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TECHNICAL GUIDE
OPTICAL COATINGS & MATERIALS
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MATERIAL PROPERTIES
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OPTICAL SPECIFICATIONS
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FUNDAMENTAL OPTICS
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GAUSSIAN BEAM OPTICS
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MACHINE VISION GUIDE
A173
LASER GUIDE
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Technical Guide
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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
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ULTRAFAST COATING (TLMB)
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OPTICAL FILTER COATINGS
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NEUTRAL DENSITY FILTERS
A53
LASER-INDUCED DAMAGE
A55
OEM AND SPECIAL COATINGS
A56
Gaussian Beam Optics
MAXBRITE™ COATINGS (MAXB)
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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
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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
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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
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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.
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The Reflection of Light
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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.
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This leads to the expression for Brewster’s angle, θB:
θ1 = θB= arctan (n2 / n1 )
(1.5)
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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 = 
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In the limit of normal incidence in air, Fresnel’s laws
reduce to the following simple equation:
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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
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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
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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.
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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
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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
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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.
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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.
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Figure 1.6 A simple representation of constructive and
destructive wave interference
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The Reflection of Ligh
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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
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Optical Coatings
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Optical Specifications
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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
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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 . . .
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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
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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)
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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.
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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
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 n (l ) sin v1 
.
v3 = arcsin  1

 n3 (l ) 
A15
OPTICAL COATINGS
Optical Coatings and Materials
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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
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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
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A17
OPTICAL COATINGS
Optical Coatings and Materials
a skewed V shape with a reflectance minimum at the
design wavelength.
TXDUWHUTXDUWHUDQWLUHIOHFWLRQFRDWLQJ
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&
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
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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
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Optical Coatings
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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
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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
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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
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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.
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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.
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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
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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)
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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
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Fundamental Optics
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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.
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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
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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
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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.
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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
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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
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There are two families of broadband antireflection
coatings from CVI Laser Optics: HEBBAR™ and BBAR.
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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
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X HEBBAR™
Gaussian Beam Optics
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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.
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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
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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
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X HEBBAR™
HEBBAR™ coating for 300 – 500 nm
threshold: 3.2 J/cm2, 10 nsec pulse
at 355 nm typical
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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
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A30
X Specialty
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X Specialty
HEBBAR™ coating for 425 – 670 nm optimized for 45°
X R
avg
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X Damage
threshold: 3.8 J/cm2, 10 nsec pulse
at 532 nm typical
1-505-298-2550
OPTICAL COATINGS
Optical Coatings
& Materials
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lasers
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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
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BandHEBBAR™ coating for 780 – 830 nm
and 1300 nm
X R
Band HEBBAR™ coating for 450 – 700 nm and 1064 nm
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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
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Optical Specifications
X Specialty
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Material Properties
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< 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
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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
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Optical Specifications
BBAR/45 425 – 675 coating for the visible region
(45° incidence)
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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
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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.
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BBAR/45 1050 – 1600 coating for the NIR region
(45° incidence)
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BBAR 193 – 248 coating for the UV region (0° incidence)
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Machine Vision Guide
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Gaussian Beam Optics
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BBAR 248 – 355 coating for the UV region (0° incidence)
Laser Guide
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A33
OPTICAL COATINGS
Optical Coatings and Materials
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BBAR 355 – 532 coating for the UV region (0° incidence)
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BBAR 425 – 675 coating for VIS and NIR regions (0° incidence)
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BBAR 670 – 1064 coating for VIS and NIR regions (0° incidence)
A34
CVI Laser Optics Antireflection Coatings
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BBAR 3500 – 5000 coating for the IR region (0° incidence)
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BBAR 1050 – 1600 coating for VIS and NIR regions
(0° incidence)
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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
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Material Properties
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V-Coating Center Wavelengths
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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
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3HUFHQW5HIOHFWDQFH
QRUPDOLQFLGHQFH
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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
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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
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VSODQH
SSODQH
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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
:DYHOHQJWK—P
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
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:DYHOHQJWK—P
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.
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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
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MAXBRIte™ Coatings (MAXB)
A43
OPTICAL COATINGS
Optical Coatings and Materials
7\SLFDOUHIOHFWDQFHFXUYH
QRUPDOLQFLGHQFH
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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
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MAXB-630-850 COATING
> 98% from 420 – 700 nm
X Damage
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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.
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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
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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
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1-505-298-2550
OPTICAL COATINGS
Optical Coatings
& Materials
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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
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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
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Optical Specifications
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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
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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
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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
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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.
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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
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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
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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.
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OEM and Special Coatings
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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
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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
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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
t1t2n =− 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
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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
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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
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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.
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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
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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.
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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
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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
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E8SSHU/LPLWV
1%.
89
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*6
)6
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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
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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%.
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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
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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
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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
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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.
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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
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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.
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External transmittance for 1 mm thick uncoated sapphire
Laser Guide
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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
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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
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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
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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
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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
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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
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External transmittance for 10 mm thick uncoated
germanium
A80
Refractive Index of Germanium
Germanium
1-505-298-2550
MATERIAL PROPERTIES
Optical Coatings
& Materials
MATERIAL PROPERTIES OVERVIEW
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Gaussian Beam Optics
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Fundamental Optics
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Optical Specifications
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Material Properties
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Laser Guide
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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
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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.
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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
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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
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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
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OPTICAL SPECIFICATIONS
Optical Coatings
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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.
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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
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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
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Surface Accuracy
A91
OPTICAL SPECIFICATIONS
Optical Specifications
A92
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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
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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
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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.
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Gaussian Beam Optics
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Machine Vision Guide
s
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Figure 4.1 Sign conventions
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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)
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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
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FUNDAMENTAL OPTICS
Optical Coatings
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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.
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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.
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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
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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.
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Machine Vision Guide
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4
4
4
4
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Figure 4.23 Aberrations of positive singlets at infinite conjugate ratio as a function of shape
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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
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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
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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
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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
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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
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1.7
0.8
0.5
FUNDAMENTAL OPTICS
Optical Coatings
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Material Properties
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Gaussian Beam Optics
Figure 4.28 Fraunhofer diffraction pattern of a singlet slit
superimposed on the Fraunhofer diffraction pattern of a
circular aperture
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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
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Figure 4.29 Collimating an incandescent source
A118
Lens Selection
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Optical Coatings
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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
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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
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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 /# =
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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
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FUNDAMENTAL OPTICS
Optical Coatings
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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
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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
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FUNDAMENTAL OPTICS
Optical Coatings
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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
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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
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VHFRQGDU\SULQFLSDOSRLQW
tH
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UD\IURPREMHFWDWLQILQLW\
RSWLFDOD[LV
r
SULPDU\YHUWH[ $ )
+
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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
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FUNDAMENTAL OPTICS
Optical Coatings
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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)
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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
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FUNDAMENTAL OPTICS
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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
>
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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)
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FUNDAMENTAL OPTICS
Optical Coatings
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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)
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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
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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.
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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
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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
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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:
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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
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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
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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
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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
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(+ = 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
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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
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Optical Coatings
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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.
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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
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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
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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
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FUNDAMENTAL OPTICS
Optical Coatings
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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
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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
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Optical Coatings
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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
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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
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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.
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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
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Optical Coatings
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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)
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Ultrafast Theory
A151
FUNDAMENTAL OPTICS
Fundamental Optics
6)
6)
6)
6)
6)
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72'IVñFP
*9'IVðFP
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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
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Optical Coatings
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ƒ6
ƒ3
3mm crystal
negative
GVD mirror
pump
7/0
output coupler
:DYHOHQJWKQP
ƒ3
negative
GVD mirror
7/0
Figure 4.65 Typical optical setup incorporating low GVD
and Negative GVD mirrors in an ultrafast application
ƒ6
200 mW
80 fs pulses
centered at 785 nm
Optical Specifications
*'' F´qIVHF
low GVD
mirror
ƒ
Material Properties
5HIOHFWLYLW\
ƒ
:DYHOHQJWKQP
ƒ
ƒ6
ƒ3
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.
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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.
ƒ6
7UDQVPLWWHGƒ3*''IVHFð
7UDQVPLWWHGƒ3&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)
)$%63
›F
/
η = refractive index of the prisms (assuming the same
material)
7)3.
ȦȦO
ȜO
where
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A154
G
ȥ
GȦ
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Very-high-quality isosceles Brewster’s angle prisms for
intra and extracavity use are available from CVI Laser
Optics. The design of these prisms satisfies the condition
of minimum loss due to entrance and exit at Brewster’s
angle. To calculate GVD at Brewsters angle, refer to
Figure 4.70 and use the following equation:
5HIOHFWHG%HDP*''IVHFð
7UDQVPLVVLRQ
PRISMS
5HIOHFWHG%HDP&8%,&IVHFñ
Fundamental Optics
Power transmission curves for the s- and p-polarized
versions of the FABS, along with the corresponding
reflected phase characteristics for beamsplitters
optimized at 800 nm, are shown in Figures 4.68 and
4.69. The linear pulse propagation properties of these
beamsplitters are dominated by the substrate material
dispersion. As with virtually all dielectric coated optics,
the s-polarized version is broader than p-polarized
version. CVI Laser Optics can produce FABS in other than
50:50 with excellent phase characteristics.
)$%63
)$%63
:DYHOHQJWKQP
Figure 4.68 Transmission characteristics for FABS series
polarizers with p-polarized light
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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
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A155
FUNDAMENTAL OPTICS
Fundamental Optics
A156
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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
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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
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(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  


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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
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GAUSSIAN BEAM OPTICS
Optical Coatings
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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
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Figure 5.4 Changes in wavefront radius with propagation distance
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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
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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)
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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
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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
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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
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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
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TEM00
TEM01
TEM10
TEM11
TEM02
Figure 5.13 Low‑order Hermite‑Gaussian resonator modes
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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
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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)
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( 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
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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
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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
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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).
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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
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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.
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Gaussian Beam Optics
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Optical Specifications
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Machine Vision Guide
Anatomy of a machine vision system
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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
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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.
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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.
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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)
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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.
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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)
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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 .
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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
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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)
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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
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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.
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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
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(b) telecentric lens
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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.
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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.
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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
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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.
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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
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hc
l=
DE
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hc
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where
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Figure 7.2 Spontaneous and stimulated emission
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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)
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Figure 7.3 Amplification by stimulated emission
A200
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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.
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Material Properties
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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.
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Figure A four-level laser pumping system
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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
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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)
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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
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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
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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   0Z1 + 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., 2Wİf/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,
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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.
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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
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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
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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
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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.
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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.
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Figure 7.12 Frequency stabilization for a helium neon laser
A212
Single Axial Longitudinal Mode Operation
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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.
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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.”
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Frequency and Amplitude Fluctuations
A213
LASER GUIDE
Laser Guide
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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
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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.
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Figure 7.15 Reflectivity of a glass surface vs. incidence angle
for s and p polarization
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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.
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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.
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Short- and long-term frequency stability of a frequency
stabilized helium neon laser
Machine Vision Guide
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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.
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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
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TYPES OF LASERS
GAS-DISCHARGE LASERS
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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.
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Figure 7.17 Typical HeNe laser construction
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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
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Figure 7.18 Construction of a HeCd laser
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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).
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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
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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
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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.
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A219
LASER GUIDE
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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.
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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.
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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
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p-type active
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Figure 7.21 Schematic of a double heterostructure
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Gaussian Beam Optics
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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).
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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
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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,
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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
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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
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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
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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
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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.
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Figure 7.24 Schematic of a lamp-pumped laser
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Figure 7.25 Schematic of a laser-pumped laser
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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
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Gaussian Beam Optics
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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.
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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
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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
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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.
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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.
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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
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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
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440
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457
465
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476
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MAXIMUM POWER (mW)
Figure 7.29 Melles Griot solid-state laser optical trains for producing five different visible output wavelengths
DPSS
DIODE
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Available Wavelengths, Technology and spectral offering
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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.
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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.
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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.
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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
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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.
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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
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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
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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.
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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
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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.
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where nt is the index of the transmitting (lower index)
material and ni of the incident material.
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Figure 7.35 An example of TIR spectroscopy
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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
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Figure 7.37 Schematic of a laser cell sorter
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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.
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A238
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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
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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
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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
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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.
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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
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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