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Aberrations of Anamorphic Optical Systems
Item Type text; Electronic Dissertation
Authors Yuan, Sheng
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/195267
ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS
By
Sheng Yuan
A Dissertation Submitted to the Faculty of the
COLLEGE OF OPTICAL SCIENCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2008
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THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Sheng Yuan entitled Aberrations of Anamorphic Optical Systems and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy _______________________________________________________________________ Date: 11/05/08
Jose M. Sasian, Dissertation Director _______________________________________________________________________ Date: 11/05/08 Tom Milster, Committee Member _______________________________________________________________________ Date: 11/05/08
Hong Hua, Committee Member _______________________________________________________________________ Date: _______________________________________________________________________ Date: Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: 11/05/08 Dissertation Director: Jose M. Sasian
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STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Sheng Yuan
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ACKNOWLEDGMENTS
I am deeply grateful to my advisor, Dr. José Sasián, for supporting me in the study of this work. I appreciate your patient guidance which kept me on track throughout my thesis studies. I am also deeply grateful to the other members of my dissertation committee, Dr. Tom Milster and Dr. Hong Hua, for reviewing my thesis. Additional thanks go to Bruce Pixton, Peng Su and Lirong Wang for their help in editing my thesis. To my parents Zai-chao Yuan and Sie Wang, thank you for always encouraging and supporting me over the years of my study. Finally, to my lovely wife Grace Yuan, thank you for your unconditional love and support during my years of graduate study. Without you this thesis would not have been possible.
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TABLE OF CONTENTS
LIST OF FIGURES ............................................................................................................ 8
LIST OF TABLES............................................................................................................ 10
ABSTRACT...................................................................................................................... 11
CHAPTER 1 ..................................................................................................................... 12 INTRODUCTION ............................................................................................................ 12
1.1 What is an anamorphic system?.............................................................................. 13 1.2 Why is it a difficult problem? ................................................................................. 14 1.3 Historical background............................................................................................. 16 1.4 Dissertation content ................................................................................................ 18
CHAPTER 2 ..................................................................................................................... 20 FIRST-ORDER THEORY FOR ANAMORPHIC SYSTEMS........................................ 20
2.1 Direction cosines and their paraxial approximations.............................................. 20 2.2 Three dimensional ray transfer and refraction equations........................................ 22 2.3 Double curvature surface types and their surface normal—general theory............ 25 2.4 Double curvature surfaces and their surface normal—paraxial approximations.... 28 2.5 Ideal (first-order) image model for anamorphic imaging ....................................... 30 2.6 The paraxial ray tracing equations for anamorphic systems................................... 33 2.7 Paraxial image properties of anamorphic systems---part one................................. 35 2.8 Paraxial image properties of anamorphic systems---part two ................................ 37 2.9 Paraxial image properties of anamorphic systems---part three .............................. 40 2.10 Paraxial image properties of anamorphic systems---part four.............................. 43 2.11 Paraxial image properties of anamorphic systems---part five .............................. 48 2.12 Summary............................................................................................................... 49
CHAPTER 3 ..................................................................................................................... 51 GENERAL ABERRATION THEORY FOR ANAMORPHIC SYSTEMS .................... 51
3.1 Fermat’s principle and Sir Hamilton’s characteristic function............................... 51 3.2 Aberration function and ray aberrations for anamorphic systems.......................... 53 3.3 Power series expansion of aberration function ....................................................... 58 3.4 Summary................................................................................................................. 63
CHAPTER 4 ..................................................................................................................... 64 METHOD OF ANAMORPHIC PRIMARY ABERRATIONS CALCULATION.......... 64
4.1 The total ray aberration equations for anamorphic systems ................................... 64 4.2 Preparation for the anamorphic primary ray aberration equations deduction ........ 70 4.3 The anamorphic primary ray aberration calculation-part one ................................ 74 4.4 The anamorphic primary ray aberration calculation-part two ................................ 75 4.5 Summary................................................................................................................. 76
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TABLE OF CONTENTS-Continued CHAPTER 5 ..................................................................................................................... 77 PRIMARY ABERRATION THEORY FOR PARALLEL CYLINDRICAL ANAMORPHIC ATTACHMENT SYSTEMS ................................................................ 77
5.1 The primary ray aberration equations for cylindrical anamorphic systems............ 79 5.2 Primary ray aberration coefficients for parallel cylindrical anamorphic systems .. 81 5.3 Primary wave aberration coefficients for parallel cylindrical anamorphic systems86 5.3 Simplification of the results.................................................................................... 88 5.5 Summary................................................................................................................. 95
CHAPTER 6 ..................................................................................................................... 99 PRIMARY ABERRATION THEORY FOR CROSS CYLINDRICAL ANAMORPHIC SYSTEMS......................................................................................................................... 99
6.1 Primary ray aberration coefficients for cross cylindrical anamorphic systems ...... 99 6.2 Primary wave aberration coefficients for cross cylindrical anamorphic systems. 104 6.3 Simplification of the results.................................................................................. 106 6.4 Summary............................................................................................................... 111
CHAPTER 7 ................................................................................................................... 114 PRIMARY ABERRATION THEORY FOR TOROIDAL ANAMORPHIC SYSTEMS......................................................................................................................................... 114
7.1 The primary ray aberration coefficients for toroidal anamorphic systems........... 114 7.2 The primary wave aberration coefficients for toroidal anamorphic systems........ 119 7.3 Simplification of the results.................................................................................. 121 7.4 Summary............................................................................................................... 125
CHAPTER 8 ................................................................................................................... 127 PRIMARY ABERRATION THEORY FOR GENERAL ANAMORPHIC SYSTEMS127
8.1 The primary ray aberration coefficients for general anamorphic systems............ 127 8.2 The primary wave aberration coefficients for general anamorphic systems ........ 133 8.3 Simplification of the results.................................................................................. 135 8.4 Summary............................................................................................................... 139
CHAPTER 9 ................................................................................................................... 141 TESTING OF THE RESULTS....................................................................................... 141
9.1 The idea of data fitting.......................................................................................... 142 9.2 The detailed primary aberration coefficients data fitting steps............................. 144 9.3 A testing example ................................................................................................. 148 9.4 Summary............................................................................................................... 154
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TABLE OF CONTENTS-Continued CHAPTER 10 ................................................................................................................. 155 DESIGN EXAMPLES.................................................................................................... 155
10.1 An anamorphic singlet ........................................................................................ 156 10.2 An afocal anamorphic attachment ...................................................................... 161 10.3 An anamorphic Triplet design ............................................................................ 167
10.3.1 General methods for discovering an anamorphic beginning design............ 169 10.3.2 An anamorphic Triplet with an anamorphic ratio of 1.22 ........................... 171 10.3.3 Another anamorphic Triplet with an anamorphic ratio of 1.35 ................... 175
10.4 An anamorphic field lens design ........................................................................ 178 10.5 An anamorphic Double Gauss design with anamorphic ratio 1.5 ...................... 181 10.6 An anamorphic fisheye lens design with anamorphic ratio 3:4.......................... 188
10.6.1 An anamorphic fisheye design without field lens........................................ 190 10.6.2 An anamorphic fisheye design with field lens............................................. 194
10.7 Summary............................................................................................................. 197 CHAPTER 11 ................................................................................................................. 198 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK................................ 198
11.1 Conclusions......................................................................................................... 198 11.2 Suggestions of future work ................................................................................. 199
APPENDIX A................................................................................................................. 200 THE TYPICAL SHAPE OF ANAMORPHIC PRIMARY WAVE ABERRATIONS.. 200 REFERENCES ............................................................................................................... 203
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LIST OF FIGURES
Figure 1-1 Example of a double curvature surface………………………………………13
Figure 1-2 Example of an anamorphic system--a system made from cylindrical
lenses……………………………………………………………………………………..14
Figure 1-3 Constant astigmatism for a single double curvature surface………...……….15
Figure 2-1 Direction cosines of a ray………………………...……………………...…...21
Figure 2-2 Three dimensional ray tracing………………………………………………..23
Figure 2-3 Gaussian optics properties for the associated x-RSOS……...….……………36
Figure 2-4 The object and stop plane…………………………………………………….46
Figure 3-1 Wavefront error………………………………………………………………54
Figure 4-1 Refraction on surface j ……………………………………………………...66
Figure 5-1 A parallel cylindrical anamorphic attachment system……………………….81
Figure 5-2 The y-z (top) and x-z (bottom) symmetry planes of a parallel cylindrical
attachment system………………………………………………………………………..82
Figure 6-1 A cross cylindrical anamorphic system example…………………………….99
Figure 6-2 The two principal sections of a cross cylindrical anamorphic system……...100
Figure 9-1 Layout of a simple anamorphic system in the y-z (left) and x-z principal
sections………………………………………………………………………………….149
Figure 9-2 Grid distortion map…………………………………………………………150
Figure 10-1 Layout in the y-z (left) and x-z (right) principal sections…………………159
Figure 10-2 On-axis system performance………………………………………………159
Figure 10-3 Layout in the y-z (left) and x-z (right) principal sections…………………163
Figure 10-4 Layout in the y-z (left) and x-z (right) principal sections…………………164
Figure 10-5 Spot diagram………………………………………………………………164
Figure 10-6 Ray fan…………………………………………………………………….165
Figure 10-7 Layout in the y-z (left) and x-z (right) principal sections…………………173
Figure 10-8 Spot diagram………………………………………………………………173
Figure 10-9 Ray fan…………………………………………………………………….174
Figure 10-10 Layout in the y-z (left) and x-z (right) principal sections………………..176
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LIST OF FIGURES-Continued
Figure 10-11 Spot diagram……………………………………………………………..176
Figure 10-12 Ray fan…………………………………………………………………...177
Figure 10-13 Layout in the y-z (left) and x-z (right) principal sections………………..179
Figure 10-14 Spot diagram……………………………………………………………..180
Figure 10-15 Layout of the initial design in the y-z (left) and x-z (right) principal
sections………………………………………………………………………………….183
Figure 10-16 Spot diagram for the initial design……………………………………….184
Figure 10-17 Spot diagram and aberrations coefficients after the first stage optimization
…………………………………………………………………………………………..185
Figure 10-18 Layout of the anamorphic Double Gauss design………………………...186
Figure 10-19 Spot diagram……………………………………………………………..187
Figure 10-20 OPD fan…………………………………………………………………..187
Figure 10-21 A RSOS Double Gauss type fish-eye Lens………………………………189
Figure 10-22 Layout in the y-z (left) and x-z (right) principal sections………………..190
Figure 10-23 Grid distortion for the initial anamorphic fisheye design………………..191
Figure 10-24 Layout of the final design in the y-z (left) and x-z (right) principal
sections………………………………………………………………………………….192
Figure 10-25 Grid distortion map of the final design.………………………………….192
Figure 10-26 Spot diagram of the final design…………………………………………193
Figure 10-27 Ray fan of the final design……………………………………………….193
Figure 10-28 Layout of the final design in the y-z (left) and x-z (right) principal sections
…………………………………………………………………………………………..195
Figure 10-29 Vignetting plot…………………………………………………………...195
Figure 10-30 Spot diagram……………………………………………………………..196
Figure 10-31 Ray Fan…………………………………………………………………..196
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LIST OF TABLES
Table 9-1 Lens data……………………………………………………………………..150
Table 9-2 Full aperture is 10mm, HFOV is 10 degree…………………………………151
Table 9-3 Full Aperture is 4mm, HFOV is 4 degree…………………………………...152
Table 9-4 Full Aperture is 1mm, HFOV is 1 degree…………………………………...153
Table 10-1 Specifications of the anamorphic singlet…………………………………...159
Table 10-2 Lens data and remaining primary aberration coefficients………………….160
Table 10-3 Specifications of the anamorphic system…………………………………..164
Table 10-4 Lens data and remaining primary aberration coefficients………………….165
Table 10-5 Specifications of the anamorphic triplet with anamorphic ratio 1.22……...173
Table 10-6 Lens data and remaining primary aberration coefficients………………….174
Table 10-7 Specifications of the anamorphic triplet with anamorphic ratio 1.35……...176
Table 10-8 Lens data and remaining primary aberration coefficients………………….177
Table 10-9 Lens data and remaining primary aberration coefficients………………….180
Table 10-10 Specifications of the anamorphic Double Gauss………………………….183
Table 10-11 Lens data and remaining primary aberration coefficients………………...188
Table 10-12 Initial design specifications of the anamorphic Double Gauss …………..190
Table 10-13 Final design specifications of the anamorphic Double Gauss…………….191
Table 10-14 Lens data of the final design………………………………………………192
Table 10-15 Specifications of the anamorphic Double Gauss with field lens………….194
Table 10-16 Lens data of the final design with field lens………………………………195
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ABSTRACT
A detailed study of the aberrations of anamorphic optical systems is presented. This
study has been developed with a theoretical structure similar to that of rotationally
symmetric optical systems (RSOS) and can be considered a generalization.
A general method of deriving the monochromatic primary aberration coefficient
expressions for any anamorphic system types with double plane symmetry has been
provided.
The complete monochromatic primary aberration coefficient expressions for
cylindrical anamorphic systems, toroidal anamorphic systems and general anamorphic
systems with aspheric departure have been presented, in a form similar to the Seidel
aberrations of RSOS.
Some anamorphic image system design examples are provided that illustrate the use
and value of the theory developed.
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CHAPTER 1
INTRODUCTION
Optical systems can be classified into groups according to their symmetry [1], for
example, rotationally symmetric optical systems (RSOS) [2], double plane symmetric
optical systems (DPSOS) [3], and single plane symmetric optical systems (SPSOS) [4,5],
etc. Currently most optical systems are RSOS, and a tremendous amount of research is
going on in this field.
However, in recent years an interest in understanding DPSOS has arisen [6-9],
mainly because this type of system can offer a unique solution to anamorphic image
formation. For this reason, a DPSOS is also called an anamorphic system.
When optical designers want to use an anamorphic system, they will find that there
are not enough well established theoretical models and results as are found for RSOS,
such as the first-order theories and the Seidel third-order aberrations, etc. To date, several
papers and books have been published explaining some aspects of the behavior of
anamorphic systems [10-18], but most of these works lack enough theoretical structure,
and accordingly the results obtained are far from complete. The current situation on
anamorphic systems research can be summarized by the following statement —we do not
have the complete primary (third-order) aberration coefficients for anamorphic systems
that would provide necessary design insight, except for the simplest case [13].
In the work that is offered here, we will present a detailed study for the
monochromatic primary aberrations in anamorphic systems. This study will be developed
13
with a theoretical structure similar to that of the RSOS. Even though the approach
method is different from the traditional wave aberration approach for RSOS, it can be
considered a generalization of RSOS.
This work will provide a clear understanding of anamorphic image formation with
first-order optics and the primary (third-order) aberration features for anamorphic
systems. And we will present the primary aberration coefficient expressions for the most
common types of anamorphic systems, in a form similar to the famous Seidel aberrations
of RSOS.
1.1 What is an anamorphic system?
An anamorphic system is an imaging system contains double curvature surfaces
which have two mutually perpendicular planes of symmetry. By a double curvature
surface we mean a surface which has different radii of curvature in two perpendicular
cross sections. Figure 1-1 shows an example of a double curvature surface.
Figure 1-1 Example of a double curvature surface
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By keeping the two symmetry planes mutually perpendicular to each other
throughout the system, an anamorphic system will have double plane symmetry. These
two symmetry planes are also called the principal sections of the anamorphic system.
Because the optical power is tied to the curvatures, an anamorphic system has
different optical powers in each principal section, thus forming an anamorphic image.
The effective focal lengths in both principal sections will determine the anamorphic ratio,
which is the ratio of the two magnifications, each associated with one principal section.
A common example of an anamorphic imaging system is a system made from cross
cylindrical lenses which can map a square object field into a rectangular image field.
Figure 1-2 shows an example of this configuration.
Figure 1-2 An example of anamorphic systems--a system made from cylindrical lenses
1.2 Why is it a difficult problem?
Anamorphic systems are different from RSOS from the first-order optics sense. If a
ray starts in one of the symmetry planes (principal sections) of an anamorphic system,
namely the x-z plane or the y-z plane, it will always stay in this plane as it is traced
through the system. For any ray not lying in one of the symmetry planes, it will be a skew
ray and will not be contained in any single plane as it is traced through the system.
15
Due to the double curvature nature of the elements in an anamorphic system, we do
not have unique object and image points in each intermediate space. Instead, we have one
set of intermediate object and image points associated with the x-z symmetry plane and
another set of points associated with the y-z symmetry plane. Thus, a single double
curvature surface is not an imaging system because it can not form a single image point
for an on-axis object point. As a result, we intrinsically have constant astigmatism for any
single double curvature surface in the system, regardless of where we might locate our
observation plane (see Figure 1-3 below).
Figure 1-3 Constant astigmatism for a single double curvature surface
Even more, we have two sets of intermediate paraxial object and image planes
floating in space, each associated with one symmetry plane. In the final image space, we
will let the two image planes coincide with each other to complete the imaging formation.
In other words, an anamorphic imaging system will generally be constrained to have
16
unique object and image planes in the object and final image space, but will not in the
intermediate spaces.
The same is true for pupils—generally we do not have a unique entrance pupil and
exit pupil in every intermediate space, except in the space where the stop is located.
Instead, we will have one set of unique pupils for each symmetry plane.
Due to these features, when we discuss optical path difference error (OPD) or ray
error, in each intermediate space, it is not clear which image point we are referring the
errors to. In the calculation of the OPD, in each intermediate space, which Gaussian
image point should be used to center our reference sphere on? And since generally we do
not have one unique exit pupil in the final image space if the system stop is not located in
this space, which coordinates are we using while we write out the wave aberration
function ? These difficulties might explain why more than 150 years have passed
since Seidel’s first description of his five Seidel aberrations, yet nobody has provided a
similar set of complete primary aberration coefficients for general anamorphic systems,
except in the simplest case [13] of a parallel-cylindrical anamorphic attachment system.
( , )W x y
1.3 Historical background
Part of the first-order theory of anamorphic systems was developed by Ernst Abbe at
the end of nineteenth century [19]. The research was continued by George J. Burch
(1904) [11], H. Chretien (1929) [12], C. G. Wynne (1954) [13], H. Kohler (1956) [15], K.
Bruder (1960) [16] and G. G. Slyusarev (1984) [18], who studied the aberrations of this
kind of system. Other authors have studied the general aberration features of anamorphic
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systems using the symmetry theory. For example, J. C. Burfoot (1954) [14], R. Barakat
and A. Houston (1966) [20], H. A. Buchdahl (1970) [1], P. J. Sands (1972) [3], and T. A.
Kuz'mina (1974) [21] considered the general aberration theory of double plane
symmetrical systems.
Among all the works on anamorphic system research mentioned above, special
attention is paid to C. G. Wynne’s paper "The Primary Aberrations of Anamorphic Lens
Systems" [13], published in 1954. In this paper, Wynne applied a modified traditional
trigonometric calculation method onto one of the simplest anamorphic systems, an
anamorphic attachment system composed of parallel-cylindrical lenses (with further
restriction on location of the system stop in the object space or final image space), and he
obtained the 16 primary aberration coefficient expressions for this attachment system, in
a form similar to the Seidel aberrations in RSOS. Other than this simplest case, K. Bruder
and G. G. Slyusarev [16,18] applied an angular eikonal method onto toroidal anamorphic
systems, but their development lack adequate theoretical structure, and accordingly the
results obtained are incomplete. Prior to the current work, the anamorphic primary
aberration theory remains a challenge for geometrical optics research.
From an optical design perspective, the primary aberration coefficient expressions
are extremely important in understanding the corresponding optical system because they
will help answer the following important questions:
1) What are the major errors in the system?
2) More importantly, what are the functional dependences of these errors so that we can
correct them?
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Additionally, we want all parameters in the primary aberration coefficient
expressions are first-order quantities so that the coefficients can be easily calculated. In
other words, we want the expressions are in a form similar to the Seidel aberrations.
The fact that we do not have a complete primary aberration theory for anamorphic
systems greatly limited our abilities to research the capacity of anamorphic image
formation. It also prevented us from obtaining insights in anamorphic imaging system
design research. The current work is dedicated to this important research area and will
address many questions arising from the above issues.
1.4 Dissertation content
This work will derive the complete monochromatic primary aberration coefficient
expressions for most anamorphic system types with double plane symmetry.
In chapter 2, we will explain the first-order theory needed for the development–these
will be the building blocks for the whole work.
Chapter 3 will be devoted to explicating the general aberration theory and the
aberration function model. In addition, it will identify which primary aberrations are
allowed for anamorphic systems.
In chapter 4, a general method used to find the primary aberration coefficients for
anamorphic systems will be built up.
Chapters 5 and 6 will be focusing on the most commonly used anamorphic systems:
systems made from cylindrical lenses. In chapter 5, the primary aberration coefficient
expressions for parallel cylindrical anamorphic attachment systems will be presented, and
19
in chapter 6, the primary aberration coefficient expressions for cross cylindrical
anamorphic systems will be presented.
In chapter 7, the primary aberration coefficient expressions for toroidal anamorphic
systems will be presented. The primary aberration coefficient expressions for anamorphic
systems made from general double curvature surfaces, including fourth-order aspheric
departures, will be discussed in chapter 8.
In chapter 9, we will provide a testing scheme for the results obtained from chapters
5 through 8. In chapter 10, several design examples will be provided.
The conclusions and suggestions for future work will be given in chapter 11.
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CHAPTER 2
FIRST-ORDER THEORY FOR ANAMORPHIC SYSTEMS
The first-order theory is fundamental to understanding anamorphic systems. This
chapter will present the direction cosines of a ray and their paraxial approximations in
section 2.1; the three dimensional ray transfer and refraction equations in section 2.2; the
general theory of double curvature surfaces in section 2.3; the paraxial approximations of
double curvature surfaces in section 2.4; the ideal (first-order) image model for
anamorphic imaging in section 2.5; the paraxial ray tracing equations for anamorphic
systems in section 2.6; and the paraxial imaging properties of anamorphic systems in
sections 2.7 through 2.11.
2.1 Direction cosines of a ray and their paraxial approximations
The most convenient way to specify a ray, which is a straight line in homogenous
media, will be its direction cosines. Suppose that we have a system of Cartesian
coordinates with origin O as shown in Figure 2-1 below. Through the point O, we can
draw a line parallel to the ray whose direction is to be specified. Let us choose an
arbitrary point P on this parallel line through O, and project line OP onto the x, y, and z
axes, at points A, B, and C respectively. Then the three direction cosines components
of the ray to be specified are defined by ( , , )L M N
cos( ) ; cos( ) ; cos( )OA OB OCL AOP M BOP N COPOP OP OP
= ∠ = = ∠ = = ∠ = . (2-1)
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Figure 2-1 Direction cosines of a ray
Notice that of the three direction cosines components ( , of any ray, there are
only two components which are independent of each other, since we have the relationship
, )L M N
2 2 2 1L M N+ + = . (2-2)
From equation (2-2), we have
. (2-3) 2 2 1[1 ( )]N L M= − + / 2
Suppose axis z is the optical axis, and also suppose the ray to be specified lies in the
paraxial region close to the z axis. The domain of paraxial optics or first-order optics is
defined to be close enough to the optical axis to ensure that the ray angle and height
( ) are small quantities by first-order standards, whose squares and cross-
products are negligibly small for all surfaces
, , ,L M x y
j of the anamorphic optical system [22].
In the paraxial region, since and L M are small, we can expand equation (2-3) as a
binomial series
2 2( )1
2L MN + ...= − + . (2-4)
22
To the first-order approximation, the quadratic terms in equation (2-4) can be ignored,
which shows andOC . 1N = OP=
Thus, in the paraxial region, equation (2-1) becomes
' 'cos( ) tan( ') ,
" "cos( ) tan( ") ,
1,
x
y
CP CPL AOP COP uOP OCCP CPM BOP COP uOP OC
N
⎧ = ∠ = = = =⎪⎪⎪ = ∠ = = = =⎨⎪
=⎪⎪⎩
(2-5)
here ,x yu u are tangent of the angles of the projections , made with the optical
axis z, respectively.
'OP ''OP
From the discussion above, we can draw the following conclusion: In the paraxial
region, the direction cosines of any ray are equal to the tangent of the angles formed by
the z axis and the projections of the ray in the respective x-z and y-z sections.
2.2 Three dimensional ray transfer and refraction equations
In optics, to trace an arbitrary ray through a system, we need two basic three
dimensional equations: The transfer equation and the refraction equation.
The transfer equation for a ray from a surface 1j − to the following surface j is
1 1 1
1 1 1
1j j j j j j
j j j
jx x y y z z tL M N
− − −
− − −
−− − − += = , (2-6)
here ( 1 1, , 1j j jx y z− − − ), ( , , )j j jx y z are the points where the ray intersects the previous
surface 1j − and the next surface j , respectively. 1jt − is the on-axis distance between the
23
two surface vertices, and ( 1 1, , 1j j jL M N− − − ) are the ray direction cosines between the two
surfaces, as shown in Figure 2-2 below.
Figure 2-2 Three dimensional ray tracing
The three dimensional ray refraction equations are derived from Snell’s law [2].
Snell’s law states that the incident and refracted rays are coplanar with the surface normal
at the point of incidence, and they are related by
' sin ' sinn I n I= , (2-7)
here are the refractive indexes of the material before and after the refraction surface,
and
, 'n n
, 'I I are the angles that the incident and the refracted rays make with the surface
normal.
Snell’s law can be written in vector form as
'( ' ) ( )n n× = ×r n r n , (2-8)
here are unit vectors along the incident and refracted rays and n is the unit vector
along the normal of the surface at the point of incidence. By vectorially multiply on
both sides of Snell’s law, we get
r,r'
n
'[ ( ' )] [ ( )]n n× × = × ×n r n n r n . (2-9)
24
Using the vector identity ( ) ( ) ( )× × = ⋅ − ⋅a b c b a c c a b we can rewrite equation (2-9)
as
'[ ' ( ' )] [ ( )]n n− ⋅ = − ⋅r n r n r n r n , (2-10)
this can be expanded in scalar form by setting ( , ,L M N ), ( ) and (', ', 'L M N , ,α β γ ) as
the components of and respectively, so that these quantities are direction cosines,
giving
, 'r r n
' ' ,' '' ' ,
n L nL kn M nM kn N nN k
α,β
γ
− =⎧⎪ − =⎨⎪ − =⎩
(2-11a)
where . '( ' ) ( ) 'cos ' cosk n n n I n I= ⋅ − ⋅ = −r n r n
It is often convenient to introduce the refraction operator , which signifies
refraction of the quantity operated upon, i.e.
Δ
( ) ' 'x xnu n u nuxΔ = − . If the quantity that
follows is a constant on refraction, then we will get zero. Taking the Lagrange
invariant as a common example, we will have
Δ
Ψ ( ) ' 0Δ Ψ = Ψ −Ψ = because Ψ is a
constant on refraction.
By using the refraction operator, equation (2-11a) can be rewritten as
( ) ,( )( ) .
nL knM knN k
,αβγ
Δ =⎧⎪Δ =⎨⎪Δ =⎩
(2-11b)
Suppose now the ray is refracted by surface j . Using equations (2-11a) and (2-11b),
and also noticing that 1' j jn n− = , 1' j jI I− = and ( 1 1' , ' , ' 1j jL M N j− − − ) = ( , ,j jL M N j ), we can
write the ray refraction equation on surface j as
25
1 1 1 1 1 1
1 1cos cos
j j j j j j j j j j j jj
j j j
j j j j j
n L n L n M n M n N n Nk
k n I n I
α β γ− − − − − −
− −
− − −= = =
= −, (2-12)
here ( , ,j j jα β γ ) are the direction cosines of the thj surface normal. This is the ray
refraction formula we are going to use throughout this work, together with the ray
transfer equation (2-6).
2.3 Double curvature surface types and their surface normal—general theory
To form an anamorphic image, we need refractive (or reflective) surfaces with
double curvature inside our imaging systems. Additionally we need the double curvature
surfaces to be so aligned that the symmetry planes of the surfaces coincide with x-z and
y-z planes. This ensures our optical system will possess double plane symmetry, and we
can thus achieve different magnifications in the x-z and y-z symmetry planes. The optical
axis will be chosen as the line of intersection of the symmetry planes.
Now let us list several examples of existing double curvature surface types [23] so
that we can examine the basic concept of this kind of non-rotationally-symmetrical
surfaces.
From a mathematical point of view, the simplest double curvature surface type might
be an elliptical paraboloid with surface sag z expressed in Cartesian coordinates as
2 21 (
2)
x y
x yzr r
= + , (2-13)
26
where xr and yr are the principal radii of curvatures in the x-z and y-z symmetry planes,
respectively. Cross sections of this surface parallel to the x-y plane are ellipses, and cross
sections perpendicular to that plane are parabolic.
From a fabrication point of view, the simplest double curvature surface type might be
a toroidal surface, described as
1 1
2 2 2 22[( ( ) ) ]x x y yz r r r r y x= − − + − − 2 , (2-14)
Again, xr and yr are the principal radii of curvature of the symmetry planes. The sag of
the toroidal surface to the fourth-order approximation is
2 2 4 2 2 4
3 21 1 2( ) (2 8x y x x y y
3 )x y x x y yzr r r r r r
= + + + + . (2-15a)
Notice that the most widely used double curvature surface type in current anamorphic
system design is the cylindrical surface, which is a special case of toroidal surfaces,
where one principal radius of curvature is equal to infinity. The surface sag equation for
cylindrical surfaces up to the fourth-order approximation is
2 4
3
1 12 8x x
x xzr r
= + . (2-15b)
Or,
2 4
31 12 8y y
yzr r
= +y . (2-15c)
Depending on which principal radius equals infinity.
From an optical testing perspective, the simplest double curvature surface type might
be the ellipsoid, as described by
27
2 2 2
2 2 1x x y y
x y zr r r r
+ + = . (2-16)
The sag equation of this surface described for an oblate section and to the fourth-order
approximation is
2 2 4 2 2 4
3 2 21 1 2( ) (2 8x y x x y x y
)x y x x y yzr r r r r r r
= + + + + . (2-17)
From the above examples of existing double curvature surface types, we can see that
for this kids of surface, the surface sag equation can generally be written up to the fourth-
order approximation in a form
2 2 4 2 2 4
3 33 4 5
1 1 2( ) (2 8x y
3 )x y x x yzr r r r r
= + + + +y , (2-18)
here ,x yr r are the radii of curvature of the surface in the two principal sections, and
are certain coefficients with dimension of length. Notice that in this equation, we
permit the fourth-order aspheric departure in both principal sections by allowing
being different from
3 4 5, ,r r r
3 4 5, ,r r r
,x yr r .
We should notice that when the principal sections are in the form of circles (e.g. a
toroidal surface), we have 3 xr r= , 5 yr r= and . When , we
return to the special case of a spherical surface. Thus from equation (2-18), we can
achieve any double curvature surface types’ sag equation, up to the fourth-order
approximation, by properly choosing and .
2 1/ 34 ( )x yr r r= 3 4x yr r r r r= = = = 5
3 4, , ,x yr r r r 5r
28
Now with the general double curvature surface sag-equation complete, it is necessary
to have convenient expressions for the direction cosines of the surface normal at the point
of incidence in order that the general ray refraction equation (2-12) can be applied.
Let us write the surface equation as ( , , ) 0F x y z = , then for a neighboring point
( , ,x x y y z zδ δ+ + +δ
0
) which is also on this surface [2], we will have
( , , )F x x y y z zδ δ δ+ + + = . (2-19)
By Taylor’s expansion, we can rewrite equation (2-19) as
( , , ) ... 0
0.
F F FF x y z x y zx y z
F F Fx y zx y z
δ δ δ
δ δ δ
∂ ∂ ∂+ + + +∂ ∂ ∂
∂ ∂ ∂⇒ + +
∂ ∂ ∂
= (2-20)
From this equation we see that the vector ( , ,F F F )x y z
∂ ∂ ∂∂ ∂ ∂
is perpendicular to the vector
( , , )x y zδ δ δ , and since the latter is restricted to lie in the surface, the
vector ( , ,F F F )x y z
∂ ∂ ∂∂ ∂ ∂
must be a vector along the surface normal at point ( , ,x y z ).
Therefore, the direction cosines of the surface normal are
12 2 2 2
( , , )( , , )
{( ) ( ) ( ) }
F F Fx y z
F F Fx y z
α β γ
∂ ∂ ∂∂ ∂ ∂=
∂ ∂ ∂+ +
∂ ∂ ∂
. (2-21)
2.4 Double curvature surfaces and their surface normal—paraxial approximations
Let us now consider the paraxial approximation of the general double curvature
surface equation (2-18) and the direction cosines of its surface normal.
29
By ignoring terms higher than order three, we can rewrite the general double
curvature surface sag equation (2-18) as
2 21 (
2)
x y
x yzr r
= + . (2-22)
This equation shows that z is a quantity of the second order in ,x y which can be
neglected under the first-order approximation, so in the paraxial region, we have
0z = . (2-23)
This means in the paraxial region, surface sag can be ignored.
However, we need to remember that even though the surface sag can be taken as zero
in the paraxial domain, the surface still has curvature so that it has power to refract the
rays, thus we need to use equation (2-22) in the direction cosines of surface normal
calculation.
To find the direction cosines of surface normal in the paraxial region, we can rewrite
equation (2-22) as
2 21( , , ) ( ) 0
2 x y
x yF x y z zr r
= − + ≡ . (2-24)
By putting the second-order surface equation (2-24) into equation (2-21) and
ignoring any terms higher than order one in x and , we find the normal of the general
double curvature surface in the paraxial region as
y
(2-25) ,,
1,
x
y
c xc y
αβ
γ
⎧ = −⎪ = −⎨⎪ =⎩
30
here 1/x xc r= , 1/y yc = r are the curvatures in both principal sections.
2.5 Ideal (first-order) image model for anamorphic imaging
Now let us see what kind of image the anamorphic system can provide. To do so we
will define an image model for the ideal behavior of anamorphic systems. This model
will be important because it will simplify the description of such systems by establishing
a reference. The model should be in accord with how an anamorphic system is ideally
meant to perform, and should be simple enough to provide insight.
A geometrical model for optical imagery that meets the above criteria would account
for the main features of the image. The fine details that are the departures from ideal
behavior can be described simply by a function depending on the aperture (stop) and field
(object) variables. This function, called the aberration function, is represented as a Taylor
series, and each term in the series represents a particular type of departure from ideal
behavior called an aberration. We will describe this function in Chapter 3.
To build our ideal imaging model, we will follow Abbe’s collinear mapping [22,24]
between two spaces: The object and the image space. The collinear mapping has the
following properties
1) Every object point will be mapped to a unique image point;
2) Every object plane will be mapped to a unique image plane.
From 1) and 2), we come to the conclusion that every object line will be mapped into a
unique image line. This result follows from the fact that a straight line is generated by the
intersection of two planes.
31
Since imaging systems almost perform these functions, we will assume that their
behavior, including that of anamorphic systems, can be described by a collinear mapping.
The expressions that relate a point P ( , ,x y z ) in the object space to a point
('P ', ', 'x y z ) in the image space on a collinear mapping are
0000
1111'dzcybxadzcybxax
++++++
= , (2-26a)
0000
2222'dzcybxadzcybxay
++++++
= , (2-26b)
0000
3333'dzcybxadzcybxaz
++++++
= . (2-26c)
The spatial variables , ,x y z and ', ', 'x y z are positions in a Cartesian coordinate system
and the parameters are constants. Primed quantities refer to image space
quantities.
, , ,a b c d
Since an anamorphic system has double plane symmetry, we choose the z and 'z
axes to lie along the line of intersection of the symmetry planes and to act as our optical
axis, thus every surface will be centered about it. We also let the 'x axis be parallel to the
x axis and the axis to be parallel to the axis. In this manner, we can compose our
coordinate systems in both spaces. The only thing that has not been decided yet is the
location of the coordinate origins, and this will be discussed in Chapter 3.
'y y
We can now use the double plane symmetry condition to simplify the above collinear
mapping equations. Considering equation (2-26a), from the double plane symmetry
requirement, 'x remains unchanged when x stays the same but y changes in sign. This is
32
true for all values of x and y, as long as 0 1 0b b= = . Also 'x must change sign when x
changes sign because of the double plane symmetry, hence 0 1 1 0a c d= = = . For equation
(2-26b), similarly we can show that of necessity 0 0 2 2 2 0a b a c d= = = = = . From equation
(2-26c), we notice that because of symmetry 'z does not change when x and y change
sign ( 'z is independent of x and y), and this requires that 0 0 3 3 0a b a b= = = = . Taking all
these requirements into account, equations (2-26) reduce to
1
0 0
' a xxc z d
=+
, (2-27a)
2
0 0
' b yyc z d
=+
, (2-27b)
3
0 0
' c z dzc z d
3+=
+. (2-27c)
By factoring out the constant and redefining the constants, we get 0d
1
0
2
0
3 3
0
' ,1
'1
' .1
a xxc z
b yyc zc z dzc z
⎧=⎪ +⎪
⎪=⎨ +⎪
⎪ +=⎪
,
+⎩
(2-28)
Equation (2-28) is the ideal (first-order) image model for anamorphic systems. From this
model, we know unless = , the imaging is anamorphic. The magnification in the x-
direction is given by
1a 2b
1
0 1xam
c z=
+, (2-29a)
33
and the magnification in the y-direction is given by
2
0 1ybm
c z=
+. (2-29b)
The ratio of the two magnifications is often called the anamorphic ratio of the system.
So we have shown that anamorphic systems can indeed form anamorphic images
while satisfying the collinear mapping condition. We should remember that in the above
simplification, we did not put any restriction on where to locate the coordinate origins in
both spaces, and the origins in both spaces are not necessarily conjugate to each other.
The general feature of the ideal image model developed in this section is to allow
point to point mapping yet with an anamorphic image.
2.6 The paraxial ray tracing equations for anamorphic systems
From the discussions in the above sections, we know the passage of a ray through an
anamorphic system is governed by the refraction equation (2-12), the transfer equation
(2-6) and the surface equation (2-18). For any surface j in the anamorphic system under
consideration, in summary, we have
1 1 1
1 1 1
j j j j j j j
j j j
1x x y y z z tL M N
− − −
− − −
−− − − += = , (2-30a)
1 1 1 1 1 1j j j j j j j j j j j j
j j
n L n L n M n M n N n Nα β γ
− − − − − −− − −= =
j
, (2-30b)
2 2 4 2 2 4
3 3, , 3, 4, 5,
21 1( ) (2 8
j j j j j jj
x j y j j j j
x y x x y yz
r r r r r= + + + + 3 ). (2-30c)
From equations (2-5), (2-23) and (2-25), we know in the paraxial region,
34
, ,
, ,
1
( , , ) ( , ,1),
( , , ) ( , ,1),
0.
j j j x j y j
j j j x j j y j j
j j
L M N u u
c x c y
z zα β γ
−
⎧ =⎪
= − −⎨⎪ = =⎩
Thus in this region, equations (2-30a)-(2-30b) can be rewritten as
1 11
, 1 , 1
j j j jj
x j y j
x x y yt
u u− −
−− −
− −= = , (2-31a)
, 1 , 1 , 1 , 11
, ,
(j x j j x j j y j j y jj j
x j j y j j
n u n u n u n un n
c x c y− − − −
−
− −= = − )− . (2-31b)
Equations (2-31) tell us an important fact— in the paraxial region surrounding the
axis of an anamorphic system, the ( , xx u ) and ( , yy u ) components of any paraxial ray
traced through the system are independent of one another, and each component can be
considered as if it is an independent paraxial ray traced in the x-z symmetry plane or y-z
symmetry plane of the system alone.
The conclusion is that in the paraxial region, rays can be traced by projecting into
two planes of symmetry, while the path of the project is governed entirely by the normal
law of paraxial ray tracing and the paraxial curvature ,x yc c in both symmetry planes.
To clearly emphasize this, we can write equations (2-31a)-(2-31b) into the
independent ray trace equations separately. For ( , xx u ) components of this paraxial ray,
we have
1 1 , 1j j j x jx x t u− − −− = , (2-32a)
, 1 , 1 1( ) ,j x j j x j j j j x jn u n u x n n c− − −− = − − . (2-32b)
35
Notice that these two equations are in exactly the same form of the paraxial meridian ray
tracing equations for a RSOS made from spherical surfaces 2 2, ( ) / 2 ...j x j j jz c x y= + + , in
which the x-z symmetry plane will be a meridian section. So we can imagine that we
have a RSOS associated with the x-z symmetry plane of the anamorphic system, and we
call it the associated x-RSOS.
For ( , yy u ) components of this paraxial ray, we have
1 1 , 1j j j y jy y t u− − −− = , (2-33a)
, 1 , 1 1( ) ,j y j j y j j j j y jn u n u y n n c− − −− = − − . (2-33b)
We see that these two equations are in exactly the same form of the paraxial meridian ray
tracing equations for another RSOS made from spherical surfaces
, in which the y-z symmetry plane will be a meridian section. So we can imagine that
we have another RSOS associated with the y-z symmetry plane of the anamorphic system,
and we can call it the associated y-RSOS.
2 2, ( )j y j j jz c x y= + / 2
...+
Equations (2-32) and (2-33) are the basic paraxial ray tracing equations for an
anamorphic system.
2.7 Paraxial image properties of anamorphic systems---part one
From the discussion in section 2.6, since for any arbitrary paraxial ray (either skewed
or non-skewed) in an anamorphic system, its ( ,,j x jx u ) component and ( ,,j y jy u )
component are completely independent of each other and are traced through the system
by projecting into the x-z and y-z symmetry planes according to their own ray trace
36
equations (2-32) and (2-33) , we arrive at a very important conclusion: whenever we are
working with an anamorphic paraxial ray’s ( , xx u ) component, we can imagine that we
are dealing with the projection of this paraxial ray in the x-z symmetry plane. This
projection can be further imagined as a paraxial ray, staying in the x-z meridian plane of
the associated x-RSOS. Thus, all results from Gaussian optics for the associated x-RSOS,
can be applied to the ( , xx u ) component of this anamorphic paraxial ray directly except
that every quantity will now have a subscript x , including the x-paraxial object plane
location xl , the x-paraxial entrance pupil location xl , the x-paraxial marginal ray angle xu
and height xh , the x-paraxial chief ray angle xu and height xh , etc [2]. Figure 2-3 shows
these quantities in an intermediate space.
Figure 2-3 Gaussian optics properties for the associated x-RSOS (in an intermediate space)
Similarly, whenever we are dealing with the ( , yy u ) component of the same
anamorphic paraxial ray, we can imagine that we are dealing with the projection of this
37
paraxial ray in the y-z symmetry plane of our anamorphic system. Again, this projection
can be further imagined as a paraxial ray, staying in the y-z meridian plane of the
associated y-RSOS. Thus all results from Gaussian optics of the associated y-RSOS can
be applied to ( , yy u ) component of this anamorphic paraxial ray directly except every
quantity will now have a lower subscript . y
2.8 Paraxial image properties of anamorphic systems---part two
It is well know that in RSOS there are only two independent paraxial rays. Normally
they are taken to be the marginal ray and chief ray, and any third paraxial ray can be
written as a linear combination of these two [22,25].
Similarly, in an anamorphic system, from equations (2-32) and (2-33), we can prove
there are only two linearly independent paraxial skew rays too.
To prove this, suppose we have two known paraxial skew rays with components on
surface j as ( 1, 1 ,,j x jx u ), ( 1, 1 ,,j y jy u ) and ( 2, 2 ,,j x jx u ), ( 2, 2 ,,j y jy u ). The paths of these two
skew rays through the system have been completely determined by equations (2-32) and
(2-33). Suppose that we also have a third unknown paraxial ray, for which we denote the
relevant components as ,( , )j x jx u , ,( , )j y jy u on surface j .
Suppose we can write the third unknown paraxial ray’s ,( , )j x jx u component as
combinations of the two known paraxial rays’ ( , xx u ) components in the form
, 1, , 2,j x j j x j jx C x D x= + , (2-34a)
, , 1 , , 2 ,x j x j x j x j xu C u D u j= + , (2-34b)
38
here , , ,x j xC D j are proportional constants on surface j and we can obtain the values of
them by solving these equations.
If we can prove that , , ,x j xC D j have no dependence on surface number j and are
constants throughout the anamorphic system, then we know for this third unknown
paraxial ray, its ( , )xx u component can always be expressed as a linear combination of
the two known paraxial skew rays. And if the same thing is true for its ( , )yy u component,
we immediately know the third ray can not be independent to the two known paraxial
rays from linear algebra theory. By this way, we can prove that there are only two
linearly independent paraxial skew rays in any anamorphic system.
To prove it, let us rewrite paraxial ray trace equations (2-32a), (2-32b) into the
following form
1 , 1 1j j x j jx x u t− − −= + , (2-35a)
, 1 , 1 1( ) ,j x j j x j j j j x jn u n u x n n c− − −= − − . (2-35b)
Equations (2-35) hold for ,( , )j x jx u also because the third ray is also a paraxial ray in our
anamorphic system, thus by substituting equations (2-34) into equations (2-35), we have
1 , , 1, , 2, , 1 , , 2 ,
, 1, 1 , , 2, 2 ,
, 1, 1 , 2, 1
( ) (
( ) ( )
,
j j x j j x j j x j j x j x j x j x j j
x j j x j j x j j x j j
x j j x j j
)x x u t C x D x C u D u t
C x u t D x u t
C x D x
+
+ +
= + = + + +
= + + +
= +
1 , 1 , 1 1 , 1
, 1 , , 2 , , 1, 1 , 2, 1 1 , 1
, 1 , 1, 1 1 , 1 , 2 , 2, 1 1 ,
, 1 1 , 1 , 1 2
( )( ) ( )( )
[ ( ) ] [ ( ) ]
j x j j x j j j j x j
j x j x j x j x j x j j x j j j j x j
x j j x j j j j x j x j j x j j j j x j
x j j x j x j j x
n u n u x n n cn C u D u C x D x n n cC n u x n n c D n u x n n cC n u D n u
+ + + + +
+ + + +
1+ + + + +
+ + +
= − −
= + − + −
= − − + − −
= + , 1j+
+
39
1 , 1 , 1 , 2 , 1( )j x j x j x j x jn C u D u+ += + .+
Thus we find
1 , 1, 1 , 2, 1j x j j x j jx C x D x+ + += + , (2-36a)
, 1 , 1 , 1 , 2 , 1x j x j x j x j xu C u D u j+ + += + . (2-36b)
Notice that the same process can be continued to the next surface and so on.
By comparing equations (2-36) with equations (2-34), we see the proportionality
constants , , ,x j xC D j do not change with the passage of the third paraxial ray through the
anamorphic system. Hence they are indeed constant throughout the anamorphic system
and we will denote them as ,x xC D because their values do not depend on surface
number j .
Thus, we have proved that the third ray’s ,( , )j x jx u component can be written as a
liner combination of the two known paraxial skew rays’ ( 1, 1 ,,j x jx u ), ( 2, 2 ,,j x jx u )
components for any surface number j in the anamorphic system.
Exactly the same way, we can prove that we can always write the unknown third
paraxial ray’s ( ,j jy u ) component as a linearly combination of the two known paraxial
skew rays’ ( 1, 1 ,,j y jy u ), ( 2, 2 ,,j y jy u ) components on surface j , as
1, 2,j y j y jy C y D y= + , (2-36c)
, 1 , 2 ,y j y y j y yu C u D u j= + . (2-36d)
Again, here ,y yC D are proportional constants and they are constants throughout the
system.
40
These four equations in (2-36) show there are only two linearly independent paraxial
skew rays in an anamorphic system, all other paraxial rays can be expressed as linear
combinations of these two known paraxial rays, with different proportionality
constants , , ,x x y yC D C D .
In practice, the two known paraxial skew rays are often taken as a paraxial skew
marginal ray (which comes from the on-axis object point and passes through a point on
the edge of the system stop) and a paraxial skew chief ray (which comes from a point on
the maximum object field and passes through the center of the stop). Once we know these
two rays, we can use their different linear combinations to form all other paraxial rays in
the anamorphic system.
2.9 Paraxial image properties of anamorphic systems---part three
In section 2.8, we obtained the linear combination relationships between any third
paraxial rays with the two known paraxial skew rays in an anamorphic system. There also
exists particular relationships between the two known paraxial rays—the anamorphic
Lagrange invariants [1], similar to the Lagrange invariant relationship in RSOS [2,22].
For the two known linearly independent paraxial skew rays’ ( ,,j x jx u ) components,
from equations (2-35), we have
1, 1, 1 1 , 1 1j j x j jx x u t− − −= + , (2-37a)
1 , 1 1 , 1 1, 1 ,( )j x j j x j j j j x jn u n u x n n c− − −= − − , (2-37b)
2, 2, 1 2 , 1 1j j x j jx x u t− − −= + , (2-37c)
41
2 , 1 2 , 1 2, 1 ,( )j x j j x j j j j x jn u n u x n n c− − −= − − . (2-37d)
From equations (2-37), we have
2 , 1, 1 , 2, 2 , 1, 1 , 2, 1, 2, 1 , 1, 2, 1 ,
2 , 2, 1 , 1, 1 , 1, 1 , 2,
1 2 , 1 1, 1 1 , 1 2,
1 2 , 1
( ) ( )
[ ( ) ] [ ( ) ]
(
j x j j j x j j j x j j j x j j j j j j x j j j j j x j
j x j j j j x j j j x j j j j x j j
j x j j j x j j
j x j
n u x n u x n u x n u x x x n n c x x n n c
n u x n n c x n u x n n c xn u x n u x
n u
− −
− −
− − − −
− −
− = − + − − −
= + − − + −
= −
= 1, 1 1 , 1 1 1 1 , 1 2, 1 2 , 1 1
1 2 , 1 1, 1 1 1 , 1 2, 1
) ( )
...tan .
j x j j j x j j x j j
j x j j j x j j
x
x u t n u x u t
n u x n u x
cons t λ
− − − − − − − −
− − − − − −
+ − +
= −
== =
Hence for all surfaces, we have
1 2 2 1( ) tanx xn x u x u cons t xλ− = = . (2-38a)
Equation (2-38a) shows the connection between the projections of the two known
paraxial skew rays in x-z symmetry plane of the anamorphic system, and it is very similar
to the Lagrange invariant relationship in the associated x-RSOS.
Exactly the same way, we can find
2 , 1, 1 , 2, 1 2 , 1 1, 1 1 1 , 1 2, 1
...tan .
j y j j j y j j j y j j j y j j
y
n u y n u y n u y n u y
cons t λ
− − − − − −− = −
== =
Hence for all surfaces, we also have
1 2 2 1( ) tany yn y u y u cons t yλ− = = . (2-38b)
Equation (2-38b) shows the connection between the projections of the two known
paraxial skew rays in y-z symmetry plane, and it is very similar to the Lagrange invariant
relationship in the associated y-RSOS.
42
When the constants ,x yλ λ are at their maximum possible value, we will replace them
as ,x yΨ Ψ (the Lagrange invariants associated with the two associated RSOS), called the
x-Lagrange invariant and the y-Lagrange invariant, respectively, which differ from ,x yλ λ
by some proportional constants.
Before we go to the next step, it might be an appropriate time to summarize the
difference between the paraxial optics for RSOS and for anamorphic systems.
For RSOS, all possible paraxial marginal rays from an on-axis object point are the
same because of the rotational symmetry, thus we can take the paraxial marginal ray as
the one lying in the meridian plane (the plane containing the optical axis and the object
point). Similarly, we can take the paraxial chief ray as the one staying in the meridian
plane also, thus RSOS ray tracing can be reduced to ray tracing in the meridian plane.
But for anamorphic systems, in general, if a paraxial marginal ray does not stay in
one of the symmetry planes, it will be a skew ray whose passage through the system will
not be confined in any single plane. Similarly, a paraxial chief ray will generally be a
skew ray unless the object point stays in one of the symmetry planes. Because of these
complications, we cannot reduce the anamorphic paraxial ray tracing into a ray trace
within a single meridian plane. Instead, we need to trace a skew paraxial marginal ray
and a skew paraxial chief ray in order to fully specify the paraxial anamorphic system.
In practice, it is not so convenient to fully specify the paraxial anamorphic system
using two skew paraxial rays. Thus we need to go a step further.
From the discussion in section 2.6, we know for the skew paraxial marginal ray or
chief ray, the ray tracing can be done by projecting onto the x-z symmetry plane and the
43
y-z symmetry plane of the anamorphic system. These projections can be considered as
independent paraxial rays traced in the associated x-RSOS and y-RSOS.
In each RSOS, we know there are only two independent paraxial rays [22,25],
normally they are taken to be the non-skew marginal ray and chief ray traced in the
meridian plane, and any third paraxial ray can be written as a linear combination of these
two.
Because each projection of the skew paraxial marginal ray or chief ray in the
corresponding symmetry plane can be expressed by the non-skew marginal ray and chief
ray in the corresponding associated RSOS, the skew paraxial ray can be fully specified by
four non-skew paraxial rays, namely the x-marginal ray, the x-chief ray, the y-marginal
ray, and the y-chief ray, in the corresponding x-RSOS and y-RSOS.
So we can trace the x-marginal ray and x-chief ray in the x-z symmetry plane and the
y-marginal ray and y-chief ray in the y-z symmetry plane, and then use all four non-skew
paraxial rays to fully specify the paraxial anamorphic system.
2.10 Paraxial image properties of anamorphic systems---part four
From the discussion in section 2.9, we know that even though there are only two
linearly independent paraxial skew rays in an anamorphic system and any other paraxial
ray can be written as a linear combination of these two rays, it is actually more
convenient to make use of four separately known non-skew paraxial rays—the paraxial
marginal and chief rays associated with the x-RSOS, traced in the x-z meridian plane;
44
and the paraxial marginal and chief rays associated with the y-RSOS, traced in the y-z
meridian plane.
When we are dealing with an arbitrary anamorphic paraxial (either skewed or non-
skewed) ray’s ( , xx u ) components, we will make use of the x-marginal and x-chief rays
which lie in the x-z symmetry plane of our anamorphic system. Similarly, whenever we
are dealing with the ( , yy u ) component of the same anamorphic paraxial ray, we will
make use of the y-marginal and y-chief rays which stay in the y-z symmetry plane of the
anamorphic system.
Let us write , ,( , )x j x jh u , , ,( , )x j x jh u as the parameters associated with the x-marginal
and x-chief ray in x-RSOS, and also write , ,( , )y j y jh u , , ,( , )y j y jh u as the parameters
associated with the y-marginal and y-chief rays in y-RSOS, on surface j . From equations
(2-36) and (2-38), we have
,j x x j x x , jx C h D h= + , (2-39a)
, , ,x j x x j x xu C u D u j= + , (2-39b)
,j y y j y yy C h D h= + , j , (2-39c)
, , ,y j y y j y yu C u D u j= + , (2-39d)
, , , ,( )j x j x j x j x j xn h u h u− = Ψ , (2-39e)
, , , ,(j y j y j y j y j yn h u h u )− = Ψ . (2-39f)
, , ,x y xC C D Dy are proportionality constants throughout the system, and they can be found
by the arbitrary anamorphic paraxial ray’s initial condition. xΨ is the x-Lagrange
45
invariant associated with the x-RSOS made from imaginary spherical
surfaces . 2 2, ( ) / 2 ...j x j j jz c x y= + + yΨ is the y-Lagrange invariant associated with the
corresponding y-RSOS made from imaginary spherical surfaces
.
2 2, ( )j y j j jz c x y= + / 2
...+
Now let us find the value of the proportionality constants , ,x yC C Dx and yD . Suppose
the arbitrary anamorphic paraxial ray cuts the paraxial object plane ( 0j = ) at point
( 0 0,ξ η ). Let be the distance from this object point to the optical axis,
and let the polar angle be
2 2 1/0 0(d ξ η= + 2)
φ . It then follows 0 0cos , sind dξ φ η φ= = , as shown in Figure
2-4 below. Suppose the maximum object field is , which is given by the radius of the
object point farthest from the origin in object coordinates, and we define the
quantity as the fractional field (object).
maxd
max/H d d=
Let this ray cut the system stop plane ( j p= ) at point ( ,p px y ), let
be the distance from this point to the optical axis, and let the polar angle be
2 2 1( )p pe x y= + / 2
θ . It then
follows that cos , sinp px e y eθ θ= = , as shown in Fig 2-4 below. Suppose the maximum
aperture is , and we define the quantitymaxe max/e eρ = as the fractional aperture (stop).
From equations (2-39), we know, at the paraxial object plane ( 0j = ), we have
0 ,0 ,0x x x xx C h D h= + .
In this plane, the x-marginal ray height ,0xh =0, the x-chief ray height ,0 maxxh d= and
0 0 cosx dξ φ= = . So we find that
46
0 ,0 max/ cos / cosx x xD h d d Hξ φ φ= = = = H ,
here cosxH H φ= is the projection of the fractional field in the x-z symmetry plane of
the anamorphic system, for this arbitrary paraxial ray.
H
Figure 2-4 The object and stop plane
At the system’s stop plane ( j p= ), we will have
, ,p x x p x x px C h D h= + .
In this plane, the x-marginal ray height , mx ph e ax= , the x-chief ray height ,x ph =0 and
cosp px x e θ= = . So we find that
, max/ cos / cosx p x p xC x h e eθ ρ θ ρ= = = = ,
here cosxρ ρ= θ is the projection of the fractional aperture ρ in the x-z symmetry plane,
for this ray.
Thus, we have found the proportionality constants ,x x xC D xHρ= = , which are
constants throughout the system. For any surface j , equations (2-39a)-(2-39b) become
, ,j x x j x x jx h H hρ= + , (2-40a)
47
, , ,x j x x j x xu u H u jρ= + . (2-40b)
Similarly, we can find the proportional constants ,y yC ρ= y yD H= . For any
surface j , equations (2-39c)-(2-39d) become
, ,j y y j y yy h H hρ= + j , (2-40c)
, , ,y j y y j y yu u H u jρ= + , (2-40d)
here sinyρ ρ= θ is the projection of the fractional aperture ρ in the y-z symmetry plane,
and sinyH H φ= is the projection of the fractional field in the y-z symmetry plane, for
this ray.
H
We recognize that ( ,x yH H ) and ( ,x yρ ρ ) are actually the normalized object and
aperture coordinates of this arbitrary anamorphic paraxial ray.
Equations (2-40) will serve as the foundation of primary aberration coefficients
derivation for anamorphic systems. These equations can be understood this way:
1) In an anamorphic system, the ray trace data of any anamorphic paraxial ray (either
skewed or non-skewed) can be composed by the linear combinations of the four known
non-skewed paraxial marginal and chief rays’ tracing data in the two associated RSOS.
2) Further more, the proportionality constants are the normalized object and stop
coordinates of this arbitrary anamorphic paraxial ray under study. Also notice that when
we are exploring the object and stop planes, these coordinates because variables.
Thus, all paraxial quantities in an anamorphic system can be written in terms of the
four known non-skewed paraxial rays’ tracing data in the two associated RSOS, together
with object and stop variables.
48
2.11 Paraxial image properties of anamorphic systems---part five
In the above sections, we have already composed the essential equations of the
paraxial optics for anamorphic systems. In this section, we are going to build up the
necessary paraxial definitions which will be extensively used in primary aberration
calculations throughout this work.
From equation (2-32b) we can get the refraction invariant of the x-marginal ray, for
the associated x-RSOS. To show this, we rewrite (2-32b) for the marginal ray as
1 , 1 1 , , , , , ,j x j j x j x j j x j j x j x j x jn u n h c n u n h c A− − −+ = + = . (2-41a)
Thus, we see that the quantity ,x jA is an invariant on refraction for the x-marginal ray, in
the associated x-RSOS, on surface j . Similarly, the refraction invariant for the x-chief
ray on surface j is
1 , 1 1 , , , , , ,j x j j x j x j j x j j x j x j x jn u n h c n u n h c A− − −+ = + = . (2-41b)
Similarly, from the equation (2-33b), we know the refraction invariants for the
associated y-RSOS, on surface j , are
1 , 1 1 , , , , , ,j y j j y j y j j u j j y j y j y jn u n h c n u n h c A− − −+ = + = , (2-41c)
1 , 1 1 , , , , , ,j y j j y j y j j y j j y j y j y jn u n h c n u n h c A− − −+ = + = . (2-41d)
By applying equations (2-41), from equations (2-39e)-(2-39f), the Lagrange
invariants associated with the two RSOS become
, , , , , , , ,( )x j x j x j x j x j x j x j x j x jn h u h u A h A hΨ = − = − , (2-42a)
, , , , , , , ,( )y j y j y j y j y j y j y j y j y jn h u h u A h A hΨ = − = − . (2-42b)
49
2.12 Summary
In section 2.1, we found that in the paraxial region, we have the direction
cosines . In section 2.2, we provided the three dimensional ray-
tracing equations. In section 2.3, we found that for any double curvature surface, its’ sag
equation can generally be written in a form as
, ,x yL u M u N= = 1=
2 2 4 2 2 4
3 3 33 4 5
1 1 2( ) ( ) ...2 8x y
x y x x y yzr r r r r
= + + + + + .
In section 2.4, we found that in the paraxial region, the direction cosines of the
surface normal for any double curvature surface are , ,x yc x c y 1α β γ= − = − =
) ;
. In section
2.5, we found from the collinear mapping point of view, anamorphic imaging with point-
to-point mapping is possible. In section 2.6, we found that in the paraxial region, the 3D
skew ray tracing equations for anamorphic systems can be replaced by the four 2D non-
skew ray-tracing equations in the two symmetry planes
1 1 , 1 , 1 , 1 1 ,, (j j j x j j x j j x j j j j x jx x t u n u n u x n n c− − − − − −− = − = − −
1 1 , 1 , 1 , 1 1 ,, ( )j j j y j j y j j y j j j j yy y t u n u n u y n n c− − − − − −− = − = − − j .
From section 2.7 to section 2.11, we found that we can think of a paraxial
anamorphic system as the two associated RSOS, thus all results we known for the two
RSOS can be applied to the anamorphic system directly. We also found that there are
only two independent paraxial skew rays in an anamorphic system, but we prefer using
the four non-skew marginal and chief rays traced in the associated x-RSOS and y-RSOS
to fully specify the system. We found that by using these four non-skew paraxial rays, we
can get all paraxial quantities associated with the anamorphic system.
50
All results built up in this chapter will be integrated into the anamorphic primary
aberrations development in chapters 3 through 8.
51
CHAPTER 3
GENERAL ABERRATION THEORY FOR ANAMORPHIC SYSTEMS
In chapter 2.5, we constructed the ideal (first-order) image model for an anamorphic
imaging system which accounts for the main features of the image. In this chapter, we
will build the aberration function, which is the departure from ideal behavior, and thus
accounts for the fine details. This departure can be described by a function depending on
the aperture (stop) and field (object) variables. We will write the aberration function into
its power series expansion form and show that there are sixteen primary aberration types
in an anamorphic system. We will also demonstrate the connection between anamorphic
primary wave error and ray errors.
We will present Fermat’s principle and Sir Hamilton’s characteristic function in
section 3.1, aberration function and ray aberrations for anamorphic systems in section 3.2,
and the power series expansion of aberration function in section 3.3.
3.1 Fermat’s principle and Sir Hamilton’s characteristic function
One of the basic laws of Geometrical optics is Fermat’s principle, which states: a
light ray traveling from point P to point P’ must traverse an optical path length that is
stationary with respect to variations of that path [26]. From Fermat’s principle, we can
draw an important conclusion: for any two non-conjugate points P and P’ in an optical
system, there will be one and only one ray that passes through both points. This
52
conclusion is invalid if P and P’ are conjugate points since all rays passing through the
conjugate points will have the same optical path length.
The theoretical importance of this conclusion is that for any ray which comes from a
point in the object plane and passes through a point in the system stop plane, this ray will
be completely defined by these two points, thus every parameter of this unique ray can be
written as a function of the object and the stop coordinates, and can be further expressed
as a power series expansion using these four variables. We will use this concept soon in
this chapter.
Now let us assume P( , ,x y z ) and P’( ', ', 'x y z ) are any two non-conjugate points in
our anamorphic optical system. We know there will be a unique ray that passes through
both points. From Sir Hamilton’s characteristic function theory [1-2,27], we know the
Hamilton’s point characteristic function V is defined as the optical path length along this
unique ray from P to P’ and it can be written as
( , ') ( , , , ', ' ')V P P V x y z x y z= . (3-1)
This function is of great theoretic value since the direction cosines of this ray
are totally determined by it via the following relationships
( , ')V P P
' ''
Vn Lx∂
=∂
, (3-2a)
' ''
Vn My∂
=∂
, (3-2b)
' ''
Vn Nz∂
=∂
, (3-2c)
here ' is the associated refractive index in the space where point P’ is located. n
53
3.2 Aberration function and ray aberrations for anamorphic systems
In chapter 2.5 we obtained the ideal image model for anamorphic systems. We will
now construct the aberration function to account for departures from the ideal behavior.
To do this, the symmetry of the system will be invoked to determine the possible
aberration form.
Remember in our ideal anamorphic imaging model, we did not put any restriction on
where to locate the coordinate origins in both object and image spaces. In RSOS, the
common practice is to locate the coordinate origins at the system’s entrance and exit
pupil, and the pupil coordinates are then used to define the system aberration function.
But in an anamorphic imaging system, as discussed in chapter 1, because the x-pupil and
y-pupil generally will not coincide with each other, we do not have such natural choices
to serve as our coordinate origins.
In this work, we will arbitrarily define the plane tangential to the last refraction
surface in the final image space as our image space reference plane, and it will play the
same role as the exit pupil plane played in RSOS. On this plane, we will build up our x-y
coordinates, which are centered on the system optical axis at point O. In object space, we
choose the reference plane as the object plane itself.
Using the above defined coordinate origins, consider the following anamorphic
imaging system: Suppose we have an object point , in the paraxial object plane.
Suppose point
0 0( , )P x y
0 0 0'( , )P ξ η is the ideal image point in the final image space. Let 'Σ be the
wavefront of the ray bundle from P which passes through the coordinate origin O, and
54
let S be a reference sphere with center and radius . Let another ray r of the ray
bundle from meet and
0 'P 0 'P O
P S 'Σ at the points , respectively, and let it meet the final
image plane at point . The coordinates of will be ( ,
0,Q Q
'P 0, 'Q P , )x y z , ( ,ξ η ) respectively,
and the direction cosines of the ray r will be . ( , , )L M N pz is the distance from the last
surface to the final image plane, as shown in Figure 3-1 below.
Figure 3-1 Wavefront error
Notice that the skew paraxial chief ray (which pass through the center of system stop)
no longer passes through pointO since the x-y plane is arbitrarily chosen to lie on the
tangential to the final refraction surface, and since the chief ray will be skewed with
respect to the optical axis if it does not stay in one of the symmetry planes.
The wave aberration function, is defined as the optical path length from the
reference sphere to the wavefront
( , )W x y
S 'Σ , measured along the ray as a function of the
transverse coordinates ( ,x y ) of the ray intersection with a reference sphere centered on
the ideal image point [2].
55
From Sir Hamilton’s characteristic function theory, we know
0 0 0
0
( , ; , ) ( , ) ( , )( , ) ( , ),
W x y x y V P Q V P QV P O V P Q
= −= −
(3-3a)
because Q and O are on the same wavefront and are therefore at the same optical distance
from P. Differentiating equation (3-3a) and noticing that the point characteristic function
is a constant with respect to reference sphere variables ( , )V P O x and , we get y
W V V zx x zW V V
xz
y y z y
∂ ∂ ∂ ∂= − − ⋅
∂ ∂ ∂ ∂∂ ∂ ∂ ∂
= − − ⋅∂ ∂ ∂ ∂
(3-3b)
The equation of the reference sphere is
2 2 2 2 2
0 0 0 0
2 2 20 0
( ) ( ) ( )
2 2 2 0.p
p
2px y z z
x y z x y z z
ξ η ξ η
ξ η
− + − + − = + +
⇒ + + − − − =
z (3-3c)
From the reference sphere equation we can find
0
0
( ) ,
( ) .
p
p
z xx z zz yy z z
ξ
η
∂ −⎧ =⎪∂ −⎪⎨∂ −⎪ =⎪∂ −⎩
(3-3d)
By inserting equation (3-3d) into equation (3-3b) and taking equations (3-2) into account,
we get
0
0
1 '([ ' ],'
1 '([ ' ]'
p
p
W N xLn x z z
W N yMn y z z
ξ
η
∂ −⎧ = − +⎪ ∂ −⎪⎨ ∂ −⎪ = − +⎪ ∂ −⎩
)
) . (3-3e)
But from the definition of direction cosines, we know
56
0
( , ,( ', ', ')
'p )x y z z
L M NQ P
ξ η− − −= . (3-3f)
This simplifies equation (3-3e) into
0 0
0 0 0 0
0 0
0 0 0 0
1 )( )' ' ' '
1 ( )' ' ' '
W x xn x Q P Q P Q P Q P
W y yn y Q P Q P Q P Q P
,'
,'
ξ ξ ξ ξ δξ
η η η η δη
∂ − − −⎧ = − + = − = −⎪ ∂⎪⎨ ∂ − − −⎪ = − + = − = −⎪ ∂⎩
(3-3g)
here (δξ ,δη ) are ray errors, by definition.
From above deduction, we get the relationship between wavefront error and ray
errors in reference sphere coordinates as
0
0
' ,'
' .'
Q P Wn x
Q P Wn y
δξ
δη
∂⎧ = −⎪ ∂⎪⎨ ∂⎪ = −
∂⎪⎩
(3-4a)
When we are dealing with primary (third-order) aberrations, we can replace the
unknown distance by0 'Q P R , the radius of the reference sphere. And we can replace
reference sphere coordinates by reference plane coordinates [2,28]. Thus equation (3-4a)
becomes
,
'
.'
R Wn xR Wn y
δξ
δη
∂⎧ = −⎪ ∂⎪⎨ ∂⎪ = −
∂⎪⎩
(3-4b)
Thus we see in anamorphic systems, the ray aberrations are proportional to the derivative
of wave aberration function , which is the same relationship as found in RSOS
with image space coordinate origin located on the system exit pupil plane.
( , )W x y
57
Suppose the reference plane is surface k in the optical system above. We then can
express the aberration functionW in the coordinates of the reference plane in the image
space as . ( , )k kW x y
Suppose now the ray r as shown in Figure 3-1 meets the system stop at a point of
fractional aperture ( ,x yρ ρ ) and also suppose the fractional field is ( ,x yH H ). From the
conclusion of Fermat’s principal, as discussed in Section 3.1, we know all parameters
associated with this ray r can be expressed as a function of ( ,x yH H , ,x yρ ρ ) and can be
further written as the Taylor expansion of these four quantities. In other words, kx , ky
and the aberration function W can be written as a Taylor series expansion of
( ,x yH H , ,x yρ ρ ).
For the primary aberration calculation, we can evaluate kx and ky as their paraxial
equivalents [22,28], thus according to equations (2-40), we have
, ,
, ,
,
,k x x k x x k
k y y k y y k
x h H h
y h H h
ρ
ρ
⎧ = +⎪⎨
= +⎪⎩
here ( , ,, ,x k y kh h ), ( , ,,x k y kh h ) are paraxial marginal and chief ray data in the two
corresponding associated RSOS, as defined in chapter 2.10.
Notice that the parameters ,x yH H can be thought of as constants when we are
exploring the aperture (stop), thus we have
,
,
1 ,
1 .
x
k x k x k
y
k y k y k
W W Wx x h
W W Wy y h
x
y
ρρ ρ
ρρ ρ
∂ ∂ ∂ ∂= ⋅ = ⋅
∂ ∂ ∂ ∂
∂∂ ∂ ∂= ⋅ = ⋅
∂ ∂ ∂ ∂
58
By putting these results into equation (3-4b), we get
, , ,
, , ,
1 1 ,' '( / ) ' '
1 1 .' '( / ) ' '
x k x x k x x k x
y k y y k y y k y
R W Wn h n h R n u
W
R W Wn h n h R n u
δξ
Wρ ρ ρ
δηρ ρ ρ
∂ ∂⎧ = − = − =⎪ ∂ ∂⎪⎨ ∂ ∂⎪ = − = − =⎪ ∂ ∂⎩
∂∂
∂∂
These equations are accurate for primary aberration calculation purpose. After regrouping,
we get the finally relationship between primary wavefront error and ray error, in stop
plane coordinates as
,
,
' ' ,
' ' .
x kx
y ky
Wn u
Wn u
δξρ
δηρ
∂⎧ =⎪ ∂⎪⎨ ∂⎪ =
∂⎪⎩
(3-4c)
Equation (3-4c) tell us that we can now calculate the anamorphic primary ray errors
using the stop coordinates, instead of the coordinates of the arbitrarily defined image
space reference plane.
3.3 Power series expansion of aberration function
From the discussion in section 3.2, all parameters with the ray path r will be a
function of the normalized object and stop variables ( ,x yH H , ,x yρ ρ ), and thus the wave
aberration function W in general will be a function of the four
variables ( , , , )x y xH H yρ ρ
)
also. As a result, we can either write the wave aberration
function as or ( , )k kW x y ( , , ,x y x yW H H ρ ρ . If we expand W into a power series of the
59
four variables ( , , , )x y x yH H ρ ρ , following Sir Hamilton, let us assume that W can be
expressed by a series expansion of the form [20]
=W +W +W +W +W +…, W (0) (1) (2) (3) (4)
here W is of degree in ( ,( )n n , , )x y x yH H ρ ρ .
Using the condition of double plane symmetry, we can easily verify that only certain
combinations of these four coordinate variables can exist [14]. More precisely, only 6
double plane symmetry invariants can exist in the power series expansion. These double
plane symmetry invariants are 2 2 2 2, , ,x y xa H b H c d yρ ρ= = = = , x xe H ρ= , y yf H ρ= .
The combinations of all these six invariants will give the remaining aberrations for
different orders. It is obvious that anamorphic systems do not have W (1) , W (3 and all
other odd order terms because any combinations of are of even order
in ( ,
)
, , , , ,a b c d e f
, , )x y x yH H ρ ρ .
The order W (0 is a constant piston term and has no aperture dependence, thus it is
commonly ignored.
)
For W (2 , it will have six terms in total, among which there are two piston terms that
are ignored. Thus
)
=W (2)1 2 3 4B c B e B f B d+ + + = 2
1 2 3 42
x x x y y yB B H B H Bρ ρ ρ ρ+ + + (3-5)
These are the first-order optics terms for an anamorphic system, which define the ideal
image properties. For reference sphere centered on the ideal image point, they will all
disappear [2,28].
60
ForW , it will have 21 fourth-order combinations, of which three are piston terms
and two terms are identical with other terms ( ). Thus there will be
sixteen different anamorphic primary aberration types, which can be classified into four
general types according to their aperture and field dependences
(4)
2 2,e ac f bd= =
(4) 2 21 2 3
4 5 6 7
8 9 10 11 12
13 14 15 16
{ },{ },{ },{ }, .
W D c D d D cd SphericalD ce D cf D de D df ComaticD ac D bd D bc D ad D ef Astigmatics FCD ae D bf D af D be Distortion
= + ++ + + ++ + + + + ++ + + +
It can be rewritten as
(4) 4 4 2 21 2 3
3 2 2 34 5 6 7
2 2 2 2 2 2 2 28 9 10 11 12
3 3 2 213 14 15 16
W =
.
x y x y
x x y x y x x y y y
x x y y y x x y x y x
x x y y x y x x y y
D D D
D H D H D H D H
D H D H D H D H D H H
D H D H D H H D H H
ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
yρ ρ ρ ρ
ρ ρ ρ ρ
+ +
+ + + +
+ + + + +
+ + + +
ρ ρ (3-6a)
Equation (3-6a) is the primary wave aberration expansion for anamorphic systems,
and though are the anamorphic primary aberration coefficients. It should be
noticed that anamorphic systems are much more complex than RSOS which have five
primary (third-order) aberrations types only.
1D 16D
Considering cos , sin , cos , sin ,x y x yH H H Hφ φ ρ ρ θ ρ ρ θ= = = = we can rewrite
equation (3-6a) into its polar form. By regrouping the terms according to their field and
aperture dependences, we have
(4) 4 4 2 2 41 2 3
34 5
2 36 7
W ={ cos sin sin cos }
{ cos cos sin sin cos
cos sin cos sin sin }
D D D
D D
D D
2
3H
θ θ θ θ
φ θ φ θ θ
ρ
φ θ θ φ θ ρ
+ +
+ +
+ +
(3-6b)
61
2 2 2 2 2 28 9 10
2 2 2 211 12
3 313 14
2 215 16
{ cos cos sin sin sin cos
cos sin sin cos sin cos }
{ cos cos sin sin
sin cos cos sin cos sin } .
D D D
D D H
D D
D D H 3
φ θ φ θ φ
φ θ φ φ θ θ ρ
φ θ φ θ
φ φ θ φ φ θ ρ
+ + +
+ +
+ +
+ +
θ
The typical wavefront shape of the anamorphic primary wave aberration types are
shown in Appendix A.
By taking the derivative of equation (3-6a) with respect to ,x yρ ρ while applying
equations (3-4c), we get the primary ray error expansions for anamorphic systems as
3 2 2 2
1 3 4 5 6 8
2 3 210 12 13 15 ,
(4 2 3 2 2
2 )
2
/ ' ' ,x x y x x y x y x y x x
y x x y y x x y x k
D D D H D H D H D H
D H D H H D H D H H n u
δξ ρ ρ ρ ρ ρ ρ ρ
ρ ρ
= + + + + +
+ + + +
ρ
2
' ' .y
(3-7a)
3 2 2 2
2 3 5 6 7 9
2 3 211 12 14 16 ,
(4 2 2 3 2
2 ) /y x y y x x x y y y y
x y x y x y x y y k
D D D H D H D H D H
D H D H H D H D H H n u
δη ρ ρ ρ ρ ρ ρ ρ
ρ ρ
= + + + + +
+ + + +
ρ (3-7b)
From equations (3-7), we can separate the corresponding ray errors for each
anamorphic primary aberration type, as shown below.
For Spherical Aberration-like aberration types
1 :D3
, 1
,
' ' 4' ' 0
x k x
y k
n u Dn u
δξ ρδη
⎫= ⎪⎬
= ⎪⎭ , (3-8a)
2 :D,
3, 2
' ' 0
' ' 4x k
y k y
n u
n u D
δξ
δη ρ
= ⎫⎪⎬
= ⎪⎭ , (3-8b)
3 :D2
, 3
2, 3
' ' 2
' ' 2x k x y
y k x
n u D
n u D
δξ ρ ρ
yδη ρ ρ
⎫= ⎪⎬
= ⎪⎭ . (3-8c)
For Coma-like aberration types
62
4 :D2
, 4
,
' ' 3' ' 0
x k x x
y k
n u D Hn u
δξ ρδη
⎫= ⎪⎬
= ⎪⎭ , (3-8d)
5 :D, 5
2, 5
' ' 2
' 'x k y
y k y x
n u D H
n u D Hx yδξ ρ
δη ρ
= ⎫ρ ⎪⎬
= ⎪⎭, (3-8e)
6 :D2
, 6
, 6
' '' ' 2
x k x y
y k x x y
n u D Hn u D H
δξ ρ
δη ρ
⎫=
ρ⎪⎬
= ⎪⎭, (3-8f)
7 :D,
2, 7
' ' 0
' ' 3x k
y k y y
n u
n u D H
δξ
δη ρ
= ⎫⎪⎬
= ⎪⎭. (3-8g)
For Astigmatism and Field curvature-like aberration types
8 :D2
, 8
,
' ' 2' ' 0
x k x
y k
n u D Hn u
xδξ ρδη
⎫= ⎪⎬
= ⎪⎭, (3-8h)
9 :D,
2, 9
' ' 0
' ' 2x k
y k y y
n u
n u D H
δξ
δη ρ
= ⎫⎪⎬
= ⎪⎭, (3-8i)
10 :D2
, 10
,
' ' 2' ' 0
x k y
y k
n u D Hn u
xδξ ρ
δη
⎫= ⎪⎬
= ⎪⎭, (3-8j)
11 :D,
2, 11
' ' 0
' ' 2x k
y k x y
n u
n u D H
δξ
δη ρ
= ⎫⎪⎬
= ⎪⎭, (3-8k)
12 :D , 12
, 12
' '' '
x k x y
y k x y x
n u D H Hn u D H H
yδξ ρ
δη ρ
= ⎫⎪⎬= ⎪⎭
. (3-8 l )
For Distortion-like aberration types
13 :D3
, 13
,
' '' ' 0
x k x
y k
n u D Hn u
δξδη
⎫= ⎪⎬
= ⎪⎭, (3-8m)
63
14 :D,
3, 14
' ' 0
' 'x k
y k y
n u
n u D H
δξ
δη
= ⎫⎪⎬
= ⎪⎭, (3-8n)
15 :D2
, 15
,
' '' ' 0
x k x y
y k
n u D H Hn u
δξ
δη
⎫= ⎪⎬
= ⎪⎭, (3-8o)
16 :D,
2, 16
' ' 0
' 'x k
y k x y
n u
n u D H H
δξ
δη
= ⎫⎪⎬
= ⎪⎭. (3-8p)
These 16 equations will serve as our models and the 16 anamorphic primary
aberration coefficients though can be derived by comparing with them, as will be
shown in chapter 5 through 8.
1D 16D
3.4 Summary
In this chapter, we constructed the aberration function for anamorphic systems with
object and stop variables, and showed that there were 16 primary aberration types,
indicating a greater complexity than RSOS. We also built up the relationship between the
primary wave aberrations and the primary ray aberrations.
64
CHAPTER 4
METHOD OF ANAMORPHIC PRIMARY ABERRATIONS CALCULATION
In chapter 2, we built the first-order theory for anamorphic imaging systems. In
chapter 3, we built the anamorphic aberration function and its power series expansion; we
shown there were sixteen primary aberration types in anamorphic imaging systems; we
also built the connection between primary wave error and ray errors, for each primary
aberration type. In this chapter, we will build the actual method for deriving the sixteen
anamorphic primary aberration coefficient expressions.
We will present the total ray aberration equations for anamorphic systems in section
4.1, and we will reduce them into the anamorphic primary ray aberration equations in
sections 4.2 through 4.4.
4.1 The total ray aberration equations for anamorphic systems
In geometric optics for RSOS, there is a so-called Aldis theorem [2,22], which gives
expressions for the traverse ray aberration components 'kδξ and '
kδη of a finite (real) ray
with respect to the ideal (paraxial) image location. The idea of the Aldis theorem in
RSOS can be described as follows
1) First suppose the image system is ideal and has no error;
2) Now suppose the aberrations can be considered to arise intrinsically on a specific
surface, with this surface error producing a specific distribution of errors in the
intermediate image plane of this surface;
65
3) The function of the remainder of the lenses in the system can be considered as
transferring that intermediate error distribution to the final image plane.
The same is true for other surfaces. Therefore, if we can calculate the contribution of
the initial error 'jε , then we can sum up the contributions from each surface in the final
image plane and get the total ray aberration for the system.
In the 1960s, A. Cox has found that the primary aberration coefficients for RSOS can
be deducted from the Aldis theorem [22]. However, we realize that a generalized idea can
be applied outside of RSOS, and will be able to get the primary aberration coefficients for
anamorphic systems.
Remember that in an anamorphic system, we have two sets of intermediate paraxial
image planes floating in space, namely the x-intermediate image planes and the y-
intermediate image planes, each associated with one principal section of the system.
Consider a system with k surfaces that are a mixture of spherical surfaces and double
curvature surfaces. Let ( , ,j j jNL M ) be the direction cosines of a finite (real) ray and let
( , ,j j jx y z ) be the coordinates of the point of incidence on surface j of this ray. For the
chosen object field ( ,x yH H ), let 0, jξ be the ideal x-image height in the x-intermediate
image plane of surface j . Similarly, let 0, jη be the ideal y-image height in the y-
intermediate image plane of surface j . And let ,x jA and ,y jA be the refraction invariants
associate with the corresponding x and y marginal rays, as defined in chapter 2.11.
Notice that the Lagrange invariant is calculated with the paraxial chief ray data from
the maximum field point. For an object point at fractional field, we should use the
66
fractional Lagrange invariant instead. For the anamorphic system described above, we
have the x-fractional Lagrange invariant , 0,x x j x jH n u jψ ξ= , and the y-fractional Lagrange
invariant as , 0,y y j y jH n u jψ η= .
Now let us just consider surface j of the system, as shown in Figure 4-1 below.
Suppose a finite skew ray from the fractional object field ( ,x yH H ), which intersects the
refraction surface at point ( , ,x y z ), then crosses the x-intermediate image plane at
point ( , )ξ and the y-intermediate image plane at point ( , )η . Here symbol “ ” means
we do not care the value of the corresponding coordinates.
Figure 4-1 Refraction on surface j
From equations (2-6) we have the ray transfer equations
( xL )x l zN
ξ = + − , (4-1a)
( yMy lN
η )z= + − , (4-1b)
67
where xl and yl are the distance from the vertex of the refraction surface to the x-
intermediate image plane and y-intermediate image plane, respectively, and in
general x yl l≠ .
From equation (4-1a), multiplying through by , we get xnu N
x x x xnu N nu Nx nu Ll nu Lzxξ = + − . (4-2)
By applying equation (2-41a), we can rewrite equation (4-2) as
( ) ( )x x x x x x x xnu N A nh c Nx A nh c Lz nu Llxξ = − − − + . (4-3)
From the fraction x-Lagrange invariant 0x x xnu Hξ ψ= , we have
0x x xnu N NHξ ψ= , (4-4)
here 0ξ is the ideal x-image height at the x-intermediate image plane.
Subtracting equation (4-4) from equation (4-3), we have
( ) ( )
( ) (x x x x x x x
x x x x x x x x x
nu N A xN zL nh c xN zL nh L H NA x H N A zL h c xnN h c z h nL) ,
δξ ψψ
= − − − − −= − − − + −
(4-5a)
here we have made use of the paraxial definition /x x xu h l= − . And by definition
0δξ ξ ξ= − is the x-ray error in this space. Similarly, we can find
( ) ( )
( ) (y y y y y y y
y y y y y y y y y
nu N A yN zM nh c yN zM nh M H N
) .A y H N A zM h c ynN h c z h nM
δη ψ
ψ
= − − − − −
= − − − + − (4-5b)
Till this step, we have got the corresponding specific distribution of errors in the
corresponding intermediate image planes of surface j . We now need to transfer these
error distributions through the system to the final image space.
68
By applying the refraction operator Δ on equation (4-5a), we get the increment of
this quantity on refraction as
{ } {( ) ( ) }
( ) ( )x x x x x x x x x x
x x x x x x x x x
nu N A x H N A zL h c xnN h c z h nLA x H N A z L h c x nN h c z h nL.
δξ ψψ
Δ = Δ − − − + −= − Δ − Δ − Δ + − Δ
(4-6a)
Notice that all refraction constants on this surface come out of the refraction operator
because they do not change their value on refraction. Similarly, for the y component, we
will get
{ } ( ) ( )y y y y y y y y y ynu N A y H N A z M h c y nN h c z h nMδη ψΔ = − Δ − Δ − Δ + − Δ . (4-6b)
From equation (2-11b), we can find
nL nNαγ
Δ = Δ , (4-7a)
nM nNβγ
Δ = Δ . (4-7b)
Substituting equation (4-7a) into equation (4-6a), we get
{ } ( ) [( ) ]x x x x x x x x x xnu N A x H N A z L h c z h h c x nNαδξ ψγ
Δ = − Δ − Δ + − − Δ . (4-8)
Notice that equation (4-8) is valid for every intermediate space. Also notice that on
surface j , '{ } { } { }x j x j xnu N nu N nu N jδξ δξΔ = − δξ } and 1{ }' {x xnu N nu N jδξ δξ += . Thus
we are able to sum equation (4-8) through all surfaces to the final image space of the
system. By taking into account the fact that most likely in object space, 1 1xn u N 1δξ and
1 ,1 1 1yn u N δη are zero because the object have no aberration, we get the expression for the
x-component of the system ray aberration to be
69
' ' ' ', ,
1 1
, , , , ,1
( )
[( ) ] .
k k
k x k k k x j j j j j x x jj j
kj
x j x j j x j x j x j j j jj j
n u N A x N z L H N
h c z h h c x n N
δξ ψ
αγ
= =
=
= Δ − Δ − Δ
+ − − Δ
∑ ∑
∑ (4-9a)
Similarly, we get the expression for the y-component of system ray aberration to be
' ' ' ', ,
1 1
, , , , ,1
( )
[( ) ] .
k k
k y k k k y j j j j j y y jj j
kj
y j y j j y j y j y j j j jj j
n u N A y N z M H N
h c z h h c y n N
δη ψ
βγ
= =
=
= Δ − Δ − Δ
+ − − Δ
∑ ∑
∑ (4-9b)
Equations (4-9a) and (4-9b) are the anamorphic total ray aberration equations.
Notice that an RSOS made from spherical surfaces is a special case of an anamorphic
system with the anamorphic ratio being one, thus equation (4-9a) and (4-9b) are valid for
RSOS also. We can easily verify that in the RSOS case these equations will be in a
simpler form
' ' ' ', ,
1 1
( )k k
k x k k k x j j j j j x x jj j
n u N A x N z L H Nδξ ψ= =
= Δ − Δ − Δ∑ ∑ , (4-9c)
' ' ' ', ,
1 1
( )k k
k y k k k y j j j j j y y jj j
n u N A y N z M H Nδη ψ= =
= Δ − Δ − Δ∑ ∑ . (4-9d)
Remember that equations (4-9) are total ray error equations, in that they include ray
errors of all orders. For primary aberration derivation purpose, we only need the third-
order ray error (corresponding to the fourth-order wave error), thus we need to reduce the
anamorphic total ray aberration equations into their third-order equivalents—the
anamorphic primary ray aberration equations.
70
4.2 Preparation for the anamorphic primary ray aberration equations deduction
We now need to find ,α βγ γ
as shown up in the basic equations (4-9a)-(4-9b). We can
rewrite the general double curvature surface sag equation (2-18) as
2 2 4 2 2 4
3 3 33 4 5
1 1 2( , , ) ( ) ( ) 02 8x y
x y x x y yF x y z zr r r r r
= − + − + + = . (4-10)
Following the method described in chapter 2, section 3, we find
3 2
33 42 2x
F x x xy3x r r r
∂= − − −
∂,
3 2
35 42 2y
F y y xy r r r
∂= − − −
∂ 3y ,
1Fz
∂=
∂.
We define the quantity1
2 2 2{( ) ( ) ( ) }F F FGx y z
∂ ∂ ∂= + +
∂ ∂ ∂2 . From the direction cosines of the
surface normal equation (2-21), we have
( / ) 1F zG G
γ ∂ ∂= = .
This gives ( / ) / ( /F x G F x)α γ= ∂ ∂ = ∂ ∂ and ( / ) / ( /F y G F y)β γ= ∂ ∂ = ∂ ∂ , which result in
3 2
33 4
(2 2x
F x x xy3 )
x r r rαγ
∂= = − + +∂
, (4-11a)
3 2
35 4
(2 2y
F y y xy r r r
βγ
∂= = − + +∂ 3 )y . (4-11b)
These equations are accurate up to the third-order.
71
For the primary aberration calculation, we can take , , ,j j j jx y L M as their paraxial
equivalents [2,28]. Hence we have
1j jx x= , (4-12a)
1j jy y= , (4-12b)
1 ,j j xL L u j= = , (4-12c)
1 ,j j y jM M u= = , (4-12d)
here the subscript 1 denotes the above quantities are of order one in paraxial ray heights
and angles. The double bar has the same meaning as in chapter 2, which shows we are
dealing with an arbitrary paraxial ray other than the known marginal and chief rays in the
two associated RSOS.
Equation (2-4) can now be rewritten as
2 2
1 ... 12
j jj j
L MN Nδ
+= − + = − ,
here
2 2
...2
j jj
L MNδ
+= + .
By putting equations (4-12c) and (4-12d) into jNδ and ignoring all terms higher than
third-order (because they only contribute to higher order aberrations), we get
2 2
12 2
1j jj j
L MN Nδ δ
+= = . (4-12e)
Thus we have
2 1 2j jN N N jδ= = − , (4-13a)
72
And
2 2j j j jn N n n N jδ= − , (4-13b)
here subscript 2 denotes that quantity is of second-order in ray heights and angles.
By applying equation (4-13a), if we move the on the left-hand side to the right-
hand side of equations (4-9), up to third-order accuracy, we will have
'kN
'2' '
2
1 1 11 k
k k
NN N
δδ
= = +−
. (4-13c)
By putting equations (4-12a) and (4-12b) into the general double curvature surface
sag equation (2-18) and ignoring all terms higher than order three, we get
2 2
1 12
, ,
1 (2
j jj j )
x j y j
x yz z
r r= = + . (4-13d)
By putting equations (4-12a) and (4-12b) into equations (4-11a) and (4-11b), and
ignoring all terms higher than third-order, we get
3 21 1 1 1
3 3, 3, 4,
3 21 1 1
3, 3, 4,
1 3
1 3
( )2 2
( ) (2 2
,
j j j j j
j x j j j
13 )j j j j
x j j j
j j
j j
x x x yr r r
x x x yr r r
αγ
α αγ γ
= − + +
= − − +
= +
(4-14a)
here3 2
1 1 3 1 13
1 , 3 3, 4,
( ), (2 2
13 )j j j j j j
j x j j j j
x x x yr r
α αγ γ
= − = − +r
. Subscript 3 denotes that quantity is of third-
order in ray heights and angles.
Similarly, we can find
73
1
1 3
3j j j
j j j
β β βγ γ γ
= + , (4-14b)
here3 2
1 1 3 1 13 3
1 , 3 5, 4,
( ), (2 2
1 )j j j j j j
j y j j j j
y y xr r
β βγ γ
= − = − +y
r.
Because the third summation in the R.H.S of equations (4-9a)-(4-9b) is quite lengthy,
let us treat them separately. For the one in equation (4-9a)
, , , , ,1
[( ) ]k
jx j x j j x j x j x j j j j
j j
h c z h h c x n Nαγ=
− − Δ∑ ,
by putting equations (4-13) though (4-14) into it and ignoring all terms higher than third-
order, also using the relation ,1/ ,x j xr c j= , we get
, , , , ,1
1 3, , , , ,
1 , 3
1 3 ' ' ', , 2 , 2 2
1 , 3
' 2, , 2
1
[( ) ]
[( )( ) ]
[ ( ) ][( ) ( )]
( )(
kj
x j x j j x j x j x j j j jj j
kj j
x j x j j x j x j x j j j jj x j j
kj j
x j x j j x j j j j j j jj x j j
k
x j j j x j j jj
h c z h h c x n N
xh c z h h c x n N
rx
h c z h n n n N n Nr
h n n c z x
αγ
αγ
αδ δ
γ
=
=
=
=
− − Δ
= − − + − Δ
= − − − − −
= − −
∑
∑
∑
∑ 31
3
).j
j
αγ
+
(4-15a)
Similarly, we can simplify the third summation in equation (4-9b) as
, , , , ,
1
3' 2, , 2 1
1 3
[( ) ]
( )( ).
kj
y j y j j y j y j y j j j jj j
kj
y j j j y j j jj j
h c z h h c y n N
h n n c z y
βγ
βγ
=
=
− − Δ
= − − +
∑
∑ (4-15b)
Again, these equations are accurate up to the third-order.
74
4.3 The anamorphic primary ray aberration calculation-part one
With all approximations in place, we can now reduce the anamorphic total ray
aberration equations (4-9) to their third-order equivalents—the anamorphic primary ray
aberration equations.
Let us put equations (4-12) through (4-15) into the R.H.S of equations (4-9a)-(4-9b),
by ignoring all terms higher than order three, we get
' ' ', 3 , 1 2 2 1
1
322 , , 2 1
1 1 3
( )
( )
k
k x k k x j j j j jj
k kj ,x x j x j x j j j j
j j j
n u A x N z L
H N h c z x
δξ δ
αψ δ
γ
=
= =
= − Δ + Δ
+ Δ − + Δ
∑
∑ ∑ n (4-16a)
' ' ', 3 , 1 2 2 1
1
322 , , 2 1
1 1 3
( )
( )
k
k y k k y j j j j jj
k kj
y y j y j y j j j jj j j
n u A y N z M
H N h c z y
δη δ
βψ δ
γ
=
= =
= − Δ + Δ
+ Δ − + Δ
∑
∑ ∑ .n (4-16b)
By pulling the refraction operator out of the expressions, we finally obtain
' ' ', 3 , 1 2 2 1
1
322 , , 2 1
1 1 3
[ ( )
( )
k
k x k k x j j j j jj
k kj ],x x j x j x j j j j
j j j
n u A x N z L
H N h c z x n
δξ δ
αψ δ
γ
=
= =
= Δ − +
+ − +
∑
∑ ∑ (4-17a)
' ' ', 3 , 1 2 2 1
1
322 , , 2 1
1 1 3
[ ( )
( )
k
k y k k y j j j j jj
k kj
y y j y j y j j j jj j j
n u A y N z M
H N h c z y
δη δ
βψ δ
γ
=
= =
= Δ − +
+ − +
∑
∑ ∑ ].n (4-17b)
These two equations are the anamorphic primary ray aberration equations, and they will
serve as our basic equations for the anamorphic aberration coefficients derivation.
75
Again, equations (4-17a) and (4-17b) are valid for RSOS too. We can easily verify
that in RSOS case, these equations will be in a simpler form
, (4-18a) ' ' ', 3 , 1 2 2 1 2
1 1
[ ( )k k
k x k k x j j j j j x x jj j
n u A x N z L H Nδξ δ ψ δ= =
= Δ − + +∑ ]∑
]
3
. (4-18b) ' ' ', 3 , 1 2 2 1 2
1 1
[ ( )k k
k y k k y j j j j j y y jj j
n u A y N z M H Nδη δ ψ δ= =
= Δ − + +∑ ∑
4.4 The anamorphic primary ray aberration calculation-part two
Now what we need to do is to replace 1 1 1 1 2 2 3 3 3, , , , , , / , /j j j j j j j j j jx y L M N zδ α γ β γ
with the four known paraxial marginal and chief rays’ tracing data in the two associated
RSOS, together with aperture and field variables.
Since 1 1 1, , , 1j j j jx y L M are all paraxial quantities associated with an arbitrary ray, as
indicated by equations (4-12a) through (4-12d), we can make use of equations (2-40) to
write them as
1 ,j x x j x x , jx h H hρ= + , (4-19a)
1 ,j y y j y yy h H hρ= + , j , (4-19b)
1 , ,j x x j x x jL u H uρ= + , (4-19c)
1 , ,j y y j y y jM u H uρ= + , (4-19d)
By putting equations (4-19c)-(4-19d) into equation (4-12e), we get
2 2
1 1 22 , , ,
1 [( ) ( ) ]2 2
j jj x x j x x j y y j
L MN u H u uδ ρ ρ
+= = + + + 2
,y y jH u . (4-19e)
76
By putting equations (4-19a) and (4-19b) into the corresponding expressions of
2 3 3 3, / , / 3j j j j jz α γ β γ , we know that all parameters in equations (4-17) can indeed be
written as combinations of aperture and field variables. This, together with the ray tracing
data of the marginal and chief rays in the two associated RSOS, allows us to actually
calculate the 16 primary aberration coefficients for anamorphic systems.
4.5 Summary
In this chapter, by applying the generalized Aldis idea onto anamorphic imaging
systems, we built up the anamorphic total ray aberration equations. We then reduced
these equations into their third-order equivalents: The anamorphic primary ray aberration
equations. We then wrote all parameters in the anamorphic primary ray aberration
equations in terms of the paraxial ray trace data in the two associated RSOS, together
with field and aperture variables.
With these steps, we lay the groundwork of actually obtaining the 16 primary
aberration coefficients for different types of anamorphic systems, as will be described in
the chapters to follow.
77
CHAPTER 5
PRIMARY ABERRATION THEORY FOR PARALLEL CYLINDRICAL
ANAMORPHIC ATTACHMENT SYSTEMS
From chapters 2 through 4, we have detailed the general method of deriving the
anamorphic primary aberration coefficient expressions for any anamorphic system types
with double plane symmetry.
We will want to consider one very important difference between RSOS and
anamorphic systems. In RSOS, due to the rotational symmetry, all surfaces inside the
systems will be spherical or spherical with even aspheric departure, thus the primary
aberration coefficient expressions are basically in the same form for all RSOS. In other
words, there are different rotation symmetric systems but they will all be the same type.
For anamorphic systems, however, there are many different surface types which can
be included in the systems, such as cylindrical, toroidal, ellipsoidal, etc. These different
surface types all possess of double plane symmetry but will have different surface
equations. From paraxial image formation perspective, these different surfaces types are
the same in the paraxial region. But from aberration perspective, anamorphic systems
made from different surface types will have different primary aberration coefficient
expressions, hence they need to be treated as different anamorphic system types, namely
cylindrical anamorphic systems, toroidal anamorphic systems, etc.
Under this complication, we are facing a choice here. We can choose to group all
types of double curvature surfaces together and write them into a general surface type and
78
then come up with a set of lengthy but generally applicable anamorphic primary
aberration coefficient expressions. The problem with this choice is that the resulting
analytical expressions might be so complex that they actually obscure any insight of the
results. Or we can explore different types of anamorphic systems separately, with
increasing complexity. In this manner, for some simple yet important anamorphic system
types, the analytical results will be much easier to follow and can provide much better
design insight.
In this work, we will explore different types of anamorphic systems with increasing
complexity. We will develop the primary aberration coefficients for parallel cylindrical
anamorphic systems, cross cylindrical anamorphic systems, and toroidal anamorphic
systems separately in chapters 5 through 7. In chapter 8, we will present the general
analytical expressions which are applicable to any type of anamorphic system. This will
allow the reader to reference the specific chapters which will be of interest and find the
corresponding primary aberration coefficients without being lost in the math details.
In the following chapters, since the method we developed is generally valid for any
type of anamorphic system, its application onto different anamorphic system types is
quite mechanical which may make the derivation procedures seem somewhat boring. But
this is actually an advantage of our method since it eliminates the need for exceptions to
the general method because of special differences in the systems. To illustrate this
important point, in the derivation process that follows, we will purposely use the same
structure for each chapter:
79
a) We will first apply our method to find the primary ray errors for the specified
anamorphic system type under study;
b) Once we get the primary ray errors, by comparing with the relationship between
the primary wave error and ray errors obtained in chapter 3.3, we can immediately
get the primary wave aberration coefficients for the anamorphic system under
study;
c) We then simplify the results to make them in a form as similar to the Seidel
aberrations for RSOS as possible.
Following this pattern for the rest of this chapter, we will present the primary ray
aberration equations for cylindrical anamorphic systems in section 5.1, the primary ray
aberration coefficients for parallel cylindrical anamorphic attachment systems in section
5.2, and the primary wave aberration coefficients for parallel cylindrical anamorphic
attachment systems and their simplification in sections 5.3 and 5.4.
5.1 The primary ray aberration equations for cylindrical anamorphic systems
From a manufacturing perspective, the most easily made double curvature surfaces
are cylindrical surfaces with one radius of curvature equal to infinity, so it is not
unexpected that the most commonly used anamorphic system are made from cylindrical
lens. For the same reason, our first research subject is an anamorphic system made from
cylindrical lenses.
For a cylindrical refracting surface with yr = ∞ , the surface sag equation with a
fourth-order approximation is
80
2 4
31 12 8x x
x xzr r
= + . (5-1a)
For this case, equation (4-13d) becomes
22 ,
12 1j x j jz c x= . (5-1b)
Since , from equations (4-14a) and (4-14b), we have 3 4 5,xr r r r= = = ∞
3 3, 1
3
12
j 3x j j
j
c xαγ
= − , (5-1c)
3
3
0j
j
βγ
= . (5-1d)
For a cylindrical refracting surface with xr = ∞ , the surface sag equation is similarly
2 4
31 12 8y y
yzr r
= +y . (5-2a)
Now equation (4-13d) becomes
22 ,
12 1j y j jz c y= . (5-2b)
And in this case 3 4 5, yr r r r= = ∞ = , from equations (4-14a) and (4-14b) we have
3
3
0j
j
αγ
= , (5-2c)
3 3 3, 1
3
12
jy j j
j
c yβγ
= − . (5-2d)
In both cases, we can verify that the general anamorphic primary ray aberration
equations (4-17a) and (4-17b) can be reduced into
81
, (5-3a) ' ' ', 3 , 1 2 2 1 2
1 1
[ ( )k k
k x k k x j j j j j x x jj j
n u A x N z L H Nδξ δ ψ δ= =
= Δ − + +∑ ]∑
] . (5-3b) ' ' ', 3 , 1 2 2 1 2
1 1
[ ( )k k
k y k k y j j j j j y y jj j
n u A y N z M H Nδη δ ψ δ= =
= Δ − + +∑ ∑
Thus, equations (5-3) are the primary ray aberration equations for any kind of cylindrical
anamorphic system.
For cylindrical anamorphic systems, we encounter two configurations which should
be considered separately because of their importance.
5.2 Primary ray aberration coefficients for parallel cylindrical anamorphic systems
The first configuration is an anamorphic attachment made from parallel cylindrical
lenses, which is often combined with a standard optical imaging system for spherical
power [13]. Here parallel means the generation lines of the cylindrical surfaces are
parallel to each other. An example of this configuration is shown in Figure 5-1 below.
Figure 5-1 A parallel cylindrical anamorphic attachment system
For this kind of configuration, rays will behave differently depending on how they
stay in the system, as shown in Figure 5-2 below. In Figure 5-2, rays that lie in the y-z
82
symmetry plane will pass through the cylindrical attachment as if the system consists
only of parallel plates. Rays staying in the x-z symmetry plane will pass through the
attachment as if the system is an x-RSOS. Notice that there is no unique Gaussian image
plane in each intermediate space because the Gaussian image planes in both principal
sections normally do not coincide. The same is true for intermediate pupils.
Figure 5-2 The y-z (top) and x-z (bottom) symmetry planes of a parallel cylindrical
attachment system
Rays that do not travel in above symmetry planes will be skew rays and their passage
through the system can be described by the three-dimensional ray refraction and transfer
equations, as described in chapter 2.
Suppose we have for any cylindrical surface, 0x jc = j , by putting equation (4-19b)
into equation (5-2b), we get
22 , 1 , , ,
1 1 (2 2
2)j y j j y j y y j y y jz c y c h H hρ= = + . (5-4)
By putting equation (5-4) together with equations (4-19) into equations (5-3a) and (5-3b),
we get
83
' ' ', 3 , 1 2 2 1 2
1 1
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , , , ,
1
,1
[ ( ) ]
1 [( )( 2 )2
1 [( )( 2 )2
1 [2
k k
k x k k x j j j j j x x jj j
k
x j x x j x x j x x j x x j x x x j x jj
k
x j x x j x x j y y j y y j y y y j y jj
k
x jj
n u A x N z L H N
A h H h u H u H u u
A h H h u H u H u u
A c
δξ δ ψ δ
ρ ρ ρ
ρ ρ ρ
= =
=
=
=
= Δ − + +
= − + Δ + Δ + Δ
− + Δ + Δ + Δ
−
∑ ∑
∑
∑
∑ 2 2 2 2, , , , , , ,
2 2 2 2, , , ,
1
2 2 2 2, , , ,
1
( 2 )( )]
1 ( 2 )2
1 ( 2 ),2
y j y j y y j y j y y y j y x x j x x j
k
x x x x j x x j x x x j x jj
k
x x y y j y y j y y y j y jj
h h h H h H u H u
H u H u H u u
H u H u H u u
ρ ρ ρ
ψ ρ ρ
ψ ρ ρ
=
=
+ + Δ +
+ Δ + Δ + Δ
+ Δ + Δ + Δ
∑
∑
Δ (5-5a)
And,
' ' ', 3 , 1 2 2 1 2
1 1
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , , , ,
1
,1
[ ( ) ]
1 [( )( 2 )2
1 [( )( 2 )2
1 [2
k k
k y k k y j j j j j y y jj j
k
y j y y j y y j x x j x x j x x x j x jj
k
y j y y j y y j y y j y y j y y y j y jj
k
y jj
n u A y N z M H N
A h H h u H u H u u
A h H h u H u H u u
A c
δη δ ψ δ
ρ ρ ρ
ρ ρ ρ
= =
=
=
=
= Δ − + +
= − + Δ + Δ + Δ
− + Δ + Δ + Δ
−
∑ ∑
∑
∑
∑ 2 2 2 2, , , , , , ,
2 2 2 2, , , ,
1
2 2 2 2, , , ,
1
( 2 )( )]
1 ( 2 )2
1 ( 2 ).2
y j y j y y j y j y y y j y y y j y y j
k
y y x x j x x j x x x j x jj
k
y y y y j y y j y y y j y jj
h h h H h H u H u
H u H u H u u
H u H u H u u
ρ ρ ρ
ψ ρ ρ
ψ ρ ρ
=
=
+ + Δ +
+ Δ + Δ + Δ
+ Δ + Δ + Δ
∑
∑
Δ (5-5b)
These two equations are quite lengthy, but they can actually be expanded and
regrouped according to their fields and aperture dependences. Rewriting them in a form
parallel to equations (3-8), we can obtain the 16 anamorphic primary aberration
84
coefficient expressions from general anamorphic aberration theory. After the regrouping,
we get the corresponding primary ray aberration terms as:
For Spherical Aberration-like aberration types
1 :D' ' ' 2 3
, 3 , , ,1
' ' ', 3
1 ( )2
0
k
k x k k x j x j x j xj
k y k k
n u A h u
n u
δξ ρ
δη=
⎫= − Δ ⎪
⎬⎪= ⎭
∑ (5-6a)
2 :D
' ' ', 3
' ' ' 2 2 3, 3 , , , , , ,
1
0
1 [ ( )]2
k x k k
k
k y k k y j y j y j y j y j y j yj
n u
n u A h u c h u
δξ
δη ρ=
⎫=⎪⎬
= − Δ + Δ ⎪⎭
∑ (5-6b)
3 :D
' ' ' 2 2 2, , , , , , ,
1
' ' ' 2 2, , , ,
1
1 [ ( )]2
1 ( )2
k
k x k k x j x j y j y j y j x j x yj
k
k y k k y j y j x j x yj
n u A h u c h u
n u A h u
δξ ρ ρ
δη ρ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎬⎪= − Δ⎪⎭
∑
∑ (5-6c)
For Coma-like aberration types
4 :D' ' ' 2 2 2
, 3 , , , , , , ,1
' ' ', 3
1 [ ( 2 ) ]2
0
k
k x k k x j x j x j x j x j x j x x j x xj
k y k k
n u A h u h u u u H
n u
δξ ψ ρ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (5-6d)
5 :D
' ' ', 3 , , , , , , , ,
1
' ' ' 2 2 2, 3 , , , ,
1
( )
1 ( )2
k
k x k k x j x j y j y j y j y j y j x j y x yj
k
k y k k y j y j x j y x j y xj
n u A h u u c h h u H
n u A h u u H
δξ ρ ρ
δη ψ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎬⎪= − Δ − Δ⎪⎭
∑
∑ (5-6e)
6 :D
' ' ' 2 2 2 2, 3 , , , , , , ,
1
' ' ', 3 , , , ,
1
1 [ ( ) ]2
( )
k
k x k k x j x j y j y j y j x j x y j x yj
k
k y k k y j y j x j x j x x yj
n u A h u c h u u H
n u A h u u H
δξ ρ
δη ρ ρ
=
=
⎫= − Δ + Δ −Ψ Δ ⎪
⎪⎬⎪= − Δ⎪⎭
∑
∑ (5-6f)
85
7 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 2, , , , , , , ,
0
1 [ (2
2 2 ) ]
k x k k
k
k y k k y j y j y j y j y j y jj
y j y j y j y j y j y j y j y y j y y
n u
n u A h u c h u
h u u c h h u u H
δξ
δη
ψ ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭
∑ (5-6g)
For Astigmatism and Field curvature-like aberration types
8 :D' ' ' 2 2
, 3 , , , , , , , ,1
' ' ', 3
1 [ ( 2 ) 2 ]2
0
k
k x k k x j x j x j x j x j x j x x j x j x xj
k y k k
n u A h u h u u u u H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ρ ⎪
⎬⎪= ⎭
∑ (5-6h)
9 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2, , , , , , , , ,
0
1 [ (2
2 2 ) 2
k x k k
k
k y k k y j y j y j y j y j y jj
y j y j y j y j y j y j y j y y j y j y y
n u
n u A h u c h u
c h h u h u u u u H
δξ
δη
]ψ ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭
∑ (5-6i)
10 :D' ' ' 2 2 2
, 3 , , , , , ,1
' ' ', 3
1 [ ( )]2
0
k
k x k k x j x j y j y j y j x j y xj
k y k k
n u A h u c h u H
n u
δξ ρ
δη=
⎫= − Δ + Δ ⎪
⎬⎪= ⎭
∑ (5-6j)
11 :D
' ' ', 3
' ' ' 2 2, 3 , , ,
1
0
1 ( )2
k x k k
k
k y k k y j y j x j x yj
n u
n u A h u H
δξ
δη ρ=
⎫=⎪⎬
= − Δ ⎪⎭
∑ (5-6k)
12 :D
' ' ', 3 , , , , , , , ,
1
, ,
' ' ', 3 , , , , , ,
1
[ ( )
]
( )
k
k x k k x j x j y j y j y j y j y j x jj
x y j y j x y y
k
k y k k y j y j x j x j y x j x j x y xj
n u A h u u c h h u
u u H H
n u A h u u u u H H
δξ
ψ ρ
δη ψ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪− Δ ⎬⎪⎪= − Δ − Δ⎪⎭
∑
∑
(5-6 l )
For Distortion-like aberration types
86
13 :D' ' ' 2 2 3
, 3 , , , ,1
' ' ', 3
1 ( )2
0
k
k x k k x j x j x j x x j xj
k y k k
n u A h u u H
n u
δξ ψ
δη=
⎫= − Δ − Δ ⎪
⎬⎪= ⎭
∑ (5-6m)
14 :D
' ' ', 3
' ' ' 2 2 2 3, 3 , , , , , , ,
1
0
1 [ ( ) ]2
k x k k
k
k y k k y j y j y j y j y j y j y y j yj
n u
n u A h u c h u u H
δξ
δη ψ=
⎫=⎪⎬
= − Δ + Δ − Δ ⎪⎭
∑ (5-6n)
15 :D' ' ' 2 2 2 2
, 3 , , , , , , ,1
' ' ', 3
1 [ ( ) ]2
0
k
k x k k x j x j y j y j y j x j x y j x yj
k y k k
n u A h u c h u u H H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (5-6o)
16 :D
' ' ', 3
' ' ' 2 2 2, 3 , , , ,
1
0
1 ( )2
k x k k
k
k y k k y j y j x j y x j x yj
n u
n u A h u u H H
δξ
δη ψ=
⎫=⎪⎬
= − Δ − Δ ⎪⎭
∑ (5-6p)
In all primary aberration types listed above, we notice and will
contribute to ray error in both x and y directions at any surface
3 5, ,D D D6 12D
j . For any single surface,
the coefficients for x-component of ray error and y-component of ray error may not be
equal, but the system aberration coefficients that come from summation through all
surfaces will be equal.
5.3 Primary wave aberration coefficients for parallel cylindrical anamorphic systems
By comparing equations (5-6) with equations (3-8), we immediately get the primary
wave aberration coefficients though as 1D 16D
21 , ,
1
18
k
,x j x j x jj
D A h=
= − Δ∑ u (5-7a)
87
2 22 , , , , ,
1
1 (8
k
y j y j y j y j y j y jj
D A h u c h=
= − Δ + Δ∑ , )u (5-7b)
23 , ,
1
14
k
y j y j x jj
D A h=
= − Δ∑ ,u (5-7c)
2 24 , , , , , ,
1
1 [ ( 2 )6
k
, ]x j x j x j x j x j x j x x jj
D A h u h u u uψ=
= − Δ + Δ − Δ∑ (5-7d)
2 25 , , ,
1
1 (2
k
y j y j x j y x jj
D A h u uψ=
= − Δ − Δ∑ , ) (5-7e)
6 , ,1
12
k
y j y j x j x jj
D A h u=
= − Δ∑ , ,u (5-7f)
2 2
7 , , , , , , ,1
2, , , , ,
1 [ ( 26
2 ) ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
D A h u c h u h u
c h h u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (5-7g)
28 , , , , , ,
1
1 [ ( 2 ) 24
k
, , ]x j x j x j x j x j x j x x j x jj
D A h u h u u uψ=
= − Δ + Δ − Δ∑ u (5-7h)
2 2
9 , , , , , , ,1
, , , , , ,
1 [ ( 24
2 ) 2 ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j
D A h u c h u h u
c h h u u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (5-7i)
2 210 , , , , , ,
1
1 (4
k
)x j x j y j y j y j x jj
D A h u c h=
= − Δ + Δ∑ u (5-7j)
211 , , ,
1
14
k
y j y j x jj
D A h=
= − Δ∑ u (5-7k)
12 , , , , , ,1
( )k
y j y j x j x j y x j x jj
D A h u u uψ=
= − Δ − Δ∑ u (5-7 l )
2 213 , , , ,
1
1 (2
k
)x j x j x j x x jj
D A h u ψ=
= − Δ − Δ∑ u (5-7m)
88
2 214 , , , , , , ,
1
1 [ ( )2
k
y j y j y j y j y j y j y y jj
D A h u c h u ψ=
= − Δ + Δ − Δ∑ 2 ]u (5-7n)
2 215 , , , , , , ,
1
1 [ ( )2
k2 ]x j x j y j y j y j x j x y j
j
D A h u c h u ψ=
= − Δ + Δ − Δ∑ u (5-7o)
2 216 , , , ,
1
1 (2
k
y j y j x j y x jj
D A h u ψ=
= − Δ − Δ∑ )u (5-7p)
Again, Notice that for , we can either choose the coefficient in the x-
component of the ray error expression or y-component of the ray error expression. Their
numerical values may differ for any single surface
3 5 6 12, , ,D D D D
j , but the summation in the final
image space will be the same. In our treatment above, the coefficients in y-component of
the ray error expression were arbitrarily chosen.
5.3 Simplification of the results
Now we have all the primary wave aberration coefficients for parallel cylindrical
anamorphic systems in terms of the paraxial marginal and chief ray trace data in both
associated RSOS. Next let us use the corresponding paraxial definitions from chapter
2.11 to simplify the results so that we can rewrite equations (5-7) in a form as similar to
the Seidel aberrations for RSOS as possible. For the current system, because , 0x jc = , we
have
, , , , ,x j j x j j x j x j j x jA n u n h c n u= + = ,
, , , , ,x j j x j j x j x j j x jA n u n h c n u= + = ,
, , , ,y j j y j j y j y jA n u n h c= + ,
89
, , ,y j j y j j y j y j,A n u n h c= + ,
, , , , , , , ,( )x j x j x j x j x j x j x j x j x jn h u h u A h A hΨ = − = − ,
, , , , , , , ,( )y j y j y j y j y j y j y j y j y jn h u h u A h A hΨ = − = − .
Notice that all these parameters are refraction invariants. From the definition of refraction
constants, we know that whenever the refraction operator Δ acts on a refraction constant,
the result will be zero. We will make use of this convenient property in our simplification
process below.
We also define
,,
,,
,
.
x jx j
j
y jy j
j
cP
nc
Pn
= Δ
= Δ
Simplifying equations (5-7) using the above paraxial definitions, for , we find 1D
21 , ,
1
,2, ,
1
18
181 .8
k
,x j x j x jj
kx j
x j x jj j
Ix
D A h
uA h
n
S
=
=
= − Δ
= − Δ
= −
∑
∑
u
(5-8a)
IxS means the result is in the same form as the Seidel spherical aberration for x-RSOS,
where all spherical surfaces in the x-RSOS are plane surfaces with radii of curvature
equal to infinity.
For , we find 2D
90
2 22 , , , , ,
1
, , , , , ,1
,, , , , ,
1
,2, ,
1
1 ( )8
1 ( )8
1 ( )8
181 .8
k
y j y j y j y j y j y jj
k
y j y j y j y j y j y jj
ky j
y j y j j y j j y j y jj j
ky j
y j y jj j
Iy
D A h u c h u
A h u u c h
uA h n u n c h
nu
A hn
S
=
=
=
=
= − Δ + Δ
= − Δ +
= − Δ +
= − Δ
= −
∑
∑
∑
∑
,
(5-8b)
Similarly, here IyS is the Seidel spherical aberration for y-RSOS.
For , we find 3D
23 , , ,
1
,, , ,
1
14
1 .4
k
y j y j x jj
kx j
x j y j y jj j
D A h u
uA A h
n
=
=
= − Δ
= − Δ
∑
∑ (5-8c)
For , we find 4D
2 24 , , , , , ,
1
2, , , , , , ,
1
2, , , , , , ,
1
, ,, , ,
1
, , ,
1 [ ( 2 )6
1 [( ) 2 ]6
1 ( 2 )6
1 ( 2 )6
12
k
, ]x j x j x j x j x j x j x x jj
k
x j x j x x j x j x j x j x jj
k
x j x j x j x j x j x j x jj
kx j x j
x j x j x jj j j
x j x j x j
D A h u h u u u
A h u A h u u
A h u A h u u
u uA A h
n nu
A A h
ψ
ψ
=
=
=
=
= − Δ + Δ − Δ
= − Δ − +
= − Δ +
= − Δ +
= − Δ
∑
∑
∑
∑
,
1
1 ,2
kx j
j j
IIx
n
S
=
= −
∑
(5-8d)
here is the Seidel coma for x-RSOS. IIxS
91
For , we find 5D
2 25 , , ,
1
,, , ,
1
,, , ,
1
1 ( )2
1 ( )( )(2
1 .2
k
y j y j x j y x jj
k
,
)x jy j y j y j x j
j j
kx j
x j y j y jj j
D A h u u
uA h n u
nu
A A hn
ψ
ψ
=
=
=
= − Δ − Δ
= − Δ −
= − Δ
∑
∑
∑
(5-8e)
For , we find 6D
6 , , ,
1
,, , ,
1
12
1 .2
k
y j y j x j x jj
k
,
x jx j y j y j
j j
D A h u u
uA A h
n
=
=
= − Δ
= − Δ
∑
∑ (5-8f)
For , we find 7D
2 27 , , , , , , , , ,
1
2, , , , ,
2, , , , , , , , , , ,
1
2, , ,
, , , , ,
1 [ ( 26
2 ) ]
1 [ ( 2 2 )6
( ) ]
1 [2 (6
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
k
y j y j y j y j y j y j y j y j y j y j y jj
y j y j y y j
y j y j y j y j y j y
D A h u c h u h u u
c h h u u
A c h u h u u c h h u
A h u
A h u u c h
ψ
ψ
=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + +
+ −
= − Δ +
∑
∑
,2 2, , , , , , , , ,
1
, , ,2, , , , , , , , , , ,
1
, ,2 2, , , , , , , , , ,
) ( )
1 [2 ( )]6
1 (3 )6
ky j ]j y j y j y j y j y j y j y j y j
j j
ky j y j y j
y j y j y j y j y j y j y j y j y j y j y jj j j j
y j y jy j y j y j y j y j y j y j y j y j y j
j j j
AA h c h c A h u
n
u c AA A h A A h A h u h c
n n n
u AA A h A h c A h c u
n n
=
=
+ − +
= − Δ + + −
= − Δ + −
∑
∑
1
, ,2, , , , , , ,
1
, 3 2, , , , , ,
1
1 [3 ( )]6
1 (3 )6
k
ky j y j
y j y j y j y j y j y j y jj j j
ky j
y j y j y j y j y j y jj j
u AA A h A h c u
n n
uA A h A h c
n
=
=
=
= − Δ + −
= − Δ +
∑
∑
∑
92
,, , ,
1
121 ,2
ky j
y j y j y jj j
IIy
uA A h
n
S
=
= − Δ
= −
∑ (5-8g)
here IIyS is the Seidel coma for y-RSOS. In the above deduction, we have made use of the
relationship 3 2, , ,( )y j y j y jA h cΔ = 0 since 3 2
, , ,y j y j y jA h c is a refraction constant. We will no
longer mention this kind of detail in later deduction procedures.
For , we find 8D
28 , , , , , ,
1
2, , , , , , ,
1
, ,2 2, , ,
1
,2, ,
1
1 [ ( 2 ) 24
1 ( 2 )4
1 [ ( 2 )]4
343 ,4
k
, , ]x j x j x j x j x j x j x x j x jj
k
x j x j x j x j x j x j x jj
kx j x j
x j x j x jj j j
kx j
x j x jj j
IIIx
D A h u h u u u
A h u A h u u
u uh A A
n nu
A hn
S
ψ=
=
=
=
= − Δ + Δ − Δ
= − Δ +
= − Δ +
= − Δ
= −
∑
∑
∑
∑
u
(5-8h)
here is the Seidel astigmatism for x-RSOS. IIIxS
For , we find 9D
2 29 , , , , , , , , , , , , ,
1
, ,
2 2, , , , , , , , , , , , , ,
1
2 2, , , , ,
1 [ ( 2 2 )4
2 ]
1 [ ( 2 ) 2 ]4
1 [ (4
k
y j y j y j y j y j y j y j y j y j y j y j y j y jj
y y j y j
k
y j y j y j y j y j y j y j y j y j y j y j y j y j y jj
y j y j y j y j y j y
D A h u c h u h u u c h h u
u u
A h u c h u c h h u A h u u
A h u c h u
ψ=
=
= − Δ + Δ + Δ + Δ
− Δ
= − Δ + + +
= − Δ +
∑
∑
,, , , , , , , , , ,
1
2 ) 2 (k
y j )]j y j y j y j y j y j y j y j y j y jj j
Ac h h u A h u h c
n=
+ + −∑
93
, ,2 2 2, , , , , , , , , , , , ,
1
,, , , , , ,
, ,2 2, , , , , ,
1
,, , ,
1 {2 [ 2 ( )]4
2 ( )}
1 {2 [ ( )4
2 (
ky j y j
y j y j y j y j y j y j y j y j y j y j y j y j y jj j j
y jy j y j y j y j y j y j
j
ky j y j
y j y j y j y j y j y jj j j
y jy j y j y j
j
u AA h A h u c h h u c h h c
n nA
A h h c h cn
u AA h A h h c
n n
Ac h h
n
=
=
= − Δ + + + −
− −
= − Δ + −
+
∑
∑
, ,2 2, , , , , , , ,
, ,2 2 2 2, , , , , , , , , , ,2
1
, , , ,2 2 2 2, , , , , ,
)] 2 }
1 1(2 2 )4
( )1 [24
y j y jy j y j y j y j y j y j y j y j
j j
ky j y j y j
y j y j y j y j y j y j y j y j y j y j y jj j j j
y j y j y j y jy j y j y j y j y j y j
j j
c ch c A h A A h h
n nu c
A h A A h A h A A h hn n n
u u h cA h A h A h
n n
=
− + −
= − Δ + + −
+= − Δ + +
∑ ,
j
cn
,
1
,, , , ,
, ,2 2 2 2 2, , , , , , , , , ,
1
,2 2, , ,
1
2 ]
1 [3 ( 2 ) ]4
1 (3 )41 (3 ),4
ky j
j j
y jy j y j y j y j
j
ky j y j
y j y j y j y j y j y j y j y j y j y jj j j
ky j
y j y j y y jj j
IIIy IVy
cn
cA A h h
nu c
A h A h A h A A h hn n
uA h P
n
S S
=
=
=
−
= − Δ + + −
= − Δ + Ψ
= − +
∑
∑
∑ (5-8i)
here IIIyS and IVyS are the Seidel astigmatism and Petzval sums for y-RSOS. The
combination of both terms is often called field curvature.
For and , we will keep the terms unchanged as 10D 11D
2 210 , , , , , ,
1
1 (4
k
).x j x j y j y j y j x jj
D A h u c h=
= − Δ + Δ∑ u (5-8j)
211 , , ,
1
1 .4
k
y j y j x jj
D A h=
= − Δ∑ u (5-8k)
For , we find 12D
94
12 , , , , , ,
1
,, , ,
1
( )
.
k
y j y j x j x j y x j x jj
kx j
x j y j y jj j
D A h u u u
uA A h
n
ψ=
=
= − Δ − Δ
= − Δ
∑
∑
u (5-8 l )
For , we find 13D
2 213 , , , ,
1
2, , ,
1
3, , 2
1
1 ( )2
12
1 12
1 ,2
k
x j x j x j x x jj
k
x j x j x jj
k
x j x jj j
Vx
D A h u
A h u
A hn
S
ψ=
=
=
= − Δ − Δ
= − Δ
= − Δ
= −
∑
∑
∑
u
(5-8m)
here is the Seidel distortion for x-RSOS. VxS
For , we find 14D
2 2 214 , , , , , , ,
1
2 2, , , , , , ,
1
, ,2 2, , , , , , , , ,
1
3 2, , , , ,
2
1 [ ( ) ]2
1 ( )2
1 [ ( ) (2
21 (2
k
y j y j y j y j y j y j y y jj
k
y j y j y j y j y j y j y jj
ky j y j
y j y j y j y j y j y j y j y j y jj j j
y j y j y j y j y j
j
D A h u c h u u
A c h u A h u
A AA c h h c A h h c
n n
A h A h h cn
ψ=
=
=
= − Δ + Δ − Δ
= − Δ +
= − Δ − + −
= − Δ −
∑
∑
∑ ) ]
, ,2, , ,
1
, ,3 2 2, , , , , , , ,2
1
3, , , , , , , , ,2
1
3, , , , , ,2
)
1 1( 22
1 1[ ( 2 )2
1 1[ ( )2
ky j y j
y j y j y jj j j
ky j y j
y j y j y j y j y j y j y j y jj j j
k
y j y j y j y j y j y j y j y j y jj j
y j y j y j y j y y j y j yj
cA A h
n nc c
A h A h h A A hn n
A h A h A h A h Pn
A h A h A h Pn
=
=
=
+
= − Δ − Δ + Δ
= − Δ + −
= − Δ + Ψ −
∑
∑
∑
)
]
jn
,1
]
1 ,2
k
jj
VyS
=
= −
∑
(5-8n)
95
here is the Seidel distortion for y-RSOS [28]. VyS
For , we find 15D
2 215 , , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )2
1 ( )2
k2 ]
.
x j x j y j y j y j x j x y jj
k
x j x j y j x j y j y j x jj
D A h u c h u
A h u A c h u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
u (5-8o)
For , we find 16D
2 216 , , , ,
1
2, , ,
1
1 ( )2
1 .2
k
y j y j x j y x jj
k
y j y j x jj
D A h u
A h u
ψ=
=
= − Δ − Δ
= − Δ
∑
∑
u (5-8p)
5.5 Summary
In this chapter, we showed the primary wave aberration coefficients for parallel
cylindrical anamorphic attachment systems in subgroups as:
Primary wave aberration coefficients associated with x-RSOS
,21 , ,
1
1 18 8
kx j
x j x j Ixj j
uD A h
n=
= − Δ = −∑ S ,
,4 , , ,
1
1 12 2
kx j
x j x j x j IIxj j
uD A A h
n=
= − Δ = −∑ S ,
,28 , ,
1
3 34 4
kx j
x j x j IIIxj j
uD A h
n=
= − Δ = −∑ S ,
313 , , 2
1
1 12 2
k 1x j x j Vx
j j
D A hn=
= − Δ = −∑ S .
96
In the above expression etc. mean the results are in the same form as the Seidel
aberration terms corresponding to the rays trace in the x-z cross section of the associated
x-RSOS, which are parallel planes in the current case. Thus for rays staying in the x-z
symmetry plane of the parallel cylindrical anamorphic attachment system, their
aberration behavior is the same as rays staying in the x-z meridian section of the
associated x-RSOS.
,Ix IIxS S
Primary wave aberration coefficients associated with y-RSOS
,22 , ,
1
1 18 8
ky j
y j y j Iyj j
uD A h
n=
= − Δ = −∑ S ,
,7 , , ,
1
1 12 2
ky j
y j y j y j IIyj j
uD A A h
n=
= − Δ = −∑ S ,
,2 29 , , ,
1
1 1(3 ) (3 )4 4
ky j
y j y j y y j IIIy IVyj j
uD A h P S
n=
= − Δ + Ψ = − +∑ S ,
314 , , , , , , ,2
1
1 1[ ( )2 2
k
y j y j y j y j y y j y j y j Vyj j
D A h A h A h Pn=
= − Δ + Ψ − = −∑ 1] S .
Similarly, in the above expression ,Iy IIyS S etc. mean the results are in the same form as
the Seidel aberration terms corresponding to the rays trace in the y-z cross section of the
associated y-RSOS. Thus for rays staying in the y-z symmetry plane of the parallel
cylindrical anamorphic attachment system, their aberration behavior is the same as rays
staying in the y-z meridian section of the associated y-RSOS. Actually, these conclusions
are generally applicable for any types of anamorphic systems because all rays in the two
symmetry planes are non-skewed.
Additional terms for skew rays
97
,3 , , ,
1
14
kx j
x j y j y jj j
uD A A h
n=
= − Δ∑ ,
,5 , , ,
1
12
kx j
x j y j y jj j
uD A A h
n=
= − Δ∑ ,
,6 , , ,
1
12
kx j
x j y j y jj j
uD A A h
n=
= − Δ∑ ,
2 210 , , , , , ,
1
1 ( )4
k
x j x j y j y j y j x jj
D A h u c h=
= − Δ + Δ∑ u ,
211 , , ,
1
14
k
y j y j x jj
D A h=
= − Δ∑ u ,
,12 , , ,
1
kx j
x j y j y jj j
uD A A h
n=
= − Δ∑ ,
2 215 , , , , , , ,
1
1 ( )2
k
x j x j y j x j y j y j x jj
D A h u A c h=
= − Δ + Δ∑ u ,
216 , , ,
1
12
k
y j y j x jj
D A h=
= − Δ∑ u .
These expressions tell us that for skew rays not staying in one of the symmetry planes of
the parallel cylindrical anamorphic system, their aberration behavior will be much more
complex than in the RSOS case and will possess of all 16 anamorphic primary aberration
types.
We emphasize that all parameters in the anamorphic primary aberration coefficient
expressions are the paraxial marginal and chief rays’ tracing data in the two associated
RSOS, together with some first-order constants and definitions. Thus, for anamorphic
98
primary aberration calculation purpose, we only need to trace the four non-skew marginal
and chief rays, in the associated x-RSOS and y-RSOS, respectively.
If we compare the above expressions with what C. G. Wynne reports in his 1954
paper [13], we will see the results are the same, but our development did not put any
restriction on location of the system stop, thus is more general than Wynne’s treatment.
Notice that parallel cylindrical anamorphic attachment systems themselves are not
imaging systems due to the fact that all elements have no power in one principal section.
To form an anamorphic image, we can use the above cylindrical elements as an
anamorphic attachment and combine it with some ordinary imaging lenses to obtain
optical power in the natural plane, as shown in Figure 5-1 above.
We also notice that the parallel cylindrical anamorphic attachment have an
interesting property. As Wynne noted in his 1954 paper, for an object at infinity, because
, ,x j j x jA n u= will be zero for all surfaces in the attachment, aberration types
1 3, 4 5 6 8 1, , , , , 0D D D D D D D and will disappear automatically. The remaining aberrations
term are the four y-RSOS aberrations
12D
2 7, 9 14, ,D D D D , together with three distortion terms
13 15, 16,D D D and one astigmatism term . 11D
However, if the object distance is finite ( 0xu ≠ ), all sixteen primary aberration types
will survive.
99
CHAPTER 6
PRIMARY ABERRATION THEORY FOR CROSS CYLINDRICAL
ANAMORPHIC SYSTEMS
In chapter 5, the 16 primary wave aberration coefficients for parallel cylindrical
anamorphic systems were obtained. In this chapter, we will apply the same method onto
cross cylindrical anamorphic systems.
Following this same development pattern as described in chapter 5, we will present
primary ray aberration coefficients for cross cylindrical anamorphic systems in section
6.1, primary wave aberration coefficients for cross cylindrical anamorphic systems and
their simplification in sections 6.2 and 6.3.
6.1 Primary ray aberration coefficients for cross cylindrical anamorphic systems
For cylindrical anamorphic systems, the next configurations to consider is a system
made from cross cylindrical lenses, as shown in the Figure 6-1 below.
Figure 6-1 A cross cylindrical anamorphic system example
100
The two principal sections of this kind of system are shown in Figure 6-2 below.
Rays propagating in one of the principal sections will pass through the cylindrical system
as if the system were made from a mixture of spherical surfaces and plane surfaces.
Figure 6-2 The two principal sections of a cross cylindrical anamorphic system
Notice that a plane surface is a special case of a spherical surface with radius r = ∞ ,
thus we can write the surface equation for any surface j up to the second order as
2 22 , 1 , 1
1 ( )2j x j j y j jz c x c y= + , (6-1)
of which either ,x jc or ,y jc will be zero. Thus any term containing , ,x j yc c× j will be zero.
From Chapter 4.4, we know
1 ,j x x j x x , jx h H hρ= + ,
1 ,j y y j y yy h H hρ= + , j ,
1 , ,j x x j x x jL u H uρ= + ,
1 , ,j y y j y y jM u H uρ= + ,
2 2
1 1 22 , , ,
1 [( ) ( ) ]2 2
j jj x x j x x j y y j
L MN u H u uδ ρ ρ
+= = + + + 2
,y y jH u .
101
By putting equation (6-1) together with the above equations into equations (5-3), we get:
' ' ', 3 , 1 2 2 1 2
1 1
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , , , ,
1
,1
[ ( ) ]
1 [( )( 2 )2
1 [( )( 2 )2
1 [2
k k
k x k k x j j j j j x x jj j
k
x j x x j x x j x x j x x j x x x j x jj
k
x j x x j x x j y y j y y j y y y j y jj
k
x jj
n u A x N z L H N
A h H h u H u H u u
A h H h u H u H u u
A c
δξ δ ψ δ
ρ ρ ρ
ρ ρ ρ
= =
=
=
=
= Δ − + +
= − + Δ + Δ + Δ
− + Δ + Δ + Δ
−
∑ ∑
∑
∑
∑ 2 2 2 2, , , , , , ,
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
2 2 2 2, ,
( 2 )( )]
1 [ ( 2 )( )2
1 ( 2 )2
1 ( 22
x j x j x x j x j x x x j x x x j x x j
k
]x j y j y j y y j y j y y y j y x x j x x jj
k
x x x x j x x j x x x j x jj
x x y y j y y j
h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
H u H u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
ψ ρ ρ
=
=
+ + Δ + Δ
− + + Δ +
+ Δ + Δ + Δ
+ Δ + Δ +
∑
∑
Δ
, ,1
),k
y y y j y jj
H u u=
Δ∑
(6-2a)
And,
' ' ', 3 , 1 2 2 1 2
1 1
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , , , ,
1
,1
[ ( ) ]
1 [( )( 2 )2
1 [( )( 2 )2
1 [2
k k
k y k k y j j j j j y y jj j
k
y j y y j y y j x x j x x j x x x j x jj
k
y j y y j y y j y y j y y j y y y j y jj
k
y jj
n u A y N z M H N
A h H h u H u H u u
A h H h u H u H u u
A c
δη δ ψ δ
ρ ρ ρ
ρ ρ ρ
= =
=
=
=
= Δ − + +
= − + Δ + Δ + Δ
− + Δ + Δ + Δ
−
∑ ∑
∑
∑
∑ 2 2 2 2, , , , , , ,
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
2 2 2 2, ,
( 2 )(
1 [ ( 2 )( )2
1 ( 2 )2
1 ( 22
x j x j x x j x j x x x j x y y j y y j
k
y j y j y j y y j y j y y y j y y y j y y jj
k
y y x x j x x j x x x j x jj
y y y y j y y j
h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
H u H u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
ψ ρ ρ
=
=
+ + Δ + Δ
− + + Δ +
+ Δ + Δ + Δ
+ Δ + Δ +
∑
∑
)]
]Δ
, ,1
).k
y y y j y jj
H u u=
Δ∑
(6-2b)
102
By regrouping the above equations by their field and aperture dependences, we get
the corresponding primary ray aberrations terms in a form similar to equations (3-8) as:
For Spherical Aberration-like aberration types
1 :D' ' ' 2 2 3
, 3 , , , , , ,1
' ' ', 3
1 [ ( )]2
0
k
k x k k x j x j x j x j x j x j xj
k y k k
n u A h u c h u
n u
δξ ρ
δη=
⎫= − Δ + Δ ⎪
⎬⎪= ⎭
∑ (6-3a)
2 :D
' ' ', 3
' ' ' 2 2 3, 3 , , , , , ,
1
0
1 [ ( )]2
k x k k
k
k y k k y j y j y j y j y j y j yj
n u
n u A h u c h u
δξ
δη ρ=
⎫=⎪⎬
= − Δ + Δ ⎪⎭
∑ (6-3b)
3 :D
' ' ' 2 2 2, , , , , , ,
1
' ' ' 2 2 2, , , , , ,
1
1 [ ( )]2
1 [ ( )]2
k
k x k k x j x j y j y j y j x j x yj
k
k y k k y j y j x j x x j y j x yj
n u A h u c h u
n u A h u c h u
δξ ρ ρ
δη ρ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎬⎪= − Δ + Δ⎪⎭
∑
∑ (6-3c)
For Coma-like aberration types
4 :D
' ' ' 2 2, 3 , , , , , ,
1
2, , , , , , , ,
' ' ', 3
1 [ (2
2 2 ) ]
0
k
k x k k x j x j x j x j x j x jj
x j x j x j x j x j x j x j x x j x x
k y k k
n u A h u c h u
h u u c h h u u H
n u
δξ
ψ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪+ Δ + Δ − Δ ⎬⎪
= ⎪⎪⎭
∑2ρ (6-3d)
5 :D
' ' ', 3 , , , , , , , ,
1
' ' ' 2 2 2 2, 3 , , , , , , ,
1
[ ( )]
1 [ ( ) ]2
k
k x k k x j x j y j y j y j y j y j x j y x yj
k
k y k k y j y j x j x j x j y j y x j y xj
n u A h u u c h h u H
n u A h u c h u u H
δξ ρ ρ
δη ψ
=
=
⎫= − Δ + Δ
ρ
⎪⎪⎬⎪= − Δ + Δ − Δ⎪⎭
∑
∑ (6-3e)
103
6 :D
' ' ' 2 2 2 2, 3 , , , , , , ,
1
' ' ', 3 , , , , , , , ,
1
1 [ ( ) ]2
[ ( )]
k
k x k k x j x j y j y j y j x j x y j x yj
k
k y k k y j y j x j x j x j x j x j y j x x yj
n u A h u c h u u H
n u A h u u c h h u H
δξ ρ
δη ρ ρ
=
=
⎫= − Δ + Δ −Ψ Δ ⎪
⎪⎬⎪= − Δ + Δ⎪⎭
∑
∑ (6-3f)
7 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 2, , , , , , , ,
0
1 [ (2
2 2 ) ]
k x k k
k
k y k k y j y j y j y j y j y jj
y j y j y j y j y j y j y j y y j y y
n u
n u A h u c h u
h u u c h h u u H
δξ
δη
ψ ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭
∑ (6-3g)
For Astigmatism and Field curvature-like aberration types
8 :D
' ' ' 2 2, 3 , , , , , ,
1
2, , , , , , , , ,
' ' ', 3
1 [ (2
2 2 ) 2 ]
0
k
k x k k x j x j x j x j x j x jj
x j x j x j x j x j x j x j x x j x j x x
k y k k
n u A h u c h u
h u u c h h u u u H
n u
δξ
ψ ρ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪+ Δ + Δ − Δ ⎬⎪
= ⎪⎪⎭
∑ (6-3h)
9 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2, , , , , , , , ,
0
1 [ (2
2 2 ) 2 ]
k x k k
k
k y k k y j y j y j y j y j y jj
y j y j y j y j y j y j y j y y j y j y y
n u
n u A h u c h u
h u u c h h u u u H
δξ
δη
ψ ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭
∑ (6-3i)
10 :D' ' ' 2 2 2
, 3 , , , , , ,1
' ' ', 3
1 [ ( )]2
0
k
k x k k x j x j y j y j y j x j y xj
k y k k
n u A h u c h u H
n u
δξ ρ
δη=
⎫= − Δ + Δ ⎪
⎬⎪= ⎭
∑ (6-3j)
11 :D
' ' ', 3
' ' ' 2 2 2, 3 , , , , , ,
1
0
1 [ ( )]2
k x k k
k
k y k k y j y j x j x j x j y j x yj
n u
n u A h u c h u H
δξ
δη ρ=
⎫=⎪⎬
= − Δ + Δ ⎪⎭
∑ (6-3k)
104
12 :D
' ' ', 3 , , , , , , , ,
1
, ,
' ' ', 3 , , , , , , , ,
1
, ,
[ ( )
]
[ ( )
]
k
k x k k x j x j y j y j y j y j y j x jj
x y j y j x y y
k
k y k k y j y j x j x j x j x j x j y jj
y x j x j x y x
n u A h u u c h h u
u u H H
n u A h u u c h h u
u u H H
δξ
ψ ρ
δη
ψ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪− Δ ⎪⎬⎪= − Δ + Δ⎪⎪
− Δ ⎪⎭
∑
∑ (6-3l)
For Distortion-like aberration types
13 :D' ' ' 2 2 2 3
, 3 , , , , , , ,1
' ' ', 3
1 [ ( ) ]2
0
k
k x k k x j x j x j x j x j x j x x j xj
k y k k
n u A h u c h u u H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (6-3m)
14 :D
' ' ', 3
' ' ' 2 2 2 3, 3 , , , , , , ,
1
0
1 [ ( ) ]2
k x k k
k
k y k k y j y j y j y j y j y j y y j yj
n u
n u A h u c h u u H
δξ
δη ψ=
⎫=⎪⎬
= − Δ + Δ − Δ ⎪⎭
∑ (6-3n)
15 :D' ' ' 2 2 2 2
, 3 , , , , , , ,1
' ' ', 3
1 [ ( ) ]2
0
k
k x k k x j x j y j y j y j x j x y j x yj
k y k k
n u A h u c h u u H H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (6-3o)
16 :D
' ' ', 3
' ' ' 2 2 2 2, 3 , , , , , , ,
1
0
1 [ ( ) ]2
k x k k
k
k y k k y j y j x j x j x j y j y x j x yj
n u
n u A h u c h u u H H
δξ
δη ψ=
⎫=⎪⎬
= − Δ + Δ − Δ ⎪⎭
∑ (6-3p)
6.2 Primary wave aberration coefficients for cross cylindrical anamorphic systems
By comparing equations (6-3) with equations (3-8), we immediately get the primary
wave aberration coefficients though as 1D 16D
2 21 , , , , ,
1
1 (8
k
, )x j x j x j x j x j x jj
D A h u c h=
= − Δ + Δ∑ u (6-4a)
105
2 22 , , , , ,
1
1 (8
k
y j y j y j y j y j y jj
D A h u c h=
= − Δ + Δ∑ , )u (6-4b)
2 23 , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u c h=
= − Δ + Δ∑ , )u (6-4c)
2 2
4 , , , , , , ,1
2, , , , ,
1 [ ( 26
2 ) ]
k
, ,x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j
D A h u c h u h u
c h h u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ u (6-4d)
2 25 , , , , , ,
1
1 [ ( )2
k
y j y j x j x j x j y j y x jj
D A h u c h u uψ=
= − Δ + Δ − Δ∑ 2, ] (6-4e)
6 , , , , , , , ,1
1 (2
k
y j y j x j x j x j x j x j y jj
D A h u u c h h=
= − Δ + Δ∑ )u (6-4f)
2 2
7 , , , , , , ,1
2, , , , ,
1 [ ( 26
2 ) ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
D A h u c h u h u
c h h u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (6-4g)
2 2
8 , , , , , , ,1
, , , , , ,
1 [ ( 24
2 ) 2 ]
k
, ,x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j
D A h u c h u h u
c h h u u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ u (6-4h)
2 2
9 , , , , , , ,1
, , , , , ,
1 [ ( 24
2 ) 2 ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j
D A h u c h u h u
c h h u u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (6-4i)
2 210 , , , , , ,
1
1 (4
k
)x j x j y j y j y j x jj
D A h u c h=
= − Δ + Δ∑ u (6-4j)
2 211 , , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u c h=
= − Δ + Δ∑ )u (6-4k)
12 , , , , , , , , , ,1[ ( )
k
y j y j x j x j x j x j x j y j y x j x jj
D A h u u c h h u uψ=
= − Δ + Δ − Δ∑ ]u (6-4l)
106
2 213 , , , , , , ,
1
1 [ ( )2
k2 ]x j x j x j x j x j x j x x j
jD A h u c h u ψ
=
= − Δ + Δ − Δ∑ u (6-4m)
2 214 , , , , , , ,
1
1 [ ( )2
k
y j y j y j y j y j y j y y jj
D A h u c h u ψ=
= − Δ + Δ − Δ∑ 2 ]u (6-4n)
2 215 , , , , , , ,
1
1 [ ( )2
k2 ]x j x j y j y j y j x j x y j
jD A h u c h u ψ
=
= − Δ + Δ − Δ∑ u (6-4o)
2 216 , , , , , , ,
1
1 [ ( )2
k
y j y j x j x j x j y j y x jj
D A h u c h u ψ=
= − Δ + Δ − Δ∑ 2 ]u (6-4p)
Again, for , we can either choose the coefficient in the x-ray error
expression or the coefficient in the y-ray error expression. The numerical values may
differ for any single surface
3 5 6 12, , ,D D D D
j , but the summation in the final image space will be the
same. In our treatment above, the coefficient in the y-ray error expression was arbitrarily
chosen.
6.3 Simplification of the results
Now we have all the primary aberration coefficients for a cross cylindrical
anamorphic system in terms of the paraxial marginal and chief ray trace data in both
associated RSOS. Similar to the treatment in chapter 5, let us use the corresponding
paraxial definitions from chapter 2.11 to simplify the results so that we can rewrite
equations (6-4) in a form as similar to the Seidel aberrations for RSOS as possible. For
the current configuration, we have
, , , ,x j j x j j x j x jA n u n h c= + ,
107
, , , ,x j j x j j x j x jA n u n h c= + ,
, , , ,y j j y j j y j y jA n u n h c= + ,
, , ,y j j y j j y j y j,A n u n h c= + .
Again, notice that either ,x jc or ,y jc will be zero for surface j , thus any term containing
, ,x j yc c× j will be zero. Other paraxial definitions are the same of chapter 5.
For , we find 1D
2 21 , , , , ,
1
, , , , , ,1
,2, ,
1
1 ( )8
1 ( )8
181 ,8
k
,x j x j x j x j x j x jj
k
x j x j x j x j x j x jj
kx j
x j x jj j
Ix
D A h u c h
A h u u h c
uA h
n
S
=
=
=
= − Δ + Δ
= − Δ +
= − Δ
= −
∑
∑
∑
u
(6-5a)
here and below ,IxS IyS , etc. will have the same meaning as in chapter 5.
For , we find 2D
2 22 , , , , ,
1
, , , , , ,1
,2, ,
1
1 ( )8
1 ( )8
18
1 .8
k
y j y j y j y j x j y jj
k
y j y j y j y j y j y jj
ky j
y j y jj j
Iy
D A h u c h
A h u u c h
uA h
n
S
=
=
=
= − Δ + Δ
= − Δ +
= − Δ
= −
∑
∑
∑
,u
(6-5b)
For , we will keep it unchanged as 3D
2 23 , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u h c u=
= − Δ + Δ∑ , ). (6-5c)
108
For , we find 4D
2 24 , , , , , , , , ,
1
2, , , , ,
2 2, , , , , , , , , , , , , ,
1
2 2, , , , ,
1
1 [ ( 26
2 ) ]
1 [ ( 2 26
1 [6
k
x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j
k
)]x j x j x j x j x j x j x j x j x j x j x j x j x j x jj
k
x j x j x j x j x j xj
D A h u c h u h u u
c h h u u
A h u A c h u c h h u h u u
A h u A c h
ψ=
=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + + +
= − Δ +
∑
∑
∑ , , , , , , , ,
, ,2, , , , , , , , , ,
1
,, , ,
, 2 2, , , , , , , , , , ,
1
2 ( )
1 [ ( ) ( )6
2 ]
1 1(3 )6
j x j x j x j x j x j x j x j
kx j x j
x j x j x j x j x j x j x j x j x j x jj j j
x jx j x j x j
j
kx j
x j x j x j x j x j x j x j x j x j x j x jj j j
u A h u u c h
A AA h u c h A c h c h
n nu
A A hn
uA A h A A c h A h c u
n n
=
=
+ +
= − Δ − + −
+
= − Δ + −
= −
∑
∑
]
, ,2, , , , , , ,
1
, 2 3, , , , , ,
1
,, , ,
1
1 [3 ( )]6
1 (3 )6
121 .2
kx j x j
x j x j x j x j x j x j x jj j j
kx j
x j x j x j x j x j x jj j
kx j
x j x j x jj j
IIx
u AA A h A c h u
n nu
A A h A c hn
uA A h
n
S
=
=
=
Δ + −
= − Δ +
= − Δ
= −
∑
∑
∑ (6-5d)
For , we find 5D
2 2 25 , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )2
1 ( )2
k
y j y j x j x j x j y j y x jj
k
y j y j x j y j x j x j y jj
D A h u c h u u
A h u A h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
, ]
. (6-5e)
For , we will keep it unchanged as 6D
6 , , , , , , ,1
1 ( )2
k
y j y j x j x j x j x j x j y jj
D A h u u h h c u=
= − Δ + Δ∑ , . (6-5f)
109
For , by noticing that equation (6-4g) is exactly the same as equation (5-7g), we
know
7D
2 27 , , , , , , ,
1
2, , , , ,
,, , ,
1
1 [ ( 26
2 ) ]
121 .2
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
ky j
y j y j y jj j
IIy
D A h u c h u h u
c h h u uu
A A hn
S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ
= −
∑
∑
, ,u
(6-5g)
For , we find 8D
2 28 , , , , , , , , ,
1
, , , , , ,
2 2, , , , , , , , , , , , , ,
1
,, , , ,
1 [ ( 24
2 ) 2 ]
1 [ ( 2 ) 2 ]4
1 { [ ( )4
k
x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j
k
x j x j x j x j x j x j x j x j x j x j x j x j x j x jj
x jx j x j x j x j
j
D A h u c h u h u u
c h h u u u
A h u c h u c h h u A h u u
AA h h c
n
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + + +
= − Δ −
∑
∑
,2 2, , , ,
1
, , ,, , , , , , , , , , ,
,2 2 2 2 2, , , , , , , , , , ,2
1
,
( )
2 ( )] 2 ( )( )}
1 1[3 ( 2 2 )]4
1 {34
kx j
x j x j x j x jj j
x j x j x jx j x j x j x j x j x j x j x j x j x j x j
j j j
kx j
x j x j x j x j x j x j x j x j x j x j x jj j j
x j
Ac h h c
n
A A Ac h h h c A h h c h c
n n n
cA A h A h A h A A h h
n n
A
=
=
+ −
+ − + − −
= − Δ + − −
= − Δ
∑
∑
,2 2, , , , , , , ,2
1
, ,2 2 2, , , , , ,2
1
,2 2, , ,
1
1 [( ) 3 ]}
1 1 1[3 3 ( )]4
1 (3 )4
1 (3 ).4
kx j
x j x j x j x j x j x j x j x jj j j
kx j x j
x j x j x j x x j x j x jj j j j j
kx j
x j x j x x jj j
IIIx IVx
cA h A h A h A h
n n
c AA A h A h u
n n n n
uA h P
n
S S
=
=
=
+ − −
= − Δ + Ψ − −
= − Δ + Ψ
= − +
∑
∑
∑
2 2
(6-5h)
110
For , by noticing that equation (6-4i) is exactly the same as equation (5-7i), we
know:
9D
2 29 , , , , , , ,
1
, , , , , ,
,2 2, , ,
1
1 [ ( 24
2 ) 2 ]
1 (3 )41 (3 ).4
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j
ky j
y j y j y y jj j
IIIy IVy
D A h u c h u h u
c h h u u uu
A h Pn
S S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + Ψ
= − +
∑
∑
, ,u
(6-5i)
For and , we will keep them unchanged as 10D 11D
2 210 , , , , , ,
1
1 (4
k
).x j x j y j y j y j x jj
D A h u h c=
= − Δ + Δ∑ u (6-5j)
2 211 , , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u h c u=
= − Δ + Δ∑ ). (6-5k)
For , we find 12D
12 , , , , , , , , , ,
1
, , , , , , , , ,1
[ ( )
( )
k
y j y j x j x j x j x j x j y j y x j x jj
k
y j y j x j x j y j x j x j x j y jj
D A h u u c h h u u
A h u u A h h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
]
.
u (6-5 l )
For , we find 13D
2 2 213 , , , , , , ,
1
2 2, , , , , , ,
1
, ,2 2, , , , , , , , ,
1
3 2, , , , ,2
1 [ ( ) ]2
1 ( )2
1 [ ( ) (2
1 1( 22
k
x j x j x j x j x j x j x x jj
k
x j x j x j x j x j x j x jj
kx j x j )]x j x j x j x j x j x j x j x j x j
j j j
x j x j x j x j x jj
D A h u c h u u
A h u A c h u
A AA h h c A c h h c
n n
A h A h hn
ψ=
=
=
= − Δ + Δ − Δ
= − Δ +
= − Δ − + −
= − Δ −
∑
∑
∑
, ,2, , ,
1
)k
x j x jx j x j x j
j j j
c cA A h
n n=
+∑
(6-5m)
111
3, , , , , , , , ,2
1
3, , , , , , ,2
1
1 1[ ( 22
1 1[ ( )2
1 .2
k
) ]
]
x j x j x j x j x j x j x j x j x jj j
k
x j x j x j x j x x j x j x jj j
Vx
A h A h A h A h Pn
A h A h A h Pn
S
=
=
= − Δ + −
= − Δ + Ψ −
= −
∑
∑
For , by noticing that equation (6-4n) is exactly the same as equation (5-7n), we
know
14D
2 2 214 , , , , , , ,
1
3, , , , , , ,2
1
1 [ ( ) ]2
1 1[ (21 .2
k
y j y j y j y j y j y j y y jj
k
y j y j y j y j y y j y j y jj j
Vy
D A h u c h u
A h A h A h Pn
S
ψ=
=
= − Δ + Δ − Δ
= − Δ + Ψ −
= −
∑
∑ ) ]
u
(6-5n)
For , we find 15D
2 215 , , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )2
1 ( )2
k2 ]
.
x j x j y j y j y j x j x y jj
k
x j x j y j x j y j y j x jj
D A h u c h u
A h u A h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
u (6-5o)
For , we find 16D
2 216 , , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )2
1 ( )2
k
y j y j x j x j x j y j y x jj
k
y j y j x j y j x j x j y jj
D A h u c h u
A h u A h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
2 ]
.
u (6-5p)
6.4 Summary
In this chapter, we have derived the primary wave aberration coefficients for cross
cylindrical anamorphic systems as:
112
Primary wave aberration coefficients associated with x-RSOS
,21 , ,
1
18 8
kx j 1
x j x j Ixj j
uD A h
n=
= − Δ = −∑ S ,
,4 , , ,
1
12 2
kx j 1
x j x j x j IIxj j
uD A A h
n=
= − Δ = −∑ S ,
,2 28 , , ,
1
1 (3 ) (3 )4 4
kx j 1
x j x j x x j IIIx IVxj j
uD A h P S
n=
= − Δ + Ψ = − +∑ S ,
313 , , , , , , ,2
1
1 1[ ( ) ]2 2
k 1x j x j x j x j x x j x j x j Vx
j j
D A h A h A h Pn=
= − Δ + Ψ − = −∑ S .
Primary wave aberration coefficients associated with y-RSOS
,22 , ,
1
18 8
ky j
y j y j Iyj j
uD A h
n=
= − Δ = −∑ 1 S ,
,7 , , ,
1
12 2
ky j
y j y j y j IIyj j
uD A A h
n=
= − Δ = −∑ 1 S ,
,2 29 , , ,
1
1 (3 ) (3 )4 4
ky j
y j y j y y j IIIy IVyj j
uD A h P S
n=
= − Δ + Ψ = − +∑ 1 S ,
314 , , , , , , ,2
1
1 1[ ( )2 2
k
y j y j y j y j y y j y j y j Vyj j
D A h A h A h Pn=
= − Δ + Ψ − = −∑ 1] S .
Additional terms for skew rays
2 23 , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u h c=
= − Δ + Δ∑ , )u ,
2 25 , , , , , ,
1
1 (2
k
y j y j x j y j x j x j y jj
D A h u A h c=
= − Δ + Δ∑ , )u ,
113
6 , , , , , , ,1
1 (2
k
y j y j x j x j x j x j x j y jj
D A h u u h h c u=
= − Δ + Δ∑ , ) ,
2 210 , , , , , ,
1
1 (4
k
)x j x j y j y j y j x jj
D A h u h c=
= − Δ + Δ∑ u ,
2 211 , , , , , ,
1
1 (4
k
y j y j x j x j x j y jj
D A h u h c=
= − Δ + Δ∑ )u ,
12 , , , , , , , , ,1
(k
y j y j x j x j y j x j x j x j y jj
D A h u u A h h c u=
= − Δ + Δ∑ ) ,
2 215 , , , , , , ,
1
1 (2
k
)x j x j y j x j y j y j x jj
D A h u A h c=
= − Δ + Δ∑ u ,
2 216 , , , , , , ,
1
1 (2
k
y j y j x j y j x j x j y jj
D A h u A h c=
= − Δ + Δ∑ )u .
If we compare the results with what we got in chapter 5, we will see the primary
aberration coefficients for cross cylindrical anamorphic systems are more general than
those of parallel cylindrical anamorphic systems, and the former can be reduced into the
latter by taking all radii of curvature in one principal section as zero ( in chapter 5, it is
). This makes sense because parallel cylindrical anamorphic systems are special
cases of cross cylindrical anamorphic systems.
, 0x jc =
Thus, C. G. Wynne’s result [13] is just a special case of ours. Furthermore, our
treatment does not have any restriction on location of the system stop.
114
CHAPTER 7
PRIMARY ABERRATION THEORY FOR TOROIDAL ANAMORPHIC
SYSTEMS
In chapters 5 and 6, the primary aberration coefficients for two types of cylindrical
anamorphic systems were obtained using the method we developed in chapter 2 through
chapter 4. Now, let us go a step further and consider another group of anamorphic
systems made from toroidal lenses.
To preserve structure and organization, and since we are applying the same method
onto different anamorphic system types, the development procedures in this chapter
resemble those in chapters 5 and 6.
This chapter presents the primary ray aberration coefficients for toroidal anamorphic
systems in section 7.1, and the primary wave aberration coefficients for toroidal
anamorphic systems and their simplifications in sections 7.2 and 7.3.
Notice that since a cylindrical surface is a special case of a toroidal surface with one
principal radius of curvature equal to infinity, all major results presented in this chapter
will be applicable to cylindrical anamorphic systems also.
7.1 The primary ray aberration coefficients for toroidal anamorphic systems
From chapter 2.3, we know the surface sag of the toroidal surface to a fourth-order
approximation is
115
2 2 4 2 2 4
3 21 1 2( ) (2 8x y x x y y
3 )x y x x y yzr r r r r r
= + + + + . (7-1)
From equation (4-13d) and equations (4-14), we have
22 , 1 ,
1 (2
21)j x j j y j jz c x c y= + , (7-2a)
3 3 3 2 2, 1 , , 1 1
3
1 (2
j )x j j x j y j j jj
c x c c x yαγ
= − + , (7-2b)
3 3 3 2 2, 1 , , 1 1
3
1 (2
jy j j x j y j j j
j
c y c c x y )βγ
= − + . (7-2c)
By putting equations (7-2) and equations (4-19) into equations (4-17)), we get
3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1
1 1 1 3
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , ,
[ ( ) ( ) ]
1 [( )( 2 )2
1 [( )( 22
k k kj
k x k k x j j j j j x x j x j x j j j jj j j j
k
x j x x j x x j x x j x x j x x x j x jj
x j x x j x x j y y j y y j
n u A x N z L H N h c z x n
A h H h u H u H u u
A h H h u H u
αδξ δ ψ δ
γ
ρ ρ ρ
ρ ρ ρ
= = =
=
= Δ − + + − +
= − + Δ + Δ + Δ
− + Δ + Δ +
∑ ∑ ∑
∑
, ,1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
)
1 [ ( 2 )( )]2
1 [ ( 2 )( )]2
1 ( 2 )2
k
y y y j y jj
k
x j x j x j x x j x j x x x j x x x j x x jj
k
x j y j x j y y j y j y y y j y x x j x x jj
x x x x j x x j x x x j x jj
H u u
A c h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
=
=
=
=
Δ
− + + Δ + Δ
− + + Δ + Δ
+ Δ + Δ + Δ
∑
∑
∑
2 2 2 2, , , ,
1
2 2, , , , , , , , , ,
1
3 3 2, , , , , , , , , ,
1
1 ( 2 )2
1 ( )[ ( ) (2
1 [ ( ) ( )(2
k
k
x x x y j y y j y y y j y jj
k2) ]x j x j x x j x x j x j x x j x x j y j y y j y y j j
j
k
x j x j x x j x x j x j y j x x j x x j y y j y yj
H u H u H u u
h c h H h c h H h c h H h n
h c h H h c c h H h h H h
ψ ρ ρ
ρ ρ ρ
ρ ρ ρ
=
=
=
+ Δ + Δ + Δ
− + + + +
+ + + + +
∑
∑
∑
∑ 2) ]
Δ
j jnΔ
(7-3a)
116
3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1
1 1 1 3
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , ,
[ ( ) ( ) ]
1 [( )( 2 )2
1 [( )( 22
k k kj
k y k k y j j j j j y y j y j y j j j jj j j j
k
y j y y j y y j x x j x x j x x x j x jj
y j y y j y y j y y j y y j
n u A y N z M H N h c z y n
A h H h u H u H u u
A h H h u H u
βδη δ ψ δ
γ
ρ ρ ρ
ρ ρ ρ
= = =
=
= Δ − + + − +
= − + Δ + Δ + Δ
− + Δ + Δ +
∑ ∑ ∑
∑
, ,1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
)
1 [ ( 2 )( )]2
1 [ ( 2 )( )]2
1 ( 2 )2
k
y y y j y jj
k
y j x j x j x x j x j x x x j x y y j y y jj
k
y j y j y j y y j y j y y y j y y y j y y jj
y y x x j x x j x x x j x jj
H u u
A c h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
=
=
=
=
Δ
− + + Δ + Δ
− + + Δ + Δ
+ Δ + Δ + Δ
∑
∑
∑
2 2 2 2, , , ,
1
2 2, , , , , , , , , ,
1
3 3 2 2, , , , , , , , ,
1
1 ( 2 )2
1 ( )[ ( ) (2
1 [ ( ) ( ) (2
k
k
y y y y j y y j y y y j y jj
k
y j y j y y j y y j x j x x j x x j y j y y j y y j jj
k
y j y j y y j y y j x j y j x x j x x j y y j y yj
H u H u H u u
h c h H h c h H h c h H h n
h c h H h c c h H h h H h
ψ ρ ρ
ρ ρ ρ
ρ ρ ρ
=
=
=
+ Δ + Δ + Δ
− + + + +
+ + + + +
∑
∑
∑
∑ , )]
2) ]Δ
j jnΔ
(7-3b)
By expanding and regrouping the above equations by their field and aperture
dependences, as we did in chapters 5 and 6, we get the corresponding primary ray
aberrations terms in a form similar to equations (3-8):
For Spherical Aberration-like aberration types
1 :D' ' ' 2 2 3
, 3 , , , , , ,1
' ' ', 3
1 [ ( )]2
0
k
k x k k x j x j x j x j x j x j xj
k y k k
n u A h u c h u
n u
δξ ρ
δη=
⎫= − Δ + Δ ⎪
⎬⎪= ⎭
∑ (7-4a)
2 :D
' ' ', 3
' ' ' 2 2 3, 3 , , , , , ,
1
0
1 [ ( )]2
k x k k
k
k y k k y j y j y j y j y j y j yj
n u
n u A h u c h u
δξ
δη ρ=
⎫=⎪⎬
= − Δ + Δ ⎪⎭
∑ (7-4b)
117
3 :D
' ' ' 2 2 2, , , , , , ,
1
' ' ' 2 2, , , , , , ,
1
2 2 2 2 2, , , , , ,
1 [ ( )]2
1 {[ ( )]2
( ) }
k
k x k k x j x j y j y j y j x j x yj
k
k y k k y j y j x j x j x j y jj
x j y j x j y j x j y j j x y
n u A h u c h u
n u A h u c h u
c c c c h h n
δξ ρ ρ
δη
ρ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ⎪⎪⎭
∑
∑ (7-4c)
For Coma-like aberration types
4 :D
' ' ' 2 2, 3 , , , , , ,
1
2, , , , , , , ,
' ' ', 3
1 [ (2
2 2 ) ]
0
k
k x k k x j x j x j x j x j x jj
x j x j x j x j x j x j x j x x j x x
k y k k
n u A h u c h u
h u u c h h u u H
n u
δξ
ψ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪+ Δ + Δ − Δ ⎬⎪
= ⎪⎪⎭
∑2ρ (7-4d)
5 :D
' ' ', 3 , , , , , , , ,
1
' ' ' 2 2, 3 , , , , , ,
1
2 2 2 2, , , , , , , ,
[ ( )]
1 [ ( )2
( ) ]
k
k x k k x j x j y j y j y j y j y j x j y x yj
k
k y k k y j y j x j x j x j y jj
y x j x j y j x j y j x j y j y j j y x
n u A h u u c h h u H
n u A h u c h u
u c c c c h h h n H 2
δξ ρ
δη
ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪⎪= − Δ + Δ ⎬⎪⎪−Ψ Δ + − Δ⎪⎪⎭
∑
∑
ρ
(7-4e)
6 :D
' ' ' 2 2 2 2, 3 , , , , , , ,
1
' ' ', 3 , , , , , , , ,
1
2 2 2, , , , , , ,
1 [ ( ) ]2
[ ( )
( ) ]
k
k x k k x j x j y j y j y j x j x y j x yj
k
k y k k y j y j x j x j x j x j x j y jj
x j y j x j y j x j x j y j j x x y
n u A h u c h u u H
n u A h u u c h h u
c c c c h h h n H
δξ ρ
δη
ρ ρ
=
=
⎫= − Δ + Δ −Ψ Δ ⎪
⎪⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ⎪⎪⎭
∑
∑ (7-4f)
7 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , , , , ,
1
2 2, , , , ,
0
1 [ ( 22
2 ) ]
k x k k
k
k y k k y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y y
n u
n u A h u c h u h u u
c h h u u H
δξ
δη
ρ=
⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ −Ψ Δ ⎭
∑ (7-4g)
118
For Astigmatism and Field curvature-like aberration types
8 :D
' ' ' 2 2, 3 , , , , , , , , ,
1
2, , , , , ,
' ' ', 3
1 [ ( 22
2 ) 2 ]
0
k
k x k k x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j x x
k y k k
n u A h u c h u h u u
c h h u u u H
n u
δξ
ψ ρ
δη
=
⎫= − Δ + Δ + Δ ⎪
⎪⎪+ Δ − Δ ⎬⎪
= ⎪⎪⎭
∑ (7-4h)
9 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , , , , ,
1
2, , , , , ,
0
1 [ ( 22
2 ) 2 ]
k x k k
k
k y k k y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j y y
n u
n u A h u c h u h u u
c h h u u u H
δξ
δη
ψ ρ=
⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ − Δ ⎭
∑ (7-4i)
10 :D' ' ' 2 2 2
, 3 , , , , , ,1
' ' ', 3
1 [ ( )]2
0
k
k x k k x j x j y j y j y j x j y xj
k y k k
n u A h u c h u H
n u
δξ ρ
δη=
⎫= − Δ + Δ ⎪
⎬⎪= ⎭
∑ (7-4j)
11 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 2 2 2 2, , , , , ,
0
1 [ ( )2
( ) ]
k x k k
k
k y k k y j y j x j x j x j y jj
x j y j x j y j x j y j j x y
n u
n u A h u c h u
c c c c h h n H
δξ
δη
ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭
∑ (7-4k)
12 :D
' ' ', 3 , , , , , , , ,
1
, ,
' ' ', 3 , , , , , , , , , ,
1
2 2, , , , , , , ,
[ ( )
]
[ ( )
( ) ]
k
k x k k x j x j y j y j y j y j y j x jj
x y j y j x y y
k
k y k k y j y j x j x j x j x j x j y j y x j x jj
x j y j x j y j x j x j y j y j j x y x
n u A h u u c h h u
u u H H
n u A h u u c h h u u u
c c c c h h h h n H H
δξ
ψ ρ
δη ψ
ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪− Δ⎬
= − Δ + Δ − Δ
+ − Δ
∑
∑⎪
⎪⎪⎪⎪⎭
(7-4 ) l
For Distortion-like aberration types
13 :D' ' ' 2 2 2 3
, 3 , , , , , , ,1
' ' ', 3
1 [ ( ) ]2
0
k
k x k k x j x j x j x j x j x j x x j xj
k y k k
n u A h u c h u u H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (7-4m)
119
14 :D
' ' ', 3
' ' ' 2 2 2 3, 3 , , , , , , ,
1
0
1 [ ( ) ]2
k x k k
k
k y k k y j y j y j y j y j y j y y j yj
n u
n u A h u c h u u H
δξ
δη ψ=
⎫=⎪⎬
= − Δ + Δ − Δ ⎪⎭
∑ (7-4n)
15 :D' ' ' 2 2 2 2
, 3 , , , , , , ,1
' ' ', 3
1 [ ( ) ]2
0
k
k x k k x j x j y j y j y j x j x y j x yj
k y k k
n u A h u c h u u H H
n u
δξ ψ
δη=
⎫= − Δ + Δ − Δ ⎪
⎬⎪= ⎭
∑ (7-4o)
16 :D
' ' ', 3
' ' ' 2 2 2, 3 , , , , , , ,
1
2 2 2 2, , , , , , ,
0
1 [ ( )2
( ) ]
k x k k
k
k y k k y j y j x j x j x j y j y x jj
x j y j x j y j x j y j y j j x y
n u
n u A h u c h u u
c c c c h h h n H H
δξ
δη ψ=
⎫=⎪⎪= − Δ + Δ − Δ ⎬⎪⎪+ − Δ ⎭
∑ (7-4p)
From chapter 6.1, we know in cross cylindrical anamorphic systems, for any
surface j , either ,x jc or ,y jc will be zero, thus any term containing , ,x j yc c× j will be zero.
With this point in mind, by taking 2, , 0x j y jc c = and 2
, , 0x j y jc c = , we can reduce equations
(7-4) into equations (6-3).
This means cylindrical anamorphic systems are subgroups of toroidal anamorphic
systems and all equations developed in this chapter will be valid for cylindrical
anamorphic system also.
7.2 The primary wave aberration coefficients for toroidal anamorphic systems
By comparing equations (7-4) with equations (3-8), we immediately get the primary
wave aberration coefficients though as 1D 16D
2 21 , , , , ,
1
1 (8
k
, )x j x j x j x j x j x jj
D A h u c h=
= − Δ + Δ∑ u (7-5a)
120
2 22 , , , , ,
1
1 (8
k
y j y j y j y j y j y jj
D A h u c h=
= − Δ + Δ∑ , )u (7-5b)
2 23 , , , , ,
1
1 [ ( )4
k
, ]x j x j y j y j y j x jj
D A h u c h u=
= − Δ + Δ∑ (7-5c)
2 2
4 , , , , , , ,1
2, , , , ,
1 [ ( 26
2 ) ]
k
, ,x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j
D A h u c h u h u
c h h u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ u (7-5d)
5 , , , , , , , ,1
1 [ ( )2
k
]x j x j y j y j y j y j y j x jj
D A h u u c h h u=
= − Δ + Δ∑ (7-5e)
2 26 , , , , , ,
1
1 [ ( )2
k2
, ]x j x j y j y j y j x j x y jj
D A h u c h u=
= − Δ + Δ −Ψ Δ∑ u (7-5f)
2 2
7 , , , , , , ,1
2, , , , ,
1 [ ( 26
2 ) ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
D A h u c h u h u
c h h u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (7-5g)
2 2
8 , , , , , , ,1
, , , , , ,
1 [ ( 24
2 ) 2 ]
k
, ,x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j
D A h u c h u h u
c h h u u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ u (7-5h)
2 2
9 , , , , , , ,1
, , , , , ,
1 [ ( 24
2 ) 2 ]
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j
D A h u c h u h u
c h h u u uψ=
= − Δ + Δ + Δ
+ Δ − Δ
∑ , ,u (7-5i)
2 210 , , , , , ,
1
1 (4
k
)x j x j y j y j y j x jj
D A h u c h=
= − Δ + Δ∑ u (7-5j)
2 2
11 , , , , , ,1
2 2 2 2, , , , , ,
1 [ (4
( ) ]
k
y j y j x j x j x j y jj
x j y j x j y j x j y j j
D A h u c h
c c c c h h n=
= − Δ + Δ
+ − Δ
∑ )u (7-5k)
121
12 , , , , , , , , , ,1
[ ( )k
]x j x j y j y j y j y j y j x j x y j y jj
D A h u u c h h u uψ=
= − Δ + Δ − Δ∑ u (7-5l)
2 213 , , , , , , ,
1
1 [ ( )2
k2 ]x j x j x j x j x j x j x x j
jD A h u c h u ψ
=
= − Δ + Δ − Δ∑ u (7-5m)
2 214 , , , , , , ,
1
1 [ ( )2
k
y j y j y j y j y j y j y y jj
D A h u c h u ψ=
= − Δ + Δ − Δ∑ 2 ]u (7-5n)
2 215 , , , , , , ,
1
1 [ ( )2
k2 ]x j x j y j y j y j x j x y j
jD A h u c h u ψ
=
= − Δ + Δ − Δ∑ u (7-5o)
2 2
16 , , , , , , ,1
2 2 2, , , , , , ,
1 [ ( )2
( ) ]
k
y j y j x j x j x j y j y x jj
x j y j x j y j x j y j y j j
D A h u c h u
c c c c h h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ 2u (7-5p)
Again, for , we can either choose the coefficient in the x-ray error
expression or the coefficient in the y-ray error expression. Their numerical values may
differ for any single surface
3 5 6 12, , ,D D D D
j , but the summation in the final image space is the same. In
our treatment above, the coefficient in the x-ray error expression was chosen because it
contains fewer terms.
7.3 Simplification of the results
We now have all of the primary wave aberration coefficients for a toroidal
anamorphic system in terms of the paraxial marginal and chief rays tracing data in both
symmetry planes. Again, let us use the corresponding paraxial definitions from chapter
2.11 to simplify the results we have obtained so that we can rewrite equations (7-5) in a
form as similar to the Seidel aberrations for RSOS as possible.
122
For , we find 1D
2 21 , , , , ,
1
, , , , , ,1
,2, ,
1
1 ( )8
1 ( )8
181 .8
k
,x j x j x j x j x j x jj
k
x j x j x j x j x j x jj
kx j
x j x jj j
Ix
D A h u c h
A h u u h c
uA h
n
S
=
=
=
= − Δ + Δ
= − Δ +
= − Δ
= −
∑
∑
∑
u
(7-6a)
For , we find 2D
2 22 , , , , ,
1
, , , , , ,1
,2, ,
1
1 ( )8
1 ( )8
181 .8
k
y j y j y j y j y j y jj
k
y j y j y j y j y j y jj
ky j
y j y jj j
Iy
D A h u c h
A h u u c h
uA h
n
S
=
=
=
= − Δ + Δ
= − Δ +
= − Δ
= −
∑
∑
∑
,u
(7-6b)
For , we will keep it unchanged as 3D
2 23 , , , , ,
1
1 (4
k
, ).x j x j y j y j y j x jj
D A h u h c u=
= − Δ + Δ∑ (7-6c)
For , by comparison with equation (6-4d), we know 4D
2 24 , , , , , , ,
1
2, , , , ,
,, , ,
1
1 [ ( 26
2 ) ]
121 .2
k
, ,x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j
kx j
x j x j x jj j
IIx
D A h u c h u h u
c h h u uu
A A hn
S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ
= −
∑
∑
u
(7-6d)
123
For , we will keep it unchanged as 5D
5 , , , , , , ,1
1 (2
k
, ).x j x j y j y j y j y j y j x jj
D A h u u h h c u=
= − Δ + Δ∑ (7-6e)
For , we find 6D
2 26 , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )21 ( )2
k2
, ]
.
x j x j y j y j y j x j x y jj
k
x j x j y j x j y j y j x jj
D A h u c h u
A h u A h c u
=
=
= − Δ + Δ −Ψ Δ
= − Δ + Δ
∑
∑
u (7-6f)
For , by comparison with equations (6-5g) through (6-5i), we obtain 7 8 9, ,D D D
2 27 , , , , , , ,
1
2, , , , ,
,, , ,
1
1 [ ( 26
2 ) ]
121 .2
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j
ky j
y j y j y jj j
IIy
D A h u c h u h u
c h h u uu
A A hn
S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ
= −
∑
∑
, ,u
(7-6g)
2 28 , , , , , , ,
1
, , , , , ,
,2 2, , ,
1
1 [ ( 24
2 ) 2 ]
1 (3 )41 (3 ).4
k
, ,x j x j x j x j x j x j x j x j xj
x j x j x j x j x x j x j
kx j
x j x j x x jj j
IIIx IVx
D A h u c h u h u
c h h u u uu
A h Pn
S S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + Ψ
= − +
∑
∑
ju
(7-6h)
2 29 , , , , , , ,
1
, , , , , ,
,2 2, , ,
1
1 [ ( 24
2 ) 2 ]
1 (3 )41 (3 ).4
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j
ky j
y j y j y y jj j
IIIy IVy
D A h u c h u h u
c h h u u uu
A h Pn
S S
ψ=
=
= − Δ + Δ + Δ
+ Δ − Δ
= − Δ + Ψ
= − +
∑
∑
, ,u
(7-6i)
124
For and , no change is needed, as 10D 11D
2 210 , , , , , ,
1
1 (4
k
).x j x j y j y j y j x jj
D A h u h c=
= − Δ + Δ∑ u (7-6k)
2 2 2 2 2 211 , , , , , , , , , , , ,
1 1
1 1( ) ( )4 4
k k
y j y j x j x j x j y j x j y j x j y j x j y j jj j
D A h u h c u c c c c h h n= =
= − Δ + Δ − − Δ∑ ∑ . (7-6k)
For , we find 12D
12 , , , , , , , , , ,
1
, , , , , , , , ,1
[ ( )
( )
k
]
.
x j x j y j y j y j y j y j x j x y j y jj
k
x j x j y j y j x j y j y j y j x jj
D A h u u c h h u u
A h u u A h h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
u (7-6 l )
For , , by comparing with equation (6-5m) and (6-5n), we immediately get 13D 14D
2 213 , , , , , , ,
1
3, , , , , , ,2
1
1 [ ( )2
1 1[ (21 .2
k2 ]
) ]
x j x j x j x j x j x j x x jj
k
x j x j x j x j x x j x j x jj j
Vx
D A h u c h u
A h A h A h Pn
S
ψ=
=
= − Δ + Δ − Δ
= − Δ + Ψ −
= −
∑
∑
u
(7-6m)
2 2 214 , , , , , , ,
1
3, , , , , , ,2
1
1 [ ( ) ]2
1 1[ (21 .2
k
y j y j y j y j y j y j y y jj
k
y j y j y j y j y y j y j y jj j
Vy
D A h u c h u
A h A h A h Pn
S
ψ=
=
= − Δ + Δ − Δ
= − Δ + Ψ −
= −
∑
∑ ) ]
u
(7-6n)
For , we find 15D
2 215 , , , , , , ,
1
2 2, , , , , , ,
1
1 [ ( )21 ( )2
k2 ]x j x j y j y j y j x j x y j
j
k
x j x j y j x j y j y j x jj
D A h u c h u
A h u A h c u
ψ=
=
= − Δ + Δ − Δ
= − Δ + Δ
∑
∑
u (7-6o)
125
For , we find 16D
2 216 , , , , , , ,
1
2 2 2, , , , , , ,
2 2, , , , , , ,
1
2 2 2, , , , , , ,
1
1 [ ( )2
( ) ]
1 ( )2
1 ( ) .2
k
y j y j x j x j x j y j y x jj
x j y j x j y j x j y j y j j
k
y j y j x j y j x j x j y jj
k
x j y j x j y j x j y j y j jj
D A h u c h u
c c c c h h h n
A h u A h c u
c c c c h h h n
ψ=
=
=
= − Δ + Δ − Δ
+ − Δ
= − Δ + Δ
− − Δ
∑
∑
∑
2u
(7-6p)
7.4 Summary
In this chapter, we have found the primary wave aberration coefficients for toroidal
anamorphic systems in subgroups as:
Primary wave aberration coefficients associated with x-RSOS
,21 , ,
1
18 8
kx j 1
x j x j Ixj j
uD A h
n=
= − Δ = −∑ S
,4 , , ,
1
12 2
kx j 1
x j x j x j IIxj j
uD A A h
n=
= − Δ = −∑ S
,2 28 , , ,
1
1 (3 ) (3 )4 4
kx j 1
x j x j x x j IIIx IVxj j
uD A h P S
n=
= − Δ + Ψ = − +∑ S
313 , , , , , , ,2
1
1 1[ ( ) ]2 2
k 1x j x j x j x j x x j x j x j Vx
j j
D A h A h A h Pn=
= − Δ + Ψ − = −∑ S
Primary wave aberration coefficients associated with y-RSOS
,22 , ,
1
18 8
ky j
y j y j Iyj j
uD A h
n=
= − Δ = −∑ 1 S
,7 , , ,
1
12 2
ky j
y j y j y j IIyj j
uD A A h
n=
= − Δ = −∑ 1 S
126
,2 29 , , ,
1
1 (3 ) (3 )4 4
ky j
y j y j y y j IIIy IVyj j
uD A h P S
n=
= − Δ + Ψ = − +∑ 1 S
314 , , , , , , ,2
1
1 1[ ( )2 2
k
y j y j y j y j y y j y j y j Vyj j
D A h A h A h Pn=
= − Δ + Ψ − = −∑ 1] S
Additional terms for skew rays
2 23 , , , , ,
1
1 (4
k
, )x j x j y j y j y j x jj
D A h u h c=
= − Δ + Δ∑ u
5 , , , , , , ,1
1 (2
k
, )x j x j y j y j y j y j y j x jj
D A h u u h h c u=
= − Δ + Δ∑
2 26 , , , , , ,
1
1 (2
k
, )x j x j y j x j y j y j x jj
D A h u A h c=
= − Δ + Δ∑ u
2 210 , , , , , ,
1
1 (4
k
)x j x j y j y j y j x jj
D A h u h c=
= − Δ + Δ∑ u
2 2 2 2 2 211 , , , , , , , , , , , ,
1 1
1 1( ) ( )4 4
k k
y j y j x j x j x j y j x j y j x j y j x j y j jj j
D A h u h c u c c c c h h n= =
= − Δ + Δ − − Δ∑ ∑
12 , , , , , , , , ,1
(k
)x j x j y j y j x j y j y j y j x jj
D A h u u A h h c u=
= − Δ + Δ∑
2 215 , , , , , , ,
1
1 (2
k
)x j x j y j x j y j y j x jj
D A h u A h c=
= − Δ + Δ∑ u
2 2 2 2 216 , , , , , , , , , , , , , ,
1 1
1 1( ) ( )2 2
k k
y j y j x j y j x j x j y j x j y j x j y j x j y j y j jj j
D A h u A h c u c c c c h h h n= =
= − Δ + Δ − − Δ∑ ∑
If we compare the results with what we got in chapter 6, we will see the primary
aberration coefficients for the two associated RSOS are the same for both anamorphic
system types. However, the primary aberration coefficients for the skew rays are different.
127
CHAPTER 8
PRIMARY ABERRATION THEORY FOR GENERAL ANAMORPHIC
SYSTEMS
Chapters 5, 6, and 7 give the primary aberration coefficients for anamorphic systems
made from specified types of surfaces, such as cylindrical surfaces and toroidal surfaces.
In this chapter, we will consider the general anamorphic systems constructed with
general double curvature surface type, with the allowance of fourth-order aspheric
departures in the two principal sections.
Following the same development pattern as used from chapters 5 through 7, we will
present the primary ray aberration coefficients for general anamorphic systems in section
8.1, and the primary wave aberration coefficients for general anamorphic systems and
their simplifications in sections 8.2 and 8.3.
8.1 The primary ray aberration coefficients for general anamorphic systems
From chapter 2.3, we know in general that we can write the sag equation for any
double curvature surface with a fourth-order approximation as
2 2 4 2 2 4
3 33 4 5
1 1 2( ) (2 8x y
3 )x y x x yzr r r r r
= + + + +y . (8-1)
Notice that in this equation, we permit the fourth-order aspheric departure in both
principal sections by allowing to be different from3 4 5, ,r r r ,x yr r or their combinations.
In this case, from equation (4-13d) and equations (4-14), we have
128
22 , 1 ,
1 (2j x j j y j j
21 )z c x c y= + , (8-2a)
3 2
3 1 13
3 3, 4,
(2 2
j j j
j j
13 )j
j
x x yr r
αγ
= − + , (8-2b)
3 2
3 1 13
3 5, 4,
(2 2
j j j
j j
y x yr r
βγ
= − + 13 )j
j
. (8-2c)
By putting equations (8-2) together with equations (4-19) into the anamorphic
primary ray aberration equations (4-17a) and (4-17b), we get
3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1
1 1 1 3
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , ,
[ ( ) ( ) ]
1 [( )( 2 )2
1 [( )( 22
k k kj
k x k k x j j j j j x x j x j x j j j jj j j j
k
x j x x j x x j x x j x x j x x x j x jj
x j x x j x x j y y j y y j
n u A x N z L H N h c z x n
A h H h u H u H u u
A h H h u H u
αδξ δ ψ δ
γ
ρ ρ ρ
ρ ρ ρ
= = =
=
= Δ − + + − +
= − + Δ + Δ + Δ
− + Δ + Δ +
∑ ∑ ∑
∑
, ,1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
)
1 [ ( 2 )( )]2
1 [ ( 2 )( )]2
1 ( 2 )2
k
y y y j y jj
k
x j x j x j x x j x j x x x j x x x j x x jj
k
x j y j y j y y j y j y y y j y x x j x x jj
x x x x j x x j x x x j x jj
H u u
A c h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
=
=
=
=
Δ
− + + Δ + Δ
− + + Δ + Δ
+ Δ + Δ + Δ
∑
∑
∑
2 2 2 2, , , ,
1
2 2, , , , , , , , , ,
1
3 3 3 2, 3, , , 4, , , , ,
1
1 ( 2 )2
1 ( )[ ( ) (2
1 [ ( ) ( )( ) ]2
k
k
x x y y j y y j y y y j y jj
k2) ]x j x j x x j x x j x j x x j x x j y j y y j y y j j
j
k
x j j x x j x x j j x x j x x j y y j y y jj
H u H u H u u
h c h H h c h H h c h H h n
h c h H h c h H h h H h
ψ ρ ρ
ρ ρ ρ
ρ ρ ρ
=
=
=
+ Δ + Δ + Δ
− + + + +
+ + + + +
∑
∑
∑
∑ jnΔ
Δ
(8-3a)
129
3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1
1 1 1 3
2 2 2 2, , , , , , ,
1
2 2 2 2, , , , ,
[ ( ) ( ) ]
1 [( )( 2 )2
1 [( )( 22
k k kj
k y k k y j j j j j y y j y j y j j j jj j j j
k
y j y y j y y j x x j x x j x x x j x jj
y j y y j y y j y y j y y j
n u A y N z M H N h c z y n
A h H h u H u H u u
A h H h u H u
βδη δ ψ δ
γ
ρ ρ ρ
ρ ρ ρ
= = =
=
= Δ − + + − +
= − + Δ + Δ + Δ
− + Δ + Δ +
∑ ∑ ∑
∑
, ,1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , , , , , ,
1
2 2 2 2, , , ,
1
)
1 [ ( 2 )( )]2
1 [ ( 2 )( )]2
1 ( 2 )2
k
y y y j y jj
k
y j x j x j x x j x j x x x j x y y j y y jj
k
y j y j y j y y j y j y y y j y y y j y y jj
y y x x j x x j x x x j x jj
H u u
A c h h h H h H u H u
A c h h h H h H u H u
H u H u H u u
ρ ρ ρ
ρ ρ ρ
ψ ρ ρ
=
=
=
=
Δ
− + + Δ + Δ
− + + Δ + Δ
+ Δ + Δ + Δ
∑
∑
∑
2 2 2 2, , , ,
1
2 2, , , , , , , , , ,
1
3 3 3 2, 5, , , 4, , , , ,
1
1 ( 2 )2
1 ( )[ ( ) (2
1 [ ( ) ( ) ( )]2
k
k
y y y y j y y j y y y j y jj
k
y j y j y y j y y j x j x x j x x j y j y y j y y j jj
k
y j j y y j y y j j x x j x x j y y j y y jj
H u H u H u u
h c h H h c h H h c h H h n
h c h H h c h H h h H h
ψ ρ ρ
ρ ρ ρ
ρ ρ ρ
=
=
=
+ Δ + Δ + Δ
− + + + +
+ + + + +
∑
∑
∑
∑ jnΔ
2) ]Δ
(8-3b)
By expanding and regrouping the results according to their field and aperture
dependences, as we did in chapters 5, 6 and 7, we obtain the corresponding primary ray
aberration terms in a form similar to equations (3-8) as:
For Spherical Aberration-like aberration types
1 :D
' ' ' 2 2, 3 , , , , , ,
1
3 3 4 3, 3, ,
' ' ', 3
1 {[ ( )]2
( ) }
0
k
k x k k x j x j x j x j x j x jj
x j j x j j x
k y k k
n u A h u c h u
c c h n
n u
δξ
ρ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪+ − Δ ⎬⎪
= ⎪⎪⎭
∑ (8-4a)
130
2 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
3 3 4 3, 5, ,
0
1 {[ ( )]2
( ) }
k x k k
k
k y k k y j y j y j y j y j y jj
y j j y j j y
n u
n u A h u c h u
c c h n
δξ
δη
ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭
∑ (8-4b)
3 :D
' ' ' 2 2, , , , , ,
1
2 3 2 2 2, , 4, , ,
' ' ' 2 2, , , , , ,
1
2 3 2 2 2, , 4, , ,
1 {[ ( )]2
( ) }
1 {[ ( )]2
( ) }
k
k x k k x j x j y j y j y j x jj
x j y j j x j y j j x y
k
k y k k y j y j x j x j x j y jj
x j y j j x j y j j x y
n u A h u c h u
c c c h h n
n u A h u c h u
c c c h h n
δξ
ρ ρ
δη
ρ ρ
=
=
⎫= − Δ + Δ ⎪
⎪⎪+ − Δ ⎪⎬⎪= − Δ + Δ⎪⎪
+ − Δ ⎪⎭
∑
∑
,
,
(8-4c)
For Coma-like aberration types
4 :D
' ' ' 2 2, 3 , , , , , , , , ,
1
2 3 3 3, , , , , , 3, , ,
' ' ', 3
1 [ ( 22
2 ) 3( ) ]
0
k
k x k k x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j j x j x j j x x
k y k k
n u A h u c h u h u u
c h h u u c c h h n H
n u
δξ
ψ ρ
δη
=
⎫= − Δ + Δ + Δ
2
⎪⎪⎪+ Δ − Δ + − Δ ⎬⎪
= ⎪⎪⎭
∑(8-4d)
5 :D
' ' ', 3 , , , , , , , ,
1
2 3 2, , 4, , , ,
' ' ' 2 2, 3 , , , , , ,
1
2 2 3 2, , , 4, , , ,
[ ( )
( ) ]
1 [ ( )2
( ) ]
k
k x k k x j x j y j y j y j y j y j x jj
x j y j j x j y j y j j y x y
k
k y k k y j y j x j x j x j y jj
y x j x j y j j x j y j y j j y
n u A h u u c h h u
c c c h h h n H
n u A h u c h u
u c c c h h h n H
δξ
ρ ρ
δη
ρ
=
=
= − Δ + Δ
+ − Δ
= − Δ + Δ
−Ψ Δ + − Δ
∑
∑2
x
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
(8-4e)
6 :D
' ' ' 2 2, 3 , , , , , ,
1
2 2 3 2, , , 4, , , ,
' ' ', 3 , , , , , , , ,
1
2 3 2, , 4, , , ,
1 [ ( )2
( ) ]
[ ( )
( ) ]
k
k x k k x j x j y j y j y j x jj
2x y j x j y j j x j x j y j j x y
k
k y k k y j y j x j x j x j x j x j y jj
x j y j j x j x j y j j x x
n u A h u c h u
u c c c h h h n H
n u A h u u c h h u
c c c h h h n H
δξ
ρ
δη
ρ
=
=
= − Δ + Δ
−Ψ Δ + − Δ
= − Δ + Δ
+ − Δ
∑
∑
yρ
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
(8-4f)
131
7 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , , , , ,
1
2 3 3 3, , , , , , 5, , ,
0
1 [ ( 22
2 ) 3( ) ]
k x k k
k
k y k k y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j j y j y j j y y
n u
n u A h u c h u h u u
c h h u u c c h h n H
δξ
δη
ρ=
⎫=
2
⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ −Ψ Δ + − Δ ⎭
∑ (8-4g)
For Astigmatism and Field curvature-like aberration types
8 :D
' ' ' 2 2, 3 , , , , , , , , ,
1
3 3 2 2 2, , , , , , , 3, , ,
' ' ', 3
1 [ ( 22
2 ) 2 3( ) ]
0
k
k x k k x j x j x j x j x j x j x j x j x jj
x j x j x j x j x x j x j x j j x j x j j x x
k y k k
n u A h u c h u h u u
c h h u u u c c h h n H
n u
δξ
ψ ρ
δη
=
⎫= − Δ + Δ + Δ ⎪
⎪⎪+ Δ − Δ + − Δ ⎬⎪
= ⎪⎪⎭
∑(8-4h)
9 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , , , , ,
1
3 3 2 2 2, , , , , , , 5, , ,
0
1 [ ( 22
2 ) 2 3( ) ]
k x k k
k
k y k k y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j y j j y j y j j y y
n u
n u A h u c h u h u u
c h h u u u c c h h n H
δξ
δη
ψ ρ=
⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ − Δ + − Δ ⎭
∑ (8-4i)
10 :D
' ' ' 2 2, 3 , , , , , ,
1
2 3 2 2 2, , 4, , ,
' ' ', 3
1 [ ( )2
( ) ]
0
k
k x k k x j x j y j y j y j x jj
x j y j j x j y j j y x
k y k k
n u A h u c h u
c c c h h n H
n u
δξ
ρ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪+ − Δ ⎬⎪
= ⎪⎪⎭
∑ (8-4j)
11 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 3 2 2 2, , 4, , ,
0
1 [ ( )2
( ) ]
k x k k
k
k y k k y j y j x j x j x j y jj
x j y j j x j y j j x y
n u
n u A h u c h u
c c c h h n H
δξ
δη
ρ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭
∑ (8-4k)
132
12 :D
' ' ', 3 , , , , , , , ,
1
2 3, , , , 4, , , , ,
' ' ', 3 , , , , , , , ,
1
2, , , , 4,
[ ( )
( ) ]
[ ( )
(
k
k x k k x j x j y j y j y j y j y j x jj
x y j y j x j y j j x j x j y j y j j x y y
k
k y k k y j y j x j x j x j x j x j y jj
y x j x j x j y j j
n u A h u u c h h u
u u c c c h h h h n H H
n u A h u u c h h u
u u c c c
δξ
ψ ρ
δη
ψ
=
=
= − Δ + Δ
− Δ + − Δ
= − Δ + Δ
− Δ + −
∑
∑3
, , , ,) ]x j x j y j y j j x y xh h h h n H H ρ
⎫⎪⎪⎪⎪⎬⎪⎪⎪
Δ ⎪⎭
(8-4 ) l
For Distortion-like aberration types
13 :D
' ' ' 2 2, 3 , , , , , ,
1
2 3 3 3 3, , 3, , ,
' ' ', 3
1 [ ( )2
( ) ]
0
k
k x k k x j x j x j x j x j x jj
x x j x j j x j x j j x
k y k k
n u A h u c h u
u c c h h n H
n u
δξ
ψ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪− Δ + − Δ ⎬⎪
= ⎪⎪⎭
∑ (8-4m)
14 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 3 3 3 3, , 5, , ,
0
1 [ ( )2
( ) ]
k x k k
k
k y k k y j y j y j y j y j y jj
y y j y j j y j y j j y
n u
n u A h u c h u
u c c h h n H
δξ
δη
ψ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪− Δ + − Δ ⎭
∑ (8-4n)
15 :D
' ' ' 2 2, 3 , , , , , ,
1
2 2 3 2 2, , , 4, , , ,
' ' ', 3
1 [ ( )2
( ) ]
0
k
k x k k x j x j y j y j y j x jj
x y j x j y j j x j x j y j j x y
k y k k
n u A h u c h u
u c c c h h h n H H
n u
δξ
ψ
δη
=
⎫= − Δ + Δ ⎪
⎪⎪− Δ + − Δ ⎬⎪
= ⎪⎪⎭
∑ (8-4o)
16 :D
' ' ', 3
' ' ' 2 2, 3 , , , , , ,
1
2 2 3 2 2, , , 4, , , ,
0
1 [ ( )2
( ) ]
k x k k
k
k y k k y j y j x j x j x j y jj
y x j x j y j j x j y j y j j x y
n u
n u A h u c h u
u c c c h h h n H H
δξ
δη
ψ=
⎫=⎪⎪= − Δ + Δ ⎬⎪⎪− Δ + − Δ ⎭
∑ (8-4p)
133
By taking , we can reduce equations (8-4) into
equations (7-4). This makes sense because toroidal anamorphic systems are special cases
of general anamorphic systems.
3 23, , 4, , , 5, ,, ,j x j j x j y j j yc c c c c c c= = = j
8.2 The primary wave aberration coefficients for general anamorphic systems
By comparing equations (8-4) with equations (3-8), we immediately obtain the
primary wave aberration coefficients though as 1D 16D
2 2 3 3 41 , , , , , , , 3, ,
1
1 {[ ( )] ( ) }8
k
x j x j x j x j x j x j x j j x j jj
D A h u c h u c c h=
= − Δ + Δ + − Δ∑ n (8-5a)
2 2 3 3 42 , , , , , , , 5, ,
1
1 {[ ( )] ( ) }8
k
y j y j y j y j y j y j y j j y j jj
D A h u c h u c c h=
= − Δ + Δ + − Δ∑ n (8-5b)
2 2 2 3 2 23 , , , , , , , , 4, , ,
1
1 {[ ( )] ( ) }4
k
x j x j y j y j y j x j x j y j j x j y j jj
D A h u c h u c c c h h=
= − Δ + Δ + − Δ∑ n (8-5c)
2 2
4 , , , , , , , ,1
2 3 3 3, , , , , , 3, , ,
1 [ ( 26
2 ) 3( )
k
x j x j x j x j x j x j x j x j x jj
,
]x j x j x j x j x x j x j j x j x j j
D A h u c h u h u u
c h h u u c c h h nψ=
= − Δ + Δ + Δ
+ Δ − Δ + −
∑Δ
(8-5d)
5 , , , , , , , ,
1
2 3 2, , 4, , , ,
1 [ (2
( ) ]
k
)x j x j y j y j y j y j y j x jj
x j y j j x j y j y j j
D A h u u c h h
c c c h h h n=
= − Δ + Δ
+ − Δ
∑ u (8-5e)
2 2
6 , , , , , ,1
2 3 2, , 4, , , ,
1 [ ( )2
( ) ]
k2
,x j x j y j y j y j x j x y jj
x j y j j x j x j y j j
D A h u c h u
c c c h h h n=
= − Δ + Δ −Ψ Δ
+ − Δ
∑ u (8-5f)
134
2 2
7 , , , , , , , ,1
2 3 3 3, , , , , , 5, , ,
1 [ ( 26
2 ) 3( )
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j j y j y j j
D A h u c h u h u u
c h h u u c c h h n=
= − Δ + Δ + Δ
+ Δ −Ψ Δ + −
∑ ,
]Δ
(8-5g)
2 2
8 , , , , , , , , ,1
3 3 2 2, , , , , , , 3, , ,
1 [ ( 24
2 ) 2 3( )
k
x j x j x j x j x j x j x j x j x jj
]x j x j x j x j x x j x j x j j x j x j j
D A h u c h u h u u
c h h u u u c c h h nψ=
= − Δ + Δ + Δ
+ Δ − Δ + −
∑Δ
(8-5h)
2 2
9 , , , , , , , , ,1
3 3 2 2, , , , , , , 5, , ,
1 [ ( 24
2 ) 2 3( )
k
y j y j y j y j y j y j y j y j y jj
y j y j y j y j y y j y j y j j y j y j j
D A h u c h u h u u
c h h u u u c c h h nψ=
= − Δ + Δ + Δ
+ Δ − Δ + −
∑]Δ
(8-5i)
2 2 2 3 2 210 , , , , , , , , 4, , ,
1
1 [ ( ) ( )4
k
]x j x j y j y j y j x j x j y j j x j y j jj
D A h u c h u c c c h h=
= − Δ + Δ + − Δ∑ n (8-5j)
2 2 2 3 2 211 , , , , , , , , 4, , ,
1
1 [ ( ) ( )4
k
y j y j x j x j x j y j x j y j j x j y j jj
D A h u c h u c c c h h=
= − Δ + Δ + − Δ∑ ]n (8-5k)
12 , , , , , , , , , ,
1
2 3, , 4, , , , ,
[ ( )
( ) ]
k
x j x j y j y j y j y j y j x j x y j y jj
x j y j j x j x j y j y j j
D A h u u c h h u u
c c c h h h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ u (8-5 l )
2 2
13 , , , , , , ,1
3 3 3, 3, , ,
1 [ ( )2
( ) ]
k2
x j x j x j x j x j x j x x jj
x j j x j x j j
D A h u c h u
c c h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ u (8-5m)
2 2
14 , , , , , , ,1
3 3 3, 5, , ,
1 [ ( )2
( ) ]
k
y j y j y j y j y j y j y y jj
y j j y j y j j
D A h u c h u
c c h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ 2u (8-5n)
2 2
15 , , , , , , ,1
2 3 2, , 4, , , ,
1 [ ( )2
( ) ]
k2
x j x j y j y j y j x j x y jj
x j y j j x j x j y j j
D A h u c h u
c c c h h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ u (8-5o)
135
2 2
16 , , , , , , ,1
2 3 2, , 4, , , ,
1 [ ( )2
( ) ]
k
y j y j x j x j x j y j y x jj
x j y j j x j y j y j j
D A h u c h u
c c c h h h n
ψ=
= − Δ + Δ − Δ
+ − Δ
∑ 2u (8-5p)
Again, for , we can either choose the coefficient in the x-ray error
expression or the coefficient in the y-ray error expression. Their numerical values may
differ for any single surface
3 5 6 12, , ,D D D D
j , but the summation in the final image space is the same. In
our treatment above, the coefficient in x-ray error expression was chosen because it
contains fewer terms.
8.3 Simplification of the results
We now have all of the primary wave aberration coefficients for general anamorphic
systems in terms of the paraxial marginal and chief ray trace data in the two associated
RSOS. Again, let us use the corresponding paraxial definitions from chapter 2.11 to
simplify them so that we can rewrite equations (8-5) in a form as similar to the Seidel
aberrations for RSOS as possible.
For , by comparing with equation (7-6a), we have 1D
2 2 3 3 41 , , , , , , , 3, ,
1
3 3 4, 3, ,
1
1 {[ ( )] ( ) }8
1 1 ( ) .8 8
k
x j x j x j x j x j x j x j j x j jj
k
Ix x j j x j jj
D A h u c h u c c h
S c c h n
=
=
= − Δ + Δ + − Δ
= − − − Δ
∑
∑
n (8-6a)
Notice that the additional aspheric term is exactly the same structure as the aspheric
contribution in the Seidel aberrations for x-RSOS.
For , by comparing with equation (7-6b), we immediately get 2D
136
2 2 3 3 42 , , , , , , , 5, ,
1
3 3 4, 5, ,
1
1 {[ ( )] ( ) }8
1 1 ( ) .8 8
k
y j y j y j y j y j y j y j j y j jj
k
Iy y j j y j jj
D A h u c h u c c h
S c c h n
=
=
= − Δ + Δ + − Δ
= − − − Δ
∑
∑
n (8-6b)
Again, notice that the additional aspheric term is the same structure as the aspheric
contribution in the Seidel aberrations for y-RSOS.
For , we have 3D
2 2 2 3 2 23 , , , , , , , , 4, , ,
1 1
1 1( ) ( )4 4
k k
.x j x j y j y j y j x j x j y j j x j y j jj j
D A h u h c u c c c h h= =
= − Δ + Δ − − Δ∑ ∑ n (8-6c)
For , by comparison with equation (7-6d), we have 4D
2 24 , , , , , , , , , , , , ,
1
2 3 3 3, , 3, , ,
3 3 3, 3, , ,
1
1 [ ( 2 26
3( ) ]
1 1 ( ) .2 2
k
)x j x j x j x j x j x j x j x j x j x j x j x j x jj
x x j x j j x j x j j
k
IIx x j j x j x j jj
D A h u c h u h u u c h h u
u c c h h n
S c c h h n
ψ=
=
= − Δ + Δ + Δ + Δ
− Δ + − Δ
= − − − Δ
∑
∑
(8-6d)
For , we have 5D
5 , , , , , , ,
1
2 3 2, , 4, , , ,
1
1 ( )2
1 ( ) .2
k
,x j x j y j y j y j y j y j x jj
k
x j y j j x j y j y j jj
D A h u u h h c u
c c c h h h n
=
=
= − Δ + Δ
− − Δ
∑
∑ (8-6e)
For , we have 6D
2 2
6 , , , , , ,1
2 2 3 2, , , 4, , , ,
1 [ ( )2
( )
k
x j x j y j y j y j x jj
]x y j x j y j j x j x j y j j
D A h u c h u
u c c c h h h n=
= − Δ + Δ
−Ψ Δ + − Δ
∑ (8-6f)
137
2 2, , , , , , ,
1
2 3 2, , 4, , , ,
1
1 ( )2
1 ( )2
k
.
x j x j y j x j y j y j x jj
k
x j y j j x j x j y j jj
A h u A h c u
c c c h h h n
=
=
= − Δ + Δ
− − Δ
∑
∑
For , by comparison with equations (7-6g) through (7-6i), we obtain 7 8 9, ,D D D
2 27 , , , , , , , , , , , , ,
1
2 3 3 3, , 5, , ,
3 3 3, 5, , ,
1
1 [ ( 2 26
3( ) ]
1 1 ( ) .2 2
k
y j y j y j y j y j y j y j y j y j y j y j y j y jj
y y j y j j y j y j j
k
IIy y j j y j y j jj
D A h u c h u h u u c h h u
u c c h h n
S c c h h n
=
=
= − Δ + Δ + Δ + Δ
−Ψ Δ + − Δ
= − − − Δ
∑
∑
)
(8-6g)
2 28 , , , , , , , , , , , ,
1
3 3 2 2, , , 3, , ,
3 3 2 2, 3, , ,
1
1 [ ( 2 24
2 3( ) ]
1 3(3 ) ( ) .4 4
k
, )x j x j x j x j x j x j x j x j x j x j x j x j x jj
x x j x j x j j x j x j j
k
IIIx IVx x j j x j x j jj
D A h u c h u h u u c h h u
u u c c h h n
S S c c h h n
ψ=
=
= − Δ + Δ + Δ + Δ
− Δ + − Δ
= − + − − Δ
∑
∑
(8-6h)
2 29 , , , , , , , , , , , , ,
1
3 3 2 2, , , 5, , ,
3 3 2 2, 5, , ,
1
1 [ ( 2 24
2 3( ) ]
1 3(3 ) ( ) .4 4
k
y j y j y j y j y j y j y j y j y j y j y j y j y jj
y y j y j y j j y j y j j
k
IIIy IVy y j j y j y j jj
D A h u c h u h u u c h h u
u u c c h h n
S S c c h h n
ψ=
=
= − Δ + Δ + Δ + Δ
− Δ + − Δ
= − + − − Δ
∑
∑
)
(8-6i)
For , we have 10D
2 2 2 3 2 210 , , , , , , , , 4, , ,
1 1
1 1( ) ( )4 4
k k
.x j x j y j y j y j x j x j y j j x j y j jj j
D A h u h c u c c c h h= =
= − Δ + Δ − − Δ∑ ∑ n (8-6j)
For , we have 11D
2 2 2 3 2 211 , , , , , , , , 4, , ,
1 1
1 1( ) ( )4 4
k k
y j y j x j x j x j y j x j y j j x j y j jj j
D A h u h c u c c c h h n= =
= − Δ + Δ − − Δ∑ ∑ . (8-6k)
For , we have 12D
138
12 , , , , , , , , , ,1
2 3, , 4, , , , ,
, , , , , , , , ,1
2 3, , 4, , , , ,
1
[ ( )
( ) ]
( )
( ) .
k
x j x j y j y j y j y j y j x j x y j y jj
x j y j j x j x j y j y j j
k
x j x j y j y j x j y j y j y j x jj
k
x j y j j x j x j y j y j jj
D A h u u c h h u u
c c c h h h h n
A h u u A h h c u
c c c h h h h n
ψ=
=
=
= − Δ + Δ − Δ
+ − Δ
= Δ + Δ
− − Δ
∑
∑
∑
u
(8-6l)
For , by comparing with equations (7-6m) and (7-6n), we get 13 14,D D
2 2 2 3 3 313 , , , , , , , , 3, , ,
1
3 3 3, 3, , ,
1
1 [ ( ) ( )2
1 1 ( ) .2 2
k
]x j x j x j x j x j x j x x j x j j x j x j jj
k
Vx x j j x j x j jj
D A h u c h u u c c h h
S c c h h n
ψ=
=
= − Δ + Δ − Δ + − Δ
= − − − Δ
∑
∑
n (8-6m)
2 2 2 3 3 314 , , , , , , , , 5, , ,
1
3 3 3, 5, , ,
1
1 [ ( ) ( )2
1 1 ( ) .2 2
k
y j y j y j y j y j y j y y j y j j y j y j jj
k
Vy y j j y j y j jj
D A h u c h u u c c h h
S c c h h n
ψ=
=
= − Δ + Δ − Δ + − Δ
= − − − Δ
∑
∑
]n (8-6n)
For , we have 15D
2 2 215 , , , , , , ,
1
2 3 2, , 4, , , ,
2 2 2 3 2, , , , , , , , , 4, , , ,
1 1
1 [ ( )2
( ) ]
1 1( ) ( )2 2
k
x j x j y j y j y j x j x y jj
x j y j j x j x j y j j
k k
x j x j y j x j y j y j x j x j y j j x j x j y j jj j
D A h u c h u u
c c c h h h n
A h u A h c u c c c h h h n
ψ=
= =
= − Δ + Δ − Δ
+ − Δ
= − Δ + Δ − − Δ
∑
∑ ∑
(8-6o)
For , we have 16D
2 2 216 , , , , , , ,
1
2 3 2, , 4, , , ,
2 2 2 3 2, , , , , , , , , 4, , , ,
1 1
1 [ ( )2
( ) ]
1 1( ) ( )2 2
k
y j y j x j x j x j y j y x jj
x j y j j x j y j y j j
k k
y j y j x j y j x j x j y j x j y j j x j y j y j jj j
D A h u c h u u
c c c h h h n
A h u A h c u c c c h h h n
ψ=
= =
= − Δ + Δ − Δ
+ − Δ
= − Δ + Δ − − Δ
∑
∑ ∑
(8-6p)
139
8.4 Summary
In this chapter, we have found the primary wave aberration coefficients for general
anamorphic systems in subgroups as:
Primary wave aberration coefficients associated with x-RSOS
3 3 41 , 3,
1
1 1 ( )8 8
k
Ix x j j x j jj
D S c c h=
, n= − − − Δ∑ ,
3 3 34 , 3,
1
1 1 ( )2 2
k
IIx x j j x j x j jj
D S c c h h=
= − − − Δ∑ , , n ,
3 3 2 28 ,
1
1 3(3 ) ( )4 4
k
IIIx IVx x j j x j x j jj
D S S c c h h=
= − + − − Δ∑ 3, , , n ,
3 3 313 , 3, , ,
1
1 1 ( )2 2
k
Vx x j j x j x j jj
D S c c h h=
= − − − Δ∑ n .
Primary wave aberration coefficients associated with y-RSOS
3 3 42 , 5,
1
1 1 ( )8 8
k
Iy y j j y j jj
D S c c h=
, n= − − − Δ∑ ,
3 3 37 , 5,
1
1 1 ( )2 2
k
IIy y j j y j y j jj
D S c c h h=
= − − − Δ∑ , , n ,
3 3 2 29 ,
1
1 3(3 ) ( )4 4
k
IIIy IVy y j j y j y j jj
D S S c c h h=
= − + − − Δ∑ 5, , , n ,
3 3 314 , 5, , ,
1
1 1 ( )2 2
k
Vy y j j y j y j jj
D S c c h h=
= − − − Δ∑ n .
Additional terms for skew rays
2 2 2 3 2 23 , , , , , , , , 4, ,
1 1
1 1( ) ( )4 4
k k
,x j x j y j y j y j x j x j y j j x j y j jj j
D A h u h c u c c c h h= =
= − Δ + Δ − − Δ∑ ∑ n ,
140
2 3 25 , , , , , , , , , , 4, , , ,
1 1
1 1( ) ( )2 2
k k
x j x j y j y j y j y j y j x j x j y j j x j y j y j jj j
D A h u u h h c u c c c h h h= =
= − Δ + Δ − − Δ∑ ∑ n ,
2 2 2 3 26 , , , , , , , , , 4, , , ,
1 1
1 1( ) ( )2 2
k k
x j x j y j x j y j y j x j x j y j j x j x j y j jj j
D A h u A h c u c c c h h h n= =
= − Δ + Δ − − Δ∑ ∑ ,
2 2 2 3 2 210 , , , , , , , , 4, , ,
1 1
1 1( ) ( )4 4
k k
x j x j y j y j y j x j x j y j j x j y j jj j
D A h u h c u c c c h h= =
= − Δ + Δ − − Δ∑ ∑ n ,
2 2 2 3 2 211 , , , , , , , , 4, , ,
1 1
1 1( ) ( )4 4
k k
y j y j x j x j x j y j x j y j j x j y j jj j
D A h u h c u c c c h h n= =
= − Δ + Δ − − Δ∑ ∑ ,
2 312 , , , , , , , , , , , 4, , , , ,
1 1
( ) ( )k k
x j x j y j y j x j y j y j y j x j x j y j j x j x j y j y j jj j
D A h u u A h h c u c c c h h h h n= =
= Δ + Δ − −∑ ∑ Δ ,
2 2 2 3 215 , , , , , , , , , 4, , , ,
1 1
1 1( ) ( )2 2
k k
x j x j y j x j y j y j x j x j y j j x j x j y j jj j
D A h u A h c u c c c h h h= =
= − Δ + Δ − − Δ∑ ∑ n ,
2 2 2 3 216 , , , , , , , , , 4, , , ,
1 1
1 1( ) ( )2 2
k k
y j y j x j y j x j x j y j x j y j j x j y j y j jj j
D A h u A h c u c c c h h h= =
= − Δ + Δ − − Δ∑ ∑ n .
If we compare the results with chapter 7, we will find that the newly added terms
have similar structures as the aspheric terms of an RSOS.
For all the anamorphic primary aberration coefficient expressions derived from
chapters 5 through 8, we can not emphasize more on the fact that all parameters shown
up in these expressions are the paraxial marginal and chief rays’ tracing data in the two
associated RSOS, together with other first-order constants and definitions. Thus, for
anamorphic primary aberration calculation purpose, we only need to trace the four non-
skew marginal and chief rays, in the associated x-RSOS and y-RSOS. Hence, our results
are indeed similar to the Seidel aberrations of RSOS.
141
CHAPTER 9
TESTING OF THE RESULTS
In chapters 5 through 8, we found both the primary ray aberration coefficients and
the primary wave aberration coefficients for the most common types of anamorphic
systems: from the simplest parallel cylindrical anamorphic systems to the most general
anamorphic systems made from general double curvature surfaces with the allowance of
four order aspheric departures. In this chapter, we will provide a testing scheme for the
primary aberration coefficient expressions developed.
Generally speaking, there are two common methods of testing the validity of a
theoretical result. The first one is to check the analytical development procedures. If none
of the steps in getting the final result can logically be disputed, the analytical result is
mathematically sound. The second method is to verify the results numerically. Nowadays,
it is very easy to setup different anamorphic systems using computer simulation software,
like ZEMAX or CODE V. The simulation software has the ability to trace real rays thus
providing the actual aberrations, and we can then compare the theoretical results
calculated from our expressions with the actual ray trace data and verify the validity of
our results.
Here we will make use of the ZEMAX program for the follows:
1) We will first assume that the analytical expressions we obtained are exact, and we will
use those expressions to calculate the primary wave aberration coefficients for any
arbitrary given anamorphic system;
142
2) We will then calculate the same primary wave aberration coefficients for the given
anamorphic system using numerical real ray-tracing and data fitting;
3) Once we get both results, we will compare them and check the percentage errors
between the theoretical results and the real ray trace fitted results, thus verify the
validation of our analytical expressions.
This chapter presents the idea of data fitting in section 9.1, the detailed primary
aberrations coefficients data fitting steps in section 9.2, and an actual testing example in
section 9.3.
9.1 The idea of data fitting
From the general aberration theory described in chapter 3.3, we know the aberration
function can be expanded into a power series with respect to aperture and field
variables. The expansion expression is actually the aberration data fitting formula, which
tells that we can use the first-order, the third-order, the fifth order, and higher order terms
to represent the total aberration of a ray in an anamorphic system.
W
Since primary ray aberrations and primary wave aberrations are related by the
derivatives of the aperture variables, we can choose to either fit the real ray errors or the
real OPD error. Here we choose to fit the real ray errors since they are more
straightforward to calculate in ray tracing programs.
When a ray is traced through a given anamorphic system form a choosing object field,
as it meets the final ideal image plane k , we can easily obtain the coordinates ( ,k kξ η ) of
the meeting point using the ray tracing program.
143
It is also easy to obtain the coordinates ( 0,k k 0ξ η ) of the ideal final image point for the
choosing object field by tracing the paraxial chief ray, which passes through the center of
the system stop.
Once we obtain these data from ray tracing, we can calculate the numerical value of
the x-component and y-component of the ray error in the final ideal image plane using
0k kδξ ξ ξ= − , (9-1a)
0k kδη η η= − . (9-1b)
Notice that equations (9-1a) and (9-1b) are total ray errors, which means that besides the
primary ray errors, the higher order ray errors are also included in these equations.
Theoretically, we can now use the first-order, the third-order, the fifth order and
higher order aberration terms to fit the data calculated in equations (9-1a) and (9-1b). But
this task is harder than we might have thought. In an anamorphic system, for the third-
order aberrations alone, we will have 16 terms already. Then how many terms will there
be for the fifth order, seventh order, ninth order and even higher orders? The answer
might be hundreds or even thousands. Thus it is not so practical to apply the data fitting
idea on equations (9-1) directly, and we need to go one step further.
In the deduction of this work, since we always use the ideal image point as our
reference, we know the first-order terms in the fitting expressions must always be zero.
We also know that, due to the higher order dependence of aperture and field, as we
scale both the system aperture and field down, the fifth and higher order aberration terms
will vanish very rapidly and will leave equations (9-1a) and (9-1b) with very little
contribution from the higher order aberration terms. Thus, under small aperture and field
144
condition, we can treat equations (9-1) as if all contributions are from the third-order
aberration terms only.
From equations (3-7a) and (3-7b), we know the primary ray error fitting equations
are
3 2 2 2
1 3 4 5 6 8
2 3 210 12 13 15 ,
(4 2 3 2 2
2 )
2
/ ' ' ,x x y x x y x y x y x x
y x x y y x x y x k
D D D H D H D H D H
D H D H H D H D H H n u
δξ ρ ρ ρ ρ ρ ρ ρ
ρ ρ
= + + + + +
+ + + +
ρ
2
' ' .y
(9-2a)
3 2 2 2
2 3 5 6 7 9
2 3 211 12 14 16 ,
(4 2 2 3 2
2 ) /y x y y x x x y y y y
x y x y x y x y y k
D D D H D H D H D H
D H D H H D H D H H n u
δη ρ ρ ρ ρ ρ ρ ρ
ρ ρ
= + + + + +
+ + + +
ρ (9-2b)
From the discussion above, for any giving anamorphic imaging system, we can scale
down the system aperture and field so that we can treat the ray errors calculated from
equations (9-1) as if they are all primary ray errors. Thus we can use equations (9-2a)
and (9-2b) to fit them and obtain the primary aberration coefficients through with
high accuracy.
1D 16D
We know as we scale down the system aperture and field, the primary aberration
coefficients will be scaled down too, but it is not a concern since they are just linearly
scaled. What important to us is that as the aperture and field getting smaller, the higher
order aberration terms will be vanishing rapidly, thus the total ray errors are approaching
the primary ray errors rapidly. And the net result should be that the fitted aberration
coefficients are approaching the theoretical calculated primary aberration coefficients—if
the analytical expressions we developed in chapters 5 through 8 are accurate.
9.2 The detailed primary aberration coefficients data fitting steps
145
Now we will describe the primary aberration coefficients data fitting steps in great
details. For any given anamorphic system, it is very easy to trace the four paraxial rays,
namely the x-marginal ray, the x-chief ray, the y-marginal ray, and the y-chief ray, in the
corresponding associated x-RSOS and y-RSOS, and find the corresponding paraxial
quantities, like ,' , ' ,x k yu u k used in equations (9-2), etc.
By examining equations (9-2a) and (9-2b) carefully, we can find the following
optimal steps in calculating the primary aberration coefficients, one by one.
1) By taking and(0,0)H = (1,0)ρ = , from equation (9-2a), we have
3
1 , 1
1 ,
4 / ' ' 4 / ' '( ' ' ) / 4.
,x x k x k
x k
D n u D n uD n u
δξ ρδξ
= =
⇒ = (9-3a)
For this specified field and aperture, we trace the real ray though the anamorphic system
and calculate δξ from equations (9-1) and then we can calculate the value of . 1D
2) Similarly, by taking (0,0)H = and (0,1)ρ = , from equation (9-2b), we have
(9-3b) 3
2 , 2
2 ,
4 / ' ' 4 / ' '
( ' ' ) / 4.y y k
y k
D n u D n u
D n u
δη ρ
δη
= =
⇒ =,y k
By applying exactly the same method as described in the calculation of , we can
calculate the value of .
1D
2D
3) By taking , (0,0)H = ( 2 / 2, 2 / 2)ρ = , we have
3 21 3 , 1 3(4 2 ) / ' ' (4 2 ) 2 / 4 ' ' ,x x y x k x kD D n u D D n uδξ ρ ρ ρ= + = + .
Since we already know from step 1), we can calculate 1D
3 ,2 ' ' 2x kD n u δξ= 1D− . (9-3c)
146
4) By taking = (1,0), H ρ = (0,0) ,we have
313 , 13 ,/ ' ' / ' 'x x k x kD H n u D n uδξ = = ,
so we can calculate
13 ,' 'x kD n u δξ= . (9-3d)
5) By taking = (0,1), H ρ = (0,0), we have
, 314 , 14 ,/ ' ' / ' 'y y kD H n u D n uδη = = y k
so we can calculate
14 ,' ' y kD n u δη= . (9-3e)
6) By taking = (H 2 / 2 , 2 / 2 ), ρ = (0,0) , we have
3 213 15 , 13 15 ,( ) 2 / 4 / ' ' ( ) 2 / 4 ' 'x x y x k x kD H D H H n u D D n uδξ = + = + ,
3 214 16 , 14 16 ,( ) 2 / 4 / ' ' ( ) 2y x y y kD H D H H n u D D n uδη = + = + / 4 ' ' y k .
Since we already know , in steps 4) and 5), we can calculate 13D 14D
15 , 132 2 ' 'x kD n u δξ= D− , (9-3f)
16 , 142 2 ' ' y kD n u δη= D− . (9-3g)
7) By taking = (0, 1), H ρ = (1, 0), we have
3 21 10 , 1 10(4 2 ) / ' ' (4 2 ) / ' ' ,x y x x k x kD D H n u D D n uδξ ρ ρ= + = + ,
. 2 35 14 , 5 14( ) / ' ' ( )y x y y k y kD H D H n u D D n uδη ρ= + = + ,/ ' '
14
And since we know already, we can calculate 1 14,D D
5 ,' ' y kD n u Dδη= − , (9-3h)
147
10 , 1( ' ' 4 ) / 2x kD n u Dδξ= − . (9-3i)
8) By taking = (1, 0), H ρ = (0, 1), we have
2 36 13 , 6 13( ) / ' ' ( ) ,/ ' 'x y x x k x kD H D H n u D D n uδξ ρ= + = + ,
. 3 22 11 , 2 11(4 2 ) / ' ' (4 2 ) / ' 'y x y y kD D H n u D D n uδη ρ ρ= + = + ,y k
3
13
And since we know already, we can calculate 2 1,D D
6 ,' 'x kD n u Dδξ= − , (9-3j)
11 , 2( ' ' 4 ) / 2y kD n u Dδη= − . (9-3k)
9) By taking = (H 2 / 2 , 2 / 2 ), ρ = (1, 0), we have
2 3 2
5 12 14 16
5 12 14 16 ,
( )
( 2 / 2 / 2 2 / 4 2 / 4) / ' ' .y x x y x y x y y k
y k
D H D H H D H D H H n u
D D D D n u
δη ρ ρ= + + +
= + + +
,/ ' '
And since we know already, we can calculate 5 14 16, ,D D D
12 , 5 14 162 ' ' 2 ( ) 2 / 2y kD n u D D Dδη= − − + . (9-3 l )
The calculations of the remaining 4 terms are not as simple as the
calculation of other terms because they are always coupled together. To separate them,
we need to make use of multiple field points.
4 7 8, , ,D D D D9
10) By taking = (1, 0), H ρ = (1, 0), we have
3 2 2 3
1 4 8 13
1 4 8 13 ,
(4 3 2 ) / ' '(4 3 2 ) / ' ' .
,x x x x x x x k
x k
D D H D H D H n uD D D D n u
δξ ρ ρ ρ= + + +
= + + + (9-4a)
Since we know already, equation (9-4a) is a linear equation of variables .
Now by taking = (0.5, 0),
1 13,D D 4 8,D D
H ρ = (1, 0), we have
148
3 2 2 3
1 4 8 13
1 4 8 13 ,
(4 3 2 ) / ' '(4 3 / 2 / 2 /8) / ' ' .
,x x x x x x x k
x k
D D H D H D H n uD D D D n u
δξ ρ ρ ρ= + + +
= + + + (9-4b)
Equation (9-4b) is a linear equation of variables too, so by combining with
equation (9-4a), we can calculate .
4 ,D D8
4 8,D D
11) By taking = (0, 1), H ρ = (0, 1), we have
(9-5a) 3 2 2 3
2 7 9 14
2 7 9 14 ,
(4 3 2 ) / ' '
(4 3 2 ) / ' ' .y y y y y y
y k
D D H D H D H n u
D D D D n u
δη ρ ρ ρ= + + +
= + + +,y k
Since we know already, equation (9-5a) is a linear equation of variables .
Now by taking = (0, 0.5),
2 14,D D 7 9,D D
H ρ = (0, 1), we have
(9-5b) 3 2 2 3
2 7 9 14
2 7 9 14 ,
(4 3 2 ) / ' '
(4 3 / 2 / 2 /8) / ' 'y y y y y y
y k
D D H D H D H n u
D D D D n u
δη ρ ρ ρ= + + +
= + + +,y k
Equation (9-5b) is a linear equation of variables too, so by combining with
equation (9-5a), we can calculate .
7 9,D D
7 9,D D
Following the above detailed steps, we can find the fitted expansion coefficients
though using real ray tracing data from different field and aperture point, providing
that the aperture and field are small.
1D
16D
9.3 A testing example
From the discussion presented in sections 9.1 and 9.2, we know that if the two
conditions described below are satisfied, the analytical expressions under test must be
accurate:
149
1) The primary aberration coefficients though calculated from the theoretical
expressions developed in chapters 5 through 8 are very close to the fitted though ,
for any corresponding anamorphic system type with small aperture and field;
1D 16D
1D 16D
2) Furthermore, as we gradually decrease the aperture and field, the fitted primary
aberration coefficients are approaching the theoretically calculated results.
In practice, the calculation of the primary aberration coefficients from the theory and
from the data fitting can be written into a ZEMAX macro and be applied onto different
anamorphic systems easily. The author of this work has tested many different anamorphic
systems using such a macro and proved that the expressions developed in chapters 5
through 8 are accurate.
Here, we will provide a simple testing example to illustrate the ideas. Consider an
anamorphic system in which two toroidal lenses together with a spherical lens, forms an
anamorphic image with an anamorphic ratio of 4:3. The effective focal length in the x-z
principal section is 40mm, and in the y-z principal section it is 30mm. Figure 9-1 below
illustrates the layouts in both principal sections. Notice that the system is uncorrected
with large aberrations. Table 9-1 lists the lens data. Figure 9-2 is the grid distortion map,
which shows the 4:3 image aspects.
Figure 9-1 Layout of a simple anamorphic system in the y-z (left) and x-z principal sections
150
Table 9-1 Lens data
Figure 9-2 Grid distortion map
We will explore the effect of scaling down the system aperture and field while
keeping all other system parameters fixed.
In Table 9-2, the full system aperture is 10mm, and the half field of view (HFOV) is
10 degrees.
In Table 9-3, the system aperture is decreased to 4mm and the HFOV is 4 degrees.
In Table 9-4, the system aperture is further decreased to 1mm and the HFOV is 1
degree.
In each case, we will compare the fitted primary aberration coefficients with the
theoretic results calculated from the analytical aberration coefficient expressions.
151
Table 9-2 Full aperture is 10mm, HFOV is 10 degree
From table 9-2, we see that for this system, even though the field and aperture are not
really small, the fitted data are actually not too far away from their corresponding
theoretical results, with a minimum accuracy of 94.9%.
152
Table 9-3 Full Aperture is 4mm, HFOV is 4 degree
From table 9-3, we clearly see that as the field and aperture are getting smaller, the
fitted results are approaching the theoretical results very rapidly, with a minimum
accuracy of 99.18% now.
153
Table 9-4 Full Aperture is 1mm, HFOV is 1 degree
From table 9-4, we see that for a small field and aperture, the fitted data can hardly
be distinguished from the theoretical results, with a minimum accuracy of 99.948% now.
154
From this arbitrarily chosen example presented above, we indeed see that as the
aperture and field reduces in each instance, the theoretical results and the data fitted
results approach each other very rapidly and they will finally reach a level with extremely
high accuracy. This validates our theoretical expressions.
9.4 Summary
In this chapter, we provided a testing scheme for the expressions we had obtained in
chapters 5 through 8.
We applied the testing method described above to many kinds of arbitrary
anamorphic systems and the conclusions coming out are always similar. It is clear then
that the theoretical expressions we developed in chapters 5 through 8 must be exact.
155
CHAPTER 10
DESIGN EXAMPLES
In the previous chapters, the complete monochromatic primary aberration theories for
most common types of anamorphic systems were presented. In this chapter, we will
present some design examples that illustrate the use and value of the theoretical results
developed in this work. The design examples given here will only cover a small part of
all of the possible anamorphic designs. However, they provide some insight into the
usefulness of the theoretical results.
In current practice, almost all anamorphic systems are designed using cylindrical
lenses because they are easy to manufacture, and as a result, they are cost-efficient.
However, since a cylindrical surface is a special case of toroidal surfaces, this chapter
will primarily use toroidal surfaces to illustrate the general design idea instead of being
limited to cylindrical anamorphic systems.
We will explore the design possibilities with increasing complexity. In section 10.1,
we will discuss the simplest anamorphic imaging system—an anamorphic singlet. In this
section, the basic ideas of anamorphic image formation will be presented.
In section 10.2, the idea of afocal attachments will be discussed. The simplest single
lens afocal attachment working with a RSOS landscape lens will be presented.
In section 10.3, we will make the renowned RSOS Cooke Triplet into its anamorphic
form.
In section 10.4, we will discuss the design idea of anamorphic field lens.
156
In section 10.5, we will make the classic Double Gauss lens into its anamorphic form,
with the help of an anamorphic field lens. And finally, in section 10.6, we will explore
the design of an anamorphic fisheye lens.
As described in chapter 1, since there is no unique entrance and exit pupil in an
anamorphic system, in all the design examples presented in this chapter, "ray aiming" to
the real stop are always turned on and the system aperture is set to "float by stop size"
[29].
Together with the layouts in both principal sections, we will primarily use the spot
diagrams to assess system performance because they are generated via sampling all over
the aperture, rather than sampling two cross-sections of the aperture only, as in the ray
fan plot case. This kind of all aperture sampling is very important for an anamorphic
system due to the lacking of rotational symmetry.
Sometimes the ray fan plot, the OPD fan, and the modulation transfer function (MTF)
will also be utilized as system performance assessment tools even though they can only
partially represent the non-symmetrical system performance.
All examples shown in this chapter will be designed using the ZEMAX program.
However, the design ideas can be applied to any other lens design program as well.
10.1 An anamorphic singlet
From the discussion in chapter 1, we know that a single double curvature surface can
not form a unique image, and thus is not an imaging system. The simplest anamorphic
system we can imagine is an anamorphic singlet which has two double curvature surfaces.
157
For an anamorphic singlet with the stop at the lens, there are only four independent
degrees of freedom that are effective in controlling aberrations and focal length. These
variables are the four lens curvatures, or alternatively, two powers and two bending
factors. To minimize longitudinal color, a low dispersion glass will be used. Lens
thickness is an ineffective variable, therefore a reasonably thin lens is assumed.
With four degrees of freedom, only four optical properties can be controlled. The two
powers are always used to fix the effective focal lengths in both principal sections thus
determining the anamorphic ratio.
Normally, we want to use the two elemental bending factors to control and ,
which are the spherical aberrations in the two principal sections.
1D 2D
But for the anamorphic system to form a unique image, we need a special first-order
condition which requires that the back focal lengths (BFL) in both principal sections
should equal each other so that the final image planes in both principal sections can
coincide. This image forming condition is also accomplished by proper bending in both
principal sections, thus it will make the two bending variables no longer independent,
which means we can not control and simultaneously. 1D 2D
However, we will have some flexibility here because the BFL requirement and the
spherical aberrations controlling are both done by lens bending. Thus the two bending
variables can be so selected to satisfy the BFL requirement and to achieve balanced
reasonably small spherical aberrations in both principal sections. By this way, even
though we do not have enough degrees of freedom to control and simultaneously, 1D 2D
158
we will minimize their cumulative effect while satisfying the unique image forming
requirement.
From the discussion above, we see that for the four degrees of freedom, the best
result one can achieve is an anamorphic imaging system with specified anamorphic ratio,
with somewhat well balanced spherical aberrations in both principal sections.
In the example presented here, the anamorphic singlet lens with the stop at the lens
will be optimized to minimize the RMS spot size monochromatically, on axis only. The
glass is Schott BK7, which has low dispersion to reduce the color effects. The Object is at
infinity. The wavelengths are the visible F, d, and C lights. The reference wavelength is
the d light. Other specifications are as listed in Table 10-1 below.
Figure 10-1 below is the layout of the optimized lens in the two principal sections.
Notice that BFL in both principal sections must be exactly the same if we are to assure a
unique image is formed. Also notice that the off-axis ray bundles indicate a large amount
of field curvature in both principal sections.
Figure 10-2 is the on-axis ray fan and spot diagram. Notice that from the ray fan plot,
the ray errors are the same in both Sagittal and Tangential directions, thus the on-axis
spot should be circular if we were considering an RSOS case. But the actually on-axis
spot is in a star shape due to the non-RSOS nature of our system. This example clearly
illustrates why we will primarily use the spot diagram instead of ray fan (or OPD fan or
MTF) as the major system performance assessment tool in this chapter.
Also notice the fact that since lens bending is done in such a way that the spherical
aberrations in both principal sections are well balanced, the on-axis spot is quite uniform.
159
Table 10-1 Specifications of the anamorphic singlet
Figure 10-1 Layout in the y-z (left) and x-z (right) principal sections
Figure 10-2 On-axis system performance
As we can see from the spot diagram, the image quality for this anamorphic singlet is
not very good even on-axis. For this slow system, the RMS spot size is on the scale of
160
70 mμ . This is a consequence of insufficient variables to correct all three on-axis
spherical-type aberrations and the longitudinal color. 1 2, ,D D D3
The lens data and the remaining primary aberration coefficients (at a half field of 10
degrees) are shown in Table 10-2 below.
Table 10-2 Lens data and remaining primary aberration coefficients
From the primary aberration coefficients presented above, it is quite clear that the
various off-axis astigmatism and field curvature terms are limiting the off-axis
performance.
Notice that each toroidal surface presented above has curvatures which are different
in sign in each principal section. This kind of surface may be difficult to make, but they
serve well in illustrating the design ideas.
161
Recall that in the classical landscape lens design, the stop location is used as an
effective variable to control the tangential field curvature. However, in this anamorphic
singlet design, since the field curvatures for either the principal sections or the skew rays
are not the same, varying the stop location can not correct the various field terms
simultaneously. Thus, the landscape lens design idea will not work effectively here.
10.2 An afocal anamorphic attachment
Ever since Chrétien published his afocal anamorphic attachment design patent in
1934, considerable research interest has focused on this group of design configuration. To
date, the majority of the existing anamorphic systems can be classified under the same
heading—an anamorphic afocal attachment combined with a standard optical imaging
system for spherical power [30-37].
There are several reasons for this phenomenon. One reason may be that most
anamorphic systems have their applications in Cinemascope, where anamorphic
attachments with different anamorphic ratios might be required.
Another reason lies in aberration controlling. In the early years of optics, people did
not have a full understanding of the aberration behavior in anamorphic systems. An
afocal attachment working in substantially collimated space with weak power will
introduce less aberration from the very beginning. Thus, this configuration can benefit the
aberration controlling process, as described by C. G. Wynne in his 1956 paper [13].
Afocal anamorphic attachments have another advantage. When pointing at an object
in infinite, the entering collimated beam will exit the anamorphic attachment lens as a
162
substantially collimated beam having a different beam shape. This allows the anamorphic
attachment lens to be attached to an ordinary imaging lens of a camera without affecting
the position of the image plane. Thus, no refocusing of the camera is necessary when the
anamorphic attachment lens is applied or when the attachment is rotated relative to the
camera [38].
Some nice discussions on this common configuration can be found in G. H. Cook’s
book [17], R. Kingslake’s pattern [39], and other place [40].
In this section, we will present a simple yet interesting enough single lens afocal
anamorphic attachment design example to illustrate the idea.
We design the attachment as a single toroidal lens, using Schott BK7 glass. Again,
we have four independent degrees of freedom for the lens (two powers and two bending
factors). We start the design by separately bending the front and rear surfaces in each
principal section to make them afocal, respectively. This will consume two degrees of
design freedom, one for each section. The afocal property in each principal section is
achieved by requiring that the corresponding paraxial marginal ray exit angle be equal to
zero, for a collimated incident beam.
We also need to control the exit beam diameter in each section so that we can control
the required anamorphic ratio. This will consume two degrees of design freedom too,
one for each section. The exit beam diameter in each principal section is controlled by its
corresponding paraxial marginal ray exit height at the rear surface of the attachment.
From the discussion presented above, we see that all four design degrees of freedom
are utilized in order to control the first-order properties of the attachment. We do not have
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any degree of freedom left to control aberrations in the attachment design. However, a
weak powered afocal design itself will introduce very little aberration so long as the
surface curvatures are not too strong.
Normally the glass thickness is not an effective design variable, but in the afocal
design example described here, glass thickness will have a strong effect on the resulting
surface curvatures: For the same exit beam diameter, the thicker the glass, the less
curvature will be required for the surfaces. Thus, the glass thickness is actually of critical
importance for aberration controlling in this sense. But as the glass thickens, the weight
of the attachment will increase rapidly. Thus, under practical consideration, the glass
thickness is chosen to be 5mm here.
Figure 10-3 shows the attachment in both principal sections. Notice that the exiting
ray bundle in each section is collimated with different diameters.
Figure 10-3 Layout in the y-z (left) and x-z (right) principal sections
To make the system more practical, we choose the major spherical objective to be a
front stop landscape lens with a 20 degrees half field of view, and with an EFFL (the
effective focal length) of 50mm. The landscape lens is working at F/10. The glass is
again Schott BK7, and the glass thickness is chosen to be 3mm.
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In the combined system, the object distance is infinity. The stop is located on the
back surface of the toroidal lens. The wavelengths are the visible F, d, and C lights. The
reference wavelength is the d light. Other system specifications are listed in Table 10-3.
Table 10-3 Specifications of the anamorphic system
Figure 10-4 Layout in the y-z (left) and x-z (right) principal sections
Figure 10-5 Spot diagram
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Figure 10-6 Ray fan
Table 10-4 Lens data and remaining primary aberration coefficients
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Figure 10-4 above shows the system layout after the afocal toroidal lens is attached to
the landscape lens. Figure 10-5 is the spot diagram. Figure 10-6 is the ray fan. Table 10-4
is the lens data and the remaining primary aberration coefficients.
System distortion is less than 2.3% at maximum field. Since distortion does not
decrease the image sharpness, for a simple two-element anamorphic landscape lens, this
distortion level should be acceptable.
From the system layout, we see that the field curvatures in both principal sections are
not as severe as in the anamorphic singlet design presented in section 10.1. However,
from the spot diagram, it is clear that as the field size increases, various astigmatism and
field curvature terms will increase rapidly, and will thus limit the off-axis performance.
This issue can also be seen from the primary aberration coefficients listed above.
Therefore, this design example actually reveals a critically important aspect of
anamorphic system design: Because the system powers, together with the power
distribution for both principal sections and for the skew rays, are quite different in
anamorphic systems, it is most likely the various astigmatism and field curvature terms
will be the limiting off-axis aberrations for this kind of system.
Again, notice that each toroidal surface presented above has curvatures which are
different in sign in each principal section. This kind of surface may be difficult to make,
but they serve well in illustrating the design idea.
If we allow the attachment to be slightly focal and also allow some defocus of the
system, we can get a slightly better design result than the one presented above.
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10.3 An anamorphic Triplet design
In sections 10.1 and 10.2, the design examples presented are uncomplicated, and yet
they illustrate the anamorphic design ideas well. The biggest problem with single element
anamorphic systems is that the anamorphic ratio is controlled by the bending of only two
surfaces. For a low anamorphic ratio, such as 1.1 to 1.2, it may work fine. However, as
the anamorphic ratio increases, the curvature of each surface will soon becomes too
strong, thereby increasing the amount of aberrations it generates to an unacceptable level.
To achieve a higher anamorphic ratio and better image quality, we have to develop
more complex configurations so that the system can have greater degrees of freedom for
aberration controlling. In this section, we will expand our design ideas to the anamorphic
triplet design.
The RSOS Cooke Triplet is a very famous design form invented by H. D. Taylor in
1893.It consists of two positive singlet elements and one negative singlet element, all of
which can be thin. Two sizable airspaces separate the three elements. The negative
element is located in the middle about halfway between the positive elements, thus
maintaining a large amount of symmetry [41,42].
The RSOS Cooke Triplet is an optical configuration that has eight degrees of
freedom. The major variables are six lens surface curvatures and two inter-element air
spacing.
Stop shift is not a degree of freedom for the classical Cooke Triplet design because
the design is nearly symmetrical about the middle element, which makes aberration
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control much easier. To retain as much symmetry as possible, the stop is normally located
at the middle element.
The aberrations that need to be controlled for RSOS are the five monochromatic
third-order Seidel aberrations, together with the first-order longitudinal and lateral color.
Thus, the RSOS Cooke Triplet has just enough degrees of freedom available to correct all
first and third-order aberrations while obtaining the desired system focal length.
But for an anamorphic triplet design, we will not have enough degrees of freedom to
correct all sixteen anamorphic primary aberrations plus the chromatic aberrations. To be
more precise, we do not even have enough degrees of freedom to correct the primary
aberrations associated with the two principal sections.
Here is the reasoning: Suppose we say one of the principal sections has been fully
corrected. This corrected principal section will now fix both the two inter-element
spacing and the BFL. Thus, for the other principal section, we only have six surface
curvature variables available now. Unfortunately, for these six degrees of freedom, we
need to control BFL in this section to guarantee that a unique image is formed, and we
also need to control EFFL in this section to satisfy the required anamorphic ratio. Thus,
in general, we will have only four degrees of freedom left, which is simply not enough
for aberration control in the second principal section alone. This is said without
mentioning the skew aberrations yet! The difficulty seems insurmountable.
Due to the lack of variables in the anamorphic triplet design procedure, we must pay
close attention to the system performance balancing so that the aberrations for rays lying
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in the symmetry planes and the aberrations for skew rays will not adversely influence
each other too much.
It is also clear that due to the lack of variables, as the system departs further from
rotational symmetry (as the anamorphic ratio increases), the system performance will
generally decrease. In this section, we will briefly explore the effect of increasing
anamorphic ratio on system performance.
10.3.1 General methods for discovering an anamorphic beginning design
When designing a lens, the first step that involves the computer is entering the
starting design. In computer aided lens design, the starting point is of crucial importance
since if the starting point is far from a good solution, the merit function will soon fall into
a local minimum and stop, often leaving the designer with a less than optimal result.
Generally speaking, for any toroidal anamorphic system with a specified
configuration and anamorphic ratio, there are two methods for establishing a reasonably
good starting design. The first method can be described as follows:
1) Start from a well-corrected RSOS design with the specified configuration;
2) Set the BFL in both principal sections to the same value, which will guarantee that a
unique final image is formed;
3) Increase the EFFL target in one principal section by a small amount, and decrease the
EFFL target in the other principal section by the same amount. By doing so, the overall
system effective F/No is substantially fixed, yet the anamorphic ratio is slightly altered;
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4) Optimize the system using the default merit function to shrink the RMS spot size. At
this step, we may also consider adding some special operands to control specified
aberrations, such as distortion, etc;
5) Repeat steps 3 and 4 several iterations. Each time the EFFL targets in both principal
sections will be slightly changed in the same direction as in the previous steps, until the
specified anamorphic ratio is finally met. Once the starting design is achieved, the further
optimization process can begin.
The above iterative process has the effect of keeping us in the same general solution
region, avoiding radical departures from the general design form. By doing so, we can
find a reasonably good starting anamorphic design from a well-corrected RSOS, and
normally this starting design will not be too far away from a good final design.
The second method for discovering a good starting point for any toroidal anamorphic
design with a specified configuration and anamorphic ratio can be described by the
follows:
1) Since a well corrected anamorphic system must also be well corrected in the two
principal sections;
2) Thus we can design two separate RSOS with the same glass types, the same inter-
element separations, the same stop size and location, and the same back focal lengths.
What will differ in both RSOS are the surface curvatures and the effective focal lengths,
which controls the specified anamorphic ratio;
3) We correct both RSOS separately, then put them together to form an anamorphic
system made from toroidal surfaces;
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4) We then require the BFL in both principal sections be equal and start further
optimization on the integrated system.
By designing the two associated RSOS separately then putting them together, we can
find a reasonably good starting design with the specified anamorphic ratio, which is
normally not too far away from a desirable final design.
In practice, both methods work fine. For illustration purpose, in the anamorphic
Triplet design example presented in this section, we will make use of the first method to
find a starting design. In the anamorphic Double Gauss design example presented in
section 10.5 below, we will make use of the second method to find a starting design.
10.3.2 An anamorphic Triplet with an anamorphic ratio of 1.22
To design an anamorphic Triplet with an anamorphic ratio of 1.22, we begin with a
ZEMAX sample RSOS triplet design file. The starting specifications are as follows: The
EFFL is 50mm. The stop size is fixed at 7.732mm with the ray aiming function in use.
The stop is on the rear surface of the negative element. The full field size is 40 degrees.
The wavelengths are at 0.48, 0.55 and 0.65 microns. The reference wavelength is at 0.55
microns. For the two positive elements, the glass is Schott Sk16, and for the negative
element, the glass is Schott F2.
To find a reasonably good starting design with the specified anamorphic ratio, we
will employ the first method here to illustrate its usage.
We require the BFL in both principal sections be equal. We then increase the EFLX
target by 2mm and decrease the EFLY target by 2mm in the merit function. We also add
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several special operands into the merit function to control spherical aberration, axial color,
lateral color, and distortion associated with each principal section. Other aberrations are
controlled by shrinking the polychromatic spot size with the default merit function
operands.
We optimize the system using the above merit function and get an anamorphic
system with EFLX=52mm and EFLY=48mm. We repeat several times the same iterating
process with a small EFLX increases and EFLY decreases, until we finally reach the
desired anamorphic ratio with EFLX= 55mm and EFLY=45mm. Other specifications are
given in Table 10-5 below.
The intermediate solution for the anamorphic Triplet obtained from the above
iterating method is actually quite good in performance, and the final solution is a
refinement to it.
In the final optimization process, we will allow defocus at the image plane. The only
aberration type we will control using special operands are various distortions. The reason
lies in the fact that distortion does not decrease image quality, thus a shrinking spot size
may yield a solution with unacceptable distortion levels.
After some minor adjustments, we find a final solution by using a default merit
function which shrinks polychromatic spot size for all field positions and all wavelengths,
while continuing to correct EFLX, EFLY, BFL and distortion with special operands.
Figure 10-7 below is the layout of the final solution in both principal sections.
Figures 10-8 and 10-9 are the corresponding spot diagram and ran fan. Table 10-6 has the
lens data and the remaining primary aberration coefficients.
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Table 10-5 Specifications of the anamorphic triplet with anamorphic ratio 1.22
Figure 10-7 Layout in the y-z (left) and x-z (right) principal sections
Figure 10-8 Spot diagram
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Figure 10-9 Ray fan
Table 10-6 Lens data and remaining primary aberration coefficients
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From the spot diagram shown above, we can see that this anamorphic Triplet design
is acceptable with an RMS spot size of smaller than 47 mμ over the fields, and the image
quality is quite uniform over the field of view.
From the aberration coefficients list above, together with the ray fans, we see that the
defocus is not primarily used to balance on-axis spherical aberration. Since at bigger
fields, the various astigmatism and field curvature terms are the limiting aberrations,
ZEMAX actually utilized defocus to primarily balance off-axis field curvatures instead,
thus resulting in a relatively large on-axis spot.
10.3.3 Another anamorphic Triplet with an anamorphic ratio of 1.35
Now let us explore the effect on image quality of increasing the anamorphic ratio.
Following the same design procedures as described in section 10.3.2, we arrive at another
final design result with EFLX= 57.5mm and EFLY=42.5mm, and thus the anamorphic
ratio is now 1.35. All other specifications are listed in Table 10-7 below.
Figure 10-10 below is the layout of the final solution in both principal sections.
Figures 10-11 and 10-12 show the corresponding spot diagram and ray fan. Table 10-8 is
the lens data and the remaining primary aberration coefficients.
From the spot diagram, together with the remaining primary aberration coefficients
listed in Table 10-8, we see that as the anamorphic ratio increases, image quality is
deteriorating very rapidly, with a maximum RMS spot size 75 mμ now. And again, the
most problematic aberrations are the various astigmatism and field curvature terms. The
maximum system distortion is about 4.45%.
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Table 10-7 Specifications of the anamorphic triplet with anamorphic ratio 1.35
Figure 10-10 Layout in the y-z (left) and x-z (right) principal sections
Figure 10-11 Spot diagram
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Figure 10-12 Ray fan
Table 10-8 Lens data and remaining primary aberration coefficients
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If we increase the anamorphic ratio even more, the image quality will deteriorate so
rapidly that the resulting systems are barely usable. For an anamorphic Triplet design that
does not have enough degrees of freedom to control all primary aberrations, an
appropriate choice might be to limit the anamorphic ratio to less than 1.35.
10.4 An anamorphic field lens design
In the design examples presented in the above sections, the system powers together
with the power distributions are different for the two principal sections and for the skew
rays. Because of this, the various anamorphic astigmatism and field curvature terms are
normally the limiting off-axis aberrations. This is an inherent feature of anamorphic
system design.
However, this feature can lead to the idea of anamorphic field lens design, which is
quite useful in many cases. We will show the anamorphic field lens design idea via a real
design example.
Suppose we want to design a classical RSOS Cooke Triplet to take a picture of a
human thumb, which may be useful in security device. Also suppose the thumb is placed
50mm before the triplet lens. Since the thumb is a curved object with a different radius of
curvature in different principal sections, we can model it is as a toroidal object.
For an RSOS Cooke Triplet, we know its field curvature will be rotationally
symmetric, so we cannot map a toroidal object surface onto a flat image plane without the
help of one or more elements with anamorphic power inside the system.
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After some thinking, we come to the idea of introducing a toroidal field lens near the
image plane to provide different field curvatures in different principal sections.
We start from the same ZEMAX sample RSOS Triplet design used in section 10.3.
We scale the lens down so that the EFFL equals 15mm and the system will be working
with a 2/3 inch CCD. We optimize the Triplet design for an object distance of 50mm
from the first lens surface. The maximum half object height is 14mm, which matches the
size of the thumb. The working F number of the Triplet is 6.67. The wavelengths are 0.48,
0.55 and 0.65 microns. The reference wavelength is at 0.55 microns.
After obtaining a reasonable starting RSOS design, we now change the object to a
toroidal surface with mm and60xr = − 30yr = − mm, which are roughly the curvatures of
a human thumb. We then introduce a toroidal field lens right before the paraxial image
plane and let the computer program shrink the polychromatic spot size over all fields
using a default merit function. The glass of the toroidal field lens is Schott BK7.
Figure 10-13 below is the layout of the final solution. Figure 10-14 is the spot
diagram. Table 10-9 lists the lens data and the remaining primary aberration coefficients.
Figure 10-13 Layout in the y-z (left) and x-z (right) principal sections
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Figure 10-14 Spot diagram
Table 10-9 Lens data and remaining primary aberration coefficients
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As we can see from the spot diagram, this system is very well-corrected, with spot
sizes close to the airy disk, which is the diffraction limited case.
From the remaining primary aberration coefficients listed above, it is clear that the
anamorphic field lens indeed works really well in reducing the various astigmatism and
field curvature terms in the system.
10.5 An anamorphic Double Gauss design with anamorphic ratio 1.5
In section 10.3, we explored the anamorphic Triplet design. Due to the lack of design
degrees of freedom, we noticed that as the anamorphic ratio increases, the image quality
deteriorates rapidly. Thus, for an anamorphic Triplet design, we should limit ourselves to
a relatively small anamorphic ratio. We also found that the various astigmatism and field
curvature terms will most likely be the limiting off-axis aberrations, which is an inherent
aberration feature for anamorphic image systems due to different powers and power
distributions in different principal sections.
In section 10.4, we introduced the idea of adding an anamorphic field lens to achieve
a flat field in both principal sections. Even tough the example presented in section 10.4 is
an RSOS system with a toroidal object surface, the idea is quite revealing and should be
applicable to general anamorphic imaging system design.
To achieve an anamorphic design with a higher anamorphic ratio and better image
quality, we need to have more design degrees of freedom, and we also need a way to
control the inherent field curvature terms. We will now combine the design ideas
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presented in the previous sections and extend our design example to the anamorphic
Double Gauss design with a field lens.
The RSOS Double Gauss is a very famous design form which was originally used for
wide angle lenses. It was originally constructed by placing two Gauss type achromatic
doublets back-to-back symmetrically about a stop. Later on a cemented buried surface
was added inside each negative meniscus, with the glasses on each side of the buried
surface having similar indices but different dispersions. Their purpose was to simulate a
glass with greater dispersion than any available. Finally, the system was made slightly
unsymmetrical about the stop and the ultimate design configuration was achieved (G. H.
Smith, 1998).
Now let us design an anamorphic Double Gauss with a moderate anamorphic ratio of
1.5. We will start from a scaled ZEMAX sample RSOS Double Gauss design. The
starting specifics are as follows: The EFFL is 50mm. The stop size is fixed at 10mm with
ray aiming in use. The stop is in the middle air space between the front and rear lens
groups. The full field size is 28 degrees. The wavelengths are the visible F, d, and C light.
For the two positive elements, the glasses used are Schott Sk2 and Sk16. The glass for the
negative element is Schott F5.
To start the anamorphic design, we must achieve a reasonably good starting point
with the specified configuration and the specified anamorphic ratio. As described in
section 10.3.2, generally speaking, there are two methods of approaching a good starting
design. In the anamorphic Triplet design example presented in section 10.3, we applied
the first approach. In this section, we will apply the second approach to illustrate its usage.
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Starting from the ZEMAX sample file, we design two RSOS Double Gauss lenses
with the same BFL of 30mm. The glasses, the inter-element separations, the stop size and
location are all the same. The things different in both systems are the surface curvatures
and the effective focal lengths, which control the anamorphic ratio. We design the first
RSOS with EFFL=50mm and the second RSOS with EFFL=75mm, thus the anamorphic
ratio is 1.5.
We optimize each RSOS separately then combine them to form an anamorphic
system made from toroidal surfaces. In this way, we find a reasonably good starting point
with the specified anamorphic ratio of 1.5.
Table 10-10 below is the design specifications. Figure 10-15 shows the starting
configuration in both principal sections. Figure 10-16 shows the spot diagram at this
stage.
Table 10-10 Specifications of the anamorphic Double Gauss
Figure 10-15 Layout of the initial design in the y-z (left) and x-z (right) principal sections
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Figure 10-16 Spot diagram for the initial design
From the system layout and spot diagram, it is clear that this initial design is
acceptable and is capable of serving as the starting point.
Now we require the BFL in both principal sections always be equal to ensure a
unique final image is formed. We then add some special operands into the default merit
function while shrinking the polychromatic spot size across the field. The special
operands are used to control all three spherical type aberrations and all coma type
aberrations. We will not specially control the various field curvature terms at this stage
because we will add a field lens to control them later.
After this first stage optimization, the spot diagram together with the remaining
primary aberrations is shown in Figure 10-17 below.
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Figure 10-17 Spot diagram and aberrations coefficients after the first stage optimization
From the spot diagram and the primary aberration coefficients table, we see the
various field Curvature terms are now indeed the dominant aberrations (ignore distortions
because they do not decrease image sharpness). Thus, it is the right time for the field lens
to be added.
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The field lens is added right in front of the paraxial focus plane with a center
thickness of 1mm and a glass type of Schott BK7. The first surface of the field lens is
toroidal and the second surface is flat. We get the initial radii of curvature for the toroidal
field lens by manually adjusting its radii while examining the spot diagram, and then
setting the surface curvatures free to vary.
After adding the field lens, we are approaching the final design. At this stage, image
plane defocus is set as a variable, all special operands are removed, and we let the
program shrink the polychromatic spot size across the field using a default merit function.
Of course, the operands for EFLX and EFLY and the operands for controlling BFL are
untouched. After the optimization and some manual refinement, we get a final design
with a total of 9 toroidal surfaces used.
Figure 10-18 below is the layout of the final solution. Figures 10-19 and 20 reflect
the system spot diagram and OPD fan. The lens data and the remaining primary
aberrations are shown in Table 10-11.
Figure 10-18 Layout of the anamorphic Double Gauss design
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Figure 10-19 Spot diagram
Figure 10-20 OPD fan
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Table 10-11 Lens data and remaining primary aberration coefficients
From the aberration coefficients list above, it is clear that the various field curvature
terms are controlled. System maximum distortion is controlled for less than 2.6%.
From the spot diagram and OPD fan, we can see clearly that the image quality of this
moderate anamorphic ratio Double Gauss design is very good with an RMS spot size of
less than 29 mμ across the field.
10.6 An anamorphic fisheye lens design with anamorphic ratio 3:4
In this section, we will explore the design of an anamorphic fisheye lens, which will
be working over a 140 degree full field of view. The anamorphic ratio will be 3:4.
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First of all, let us find an RSOS fisheye design as the beginning point. Starting from
the classical wide-angle Double Gauss configuration, we form the first and second
elements into meniscus shapes, which are common in fisheye type lenses. We also
gradually increase the front group elements separation during the optimization process so
that the incident ray height can be gradually decreased before it reaches the system stop.
After some trials, we arrived at a modified wide-angle Double Gauss configuration with
separated front elements.
We then play with the glass types and gradually increase the half field of view to 70
degrees. After some detailed adjustments, we arrive at a RSOS fisheye design as shown
in Figure 10-21 below, which will serve as our starting RSOS design.
Figure 10-21 A RSOS Double Gauss type fish-eye Lens
This is a modified Double Gauss configuration with separated front elements. The
EFFL is 50mm. The system is fixed at F/5 with ray aiming applied. The stop is in the
middle air space between the front and rear lens groups. The full field size is 140 degrees.
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The wavelengths are the visible F, d and C lights. In the above layout, from left to right,
the glasses used are Schott N-Sk14, Lak11, Sf1, N-Sk11, Sf57 and Laf22a.
10.6.1 An anamorphic fisheye design without field lens
From the starting RSOS fisheye design, utilizing the first design method described in
section 10.3, we gradually altered the EFLX and EFLY target while using the default
merit function to shrink the polychromatic spot size across the field. After several
iterating steps, we find a reasonably good starting anamorphic fisheye design, with the
specified anamorphic ratio of 3:4. The design specifications at this stage are listed in
Table 10-12 below.
Table 10-12 Initial design specifications of the anamorphic Double Gauss
Figure 10-22 below shows the initial anamorphic fisheye design in both principal
sections. Figure 10-23 shows the grid distortion at this stage.
Figure 10-22 Layout in the y-z (left) and x-z (right) principal sections
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Figure 10-23 Grid distortion for the initial anamorphic fisheye design
An examination of the grid distortion map reveals the image field size is too large,
thus we must scale the lens down so that the image can fit a standard 35 mm film format.
After the scaling, defocus is turned on as a variable, all special operands other than
EFLX, EFLY, and BFL are removed from the default merit function, and we let the
program shrink the polychromatic spot size across the field. After some refinement trials,
we achieve a final design. The final design specifications are listed in Table 10-13 below.
Figure 10-24 illustrates the final anamorphic fisheye design in both principal sections.
Table 10-14 lists the lens data. Figure 10-25 shows the grid distortion. Figures 10-26 and
27 represent the spot diagram and ray fan.
Table 10-13 Final design specifications of the anamorphic Double Gauss
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Figure 10-24 Layout of the final design in the y-z (left) and x-z (right) principal sections
Table 10-14 Lens data of the final design
Figure 10-25 Grid distortion map of the final design
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Figure 10-26 Spot diagram of the final design
Figure 10-27 Ray fan of the final design
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From the spot diagram, we see that system performance is very good with an RMS
spot size of less than 31 mμ over all fields. The maximum system distortion is 36.7%,
which is at the normal distortion level for a fisheye lens.
10.6.2 An anamorphic fisheye design with field lens
From the discussion presented in previous sections, we know that by adding a field
lens to reduce the effect of various field curvature terms, we can normally achieve a
design with better system performance.
Now let us add a cylindrical field lens near the final image plane. The glass type of
the cylindrical field lens is again Schott Bk7. We also introduce a little bit of vignetting
by limiting the semi-diameter of the surface right after the system stop. After some
detailed refinement, we achieve a final design. The specifications of the final design are
listed in Table 10-15 below.
Figure 10-28 below shows the layout of the final design in both principal sections.
Table 10-16 shows the lens data. Figure 10-29 shows the vignetting plot. Figure 10-30
shows the spot diagram. Figure 10-31 shows the ray fan.
Table 10-15 Specifications of the anamorphic Double Gauss with field lens
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Figure 10-28 Layout of the final design in the y-z (left) and x-z (right) principal sections
Table 10-16 Lens data of the final design with field lens
Figure 10-29 Vignetting plot
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Figure 10-30 Spot diagram
Figure 10-31 Ray Fan
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Examining the spot diagram reveals that system performance is much improved with
a new RMS spot size across the field less than 20 mμ . The vignetting we introduced is
very small with a relative geometrical transmission at the edge of the field larger than
86%. The maximum system distortion is now 39.2%.
10.7 Summary
In this chapter, a sampling of some different anamorphic designs has been presented.
We started with the simplest anamorphic singlet design and finished with a complex
anamorphic fisheye design. We introduced the important methods for approaching a
starting design, and we explained the limiting aberrations in each configuration and how
they are controlled.
From the examples presented in this chapter, together with the theoretical structures
presented in previous chapters, the reader of this work should have a better understanding
of anamorphic imaging system design.
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CHAPTER 11
CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
11.1 Conclusions
In this work, a theoretical structure has been developed to describe the aberrations of
anamorphic optical systems that can be considered a generalization of the aberration
theory of rotationally symmetric optical system (RSOS).
In chapter 2, we found that we can think of a paraxial anamorphic system as two
RSOS, each associated with one symmetry plane. Thus all known results for the two
RSOS can be applied to the anamorphic system directly. We also found that there are
only two independent paraxial skew rays in an anamorphic system, but we prefer using
the four non-skew marginal and chief rays, traced in the two associated RSOS, to fully
specify the system. We found that by using these four paraxial rays, we can get all
paraxial quantities associated with the anamorphic system.
In chapter 4, by applying the generalized Aldis idea onto anamorphic systems, we
built up the anamorphic total ray aberration equations. We then reduced these equations
into the anamorphic primary ray aberration equations. We then wrote all parameters in
the anamorphic primary ray aberration equations in terms of the ray-tracing data of the
four non-skew paraxial rays in the two associated RSOS, together with object and stop
variables. By these steps, we built up a general method of deriving the primary aberration
coefficients for any anamorphic system types.
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In chapters 6, the monochromatic primary aberration coefficient expressions for
cross cylindrical anamorphic systems were found by applying the method we developed.
In chapters 7, the monochromatic primary aberration coefficient expressions for
toroidal anamorphic systems were found by applying the method we developed.
In chapters 8, the monochromatic primary aberration coefficient expressions for
general anamorphic systems were found by applying the method we developed.
All results listed above are novel and can not be found in existing literatures. Thus
our work greatly expanded the scope of current anamorphic imaging systems research.
11.2 Suggestions of future work
In this work, the ground work (first-order optics), the relevant concepts and the
primary aberration theory for understanding the imagery of the most common types of
anamorphic systems have been given. But of course, there is room for future work.
For example, the chromatic variation of the aberrations needs to be investigated. The
thin lens contributions, and the stop shift formulae for anamorphic systems need to be
derived, and the higher order aberrations need to be explored, etc.
More importantly, we need to apply the theoretical results to all kinds of actual
anamorphic designs so that the design experiences can be accumulated, as we did in the
past 150 years since Seidel presented the Seidel aberrations for RSOS.
Serious applications of the results presented in this work onto different anamorphic
system designs, together with other theoretical research should be able to further expand
the research in this field.
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APPENDIX A
THE TYPICAL SHAPE OF ANAMORPHIC PRIMARY WAVE ABERRATIONS
4 4
1 1 cosxW D D 4ρ ρ θ= ⋅ = ⋅ 4 42 2 sinyW D D 4ρ ρ θ= ⋅ = ⋅
2 2 4 2 23 3 sin cosx yW D Dρ ρ ρ θ= ⋅ = ⋅ θ 3 3 3
4 4 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅
2 35 5 sin cosy x y yW D H D H 2ρ ρ ρ θ= ⋅ = ⋅ 2 3 2
6 6 sin cosx x y xW D H D Hθ
ρ ρ ρ θ θ= ⋅ = ⋅
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3 37 7 siny y yW D H D H 3ρ ρ θ= ⋅ = ⋅ 2 2 2 2 2
8 8 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅
2 2 2 2 29 9 siny y yW D H D Hρ ρ θ= ⋅ = ⋅ 2 2 2 2 2
10 10 cosy x yW D H D Hρ ρ θ= ⋅ = ⋅
2 2 2 2 211 11 sinx y xW D H D Hρ ρ θ= ⋅ = ⋅ 12
212 sin cos
x y x y
x y
W D H H
D H H
ρ ρ
ρ θ θ
= ⋅
= ⋅
202
3 313 13 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅ 3 3
14 14 siny y yW D H D Hρ ρ θ= ⋅ = ⋅
2 215 15 cosx y x x yW D H H D H Hρ ρ θ= ⋅ = ⋅ 2 2
16 16 sinx y y x yW D H H D H Hρ ρ θ= ⋅ = ⋅
Notice that even though are of the same shape in aperture dependences, but due to their different field dependences, they are different aberration types. The same is true for .
13 15,D D
14 16,D D
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