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Optics 479 Magnification An introduction to the use of lenses to solve optical applications can begin with the elements of ray tracing. Figure 1 demonstrates an elementary ray trace showing the formation of an image, using an ideal thin lens. The object height is y1 at a distance s1 from an ideal thin lens of focal length f. The lens produces an image of height y2 at a distance s2 on the far side of the lens. We can use basic geometry to look at the magnification of a lens. In Figure 2, we have the same ray tracing figure with some particular line segments highlighted. The ray through the center of the lens and the optical axis intersect at an angle φ. Recall that the opposite angles of two intersecting lines are equal. Therefore, we have two similar triangles. Taking the ratios of the sides, we have φ= y1/s1 = y2/s2 Rearranging one more time, we finally arrive at 1/f = 1/s1 + 1/s2. This is the Gaussian lens equation. This equation provides the fundamental relation between the focal length of the lens and the size of the optical system. A specification of the required magnification and the Gaussian lens equation form a system of two equations with three unknowns: f, s1, and s2. The addition of one final condition will fix these three variables in an application. LENS SELECTION GUIDE Optical Ray Tracing TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Focusing and Collimating This can then be rearranged to give y2/y1 = s2/s1 = M. Gaussian Lens Equation Let’s now go back to our ray tracing diagram and look at one more set of line segments. In Figure 3, we look at the optical axis and the ray through the front focus. Again looking at similar triangles sharing a common vertex and, now, angle η, we have y2/f = y1/(s1-f). Now we are ready to look at what happens to an arbitrary ray that passes through the optical system. Figure 4 shows such a ray. In this figure, we have chosen the maximal ray, that is, the ray that makes the maximal angle with the optical axis as it leaves the object, passing through the lens at its maximum clear aperture. This choice makes it easier, of course, to visualize what is happening in the system, but this maximal ray is also the one that is of most importance in designing an application. While the figure is drawn in this fashion, the choice is completely arbitrary and the development shown here is true regardless of which ray is actually chosen. OPTICAL SYSTEMS Rearranging and using our definition of magnification, we find y2/y1 = s2/s1 = f/(s1-f). Email: [email protected] • Web: newport.com MIRRORS In addition to the assumption of an ideally thin lens, we also work in the paraxial approximation. That is, angles are small and we can substitute θ in place of sin θ. Optical Invariant This puts a fundamental limitation on the geometry of an optics system. If an optical system of a given size is to produce a particular magnification, then there is only one lens position that will satisfy that requirement. On the other hand, a big advantage is that one does not need to make a direct measurement of the object and image sizes to know the magnification; it is determined by the geometry of the imaging system itself. KITS Three rays are shown in Figure 1. Any two of these three rays fully determine the size and position of the image. One ray emanates from the object parallel to the optical axis of the lens. The lens refracts this beam through the optical axis at a distance f on the far side of the lens. A second ray passes through the optical axis at a distance f in front of the lens. This ray is then refracted into a path parallel to the optical axis on the far side of the lens. The third ray passes through the center of the lens. Since the surfaces of the lens are normal to the optical axis and the lens is very thin, the deflection of this ray is negligible as it passes through the lens. Figure 3 Figure 2 CYLINDRICAL LENSES By ideal thin lens, we mean a lens whose thickness is sufficiently small that it does not contribute to its focal length. In this case, the change in the path of a beam going through the lens can be considered to be instantaneous at the center of the lens, as shown in the figure. In the applications described here, we will assume that we are working with ideally thin lenses. This should be sufficient for an introductory discussion. Consideration of aberrations and thick-lens effects will not be included here. The quantity M is the magnification of the object by the lens. The magnification is the ratio of the image size to the object size, and it is also the ratio of the image distance to the object distance. SPHERICAL LENSES Figure 1 This additional condition is often the focal length of the lens, f, or the size of the object to image distance, in which case the sum of s1 + s2 is given by the size constraint of the system. In either case, all three variables are then fully determined. TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 480 Optics θ1 y1 θ2 y2 reciprocal relation. For example, to improve the collimation by a factor of two, you need to increase the beam diameter by a factor of two. f Figure 4 Figure 5 θ2 y1 θ1 y2 LENS SELECTION GUIDE f This arbitrary ray goes through the lens at a distance x from the optical axis. If we again apply some basic geometry, we have, using our definition of the magnification, θ1 = x/s1 and θ2 = x/s2 = (x/s1)(y1/y2). KITS CYLINDRICAL LENSES SPHERICAL LENSES Rearranging, we arrive at y2θ2 = y1θ1. This is a fundamental law of optics. In any optical system comprising only lenses, the product of the image size and ray angle is a constant, or invariant, of the system. This is known as the optical invariant. The result is valid for any number of lenses, as could be verified by tracing the ray through a series of lenses. In some optics textbooks, this is also called the Lagrange Invariant or the Smith-Helmholz Invariant. This is valid in the paraxial approximation in which we have been working. Also, this development assumes perfect, aberration-free lenses. The addition of aberrations to our consideration would mean the replacement of the equal sign by a greater-than-or-equal sign in the statement of the invariant. That is, aberrations could increase the product but nothing can make it decrease. MIRRORS OPTICAL SYSTEMS Application 1: Focusing a Collimated Laser Beam As a first example, we look at a common application, the focusing of a laser beam to a small spot. The situation is shown in Figure 5. Here we have a laser beam, with radius y1 and divergence θ1 that is focused by a lens of focal length f. From the figure, we have θ2 = y1/f. The optical invariant then tells us that we must have y2 = θ1f, because the product of radius and divergence angle must be constant. As a numerical example, let’s look at the case of the output from a Newport R-31005 HeNe laser focused to a spot using a Newport KPX043 plano-convex lens. This laser has a beam diameter of 0.63 mm and a divergence of 1.3 mrad. Note that these are beam diameter and full divergence, so in the notation of our figure, y1 = 0.315 mm and θ1 = 0.65 mrad. The KPX043 lens has a focal length of 25.4 mm. Thus, at the focused spot, we have a radius θ1f = 16.5 µm. So, the diameter of the spot will be 33 µm. This is a fundamental limitation on the minimum size of the focused spot in this application. We have already assumed a perfect, aberration-free lens. No improvement of the lens can yield any improvement in the spot size. The only way to make the spot size smaller is to use a lens of shorter focal length or expand the beam. If this is not possible because of a limitation in the geometry of the optical system, then this spot size is the smallest that could be achieved. In addition, diffraction may limit the spot to an even larger size (see Gaussian Beam Optics section beginning on page 484), but we are ignoring wave optics and only considering ray optics here. Application 2: Collimating Light from a Point Source Another common application is the collimation of light from a very small source, as shown in Figure 6. The problem is often stated in terms of collimating the output from a “point source.” Unfortunately, nothing is ever a true point source and the size of the source must be included in any calculation. In figure 6, the point source has a radius of y1 and has a maximum ray of angle θ1. If we collimate the output from this source using a lens with focal length f, then the result will be a beam with a radius y2 = θ1f and divergence angle θ2 = y1/f. Note that, no matter what lens is used, the beam radius and beam divergence have a Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Figure 6 Since a common application is the collimation of the output from an optical fiber, let’s use that for our numerical example. The Newport F-MBB fiber has a core diameter of 200 µm and a numerical aperture (NA) of 0.37. The radius y1 of our source is then 100 µm. NA is defined in terms of the half-angle accepted by the fiber, so θ1 = 0.37. If we again use the KPX043, 25.4 mm focal length lens to collimate the output, we will have a beam with a radius of 9.4 mm and a half-angle divergence of 4 mrad. We are locked into a particular relation between the size and divergence of the beam. If we want a smaller beam, we must settle for a larger divergence. If we want the beam to remain collimated over a large distance, then we must accept a larger beam diameter in order to achieve this. Application 3: Expanding a Laser Beam It is often desirable to expand a laser beam. At least two lenses are necessary to accomplish this. In Figure 7, a laser beam of radius y1 and divergence θ1 is expanded by a negative lens with focal length –f1. From Applications 1.1 and 1.2 we know θ2 = y1/|–f1|, and the optical invariant tells us that the radius of the virtual image formed by this lens is y2 = θ1|–f1|. This image is at the focal point of the lens, s2 = –f1, because a wellcollimated laser yields s1 ~ ∞, so from the Gaussian lens equation s2 = f. Adding a second lens with a positive focal length f2 and separating the two lenses by the sum of the two focal lengths –f1 +f2, results in a beam with a radius y3 = θ2f2 and divergence angle θ3 = y2/f2. Figure 7 Optics 2y3 y3/y1 = θ2f2/θ2|–f1| = f2/| –f1|, = 2y1f2/|–f1| = 2(0.315 mm)(250 mm)/|–25 mm| or the ratio of the focal lengths of the lenses. The expanded beam diameter = 6.3 mm. located at a distance s1 from a lens of focal length f. The figure shows a ray incident upon the lens at a radius of R. We can take this radius R to be the maximal allowed ray, or clear aperture, of the lens. TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS The expanded beam diameter The expansion ratio 481 The divergence angle 2y3 = 2θ2f2 = 2y1f2/|–f1|. = θ1|–f1|/f2 = (0.65 mrad)|–25 mm|/250 mm θ3 = y2/f2 = θ1|–f1|/f2 = 0.065 mrad. Application 4: Focusing an Extended Source to a Small Spot If s1 is large, then s2 will be close to f, from our Gaussian lens equation, so for the purposes of approximation we can take θ2 ~ R/f. Then from the optical invariant, we have y2 = y1θ1/θ2 = y1(R/s1)(f/R) or y2 = 2y1(R/s1)f/#. where f/2R = f/D is the f-number, f/#, of the lens. In order to make the image size smaller, we could make f/# smaller, but we are limited to f/# = 1 or so. That leaves us with the choice of decreasing R (smaller lens or aperture stop in front of the lens) or increasing s1. However, if we do either of those, it will restrict the light gathered by the lens. If we either decrease R by a factor of two or increase s1 by a factor of two, it would decrease the total light focused at s2 by a factor of four due to the restriction of the solid angle subtended by the lens. KITS This application is one that will be approached as an imaging problem as opposed to the focusing and collimation problems of the previous applications. An example might be a situation where a fluorescing sample must be imaged with a CCD camera. The geometry of the application is shown in Figure 8. An extended source with a radius of y1 is Figure 8 CYLINDRICAL LENSES As an example, consider a Newport R-31005 HeNe laser with beam diameter 0.63 mm and a divergence of 1.3 mrad. Note that these are beam diameter and full divergence, so in the notation of our figure, y1 = 0.315 mm and θ1 = 0.65 mrad. To expand this beam ten times while reducing the divergence by a factor of ten, we could select a plano-concave lens KPC043 with f1 = –25 mm and a planoconvex lens KPX109 with f2 = 250 mm. Since real lenses differ in some degree from thin lenses, the spacing between the pair of lenses is actually the sum of the back focal lengths BFL1 + BFL2 = –26.64 mm + 247.61 mm = 220.97 mm. For minimal aberrations, it is best to use a plano-concave lens for the negative lens and a plano-convex lens for the positive lens with the plano surfaces facing each other. To further reduce aberrations, only the central portion of the lens should be illuminated, so choosing oversized lenses is often a good idea. This style of beam expander is called Galilean. Two positive lenses can also be used in a Keplerian beam expander design, but this configuration is longer than the Galilean design. SPHERICAL LENSES is reduced from the original divergence by a factor that is equal to the ratio of the focal lengths |–f1|/f2. So, to expand a laser beam by a factor of five we would select two lenses whose focal lengths differ by a factor of five, and the divergence angle of the expanded beam would be 1/5th the original divergence angle. LENS SELECTION GUIDE θ3 The divergence angle of the resulting expanded beam Fiber Optic Coupling Application 5: Coupling Laser Light into a Multimode Fiber The objective lens has an effective focal length of 9 mm. In this case, the focused beam will have a diameter of 9 µm and a maximal ray of angle 0.05, so both the Email: [email protected] • Web: newport.com MIRRORS When we look at coupling light from a well-collimated laser beam into a multimode optical fiber, we return to the situation that was illustrated in Figure 5. The radius of the fiber core will be our y2. We will have to make sure that the lens focuses to a spot size less than this parameter. An even more important restriction is that the angle from the lens to the fiber θ2 must be less than the NA of the optical fiber. Let’s consider coupling the light from a Newport R-30990 HeNe laser into an F-MSD fiber. The laser has a beam diameter of 0.81 mm and divergence .0 mrad. The fiber has a core diameter of 50 µm and an NA of 0.20. Let’s look at the coupling from the beam into the fiber when a Newport M-20X objective lens is used in an F-915 or F-915T fiber coupler. OPTICAL SYSTEMS The problem of coupling light into an optical fiber is really two separate problems. In one case, we have the problem of coupling into multimode fibers, where the ray optics of the previous section can be used. In the other case, coupling into single-mode fibers, we have a fundamentally different problem. In this case, one must consider the problem of matching the mode of the incident laser light into the mode of the fiber. This cannot be done using the ray optics approach, but must be done using the concepts of Gaussian beam optics (see page 484). Optics Gaussian Beam Optics The Gaussian is a radially symmetrical distribution whose electric field variation is given by the following equation : Its Fourier Transform is also a Gaussian distribution. If we were to solve the Fresnel integral itself rather than the Fraunhofer approximation, we would find that a Gaussian source distribution remains Gaussian at every point along its path of propagation through the optical system. This makes it particularly easy to visualize the distribution of the fields at any point in the optical system. The intensity is also Gaussian: This relationship is much more than a mathematical curiosity, since it is now easy to find a light source with a Gaussian intensity distribution: the laser. Most lasers automatically oscillate with a Gaussian distribution of electrical field. The basic Gaussian may also take on some particular polynomial multipliers and still remain its own transform. These field distributions are known as higherorder transverse modes and are usually avoided by design in most practical lasers. The Gaussian has no obvious boundaries to give it a characteristic dimension like the diameter of the circular aperture, so the definition of the size of a Gaussian is somewhat arbitrary. Figure 1 shows the Gaussian intensity distribution of a typical HeNe laser. MIRRORS OPTICAL SYSTEMS KITS CYLINDRICAL LENSES SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 484 Figure 1 The parameter ω0, usually called the Gaussian beam radius, is the radius at which the intensity has decreased to 1/e2 or 0.135 of its axial, or peak value. Another point to note is the radius of half maximum, or 50% intensity, which is 0.59ω0. At 2ω0, or twice the Gaussian radius, the intensity is 0.0003 of its peak value, usually completely negligible. The power contained within a radius r, P(r), is easily obtained by integrating the intensity distribution from 0 to r: When normalized to the total power of the beam, P(∞) in watts, the curve is the same as that for intensity, but with the ordinate inverted. Nearly 100% of the power is contained in a radius r = 2ω0. One-half the power is contained within 0.59ω0, and only about 10% of the power is contained with 0.23ω0, the radius at which the intensity has decreased by 10%. The total power, P(∞) in watts, is related to the on-axis intensity, I(0) (watts/m2), by: The on-axis intensity can be very high due to the small area of the beam. Care should be taken in cutting off the beam with a very small aperture. The source distribution would no longer be Gaussian, and the far-field intensity distribution would develop zeros and other non-Gaussian features. However, if the aperture is at least three or four ω0 in diameter, these effects are negligible. Propagation of Gaussian beams through an optical system can be treated almost as simply as geometric optics. Because of the unique self-Fourier Transform characteristic of the Gaussian, we do not need an integral to describe the evolution of the intensity profile with distance. The transverse distribution intensity remains Gaussian at every point in the system; only the radius of the Gaussian and the radius of curvature of the wavefront change. Imagine that we somehow create a coherent light beam with a Gaussian distribution and a plane wavefront at a position x=0. The beam size and wavefront curvature will then vary with x as shown in Figure 2. Figure 2 The beam size will increase, slowly at first, then faster, eventually increasing proportionally to x. The wavefront radius of curvature, which was infinite at x = 0, will become finite and initially decrease with x. At some point it will reach a minimum value, then increase with larger x, eventually becoming proportional to x. The equations describing the Gaussian beam radius w(x) and wavefront radius of curvature R(x) are: where ω0 is the beam radius at x = 0 and λ is the wavelength. The entire beam behavior is specified by these two parameters, and because they occur in the same combination in both equations, they are often merged into a single parameter, xR, the Rayleigh range: In fact, it is at x = xR that R has its minimum value. Note that these equations are also valid for negative values of x. We only imagined that the source of the beam was at x = 0; we could have created the same beam by creating a larger Gaussian beam with a negative wavefront curvature at some x < 0. This we can easily do with a lens, as shown in Figure 3. Figure 3 Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Optics where f/# is the photographic f-number of the lens. Equating these two expressions allows us to find the beam waist diameter in terms of the input beam parameters (with some restrictions that will be discussed later): We can also find the depth of focus from the formulas above. If we define the Email: [email protected] • Web: newport.com MIRRORS We can see from the expression for q that at a beam waist (R = ∞ and ω = ω0), q is pure imaginary and equals ixR. If we know where one beam waist is and its size, we can calculate q there and then use the bilinear ABCD relation to find q anywhere OPTICAL SYSTEMS where the quantity q is a complex composite of ω and R: We have invoked the approximation tanθ ≈ θ since the angles are small. Since the origin can be approximated by a point source, θ is given by geometrical optics as the diameter illuminated on the lens, D, divided by the focal length of the lens. or about 160 µm. If we were to change the focal length of the lens in this example to 100 mm, the focal spot size would increase 10 times to 80 µm, or 8% of the original beam diameter. The depth of focus would increase 100 times to 16 mm. However, suppose we increase the focal length of the lens to 2,000 mm. The “focal spot size” given by our simple equation would be 200 times larger, or 1.6 mm, 60% larger than the original beam! Obviously, something is wrong. The trouble is not with the equations giving ω(x) and R(x), but with the assumption that the beam waist occurs at the focal distance from the lens. For weakly focused systems, the beam waist does not occur at the focal length. In fact, the position of the beam waist changes contrary to what we would expect in geometric optics: the waist moves toward the lens as the focal length of the lens is increased. However, we could easily believe the limiting case of this behavior by noting that a lens of infinite focal length such as a flat piece of glass placed at the beam waist of a collimated beam will produce a new beam waist not at infinity, but at the position of the glass itself. KITS It turns out that we can put these laws in a form as convenient as the ABCD matrices used for geometric ray tracing. But there is a difference: ω(x) and R(x) do not transform in matrix fashion as r and u do for ray tracing; rather, they transform via a complex bi-linear transformation: or about 8 µm. The depth of focus for the beam is then: CYLINDRICAL LENSES At large distances from a beam waist, the beam appears to diverge as a spherical wave from a point source located at the center of the waist. Note that “large” distances mean where x»xR and are typically very manageable considering the small area of most laser beams. The diverging beam has a full angular width θ (again, defined by 1/e2 points): Using these relations, we can make simple calculations for optical systems employing Gaussian beams. For example, suppose that we use a 10 mm focal length lens to focus the collimated output of a helium-neon laser (632.8 nm) that has a 1 mm diameter beam. The diameter of the focal spot will be: SPHERICAL LENSES These equations, with input values for ω and R, allow the tracing of a Gaussian beam through any optical system with some restrictions: optical surfaces need to be spherical and with not-too-short focal lengths, so that beams do not change diameter too fast. These are exactly the analog of the paraxial restrictions used to simplify geometric optical propagation. Fortunately, simple approximations for spot size and depth of focus can still be used in most optical systems to select pinhole diameters, couple light into fibers, or compute laser intensities. Only when f-numbers are large should the full Gaussian equations be needed. depth of focus (somewhat arbitrarily) as the distance between the values of x where the beam is √2 times larger than it is at the beam waist, then using the equation for ω(x) we can determine the depth of focus: LENS SELECTION GUIDE In the free space between lenses, mirrors and other optical elements, the position of the beam waist and the waist diameter completely describe the beam. When a beam passes through a lens, mirror, or dielectric interface, the diameter is unchanged but the wavefront curvature is changed, resulting in new values of waist position and waist diameter on the output side of the interface. else. To determine the size and wavefront curvature of the beam everywhere in the system, you would use the ABCD values for each element of the system and trace q through them via successive bilinear transformations. But if you only wanted the overall transformation of q, you could multiply the elemental ABCD values in matrix form, just as is done in geometric optics, to find the overall ABCD values for the system, then apply the bilinear transform. For more information about Gaussian beams, see Anthony E. Siegman’s book, Lasers (University Science Books, 1986). TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS The input to the lens is a Gaussian with diameter D and a wavefront radius of curvature which, when modified by the lens, will be R(x) given by the equation above with the lens located at -x from the beam waist at x = 0. That input Gaussian will also have a beam waist position and size associated with it. Thus we can generalize the law of propagation of a Gaussian through even a complicated optical system. 485 Optics 491 CaF2 Crystal Quartz BK 7 is one of the most common borosilicate crown glasses used for visible and near infrared optics. Its high homogeneity, low bubble content, and straightforward manufacturability make it a good choice for transmissive optics. The transmission range for BK 7 is 380–2100 nm. It is not recommended for temperature sensitive applications, such as precision mirrors. Calcium Fluoride is a cubic single crystal material grown using the vacuum Stockbarger Technique with good vacuum UV to infrared transmission. CaF2’s excellent UV transmission, down to 170 nm, and non-birefringent properties make it ideal for deep UV transmissive optics. Material for IR use is grown using naturally mined fluroite, at much lower cost. CaF2 is sensitive to thermal shock, so care must be taken during handling. Crystal Quartz is a positive uniaxial birefringent single crystal grown using a hydrothermal process. It has good transmission from the vacuum UV to the near infrared. Due to its birefringent nature, crystal quartz is commonly used for wave plates. BK 7 CaF2 LENS SELECTION GUIDE BK 7 TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Optical Materials Crystal Quartz SPHERICAL LENSES Pyrex® UV Grade Fused Silica is synthetic amorphous silicon dioxide of extremely high purity. This non-crystalline, colorless silica glass combines a very low thermal expansion coefficient with good optical qualities, and excellent transmittance in the ultraviolet. Transmission and homogeneity exceed those of crystalline quartz without the problems of orientation and temperature instability inherent in the crystalline form. Fused silica is used for both transmissive and reflective optics, especially where high laser damage threshold is required. Magnesium Fluoride is a positive birefringent crystal grown using the vacuum Stockbarger Technique with good vacuum UV to infrared transmission. It is typically oriented with the c axis parallel to the optical axis to reduce birefringent effects. High vacuum UV transmission, down to 150 nm, and its proven use in fluorine environments make it ideal for lenses, windows, and polarizers for Excimer lasers. MgF2 is resistant to thermal and mechanical shock. Pyrex® is a borosilicate glass with a low coefficient of thermal expansion. It is mainly used for non-transmissive optics, such as mirrors, due to its low homogeneity and high bubble content. MgF2 UV Fused Silica Zerodur® Zerodur® is a glass ceramic material that has a coefficient of thermal expansion approaching zero, as well as excellent homogeneity of this coefficient throughout the entire piece. This makes Zerodur ideal for mirror substrates where extreme thermal stability is required. Zerodur should not be used for transmissive optics due to inclusions in the material. KITS MgF2 CYLINDRICAL LENSES UV Grade Fused Silica OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com Optics Index of Refraction Wavelength (nm) Source BK 7 SF 2 UV Fused Silica CaF2 MgF2 no MgF2 ne 193 ArF excimer laser 1.65528 1.52127 1.56077 1.50153 1.42767 1.44127 1.66091 244 Ar-Ion laser 1.58265 1.98102 1.51086 1.46957 1.40447 1.41735 1.60439 1.61562 248 KrF excimer 1.57957 1.93639 1.50855 1.46803 1.40334 1.41618 1.60175 1.61289 Crystal Quartz Crystal Quartz no ne 1.67455 257 Ar-Ion laser 1.57336 1.86967 1.50383 1.46488 1.40102 1.41377 1.59637 1.60731 266 Nd:YAG laser 1.56796 1.82737 1.49968 1.46209 1.39896 1.41164 1.59164 1.60242 308 XeCl excimer laser 1.55006 1.73604 1.48564 1.45255 1.39188 1.40429 1.57556 1.58577 325 HeCd laser 1.54505 1.71771 1.48164 1.44981 1.38983 1.40216 1.57097 1.58102 337.1 N2 laser 1.54202 1.70749 1.47919 1.44813 1.38858 1.40085 1.56817 1.57812 351 XeF excimer laser 1.53896 1.69778 1.47672 1.44642 1.38730 1.39952 1.56533 1.57518 351.1 Ar-Ion laser 1.53894 1.69771 1.47671 1.44641 1.38729 1.39951 1.56531 1.57516 354.7 Nd:YAG laser 1.53821 1.69548 1.47612 1.44601 1.38699 1.39920 1.56463 1.57446 363.8 Ar-Ion laser 1.53649 1.69029 1.47472 1.44504 1.38626 1.39844 1.56302 1.57279 404.7 Mercury arc, h line 1.53023 1.67263 1.46961 1.44151 1.38360 1.39567 1.55714 1.56670 416 Kr-Ion laser 1.52885 1.66893 1.46847 1.44072 1.38301 1.39505 1.55583 1.56535 435.8 Mercury arc,g line 1.52669 1.66331 1.46670 1.43949 1.38207 1.39408 1.55379 1.56323 441.6 HeCd laser 1.52611 1.66184 1.46622 1.43916 1.38183 1.39382 1.55324 1.56266 457.9 Ar-Ion laser 1.52461 1.65807 1.46498 1.43830 1.38118 1.39314 1.55181 1.56119 465.8 Ar-Ion laser 1.52395 1.65641 1.46443 1.43792 1.38088 1.39284 1.55118 1.56053 472.7 Ar-Ion laser 1.52339 1.65505 1.46397 1.43760 1.38064 1.39258 1.55065 1.55998 476.5 Ar-Ion laser 1.52309 1.65432 1.46372 1.43744 1.38051 1.39245 1.55036 1.55969 480 Cadmium arc, F’ line 1.52283 1.65367 1.46350 1.43728 1.38040 1.39233 1.55011 1.55943 486.1 Hydrogen arc, F line 1.52238 1.65258 1.46313 1.43703 1.38020 1.39212 1.54968 1.55898 488 Ar-Ion laser 1.52224 1.65225 1.46301 1.43695 1.38014 1.39206 1.54955 1.55885 496.5 Ar-Ion laser 1.52165 1.65083 1.46252 1.43661 1.37988 1.39179 1.54898 1.55826 501.7 Ar-Ion laser 1.52130 1.65000 1.46223 1.43641 1.37973 1.39163 1.54865 1.55792 510.6 Cu vapor laser 1.52073 1.64865 1.46176 1.43609 1.37948 1.39137 1.54810 1.55735 514.5 Ar-Ion laser 1.52049 1.64808 1.46156 1.43595 1.37937 1.39126 1.54787 1.55711 532 Nd:YAG laser 1.51947 1.64570 1.46071 1.43537 1.37892 1.39079 1.54689 1.55610 543.5 HeNe laser 1.51886 1.64427 1.46019 1.43502 1.37865 1.39051 1.54630 1.55549 546.1 Mercury arc, e line 1.51872 1.64397 1.46008 1.43494 1.37859 1.39044 1.54617 1.55535 578.2 Cu vaport laser 1.51720 1.64053 1.45880 1.43408 1.37792 1.38974 1.54470 1.55383 587.6 Helium arc, d line 1.51680 1.63963 1.45846 1.43385 1.37774 1.38956 1.54431 1.55343 589.3 Sodium arc, D line 1.51673 1.63947 1.45840 1.43381 1.37771 1.38952 1.54424 1.55336 594.1 HeNe laser 1.51653 1.63904 1.45824 1.43370 1.37762 1.38943 1.54405 1.55316 611.9 HeNe laser 1.51584 1.63752 1.45765 1.43331 1.37732 1.38911 1.54337 1.55247 628 Ruby laser 1.51526 1.63626 1.45716 1.43298 1.37706 1.38884 1.54281 1.55188 632.8 HeNe laser 1.51509 1.63590 1.45702 1.43289 1.37698 1.38876 1.54264 1.55171 635 Laser diode 1.51501 1.63574 1.45695 1.43284 1.37695 1.38873 1.54257 1.55164 643.8 Cadmium arc, C' line 1.51472 1.63512 1.45671 1.43268 1.37682 1.38859 1.54228 1.55134 647.1 Kr-Ion laser 1.51461 1.63489 1.45661 1.43262 1.37677 1.38854 1.54218 1.55123 650 Laser diode 1.51452 1.63469 1.45653 1.43257 1.37673 1.38850 1.54209 1.55114 656.3 Hydrogen arc, C line 1.51432 1.63427 1.45637 1.43246 1.37664 1.38840 1.54189 1.55093 670 Laser diode 1.51391 1.63340 1.45601 1.43223 1.37646 1.38821 1.54148 1.55051 676.4 Kr-Ion laser 1.51372 1.63301 1.45585 1.43212 1.37637 1.38812 1.54130 1.55032 694.3 Ruby laser 1.51322 1.63198 1.45542 1.43185 1.37615 1.38789 1.54080 1.54981 750 Laser diode 1.51184 1.62922 1.45424 1.43109 1.37553 1.38724 1.53943 1.54839 MIRRORS OPTICAL SYSTEMS KITS CYLINDRICAL LENSES SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 492 Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Optics 493 Source BK 7 SF 2 UV Fused Silica CaF2 MgF2 no MgF2 ne 780 Laser diode 1.51118 1.62796 1.45367 1.43074 1.37524 1.38693 1.53878 1.54771 830 Laser diode 1.51020 1.62613 1.45282 1.43023 1.37480 1.38647 1.53779 1.54668 850 Laser diode 1.50984 1.62548 1.45250 1.43004 1.37464 1.38630 1.53742 1.54630 852.1 Cesium arc, s line 1.50980 1.62541 1.45247 1.43002 1.37462 1.38628 1.53739 1.54626 905 Laser diode 1.50892 1.62387 1.45168 1.42957 1.37422 1.38586 1.53648 1.54532 980 Laser diode 1.50779 1.62202 1.45067 1.42902 1.37371 1.38533 1.53531 1.54409 1014 Mercury arc, t line 1.50731 1.62128 1.45024 1.42879 1.37350 1.38510 1.53481 1.54357 Crystal Quartz Crystal Quartz no ne Nd:YLF laser 1.50678 1.62049 1.44976 1.42854 1.37326 1.38485 1.53425 1.54299 1060 Nd:Glass laser 1.50669 1.62035 1.44968 1.42850 1.37322 1.38480 1.53415 1.54288 1064 Nd:YAG laser 1.50663 1.62028 1.44963 1.42848 1.37319 1.38478 1.53410 1.54282 1300 Laser diode 1.50370 1.61644 1.44692 1.42721 1.37188 1.38338 1.53094 1.53950 1320 Nd:YAG laser 1.50346 1.61616 1.44669 1.42711 1.37177 1.38327 1.53068 1.53922 1550 Laser diode 1.50065 1.61312 1.44402 1.42602 1.37052 1.38194 1.52761 1.53596 1970.1 Mercury arc 1.49495 1.60780 1.43852 1.42401 1.36803 1.37928 1.52138 1.52932 2100 Ho:YAG laser 1.49296 1.60608 1.43659 1.42334 1.36718 1.37837 1.51924 1.52703 2325.4 Mercury arc 1.48921 1.60291 1.43293 1.42212 1.36559 1.37667 1.51524 1.52277 2940 Er:YAG laser 1.47670 1.59273 1.42065 1.41827 1.36051 1.37123 1.50246 1.50908 Properties of Optical Materials Abbe Number vd Coefficient of Thermal Expansion (10-6/°C) Conductivity (W/m°C) Heat Capacity (J/gm°C) Density at 25°C (gm/cm3) Knoop Hardness (kg/mm2) Young’s Modulus (GPa) BK 7 64.17 7.1 1.114 0.858 2.51 610 81.5 SF 2 33.85 8.4 0.735 0.498 3.86 410 55 UV Fused Silica 67.8 0.52 1.38 0.75 2.202 600 73 94.96 18.85 9.71 0.85 3.18 158 75.8 106.18 13.7 || to c axis 8.48 ⊥ to c axis 21 || to c axis 30 to ⊥ c axis 1.024 3.177 415 138.5 Crystal Quartz 69.87 7.1 to || c axis 13.2 ⊥ to c axis 10.4 || to c axis 6.2 ⊥ to c axis 0.74 2.649 740 97 || to c axis 76.5 ⊥ to c axis Pyrex® 66 3.25 1.13 0.75 2.23 418 65.5 Zerodur® 56.09 0 ± 0.1 1.46 0.80 2.53 620 90.3 CYLINDRICAL LENSES CaF2 MgF2 SPHERICAL LENSES 1053 LENS SELECTION GUIDE Wavelength (nm) TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Index of Refraction (continued) KITS OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 494 Optics Optics Formulas Light Right-Hand Rule Light is a transverse electromagnetic wave. The electric E and magnetic M fields are perpendicular to each other and to the propagation vector k, as shown below. Light Intensity Energy Conversions The light intensity, I is measured in Watts/m2, E in Volts/m, and H in Amperes/m. The equations relating I to E and H are quite analogous to OHMS LAW. For peak values these equations are: Power density is given by Poynting’s vector, P, the vector product of E and H. You can easily remember the directions if you “curl” E into H with the fingers of the right hand: your thumb points in the direction of propagation. Wavelength Conversions SPHERICAL LENSES 1 nm Snell’s Law The quantity η0 is the wave impedance of vacuum, and η is the wave impedance of a medium with refractive index n. OPTICAL SYSTEMS KITS CYLINDRICAL LENSES Wave Quantity Relationship k: wave vector [radians/m] ν: frequency [Hertz] ω: angular frequency [radians/sec] λ: wavelength [m] λ0: wavelength in vacuum [m] MIRRORS = 10 Angstroms(Å) = 10–9m = 10–7cm = 10–3µm n: refractive index Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Snell’s Law describes how a light ray behaves when it passes from a medium with index of refraction n1, to a medium with a different index of refraction, n2. In general, the light will enter the interface between the two medii at an angle. This angle is called the angle of incidence. It is the angle measured between the normal to the surface (interface) and the incoming light beam (see figure). In the case that n1 is smaller than n2, the light is bent towards the normal. If n1 is greater than n2, the light is bent away from the normal (see figure below). Snell’s Law is expressed as n1sinθ1 = n2sinθ2. Optics 495 Beam Deviation For plane-polarized light the E and H fields remain in perpendicular planes parallel to the propagation vector k, as shown below. A flat piece of glass can be used to displace a light ray laterally without changing its direction. The displacement varies with the angle of incidence; it is zero at normal incidence and equals the thickness h of the flat at grazing incidence. Both displacement and deviation occur if the media on the two sides of the tilted flat are different — for example, a tilted window in a fish tank. The displacement is the same, but the angular deviation δ is given by the formula. Note:δ is independent of the index of the flat; it is the same as if a single boundary existed between media 1 and 3. (Grazing incidence: light incident at almost or close to 90° to the normal of the surface). Example: The refractive index of air at STP is about 1.0003. The deviation of a light ray passing through a glass Brewster’s angle window on a HeNe laser is then: LENS SELECTION GUIDE Beam Displacement TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Plane-Polarized Light δ= (n3 - n1) tan θ δ= (0.0003) x 1.5 = 0.45 mrad Both E and H oscillate in time and space as: At 10,000 ft. altitude, air pressure is 2/3 that at sea level; the deviation is 0.30 mrad. This change may misalign the laser if its two windows are symmetrical rather than parallel. sin (ωt-kx) CYLINDRICAL LENSES The relationship between the tilt angle of the flat and the two different refractive indices is shown in the graph below. SPHERICAL LENSES At Brewster’s angle, tan θ= n2 KITS OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com Optics Angular Deviation of a Prism Angular deviation of a prism depends on the prism angle α, the refractive index, n, and the angle of incidence θi. Minimum deviation occurs when the ray within the prism is normal to the bisector of the prism angle. For small prism angles (optical wedges), the deviation is constant over a fairly wide range of angles around normal incidence. For such wedges the deviation is: δ ≈ (n - 1)α optical path. Although effects are minimal in laser applications, focus shift and chromatic effects in divergent beams should be considered. two sides of the boundary. The intensities (watts/area) must also be corrected by this geometric obliquity factor: Fresnel Equations: It = T x Ii(cosθi/cosθt) i - incident medium Conservation of Energy: t - transmitted medium R+T=1 use Snell’s law to find θt Normal Incidence: This relation holds for p and s components individually and for total power. r = (ni-nt)/(ni + nt) Polarization t = 2ni/(ni + nt) Only s-polarized light reflected. To simplify reflection and transmission calculations, the incident electric field is broken into two plane-polarized components. The “wheel” in the pictures below denotes plane of incidence. The normal to the surface and all propagation vectors (ki, kr, kt) lie in this plane. Total Internal Reflection (TIR): E parallel to the plane of incidence; ppolarized. Brewster's Angle: SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 496 θβ = arctan (nt/ni) CYLINDRICAL LENSES θTIR > arcsin (nt/ni) nt < ni is required for TIR Field Reflection and Transmission Coefficients: The field reflection and transmission coefficients are given by: r = Er/Ei t = Et/Ei KITS Non-Normal Incidence: rs = (nicosθi -ntcosθt)/(nicosθi + ntcosθt) Prism Total Internal Reflection (TIR) rp = (ntcos θi -nicosθt)/ntcosθi + nicosθt) MIRRORS OPTICAL SYSTEMS ts = 2nicosθi/(nicosθi + ntcosθt) TIR depends on a clean glass-air interface. Reflective surfaces must be free of foreign materials. TIR may also be defeated by decreasing the incidence angle beyond a critical value. For a right angle prism of index n, rays should enter the prism face at an angle θ: tp = 2nicosθi/(ntcosθi + nicosθt) Power Reflection: The power reflection and transmission coefficients are denoted by capital letters: θ < arcsin (((n2-1)1/2-1)/√2) R = r2 T = t2(ntcosθt)/(nicosθi) In the visible range, θ = 5.8° for BK 7 (n = 1.517) and 2.6° for fused silica (n = 1.46). Finally, prisms increase the The refractive indices account for the different light velocities in the two media; the cosine ratio corrects for the different cross sectional areas of the beams on the Phone: 1-800-222-6440 • Fax: 1-949-253-1680 E normal to the plane of incidence; s-polarized. Optics Magnification: Power reflection coefficients Rs and Rp are plotted linearly and logarithmically for light traveling from air (ni = 1) into BK 7 glass (nt = 1.51673). Brewster’s angle = 56.60°. Transverse: If a lens can be characterized by a single plane then the lens is “thin”. Various relations hold among the quantities shown in the figure. MT < 0, image inverted Longitudinal: TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Power Reflection Coefficients Thin Lens Equations 497 Gaussian: Sign Conventions for Images and Lenses Thick Lenses Quantity + virtual real real virtual F convex lens concave lens Lens Types for Minimum Aberration | s2/s1 | Best lens <0.2 plano-convex/concave >5 plano-convex/concave >0.2 or <5 bi-convex/concave CYLINDRICAL LENSES s1 s2 A thick lens cannot be characterized by a single focal length measured from a single plane. A single focal length F may be retained if it is measured from two planes, H1, H2, at distances P1, P2 from the vertices of the lens, V1, V2. The two back focal lengths, BFL1 and BFL2, are measured from the vertices. The thin lens equations may be used, provided all quantities are measured from the principal planes. SPHERICAL LENSES ML <0, no front to back inversion LENS SELECTION GUIDE The corresponding reflection coefficients are shown below for light traveling from BK 7 glass into air Brewster’s angle = 33.40°. Critical angle (TIR angle) = 41.25°. Newtonian: x1x2 = -F2 KITS OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com Optics Lens Nomogram: The Lensmaker’s Equation Numerical Aperture Convex surfaces facing left have positive radii. Below, R1>0, R2<0. Principal plane offsets, P, are positive to the right. As illustrated, P1>0, P2<0. The thin lens focal length is given when Tc = 0. φMAX is the full angle of the cone of light rays that can pass through the system (below). Constants and Prefixes Speed of light in vacuum c = 2.998108 m/s Planck’s const. h = 6.625 x 10-34Js Boltzmann’s const. k = 1.308 x 10-23 J/K Stefan-Boltzmann σ = 5.67 x 10-8 W/m2 K4 1 electron volt eV = 1.602 x 10-19 J exa (E) 1018 peta (P) 1015 tera (T) 1012 KITS CYLINDRICAL LENSES SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 498 For small φ: MIRRORS OPTICAL SYSTEMS Both f-number and NA refer to the system and not the exit lens. Phone: 1-800-222-6440 • Fax: 1-949-253-1680 giga (G) 109 mega (M) 106 kilo (k) 103 milli (m) 10-3 micro (µ) 10-6 nano (n) 10-9 pico (p) 10-12 femto (f) 10-15 atto (a) 10-18 Wavelengths of Common Lasers Source (nm) ArF 193 KrF 248 Nd:YAG(4) 266 XeCl 308 HeCd 325, 441.6 N2 337.1, 427 XeF 351 Nd:YAG(3) 354.7 Ar 488, 514.5, 351.1, 363.8 Cu 510.6, 578.2 Nd:YAG(2) 532 HeNe 632.8, 543.5, 594.1, 611.9, 1153, 1523 Kr 647.1, 676.4 Ruby 694.3 Nd:Glass 1060 Nd:YAG 1064, 1319 Ho:YAG 2100 Er:YAG 2940 Optics Focusing a Collimated Gaussian Beam Depth of Focus (DOF) The Gaussian intensity distribution: In the figure below the 1/e radius, ω(x), and the wavefront curvature, R(x), change with x through a beam waist at x = 0. The governing equations are: DOF = (8λ/π)(f/#)2 2 I(r) = I(0) exp(-2r2/ω02) is shown below. ( New Waist Diameter ) ω2(x) = ω20 ⎡1 + λx /πω20 ⎤⎥ ⎢⎣ ⎦ 2 2 ⎡ ⎤ R(x) = x 1 + πω 0 / λx ⎥⎦ ⎢⎣ 2 ) Beam Spread 2ω0 is the waist diameter at the 1/e2 intensity points. The wavefronts are planar at the waist [R(0) = ∞]. At the waist, the distance from the lens will be approximately the focal length: s2≈F. D = collimated beam diameter or diameter illuminated on lens. The total beam power, P(∞) [watts], and the on-axis intensity I(0) [watts/area] are related by: KITS The figure below compares the far-field intensity distributions of a uniformly illuminated slit, a circular hole, and Gaussian distributions with 1/e2 diameters of D and 0.66D (99% of a 0.66D Gaussian will pass through an aperture of diameter D). The point of observation is Y off axis at a distance X>Y from the source. CYLINDRICAL LENSES Diffraction SPHERICAL LENSES The right hand ordinate gives the fraction of the total power encircled at radius r: LENS SELECTION GUIDE ( Only if DOF <F, then: TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Gaussian Intensity Distribution 499 OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com KITS CYLINDRICAL LENSES SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 500 Optics Optics Glossary Abbe Number: The constant of an optical medium that describes the ratio of its refractivity to its dispersion. Back Focal Length (BFL): The distance between the last surface of a lens to its image focal plane. Specifically, Vd = (nd-1)/(nF-nC), where n is the index of refraction at the Fraunhofer d, F, and C lines, respectively. Bandpass: The range of wavelengths that passes through a filter or other optical component. Aberration: An optical defect resulting from design or fabrication error that prevents the lens from achieving precise focus. The primary aberrations are spherical, coma, astigmatism, field curvature, distortion, and chromatic aberration. Bandwidth: Range of wavelengths over which the specified transmission or reflection occurs. Achromatic Lens: Lens in which chromatic aberration has been corrected at a minimum of two wavelengths. Airy Disc: A pattern of illumination caused by diffraction at the edge of a circular aperture, consisting of a central core of light surrounded by concentric rings of gradually decreasing intensity. Anamorphic: Distorted, as in an optical system with different magnification levels or with focal lengths perpendicular to the optical axis. Angle of Incidence: The angle formed by a ray of light striking a surface and the normal to that surface. Antireflection (AR) Coating: A thin layer of material that, when applied to a lens or window, increases its transmittance by reduction of its reflectance. AR coatings may be multilayer or single layer coatings. OPTICAL SYSTEMS Aperture: An opening through which light may pass. The clear aperture is that area in an optical system limiting the bundle of light able to pass through the system. Aspheric: Not spherical. To reduce spherical aberration, a lens may be altered slightly so that one or more surfaces are Aspheric. Beamsplitter: An optical device that divides an incident beam into at least two distinct beams. Bi-Concave: Having two outer surfaces that curve inward. Bi-Convex: Having two outer surfaces that curve outward. Birefringence: The change in refractive index with the polarization of light. A birefringent crystal, such as calcite or quartz, will divide an unpolarized beam into two beams (ordinary and extraordinary) having opposite polarization. Blocking: Refers to filter transmittance outside the bandpass region. It is the rejection of out-of-band wavelengths by a filter. Blur Circle: The image of a point-source object formed by an optical system on its focal surface. The precision level of the lens and its state of focus determine the size of the blur. Borosilicate Glass: An optical glass containing boric oxide, along with silica and other ingredients. BK 7 and Pyrex® are examples of borosilicate glasses. Brewster’s Angle: For light incident on a plano boundary between two materials having different index of refraction; that angle of incidence at which the reflectance is zero for light that has its electrical field vector in the plane defined by the direction of propagation and the normal to the surface. For propagation from material 1 to material 2, Brewster’s angle is given as tan-1(n2/n1). MIRRORS Astigmatism: An aberration in a lens in which the tangential and sagittal (horizontal and vertical) lines are focused at two different points along the optical axis. Beam Deviation: See Deviation. Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Broadband Coating: A multilayer coating with specified reflection or transmission over a broad spectral band. Newport’s AR.14 is a broadband AR coating, while Newport BD.1 is broadband mirror coating. Cavity: A periodic structure of thin films comprised of two quarter-wave stack reflectors separated by a dielectric spacer. Cavities are the building blocks of bandpass filters. Center Wavelength: The center of the wavelength band of a coating. Centration: The deviation between the optical axis and the mechanical axis of a lens. Centration is specified in terms of the deflection of a beam directed along the mechanical axis of the lens. Chromatic Aberration: An optical defect in a lens resulting in different wavelengths of light focusing at different distances from the lens. Corrected by achromatic lenses. Circle of Least Confusion: The smallest cross-section of a focused beam of light at the point of best focus for the image. Clear Aperture: The area of an optical component that controls the amount of light incident on a given surface. In Newport lenses and mirrors, the clear aperture gives the diameter over which specifications are guaranteed. Coefficient of Thermal Expansion: A material property defined as the fractional change in length per original length (or fractional change in volume) with a change in temperature. Collimated Beam: A beam of light in which all of the rays are parallel to each other. Coma: An aberration that occurs in a lens when rays emanating from points not on the optical axis do not converge, causing the image of a point to appear comet-shaped. Cone Angle: The central angle of a cone of rays converging to or diverging from a point. See Numerical Aperture. Optics Continuous Wave Irradiation: Emission of radiant energy (light) in a continuous wave, rather than pulsed. Converging: The bending of light rays toward one another, achieved with a positive (convex) lens. Cut-Off Wavelength: For a filter, the wavelength where the transmission falls below 50%. Deviation: The angle between the paths of a ray of light before and after passing through one or more optics. Extinction Ratio: The ratio of the intensities along the polarization axes of a plane-polarized beam that is transmitted through a polarizer; expressed as Tp/Ts. Dielectric Coatings: Thin-film optical coatings made up of alternating layers of non-conductive material. The key factor in whether one uses a dielectric coating or another technology to accomplish the filtering effect is whether or not absorption is desired. Dielectric coatings typically have low to non-existent absorption whereas coatings using metals often exhibit some level of absorption. Diffraction Limited: Describes an optical system in which the quality of the image is determined only by the effects of diffraction and not by lens aberrations. Cylindrical Lens: A lens with at least one surface shaped like a portion of a cylinder. A typical application is reducing the astigmatism of laser diodes. Diverging: The bending of light rays away from each other, achieved with a negative (concave) lens. Edging: Grinding, or finishing, the edge of an optical element or lens. Field Curvature: An aberration in which the edges of a field seem to be out of focus when the center is focused clearly. Field of View: The maximum visible space seen through a lens or optical instrument. Figure: See Surface Figure. Flatness: See Surface Flatness. Flint Glass: An optical glass with higher dispersion and higher refractive index than crown glass; a heavy, brilliant glass, softer than crown glass. For example, SF Series glasses are used in Newport achromatic lenses. Focal Length (FL): See Effective Focal Length. Front Focal Length (FFL): The distance from the objective plane of a lens to its first surface. Fused Silica: Crystal quartz melted at a high temperature to make an amorphous, non-birefringent glass of low refractive index. Used in high-energy components and optical components designed for UV. It can be used down to 195 nm. OPTICAL SYSTEMS Distortion: Variations in magnification from the center to the edge of an image, making straight lines look curved. Barrel, or negative, distortion causes a square grid to appear barrel-shaped; pincushion, or positive, distortion increases in proportion to the distance from the center of the image. F-Number: A measure of the ability of a lens to gather light. Represented by f/# and also called its ”speed”. The ratio of the focal length of the lens to its effective aperture. Related to numerical aperture by f/#=1/(2NA). KITS Dispersion: The separation of a beam into its various wavelength components due to wavelength dependent speed of propagation in the material. Cut-On Wavelength: For a filter, the wavelength where the transmission increases above 50%. Decentration: The failure of one or more lens surfaces to align their centers of curvature with the geometric axis of a lens system. Erect Image: An image whose spatial orientation is the same as that of the object. Diffraction: The sidewise or sideways spread of light as it passes the edge of an object or emerges from a small aperture; causes halos or blurring of the image. Crystal Quartz: Crystalline form of silicon dioxide; used in wave plates. Damage Threshold: The maximum energy density to which an optical surface may be subjected without failure. Depth of Focus: The distance along the optical axis through which an image can be clearly focused. CYLINDRICAL LENSES Crown Glass: A silicate glass containing oxides of sodium and potassium, used in lenses and windows. Harder than flint glass, it has low index and low dispersion, such as BK 7. Entrance Pupil: The image of the aperture stop as viewed through the object side of the lens. SPHERICAL LENSES Critical Angle: The smallest angle of incidence at which total internal reflectance takes place. Maximum angle of incidence formed by a ray of light as it passes from a denser to a less dense medium. When the critical angle is exceeded, total internal reflection occurs, and all the incident light reflects back in to the more dense media. Depth of Field: The distance along the optical axis through which an object can be located and clearly defined when the lens is in focus. LENS SELECTION GUIDE Contrast: The difference in light intensity in an object or image; defined as (Imax - Imin)/(Imax + Imin), where Imax and Imin are the maximum and minimum intensities. Effective (or Equivalent) Focal Length (EFL): The focal length of an infinitely thin lens having the same paraxial imaging properties as a thick lens or multiple-element lens system. TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Density, Optical: A measure of the transmittance (T) through an optical medium; expressed as D = -log (T) or T = 10-D. Conjugate Ratio: The ratio of the object distance to the image distance. 501 MIRRORS Email: [email protected] • Web: newport.com Optics filters can be constructed using interference, including bandpass, beamsplitter, dichroic, and edge filters. Metallic Coating: A thin layer of metal applied to a substrate by evaporation to create a mirrored surface. Gaussian Optics: Optical characteristics limited to infinitesimally small pencils of light; also called paraxial or firstorder optics. Interferometer: An instrument that uses the interference of light waves to measure small displacements or deformation. Iris Diaphragm: A mechanical device for varying the effective diameter of an optical system. Micro Optics: A term referring to small (less than 5 mm in size) lenses, beamsplitters, prisms, cylinders or other optical components commonly found in endoscopes or microscopes. Micro optics are also used to focus light in semiconductor laser and fiber optic applications. Geometric Optics: That branch of optics dealing with the tracing of ray paths through optical systems. Geometric optics ignores the nature of the electromagnetic modes of light. High-Efficiency Coating: Specialized coating applied to optics to improve transmission or reflection. Irregularity: Refers to figure deviations that are not spherical in nature. Using a test plate, irregularity is measured by counting the difference in the number of fringes in two orthogonal axes. Homogeneity: The state in which all volume components of a substance are identical in optical properties and composition. Knoop Hardness: A measure of hardness determined by the depth of penetration of a diamond stylus under a specified load. Similar to the Rockwell hardness test. Hybrid: Anything formed out of heterogeneous elements. Lateral Color: A chromatic aberration resulting in image size variation as a function of wavelength. Also known as chromatic difference of magnification. CYLINDRICAL LENSES LENS SELECTION GUIDE FWHM: Full Width Half Maximum. The bandwidth of an optical instrument as measured at the half-power points. SPHERICAL LENSES TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 502 Image Circle: The circular image field over which image quality is acceptable; can be defined in terms of its angular subtense. Alternately known as circle of coverage. Image Inversion: Change in the orientation of an image in one meridian. MIRRORS OPTICAL SYSTEMS KITS Image Plane: The plane perpendicular to the optical axis at the image point. Image Transposition: The flipping of an image’s orientation, such as inversion of an image’s orientation in one axis or the reversion of an image’s orientation in two axes. Index of Refraction: The ratio of the speed of light in air to its velocity in another medium; determines how much light bends as it passes through a lens, e.g., high-index flint glass bends light more than low-index crown glass does. Infrared: The long wavelength portion of the spectrum whose wavelengths are invisible to the human eye (the range is approximately 780 nm and longer wavelengths). Interference Filter: A filter that controls the spectral composition of transmitted energy by interference. Several types of Limit of Resolution: The limit to the performance of a lens imposed by the diffraction pattern resulting from the finite aperture of the optical system. Long Pass: Filter that efficiently passes radiation whose wavelengths are longer than a specific wavelength, but not shorter. Longitudinal Color: The longitudinal variation of focus (or image position) with wavelength; often referred to as axial chromatic aberration. Magnesium Fluoride: Material used as antireflection coating for lenses because of its low refractive index. Also used as an optical substrate material for UV and infrared applications. Magnification: The enlargement of an object by an optical instrument; ratio of the size of the image to the actual size of the object. Meniscus: Describes a lens having one convex and one concave surface. Meridional Plane: The plane in an optical system containing its optical axis and the chief ray. Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Microscope Eyepiece: An eyepiece located at the near end of the microscope tube. Often a simple Huygen’s eyepiece, though other varieties (negative eyepieces, flat field projection eyepieces) are common, depending on application. Microscope Objective: The lens located at the object end of a microscope tube. Many types of objectives are used in microscopy; simple achromats and colorcorrected apochromats are popular choices. MIL-C-675: Specifies that a coating will not show degradation to the naked eye after 20 strokes with a rubber pumice eraser. Coatings meeting MIL-C-675 can be cleaned repeatedly and survive moderate to severe handling. MIL-C-14806: Specifies durability of surfaces under environmental stress. Coatings are tested at high humidity, or in brine solutions to determine resistance to chemical attack. These coatings can survive in humid or vapor filled areas. MIL-M-13508: Sets the durability standards for metallic coatings. Coatings will not peel away from the substrate when pulled with cellophane tape. Further, no damage visible to the naked eye will appear after 50 strokes with a dry cheesecloth pad. Gentle, nonabrasive cleaning is advised. Modulation Transfer Function (MTF): A measure of the ability of an optical lens or system to transfer detail of the object to the image. Given as degree of contrast (or modulation depth) in the image as a function of spatial frequency. Optics Optical Density: See Density, Optical. Narrowband Coating: A coating designed to provide transmittance (or reflectance) over a very restricted band of wavelengths. Paraxial Image Plane: Image plane located by using first-order geometric optics. See Gaussian Optics. Pinhole Aperture: A small, sharp-edged hole that functions as an aperture, for example, in a spatial filter. Plane of Incidence: The plane that is defined by the incident and reflected rays. Plano-Concave: A lens with one flat (plano) surface and one inward-curved (concave) surface. Plano-Convex: A lens with one flat (plano) surface and the other outwardcurved (convex) surface. Plano Elements: Lenses or mirrors with flat surfaces. Object-to-Image Distance: Also known as the total conjugate distance or track length. Can be finite or infinite depending on the application. Polarized, Circularly: Light whose electric field vector describes a circle as a function of time. Objective: The optical element that receives light from the object and forms the first or primary image in telescopes, microscopes, and other optical systems. Principal Planes: In a thick lens or multiple-lens system, the plane at which the entering rays and exiting rays appear to intersect the position of the equivalent thick lens. Pulse Modulation: The process of periodically or intermittently varying the amplitude of a pulse of light. Q: The Q of a resonator is defined as: (2π x average energy stored in the resonator)/(energy dissipated per cycle) Q-Switched: In an optical resonator, the higher the reflectivity of its surfaces, the higher the Q. A Q-switch rapidly changes the Q in the optical resonator of a laser to prevent lasing until a high level of optical gain and energy storage has been reached in the lasing medium; a giant pulse is generated when the Q is rapidly decreased. Quarter Wave Optical Thickness: Common thin-film term. The QWOT (Quarter Wave Optical Thickness) is the wavelength at which the optical thickness, defined as the index of refraction, n, multiplied by the physical thickness, d, of a coating evaporant layer; is one quarter wavelength, or n x d=λ/4. OPTICAL SYSTEMS Numerical Aperture: Defines the maximum cone angle of light accepted or emitted by an optical system. Given by sine of the half-angle of the maximum angle. Related to f-number by NA = 1/(2f/#). Orthogonal: Mutually perpendicular. Outof-Band Blocking; See Blocking. Primary Reflections: The principal, intended reflections at optical surfaces, as differentiated from secondary, usually unintended or unwanted reflections occurring in an optical system. KITS Nodal Points: The two points at which the nodal planes appear to intersect with the optical axis. When a ray is directed at the first nodal point in an optical system, it appears to emerge from a second nodal point on the optical axis with no deviation in its angle. Optical Path Difference: For a perfect optical system, the optical path or distance from an object point to a corresponding image point will be equal for all rays. In near-perfect systems, slight differences will exist between rays resulting in an optical path difference, usually expressed in fractions of the wavelength being analyzed. CYLINDRICAL LENSES Newton’s Rings: Used to measure the fit of a lens surface against the surface of a test glass. The rings result when two adjacent polished surfaces are placed together with an air space between them and the light beams they reflect interfere. Optical Interference: The additive process, whereby the amplitudes of two or more overlapping light waves are systematically attenuated and reinforced. Power: 1) Lens, See Magnification (magnification power). 2) Refers to figure deviations that are spherical in nature. Using a test plate, power is measured by counting the number of fringes in two orthogonal axes. Power comprises the majority of figure deviations in a lens. Sometimes called Spherical Error. SPHERICAL LENSES Neutral Density: A coating or absorbing glass, which has a flat or nearly flat absorption curve throughout a specified spectrum. Neutral density filters decrease the intensity of light without changing the relative spectral distribution of energy. Optical Flat: A piece of glass with one or both surfaces polished flat. Also known as a test plate, test glass or reference flat. LENS SELECTION GUIDE Multilayer Coating: Coating composed of several layers of coating material. Different multilayer designs are used to produce a variety of coating components such as mirrors, AR coatings, bandpass coatings, dichroic coatings, and beamsplitters. Polychromatic Aberrations: The separation of an image into planes of distinct color, caused by the variation of the index of refraction of glass, and the focal length of a lens, with the wavelength of light; in a given plane, all colors but one are unfocused. TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Optical Axis: A line passing through the centers of curvature of a lens or other optical components. Multi-Element System: An assembly of single and/or compound lenses optimized to provide certain optical characteristics. 503 Polarized, Linearly: See Polarized, Plane. Radius of Curvature: One-half the diameter of a circle defining the convex or concave shape of a lens. Email: [email protected] • Web: newport.com MIRRORS Oblique Ray: A ray of light that is neither perpendicular nor parallel, but inclined. Polarized, Plane: Light whose electric field vector vibrates in only one plane. CYLINDRICAL LENSES SPHERICAL LENSES LENS SELECTION GUIDE TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS 504 Optics Real Image: Light rays reproduce an object, called an image, by gathering a beam of light diverging from an object point and transforming it into a beam converging toward another point. If the beam is converging, it produces a real image. Reference Flat: An optical flat used as a test glass. Refraction: The change in direction of a ray of light as it passes from one optical medium to another with a different optical density. See Snell’s Law. Refractive Index: The ratio between the speed of light through vacuum to the speed of light through the particular medium. The index determines how much a ray of light will bend as it passes from one given medium to another. See Snell’s law. Resolution: The ability of a lens to image the points, lines, and surfaces of an object so they are perceived as discrete entities. Reticle: An optical element containing a pattern placed at the image plane of a system. The reticle facilitates system alignment or the measurement of target characteristics. MIRRORS OPTICAL SYSTEMS KITS Reverted Image: An image in which left and right seem to be reversed. Rockwell Hardness: Resistance of a substance to penetration by a pyramidal stylus pressed in under a specific load; also see Knoop hardness. Sag: An abbreviation for “sagitta,” the Latin word for “arrow.” Used to specify the distance on the normal from the surface of a concave lens to the center of the curvature. It refers to the height of a curve measured from the chord, Scratch-Dig: A measure of the visibility of surface defects as defined by several U.S. military standards including MIL-PRF-13830B, MIL-F-48616, and MIL-C-48497. Unless otherwise noted, specifications for surface quality of our products are in accordance with MIL-PRF-13830. Using MIL-PRF-13838B, the ratings consist of two numbers, the first denoting the visibility of scratches, the second, of digs (small pits). A 0-0 scratch-dig number indicates a surface free of visible defects. Numbers increase as the visibility of blemishes increases. Scratches and digs are evaluated for size by comparison to standards fabricated in accordance with US Army ARDEC drawing C7641866. No absolute measurement of defect size is made or implied by the scratch-dig standard. MIL-F-48616 and MIL-C-48497 use alphabetical notations to designate defect size and prescribe physical measurement of defects to determine conformance. A specification of F/F using MIL-C or MIL-F is approximately equivalent to 80/50 with the exception that measurement is used to characterize defects rather than comparison to a set of standards. Short Pass: Filter that efficiently passes radiation whose wavelengths are shorter than a specific wavelength, but not longer. Slit: An aperture, typically rectangular in shape, whose length is large compared to its width. Snell’s Law of Refraction: Gives the ratio of bend angles as light passes from one medium to another; expressed as n1sinθ1=n2sinθ2, where n is the index of refraction. Spatial Filtering: Enhancing an image by increasing or decreasing its spatial frequencies. Spectrophotometry: Measuring the reflection or transmission of light for each component wavelength in the spectrum of a specimen. where R = radius of curvature of the surface and Y = radius of the aperture of the surface. Sagittal Focus: The focus of rays lying in the sagittal plane, which is the plane perpendicular to the meridional plane. Spherical Error: See Power. Spot Size: Minimum image size to which a lens may focus a collimated beam. Phone: 1-800-222-6440 • Fax: 1-949-253-1680 Striae: An imperfection in optical glass characterized by streaks of transparent material of a different refractive index than the body. Substrate: The underlying material to which an optical coating is applied. Surface Contour: The outline or profile of a surface. Surface Figure: A measure of how closely the surface of an optical element matches a reference surface. Since geometrical errors will cause distortion of a transmitted or reflected wave, deviations from the ideal are measured in terms of wavelengths of light. Surface Flatness: The amount by which an optical surface differs from a perfect plane. It is typically measured by an interferometric technique. Surface Roughness: A measure of the texture of a surface on a microscopic scale. It is usually denoted as a root mean square (rms) value and measured in units of length, such as angstroms. Surface Quality: See Scratch-Dig. Total Internal Reflection (TIR): When the angle of incidence of light striking the boundary surface of a substance exceeds the critical angle, the result is total internal reflection. Transmission: Amount of light that is passed through an optical component or system. Given as fraction or percentage of input light. Truncation Ratio: The dimensionless ratio of the Gaussian beam diameter at the 1/e2 intensity point to the limiting aperture of the lens. Ultraviolet: The short wavelength of the electromagnetic spectrum invisible to the human eye. The range is approximately 400 nm and shorter wavelengths. V-Coating: A narrowband coating for specific laser wavelengths. This term is usually applied in reference to AR coatings. Optics LENS SELECTION GUIDE Virtual Image: Light rays that diverge from an object point can be captured by an optical system to form an image. Depending on the optical system, the light beam can either converge to another point or diverge from another point. In the case that the light converges, it will form a real image. In the case that the light diverges it will form a virtual image. TECHNICAL REFERENCE AND FUNDAMENTAL APPLICATIONS Vignetting: The gradual reduction of image illuminance with an increasing offaxis angle, resulting from limitations of the clear apertures of elements within an optical system. 505 V-Value: See Abbe Number. Wavelength: The distance light travels in one cycle of its electromagnetic wave. CYLINDRICAL LENSES Wedge: An optical element with its faces inclined toward each other at very small angles, diverting light toward the thicker parts of the element. SPHERICAL LENSES Wavefront Distortion: Departure of a wavefront from ideal (usually spherical or planar) caused by surface errors or design limitations. Young’s Modulus: Modulus of elasticity; the amount of stress required to produce a unit change in length (strain); expressed in pounds per square inch (PSI) or dynes per square cm. KITS OPTICAL SYSTEMS MIRRORS Email: [email protected] • Web: newport.com