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Important Points from Course:
Overview:
1. The basic methods of optical design:
 Historical Research – What has been done before that you can use?
While it is necessary to understand the past, it is wasteful to re-invent it.
 First Order Layout – Use your knowledge of previous designs and the
behavior of optical components (lenses, prisms, etc) to create a rough
layout of idealized components that will approximately do the job.
 Gaussian Optics – Use the theorems and methods developed in first order
optics to refine the layout and determine the basic parameters (focal
lengths, aperture sizes, stops).
 Degrees of Freedom – When transitioning to finite optical elements, put
in enough elements so that it will be possible to refine the system
adequately. For example:
i. Field curvature cannot be corrected without a mix of positive and
negative lenses or mirrors – this is from first order optics (Petzval
Theorem).
ii. Color aberrations cannot be corrected (except for an all-mirror
system) without using glasses with different V-numbers.
 Heuristic Refinement – The system is evaluated by simulating the
propagation of light (by ray tracing or beam propagation algorithms).
Successive changes in the system are made and evaluated to search for a
better system. While the tedious part of this job is now done by computer,
a good design does not often come automatically. The designer still must:
i. Create a merit function defining the analysis algorithms and results
that constitute a “good” design. This often changes throughout the
process.
ii. Decide what parts of the system can be varied in the heuristic
search and how those variables should be constrained. This is also
subject to change during the design process.
iii. Shepard the process through sticking points to reach an acceptable
solution. This often requires changes in the merit function and
sometimes changes in the basic system layout.
2. The Tools of Optical Design:
 General knowledge of Optical Components and their approximations:
i. The thin lens approximation, imaging equation and lensmaker’s
formula (including mirrors).
ii. A general knowledge of various optical methodologies, such as
holography, coded-aperture imaging, computed tomography,
interferometry, etc. When you need one of these, you can then
research it in more detail – but if you never heard of it before, you
will not know that you need to look into it.
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(It is worth ordering a catalog from a major company such as Melles
Griot – not only is it a valuable reference on the variety of optical
components and systems available, but they also include enough
tutorial text to constitute a good, condensed textbook.)


Knowledge of Light Propagation Approximations:
i. The ray approximation to light propagation (Geometrical Optics);
Postulates, theorems and limitations.
ii. The methods of graphical ray-tracing which allow you to quickly
sketch out optical systems without doing calculations.
iii. The postulates, basic equations and theorems of Paraxial Optics
(Gaussian Optics). In general, optical systems must be designed to
work in the paraxial approximation before they can be made to
work with real components.
iv. The basics of Fourier optics and beam propagation – where they
can be used with geometrical optics (PSF, MTF calculations for
example), and when they need to replace ray optics (zone plates
and holography, for example).
Familiarity with computer optical design programs:
In today’s world, no one will ask you to design an optical system
without access to such a program. You need to be aware of the methods
of use, the limitations, and pitfalls of such programs. In particular, you
need to be aware that these programs are not “automatic”, but need to be
carefully guided, debugged, and prodded to reach a satisfactory solution.
A general knowledge of how they work, and a willingness to read the
manual is generally more important than skill with a particular program.
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Paraxial Optics Review
h
Coordinates:
2
1
z
-3
-2
0
-1
1
2
3
-1
-2

Coordinates for ray tracing are local Cartesian coordinates, centered on
the intersection of the surface with the z-axis.

All distances are directed (signed), and measured from the coordinate
origin. (i.e., points above and to the right of the origin are positive
distances, points to the left and below are negative distances.)

Rays are always directed lines. Ray angles are either measured w.r.t. the
z-axis, or the surface normal (as appropriate).

All angles are acute. If the rotation direction from the reference (z-axis or
surface normal) to the ray is CCW, the angle is positive. If CW, the ray is
negative.

All angles are equivalent to slopes (tangents) for purposes of ray
propagation, and sines, for purposes of refraction.

The distance from surface n to surface n+1 is a property of surface n and
is a positive number designated 𝑑𝑛 .
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Example:
n
n’
P
i
i’
h
u
u’
a
O
A
z
A’
R
l
l’
Notation Rules:
Unprimed rays, angles and distances are before the ray is refracted by the surface, primed
values are after.
Paraxial Assumptions:
1. Surface sag is ignored as negligible.
2. angles  sines  tangents
Positive Values:
 Angles: 𝑢, 𝑖, 𝑖 ′
 Lengths: 𝑙 ′ , ℎ, 𝑅
Negative Values:
 Angles: 𝑎, 𝑢′
 Lengths: 𝑙
Geometric relationships (taking the sign rules into account):
𝑖 = |𝑎| + |𝑢| = 𝑢 − 𝑎,
𝑖 ′ = |𝑎| − |𝑢′ | = 𝑢′ − 𝑎
(𝑠𝑖𝑛𝑐𝑒 𝑢 > 0,
(𝑠𝑖𝑛𝑐𝑒 𝑢′ < 0,
𝑎 < 0)
𝑎 < 0)
|ℎ|
ℎ
=−
|𝑅|
𝑅
|ℎ|
ℎ
𝑢=
=−
|𝑙|
𝑙
𝑎=−
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|ℎ|
ℎ
=
−
|𝑙 ′ |
𝑙′
Snell’s Law (Paraxial)
Relates the quantities to the left of the surface with those to the right:
𝑛𝑖 = 𝑛′ 𝑖 ′
𝑛(𝑢 − 𝑎) = 𝑛′ (𝑢′ − 𝑎)
Ray Trace Equations:
If we substitute for the incident and refracted angles in Snell’s Law their
expressions in terms of lengths, but leave the ray angles, we arrive at:
𝑛′ 𝑢′ − 𝑛𝑢 = −ℎ𝐾,
Where 𝐾 ≡ 𝑐(𝑛′ − 𝑛) is the surface power
1
and 𝑐 ≡ 𝑅 is the surface curvature
If we trace through sequential surfaces numbered 𝑖 = 1,2, ⋯ we can derive the
ray transfer equation from the ray’s direction and height and the distance, di,
between surface i and surface i+1:
ℎ𝑖+1 = ℎ𝑖 + 𝑑𝑖 𝑢𝑖′
Thin Lens in Air: If we trace a ray through two surfaces, assuming the distance
between them is negligible (d=0), we get the thin lens equation:
𝑢′ − 𝑢 = −ℎ𝐾
Where 𝐾 = (𝑛 − 1)(𝑐1 − 𝑐2 ) is the power of the thin lens
Imaging Equations:
If we again start with the Paraxial Snell’s Law expression, but substitute for all
the angles in terms of the relevant distances, we can derive a similar set of
equations which trace the position of images through a surface, lens, or series of
lenses:
Imaging through a surface:
𝑛′ 𝑛
− =𝐾
𝑙′ 𝑙
Imaging through a thin lens:
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1 1
− =𝐾
𝑙′ 𝑙
and image transfer between lenses:
𝑙𝑖+1 = 𝑙𝑖′ − 𝑑𝑖
Analysis of the image equations leads to the concept of focal length (=1/K).
Multiple Lenses, Thick Lenses, etc:
Iteratively applying the above equations to multiple surfaces or lenses allows the
calculation of the thick lens equation, the lens combination equation, and the
derivation of the concept and usage of Principle Planes.
See notes from 1/18/2013 and 1/23/2013 for details.
Summary of Paraxial Optics:
Tracing Equations:
Refract at a surface
nu  nu  hK
Refract through a thin lens in air
u  u  hK
hi 1  hi  diui
Transfer
Imaging Equations:
n n
 K
l l
1 1
 K
l l
li 1  li  di
Image location from a surface
Image location from a thin lens in air
Image transfer
Definitions:
K  cn  n
K  n  1c1  c2 
1
f 
K
1
c
R
Power of a surface
Power of a thin lens in air
Focal length
curvature is inverse radius of surface
Tracing Graphical Rays:
Ray parallel to axis:
u  u  hK   hK , since u  0
h
h 1
u  
 l     f
l
u K
Hence the ray passes through the focal point.
Ray passing through focal point:
h
h h
u  u  hK   hK  0 , since u   
f
l
f
Hence the ray leaves the lens parallel to the axis.
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Ray passing through lens vertex:
u  u  hK  u , since h  0
Hence the ray is undeviated.
Fourier Optics, The Plane Wave Spectrum and Beam
Propagation:
The (scalar) plane wave solution to Maxwell’s equations has useful properties for
describing the propagation of EM fields:
 The plane wave solution has the same amplitude, frequency, and propagation
direction (in free space) everywhere.
 “Propagation” of a plane wave involves simply changing it’s phase by inserting a
different position vector, r  [ x, y, z ] into the expression
E  ReA exp ik  r    t 
2
, where k 
.
 A exp i k x x  k y y  k z z    t 



Thus, an EM field can be simply calculated at any point in space, provided it can be
described as a collection of plane waves – the “Plane Wave Spectrum” (PWS).
A connection between plane waves and the Fourier Transform can be made by looking at
the intersection of a plane wave with a plane surface. For simplicity, we will set t = 0 and
look at the intersection at the x,y,z = 0 plane. The scalar expression for a plane wave at
this plane and time is:
E  A exp i k x x  k y y  .
When (the real part of) this function is plotted, it is seen to be a 2-D spatial frequency:
This can be made explicit by an appropriate variable change, u kx/2, v ky/2 ,
resulting in the expression:
E  A expi 2 ux  vy
where u, v are spatial frequencies.
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The Fourier Transform decomposes functions on a plane into a set of spatial frequencies:
Au , v    E  x, y  exp  i 2 ux  vy dx dy
E  x, y    Au , v  exp i 2 ux  vy du dv
Hence, the Fourier Transform can decompose a EM field (at t = 0) on an x-y plane into a
set of spatial frequencies, each of which can be associated with a specific plane wave
(given the wavelength, ) with amplitude equal to the spatial frequency’s amplitude and
wave vector k  [k x , k y , k z ] , where:
k x  2u
k y  2v,
and
  2  2

kz  
 k x2  k y2 

  



The spatial frequency that represents this plane wave’s intersection with any other x-y
plane down the z-axis is simply:
E  A exp ik x x  k y y  k z z 


 Aei kz z  exp i 2 ux  vy
Hence, any monochromatic E-M field defined on a x-y plane (such as a pupil function or
an illuminated aperture) can be propagated to another parallel x-y plane at distance z
from the first plane, by:
1. Decomposing the function into spatial frequencies by the Fourier Transform.
2. Finding the k vector of the plane wave associated with each spatial frequency (the
plane wave whose intersection with the plane is the given spatial frequency).
3. Changing the phase of each spatial frequency by the factor e ik z .
4. Inverse Fourier Transforming the re-phased spatial frequencies to find the new EM field distribution.
(Of, course, taking due account of the organization and characteristics of the FFT, if done
discretely.)
z
This technique can be used for many optical design and evaluation tasks:
 Evaluation and design of zone plates.
 Decoding optical holograms.
 Evaluation and design of very small lenses and optical elements – too small for
the ray approximations of geometrical optics to apply.
 Evaluation of diffraction effects of apertures in optical systems.
Digital Filtering:
Most imaging today eventually produces pixilated images that can be manipulated by a
computer. The field of digital image enhancement is extremely broad, but some basic
principles hold throughout:
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

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An image filter is easiest to visualize in the frequency (Fourier) domain: Any
digital image can be decomposed by the DFT into a finite set of spatial
frequencies. Each of these frequencies can be multiplied by a scalar gain factor
and, potentially, a phase factor to adjust their phase. A filter in the frequency
domain, therefore, is simply a matrix of (possibly complex) gain factors – one for
each spatial frequency in the image. (Note that the spatial frequencies of an
image have no relationship to the Plane Wave Spectrum – they are simply
intensity distributions that an intensity image can be decomposed into. The PWS
is based upon amplitude and phase distributions of EM fields in space.)
A space-domain (convolution) filter can be derived from the frequency filter by
inverse Fourier Transforming the frequency-domain filter array. If the space filter
consists of a few finite values surrounded by near-zero values, there is no penalty
(and a speed advantage) to truncating it to the non-zero area.
Random noise in the image will result in spurious data in the spatial frequency
domain. Any filter that amplifies spatial frequencies will also amplify noise.
Linear filters will not increase the SNR of an image.
Spatial frequencies missing from the image cannot be recovered by linear
filtering. This kind of “image restoration” is done by using a heuristic search to
find an object “consistent with” the partial information. There is no guarantee
that a “restored image” is correct.
We covered a specific example of a linear filter designed to boost the MTF of an optical
system with a cubic wavefront distortion designed to produce an extended depth of field:
To invert the imaging process, O  H  I , using the filter operation O  G  I , we
H
calculate the filter by: G  W
, where W is a band limiting matrix (maximum
2
H 
limits on each spatial frequency boost, determined by the physical band limit of the
optics) and  is a small number designed to prevent division by zero.
Aberrations in Optical Systems
Key Points:
1. Aberrations are deviations from the ideal optical system.
2. Aberrations can be described by ray trace errors – Transverse Ray
Aberration is the distance that a ray strikes the image plane from the ideal
image point.
3. Aberrations can be described by wavefront errors at the exit pupil – The
Zemax Wavefront map is the deviation of the wavefront at the exit pupil (in
waves) from a perfect sphere centered on the ideal image point.
4. Since rays travel perpendicular to wavefronts, the Transverse Ray
Aberration is directly proportional to the difference in slope between the exit
pupil wavefront and the ideal sphere.
NOTE: Wavefront and ray errors are in some ways equivalent and Zemax’s Default Merit
Function gives you the choice to minimize either ray errors or wavefront errors. Highly
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aberrated systems, however, will not optimize nearly as well using wavefront errors as
when using ray errors.
Aberrations in First-Order Optics:

First order (or Paraxial) Optics describes optical systems in which the sine
function can be well approximated by the angle itself. In this approximation,
optical systems are linear systems and hence many useful theorems and formulas
can be derived.

First order optics can be extended to finite ray angles by a suitable re-definition of
the angle variable of a ray with respect to the z-axis as the tangent of the angle
w.r.t. the z-axis. This allows first order optics to describe ideal (or perfect) optical
systems and is extremely useful for deciding on the general layout of an optical
system.

Certain aberrations arise in the first order approximation, and must be corrected in
the first order system before there is any hope of producing a corrected real
system. These are:
1. Chromatic aberrations. Since the index of refraction changes the
same in first order optics as in finite optics, the same changes of focal
length and magnification with wavelength occur. These aberrations
must be corrected at the first order (paraxial) level before there is any
hope of correcting them in a finite ray-angle system. Except for
mirrors (which have no chromatic aberrations), this correction requires
use of at least two glasses with different V-numbers (rates of index
change with wavelength).
2. Image plane curvature (Petzval curvature). A quasi-paraxial
derivation shows that an optical system can only image a plane onto a
plane if it contains equal powers of negative and positive elements –
otherwise, a plane will be imaged onto a curved surface. It is still
possible for a system to have net positive or negative power, even if
there is exactly the same total negative and positive power in the
system – this is because of the theorem that the power a surface
contributes to the whole system is proportional to its power times the
height of the marginal ray at that surface. (The marginal ray is the ray
from a given object point that just touches the edge of the stop.)
Hence, if the marginal ray has less height at the negative surfaces than
the positive ones, the system can have net positive power.
Third-Order Optics:
“Third Order Optics” describes optical systems where the ray angles of incidence don’t
exceed values for which the third-order Taylor expansion is a good approximation to the
sin function: sin     
3
. In the third-order approximation, families of aberrations
6
can be described that are independent of each other, and the total aberration of the optical
system can be described as the sum of the aberrations of each element or surface.
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Even in systems where the third-order approximation isn’t valid, the third-order
aberrations must be corrected for there to be any hope of designing a good system. In
other words, “higher-order” aberrations generally cannot be used to compensate for
uncorrected third-order ones.
The third-order aberrations are generally called Seidel aberrations after Ludwig von
Seidel, who first described them in 1857. Historically, the Seidel aberrations were very
important because they could be derived from Paraxial ray traces. With today’s
computers, this is much less important, and few will ever perform these calculations
manually.
The Seidel aberrations are:
1. Spherical Aberration: The change in power of a surface with ray height
2. Coma: The power asymmetry in an off-axis pencil or rays – the rays further from
the z-axix than the chief ray see a different rate of power change than the rays to
the side near the z-axis, for each surface.
3. Astigmatism: A fan of off-axis rays perpendicular to the z-axis (tangential) has a
different focal length then a fan pointing at the z-axis (sagittal) .
4. Field Curvature: While this is a quasi-first-order aberration, it also can be
described as a sum over all surfaces in the system.
5. Distortion: The change in magnification with object distance from the z-axis.
Zemax will produce a very useful diagram of the surface-by-surface and total sum of
these aberrations, using the “Analysis – Aberration Coefficients – Seidel Diagram”
choice of analysis window. This diagram also includes “lateral color” and “axial color”
aberration, which can also be separated into individual surface contributions.
An unbalanced Seidel sum will alert the designer that he or she must change the system
in order to achieve third-order aberration correction before significant progress can be
made. Possible changes are:
 Change the stop position.
 Add elements with opposing values of the offending aberrations. In general, this
means negative elements if the offending aberrations are primarily from positive
elements, and vice-versa. Total system power can be maintained by use of the
power vs. marginal ray height theorem – the power an element contributes to the
system is proportional to its own power times the height of the marginal ray at
that element.
The system must be re-optimized and re-evaluated after any changes.
Often, adding operands that limit third-order aberrations to the Merit Function is a
valuable way to constrain computerized optimizations to desired forms. There is a place
for “no-holds-barred” global computer searches (if you have plenty of time and are out
of ideas, say), but often all that is needed is to refine a system without making radical
changes. Constraining the Seidel aberrations (combined with the default ray-based merit
function) is a good way to do that.
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Seidel Diagram from Zemax for a camera lens:
Lens:
Seidel Diagram:
pg. 12