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