) asymptotically approaches z for large z, w(z) asymptotically approaches the value lz w (z) = (5.7) p w0 where z is presumed to be much larger than πw0/λ so that the 1/e2 irradiance contours asymptotically approach a cone of angular radius A160 Gaussian Beam Propagation w w0 w0 1 e2 irradiance surface ic co ptot asym ne v z w0 Figure 5.3 Growth in 1/e2 radius with distance propagated away from Gaussian waist 1-505-298-2550 GAUSSIAN BEAM OPTICS Optical Coatings & Materials The beam radius at 100 m reaches a minimum value for a starting beam radius of about 4.5 mm. Therefore, if we wanted to achieve the best combination of minimum beam diameter and minimum beam spread (or best collimation) over a distance of 100 m, our optimum starting beam radius would be 4.5 mm. Any other starting value would result in a larger beam at z = 100 m. Typically, one has a fixed value for w0 and uses the expression lz w0 (optimum ) = p We can find the general expression for the optimum starting beam radius for a given distance, z. Doing so yields 1/ 2 (5.10) )LQDO%HDP5DGLXVPP Fundamental Optics to calculate w(z) for an input value of z. However, one can also utilize this equation to see how final beam radius varies with starting beam radius at a fixed distance, z. Figure 4.5 shows the Gaussian beam propagation equation plotted as a function of w0, with the particular values of λ = 632.8 nm and z = 100 m. 1/ 2 Optical Specifications lz 2 w ( z ) = w0 1 + p w02 Material Properties divergence, is the distance from the waist at which the wavefront curvature is a maximum. Far-field divergence (the number quoted in laser specifications) must be measured at a distance much greater than zR (usually >10 x zR will suffice). This is a very important distinction because calculations for spot size and other parameters in an optical train will be inaccurate if near- or mid-field divergence values are used. For a tightly focused beam, the distance from the waist (the focal point) to the far field can be a few millimeters or less. For beams coming directly from the laser, the far-field distance can be measured in meters. 6WDUWLQJ%HDP5DGLXV w PP Figure 5.5 Beam radius at 100 m as a function of starting beam radius for a HeNe laser at 632.8 nm Gaussian Beam Optics Machine Vision Guide Figure 5.4 Changes in wavefront radius with propagation distance Laser Guide marketplace.idexop.com Gaussian Beam Propagation A161 GAUSSIAN BEAM OPTICS Gaussian Beam Optics Using this optimum value of w0 will provide the best combination of minimum starting beam diameter and minimum beam spread [ratio of w(z) to w0] over the distance z. For z = 100 m and λ = 632.8 nm, w0 (optimum) = 4.48 mm (see example above). If we put this value for w0 (optimum) back into the expression for w(z), minimized, as illustrated in figure 5.6. By focusing the beam-expanding optics to place the beam waist at the midpoint, we can restrict beam spread to a factor of √2 over a distance of 2zR, as opposed to just zR. w (100) = 2 ( 4.48) = 6.3 mm This result can now be used in the problem of finding the starting beam radius that yields the minimum beam diameter and beam spread over 100 m. Using 2(zR) = 100 m, or zR = 50 m, and λ = 632.8 nm, we get a value of w(zR) = (2λ/π)½ = 4.5 mm, and w0 = 3.2 mm. Thus, the optimum starting beam radius is the same as previously calculated. However, by focusing the expander we achieve a final beam radius that is no larger than our starting beam radius, while still maintaining the √2 factor in overall variation. By turning this previous equation around, we find that we once again have the Rayleigh range (zR), over which the beam radius spreads by a factor of √2 as Alternately, if we started off with a beam radius of 6.3 mm, we could focus the expander to provide a beam waist of w0 = 4.5 mm at 100 m, and a final beam radius of 6.3 mm at 200 m. w ( z ) = 2 (w0 ) (5.11) Thus, for this example, zR = p w02 l APPLICATION NOTE 2 with withz = p w0 R l 2w . w ( zR ) = 0 with Location of the beam waist w ( zR ) = 2w0 . If we use beam-expanding optics that allow us to adjust the position of the beam waist, we can actually double the distance over which beam divergence is The location of the beam waist is required for most Gaussian-beam calculations. CVI Laser Optics lasers are typically designed to place the beam waist very close to the output surface of the laser. If a more accurate location than this is required, our applications engineers can furnish the precise location and tolerance for a particular laser model. beam waist 2w0 beam expander w(–zR) = 2w0 w(zR) = 2w0 zR zR Figure 5.6 Focusing a beam expander to minimize beam radius and spread over a specified distance A162 Gaussian Beam Propagation 1-505-298-2550 GAUSSIAN BEAM OPTICS Optical Coatings & Materials TRANSFORMATION AND MAGNIFICATION BY SIMPLE LENSES The main differences between Gaussian beam optics and geometric optics, highlighted in such a plot, can be summarized as follows: X There is a maximum and a minimum image distance for Gaussian beams. maximum image distance occurs at s = f=z/R, rather than at s = f. X The X There lens appears to have a shorter focal length as zR/f increases from zero (i.e., there is a Gaussian focal shift). X A s df ,PDJH'LVWDQFH 4 SDUDPHWHU 4 zf 5 4 In the regular form, (5.13) . . In the far-field limit as z/R approaches 0 this reduces to the geometric optics equation. A plot of s/f versus s"/f for various values of zR/f is shown in figure 5.7. For a positive thin lens, the three distinct regions of interest correspond to real object and real image, real object and virtual image, and virtual object and real image. sf 2EMHFW'LVWDQFH Figure 5.7 Plot of lens formula for Gaussian beams with normalized Rayleigh range of the input beam as the parameter Self recommends calculating zR, w0, and the position of w0 for each optical element in the system in turn so that the overall transformation of the beam can be calculated. To carry this out, it is also necessary to consider magnification: w0"/w0. The magnification is given by m= w0 ″ = w0 { 1 1 − ( s / f ) + ( zR / f ) 2 2 } Machine Vision Guide (5.14) 4 4 4 4 Gaussian Beam Optics 4 4 dimensionless form, form, or,or, in in dimensionless s / f zR / f / s / f s Ǝ / f s / f zR / f / s / f s Ǝ / f Fundamental Optics where s is the object distance, s" is the image distance, and f is the focal length of the lens. For Gaussian beams, Self has derived an analogous formula by assuming that the waist of the input beam represents the object, and the waist of the output beam represents the image. The formula is expressed in terms of the Rayleigh range of the input beam. Optical Specifications is a common point in the Gaussian beam expression at s/f = s"/f = 1. For a simple positive lens, this is the point at which the incident beam has a waist at the front focus and the emerging beam has a waist at the rear focus. 1 1 + = 1. (5.12) s / f s″ / f s zR / s f s Ǝ f Ǝ f zR / s f sform, or, sindimensionless Material Properties It is clear from the previous discussion that Gaussian beams transform in an unorthodox manner. Siegman uses matrix transformations to treat the general problem of Gaussian beam propagation with lenses and mirrors. A less rigorous, but in many ways more insightful, approach to this problem was developed by Self (S. A. Self, “Focusing of Spherical Gaussian Beams”). Self shows a method to model transformations of a laser beam through simple optics, under paraxial conditions, by calculating the Rayleigh range and beam waist location following each individual optical element. These parameters are calculated using a formula analogous to the well-known standard lens-maker’s formula. The standard lens equation is written as . (5.15) Laser Guide marketplace.idexop.com Transformation and Magnification by Simple Lenses A163 GAUSSIAN BEAM OPTICS Gaussian Beam Optics The Rayleigh range of the output beam is then given by 1 1 1 + = .(5.16) s s ″ + zR″ 2 /( s ″ − f ) f All the above formulas are written in terms of the Rayleigh range of the input beam. Unlike the geometric case, the formulas are not symmetric with respect to input and output beam parameters. For back tracing beams, it is useful to know the Gaussian beam formula in terms of the Rayleigh range of the output beam: 1 1 1 + = .(5.17) s s ″ + zR″ 2 /( s ″ − f ) f BEAM CONCENTRATION The spot size and focal position of a Gaussian beam can be determined from the previous equations. Two cases of particular interest occur when s = 0 (the input waist is at the first principal surface of the lens system) and s = f (the input waist is at the front focal point of the optical system). For s = 0, we get s ′′ = ( f 1 + l f / pw02 and s ′′ = ( f ) ) 2 and 1 + llf f/ p ww /p 00 w= and w= ( 2 (5.18) 2 ) 1/ 2 ) 1/ 2 1 + l f / pw 2 2 0 l f / pw0 ( 1 + l f / pw 2 2 0 e2 Dbeam Figure 5.8 Concentration of a laser beam by a laser-line focusing singlet Substituting typical values into these equations yields nearly identical results, and for most applications, the simpler, second set of equations can be used. In many applications, a primary aim is to focus the laser to a very small spot, as shown in figure 5.8, by using either a single lens or a combination of several lenses. If a particularly small spot is desired, there is an advantage to using a well-corrected high-numericalaperture microscope objective to concentrate the laser beam. The principal advantage of the microscope objective over a simple lens is the diminished level of spherical aberration. Although microscope objectives are often used for this purpose, they are not always designed for use at the infinite conjugate ratio. Suitably optimized lens systems, known as infinite conjugate objectives, are more effective in beam-concentration tasks and can usually be identified by the infinity symbol on the lens barrel. DEPTH OF FOCUS (5.19) For the case of s=f, the equations for image distance and waist size reduce to the following: s″ = f ands ″ = f and and w = l f / p w0 . w = l f / p w0 . A164 1 2w 0 w Transformation and Magnification by Simple Lenses Depth of focus (±Δz), that is, the range in image space over which the focused spot diameter remains below an arbitrary limit, can be derived from the formula lz 2 w ( z ) = w0 1 + p w02 1/ 2 . The first step in performing a depth-of-focus calculation is to set the allowable degree of spot size variation. If we choose a typical value of 5%, or w(z)0 = 1.05w0, and solve for z = Δz, the result is 1-505-298-2550 GAUSSIAN BEAM OPTICS Optical Coatings & Materials Dz ≈ ± 0.32p w02 . l TRUNCATION d = K × l × f /#(5.20) In the case of the Airy disc, the intensity falls to zero at the point dzero = 2.44 x λ x f/#, defining the diameter of the spot. When the pupil illumination is not uniform, the image spot intensity never falls to zero, making it necessary to define the diameter at some other point. This is commonly done for two points: dFWHM = 50% intensity point and the Fundamental Optics where K is a constant dependent on truncation ratio and pupil illumination, λ is the wavelength of light, and f/# is the speed of the lens at truncation. The intensity profile of the spot is strongly dependent on the intensity profile of the radiation filling the entrance pupil of the lens. For uniform pupil illumination, the image spot takes on the Airy disc intensity profile shown in figure 5.9. When the pupil illumination is between these two extremes, a hybrid intensity profile results. Optical Specifications In a diffraction-limited lens, the diameter of the image spot is Material Properties Since the depth of focus is proportional to the square of focal spot size, and focal spot size is directly related to f-number (f/#), the depth of focus is proportional to the square of the f/# of the focusing system. If the pupil illumination is Gaussian in profile, the result is an image spot of Gaussian profile, as shown in figure 5.10. d1/e2 = 13.5% intensity point. It is helpful to introduce the truncation ratio LQWHQVLW\ T= Db (5.21) Dt Gaussian Beam Optics ,QWHQVLW\ LQWHQVLW\ lIQXPEHU Figure 5.9 Airy disc intensity distribution at the image plane ,QWHQVLW\ LQWHQVLW\ LQWHQVLW\ Machine Vision Guide where Db is the Gaussian beam diameter measured at the 1/e2 intensity point, and Dt is the limiting aperture diameter of the lens. If T = 2, which approximates uniform illumination, the image spot intensity profile approaches that of the classic Airy disc. When T = 1, the Gaussian profile is truncated at the 1/e2 diameter, and the spot profile is clearly a hybrid between an Airy pattern and a Gaussian distribution. When T = 0.5, which approximates the case for an untruncated Gaussian input beam, the spot intensity profile approaches a Gaussian distribution. Calculation of spot diameter for these or other truncation ratios requires that K be evaluated. This is done by using the formulas lIQXPEHU Figure 5.10 Gaussian intensity distribution at the image plane Laser Guide marketplace.idexop.com Transformation and Magnification by Simple Lenses A165 GAUSSIAN BEAM OPTICS Gaussian Beam Optics K FWHM = . + . (T − .) − . (T − .) (5.22) and . . − . K / e = . + (T − .) (T − .) (5.23) The K function permits calculation of the on-axis spot diameter for any beam truncation ratio. The graph in figure 5.11 plots the K factor vs T(Db/Dt). The optimal choice for truncation ratio depends on the relative importance of spot size, peak spot intensity, and total power in the spot as demonstrated in the table below. The total power loss in the spot can be calculated by using and hence is spatially separated at a lens focal plane. By centering a small aperture around the focal spot of the direct beam, as shown in figure 2.12, it is possible to block scattered light while allowing the direct beam to pass unscathed. The result is a cone of light that has a very smooth irradiance distribution and can be refocused to form a collimated beam that is almost uniformly smooth. As a compromise between ease of alignment and complete spatial filtering, it is best that the aperture diameter be about two times the 1/e2 beam contour at the focus, or about 1.33 times the 99% throughput contour diameter. VSRWPHDVXUHGDWLQWHQVLW\OHYHO 2 −2 D / D PL = e ( t b ) (5.24) K )DFWRU VSRWPHDVXUHGDWLQWHQVLW\OHYHO for a truncated Gaussian beam. A good compromise between power loss and spot size is often a truncation ratio of T = 1. When T = 2 (approximately uniform illumination), fractional power loss is 60%. When T = 1, d1/e2 is just 8% larger than when T = 2, whereas fractional power loss is down to 13.5%. Because of this large savings in power with relatively little growth in the spot diameter, truncation ratios of 0.7 to 1.0 are typically used. Ratios as low as 0.5 might be employed when laser power must be conserved. However, this low value often wastes too much of the available clear aperture of the lens. SPATIAL FILTERING Laser light scattered from dust particles residing on optical surfaces may produce interference patterns resembling holographic zone planes. Such patterns can cause difficulties in interferometric and holographic applications where they form a highly detailed, contrasting, and confusing background that interferes with desired information. Spatial filtering is a simple way of suppressing this interference and maintaining a very smooth beam irradiance distribution. The scattered light propagates in different directions from the laser light A166 Transformation and Magnification by Simple Lenses VSRWGLDPHWHU K ! l ! IQXPEHU TDE DW Figure 5.11 K factors as a function of truncation ratio Spot Diameters and Fractional Power Loss for Three Values of Truncation Truncation Ratio dFWHM d1/e2 dzero PL(%) Infinity 1.03 1.64 2.44 100 2.0 1.05 1.69 — 60 1.0 1.13 1.83 — 13.5 0.5 1.54 2.51 — 0.03 focusing lens pinhole aperture Figure 5.12 Spatial filtering smoothes the irradiance distribution 1-505-298-2550 GAUSSIAN BEAM OPTICS Optical Coatings & Materials REAL BEAM PROPAGATION To address the issue of non-Gaussian beams, a beam quality factor, M2, has come into general use. The mode, TEM01, also known as the “bagel” or “doughnut” mode, is considered to be a superposition of the Hermite-Gaussian TEM10 and TEM01 modes, locked in phase quadrature. In real-world lasers, the Hermite-Gaussian modes predominate since strain, slight misalignment, or contamination on the optics tends to drive the system toward rectangular coordinates. Nonetheless, the Laguerre-Gaussian TEM10 “target” or “bulls-eye” mode is clearly observed in well-aligned gas-ion and helium neon lasers with the appropriate limiting apertures. Fundamental Optics In Laser Modes, we will illustrate the higher-order eigensolutions to the propagation equation, and in The Propagation Constant, M2 will be defined. The section Incorporating M2 into the Propagation Equations defines how non-Gaussian beams propagate in free space and through optical systems. The propagation equation can also be written in cylindrical form in terms of radius (ρ) and angle (Ø). The eigenmodes (EρØ) for this equation are a series of axially symmetric modes, which, for stable resonators, are closely approximated by Laguerre-Gaussian functions, denoted by TEMρØ. For the lowest-order mode, TEM00, the Hermite-Gaussian and Laguerre-Gaussian functions are identical, but for higher-order modes, they differ significantly, as shown in figure 5.14. Optical Specifications For a typical helium neon laser operating in TEM00 mode, M2 <1.1. Ion lasers typically have an M2 factor ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 10 or more. In all cases, the M2 factor affects the characteristics of a laser beam and cannot be neglected in optical designs, and truncation, in general, increases the M2 factor of the beam. x and y directions. In each case, adjacent lobes of the mode are 180 degrees out of phase. Material Properties In the real world, truly Gaussian laser beams are very hard to find. Low-power beams from helium neon lasers can be a close approximation, but the higher the power of the laser is, the more complex the excitation mechanism (e.g., transverse discharges, flash-lamp pumping), and the higher the order of the mode is, the more the beam deviates from the ideal. THE PROPAGATION CONSTANT LASER MODES The propagation of a pure Gaussian beam can be fully specified by either its beam waist diameter or its far-field divergence. In principle, full characterization of a beam can be made by simply measuring the waist diameter, 2w0, or by measuring the diameter, 2w(z), at a known and specified distance (z) from the beam waist, using the equations Gaussian Beam Optics The fundamental TEM00 mode is only one of many transverse modes that satisfy the round-trip propagation criteria described in Gaussian Beam Propagation. Figure 5.13 shows examples of the primary lower-order HermiteGaussian (rectangular) solutions to the propagation equation. Note that the subscripts n and m in the eigenmode TEMnm are correlated to the number of nodes in the Machine Vision Guide TEM00 TEM01 TEM10 TEM11 TEM02 Figure 5.13 Low‑order Hermite‑Gaussian resonator modes Laser Guide marketplace.idexop.com Real Beam Propagation A167 GAUSSIAN BEAM OPTICS Gaussian Beam Optics TEM00 TEM01* very close, as does the beam from a few other gas lasers. However, for most lasers (even those specifying a fundamental TEM00 mode), the output contains some component of higher-order modes that do not propagate according to the formula shown above. The problems are even worse for lasers operating in highorder modes. TEM10 Figure 5.14 Low‑order axisymmetric resonator modes The need for a figure of merit for laser beams that can be used to determine the propagation characteristics of the beam has long been recognized. Specifying the mode is inadequate because, for example, the output of a laser can contain up to 50% higher-order modes and still be considered TEM00. 1/ 2 lz 2 w ( z ) = w0 1 + pw02 and pw2 R ( z ) = z 1 + 0 lz The concept of a dimensionless beam propagation parameter was developed in the early 1970s to meet this need, based on the fact that, for any given laser beam (even those not operating in the TEM00 mode) the product of the beam waist radius (w0) and the far-field divergence (θ) are constant as the beam propagates through an optical system, and the ratio 2 where λ is the wavelength of the laser radiation, and w(z) and R(z) are the beam radius and wavefront radius, respectively, at distance z from the beam waist. In practice, however, this approach is fraught with problems – it is extremely difficult, in many instances, to locate the beam waist; relying on a single-point measurement is inherently inaccurate; and, most important, pure Gaussian laser beams do not exist in the real world. The beam from a well-controlled helium neon laser comes M2 = w0R vR (5.25) w0 v where w0R and θR, the beam waist and far-field divergence of the real beam, respectively, is an accurate indication of the propagation characteristics of the beam. For a true Gaussian beam, M2 = 1. Mv w5 Mw HPEHGGHG *DXVVLDQ PL[HG PRGH v w z z Z5R M>wR@ Mwy z R Figure 5.15 The embedded Gaussian A168 Real Beam Propagation 1-505-298-2550 GAUSSIAN BEAM OPTICS Optical Coatings & Materials 1/ 2 zlM 2 2 wR ( z ) = w0 R 1 + pw0R2 and EMBEDDED GAUSSIAN INCORPORATING M2 INTO THE PROPAGATION EQUATIONS In the previous section we defined the propagation constant M2 l zM 2 p w0 (optimum ) = 1/2 (5.29) The definition for the Rayleigh range remains the same for a real laser beam and becomes zR = pwR (5.30) M l where w0R and θR are the beam waist and far-field divergence of the real beam, respectively. For a pure Gaussian beam, M2 = 1, and the beam-waist beam-divergence product is given by It follows then that for a real laser beam, (5.26) 1/ 2 In a like manner, the lens equation can be modified to incorporate M2. The standard equation becomes ( 1 s + zR / M The propagation equations for a real laser beam are now written as zlM 2 wR ( z ) = w0 R 1 + pw0R2 andand and and lz 2 w ( z ) = w0 1 + p w02 2 1 / 2 lz w ( z ) = w0 1 + p w02 ) 2 2 / (s − f ) + 1 1 = s ′′ f Machine Vision Guide M 2l l > p p pw2 2 R ( z ) = z 1 + 0 l z2 2 pw R ( z ) = z 1 + 0 lz and Gaussian Beam Optics w0 v = l / p Fundamental Optics For M2 = 1, these equations reduce to the Gaussian beam propagation equations. w v M = 0R R w0 v 2 w0R vR = where wR(z) and RR(z) are the 1/e2 intensity radius of the beam and the beam wavefront radius at z, respectively. The equation for w0(optimum) now becomes Optical Specifications A mixed-mode beam that has a waist M (not M ) times larger than the embedded Gaussian will propagate with a divergence M times greater than the embedded Gaussian. Consequently the beam diameter of the mixed-mode beam will always be M times the beam diameter of the embedded Gaussian, but it will have the same radius of curvature and the same Rayleigh range (z = R). 2 (5.28) Material Properties The concept of an “embedded Gaussian,” shown in figure 5.15, is useful as a construct to assist with both theoretical modeling and laboratory measurements. pw 2 2 RR ( z ) = z 1 + 0R 2 zlM (5.31) and the normalized equation transforms to 2 1/ 2 (5.27) marketplace.idexop.com ( s / f ) + ( zR / M 2 f ) 2 / ( s / f − 1) + 1 ( s ′′ / f ) = 1. (5.32) Laser Guide pw 2 2 RR ( z ) = z 1 + 0R 2 zlM 1 Real Beam Propagation A169 GAUSSIAN BEAM OPTICS LENS SELECTION Gaussian Beam Optics The most important relationships that we will use in the process of lens selection for Gaussian-beam optical systems are focused spot radius and beam propagation. FOCUSED SPOT RADIUS wF = waist radius that minimizes the beam radius at distance z, and is obtained by differentiating the previous equation with respect to distance and setting the result equal to zero. Finally, l fM 2 (5.33) pwL where wF is the spot radius at the focal point, and wL is the radius of the collimated beam at the lens. M2 is the quality factor (1.0 for a theoretical Gaussian beam). BEAM PROPAGATION zR = pw l where zR is the Raleigh range. We can also utilize the equation for the approximate on-axis spot size caused by spherical aberration for a plano-convex lens at the infinite conjugate: 1/ 2 zlM 2 2 wR ( z ) = w0 R 1 + pw0R2 2 1 / 2 zlM 2 andwR ( z ) = w0 R 1 + pw0R2 and 2 pw 2 andRR ( z ) = z 1 + 0R 2 zlM 2 pw 2 RR ( z ) = z 1 + 0R 2 zlM spot diameter (3rd -order spherical aberration) = l zM 2 p EXAMPLE: OBTAIN AN 8 MM SPOT AT 80 m 1/2 where w0R is the radius of a real (non-Gaussian) beam at the waist, and wR (z) is the radius of the beam at a distance z from the waist. For M2 = 1, the formulas reduce to that for a Gaussian beam. w0(optimum) is the beam Using the CVI Laser Optics HeNe laser 25 LHR 151, produce a spot 8 mm in diameter at a distance of 80 m, as shown in figure 5.16 The CVI Laser Optics 25 LHR 151 helium neon laser has an output beam radius of 0.4 mm. Assuming a collimated 8 mm 0.8 mm 45 mm 80 m Figure 5.16 Lens spacing adjusted empirically to achieve an 8 mm spot size at 80 m A170 Lens Selection (f /#)3 This formula is for uniform illumination, not a Gaussian intensity profile. However, since it yields a larger value for spot size than actually occurs, its use will provide us with conservative lens choices. Keep in mind that this formula is for spot diameter whereas the Gaussian beam formulas are all stated in terms of spot radius. and w0 (optimum ) = 0.067 f 1-505-298-2550 GUASSIAN BEAM OPTICS Optical Coatings & Materials beam, we use the propagation formula l zM 2 p 1/2 Material Properties w0 (optimum ) = overall length = f1 + f2 and the magnification is given by to determine the spot size at 80 m: magnification = 1/ 2 ( ) = 40.3-mm beam radius or 80.6-mm beam diameter. This is just about exactly a factor of 10 larger than we wanted. We can use the formula for w0 (optimum) to determine the smallest collimated beam diameter we could achieve at a distance of 80 m: 1/ 2 = 4.0 mm. In this case, using a negative value for the magnification will provide us with a Galilean expander. This yields values of f2 = 55.5 mm] and f1 = 45.5 mm. f1 + f2 = 50 mm and f2 = −10. f1 Gaussian Beam Optics This tells us that if we expand the beam by a factor of 10 (4.0 mm/0.4 mm), we can produce a collimated beam 8 mm in diameter, which, if focused at the midpoint (40 m), will again be 8 mm in diameter at a distance of 80 m. This 10x expansion could be accomplished most easily with one of the CVI Laser Optics beam expanders, such as the 09 LBX 003 or 09 LBM 013. However, if there is a space constraint and a need to perform this task with a system that is no longer than 50 mm, this can be accomplished by using catalog components. In order to determine necessary focal lengths for an expander, we need to solve these two equations for the two unknowns. Fundamental Optics 0.6328 × 10 × 80, 000 w0 (optimum ) = p −3 where a negative sign, in the Galilean system, indicates an inverted image (which is unimportant for laser beams). The Keplerian system, with its internal point of focus, allows one to utilize a spatial filter, whereas the Galilean system has the advantage of shorter length for a given magnification. Optical Specifications 2 0.6328 × 10-3 × 80, 000 w(80 m ) = 0.4 1 + p 0.42 f2 f1 Keplerian beam expander f1 f2 Machine Vision Guide Figure 5.17 illustrates the two main types of beam expanders. Galilean beam expander The Keplerian type consists of two positive lenses, which are positioned with their focal points nominally coincident. The Galilean type consists of a negative diverging lens, followed by a positive collimating lens, again positioned with their focal points nominally coincident. In both cases, the overall length of the optical system is given by f1 f2 Figure 5.17 Two main types of beam expanders Laser Guide marketplace.idexop.com Lens Selection A171 GAUSSIAN BEAM OPTICS Gaussian Beam Optics plano-convex lens is acceptable, check the spherical aberration formula. The spot diameter resulting from spherical aberration is [ P The spot diameter resulting from diffraction (2Z ) is [ P Clearly, a plano-convex lens will not be adequate. The next choice would be an achromat, such as the LAO-50.0-18.0. The data in the spot size charts indicate that this lens is probably diffraction limited at this f-number. Our final system would therefore consist of the LDK-5.0-5.5-C spaced about 45 mm from the LAO-50.0-18.0, which would have its flint element facing toward the laser. REFERENCES A. Siegman. Lasers (Sausalito, CA: University Science Books, 1986). S. A. Self. “Focusing of Spherical Gaussian Beams.” Appl. Opt. 22, no. 5 (March 1983): 658. H. Sun. “Thin Lens Equation for a Real Laser Beam with Weak Lens Aperture Truncation.” Opt. Eng. 37, no. 11 (November 1998). R. J. Freiberg, A. S. Halsted. “Properties of Low Order Transverse Modes in Argon Ion Lasers.” Appl. Opt. 8, no. 2 (February 1969): 355-362. W. W. Rigrod. “Isolation of Axi-Symmetric Optical-Resonator Modes.”Appl.Phys. Let. 2, no. 3 (February 1963): 51-53. M. Born, E. Wolf. Principles of Optics Seventh Edition (Cambridge, UK: Cambridge University Press, 1999). A172 Lens Selection 1-505-298-2550 Optical Coatings & Materials MACHINE VISION GUIDE A174 CHOOSING A CUSTOM LENS A176 CREATING A CUSTOM SOLUTION A177 Material Properties INTRODUCTION AND OVERVIEW MACHINE VISION LENS FUNDAMENTALS A178 FREQUENTLY ASKED QUESTIONS Optical Specifications VIDEO CAMERAS FOR MACHINE VISION A192 A194 Fundamental Optics Gaussian Beam Optics Machine Vision Guide Laser Guide marketplace.idexop.com A173 MACHINE VISION GUIDE INTRODUCTION AND OVERVIEW Machine Vision Guide Today, more and more manufacturers are using machine vision technology to improve productivity and reduce costs. Machine vision integrates optical components with computerized control systems to achieve greater productivity from existing automated manufacturing equipment. At the core of this growth in machine vision are manufacturers’ ever-increasing demands for finer control over the quality of manufactured parts. Whether it is the medical industry’s desire to reduce liability, or the consumer market’s need to lower costs, 100% part inspection is becoming the norm. When a single bad part can jeopardize a customer relationship or spur a lawsuit, manufacturers seek to meet quality standards that far exceed the capabilities of older technologies. Machine vision systems come in many forms. Some systems use an analog camera and digitize the image with a frame grabber. An increasing number of systems use digital cameras which, like any other peripheral device, send data directly to PC memory. For some applications, “smart cameras” provide complete vision systems in a single box. Despite their differences, all these systems depend on the front-end optics to provide a high-quality image to the sensor. Machine vision continues to expand into new applications. Camera size and cost have decreased. High-resolution digital cameras are in common use. Smart cameras make entire vision systems available for less than a processor cost only a few years ago. Geometrical pattern matching software has improved the precision and robustness of object location. Each new development leads to new requirements for highperformance optics. Melles Griot, a leader in optics technology for three decades, is a major resource for professionals in many fields who are working with machine vision systems. Our engineers understand the precision and accuracy required for the most critical components of vision systems – the optics. However complex your requirements, Melles Griot has the expertise and the experience to maximize the power of your machine vision system. The image is the only source of information for a machine vision system. The quality of the analysis is dependent on the quality of the image, and the quality of the image is determined by the appropriate choice of optics. Software cannot correct for poor image quality. Nonetheless, optics are the most neglected aspect of a vision system. The lighting and lens must work together to collect the relevant information about the object. The lighting must illuminate each feature, provide good contrast, and minimize confusing artifacts. The lens must resolve features over the entire object and a range of working distances. For alignment and gauging applications, the lens must present the image in a fixed geometry, so that the image location is precisely calibrated to the object’s position in space. The lens can image only the rays launched by the lighting; the lighting must launch only rays that contribute to the desired image. The success of the machine vision system depends on the performance of the optics. A174 Introduction and Overview 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials Material Properties 2SWLFV /LJKWLQJOHQVHV ,QWHUIDFH3URFHVVRU&DUG &ROOHFWVDQGSURFHVVHV FDPHUDGDWD 3URFHVVRU&DUG 'HGLFDWHGLPDJH SURFHVVLQJ 3& Gaussian Beam Optics )UDPH*UDEEHU 'LJLWL]HVYLGHRVLJQDO DQGVWRUHVLW 6PDUW&DPHUD 6HOIFRQWDLQHG YLVLRQV\VWHP Fundamental Optics 'LJLWDO&DPHUD &RQYHUWVLPDJH LQWRGLJLWDOGDWD Optical Specifications $QDORJ&DPHUD &RQYHUWVLPDJHWR DQDORJYLGHRVLJQDO ,QVRPHV\VWHPV 3& Machine Vision Guide Anatomy of a machine vision system Laser Guide marketplace.idexop.com Introduction and Overview A175 MACHINE VISION GUIDE Machine Vision Guide In many cases, the front-end optics for machine vision can be built using off-the-shelf components. Camera, lens, and lighting manufacturers offer a variety of standard products, many of which are specifically designed for machine vision applications. The following pages contain useful information that will help you choose the appropriate components for your front-end optics. The decision to design custom front-end optics, especially using a custom lens, should be considered very carefully, because the time and cost can be significant. However, since the lens is critical to system success and custom lenses can provide performance, size, and stability that are not available from standard commercial lenses, the extra expense is often easily justified. Here are some reasons for choosing a custom lens: X The field of view required to attain the necessary resolution cannot be achieved using a standard lens. X The space available is too small to fit a standard lens. X In addition to precise centering, a change of magnification is required. A clear understanding of the system’s optical requirements is key to choosing a lens. The flowchart below can be used to determine whether a custom lens is needed. X The required depth of field cannot be achieved with a stationary lens. X The area of interest is not readily accessible with a standard lens. Determine basic lens parameters, such as field of view, magnification, and working distance Decide if special lens properties are required: low distortion, telecentricity, special wavelengths, etc. Yes Choose a custom machine vision lens that meets requirements No Determine standard lens type Find the commercial lens that meets basic requirements No Yes Modify system requirements Lens meets system requirements: cost, size, stability, etc. Yes No Consider custom design Test lens with vision application Determining the need for a custom lens A176 Choosing a Custom Lens 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials CREATING A CUSTOM SOLUTION It is often desirable to build a small number of prototypes of a new system before committing to full production. This allows the customer to test the system in operation and perhaps identify some refinements. It also allows Melles Griot to prove that the design package is correct. Furthermore, we develop assembly, test, and calibration procedures for the system. PRODUCTION Melles Griot produces custom systems in quantities ranging from one to hundreds. During production, we continue to work with the customer to be sure that the system is meeting the requirements. Melles Griot recognizes that requirements evolve as products and production techniques improve. We are committed to supporting our customers through this evolution with continuing design refinements. Machine Vision Guide Occasionally, we may need to spend significant engineering effort to determine whether we can build a system that meets the requirements. In these cases we propose a design study, which includes calculations, layouts, and lens designs, as well as research into similar systems and available components. We may also test important concepts in the lab. For example, we may make a mock-up of the lighting configuration and make test images of a customer’s part to prove that it can be adequately illuminated. The output of a design study is a written report, which serves as a basis for customer review and includes a set of specifications and layouts for the design stage. A principal goal of our custom system design process is PROTOTYPE Gaussian Beam Optics DESIGN STUDY During the detailed design phase, we develop a complete drawing package for the optical system. If we have not done a design study, we start by developing complete specifications and a system layout. In either case, we make all glass and metal drawings, assembly drawings, and procedures. At the conclusion of the detailed design, we review the system drawings and final specifications with the customer to ensure the system, as detailed, will meet the requirements. Fundamental Optics The development process begins with the identification of requirements. At this point, Melles Griot takes the widest possible view of the system, including the measurements it is required to make, the environment in which it will work, the systems with which it will interface, the people who will support it, the time available for development, cost targets, and more. Our goal is to develop a solution to the whole problem, not just some of its aspects. It costs far less to deal with conflicts and challenges at this stage than to discover them later. The vision system planning worksheet is one tool we use for this process. We frequently exchange drawings, photographs, or product samples with our customers. In some cases, we visit the customer’s site to better understand the requirements. DETAILED DESIGN Optical Specifications IDENTIFICATION OF REQUIREMENTS to maximize the probability of success. A design study allows the customer to see our designs and rationale before committing to production of hardware. It also allows Melles Griot to plan the prototype hardware carefully, reducing the technical risk, and enables us to quote a lower price for the finished product. Material Properties Successful development of a custom machine vision system requires a partnership between customer and vendor. The customer knows his or her manufacturing process and requirements. Melles Griot knows how to build production-ready optics. Depending on the application and on customer needs, developing a custom machine vision system includes some or all of the following stages: identification of requirements, a design study, the detailed design, creation of a prototype, and, finally, production. As the following pages demonstrate, Melles Griot works with the customer throughout the development process to achieve the best fit between the optics and the customer’s requirements. Laser Guide marketplace.idexop.com Creating a Custom Solution A177 MACHINE VISION GUIDE MACHINE VISION LENS FUNDAMENTALS Machine Vision Guide All of the information collected by a machine vision system comes through the lens. The correct choice of lens can reduce image-processing requirements and improve system performance and robustness. Software cannot correct the effects of a poorly chosen lens. This primer provides the technical and practical information needed to choose a lens for a machine vision system. First we review design principles, providing simple formulas that form the basis of further calculations. From models, we proceed to a discussion of real-world lenses and practical parameters. A discussion of special lenses completes this section. FIRST-ORDER DESIGN THEORY To establish an understanding of theoretical principles, we will first review a few basic lens definitions and parameters. We then examine the thin-lens model. The thin-lens model describes a lens with no limitations – one that can be used at any magnification and work at any conjugate. However, since real lenses do have limitations, the thin-lens model does not provide the complete picture. Following this theoretical discussion, we will examine real lenses and their parameters, as well as special lenses. CAMERA FORMAT The camera format defines the dimensions of the image sensor. Lenses, by design, provide images over a limited area. Be sure the lens covers an area as large or larger than the camera format. FIELD OF VIEW The field of view (FOV) is the object area that is imaged by the lens onto the image sensor. It must cover all features to be measured, with additional tolerance for alignment errors. It is also good practice to allow some margin (e.g., 10%) for uncertainties in lens magnification. Features within the FOV must appear large enough to be measured. This minimum feature size depends on the application. As an estimate, each feature must have three pixels across its width, and three pixels between features. If there are more than 100 features across a standard camera field, consider using multiple cameras. A178 Machine Vision Lens Fundamentals MAGNIFICATION The required magnification (m) is m= Wcamera WFOV 6.1) where Wcamera is the width of the camera sensor and WFOV is the width of the FOV. Note that the required magnification depends on the camera sensor size. WORKING DISTANCE The working distance is the distance from the front of the lens to the object. In machine vision applications, this space is often needed for equipment or access. In general, a lens that provides a long working distance will be larger and more expensive than one that provides a shorter working distance. The back working distance is the distance from the rear-most lens surface to the sensor. THIN-LENS MODEL To understand machine vision lenses, we start with the thin-lens model. It is not an exact description of any real lens but illustrates lens principles. It also provides terms with which to discuss lens performance. A ray, called the chief ray, follows a straight line from a point on the object, through the center of the lens, to the corresponding point on the image (figure 6.1). The lens causes all other rays that come from this same object point and that reach the lens to meet at the same image point as the chief ray. Those rays which pass through the edge of the lens are called marginal rays. The distance from the object plane to the lens (s1) is called the object conjugate. Likewise, the distance from the lens to the sensor plane (s2) is called the image conjugate. These conjugates are related by: 1 1 1 = + f s1 s2 . (6.2) 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials object conjugate (s 1) chief ray marginal ray cone half-angle for infinite conjugates (vcone ): f/# infinite cone half-angle for 1:1 conjugates: f/# image Material Properties object plane cone half-angle for 1:1 conjugates: lens f/# object aperture (f ) image conjugate ( s 2) focal length ( f ) object conjugate ( s 1) Figure 6.1 Thin‑lens model FOCAL LENGTH If we let the object conjugate get very large, we see . (6.3) m= s2 . (6.4) s1 APPLICATION NOTE Thin Lens Example We need a magnification of 0.5x, with a working distance of 50 mm. We want to find the correct lens focal length and total system length (TSL). Substituting equation 6.4 into equation 6.2 and solving for f, we get: Gaussian Beam Optics In other words, the focal length is the distance between the lens and the sensor plane when the object is at infinity. For photographic lenses, the objects are usually far away, so all images are formed in nearly the same plane, one focal length behind the lens. From geometry, we can see that. Figure 6.2 f/number (f/#) Fundamental Optics 1 1 ≈ ⇒ s2 ≈ f f s2 image conjugate ( s 2) Optical Specifications camera focal length ( f ) plane so F-NUMBER The f-number (f/#) describes the cone angle of the rays that form an image (figure 6.2). The f-number of a lens determines three important parameters: X The brightness of the image X The depth of field X The resolution of the lens Machine Vision Guide The magnification is the ratio of the image to the object conjugates. If the focal length of a lens increases for a specified magnification, both object and image conjugates increase by the same ratio. Therefore, we need a lens with focal length of approximately 17 mm. The total system length is approximately 75 mm. Laser Guide marketplace.idexop.com Machine Vision Lens Fundamentals A179 MACHINE VISION GUIDE Machine Vision Guide For photographic lenses, where the object is far away, the f-number is the ratio of the focal length of the lens to the diameter of the aperture. The larger the aperture is, the larger the cone angle and the smaller the f-number will be. A lens with a small f-number (large aperture) is said to be “fast” because it gathers more light, and photographic exposure times are shorter. A wellcorrected fast lens forms a high-resolution image but with a small depth of field. A lens with a large f-number is said to be “slow.” It requires more light but has a larger depth of field. If the lens is very slow, its resolution may be limited by diffraction effects. In this case, the image is blurred even at best focus. NA is related to f-number by these exact relationships: 1 NA = sin arctan 2 × f/# ~(6.8) 1 . f/# = 2 × tan arcsin ( NA ) For NA < 0.25 (f-number >2), these simplify to: NA ≅ (6.9z) f/# ≅ The f-number printed on a photographic lens is the infinite conjugate f-number, defined as f/#∞ = f f (6.5) 1 2 × f/#∞ . (6.6) This infinite conjugate f-number is applicable only when the lens is imaging an object far away. For machine vision applications, the object is usually close, and the cone half angle is calculated from the working f-number. NUMERICAL APERTURE For lenses designed to work at magnifications greater than 1 (for example, microscope objectives), the cone halfangle on the object side is used as the performance measure. By convention, this angle is given as a numerical aperture (NA). The NA (figure 6.3) is given by NA = sin( vcone ) .(6.7) A180 Machine Vision Lens Fundamentals 1 2 × NA . APPLICATION NOTE where f is the focal length of the lens and Ø is the diameter of the lens aperture. When the lens is forming an image of a distant object, the cone half angle (CHA) of the rays forming the image is vcone = arctan 1 2 × f/# Working f-number In machine vision, the working f‑number describes lens performance: I LPDJH I REMHFW V V where s2 and s1 are the image and object conjugates, respectively. f/#image is called the working f‑number in image space, or simply the image‑side f‑number. Similarly, f/#object is the object‑side f‑number. For close objects, f/#image is larger than f/#infinity, so the lens is “slower” than the number given on the barrel. For example, a lens shown as f/4 on its barrel (i.e., an f‑number of 4) will act like an f/8 lens when used at a magnification of 1. The object‑side f‑number determines the depth of field. It is given by I REMHFW m I LPDJH . 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials microscope objective object cone half-angle: sin(vcone) = NA image conjugate ( s 2) REAL-WORLD LENSES entrance pupil real-world lens exit pupil Machine Vision Guide For many lenses, the entrance and exit pupils are located near each other and within the physical lens. The exit pupil may be in front of or behind the entrance pupil. For certain special lens types, the pupils are deliberately placed far from their “natural” positions. For example, a telephoto lens has its exit pupil far in front of its entrance pupil (figure 6.5). In this way, a long-focal-length lens fits into a short package. A telecentric lens has its entrance pupil at infinity, well behind its exit pupil (figure 6.6). STANDARD LENSES Commercial lenses, produced in high volume, are by far the best value in terms of performance for the price. Finding a suitable stock lens is the most cost-effective Gaussian Beam Optics In object space, we think of the real-world lens as a thin lens located at the entrance pupil. The entrance pupil is generally located within the physical lens, but not always. Wherever it is located, light rays in object space proceed in straight lines to the entrance pupil. The effects of any elements in front of this position are taken into account when the entrance pupil position is calculated. In the same way, we think of the real-world lens as a thin lens located at the exit pupil in image space. A lens that is corrected for one set of conditions may show significant aberrations when used under a different set of conditions. For example, a surveillance lens with a magnification of 1/10 is corrected for distant objects. By using extension tubes, the image conjugate of the lens can be extended so that the lens forms an image at a magnification of 1. This image may, however, show significant aberrations because the lens was not corrected to work at these conjugates. Fundamental Optics THICK-LENS MODEL The thin-lens model treats a lens as a plane with zero thickness. To model a real-world lens, we divide this thin-lens plane into two planes (figure 6.4). These planes contain the entrance and the exit pupils of the lens. Everything in front of the entrance pupil is said to be in object space. Everything behind the exit pupil is said to be in image space. How light gets from the entrance pupil to the exit pupil is not considered in this model. The job of the lens designer is to choose glasses, curvatures, and thicknesses for the lens’ elements that keep its overall aberrations within acceptable limits. Such a lens is said to be well corrected. It is impossible to design a lens that is well corrected for all conjugates, FOVs, and wavelengths. The lens designer works to correct the lens over the small range of operating conditions at which the lens must function. The smaller the range is, the simpler the design can be. Optical Specifications Figure 6.3 Numerical aperture (NA) Material Properties object conjugate ( s 1) ABERRATIONS If real lenses followed first- order theory, lens design would be easy. Unfortunately, it is difficult to make a real lens approximate this behavior. Diffraction sets a lower limit on image spot size. The differences between ideal “diffraction-limited” behavior and real-lens behavior are called aberrations. object space image space Figure 6.4 Thick-lens model Laser Guide marketplace.idexop.com Machine Vision Lens Fundamentals A181 MACHINE VISION GUIDE Machine Vision Guide entrance pupil design and many thousands of dollars to manufacture in small quantities. It is always best to consider commercial lens options before initiating a custom lens design. real-world lens exit pupil REAL LENS PARAMETERS object space image space Figure 6.5 Telephoto lens real-world lens ∞ entrance pupil exit pupil image space object space Figure 6.6 Telecentric lens solution to a machine vision problem. The accompanying table lists various lens types and their range of operating conditions. Commercial lenses incorporate design and manufacturing techniques that are not available in custom designs. For example, a lens for a 35 mm, singlelens reflex (SLR) camera that costs one hundred dollars at the local camera store would cost ten thousand dollars to RESOLUTION Resolution is the ability of an optical system to distinguish between two features that are close together. For example, if a lens images a row of pins on an electrical connector, it must have sufficient resolution to see each pin as separate from its neighbors. A lens imaging a lot code on a pharmaceutical bottle must have sufficient resolution to distinguish one character from another. Resolution is also required to make sharp images of an edge. A lens with high resolution will show an edge transition in fewer pixels than a lens with low resolution. There are many different definitions of lens resolution. They differ by what type of test object is measured (points, bars, sine patterns, or other objects) and by the criteria for determining when two objects are “resolved.” A practical measurement for machine vision uses threebar targets of various spatial frequencies. A chromeon-glass USAF-1951 target is a good test object. If the contrast between bar and space is greater than 20%, the bars are considered to be resolved. Resolution does not determine the dimensional accuracy to which objects can be measured. The position of a large object can be determined to within a fraction of a resolution spot under suitable conditions. Many Common Commercial Lens Types A182 Lens Type Magnification Image Format Object FOV (mm) Focal Length (mm) Working f-number Range (Object Side) Surveillance <0.1 1.0” CCD format Large 2 – 50 >20 (adjustable) Standard Machine Vision .05 – 5 2/3” CCD 2 – 200 25 – 75 >4 (adjustable) Telecentric Machine Vision .07 – 5 2/3” CCD 2 – 170 N/A >6 (adjustable) F-Mount Lenses <1 45 mm Large 35 – 100 >4 (adjustable) Large/Medium Format Photographic <1 80 mm Large 50 – 250 >4 (adjustable) Photographic Enlarger 2 – 20 500 mm 50 40 – 150 >4 (adjustable) Microscope 5 – 100 Requires additional lens <2 5 – 40 0.1 – 0.95 NA (fixed) Machine Vision Lens Fundamentals 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials Factors other than lens resolution can affect contrast. Stray light from the environment, as well as glare from uncoated or poorly polished optics, reduce contrast. The angles of the lens and of the illumination have a great effect on contrast. The contrast of some objects is dependent on the color of the illumination. Optical Specifications DIFFRACTION Diffraction limits the resolution possible with any lens. In most machine vision calculations, we consider light as traveling in straight lines (rays) from object points to image points. In reality, diffraction spreads each image point to a spot whose size depends on the f-number of the lens and the wavelength of the light. This spot pattern is called an Airy disk. Its diameter is given by feature, and “dark” is the gray level of the darkest pixel. A contrast of 1 means modulation from full light to full dark; a contrast of 0 means the image is gray with no features. Finer (higher spatial frequency) features are imaged with less contrast than larger features. A highresolution lens not only resolves finer features, but it also generally images medium-scale features at higher contrast. A high-contrast image appears “sharper” than a lower contrast image, even at the same resolution. Material Properties vision systems determine positions to one-quarter pixel. On the other hand, if the lens has distortion, or if its magnification is not known accurately, then the measured position of a feature may be in error by many resolution spot widths. DAiry = 2.44 × l × f/#(6.10) contrast = light − dark . (6.11) light + dark Here, “light” is the gray level of the brightest pixel of a DOF = 2 × f/#object × blur(6.12) blur diameter object lens f/number object Machine Vision Guide CONTRAST Contrast is the amount of difference between light and dark features in an image. Contrast (also called modulation) is defined by: In general, the geometrical DOF (figure 6.7) is given by Gaussian Beam Optics For example, a typical CCD camera has pixels that are 10 µm square. To form a diffraction-limited spot of this diameter, the working f-number on the image side should be approxiamtely f/10. An f/22 lens forms an image spot larger than a pixel. Its image therefore appears less sharp than that of the f/10 image. An f/2 lens image will not appear sharper than an f/10 image, since the camera pixel size limits the resolution. In this case, the system is said to be detector-limited. Fundamental Optics where DAiry is the diameter of the inner bright spot, λ is the wavelength of light, and the f-number is the image side f-number. Since the wavelength of visible light is ~ 0.5 µm, this means the diameter of the diffractionlimited spot (in mm) is approximately equal to the working f-number. DEPTH OF FIELD The depth of field (DOF) is the range of lens-to-object distances over which the image will be in sharp focus. The definition of “sharp” focus depends on the size of the smallest features of interest. Because this size varies between applications, DOF is necessarily subjective. If very fine features are important, the DOF will be small. If only larger features are important, so that more blur is tolerable, the DOF can be larger. The system engineer must choose the allowable blur for each application. depth of field Figure 6.7 Depth of field Laser Guide marketplace.idexop.com Machine Vision Lens Fundamentals A183 MACHINE VISION GUIDE Machine Vision Guide where blur is the diameter of the allowable blur in object space. A larger blur or larger f-number increases the DOF. To find the DOF for detector-limited resolution, we choose the diffraction spot size created by the lens to be one pixel width in diameter, and the geometric blur caused by defocus also to be one pixel width in diameter. With these assumptions: Wpixel (mm ) DOF (mm ) = 2 × m 2 . (6.13) Here, we set the image side f-number of the lens equal to the pixel width in micrometers. Wpixel is the pixel width in micrometers; m is the lens magnification. Thus, for a camera with 10 µm pixels, operating at 0.5x magnification, with an image side f-number of f/10, the DOF is 800 µm, or 0.8 mm. These assumptions are very conservative. Using a higher f-number reduces the resolution of the lens slightly but greatly increases the DOF. For example, with the lens operating at f/22 and allowing a geometric blur of two pixel widths, the DOF is 3.2 mm, which is four times larger. This is a better estimate if the important image features are larger than two pixels (40 µm). The choice of f-number and allowable blur depends on the requirements of the particular application. objects, or objects whose distance from the lens is not known precisely. A telecentric lens views the whole field from the same perspective angle. Thus, deep, round holes look round over the entire field instead of appearing elliptical near the edge of the field. Objects at the bottom of deep holes are visible throughout the field. The degree of telecentricity is measured by the chief ray angle in the corner of the field (figure 6.8). In machine vision, a standard commercial lens may have chief ray angles of 10º or more. Telecentric lenses have chief ray angles of less than 0.5º, in fact, some telecentric lenses have chief ray angles of less than 0.1º. Telecentricity is a measure of the angle of the chief ray in object space and does not affect the DOF. DOF is determined by the angles of the marginal rays. Chief ray and marginal ray angles are independent of each other. FKLHIUD\ WHOHFHQWULF DQJOH REMHFW TELECENTRICITY Telecentricity determines the amount that magnification changes with object distance. Standard lenses produce images with higher magnification when the object is closer to the lens. We experience this with our eyes. A hand held up near your face looks larger than it does when it is moved farther away. For the same field size, a longer focal length shows less magnification change than a short focal length lens. A telecentric lens acts as if it has an infinite focal length. Magnification is independent of object distance. An object moved from far away to near the lens goes into and out of sharp focus, but its image size is constant. This property is very important for gauging three-dimensional A184 Machine Vision Lens Fundamentals REMHFW DSSHDUVODUJHU ZKHQFORVHUWROHQV LPDJH DFRQYHQWLRQDOFDPHUDOHQV FKLHIUD\ WHOHFHQWULF DQJOHa REMHFW ± QR VL]H FKDQJH LPDJH REMHFW EWHOHFHQWULFOHQV Figure 6.8 Telecentricity: (a) conventional camera (b) telecentric lens 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials DISTORTION In optics, distortion is a particular lens aberration which object lens: chief ray angle MICROSCOPE OBJECTIVES Figure 6.9 Gauging depth of field pincushion distortion Figure 6.10 Pincushion distortion CHOOSING AN OBJECTIVE The most important parameter for choosing a microscope objective is its NA. The larger the NA, the higher the resolving power, which means that the objective can distinguish closely spaced features from each other. The NA is related to the magnification; a higher magnification objective usually has a larger NA. The objective provides its specified magnification when used in a microscope with the proper tube length, or with the proper decollimating lens. The objective can also be used at different magnifications; the specified magnification provides an approximate guide. Both NA and magnification are usually printed on the barrel of the objective. An objective with a larger NA gathers Machine Vision Guide object SPECTRAL RANGE Most machine vision lenses are color corrected throughout the visible range. Filters that narrow the spectral range to a single color sometimes improve lens resolution. CCD cameras are inherently sensitive to nearinfrared (NIR) light. In most cases, an NIR filter should be included in the system to reduce this sensitivity. In many cameras, the NIR filters are built in. Gaussian Beam Optics gauging depth of field Lens distortion errors are often small enough to ignore. Because distortion is fixed, these errors can also be removed by software calibration. Lenses designed to have low distortion are available. Fundamental Optics gauging position accuracy Distortion is generally specified in relative terms. A lens that exhibits 2% distortion over a given field will image a point in the corner of its field 2% too far from the optical axis. If this distance should be 400 pixels, it will be measured as 408 pixels. Optical Specifications GAUGING DEPTH OF FIELD The gauging depth of field (GDOF) is the range of distances over which the object can be gauged to a given accuracy (figure 6.9). A change in object distance changes the image magnification and therefore the measured lateral position of the object. The GDOF describes how precisely the object distance must be controlled to maintain a given measurement accuracy. Telecentric lenses provide larger GDOFs than do conventional lenses. causes objects to be imaged farther or closer to the optical axis than for a perfect image. It is a property of the lens design and not the result of manufacturing errors. Most machine vision lenses have a small amount of pincushion distortion (figure 6.10). Because relative distortion increases as the square of the field, it is important to specify the field over which field distortion is measured. Material Properties The objective element of a telecentric lens must be larger than the FOV. The lens must “look straight down” on all portions of the field. Telecentric lenses designed for very large fields are thus large and expensive. Most telecentric lenses cover fields of less than 150 mm in diameter. Laser Guide marketplace.idexop.com Machine Vision Lens Fundamentals A185 MACHINE VISION GUIDE Machine Vision Guide more light but provides a smaller DOF, shorter working distance, and higher cost than an objective with a smaller NA. These tradeoffs are crucial to the success of the application, the objective NA must be chosen carefully. The FOV is the sensor size divided by the magnification. The magnification (and FOV) can be adjusted by changing tube length or the focal length of the decollimating lens. Using a magnification greatly different from the one printed on the objective generally results in a poorly optimized system. objective mounting flange to the object is the same for each objective in the family. On a microscope, this means the the objective (and magnification) can be switched without a large refocus motion. Microscope objectives have a small working distance (WD), the distance from the tip of the objective barrel to the object. This is a problem in machine vision, where there are often fixtures that must fit between the objective and the object. For those applications, there are objectives with longer working distance, called LWD or ELWD lenses. These objectives are larger and more expensive than standard objectives. TYPES OF OBJECTIVES Objectives are classified into groups depending on how well they are corrected for the dominant aberrations: chromatic aberration (color), spherical aberration, and field curvature. The simplest objectives (achromats) are corrected for color in the red and blue and for spherical aberration in the green. More complex objectives (apochromats) are color corrected in the red, yellow, and blue and corrected for spherical aberration at two to three different wavelengths. For applications that require good image quality across a wide FOV, “plan” objectives (plan achromats and plan apochromats) are also corrected for field curvature. Plan objectives generally have longer working distances than simple designs. There are several different and incompatible standards for microscope mounting threads (DIN, JIS, RMS, and others). It is usually not possible to adapt from one thread to another. Within a single family, objectives are usually “parfocal”, which means the distance from the Each objective is designed to be used with a specific type of microscope. Biological objectives are corrected to view the object through a glass coverslip. If a biological objective, particularly one with a large NA, is used without a coverslip, the image will not be sharp. Objective Type Designation A186 Designation Meaning Application Achro, Achromat Color corrected at 2 colors, with nominal spherical aberration correction Low cost, less demanding applications Fluor, Fl, Fluar, Neofluar, Fluotar Color and spherical aberration corrected with fluorite element Intermediate between achro and apo perfor‑ mance Apo, Apochromat Color corrected at 3 or more colors, with superior spherical aberration correction Best polychromatic imaging EF, Achroplan Extended field (but less than plan) Wider field than achroplan Plan, Pl, Achroplan Corrected for field curvature; wide field of view longer working distance Sharp images across the field of view ELWD SLWD ULWD Extra-long working distance Super-long working distance Ultra-long working distance Plan objectives with greatly increased working distances I, Iris, W/Iris Includes an iris to adjust numerical aperture Useful to adjust depth of field and resolution Machine Vision Lens Fundamentals 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials Similarly, nonbiological objectives will not function optimally if there is glass between the objective and the object. flens . (6.14) fattachment TELECONVERTERS (EXTENDERS) Teleconverters are short relay optics that fit between the lens and the camera and increase the effective lens focal length. They are usually available at 1.5x and 2x power. The penalties for their use are increased image-side f-number (by the power factor) and increased distortion. REVERSE MOUNTING For magnifications greater than 1, a camera lens can be used in reverse, with the object held at the usual camera plane and the camera in the usual object plane. In this case, the object distance will be short, whereas the lensto-camera distance is long. Adaptors are available to hold camera lenses in this orientation. Gaussian Beam Optics Machine Vision Guide ZOOM LENSES Zoom lenses have focal lengths that are adjustable over some range. They are useful for prototypes in which the focal length requirement has not yet been determined. They can be set at focal lengths between those available with fixed lenses. Zoom lenses are larger, less robust, more expensive, and have smaller apertures than similar fixed-focal-length lenses. Also, they frequently have more distortion. m= Fundamental Optics SPECIAL LENSES CLOSE-UP LENSES Close-up attachment lenses reduce the object distance of a standard lens. The nominal magnification of a lens with a close-focusing attachment is Optical Specifications Many special-purpose objectives are available. Some are color corrected for wavelengths in the infrared or ultraviolet regions. Low-fluorescence objectives are available for ultraviolet fluorescence applications. Strain-free objectives are used for applications where the polarization of the image light must be maintained. Material Properties Older microscope objectives (before 1980) were designed to form an image at a given distance (the tube length) behind the objective flange. This distance varied between 160 mm and 210 mm depending on the manufacturer and the application. At the proper tube length, the objectives formed images at their nominal magnifications. Modern microscope objects are “infinity corrected.” They are optimized to provide collimated light on their image side. A separate decollimating or tube lens then forms the image. This design gives microscope manufacturers flexibility to insert lighting and beamsplitters in the collimated space behind the objective. The proper focal length tube lens is required to form an image at the objective nominal magnification. any object distance (see Telecentricity). They are useful for precision gauging or application where a constant perspective angle across the field is desirable. Their object distance is generally less than that of standard lenses. The magnification of a telecentric lens is fixed by its design. Because the first element must be as large as the field width, telecentric lenses tend to be larger and more expensive than standard lenses. MACRO LENSES A camera lens optimized to work at magnifications near 1 is called a macro lens. A macro lens provides better image quality than a standard camera lens used with extension tubes. TELECENTRIC LENSES Telecentric lenses provide constant magnification for Laser Guide marketplace.idexop.com Machine Vision Lens Fundamentals A187 MACHINE VISION GUIDE MACHINE VISION LIGHTING FUNDAMENTALS Machine Vision Guide There are well-established design rules for choosing a lens. There are fewer such rules for lighting, yet proper lighting is as important as using the correct lens to form useful images. For a feature to appear in an image, light must come from the illuminator, reflect off the object, and be collected by the lens (figure 6.11). If the light to populate a given ray is not available from the illuminator, that ray will not be part of the image. In our daily experience, we use light from the environment to see. In machine vision applications, light from the environment is undesirable, because it may change when we least expect it. We need to provide controlled light in a manner that accentuates features we care about and minimizes distracting features. Vision lighting and imaging optics are best designed together as a system. The illuminator should launch all rays that can be collected by the lens as part of an image. At the same time, it should not launch rays that will never be part of an image (e.g., those rays that fall outside the FOV of the lens). These extra rays only contribute to glare, which reduces image contrast. Unless the lighting and imaging optics are designed together, it is difficult to achieve a match between them. TYPES OF REFLECTION SPECULAR REFLECTIONS Specular reflections are bright but unreliable. They are bright because the intensity of the reflection is comparable to the intensity of the light source. In many cases, a specular reflection saturates the camera. Specular reflections are unreliable because a small change in the angle between the illuminator, the object, and the lens may cause the specular reflection to disappear completely. Unless these angles are well controlled, it is best to avoid depending on specular reflections. The best method for lighting specular parts is with diffuse lighting (figure 6.13). The large illumination solid angle means that the image remains almost constant as the reflection angle changes. DIFFUSE REFLECTIONS Diffuse reflections are dim but stable. The intensity of the reflection is reduced from that of the source by a factor ranging from 10 to 1000. The reflected intensity changes slowly with the angle (figure 6.14). Diffuse surfaces can be lit successfully with either diffuse or point-like illuminators. Other considerations, such as specular elements on the object or the influence of shadows, determine the best approach. lens and camera light source Objects reflect light in two ways. In specular reflection, light from each incoming ray reflects in a single direction (figure 6.12). A tinned circuit board trace or a mirror exhibits specular reflection. In diffuse reflection, light from each incoming ray is scattered over a range of outgoing angles. A piece of copier paper is a diffuse reflector. In reality, objects exhibit the whole range of behaviors between the specular and diffuse extremes. A machined metal surface scatters light over a small range of angles, and it scatters differently in directions parallel and perpendicular to the turning marks. Paper exhibits some specular properties, as anyone who has ever tried to read with a high-intensity lamp can attest. Many objects have components that reflect differently. An electrical connector includes both shiny (specular) metal pins and dull (diffuse) plastic housing parts. object Figure 6.11 Lighting an object specular reflection diffuse reflection Figure 6.12 Types of reflection A188 Machine Vision Lighting Fundamentals 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials LIGHTING TECHNIQUES POINT-LIKE LIGHTING Point-like lighting is generally easy to implement because the illuminators are small and can be located at a distance from the object. Incandescent lamps, optical diffuse lighting object location Figure 6.15 Solid angle fiber bundles, ring lights, and LEDs are examples of point-like illuminators. Some, like fiber optic bundles, are directional, so light can be directed onto the object from a distance. Point-like illumination provides high intensity and light efficiency. It is good for creating sharp image edges, casting shadows, and accenting surface features. Their small size makes the illuminators easier to mount and integrate than diffuse sources. Fundamental Optics lens and camera sphere with unit radius: total area 4p Optical Specifications LIGHTING SOLID ANGLE: POINT OR DIFFUSE Lighting solid angle is the area of a unit sphere, centered on the object, that the illumination occupies (figure 6.15). Just as angles are measured in radians, with 2π radians in a full circle, solid angles are measured in steradians, with 4π steradians in a full sphere. Illumination from a small solid angle is called point-like; illumination from a large solid angle is called diffuse. solid angle: area of unit sphere through which light enters Material Properties The basic approach to lighting for a particular application is easily determined. It is a function of the type of object and the features to be measured. The more detailed lighting design builds on this basic technique. For examples, see the accompanying table. specular object Figure 6.13 Specular objects viewed with diffuse lighting lens and camera light source Gaussian Beam Optics The same shadows and surface features that are useful in some applications can be distractions in others. With specular objects, point-like illumination creates very bright reflections which may saturate video cameras. Away from these reflections, specular objects appear dark. Machine Vision Guide DIFFUSE LIGHTING By definition, diffuse lighting must cover a large solid angle around the object. Fluorescent lamps (both straight tubes and ring lights) are inherently diffuse. Diffusers in front of point-like sources make them more diffuse. Diffuse illumination of specular surfaces allows imaging without bright reflections. Surface texture is minimized, and there is less sensitivity to surface angles on parts. diffuse object Figure 6.14 Diffuse objects illuminated with point-like source Diffuse illumination can be difficult to implement, because the illuminator must surround much of the object. For example, when reading characters stamped on textured foil, sources with solid angles approaching Laser Guide marketplace.idexop.com Machine Vision Lighting Fundamentals A189 MACHINE VISION GUIDE Machine Vision Guide Comparison Table for Different Lighting Techniques Illumination Solid Angle Direction Advantages Disadvantages Direct Front Illumination Incandescent lamp or fiber bundle illuminates object from the top Point Front Easy to implement; good for casting shadows; fiber-optic delivery available in many configurations May create unwanted shadows; illumination is uneven Coaxial Lighting Illumination from the precise direction of the imaging lens, either through the lens or with a beamsplitter in front of the lens Point Front Eliminates shadows; uniform across field of view Complicated to implement; intense relfection from specular surfaces Diffuse Front Illumination Fluorescent lamp fiber illuminator with diffuser, or incandescent lamp with diffuser; illuminates object from the front Diffuse Front Soft; relatively nondirectional; reduces glare on specular surfaces; relatively easy to implement Illuminator relatively large; edges of parts may be fuzzy; low contrast on monocolor parts Light Tent Diffuse illuminator surrounds object Diffuse Front Eliminates glare; eliminates shadows Must surround object; illuminator is large; can be costly Dark-Field Illumination Point-like source at near right angle to object surface Point Side Illuminates defects; provides a high-contrast image in some applications Does not illuminate flat smooth surfaces Diffuse Backlighting Source with diffuser behind object Diffuse Back Easy to implement; creates silhouette of part; very highcontrast image; low cost Edges of parts may be fuzzy; must have space available behind object for illuminator Collimated Backlighting Point source with collimating lens behind object Point Back Produces sharp edges for gauging object for illuminator Must have space available behind Polarized Front Illumination Point-like or diffuse front illumination; polarizer on illuminator; analyzer in front of imaging lens Point or Diffuse Front Reduces glare Reduces light to lens Polarized Backlighting Diffuse backlight; polarizer on illuminator; analyzer in front of imaging lens Diffuse Back Highlights birefringent defects; relatively easy to implement Only useful for birefringent defects; edges of parts may be fuzzy; must have space available behind object for illuminator 2π steradians are required. These “light tents” are difficult to construct effectively because the lens, camera, and handling equipment must be mounted around the illuminator. Diffuse illumination can also cause blurred edges in images. In general, a diffuse illuminator is more complex than a point-like illuminator. LIGHTING DIRECTION – BRIGHT FIELD In bright-field illumination, the light comes in approximately perpendicular to the object surface (figure 6.16). The whole object appears bright, with features displayed as a continuum of gray levels. Normal room lighting is bright-field illumination. This sort of illumination is used for most general-vision applications. A190 Machine Vision Lighting Fundamentals An important special case of bright-field illumination is coaxial illumination. Here, the object is illuminated from precisely the direction of the imaging lens. This requires a beamsplitter, either within or in front of the imaging lens. Coaxial illumination is used to inspect features on flat, specular surfaces, to image within deep features, and to eliminate shadows. LIGHTING DIRECTION – DARK FIELD If the object is illuminated from a point parallel to its surface, texture and other high-angle features appear bright while most of the object appears dark. This low-angle illumination is called dark-field illumination. Dark-field illumination is useful for imaging surface contamination, scratches, and other small raised features. 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials coaxial bright field object Material Properties because they are inherently two dimensional and binary. Flexible parts feeders frequently use backlit images to determine the orientation of mechanical parts to be picked up by a robot for assembly. lens and camera dark field backlight Figure 6.16 Lighting angles LIGHTING DIRECTION – BACKLIGHT Backlight illumination means the illuminator is behind the object. It can be either point-like or diffuse. Pointlike lighting, projected through a collimator whose axis is parallel to the lens axis, is similar to coaxial lighting. There are two distinct uses of backlighting: viewing translucent objects in transmission and silhouetting opaque objects. Optical Specifications LIGHTING COLOR Most machine vision applications use unfiltered light; however, in some cases, monochromatic illumination provides better feature contrast. A narrow spectrum also reduces the effect of any chromatic aberration in the imaging lens and therefore provides improved resolution. Filtering does, however, reduce the amount of illumination and may be unsuitable for applications in which there is a shortage of light. Sheet glass is an example of a translucent product that is inspected by using backlight. Point-like lighting that is not coaxial with the lens highlights surface defects (scratches, gouges) as well as internal defects (bubbles, inclusions). Fundamental Optics POLARIZATION Polarized illumination is used to reduce glare from specular surfaces. A polarizer is placed in front of the illuminator, and another polarizer (called the analyzer), whose polarization axis is perpendicular to that of the first, is placed in front of the imaging lens. Light that is specularly reflected from the object retains its polarization direction and is therefore blocked by the analyzer. Light scattered from the object is randomly polarized and is passed by the analyzer. Backlighting is more commonly used to silhouette opaque parts. Silhouettes are easy images to process Gaussian Beam Optics LIGHT SOURCES Several types of light sources and illuminators are available for machine vision applications; their properties are summarized in the accompanying table. Advantages and Disadvantages of Different Light Sources Advantages Disadvantages LED Array of light-emitting diodes Can form many configurations within the arrays; single color source can be useful in some applications; can strobe LEDs at high power and speed Some features hard to see with single color source; large array required to light large area Fiber-Optic Illuminators Incandescent lamp in housing; light carried by optical fiber bundle to application Fiber bundles available in many configura‑ tions; heat and electrical power remote from application; easy access for lamp replacement Incandescent lamp has low efficiency, especially for blue light Fluorescent High-frequency tube or ring lamp Diffuse source; wide or narrow spectral range available; lamps are efficient and long lived Limited range of configurations; intensity control not available on some lamps Strobe Xenon arc strobe lamp with either direct or fiber bundle light delivery Freezes rapidly moving parts; high peak illumination intensity Requires precise timing of light source and image capture electronics; may require eye protection for persons working near the application Machine Vision Guide Light Source Type Laser Guide marketplace.idexop.com Machine Vision Lighting Fundamentals A191 MACHINE VISION GUIDE VIDEO CAMERAS FOR MACHINE VISION Machine Vision Guide Many cameras are available for machine vision. They incorporate different sensors and different interface electronics, and they come in many sizes. Together, the camera and lens determine the FOV, resolution, and other properties of the image. Many cameras are designed specifically for machine vision applications. This section outlines key issues that should be addressed when choosing a camera lens. CAMERA TYPES SENSORS Most machine vision cameras use charge-coupled device (CCD) image sensors. Charge from each line of pixels is transferred down the line, pixel by pixel and row by row, to an amplifier where the video signal is formed. CCD cameras are available in a wide variety of formats, resolutions, and sensitivities. They provide the best performance for most applications. Complementary metal-oxide semiconductor (CMOS) sensors are becoming available for some applications. Because they are made using the same processes used to fabricate computer chips, they can be produced very inexpensively. Low-cost CMOS cameras are already used in toys and in webcams. Unlike CCD sensors, which must be read out one full line at a time, CMOS sensors can be read pixel by pixel, in any order. This is useful for timecritical applications in which only part of the image is of interest. At present, the noise performance of CMOS sensors is inferior to that of CCDs. INTERFACES Two types of camera interfaces are in use: analog and digital. In an analog camera, the signal from the sensor is turned into an analog voltage and sent to the framegrabber board in the vision-system computer. EIA, RS-170, NTSC, CCIR, and PAL are all common analog interface standards. Analog cameras are inexpensive, but they are subject to noise and timing problems. Most new machine vision cameras use a digital interface. The signal from each pixel is digitized by the camera, and the data are sent in digital form directly to the computer. CameraLink® and Firewire® are two popular digital A192 Video Cameras for Machine Vision interface standards. The digital signal is not subject to noise, and there is a perfect correspondence between each pixel on the sensor and in the image. Digital cameras support a wide variety of image resolutions and frame rates. Since the signal is already digitized, a simple interface board replaces the frame-grabber. REMOTE-HEAD CAMERAS Machine vision cameras are now quite compact; many are smaller than 50 mm cubes. Remote-head cameras have an even smaller camera “head” consisting of the sensor chip in a protective enclosure, connected to the camera body by a short (<1 m) length of cable. Microhead or “lipstick” cameras can be very small, but they are also much more expensive than single-piece cameras. COLOR CAMERAS Most color CCD cameras use a single sensor with an array of color filters printed over their pixels. Adjacent pixels sense different colors, so the resolution at each color is lower than that of a similar monochrome sensor. Some high-performance cameras use a color-separation prism to send light to three separate CCDs. These cameras provide full resolution at each color. Lenses for these “three-chip” cameras must have sufficient back working distance to allow room for the prism. LINE-SCAN CAMERAS Line-scan cameras have a single row of pixels, which may be 1000, 2000, 4000, or more pixels long. They record images one row at a time. Often the object moves past the camera to provide the second dimension (e.g., a web of paper being inspected during manufacture). Line-scan cameras provide high-resolution images at very high data rates. Long line-scan sensors require large-format lenses to cover their length. In addition, because each line of pixels is exposed for only a very short time, linescan cameras require intense lighting and large-aperture lenses. CAMERA FORMATS The size of an image sensor is called its format (figure 6.17). The name of a format does not correspond to any dimension. Historically, a ½-inch format is the size of the sensing area of a vidicon tube, which is ½ inch in diameter. It is important to choose a lens that covers 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials the camera format. For a given FOV, the camera format determines the required magnification. A larger sensor requires a larger magnification for a given FOV. Material Properties Optical Specifications LENSES FOR HIGH-RESOLUTION CAMERAS To improve sensitivity, many high-resolution CCD sensors include microlens arrays on their surfaces. These arrays make the active area of the pixels appear larger, so that the active-area fraction (fill factor) appears to be near 100%. Unfortunately, this is true only for light that is nearly normal to the sensor surface. Light reaching the sensor at greater angles (e.g., >5º) misses the active area and is lost. This means that lenses used with these sensors must have a long exit-pupil distance and should not have a very small f-number; otherwise the edges of the image will appear dark. 6.4 4.8 3.2 6.6 4.8 3.6 2.4 1 / / / / 4 1 3 1 2 2 3 Fundamental Optics 12.7 9.5 8.8 inch inch inch inch Gaussian Beam Optics 1 inch format (4:3 aspect ratio) 35.0 9.2 9.2 Machine Vision Guide Kodak MegaPlus 23.0 ® 35-mm camera format dimensions in mm Figure 6.17 Camera formats Laser Guide marketplace.idexop.com Video Cameras for Machine Vision A193 MACHINE VISION GUIDE FREQUENTLY ASKED QUESTIONS Machine Vision Guide Why are custom lenses so much more expensive than lenses from stock? A: The cost of manufacturing optics is extremely volume dependent. Mass-produced lenses provide excellent performance at low cost. Lenses produced in small quantity can cost five to twenty times as much. It is always worth attempting to use or adapt a mass-produced lens for an application before designing a custom lens. ration inherent in lens designs. Telecentric lenses offered by Melles Griot have low distortion. Low distortion and telecentricity are separate, unrelated lens parameters. How can I prevent my vision optics from moving out of adjustment? A: Optical mounts for on-line applications should be rigid, have positive locks, and have no more than the required adjustments. Laboratory mounting fixtures are generally not rugged enough for permanent on-line installations. Is there a telecentric lens with a very large FOV? A: Because the first element of a telecentric lens must be larger than its FOV, telecentric lenses are generally restricted to fields of less than 150 mm. Larger FOVs are possible in some applications, including web inspection, using line-scan cameras. How should I mount my video camera and lens? A: Machine vision optics should be mounted firmly but not stressed by excessive force. Do not rely on the camera C-mount thread to support heavy lenses. Either mount the lens and let the camera be supported by the lens, or provide support for both. Avoid overtightening the lens mounting clamps. How can I increase the DOF of my lens? A: Increase the f-number (decrease the aperture size). This may require increased lighting. However, very large f-numbers (>f/22 image side working f-number) will significantly degrade the lens resolution. Do telecentric lenses have larger DOFs than other lenses? A: No. The image from a telecentric lens remains in focus over the same DOF as that of a conventional lens working at the same f-number. Telecentric lenses provide constant magnification at any object distance. Therefore, they make accurate dimensional measurements over a larger range of object distances than a conventional lens. Do telecentric lenses have less distortion than other lenses? A: In optics, “distortion” is the name of a specific aber- A194 Frequently Asked Questions Can I change the magnification of my telecentric lens? A: No. By definition, a telecentric lens has a fixed magnification. Melles Griot offers a variety of telecentric lenses with a large selection of magnifications. OEM and Special Coatings Melles Griot maintains advanced coating capabilities. In the last few years, Melles Griot has expanded and improved these coating facilities to take advantage of the latest developments in thin-film technology. The resulting operations can provide high-volume coatings at competitive prices to OEM customers, as well as specialized, high-performance coatings for the most demanding user. The most important aspect of our coating capabilities is our expert design and manufacturing staff. This group blends years of practical experience with recent academic research knowledge. With a thorough understanding of both design and production issues, Melles Griot excels at producing repeatable, high-quality coatings at competitive prices. User-Supplied Substrates Melles Griot not only coats catalog and custom optics with standard and special coatings but also applies these coatings to user-supplied substrates. A significant portion of our coating business involves applying standard or slightly modified catalog coatings to special substrates. 1-505-298-2550 MACHINE VISION GUIDE Optical Coatings & Materials High Volume Technical Support Laser-Induced Damage Machine Vision Guide Melles Griot conducts laser-induced damage testing of our optics. Although our damage thresholds do not constitute a performance guarantee, they are representative of the damage resistance of our coatings. Occasionally, in the damage-threshold specifications, a reference is made to another coating because a suitable highpower laser is not available to test the coating within its design wavelength range. The damage threshold of the referenced coating should be an accurate representation of the coating in question. When choosing a coating for its power-handling capabilities, some simple guidelines can be followed to make the decision process easier. First, the substrate material is very important. Higher damage thresholds can be achieved using fused silica instead of N-BK7. Second, consider the coating. Metal coatings have the lowest damage thresholds. Broadband dielectric coatings, such as the HEBBAR™ and MAXBRIte™ are better, but single-wavelength or laser-line coatings, such as the V coatings and the MAX-R™ coatings, are better still. If even higher thresholds are needed, then high-energy laser (HEL) coatings are required. If you have any questions or concerns regarding the damage levels involved in your applications, please contact a Melles Griot applications engineer. Gaussian Beam Optics Expert Melles Griot applications engineers are available to discuss your system requirements. Often a simple modification to a system design can enable catalog components or coatings to be substituted for special designs at a reduced cost, without affecting performance. The damage resistance of any coating depends on substrate, wavelength, and pulse duration. Improper handling and cleaning can also reduce the damage resistance of a coating, as can the environment in which the optic is used. These damage threshold values are presented as guidelines and no warranty is implied. Fundamental Optics A large portion of the work done at the Melles Griot coating facilities involves special coatings designed and manufactured to customer specifications. These designs cover a wide range of wavelengths, from the infrared to deep ultraviolet, and applications ranging from basic research through the design and manufacture of industrial and medical products. The most common special coating requests are for modified catalog coatings, which usually involve a simple shift in the design wavelength. Optical Specifications Custom Designs Material Properties The high-volume output capabilities of the Melles Griot coating departments result in very competitive pricing for large-volume special orders. Even the small-order customer benefits from this large volume. Small quantities of special substrates can be cost-effectively coated with popular catalog coatings during routine production runs. For each damage-threshold specification, the information given is the peak fluence (energy per square centimeter), pulse width, peak irradiance (power per square centimeter), and test wavelength. The peak fluence is the total energy per pulse, the pulse width is the full width at half maximum (FWHM), and the test wavelength is the wavelength of the laser used to incur the damage. The peak irradiance is the energy of each pulse divided by the effective pulse length, which is from 12.5 - 25% longer than the pulse FWHM. All tests are performed at a repetition rate of 20 Hz for 10 seconds at each test point. This is important because longer durations can cause damage at lower fluence levels, even at the same repetition rate. Laser Guide marketplace.idexop.com Frequently Asked Questions A195 MACHINE VISION GUIDE Machine Vision Guide Leybold SYRUSpro™ PIAD Coating System Our Leybold SYRUSpro™ plasma ion-assisted deposition (PIAD) coating system features an advanced plasma source (APS) composed of a hot cathode, a cylindrical anode tube, and a solenoid magnet. During operation, the plasma system is energized by applying a dc voltage between the large heated cathode and surrounding anode. The dc voltage between the cathode and anode creates a glow discharge plasma, which is supplied with a noble gas such as argon. The energetic plasma is extracted in the direction of the substrate holder and fills the evaporation chamber. A high flux of energetic ions bombard the growing thin film increasing the packing density of the condensed molecules via momentum transfer. Simultaneously, electrons in the plasma provide an effective source of charge neutralization allowing excellent plasma process uniformity through the chamber. To ensure the growth of fully stoichiometric films, reactive gases such as oxygen can be introduced in the plasma facilitating film growth rates as high as 1.5 nm/ sec depending upon the evaporant material. A196 Frequently Asked Questions Expert use of the PIAD process allows fine control of packing density, stoichiometry, refractive index, and film stress. Unlike conventionally coated substrates, PIAD can produce thin films that are highly insensitive to changes in environmental humidity levels and temperature. The Leybold SYRUSpro coating system’s advanced optical monitoring system works in conjunction with a redundant quartz-crystal thickness-measurement system to enable complex multiwavelength monitoring strategies, trigger-point control (non-quarter wavelength layers), and multiple redundant test or witness sample strategies. In quartz-crystal film-thickness monitoring systems, deposition rate is based upon the measurement of the changing frequency of oscillation of a natural quartz crystal as the accumulating evaporated thin-film increases the crystal’s mass, changing its natural frequency (oscillation frequency decreases as the film thickness increases). This changing frequency is typically monitored every 100 msec, providing the utmost in accuracy and product-to-product coating uniformity. 1-505-298-2550 Optical Coatings & Materials LASER GUIDE A203 TRANSVERSE MODES AND MODE CONTROL A208 SINGLE AXIAL LONGITUDINAL MODE OPERATION A210 FREQUENCY AND AMPLITUDE FLUCTUATIONS A213 TUNABLE OPERATION A216 TYPES OF LASERS A217 LASER APPLICATIONS A232 Fundamental Optics PROPAGATION CHARACTERISTICS OF LASER BEAMS Optical Specifications A198 Material Properties BASIC LASER PRINCIPLES Gaussian Beam Optics Machine Vision Guide Laser Guide marketplace.idexop.com A197 LASER GUIDE BASIC LASER PRINCIPLES The basic operating principles of the laser were put forth by Charles Townes and Arthur Schalow from the Bell Telephone Laboratories in 1958, and the first actual laser, based on a pink ruby crystal, was demonstrated in 1960 by Theodor Maiman at Hughes Research Laboratories. Since that time, literally thousands of lasers have been invented (including the edible “Jello” laser), but only a much smaller number have found practical applications in scientific, industrial, commercial, and military applications. The helium neon laser (the first continuous-wave laser), the semiconductor diode laser, and air-cooled ion lasers have found broad OEM application. In recent years the use of diodepumped solid-state (DPSS) lasers in OEM applications has been growing rapidly. `The term “laser” is an acronym for (L)ight (A)mplification by (S)timulated (E)mission of (R)adiation. To understand the laser, one needs to understand the meaning of these terms. The term “light” is generally accepted to be electromagnetic radiation ranging from 1 nm to 1000 µm in wavelength. The visible spectrum (what we see) ranges from approximately 400 to 700 nm. The wavelength range from 700 nm to 10 µm is considered the near infrared (NIR), and anything beyond that is the far infrared (FIR). Conversely, 200 to 400 nm is called ultraviolet (UV); below 200 nm is the deep ultraviolet (DUV). model became the basis for the field of quantum mechanics and, although not fully accurate by today’s understanding, still is useful for demonstrating laser principles. In Bohr’s model, shown in figure 7.1, electrons orbit the nucleus of an atom. Unlike earlier “planetary” models, the Bohr atom has a limited number of fixed orbits that are available to the electrons. Under the right circumstances an electron can go from its ground state (lowest-energy orbit) to a higher (excited) state, or it can decay from a higher state to a lower state, but it cannot remain between these states. The allowed energy states are called “quantum” states and are referred to by the principal “quantum numbers” 1, 2, 3, etc. The quantum states are represented by an energy level diagram. ( ( ( (QHUJ\H9 Laser Guide Lasers are devices that produce intense beams of light which are monochromatic, coherent, and highly collimated. The wavelength (color) of laser light is extremely pure (monochromatic) when compared to other sources of light, and all of the photons (energy) that make up the laser beam have a fixed phase relationship (coherence) with respect to one another. Light from a laser typically has very low divergence. It can travel over great distances or can be focused to a very small spot with a brightness which exceeds that of the sun. Because of these properties, lasers are used in a wide variety of applications in all walks of life. LRQL]HGFRQWLQXXP VWH[FLWHGVWDWH Q Q JURXQGVWDWH Q Figure 7.1 The Bohr atom and a simple energy-level diagram For an electron to jump to a higher quantum state, the atom must receive energy from the outside world. This can happen through a variety of mechanisms such as inelastic or semielastic collisions with other atoms and absorption of energy in the form of electromagnetic radiation (e.g., light). Likewise, when an electron drops from a higher state to a lower state, the atom must give off energy, either as kinetic activity (nonradiative transitions) or as electromagnetic radiation (radiative transitions). For the remainder of this discussion we will consider only radiative transitions. PHOTONS AND ENERGY THE BOHR ATOM In 1915, Neils Bohr proposed a model of the atom that explained a wide variety of phenomena that were puzzling scientists in the late 19th century. This simple A198 Basic Laser Principles In the 1600s and 1700s, early in the modern study of light, there was a great controversy about light’s nature. Some thought that light was made up of particles, while others thought that it was made up of waves. Both concepts 1-505-298-2550 LASER GUIDE Optical Coatings & Materials E = hn (7.1) ln = c(7.2) where λ is the wavelength of the light and c is the speed of light in a vacuum, equation 7.1 can be rewritten as hc .(7.3) l It is evident from this equation that the longer the wavelength of the light, the lower the energy of the photon; consequently, ultraviolet light is much more “energetic” than infrared light. In general, when an electron is in an excited energy state, it must eventually decay to a lower level, giving off a photon of radiation. This event is called “spontaneous emission,” and the photon is emitted in a random direction and with a random phase. The average time it takes for the electron to decay is called the time constant for spontaneous emission, and is represented by τ. On the other hand, if an electron is in energy state E2, and its decay path is to E1, but, before it has a chance to spontaneously decay, a photon happens to pass by whose energy is approximately E2–E1, there is a probability that the passing photon will cause the electron to decay in such a manner that a photon is emitted at exactly the same wavelength, in exactly the same direction, and with exactly the same phase as the passing photon. This process is called “stimulated emission.” Absorption, spontaneous emission, and stimulated emission are illustrated in figure 7.2. ( ( Gaussian Beam Optics ( ( DEVRUSWLRQ ( ( ( ( Machine Vision Guide Returning to the Bohr atom: for an atom to absorb light (i.e., for the light energy to cause an electron to move from a lower energy state En to a higher energy state Em), the energy of a single photon must equal, almost exactly, the energy difference between the two states. Too much energy or too little energy and the photon will not be absorbed. Consequently, the wavelength of that photon must be Fundamental Optics E= SPONTANEOUS AND STIMULATED EMISSION Optical Specifications where ν is the frequency of the light and h is Planck’s constant. Since, for a wave, the frequency and wavelength are related by the equation Likewise, when an electron decays to a lower energy level in a radiative transition, the photon of light given off by the atom must also have an energy equal to the energy difference between the two states. Material Properties explained some of the behavior of light, but not all. It was finally determined that light is made up of particles called “photons” which exhibit both particle-like and wave-like properties. Each photon has an intrinsic energy determined by the equation VSRQWDQHRXVHPLVVLRQ hc l= DE where DE = E m − E n . where(7.4) hc l= DE where DE = E m − E n . ( ( ( ( VWLPXODWHGHPLVVLRQ Figure 7.2 Spontaneous and stimulated emission Laser Guide marketplace.idexop.com Basic Laser Principles A199 LASER GUIDE Laser Guide Now consider the group of atoms shown in figure 7.3: all begin in exactly the same excited state, and most are effectively within the stimulation range of a passing photon. We also will assume that τ is very long, and that the probability for stimulated emission is 100%. The incoming (stimulating) photon interacts with the first atom, causing stimulated emission of a coherent photon; these two photons then interact with the next two atoms in line, and the result is four coherent photons, on down the line. At the end of the process, we will have eleven coherent photons, all with identical phases and all traveling in the same direction. In other words, the initial photon has been “amplified” by a factor of eleven. Note that the energy to put these atoms in excited states is provided externally by some energy source which is usually referred to as the “pump” source. Of course, in any real population of atoms, the probability for stimulated emission is quite small. Furthermore, not all of the atoms are usually in an excited state; in fact, the opposite is true. Boltzmann’s principle, a fundamental law of thermodynamics, states that, when a collection of atoms is at thermal equilibrium, the relative population of any two energy levels is given by N2 E − E1 = exp − 2 N1 kT (7.5) H ODW WLPX V VWLP XODW HGH PLV VLRQ ]RQ 7,0( H[FLWHG GHFD\HGYLD VSRQWDQHRXVHPLVVLRQ Basic Laser Principles H GHFD\HGYLD VWLPXODWHGHPLVVLRQ Figure 7.3 Amplification by stimulated emission A200 H ]RQ VLRQ LV GHP where N2 and N1 are the populations of the upper and lower energy states, respectively, T is the equilibrium temperature, and k is Boltzmann’s constant. Substituting hν for E2–E1 yields DN ≡ N1 − N 2 = (1 − e − hv / kT ) N1. (7.6) For a normal population of atoms, there will always be more atoms in the lower energy levels than in the upper ones. Since the probability for an individual atom to absorb a photon is the same as the probability for an excited atom to emit a photon via stimulated emission, the collection of real atoms will be a net absorber, not a net emitter, and amplification will not be possible. Consequently, to make a laser, we have to create a “population inversion.” The electron is pumped (excited) into an upper level E4 by some mechanism (for example, a collision with another atom or absorption of high-energy radiation). It then decays to E3, then to E2, and finally to the ground state E1. Let us assume that the time it takes to decay from E2 to E1 is much longer than the time it takes to decay from E2 to E1. In a large population of such atoms, at equilibrium and with a continuous pumping process, a population inversion will occur between the E3 and E2 energy states, and a photon entering the population will be amplified coherently. THE RESONATOR Although with a population inversion we have the ability to amplify a signal via stimulated emission, the overall single-pass gain is quite small, and most of the excited atoms in the population emit spontaneously and do not contribute to the overall output. To turn this system into a laser, we need a positive feedback mechanism that will cause the majority of the atoms in the population to contribute to the coherent output. This is the resonator, a system of mirrors that reflects undesirable (off-axis) photons out of the system and reflects the desirable (on-axis) photons back into the excited population where they can continue to be amplified. 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Material Properties H[FLWDWLRQ PHFKDQLVP SDUWLDO UHIOHFWRU ODVLQJPHGLXP KLJK UHIOHFWRU UHVRQDWRUVXSSRUWVWUXFWXUH Optical Specifications Now consider the laser system shown in figure 7.5. The lasing medium is pumped continuously to create a population inversion at the lasing wavelength. As the excited atoms start to decay, they emit photons spontaneously in all directions. Some of the photons travel along the axis of the lasing medium, but most of the photons are directed out the sides. The photons traveling along the axis have an opportunity to stimulate atoms they encounter to emit photons, but the ones radiating out the sides do not. Furthermore, the photons traveling parallel to the axis will be reflected back into the lasing medium and given the opportunity to stimulate more excited atoms. As the on-axis photons are reflected back and forth interacting with more and more atoms, spontaneous emission decreases, stimulated emission along the axis predominates, and we have a laser. Figure 7.5 Schematic diagram of a basic laser Fundamental Optics Finally, to get the light out of the system, one of the mirrors has a partially transmitting coating that couples out a small percentage of the circulating photons. The amount of coupling depends on the characteristics of the laser system and varies from a fraction of a percent for helium neon lasers to 50% or more for high-power lasers. SXPSLQJSURFHVV Gaussian Beam Optics TXDQWXPHQHUJ\OHYHOV ( ( ( ` SRSXODWLRQ LQYHUVLRQ Machine Vision Guide ( ODVHU DFWLRQ JURXQG HQHUJ\OHYHO OHYHOSRSXODWLRQV Figure A four-level laser pumping system Laser Guide marketplace.idexop.com Basic Laser Principles A201 LASER GUIDE Laser Guide Practical Optical Coatings In the design of a real-world laser, the optical resonator is often the most critical component, and, particularly for low-gain lasers, the most critical components of the resonator are the mirrors themselves. The difference between a perfect mirror coating (the optimum transmission and reflection with no scatter or absorption losses) and a real-world coating, capable of being produced in volume, can mean a 50% (or greater) drop in output power from the theoretical maximum. Consider the 543 nm green helium neon laser line. It was first observed in the laboratory in 1970, but, owing to its extremely low gain, the mirror fabrication and coating technology of the day was incapable of producing a sufficiently low-loss mirror that was also durable. Not until the late 1990s had the mirror coating technology improved sufficiently that these lasers could be offered commercially in large volumes. performance. Likewise, ion-beam sputtering and next-generation ion-assisted ion deposition has increased the packing density of laser coatings, thereby reducing absorption, increasing damage thresholds, and enabling the use of new and exotic coating materials. The critical factors for a mirror, other than transmission and reflection, are scatter, absorption, stress, surface figure, and damage resistance. Coatings with low damage thresholds can degrade over time and cause output power to drop significantly. Coatings with too much mechanical stress not only can cause significant power loss, but can also induce stress birefringence, which can result in altered polarization and phase relationships. The optical designer must take great care when selecting the materials for the coating layers and the substrate to ensure that the mechanical, optical, and environmental characteristics are suitable for the application. The equipment used for both substrate polishing and optical coating is a critical factor in the end result. Coating scatter is a major contributor to power loss. Scatter arises primarily from imperfections and inclusions in the coating, but also from minute imperfections in the substrate. Over the last few years, the availability of “super-polished” mirror substrates has led to significant gains in laser A202 Basic Laser Principles 1-505-298-2550 LASER GUIDE Optical Coatings & Materials PROPAGATION CHARACTERISTICS OF LASER BEAMS BEAM WAIST AND DIVERGENCE and w (z) = lz (7.10) p w0 where z is presumed to be much larger than πw02/λ so that the 1/e2 irradiance contours asymptotically approach a cone of angular radius v= w (z) l = . z p w0 (7.11) Fundamental Optics (7.7) where w = w(z) and P is the total power in the beam, is the same at all cross sections of the beam. The invariance of the form of the distribution is a special consequence of the presumed Gaussian distribution at z = 0. Simultaneously, as R(z) asymptotically approaches z for large z, w(z) asymptotically approaches the value Optical Specifications Even if a Gaussian TEM00 laser beam wavefront were made perfectly flat at some plane, with all rays there moving in precisely parallel directions, it would acquire curvature and begin spreading in accordance with (7.9) Material Properties Diffraction causes light waves to spread transversely as they propagate, and it is therefore impossible to have a perfectly collimated beam. The spreading of a laser beam is in accord with the predictions of diffraction theory. Under ordinary circumstances, the beam spreading can be so small it can go unnoticed. The following formulas accurately describe beam spreading, making it easy to see the capabilities and limitations of laser beams. The notation is consistent with much of the laser literature, particularly with Siegman’s excellent Lasers (University Science Books). This value is the far-field angular radius (half-angle divergence) of the Gaussian TEM00 beam. The vertex of the cone lies at the center of the waist (see figure 7.6). (7.8) NEAR-FIELD VS. FAR-FIELD DIVERGENCE Unlike conventional light beams, Gaussian beams do not diverge linearly, as can be seen in figure 7.6. Near the laser, the divergence angle is extremely small; far from the laser, the divergence angle approaches the asymptotic limit described in equation 7.11 above. The Raleigh range (zR), defined as the distance over which the beam radius spreads by a factor of √2, is given by zR = The irradiance distribution of the Gaussian TEM00 beam, namely, pw l Machine Vision Guide The plane z = 0 marks the location of a beam waist, or a place where the wavefront is flat, and w0 is called the beam waist radius. Gaussian Beam Optics where z is the distance propagated from the plane where the wavefront is flat, λ is the wavelength of light, w0 is the radius of the 1/e2 irradiance contour at the plane where the wavefront is flat, w(z) is the radius of the 1/e2 contour after the wave has propagated a distance z, and R(z) is the wavefront radius of curvature after propagating a distance z. R(z) is infinite at z = 0, passes through a minimum at some finite z, and rises again toward infinity as z is further increased, asymptotically approaching the value of z itself. It is important to note that, for a given value of λ, variations of beam diameter and divergence with distance z are functions of a single parameter, w0, the beam waist radius. (7.12) Laser Guide marketplace.idexop.com Propagation Characteristics of Laser Beams A203 LASER GUIDE Laser Guide 1 w w0 e2 irradiance surface ic co ptot asym w0 ne v z w0 Figure 7.6 Growth in beam diameter as a function of distance from the beam waist The Raleigh range is the dividing line between nearfield divergence and mid-range divergence. Far-field divergence (the number quoted in laser specifications) must be measured at a point >zR (usually 10zR will suffice). This is a very important distinction because calculations for spot size and other parameters in an optical train will be inaccurate if near- or mid-field divergence values are used. For a tightly focused beam, the distance from the waist (the focal point) to the far field can be a few millimeters or less. For beams coming directly from the laser, the far-field distance can be measured in meters. z2 are the distances from the beam waist of mirrors 1 and 2, respectively. (Note that distances are measured from the beam waist, and that, by convention, mirror curvatures that are concave when viewed from the waist are considered positive, while those that are convex are considered negative.) In any case but that of a flat output mirror, the beam waist is refracted as it passes through the mirror substrate. If the output coupler’s second surface is flat, the effective waist of the refracted beam is moved toward the output coupler and is reduced in diameter. However, by applying a spherical correction to the second surface of the output coupler, the location of the beam waist can be moved to the output coupler itself, increasing the beam waist diameter and reducing far-field divergence. (See Calculating a Correcting Surface.) R a. z1 = w w 125 LOCATING THE BEAM WAIST For a Gaussian laser beam, the location (and radius) of the beam waist is determined uniquely by the radius of curvature and optical spacing of the laser cavity mirrors because, at the reflecting surfaces of the cavity mirrors, the radius of curvature of the propagating beam is exactly the same as that of the mirrors. Consequently, for the flat/curved cavity shown in figure 7.7 (a), the beam waist is located at the surface of the flat mirror. For a symmetric cavity (b), the beam waist is halfway between the mirrors; for non-symmetric cavities (c and d), the beam waist is located by using the equation R b. w w R R c. w w w R R1 + R2 − 2L R d. (7.13) w w w z1 + z2 = L where L is the effective mirror spacing, R1 and R2 are the radii of curvature of the cavity mirrors, and z1 and A204 R L (R2 − L ) and R Propagation Characteristics of Laser Beams GLPHQVLRQVLQPP Figure 7.7 Location of beam waist for common cavity geometries 1-505-298-2550 LASER GUIDE Optical Coatings & Materials This is illustrated by the case of a typical helium neon laser cavity consisting a flat high reflector and an output mirror with a radius of curvature of 20 cm separated by 15 cm. If the laser is operating at 633 nm, the beam with the appropriate sign convention and assuming that n = 1.5, we get a convex correcting curvature of approximately 5.5 cm. At this point, the beam waist has been transferred to the output coupler, with a radius of 0.26 mm, and the far-field half-angle divergence is reduced to 0.76 mrad, a factor of nearly 4. Correcting surfaces are used primarily on output couplers whose radius of curvature is a meter or less. For longer radius output couplers, the refraction effects are less dramatic, and a correcting second surface radius is unnecessary. FXUYHGPLUURUwz pw lz IODWPLUURUw0 5DGLXVRI&XUYHG0LUURU0LUURU6SDFLQJR/z Figure 7.8 Beam waist and output diameter as a function of mirror radius and separation Machine Vision Guide In the real world, the truly 100%, single transverse mode, Gaussian laser beam (also called a pure or fundamental mode beam) described by equations 7.7 and 7.8 is very hard to find. Low-power beams from helium neon lasers can be a close approximation, but the higher the power of the laser, and the more complex the excitation mechanism (e.g., transverse discharges, flash-lamp pumping), or the higher the order of the mode, the more the beam deviates from the ideal. Gaussian Beam Optics HIGHER ORDER GAUSSIAN LASER BEAMS (7.14) Fundamental Optics A laser beam is refracted as it passes through a curved output mirror. If the mirror has a flat second surface, the waist of the refracted beam moves closer to the mirror, and the divergence is increased. To counteract this, laser manufacturers often put a radius on the output coupler’s second surface to collimate the beam by making a waist at the output coupler. 1 1 1 = ( n − 1) − f R1 R2 Optical Specifications CALCULATING A CORRECTING SURFACE waist radius, beam radius at the output coupler, and beam half-angle divergence are w0 = 0.13 mm, w200 = 0.26 mm, and θ = 1.5 mrad, respectively; however, with a flat second surface, the divergence nearly doubles to 2.8 mrad. Geometrical optics would give the focal length of the lens formed by the correcting output coupler as 15 cm; a rigorous calculation using Gaussian beam optics shows it should be 15.1 cm. Using the lens-makers formula Material Properties It is useful, particularly when designing laser cavities, to understand the effect that mirror spacing has on the beam radius, both at the waist and at the curved mirror. Figure 7.8 plots equations 7.7 and 7.8 as a function of R/z (curved mirror radius divided by the mirror spacing). As the mirror spacing approaches the radius of curvature of the mirror (R/z = 1), the beam waist decreases dramatically, and the beam radius at the curved mirror becomes very large. On the other hand, as R/z becomes large, the beam radius at the waist and at the curved mirror are approximately the same. To address the issue of higher order Gaussian beams and mixed mode beams, a beam quality factor, M2, has come into general use. A mixed mode is one where several modes are oscillating in the resonator at the same time. A common example is the mixture of the lowest order single transverse mode with the doughnut mode, before the intracavity mode limiting aperture is critically set to Laser Guide marketplace.idexop.com Propagation Characteristics of Laser Beams A205 LASER GUIDE Laser Guide select just the fundamental mode. Because all beams have some wavefront defects, which implies they contain at least a small admixture of some higher order modes, a mixed mode beam is also called a “real” laser beam. For a theoretical single transverse mode Gaussian beam, the value of the waist radius–divergence product is (from equation 7.11): w0θ = λ /π. (7.15) It is important to note that this product is an invariant for transmission of a beam through any normal, high-quality optical system (one that does not add aberrations to the beam wavefront). That is, if a lens focuses the single mode beam to a smaller waist radius, the convergence angle coming into the focus (and the divergence angle emerging from it) will be larger than that of the unfocused beam in the same ratio that the focal spot diameter is smaller: the product is invariant. For a real laser beam, we have W0Θ = M2λ/π (7.16) where W0 and Θ are the 1/e2 intensity waist radius and the far-field half-divergence angle of the real laser beam, respectively. Here we have introduced the convention that upper case symbols are used for the mixed mode and lower case symbols for the fundamental mode beam coming from the same resonator. The mixed-mode beam radius W is M times larger than the fundamental mode radius at all propagation distances. Thus the waist radius is that much larger, contributing the first factor of Min equation 7.16. The second factor of M comes from the half-angle divergence, which is also M times larger. The waist radius–divergence half-angle product for the mixed mode beam also is an invariant, but is M2 larger. The fundamental mode beam has the smallest divergence allowed by diffraction for a beam of that waist radius. The factor M2 is called the “times-diffraction-limit” number or A206 Propagation Characteristics of Laser Beams (inverse) beam quality; a diffraction- limited beam has an M2 of unity. For a typical helium neon laser operating in TEM00 mode, M2 < 1.05. Ion lasers typically have an M2 factor ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 30 or 40. The M2 factor describes the propagation characteristics (spreading rate) of the laser beam. It cannot be neglected in the design of an optical train to be used with the beam. Truncation (aperturing) by an optic, in general, increases the M2 factor of the beam. The propagation equations (analogous to equations 7.7 and 36.8) for the mixed-mode beam W(z) and R(z) are as follows: 1/ 2 zM 2l 2 z 2 (7.17) = W ( z ) = W0 1 + W 0 1+ pW02 ZR 1/ 2 and zM 2l 2 z 2 2 = W z = W 1 + W ( ) 0 p W 2 2 0Z1 + 2 and 0 0 R Z R ( z ) = z 1+ pW = z 1 + z . R zM 2l and pW 2 2 Z 2 0 R ( z ) = z 1 + = z 1 + R . (7.18) 2 zM l z The Rayleigh range remains the same for a mixed mode laser beam: ZR = pW02 pw02 = = zR . (7.19) l M 2l Now consider the consequences in coupling a high M2 beam into a fiber. Fiber coupling is a task controlled by the product of the focal diameter (2Wƒ) and the focal convergence angle (θƒ). In the tight focusing limit, the focal diameter is proportional to the focal length f of the lens, and is inversely proportional to the diameter of the beam at the lens (i.e., 2Wƒ∞f/Dlens). The lens-to-focus distance is ƒ, and, since ƒxθƒ is the beam diameter at distance f in the far field of the focus, 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Dlens ∞ ƒθƒ. Combining these proportionalities yields APPLICATION NOTE Stable vs Unstable Resonator Cavities for the fiber-coupling problem as stated above. The diameter-divergence product for the mixed-mode beam is M2 larger than the fundamental mode beam in accordance with equations 7.15 and 7.16. A stable resonator cavity is defined as one that self-focuses energy within the cavity back upon itself to create the typical Gaussian modes found in most traditional lasers. The criterion for a stable cavity is that Optical Specifications where Fundamental Optics There is a threefold penalty associated with coupling a beam with a high M2 into a fiber: 1) the focal length of the focusing lens must be a factor of 1/M2 shorter than that used with a fundamental-mode beam to obtain the same focal diameter at the fiber; 2) the numerical aperture (NA) of the focused beam will be higher than that of the fundamental beam (again by a factor of 1/M2) and may exceed the NA of the fiber; and 3) the depth of focus will be smaller by 1/M2 requiring a higher degree of precision and stability in the optical alignment. Material Properties Wƒθƒ = constant where R1 and R2 are the radii of the cavity mirrors and L is the mirror separation. Gaussian Beam Optics The mode volumes of stable resonator cavities are relatively small. This is fine when the excitation regions of a laser are also relatively small, as is the case with a HeNe or DPSS laser. However, for large-format high-energy industrial lasers, particularly those with high single-pass gain, stable resonators can limit the output. In these cases, unstable resonators, like the one shown in the illustration below, can generate higher output with better mode quality. In this case, the output coupling is determined by the ratio of the diameters of the output and high-reflecting mirrors, not the coating reflectivity. In the near field, the output looks like a doughnut, because the center of the beam is occluded by the output mirror. At a focus, however, the beam has most of the propagation characteristics of a fundamentalmode stable laser. Machine Vision Guide Laser Guide marketplace.idexop.com Propagation Characteristics of Laser Beams A207 LASER GUIDE TRANSVERSE MODES AND MODE CONTROL Laser Guide The fundamental TEM00 mode is only one of many transverse modes that satisfies the condition that it be replicated each round-trip in the cavity. Figure 7.9 shows examples of the primary lower-order Hermite-Gaussian (rectangular) modes. Note that the subscripts m and n in the mode designation TEMmn are correlated to the number of nodes in the x and y directions. The propagation equation can also be written in cylindrical form in terms of radius (ρ) and angle (Ø). The eigenmodes (EρØ) for this equation are a series of axially symmetric modes, which, for stable resonators, are closely approximated by Laguerre-Gaussian functions, denoted by TEMρØ. For the lowest-order mode, TEM00, the Hermite-Gaussian and Laguerre- Gaussian functions are identical, but for higher-order modes, they differ significantly, as shown in figure 7.10. The mode, TEM01*, also known as the “bagel” or “doughnut” mode, is considered to be a superposition of the Hermite-Gaussian TEM10 and TEM01 modes, locked in phase and space quadrature. (See W.W. Rigrod, “Isolation of Axi-Symmetric Optical-Resonator Modes,” Applied Physics Letters, Vol. 2 (1 Feb. ‘63), pages 51–53.) TEM01 MODE CONTROL The transverse modes for a given stable resonator have different beam diameters and divergences. The lower the order of the mode is, the smaller the beam diameter, the narrower the far-field divergence, and the lower the M2 value. For example, the TEM01* doughnut mode is approximately 1.5 times the diameter of the fundamental TEM00 mode, and the Laguerre TEM10 target mode is twice the diameter of the TEM00 mode. The theoretical M2 values for the TEM00, TEM01*, and TEM10 modes are 1.0, 2.3, and 3.6, respectively (R. J. Freiberg et al., “Properties of Low Order Transverse Modes in Argon Ion Lasers”). Because of its smooth intensity profile, low divergence, and ability to be focused to a diffractionlimited spot, it is usually desirable to operate in the lowest-order mode possible, TEM00. Lasers, however, tend to operate at the highest-order mode possible, either in addition to, or instead of, TEM00 because the larger beam diameter may allow them to extract more energy from the lasing medium. The primary method for reducing the order of the lasing mode is to add sufficient loss to the higher-order modes so that they cannot oscillate without significantly increasing the losses at the desired lower-order mode. In most lasers this is accomplished by placing a fixed or In real-world lasers, the Hermite-Gaussian modes predominate since strain, slight misalignment, or contamination on the optics tends to drive the system toward rectangular coordinates. Nonetheless, the TEM00 Laguerre-Gaussian TEM10 “target” or “bulls-eye” mode is clearly observed in well-aligned gas-ion and helium neon lasers with the appropriate limiting apertures. TEM10 TEM11 TEM02 Figure 7.9 Low-order Hermite-Gaussian resonator modes A208 Transverse Modes and Mode Control 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Material Properties variable aperture inside the laser cavity. Because of the significant differences in beam diameter, the aperture can cause significant diffraction losses for the higherorder modes without impacting the lower-order modes. As an example, consider the case of a typical argon-ion laser with a long-radus cavity and a variable modeselecting aperture. Optical Specifications When the aperture is fully open, the laser oscillates in the axially symmetric TEM10 target mode. As the aperture is slowly reduced, the output changes smoothly to the TEM01* doughnut mode, and finally to the TEM00 fundamental mode. TEM01* Gaussian Beam Optics TEM00 Fundamental Optics In many lasers, the limiting aperture is provided by the geometry of the laser itself. For example, by designing the cavity of a helium neon laser so that the diameter of the fundamental mode at the end of the laser bore is approximately 60% of the bore diameter, the laser will naturally operate in the TEM00 mode. TEM10 Figure 7.10 Low-order axisymmetric resonator modes Machine Vision Guide Laser Guide marketplace.idexop.com Transverse Modes and Mode Control A209 LASER GUIDE SINGLE AXIAL LONGITUDINAL MODE OPERATION Laser Guide THEORY OF LONGITUDINAL MODES In a laser cavity, the requirement that the field exactly reproduce itself in relative amplitude and phase each round-trip means that the only allowable laser wavelengths or frequencies are given by l= P Nc or n = N P (7.20) where λ is the laser wavelength, v is the laser frequency, c is the speed of light in a vacuum, N is an integer whose value is determined by the lasing wavelength, and P is the effective perimeter optical path length of the beam as it makes one round-trip, taking into account the effects of the index of refraction. For a conventional twomirror cavity in which the mirrors are separated by optical length L, these formulas revert to the familiar l= 2L Nc or n = . N 2L (7.21) These allowable frequencies are referred to as longitudinal modes. The frequency spacing between adjacent longitudinal modes is given by Dn = c . P (7.22) As can be seen from equation 7.22, the shorter the laser cavity is, the greater the mode spacing will be. By differentiating the expression for ν with respect to P we arrive at dn = − Nc Nc dP or dn = − 2 dL. 2L P2 (7.23) Consequently, for a helium neon laser operating at 632.8 nm, with a cavity length of 25 cm, the mode spacing is approximately 600 MHz, and a 100 nm change in cavity length will cause a given longitudinal mode to shift by approximately 190 MHz. A210 Single Axial Longitudinal Mode Operation The number of longitudinal laser modes that are present in a laser depends primarily on two factors: the length of the laser cavity and the width of the gain envelope of the lasing medium. For example, the gain of the red helium neon laser is centered at 632.8 nm and has a full width at half maximum (FWHM) of approximately 1.4 GHz, meaning that, with a 25 cm laser cavity, only two or three longitudinal modes can be present simultaneously, and a change in cavity length of less than one micron will cause a given mode to “sweep” completely through the gain. Doubling the cavity length doubles the number of oscillating longitudinal modes that can fit under the gain curve. The gain of a gas-ion laser (e.g., argon or krypton) is approximately five times broader than that of a helium neon laser, and the cavity spacing is typically much greater, allowing many more modes to oscillate simultaneously. A mode oscillating at a frequency near the peak of the gain will extract more energy from the gain medium than one oscillating at the fringes. This has a significant impact on the performance of a laser system because, as vibration and temperature changes cause small changes in the cavity length, modes sweep back and forth through the gain. A laser operating with only two or three longitudinal modes can experience power fluctuations of 10% or more, whereas a laser with ten or more longitudinal modes will see modesweeping fluctuations of 2% or less. SELECTING A SINGLE LONGITUDINAL MODE A laser that operates with a single longitudinal mode is called a single frequency laser. There are two ways to force a conventional two-mirror laser to operate with a single longitudinal mode. The first is to design the laser with a short enough cavity that only a single mode can be sustained. For example, in the helium neon laser described above, a 10 cm cavity would allow only one mode to oscillate. This is not a practical approach for most gas lasers because, with the cavity short enough to suppress additional modes, there may be insufficient energy in the lasing medium to sustain any 1-505-298-2550 LASER GUIDE Optical Coatings & Materials lasing action at all, and if there is lasing, the output will be very low. VLJQDO GHWHFWRU HWDORQ SLFNRII EHDPVSOLWWHU FDYLW\ OHQJWK FRQWURO OHQJWKWXQDEOHODVHU Figure 7.11 Laser frequency stabilization scheme changes in power. By “locking” the discriminant ratio at a specific value (e.g., 50%) and providing negative feedback to the device used to control cavity length, output frequency can be controlled. If the frequency increases from the preset value, the length of the laser cavity is increased to drive the frequency back to the set point. If the frequency decreases, the cavity length is decreased. The response time of the control electronics is determined by the characteristics of the laser system being stabilized. Machine Vision Guide The frequency output of a single-longitudinal-mode laser is stabilized by precisely controlling the laser cavity length. This can be accomplished passively UHIHUHQFH GHWHFWRU Gaussian Beam Optics FREQUENCY STABILIZATION If the laser frequency decreases, the ratio decreases. In other words, the etalon is used to create a frequency discriminant that converts changes in frequency to Fundamental Optics THE RING LASER The discussions above are limited to two-mirror standingwave cavities. Some lasers operate naturally in a single longitudinal mode. For example, a ring laser cavity, (used in many dye and Ti:Sapphire lasers as well as in gyroscopic lasers) that has been constrained to oscillate in only one direction produces a traveling wave without the fixed nodes of the standing-wave laser. The traveling wave sweeps through the laser gain, utilizing all of the available energy and preventing the buildup of adjacent modes. Other lasers are “homogeneously broadened” allowing virtually instantaneous transfer of energy from one portion of the gain curve to another. A typical stabilization scheme is shown in figure 7.11. A portion of the laser output beam is directed into a low-finesse Fabry-Perot etalon and tuned to the side of the transmission band. The throughput is compared to a reference beam, as shown in the figure. If the laser frequency increases, the ratio of attenuated power to reference power increases. Optical Specifications Once the mode is selected, the challenge is to optimize and maintain its output power. Since the laser mode moves if the cavity length changes slightly, and the etalon pass band shifts if the etalon spacing varies slightly, it important that both be stabilized. Various mechanisms are used. Etalons can be passively stabilized by using zero-expansion spacers and thermally stabilized designs, or they can be thermally stabilized by placing the etalon in a precisely controlled oven. Likewise, the overall laser cavity can be passively stabilized, or, alternatively, the laser cavity can be actively stabilized by providing a servomechanism to control cavity length, as discussed in Frequency Stabilization. Material Properties The second method is to introduce a frequency-control element, typically a low-finesse Fabry-Perot etalon, into the laser cavity. The free spectral range of the etalon should be several times the width of the gain curve, and the reflectivity of the surfaces should be sufficient to provide 10% or greater loss at frequencies half a longitudinal mode spacing away from the etalon peak. The etalon is mounted at a slight angle to the optical axis of the laser to prevent parasitic oscillations between the etalon surfaces and the laser cavity. by building an athermalized resonator structure and carefully controlling the laser environment to eliminate expansion, contraction, and vibration, or actively by using a mechanism to determine the frequency (either relatively or absolutely) and quickly adjusting the laser cavity length to maintain the frequency within the desired parameters. Laser Guide marketplace.idexop.com Single Axial Longitudinal Mode Operation A211 LASER GUIDE Laser Guide Other techniques can be used to provide a discriminant. One common method used to provide an ultrastable, long-term reference is to replace the etalon with an absorption cell and stabilize the system to the saturated center of an appropriate transition. Another method, shown in figure 7.12, is used with commercial helium neon lasers. It takes advantage of the fact that, for an internal mirror tube, the adjacent modes are orthogonally polarized. The cavity length is designed so that two modes can oscillate under the gain curve. The two modes are separated outside the laser by a polarization-sensitive beamsplitter. Stabilizing the relative amplitude of the two beams stabilizes the frequency of both beams. The cavity length changes needed to stabilize the laser cavity are very small. In principle, the maximum adjustment needed is that required to sweep the frequency through one free spectral range of the laser cavity (the cavity mode spacing). For the helium neon laser cavity described earlier, the required change is only 320 nm, well within the capability of piezoelectric actuators. Commercially available systems can stabilize frequency output to 1 MHz or less. Laboratory systems that stabilize the frequency to a few kilohertz have been developed. FRPSDUDWRU FRQWURO HOHFWURQLFV YHUWLFDO SRODUL]DWLRQ UHIHUHQFH EHDP KRUL]RQWDO SRODUL]DWLRQ UHIHUHQFH EHDP KLJKUHIOHFWLQJ PLUURU +H1HODVHUGLVFKDUJHWXEH DGMXVWDEOHFDYLW\VSDFHU RXWSXWFRXSOHU Figure 7.12 Frequency stabilization for a helium neon laser A212 Single Axial Longitudinal Mode Operation 1-505-298-2550 LASER GUIDE Optical Coatings & Materials FREQUENCY AND AMPLITUDE FLUCTUATIONS The major sources of noise in a laser are fluctuations in the pumping source and changes in length or alignment caused by vibration, stress, and changes in temperature. For example, unfiltered line ripple can cause output fluctuations of 5% to 10% or more. c c = . 2L 2P (7.24) Mode beating can cause peak-to-peak power fluctuations of several percent. The only way to eliminate this noise component is to limit the laser output to a single transverse and single longitudinal mode. Not all continuous-wave lasers are amenable to APC as described above. For the technique to be effective, there must be a monotonic relationship between output power and a controllable parameter (typically current or voltage). For example, throughout the typical operating range of a gas ion laser, an increase in current will increase the output power and vice versa. This is not the case for some lasers. The output of a helium neon laser is very insensitive to discharge current, and an increase in current may increase or decrease laser output. In a helium cadmium laser, where electrophoresis determines Machine Vision Guide Finally, when all other sources of noise have been eliminated, we are left with quantum noise, the noise generated by the spontaneous emission of photons from the upper laser level in the lasing medium. In most applications, this is inconsequential. Automatic current control effectively reduces amplitude fluctuations caused by the driving electronics, but it has no effect on amplitude fluctuations caused by vibration or misalignment. Automatic power control can effectively reduce power fluctuations from all sources. Neither of these control mechanisms has a large impact on frequency stability. Gaussian Beam Optics Dn longitudinal = With APC, instead of monitoring the voltage across a sensing resistor, a small portion of the output power in the beam is diverted to a photodetector, as shown in figure 10.14, and the voltage generated by the detector circuitry is compared to a reference. As output power fluctuates, the sensing circuitry generates an error signal that is used to make the appropriate corrections to maintain constant output. Fundamental Optics High-frequency noise (>1 MHz) is caused primarily by “mode beating.”Transverse Laguerre-Gaussian modes of adjacent order are separated by a calculable fraction of the longitudinal mode spacing, typically ~17 MHz in a 1 m resonator with long radius mirrors. If multiple transverse modes oscillate simultaneously, heterodyne interference effects, or “beats,” will be observed at the difference frequencies. Likewise, mode beating can occur between longitudinal modes at frequencies of Two primary methods are used to stabilize amplitude fluctuations in commercial lasers: automatic current control (ACC), also known as current regulation, and automatic power control (APC), also known as light regulation. In ACC, the current driving the pumping process passes through a stable sensing resistor, as shown in figure 7.13, and the voltage across this resistor is monitored. If the current through the resistor increases, the voltage drop across the resistor increases proportionately. Sensing circuitry compares this voltage to a reference and generates an error signal that causes the power supply to reduce the output current appropriately. If the current decreases, the inverse process occurs. ACC is an effective way to reduce noise generated by the power supply, including line ripple and fluctuations. Optical Specifications Likewise, a 10 µrad change in alignment can cause a 10% variation in output power, and, depending upon the laser, a 1 µm change in length can cause amplitude fluctuations of up to 50% (or more) and frequency fluctuations of several gigahertz. METHODS FOR SUPPRESSING AMPLITUDE NOISE AND DRIFT Material Properties The output of a freely oscillating laser will fluctuate in both amplitude and frequency. Fluctuations of less than 0.1 Hz are commonly referred to as “drift”; faster fluctuations are termed “noise” or, when talking about sudden frequency shifts, “jitter.” Laser Guide marketplace.idexop.com Frequency and Amplitude Fluctuations A213 LASER GUIDE Laser Guide ODVHURXWSXW ODVHU ODVHUKHDG ODVHURXWSXW ODVHU EHDPVDPSOHU DPSOLILHU OSKRWRGHWHFWRU IHHGEDFNFLUFXLW IHHGEDFNFLUFXLW SRZHUFLUFXLW SRZHUVXSSO\FRQWUROOHU SRZHUVXSSO\FRQWUROOHU Figure 7.13 Automatic current control schematic Figure 7.14 Automatic power control schematic the density and uniformity of cadmium ions throughout the discharge, a slight change in discharge current in either direction can effectively kill lasing action. One consideration that is often overlooked in an APC system is the geometry of the light pickoff mechanism itself. One’s first instinct is to insert the pickoff optic into the main beam at a 45º angle, so that the reference beam exits at a 90º angle. However, as shown in figure 7.15, for uncoated glass, there is almost a 10% difference in reflectivity for s and p polarization. In a randomly polarized laser, the ratio of the s and p components is not necessarily stable, and using a 90º reference beam can actually increase amplitude fluctuations. This is of much less concern in a laser with a high degree of linear polarization (e.g., 500:1 or better), but even then there is a slight presence of the orthogonal polarization. Good practice dictates that the pickoff element be inserted at an angle of 25º or less. A214 Frequency and Amplitude Fluctuations 3HUFHQW5HIOHFWDQFH If traditional means of APC are not suitable, the same result can be obtained by placing an acousto-optic modulator inside the laser cavity and using the error signal to control the amount of circulating power ejected from the cavity. sSODQH pSODQH vS $QJOHRI,QFLGHQFHLQ'HJUHHV Figure 7.15 Reflectivity of a glass surface vs. incidence angle for s and p polarization 1-505-298-2550 LASER GUIDE Optical Coatings & Materials APPLICATION NOTE Material Properties Measuring Frequency Stability Optical Specifications The accepted method of measuring long-term frequency stability is to heterodyne the laser to be tested with another laser of equal or greater stability. By observing the variation of the resulting beat frequencies, the combined drift of the two lasers can be measured. The results will be no better than the sum of the two instabilities and will, therefore, provide a conservative measure of frequency drift. ,QWHQVLW\ VHFRQGV 7LPH Gaussian Beam Optics )UHTXHQF\ 0+] ,QWHQVLW\ Fundamental Optics )UHTXHQF\ 0+] In the charts below, a frequency-stabilized HeNe was hetrodyned with the output from a Zeemanstabilized laser.The charts show the performance over one minute and over an eight-hour typical workday. The laser can be cycled over a 20°C temperature range without mode hopping. KRXUV 7LPH Short- and long-term frequency stability of a frequency stabilized helium neon laser Machine Vision Guide Laser Guide marketplace.idexop.com Frequency and Amplitude Fluctuations A215 LASER GUIDE TUNABLE OPERATION Laser Guide Many lasers can operate at more than one wavelength. Argon and krypton lasers can operate at discrete wavelengths ranging from the ultraviolet to the near infrared. Dye lasers can be continuously tuned over a spectrum of wavelengths determined by the fluorescence bandwidths of the specific dyes (typically about 150 nm). Alexandrite and titanium sapphire lasers can be tuned continuously over specific spectral regions. around the normal to its face shifts the transmission bands, tuning the laser. Since there are no coatings and the filter is at Brewster’s angle (thereby polarizing the laser), there are no inherent cavity reflection losses at the peak of the transmission band. A single filter does not have as significant a line-narrowing effect as does a grating, but this can be overcome by stacking multiple filter plates together, with each successive plate having a smaller free spectral range. To create a tunable laser, the cavity coatings must be sufficiently broadband to accommodate the entire tuning range, and a variable-wavelength tuning element must be introduced into the cavity, either between the cavity optics or replacing the high-reflecting optic, to introduce loss at undesired wavelengths. Three tuning mechanisms are in general use: Littrow prisms, diffraction gratings, and birefringent filters. Littrow prisms (see figure 7.16) and their close relative, the full-dispersing prism, are used extensively with gas lasers that operate at discrete wavelengths. In its simplest form, the Littrow prism is a 30º-60º-90º prism with the surface opposite the 60º angle coated with a broadband high-reflecting coating. The prism is oriented so that the desired wavelength is reflected back along the optical axis, and the other wavelengths are dispersed off axis. By rotating the prism the retroreflected wavelength can be changed. In laser applications, the prism replaces the high-reflecting mirror, and the prism’s angles are altered (typically to 34º, 56º, and 90º) to minimize intracavity losses by having the beam enter the prism exactly at Brewster’s angle. For higher-power lasers which require greater dispersion to separate closely spaced lines, the Littrow prism can be replaced by a full-dispersing prism coupled with a high reflecting mirror. RXWSXWFRXSOHU SODVPDWXEH /LWWURZSULVP Figure 7.16 Littrow prism used to select a single wavelength Gratings are used for laser systems that require a higher degree of dispersion than that of a full-dispersing prism. Birefringent filters have come into general use for continuously tunable dye and Ti:Sapphire lasers, since they introduce significantly lower loss than do gratings. The filter is made from a thin, crystalline-quartz plate with its fast axis oriented in the plane of the plate. The filter, placed at Brewster’s angle in the laser beam, acts like a weak etalon with a free spectral range wider than the gain curve of the lasing medium. Rotating the filter A216 Tunable Operation 1-505-298-2550 LASER GUIDE Optical Coatings & Materials TYPES OF LASERS GAS-DISCHARGE LASERS SUHFLVLRQPLUURUDGMXVWPHQW PHWDOOLFVSLGHUIRUERUHFHQWUDWLRQ JHWWHU RXWSXWEHDPFRD[LDOZLWK F\OLQGULFDOKRXVLQJ ERUH JDVUHVHUYRLU RSWLRQDO%UHZVWHU VZLQGRZ IRUOLQHDUSRODUL]DWLRQ KLJKUHIOHFWLQJPLUURU Machine Vision Guide JODVVWRPHWDOVHDOV FROOLPDWLQJRXWSXWPLUURU Gaussian Beam Optics FDWKRGHFRQQHFWLRQWKURXJKKRXVLQJ Fundamental Optics In principle, gas-discharge lasers are inherently simple— fill a container with gas, put some mirrors around it, and strike a discharge. In practice, they are much more complex because the gas mix, discharge parameters, and container configuration must be specifically and carefully designed to create the proper conditions for a population inversion. Furthermore, careful consideration Figure 7.17 below shows a cutaway of a helium neon laser, one of the simplest gas-discharge lasers. An electrical discharge is struck between the anode and cathode. The laser bore confines the discharge, creating the current densities needed to create the inversion. In this example, the laser mirrors are mounted to the ends of the tube and are effectively part of the gas container. In other cases, the mirrors are external to the container, and light enters and exits the chamber through Brewster’s windows or extremely low-loss antireflectioncoated normal windows. Because most gas-discharge lasers are operated at extremely low pressures, a getter is needed to remove the impurities generated by outgassing in the walls of the container or by erosion of the electrodes and bore caused by the discharge. The Brewster’s window is used to linearly polarize the output of the laser. Optical Specifications Lasers can be broadly classified into four categories: gas discharge lasers, semiconductor diode lasers, optically pumped lasers, and “other,” a category which includes chemical lasers, gas-dynamics lasers, x-ray lasers, combustion lasers, and others developed primarily for military applications. These lasers are not discussed further here. must be given to how the discharge will react with its container and with the laser optics. Finally, since the temperature of the gas can affect the discharge conditions, questions of cooling must be addressed. Material Properties Since the discovery of the laser, literally thousands of types of lasers have been discovered. As Arthur Schawlow is purported to have said, “Hit it hard enough and anything will lase.” However, only a relative few of these lasers have found broadly based, practical applications. FXUUHQWUHJXODWHG SRZHUVXSSO\ VKRUWDQRGHOHDGDQGSRWWHGEDOODVW IRUORZDQRGHFDSDFLWDQFH Figure 7.17 Typical HeNe laser construction Laser Guide marketplace.idexop.com Types of Lasers A217 LASER GUIDE Laser Guide The most common types of gas-discharge lasers are helium neon lasers, helium cadmium lasers (a metalvapor laser), noble-gas ion lasers (argon, krypton), carbon-dioxide lasers, and the excimer-laser family. Each of these will be discussed briefly below. HELIUM NEON LASERS The helium neon (HeNe) laser, shown in figure 7.17, the second laser to be discovered, was the first to be used in volume applications. Today, millions of these lasers are in the field, and only semiconductor diode lasers are sold in greater quantity. The HeNe laser operates in a high-voltage (kV), lowcurrent (mA) glow discharge. Its most familiar output wavelength is 633 nm (red), but HeNe lasers are also available with output at 543 nm (green), 594 nm (yellow), 612 nm (orange), and 1523 nm (near infrared). Output power is low, ranging from a few tenths to tens of milliwatts, depending on the wavelength and size of the laser tube. Helium is the major constituent (85%) of the gas mixture, but it is the neon component that is the actual lasing medium. The glow discharge pumps the helium atoms to an excited state that closely matches the upper energy levels of the neon atoms. This energy is then transferred to the neon atoms via collisions of the second kind (i.e., exciting the neon to a higher energy level as opposed to transferring the energy as kinetic motion). One characteristic of the glow discharge is its negative impedance (i.e., increasing the voltage decreases the current); consequently, to function with a standard current-regulated power supply, a ballast resistor must be used in series with the laser to make the overall impedance positive. The popularity (and longevity) of the HeNe laser is based on five factors: they are (relative to other lasers) small and compact; they have the best inherent beam quality of any laser, producing a virtually pure single transverse mode beam (M2 < 1.05); they are extremely long lived, with many examples of an operating life of 50,000 hours or more; they generate relatively little heat and are convection cooled easily in OEM packages; and they have a relatively low acquisition and operating cost. A218 Types of Lasers HELIUM CADMIUM LASERS Helium cadmium (HeCd) lasers are, in many respects, similar to the HeNe laser with the exception that cadmium metal, the lasing medium, is solid at room temperature. The HeCd laser is a relatively economical, cw source for violet (442 nm) and ultraviolet (325 nm) output. Because of its excellent wavelength match to photopolymer and film sensitivity ranges, it is used extensively for three-dimensional stereolithography and holographic applications. As mentioned above, cadmium, a metal, is solid at room temperature. For lasing to occur, the metal must be evaporated from a reservoir, as shown in figure 7.18, and then the vapor must be distributed uniformly down the laser bore. This is accomplished through a process called electrophoresis. Because cadmium will plate out on a cool surface, extreme care must be taken in the design of the laser to contain the cadmium and to protect the optics and windows from contamination, since even a slight film will introduce sufficient losses to stop lasing. The end of life usually occurs when cadmium is depleted from its reservoir. NOBLE-GAS ION LASERS The noble-gas ion lasers (argon-ion and krypton-ion), have been the mainstay of applications requiring high cw power in the visible, ultraviolet, and near-infrared spectral regions. High-power water-cooled systems can be found in research laboratories around the world; lower power air-cooled systems are used in a wide variety of OEM applications. Argon-ion lasers are available with output %UHZVWHUZLQGRZ KHDWHU FDGPLXPUHVHUYRLU ERUH PDLQDQRGH FDWDSKRUHWLFDQRGH WKHUPLRQLF FDWKRGH ERUHVXSSRUW KHOLXPSXPS Figure 7.18 Construction of a HeCd laser 1-505-298-2550 FDGPLXP WUDS LASER GUIDE Optical Coatings & Materials The main features of both lasers are the same. Both use a coiled, directly heated dispenser cathode to supply the current; both have a gas return path that counteracts gas pumping (non-uniform gas pressure throughout the length of the tube caused by the charged particles moving toward the electrodes). ILQVIRUDLUFRROLQJ WKHUPLRQLF FDWKRGH LQWHUQDO PLUURUV $LU&RROHG,RQ/DVHU %UHZVWHU V DQJOHZLQGRZV GLVFERUHVWUXFWXUH H[WHUQDOVROHQRLG HOHFWURPDJQHW DQRGH ZDWHUFRROLQJSDWK WKHUPLRQLF FDWKRGH +LJK3RZHU:DWHU&RROHG,RQ/DVHU Figure 7.19 Air-cooled and water-cooled ion lasers Unlike atomic lasers, CO2 lasers work with molecular transitions (vibrational and rotational states) which lie at low enough energy levels that they can be populated thermally, and an increase in the gas temperature, caused by the discharge, will cause a decrease in the inversion level, reducing output power. To counter this effect, high-power cw CO2 lasers use flowing gas technology to remove hot gas from the discharge region and replace it with cooled (or cooler) gas. With pulsed CO2 lasers that use transverse excitation, the problem is even more severe, because, until the heated gas between the electrodes is cooled, a new discharge pulse cannot form properly. Machine Vision Guide H[WHUQDO PLUURUV CARBON DIOXIDE LASERS Because of their ability to produce very high power with relative efficiency, carbon dioxide (CO2) lasers are used primarily for materials-processing applications. The standard output of these lasers is at 10.6 µm, and output power can range from less than 1 W to more than 10 kW. Gaussian Beam Optics VROLG%H2ERUH DQRGH The main life-limiting factors in ion lasers are cathode depletion and gas consumption. The intense discharge drives atoms into the walls of the discharge tube where they are lost to the discharge. Over time the tube pressure will decrease, causing the discharge to become unstable. This is particularly a problem with krypton-ion lasers. Water-cooled systems typically have some refill mechanism to keep the pressure constant. Air cooled systems typically do not, limiting their practical operating life to approximately 5000 operating hours. Fundamental Optics Ion lasers can be broken into two groups: high-power (1 to 20+W) water-cooled lasers and low-power aircooled lasers. Both are shown schematically in figure 7.19. Water-cooled systems are available with either BeO bores or a construction wherein tungsten discs are attached to a thin-walled ceramic tube surrounded by a water jacket. The heat from the discs is conducted through the walls of the tube to the surrounding water. The entire bore structure is surrounded by a solenoid electromagnet, which compresses the discharge to increase current density and minimize bore erosion. Optical Specifications Unlike the HeNe laser, ion lasers operate with a highintensity low-pressure arc discharge (low voltage, high current). A 20 W visible laser will require 10 kW or more power input, virtually all of which is deposited in the laser head as heat which must be removed from the system by some cooling mechanism. Furthermore, the current densities in the bore, which can be as high as 105 A/cm2, place large stresses on the bore materials. The bore of an air-cooled system is always made of beryllium oxide (BeO), a ceramic known for its ability to conduct heat. A fin structure is attached to the outside of the ceramic bore, and a blower removes the generated heat, typically less than 1 kW. Material Properties up to 7 W in the ultraviolet (333 – 354 nm) and 25 W or more in the visible regions (454 – 515 nm), with primary output at 488 nm (blue) and 514 nm (green). Krypton-ion lasers have their primary output at 568 nm (yellow), 647 nm (red), and 752 nm (near infrared). Mixed-gas lasers combine both argon and krypton to produce lasers with a wider spectral coverage. Laser Guide marketplace.idexop.com Types of Lasers A219 LASER GUIDE Laser Guide A variety of types of CO2 lasers are available. High-power pulsed and cw lasers typically use a transverse gas flow with fans which move the gas through a laminar-flow discharge region, into a cooling region, and back again (see figure 7.20). Low-power lasers most often use waveguide structures, coupled with radio-frequency excitation, to produce small, compact systems. The principal advantage of an excimer laser is its very short wavelength. The excimer output beam can be focused to a spot diameter that is approximately 40 times smaller than the CO2 laser beam with the same beam quality. Furthermore, whereas the long CO2 wavelength removes material thermally via evaporation (boiling off material), the excimer lasers with wavelengths near 200 nm remove material via ablation (breaking molecules apart), without any thermal damage to the surrounding material. HOHFWURGH SEMICONDUCTOR DIODE LASERS The means of generating optical gain in a diode laser, the recombination of injected holes and electrons (and consequent emission of photons) in a forward-biased semiconductor p-n junction, represents the direct conversion of electricity to light. This is a very efficient process, and practical diode laser devices reach a 50% electrical-to-optical power conversion rate, at least an order of magnitude larger than most other lasers. Over the past 20 years, the trend has been one of a gradual replacement of other laser types by diode laser based– solutions, as the considerable challenges to engineering with diode lasers have been met. At the same time the compactness and the low power consumption of diode lasers have enabled important new applications such as storing information on compact discs and DVDs, and the practical high-speed, broadband transmission of information over optical fibers, a central component of the internet. ODPLQDUIORZ GLVFKDUJHUHJLRQ HOHFWURGH FRROLQJFRLOV EORZHU Figure 7.20 Schematics of transverse flow CO2 laser system EXCIMER LASERS The term excimer or “excited dimer” refers to a molecular complex of two atoms which is stable (bound) only in an electronically excited state. These lasers, which are available only as pulsed lasers, produce intense output in the ultraviolet and deep ultraviolet. The lasers in this family are XeFl (351 nm), XeCl (308 nm), KrF (248 nm), KrCl (222 nm), ArF (193 nm), and F2 (157 nm). They are used extensively in photolithography, micromachining, and medical (refractive eye surgery) applications. At first glance, the construction of an excimer laser is very similar to that of a transverse-flow, pulsed CO2 laser. However, the major difference is that the gases in the system are extremely corrosive and great care must be taken in the selection and passivation of materials to minimize their corrosive effects. A system built for CO2 would fail in minutes, if not seconds. A220 Types of Lasers CONSTRUCTION OF A DOUBLE-HETEROSTRUCTURE DIODE LASER In addition to a means to create optical gain, a laser requires a feedback mechanism, a pair of mirrors to repeatedly circulate the light through the gain medium to build up the resulting beam by stimulated emission. The stripe structures needed to make a laser diode chip are formed on a single crystal wafer using the standard photolithographic patterning techniques of the semiconductor industry. The substrate crystal axes are first oriented relative to the patterning such that, after fabrication, a natural cleavage plane is normal to the stripe direction, and cleaving both ends of the chip provides a pair of plane, aligned crystal surfaces that act as a Fabry-Perot resonator for optical feedback. These 1-505-298-2550 LASER GUIDE Optical Coatings & Materials p-type active layer output facet vx@10º 0.25 typical oxide insulator n-cladding bottom p-cladding contact (5) n-cladding Pout vy@30º dimensions in mm Figure 7.21 Schematic of a double heterostructure index-guided diode laser Gaussian Beam Optics DPout laser output DI Pout , optical output power Machine Vision Guide To make a planar waveguide that concentrates the light in the junction region (confinement between the top and bottom horizontal planes of the active region in figure 7.21), the cladding layers are made of an alloy of lower refractive index (larger band gap) than the active junction region. This is then termed a doubleheterostructure (DH) laser. The output power of the laser is horizontally polarized because the reflectivity of the planar waveguide is higher for the polarization direction parallel to the junction plane. Because the junction region is thin for efficient recombination (typically 0.1 µm), some light spreads into the cladding layers which are therefore made relatively thick (typically 1 µm) for adequate light confinement. p-cladding top contact (1) Fundamental Optics These compounds are direct band-gap semiconductors with efficient recombination of injected holes and electrons because no phonons (lattice vibrations) are required to conserve momentum in the recombination interaction. The injection layers surrounding the junction, the cladding layers, can be indirect band-gap semiconductors (where phonons are involved). To confine the light laterally (between planes perpendicular to the junction plane), two main methods (with many variants) are used. The first and simplest puts a narrow conductive stripe on the p-side of the device to limit the injected current to a line, giving a gain-guided laser. There is some spreading of current under the stripe, and the light is restricted only by absorption in the unpumped regions of the junction. The transverse mode of the laser light is therefore not tightly controlled. Optical Specifications The semiconductor crystal must be defect free to avoid scattering of carriers and of light. To grow crystal layers without defects, commercial semiconductor lasers use III-V compounds, elements taken from those columns of the periodic table. These form varying alloys with the addition of dopants that can be lattice-matched to each other and to the initial crystal substrate. The band gap of the semiconductor chosen determines the lasing wavelength region. There are three main families: GaN-based lasers with UV-blue outputs, GaAs-based lasers with red-near infrared outputs, and InP-based lasers with infrared outputs. These base crystals are precisely doped with Ga, Al, In, As, and P to precisely control the band gap and index of refraction of the layers in the diode structure. GAIN GUIDING AND INDEX GUIDING IN DIODE LASERS Material Properties mirrors use either the Fresnel reflectivity of the facet (often sufficient because of the high gain of diode lasers), or they can be dielectric coated to other reflectivities. This might be desired, for instance, to protect against damage from the high irradiance at the facets. This geometry gives the familiar edge-emitting diode laser (see figure 7.21). spontaneous emission Ith Imax I, forward drive current Figure 7.22 Definition of threshold current, Ith, and slope efficiency from the curve of light output, Pout vs drive current I Laser Guide marketplace.idexop.com Types of Lasers A221 LASER GUIDE Laser Guide Many high-power diode lasers, used for instance in sidepumping another solid-state laser (where mode control is less critical), are gain guided. More efficient lateral laser mode control is achieved by fabricating, with multiple photolithographic, epitaxial, and etching steps, regions of low index of refraction on either side of the lasing stripe (the two lateral n-cladding regions in the upper half of figure 7.21). This confines the light by waveguiding between planes perpendicular to the junction plane as well giving an index-guided laser. These lasers produce a stable single transverse mode of lowest order, as required in data storage applications to read compact discs, and telecommunications applications where coupling into a fiber optic is important. THRESHOLD CURRENT AND SLOPE EFFICIENCY DEFINITIONS Output power from a diode laser increases linearly with the drive current excess above the threshold current (see figure 7.22). This steeply rising light output curve is extrapolated backward to the zero light output intercept to define the threshold current; the weak incoherent light emission for currents below threshold is due to the spontaneous recombination of carriers such as occurs in LEDs. When divided by the drive voltage V, the slope of the output vs current curve yields the differential (above threshold) electrical-to-optical power conversion efficiency (also termed the slope or quantum efficiency) which ranges from 50 to 80 percent for various devices. Slope efficiency = DPout .(7.25) V DI FABRICATION METHODS AND QUANTUM WELLS Three types of epitaxial crystal growth are employed in fabricating the layers of semiconductor alloys for diode laser chips: liquid phase epitaxy (LPE), metal-organic chemical vapor deposition (MOCVD), and molecular beam epitaxy (MBE). Most early diode lasers were made by the LPE process, and it is still in use for many commercial diode lasers A222 Types of Lasers and LEDs. In this process, a heated, saturated solution is placed in contact with the substrate, and it is cooled, leaving an epitaxial film grown on the substrate. Highquality crystal layers are readily produced by this technique, but it is hard to control alloy composition. Furthermore, making thin layers is difficult. Because quantum well (QW) structures, discussed below, require very thin layers, the LPE process is not appropriate for these devices; they are fabricated using the MOCVD or MBE process. In the MOCVD process, gases transport the reactants to the heated substrate, where they decompose and the epitaxial layer slowly grows. In the high-vacuum MBE process, the reactants are evaporated onto the substrate, giving a very slow, controlled epitaxial growth. The equipment for MBE is more expensive, and the process is slower, making this process most suitable for critical and complex devices of low production volume. The emergence of the MOCVD and MBE processes made possible improved diode lasers employing quantum well structures as their active regions. A quantum well is a layer of semiconductor of low electron (or hole) potential energy between two other layers of higher potential energy. The well layer is made thin enough, typically less than 0.01 µm, to be comparable in size to the Bohr radius of the electron (or hole) in the material. This brings in quantum effects—the confined carrier acts, in the direction perpendicular to the layer plane, as a one-dimensional particle in a potential well. In practical terms, the density of carriers is greatly increased in this QW structure, and the laser threshold current decreases by an order of magnitude. The laser’s active region is effectively an engineered, manmade material whose properties can be designed. There is a disadvantage to QW lasers: the active region is too thin to make a reasonable waveguide. This problem is solved by inserting intermediate layers of graded index between the QW and both cladding layers. This is termed the graded-index separate-confinement heterostructure (GRINSCH) since the carriers are confined to the QW while the laser mode is confined by the surrounding layers. The electrical and optical confinements are separate. For higher output power, 1-505-298-2550 LASER GUIDE Optical Coatings & Materials The epitaxial growth process of this structure is more difficult than that for edge emitters This is because provision must be made to channel current flow around the mirrors to reduce device resistance (for clarity, the bypass channels are omitted in figure 10.23) and because precise control of the mirror layer thicknesses is needed to locate standing wave peaks at the QW active layer(s). Countering these drawbacks, by having no facets to cleave, these lasers have a similar topology to LEDs. They can be tested at the wafer level and packaged using similar low-cost manufacturing methods. In addition, VCSELs have large-area circular beams (defined by the circular limiting aperture of the mirrors) and low threshold currents coupling well into optical fibers for low-power (~1 mW) communication applications. Gaussian Beam Optics Machine Vision Guide To address this issue, gratings are fabricated into the laser, either at the ends of the gain stripe to create a distributed Bragg reflector (DBR) structure, or along the whole length of the gain region to create a distributed feedback (DFB) structure. The grating has a period on the order of 200 nm and is fabricated using interferometric techniques. (The beam from an argon or HeCd laser is split into two; the beams are are then overlapped to create fringes, which in turn are used to expose the photoresist in the photolithography process.) The gratings work by providing a small reflected Recently, another surface-emitting structure, the verticalcavity surface-emitting laser (VCSEL), has come into use in telecommunication links. In this structure (see figure 7.23) multilayer mirrors are fabricated on the top and bottom of the QW gain region to give feedback. Consequently, the laser output is perpendicular to the active QW plane. Fundamental Optics WAVELENGTH STABILIZATION WITH DISTRIBUTED, SURFACE-EMITTING OUTPUT GEOMETRIES The wavelength of a AlGaAs diode laser tunes with substrate temperature at a rate of about 0.07 nm/°C, a rapid enough rate that many applications require the baseplate of the device to be mounted on a temperature controlled thermoelectric cooler to maintain wavelength stability. Wavelength, threshold current, and efficiency are all sensitive to changes in temperature. If the laser baseplate temperature is allowed to drift, in addition to this long-term shift in wavelength, the output oscillation will jump between drifting longitudinal cavity modes and thus exhibit small, rapid, discontinuous changes in wavelength and/or output power which often are undesirable. The DFB laser is an edge emitter. In the second-order gratings fabricated in both DFB and DBR lasers, the firstorder diffraction is perpendicular to the surface of the grating. By providing an output window on one of the gratings in a DBR laser, the output can be brought out through the surface of the chip, i.e., a surface-emitting laser. Optical Specifications The lasing wavelength in QW lasers is determined by both the bulk band gap and the first quantized energy levels; it can be tuned by varying the QW thickness. Further adjustment of the wavelength is possible with strained quantum QWs. If an epitaxial layer is kept below a critical thickness, an alloy with a lattice mismatch to the substrate will distort its lattice (in the direction normal to the substrate) to match the substrate lattice instead of causing misfit dislocations. The strain in the lattice of the resulting QW changes its band gap, an effect taken advantage of to push the lasing wavelength into a desired region. feedback at each index step. The single frequency whose multiple feedback reflections add up in phase determines the lasing wavelength and stabilizes it against changes in drive current and baseplate temperature. Because the laser operates in a single frequency, noise is also reduced. DBR and DFB lasers are used extensively as telecommunication light sources. Material Properties several QWs separated by buffer layers can be stacked on top of one another,—a multiple quantum well (MQW) structure. A structure with only one quantum layer is designated a single quantum well (SQL) to distinguish it from a MQW. DIODE LASER BEAM CONDITIONING Because the emitting aperture is small on a typical diode laser, beam divergences are large. For example, the emitting area for the index-guided laser shown in figure 7.21 might be as small as 3x1 µm, resulting in divergence of 10ºx30º. The optics needed to collimate this beam Laser Guide marketplace.idexop.com Types of Lasers A223 LASER GUIDE Laser Guide or to focus it into a fiber must work at a high numerical aperture, resulting in potential lens aberrations, and requiring critical focusing because of short depth of field. Focal lengths must be kept short as well or the optics rapidly become large. The beam itself is elliptical and may be astigmatic. It is often desirable to first circularize the beam spot with an anamorphic prism pair or cylinder lens before coupling the laser output into an optical train. Higher-power lasers with high-order modes cause additional problems when coupling their beams into a fiber or optical system. A wide variety of specialized components are available to address these issues, from molded miniature aspherical lenses to hyperbolic profile fiber cylinder lenses, but all require critical focusing adjustment in their mounting into the laser diode housing. For these reasons many diode lasers are offered with beam-correcting optics built in by the manufacturer who has the appropriate tooling for the task. Typically these lasers are available as collimated units, or as fibercoupled (“pigtailed”) devices. HIGH-POWER DIODE LASERS Single transverse mode diode lasers are limited to 200 mW or less of output power by their small emitting aperture. The facet area is so small (about 3x1 µm) that this power still represents a high irradiance (~7 MW/cm2). The output is limited to this level to stay safely under the irradiance that would cause damage to the facet. 3RXW +5 PLUURUVWDFNV GLHOHFWULF RU'%5V WRSFRQWDFW 4: DFWLYHOD\HU VXEVWUDWH ERWWRP FRQWDFW Figure 7.23 Schematic of the VCSEL structure, with light emitted perpendicular to the active layer A224 Types of Lasers Enlarging the emitting area with an increase of the lateral width of the active stripe is the most common method of increasing the laser output power, but this also relaxes the single transverse mode constraint. Multiple transverse lateral modes, filaments, and lateral mode instabilities arise as the stripe width increases. For example, in a GaAs laser running at 808 nm, the output power rises linearly from 500 mW to 4 W as the lateral width of the emitting aperture increases from 50 to 500 µm. However, the M2 value of the beam in this plane increases from 22 to 210. The M2 increase makes it difficult to couple these devices to fibers, but they find considerable application in pumping solid-state laser chips designed to accept a high-numerical-aperture focus. The pump diode lasers for even higher-power DPSS lasers are made as linear arrays of 20 or more stripe emitters integrated side by side on a 1 cm long semiconductor bar. The bar is mounted in a water-cooled housing to handle the heat load from the high drive current. These diode laser arrays provide from 20 to 40 W of continuous output power at wavelengths matching the absorption bands of different laser crystals (e.g., 808 nm for pumping Nd:YAG lasers). The individual stripe emissions are not coherently related, but bars can be used to side pump a laser rod, just as the arc lamps they replace formerly did. Another common delivery geometry is a bundle of multimode fibers, fanned into a line of fibers on one end with each fiber butted against an individual stripe on the bar, with the other end of the bundle gathered into a circular grouping. This converts the bar output into a round spot focusable onto the end of the crystal to be pumped. Finally, for even more output, a few to a dozen bars are mounted like a deck of cards one on top of another in a water-cooled package, connected in series electrically, and sold as a stacked array. These can deliver in excess of 500 W output power from one device. PACKAGING, POWER SUPPLIES, AND RELIABILITY For low-power lasers, the industry uses standard semiconductor device package designs, hermetically sealed with an output window. Lasers with higher power dissipation come with a copper baseplate for attachment to a finned heat sink or thermoelectric 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Machine Vision Guide SUMMARY OF APPLICATIONS The applications mentioned in the discussion above, and a few others, are summarized in the following table and ordered by wavelength. The newer GaN lasers provide low power (10 – 100 mW) blue and UV wavelengths In general, ignoring the efficiency of the pump laser itself, laser pumping is a much more efficient mechanism than lamp pumping because the wavelength of the pump laser can be closely matched to specific absorption bands of the lasing medium, whereas most of the light from a broad spectrum lamp is not usefully absorbed in the gain medium and merely results in heat that must be removed from the system. Furthermore, the size of the laser pump beam can be tightly controlled, serving as a gain aperture for improving the output mode characteristics of the pumped laser medium. On the other hand, laser pumping is often not suitable for high-energy applications where large laser crystals are required. Diode-pumped solid-state (DPSS) lasers, a class of laser-pumped lasers, will be discussed in detail below. Gaussian Beam Optics Diode lasers degrade with high power and long operating hours as crystal defects migrate and grow, causing dark lines or spots in the output mode pattern and increases in threshold current or decreases in slope efficiency. The best way to prolong life is to keep the laser baseplate running cool. Remember that accelerated life tests are run by operating at high baseplate temperature. Expectations for the median life of a device are set from measurements of large populations— individual devices can still suddenly fail. Nevertheless, the industry expectations today for standard diode lasers run within their ratings is ~105 hours of operation for lowpower diode lasers and perhaps an order of magnitude less for the high-power versions. Optically pumped lasers use photons of light to directly pump the lasing medium to the upper energy levels. The very first laser, based on a synthetic ruby crystal, was optically pumped. Optically pumped lasers can be separated into two broad categories: lamp-pumped and laser-pumped. In a lamp-pumped laser, the lasing medium, usually a solid-state crystal, is placed near a high-intensity lamp and the two are surrounded by an elliptical reflector that focuses the light from the lamp into the crystal, as shown in figure 7.24. In laser-pumped systems, the light from another laser is focused into a crystal (or a stream of dye), as shown in figure 7.25. Fundamental Optics Diode lasers are susceptible to permanent damage from static electricity discharges or indeed any voltage transient. Their low operating voltage (~2 V) and ability to respond at high speed means that a static discharge transient can be a drive current spike above the maximum safe level and result in catastrophic facet damage. All the usual antistatic electricity precautions should be taken in working with diode lasers: cotton gloves, conductive gowns, grounded wrist straps, work tables, soldering irons, and so on. Correspondingly, the drive current power supply should be filtered against surges and include “slow starting” circuitry to avoid transients. OPTICALLY PUMPED LASERS Optical Specifications Careful heat sinking is very important because all the major device parameters— wavelength, threshold current, slope efficiency, and lifetime— depend on device temperature (the cooler, the better). Temperatureservoed TECs are preferred for stable operation with the temperature sensor for feedback mounted close to the diode laser. finding applications as excitation sources for biomedical fluorescence studies (DNA sequencing, confocal microscopy). The dominant application for diode lasers is as readouts for optical data storage, followed by growing numbers in use in telecommunications. For high-power (>1 W) diode lasers, the main application is as optical pumps for other solid state lasers. Material Properties cooler (TEC). Many are offered coupled into a fiber at the manufacturing plant in a pigtailed package (with an output fiber attached) because of the criticality in mounting the coupling optics as mentioned above. DIODE-PUMPED SOLID STATE LASERS THE DPSS LASER REVOLUTION The optical difficulties encountered with diode lasers— difficulty in coupling to the high divergence light, poor mode quality in the slow axis of wide-stripe lasers, low output power from single-transverse-mode lasers— led Laser Guide marketplace.idexop.com Types of Lasers A225 LASER GUIDE Laser Guide to a new philosophy (figure 7.26) about how best to use these efficient, long-lived, compact light sources. This concept, championed in the 1980s by a group at Stanford University headed by Prof. Bob Byer, has been termed the diode-pumped solid state (DPSS) laser revolution. The logic is simple. The primary light source (the diode laser) pumps another laser (an infrared crystal laser) to convert to a good mode, the beam of which is wavelength converted (by nonlinear optics techniques) to a visible output. The diode laser source replaces the discharge lamp for optically pumping the gain crystal in a traditional, high-efficiency, infrared laser. The infrared beam is generated in that independent resonator with a good mode, and consequently it can be efficiently converted with an intracavity nonlinear crystal to a visible beam with a good mode. Though power is lost at each step, the result is still a single-mode visible beam generated with a total electrical-to-optical conversion efficiency of several percent. These DPSS lasers are replacing the older visible gas lasers whose conversion efficiencies rarely reach 0.1%. END- AND SIDE-PUMPING GEOMETRIES The first DPSS lasers were made by focusing the diode light from a single laser diode emitter through the highreflector coating (at 1.06 µm) on the end of the Nd:YAG rod. This “end-pumping” geometry provided good overlap between the pumped volume and the lasing volume, but it limited the pump power to that available from single-mode diode emitters. In order to increase laser output and reduce cost, diode arrays were mounted along the length of the laser rod (diode lasers suitable for end pumping are twice as expensive as diode laser arrays). However, because of poor overlap of the pump beam with the 1.06 µm beam, the efficiency of this “sidepumping” technique was only half that of end-pumping geometries. No pump diode cost savings resulted. Then in the late 1980s two advances were made. First, a variety of new laser rod materials, better tailored to take advantage of diode laser pumps, were introduced. Nd:YVO4 crystals have five times the gain cross section of Nd:YAG, and the Nd can be doped into this crystal at much higher concentrations. This decreases the absorption depths in the crystal from cm to mm, easing A226 Types of Lasers the collimation or focusing quality required of the pump beam. This crystal had been known, but could be grown only to small dimensions, which is acceptable for diodepumped crystals. Another crystal introduced was Yb:YAG, pumped at 980 nm and lasing at 1.03 µm, leaving very little residual heat in the crystal per optical pumping cycle and allowing small chips of this material to be pumped at high levels. Second, means were devised to make micro-cylindrical lenses (focal lengths less than 1 mm) with the correct surfaces (one type is a hyperbolic profile) for collimating or reducing the fast-axis divergence of the diode laser output. With good tooling and beam characterization these are correctly positioned in the diode beam and bonded in place to the diode housing. This allows more conventional lenses, of smaller numerical aperture, to be used in subsequent pump light manipulations. HOOLSWLFDO UHIOHFWRU KLJKUHIOHFWRU IODVKODPS ODVLQJFU\VWDO RXWSXWFRXSOHU Figure 7.24 Schematic of a lamp-pumped laser SXPS ODVHU KLJKUHIOHFWRU WUDQVPLWVSXPSZDYHOHQJWK ODVLQJFU\VWDO RXWSXW FRXSOHU PRGHPDWFKLQJ RSWLFV Figure 7.25 Schematic of a laser-pumped laser 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Diode Laser Applications Wavelength λ (nm) Lattice Material* 375 GaN Biomedical fluorescence Application GaN Biomedical fluorescence, DVD mastering, direct-to-plate GaN Biomedical fluorescence, HeCd laser replacement Material Properties 405 440 GaN Biomedical fluorescence GaN Biomedical fluorescence 635 – 640 GaAs Pointers, alignment, HeNe laser replacement 650 – 680 GaAs Biomedical fluorescence, barcode scanners, pointers, alignment, surgical 780 GaAs Audio CD readouts 785 GaAs Raman spectroscopy 808 GaAs Optical pumps for Nd:YAG lasers, thermal printing 940 InP Optical pumps for Yb:YAG lasers InP Optical pumps for Er fiber telecom amplifiers InP Input source for telecom short-wavelength channels, OCT 1455 InP Optical pump for Raman gain in standard telecom fiber 1550 InP Input source for telecom long-wavelength channels PRGH FRQYHUWHU ZDYHOHQJWK FRQYHUWHU SXPSGLRGH ODVHU ,5ODVHU FU\VWDO QRQOLQHDU FU\VWDO KLJKHII h3 JRRGHII hM # JRRGHII hl # SRRUPRGH M JRRGPRGH M JRRGPRGH M h7 h3hMhl # Gaussian Beam Optics SULPDU\OLJKW VRXUFH Fundamental Optics 980 1310 Optical Specifications 473 488 Figure 7.26 The logic for DPSS lasers Figure 7.27 shows the example previously mentioned, delivering the array light through a fiber bundle, with the fibers at one end spread out to butt-align with the linear stripes of an array, and the other end of the bundle gathered to an approximately circular spot. Machine Vision Guide END-PUMPING WITH BARS With these two new degrees of freedom, laser designers realized they could create optical trains that would give them end-pumping system efficiencies (achieve good overlap between pump and lasing modes) with diode arrays as pump sources to obtain a lower diode cost per watt in their systems. This produced an explosion of unique DPSS laser designs generically described as “end-pumping with bars.” Although the circular spot is large, its focal image, formed with high numerical aperture (NA) optics, is small enough to satisfactorily overlap the IR cavity laser mode. The small depth of focus, from the high NA optics, is inconsequential here because of the short absorption depth in the Nd:YVO4 laser crystal. The laser head can be disconnected from the diode modules at the fiber coupler without loss of alignment. Laser Guide marketplace.idexop.com Types of Lasers A227 LASER GUIDE Laser Guide In another example, an even higher NA optic (comprising a cylindrical lens and a molded aspheric lens) was used to directly focus the 1 cm width of a micro-lensed array bar onto the end of a Nd:YVO4 gain crystal. This produced an oblong pump spot, but good overlap with the IR cavity mode was achieved by altering the infrared cavity (inserting two intracavity beam expansion prisms in that arm) to produce a 5:1 elliptical cavity mode in the gain crystal. Another design used a nonimaging pyramidal “lens duct” to bring in the pump light from a diode laser stack to the end of a gain crystal. Yet another brought light from several arrays into a lasing rod centered in a diffuse-reflecting cavity by means of several planar (glassslide) waveguides, each piercing a different sector of the reflector sidewall. These are but a few of the design approaches that have been successfully taken. 488 NM SOLID-STATE LASERS A variety of solid-state approaches are used to achieve 488 nm output, a popular wavelength used for excitation of fluorophores in biotech applications. One approach, optically pumped semiconductor lasers (OPSL), utilizes a diode to pump a solid state gain chip atop a Bragg reflector which then passes through an external non linear crystal to double the 976 nm fundamental radiation down to 488 nm. The OPSL approach typically requires a large number of components and more complex fabrication, coating and assembly compared to other approaches that follow. True DPSS approaches utilize a pump diode, two gain mediums and a frequency mixing of 914 and 1064 nm radiation along with a single frequency generator to produce 491 nm, a wavelength close but not quite optimum for the narrow bandwidth filters and fluorophores used in these applications. Directly doubled diodes (DDD) offer the best of both worlds by generating the optimum 488 nm wavelength and using a minimal number of components to achieve it (see figure 7.28). DDD’s use a pump diode at 976 nm and, unlike other doubled diode laser approaches which use external cavity components, an integrated nonlinear crystal/cavity. This approach results in a laser that is highly efficient, robust and volume manufacturable. Microchip lasers Another procedure that can be used to make available potentially inexpensive, mass produced, low power, A228 Types of Lasers visible output DPSS lasers is mimicking semiconductor chip processing methods. In the late 1980s, MIT Lincoln Labs took this approach and created the “microchip” laser. A thin Nd:YVO4 plate is polished flat and then diced into ~2 mm square chips. Each of these chips is then optically contacted to similar, flat, diced KTP doubling crystal plates to make a cube. Prior to dicing, the surfaces that will become the outer cube surfaces are coated for high reflectivity at 1.06 µm. When single-mode diode laser pump light is focused through one mirrored end of the cube, the heat produced makes a thermallyinduced waveguide that creates a stable cavity for IR lasing. Since the KTP crystal is within this cavity, the IR lasing is converted to a 532 nm (green) output beam with 10’s of milliwats of output. The diode temperature must be controlled to maintain a stable pump wavelength and RXWSXW : +5 1G<92 ILEHU EXQGOH IRFXVLQJ RSWLFV GLRGHEDU : GLRGHEDU Figure 7.27 Schematic of an “end-pumping with bars” geometry using fiber bundle delivery, one of many variants on the DPSS laser theme fundamental radiation ~976 nm SHG output ~488 nm pump diode nonlinear crystal Figure 7.28 Directly doubled diodes 1-505-298-2550 LASER GUIDE Optical Coatings & Materials X the means for optically coupling the pumping light into the gain medium, X the management of the thermal lens produced by absorption of the pump light in the cavity, X the control of green noise, X the strain-free mounting, heat sinking, and placement of the small lasing and nonlinear crystals in the laser cavity, and Fundamental Optics X the hermetic sealing of the laser cavity to protect the often delicate crystals and critical alignments of components Note that because the intracavity space must be hermetically sealed there usually is no field repair, maintenance, or adjustment of a DPSS laser head. If it fails, it is returned to the manufacturer. Gaussian Beam Optics It is evident that DPSS lasers are a lot less generic than the gas lasers they replace. For a problem with a particular laser model, there may be no standard solution available in the technical literature. With so many variables, there often are surprises when new designs are first manufactured and introduced. Under these circumstances, the user is advised to pick a supplier with a record of years of consistent manufacture, who has over time dealt with their own unique set of component and assembly problems. If this advice is followed, then the expectation with current products is that a new DPSS laser will operate reliably for 10,000 hours or more. Machine Vision Guide Two early solutions to this problem emerged. The first is to make the IR cavity long enough to give hundreds of oscillating modes, so that the noise terms average to insignificance as in a long gas laser. The second is to make the IR oscillation run on a single longitudinal mode so that there are no SFM terms. This can be done by using intracavity frequency control elements such as an etalon, or by using a ring cavity (with a Faraday effect biasing element to maintain the direction of light travel around the ring). Ring cavities eliminate the standingwave interference effect of linear cavities, termed “spatial hole burning,” and the laser runs single frequency when this is done. As more experience was gained with DPSS laser design, other clever solutions to the “green problem” were found, tailored to each particular device UNIQUENESS OF DPSS LASER DESIGNS AND LASER RELIABILITY Unlike the gas lasers they replace, no universal approach is applied in the details of different DPSS laser designs and laser models. There is a large variety of solutions to the major problems, many solutions are unique, and many are held as proprietary. Major design differences are found in: Optical Specifications THE “GREEN NOISE” PROBLEM As the early DPSS laser designs giving visible-output beams were being introduced, it became apparent that there was a problem unique to this architecture. The visible output power, 532 nm in the green spectrum, could break into high-frequency chaotic oscillations of nearly 100% peak-to-peak amplitude. This was named the “green problem” by Tom Baer (then at SpectraPhysics), who in 1986 showed the effect to be due to the dominance of sum-frequency-mixing (SFM) terms coupling different longitudinal modes over secondharmonic-generation (SHG) terms, in the nonlinear conversion step from IR to visible output. Several conditions (all met by the new laser designs) lead to this effect: (1) the IR laser cavity is short (~10 cm or less) with only a few longitudinal modes oscillating, (2) nonlinear conversion efficiencies are high (20% or more), and (3) nonlinear phase-matching bandwidths span several longitudinal mode spacings (true of the commonly used KTP doubling crystal). Then the sum frequency mixing output losses couple the longitudinal modes in relaxation oscillations where the turn on of one mode turns off another. and often held as trade secrets. It can be surmised that these involve precise control of wavelength, spatial hole burning, beam polarization, and cavity-element optical path differences to reduce the strength of longitudinal mode SFM terms. Material Properties thermal waveguide. In addition, the cube temperature should be stabilized. Because of the short cavity, the IR laser operates at a single longitudinal mode, and the cavity length must be thermally tuned to keep the mode at the peak of the gain curve. Laser operation in a single frequency suppresses “green noise,” discussed next. Laser Guide marketplace.idexop.com Types of Lasers A229 LASER GUIDE Laser Guide 561 nm DPSS LASER The newest addition to the Melles Griot laser product line is a DPSS laser with yellow output at 561 nm, an ideal excitation wavelength for biomedical fluorescence. This 16.5-cm-long laser head delivers up to 75 mW of output power, and consumes less than 10 W of wall plug power at 25 mW and less than 18 W at 75 mW. The laser is pumped by a single stripe diode laser. Frequencyselective elements in the cavity limit IR oscillation to the 1.123 µm Nd line (one of the weaker lines in the 1.064 µm manifold) and constrain this oscillation to a single longitudinal mode. The output is low noise (0.5% rms) with excellent mode quality (M2< 1.2). Polarization is vertical with respect to the mounting surface with an extinction ratio of >100:1. AN EXAMPLE OF A DPSS LASER PRODUCT LINE— THE MELLES GRIOT VISIBLE OUTPUT LASERS Figure 7.29 depicts the mix of laser crystals, laser operating wavelengths, and doubling crystals generating the four visible output wavelengths of the present Melles Griot product line of continuous wave DPSS lasers. OTHER NOTABLE DPSS LASERS A brief discussion of three other significant DPSS laser developments conclude this section. The Er-doped fiber amplifier (EDFA) is not a laser, but it is an optically pumped amplifier for the 1550 nm longwavelength long-haul fiberoptic channels that make modern telecommunications possible. Pumping an Er-doped silica fiber with 980 nm diode laser light inverts the populations of energy levels in the Er ions to provide gain for optical telecommunication signals run through the same fiber. This optical amplifier is much simpler than the discrete electronic repeaters it replaced. A Lucent Technologies executive expressed the importance of this when he said: “What broke [wavelength division multiplexing telecommunications] free was the invention of the [EDFA] optical amplifier.” mixing the green beam with the residual transmitted infrared to 355 nm. This process is straightforward in a high-peak-power pulsed beam—just a matter of inserting the appropriate doubling and tripling crystals. What is remarkable is that DPSS laser designs have matured sufficiently to make this possible in a hands-off, longlived system rugged enough to survive and be useful in an industrial environment. The double-clad fiber laser is shown in figure 7.30. Fiber lasers work by optically pumping (with a diode laser) a doped fiber and adding mirrors for feedback at either end of the fiber. In the dual-clad fiber, the Yb-doped single-mode fiber core is surrounded by a large diameter cladding (with a corrugated star-shaped cross section in the figure) that is itself clad by a low-index polymer coating. Diode laser light at 940 nm is readily launched into and guided in the large diameter outer cladding, and the corrugated cross-section of this fiber suppresses the helical ray modes of propagation that would have poor overlap with the inner core. Over the length of the fiber, the pump light is absorbed by the singlemode core, and high- power lasing near 1.03 µm in a low- order mode is produced. The quantum efficiency of the Yb lasing cycle (ratio of pump wavelength to lasing wavelength) is 91%, which leaves little heat deposited in the fiber. Over 1 kW of output at 80% slope efficiency has been produced in such a fiber laser. These will become important laser sources for industrial applications. VLQJOHPRGH ODVHURXWSXW ODVHU GLRGHDUUD\ VLOLFDFODGGLQJ VLQJOHPRGHFRUH<E ORZLQGH[SRO\PHU The Q-switched industrial DPSS laser is a 1 W average power, ultraviolet (355 nm), high-repetition-rate (30 kHz) system. Output is obtained by doubling the 1.064 µm Q-switched fundamental to green at 532 nm, and then A230 Types of Lasers Figure 7.30 Schematic diagram of the structure of a double-clad fiber, and the method of pumping the inner core by direct illumination into the large diameter of the outer cladding 1-505-298-2550 LASER GUIDE Optical Coatings & Materials pump diode 808 nm coupling optics SHG non-linear crystal l output LBO/KTP 532 nm Nd:YVO4 914 nm BBO 457 nm Nd:YAG 946 nm LBO 473 nm Nd:YAG 1123 nm LBO 561 nm pump diode 976 nm nonlinear crystal ~488 nm Optical Specifications Nd:YVO4 1064 nm Material Properties IR laser Crystal Fundamental Optics 10,000 AVAILABLE LASER WAVELENGTHS (nm) 1000 Gaussian Beam Optics 100 10 514 520 532 543 561 568 594 633 635 640 647 650 660 670 676 685 780 830 502 505 1 405 425 440 454 457 465 473 476 483 488 496 MAXIMUM POWER (mW) Figure 7.29 Melles Griot solid-state laser optical trains for producing five different visible output wavelengths DPSS DIODE GAS Machine Vision Guide Available Wavelengths, Technology and spectral offering Laser Guide marketplace.idexop.com Types of Lasers A231 LASER GUIDE LASER APPLICATIONS Laser Guide Lasers have become so much a part of daily life that many people may not realize how ubiquitous they are. Every home with a CD player has a laser; hardware stores are now selling a wide variety of laser levels; many, if not most, computers, printers, and copiers are using laser technology. Laser applications are so numerous that it would be fruitless to try to list them all; however, one can give some illustrative examples of how lasers are used today. x-y VFDQQLQJPLUURU 89ODVHU IODWILHOG IRFXVLQJOHQV SURWRW\SHSDUW FKHPLFDOEDWK PRYLQJYHUWLFDOVWDJH INDUSTRIAL APPLICATIONS High-power lasers have long been used for cutting and welding materials. Today the frames of automobiles are assembled using laser welding robots, complex cardboard boxes are made with laser-cut dies, and lasers are routinely used to engrave numbers and codes on a wide variety of products. Some less well-known applications include three-dimensional stereolithography and photolithography. THREE-DIMENSIONAL STEREOLITHOGRAPHY Often a designer, having created a complex part in CAD software, needs to make a prototype component to check out the dimensions and fit. In many cases, it is not necessary for the prototype to be made of the specified (final) material for this checking step, but having a part to check quickly is important. This is where rapis prototyping comes in, i.e., three-dimensional stereolithography. The stereolithography machine consists of a bath of liquid photopolymer, an ultraviolet laser, beam-handling optics, and computer control (see figure 7.31). When the laser beam is absorbed in the photopolymer, the polymer solidifies at the focal point of the beam. The component design is fed directly from the CAD program to the stereolithography computer. The laser is scanned through the polymer, creating, layer by layer, a solid, three-dimensional model of the part. PHOTOLITHOGRAPHY Lasers are used throughout the manufacture of semiconductor devices, but nowhere are they more important than in exposing photoresist through the masks used for creating the circuits themselves. Originally, ultraviolet mercury lamps were used as the light sources to expose the photoresist, but as features became smaller and more complex devices were put A232 Laser Applications Figure 7.31 A laser stereolithography system for rapid prototyping of three-dimensional parts on a single wafer, the mercury lamp’s wavelengths were too long to create the features. In the 1990’s, manufacturers started to switch to ultraviolet lasers operating at approximately 300 nm to expose the photoresist. Manufacturers are now using wavelengths as short as 193 nm to get the resolution needed for today’s semiconductor integrated circuit applications. MARKING AND SCRIBING Lasers are used extensively in production to apply indelible, human and machine-readable marks and codes to a wide variety of products and packaging. Typical applications include marking semiconductor wafers for identification and lot control, removing the black overlay on numeric display pads, engraving gift items, and scribing solar cells and semiconductor wafers. The basic marking system consists of a laser, a scanning head, a flat-field focusing lens, and computer control. The computer turns the laser beam on and off (either directly or through a modulator) as it is scanned over the surface to make the mark. Depending upon the application, scanning may occur in a raster pattern (typical for making dot-matrix marks) or in a cursive pattern, with the beam creating letters one at a time. The mark itself results either from ablation of the surface of the material, or by a photochemically induced change in the color of the material. Another marking technique, used with high-energy pulsed CO2 and excimer lasers, is to shine the light through a mask containing the marking pattern and focusing the resulting image onto the marking surface. 1-505-298-2550 LASER GUIDE Optical Coatings & Materials Lasers used in this application have to have excellent pointing stability, constant wavelength and power Interferometric Measurement: Interferometric measurement can be used for high-resolution position measurement as well as for measuring waveform deformation of optical beams as they pass through a component or system (see figure 7.33). The technique uses the wave periodicity of the light beam as a very fine ruler. The position of an object in the path of the beam is computed from the phase of the light reflected from it. Interference between the object beam and a reference beam provides measureable intensity variations which yield this phase information. Distance and velocity measurement can be performed for moving objects as long as the fringe-recording mechanism is paced with it. Machine Vision Guide Scatter Measurement: In the semiconductor industry, patterns of material are deposited on a wafer substrate using photolithographic processes. Defects on the wafer can result in poor reliability, disconnects in circuitry, or complete circuit failure. Consequently manufacturers need to map the wafer to determine the defects’ location and size so that they can either be eliminated or avoided. To do this, they scan the wafer with a laser and measure backscatter with a very sensitive photodetector array. Gaussian Beam Optics NONCONTACT MEASUREMENT There are many types of laser-based noncontact measurement techniques in use today, including scatter measurement, polarimetry and ellipsometry, and interferometric measurement. Polarimetry and Ellipsometry: The optical phase thickness of a thin film can be carefully measured using polarimetry or ellipsometry (see figure 7.32). A beam of known polarization and phase state enters the thin film layer at an angle. The thin film has a known index of refraction. The measured phase change in the reflected beam is then correlated to an optical phase thickness for that layer using the known index of refraction. This technique can also be used with a thicker transparent media, such as glass, where changes in the polarization and phase state of a beam scanned across the substrate indicate variations in index of refraction due to inclusions or stress-induced birefringence. The most common lasers used in these applications are violet, red and near infrared single-emitter laser diodes and mid-visible diode-pumped solid-state lasers owing to their cw output, low noise, and compact sizes. Fundamental Optics Currently, most volume marking applications are performed with lamp-pumped YAG-based pulsed or Q-switched lasers. Pulsed and cw CO2 lasers make up the bulk of the remainder. However, DPSS and fiber lasers are encroaching on this field owing to their higher reliability and lower operating cost. Because of their very short wavelengths (100 – 300 nm), excimer lasers are used in applications requiring extremely high resolution, or whose materials would thermally damage at longer wavelengths. Optical Specifications A wide variety of materials, including metal, wood, glass, silicon, and rubber, are amenable to laser marking and scribing. Each material has different absorption and thermal characteristic, and some even have directional preferences due to crystalline structure. Consequently, the type of laser used depends, to some extent, on the material to be marked (e.g., glass transmits the 1.06 µm output from a YAG laser but absorbs the 10.6 µm output from a CO2 laser). Other considerations are the size of the pattern, the speed of the scan, cosmetic quality, and cost. stability to calculate the correct size of the defects through complex algorithms, and low noise so the little scatter the defect makes can be distinguished from the background laser light. Blue 488 nm argon ion lasers have been the laser of choice for many years. As lithography has shifted to shorter and shorter ultraviolet wavelengths, however, we are beginning to see the metrologic techniques for wafer defect measurement also moving to shorter wavelengths. Ultraviolet diode and solid-state lasers are likely to replace the ion laser in the next generation of instruments. Material Properties Laser scribing is similar to laser marking, except that the scan pattern is typically rectilinear, and the goal is to create microscoring along the scan lines so that the substrate can be easily broken apart. Laser Guide marketplace.idexop.com Laser Applications A233 LASER GUIDE Laser Guide Typical applications of this technique include positioning of masks for the lithography process, mirror distance correlation within an FTIR spectrometer, optical feedback in many high-resolution positioning systems, and determining the alignment and flatness of hard disk drive heads. For applications requiring measurement over a long path length, lasers with a single longitudinal mode and long coherence length are often required. In these cases, frequency-stabilized helium neon lasers or a solid-state lasers with frequency selective elements are used. SCIENTIFIC APPLICATIONS Lasers are used extensively in the scientific laboratory for a wide variety of spectroscopic and analytic tasks. Two interesting examples are confocal scanning microscopy and time-resolved spectroscopy. TIME-RESOLVED SPECTROSCOPY Time-resolved spectroscopy is a technique used to observe phenomena that occur on a very short time scale. This technique has been used extensively to +H1H ODVHU UHFHLYHU UHFHLYHU HOOLSWLFDOO\ SRODUL]HGEHDP SRODUL]LQJ EHDPVSOLWWHU VXUIDFHILOP Figure 7.32 Surface film thickness measurement HeNe laser Twyman-Green interferometer surface of interest detector detector Figure 10.33 Interferometric measurement Laser Applications CONFOCAL SCANNING MICROSCOPY Scanning microscopy is used to build up a threedimensional image of a biological sample. In standard light microscopy, a relatively large volume of the sample is illuminated, and the resultant light gathered by the objective lens comes not only from the plane in focus itself, but also from below and above the focal plane. This results in an image that contains not only the infocus light, but also the haze or blur resulting from the light from the out-of-focus planes. The basic principle of confocal microscopy is to eliminate the out-of-focus light, thus producing a very accurate, sharp, and highresolution image. A schematic of a confocal microscope is shown in figure 7.34. A visible laser is used as the light source to produce a distinct and spatially constrained point source of illumination. This light is then focused on the sample. A pinhole is placed in front of the detector at an optical distance that is exactly the same as the optical distance between the focus point and the illuminating source point (the confocal condition). Consequently, only the light generated at the illuminating point will, upon reflection or scattering from the sample, pass through the pinhole in front of the detector; out-of-focus light will be blocked by the pinhole. The signal from the detector is then digitized and passed to a computer. The complete image is digitally built up by scanning the sample in the x and y directions. beamsplitter ratio amp A234 understand biological processes such a photosynthesis, which occur in picoseconds (10–12 seconds) or less. A fluorescing sample is excited by a laser whose pulse length is much shorter than the time duration of the effect being observed. Then, using conventional fluorescence spectroscopy measurement techniques, the time domain of the fluorescence decay process can be analyzed. Because of the speed of the processes, modelocked lasers are used as the exciting source, often with pulse compression schemes, to generate pulses of the femtosecond (10–15 sec) time scale, very much faster than can be generated by electronic circuitry. TIR AND FLUORESCENCE CORRELATION SPECTROSCOPY Fluorescence correlation spectroscopy measures the variation in fluorescence emission at the molecular level as fluorochromes travel through a defined field. The 1-505-298-2550 LASER GUIDE Optical Coatings & Materials MICROARRAY SCANNING In DNA research, a microarray is a matrix of individual DNA molecules attached, in ordered sets of known sequence, to a substrate which is approximately the GHWHFWRU FRQIRFDODSHUWXUH SRVLWLRQFULWLFDO Optical Specifications LOOXPLQDWLQJ DSHUWXUH EHDPVSOLWWHU FZODVHU VFDQHQJLQH LQIRFXVOLJKW RXWRIIRFXVOLJKW Fundamental Optics One means of reducing the excitation volume is to use total internal reflection (TIR) techniques (see figure 10.35). If a laser beam, passing through a high index material (e.g., glass at n≅1.5) strikes an interface with a lower index sample material (e.g., an aqueous solution at n≅1.3) at an oblique angle, there is an angle of incidence (the critical angle) at which all of the light will be completely reflected at the interface, and none will pass into the lower-index material. The critical angle is given by observing microscope itself, and then filtering out the returning beam with a dichroic mirror. Material Properties data can then be used to determine binding and fusion constants for various molecular interactions. Because the measured volumes are so small, measurements are typically made using single-photon or two-photon confocal microscopy techniques. In many cases, the region of interest for fluorescence correlation spectroscopy is the first 100 to 200 nm of the sample’s surface. However, the excitation depth (vertical resolution) for conventional confocal spectroscopy is 1 to 1.5 µm, leading to low signal-to-noise ratios and diminished accuracy. VSHFLPHQVWDJH &RQIRFDO6HWXS (7.26) Figure 7.34 Optical schematic of a confocal microscope Various techniques have been used to obtain TIR. Most commonly, the laser beam is brought in through a prism, as shown in figure 7.35. Another technique is to bring the beam in through the steeply curved edge of the ODVHU EHDP SULVP VDPSOH IRFXVLQJOHQV 7,5IRURSDTXHVDPSOH PLFURVFRSH REMHFWLYH Machine Vision Guide Because the beam is completely reflected at the interface, there is no energy flux across the interface; there is, however, an electromagnetic field generated in the lower index material, determined by the boundary conditions on the electric and magnetic fields at the interface. This transmitted wave is evanescent, propagating along the surface of the interface, but decaying in intensity exponentially with depth, limiting excitation to a few hundred nanometers—five to ten times better resolution than with confocal techniques alone. PLFURVFRSH REMHFWLYH IOXRUHVFHQFH Gaussian Beam Optics where nt is the index of the transmitting (lower index) material and ni of the incident material. IOXRUHVFHQFH WUDQVPLWWLQJ VDPSOH ODVHU EHDP SULVP IRFXVLQJOHQV 7,5IRUWUDQVPLWWLQJVDPSOH Figure 7.35 An example of TIR spectroscopy Laser Guide marketplace.idexop.com Laser Applications A235 LASER GUIDE Laser Guide size of a microscope slide. A single array can contain thousands of molecules each tagged with a specific fluorochrome. The array is then put into a microarray reader where each individual site of the matrix is individually probed by a variety of laser wavelengths at, or near, the excitation band of specific protein tags. The resulting fluorescence is measured and the fluorescence, position, and sequence data are stored in a computer database for later analysis. Microarrays and microarray readers have had a dramatic impact on the speed at which data can be taken. Previously experiments were conducted one or two molecules at a time; preparation and setting up could take hours. With microarray readers, the raw data for analysis of thousands of molecules can be taken in minutes. The main driver for microarrays is the pharmaceutical industry. If one can identify the differences in the way genes are expressed in a healthy organ and in a diseased organ, and then determine the genes and associated proteins that are part of the disease process, it may be possible to synthesize a drug that would interact with the proteins and cure or reduce the effect of the disease. The optical system for a typical microarray scanner is shown in figure 7.36. The beam from a laser is focused onto a well (molecule) on the molecular matrix. If the appropriate fluorescent tag is present, the resulting fluorescence is measured by a detector. A filter in front of the detector separates the laser wavelength from the fluorescence signal. The laser beam is then moved to the next well. Today’s microarray scanner systems use two or more cw lasers, each with a different wavelength. Output power typically ranges from 10 to 50 mW, a power level that allows scanning without damaging or changing the material under test. Laser pointing stability is important as the microarray wells are quite small and repeatability is needed to relocate cells. Power stability and low noise are also extremely important due to the small sample size and the resulting weak fluorescence signal. The most common lasers in use today for excitation are the blue solid-state (473 – 488 nm), green solid-state A236 Laser Applications (532 nm) and red diode (650 – 690 nm) lasers. Solidstate and semiconductor laser technology is chosen primarily for its compact size, reliability, and power efficiency. Other wavelengths, including violet (405 nm) and ultraviolet (375 nm) from diode lasers, are currently being tested for application in microarray-reading applications. CLINICAL AND MEDICAL APPLICATIONS One of the earliest applications of lasers in medicine was photocoagulation, using an argon-ion laser to seal off ruptured blood vessels on the retina of the eye. The laser beam passed through the lens and vitreous humor in the eye and focused on the retina, creating scar tissue that effectively sealed the rupture and staunched the bleeding. Today, lasers are used extensively in analytical instrumentation, ophthalmology, cellular sorting, and of course, to correct vision. Many types of lasers are used in clinical applications including CO2 , solid state, and diode lasers, as well as an array of gas lasers covering the spectrum from the ultraviolet to the infrared. FLOW CYTOMETRY Flow cytometry is a technique used for measuring single cells. Not only is it a key research tool for cancer and ODVHU EORFNLQJ ILOWHU IOXRUHVFHQFH IURPPDUNHGFHOO IRFXVLQJ OHQV GHWHFWRU ODVHU FRXQWHU PDUNHG FHOOV VHSDUDWLRQILHOG XQPDUNHG FHOOV Figure 7.37 Schematic of a laser cell sorter 1-505-298-2550 LASER GUIDE Optical Coatings & Materials immunoassay disease research, but it is also used in the food industry for monitoring natural beverage drinks for bacterial content or other disease-causing microbes. Lasers are also used to treat macular degeneration, an overgrowth of veins and scar tissue in the retinal region, a condition associated with advancing age. In this procedure, the patient is injected with a selective dye, which enhances the absorption of laser light by the blood in the blood vessels. When the blood vessels absorb laser energy, they wither in size, uncovering the active retina. A multiwatt green DPSS laser is most commonly used for this application because the green wavelength is not absorbed by the lens or aqueous portion of the eye, which allows the laser to affect only the targeted veins. Optical Specifications Fundamental Optics The most popular lasers used in flow cytometry are the 488 nm (blue) argon-ion laser and the 632 nm (red) and 594 nm (yellow) HeNe lasers. However, new violet, blue and red diode lasers and a variety of new DPSS lasers are entering the field. Material Properties In a basic cytometer, the cells flow, one at a time, through a capillary or flow cell where they are exposed to a focused beam of laser light (see figure 7.37). The cell then scatters the light energy onto a detector or array of detectors. The pattern and intensity of the scattered energy helps to determine the cell size and shape. In many cases the cells are tagged with a variety of fluorochromes designed to selectively adhere to cells or cell components with specific characteristics. When exposed to the laser light, only those with the tag fluoresce. This is used in many systems to assist with separation or sorting of cells or cellular components. Cosmetic treatment of wrinkles, moles, warts, and discolorations (birth marks) is often accomplished with near infrared and infrared lasers. These procedures are often assisted by topical or injected photosensitive chemicals that assist with selective absorption at specific sites. Gaussian Beam Optics SURGICAL APPLICATIONS Lasers are used in a variety of surgical and dental procedures, including cutting tissue, vaporizing tumors, removing tattoos, removing plaque, removing cavities, removing hair and follicles, resurfacing of skin and of course, correcting vision. In many ways, medical applications are like materials processing applications. In some cases material is ablated. In others tissue is cut or welded, and in yet others, photochemical changes are caused in blood vessels to encourage shrinkage and absorption. Understanding tissue absorption characteristics and reaction to wavelength and power are key. Machine Vision Guide Ultraviolet excimer lasers are used for vision correction because they can ablate material from the lens of the eye without causing thermal damage which could blur vision or make the lens opaque. Ruby lasers are used for tattoo removal because many of the dyes break down when exposed to 694 nm radiation, yet the skin tissue is left undamaged. Laser Guide marketplace.idexop.com Laser Applications A237 LASER GUIDE Laser Guide A238 1-505-298-2550 APPENDIX & INDICES Appendix and Indices OPTICAL SPECIFICATIONS A240 INDEX OF REFRACTION A242 DISPERSION EQUATIONS A244 SURFACE QUALITY & SURFACE FIGURE A246 LASER INDUCED DAMAGE THRESHOLD A247 RADIUS OF CURVATURE A248 STANDARD SIZE CODES AND RADII OF CURVATUREA249 MARKINGA250 marketplace.idexop.com GENERAL PRODUCT INDEX A251 PRODUCT CODE INDEX A252 TECHNICAL GUIDE INDEX A254 A239 OPTICAL SPECIFICATIONS OPTICAL SPECIFICATIONS Appendix and Indices SUBSTRATE MATERIAL LASER GRADE The material from which an optic is made. Laser grade is the highest level of inspection criteria in the optics industry. The most common materials are N-BK7, UV grade fused silica, MgF2, and CaF2. CVI Laser Optics has experience Laser grade optics are virtually defect-free. High power with a wide variety of glasses, fused silicas, and crystalline magnification is used to detect and measure defects. materials. WEDGE SURFACE FIGURE The angle between the two surfaces of an optical The deviation from the ideal surface. element. CVI Laser Optics specifies surface figure in terms of This can also be expressed as the difference in edge waves peak-to-valley (p-v) at 633 nm, prior to coating. thickness around the part, for example a 25.4 mm The peak-to-valley specification is more stringent than an diameter optic with an edge thickness variation (ETV) of RMS or average surface specification and assures high 0.025 mm has a wedge of 3.44 minutes of arc. quality parts for all applications. We manufacture flats to RADIUS OF CURVATURE λ/20 and spherical surfaces to λ/10 accuracy on a routine basis. A coated surface figure may also be specified. COSMETIC SURFACE QUALITY Surface quality describes a level of defects visually detected on the surface of an optical component. The radius of the sphere coincident with the optical surface. A flat has radius of curvature equal to infinity.The reciprocal of the radius is called the curvature of the surface. CVI Laser Optics can manufacture a wide 100% visual inspection is performed on all optics. Surface range of curvatures using its existing tooling and test quality becomes critically important in high energy laser plates and has the capability to make new test plates if applications or where scatter must be reduced for better required. signal to noise performance. MIL-PRF-13830B This inspection criteria was adopted in 1997.The first number denotes the size and concentration of scratches as compared to a known NIST standard. The second number defines the largest pit by its diameter in hundredths of millimeters. For example: 10 dig = ½ (Length of dig =Width of dig) = 0.1 mm diameter. CVI LASER OPTICS LASER QUALITY We have extensive experience in delivering high laser damage threshold optics and have developed a proprietary inspection method to consistently meet our customers’ laser induced damage requirements. This proprietary inspection method utilizes significantly CVI Laser Optics’s standard radius tolerance is 80.5% and 80.1% is available for selected radii. CONCENTRICITY/CENTRATION The deviation between the optical and mechanical axes of a lens. Concentricity error is the measured maximum edge thickness variation. CVI Laser Optics’s standard concentricity is ≤ 0.05 mm edge thickness variation and the standard centration error is ≤ 3 arc minutes. CLEAR APERTURE The central area over which the optical specifications apply. brighter light sources than those specified in MIL-PRF- CVI Laser Optics specifies clear aperture in terms of the 13830B. diameter or linear dimensions of this central area. CVI Laser Optics also utilizes magnification when required to detect scratches, digs and other defects. A240 Optical Specifications 1-505-296-9541 OPTICAL SPECIFICATIONS Appendix and Indices ANGLE AND PLANE OF INCIDENCE The angle formed between the normal to the optical surface and the incident ray. An incidence angle of zero degrees is referred to as normal incidence.The plane of incidence is the plane containing the incident ray and the normal. POLARIZATION The orientation and phase shift of the electric field when resolved into components parallel and perpendicular to the plane of incidence. Light that is p-polarized has the electric field polarized parallel to the plane of incidence. Light that is s-polarized has the electric field polarized perpendicular to the plane of incidence. UNP refers to unpolarized light, which is a random mixture of equal amounts of s- and p-polarization states. Specify the polarization state whenever ordering an optic for use at non-normal incidence angle. marketplace.idexop.com Optical Specifications A241 INDEX OF REFRACTION ULTRAVIOLET AND VISIBLE MATERIALS Appendix and Indices Ultraviolet Materials Wavelength (nm) MgF2 ηe MgF2 ηo CaF2 Sapphire ηe Sapphire ηo Crystal Quartz ηe Crystal Quartz ηo Fused Silica 193 1.44127 1.42767 1.50153 1.91743 1.92879 1.67455 1.66091 1.56077 213 1.42933 1.41606 1.48544 1.87839 1.88903 1.64452 1.63224 1.53539 222 1.42522 1.41208 1.47996 1.86504 1.87540 1.63427 1.62238 1.52669 226 1.42358 1.41049 1.47779 1.85991 1.87017 1.63033 1.61859 1.52335 244 1.41735 1.40447 1.46957 1.84075 1.85059 1.61562 1.60439 1.51086 248 1.41618 1.40334 1.46803 1.83719 1.84696 1.61289 1.60175 1.50855 257 1.41377 1.40102 1.46488 1.82972 1.83932 1.60714 1.59620 1.50368 266 1.41164 1.39896 1.46209 1.82358 1.83304 1.60242 1.59164 1.49968 280 1.40877 1.39620 1.45836 1.81509 1.82437 1.59589 1.58533 1.49416 308 1.40429 1.39188 1.45255 1.80198 1.81096 1.58577 1.57556 1.48564 325 1.40216 1.38983 1.44981 1.79582 1.80467 1.58102 1.57097 1.48164 337 1.40086 1.38859 1.44814 1.79206 1.80082 1.57812 1.56817 1.47919 351 1.39952 1.38730 1.44642 1.78825 1.79693 1.57518 1.56533 1.47672 355 1.39917 1.38696 1.44597 1.78732 1.79598 1.57446 1.56463 1.47612 Visible Materials A242 Wavelength (nm) Schott N-BK7 Schott N-F2 Schott N-SF2 Schott N-SF10 Schott N-SF11 Schott N-BaK4 Crystal Quartz ηe Crystal Quartz ηo Fused Silica 400 1.53085 1.65243 1.68453 1.77826 1.84542 1.58695 1.56730 1.55772 1.47012 442 1.52607 1.64063 1.67098 1.75964 1.82254 1.58069 1.56266 1.55324 1.46622 458 1.52461 1.63718 1.66704 1.75428 1.81602 1.57879 1.56119 1.55181 1.46498 488 1.52224 1.63178 1.66091 1.74597 1.80595 1.57574 1.55885 1.54955 1.46301 515 1.52046 1.62783 1.65644 1.73996 1.79871 1.57346 1.55711 1.54787 1.46156 532 1.51947 1.62569 1.65403 1.73672 1.79482 1.57220 1.55610 1.54690 1.46071 590 1.51670 1.61984 1.64746 1.72797 1.78435 1.56870 1.55333 1.54421 1.45838 633 1.51508 1.61656 1.64378 1.72312 1.77858 1.56669 1.55171 1.54264 1.45702 670 1.51391 1.61424 1.64119 1.71971 1.77454 1.56524 1.55051 1.54148 1.45601 694 1.51323 1.61293 1.63973 1.71780 1.77228 1.56441 1.54981 1.54080 1.45542 755 1.51172 1.61010 1.63659 1.71374 1.76749 1.56260 1.54827 1.53932 1.45414 780 1.51118 1.60911 1.63550 1.71233 1.76583 1.56196 1.54771 1.53878 1.45367 800 1.51078 1.60838 1.63469 1.71130 1.76462 1.56148 1.54729 1.53837 1.45332 820 1.51039 1.60770 1.63394 1.71033 1.76349 1.56102 1.54688 1.53798 1.45298 860 1.50966 1.60644 1.63256 1.70858 1.76144 1.56018 1.54612 1.53724 1.45234 980 1.50779 1.60335 1.62919 1.70438 1.75655 1.55807 1.54409 1.53531 1.45067 1064 1.50663 1.60159 1.62730 1.70207 1.75390 1.55682 1.54282 1.53410 1.44963 1320 1.50346 1.59723 1.62268 1.69667 1.74776 1.55358 1.53922 1.53068 1.44669 1550 1.50065 1.59380 1.61914 1.69275 1.74340 1.55087 1.53596 1.52761 1.44402 2010 1.49435 1.58680 1.61204 1.68531 1.73528 1.54507 1.52863 1.52073 1.43794 Index of Refraction 1-505-296-9541 INDEX OF REFRACTION INFRARED MATERIALS Wavelegnth (nm) Zinc Selenide (ZnSe) Calcium Fluoride (CaF2) Germanium (Ge) Silicon (Si) 0.6328 2.590 1.43289 5.3900 3.9200 1.40 2.461 1.42673 4.3400 3.4900 1.50 2.458 1.42626 4.3500 3.4800 1.66 2.454 1.42551 4.3300 3.4700 1.82 2.449 1.42475 4.2900 3.4600 2.05 2.446 1.42360 4.2500 3.4500 2.06 2.446 1.42355 4.2400 3.4900 2.15 2.444 1.42308 4.2400 3.4700 2.44 2.442 1.42146 4.0700 3.4700 2.50 2.441 1.42110 4.2200 3.4400 2.58 2.440 1.42062 4.0600 3.4364 2.75 2.439 1.41954 4.0526 3.4335 3.00 2.438 1.41785 4.0540 3.4307 3.42 2.436 1.41469 4.0370 3.4277 3.50 2.435 1.41404 4.0356 3.4272 4.36 2.432 1.40609 4.0227 3.4223 5.00 2.430 1.39896 4.0177 3.4203 6.00 2.426 1.38560 4.0138 3.4188 6.24 2.425 1.38197 4.0100 3.4185 7.50 2.420 1.36000 4.0095 3.4171 8.66 2.414 1.33504 4.0071 3.4161 9.50 2.410 1.31375 4.0064 3.4158 9.72 2.409 1.30768 4.0062 3.4155 10.60 2.400 1.28116 4.0058 3.4155 11.00 2.400 1.26783 4.0059 3.4155 11.04 2.400 1.26645 4.0059 3.4155 12.50 2.390 1.20951 4.0000 3.4155 13.02 2.385 1.18573 4.0000 3.4155 13.50 2.380 1.16187 4.0000 3.4155 15.00 2.370 1.07290 4.0000 3.4155 16.00 2.360 0.99783 4.0000 3.4155 16.90 2.350 0.91507 4.0000 3.4155 17.80 2.340 0.81173 4.0000 3.4155 18.60 2.330 0.69336 4.0000 3.4155 19.30 2.320 0.55456 4.0000 3.4155 20.00 2.310 0.34029 4.0000 3.4155 marketplace.idexop.com Index of Refraction Appendix and Indices Infrared Materials A243 DISPERSION EQUATIONS DISPERSION EQUATIONS Appendix and Indices DISPERSION EQUATIONS FOR OPTICAL MATERIALS Typically either a Sellmeier or Laurent series equation is used to describe glass dispersion. SELLMEIER SERIES EQUATION The Sellmeier series equation is: η2 = 1 + B1λ2 B λ2 B λ2 + 22 + 23 2 λ - C1 λ - C2 λ - C3 where the wavelength, λ, is expressed in µm. Dispersion Equation Constants - Sellmeier series equation A244 B1 B2 B3 C1 C2 C3 MgF2 ηe 4.13440230E-01 5.04974990E-01 2.49048620E+00 1.35737865E-03 8.23767167E-03 5.65107755E+02 MgF2 ηo 4.87551080E-01 3.98750310E-01 2.31203530E+00 1.88217800E-03 8.95188847E-03 5.66135591E+02 Sapphire ηe 1.50397590E+00 5.50691410E-01 6.59273790E+00 5.48041129E-03 1.47994281E-02 4.02895140E+02 Sapphire ηo 1.43134930E+00 6.50547130E-01 5.34140210E+000 5.27992610E-03 1.42382647E-02 3.25017834E+02 CaF2 5.67588800E-01 4.71091400E-01 3.84847230E+00 2.52642999E-03 1.00783328E-02 1.20055597E+03 Fused Silica 6.96166300E-01 4.07942600E-01 8.97479400E-01 4.67914826E-03 1.35120631E-02 9.79340025E+01 Schott N-BK7 1.03961212E+00 2.31792344E-01 1.01046945E+00 6.00069867E-03 2.00179144E-02 1.03560653E+02 Schott F2 1.34533359E+00 2.09073118E-01 9.37357162E-01 9.97743871E-03 4.70450767E-02 1.11886764E+02 Schott N-F2 1.39757037E+00 1.59201403E-01 1.26865430E+00 9.95906143E-03 5.46931752E-02 1.19248346E+02 Schott SF2 1.40301821E+00 2.09073176E-01 9.39056586E-01 1.05795466E-02 4.93226978E-02 1.12405955E+02 Schott N-SF2 1.47343127E+00 1.63681849E-01 1.36920899E+00 1.09019098E-02 5.85683687E-02 1.27404933E+02 Schott SF5 1.46141885E+00 2.47713019E-01 9.49995832E-01 1.11826126E-02 5.08594669E-02 1.12041888E+02 Schott N-SF5 1.52481889E+00 1.87085527E-01 1.42729015E+00 1.1254756E-02 5.88995392E-02 1.29141675E+02 Schott SF10 1.61625977E+00 2.59229334E-01 1.07762317E+00 1.27534559E-02 5.81983954E-02 1.16607680E+02 Schott N-SF10 1.62153902E+00 2.56287842E-01 1.64447552E+00 1.22241457E-02 5.95736775E-02 1.47468793E+02 Schott SF11 1.73848403E+00 3.11168974E-01 1.17490871E+00 1.36068604E-02 6.15960463E-02 1.21922711E+02 Schott N-SF11 1.73759695E+00 3.13747346E-01 1.89878101E+00 1.13188707E-02 6.23068142E-02 1.55236290E+02 Schott N-SK11 1.17963631E+00 2.29817295E-01 9.35789652E-01 6.80282081E-03 2.19737205E-02 1.01513232E+02 Schott N-BaK1 1.12365662E+00 3.09276848E-01 8.81511957E-01 6.44742752E-03 2.22284402E-02 1.07297751E+02 Schott N-BaK4 1.28834642E+00 1.32817724E-01 9.45395373E-01 7.79980626E-03 3.15631177E-02 1.05965875E+02 Dispersion Equations 1-505-296-9541 DISPERSION EQUATIONS Appendix and Indices LAURENT SERIES EQUATION The Laurent series equation is: η 2 = A0 + A1λ2 + A2 A A A + 43 + 64 + 85 λ2 λ λ λ where the wavelength, λ, is expressed in µm. Dispersion Equation Constants - Laurent series equation A0 A1 A2 A3 A4 A5 Crystal Quartz ηe 2.38490000E+00 -1.25900000E-02 1.07900000E-02 1.65180000E-04 -1.94741000E-06 9.36476000E-08 Crystal Quartz ηo 2.35728000E+00 -1.17000000E-02 1.05400000E-02 1.34143000E-04 -4.45368000E-07 5.92362000E-08 marketplace.idexop.com Dispersion Equations A245 SURFACE QUALITY & SURFACE FIGURE SURFACE QUALITY & SURFACE FIGURE Appendix and Indices SURFACE QUALITY Scratch-Dig Relative Cost Applications 60-40 Very Low 40-20 Low 20-10 Moderate For laser and imaging applications with focused beams where minimizing scattered light is more critical. Best quality offered by typical catalog houses. 20-10 CVI LQ Moderate CVI Laser Quality level of inspection. For laser and imaging applications with focused beams where minimizing scattered light is more critical. 10-15 CVI LQ Moderately High Used in low power laser and imaging applications where scattered light is not as critical as cost. For laser and imaging applications with focused beams that tolerate little scattered light. CVI Laser Quality level of inspection. Required for high damage threshold in high laser energy applications. Best performance for laser material processing applications and laser cavity optics. SURFACE FIGURE/TRANSMITTED WAVEFRONT DISTORTION A246 Surface Quality Relative Cost λ/2 Very Low λ /4 Low For general laser and imaging applications where wavefront performance is balanced with cost. λ/8 Moderate For laser and imaging applications with low wavefront distortion requirements, especially in multi-element systems. Best quality offered by typical catalog houses. λ /10 Moderate High Surface Quality & Surface Figure Applications Cost optimization solution. Also used with very fast or short radius singlets. CVI Laser Optics signature wavefront quality level. Required for best performance in ultraviolet and performance critical applications 1-505-296-9541 LASER INDUCED DAMAGE THRESHOLD LASER INDUCED DAMAGE THRESHOLD Appendix and Indices LASER INDUCED DAMAGE THRESHOLD Typical Laser Induced Damage Threshold Data 1064 nm, 20 NSEC, 20 HZ DATA X Antireflection coatings on fused silica > 15 J/cm2 X Antireflection coatings on N-BK7 > 10 J/cm2 X Antireflection coatings on N-SF11 > 4 J/cm2 X Optical X High Cement > 2 J/cm2 reflection coatings > 20 J/cm2 For higher damage thresholds call CVI Laser Optics to optimize the various material parameters and provide certification. CVI Laser Optics is a leader in damage resistant coatings for excimer and other high energy lasers. UV DATA X Antireflection coatings on fused silica X 193 nm > 2 J/cm2, 20 nsec, 20 Hz X 266 nm, 355 nm > 5 J/cm2, 10 nsec, 10 Hz LASER INDUCED DAMAGE THRESHOLD TESTING LIDT (Laser Induced Damage Threshold) is defined as any laser-induced permanent change which is observable at high magnification at the lowest power sufficient to induce damage at any test site. LIDT depends on test wavelength, pulse width, repetition rate, and inspection method. To determine the damage threshold, CVI Laser Optics tests a number of samples at varied settings using increasing power. Visible observation is performed using a 20x microscope immediately before and after the optic is subjected to the laser. The test samples are then characterized for laser induced damage and any changes in beam scatter are documented. LIDT test procedures are subject to change, and can be changed upon the request of a customer. The two main mechanisms that cause laser damage to an optical coating are dielectric breakdown and thermal absorption. Factors which significantly reduce the LIDT are scratches, pores, inclusions, and impurities. marketplace.idexop.com Laser Induced Damage Threshold A247 RADIUS OF CURVATURE RADIUS OF CURVATURE 5DGLXVPP Appendix and Indices For short radius lenses, spherical aberration is the major contributor to wavefront distortion.This transition happens at or about f/10. CVI Laser Optics uses this as a practical limit to optimize the manufacturing process. Further improvements to surface figure do not result in measurable improvements to the overall wavefront distortion and will only increase cost. Use the charts as a guide for when spherical aberration dominates the wavefront distortion. 7ROHUDQFH Radius vs. Tolerance Radii Tolerance Tolerance Radius % (mm) 0.5 3500 If R ≤ 3500 mm then Radius Tolerance is 0.5% Explanation 1.0 7000 If 3501 mm ≤ R ≤ 7000 mm then Radius Tolerance is 1% 1.5 10500 If 7501 mm ≤ R ≤ 10500 mm then Radius Tolerance is 1.5% 2.0 14300 If 10501 mm ≤ R ≤ 14300 mm then Radius Tolerance is 2.0% 2.5 18000 If 14301 mm ≤ R ≤ 18000 mm then Radius Tolerance is 2.5% 3.0 21750 If 18001 mm ≤ R ≤ 21750 mm then Radius Tolerance is 3.0% 3.5 25500 If 21751 mm ≤ R ≤ 25500 mm then Radius Tolerance is 3.5% 4.0 29000 If 25501 mm ≤ R ≤ 29000 mm then Radius Tolerance is 4.0% 4.5 33000 If 29001 mm ≤ R ≤ 33000 mm then Radius Tolerance is 4.5% 5.0 37000 If 33001 mm ≤ R ≤ 37000 mm then Radius Tolerance is 5.0% 5.5 40000 If 37001 mm ≤ R ≤ 40000 mm then Radius Tolerance is 5.5% 6.0 44800 If 40001 mm ≤ R ≤ 44800 mm then Radius Tolerance is 6.0% 6.5 48800 If 44801 mm ≤ R ≤ 48800 mm then Radius Tolerance is 6.5% – – Greater than 48800 mm will need special consideration. Practical Limits to Surface Figure for Short Radius Lenses Radius of Curvature if ≤ Diameter if ≤ Surface Figure 31.00 mm 12.70 mm λ/4 75.00 mm 25.40 mm λ/4 125.00 mm 38.10 mm λ/4 185.00 mm 50.80 mm λ/4 315.00 mm 76.20 mm λ/4 31.00 mm 12.70 mm λ 75.00 mm 15.00 mm λ 125.00 mm 20.00 mm λ 185.00 mm 25.40 mm λ 315.00 mm 30.00 mm λ Spherical Lenses Cylindrical Lenses A248 Radius of Curvature 1-505-296-9541 STANDARD SIZE CODES AND RADII OF CURVATURE STANDARD SIZE CODES AND RADII OF CURVATURE Radii (m) Standard radius Appendix and Indices Standard Size Codes and Radii of Curvature Optional radius 0525 0537 0737 1025 1037 2037 c Size Code 0.500 in 0.500 in 0.375 in 1.000 in 0.250 in 1.000 in 0.375 in 2.000 in 0.375 in c Diameter 0.250 in 0.750 in 0.375 in c Thickness 0.010 0.025 0.050 0.075 0.10 0.15 0.20 0.25 CONCAVE 0.30 0.40 0.50 0.75 1.00 1.20 1.50 2.00 3.00 4.00 5.00 10.0 20.0 0.30 0.50 CONVEX 0.75 1.00 2.00 5.00 10.0 20.0 marketplace.idexop.com Standard Size Codes and Radii of Curvature A249 MARKING MARKING Appendix and Indices flat indicates slow axis (aligned with V-groove on housing) V-groove collimated light real focus optional housing with retaining ring retaining ring collimated light QWPM waveplate assembly point source of light at the focal point of lens radial groove indicates QWPO V-groove slow axis indicated with black flat (aligned with V-groove on housing) fast axis indicated with orange flat Positive Systems (Convergent): The arrow marking on the housing always points to the collimated light. optional housing with retaining ring retaining ring QWPO-series zero-order quartz waveplate with ring mount collimated light virtual focus slow axis indicated with small hole collimated light collimated light housing with retaining ring virtual focus example positive lens Negative Systems (Divergent): The arrow marking on the housing always points to the collimated light. A250 Markings retaining ring QWPO air-spaced waveplate assembly 1-505-296-9541