Optics 479

Focusing and Collimating

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Optical Ray Tracing Magnification Rearranging one more time, we finally
arrive at
An introduction to the use of lenses to We can use basic geometry to look at the
solve optical applications can begin with magnification of a lens. In Figure 2, we 1/f = 1/s1 + 1/s2.
the elements of ray tracing. Figure 1 have the same ray tracing figure with
demonstrates an elementary ray trace some particular line segments This is the Gaussian lens equation.
showing the formation of an image, using highlighted. The ray through the center of This equation provides the fundamental
an ideal thin lens. The object height is y1 the lens and the optical axis intersect at relation between the focal length of the
an angle φ. Recall that the opposite

LENS SELECTION GUIDE

at a distance s1 from an ideal thin lens of lens and the size of the optical system.
focal length f. The lens produces an angles of two intersecting lines are equal. A specification of the required
image of height y2 at a distance s2 on the Therefore, we have two similar triangles. magnification and the Gaussian lens
far side of the lens. Taking the ratios of the sides, we have equation form a system of two equations
with three unknowns: f, s1, and s2. The
φ= y1/s1 = y2/s2 addition of one final condition will fix
these three variables in an application.
This can then be rearranged to give
This additional condition is often the
y2/y1 = s2/s1 = M. focal length of the lens, f, or the size of
the object to image distance, in which

SPHERICAL LENSES
The quantity M is the magnification of case the sum of s1 + s2 is given by the
Figure 1
the object by the lens. The magnification size constraint of the system. In either
is the ratio of the image size to the object case, all three variables are then fully
By ideal thin lens, we mean a lens whose size, and it is also the ratio of the image determined.
thickness is sufficiently small that it does distance to the object distance.
not contribute to its focal length. In this
case, the change in the path of a beam
going through the lens can be considered
to be instantaneous at the center of the

CYLINDRICAL LENSES
lens, as shown in the figure. In the
applications described here, we will
assume that we are working with ideally
Figure 3
thin lenses. This should be sufficient for
Figure 2
an introductory discussion. Consideration
of aberrations and thick-lens effects will Optical Invariant
not be included here. This puts a fundamental limitation on the
geometry of an optics system. If an optical Now we are ready to look at what
Three rays are shown in Figure 1. Any two system of a given size is to produce a happens to an arbitrary ray that passes
of these three rays fully determine the particular magnification, then there is through the optical system. Figure 4
size and position of the image. One ray only one lens position that will satisfy shows such a ray. In this figure, we have
emanates from the object parallel to the that requirement. On the other hand, a chosen the maximal ray, that is, the ray

KITS
optical axis of the lens. The lens refracts big advantage is that one does not need that makes the maximal angle with the
this beam through the optical axis at a to make a direct measurement of the optical axis as it leaves the object,
distance f on the far side of the lens. A object and image sizes to know the passing through the lens at its maximum
second ray passes through the optical magnification; it is determined by the clear aperture. This choice makes it
axis at a distance f in front of the lens. geometry of the imaging system itself. easier, of course, to visualize what is
This ray is then refracted into a path happening in the system, but this
parallel to the optical axis on the far side Gaussian Lens Equation maximal ray is also the one that is of
of the lens. The third ray passes through most importance in designing an
OPTICAL SYSTEMS

the center of the lens. Since the surfaces Let’s now go back to our ray tracing application. While the figure is drawn in
of the lens are normal to the optical axis diagram and look at one more set of line this fashion, the choice is completely
and the lens is very thin, the deflection of segments. In Figure 3, we look at the arbitrary and the development shown
this ray is negligible as it passes through optical axis and the ray through the front here is true regardless of which ray is
the lens. focus. Again looking at similar triangles actually chosen.
sharing a common vertex and, now, angle
In addition to the assumption of an η, we have y2/f = y1/(s1-f).
ideally thin lens, we also work in the
paraxial approximation. That is, angles Rearranging and using our definition of
are small and we can substitute θ in magnification, we find
place of sin θ.
MIRRORS

y2/y1 = s2/s1 = f/(s1-f).

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 480 Optics
FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

θ1
reciprocal relation. For example, to
improve the collimation by a factor of
y1 two, you need to increase the beam
y2
θ2 diameter by a factor of two.
f
θ2

y2
Figure 4 Figure 5 y1
θ1
f
LENS SELECTION GUIDE

This arbitrary ray goes through the lens at As a numerical example, let’s look at the
a distance x from the optical axis. If we case of the output from a Newport Figure 6
again apply some basic geometry, we R-31005 HeNe laser focused to a spot
have, using our definition of the using a Newport KPX043 plano-convex Since a common application is the
magnification, lens. This laser has a beam diameter of collimation of the output from an optical
0.63 mm and a divergence of 1.3 mrad. fiber, let’s use that for our numerical
θ1 = x/s1 and θ2 = x/s2 = (x/s1)(y1/y2). Note that these are beam diameter and example. The Newport F-MBB fiber has a
full divergence, so in the notation of our core diameter of 200 µm and a numerical
Rearranging, we arrive at figure, y1 = 0.315 mm and θ1 = 0.65 mrad. aperture (NA) of 0.37. The radius y1 of our
The KPX043 lens has a focal length of source is then 100 µm. NA is defined in
y2θ2 = y1θ1. 25.4 mm. Thus, at the focused spot, we terms of the half-angle accepted by the
SPHERICAL LENSES

have a radius θ1f = 16.5 µm. So, the fiber, so θ1 = 0.37. If we again use the
This is a fundamental law of optics. In diameter of the spot will be 33 µm. KPX043, 25.4 mm focal length lens to
any optical system comprising only collimate the output, we will have a beam
lenses, the product of the image size and This is a fundamental limitation on the with a radius of 9.4 mm and a half-angle
ray angle is a constant, or invariant, of minimum size of the focused spot in this divergence of 4 mrad. We are locked into
the system. This is known as the optical application. We have already assumed a a particular relation between the size and
invariant. The result is valid for any perfect, aberration-free lens. No divergence of the beam. If we want a
number of lenses, as could be verified by improvement of the lens can yield any smaller beam, we must settle for a larger
tracing the ray through a series of lenses. improvement in the spot size. The only divergence. If we want the beam to
CYLINDRICAL LENSES

In some optics textbooks, this is also way to make the spot size smaller is to remain collimated over a large distance,
called the Lagrange Invariant or the use a lens of shorter focal length or then we must accept a larger beam
Smith-Helmholz Invariant. expand the beam. If this is not possible diameter in order to achieve this.
because of a limitation in the geometry of
This is valid in the paraxial the optical system, then this spot size is Application 3: Expanding a
approximation in which we have been the smallest that could be achieved. In Laser Beam
working. Also, this development assumes addition, diffraction may limit the spot to
perfect, aberration-free lenses. The an even larger size (see Gaussian Beam It is often desirable to expand a laser
addition of aberrations to our Optics section beginning on page 484), beam. At least two lenses are necessary
consideration would mean the but we are ignoring wave optics and only to accomplish this. In Figure 7, a laser
replacement of the equal sign by a considering ray optics here. beam of radius y1 and divergence θ1 is
greater-than-or-equal sign in the expanded by a negative lens with focal
KITS

statement of the invariant. That is, length –f1. From Applications 1.1 and 1.2
aberrations could increase the product
Application 2: Collimating we know θ2 = y1/|–f1|, and the optical
but nothing can make it decrease. Light from a Point Source invariant tells us that the radius of the
virtual image formed by this lens is y2 =
Application 1: Focusing a Another common application is the θ1|–f1|. This image is at the focal point of
Collimated Laser Beam collimation of light from a very small the lens, s2 = –f1, because a well-
source, as shown in Figure 6. The problem collimated laser yields s1 ~ ∞, so from
As a first example, we look at a common is often stated in terms of collimating the the Gaussian lens equation s2 = f. Adding
OPTICAL SYSTEMS

application, the focusing of a laser beam output from a “point source.” a second lens with a positive focal length
to a small spot. The situation is shown in Unfortunately, nothing is ever a true point f2 and separating the two lenses by the
Figure 5. Here we have a laser beam, with source and the size of the source must be sum of the two focal lengths –f1 +f2,
radius y1 and divergence θ1 that is included in any calculation. In figure 6, results in a beam with a radius y3 = θ2f2
focused by a lens of focal length f. From the point source has a radius of y1 and and divergence angle θ3 = y2/f2.
the figure, we have θ2 = y1/f. The optical has a maximum ray of angle θ1. If we
invariant then tells us that we must have collimate the output from this source
y2 = θ1f, because the product of radius using a lens with focal length f, then the
and divergence angle must be constant. result will be a beam with a radius y2 =
θ1f and divergence angle θ2 = y1/f. Note
that, no matter what lens is used, the
MIRRORS

beam radius and beam divergence have a

Figure 7

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 Optics 481

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
The expansion ratio The expanded beam diameter located at a distance s1 from a lens of
2y3 = 2y1f2/|–f1| focal length f. The figure shows a ray
y3/y1 = θ2f2/θ2|–f1| = f2/| –f1|, incident upon the lens at a radius of R.
= 2(0.315 mm)(250 mm)/|–25 mm| We can take this radius R to be the
or the ratio of the focal lengths of the maximal allowed ray, or clear aperture, of
lenses. The expanded beam diameter = 6.3 mm. the lens.
The divergence angle
2y3 = 2θ2f2 = 2y1f2/|–f1|.
θ3 = θ1|–f1|/f2

LENS SELECTION GUIDE

The divergence angle of the resulting
expanded beam = (0.65 mrad)|–25 mm|/250 mm

θ3 = y2/f2 = θ1|–f1|/f2 = 0.065 mrad.

is reduced from the original divergence by For minimal aberrations, it is best to use Figure 8
a factor that is equal to the ratio of the a plano-concave lens for the negative lens
focal lengths |–f1|/f2. So, to expand a laser and a plano-convex lens for the positive If s1 is large, then s2 will be close to f,
beam by a factor of five we would select lens with the plano surfaces facing each from our Gaussian lens equation, so for
two lenses whose focal lengths differ by a other. To further reduce aberrations, only the purposes of approximation we can
factor of five, and the divergence angle of the central portion of the lens should be take θ2 ~ R/f. Then from the optical

SPHERICAL LENSES
the expanded beam would be 1/5th the illuminated, so choosing oversized lenses invariant, we have
original divergence angle. is often a good idea. This style of beam
expander is called Galilean. Two positive y2 = y1θ1/θ2 = y1(R/s1)(f/R) or
As an example, consider a Newport lenses can also be used in a Keplerian
R-31005 HeNe laser with beam diameter beam expander design, but this y2 = 2y1(R/s1)f/#.
0.63 mm and a divergence of 1.3 mrad.
configuration is longer than the
Note that these are beam diameter and
full divergence, so in the notation of our Galilean design. where f/2R = f/D is the f-number, f/#, of
figure, y1 = 0.315 mm and θ1 = 0.65 mrad. the lens. In order to make the image size
To expand this beam ten times while Application 4: Focusing an smaller, we could make f/# smaller, but we

CYLINDRICAL LENSES
reducing the divergence by a factor of Extended Source to a are limited to f/# = 1 or so. That leaves us
ten, we could select a plano-concave lens with the choice of decreasing R (smaller
KPC043 with f1 = –25 mm and a plano- Small Spot lens or aperture stop in front of the lens)
convex lens KPX109 with f2 = 250 mm. This application is one that will be or increasing s1. However, if we do either
Since real lenses differ in some degree approached as an imaging problem as of those, it will restrict the light gathered
from thin lenses, the spacing between the opposed to the focusing and collimation by the lens. If we either decrease R by a
pair of lenses is actually the sum of the
problems of the previous applications. An factor of two or increase s1 by a factor of
back focal lengths BFL1 + BFL2 = –26.64
mm + 247.61 mm = 220.97 mm. example might be a situation where a two, it would decrease the total light
fluorescing sample must be imaged with a focused at s2 by a factor of four due to the
CCD camera. The geometry of the restriction of the solid angle subtended
application is shown in Figure 8. An by the lens.
extended source with a radius of y1 is

KITS
Fiber Optic Coupling
The problem of coupling light into an Application 5: Coupling Laser Let’s consider coupling the light from a
optical fiber is really two separate Light into a Multimode Fiber Newport R-30990 HeNe laser into an
problems. In one case, we have the F-MSD fiber. The laser has a beam
OPTICAL SYSTEMS

problem of coupling into multimode When we look at coupling light from a diameter of 0.81 mm and divergence
fibers, where the ray optics of the well-collimated laser beam into a .0 mrad. The fiber has a core diameter of
previous section can be used. In the multimode optical fiber, we return to the 50 µm and an NA of 0.20. Let’s look at the
other case, coupling into single-mode situation that was illustrated in Figure 5. coupling from the beam into the fiber
fibers, we have a fundamentally different The radius of the fiber core will be our y2. when a Newport M-20X objective lens is
problem. In this case, one must consider We will have to make sure that the lens used in an F-915 or F-915T fiber coupler.
the problem of matching the mode of the focuses to a spot size less than this
incident laser light into the mode of the parameter. An even more important The objective lens has an effective focal
fiber. This cannot be done using the ray restriction is that the angle from the lens length of 9 mm. In this case, the focused
optics approach, but must be done using to the fiber θ2 must be less than the NA beam will have a diameter of 9 µm and a
the concepts of Gaussian beam optics of the optical fiber. maximal ray of angle 0.05, so both the
MIRRORS

(see page 484).

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 484 Optics

Gaussian Beam Optics

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

The Gaussian is a radially symmetrical The parameter ω0, usually called the radius of curvature of the wavefront
distribution whose electric field variation Gaussian beam radius, is the radius at change. Imagine that we somehow create
is given by the following equation : which the intensity has decreased to 1/e2 a coherent light beam with a Gaussian
or 0.135 of its axial, or peak value. distribution and a plane wavefront at a
Another point to note is the radius of half position x=0. The beam size and
maximum, or 50% intensity, which is wavefront curvature will then vary with x
0.59ω0. At 2ω0, or twice the Gaussian as shown in Figure 2.
radius, the intensity is 0.0003 of its peak
LENS SELECTION GUIDE

Its Fourier Transform is also a Gaussian value, usually completely negligible.

distribution. If we were to solve the
Fresnel integral itself rather than the The power contained within a radius r,
Fraunhofer approximation, we would find P(r), is easily obtained by integrating the
that a Gaussian source distribution intensity distribution from 0 to r: Figure 2
remains Gaussian at every point along its
path of propagation through the optical The beam size will increase, slowly at
system. This makes it particularly easy to first, then faster, eventually increasing
visualize the distribution of the fields at proportionally to x. The wavefront radius
any point in the optical system. The of curvature, which was infinite at x = 0,
intensity is also Gaussian: When normalized to the total power of will become finite and initially decrease
SPHERICAL LENSES

the beam, P(∞) in watts, the curve is the with x. At some point it will reach a
same as that for intensity, but with the minimum value, then increase with larger
ordinate inverted. Nearly 100% of the x, eventually becoming proportional to x.
power is contained in a radius r = 2ω0. The equations describing the Gaussian
One-half the power is contained within beam radius w(x) and wavefront radius of
This relationship is much more than a 0.59ω0, and only about 10% of the power curvature R(x) are:
mathematical curiosity, since it is now is contained with 0.23ω0, the radius at
easy to find a light source with a Gaussian which the intensity has decreased by 10%.
intensity distribution: the laser. Most The total power, P(∞) in watts, is related
CYLINDRICAL LENSES

lasers automatically oscillate with a to the on-axis intensity, I(0) (watts/m2),

Gaussian distribution of electrical field. by:
The basic Gaussian may also take on
some particular polynomial multipliers
and still remain its own transform. These
field distributions are known as higher- where ω0 is the beam radius at x = 0 and
order transverse modes and are usually λ is the wavelength. The entire beam
avoided by design in most practical lasers. behavior is specified by these two
parameters, and because they occur in
The Gaussian has no obvious boundaries the same combination in both equations,
to give it a characteristic dimension like The on-axis intensity can be very high due they are often merged into a single
the diameter of the circular aperture, so to the small area of the beam. parameter, xR, the Rayleigh range:
KITS

the definition of the size of a Gaussian is

somewhat arbitrary. Figure 1 shows the Care should be taken in cutting off the
Gaussian intensity distribution of a beam with a very small aperture. The
typical HeNe laser. source distribution would no longer be
Gaussian, and the far-field intensity In fact, it is at x = xR that R has its
distribution would develop zeros and minimum value.
other non-Gaussian features. However, if
the aperture is at least three or four ω0 in Note that these equations are also valid
OPTICAL SYSTEMS

diameter, these effects are negligible. for negative values of x. We only

imagined that the source of the beam
Propagation of Gaussian beams through was at x = 0; we could have created the
an optical system can be treated almost same beam by creating a larger Gaussian
as simply as geometric optics. Because of beam with a negative wavefront curvature
the unique self-Fourier Transform at some x < 0. This we can easily do with
characteristic of the Gaussian, we do not a lens, as shown in Figure 3.
need an integral to describe the evolution
of the intensity profile with distance. The
transverse distribution intensity remains
Figure 1 Gaussian at every point in the system;
MIRRORS

only the radius of the Gaussian and the

Figure 3

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 Optics 485

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
The input to the lens is a Gaussian with else. To determine the size and wavefront depth of focus (somewhat arbitrarily) as
diameter D and a wavefront radius of curvature of the beam everywhere in the the distance between the values of x
curvature which, when modified by the system, you would use the ABCD values where the beam is √2 times larger than it
lens, will be R(x) given by the equation for each element of the system and trace is at the beam waist, then using the
above with the lens located at -x from the q through them via successive bilinear equation for ω(x) we can determine the
beam waist at x = 0. That input Gaussian transformations. But if you only wanted depth of focus:
will also have a beam waist position and the overall transformation of q, you could
size associated with it. Thus we can multiply the elemental ABCD values in
generalize the law of propagation of a matrix form, just as is done in geometric

LENS SELECTION GUIDE

Gaussian through even a complicated optics, to find the overall ABCD values for
optical system. the system, then apply the bilinear
transform. For more information about Using these relations, we can make
In the free space between lenses, mirrors Gaussian beams, see Anthony E. simple calculations for optical systems
and other optical elements, the position Siegman’s book, Lasers (University Science employing Gaussian beams. For example,
of the beam waist and the waist diameter Books, 1986). suppose that we use a 10 mm focal
completely describe the beam. When a length lens to focus the collimated
beam passes through a lens, mirror, or Fortunately, simple approximations for output of a helium-neon laser (632.8 nm)
dielectric interface, the diameter is spot size and depth of focus can still be that has a 1 mm diameter beam. The
unchanged but the wavefront curvature is used in most optical systems to select diameter of the focal spot will be:
changed, resulting in new values of waist pinhole diameters, couple light into

SPHERICAL LENSES
position and waist diameter on the fibers, or compute laser intensities. Only
output side of the interface. when f-numbers are large should the full
Gaussian equations be needed.
These equations, with input values for
ω and R, allow the tracing of a Gaussian At large distances from a beam waist, the or about 8 µm. The depth of focus for the
beam through any optical system with beam appears to diverge as a spherical beam is then:
some restrictions: optical surfaces need wave from a point source located at the
to be spherical and with not-too-short center of the waist. Note that “large”
focal lengths, so that beams do not distances mean where x»xR and are

CYLINDRICAL LENSES
change diameter too fast. These are typically very manageable considering the
exactly the analog of the paraxial small area of most laser beams. The
restrictions used to simplify geometric diverging beam has a full angular width θ or about 160 µm. If we were to change the
optical propagation. (again, defined by 1/e2 points): focal length of the lens in this example to
100 mm, the focal spot size would
It turns out that we can put these laws increase 10 times to 80 µm, or 8% of the
in a form as convenient as the ABCD original beam diameter. The depth of
matrices used for geometric ray tracing. focus would increase 100 times to 16 mm.
But there is a difference: ω(x) and R(x) do We have invoked the approximation tanθ However, suppose we increase the focal
not transform in matrix fashion as r and u ≈ θ since the angles are small. Since the length of the lens to 2,000 mm. The “focal
do for ray tracing; rather, they transform origin can be approximated by a point spot size” given by our simple equation
via a complex bi-linear transformation: source, θ is given by geometrical optics as would be 200 times larger, or 1.6 mm,

KITS
the diameter illuminated on the lens, D, 60% larger than the original beam!
divided by the focal length of the lens. Obviously, something is wrong. The
trouble is not with the equations giving
ω(x) and R(x), but with the assumption
that the beam waist occurs at the focal
distance from the lens. For weakly
where the quantity q is a complex where f/# is the photographic f-number of focused systems, the beam waist does
composite of ω and R: the lens. not occur at the focal length. In fact, the
OPTICAL SYSTEMS

position of the beam waist changes

Equating these two expressions allows us contrary to what we would expect in
to find the beam waist diameter in terms geometric optics: the waist moves toward
of the input beam parameters (with some the lens as the focal length of the lens is
restrictions that will be discussed later): increased. However, we could easily
believe the limiting case of this behavior
We can see from the expression for q that by noting that a lens of infinite focal
at a beam waist (R = ∞ and ω = ω0), q is length such as a flat piece of glass placed
pure imaginary and equals ixR. If we know at the beam waist of a collimated beam
where one beam waist is and its size, we will produce a new beam waist not at
can calculate q there and then use the We can also find the depth of focus from infinity, but at the position of the glass
MIRRORS

bilinear ABCD relation to find q anywhere the formulas above. If we define the itself.

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 Optics 491

Optical Materials

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
BK 7 CaF2 Crystal Quartz
BK 7 is one of the most common Calcium Fluoride is a cubic single crystal Crystal Quartz is a positive uniaxial
borosilicate crown glasses used for material grown using the vacuum birefringent single crystal grown using a
visible and near infrared optics. Its high Stockbarger Technique with good vacuum hydrothermal process. It has good
homogeneity, low bubble content, and UV to infrared transmission. CaF2’s transmission from the vacuum UV to the
straightforward manufacturability make it excellent UV transmission, down to 170 near infrared. Due to its birefringent
a good choice for transmissive optics. nm, and non-birefringent properties make nature, crystal quartz is commonly used

LENS SELECTION GUIDE

The transmission range for BK 7 is it ideal for deep UV transmissive optics. for wave plates.
380–2100 nm. It is not recommended for Material for IR use is grown using
temperature sensitive applications, such naturally mined fluroite, at much lower
as precision mirrors. cost. CaF2 is sensitive to thermal shock,
so care must be taken during handling.

BK 7
CaF2
Crystal
Quartz

SPHERICAL LENSES
UV Grade Fused Silica MgF2 Pyrex®
UV Grade Fused Silica is synthetic Magnesium Fluoride is a positive Pyrex® is a borosilicate glass with a low
amorphous silicon dioxide of extremely birefringent crystal grown using the coefficient of thermal expansion. It is

CYLINDRICAL LENSES
high purity. This non-crystalline, colorless vacuum Stockbarger Technique with good mainly used for non-transmissive optics,
silica glass combines a very low thermal vacuum UV to infrared transmission. It is such as mirrors, due to its low
expansion coefficient with good optical typically oriented with the c axis parallel homogeneity and high bubble content.
qualities, and excellent transmittance in to the optical axis to reduce birefringent
the ultraviolet. Transmission and effects. High vacuum UV transmission, Zerodur®
homogeneity exceed those of crystalline down to 150 nm, and its proven use in
quartz without the problems of fluorine environments make it ideal for Zerodur® is a glass ceramic material that
orientation and temperature instability lenses, windows, and polarizers for has a coefficient of thermal expansion
inherent in the crystalline form. Fused Excimer lasers. MgF2 is resistant to approaching zero, as well as excellent
silica is used for both transmissive and thermal and mechanical shock. homogeneity of this coefficient
reflective optics, especially where high throughout the entire piece. This makes
laser damage threshold is required. Zerodur ideal for mirror substrates where

KITS
MgF2 extreme thermal stability is required.
Zerodur should not be used for
UV Fused transmissive optics due to inclusions in
Silica
the material.

OPTICAL SYSTEMS
MIRRORS

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 492 Optics
FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

Index of Refraction
Wavelength MgF2 MgF2 Crystal Quartz Crystal Quartz
(nm) Source BK 7 SF 2 UV Fused Silica CaF2 no ne no ne
193 ArF excimer laser 1.65528 1.52127 1.56077 1.50153 1.42767 1.44127 1.66091 1.67455
244 Ar-Ion laser 1.58265 1.98102 1.51086 1.46957 1.40447 1.41735 1.60439 1.61562
248 KrF excimer 1.57957 1.93639 1.50855 1.46803 1.40334 1.41618 1.60175 1.61289
257 Ar-Ion laser 1.57336 1.86967 1.50383 1.46488 1.40102 1.41377 1.59637 1.60731
266 Nd:YAG laser 1.56796 1.82737 1.49968 1.46209 1.39896 1.41164 1.59164 1.60242
308 XeCl excimer laser 1.55006 1.73604 1.48564 1.45255 1.39188 1.40429 1.57556 1.58577
LENS SELECTION GUIDE

325 HeCd laser 1.54505 1.71771 1.48164 1.44981 1.38983 1.40216 1.57097 1.58102
337.1 N2 laser 1.54202 1.70749 1.47919 1.44813 1.38858 1.40085 1.56817 1.57812
351 XeF excimer laser 1.53896 1.69778 1.47672 1.44642 1.38730 1.39952 1.56533 1.57518
351.1 Ar-Ion laser 1.53894 1.69771 1.47671 1.44641 1.38729 1.39951 1.56531 1.57516
354.7 Nd:YAG laser 1.53821 1.69548 1.47612 1.44601 1.38699 1.39920 1.56463 1.57446
363.8 Ar-Ion laser 1.53649 1.69029 1.47472 1.44504 1.38626 1.39844 1.56302 1.57279
404.7 Mercury arc, h line 1.53023 1.67263 1.46961 1.44151 1.38360 1.39567 1.55714 1.56670
416 Kr-Ion laser 1.52885 1.66893 1.46847 1.44072 1.38301 1.39505 1.55583 1.56535
435.8 Mercury arc,g line 1.52669 1.66331 1.46670 1.43949 1.38207 1.39408 1.55379 1.56323
441.6 HeCd laser 1.52611 1.66184 1.46622 1.43916 1.38183 1.39382 1.55324 1.56266
457.9 Ar-Ion laser 1.52461 1.65807 1.46498 1.43830 1.38118 1.39314 1.55181 1.56119
SPHERICAL LENSES

465.8 Ar-Ion laser 1.52395 1.65641 1.46443 1.43792 1.38088 1.39284 1.55118 1.56053
472.7 Ar-Ion laser 1.52339 1.65505 1.46397 1.43760 1.38064 1.39258 1.55065 1.55998
476.5 Ar-Ion laser 1.52309 1.65432 1.46372 1.43744 1.38051 1.39245 1.55036 1.55969
480 Cadmium arc, F’ line 1.52283 1.65367 1.46350 1.43728 1.38040 1.39233 1.55011 1.55943
486.1 Hydrogen arc, F line 1.52238 1.65258 1.46313 1.43703 1.38020 1.39212 1.54968 1.55898
488 Ar-Ion laser 1.52224 1.65225 1.46301 1.43695 1.38014 1.39206 1.54955 1.55885
496.5 Ar-Ion laser 1.52165 1.65083 1.46252 1.43661 1.37988 1.39179 1.54898 1.55826
501.7 Ar-Ion laser 1.52130 1.65000 1.46223 1.43641 1.37973 1.39163 1.54865 1.55792
510.6 Cu vapor laser 1.52073 1.64865 1.46176 1.43609 1.37948 1.39137 1.54810 1.55735
CYLINDRICAL LENSES

514.5 Ar-Ion laser 1.52049 1.64808 1.46156 1.43595 1.37937 1.39126 1.54787 1.55711
532 Nd:YAG laser 1.51947 1.64570 1.46071 1.43537 1.37892 1.39079 1.54689 1.55610
543.5 HeNe laser 1.51886 1.64427 1.46019 1.43502 1.37865 1.39051 1.54630 1.55549
546.1 Mercury arc, e line 1.51872 1.64397 1.46008 1.43494 1.37859 1.39044 1.54617 1.55535
578.2 Cu vaport laser 1.51720 1.64053 1.45880 1.43408 1.37792 1.38974 1.54470 1.55383
587.6 Helium arc, d line 1.51680 1.63963 1.45846 1.43385 1.37774 1.38956 1.54431 1.55343
589.3 Sodium arc, D line 1.51673 1.63947 1.45840 1.43381 1.37771 1.38952 1.54424 1.55336
594.1 HeNe laser 1.51653 1.63904 1.45824 1.43370 1.37762 1.38943 1.54405 1.55316
611.9 HeNe laser 1.51584 1.63752 1.45765 1.43331 1.37732 1.38911 1.54337 1.55247
628 Ruby laser 1.51526 1.63626 1.45716 1.43298 1.37706 1.38884 1.54281 1.55188
632.8 HeNe laser 1.51509 1.63590 1.45702 1.43289 1.37698 1.38876 1.54264 1.55171
635 Laser diode 1.51501 1.63574 1.45695 1.43284 1.37695 1.38873 1.54257 1.55164
KITS

643.8 Cadmium arc, C' line 1.51472 1.63512 1.45671 1.43268 1.37682 1.38859 1.54228 1.55134
647.1 Kr-Ion laser 1.51461 1.63489 1.45661 1.43262 1.37677 1.38854 1.54218 1.55123
650 Laser diode 1.51452 1.63469 1.45653 1.43257 1.37673 1.38850 1.54209 1.55114
656.3 Hydrogen arc, C line 1.51432 1.63427 1.45637 1.43246 1.37664 1.38840 1.54189 1.55093
670 Laser diode 1.51391 1.63340 1.45601 1.43223 1.37646 1.38821 1.54148 1.55051
676.4 Kr-Ion laser 1.51372 1.63301 1.45585 1.43212 1.37637 1.38812 1.54130 1.55032
694.3 Ruby laser 1.51322 1.63198 1.45542 1.43185 1.37615 1.38789 1.54080 1.54981
750 Laser diode 1.51184 1.62922 1.45424 1.43109 1.37553 1.38724 1.53943 1.54839
OPTICAL SYSTEMS
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FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Index of Refraction (continued)
Wavelength MgF2 MgF2 Crystal Quartz Crystal Quartz
(nm) Source BK 7 SF 2 UV Fused Silica CaF2 no ne no ne
780 Laser diode 1.51118 1.62796 1.45367 1.43074 1.37524 1.38693 1.53878 1.54771
830 Laser diode 1.51020 1.62613 1.45282 1.43023 1.37480 1.38647 1.53779 1.54668
850 Laser diode 1.50984 1.62548 1.45250 1.43004 1.37464 1.38630 1.53742 1.54630
852.1 Cesium arc, s line 1.50980 1.62541 1.45247 1.43002 1.37462 1.38628 1.53739 1.54626
905 Laser diode 1.50892 1.62387 1.45168 1.42957 1.37422 1.38586 1.53648 1.54532
980 Laser diode 1.50779 1.62202 1.45067 1.42902 1.37371 1.38533 1.53531 1.54409

LENS SELECTION GUIDE

1014 Mercury arc, t line 1.50731 1.62128 1.45024 1.42879 1.37350 1.38510 1.53481 1.54357
1053 Nd:YLF laser 1.50678 1.62049 1.44976 1.42854 1.37326 1.38485 1.53425 1.54299
1060 Nd:Glass laser 1.50669 1.62035 1.44968 1.42850 1.37322 1.38480 1.53415 1.54288
1064 Nd:YAG laser 1.50663 1.62028 1.44963 1.42848 1.37319 1.38478 1.53410 1.54282
1300 Laser diode 1.50370 1.61644 1.44692 1.42721 1.37188 1.38338 1.53094 1.53950
1320 Nd:YAG laser 1.50346 1.61616 1.44669 1.42711 1.37177 1.38327 1.53068 1.53922
1550 Laser diode 1.50065 1.61312 1.44402 1.42602 1.37052 1.38194 1.52761 1.53596
1970.1 Mercury arc 1.49495 1.60780 1.43852 1.42401 1.36803 1.37928 1.52138 1.52932
2100 Ho:YAG laser 1.49296 1.60608 1.43659 1.42334 1.36718 1.37837 1.51924 1.52703
2325.4 Mercury arc 1.48921 1.60291 1.43293 1.42212 1.36559 1.37667 1.51524 1.52277
2940 Er:YAG laser 1.47670 1.59273 1.42065 1.41827 1.36051 1.37123 1.50246 1.50908

SPHERICAL LENSES
Properties of Optical Materials
Coefficient of
Abbe Number Thermal Expansion Conductivity Heat Capacity Density at 25°C Knoop Hardness Young’s Modulus
vd (10-6/°C) (W/m°C) (J/gm°C) (gm/cm3) (kg/mm2) (GPa)
BK 7 64.17 7.1 1.114 0.858 2.51 610 81.5
SF 2 33.85 8.4 0.735 0.498 3.86 410 55
UV Fused Silica 67.8 0.52 1.38 0.75 2.202 600 73

CYLINDRICAL LENSES
CaF2 94.96 18.85 9.71 0.85 3.18 158 75.8
MgF2 106.18 13.7 || to c axis 21 || to c axis 1.024 3.177 415 138.5
8.48 ⊥ to c axis 30 to ⊥ c axis
Crystal Quartz 69.87 7.1 to || c axis 10.4 || to c axis 0.74 2.649 740 97 || to c axis
13.2 ⊥ to c axis 6.2 ⊥ to c axis 76.5 ⊥ to c axis
Pyrex® 66 3.25 1.13 0.75 2.23 418 65.5
Zerodur® 56.09 0 ± 0.1 1.46 0.80 2.53 620 90.3

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OPTICAL SYSTEMS
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 494 Optics

Optics Formulas
FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

Light Right-Hand Rule Light Intensity Energy Conversions

Light is a transverse electromagnetic
wave. The electric E and magnetic M The light intensity, I is measured in
fields are perpendicular to each other Watts/m2, E in Volts/m, and H in
and to the propagation vector k, as Amperes/m. The equations relating I to E
shown below. and H are quite analogous to OHMS LAW.
For peak values these equations are:
LENS SELECTION GUIDE

Power density is given by Poynting’s

vector, P, the vector product of E and H.
You can easily remember the directions if
you “curl” E into H with the fingers of the
right hand: your thumb points in the
direction of propagation.
Wavelength Conversions
1 nm = 10 Angstroms(Å)
= 10–9m = 10–7cm = 10–3µm
SPHERICAL LENSES

Snell’s Law
Snell’s Law describes how a light ray
The quantity η0 is the wave impedance of behaves when it passes from a medium
vacuum, and η is the wave impedance of with index of refraction n1, to a medium
a medium with refractive index n. with a different index of refraction, n2. In
general, the light will enter the interface
Wave Quantity Relationship between the two medii at an angle. This
angle is called the angle of incidence. It is
CYLINDRICAL LENSES

the angle measured between the normal

to the surface (interface) and the
incoming light beam (see figure). In the
case that n1 is smaller than n2, the light is
bent towards the normal. If n1 is greater
than n2, the light is bent away from the
normal (see figure below). Snell’s Law is
expressed as n1sinθ1 = n2sinθ2.
KITS
OPTICAL SYSTEMS

k: wave vector [radians/m]

ν: frequency [Hertz]

ω: angular frequency [radians/sec]

λ: wavelength [m]

λ0: wavelength in vacuum [m]

MIRRORS

n: refractive index

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FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Plane-Polarized Light Beam Displacement Beam Deviation
For plane-polarized light the E and H A flat piece of glass can be used to Both displacement and deviation occur if
fields remain in perpendicular planes displace a light ray laterally without the media on the two sides of the tilted
parallel to the propagation vector k, as changing its direction. The displacement flat are different — for example, a tilted
shown below. varies with the angle of incidence; it is window in a fish tank. The displacement
zero at normal incidence and equals the is the same, but the angular deviation δ is
thickness h of the flat at grazing given by the formula. Note:δ is
incidence. independent of the index of the flat; it is

LENS SELECTION GUIDE

the same as if a single boundary existed
(Grazing incidence: light incident at between media 1 and 3.
almost or close to 90° to the normal of
the surface). Example: The refractive index of air at STP
is about 1.0003. The deviation of a light
ray passing through a glass Brewster’s
angle window on a HeNe laser is then:

δ= (n3 - n1) tan θ

At Brewster’s angle, tan θ= n2

SPHERICAL LENSES
Both E and H oscillate in time and space δ= (0.0003) x 1.5 = 0.45 mrad
as:
At 10,000 ft. altitude, air pressure is 2/3
sin (ωt-kx) that at sea level; the deviation is 0.30
mrad. This change may misalign the laser
if its two windows are symmetrical rather
than parallel.

CYLINDRICAL LENSES
The relationship between the tilt angle of
the flat and the two different refractive
indices is shown in the graph below.

KITS
OPTICAL SYSTEMS
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FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

Angular Deviation of a Prism optical path. Although effects are minimal two sides of the boundary.
in laser applications, focus shift and
Angular deviation of a prism depends on chromatic effects in divergent beams The intensities (watts/area) must also be
the prism angle α, the refractive index, n, should be considered. corrected by this geometric obliquity
and the angle of incidence θi. Minimum factor:
deviation occurs when the ray within the Fresnel Equations:
prism is normal to the bisector of the It = T x Ii(cosθi/cosθt)
prism angle. For small prism angles i - incident medium
(optical wedges), the deviation is Conservation of Energy:
LENS SELECTION GUIDE

constant over a fairly wide range of t - transmitted medium

angles around normal incidence. For R+T=1
such wedges the deviation is: use Snell’s law to find θt
This relation holds for p and s components
δ ≈ (n - 1)α Normal Incidence: individually and for total power.

r = (ni-nt)/(ni + nt) Polarization

t = 2ni/(ni + nt) To simplify reflection and transmission
calculations, the incident electric field is
Brewster's Angle: broken into two plane-polarized
SPHERICAL LENSES

components. The “wheel” in the pictures

θβ = arctan (nt/ni) below denotes plane of incidence. The
normal to the surface and all propagation
Only s-polarized light reflected. vectors (ki, kr, kt) lie in this plane.

Total Internal Reflection E parallel to the plane of incidence; p-

(TIR): polarized.

θTIR > arcsin (nt/ni)

CYLINDRICAL LENSES

nt < ni is required for TIR

Field Reflection and

Transmission Coefficients:
The field reflection and transmission
coefficients are given by:

r = Er/Ei t = Et/Ei

Non-Normal Incidence:
KITS

rs = (nicosθi -ntcosθt)/(nicosθi + ntcosθt) E normal to the plane of incidence;

s-polarized.
Prism Total Internal rp = (ntcos θi -nicosθt)/ntcosθi + nicosθt)
Reflection (TIR)
ts = 2nicosθi/(nicosθi + ntcosθt)
TIR depends on a clean glass-air
interface. Reflective surfaces must be free tp = 2nicosθi/(ntcosθi + nicosθt)
OPTICAL SYSTEMS

of foreign materials. TIR may also be

defeated by decreasing the incidence Power Reflection:
angle beyond a critical value. For a right
angle prism of index n, rays should enter The power reflection and transmission
the prism face at an angle θ: coefficients are denoted by capital letters:

θ < arcsin (((n2-1)1/2-1)/√2) R = r2 T = t2(ntcosθt)/(nicosθi)

In the visible range, θ = 5.8° for BK 7 The refractive indices account for the
(n = 1.517) and 2.6° for fused silica different light velocities in the two media;
(n = 1.46). Finally, prisms increase the the cosine ratio corrects for the different
MIRRORS

cross sectional areas of the beams on the

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 Optics 497

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Power Reflection Coefficients Thin Lens Equations Magnification:

Power reflection coefficients Rs and Rp If a lens can be characterized by a single Transverse:

are plotted linearly and logarithmically plane then the lens is “thin”. Various
for light traveling from air (ni = 1) into BK relations hold among the quantities MT < 0, image inverted
7 glass (nt = 1.51673). shown in the figure.
Brewster’s angle = 56.60°. Longitudinal:
Gaussian:

LENS SELECTION GUIDE

Newtonian: x1x2 = -F2 ML <0, no front to back inversion

SPHERICAL LENSES
Sign Conventions for Images Thick Lenses
and Lenses
A thick lens cannot be characterized by a
Quantity + -
single focal length measured from a
s1 real virtual
single plane. A single focal length F may

CYLINDRICAL LENSES
s2 real virtual be retained if it is measured from two
F convex lens concave lens planes, H1, H2, at distances P1, P2 from
the vertices of the lens, V1, V2. The two
Lens Types for Minimum back focal lengths, BFL1 and BFL2, are
The corresponding reflection coefficients Aberration measured from the vertices. The thin lens
are shown below for light traveling from equations may be used, provided all
| s2/s1 | Best lens
BK 7 glass into air Brewster’s angle = quantities are measured from the
33.40°. Critical angle (TIR angle) = 41.25°. <0.2 plano-convex/concave principal planes.
>5 plano-convex/concave
>0.2 or <5 bi-convex/concave

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FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

Lens Nomogram:
LENS SELECTION GUIDE

The Lensmaker’s Equation Numerical Aperture Constants and Prefixes

SPHERICAL LENSES

Speed of light in vacuum c = 2.998108 m/s

Convex surfaces facing left have positive φMAX is the full angle of the cone of light
Planck’s const. h = 6.625 x 10-34Js
radii. Below, R1>0, R2<0. Principal plane rays that can pass through the system
offsets, P, are positive to the right. As (below). Boltzmann’s const. k = 1.308 x 10-23 J/K

illustrated, P1>0, P2<0. The thin lens Stefan-Boltzmann σ = 5.67 x 10-8 W/m2 K4

focal length is given when Tc = 0. 1 electron volt eV = 1.602 x 10-19 J

exa (E) 1018
peta (P) 1015
tera (T) 1012
CYLINDRICAL LENSES

giga (G) 109

mega (M) 106
kilo (k) 103
milli (m) 10-3
micro (µ) 10-6
nano (n) 10-9
pico (p) 10-12
femto (f) 10-15
atto (a) 10-18

Wavelengths of Common
For small φ: Lasers
KITS

Source (nm)
ArF 193
KrF 248
Nd:YAG(4) 266
XeCl 308
Both f-number and NA refer to the system
HeCd 325, 441.6
and not the exit lens.
N2 337.1, 427
OPTICAL SYSTEMS

XeF 351
Nd:YAG(3) 354.7
Ar 488, 514.5, 351.1, 363.8
Cu 510.6, 578.2
Nd:YAG(2) 532
HeNe 632.8, 543.5, 594.1, 611.9, 1153, 1523
Kr 647.1, 676.4
Ruby 694.3
Nd:Glass 1060
Nd:YAG 1064, 1319
Ho:YAG 2100
MIRRORS

Er:YAG 2940

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 Optics 499

FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Gaussian Intensity Focusing a Collimated Depth of Focus (DOF)
Distribution Gaussian Beam
DOF = (8λ/π)(f/#)2
The Gaussian intensity distribution: In the figure below the 1/e radius, ω(x),
2

and the wavefront curvature, R(x), change Only if DOF <F, then:
I(r) = I(0) exp(-2r2/ω02) with x through a beam waist at x = 0. The
governing equations are: New Waist Diameter
is shown below.
(
ω2(x) = ω20 ⎡1 + λx /πω20 ⎤⎥ )
2

⎢⎣ ⎦

LENS SELECTION GUIDE

⎡
( )
⎤
2
R(x) = x 1 + πω 0 / λx
2
⎢⎣ ⎥⎦
Beam Spread
2ω0 is the waist diameter at the 1/e2
intensity points. The wavefronts are
planar at the waist [R(0) = ∞].

At the waist, the distance from the lens

The right hand ordinate gives the fraction will be approximately the focal length:
of the total power encircled at radius r: s2≈F.

SPHERICAL LENSES
D = collimated beam diameter or
diameter illuminated on lens.

The total beam power, P(∞) [watts], and

the on-axis intensity I(0) [watts/area] are
related by:

CYLINDRICAL LENSES
Diffraction
The figure below compares the far-field
intensity distributions of a uniformly
illuminated slit, a circular hole, and
Gaussian distributions with 1/e2
diameters of D and 0.66D (99% of a 0.66D
Gaussian will pass through an aperture of
diameter D). The point of observation is Y

KITS
off axis at a distance X>Y from the source.

OPTICAL SYSTEMS
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Optics Glossary
FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND

Abbe Number: The constant of an Back Focal Length (BFL): The distance Broadband Coating: A multilayer coating
optical medium that describes the ratio between the last surface of a lens to its with specified reflection or transmission
of its refractivity to its dispersion. image focal plane. over a broad spectral band. Newport’s
AR.14 is a broadband AR coating, while
Specifically, Vd = (nd-1)/(nF-nC), where n Bandpass: The range of wavelengths that Newport BD.1 is broadband mirror coating.
is the index of refraction at the passes through a filter or other optical
Fraunhofer d, F, and C lines, respectively. component. Cavity: A periodic structure of thin films
comprised of two quarter-wave stack
LENS SELECTION GUIDE

Aberration: An optical defect resulting Bandwidth: Range of wavelengths over reflectors separated by a dielectric spacer.
from design or fabrication error that which the specified transmission or Cavities are the building blocks of
prevents the lens from achieving precise reflection occurs. bandpass filters.
focus. The primary aberrations are
spherical, coma, astigmatism, field Beam Deviation: See Deviation. Center Wavelength: The center of the
curvature, distortion, and chromatic wavelength band of a coating.
aberration. Beamsplitter: An optical device that
divides an incident beam into at least two Centration: The deviation between the
Achromatic Lens: Lens in which distinct beams. optical axis and the mechanical axis of a
chromatic aberration has been corrected lens. Centration is specified in terms of
at a minimum of two wavelengths. Bi-Concave: Having two outer surfaces the deflection of a beam directed along
SPHERICAL LENSES

that curve inward. the mechanical axis of the lens.

Airy Disc: A pattern of illumination
caused by diffraction at the edge of a Bi-Convex: Having two outer surfaces that Chromatic Aberration: An optical defect
circular aperture, consisting of a central curve outward. in a lens resulting in different
core of light surrounded by concentric wavelengths of light focusing at different
rings of gradually decreasing intensity. Birefringence: The change in refractive distances from the lens. Corrected by
index with the polarization of light. A achromatic lenses.
Anamorphic: Distorted, as in an optical birefringent crystal, such as calcite or
system with different magnification levels quartz, will divide an unpolarized beam Circle of Least Confusion: The smallest
CYLINDRICAL LENSES

or with focal lengths perpendicular to the into two beams (ordinary and cross-section of a focused beam of light
optical axis. extraordinary) having opposite at the point of best focus for the image.
polarization.
Angle of Incidence: The angle formed by Clear Aperture: The area of an optical
a ray of light striking a surface and the Blocking: Refers to filter transmittance component that controls the amount of
normal to that surface. outside the bandpass region. It is the light incident on a given surface. In
rejection of out-of-band wavelengths by a Newport lenses and mirrors, the clear
Antireflection (AR) Coating: A thin filter. aperture gives the diameter over which
layer of material that, when applied to a specifications are guaranteed.
lens or window, increases its Blur Circle: The image of a point-source
transmittance by reduction of its object formed by an optical system on its Coefficient of Thermal Expansion: A
reflectance. AR coatings may be focal surface. The precision level of the material property defined as the
KITS

multilayer or single layer coatings. lens and its state of focus determine the fractional change in length per original
size of the blur. length (or fractional change in volume)
Aperture: An opening through which with a change in temperature.
light may pass. The clear aperture is Borosilicate Glass: An optical glass
that area in an optical system limiting containing boric oxide, along with silica Collimated Beam: A beam of light in
the bundle of light able to pass through and other ingredients. BK 7 and Pyrex® which all of the rays are parallel to each
the system. are examples of borosilicate glasses. other.
OPTICAL SYSTEMS

Aspheric: Not spherical. To reduce Brewster’s Angle: For light incident on a Coma: An aberration that occurs in a
spherical aberration, a lens may be plano boundary between two materials lens when rays emanating from points
altered slightly so that one or more having different index of refraction; that not on the optical axis do not converge,
surfaces are Aspheric. angle of incidence at which the causing the image of a point to appear
reflectance is zero for light that has its comet-shaped.
Astigmatism: An aberration in a lens in electrical field vector in the plane defined
which the tangential and sagittal by the direction of propagation and the Cone Angle: The central angle of a cone
(horizontal and vertical) lines are normal to the surface. For propagation of rays converging to or diverging from a
focused at two different points along the from material 1 to material 2, Brewster’s point. See Numerical Aperture.
optical axis. angle is given as tan-1(n2/n1).
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FUNDAMENTAL APPLICATIONS
TECHNICAL REFERENCE AND
Conjugate Ratio: The ratio of the object Density, Optical: A measure of the Effective (or Equivalent) Focal Length
distance to the image distance. transmittance (T) through an optical (EFL): The focal length of an infinitely
medium; expressed as D = -log (T) or thin lens having the same paraxial
Continuous Wave Irradiation: Emission T = 10-D. imaging properties as a thick lens or
of radiant energy (light) in a continuous multiple-element lens system.
wave, rather than pulsed. Depth of Field: The distance along the
optical axis through which an object can Entrance Pupil: The image of the
Contrast: The difference in light intensity be located and clearly defined when the aperture stop as viewed through the
in an object or image; defined as lens is in focus. object side of the lens.

LENS SELECTION GUIDE

(Imax - Imin)/(Imax + Imin), where Imax and
Imin are the maximum and minimum Depth of Focus: The distance along the Erect Image: An image whose spatial
intensities. optical axis through which an image can orientation is the same as that of the
be clearly focused. object.
Converging: The bending of light rays
toward one another, achieved with a Deviation: The angle between the paths Extinction Ratio: The ratio of the
positive (convex) lens. of a ray of light before and after passing intensities along the polarization axes of
through one or more optics. a plane-polarized beam that is
Critical Angle: The smallest angle of transmitted through a polarizer;
incidence at which total internal Dielectric Coatings: Thin-film optical expressed as Tp/Ts.
reflectance takes place. Maximum angle coatings made up of alternating layers of

SPHERICAL LENSES
of incidence formed by a ray of light as it non-conductive material. The key factor F-Number: A measure of the ability of a
passes from a denser to a less dense in whether one uses a dielectric coating lens to gather light. Represented by f/#
medium. When the critical angle is or another technology to accomplish the and also called its ”speed”. The ratio of
exceeded, total internal reflection occurs, filtering effect is whether or not the focal length of the lens to its effective
and all the incident light reflects back in absorption is desired. Dielectric coatings aperture. Related to numerical aperture
to the more dense media. typically have low to non-existent by f/#=1/(2NA).
absorption whereas coatings using metals
Crown Glass: A silicate glass containing often exhibit some level of absorption. Field Curvature: An aberration in which
oxides of sodium and potassium, used in the edges of a field seem to be out of

CYLINDRICAL LENSES
lenses and windows. Harder than flint Diffraction: The sidewise or sideways focus when the center is focused clearly.
glass, it has low index and low spread of light as it passes the edge of an
dispersion, such as BK 7. object or emerges from a small aperture; Field of View: The maximum visible
causes halos or blurring of the image. space seen through a lens or optical
Crystal Quartz: Crystalline form of silicon instrument.
dioxide; used in wave plates. Diffraction Limited: Describes an optical
system in which the quality of the image Figure: See Surface Figure.
Cut-Off Wavelength: For a filter, the is determined only by the effects of
wavelength where the transmission falls diffraction and not by lens aberrations. Flatness: See Surface Flatness.
below 50%.
Dispersion: The separation of a beam Flint Glass: An optical glass with higher
Cut-On Wavelength: For a filter, the into its various wavelength components dispersion and higher refractive index

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wavelength where the transmission due to wavelength dependent speed of than crown glass; a heavy, brilliant glass,
increases above 50%. propagation in the material. softer than crown glass. For example, SF
Series glasses are used in Newport
Cylindrical Lens: A lens with at least one Distortion: Variations in magnification achromatic lenses.
surface shaped like a portion of a from the center to the edge of an image,
cylinder. A typical application is reducing making straight lines look curved. Barrel, Focal Length (FL): See Effective Focal
the astigmatism of laser diodes. or negative, distortion causes a square Length.
grid to appear barrel-shaped; pincushion,
OPTICAL SYSTEMS

Damage Threshold: The maximum or positive, distortion increases in Front Focal Length (FFL): The distance
energy density to which an optical proportion to the distance from the center from the objective plane of a lens to its
surface may be subjected without failure. of the image. first surface.

Decentration: The failure of one or more Diverging: The bending of light rays away Fused Silica: Crystal quartz melted at a
lens surfaces to align their centers of from each other, achieved with a negative high temperature to make an amorphous,
curvature with the geometric axis of a (concave) lens. non-birefringent glass of low refractive
lens system. index. Used in high-energy components
Edging: Grinding, or finishing, the edge of and optical components designed for UV.
an optical element or lens. It can be used down to 195 nm.
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FWHM: Full Width Half Maximum. The filters can be constructed using Metallic Coating: A thin layer of metal
bandwidth of an optical instrument as interference, including bandpass, applied to a substrate by evaporation to
measured at the half-power points. beamsplitter, dichroic, and edge filters. create a mirrored surface.

Gaussian Optics: Optical characteristics Interferometer: An instrument that uses Micro Optics: A term referring to small
limited to infinitesimally small pencils the interference of light waves to (less than 5 mm in size) lenses,
of light; also called paraxial or first- measure small displacements or beamsplitters, prisms, cylinders or other
order optics. deformation. optical components commonly found in
endoscopes or microscopes. Micro optics
LENS SELECTION GUIDE

Geometric Optics: That branch of optics Iris Diaphragm: A mechanical device for are also used to focus light in
dealing with the tracing of ray paths varying the effective diameter of an semiconductor laser and fiber optic
through optical systems. Geometric optical system. applications.
optics ignores the nature of the
electromagnetic modes of light. Irregularity: Refers to figure deviations Microscope Eyepiece: An eyepiece
that are not spherical in nature. Using a located at the near end of the
High-Efficiency Coating: Specialized test plate, irregularity is measured by microscope tube. Often a simple
coating applied to optics to improve counting the difference in the number of Huygen’s eyepiece, though other varieties
transmission or reflection. fringes in two orthogonal axes. (negative eyepieces, flat field projection
eyepieces) are common, depending on
Homogeneity: The state in which all Knoop Hardness: A measure of hardness application.
SPHERICAL LENSES

volume components of a substance are determined by the depth of penetration of

identical in optical properties and a diamond stylus under a specified load. Microscope Objective: The lens located
composition. Similar to the Rockwell hardness test. at the object end of a microscope tube.
Many types of objectives are used in
Hybrid: Anything formed out of Lateral Color: A chromatic aberration microscopy; simple achromats and color-
heterogeneous elements. resulting in image size variation as a corrected apochromats are popular
function of wavelength. Also known as choices.
Image Circle: The circular image field chromatic difference of magnification.
over which image quality is acceptable; MIL-C-675: Specifies that a coating will
CYLINDRICAL LENSES

can be defined in terms of its angular Limit of Resolution: The limit to the not show degradation to the naked eye
subtense. Alternately known as circle of performance of a lens imposed by the after 20 strokes with a rubber pumice
coverage. diffraction pattern resulting from the eraser. Coatings meeting MIL-C-675 can
finite aperture of the optical system. be cleaned repeatedly and survive
Image Inversion: Change in the moderate to severe handling.
orientation of an image in one meridian. Long Pass: Filter that efficiently passes
radiation whose wavelengths are longer MIL-C-14806: Specifies durability of
Image Plane: The plane perpendicular to than a specific wavelength, but not surfaces under environmental stress.
the optical axis at the image point. shorter. Coatings are tested at high humidity, or
in brine solutions to determine
Image Transposition: The flipping of an Longitudinal Color: The longitudinal resistance to chemical attack. These
image’s orientation, such as inversion of variation of focus (or image position) coatings can survive in humid or vapor
KITS

an image’s orientation in one axis or the with wavelength; often referred to as filled areas.
reversion of an image’s orientation in axial chromatic aberration.
two axes. MIL-M-13508: Sets the durability
Magnesium Fluoride: Material used as standards for metallic coatings. Coatings
Index of Refraction: The ratio of the antireflection coating for lenses because will not peel away from the substrate
speed of light in air to its velocity in of its low refractive index. Also used as when pulled with cellophane tape.
another medium; determines how much an optical substrate material for UV and Further, no damage visible to the naked
light bends as it passes through a lens, infrared applications. eye will appear after 50 strokes with a dry
OPTICAL SYSTEMS

e.g., high-index flint glass bends light cheesecloth pad. Gentle, nonabrasive
more than low-index crown glass does. Magnification: The enlargement of an cleaning is advised.
object by an optical instrument; ratio of
Infrared: The long wavelength portion of the size of the image to the actual size of Modulation Transfer Function (MTF): A
the spectrum whose wavelengths are the object. measure of the ability of an optical lens
invisible to the human eye (the range is or system to transfer detail of the object
approximately 780 nm and longer Meniscus: Describes a lens having one to the image. Given as degree of contrast
wavelengths). convex and one concave surface. (or modulation depth) in the image as a
function of spatial frequency.
Interference Filter: A filter that controls Meridional Plane: The plane in an
the spectral composition of transmitted optical system containing its optical axis
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Multi-Element System: An assembly of Optical Axis: A line passing through the Polychromatic Aberrations: The
single and/or compound lenses centers of curvature of a lens or other separation of an image into planes of
optimized to provide certain optical optical components. distinct color, caused by the variation of
characteristics. the index of refraction of glass, and the
Optical Density: See Density, Optical. focal length of a lens, with the
Multilayer Coating: Coating composed of wavelength of light; in a given plane, all
several layers of coating material. Optical Flat: A piece of glass with one or colors but one are unfocused.
Different multilayer designs are used to both surfaces polished flat. Also known
produce a variety of coating components as a test plate, test glass or reference flat. Power: 1) Lens, See Magnification

LENS SELECTION GUIDE

such as mirrors, AR coatings, bandpass (magnification power). 2) Refers to figure
coatings, dichroic coatings, and Optical Interference: The additive deviations that are spherical in nature.
beamsplitters. process, whereby the amplitudes of two Using a test plate, power is measured by
or more overlapping light waves are counting the number of fringes in two
Narrowband Coating: A coating systematically attenuated and reinforced. orthogonal axes. Power comprises the
designed to provide transmittance majority of figure deviations in a lens.
(or reflectance) over a very restricted Optical Path Difference: For a perfect Sometimes called Spherical Error.
band of wavelengths. optical system, the optical path or
distance from an object point to a Primary Reflections: The principal,
Neutral Density: A coating or absorbing corresponding image point will be equal intended reflections at optical surfaces,
glass, which has a flat or nearly flat for all rays. In near-perfect systems, slight as differentiated from secondary, usually

SPHERICAL LENSES
absorption curve throughout a specified differences will exist between rays unintended or unwanted reflections
spectrum. Neutral density filters decrease resulting in an optical path difference, occurring in an optical system.
the intensity of light without changing usually expressed in fractions of the
the relative spectral distribution of wavelength being analyzed. Principal Planes: In a thick lens or
energy. multiple-lens system, the plane at which
Orthogonal: Mutually perpendicular. Out- the entering rays and exiting rays appear
Newton’s Rings: Used to measure the fit of-Band Blocking; See Blocking. to intersect the position of the equivalent
of a lens surface against the surface of a thick lens.
test glass. The rings result when two Paraxial Image Plane: Image plane

CYLINDRICAL LENSES
adjacent polished surfaces are placed located by using first-order geometric Pulse Modulation: The process of
together with an air space between them optics. See Gaussian Optics. periodically or intermittently varying the
and the light beams they reflect interfere. amplitude of a pulse of light.
Pinhole Aperture: A small, sharp-edged
Nodal Points: The two points at which hole that functions as an aperture, for Q: The Q of a resonator is defined as:
the nodal planes appear to intersect with example, in a spatial filter. (2π x average energy stored in the
the optical axis. When a ray is directed at resonator)/(energy dissipated per cycle)
the first nodal point in an optical system, Plane of Incidence: The plane that is
it appears to emerge from a second nodal defined by the incident and reflected Q-Switched: In an optical resonator, the
point on the optical axis with no rays. higher the reflectivity of its surfaces, the
deviation in its angle. higher the Q. A Q-switch rapidly changes
Plano-Concave: A lens with one flat the Q in the optical resonator of a laser

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Numerical Aperture: Defines the (plano) surface and one inward-curved to prevent lasing until a high level of
maximum cone angle of light accepted or (concave) surface. optical gain and energy storage has been
emitted by an optical system. Given by reached in the lasing medium; a giant
sine of the half-angle of the maximum Plano-Convex: A lens with one flat pulse is generated when the Q is rapidly
angle. Related to f-number by NA = (plano) surface and the other outward- decreased.
1/(2f/#). curved (convex) surface.
Quarter Wave Optical Thickness:
Object-to-Image Distance: Also known Plano Elements: Lenses or mirrors with Common thin-film term. The QWOT
OPTICAL SYSTEMS

as the total conjugate distance or track flat surfaces. (Quarter Wave Optical Thickness) is the
length. Can be finite or infinite wavelength at which the optical
depending on the application. Polarized, Circularly: Light whose thickness, defined as the index of
electric field vector describes a circle as a refraction, n, multiplied by the physical
Objective: The optical element that function of time. thickness, d, of a coating evaporant layer;
receives light from the object and forms is one quarter wavelength, or n x d=λ/4.
the first or primary image in telescopes, Polarized, Linearly: See Polarized, Plane.
microscopes, and other optical systems. Radius of Curvature: One-half the
Polarized, Plane: Light whose electric diameter of a circle defining the convex
Oblique Ray: A ray of light that is neither field vector vibrates in only one plane. or concave shape of a lens.
perpendicular nor parallel, but inclined.
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Real Image: Light rays reproduce an Scratch-Dig: A measure of the visibility Striae: An imperfection in optical glass
object, called an image, by gathering a of surface defects as defined by several characterized by streaks of transparent
beam of light diverging from an object U.S. military standards including material of a different refractive index
point and transforming it into a beam MIL-PRF-13830B, MIL-F-48616, and than the body.
converging toward another point. If the MIL-C-48497. Unless otherwise noted,
beam is converging, it produces a real specifications for surface quality of our Substrate: The underlying material to
image. products are in accordance with which an optical coating is applied.
MIL-PRF-13830. Using MIL-PRF-13838B,
Reference Flat: An optical flat used as a the ratings consist of two numbers, the Surface Contour: The outline or profile
LENS SELECTION GUIDE

test glass. first denoting the visibility of scratches, of a surface.

the second, of digs (small pits). A 0-0
Refraction: The change in direction of a scratch-dig number indicates a surface Surface Figure: A measure of how
ray of light as it passes from one optical free of visible defects. Numbers increase closely the surface of an optical element
medium to another with a different as the visibility of blemishes increases. matches a reference surface. Since
optical density. See Snell’s Law. Scratches and digs are evaluated for size geometrical errors will cause distortion of
by comparison to standards fabricated in a transmitted or reflected wave,
Refractive Index: The ratio between the accordance with US Army ARDEC drawing deviations from the ideal are measured in
speed of light through vacuum to the C7641866. No absolute measurement of terms of wavelengths of light.
speed of light through the particular defect size is made or implied by the
medium. The index determines how scratch-dig standard. MIL-F-48616 and Surface Flatness: The amount by which
SPHERICAL LENSES

much a ray of light will bend as it passes MIL-C-48497 use alphabetical notations an optical surface differs from a perfect
from one given medium to another. See to designate defect size and prescribe plane. It is typically measured by an
Snell’s law. physical measurement of defects to interferometric technique.
determine conformance. A specification
Resolution: The ability of a lens to image of F/F using MIL-C or MIL-F is Surface Roughness: A measure of the
the points, lines, and surfaces of an approximately equivalent to 80/50 with texture of a surface on a microscopic
object so they are perceived as discrete the exception that measurement is used scale. It is usually denoted as a root
entities. to characterize defects rather than mean square (rms) value and measured
comparison to a set of standards. in units of length, such as angstroms.
CYLINDRICAL LENSES

Reticle: An optical element containing a

pattern placed at the image plane of a Short Pass: Filter that efficiently passes Surface Quality: See Scratch-Dig.
system. The reticle facilitates system radiation whose wavelengths are shorter
alignment or the measurement of target than a specific wavelength, but not Total Internal Reflection (TIR): When
characteristics. longer. the angle of incidence of light striking the
boundary surface of a substance exceeds
Reverted Image: An image in which left Slit: An aperture, typically rectangular in the critical angle, the result is total
and right seem to be reversed. shape, whose length is large compared to internal reflection.
its width.
Rockwell Hardness: Resistance of a Transmission: Amount of light that is
substance to penetration by a pyramidal Snell’s Law of Refraction: Gives the passed through an optical component or
stylus pressed in under a specific load; ratio of bend angles as light passes from system. Given as fraction or percentage of
KITS

also see Knoop hardness. one medium to another; expressed as input light.
n1sinθ1=n2sinθ2, where n is the index of
Sag: An abbreviation for “sagitta,” the refraction. Truncation Ratio: The dimensionless
Latin word for “arrow.” Used to specify ratio of the Gaussian beam diameter at
the distance on the normal from the Spatial Filtering: Enhancing an image by the 1/e2 intensity point to the limiting
surface of a concave lens to the center of increasing or decreasing its spatial aperture of the lens.
the curvature. It refers to the height of a frequencies.
curve measured from the chord, Ultraviolet: The short wavelength of the
OPTICAL SYSTEMS

Spectrophotometry: Measuring the electromagnetic spectrum invisible to the

reflection or transmission of light for human eye. The range is approximately
each component wavelength in the 400 nm and shorter wavelengths.
spectrum of a specimen.
where R = radius of curvature of the V-Coating: A narrowband coating for
surface and Y = radius of the aperture of Spherical Error: See Power. specific laser wavelengths. This term is
the surface. usually applied in reference to AR
Spot Size: Minimum image size to which coatings.
Sagittal Focus: The focus of rays lying in a lens may focus a collimated beam.
the sagittal plane, which is the plane
perpendicular to the meridional plane.
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Vignetting: The gradual reduction of
image illuminance with an increasing off-
axis angle, resulting from limitations of
the clear apertures of elements within an
optical system.

Virtual Image: Light rays that diverge

from an object point can be captured by
an optical system to form an image.

LENS SELECTION GUIDE

Depending on the optical system, the
light beam can either converge to
another point or diverge from another
point. In the case that the light
converges, it will form a real image. In
the case that the light diverges it will
form a virtual image.

V-Value: See Abbe Number.

Wavefront Distortion: Departure of a

SPHERICAL LENSES
wavefront from ideal (usually spherical or
planar) caused by surface errors or design
limitations.

Wavelength: The distance light travels in

one cycle of its electromagnetic wave.

Wedge: An optical element with its faces

inclined toward each other at very small

CYLINDRICAL LENSES
angles, diverting light toward the thicker
parts of the element.

Young’s Modulus: Modulus of elasticity;

the amount of stress required to produce
a unit change in length (strain);
expressed in pounds per square inch
(PSI) or dynes per square cm.

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OPTICAL SYSTEMS
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