Aberration Fields in Tilted Anq Decentered Optical

Aberration fields in tilted and decentered optical systems

Item Type text; Dissertation-Reproduction (electronic)

Authors Thompson, Kevin Paul

Publisher The University of Arizona.

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Link to Item http://hdl.handle.net/10150/565458 ABERRATION FIELDS IN TILTED

ANQ DECENTERED OPTICAL SYSTEMS

by-

Kevin Paul Thompson

A Dissertation Submitted to the Faculty of the

COMMITTEE ON OPTICAL SCIENCES (GRADUATE)

In Partial Fulfillment of the Requirements For the Degree of

Doctor of Philosophy '

In the Graduate College

THE UNIVERSITY OF ARIZONA

1980 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee r we certify that we have read the dissertation prepared by Kevin Paul Thompson ______entitled Aberration Fields in Tilted and Decentered Optical Systems

and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy «■

Date

YO Date 'fd Date /

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy 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.

z £ Dissertation Director Date 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 bor rowers 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 re production 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 judgment the proposed use of the material is in the in terests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: / Ueao. ACKNOWLEDGMENTS

I would like to thank Professor Roland Shack for providing me with such a ripe topic to pursue and a very firm basis from which to begin. His restrained guidance and timely encouragement made the entire project quite enjoyable. It is he who should be credited with the development of the first and third order theory and the vector formula tion of the wave aberration expansion.

To ThereSe, I would like to express my admiration for her un ending patience and support during my academic career.

Thanks also go to Nancy Arora for diligently typing the manu script and to Kathy Seeley for editing and general assistance.

Lastly, to the Perkin-Elmer Corporation for providing the funding for this research. TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ...... vi

LIST OF TABLES ...... xiv

ABSTRACT ...... " . . . xvi

1. INTRODUCTION...... , ...... 1

2. FIRST ORDER THEORY ...... 6

Summary of First Order Properties in Tilted and Decentered Systems ...... 19 Tilt Parameter ...... 19 Displacement of Centers of Image Fields and and Pupils ...... 21 Optical Axis Ray . .: ...... 21 Gaussian Image Plane Tilt ...... 21 Aberration Field Decentration ...... 22 Aspheric Contribution...... 22 Special Cases ...... 22

3. ABERRATION FIELDS IN PERTURBED OPTICAL SYSTEMS I THE THIRD ORDER ABERRATIONS ...... 23

.4. THE ANALYSIS OF PERTURBED OPTICAL SYSTEMS ...... 57

5. APPLICATION: MISALIGNMENTS IN RITCHEY-CHRETIEN TELESCOPES A NEW TECHNIQUE ...... 77

6. ABERRATION FIELDS IN PERTURBED OPTICAL SYSTEMS II THE INTERPRETATION OF THE FIFTH ORDER ABERRATIONS . . . 103

Fifth Order Spherical Aberration Wpgo ...... 108 Fifth Order Coma, ...... 109 Oblique Spherical Aberration Wg^o^, W 21+2 • • ..... 113 Focal Surface for Medial Oblique Spherical Aberration W240m • • • • • - • . • • • . . . 115 Oblique Spherical Aberration with respect to the Medial Focal Surface Associated with Oblique Spherical Aberration W 242 ...... 117

: iv - V

TABLE OF CONTENTS— Continued

Page

Linear and Field Cubed Coma W 1 3 1 , ^ 3 3 lM ...... 118 Elliptical Coma W 333 ...... 127 Medial Focal .Surface W2 2 0 M ^420M ...... 132 Astigmatism *2 2 2 , W422 . 136 Distortion W 31 %, V s i l ...... 142

7. GRAPHICAL METHODS FOR ANALYZING PERTURBED OPTICAL SYSTEMS. . 143

8 . SUMMARY AND FUTURE CONSIDERATIONS . . . ■» ...... 210

APPENDIX A: VECTOR RELATIONS ...... -214

Introduction to Vector Multiplication ...... 214 Summary of the Properties of Vector Multiplication, Squared and Cubic Vectors . 216 Vector Conjugates ...... 217 Vector Identities ...... 218 Vector Operator for Obtaining the Gradient of the Wavefront ...... 218

APPENDIX B: PLOTS FOR LOCATING THE NODES IN SYSTEMS WITH BINODAL ASTIGMATISM ...... 220

APPENDIX C: DERIVING AND SUMMARIZING THE PROPERTIES OF THE WAVE ABERRATION EXPANSION IN A PERTURBED OPTICAL SYSTEM THROUGH. FIFTH ORDER ...... 231

APPENDIX D: VECTOR GENERALIZATION OF THE ALGEBRAIC SOLUTION TO CUBIC EQUATIONS . , ...... 263

APPENDIX E: DETERMINING THE ORIENTATION OF ELLIPTICAL COMA ...... 265

APPENDIX F: EXTENDING THE EXPRESSIONS FOR ANALYZING PERTURBED SYSTEMS TO FIFTH ORDER IN THE WAVE ABERRATION EXPANSION ...... 268

LIST OF REFERENCES 272 LIST OF ILLUSTRATIONS

Figure Page

2-1„ Defining the Equivalent Local Axis and Equivalent Tilt Parameter B0 f°r a Spherical Surface ...... 9

2-2. Locating Objects and Images in a Perturbed System . . . . 10

2.3 Optical Axis Ray (OAR) ...... 12

2-4. Finding the Displacement of the Aberration Field Contribution of a Surface; a* ...... 13

2-5. Deriving the Gaussian Tilt Invariant ...... 14

2-6. The Gaussian Tilt Invariant in a Perturbed System .... 15

2-7. First Order Tilt and Displacement of the Gaussian Image P l a n e ...... 16

2 =-8 . Locating the Center of an Aspheric Contribution to the Aberration Field ...... 17

2-9. Summary of the First Order Parameters in Tilted and Decentered Systems ...... 20 i 3-1. Illustrating^the Field Vector H and the Aperture Vector p ...... 26

3-2. The Effective Field Height H.. for a Surface Contribution to the Aberration Field In the Image Plane ..... 29

3-3. In a Perturbed Optical System the Center of the Total Coma Field for the System Is Displaced to the Point Located by the Vector aigi ...... 31

3-4. Properties of the Transverse Blur for Third Order Coma Wi3 ij (a) Aligned System, (b) Perturbed System . . . 33

3-5. The Medial Focal Surface in a Perturbed Optical System May Be Displaced Transversely and Longitudinally from the Center of the Gaussian Image Plane • . . 37

3-6 . Illustrating the Special Cases for the Medial Focal Surface ...... 40

3-7. The Relation between Linear and Squared Vectors ..... 42

" ' ■ ' - ' vi LIST OF ILLUSTRATIONS--Continued

Figure Page

3-8. In a Perturbed System the Astigmatism Can Be Zero at Two Points in the Field ...... 43

3-9. Field Contours of Constant Astigmatism 44

3-10. Node Vectors for Illustrating the Properties of. Astigmatism in the Field ...... 46

3-11. Magnitude and Orientation Plots for Binodal Astigmatism. . 47

3-12. The Shape of the Two Focal Surfaces Containing Line Images in a System with Binodal A s t i g m a t i s m ...... 48 3-13. Illustrating the Special Cases for Astigmatism. Field curve profiles that contain the nodes . .... 51

.3-14. Illustrating Special Cases for Astigmatism. Magnitude and orientation plots on the focal surfaces con taining line images ...... 52

3-15. Illustrating the Four Types of Astigmatic Focal Surfaces in a Perturbed Optical System...... 54

4-1. Node Plot for Distortion ...... 65

5-1. . Node Plots for the Arbitrarily Misaligned System ..... 83

5-2. On-Axis Spot Diagram in the Arbitrarily Misaligned System ...... 83

5-3. Plotted Output for the Actual and the Change in the RMS Spot Size in the Arbitrarily Misaligned System (X and Y Field Profiles) . 85

5-4. Contour Plots for RMS Spot Size, in Micrometers ..... 87

5-5. Node Plots in the Coma-Compensated System ...... 89

5-6. Contour Plot for RMS Spot Size (Micrometers) in the Coma- Compensated System ...... 90

5-7. Simulation of a Through-Focus Star Plate Taken with a Coma-Compensated, Misaligned Ritchey-Chretien . . . . 91

5-8. Illustrating the Technique for Determining Misalignments from Through-Focus Star Plates ...... 93 . . vi'ii LIST OF ILLUSTRATIONS— Continued

Figure Page

5-9. Determining if a Telescope is Misaligned from Through-Focus Star Plates ...... 94

5-10. Node Plots for the Coma-Compensated System with respect to the Local Axis of the Primary...... 97

5-11. Measuring the Perturbation Vector b 222 from the Node Plot Corresponding to Fig. 5-8 ^ . 100

6-1. Transverse Blur for Third Order Coma W 131 ...... 109

6-2. Transverse Blur for Fifth Order Coma, ...... 110

6-3. In a Perturbed System the Center of the Aberration Field for Fifth Order Coma, 5 1 , Can Be Displaced in the Image P l a n e ...... Ill

6-4. Properties of the Transverse Blur for Fifth Order Coma, Wisi (a) Centered (b) Perturbed ... . ■...... 112

6-5. Properties of Oblique Spherical Aberration ...... 114

6 -6 . Node Plot for the Medial Surface Associated with Oblique Spherical Aberration, # 21+0^ - ...... 116

6-7. Locating the Medial Surface for Oblique Spherical Aberration in a Perturbed System ...... 117

6 -8 . In a Perturbed System the Oblique Spherical Aberration (OBSA) with respect to the Medial Surface for OBSA Is Zero at Two Points (Nodes) in the Field ...... 119

6-9. Field Curves for OBSA Illustrating Binodal Behavior . . . 119

6-10. For Any Field Point H, the Magnitude of the OBSA Is the Produce of the Distances to the Two Nodes |Ni| X I S I • • • • • v • 120

6-11. In a Perturbed System the Field Cubed Coma Contribution Can Be Zero at Three Co11inear Points in the Field. . 123

6-12. Contour of Constant Wave.Aberration Magnitude for Linear Plus Field Cubed Coma in an Aligned System . . 125 ix LIST OF ILLUSTRATIONS— Continued

Figure Page

6-13. Transverse Magnitude and Orientation Plot Corresponding to Fig. 6-12 ...... 125

6-14. Node Plot for Illustrating the Behavior of Linear Plus Field Cubed Coma in a Perturbed System . . . . . 126

6r-15 . Contour Plot of Linear Plus Field Cubed Coma in a Perturbed System with the Node Plot Shown in Fig. 6 - 1 4 ...... 128

6-16. Transverse Magnitude and Orientation Plot Corresponding to Fig. 6-15 ...... 128

6-17. The Effect of Elliptical Coma on Linear Plus Field Cubed Coma in a CenteredSystem ...... 129

6-18. In a Perturbed System Elliptical Coma Develops Three ' Nodes in the Field, i.e., Trinodal Behavior ..... 131

6-19. In a Perturbed System the Fifth Order Contribution to the Medial Surfaces Develops Three Collinear Zeroes in the F i e l d ...... 134

6-20. The Behavior of the Balance Zone for the Third Plus Fifth Order Medial Focal Surface in a Perturbed System ...... 135

6-21. In a Perturbed System the Fifth Order Contribution to the Astigmatism Generally Develops Four Nodes in the Field ...... 137

6-22. Comparing S and T with N and P Focal Surfaces in an Aligned System ...... 141

7-1. Sufface-by-Surface Perturbation Data Entry...... 148

7-2. Third Order Node Plot ...... 153

7-3. Fifth Order Node Plot I ...... 154

7-4. Fifth Order Node Plot II ...... 155

7-5. Orientation Plot for Linear Plus Field Cubed Coma. Perturbed System ...... 157 X

LIST OF ILLUSTRATIONS--Continued

7-6. Orientation Plot for Linear Plus Field Cubed Coma. Aligned System ...... 158

7-7. Magnitude Contour*Plot for Linear Plus Field Cubed Coma. Perturbed System ...... 159

7-8. Magnitude Contour Plot for Linear Plus Field Cubed Coma. Aligned System ...... 160

7-9. Magnitude Contour Plot for the Medial Focal Surface. Perturbed System ...... 161

7-10. Magnitude Contour Plot for the Medial Focal Surface. Aligned System . 162

7-11. Fifth Order Node Plot III ...... 164

7-12. Orientation Plot for Elliptical Coma. Perturbed System . 165

7-13. Orientation Plot for Elliptical Coma. Aligned System . . 166

7-14. Magnitude Contour Plot for Elliptical Coma. Perturbed System ...... 167

7-15. Magnitude Contour for Elliptical Coma. Aligned System , , , . . , . . ; ...... ' . . 168

7-16. Orientation Plot for N-Surface Astigmatism. Perturbed System ...... 170

7-17. Orientation Plot for P-Surface Astigmatism. Perturbed System ...... 171

7-18. Orientation Plot for N-Surface Astigmatism, . Aligned System ...... 172

7-19. Orientation Plot for P-Surface Astigmatism, Aligned System ...... 173

7-20. Magnitude Contour Plot for Astigmatism with respect to the Medial Focal Surface. Perturbed System ..... 174

7-21. Magnitude Contour Plot for Astigmatism with respect to the Medial. Focal Surface. Aligned System . . . , . . . 175 xi

LIST OF ILLUSTRATIONS— Continued

Figure Page

7-22. N and P Field Curves, Aligned System ...... 176

7-23. N and P Field Curves. Y-Field Profile. Perturbed System 177

7-24. N and P Field Curves. X-Field Profile. Perturbed System ...... , . . 178

7-25. Orientation Plot for Distortion. Perturbed System . . . . 179

7-26. Orientation Plot for Distortion. Aligned System ...... 180

7-27. Magnitude Contour for Distortion. Perturbed System . , . 181

7-28. Magnitude Contour for Distortion. Aligned System .... 182

7-29. List of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface ...... 185

7-30. Profile Plot of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface . . . . 186

7-31. List of the Change in the RMS Wavefront Error in the Perturbed System with respect to the Best Focal Surface. Y-Field Profile ...... 187

7-32. List of the Change in the RMS Wavefront Error in the Per turbed System with respect to the Best Focal Surface. X-Field Profile ...... 188

7-33. X and Y Field Profile Plots of the Actual and the Change in the RMS Wavefront Error in the Perturbed System with respect to the Best Focal Surface .... 189

7-34. Contour Plot of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface . . . . 190

7-35. Contour Plot of RMS Wavefront Error in the Perturbed System with respect to the Best Focal Surface .... 191

7-36. Contour Plot of the Change in the RMS Wavefront Error in the Perturbed Systems with respect to the Best Focal Surface, ...... 192

7-37. Field Focus Plot for the Aligned System 193 xii

LIST OF ILLUSTRATIONS— Continued

Figure Page

7-38. Field Focus Plot for the Perturbed System. Y-Field Profile ...... 195

7-39. Field Focus Plot .for the Perturbed System. X-Field Profile ...... 196

7-40. List of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane . . 197

7-41. Profile Plot of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane ...... 198 \ 7-42. List of the Change in the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image Plane. Y-Field Profile ...... 199

7-43. List of the Change in the RMS Wavefrone Error in the Perturbed System with respect to a Flat, Best Focused Image Plane. X-Field Profile ...... 200

7-44. X and Y Field Profile Plots of the Actual and the Change in the RMS Wavefront Error.in the Perturbed System with respect to a Flat, Best Focused Image Plane . . 201

7-45. Contour Plot of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane ...... 202

7-46. Contour Plot of the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image P l a n e ...... 203

7-47. Contour Plot of the Change in the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image Plane ...... 204

7-48. Comparing Analytic and Real Rya Data for the Change in the RMS Wavefront Error in the Perturbed System for the Triplet. Y-Field Profile ...... 205

7-49. Spot Diagrams Constructed from Real Ray Calculations . . . 207

7-50. Spot Diagrams Constructed from Analytic Calculations . . . 208 XI11

LIST OF ILLUSTRATIONS--Continued

Figure Page

7-51. Comparing Analytic and Real Ray Data for the Change in the RMS Wavefront Error due to Perturbations in an F/3.5,45° Full Field, Five Element Double Gauss. Y-Field Profile . . . C...... 209

A-l. Vector Multiplication ...... 2i5

C-l. Nodal Properties of the Comatic Aberrations in a Perturbed System ...... 256

C-2. The Nodal Properties of the Medial Focal Surface and Astigmatism in a Perturbed System ...... 259

C^3. The Nodal Properties of Oblique Spherical Aberration in a Perturbed System ...... 261

E-l. Finding the Orientation of Elliptical Coma ...... 266

•i LIST OF TABLES

Table Page

3-1. Converting Seidel Coefficients to Wave Aberration Coefficients...... 25

3-2. Special Cases for the Medial Focal Surface in a Perturbed Optical System...... 38

3-3. Special Gases for Astigmatism with Respect to the Medial Focal Surface in a Perturbed Optical System ..... 50

5-1. System Data ...... 78

5 -2 . Paraxial Ray Trace ...... 79

5-3. Wave Aberration Coefficients in Waves at A = 0.587 urn . . . 79

5-4. Perturbations for the Initial Decentered Design ...... 81

5-5. First Order Tilt and Decenter Properties in the. Arbitrarily Misaligned System ...... 81

5-6. Tabular Output for the Change in RMS Spot Size in the Arbitrarily Misaligned System (Y-Field Profile) . . . 84

5-7. Perturbations in the Misaligned, Coma-Compensated System. . 88

5-8. First Order Tilt and Decenter Properties in the Coma- Compensated System . 88

5-9. First Order Tilt and Decenter Properties in the Coma- Compensated System with respect to the Local Axis of the Primary ...... 96

6-1. Perturbation Vectors ...... 107

7-1. Lens Data ...... 145

7-2. Paraxial Ray Trace ...... 145 .

7-3. Third Order Wave Aberration Coefficients , ...... 146

7-4. Fifth Order Wave Aberration Coefficients I 146

7-5. Fifth Order Wave Aberration Coefficients II . > . , , . . . 147

xiv \ . xv

LIST OF TABLES— Continued

Table Page

7-6. Perturbation Data ...... 149

7-7. First Order Properties of the Perturbed Design ...... 150

7-8. The Third Order Perturbation Vectors ...... 150

7-9. Fifth Order Perturbation Vectors I ...... 151

7-10. Fifth Order Perturbation Vectors II ...... 151 ABSTRACT

In the process of designing a lens, the designer is faced with two major tasks. One is to find a design that provides the required performance across the specified field. The second is to find the tolerances on the fabrication and alignment of the system during manufacture.

Currently, there are two distinct approaches to lens design.

Some designers use aberration coefficients to construct a merit func tion for optimization and use the values of the coefficients to guide the design. Others use real rays to construct the merit function and ray fans to follow the progress of the design. In studying the effects of tilts and decentefs on the ensuing design in order to determine the tolerances, both groups have been forced to use real ray analysis.

This is because the wave aberration expansion used by designers assumes that the system being studied is rotationally symmetric. Tilts and decenters destroy this symmetry.

In this work a formulation of the wave aberration expansion is developed that accounts for the effects of tilts and decenters.

An emphasis is placed on interpreting the effects of tilts and decenters in the context of the aberration fields in an aligned system. Tech niques for displaying the properties of the third and fifth order terms in the wave aberration expansion in a perturbed system are developed.

The methods of display allow a designer who is well acquainted with the behavior of the terms in a rotationally symmetric system to xvii envision their behavior in a perturbed system without being well acquainted with the theory that is developed.

Once the wave aberration expansion for the perturbed systems is developed, it is applied to calculating the rms wavefront error or the rms spot size in a perturbed system. The resulting analytic expressions are considerably faster than real ray calculations. The increase in speed makes it economical, for the first time, to evaluate image quality over the entire image plane in nonsymmetric systems.

This is an important new tool for providing a complete picture of the effects of a set of arbitrarily oriented tilts and decenters on an optical design.

The theory that is developed allows the tilts and decenters to be arbitrarily oriented at any surface, i.e., they are not restricted to be either meridional or coplanar. This is accomplished by beginning the development with a vector formulation of the wave aberration expansion. This new form has some important implications even for aligned systems. In particular, it is shown that the sagittal and tangential focal surfaces located by a Coddington skew ray trace are not continuous in a misaligned system. There continue though, to be two continuous focal surfaces on which line imagery is obtained in the presence of pure astigmatism. These surfaces are found directly, in both perturbed and aligned systems, when the vector formulation is used. These surfaces are inherently more significant than the sagittal and tangential focal surfaces, xviii'

The expressions for describing the performance of a perturbed optical system are used to develop an interactive computer program that relies heavily on computer graphics to provide an efficient method of studying the effects of tilts and decenters on an optical design. The use of computer graphics greatly reduces the amount of output required to obtain a detailed picture of the response of the system to perturbations. The graphic routines are illustrated by studying the effects of tilts and decenters on a Ritchey-ChrStien telescope and a triplet design. CHAPTER 1

INTRODUCTION

Today in the field of lens design there are two distinct

approaches to the design of an optical system. One is the use of real

rays only in constructing.a merit function for optimization and

analyzing the resulting design. The other is the use of aberration

coefficients in constructing a merit function and relying on the value

of the coefficients to guide the optimization process.

Recently, in the area of tolerance analysis, a series of real ray-based programs have been developed. When combined with a statistical modeling program, these programs relieve the optical designer of much of the drudgery usually associated with finding the tolerances on the components in an optical system. There has not yet been a comparable approach to the analysis of perturbed systems based on aberration theory.

The development of the tools required to establish such an approach is the subject of this dissertation.

A perturbed optical system is taken to mean a system whose ele ments contain small errors in radius, thickness, index, and aspheric coefficients, and which may be tilted, decentered or wedged. The errors are assumed to be those encountered during the fabrication and assembly of the system and are therefore taken to be small. The optical system being analyzed must be a rotationally symmetric design which may be refractive, reflective or catadioptric.

1 In this work, we will only discuss tilt and decenter perturba tions . The centered system perturbations, i.e., changes in radius, thickness, index and aspheric coefficients may be treated by simply recomputing the values of the aberration coefficients using the standard paraxial equations. No new concepts or techniques are required to deal with these perturbations.

The developments that will be made can be traced to two fund amental roots. The first is a concept conceived by Buehroeder (1976) relating to the behavior of the conventional aberration fields in per turbed optical systems. Using two well-known properties of aberration fields in aligned systems, 1) the aberration field at the image plane is the sum of individual surface contributions, and 2 ) the aberration field contribution of a surface is centered along the line connecting the centers of the pupils for the surface with the center of curvature of the surface, he established the following premise : In a perturbed optical system, the aberration field at the image plane is still the sum of the surface contributions, but these contributions no longer have a common center at the image plane. This establishes the concept of the decentration of conventional, aligned system, aberration fields in the image plane. This concept allows the description of the behavior of a perturbed optical system within the terminology used to treat aberration fields in aligned systems.

The second major development which made this work possible is due to Shack. He established a vector expression for the wave aberration expansion that made use of a seldom-seen vector operation, vector multiplication. This expansion allows treating systems with random,

nonmeridional, noncoplanar perturbations without resorting to confusing

or cumbersome notation. This tool has made it possible to develop

an understanding of all of the terms through fifth order in the wave

aberration expansion^ •

Starting with these ;two bases, a method of analyzing perturbed

optical systems using aberration coefficients has been developed. The

techniques emphasize the use of computer graphics to display properties

of the conventional aberration fields in perturbed systems and to evaluate

the image quality over the entire image plane. Through the use of

computer graphics, the problems associated with providing information on

the status of systems without symmetry are greatly alleviated. With

a minimal amount of graphic output, a complete picture of the response of

an arbitrarily misaligned system can be constructed. This method of

displaying information greatly enhances the speed with which one can

reach an understanding of the response of a system to perturbations. -

In order for an approach to analyzing perturbed optical systems based on aberration coefficients to be successful, it must satisfy

certain criteria. One reason for using such an approach is to obtain

insight into the behavior of an optical system. Aberration coefficients

are used to provide a visualization of the overall behavior of the

system rather than a table of numbers indicating its current status.

. A method of analyzing perturbed systems using aberration co

efficients must be developed within the framework and terminology used

in the design process. The procedures and analysis techniques should be ■ : ■ 4 natural extensions of the methods a designer normally employs. The means of communicating the current condition of a perturbed system should provide a complete picture of the situation with minimal output and encourage the pursuit of a complete characterization of the response of ; the design to perturbations. These features have been the guiding principles for the work that is presented here.

As we will find, the aberration coefficients at a surface are not affected by perturbations. This is because the paraxial (first order) properties are unaffected by--tilts and decenters. Rather, it is the behavior of the aberrations in the field (i.e., the field dependence) which is modified when tilt and decenter perturbations are present. In general, a particular aberration term (i.e. coma, astigmatism..,) has its center of symmetry displaced from the center of the image plane and in many cases a term will develop multiple zeroes with specific symmetry properties in the image plane. By studying the properties of the zeroes of an aberration term, considerable insight into the re sponse of the system to perturbations may be developed. The concept of the behavior of the zeroes of an aberration term is the key to studying perturbed optical systems using aberration coefficients.

The structure of the dissertation is as follows. Chapter 2 establishes the equations required to find the first order properties in perturbed optical systems. The key parameter defined in this chapter is the aberration field decentration vector a. which locates the. center of the surface contribution to the image plane aberration field. This ; ■ '■ ■ s.; vector completely characterizes the aberrational behavior of a perturbed

optical system. Chapter 3 develops the behavior of the third order

aberration fields in misaligned systems. Here the concept of nodes and node plots is introduced as a means of displaying the properties of the aberration terms. Chapter 4 provides the equations used to analyze

the performance of a perturbed optical system to third order in the wave aberration expansion. Included here are the equations for generating ray fans„ spot diagrams and field curves, and for evaluating rms spot

size or rms wavefront error along with a discussion of certain features of distortion. Chapter 5 is an application of the computer graphics pro gram to a specific example. The effects of misalignments on Ritchey-

ChrStien telescope designs. Here, a sensitive new technique for detecting misalignment or support structure errors is presented supported by the graphics analysis package. Chapter 6 presents the nodal behavior of the fifth order terms in the wave aberration expansion, excluding distortion. It is here that the strength of the nodal approach to

interpreting aberration fields becomes apparent. Chapter 7 establishes

a methodology for analyzing perturbed optical systems using computer graphics by presenting a sample analysis of a perturbed triplet design.

Chapter 8 summarizes the results and indicates areas where this work may be continued. CHAPTER 2

FIRST ORDER THEORY

This chapter describes the first order properties of tilted and decentered optical systems. The properties of interest, are image displacement, pupil displacement, Gaussian image plane tilt, optical axis ray trace, and aberration field decentration, calculated surface by surface. By choosing the appropriate reference axes, these quanti ties may be found in a perturbed system.

In an aligned optical system there is one axis about which all of the elements are rotationally symmetric, The Gaussian and paraxial quantities are measured with respect to this axis. This axis is simultaneously the mechanical axis of the system; the path of the zero field, zero aperture ray; and the line containing the centers of curvature and vertices of all of the surfaces. In a perturbed system these no longer coincide and a terminology is needed to distinguish between them.

The mechanical axis of the aligned system is defined as the reference axis (RA) of the perturbed system. This is the axis from which the surface tilts and decenters; object, image and pupil decenters and Gaussian image plane tilt are measured. The reference axis is a fixed line in space that is unaffected by perturbations to the optical system.

The local axis (LA) of a surface is defined as the line con taining the vertex and center of curvature of a surface. The tilt of

6 ' a surface is given by the angle between the local axis of the surface

and the reference axis of the system. The decentration gives the

location of the vertex of the surface with respect to the reference

■ axis.

In an optical system, any surface (denoted by a subscript j)

has associated with it an object plane 0 ., and an entrance pupil E ., 3 ■ 3 either or both of which may be real or virtual. If surface j is an

interior surface, these planes are relayed images of the object field 0 ,

and the entrance pupil E, for the optical system. The location of 0. 3 and E . then, depend on the power of the preceding surfaces An J image plane 0.' and an exit pupil E.' are also associated with the jth 3 3 surface. They are images of 0. and E. formed by the surface j. These 3 3 planes, 0.’,E.* become the object plane and entrance pupil for the

J J . follosing surface, i.e. CK1 s (L+^ ,EJ - Ej+^.

. The optical axis ray (OAK),.by definition, passes through the

center of the object/image planes and the pupils for all of the sur

faces in the system. This ray is the paraxial equivalent to a zero

field, zero aperture ray i,n a real ray trace. Given the tilts and

decenters in a perturbed optical system, this chapter will provide the

equations necessary to trace this ray, surface by surface, through the

system. The intersection of this ray with the image plane of the un

perturbed system gives the image displacement (boresight error) which

locates, the center of the Gaussian image plane.

The aberration field contribution of a surface is centered on

the line connecting the center of the pupil, for that surface, with its 8 center of curvature. The location of the center of curvature is determined by the tilt and decenter of the surface. Since the center of the pupil is located by the OAR, the center of the aberration field contributions for each surface may be found directly.

A spherical surface does not have a unique vertex. As a result, any line through the center of curvature intersecting the sur face may be taken to be the local axis of the surface. The properties of a surface in a perturbed system are uniquely determined by the dis placement of the center of curvature from the reference axis. It is convenient to be able to express the displacement of the center of curvature in.terms of an equivalent tilt. This is accomplished by defining the equivalent local axis of a spherical surface to be the line connecting the center of curvature to the intersection of the perturbed surface with the reference axis. The equivalent tilt param eter, gg, then is related to the displacement of the center of curvature 6 c, and to the conventional tilt and decenter parameters g and 5v by

f?0 = 3 + c5v « c5c (2-1) as shown in Fig. 2r-l,

The OAR connects the centers of object/images and pupils through the system. To trace this ray through a perturbed system it is sufficient to locate the center of the object/image and pupils for each surface in the system. With the concept of a local axis, reference axis, and equivalent tilt gg, this is done by using the standard paraxial equations referred to the equivalent local axis of the sur face and then expressing the results with respect to the reference axis. 9 SURFACE

LA (equ iv) 6c RA

Fig. 2.1. Defining the Equivalent Local Axis and Equivalent Tilt Parameter 6 q for a Spherical Surface.

For any surface the relative image displacement, (SQ/y^)1,

(image displacement normalized by the image height in the centered system) is given by

Here 6 Q/y^. is the relative object displacement for the surface, measured from the reference axis, y is the marginal ray height at the surface in the centered system. An e n'-n, the difference between image space index of refraction and the object space index and jk = nuy-nuy is the

Lagrange invariant (u is the marginal ray angle preceding the surface. 10 u is the chief ray angle preceding the surface, and y is the chief ray height at the surface in the centered system). This equation may be obtained from Fig. 2-2.

REFRACTING SURFACE

IMAGE LA 6Q=0 zTjrlT" 6c

OBJECT .

Fig. 2.2. Locating Objects and Images in a Perturbed System.

The displaced image plane whose center is given by Eq. (2-2) becomes the object plane for the following surface. By successive application, the centers of the object/image fields for each surface may be found, along with the final image field displacement for the system (boresight error). Similarly, to locate the center of the entrance and exit pupils throughout the system, the following equation may be used.

(2-3)

Here (dE/y^)' is the relative exit pupil displacement (the exit pupil displacement normalized by the exit pupil radius) and (dE/y^) is the relative entrance pupil displacement.

Given the centers of the object/images and pupils for each surface in the system, the paraxial ray height y*, at a surface, and the paraxial ray angle u* preceding the surface for the OAR are given by

(2-4)

(2-5)

where both quantities are measured with respect to the reference axis as in''Fig. "2-5.

The OAR can be traced through a perturbed optical system to account for the effects of tilts and decenters on the location of the centers of object/image fields and pupils. While a chief and marginal ray, traced before the system is perturbed, characterize the aberra tions in a centered system,z the OAR is used to find the properties of these aberrations in a perturbed system. The OAR is the only additional ray that is required to account for the effects of tilts and decenters on the image plane aberration field. 12

ENTRANCE OBJECT PUPIL PLANE SURFACE PLANE

c OAR

6 E RA LA

Fig. 2.3. Optical Axis Ray (OAR).

The center of the aberration field contribution for any surface in the system is along the line that connects its center of curvature, located by $o> and the center of the pupil, located by the OAR. Re ferring to Fig. 2-4, the displacement, measured from the OAR, is given by

- * T * Yla 10 a* = ----- = - — (2-6) *i 1 13 where

i = u + yc (2-7)

Iq* = i* - 60

= (u* + y*c) - 60 • (2-8) a* is the relative (normalized by the image height) displacement of the aberration field contribution due to a particular surface measured in the

Gaussian image plane. It is this parameter alone that accounts for the effects of tilts and decenters on the aberrational properties of the

OBJECT PUPIL - r CHIEF RAY SURFACE

OAR

Fig. 2.4. Finding the Displacement of the Aberration Field Contribution of a Surface; a*. 14

optical system. It is simply the ratio of the OAR angle of incidence,

measured with respect to the local axis of the surface, in the perturbed

system, to the chief ray angle of incidence in the centered system.

The wave aberrations are measured with respect to the Gaussian

image plane. It is important then to find the tilt of the Gaussian

image plane in the perturbed system. This is done by considering first

the properties of tilted object and image planes in a centered system.

Knowing that the principal planes in a system are planes of unit mag nification and referring to Fig. 2.5 leads to an equation that is

often called the Gaussian tilt invariant

u' tan9’ = u tan9, (2-9) which has the paraxial equivalent,

u'8' = u0. (2-10)

P P'

OBJECT MAGE PLANE PLANE

Fig. 2.5. Deriving the Gaussian Tilt Invariant. 15

In a perturbed system the object and image planes associated with each surface are the relevant planes. For a single surface the principal points coincide at the surface. The principal points P,P’, like the centers of the object/images and pupils, lie on the OAR.

The principal planes are normal to the line connecting the principal points to the center of curvature of the surface. They are tilted then, at an angle 6* with respect to the reference axis, where,

6* = 6o - cy* (2-11) is the angle made by the line connecting the center of curvature to the intersection of the OAR and the surface with respect to the reference axis as in Fig. 2.6 .

SURFACE

OBJECT IMAGE PLANE PLANE

y “=0

Fig. 2.6- The Gaussian Tilt Invariant in a Perturbed System. 16

The tilt invariant measured in the rotated coordinate system is

u 1 0 q1 = u0 o• (2 -12)

When measured from the reference axis this becomes

u'(0 ' - g*) = u (0 - 0*) (2-13) u* 0 1 = u0 +. 3*A(u) .

This equation when carried along with the paraxial and OAR ray trace gives the tilt of the Gaussian image plane, 0 ’, measured from the ref erence axis. When Eqs. (2-2) and (2-13) are combined, they give the location of the coordinate system centered in the Gaussian image plane of the perturbed system relative to the Gaussian image plane in the aligned system as shown in Fig. 2.7.

GAUSSIAN IMAGE PLANE DESIGN PLANE

OAR

Fig. 2.7. First Order Tilt and Displacement of the Gaussian Image Plane. 17

When an aspheric is placed onto a surface, it provides an additional contribution to the aberration field at the image plane.

The center of the aberration contribution due to the aspheric is along the line connecting the vertex of the aspheric cap (the point of the surface about which the aspheric departure from the base curve is rotationally symmetric) with the center of the pupil. Referring to Fig.

2.8, the relative displacement in the object plane, is

yTa*A % - 7* oA* = - L - A = __d----- . (2-14) yl y

Here 5v^ is the displacement of the vertex of the aspheric cap from the reference axis and a * is the relative displacement of the aspheric

ASPHERIC ENTRANCE OBJECT SURFACE PUPIL

OAR

Fig. 2.8. Locating the Center of an Aspheric Contribution to the Aberration Field. 18 contribution to the aberration field measured from the OAR for a par ticular surface. This displacement is independent of the tilt of the surface that contains the aspheric.

There are two special cases to consider. One arises when i is zero at any surface with power and the other when y is zero at an aspheric surface. The case y = 0 corresponds physically to having an aspheric plate or surface at a pupil. In a centered system this aspheric generates only spherical aberration. It does not contribute any field dependent aberrations because it is at a pupil. If the vertex of the aspheric is displaced from the center of the pupil, other aberrations, which are uniform over the field, are generated. The magnitude of the generated aberration terms are proportional to the separation between the vertex of the aspheric and the center of the pupil to some power (i.e. squared, cubed, etc.). The properties of the aberrations in this situation are discussed further in following chapters. It suffices here to note that if y = 0 at an aspheric sur face , the^ abBrrational'pr.op’eTtaes "sare ^de,S''crib'e'd' by the separation of the vertex of the aspheric from the OAR. For an aspheric surface this relative separation is denoted by Ap^* where

<5v^ - 6 E Ap a * - ;---- ■ (2-15) /B

The case i = 0 corresponds to the center of curvature of a surface being located at a pupil. Again, in a centered system, the only aberration introduced by such a surface is spherical aberration.

Following Burch (1942) we can model the aberrational properties of any spherical surface as a zero power aspheric plate located at the sur

face's center of curvature. This case is then the same as the case y = 0 with 6 v^ replaced by 6 c. The aberrations generated in a per turbed system are described by the separation between the center of curvature and the center of the pupil, Ap*, where

Ap* = 6C ~ 5E • (2-16) E

The preceding developments were made assuming the tilts and decenters were restricted to the meridional plane for convenience. This

is an unnecessary restriction. This approach can be used with any combination of nonmeridional, noncoplanar tilts and decenters by

generalizing the equations to vector form. This generalization results

in a very powerful approach to perturbed optical systems. In the

following summary of the first order properties in tilted and decentered

systems, the vector notation is used. The surface number subscript is also included.

Summary of First Order Properties in Tilted and Decentered Systems.

The following properties are defined with reference to Fig. 2.9.

Tilt Parameter: Because a spherical surface has no inherent

vertex, a combination of tilt and decenter can always be modeled by an

equivalent tilt Bq , where 6 v

Fig. 2.9. Summary of the First Order Parameters in Tilted and Dcccntercd Systems. 21

Displacement of Centers of Image Fields and Pupils:

> i /; V': /.; * 60-

Optical Axis Ray: By definition this ray passes through the

center of the object/image fields and pupils. The paraxial ray angle preceding a surface is

u.* = u M > u M A y JVE/i

The paraxial ray height at a surface is

v ' t ) „ ' <*), ■

Gaussian Image Plane Tilt:

V = A - [ u .6 + t.*A(u )] J . -i -1 J J with J

> * - B0J - Cjy,*. . 22

Aberration Field Decentration: Locates the center of the sur face contribution to the image plane aberration. Note a* = a*’.

Base Sphere:

in • a.* = (with reference to OAR) i . J with

(u * + c.y.*) - B0 . 10 - J J J J

Aspheric Contribution:

a . . Aj

Special Cases:

6 c. - 5E. 4p.* J J Base Sphere Cxp,

6 vA ' 1 1 Aspheric Contribution = 0 '"Aj" CHAPTER 3

ABERRATION FIELDS IN PERTURBED OPTICAL SYSTEMS I

THE THIRD ORDER ABERRATIONS .

For many optical designs the response of the system to perturba tions can be completely characterized by the behavior of the third order aberrations (the Seidel aberrations). This chapter will present the behavior of the third order aberrations under the influence of tilt and decenter perturbations.

In this chapter it is assumed that the reader is thoroughly familiar with the properties of the five Seidel aberrationsj spherical aberration, coma, astigmatism, field curvature, and distortion, in a : centered, rotationally symmetric optical system. Also, some knowledge of the concept of a general wave aberration expansion for a rotationally symmetric system is assumed. Useful references on this material are

Wei ford (19 74), Kings lake (1978), Smith (1966), B o m and Wolf (1975), or Mil. Handbook 141.(1962). From this basic foundation we will find the behavior of the five Seidel aberrations in a perturbed optical system.

A method of displaying the properties will be developed and the re quired equations presented. We will find the aberration coefficients are unchanged in a tilted and decentered system. Rather, it is the

field-dependent behavior of the terms that is affected. Any particular aberration (i.e., coma, astigmatism,...) can be understood by finding the location of the zero or zeroes (nodes) of the term in the image plane.

23 . 24. There are no new aberration types introduced; rather the interpretation of the behavior of an aberration term in the image plane must be generalized to account for the properties in perturbed systems.

In this development the wave aberration coefficients; Wqi+q5 w131> w2 2 2 j W2 2 0 » W 311, are used rather than the Seidel coefficients

Sj - Sy. The paraxial equations for calculating the Seidel coefficients are provided in Table 3-1. All of the results in this chapter arise from the concept of displaced surface contributions to the aberration field applied to a vector generalization of the wave aberration expan sion for rotationally symmetric optical systems. We will begin by presenting the vector approach to the wave aberration expansion.

In a centered, rotationally symmetric optical system, the power series expansion

0 0 0 0 0 0 k = 2p + m

W = 1 1 1 I C W . H P cos 4, (3-1)

j P n m 1 = 2n + ro I. . 5 can be used to describe the wavefront deformation introduced by an optical system on an inpinging spherical wavefront. In this expression, W is the difference in optical path between the deformed wavefront leaving the system, and a reference sphere whose center of curvature is at the field point of interest and whose vertex lies in the center of the. par axial exit pupil. The forms for the subscripts k and 1 are a result of assuming rotational symmetry. The implications of this assumption on the form of the power series are covered by Radkowski (1967). In this equation is the wave aberration coefficient, H is the normalized field height (actual field height divided by the final image height), p 25

Table 3-1= Converting Seidel Coefficients to Wave Aberration Coefficients.

Spherical Aberration: Seidel Coefficients:

W04O = ^ Si Sj = A 2 y 4 ( 2

Coma:

w13i = l s n S II ’ 'I ^ ( n 3 Astigmatism: '

' 1 W222 = 2 SIII SIII = "j B2yA(n>

Petzval Curvature:

w220p " 4 SIV SIV - -I® 2 e i ( i )

Distortion:

w 311 = J sv SV = * 1 A [ S IV * S III

A = ni = nu + nyc, B = nl = nu + nyc, &(—)U\ = U f u n/ n v n 26

is the normalized aperture height (actual aperture height divided by the exit pupil radius) and gives the angular dependence in the aperture.

In an aligned system, the field points of interest lie in the meridional plane. In a perturbed system field points everywhere in the image plane must be considered. To accomplish this an absolute coordi nate system which has its y axis (vertical) in the meridional plane of the aligned system will be chosen as is done in most design programs.

A field point anywhere in the image plane is then located by the vector

H, where

H (3-2)

Similarly, in the exit pupil, the aperture point can be located by a vector p , where

P P e (3-3) These two vectors are illustrated in Fig. 3-1.

11 H y IMAGE PLANE

EXIT PUPIL

Fig. 3-1. Illustrating the Field Vector H and the Aperture Vector p. Since the planes containing p and H are parallel, one can

consider taking the dot product of the two vectors by superimposing the

pupil and image planes giving,

H°p = Hp cos(8 - ) . (3-4)

The wave aberration expansion [Eq. (3-1)] then, can be written in vector form as CO CO oo k = 2pr + m W . = I I I £(WklnPj (H'H)P (p-p)n (H»p)m . 1 = 2n + m j P n m (3-5)

This is a very important step as it allows treating nonmeridional, non-

coplanar, tilts and decenters without cumbersome, confusing notation.

For comparison, the scalar form of the wave aberration expansion for the

third order aberrations is

W = Woi+oP1* + WlSl.Hp3 cosej) + W22gH^P2 COS2^

+ W22qH2P2 + cosdi. [8 = 0] (3-6)

The corresponding vector form is

w = ’^040 (P**p ) 2 + W], 3i(H«p) (p*-p*) + W222 (H'«p) 2

+ w22 0(H°H)(^«B) + W312(H-H)(H-p). (3-7)

To account for the effects of tilts and decenters on the

aberration expansion, only one modification is necessary to Eq. (3-7). '

The decentrations of the surface contributions to the aberration field

at the image plane must be accounted for. In Chapter 2 the equations

necessary to locate the center of the surface contributions were given. " 28

The result was the definition of a vector, gj (the star superscript will be dropped), which locates the center of each surface contribution in the image plane. The vector denotes the point from which the

" ' - effective field height, j, for the surfacecontribution is measured.

To develop the theory of the aberrations at the image plane, a common center is required. This is chosen to be the center of the

Gaussian image plane (located by the OAR). The field point, measured from this common center, is located by the conventional field vector

H. The effective field height for the surface contribution can be -a. written in terms of H as,

HAj = H - Z (3-8) . as illustrated in Fig. 3-2. The wave aberration expansion in a per turbed optical system then is

j P n m

-ill h \ lmh U f - t ) • (H - t.) lp (?•?)" j p n m Klm J 1 J

t(H - (3-9)

This equation accounts completely for the effects of tilt and decenter perturbations on the wave aberration expansion for a rotationally symmetric optical system.

If the only concern is to calculate the wave aberration in per turbed systems numerically, Eq. (3-9) could be used. However, the main incentive for using aberration coefficients is to gain an 29

GAUSS IAN IMAGE PLANE

OAR

Fig. 3-2. The Effective Field Height for a Surface Contribution to the Aberration Field in the lihage Plane. understanding of the behavior of the optical system. To obtain this understanding requires writing this equation in terms of image plane quantities, i.e., performing the summations over surfaces. The remainder of this chapter will deal with performing the summations and inter preting the results for the third order aberrations. The wave aberra tion expansion, to third order, including tilts and decenters is

W = AW2o(Pep) + AW11(H*p) + pV040.(p*p)2 j

* 1*1 3U t (H " 0 , ) - p l (p-p) j 3 3

+ 1*222, [(H - a j'p ]2 + 1*220; [(H " Oj)] (p-p) j J J j

+ 1 * 3 1 1 , [(H - a ).(H - o )][(H - a )•?] . (3-10) j J J J J ■ . 3 0

It is important to realize that the aberration coefficients,

are not affected by.tilts and decentrations. This is because they are

functions of paraxially determined qualities. It is the field depen

dence that is altered through the vector H . Aj The first term to consider is the spherical aberration, •

w = Iw040 4(P’P)2 . (3-11) j J

This term is independent of field and independent of a.. This means that J the spherical aberration is unaffected by tilt and decenter pertur bations. There may be an effect due to change of conjugate with field

from tilted object and image planes, but this is a higher order effect

that will be neglected.

The next term is coma,

w = I wl 31-; t (H - o .) ° p ]( p ° p ) j

■ [((IWi3i .H) - (£w131.a.))*p](p*p) . (3-12) j J j 3 J

"The -firsT 'stlmmat ion -gives

ElV131 H = W131H, ■ (3-13) j J

where 33 is simply the coma aberration coefficient for the centered

system. The summation 31-a•> the sum of the surface contribution j 3 3 displacement vectors in the image plane each weighted by the correspond

ing surface contribution, which results in a net, unnormalized, vector

in the image plane,

A131 - 1^2 31 o.. (3-14) j J 3 31

Defining a normalized vector,

al31 - A 13l/ # 1 31, (3-15) gives for the coma aberration,

W = W131[(H - ai3i)»p](p"p). (3-16)

This is an expression for the third order coma field in a perturbed system written in terms of image plane quantities only. . The effect of tilts and decentrations has been to displace the total coma field for the centered system to the point located by the image plane vector ai3i shown in Fig. 3-3.

Fig. 3-3. In a Perturbed Optical System the Center of the Total Coma Field for the System Is Displaced to the Point Located by the Vector a%31. 32

For any field point H, measured from the center of the Gaussian image plane, the magnitude of the coma is 3i| 311. The coma is oriented

along the vector where

H 131 = H - a131. (3-17)

This behavior is shown in Fig. 3-4.

Throughout this chapter we will be concerned not only with the

general properties of the aberrations but also with the special cases.

Special cases generally arise from zero divide operations. The only

special case for coma occurs when 3i = 0 , i.e., the optical design is

corrected for coma. Then the normalization of 3i, Eq. (3-15), cannot be performed. For a system corrected for coma Eq. (3-16) becomes

W = - (Ai31’p)(p ’p) , (3-18) where A^3X is defined in Eq. (3-14). This is coma that is uniform over

the field, i.e., the magnitude and orientation of the coma are indepen

dent m-f the fieTd poin-t. The'Amargn'itude of 'the coma is -1A3 3i | and the

coma is oriented along the vector A^3^. Physically this is the limiting

case of the node vector ai3i going to infinity as the coma, 3i goes to

zero.

This completes the discussion of third order coma. The procedure

of performing the summation over surfaces, by defining image plane

vectors that are summations over the product of the surface contributions,

^klm -5 anc* some function of the surface contribution displacement vec

tors, Oj (in this case simply the vector itself), will be used to treat

all of the aberrations. This will lead to a set of image plane vectors 33

(a)

131

(b)

Fig. 3-4. Properties of the Transverse Blur for Third Order Coma W131, (a) Aligned System, (b) Perturbed System. 34 that will be called perturbation vectors (even though some will be scalars). The notation used will be explained further at the beginning of Chapter 6 where the interpretation will be more apparent. The con cept of a net decentration of the center of the total image plane aberration field of the centered system for a particular term., will apply to all of the aberrations. For many of the terms this be havior will be accompanied by the development of multiple zeroes

(nodes) in the field- as will be seen for third order astigmatism.

The field curvature and astigmatism will be treated next,

w = AW2o (p^p) + Iw220-i [ 0^ ~ t?-:) • (^ - ] (F'p3 j 3 3 3

+ IW222JCH - P.)-^]2 . (3-19) j In this development the rms wave front over the field will be used as a measure of image quality. The best focal surface, i.e., the surface that results in the minimum rms wavefront, is the medial astigmatic

focal surface. The coefficient #220 corresponds to the sagittal focal surface. The medial surface is obtained from the coefficient

W220m = W220 + V 2 W222 • (3-20)

Using the relationship between cos2-^ and cos2

Eq. (3-19) gives,

W = AW2op2 + W22gH2p2 + W222H2p2 (l/2 + 1/2 cos 2ct>)

= AW20p2 + 'V220?v]H2p2 + l/2W222H2p2 cos2^. . (3-21) . 35

In vector notation this is,

W = AW2 oCp°p) + w220]yjCH<“H) Cp'p) + 1/2W222(H2 ‘P^) • (•>-22)

Consider first the properties of the medial focal surface in a perturbed system,

-AW20 = IW220m - 0“.) »(H - o'.)] j. J J J

= ^ 220M .(>H) - 2H y ( | W 220Mj^ )

+ IW220M . . (2;-23) j ^

Defining the perturbation vectors

A220M - IW220M>O a (*5-24) j . j 3

0 22OM - iw220 Co.oa.) [a scalar] (*5-25) j j • Mj 2 gives

-AW2o = W22 0M (H-H) - 2H-A220M + B220M - 05-26)

In terms of normalized quantities,

a220M = A220M/W220M (:5-27)

b220M 2 B22 0M/W220M - a220M -a220M , (25-28) this becomes ’

-AW20 = W220iyjC CH - a220M) ° (H - a220M)+ b 22 0]vj]

-29) = % 20M [ CH220M oH 220M ) + 0 22Om ] • ^ : 36

Equation (3-29) can be interpreted as follows. The term

^22 d M ^ 220]yio^ 220^ implies that the focal surface is still quadratic with field, but now the vertex is located transversely from the OAR in the Gaussian image plane by the vector 8.220%;' That is to say, as with the coma, the center of this contribution is displaced from the center .’y of the image plane. As before, a22 oM is the normalized sum over sur faces of the product of the surface contribution *220^ and the displacement of the surface contribution a.. Because the W22 0{Vj. 3 j coefficients are not equal to the Wigy coefficients, the vectors a2 2 Q^ and aisi refer to different points in the image plane.

In addition to the transverse displacement of the medial .surface due to a22 0^> there is a longitudinal shift (i.e., focal shift) along the optical axis from the scalar term W22 b220^- This term gives the separation of the medial foca.1 surface from the Gaussian image plane at the vertex of the medial surface located by #2 2 In linear units this separation is

5z220m = - 8 ( f # ) 2W220Mb220}lj1 (3-30)

This behavior is illustrated in Fig. 3-5.

The medial surface can be pictured as a physical surface in

space. In a centered optical system it is simply a curved surface of

radius

rM = ; C 3 " 3 i : )

If one considers constructing a spherical image surface of radius r^. 37

GAUSSIAN PLANE IMAGE

22 0m OAR

MEDIAL/ FOCAL SURFACE

DESIGN PLANE

Fig. 3-5. The Medial Focal Surface in a Perturbed Optical System May Be Displaced Transversely and Longitudinally from the Center of the Gaussian Image Plane.

the image quality on this surface would correspond to the minimum rms wavefront for the system (i.e., the measurement of rms wavefront along any other focal surface would result in a larger value for the rms wavefront). In a centered system the vertex of this bowl-shaped sur face is located on the reference axis at the Gaussian image plane

(assuming no spherical aberration). In a perturbed system, with nonzero values for a22 0M and b220^> the vertex of this surface (defined to be the point of minimum departure from the Gaussian image plane) would have to be moved to the point defined by ^220^ ^220^ if the rms 38 wavefront error is to continue to be a minimum along the surface. The concept of the displacement and deformation of physical surfaces that can be associated with aberration terms is important to understanding the effects of tilts and decenters.

The special case.s- for the medial surface are summarized in

Table 3-2 and illustrated in Fig. 3-6. There are two categories to consider: those for which the system has a flat medial image plane,

W22 O14 = 0 and those for systems with a curved medial focal surface

W220M * 0 .

Table 3.2. Special Cases for the Medial Focal Surface in a Perturbed Optical System.

I. Curved Medial Surface ^ 2 2 0 ^ ? 0:

-AW20 = W22 0mI ( H - a2 20M)^(H - a220M) + b220Ml

(a) General Case: Medial surface of the centered system is de centered relative to the OAR and defocused from the Gaussian image plane (GIP).

(b) a220jij = 0 The vertex of the medial surface of the cen tered system is on the OAR but defocused from the GIP by W220M b220M-

(c) b220]yj;= 0 The vertex of the medial surface of the cen tered system is displaced in the GIP from the OAR by # 220^' There is no defocus. Table 3-2 .--Continued. Special Cases for the Medial Focal Surface in a Perturbed Optical System.

II. Flat Medial Plane W22 0^ = 0:

-AW2o = -2H»A2 2 om + B22 om (3-32)

(a) General Case: Flat medial focal plane tilted and defocused relative to the GIP.

(b) A22 om = 0 Flat medial focal plane defocused relative to the GIP, no tilt.

The astigmatism with respect to the medial surface in a per

turbed system from Eq. (3-22), is given by

W = l/2pV222 [(H - t j '

= V2[VW 222 H 2 - 2 H ( l W 2Z2 a ) - J w 222 a 2]-p2 . (3-33) j •’ j J 3 j 3 J Defining the perturbation vectors,

a222 = IW2 22.cr. (3-34) j

B^222 = IW222 o .2 (3-35) j 3 3

and the corresponding normalized vectors

3-222 = a2 2 2 /W222 (3-36)

b2222 = B2 222/W222 - a2 222 (3-37) 40 GAUSSIAN IMAGE PLANE

OAR

Ka) • H(a)

Kb) 11(b)

1 (c) 1 1 (c)

Fig. 3-6. Illustrating the Special Cases for the Medial Focal Surface. See Table 3-2. gives for Eq. (3-33)

-=» ^ W = 1/2W222 E(H - a222) + b2222]'P2 (3-38) = I /2W 222[H2 222 + b2222]°P2 •

This equation, which gives the behavior of astigmatism, looks similar to Eq. (3-29). However, an important distinction exists; the term

H22 0M 'H22 0M + b220M is a scalar quantity while H2222 + ^ 2 2 2 is a vector (or more appropriately , a squared vector). As the notation and preceding calculations indicate, these vectors have properties not normally associated with vectors and some unfamiliar vector operations result. These operations are discussed in Appendix A,

As with the preceding aberrations, astigmatism can be under stood by finding where in the image plane the aberration is zero. From

Eq. (3-38) this implies solving

0 = (H - a2 22)^ + b2 222 (3-39) for H. With the concept of vector multiplication, this is done as if these were scalar quantities, giving,

H = 3-222 + ■C"b2222^ 2. (3-40)

Using the standard method of taking the square root of negative numbers gives the astigmatism is zero at, where if

(3-42)

then

t)222 = j b2 | 2 e^^ = b222e^^ (3-43) and

(3-44)

The vector operations relating to Eqs. (3-42) through (3-44) are

illustrated in Fig. 3-7. Equation (3-41) implies that in a perturbed system, the astigmatism with respect to the medial surface is zero at two points in the image plane. These two points are located by the vectors a222 + ^222* a222 ~ it>222 as in Fig. 3-8.

H x

Fig. 3-7. The Relation between Linear and Squared Vectors. 43

222

Fig. 3-8. In a Perturbed System the Astigmatism Can Be Zero at Two Points in the Field.

For the first time an aberration term has developed multiple

zeroes in the field upon the introduction of tilt and decenter pertur bations. This will occur frequently in the treatment of the fifth order aberrations. This behavior for astigmatism has been called by

Shack, binodal astigmatism. It is illustrated in Fig. 3-9 where con tours of constant magnitude of astigmatism have been plotted for a system containing binodal astigmatism. It results, in general, when ever a system that is not corrected for astigmatism is perturbed. The centered system, where the astigmatism is zero at the center of the field, is simply a special case of binodal astigmatism where the two nodes are coincident at the center of the field. Using the concept of 44

Fig. 3-9. Field Contours of Constant Astigmatism.

Binodal astigmatic field. '45

multiple nodes to explain the aberrations will provide insights into

the behavior of many of the fifth order terms. It is this concept

that has allowed for the first time understanding of the properties of

aberration fields in perturbed optical systdms. We will then discuss

in detail how nodes may be used to understand and calculate various

features of the aberrations.

The nodes of the astigmatic field are easily located by the vectors a.222 and b^zz- Once they have been found, the magnitude and orientation of the line foci on each of the two astigmatic focal

surfaces can be determined directly from their location with respect to

the field point of interest. Rewriting Eq. (3-38),

W =• I/2W2 2 2 { [ (H - 3-22 2 ) + ^ 2 2 2 ] [ (H - 3222) “ 1^222]

E 1/2W222(N1N2) ^ 2 . (3-45)

The vectors ,N2 are called node vectors. These are vectors that connect the field point of interest, located by H, with the nodes as in Fig. 3-10. The magnitude of the astigmatism is given by the product of the distance from the field point of interest to each of the nodes, | Ni| j N2| . The orientation of the line foci on one of the focal surfaces are along the vector {N1N2}^. This is simply the line that bisects the angle subtended between the vectors and % . The

line foci on the other focal surface are at right angles to these.

Given the location of the nodes, one can construct the node vectors and find the magnitude and orientation of the astigmatism for any field point. , 46

Fig. 3-10. Node Vectors for Illustrating the Properties of Astigma tism in the Field.

Figure 3-11 is a plot of the relative magnitude and orientation of the astigmatism on each of the focal surfaces for a system with binodal astigmatism.

Astigmatism is frequently analyzed by employing sagittal and tangential field curves. These are profile plots of the sagittal and tangential focal surfaces that are rotationally symmetric in an aligned system. The equations for calculating field curves will be presented in Chapter 4. Here, the properties of field curves and more generally focal surfaces in perturbed systems will be discussed.

In a system with binodal astigmatism, the astigmatism is zero at two points in the field. At these points, the two focal surfaces 47

/ / —- — — — vxX / / — — — — — \ X X \ 'V / / — x X \ X z / Z - - - V \ \ \ z z Z » • * •> s \ \ \ / / • ' ' \ \ \ \ / / / \ \ ' X

\ \ \ \ \ \ \ \ X \\ \ \ i \ v. \ \

X X X\ X

X X X X \ ' x x X ''

x ' __

Fig. 3-11. Magnitude and Orientation Plots for Binodal Astigmatism. 48 touch, they do not cross. The surfaces (and therefore any set of field

curves) cannot cross because this would imply that there would be some

closed curve in the field where the astigmatism is zero. This behavior

is illustrated in Fig. 3-12, which is a projection plot of the two focal

surfaces that contain line foci due to astigmatism. Here a flat (i.e.,

#2 2 0 ^ = 0) medial surface has been used for convenience.

Fig. 3-12. The Shape of the Two Focal Surfaces Containing Line Images in a System with Binodal Astigmatism. 49 From Fig. 3-11 and Fig. 3-12 it can be seen that the focal sur

faces can no longer be distinguished on the basis of image orientation.

For this reason a new terminology will be introduced here. This terminology will not be completely justified until Chapter 5, where the

fifth order terms are included. The two focal surfaces will be called

N and P surfaces. This choice refers to the negative (N) and positive

(P) branches that are used in the.equations to calculate these surfaces.

With positive z left to right, the N surface is always to the left of the medial surface and the P surface is always to the right. Whether

these surfaces correspond to sagittal or tangential surfaces in an aligned system depends on the sign of the field curvature and astigmatism.

As with the medial surface, there are two types of special cases

for astigmatism that are listed in Table 3-3. One set is the special

symmetries that arise in a system that is not corrected for astigmatism.

The other is the cases in which the system is corrected for astigmatism.

For the various cases. Figs. 3-13 and 3-14 provide the corresponding

.field, curves (where the profile is chosen to contain the nodes) and

magnitude and orientation plots when appropriate. Figure 3-15 provides

projection plots of the four basic types of focal surfaces that may

result in a perturbed system. Table 3-3. Special Cases for Astigmatism with Respect to the Medial Focal Surface in a Perturbed Optical System.

I. Centered System Contains Astigmatism W222 ^ 0;

W = 1/2* 2 2 2 [(H - &222)2 + b2 222l"P2

(a) General Case Binodal astigmatism, centered at 8222 (i-e. plane symmetric about fhe point_^located_by ^ 222) with nodes at a222 + ^ 2 2 2 311d a222 ~ ib222

•(b). a222 = 0 Binodal astigmatism that is plane symmetric about the OAR.

(C) b2222 = 0 Conventional quadratic astigmatism (i.e., the nodes are coincident) decentered in the field by the vector 8^22 *

II. Centered System Corrected for Astigmatism *222 ~ ® •'

W = l/2 [-2HA 222 + ^ 2 2 2 1 °P2 (3-46)

(a) General Case: Let: a222^ = 1/26 ^ 222/^222 (3-47)

Then: W = -[(H - &222^)A222]* P2 » (3-48)

This is linear astigmatism (i.e., astigmatism that depends linearly on field) centered at a222Tj- The magnitude is - 1 [ (H - a222L)A222] I and the orientation of th^e sagittal line is along the vector -{ (H - a222L) A222^i5- Here the N mid P fields are conically shaped in three dimensions.

(b) A222 — 0 W 1/2B2 222°P2 (3-49)

This is astigmatism whose magnitude and orien tation are constant over the field and governed by the vector B2 22 2 *

N M P

1(a) H ( a )

N M P N M P

Kb) 11(b)

Kc) 11(c)

Fig. 3-13. Illustrating the Special Cases for Astigmatism.

Field curve profiles that contain the nodes. See Table 3-3. 52

/ / / / / / / / / / / / / / / / ///////X

1(c) Quadratic astigmatism

x \ X

I (a) Binodal astigmatism

Fig. 3-14. Illustrating Special Cases for Astigmatism.

Magnitude and orientation plots on the focal surfaces con taining line images. See Table 3-3. 53

m u u i H i \ \ \ \ \ \ \ \ M I w w w m \ \ \ \ \ \ \ \ M I i / ii////// \ \ \ \ A \ \ \ 1 1 I I / z z

/////III \ \ WWW m v / n \ \ m \

H (a ) Linear astigmatism

I ! i i i I

I I I

i I

11(b) Constant astigmatism

Fig. 3-14.— Continued Illustrating Special Cases for Astigmatism.

Magnitude and orientation plots on the focal surfaces containing line images. See Table 3-3. 54

(a) Binodal astigmatism (b) Decentered quadratic astigmatism

Fig. 3-15. Illustrating the Four Types of Astigmatic Focal Surfaces in a Perturbed Optical System. 55 The last of the third order aberrations is distortion,

W = IW3n [(H - a )o(H - a )][(H - a,)-p]. (5-50) A J J J J

This aberration results only in image displacement and does not result in image degradation. As a consequence it is better treated and inter preted in terms of the properties of the transverse aberration, which will be introduced in the following chapter. The discussion of distor

tion appears in Chapter 4.

As noted in tfye previous chapter there is one special case that

arises when the field-dependent aberration contributions due to a per

turbed surface are zero. In this case the aberration introduced is due

to a shearing between the pupil and the center of symmetry for the

spherical aberration contribution and is uniform over the field. The

shearing is parameterized by the displacement vector Ap. (Ap, for an J J aspheric surface). The contribution to the aberrations is found by expanding the pupil dependence for the spherical aberration,

W = EWo4oi [(p Ap .) «(p + Ap ) ]2 (5-51) j J J J doing the summations and identifying the appropriate terms. This results in the following set of perturbation vectors,

Al31u ~ -4^040. Ap (5-52) j

(5-55)

(5-54)

(5-55) , 56

These vectors are treated in the same manner as the vectors without the

u subscript. In terms of properties of aberration fields there are no new effects. We will not then distinguish these contributions from the

general case. We simply note that when y . = 0 or i. =0 at a surface, j 3 these vector calculations are substituted for those made in the general

case.

This completes the discussion of the properties of the third

order aberration terms in perturbed optical systems (excluding distor

tion) . In systems dominated by third order aberration, for example, most

rotationally symmetric mirror systems, these properties are sufficient

to account for and understand the behavior of the systems under the in

fluence of tilt and decenter perturbations. the vectors that were intro

duced Akim* ®kim* ^klm* Cklm used in the following chapters

to calculate the image degradation induced by tilts and decenters. CHAPTER 4

THE ANALYSIS OF PERTURBED OPTICAL SYSTEMS

In most optical design programs there are a variety of options that

may be used to analyze the performance of a system during the design process. These options include ray fans, field curves, distortion calcu

lations, nhs wavefront and rms spot size calculations, spot diagrams,

Strehl ratios and MTF calculations, and a variety of options that are

derived from spot diagram data. In the large design programs these cal

culations are all real ray based.

For most of these options analytic expressions can be developed

using wave aberration coefficients to provide the same information without

tracing real rays. In all cases the analytic expressions are approxima

tions to the real behavior. The success of the analytic approach for

predicting real ray behavior depends on the order (i.e., third, fifth,...)

of the coefficients used and the speed (f/No.) and field of view of the

components in the system.

The expressions for analyzing centered, rotationally symmetric

optical systems using coefficients have been available for some time. They

were developed before the advent of computers to provide efficient methods

for analyzing optical systems. In general, they depend only on image

plane aberration coefficients, so the calculation time, given the coef

ficients, is independent of the number of surfaces in the system. As com

puters have become more and more efficient, these analysis routines have

been changed to real ray-based calculations. Today all of the options , ' ' ,58 ■ listed are computed using real rays in the large design programs. The main advantage of using real rays is the accuracy obtained in the calculation.

In rotationally symmetric systems real ray analysis is generally preferable to coefficient-.based analysis. In perturbed optical systems, however, this is not the case, especially in the area of rms calculations.

This is because in a rotationally symmetric system the analysis needs to be done only over one profile of the positive field. By analyzing the system at three to five field points along the positive y axis in the field, a complete picture of the performance of the system by interpolation can generally be made. In an arbitrarily perturbed system this is no longer the case. To obtain a comparable picture of the performance of the system requires 30 to 200 points distributed across the field in the analysis. In rms calculations, which require that between 200 and 1000 rays be traced through the system for each field point, the cost rapidly becomes prohibitive.

Until now, a method of using aberration coefficients to analyze perturbed optical systems has not been available. The result is that designers have been forced to analyze perturbed systems using real rays.

The method generally is to use as many field points as economically feasible, and the result has been a gross under sampling of the field and a lack of understanding of the response of the system to perturbations.

This situation arises whenever a tolerance analysis is to be done to pro vide the manufacturing and fabrication tolerances for a design. Using the theory developed in Chapter 3, it is possible to develop analytic expressions for the rms wavefront error in perturbed optical systems.

These expressions depend only on image plane quantities, and the result

is that calculations that give information over the entire image plane

maybe made economically.

This chapter will-develop some of the expressions required to

analyze perturbed optical systems using wave aberration coefficients.

Starting with the wave aberration expansion to third order including the

effects of tilts and decenters, expressions for the transverse aberration,

rms wavefront and spot size, field curve calculation, and distortion

plotting will be given. In the next chapter these equations will be

used to investigate the behavior of a perturbed mirror system using a

program developed for a Tektronix 4051 graphics terminal.

Combining all of the results of Chapter 3, the equation for the

wave aberration expansion to third order, including tilts and decenters is

W = AW^oCpop) + W04o(p«p) 2

+ W13I[(H - a13i)-p](p-p)

+ W220]yi[ (H ~ a220]y) ° CH ~ a2 2 0 ^ + ^ 2 2 0 ^ (P°p)

+ 1/2W222[(H - t222)2 > b2222]‘P2$ (4-1) where tilt and distortion have been left out as they do not contribute

to image blur and will be treated separately. In analyzing perturbed

optical systems, it is more convenient to use the unnormalized form of

the wave aberration expansion, which is 60

W = AW2 0 (p°p) + W 04o(p °p) 2

+ [CW131H » A 131) -pKp'-p)

+ IW220M CH-H> - 2CH«A220m) + B220M](p.t)

+ 1/2[W222S 2 - 2HA222 + ^ 2 2 2 ] °P^o : ^ ^

To simplify the notation the following definitions will be made,

Lll31 - W 131H - A 131, (4-3)

[ l220M = W220m (HoH) - 2(H«A2 2 om) + B220m, (4-4)

( 12222 - - 2HA222 + B2222. (4-5)

With these definitions, Eq. (4-2) becomes

W = AW20 (p-p) + Woifo (p°P) 2 + ([ 1131°P) (p'?p)

* I 1220M(P-^ ♦ l / 2 n 2222-?- t4-6)

Starting with the wave aberration expansion for a perturbed system,

an equation for the transverse aberration may be developed. This can be

used to generate ray fans, spot diagrams, and distortion plots. In a

rotationally symmetric optical system, the transverse aberration is ob

tained from the wave aberration expansion by taking the derivative of

the wave aberration expansion, '

(n’u')t , i|f + jfp (4-7)

In more general notation, this is equivalent to taking the gradient of the

wavefront ■ (a'u' )t = VW, (4-8) where V (del) is the gradient operator,

2/ A. A r\ 7 5^ * *dy • (4-9)

Using a general vector relation established in Appendix A, the equation for the transverse aberration May be found directly. The relation is

n Ch " - ? 1) Cp’-p )” ] = ■ Cp -p )” "1?

n(?-:?)nff1tp*)n'1- (4-10)

With this relation, operating with the gradient operator on Eq. (4-6) gives, for the transverse aberration,

(n'u')e = VW = 2AW2 oP + 4Woito(P°P)P

+ 2([ ]i3i«p)p + (p°p)[ ]131

+- 2 [ ]220mP

+ t ]2222P*- (4-11)

As pointed out in Chapter 3, distortion is best treated in the transverse rather than the wavefront domain. Now that the technique for competing transverse aberration has been presented, the distortion can be treated in detail. Because distortion is an aberration that depends on field cubed it is the most complex of the third order aberrations. For completeness, the derivation will be included here even though it is considerably more involved than any of the derivations of Chapter 3. This treatment is not an integral part of the following analysis routines and can be skipped without loss of continuity. 62

For distortion, the wave aberration expansion is

> = I W s n . t C # - o.)-(H . o,)] [CH - o.J-pi'' • (4-12) 3, 3

= IWsiij (ff-fl) (£.£) - 2 ^ W 311j^ (t.p)

+ ^ " (H'H) ^1*311

+ 2lw311j (H-a.) (a.-t) - fozil) °P- . ] • . v 3 /

Before defining image plane perturbation vectors, one of the terms must be written inia different form. The term 2%W3n.(H*a.)(a.*p) cannot be j 3 3 3 summed directly. To perform the summation, a vector identity established by Shack must be applied. The identity from Appendix A is,

2 (A«"B) (A>C) = (A-A) (B»€) + A^BG. (4-14)

Applied here this gives,

2Iw31lj CH'Oj) (a^'p) = ^IWgi (c y a ^ (H»p)

■ + (4-15)

It is also desirable to retain the same pupil dependence. This is accom- . plished by applying the vector relation, again from Appendix A,

A2oBC = A2#*-? (4-16) to obtain

= Y[W311.f.2)H«.p. (4-17) 63

Now, defining image plane perturbation vectors.

b311 = 1^3114 (o4 °cr.) . [a scalar] (4-19) 3 J J J

^ 3 1 1 = 1^3114^4 . (4-20) • • 3 J 3 ■ - , ■

C 311 = Iw 3iij(Oj.Oj)cy (4-21)

gives the unnormalized equation for distortion in a perturbed system

W = W 31! (H-H) (H-£) - 2 (H-A3i 0 (H«p) + 2B311 (H-p)

- (H.°H) (A3i i °p) + B?3nH*»p - G311»p (4-22)

"= t ]311°P- (4-23)

The transverse aberration then is simply

(n'u')s = VW = [ ] a n . (4-24)

The expression for the image displacement due to distortion can be interpreted through the use of node plots by writing the normalized form,

(n’u* )e = AWiiH + W 31i{ [ (H - a 3n ) 2 + b^suKH - a 3n ) *

+ [2b3iiH - (C311: - b 2 3iia*3n ) ]} (4-25) where

.a311 = A 31l/W 311 , (4^26)

^ ^ b 311 = B31l/W311 ~ a311'a311 (4-27)

311 = ^Sll/^Sll - ? 3 1 1 (4-28)

c 311 '= C311/W 311 - (a3ii°a 3ii)a 3 n (4-29) 64 and the first order image displacement coefficient AWn has been included.

Equation (4-25) does not follow directly from Eq. (4-22), and the deri vation of this form is contained in Appendix C. The normalized form can be further simplified with the following definitions,

W 111E = AWn + 2W 311b3 11 (4-30)

BlllE = ' ^ 3 1 1 a*311^ : (4-31)

h311 = H - a311, Hiiig = H - aiiig (4-32) giving finally for'distortion,

(n'u')e = W 311 [(H^sn + b2 311)H*31i] + (4-33)

To construct a node plot for this aberration the terms W 3n , are considered separately. The term WixigHiHg, gives a displacement along the vector Hnig that is directed radially away from the first order node located in the field by the vector a% % The magnitude of this displace ment component depends linearly on the distance from the node. The nodes for the third order term W 3u , are found by noting that H 2 3n + b^sii is of the same form as H^222 + b2 22 2 $ which gave rise to binodal astigmatism. This term then gives zeroes, located at H3n + ib3n and -3b. H 3h - ib311 . There is an additional zero at the point located by the vector a3n due to the H* 3i% multiplying. The third order termthen goes to zero at three collinear points in the field.

The general node plot for distortion is shown in Fig. 4-1. To obtain the behavior of the distortion across the field, both the first and 65 third order nodes must be considered. This will not be treated in detail here. For further insights into interpreting the behavior of distortion through node plots, the reader is referred to Chapter 6 and the discussion relating to linear plus field cubed coma, which has the same nodal behavior.

311

Fig. 4-1. Node Plot for Distortion. 66 In designing and performing a tolerance analysis on an optical system, there is generally some number established to represent a boundary that limits the allowable value of some measure of image quality. Image quality can be defined in terms of MTF, rms wavefront, rms spot size, encircled energy, or a variety of other quantities generally calculated at a series of representative field points in the image plane using real rays. Usually, the cost of calculating these measures of image quality are relatively high. During the design process then, other analysis tools such as ray fans, field curves, or aberration coefficient tables are used to guide the design process. The actual measure of image quality is evaluated infrequently.

The technique for analyzing perturbed systems is generally to try to relate the behavior of one of the inexpensive analysis tools, such as ray fans, to the degradation in image quality for a few cases and attempt to predict the behavior of the system by interpreting the response of the ray fans to a large number of perturbations. Here, an expression for the rms wavefront or rms spot size calculated using aberration coefficients will be developed for perturbed systems. This analytic expression is much more efficient than the real ray techniques and allows direct evaluation of these measures of image quality for each cycle of analysis.

The use of analytic expressions in optical design for rms wavefront or spot size, especially orthonormalized expressions, has been investigated by many people including Unvala (1967), Radkowski (op. cit.), Wiese (1974),

Kreitzer (1976), and recently Robb (1980). Unvala developed the 67 orthonormalized expression for rms wavefront and demonstrated how it can be used to understand optimal aberration balancing. Radkowski. worked with generalizing this approach to noncircular pupils. Wiese applied the expressions to design problems with the ACCOS design program. Krietzer illustrated that in the design process the rms wavefront is a physically significant quality factor even in systems with large aberrations. The reader is referred to these works for detailed discussion of orthonormal ized aberration expressions and their relation to actual image quality.

The wavefront variance is given by

ti)2 = ~//W2pdpd

The rms wavefront error is simply the square root of the variance. Here the integration will be performed over a circular pupil. The procedure of evaluating this integral is unaffected by perturbations because it is the field dependence and not the pupil dependence that is affected by pertur bations. All of the techniques that have been developed to deal with vignetted and obscured apertures can be applied directly.

The integration and subsequent orthonormalization have been done by Unvala and, in the notation used here, by Wiese. Since the integration is over the pupil and the pupil dependence is unaffected by perturbations, the coefficients obtained are unchanged. All that is necessary is to account for the change in the field dependence. This is done by substi* tuting Eqs. (4-3), (4-4), (4-5), (4-22) for the field dependence. The result is that the orthonormalized expression for the rms wavefront error in a perturbed optical system with a circular aperture to third order is 68

w rms {u)'2 } = T j 2 " ^ w 20 + W040 + [ ] 2 2 0 ^4]"

1 2 + 180 W 0400

+ & [ A 222' r ]222

(AW 1 + y [ ] l3i+; [ ] 3ii)«CAWn H + [ ] i3i +[ ] 3 1 1)].

^rms^p (4-35)

The corresponding expression in a centered system is

^ rms^ c jj!aW2o + #040 + W220M (HcH)

1 2 + 180 *040

+ ^ 4 [W222H2 «W222H 2

+ ■^'^(aWjjH + W 434H + Wg^ 4 d ) « (a#i

+ Wifl3lff + W3 1 1 (t.S)t)] ■

(4-36)

I 69 . A measure of image degradation due to perturbations is the change in rms wavefront,

Ah3rms = K m s ^ ~ C(0rms}c* (4 - 37 )

Notice that this can be a negative number since at a particular field point the rms wavefront can be reduced due to perturbations. A negative value for Aiorms indicates that the system performance has improved.

Equation (4-37) is a function of image plane quantities only.

Given the aberration coefficients and the perturbation vectors, the cal culation time is independent of the number of surfaces at each field point.

It is possible then to evaluate efficiently the change in rms wavefront over the entire image plane.

For the third order aberrations the relation between the node plots and the resulting image degradation is generally clear. This can be seen from the orthonormalized form of Eq. (4-35). The first term, being a function of focus, is the equation for the best focal surface. Assuming the image plane can be adjusted, this does not contribute any image degradation as the curvature of the best focal surface is unchanged, it is simply displaced.. The second term is not a function of the perturbations and therefore does not contribute to 'the image degradation. The fourth term relates to the location of the centroid of the image and in general is not included in the calculation of image quality as it is simply an image displacement term. The only terms that contribute to the degradation in image quality are terms three and five. Term three depends only on astigmatism and term five only on coma. The coma and astigmatism node . . 70 plots then will generally explain directly the contours of constant image degradation in the field.

Whereas the rms wavefront is the appropriate image quality factor for design (even in systems with large aberrations as shown by Krietzer), in analyzing systems with.large aberrations, rms spot;size is more common.

The expression for the mean square spot size is obtained from

_ !_ i r r /sw\2- / s w V e2 pdpdcj). (4-38) ir (n'u,)z 1) Lvy/ \6 xJ

The rms spot size is the square root of this quantity. Again the inte gration is over the pupil, and therefore it is unaffected by tilts and de*

centers, The process of integrating and orthonormalizing this gives a result that is of the same form as Eq, (4-35) simply with different co efficients. The resulting Expression for the rms spot size in a perturbed system is

, ■ V, • h'11' ^rms-’p = n'u'tPp) 2 = jzplWgo + j W 04 o + [ l220M l 2

4 w2 + g "040

i rr^ 2 a 2 i J {_[ J 222 * ( J222J

[(AWnH + n i31 + M 3 11)-(AW11H + riiai + M s n ) ]

+ I [ p i 31" (4-39) . .71

The corresponding expression for an aligned system is

n,U,Cefms)c ” * |2 [aW2o + | WQ40 * W220M (if.H)j2

4 2 g" w040 (4-40)

+ ^(AWn H + W131H + Wgn.CH-HjHMAWnH + W131H

+ W3n (ii«H)H)]

The change in rffls spot size then is,

AErms = ^ r m s ’p " ^ r m s ’c' t4"41’

• In a centered optical system the behavior of the field curvature and astigmatism during the design process is often studied through the use of field curves. In perturbed systems field curves continue to be useful for studying the effects of the nodes on the astigmatism.

One problem real ray programs have had is how sagittal and tangential focal surfaces should be defined in a system without rotational symmetry. Here a consistent method of distinguishing focal surfaces in a perturbed System will be introduced. To accomplish this, vector expressions for generating field curves in an aligned system will be developed. These will then be generalized to perturbed systems. 72

Field curves are plots of particular focal surfaces. In an aligned system these surfaces are the sagittal and tangential focal sur faces fbr astigmatism. The sagittal surface is the surface along which the skew rays along the x axis in the aperture come to focus, leaving a line image oriented radially in the field. The tangential focal surface is the surface along which the meridional rays in the aperture are in focus resulting in a line image oriented perpendicularly to the sagittal line. The medial focal surface lies halfway between the sagittal and tangential surfaces, and the image blur on this surface is circular.

In an aligned system these surfaces afe fotationally symmetric and quadratic with field.

To obtain the expression for the focal surfaces in terms of vector quantities, consider the aberration terms in an aligned system in vector form,

W = AW2 0 (p'p) + w2 2 0m (H °H)(P‘P) + J W2 22H 2 ‘P2 • . (4-42)

The focal surfaces are defined by the properties of the transverse aberration, i.e., the location of the line foci, so the transverse expression

VW = 2AW2 o'T + 2W2 20M (H°H)p + W222H2^* (4-43) should be used. Recalling that

1 W220m = W220 + 2" W222 (4-44) - . - 73 and that W 220 is the coefficient” for the sagittal focal surface allows finding the expressions for the focal surfaces

Sagittal: -AW20 = W2 2 0m (H-H) - | W2 2 2 (H-H) (4-45)

n ’u't = W2 2 2 ( M ) h + W222 (H2)^* (4-46)

Medial: -AW20 = W2 2 0m (H»H) (4-47)

n'u1-? = W2 2 2 (H2 )h* (4-48)

Tangential: -AW20 = W2 2 Q^(H«H) + y W2 2 2 (H«H) (4-49)

n'u't = -W222 (H=H)p1 + W2 2 2 (H2)p*. (4-50)

To see that line foci indeed result from Eqs. (4-46) and (4-50), consider the exponential form. For the sagittal image, Eq. (4-46) can be written as

n'u'e = W 222H2p e1 + W222H 2p e1 ^26 " (4-51)

= 2W2 2 2 H2p cos (9 -

^ n* uf e = 2 W2 2 2 (H«p)H. (4-52)

The aberration then is always along H and therefore is a line aberration oriented radially away from the origin. Similarly, the expression for the 74 image on the tangential surface can be written

n'u'e = 2.W2 2 2 .(H»p)iH,\. (4-53) i.e., it is always a line image oriented at tight angles to the sagittal image. Qn the medial surface the transverse aberration is a function.of p* and results in a circular pattern.

In a perturbed system, as discussed in Chapter 3, there continue to be two focal surfaces on which line images oriented at right angles to each other at each field point are obtained. These surfaces are distinguished by whether the negative or positive sign is chosen for the departure from the medial surface. Substituting the field dependence for the aberration terms in a perturbed system into the expression for an aligned systeqi gives,

N-Surface: ■ . -AW2o ” E l22 0M " ■|'E lz2 2 ° E I222 (4-54)

n ’u* £ = (E 3222 ° E + E ]2 2 2 2 P*

= 2 ( n 222^ ) ["1222 (4-55)

Medial Surface: -a W2 o - E 1220^ (4-56)

n'u't = E^32 222P6* (4-57) .

P-Surface: -AW2.o = E 1220^ + 1.222 * [\I 222 " (4-58)

n fU?£ = -([ ] 222 'E 1222)p' + [ 12222P*

- 2 ([ ]2 2 2 "P) iE ]2 2 2 . (4-59) where image on the tangential surface can be written

n'u't = 2W2 2 2 (H°P)iH, (4-53) i.e., it is always a line image oriented at right angles to the sagittal image. On the medial surface the transverse aberration is a function of p* and results in a circular pattern.

N-Surface: ^AW2o = [ l22 0 M ~ 12226 1 I 222 (4-54) •

n'u'E = ( [ % 22-r]222)P + rt2222#*

= 2 c r ] 2 2 2 « p ) n 2 2 2 (4 -5 5 )

Medial Surface: -AW20 - [ ]2 2 0^ (4-56)

n'u’t = t^]2 2 2 2 d* (4-57)

P-Surface: -AW2 0 = [ l2 2 oM + ^rl ]222* [ 1222 (4-58)

ri’u ’e = -([ ]2 2 2 °[ ]222)p + [ ]2 222P*

= 2([ ]2 2 2 °p)i[ 1222 (4-59)

Where . 76

The inherent difference between the expressions developed using the vector expansion and those from the scalar form is that in the scalar form, the departure from the medial surface depends on the sign of the # 2 2 2 coefficient, whereas in the vector expression it does not.

The N and P surfaces then are not equivalent to the conventional S and

T surfaces but as will be discussed in Chapter 6 they are the correct approach to astigmatic focal surfaces even in aligned systems. CHAPTER 5

- APPLICATION: MISALIGNMENTS IN RITCHEY-CHRETIEN TELESCOPES. A NEW TECHNIQUE. .

This chapter presents an in-depth discussion of the effects of misalignments in the most common form of astronomical telescope, the

Ritchey-Chrdtien design, With the discovery by Shack of binodal astigmatism, some important insights into the effects of misalignments on this type of telescope may be developed. These insights may be used to deduce from througE-focus star plates what misalignments or support problems exist in a telescope and in many cases how specifically

the problems can be corrected. This new technique for detecting and

correcting errors in astronomical telescopes will be treated in detail.

The Ritchey-Chretien design is a two-mirror telescope with a

concave primary and convex secondary in which conics are placed on the

two mirrors to correct the third order spherical aberration and the

third order coma. The result is that the field of view of these systems

is limited by the third order field curvature and astigmatism.

When such a system is arbitrarily misaligned, the predominant

effect is the introduction of coma that is constant in both magnitude

and orientation across the field. This coma is usually removed by

tilting the secondary. As will be shown, this action is not sufficient

to realign the telescope and usually results in binodal astigmatism,

which can give appreciable image degradation and nonsymmetric perfor- <- • ■ ■ - - . ' ' ■ ■ ■ - ■ mance in the field.

77 This problem will be studied by simulation using a program developed for a Tektronix 4051 graphics terminal. The response of a specific Ritchey-Chrdtien design to misalignments will be studied using a variety of graphic displays to develop an understanding of the system. The techniques that will be illustrated have been successful in . - . predicting perturbations in a 2.3-m astronomical telescope. For this chapter, however, a hypothetical 4-m telescope is used as the example.

The mirror system to be studied is an f/8 , 2/3° full-field-of-

Yi.ew telescope with a 4-m aperture. The specific system data including the appropriate conics for the Ritchey-Chretien solution and the paraxial ray trace data are given in Tables 5-1 and 5-2. (All units in this discussion are in millimeters unless otherwise noted.) Table 5-3 gives the surface-by-surface listing of the wave aberration coefficients.

Table 5-1. System Data

§ CUmMTURE THICKNESS INDEX CONIC

t ' "[email protected]® -leW#® -1.0961

2 -@ 6 @@©1®4 9441.8330 1.00000 -5.1628 79

Table 5**2, Paraxial Ray Trace

FS- . 8=00 l/2 fov= @«33deg L» 11.4117 .

.8 Y ■Y.BAR y: UBAR IBAR 1 1981=2080 8.8000 0=000008 0.005760 .0=005760 • 2 ' 598=1146 43=1458 0=185714 ■' "0.085760 "0.010264 3 . 8=8882 182=5874 "8=062580 . @=©14769 0.814769

Table 5-3. Wave Aberration Coefficients in Waves at X = 0.587iam.

•R131 M228H H2 2 2 H311

1 674=935 •167=467 5=194 10.388 1A "739=796 0 . 8.000 2 ' -238.259 78.821 Sc -6=519 -1.375 2A 303=132 88=646 6=481 6.481 0 = 474' ABERRATION- TOTALS 16=726 10=350 80

As discussed in Chapter 2, the aspheric contribution to the aberration field associated with a surface must be kept separate from the spherical contribution when analyzing the perturbed system. This implies that the tilt and decenter of each surface are independent parameters here, as both.surfaces are aspheric. There are then four misalignments to contend with.However, with the proper choice of reference axis, there are in fact only two independent misalignments that can occur. If the reference axis is chosen to be the line con necting the vertex of the primary with the vertex of the secondary, then only tilts of the primary and secondary can occur. If the reference axis is chosen to be the local axis of the primary (i.e., the line connecting the vertex with the center of curvature), then the two free parameters are the tilt and decenter of the secondary. Any other choice of reference axis can be shown to result in only two free parameters by transformation of the coordinate system to one of the equivalent special cases given above, .

For this example, assume the reference axis is chosen to pass through the vertices of the two mirrors. The mirrors then can only be tilted. We will begin by introducing the "arbitrary" perturbations given in Table .5-4 into the system. These values are about the order of magnitude one would expect to encounter when aligning a large telescope.

Table 5-5 lists the first order properties (see Chap. 2) of the perturbed system.. The most important column here is the list of sigma vectors, o. The row labeling, i.e. IX, IXA, IY,..., corresponds to the subscript that appears with t, i.e. 01 = ciî + alyj, aiA = .îî + alYAj>

where A denotes the aspheric contribution.

Table 5-4. Perturbations for the Initial Decentered Design.

Tilt Magnitude Orientation (Arc Min) (deg)

Primary 1 .2 27

Secondary 1 .2 . -70

Table 5-5. First Order Tilt and Decenter Properties in the Arbitrarily Misaligned System.

—& 8 SIGMA OBJ DISP PUP DISP UY FIELD TILT

IX -0.0273 ■ 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 0 8 .0 0 0 0 U= 8 IXA 0 .0 0 0 0 0 .0 0 0 0 1Y ©=0545 8.0000 0.0000 0.080000 0 .0 0 0 0 U=0 m 0 .0 0 0 0 8 .0 0 0 0

2 X —0.0868 ■ 8=0545 8.8008 ' -8.080314. 2.3532 -0.800157 2XA -0.0545 0 .0 0 0 0 2Y 0 .1 2 1 0 : -8.1091 0 .0 0 0 0 0.008628 -4.7064 0.000314 2YA 0=1091 0 .0 0 0 0

IMAGEX 0.0888 -0.0025 0.001469 16.2207 0.802759 PLANEY -0.1217 8.0089 -0.001855 -22=2243 -0=003370 82

Figure 5-1 shows the graphic format for illustrating the nodal

properties of the perturbed system. The upper left hand quadrant

(quadrant 1) gives the boresight error and the Gaussian image plane

tilt and illustrates the location of the Gaussian image plane (dashed

axes) with respect to the reference plane (i.e., the plane centered on

the reference axis, solid axes), The upper right-hand quadrant

(quadrant 2) gives the properties of the coma field. The lower left-hand

_ quadrant (quadrant 3) gives the vertex location of the medial focal sur

face and the lower, right-hand quadrant (quadrant 4) gives the nodal

properties of the astigmatism relative to the medial surface.

Referring to quadrant 2 , the predominant effect of the two per

turbations has been to introduce uniform coma into the system. Uniform

coma arises (rather than linear) because the aligned system is

corrected for coma. The magnitude and orientation are calculated using,

: _ v/o _ ' ^ a X31 = [CWi3i)icri + (Wl3i)iA a lA + ( W ^ ^ c ^

. + CWi3i)2Aa2A]. (5-1)

fsee Bq.((3-14)]

The values shown in Fig. 5-1 may be found by inserting the appropriate

values from Tables 5-3 and 5-5.

- Figure 5-2 shows an on-axis spot diagram illustrating the re

sulting coma. In practice this coma is routinely removed by simply

_ tilting the secondary and checking the system visually until the coma

is no longer evident in the star plates. In Eq. (5-1) this corresponds

to choosing a value of cr2 that results in A 131 being zero. This is a 83

GAUSSIAN IMAGE PLANE W131* 0.00U Hi 1.00 CENTER UNIFORM 3rd YD: -0.12 ORDER COMA XD 0.09 OVER THE FIELD

TILKrad) MAGNITUDE*. Y: — 7.25 u -0.0034 X: ORIENTATION: 0.0028 144.8 deg H220M® 9.83u W222* 6.08u CENTER Ha 1.00 Ha 1.00 Y20 ^ 5 CENTER : X222: : -0 .0 i Y220M: NODES 0.10 : Yl: X220M: J . 0.08 -8.06 ...... 5 "W------Hx xi: ; -0.06 Y2: MINIMUM : 0.0i FOCAL ; X2: shift: 0.04 -0.007 nn

i Fig. 5-1. Node Plots for the Arbitrarily Misaligned System.

Fig. 5-2. On-Axis Spot Diagram in the Arbitrarily Misaligned System. 84

simple linear process as both aj and are independent of the tilt of

the secondary. This does not insure that the system is aligned, as one has simply chosen a value of 02 to offset the effects of ai and 0 2 ^

(which are linearly dependent). In general, the resulting system contains binodal astigmatism as will be shown shortly.

Before proceeding to correct the coma, some other useful features of the Tektronix program will be presented. In particular, it is of

interest to know how the perturbations affect some measure of the image quality. For an astronomical telescope, the rms spot size is usually

the quantity of interest. Table 5-6 lists the change in rms spot size

Table 5-6. Tabular Output for the Change in RMS Spot Size in the Arbitrarily Misaligned System (Y-Field Profile).

Y-FIELD X-FIELD CENTERED RMS CHANGE IN RMS FOCUS microns microns (mm) -1.89 0.00 138.83 66.78 -2.380 -9.89 0.00 59.84 72.16 -2.300 -0.60 0.00 32.96 66.23 -2.300 -0.40 0.00 67.12 39.08 -2.300 -0.20 0.00 92.94 31.94 -2.300 0.00 0.00 101.79 35.00 -2.300 0.20 0.00 92.94 43.59 -2.300 0,40 0.00 67.12 57.25 -2.300 0.60 0.00 32.96 73.65 -2.300 0.80 0.00 59,84 43.01 -2.380 1.00 0.00 138.83 0.61 -2.300 85 for a profile in the field. Figure 5-3 gives a profile plot of both the actual and the change in mis spot size for two profiles in the field, 0 and 90° . The most useful output, however, are the contour plots shown in Fig. 5-4. These plots give a much better perspective on the problem and give a more complete picture than is generally available on the

dErns nicrons 300.00

actual open change in A Y-field o X-field

FIELD 1.00

- 200.00

Fig. 5-3. Plotted Output for the Actual and the Change in the RMS Spot Size in the Arbitrarily Misaligned System (X and Y Field Profiles). 86 effects of perturbations on image quality. In Fig. 5-4(a) the contours of constant change in rms spot size are shown. In Fig. 5-4(b) contours of actual rms spot size for the perturbed system are shown. In Fig.

5-4 (c) the contours of actual rms spot size for the aligned system are shown for comparison.

Now we will introduce the appropriate secondary tilt to remove the coma. Here this is done by solving Eq. (5-1) for a2, but in practice it is usually done visually. The resulting perturbations are given in Table 5-7. The resulting first order properties are given in Table 5-8, and the node plots are illustrated in Fig. 5-5. Referring to quadrant 4, one sees that the system has binodal astigmatism. The effect on the image quality over the field is shown in Fig. 5-6. From

Fig. 5-6(b) it is clear that the effect of binodal astigmatism has been to shift the usable .field a relatively small amount in this case.

In most cases this amount of image degradation would go unnoticed. However, if a through-focus star plate were taken with

• s uc-h -a-'-syst-e m, there woul d be a notice able effe ct. A s imul at ion of a through-focus star plate is shown in Fig. 5-7. Here an "ideal" array of stars has been used at zero, 0.7, and full field in increments of

45° in the field. Five focal positions are superimposed with a small image displacement added to separate the images. This figure simulates well what a through-focus- star plate looks like under magnification when taken with a large telescope that contains binodal astigmatism. 87

IMAGE PLANE _ 1 .0 3

Hx - 1.0

60 IMAGE PLANE Hy 40

- 1.00 125

Hx -l.0l

IMAGE PLANE Hy _ 1.00 100

100 -1.0&

-i.ee

(c)

Fig. 5-4. Contour Plots for RMS Spot Size in Micrometers.

(a) Change in rms spot size in the arbitrarily misaligned system. (b) Actual rms spot size in the arbitrarily misaligned system. (c) Actual rms spot size in the aligned system. 88

Table 5-7. Perturbations in the Misaligned, Coma-Compensated System

Tilt Magnitude Orientation (arc min) (deg)

Primary ' 1.2 27

Secondary 4.5 27

Table 5-8. First Order Tilt and Decenter Properties in the Coma- Compensated System.

*>* 0 SIGMA OBJ DISP PUP DISP UY FIELD TILT IX -0.0273 0.0000 0.0000 0.000000 0.0000 U=0 1XA 8.0000 0.0000 1Y 0.0545 0.0000 0.0000 0.000080 0.0000 U=0 1YA 0.0000 0.0000 2X 0.0033 0.0545 0.0000 -0.000314 2.3532 -0.000157 2XA -0.0545 0.0000 2Y -0.0065 -0.1091 0.0000 0.000628 -4.7064 0.008314 2YA 0.1091 0.0000

IMAGE X -0.0068 0.0045 -8.000381 -1.2472 -0.800914 PLANE Y 0.0137 -0.0890 0.080763 2,4944 0.001829 89

GAUSS I AH IMAGE PLANE U 13 1 = 0.00u Hy 1.00 CENTER | UNIFORM 3rd YD: ' 0.01 ORDER COMA XD -0.01 OVER THE FIELD ------Hx rp TILTCrad) MAGNITUDE: Y: 0.01 u 0.0018 x: ORIENTATION: 1 -0.0009 153.4 deg W220M= 9.83u W222= 6.08u CENTER Hy 1.00 Hy 1-00 Y20?13 CENTER X222: -0.06 Y220M: NODES 0.06 Yl: X220M: 0.20 -0.93 ...... A ...... Hx ...... Hx xi: -0. 10 Y2: MINIMUM 0.05 FOCAL X2: shift: -0.03 -0.014 MM

Fig. 5-5. Node Plots in the Coma-Compensated System. 90

IMAGE PLANE Hy

- 1 . 0 3

IMAGE PLANE Hu

100 Hx -1.0 &

- 1.00

Fig. 5-6. Contour Plot for RMS Spot Size (Micrometers) in the Coma- Compensated System.

(a) Change in spot size. (b) Actual spot size. 91

Fig. 5-7. Simulation of a Through-Focus Star Plate Taken with a Coma- Compensated, Misaligned Ritchey-Chrctien. ... v. . -- ' 92

A star plate such as this provides a fairly sensitive test for

misalignments or support structure problems in a large telescope.

Inspection of Fig. 5-7 shows that the orientation of the astigmatic

images is not correct. In general» because the size of the image is

proportional to the magnitude of the star, information on the magnitude

of the astigmatism is not readily available. This information is hot required however, the misalignments can be determined directly knowing

only the orientation of the sagittal or tangential images at as many

field points as possible (generally 10 to 2 0 ).

The technique for reducing the - data on a through-focus star plate

is. as follows: Start with a plate similar to that in Fig. 5-7 with at

least 10 stars showing enough astigmatism to determine the orientation of

the sagittal images. Draw short lines on a piece of overlaid tracing

paper over the star images for which the sagittal orientation has been

determined. The resulting plot for Fig. 5-7 is shown in Fig. 5-8. To

determine at this point if the telescope is properly aligned, simply

extend these lines to the center of the picture and check to see if they

intersect at a common center. Fig. 5-9. If they do not, the telescope

is misaligned or improperly supported.

If the telescope is found to be misaligned or improperly sup

ported, it is possible to find what Condensation is necessary to remove

the binodal astigmatism. This is done by locating the nodes for the

astigmatic field. There are two graphical techniques for locating the

nodes from line plots such as Fig. 5-8. One is to use the plots in

Appendix B to find the closest match to the orientation found in Fig. 5-8 Fig. 5 - 8 . Illustrating the Technique for Determining Misalignments from Through-Focus Star Plates. 5-9• Determining if , Te,

— 8 W « „ S ; 95

The other is to use the knowledge that the sagittal image bisects the angle between the nodes "to find the nodes. The second technique is a bit more complicated, so the first technique will be used here. The plots in Appendix B are simply a family of hyperbolas where the separation of the foci are changed in relative increments of 0.1. Plots such as these are easily generated for use with the appropriate plate size. For

Fig. 5-8 one would find that the first plot in Appendix B gives the best agreement. This determines the location of the astigmatic nodes in the field.

Before proceeding to determine the proper compensation, some general properties of perturbed Ritchey-Chrdtien telescopes will be discussed to simplify the analysis. Referring to Fig. 5-5, one node for the astigmatism is near the center of the field. This occurs -s.2 because B2 22 is small here. This is a general property of most coma- ' compensated misaligned astronomical telescopes of Ritchey-Chrdtien design. It occurs because of the surface-by-surface aberrational prop erties of the system and the condition imposed by Eq. (5-1) . In particular it is due to

(Wi3l)2 = CWi3l)-2A and

(W222)2 =' ~(^222)2a -

As long as this is approximately true for the telescope being analyzed, ^ 2 (\j one can assume B222 - 0$ which simplifies the algebra. This assumption will be made here although we note that it is not required for the analysis to be successful in. determining the proper compensation. 96

In general, the compensation controls on a telescope, once it is

installed, are on the tilt and decenter of the secondary. This implies

that the reference axis should now be the local axis of the primary,

x.e., crl = “■ 0. Making this conversion gives the first order prop erties shown in Table 5-9 and the node plots in Fig. 5-10. If the

reference plane axes in quadrant 1 are superimposed for Fig. 5-5 and

5 -1 0 , one sees that the two choices of reference axis are equivalent.

Table 5-9. First Order Tilt and Decenter Properties in the Coma Compensated System with respect to the Local Axis of the Primary.

Q SIGMA OBJ D1SP- PUP DESP U V . . FIELO TILT ©c®©®@ 8.888® 0.008® 0.000000 0.0000 u=© i m 0.000® IY @=®8 ®© ®.800® 8.880® 0.88080© 0.0000 Ua® lYft 0.0000 2X 0.0385 8.8888 0.8088 8.880808 8.8000 8.080000 2XA -8.0273 -1.1766 2Y -@o®Sia 8.0000 0.0000 0.800800 0.0000 8.000000 2 YA 0.8545 2.3532

IMAGEK -0.0324 8.8024 “8.800627 -5.920© -0.001245 FLAMEY 8.0648 “8.8847 0.801254 11.8399 0.00249® 97

GAUSSIAN IMAGE PLANE W131- 9.00U Hy 1.99 CENTER ! UNIFORM 3rd YD: 9.96 2 ORDER COMA XD ; -9.93 OVER THE FIELD ...... Hx rp TILKrad) ! MAGNITUDE: Y: : 0.01 u 9.9925 X: : ORIENTATION: -9.9912 153.4 deg M229M* 9.83u W222- 6.08u CENTER Y222: Hy 1.99 Hy 1*00 0.07 CENTER X222: -0.04 Y229M: NODES 9.99 Yl: X229M: 0.15 0.00 ...... ______Hx ...... % ______Hx xi: -0.97 Y2: MINIMUM 0.00 FOCAL X2: shift: 9.00 -9.914 mm

Fig. 5-10. Node Plots for the Coma Compensated System with respect to the Local Axis of the Primary. . ;

98 So far only the effects of misalignments have been considered.

Another important factor in large telescopes is mirror warpage due generally to an improperly supported primary. This will introduce con stant astigmatism into the system giving a third contribution to the binodal astigmatism (besides the tilt and decentef of the secondary).

To separate the effects of mirror warpage from misalignments, one notes that mirror warpage contributes to the b^zz vector but not to the az22 vector. However, since the location of the center of the Gaussian _a.2 image plane is not known, the azzz vector cannot be found directly.

From Chapter 3, the equation describing binodal astigmatism is

1 2 W = yWzzzCCH - a2 2 2 ) 2 + ^ 2 2 2 !“P2 with -s.2 ^ 2 -i.2 ^ ... 222 . = b 22.2/w 222 " a222-

The nodes are located at

H = a222 ± 10222- 1

The node separation is

2 1^ 2 2 2 1» oriented along

± ib2 2 2 -

Once the nodes have been located, the vector b 222 can be determined directly and the compensation to remove the binodal astigmatism can be calculated. • • 99 If the binodal astigmatism is due to misalignment only, then

222 0 and > ■ ' . ■ t*222 - ±18-222' (5-2)

If it is due to the support structure, then constant astigmatism is introduced, i.e., a.222 = 0

t>222 = ±B2 2 2 / ^ 2 2 2 > (5- 3) corresponding to Fig. 3-13(b).

In either of these cases it is not necessary to know a2 2 2 , so the location of the center of the Gaussian image plane is not required.

When both misalignment and improper support are present, then

_,2 ^ 2 _.2 b 222 = ^2 22/ IV2 22 ~ a222‘ (5-4) t + due. to due to support. misalignment

The best approach if this situation is suspected is to apply a known decenter to the secondary in a known direction. By taking a through-focus star plate with this configuration and locating the nodes, one can separate out the misalignment-induced linear term from the support- induced constant term. If this is not done and one proceeds to compen sate the binodal astigmatism, the result is decentered, conventional quadratic astigmatism as illustrated in Fig. 3-13(c). Usually a tele scope that contains decentered, quadratic astigmatism is perfectly acceptable as one simply takes the center of the field to be that point at which the astigmatism is zero. We note, however, that because

0 2 j 02^ are not zero, there will be nonsymmetric distortion in the field. 100

Returning to the example, given the node locations found from

Fig. 5-8, the vector +ib222 is as illustrated in Fig. 5-11. Its mag nitude and orientation can be measured directly. This information can be used to calculate the tilt and decenter present for the secondary, thereby allowing the system to be realigned.

H 5 y

x 5

Fig. 5-11. Measuring the Perturbation Vector b 222 from the Node Plot Corresponding to Fig. 5-8. 101

Since one node appears near the center of the field, we will assume the binodal astigmatism is due to misalignment only. We have then two equations governing the current status Of the system,

Aisi = C^i 3 1 )2cr2 + CWl3i)2^2^ = 0 (5-5)

±it>222 ” a222 " W2 2 2 1 ^ 222'*02 + ^ 2 2 2 ) 2^ 2 ^ 2 • (5-6)

Solving Eq. (5-5) for 0 2 , using the values ih Table 5-3,

^ (wl3l)2 ^ _a. 0 2 ‘ '* Wm J i * *** ’ -1 -1o2 a; (5-7)

This relation holds as long as no coma is seen in the star plate.

Substituting into Eq. .(5-6) gives,

±ib222 = [(-1.1) (-6.5)a2A + ( 6 . 5 ) ^ ] -

= 1.302^. (5-8)

From Chapter 2, Eq. (2-14), cr2A is given by

^ 6 v 2 , a2 a = ~ > (5-9) 72 where 6 v 2 is the vector decenter in linear units of the secondary.

Substituting for 7 2 from Table 5-2 gives

<5v 2 = ±132.5b222* (5-10) We have established a direct relation between the decenter of the secon- -X dary and the vector b 2 2 2 , which is determined directly from the through- focus star plates.

From Fig. 5-11, the vector ib222 is given by

ib222 = 0.08 e-i(25°) (5-11) 102

Hie decenter of the secondary then is,

6v 2 = ±2.6 e_1(25°) ? (5-12) i.e. +2.6 mm at either -25° or + 155° to the Y axis. The tilt is then given by,

02 . = -1 -1^ 2A

1 32 + c 6v-2 _ l02A (5-13) 3-2 giving

6 V 2 = ±0.00037 6^2

^ ±(0.05O)e"l(-25O-) . (5-14)

. We have established a more thorough technique for detecting and correcting misalignments and mirror warpage in Ritchey-Chr61 ien tele scopes. We have shown that the removal of coma is not sufficient to ensure that the system is aligned. The correct technique for this type of telescope is the analysis of through-focus star plates as presented here. These plates are sensitive to misalignments and also, in general, contain the information necessary to determine what the errors are, except for an ambiguity in sign. The techniques presented here were used successfully to predict from star plates a known decenter for an actual 2.3-m astronomical telescope. CHAPTER 6

ABERRATION FIELDS IN PERTURBED OPTICAL SYSTEMS II.

THE INTERPRETATION OF THE FIFTH ORDER ABERRATIONS

Whereas the use of third order aberrations is sufficient to account for the effects-of perturbations to most rotationally symmetric mirror systems, they are not adequate for most refractive designs. In general, the third order terms will give the effects of centered system perturbations, i.e., changes in curvature, thickness, and index in any system. This is because the third order coefficients change more quickly than the fifth order coefficients when these parameters are changed.

Their behavior will dominate the response of the system to centered per turbations. Tilt and decenter perturbations, on the other handj require inclusion of the fifth order terms in the wave aberration expansion when the fifth order surface contributions are comparable to the third order surface contributions. This is the case in most refractive lens groups and some wide field mirror systems. These terms must be in cluded because tilt and decenter perturbations affect the field behavior and not the coefficients.

In the process of designing refractive lenses one generally balances some of the third and fifth order terms at various zones in the the field. When tilts and decenters are introduced, the zeroes of the third and fifth order contributions become displaced with respect to each other in the field. Many times this has a drastic effect on the

103 104 balancing between terms and the resulting image quality, especially at the edge of the field.

The fifth order aberration coefficients in an optical system are more difficult to calculate than the third order coefficients. They consist of intrinsic surface contributions (i.e., contributions that depend only on properties of or at the surface) plus, induced contri butions. The induced contributions depend on the sum of combinations of third order image and pupil aberrations preceding the surface of interest. Referring to Rimmer's work (1965) to clarify Buchdahl's

(1954) notation on the calculation of fifth order aberration coefficients, one notes that the coefficients continue to be functions of paraxial quantities. Paraxial quantities are unaffected by tilts and decenters, therefore the fifth order coefficients are also unaffected. Given the surface-by-surface contributions to the fifth order aberrations, one can proceed, as in Chapter 3, to find the behavior of the fifth order aberration fields in perturbed optical systems. This will be the aim of this chapter.

As might be expected, the development of the theory of fifth order aberration fields in perturbed systems is considerably more complex. However, with the concept of nodal behavior characterizing an aberration field, the interpretation is not exceedingly complicated.

This chapter will present the nodal behavior of the fifth order aber ration fields.

Special cases will not be discussed in detail for the fifth order fields for two reasons. Generally, there are two types of 105 special cases; one set when the aberration term is zero and the other set when the term is not zero. When a fifth order aberration coefficient at the image plane is zero (in a system without aspherics) s it generally implies that the surface contributions are small. This is because a designer rarely has enough degrees of freedom to balance fifth order aberrations between surfaces. The design approach is to introduce enough third order aberration to balance the higher order terms at some zone in the field. In general then, when a fifth order coefficient is zero, its effect on the perturbed system is negligible and a nodal inter pretation is not required. When a fifth order coefficient at the image plane is not zero; special cases refer to situations of special node symmetry, i.e., nodes symmetric about the center of the field, one node on axis, etc. Although some of these cases may be interesting, there are simply too many to consider discussing them in detail.

In the discussion that follows, only the final normalized form, for the equations that describe the nodal properties of the aberration fields will be presented in most cases. An attempt has been made to write this chapter in a manner such that most of the equations can be ignored without interrupting the continuity of the discussion. This is meant to aid those interested in the properties of the aberration fields rather than the techniques for calculating the properties. The equations presented here are given in Appendix C. This appendix contains a brief derivation of all of the normalized and unnormalized equations required to compute nodal locations for the aberration fields. 106

The image plane perturbation vectors required to compute the properties of the aberration field are defined in fable 6-1. The notation is interpreted as follows. The subscript klm refers to the aberration coefficient for which the vector is calculated. The arrows distinguish vectors from scalars, and the superscript denotes whether the vector is a linear, squared, or cubic vector. Cubic vectors are similar to squared vectors, which were introduced in the treatment of astigmatism in Chapter 3. A cubic vector is obtained from a linear vec4 tor by cubing the magnitude of the linear vector and multiplying its orientation angle by 3. Note thht in Table 6-1 the vector C 3^ m is not cubed because of the summation over surfaces. Further properties of cubic vectors are given in Appendix A. The letter notation for the perturbation vectors indicate the power of the o. vectors involved. The first letter. A, contains one o. vector, and fifth letter,

E, contains five ck vectors. A capital letter denotes an unnormalized vector, while a small letter denotes a normalized vector (except for the field vector H, which is always normalized).

In this chapter, as with Chapter 3, the fields will be treated term by term, grouping terms together when appropriate. For each term, some of the properties in an aligned system will be discussed first, as the properties of fifth order aberrations are not generally treated in the standard references given in Chapter 3. This will be followed by a presentation of the nodal locations and interpretation of the nodal properties. As one might expect, the complexity of a term in a Perturbation Vectors 108 perturbed system depends on the order Of the field dependence. The discussion of terms has been arranged accordingly.

Fifth Order Spherical Aberration, WpgQ

w = WoeoC^p)3. (6-1)

As with third order spherical aberration, the fifth order spherical aberration is unaffected by tilt and decenter perturbations because it is independent of field.

Fifth Order Coma, W 151

W = W ^ x CSp) (p°p)2 . . (6-2)

This aberration term is similar to third order coffla. Its symmetry prop erties are altered by the additional (p ®p) dependence. To illustrate the differences, the properties of the transverse blur due to these aberrations in an aligned system will be discussed using the vector approach introduced in Chapter 4. The equation for the transverse aberration due to third order coma is

(n'u')e = W :. <= Wjai (p-p)H + 2Wl 31 (H-p)p. (6-5)

Referring to Fig. 6-1, the first term gives a line of length W} 31 directed radially away from the center (zero) of the aberration field.

By marking this line in units of' p2, the centers of the circles gener ated by the second term may be located. The second term generates circles of radius P2Wi3^. The centers for the circles are located by 109 noting that the lower edge of the circle passes through the correspond

ing zone mark on the line. This has been done in Fig. 6-1 for zone

increments of 0.2 in the aperture. Notice that one revolution in the

aperture corresponds to two revolutions in the image plane due to the

(H•p) dep endence.

Fig. 6-1. Transverse Blur for Third Order Coma 31.

For fifth order coma, the transverse aberration is given by,

(n'u')c = VW = W15i(p *p )2H + 4W151(H*p)(p«p)p . (6-4)

This can be interpreted graphically in a similar manner. Referring to

Fig. 6-2, the first term results in a line of length W151, directed 110

radially from the center of the aberration field, which can be marked in units of p4 . The second term gives circles located as before, but now the radius of the circles is 2p 3 Wi51, which alters the symmetry of the patterns as may be seen in the figure.

Fig. 6-2. Transverse Blur for Fifth Order Coma, ^ 5 1 .

In a perturbed system the behavior of the fifth order coma

aberration can be obtained from

w = W151[(H - a151)*p](p *p)2. (6-5)

Comparing this with Eq. (3-16), the field behavior is seen to be the Ill same as that for third order coma. The transverse aberration increases

linearly with field and is oriented radially away from the node. The node is located by the vector a.151. Fig. 6-3. The effect of a dis placed node on the transverse aberration is shown in Fig. 6-4.

Fig. 6-3. In a Perturbed System the Center of the Aberration Field for Fifth Order Coma, W 151, Can Be Displaced in the Image Plane.

The displacement is given by the vector a^gi. 112

i H y II Q) O cn

H X.

(a)

151

(b)

Fig. 6-4. Properties of the Transverse Blur for Fifth Order Coma, W151 (a) Centered (b) Perturbed. . 113

Oblique Spherical Aberration #242

. — — p -

w = AW2 0 Cp.p) + w2ifpM C^°ft) Cp*p02

+ l/2W2lf2(fi2-p2)(p^ . (6-6)

In an aligned system, oblique spherical aberration exhibits properties associated with both spherical aberration and astigmatism. In a per turbed system the response of an aberration is controlled by its field dependence. Oblique spherical aberration then behaves as the medial surface and medial astigmatism do in a perturbed system. To make this analogy requires discussing the significance of sagittal, medial, and tangential (SMT) surfaces for oblique spherical aberration.

Figure 6-5 illustrates some of the predominant properties of this aberration in an aligned system. In (a) a through focus plot of the mapping of a circular zone in the aperture onto the image plane in the presence of oblique spherical aberration is shown. From this it is clear that there are. focal positions that result in images that can be used to identify SMT surfaces. For any other zone in the aperture, the mapping has the same symmetry, but the size is scaled by p 3 and the focal position for a particular symmetry has changed. Each zone has a set of SMT surfaces that depends quadratically on field as with astig matism, but now the curvature of the surfaces is a function of the zone chosen in the aperture as seen in (b).

This ambiguity can be eliminated with the following definition.

The reason for the zone dependent curvature is the p4 aperture dependence 114

IMS SIZE = p3

i*(W240M " 1/2W242)p T 0 0

24 0 cO = * JK2 • (b) 4(^240% + 1/2W242)p2 t M FOCUS 4

MEDIAL PARAXIAL .0 IMAGE IMAGE PLANE PLANE .7

(a)

Fig. 6-5. Properties of Oblique Spherical Aberration.

(a) Mapping of a circular zone in the aperture onto the image plane as a function of focus. (b) Each zone in the aperture has its own set of SMT surfaces because of the presence of spherical aberration. (c) The spherical aberration caustic. US which generates a spherical aberration caustic across the field as in (c). For spherical aberration, the medial image plane is located where the rays from the 0.7 zone in the aperture come to a focus. The medial surface for oblique spherical aberration then is defined to be that surface in the field along which the medial image for the ]p |- = 0 .7 zone is obtained. Similarly, the sagittal and tangential surfaces are obtained from the through focus mappings of the |p| = 0.7 zone in the aperture onto the image plane.

With these definitions of sagittal, medial and tangential sur faces, the properties of oblique spherical aberration in a perturbed system may be understood. The main difference between oblique spherical aberration and astigmatism is that the blur shape for astigmatism at a particular focus is independent of aperture while that for oblique spherical aberration is not.

Focal Surface for Medial Oblique Spherical Aberration Wg^Q^

In a perturbed optical system the properties of the medial focal surface associated with medial oblique spherical aberration may be found from '

-AW2o = 1/2W240m 1 (h - a240M) °(H - a240M) + • (6-7)

Comparing this with Eq. (3-29) shows that this is of the same form as

the expression for the medial focal surface associated with third order

field curvature and astigmatism. (The behavior discussed following 116

Eq. (3-29) applies here, and the reader is referred to that section

for a more detailed treatment.) In general, the vertex of this focal

surface is displaced in the Gaussian image plane by the vector a24 oM measured from the OAR as shown in Fig. 6 -6 . There is also a longitudinal

displacement due to the- b24oM term as in Fig. 6-7.

The b24 QM contribution illustrates the first case of a fifth order aberration generating third order aberration, in this case, third order spherical aberration. This behavior will be seen for many of the

fifth order terms. It results here in a change of focus to retain the medial image location in the modified spherical aberration caustic.

Fig. 6 -6 . Node Plot for the Medial Surface Associated with Oblique Spherical Aberration, W24 qm . 117

GAUSSIAN IMAGE PLANE

OAR

R.A.

MEDIAL FOCAL SURFACE DESIGN PLANE

Fig. 6-7. Locating the Medial Surface for Oblique Spherical Aberration in a Perturbed System.

Oblique Spherical Aberration with respect to the Medial Focal Surface

Associated with Oblique Spherical Aberration W242

In a perturbed system the expression

W = 1/4W242[(H - a242)2 + b 2242]-p2 (6-8) . 118 may be used to find the properties of the focal surfaces as defined for . the Ip| = 0.7 zone in the aperture. Comparing this with Eq. (3-38) shows that the field dependence is of the same form as seen for medial astig matism with respect to the medial astigmatic focal surface. (Again a more detailed discussion can be found following Eq. (3-38).) In general, this aberration develops two zeroes, or binodal behavior, in the field as illustrated in Fig. 6 -8 . At these two points in the field the focal surfaces for oblique spherical aberration touch as in Fig. 6-9: they do not cross. The magnitude and orientation of the transverse blur on either of the focal surfaces can be found from the location of the two nodes with respect to the field point of interest. The magnitude is pro portional to the product of the distance to each of the nodes. The orientation of one of the oriented images is along the line that bisects the angle subtended by the nodes. The other image is oriented at a right angle to this as illustrated in Fig. 6-10.

Linear and Field Cubed Coma W131, * 331^

W = Vli 31 (H-p) (p °p) + W 331m (H°H) (H»p) (p-p) . (6-9)

The next fifth order aberration to be treated is field cubed coma, # 331^.

The M subscript is used to account for the conversion of cos3 (e-) to cos [3(0-) ] for the elliptical coma term, W 333, which will be treated later. Here,

w 3 31m = w331 + 3/4W 3 3 3 . (6-10) 119

242

Fig. 6-8. In a Perturbed System the Oblique Spherical Aberration (OBSA) with respect to the Medial Surface for OBSA Is Zero at Two Points (Nodes) in the Field.

N M P

Fig. 6-9. Field Curves for OBSA Illustrating Binodal Behavior. 120

Fig. 6-10. For Any Field Point H, the Magnitude of the^OBSA Is^ the Product of the Distances to the Two Nodes | | x jN^j.

The image on one of the focal surfaces is^ elongated along the line that bisects the angle between N% and N2 (dashed line). The image on the other focal surface is at right angles to this. .121 The transverse aberration for field cubed coma in an aligned system is

"identical to that for third order (linear) coma. The only difference

between the two terms is that one is linearly proportional to field

height while the fifth order aberration is proportional to field height

cubed. This means that if the coefficients, W131, W33l^, have opposite

signs, there will be some circular ring in the field where these two

terms cancel each other and result in no contribution to the aberration.

In any system with appreciable field cubed coma, this will be the case if

the design is to give reasonable performance.

In an aligned system the location of the balance zone in the

field gives insight into the aberration correction of the system. In

general, analytic solutions for locating the balance zone in a perturbed

system are not practical. It is possible,'however, to relate the node

plots to the behavior of the balance zone.

In a perturbed system the field cubed coma aberration generates

terms that, are linear and constant over the field. These contributions

will be separated from the field cubed coma and combined with the linear

coma. This technique of removing terms generated by the higher order

aberration and combining them with the lower order balancing aberration

will be used for those fifth order aberrations that have lower order

balancing terms. This greatly simplifies the interpretation using

node plots.

The linear term generated by the field cubed coma changes the

effective magnitude of the linear coma. The effective wave aberration 122 coefficient for the linear coma becomes

W 1 3 1e = W131 + 2W331M b331M- (6-11)

The constant term generated by the field cubed coma changes the nodal location for the linear coma. The new node location is given by the vector

^ ■ 1 ^ ^ a131E = !V131E (W131a131 + W 331M [C331M ~ b 2 33 1 M a^ 3 1 M ] >- J6-12)

The expression for linear coma is now

W = W131E [(H - a131E) ‘Pi (P-*P) • (6-13)

In the presence of field cubed coma, the third order coma field has the same nodal properties that were described in Chapter 3.

With the linear and constant terms accounted for, the expression for field cubed coma in a perturbed system can be written as

w = W331M{[(H - a331M)2 + b 233lM](H - ^ 33^ ) **p) (p-p) . (6-14)

The nodal interpretation can be made directly from this equation. The term in square brackets is of the same form as Eq. (3-38) and therefore represents binodal behavior. The term (H - aggi^)* gives a third zero at the center of symmetry in the field located by the vector a.331^. Re call that the star notation indicates a conjugate vector and is important for obtaining the correct orientation for the transverse aberration.

Field cubed coma then, develops three collinear zeroes in the field in a perturbed system as shown in Fig. 6-11. 123

3 3 1 M

Fig. 6-11. In a Perturbed System the Field Cubed Coma Contribution Can Be Zero at Three Collinear Points in the Field.

As with third order astigmatism, the nodal locations may be used to find the magnitude and orientation of the field cubed coma contribution for any point in the field. The magnitude is the product of the magnitudes of the three node vectors, one drawn from each of the three nodes to the field point of interest (i.e., the product of the 124 distance from the field point to each of the nodes). The orientation is given by the sum of the orientation angles for the two outer node vectors minus the orientation angle of the central node vector (because it is a conjugate vector). Notice that because of the conjugate vector, the proper orientation in an aligned system results as

M * = H(HH*) = H(H-H), . i.e., the aberration is oriented along the field vector as required.

Note also that the conjugate term ensures that the resulting orientation is independent of the choice of coordinate system. The net coma contri bution at any field point is found by adding the resulting field cubed coma vector to the linear coma vector, H - a131g. The length of the resulting vector gives the magnitude of the sagittal coma and the orientation.

With an understanding of the nodal behavior of linear and field cubed coma, insights into the effects of tilts and decenters on the balance zone can be developed. To illustrate how the nodes affect the balance zone, contour plots of constant aberration magnitude in the field and plots illustrating the orientation and relative magnitude of the transverse aberration will be used.

Consider first an aligned system. Figure 6-12 shows contours of constant wave aberration magnitude for linear plus field cubed coma where W j = -0.3 ym and W331^ = 0.4 ym. The corresponding transverse orientation plot is shown in Fig. 6-13. Here the plot symbol is chosen 125

1 Hx o

1 .

Fig. 6 - 2. Contour of Constant Wave Aberration Magnitude for Linear Plus Field Cubed Coma in an Aligned System.

w131 = -0.3 ym, = 0.4 ym.

vvykui>4».« 4 VVWUAA*. . y » 4 A A 4 * * A A 4 »►**.▲<<*« » > > > • • * 4 4 * < < « < > > > > • . • ««<<<<« » > > V # . • 4 r < < < * 1 9 r v v 4 * » v r T r r T < * t ▼ -r 4Akhrrrrv'

* • * 4 f 4 t* * • *44 i» • . « 4

6 • • 4 4 k k

Fig. 6-13. Transverse Magnitude and Orientation Plot Corresponding to Fig. 6-12. 126 to represent the angular portion of a coma pattern with the length giving the magnitude of the sagittal coma. From these plots one can see that the two terms balance each other over a ring located at a relative field of 0.91.

Consider next perturbing the system. Figure 6-14 shows a possible node plot arising from an arbitrary set of perturbations to an optical system. In the figure the filled circles give the location of the three collinear zeroes for the field cubed coma contribution. The x denotes the nodal location of the linear coma accounting for the field cubed contribution. For this example the magnitude of the third order coma is unchanged, i.e., W131e = W ^ .

Fig. 6-14. Node Plot for Illustrating the Behavior of Linear Plus Field Cubed Coma in a Perturbed System.

x locates the third order node and • locates the fifth

order nodes. Figures 6-15, 6-16 show how these nodes affect.the balance zone for

linear plus field cubed coma. The. first thing to note is that the bal

ance zone is no longer present. This occurs because these are orien

tation-dependent vector aberrations. This means that, for the terms to

cancel each other, they not only must have the same magnitude, they must also be oriented in opposite directions. As soon as the centeis

of symmetry for the aberration terms are displaced with respect to each other, the zone of balance collapses to, in general, two points

in the field. These two points of balance are shown in Fig. 6-15.

Comparing Fig. 6-13 with Fig. 6-15 shows clearly the transition from a

central zero and ring balance zone to two points of balance.

Elliptical Coma, W 333

W = l/4W3 3 3(H3op). . (6-15)

The remaining coma-type aberration is elliptical coma. This aberration term, in an aligned system, results in the circular zones of the

coma blur, shown in Fig. 6-1, being mapped to ellipses . This behavior • is illustrated in Fig. 6-17. In (a) the vector arrows are the contri bution due to the elliptical coma term and illustrate the mapping of the circles to ellipses along the major and minor axes. At any point along the circular zone, determined by the orientation of the aperture vector p, the elliptical coma contribution gives an additional vector

displacement that maps to a point on the ellipse. In (b) both the

circular zones and the resulting ellipses are shown and in-(c) is the resulting blur pattern. 128

Fig. 6-15. Contour Plot of Linear Plus Field Cubed Coma in a Perturbed System with the Node Plot Shown in Fig. 6-14.

A AA A A AA A A A 7 7 7 7 4 ▼ 4 7 7 hA A A fsN A 7 7 T * > » » * 4 4 7 A 1 T 7 > > X A • * * 4 7 hs A 4 7 > > > > X XX * • • * 4 > > > > X X • • • • > > > r h 4 7 7 > > > • * • * r A -# 7 > > > > > > » X*A -f 7 *77 7 7 7 7 * f 7 "77 7 7 7 7 T » * “T ~7 ~7 ~7 7 7 7 A A AA A AA A < "^7-777 7 7 7 1 A A A AA A A < * 4 7 7 7-7 "7-7 7 7 A A A A A A A < « - 7 7 7 7 -7'I 7 7 A A A AA A A * ' 4 7 *7 7 'I 7 7 A A A A A A 4 • 4 7 7 A 7 7 A A A A A A 7 » • 4 7 -1 A 'I 7 7 A A A A 4 u » • 4 4 7 7 AA A 7 4 4 4 4 4 4 4 *

Fig. 6-16. Transverse Magnitude and Orientation Plot Corresponding to Fig. 6-15. 129

(a)

Fig. 6-17. The Effect of Elliptical Coma on Linear Plus Field Cubed Coma in a Centered System.

(a) The vector arrows are the contribution of the elliptical coma pattern.

(b) Composite showing the original aberration due to linear plus field cubed coma and the ellipses resulting from the elliptical coma term.

In a perturbed system, the expression for elliptical coma is

W = 1/4W333 [ (H - ^ 333) 3 + 3(H - £ 3 3 3 ) ^ 3 3 3 .

- (c3333 - 3b2 333&333)]°p3

= 1/4W 333 [H3333 + SHsssb2 333 - 0^ 333]»p3. (6-16)

This term does not have or generate any lower order balancing terms.

The only question then is, where in the field is this term zero? Writ ten in the second form it can be seen that the term in square brackets resembles a cubic equation. Since all of the terms involve vector multiplication (i.e., there are no dot products), it is possible to solve this equation using a vector generalization of the standard scalar solution to cubic equations. This generalization is given in Appendix D.

The result is that for elliptical coma there are in general three nodes

(zeroes) in the field. This will be called trinodal behavior. At these three points in the field the circular zones of the coma pattern remain circles; they do not map to ellipses.

From Appendix D, the nodalbehavior is controlled by" a combination of two vectors, R and S. Given these two vectors, which are calculated from perturbation vectors, the nodes are located using combinations -Zh, -Sfc of the sum and difference vectors x and x, where

¥ ; i - t l (6-17) 2

v _ R — S ' X — _ . (6-18) 131

The vector I1333 locates the centroid of the node pattern and is the point from which the vectors that locate the nodes are measured. These vectors are

2x (6-19)

-x + i/3 x (6-20)

-x - i/3 x (6-21)

This general behavior is illustrated in Fig. 6-18.

-x + i /3"

Fig. 6-18. In a Perturbed System Elliptical Coma Develops Three Nodes in the Field, i.e., Trinodal Behavior. For any field point, the magnitude Of the elliptical coma term is the product of the distance to the three nodes. The orientation of the major axes of the resulting ellipses, seen in Fig. 6-17 (c), depends on both the orientation of the linear plus field cubed coma term and the orientation of [ ] 3 3 33 (defined in Appendix C), the elliptical coma term.

The orientation relative to the linear plus field cubed coma is found in

Appendix E. In general, the major axes for the ellipses are not along the axis of the displacement for each zone.

Medial Focal Surface #220^, W42 0M '

W = AW2o (p°p) + w220m CH °H) CP°P) + W420M CH«H)2(p«p) . (6-22)

As discussed in Chapter 3, the medial fpeal surface is the surface along which the minimum fms wavefront error is obtained. As with linear plus field cubed coma, there is in general a ring in the field where the fifth order aberration, proportional to the field to the fourth power, balances the third order aberration, proportional to the field squared. Over this ring, the medial focal surface crosses the Gaussian image plane.

In a perturbed system the fifth order aberration generates terms that affect the third order aberration field. Again these terms will be combined with the third order terms to give a net third order field, which may be written as. 133

The quantities with E (effective) subscripts are defined in Appendix C.

The fifth order term then has changed both the magnitude and the nodal locations of the third order term but not the nodal behavior.

With the third order terms accounted for, the remaining terms in the fifth order aberration can be written as

W = W420MU(H - a420M )2 + ^ 4 2 0 ^ "(H - a420M ) 2> C^P) • (-6_24)

As with field cubed coma, the zeroes of this term are apparent. The quantity in square brackets is of the same form as Eq. (3-38) and exhibits binodal behavior. The other term, (H - a4 2 0^) is a doubly degenerate zero at the field point located by the vector a^QM' As with the field cubed coma, this aberration term develops three collinear zeroes in the field when perturbations are introduced. The difference here is that the central zero, located by a.4 2 0 ^ is weighted by the field squared rather than linearly with the field. This nodal behavior is illustrated in Fig. 6-19 where the larger central node indicates the field squared dependence.

Given the nodal behavior, it remains to consider the balance zone. Whereas the node plot for the medial surface is similar to that for linear plus field cubed coma, the balance zone response is not.

This is because while coma is an orientation-dependent aberration, this aberration is not. The result is that the balance zone in a perturbed system remains a closed curve that is distorted. This distortion, in terms of perturbation vectors, is illustrated in Fig. 6-20. In general, starting with the ring balance zone in the aligned system, the change in 134

Fig. 6-19. In a Perturbed System the Fifth Order Contribution to the Medial Surfaces Develops Three Collinear Zeroes in the Field.

The central zero is degenerate and the field dependence is quadratic rather than linear from this node.

the magnitude of the third order coefficient moves the ring in or out

(i.e., changes the radius). The separation of the vertices, due to

r a420M maps the ring to an oval. An oval results rather than an ellipse because the field dependence about the point located by the

vector a4 2 oM is of the fourth power (6 ^ 4 2 0 ^ = 0 ), while that about the point located by is quadratic. The two additional nodes

(£2420m ^ 0 ) contribute to further distorting the ring along the line

connecting these nodes. 135

'420

Fig. 6-20. The Behavior of the Balance Zone for the Third Plus Fifth Order Medial Focal Surface in a Perturbed System.

(a) Ring balance zone for the medial focal surface in an aligned system.

(b) Perturbations may change the radius of the ring balance zone due to to the change in the effective magnitude of the third order con tribution, W22 0J4E*

(c) Perturbations may also map the ring to an oval when the centers of the third and fifth order contributions are separated.

(d) The most general form for the ring balance zone in a perturbed system. 136

Astigmatism W2 2 2 s w^ 2 2

W = 1/2W222(H2 ‘P2) + 1/2W422(H-H)(H2 «^2). (6-25)

Fifth order astigmatism is another of the aberrations that has the same blur features as a third order term with simply a higher order field dependence. Again the result is for a well-corrected system with a significant fifth order contribution, there will be some ring.in the

field over which the third and fifth order terms cancel to give no

astigmatic contribution to the net aberration.

In a perturbed system the fifth order aberration generates terms that modify the nodal locations and magnitude of the third order aberra tion. The third order astigmatism is then written as

W = 1/2 W222e[(H - a2 2 2 E)2 * b 2 2 2 2 E] , (6-26). where the terms with E subscripts are defined in Appendix C. This contribution continues to exhibit binodal behavior with zeroes at

a222E ± ib222E -

With the lower order terms removed, the remaining terms for fifth order astigmatism can be written as,

W = 1/2W42 2 [(H3422 + 3H42 2 b 2 42 2 - C 3 422)H*422°'p2], (6-27) where

H422 = H - B422 (6-28) and c3 ' 4 2 2 is defined in Appendix C. The nodes of this term can be found by comparison to Eq. (6-16). The term in parentheses is of the same form and therefore exhibits trinodal behavior. The H*4 2 2 term gives an additional fourth zero at the point located by the vector t 422 . This general nodal behavior is shown in Fig. 6-21.

"*422 f +1/3x422

Fig. 6-21. In a Perturbed System the Fifth Order Contribution to the Astigmatism Generally Develops Four Nodes in the Field. 138

Again, the magnitude of the fifth order contribution is given by-

the product of the magnitudes of the four node vectors (the vectors from

the field point of interest to each of the nodes). The orientation of

the fifth order contribution [ ] 4 2 2 (see Appendix C) is the sum of the

orientation angles of the vectors to the three nodes in the trinodal

contribution minus the orientation angle of the vector locating the

centroid node, H*4 2 2 (because it is a conjugate vector) . To find the

orientation of the astigmatism at any field point, this contribution,

' -a* I 32422» is added to the third order contribution (whose orientation is

found as in Chapter 3) [ ]2 ,2 2 2 » to give a net astigmatism vector.

0 2a. - f t w ♦ . h 222Z. (6-29)

The line images on the N and P surfaces are then oriented along

and

i[ 3A

The effe.cts of the nodes on the balance zone remain to be dis-

cussed. Here, as with linear plus field cubed coma, the third and fifth

order terms are orientation dependent. In general, as nodes develop,

the ring zone of balance collapses to the points of balance. In the

case of astigmatism, there are six nodes to consider, which makes direct

interpretation more difficult. General properties are the inner

field region, i.e. inside the balance zone, is generally dominated

by the binodal third order term assuming the nodes are within 139 the balance zone. The balance zone generally becomes a region of con stant aberration, i.e., the terms do not tend to go to zero but rather a broad minimum develops. Unlike the coma terms that tend to continue to be zero at one or two points, the astigmatism may not have any zeroes remaining in the field because of the more complex nodal behavior.

Generally, when the fifth order nodes are inside the balance zone, the behavior outside the balance zone remains roughly the same. As the nodes move outside the balance zones, the behavior becomes more complex and orientation and contour plots are required to interpret the nodal behavior.

After studying the general behavior of the balance zone for astigmatism, it becomes clear that the definition of sagittal and tan-: gential focal surfaces breaks down completely when the fifth order terms are included, even in an aligned system. It is common to use the Cod- dington close skew ray trace to locate two focal surfaces along which sagittal and tangential line images are obtained. When both third and fifth order aberration are significant for a system, these focal surfaces cross over each other resulting in a ring intersection in the field. It has been assumed that the Coddington trace locates two continuous focal surfaces which intersect. This view, in light of the nodal properties of the aberrations, is not correct. The surfaces located by the Codding ton trace are not continuous in a perturbed system. There are, however, two continuous focal surfaces on which line images are obtained in a perturbed system. These are the N and P surfaces discussed in Chapter 3 and defined in Chapter 4. . ■ ■ 140

The crossover of focal surfaces that appears to occur in an aligned system is not a crossover, but rather a ring node. The two focal surfaces (N and P) touch over this ring in the field but do not cross it. This behavior must be correct because when small perturbations are introduced, field curves taken along any profile not containing a node show that the curves no longer touch or intersect. It is not possible to make a smooth transition between an aligned and a slightly perturbed system unless this picture is used. As with linear plus field cubed coma the ring node of the aligned system collapses to points of balance when small perturbations are introduced because the third and fifth order terms that are balancing each other are vector orientation- dependent aberrations.

The concept of associating image orientation with focal surfaces in perturbed or aligned systems is best abandoned completely. The images on the N and P surfaces are still at right angles to each other, but they can have any orientation angle. The concept of bisecting nodes that was introduced for third order astigmatism breaks down when the fifth order terms are added. There are no apparent well-defined properties on which to make a useful, consistent definition relating to image orientation.

In centered systems, N and P surfaces are not equivalent to S and T surfaces. This is because while S and T surfaces cross, N and P surfaces do not. Figure 6-22 gives plots that show image orientation and field curves for a centered system with third order astigmatism balanc ing fifth order for both the S and T and N and P interpretations.

Similar plots in a perturbed system are given in Chapter 7. 141

: : ! ^ z w //

• > X

I • - \ I • - i I • • i I • - i i

(e) (f)

Fig. 6-22. Comparing S and T with N and P Focal Surfaces in an Aligned System.

(a) S and T field curves (b) N and P field curves (c) S-surface astigmatism (d) N-surface astigmatism (e) T-surface astigmatism (f) P-surface astigmatism 142

Distortion W 3 ii, Wsii .

W = W 311 (H«H) (H«p") + W 3 1 1 (M«H) (H-H) (H-^) „ (6-30)

Distortion is- yet another case of the fifth order aberration having a lower order balancing term. Here, however, because this term depends on fifth order in the field, a nodal interpretation has not been pursued due to the limited usefulness of such an interpretation. As discussed in Chapter 4, the third order aberration generates a first order term that results in a node plot resembling linear plus field cubed coma.

If the fifth order term were to be treated with a nodal interpretation,

-there would be three sets of nodes to contend with for first, third and

fifth order contributions. There would be eight nodes with three field

dependencies to understand. Such a plot would be of little use. Gener

ally, this aberration is treated through the use of orientation plots

and magnitude contour plots. These will be shown in Chapter 7. The

expression for fifth order distortion is given in Appendix C.

Finally, we note that as with the third order aberrations, there are uniform aberration vectors for the fifth order aberrations that arise from the fifth order spherical aberration. These vectors are given in Appendix C. - CHAPTER 7

GRAPHICAL METHODS FOR ANALYZING

PERTURBED OPTICAL SYSTEMS

Using the theory presented in the preceding chapters, a method of analyzing the effects of perturbations on an optical system which relies on interactive computer graphics has been developed. This method will be demonstrated in this chapter. > The approach was guided by a desire to provide a designer familiar with aberration fields in an aligned system with insight into the behavior of a design when per turbations are introduced, without necessarily being thoroughly familiar with the theory presented here. This is accomplished by emphasizing the use of node plots with some.supporting interpretive routines and contour routines for image degradation. Standard methods of analysis, i.e., ray fans, field curves, and spot diagrams calculated using coefficients are also available.

The program accounts for the aberration terms through fifth order in the wave aberration expansion. .The extension of the equations prey : sented in Chapter 4 for analyzing perturbed systems to fifth order are given in Appendix F. No new developments are required to make these extensions.

The technique.will be illustrated by analyzing the effect of a set of arbitrary perturbations on a 20° full field, f/4 triplet. As will be shown, this lens is not well represented by the third order

v ' '' ; :V ■' - ; , 143 ■ , : : ' ' ' 144 aberration terms, but is accurately described when the fifth order terms are included. At the end of this chapter there will be a brief treatment of a 47° full field, f /3.5, five element double gauss illustrating \ that even higher order aberration terms are required for some systems.

The tables and figures in this chapter were generated by the program

(except Table 7-6 and Figs. 7-48 to 7-51).

The data pertaining to the triplet, shown in Tables 7-1 to 7-5, are entered and stored on tape as the system to be analyzed, This is referred to as the centered system data. Starting with this data, a . set of perturbations are entered, surface by surface, as in Fig. 7-1.

These perturbations are given in Table 7-6 in terms of element-by-element rather than surface-by-surface data.

Given the perturbations, the program computes and generates a table of the first order properties of the perturbed system as shown in

Table 7-7. These are the quantities described in Chapter 2, The most important column here is the listing of the aberration field displacement vectors, a. (first column). Large relative values for a . generally indicate dominant elements in the perturbation analysis, although the wave aberration coefficients associated with the surface must also be considered.

Once the aberration field displacement vectors are calculated, the full set of unnormalized image plane perturbation vectors defined in

Table 6-1 are computed. They may be displayed as in Tables 7-8 to

7-10. Once these perturbation vectors are computed, the calculation time for all of the remaining analysis routines is independent of the number of surfaces in the system. 145

Table 7-1. Lens Data

TRIPLET EXAMPLE FVgg LEMS DATA

0 CURVATURE -THICKNESS INDEX, CONIC

i @=@44835 4 = 45©@ 1=6937®. @.@@®@

2 @.@@@50® 7=442® io®@@0 ® @=@0 ®@

3 -@=@39619 1=95@@ 1=73367 ■ 8 .0 0 0 0

4 0.053245 5=481® 1 .0 0 0 0 0 @'=0 0 0 ® 5 0=017057 - ■3=100® ’ 1=72018 0=8008

6 "0.047235 38.665® 1 .0 0 0 0 0 0 .0 0 0 0

Table 7-2. Paraxial Ray Trace

EXAMPLE ' 7/2S 2>2 lFw= RAY TRACE L- i

U 0.176327 2 3=7393 *2.2943 ,114771 3 4.3@74 -@=2929 ,192397 ,280553 4. 4=2321 ,038630 0=150202 ,150202 5 4=7712 1=4273 .098353 .284745 6 4=8428 1=8649 0=141187 .053096 7 0=179426 =179426 146 Table 7-3. Third Order Wave Aberration Coefficients

TRIPLET EXAMPLE 7/23 3E» ORDER HAVE ABERRATION COEFFICIENTS HftUES © ®. 53®®©!!i 0 W131 W220M W222 H311

1 7.558 4.498 10.807 0.669 3.116 2 5.84® -33.003 ' 46.316 46.629 -65.563 3 -21.987 67.907 -61=704 -52.432 54.802 4' -12.158 .. -39.122 -43.911 . -31.472 -45.332 5 2.975 ' 18.851 33.808 29.865 59.805 6 19.067 ' -19.691 16.001 5.084 . -6.956 ABERRATION TOTALS A.295 -0.559 1.516 -1.656 -0.121

Table 7-4. Fifth Order Wave Aberration Coefficients I I i TRIPLET EXAMPLE 3 7/25 STH ORDER WAVE COEFFICIENTS I WAVES 0 0.3500011s • ' • 0 H0S© W131 . W420M W422 W511

1 0.225 0.078 -0.133 . -0.04® -0.034 | 2 0.419 0 2.862 3.012 -2.0®9 3 —1.401 4=35® . -1.627 -1.993 ' 0.925 4 -1.272 ; -2.438 -3.301 -3.773 -3.364 5 0.431 1.559 3.682 4.656 3.931 6 0.890 , -1.735 -1.028 -0.955 0.820 ABERRATION TOTALS 7 ;, -0=768 . -0.044 0.455 1.505 0.269 147

Table 7-5. Fifth Order Wave Aberration Coefficients II

TRIPLET EXAMPLE 7/25 STM ORDER WAVE ABERRATION COEFFICIENTS II

WAVES G . 8.33009* 0 / W240K: ■ M242 W331M-- ; M333 W080 '

1 ©i®2i -0.244 -0.213 ' ■ 0.057 . - 0.009 2 2.265 • 2.699. -5.781 3.680 0,029 3 v -3.448 -4.032 3.276 . -3.779 -0,106, 4 -1.366 -0.639 - -3.227 2.146 -0.174 5 : 1.286 0.931 ; . 2.461 . . -i . i88; . ' 0,070 6 : 0,307 ,' V 0. 141; '• ; 2.142 -1.436 ' ' 0.036 ABERRATION TOTALS 7 -©«936 . -1.143 0,658 -0,521 -0.136 148

TRIPLET EXAMPLE 7/23 TILT AMD DECENTER DATA' (DEGREES,LENS UNITS) SURFACE NUMBER®® TO EXIT INPUT OUTPUT LABELS DECENTERED DESIGN SURFACE NUMBER I Y-DECENTER,X-DECENTERgY-TILT,X-TILT ®,@,.1026,.1387 SURFACE NUMBER Y-DECENTER,X-DECENTER,Y-TILT,X-TILT .0080,-.0108,.1026,.1387 SURFACE NUMBER Y-DECENTER,X-DECEMTER,Y-TILT,X-TILT .0446,-.0227,0,0 SURFACE NUMBER 4 Y-DECENTER,X-DECENTER,Y-T1LT,X-TILT .0446,0227,=0831,0 SURFACE NUMBER - 3 Y-DECENTER,X-DECENTER,Y-TILT,X-TILT .0263,-.0424,-.039®$-.1394 SURFACE NUMBER 6 Y-DECENTER,X-DECENTER,Y-TILT,X-TILT .0233,-.0338,-.1044,-.2293 , ENTER RELATIVE OBJECT DISPLACEMENT:Y,X 0 , 0 ENTER RELATIVE PUPIL DISPLACEMENTSY,X 0,0 ENTER OBJECT PLANE TILYsYpX 0 ,0

Fig. 7-1. Surface-by-Surface Perturbation Data Entry. 149

Table 7-6. Perturbation Data

Element .1 2 3 Magnitude 0 0.05 0.05 Decenter (mm) Orientation 0 -25 -60 (degrees)

Magnitude 10 0 5 (arc min) Tilt •Orientation 55 0 - 1 1 0 (degrees)

Magnitude 0 5 5 (arc min) Wedge Orientation 0 0 -125 (degrees) 150

Table 7-7. First Order Properties of the Perturbed Design

TRIPLET EXAMPLE 7/23 DECENTERED DESIGN is4 ORDER TILT AND DECENTER PROPERTIES

-S - 8 0 SIGMA ■ OBJ DKSP PUP DISP U V FIELD TILT IX -0 0 0381 0 .0 0 0 0 8.8800 0 .0 0 0 0 0 0 0 .0 0 0 0 UN® IY ©.043© 0 o0 000 0 .0 0 0 0 0..00080© 0 .0 0 0 0 U=@

2X -000091 -0.0095 -0.0046 -0.000991 —8.0044 -©0 002421 2Y 0.0067 0.0070 0.0034 ©.000733 0.8033 • - ©.801791 3X 00 0026 -0.0008 . -8 .0 0 1 1 0 .0 0 0 0 0 2 -0.0044 -@,002422 3Y -0.0058 0.©006; 0.8008 -0.008002 0.0033 @.001792 4X -0.0088 0,0018 -0.0009 0.000309 -0.0038 -0.814955 4Y 0.0294 -00.0045 0.0004 -0.800695 0.0019 0.015456 3X 0.0026 0.0052 -0.0009 0.001274 0.0032 . 0.004472 5Y 0.0130■ -0.0153 0.0004 -0.003938 -0.0197 -0.000884 6 X 00 0829 0.0116 -0.0028 0.001588 0.0081 0.012508 6 Y -0.0341 -0.0171 8 .0 0 1 0 -0.002391 . -0.0271 -0.002974

IMAGE X -0.0061 0.0040 -8.801590 -0.0534 0.004774 PLANE Y -0.0078 . -0 .0 0 2 6 -0 0 001085 -0.0691 -0.004430

Table 7-8. The Third Order Perturbation Vectors

TRIPLET EXAMPLE • ?/23 3RD ORDER ABERRATION FIELD VECTORS (UNNORMALIZED> DECENTERED DESIGN-

^2 8 kin kin kin kin -S=65E- COMA -3oOE*

MEDIAL SURFACE 8=41E-0©3 -io43E-004

X: 9.73E- -9-89E-6 ASTIGMATISM YS -3.33E- -2.93E-005

X: 2.9IE- 3.1SE-005 -2.80E- DISTORTION -4.29E-005 Y: -3.20E- 7.1 7.78E- 151

Table 7-9. Fifth Order Perturbation Vectors I

TRIPLET EXAMPLE 7/23 3th ORDER ABERRATION FIELD VECTORS I (UNMORMALTZED) DECENTERED DESIGN ~2 A 8 8 C kin kin kin kin X: -So22E-0®3 63131 ' . TS "So47E—003 x: 3.5IE-0®# M240M 9o13E-007 T: &o23E-00G x: -S.G2E-007 3o3®E-0©7 W242 Y: S.@8E-00S -So S0E-007 x: Io S0E-004 -4 o78E-00S 8.39E-007 - W331M 7.14E-00S ; Y: -2ol8E-®04 -7.33E-00S -3=99E-007 xs -1.03E-004 3o48E—006 63333 YS @o 03E-003 So13E-006 x: -3.8IE-003 4e 47E—006 -3.40E-007 63420M -So12E-00E YS 4o79E-0®6 .go iSE-006 9o431-008 XS -3.84E-003 4oIIE-006 -3.27E-007 63422 — So 64E-006 YS Bo@6E-©0€ lo70E-©06 8» 94E-©©@ X S •• 7» iSE-003 =i o 24E-006 3.23E-®©? ' 63311 . 2o 04E—006 , YS -5.29E-00S , -3o6®E-©@6 -1.75E-007

i j Table 7-10 Fifth Order Perturbation Vectors II

TRIPLET EXAMPLE ' 7/23 3th ORDER ABERRATION FIELD VECTORS II (UNNORMALIZED) DECENTERED DESIGN

6tlF3 kin kin kin

i:=SS>E-i M333 -3ol7E-i

H420M Y: x: 1.98E-' 2.55E-G H422 -3.78E-' 2,28E-8 -7.95E-I -io9iE-0©8 m u 2.77E-' -2oiSE-E -2,58E-i . : 152

Figures 7-2, 7-3, 7-4, and 7-11 show the node plots for this example. These plots are probably the most important feature of this method as they provide a very efficient means for understanding the effects of perturbation on a system. Figure 7-2 gives the node plots for the third order aberration terms, neglecting the effects of the higher order aberrations, This output is generally useful for determin ing how much affect the higher order aberrations have on the lower order terms. Notice here the large displacement of the third order coma field which will result in considerable on-axis degradation in image quality. There has also been a fairly large decentration of the medial ■ " . - ■ t focal surface.

Figure 7-3 illustrates the nodal properties of fifth order coma and oblique spherical aberration. Notice here that the oblique spherical aberration has not been appreciably changed and, because the magnitude of the fifth order coma is small, it too can be neglected in the perturbation analysis. These terms then, will not be discussed in detail.

The nodal properties of linear plus field cubed coma and the medial focal surface are shown in Fig. 7-4. Here, both third and fifth order nodes are displayed simultaneously as these are balancing aber rations . In both cases, the higher order terms have changed both the the magnitude and the nodal locations of the third order terms as can be seen by comparing with the third order data in Fig. 7-2. The . . large central node in the plot for the medial surface indicates the double degeneracy of this node. 153

TRIPLET EXAMPLE 7/25 DECEHTERED DESIGN GAUSSIAN IMAGE PLANE W131* -0.31u Hy 3.90 CENTER CENTER y d : - 0.01 XD - 0.01 _.Hx Hx rp TILT(rad) -0.0044 0.0048 W220M* 0.83u W222- -0.91a CENTER Y222: Hy 1.00 0.04 CENTER X222: - 0.11 Y220M NODES -9.17 X220M 0.08 0.39 Hx Hx -0.32 Y2: MINIMUM - 0.01 FOCAL X2: SHIFT: 0.10 0.009 MM

Fig. 7-2. Third Order Node Plot. 154

TRIPLET EXAMPLE 7/23 DECENTERED DESIGN GAUSSIAN IMAGE PLANE M131- -0.02u Hy 1.00 Hy 3.00 CENTER I CENTER y d : • -0.01 Y151: XD: 0.61 -0.01 Hx ^rcttTrT.-Hx X151: rp 2.18 TILT(rad) Y: -0.0044 X: 0.0048 H240M- -0.51u H242- -0.63u CENTER Hy 1.00 Hy 1.00 Y?0?02 CENTER : X242: : 0.00 Y240M: NODES 0.00 : Yi: X240M: : -9.03 -9.01 ...... Hx ...... _.Nx xi: -0.03 Y2: MINIMUM -0.01 FOCAL ; X2: SHIFT: 0.03 0.000 MM •

Fig. 7-3. Fifth Order Node Plot I. 155

TRIPLET EXAMPLE 7/25 DECENTFRED DESIGN M31 COMA H131E- -0.31a H331M- 0.36u Hy. 2.08 • CENTER x CENTER NODE • Y131E- 0.71 Y331M* -0.33 * X131E- 1.11 X331M- . 0.44 „Hx OUTER NODES • • Yl* 0.00 • Xl= 0.02 Y2* -0.66 X2= 0.86

i MEDIAL SURFACE W20M H220ME* 0.83u H420M- 0.25a Hy; 1.00 i CENTER x CENTER VERTEX • Y220ME- -0.18 Y420M= 0.02 • X220ME- 0.41 X420M- -0.15 ...... _Hx FOCUS OUTER VERTICES • ...... >...... • ■ Yl* 0.88 « * dz« 0.010MM XI* -0.34 « Y2- -0.04 X2- 0.04 •

Fig. 7-4. Fifth Order Node Plot II. . 156

It is important to be able to relate these multiple order node . plots to the behavior of the aberration term in the field. This is done by using the graphic routines shown in Figs. 7-5 to 7-10. Figure

7-5 is a plot of the relative magnitude and orientation of the W3i

(linear plus field cubed) coma in the field. The plot symbol is chosen to represent the angular portion of the coma pattern illustrated in

Fig. 6-1. Its length gives the magnitude of the sagittal coma-. For comparison, the corresponding plot for the 'aligned system is shown in

Fig. 7-6. From these figures it is clear that a considerable amount of coma has been introduced into the system. The balance points remaining from the ring balance of the aligned system lie outside of the field of view. As a result, both the magnitude and the orientation of the coma contribution is nearly constant across the field of view.

A plot of the contours of constant aberration magnitude in the field for linear plus field cubed coma is shown in Fig. 7-7. This is another useful tool for understanding the relationship between the nodes and the resulting aberration fields. The corresponding plot for

the centered system is shown in Fig. 7-8. In Fig. 7-7, the effect of

the three collinear fifth order nodes on the magnitude of the aberration

in the field can be seen.

The contour plot of constant focal shift for the medial surface

is given in Fig. 7-9. Notice that this is completely different from

the corresponding plot for linear plus field cubed coma. This is because while coma is a vector orientation-dependent aberration, the terms

associated with the medial focal surface are scalars. The plot for the

medial surface in the centered system is shown in Fig. 7-10. The 157

TRIPLET EXAMPLE .DECENTERED DESIGN 7/25 LINEAR+FIELD CUBED COMA SCALECnn/in): 8.056 IMAGE PLANE 1..00 7 7 7 7 7 7 A/I >1 >1 'I ^ 7 7 7 7 7 7 7 A A 'I 'I 7 7 7 7 7 7 7 7 7 7 7 A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A 7 7 7 7 7 7 7 7 7 7 -7 7 7 7 7 A/t'1^1'7'777-7 7 7 7 7 7 7 7 7 7 A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A 7 -1.00 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 AA A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A A A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 77 7 7 7 7 7 7 7 7 7 7 7 7 7 A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 A 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

Fig. 7-5. Orientation Plot for Linear Plus Field Cubed Coma. Perturbed System. 158

TRIPLET EXAMPLE CENTERED DESIGN 7/25 LINEAR+FIELD CUBED COMA SCALE(mM/in>: 9.039 IMAGE PLANE 1.00

-1.00

......

Fig. 7-6. Orientation Plot for Linear Plus Field Cubed Coma. Aligned System. 159

WAVES*: 9.3309u TRIPLET EXAMPLE DEEEHTERE.D DESIGN CONTOUR OF CONSTANT WAVE ABERRATION LINEAR+FIELD CUBED COMA CIRCLE-l,RECT«2lFIELD:1 PLOT LABEL?! IMAGE PLANE CONTOUR LABEL?! Hy FIELD 1.00 Hxmin,Hxmax,Hynin,Hynax: , -1,1,-1,1

START SCAH:Hy,Hx 1,8 0 ,1 8 ,8 SCAN TYPE <1=X,2«Y,3-R,4-A>: 2 1 3 STEP SIZE - 1.06 f (relative field): .1 .85 CONTOUR VALUE(waves): 1 1.2 1.4

CONTOUR-1,CONTIHUE-2: 1 CONTINUE SCAN? 1 9 - 1.00

Fig. 7-7. Magnitude Contour Plot for Linear Plus Field Cubed Coma. Perturbed System. 160

WAVES*: 0.550011 TRIPLET EXAMPLE 7/25 CENTERED DESIGN CONTOUR OF CONSTANT LJAUE ABERRATION LINEAR+FIELD CUBED CC)MA CIRCLE31 *RECT"2|FIELD: 1 PLOT LABEL71 IMAGE >LAHE CONTOUR LABEL? 1 u'J FIELD 1.00 Hxmi n,Hxnax,Hymi n» Hynax: “1*1,-1;1 0.10

START SCAN:Hy,Hx .3,0 X 1,0 / SCAN TYPE / <1=X,2*Y,3*R,4*A): / ,0.10 \ A . r STEP SIZE -1.00 r \ " ' 1 i.be (relative field): \ .05 \ .1 \ CONTOUR VALUECveves): \ .10 \

CONTOUR-1, CONTINUE-2! 1 ---- - CONTINUE SCAN? 0 -1.00

Fig. 7-8. Magnitude Contour Plot for Linear Plus Field Cubed Coma. Aligned System. 161

WAVES9: 0.3300U TRIPLET EXAMPLE 7/25 DECENTERED DESIGH______CONTOUR OF CONSTANT WAVE ABERRATION MEDIAL SURFACE CIRCLE-1,RECT-21FIELD:1 PLOT LABEL71 IMAGE PLANE CONTOUR LABEL?I FIELD Hxnin,Hxnax,Hyiiin,Hynax

START SCAN:Hy,Hx

SCAN TYPE (1—X,2=Y,3=R,4—A)

Hx STEP SIZE (relative field)

CONTOUR VALUE(waves)

CONTOUR-1,CONTINUE-2 1 CONTINUE SCAN?

- 1.00

Fig. 7-9. Magnitude Contour Plot for the Medial Focal Surface. Perturbed System. 162

WAVES*: 0.5300U TRIPLET EXAMPLE 7/25 CENTERED DESIGN______CONTOUR OF CONSTANT WAVE ABERRATION MEDIAL SURFACE CIRCLE*!,RECT»2|FIELD:l PLOT LABEL?! IMAGE PLANE CONTOUR LABEL?! Hy FIELD 1 .00 Hxn i n,Hxnax,Hynin,Hymax: -1,1,-1,1

START SCAN:Hy,Hx 0,0

SCAN TYPE C13X,2*Y,3=R,4=A)I 2

STEP SIZE - 1.0 (relative field): .1 CONTOUR VALUE(waves): 1.5 1.0 0.5 CONTOUR-1,CONTINUE-2: 1 CONTINUE SCAN? 0 - 1.00

Fig. 7-10. Magnitude Contour Plot for the Medial Focal Surface. Aligned System. 163 decentration of the focal surface and the change in focus are both evi dent in comparing these plots. Because the fifth order term is small and additive, the third order term dominates the behavior of this aberration.

The node plots for elliptical coma and astigmatism are shown in Fig. 7-11. Notice here the large displacement of one of the nodes in the elliptical coma pattern. Because this is an unbalanced aberration to fifth order, this large nodal displacement will result in some residual disagreement between coefficient and real ray results. The node plot for astigmatism with its six nodes is beginning to become more difficult to visualize directly and one has to rely more heavily on alternate graphics to interpret the behavior.

The plot illustrating the magnitude and orientation of the elliptical coma contribution to the linear plus field cubed coma aberration is given in Fig. 7-12. Here the plot symbol is a two-headed arrow oriented along the major axis of the elliptical mapping of the

"circular zones. These are simply the mapping vectors along the major axis, shown in Fig. 6-17, placed back to back. As shown in Appendix E, the orientation of these vectors depends on both the orientation of the. elliptical coma contribution and the linear plus field cubed coma. The

corresponding plot for the aligned system is given in Fig. 7-13. In

Fig. 7-13 the change in orientation at the edge of the field is due to the change in sign of the linear plus field cubed contribution.

Figure 7-14 is a contour plot of the magnitude of the elliptical coma contribution in the field. This plot provides a clear interpretation

of the node plot in Fig. 7-11. The corresponding plot for the aligned

system is shown in Fig. 7-15. As with linear plus field cubed coma there

< - ■ - ' 164

TRIPLET EXAMPLE 7/23 DECENTERED DESIGN ELLIPTICAL COMA M33 W333- -0.29u My; 2.00 CENTER Y333- -0.28 X333- 0.36 NODES • Hx Yl- -0.84 XI- 1.03 Y2- -0.03 • X2- 0.01 Y3- 0.03 X3- 0.04

ASTIGMATISM W22 H222E- -0.93u H422- 0.83u Hy; 1.00 CENTER CENTER NODE • • • Y222E- -0.01 Y422- 0.01 • • X222E* 0.04 X422- -0.03 i NODES x OUTER NODES • ...... * - --- Hx yi- 0.02 Yl- -0.08 XI- -0.14 XI- 0.03 • Y2- -0.04 Y2- 0.02 t X2- 0.23 X2- 0.04 Y3- 0.09 X3- -0.21 i

Fig. 7-11. Fifth Order Node Plot III. 165

TRIPLET EXAMPLE 7/25 _DECENTERED DESIGN ELLIPTICAL COMA SCALECnn/in): 9.950 IMAGE PLANE 1.00 / / / / / / y* ^ / / / / / X Xxx^^«* / // // / 1 t 1 / / I I 1 1 •* t \ 1 1 I i • . • « • - • • •« x x X \ \X \ \ \ \ % ' \ x X %* • • •••••••%%% -1.00 \ N x X %• •••••••••«»% N s. • •••••••••••« ------—--- ——-- —- Xy Xy XX /

Fig. 7-12. Orientation Plot for Elliptical Coma. Perturbed System.

i 166

TRIPLET EXAMPLE 7/25 -CENTERED DESIGN ELL IPTICAL COMA SCALE(nn/in): 0.859 IMAGE PLANE 1-09

i « « i —

*% + ***» • ••####%

^ % *...... - - \ # » • • *...... - % ...... *• « • * ...... -1.00 i — ...... |

%<••••...... j ' ^ *......

%//#»**

1 1 1 » » . .

Fig. 7-13. Orientation Plot for Elliptical Coma. Aligned System. 167

WAVES#: 8.359011 TRIPLET EXAMPLE 7/23 DECEMTERED. DES1QH______CONTOUR OF CONSTANT WAVE ABERRATION ELLIPTICAL COMA CIRCLE-1,RECT-2IFIELD:1 PLOT LABEL71 IMAGE PLANE CONTOUR LABEL?1 Hy FIELD 1.00 Hxti i n, Hx«ttx, Hyn i n, Hymax: , -1,1,-1,1

START SCAH:Hy,Hx 1,9

SCAN TYPE <1-X,2»Y,3-R,4-A): 2 1 STEP SIZE (relative field): .1 CONTOUR VALUECvaves): .1 .2 .05 .15 .25 CONTOUR-1,CONTINUE-2: 1 CONTINUE SCAN? 0 - 1.00

Fig. 7-14. Magnitude contour Plot for Elliptical Goma. Perturbed System. 168

HAVES9: 0.5500a TRIPLET EXAMPLE CENTERED DESIGH CONTOUR OF CONSTANT WAVE ABERRATION ELLIPTICAL COMA CIRCLE*1,RECT-21FIELD:1 PLOT LABEL?1 IMAGE PLANE CONTOUR LABEL?1 Hy FIELD _ 1.00 Hxnin»HxnaxiHynin»Hynttx: -1*1,-1,1

START SCAN:Hy,Hx 1,0

SCAN TYPE <1=X,2-Y,3-R,4*A>: 2

STEP SIZE (relative field): .1 CONTOUR VALUE(waves): .1 .05 CONTOUR*!,CONTINUE-2: 1 CONTINUE SCAN? 0 - 1.00

Fig. 7-15. Magnitude Contour for Elliptical Coma. Aligned System. 169 has been a significant change in this aberration term due to the large displacement of one of the nodes,

For astigmatism, the magnitude and orientation plots may be made along the N or P focal surfaces as shown in Figs. 7-16 and 7-17.

On these focal surfaces,.line images which are orthogonal result as shown for this aberration contribution. The corresponding plots for the aligned system are shown in Figs. 7-18 and 7-19. A contour plot of constant aberration magnitude is given in Fig. 7-20. This illustrates the dominance of binodal astigmatism inside the balance zone as dis cussed in Chapter 6. The contour plot for the aligned system is shown in Fig. 7-21. For this example, the balance zone lies just outside of the field of view.

In studying astigmatism, field curve, profiles provide an additional valuable analysis tool. By providing orthogonal profiles of the N and P focal surfaces, relative to the design plane of the aligned system (D), the effects of the Gaussian image plane tilt, displacement of the vertex of the medial surface and the astigmatic nodes can be dis played. These profiles are shown in Figs. 7-22 and 7-23. The field curve profile for the aligned system is shown in Fig. 7-24.

Due to a lack of a nodal interpretation for fifth order dis tortion, the magnitude and orientation plots and contour plots are important for analyzing this term. These plots are shown for the per turbed and aligned system in Figs. 7-25 through 7-28. From these, one 170

TRIPLET EXAMPLE 7/25 DECENTERFD DESIGN N-SURFACE ASTIGMATISM SCALE

------— » * s / / s X — — ' / / / / X ^ • — ^ 1 / / / ✓ X — • ^ 1 / // / # # • • » 1 1 1 1 - 1.00 » t 1 1 1 • • X \ \\ X x % ^ • * \ \ \\ \ \ X ^ — - \ \ \ \ \ X -X •— — — N \ \ \ \ \ ^ — — — X X X -x X. — — — —— x X X

Fig. 7-16. Orientation Plot for N-Surface Astigmatism. Perturbed System. 171

TRIPLET EXAMPLE 7/25 -DECENTERED. DESIGN P-SURFACE ASTIGMATISM SCALE(mm/in): 0.050 IMAGE PLANE Luflfi__ / 1 1 1 / 1 // / / 1 \ \ \ 1 / 1 / / //S' « \ \ \ \ 1 1 1 / / ///S' \ \ \ \ \ 1 1 1 / / / / / / s \ \ \ \ \ 1 1 1 / / / / / / s » X X X \X \ 1 » 1 / / s / / / / X \X % « 1 1 / / X / / -- - — - - *••» x

- 1.00

v' / / / / / / I l % X X -X //%///Z / 1 I X \ \ \ /////// / I 1 \ \ W W W X ////II \ \ \ \ \ \ \ w //////• nil • i i i i i \ \ \ \ \ \

Fig. 7-17. Orientation Plot for P-Surface Astigmatism. Perturbed System. 172

TRIPLET EXAMPLE CENTERED PPSTCM 7/23 N-SURFACE ASTIGMATISM SCALE: 0.959 IMAGE PLANE 1.00

* — — X X X % / / / / — •— — ■X X X X % * / / / / / — X X X X \ % / / z z / / — — — ~- X XXX X \ \ / / / / / s - - •— X X \ \ \ \ \ X / / / / / / - - X \ \ \ \ \ \ z / / / / / X \ \ \ \ \ \ / // 1 l • * • • » % \ \ \ \I 1 - 1.00 1 1 1 1 i • 1 1 1 1 1 1 1 \ \ \ 1 t » ' • • • • 1 / / / 1 1 \\ \ \ \ \ - «• » / / / / / 1 / \\ \ \ \ \ X - - / / / / // / \ \ \ \ X X X - - / / z / / / # \ \ \ \ x v X — — / / Z z / / \ \ X x ^ —— / / / z / X X X X -V. — — / / / X X x — — ,

Fig- 7-18. Orientation Plot for N-Surface Astigmatism. Aligned System. 173

TRIPLET EXAMPLE CENTERED DPSir.M 7/25 P-SURFACE ASTIGMATISM SCALE(nn/in): 9.050 IMAGE PLANE 1^00

% \ I I I / ' \ \ \ \ I I / / ' ' N \ \ \ \ I / / / / ' ' X \ \ \ \ \ I z ///S' x X X \ \ \ \ / / ///S' ~ X X X X N \ \ / / ///S' ' X X X X X \ % # / S S S S S *•

- 1.00 —■ - ______

— — — — — - «* • • • x — —— —— — — —• — - '•% X X X. / s// r t X S X X XX * s S / s// /. / X \ NX X X X X x S / / / / / / \ \ \\ X X x ' s/ / / / / / \ \ \ \ \ X X ' // / / / / \ \ \ \\ \ X / / / / / / \ \ \ \ \ % / / / 1 1 X X %

Fig. 7-19. Orientation Plot for P- Surface Astigmatism. Aligned System. 174

WAVES#: 0.5500U TRIPLET EXAMPLE 7/25 DECEHTERED DESIGN______CONTOUR OF CONSTANT WAVE ABERRATION ASTIGMATISM WRT MEDIAL SURFACE CIRCLE-1,RECT-21FIELD:I PLOT LABEL?1 IMAGE PLANE CONTOUR LABEL?! Hy FIELD 1 .0 0 Hxnin,Hxnax,HyFiin,Hynttx: -1,1,-1,t

START SCAN:Hy,Hx • 4,0 8, • 8 -.2,.8 SCAN TYPE a*X,2«Y,3«R,4»A>: 2 1 STEP SIZE -1 (relative field): • 1 .05 CONTOUR VALUE(waves): .05 .15 .25 CONTOUR-1,CONTINUE-2: i 1 CONTINUE SCAN? 0 - 1.00

Fig. 7-20. Magnitude Contour Plot for Astigmatism with respect to the Medial Focal Surface. Perturbed System. 175

WAVES*: 0.3300U TRIPLET EXAMPLE 7/25 CENTERED DES.IQH______CONTOUR OF CONSTANT WAVE ABERRATION ASTIGMATISM WRT MEDIAL SURFACE CIRCLE-1,RECT-2JFIELD:I PLOT LABEL71 IMAGE PLANE CONTOUR LABEL?1 Hy FIELD Hxnin,HxHax,Hynin,Hynax

START SCAN:Hy,Hx 1*0 .4,0 0.15 SCAN TYPE (1=X,2-Y,3=R,4-A) 05

STEP SIZE (relative field) .05 CONTOUR VALUE(waves) .25 .15 .05 CONTOUR-1,CONTINUE-2 1 2 CONTINUE SCAN? 1.00

Fig. 7-21. Magnitude Contour Plot for Astigmatism with respect to the the Medial Focal Surface. Aligned System. 176

TRIPLET EXAMPLE 7/25 CENTER DESIGN NiP FOCAL SURFACE PROFILE PLOT

PROFILE ORIENTATION (cleg) (9-Y,90»X>: 0 FIELD PLOT SCALE 1.80 ? 1 -0.500

DRAM OAR? 1 DRAW REF AXIS? 1 DRAW DESIGN PLANE? 1 1.00

Fig. 7-22. N and P Field Curves. Aligned System. 177

TRIPLET EXAMPLE 7/25 DECENTERED DESIGN NiP FOCAL SURFACE PROFILE PLOT

PROFILE ORIENTATION: 0 FIELD PLOT SCALE 1.00 (Znin»ZnaXfHnin»Hnax>: .5;.5*1;1 NUMBER OF POINTS: 20 PLOT LABEL? 1 PLOT SYMBOL? 1 DOT CONNECT? 1 iUUL dz(MM) re DRAW AXIS ? 0 -0.500 0.500

DRAW OAR? 1 DRAW REF AXIS? 1 DRAW DESIGN PLANE? 1 1.00

Fig. 7-23. N and P Field Curves. Y-Field Profile. Perturbed System. 178

TRIPLET EXAMPLE 7/25 DECEHTERED DESIGN HIP FOCAL SURFACE PROFILE PLOT

PROFILE ORIEMTATIOHCdeg) <9-Y,99-X>: 90 FIELD PLOT SCALE 1.00 (Zmin,ZMax,Hmin,Hmax>: -.5,.5,-1;1 NUMBER OF POINTS: 20 PLOT LABEL? 1 PLOT SYMBOL? 1 DOT CONNECT? 1 jttOTL dz(nn) ra DRAW AXIS ? 0 -0.500 0.500

DRAW OAR? 1 DRAW REF AXIS? 1 DRAW DESIGN PLANE? I

Fig. 7-24. N and P Field Curves. X-FieId Profile. Perturbed System, 179

TRIPLET EXAMPLE 7/23 DECENTERED DFSir.H DISTORTION SCALE: 8.050 IMAGE PLANE 1.00

\ v x I I 1 1 I I < 4 V V \ WVXlistiiiit

-1.09 * % ^ x x x X ^ ^ ^ ' % ^ X x X > x X ^ ' ' x x >• ^ ^ X X x x x \ \\XX

Fig. 7-25. Orientation Plot for Distortion. Perturbed System. 180

TRIPLET EXAMPLE 7/23 CEHTFRED ftFSIf.H DISTORTION SCALE

-1.90

Fig. 7-26. Orientation plot for Distortion. Aligned System. 181

WAVES*: 0.55O0U TRIPLET EXAMPLE 7/25 DECENTERED DESIGN CONTOUR OF CONSTANT & 4AUE ABERRATION DISTORTION CIRCLE-1,RECT-21FIELD!1 PLOT LABEL?1 IMAGE PLANE CONTOUR LABEL?1 u FIELD ^1.00 Hxnin, Hxnax# HyFiin#Hynttx:

START SCAN:Hy,Hx ^

SCAN TYPE / / / <1»X,2-Y,3-R,4-A): / / 2 / / 1 / / STEP SIZE -1-0* / (relative field): / .1 / .15 / CONTOUR VALUE

Fig. 7-27. Magnitude Contour for Distortion. Perturbed System. 182

WAVESe: 0.3300U TRIPLET EXAMPLE 7/25 rFNTFRFD DESIGN CONTOUR OF CONSTANT k IAUE ABERRATION DISTORTION CIRCLE-1,RECT-2»FIELD:l PLOT LABEL71 IMAGE ’LANE CONTOUR LABEL?1 Hi FIELD Hxnin,HxFiox,Hynin,Hvfittx: ___ !'dll® -1,1,-1,1 ^

START SCAN:Hy,Hx / 1,0 /

SCAN TYPE / C1=X,2-Y,3-R,4-A): 2 /

STEP SIZE -1.0M .. *" 1 ' 1 ij.be (relative field): \ .1 \ CONTOUR VALUE(waves): \ 1 \ .1 \ CONTOUR-1,CONTIHUE-2: N . 1 2 CONTINUE SCAN? 1 — : -i.ee

Fig. 7-28. Magnitude Contour for Distortion. Aligned System. 183

can see that a considerable amount of distortion has been:introduced at

the edge of the field.

The node plots and supporting balance zone graphics provide

an important new tool for analyzing the effects of perturbations. From

the plots presented here.one finds that the properties of linear plus

field cubed coma and elliptical coma are dominant. The analysis then can be concentrated on finding the source of this dominance or methods of

compensating for these terms.

Whereas the use of node plots provide a rapid method of gaining

insight: into the effects of perturbations on an optical system, it is

important to be able to evaluate the image quality also. In general, a

designer is given some criterion such as the maximum allowable rms

wavefront error which the design must meet over the field of view.v.Once

the design is completed and evaluated, an error budget must be estab

lished oh the allowable tilts and decenters of the elements such that the

given criterion is met across the field. It is useful then to be able to

obtain either the actual rms wavefront in the perturbed system or the

change in rms wavefront due to the perturbations. In a system with

randomly oriented perturbations it is necessary to evaluate these

quantities over the entire image plane.

For this purpose the equations developed to find the rms

wavefront in a perturbed system, Eq. F-3, have been combined with an

interactive contour plot routine. This allows a designer to find if or

where in the field a particular value for actual or change in rms wave-

front occurs ; Because analytic calculations of rms wavefront are much

faster than real ray calculations, such plots may be made economically. ; ' : ■ 184 There are two methods for analyzing rms wavefront. In one,

the calculations are made with respect to the best focal surface. In

the other, a flat best focused image plane is used. Both methods will be shown here.

First, consider making the calculations with respect to the best focal surface. To aid in constructing contour plots, tabular and

. . ■ ; - ' . plotted output for profiles in the field are available. These are shown

in Figs. 7-29 through 7-33 for the aligned and the perturbed system. The plot labeling in Figs. 7-30 and 7-33 is found by studying the columns to the left of the plot.

The contour plots for the actual rms wavefront in the aligned system and both the actual and the change in the rms wavefront for the perturbed system are shown in Figs. 7-34, 7-35 and 7-36. From these, the

dominance of on axis coma due to the large displacement of the third or der coma field may be seen.

One additional means for analyzing perturbed systems is through

the use of field focus plots. These plots, introduced by Shack (1974) provide yet another perspective on the effects of perturbations on an

optical system. Field focus plots are profiles of the allowable defocus >

as a function of field, such that a given tolerance on image qualityis

met. Figure 7-37 shows the field focus plot for the aligned system. In

the plots, the central profile represents the shape of the medial sur

face. As the tolerance on allowable rms wavefront increases, the amount

of defocus allowable increases and the two profiles move out symmetrically

from the medial surface. There are two profiles for each tolerance 185

MAVES6: 8.5500U TRIPLET EXAMPLE 7/25 CEHTERED DESIGN CHANGE IH RMS WAVEFRONT PROFILE WRT BEST FOCAL SURFACE PROFILE ORIENTATION

Fig. 7-29. List of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface. 186

WAVES*: 0.5500U TRIPLET EXAMPLE 7/25 CENTERED DESIGN RMS WAVEFRONT PROFILE PLOT WRT BEST FOCAL SURFACE? 1 ORIENTATION(deg) is <0=Y,90»X>: wttv< es 0 0.50 CHANGE IN-1,ACTUAL-21 RMS: 2 PLOT SCALE: (Hnin»Hnax»w«in»wnax) -1,1*-.5;.5 NUMBER OF POINTS: 20 PLOT LABEL? X. pi nr SYMFOI . . . . FIELD ,4*o»5* ): 1 -1.00 1.00 DOT CONNECT? <1-SOLID,2-DASHED,3=N0>: 1 MULTIPLOT? 0 FIELD FOCUS PLOT? 0 -0.50

Fig. 7-30. Profile Plot of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface. 187

WAVES*: 0.3300U TRIPLET EXAMPLE 7/25 -BECEMTERED DESIGN CHANGE IN RMS WAVEFRONT PROFILE WRT BEST FOCAL SURFACE PROFILE ORIENTATION 0.00 Y-FIELD X-FIELD CENTERED RMS CHANGE IN RMS FOCUS waves waves (MM) -1.00 0.00 0.230 0.082 -0.082 -0.90 0.00 0.230 0.086 -0.067 -0.80 0.00 0.214 0.089 -0.055 -0.70 0.00 0.187 0.094 -0.047 -0.60 0.00 0.153 0.100 -0.042 -0.50 0.00 0.117 0.108 -0.039 -0.40 0.00 0.083 0.118 -0.038 -0.30 0.00 0.054 0.129 -0.038 -0.20 0.00 0.033 0.138 -0.040 -0.10 0.00 0.023 0.141 -0.042 0.00 0.00 0.021 0.138 -0.046 0.10 0.00 0.023 ' 0.133 -0.050 0.20 0.00 0.033 0.123 -0.055 0.30 0.00 0.054 0.106 -0.060 0.40 0.00 0.083 0.089 -0.067 0.50 0.00 0.117 0.074 -0.073 0.60 0.00 0.153 0.063 -0.085 0.70 0.00 0.187 0.062 -0.097 0.80 0.00 0.214 0.064 -0.Ill 0.90 0.00 0.230 0.073 -0.129 1.00 0.00 0.230 0.090 -0.130 PLOT? 0

Fig. 7-31. List of the Change in the RMS Wavefront Error in the Per turbed System with respect to the Best Focal Surface. Y-Field Profile. 188

HAVESO: 0.3S00U TRIPLET EXAMPLE 7/25 Ldec entfrfd DFSTr.M CHANGE IN RMS WAVEFRONT PROFILE WRT BEST FOCAL SURFACE PROFILE ORIENTATION 90.00 Y-FIELD X-FIELD CENTERED RMS CHANGE IN RMS FOCUS waves waves (mm) 0.00 -1.00 0.230 0.029 -0.180 0.00 -0.90 0.230 0.018 -0.158 0.00 -0.80 0.214 0.016 -0.140 0.00 -0.70 0.187 0.024 -0.124 0.00 -0.60 0.153 0.040 - 0.110 0.00 -0.50 0.117 0.062 -0.097 0.00 -0.40 0.083 0.089 -0.085 0.00 -0.30 0.054 0.115 -0.074 0.00 -0.20 0.033 0.134 -0.064 0.00 -0.10 0.023 0.141 -0.054 0.00 0.00 0.021 0.138 -0.046 0.00 0.10 0.023 0.129 -0.038 0.00 0.20 0.033 0.114 -0.031 0.00 0.30 0.054 0.094 -0.025 0.00 0.40 0.083 0.076 - 0.020 0.00 0.30 0.117 0.064 -0.017 0.00 0.60 0.153 0.056 -0.017 0.00 0.70 0.187 0.050 -0.019 0.00 0.80 0.214 0.046 -0.025 0.00 0.90 0.230 0.041 -0.033 0.00 1.00 0.230 0.035 -0.050 PLOT? 0

Fig. 7-32. List of the Change in the RMS Wavefront Error in the Perturbed System with respect to the Best Focal Surface. X-Field Profile. 189

WAVES*: 0.3300U TRIPLET EXAMPLE 7/25 DECENTFPED DESIGN CHANGE IN RMS MAUEFRONT PROFILE PLOT WRT BEST FOCAL SURFACE? 1 ORIENTATION "MS <0»Y,90*X>: tes 0 0 90 90 L 0.50 CHANGE IN*l,ACTUAL-21 RMS: 1 2 1 2 PLOT SCALE: (HninyHnaxf wnin»wnax> -1» l*-.5; .5 NUMBER OF POINTS: . PLOT LABEL? 00 0 o oT ? «* *1 PLOT SYMBOL T " * * ? , , ,4=Of3* >: 4 1 2 3 -1.00 1.00 DOT CONNECT? <1-SOLID,2-DASHED,3-NO): 3 1 3 1 MULTIPLOT? 1 0 FIELD FOCUS PLOT? 1 ORIENTATION: 0 ^ -0.50 TOLERANCE

Fig. 7-33. X and Y Field Profile Plots of the Actual and the Change in the RMS Wavefront Error in the Perturbed System with re spect to the Best Focal Surface. 190

WAVES*: 0.3500U TRIPLET EXAMPLE 7/25 CENTERED DESIGN RMS WAVEFRONT OVER TKIE FIELD CONTOUR PLOT WRT BEST FOCAL SURFACE? 1 CHANGE IN*1,ACTUAL-2!2 IMAGE ’LANE CIRCLE-1,RECT-2IFIELD:! Hi PLOT LABEL?! ? 1.00 CONTOUR LABEL?! FIELD Hxn i n, Hxnax, Hy n i fi' Hynax: -1*1,-1,1 ---- L9^20 START SCAM:Hy,Hx 1,6 _^10 X . SCAN TYPE / <1«X,2*Y,3*R,4*A): / / 2 I /

STEP SIZE “1*0^ ' 1 ' I* ■ M .' ]' i.be (relative field): V \ .1 \ \ CONTOUR VALUE (waves): \ .2 X .1 X. CONTOUR-1, CONTINUE-2: — CONTINUE SCAN? 0 -i.ee

Fig. 7-34. Contour Plot of the RMS Wavefront Error in the Aligned System with respect to the Best Focal Surface. 191

WAVESP: 0.5500a TRIPLET EXAMPLE 7/25 DECENTERED DESIGN RMS WAVEFRONT OVER TYIE FIELD CONTOUR PLOT MRT BEST FOCAL SURFACE?1 CHANGE IN=1,ACTUAL-2:2 IMAGE »LANE CIRCLE-1,RECT-21FIELD:I • H PLOT LABEL?1 * 1.00 CONTOUR LABEL?! 1 FTEI n Hxitin,Hxnax,Hynin,Hypiex: -1,1,-1,1 ____ l0j25 START SCAN:Hy,Hx ------1 ,8 0 . 2 0 ^ X . - . 6 , 0 ------.4,0 Z Z " SCAN TYPE / Z <1=X,2«Y,3»R,4«A>: / / 2 II -4. t he STEP SIZE - 1 . 0 b I ' V (relative field): \ V .1 .05 X .15 \ CONTOUR VALUECwavcs): X . ^ .2 .25 .3 .15 CONTOUR-1,CONTINUE-2: 1 /------CONTINUE SCAN? C 0 1 N -1.00

Fig. 7-35. Contour Plot of RMS Wavefront Error in the Perturbed System with respect to the Best Focal Surface. 192

W4VES9: 9.5500U TRIPLET EXAMPLE 7/25 DFCFNTFRFD DFSTGN RMS WAVEFRONT OVER TKIE FIELD CONTOUR PLOT MRT BEST FOCAL SURFACE?1 CHANGE IN-1, ACTUAL-2:1 IMAGE ’LANE CIRCLE-1,RECT-21FIELD:! Hi PLOT LABEL71 ? 1.00 CONTOUR LABEL?1 FIELD ^ ______0.09 Hxnin, Hxnax, Hynin,Hynax: -1,1,-1,1 START SCAN:Hy,Hx / ---- V 1,0 0,0 0, . 6 jT j 0,1 1, .2 / J J0£00 -.8,0 9, -.8 / /->. / // ■ SCAN TYPE ' / \ / / ^ T iK ( (1-X,2=Y,3«R,4=A>: / ) / /f

1 ye. 0 3 / ./e/pfi f m \ STEP SIZE -1-4 1 I V \ \ — 1 X i.be Crelstivc field): 1 / 1 TI \ vy \ \V_ CONTOUR VALUECwaves): V \ .1 .975 \ X .05 .025 \ .15 .125 / \ CONTOUR-1,CONTIHUE-2: / \ 1 0.08 1 \ J CONTINUE SCAN? \ 0 1 \ -1.00

Fig. 7-36. Contour Plot of the Change in the RMS Wavefront Error in the Perturbed Systems with respect to the Best Focal Surface. 193

WAVES*: 0.5500U TRIPLET EXAMPLE 7/25 ££MJER£D DESIGH FIELD FOCUS PLOT RMS WAVEFRONT

FIELD ORIENTATIONCdeg)' FIELD 0.00 TOLERANCE 1.00 (wnin«wnax»dw): 0.20 0.60 0.20 PLOT SCALE (Znin,Znax,Hniti,H«ax): y 1 y ~i f 1 NUMBER OF POINTS: 20 PLOT LABEL? 1 oar #- H dz(nn)

- 1.00 1.00

- 1.00

Fig. 7-37. Field Focus Plot for the Aligned System. 194 because the focus can be either positive or negative. Figures 7-38 and 7-39 show the corresponding plots for two orthogonal profiles in the field for the perturbed system. Notice in these plots that many effects may be seen simultaneously. The displacement of the center of the medial focal surface can be seen from the tilt of the medial sur face, The tolerance profiles can be seen to have moved in from the . shortened length of the closed profile and been shifted transversely in the image plane (again, most easily seen with the closed profile).

Figures 7-40 through 7-47 provide the material corresponding to making the analysis with respect to a best focused flat image plane.

In this case, best focus is chosen to give the smallest maximum value for the rms Wavefront across the field. In these figures, the behavior is dominated by the displacement of the medial focal surface rather than the coma.

The accuracy of this approach, when compared with real ray calculations for this particular system, is shown in Fig. 7-48. Here the comparison is made for a profile along the y-axis only because of the expense involved in the real ray calculations. Also included are the values of a calculation made using only third order coefficients, i.e.,

Eq. (4-37). This illustrates the importance of including the fifth order coefficients in this system.

To get a qualitative view of the agreement between the two approaches, an array of spot diagrams in the field have been constructed 195

WAVES*: 0.5580U TRIPLET EXAMPLE 7/23 DECEHTERED DESIGN FIELD FOCUS PLOT RMS WAVEFRONT

FIELD ORIENTATIOH FIELD 0.00 TOLERANCE 1.00 : 0.20 0.60 0.20 PLOT SCALE

1.00

Fig. 7-38. Field Focus Plot for the Perturbed System. Y-Field Profile. 196

WAVES#: e.330911 TRIPLET EXAMPLE 7/25 DECEHIERED DESIGN__ FIELD FOCUS PLOT RMS WAVEFRONT

FIELD ORIENTATION FIELD 99.09 TOLERANCE(waves) 1.00 (wM&n,WMax,dw>: 0.20 9.60 0.20 PLOT SCALE (ZninyZnaxyHnin»Hnax): “1y1y —I»I NUMBER OF POINTS: 20 PLOT LABEL? 1

1.00 1.00

1.00

Fig. 7-39. Field Focus Plot for the Perturbed System. X-Field Profile. 197

WAVES*: 0.5500U TRIPLET EXAMPLE 7/25 CENTERED DFSTCN CHANGE IN RMS WAVEFRONT PROFILE

PROFILE ORIENTATION(deg) 0.00 Y-FIELD X-FIELD CENTERED RMS CHANGE IN RMS FOCUS waves waves -1.00 0.00 0.239 0.000 -0.092 -0.90 0.00 0.230 0.000 -0.092 -0.80 0.00 0.226 0.000 -0.092 -0.70 0.00 0.223 0.000 -0.092 -0.60 0.00 0.220 0.000 -0.092 -0.50 0.00 0.219 0.000 -0.092 -0.40 0.00 0.221 0.000 -0.092 -0.30 0.00 0.225 0.000 -0.092 -0.20 0.00 0.230 0.000 -0.092 -0.10 0.00 0.234 0.000 -0.092 0.00 0.00 0.236 0.000 -0.092 0. 10 0.00 0.234 0.000 -0.092 0.20 0.00 0.230 0. 000 -0.092 0.30 0.00 0.225 0.000 -0.092 0.40 0.00 0.221 0.000 -0.092 0.50 0.00 0.219 0.000 -0.092 0.60 0.00 0.220 0.000 -0.092 0.70 0.00 0.223 0.000 -0.092 0.80 0.00 0.226 0.000 -0.092 0.90 0.00 0.230 0.000 -0.092 1.00 0.00 0.239 0.000 -0.092 PLOT? 1

Fig. 7-40. List of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane. 198

WAVES#: 0.5300U TRIPLET EXAMPLE 7/25 CENTERED DESIGN RMS WAVEFRONT PROFILE PLOT WRT BEST FOCAL SURFACE? 0 FOCUS(nn): -.092 ORIENTATION is (0=Y,90-X>: wov< BS 9 0.50 CHANGE IN»l,ACTUAL-2;RMS: 2 PLOT SCALE: (Hmin,Hmax,wmin,wmax) -ltl»-.5».3 „ NUMBER UE PU1NIS: * » " . 2 0 PLOT LABEL? 1 pi n r SYMnni 1 | | . | | FIELD ( 1 = . »2*x,3=>,4*Of5- >: 1 -1.00 1 . 0 0 DOT CONNECT? C1-SOLID,2-DASHED,3-NO): 1 MULTIPLOT? FIELD FOCUS PLOT? 0 -0.50

Fig. 7-41. Profile Plot of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane. 199

WAVES*: 0.5500U TRIPLET EXAMPLE 7/25 DECENTERFD DESIGN CHANGE IN RMS WAVEFRONT PROFILE

PROFILE ORIENTATION -1.00 0.00 0.239 0.082 -0.100 -0.90 0.00 0.230 0.114 -0.100 -0.80 0.00 0.226 0.128 -0.100 -0.70 0.00 0.223 0.132 -0.100 -0.60 0.00 0.220 0.127 -0.100 -0.50 0.00 0.219 0.118 -0.100 -0.40 0.00 0.221 0.103 -0.100 -0.30 0.00 0.225 0.087 -0.100 -0.20 0.00 0.230 0.070 -0.100 -0.10 0.00 0.234 0.053 -0.100 0.00 0.00 0.236 0.038 -0.100 0.10 0.00 0.234 0.025 -0.100 0.20 0.00 0.230 0.013 -0.100 0.30 0.00 0.225 0.003 -0.100 0.40 0.00 0.221 -0.002 -0.100 0.50 0.00 0.219 -0.002 -0.100 < 0.60 0.00 0.220 • 0.007 -0.100 0.70 0.00 0.223 0.026 -0.100 0.80 0.00 0.226 0.056 -0.100 0.90 0.00 0.230 0.095 -0.100 1.00 0.00 0.239 0.142 -0.100 PLOT? 0

Fig. 7-42. List of the Change in the RMS Wavefront Error in the Per turbed System with respect to a Flat, Best Focused Image Plane. Y-Field Profile. 200

WAVES*: 8.3500U TRIPLET EXAMPLE 7/25 _DECENTERED DESIGN CHANGE IN RMS WAVEFRONT PROFILE

PROFILE ORIENTATION(deg) 90.00 Y-FIELD X-FIELD CENTERED RMS CHANGE IN RMS FOCUS waves waves (mm) 0.00 -1.00 0.239 0.178 -0.100 0.00 -0.90 0.230 0.114 -0.100 0.00 -0.80 0.226 0.056 -0.100 0.00 -0.70 0.223 0.010 -0.100 0.00 -0.60 0.220 -0.023 -0.100 0.00 -0.50 0.219 -0.038 -0.100 0.00 -0.40 0.221 -0.038 -0.100 0.00 -0.30 0.225 -0.025 -0.100 0.00 -0.20 0.230 -0.007 -0.100 0.00 -0.10 0.234 0.015 -0.100 0.00 0.00 0.236 0.038 -0.100 0.00 0.10 0.234 0.064 -0.100 0.00 0.20 0.230 0.091 -0.100 0.00 0.30 0.225 0.118 -0.100 0.00 0.40 0.221 0.144 -0.100 0.00 0.50 0.219 0.166 -0.100 0.00 0.60 0.220 0.180 -0.100 0.00 0.70 0.223 0.185 -0.100 0.00 0.80 0.226 0.176 -0.100 0.00 0.90 0.230 0.149 -0.100 0.00 1.00 0.239 0.096 -0.100 PLOT? 1

Fig. 7-43. List of the Change in the RMS Wavefront Error in the Per turbed System with respect to a Flat, Best Focused Image Plane. X-Field Profile. 201

WAVES#: 9.5500U TRIPLET EXAMPLE 7/25 DECENTERED DESIGN CHANGE IN RMS WAVEFRONT PROFILE PLOT WRT BEST FOCAL SURFACE? 0 F0CUS ™ "MS <0=Y,90«X>: wa ves 0 0 90 90 0.50 CHANGE IN*l,ACTUAL-2IRMS: 1 2 1 2 \ PLOT s c a l e : (Hnin»Hnaxyunin» wnax) X . -1,1,-.5,.5 X NUMBER OF POINTS: . 20 " n PLOT LABEL? „• ° o n 1 0 . ° o 0 - * o PI OT SYMRm • . 1 - • | . * <1*.,2=x,3»>,4-o,5- >: * * * * 4 1 2 3 -1.00 1.00 DOT CONNECT? <1-SOLID,2-DASHED,3-NO): 3 1 3 1 MULTIPLOT? 1 0 FIELD FOCUS PLOT? 0 -0.50

Fig. 7-44. X and Y Field Profile Plots of the Actual and the Change in the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image Plane. 202

WAVES9: 0.3300U TRIPLET EXAMPLE 7/25 CEMTFRFD DESIGN RMS WAVEFRONT OVER J\IE FIELD CONTOUR PLOT WRT BEST FOCAL SURFACE?© FOCUS:-.092 CHANGE IN*1,ACTUAL-2:2 IMAGE PLANE CIRCLE-1,RECT-2JFIELD:1 H If PLOT LABEL71 1.00 CONTOUR LABEL?1 1 FIELD --- 9.23 Hxfiin, Hxnax, Hynin,Hynax: ~ 1,1 f—If 1 START SCAH:Hy,Hx X 1,8 / .4,0 / SCAN TYPE / <1-Xf 2=Y,3=R,4-A): / .0.23 \

STEP SIZE -1.0b 1 ' ' f I ? " ' ' ' L b e (relative field): \ X — .1 \ .05 \ CONTOUR VALUE(waves): \ .23 \ .23 \ CONTOUR* 1, CONTINUE-2: 1 CONTINUE SCAN? -- —. 0 J -i.ee

Fig. 7-45. Contour Plot of the RMS Wavefront Error in the Aligned System with respect to a Flat, Best Focused Image Plane. 203

WAVES*: 0.5500U TRIPLET EXAMPLE 7/23 DECFNTEPED DESIGN RMS WAVEFRONT OVER TIIE FIELD CONTOUR PLOT WRT BEST FOCAL SURFACE?© FOCUS(MM):-.1 CHANGE IN-1, ACTUAL-2:2 IMAGE »LANE CIRCLE-1,RECT-21FIELD!1 H 1 PLOT LABEL?! ’ 1.00 CONTOUR LABEL? 1 1 FIELD Hxn i n, Hxnex, Hyn i n, Hynax: -1,1,-1,1 / START SCAN:Hy,Hx / / -.4,1 -.4,0 -.8,-.2 / / 8,8 -.4,- 2 / / -.2,1 -.9,0 / / X ) - SCAN TYPE / / Z / (1-X,2-Y,3=R,4-A): I f /

2 i/a4 0 [ | /0r28 STEP SIZE ”1*0? I \ / / 1 1 >> Ai.be (relative field): | 1 \^/ re. 30 / \ • 05 \ i CONTOUR VALUE: \0.30 /. / J 0.40 .4 \ / .3 \ / . .2 \ / CONTOUR-1,CONTIHUE-2: \ / 1 2 X f CONTINUE SCAN? X J 8 X -i.ee

Fig. 7-46. Contour Plot of the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image Plane. 204

WAVES*: 0.5500U TRIPLET EXAMPLE 7/25 DECEHTERED DESIGN______RMS WAVEFRONT OVER THE FIELD CONTOUR PLOT WRT BEST FOCAL SURFACE?® FOCUS: 1 CHANGE IN=1,ACTUAL-2:1 IMAGE PLANE CIRCLE-1,RECT-21FIELD:1 Hy PLOT LABEL?! 1.00 CONTOUR LABEL?! 0.10 FIELD Hxn in,Hxnax*Hyni n,Hynax: -1,1,-1,1 START SCAH:Hy,Hx 1,0 #3,0 0,•6 -.8,0 .2,0 .8,.6 -.5,0 SCAN TYPE <1»X,2-Y,3«R,4-A>: 2 1 STEP SIZE (relative field): .1 .05 CONTOUR VALUECvavcs): .1 .15 .05 -.05 0 .2 CONTOUR-1,CONTINUE-2: I 2 CONTINUE SCAN? 1 0 - 1.00

Fig. 7-47, Contour Plot of the Change in the RMS Wavefront Error in the Perturbed System with respect to a Flat, Best Focused Image Plane. 205

CHANGE IN RMS WAVEFRONT ERROR

(waves at X = .55 ym)

REAL RAY THIRD AND FIFTH ORDER THIRD ORDER

Fig. 7-48. Comparing Analytic and Real Ray Data for the Change in the RMS Wavefront Error in the Perturbed System for the Triplet. Y-Field Profile. 206

using real rays. Fig. 7-49, and coefficients (Eq. (F-2)), Fig. 7-50. In both figures, the spot diagrams were made at the best focal position for the field point;. On the right side of the figure are the correspond ing spot diagrams for the aligned systems.

For some systems, particularly those with large fifth order sur face contributions ( M O waves) , even the fifth order terms are not adequate to treat theaffects of perturbations. One such system is a

47° full field, f/3.5, five element double Gauss. Figure 7-51 shows a typical plot comparing analytic to real ray calculations for the image degradation due to perturbations for such a system. The general, lack of agreement indicates, as one might expect, that the seventh order terms are important here. 207

Fig. 7-49. Spot Diagrams Constructed from Real Ray Calculations. 208

Fig. 7-50. Spot Diagrams Constructed from Analytic Calculations. 209

CHANGE IN RMS WAVEFRONT ERROR (waves at X = .55 wm)

— T" 0.5

REAL RAY

ANALYTIC

Y-FIELD

-0.5

Fig. 7-51. Comparing Analytic and Real Ray Data for the Change in the RMS Wave front Error due to Perturbations in an F/3.5, 45° Full Field, Five Element Double Gauss. Y-Field Profile. CHAPTER 8

SUMMARY AND FUTURE CONSIDERATIONS

We have now accomplished the goal of this work: to develop a designer-oriented approach to analyzing the effects of small tilts and decenters on a rotationally.symmetric optical design. The emphasis has been on developing the theory in such a way as to provide insights into the effects of perturbations on specific aberration fields associated with the centered design. This emphasis led to the introduction of multiple order node plots as a means of quickly comprehending the effects of perturbations. The other major development was the expressions for evaluating the actual or the change in the rms wavefront error or the rms spot size. These expressions allow evaluating the image, quality over the entire field economically. By plotting contours of constant change in rms in the field, a complete summary picture of the system may be obtained. The combination of node plots and image degradation plots provides a powerful tool for studying the effects of perturbations to an optical design.

The major limitation of this approach is the accuracy of the rms calculations. By carrying the equations through fifth order in the wave aberration expansion many optical designs may be adequately modeled.

However, systems with appreciable seventh, ninth, and higher order terms show significant departure between the analytic calculations and real ray results. Even in these systems though, the qualitative behavior

210 : 2ii continues to be correct and the relative sensitivity of components to perturbations may still be studied.

At this time it does not appear to be practical to extend the

nodal interpretation beyond fifth order. However, it is a straight

forward task to extend the rms calculations to seventh and possibly

ninth order in the wave aberration expansion. The evaluation of the

resulting expressions would continue to be appreciably faster than real

ray calculations. The coefficients themselves could be calculated using

the proximate ray trace approach developed by Hopkins (1976).

The path taken to arrive at this method of analyzing perturbed

. i • systems was but one of many possible from the two basic roots presented

in Chapter 1. The concept of decentered aberration fields, when com

bined with the vector generalization of the wave aberration expansion,

provides a very powerful basis to begin many important problems. Here

we have dealt with only one of these problems.

Another important area that may be pursued from this basis is

the aberration fields in eccentric aperture systems from a design

perspective. In these systems, the pupil dependence in the wave

aberration expansion is expanded rather than the field dependence. When

this is done, one finds the same behavior for the aberration fields as

presented for tilted and decentered systems. The problem in this case

is somewhat simpler because the aperture projects at the same relative

displacement at each surface. The behavior of the design then is

independent of the surface contributions, depending only on the

aberration totals. ' • 212

The success of the above development leads one to believe that the behavior of fairly general optical systems consisting of sections of

rotationally symmetric pieces which may be tilted and decentered in the

design process may be found. The success and insight obtained in

dealing with perturbed systems also indicates that the vector approach

to aberration theory using squared, cubic, etc. vectors may be more

appropriate and useful than the standard scalar approach in describing

the behavior of a centered rotationally symmetric system during the

design process.

An important point in favor of this conclusion is the importance

found to be associated with the medial focal surface. This surface,

unlike the tangential, sagittal or Petzval surfaces, remains a continuous

surface in a perturbed system. It is an inherently more significant

surface than previously thought. Also, the need for replacing the

sagittal and tangential surfaces by. N and P focal surfaces is a

significant interpretation that follows directly from the vector

approach. These results indicate that there is an important difference

between a scalar and a vector development. With the knowledge of the

response of aberration fields to small perturbations, it is clear that

the vector approach is preferable.

The developments made here only provide techniques for analyzing

the effects of small tilts and decenters on an optical design. . This

generally represents only the first step in a tolerance analysis, which

is where this.would generally be applied. An important area that re

mains to be pursued is how this theory can be incorporated with

statistical analysis techniques to provide information on the allowable , . .213 tolerances for the components of an optical system. There are indications that significant progress can be made in this area. This is because all of the perturbation vectors are functions of the single parameter a.. This parameter is, in turn, directly a function of the tilt and 3 ■ ' . _ decenter parameters. Given the distribution functions for the tilt and decenter, one may be able to obtain fairly directly the dependence of the image degradation on the parameters controlling the distributions, eg., the mean, variance, etc. Such techniques would provide a valuable new tool for the tolerance analysis. APPENDIX A

VECTOR RELATIONS

Many of the properties found.for aberration fields in perturbed systems rely on a vector operation which is rarely used; vector multiplication. This appendix contains an introduction to vector multi plication and a summary of useful vector properties and identities.

Vector multiplication requires the use of an absolute coordinate system. Two vectors multiplied in one coordinate system give a different resultant vector than the same two vectors multiplied in a rotated coordinate system. When used to develop a vector aberration expansion, vector multiplication is an intermediate step. The resulting form of the expansion is independent of the coordinate system, even though some of the intermediate operations require an absolute coordinate system.

Introduction to Vector Multiplication

The development leading to Eq. (3-38) used an operation not nor-. mallyapplied to vectors, i.e., vector multiplication. Unlike vector dot products, which result in a scalar, and vector cross products, which give an orthogonal vector, vector multiplication results in a coplanar vector. This operation is best understood by writing vectors in phasor notation. Consider two general vectors A and B. These can be written in phasor notation as

ae1" f . be16.'

214 215 Multiplying these two vectors gives

AB = abe1^ + 3) . (A-l)

The product AB is a vector whose magnitude is the product of the magni tudes of A and B and whose orientation angle is the sum of the orienta tion angles for A and B. This operation is illustrated in Fig. A-l.

For the special case of A = B,

AA = A? = |A|2ei-a, (A-2) which is a squared vector of the type seen in Eq. (3-38).

Fig. A-l. Vector Multiplication. The difference between a vector dot product and vector multi plication may be seen in the following:

Dot Product: Let A = a i + ai = ae x 7

Then A-A = a2 = a^,2 + a 2 = | A |2 (A-3)

a scalar.

By contrast

Vector Multiplication: AA = A2 = (2axa^)i + (a^ 2 - ax2)j

= |A|2 ei2a (A-4)

a vector.

Similarly, •

Dot Product: With B = b i + b j = be1^ x Y

A-B = axbx + ayby = | A| | B| cos (a - g) '

(A-5)

Vector Multiplication: AB = (a b + a b )i + (a b - a b )j r y .x x y J y y x x J

A||B|eiCa + B). (A-6)

Summary of the Properties of Vector Multiplication, Squared and Cubic Vectors

Consider two arbitrary vectors, 217

Then, a) Dot Product: A-A = a2

A°B = ab cos (a - g) = a b + a b x x y y (A-7) b) Vector Multiplication: AB = a b e ^ a + ^ = (AB) i + (AB) j x y (AB)^ = absin(a + B) = ax^y + ay^x

(AB) = abcos (a + 3) = a b - a b y y y x x (A-8) c) Squared Vector: A2 = a2 ei2ct = (A2)^i + (A2)^j

(A2) = a2 sin2a = 2a a x x y

(A2)^ = a2 cos2a = a 2 - a^ 2 (A-9) d) Cubic Vector: A3 = a3 e^^a = (A3) i + (A3) j x y

(A3) = a3 sin3a = 3a 2a - a 3 x y x x

(A3) = a3 cos3a = a 3 - 3a 2a (A-10) y y x y v J

Vector Conjugates

In some cases, to. preserve pupil dependence, it is useful to introduce conjugate vectors. This operation simply implies the sign of the exponent is changed indicating a reflection of the vector about the y (vertical) axis. 218

If +ia ^ A = ae = a i + a j , x yJ 3 then the conjugate to A, denoted by A*, is

'A* = ae~la =- -axi + a^j. (A-ll)

When used with vector products, this gives

AB* = abel(-a " = (AB*)^ + (AB*)yj

(AB*)x = axby - aybx

(AB*)y = ayby + axbx . (A-12)

Vector Identities

a) 2(A-B)(A-C) = (A-A) (#-C) + A^-BC ' (A-13)

b) A°BC = AB*eC (A-14)

c) 2(A.B)(AB-C2) = (A-A)(B2 -C2) + (B-B) (A^C2) (A-15)

d) 2 ( A . f ) ( A 2 .C2) = (A.A)CAB.C2) + A3.BC2 (A-16)

Vector Operator for Obtaining the Gradient of the Wavefront

V[(Hn.£n)Cp.£jm ]- = 2inCHn .^n)(p.p)m-1p 219

Proof for Eq. (A-17),

Chain Rule:

v[0f.p")(p.p)^ [V(p'.p)m ]

+ rvfH^iKp-p ) 111

m v(p.F)m = [fe1 +17 j_(x2 + y2)

m-l, = m(x2 + y2) [2xi + 2yj]

^m-l-* 2m(p°p) p

n = 1

Hi+Hj = f 1 = nH11^ " 1)* A y n = 2 (i#)xV(2xy) + (H2)yV(y2 - x2)

= (H2)x [2yi + 2xj] + (H2) [2yj - 2xi] y

2[(H2)xy - (H2)yx]i + 2 [ (H2) yy + (H2)xx]j

iP-t* = ntfCp11"1)* n = 3 (H3) xV (3y2x - x3) + (H3) yV (y3 - 3x2y)

(t3)x[(3y2 - 3x2)i + 6xyj]

+ (H3)y [- 6xyi + 3(y2 - x2)j]

= 3[(H3)x ( P ) y - (H3)y (^2)x]i.

+ 3[(H3)y (p2)y + (H3)x^ 2)x]j APPENDIX B

PLOTS FOR LOCATING THE NODES

IN SYSTEMS WITH BINODAL ASTIGMATISM

This appendix includes a series of plots (shown in Figs. B-la through B-lj) which may be used to locate the nodes due to binodal astigmatism in misaligned or improperly supported Ritchey-Chrdtien telescopes. These plots are simply a set of hyperbolae where the spacing between foci is increased in relative increments of .1 between plots. The foci represent the nodes of the binodal astigmatic field and the hyperbolae give the orientation of the sagittal line images at any field point for that particular node spacing. By over laying a discrete set of orientation samples, i.e.. Fig. 5-8, one can find the appropriate node spacing by finding the closest match across the field.

220 221

Fig. B-l(a) 222

Fig. B-l(b) 223

Fig. B-l(c) 224

Fig. B-l(d). Fig. B-l(e). 226

Fig. B-l(f). 227

Fig. B-l(g) 228

Fig. B-l(h). 229

Fig. B-l(i). 250

Fig. B-l(j) APPENDIX C

DERIVING AND SUMMARIZING THE PROPERTIES OF THE WAVE

ABERRATION EXPANSION IN A PERTURBED OPTICAL SYSTEM

THROUGH FIFTH ORDER

This appendix provides the derivation of the equations used in

Chapters 3 and 6 to interpret the behavior of aberration terms in perturbed systems. Starting with the wave aberration expansion in a centered system the unnormalized and then the normalized expressions for the wave aberration expansion in a perturbed system will be found.

Also included are the expansions required to include the special cases of i = 0 and y = 0, discussed at the end of Chapter 2. A brief summary of the nodal properties of the aberration terms completes this appendix.

In a centered system the wave aberration expansion to fifth order in the image plane is

W = AW2oP2 + A W n H p cos + WquoP4 + w13lHp3 cos + W220h2P 2

+ W 222h 2P 2 cos2 + W3 1 iH3p cos# + W q s o P 6 + w 1 5 1 h P 5 cos#

+ W24oH2pI* + W242H2p^ cos2# + W33iH3p 3cos#

+ W333H3p3 COS3# + W42oHlfp 2 + W4 22Http2 cos2#

+ W 5 1 iH5p cos# + Wqsop8 , (C-l)

231 232 where seventh order spherical aberration is .included as it is generally available from a lens design program. This can be written as a sum of surface contributions. The effect of tilts and decentrations is to displace the center of symmetry for each aberration contribution to the point located by the vector cy in the image plane. The resulting, expression for the wave aberration expansion to fifth order in vector notation is

w = aw2o (p°p) + A W n (il-p) + l_ Wo4o4 (p-p)2 j 3

+ I Wl 31 [ (H-0 .) °p] (p°p) + I W22 0m [ (H-o .) * (H-a .) ] (p° p) j 3 J j j . J 3

+ 4 I W 2 22.[(H-a.)2 °p2] + I W 311.[(H-a.)'(H-a.)][(H-a.)«p] J J j: J J J J

+ I W06O4 (P°P)3 + I Wisi.[(H-a )«p] (p-p)2 3 3 j 3 3

+ 1 w 2kQM I (H-a .)» (B-o..) ]

+ 7 I W242 . [(H-a.)2 »p2] (p-p) - j J 3

+ I W 33!Mj [CH-oj)•(H-0j)][(H-Oj)°p](p°p)

+ jl W33 3j [(H-aj)3»p3] + I W 420M _ [ (H-a^. ) • (H-Oj ) ] 2'(p-p)

--Continued . 233 + i I W422i e (H-^,) ] [ (H-aJ 2°P^] j 3 3 J 3

+ I W g n [(H-C.)e (H-0 . ) ] 2 [(S-a.)«p] + % Wo80-j (P°P)4V . j 3 3 3 j ... (C-2), where through the use of trigonometric identities involving cos2# and cos3# 1 3 W 220m = W 220 J W222 W 331m = % 3 1 ^ % % 3 3

1 t (C-3,

w 240m = W240 + J W242 ^420^ 9 %20 * 7^422 have been defined.

Before performing the summations to obtain an expression that depends only on image plane quantities, a set of normalized and un normalized image plane perturbation vectors needs to be defined. These are the vectors discussed at the beginning of Chapter 6 . 234

'ti. ■ Z"ki^

klm ^klm^klm ~ aklm° aklm

F 2 B2vimklm = - K4 klm. j b2klm = B2klm/Wklm ' ^klm 3 J

Cklm " Sln/Slm ~ ^aklm°aklflPaklm

e3 C3 /W - a3 ?3klm s I S l m / j klm u klm klm a klm J J

^klm ~ Sln/Slm ” ^Slm’Slm^

^ k l m = iSlrn^ (0jeaj)aj2 d2klm = D\lm/Wklm * laklm° Slm^^klm 3 3

Sim = |Slmi(aj 0 J ) 3 S i m S l m ^ S l m ^aklm° aklnP aklm J J 235

In the derivations that follow, a letter in parentheses appear

ing on the left of an equation refers to the vector identity used from

Appendix A.

Wu4 0 ;. w = I ^040. (pep )2

" W 040 Cp°p)2 (C-4)

W l 3 1 : W = I WX3i .[(H-o.)ep]Cp6?)

; / • 3 ,3 . J

= r w131j (6»p)(p«p) - (i Wi3ij^j°p)(rp) .

= w1 3 1 (S»p)(p=p) - (A131-p)(p-p)

= ICW131S - ^i3i)°p3Cp°p) (C-5)

W220m: W = I W220M-[(H-Oj)‘(H-^j)3(PeP)

= W22 oM (M) - 2H^I W220M^ j) + I W220M _(aj-aj)j(p«p)

= [ W22 0m (^°S) - 2 (S'>^220m) + b22 0m] (P°P) (C-6 )

,2 ^ 2 - W222: W = 4 I W222,[(H-oJ2e'P2] 3 1 J . *

- I [(| "mjS2 - 2fl | ♦ I itizy,2).?2]

= -j [W222®2 ■ 26^222 + ^ 22 2 ] eP2 CC-7) 236

W s r i : W = I W 311.[(H-a ) • (H-ct )] [(H-a )-p] j J J J J

= I W3iij (H“H) (H-p) - Wdllja^ (H«p) J \ 3

+ (I Wsilj (o yO j)) (H-p)

- (H°H) W 311 jO j J - p + 2 I W311j(H"Oj)(Oj-p)

- w311j (aj-Cj)c^-p

(a) 2 I W311j(H°o\)(Oj-p) = W31lj(Cj"0j)^(A-p)

* ( I Ws i i j S j 2)-?!?

(b) WsiijOj^-Hp = WaUjOj^fl-'p

w = [ W31y(H-H)H - 2(H-A311)H + 2B311H - (%-%)A3ii + ^ 311%*

C 3II ] °P (C-8)

W 06 0 : w = I W 060 • ( p - p)3 j J

= W q s q ( p ^ p )3 (G-9)

Wisi: W '= I Wisi [(H-a )-p](p«p)2 j J 3

= t 15iH-Aj 51) °p](p-p ) 2 [see W131] (C-10)

WZ4 0M : W = I W 24om [(H-a )•(H-a.)](p-p)2 j j J J

= [W240m (H-H) - 2(H-A24um) + b240m ](P’P )2

[see W22 om] (C-ll) 237

Wait2 : w = y I W2.4 2 . [(H-o-.)2”?2] Cp-p)

1 v 2 y [ (W242H^, - 2^242 + ^ 242) ° P 2 ] (P °P) [See W 222] (C-12)

w 331m : w = I W 33i.NL [(H-a.)* (H-a.)] [(H-0.)-p] (p«p)

= ([W331M CH»H)S - 2(H‘A331m)H > 2B33 1 ^ - (H»H)%d3lM

+ B3 31 ^ * - C33i.M ]*p) (p*pX [See W311] (C-13)

W 333: W = j I W 333 [(H-a ) 3»p3] j J J

" W333j»3 - + 3H^ - I Wa 33 -j'-p

= j [ W333H 3 - 3 H2A 333+ 3HB 333 - ^ 333] ° p 3 ( C - 14)

W420M : W = I W 4 20M _[(H-a )•(H-a }] 2Cp-p)

/. W 420,, [ (H °H - 2H °a . + a .• a .) (H-H - 2 H - d . + a .«a .) ] (p°p) l j J r 3 J J ^ ^

I W ^ O m (S«H) (H-H) - 4(H-H)( h -,/I W420,. a. M. j ■ BJ 3 3 ' -> -t- + 2(H-H) (I Wh20m _ Ca j ° CTj )J + 4 I WH20M CH-d.)(H-a.)

- 4H° W ^ Ca^apa.J + % Wk20M Co.'o.)(o.‘o.) Cp*p)

(a) 4 I W 4 2 0m (H* a ) (H° a ) 2(H-H)^i Wu20M _(dj-dj) . j j 3 3

* 2 p - ( i w»20h_;.=) 238

W = [W42 0MC^H) (H-H) - 4(H-FI)(fl-A420M) + 4B420M(H‘M H)

+ 2 (H2«B42om) - 4 (H»C4 20m) + D4 2 0m1 (P°P) (C-15).

W422: W = j I W422j [ (^-^j)' C^-^j) ] [ (H-Oj)2°P2] 3

= W 422-: [(H»H - 2(H»a.) + a -a ) (H2 - 2 (Sa ) + a 2) ] «p 2 £ 4 3 3 3 J 3 3 3

= Y W422j (H-Hj (H2-p 2; - 2 ni-HJWtt22jO jjj -p2^

* (S-a)((i w, 2 2 .s .2)-5 2) - 2(a-(i w ^ajSj))^2-?2)

W 422 • (H°a.) ( S a . - p 2 ) w 4 22-CS-a.) (a.2« p 2 ) + 4 Ij 3 3 3 “ 2 I j 3 3 3

+ 1^1 W 4 2 2 j (H2- p 2 ) - 2H ^ W422j °a j ^ CTj)°p2

+ W422j ( o ^ a ^ a ^ y p 2

+ 2 ( H - H ) ^ W1+22joj2)-p2

(d) -2 % W422 - (H-a.) (ct.2«p2) = -Hfc W422 • (o. ‘"a .)a . j-p 2 j J J \jJ 3 3 3 3)

- ( I W422jCTjd j-Hp2

(b) ■(l W422jaj3y 5p2 = -(l W422jOj3^S*'p2

W = 4 [W422(H»H)ii2 - 2(H-H)HA4 22 + 3(H-H)B422 - 2(H-A422)H2

-^3 -5- ->2 rr2>T-^> ->2 -> - C422H* + 3B422^Z - 3HC422 + 6422 ]eP2 239

Wsn :

- (H«H) ^ Wsll jOjj'P) + 4 (H?H) ^ W 511l (H-a^) (a,. »p)

+ 4 I w511^ (H-a.) (o.-a.) (c.-p) j J J. J J J

- W511 j (Gj«aj) (CTj»aj)a^p [See Wit20M]

4 I Wsiij (H-a^) ( H ^ K H - p ) = 2 ( H - H ) ^ W 511 j (a j - ^ ) j (H-p)

(a) + 2{ P ^ l WsilfjVJ&p)

(a) 4 (H-H)^ WgiijCH-a.) (a^plj = 2( H . H ) ^ W 511 j (a^.• a .)j (6-p)

+ 2 W 511 jaj2^ °Hp)

(b)

(a) -4 I W 511 j (H'Oj) (H-Oj) (oyp) = -2 ( H - H ) ^ W 511 ^ (a^ • a^. )ct^)

•2 I W 511.(H2-a 2)(a -p) j J J J (d) -2 I w5nj (H2-5.2) (?..?) , WSHj (I-?,)!)?)

' (l Wslljl^.H2?

“**0 (V -*» \ * —i» " = -H^n W511j(Cj°Gj)gJ °p - (I Wsl^H^.p

(a) 4 % Wsii. CH‘CT ) (a.*0.) (a.*p) = 2[l Wsil. (G."a ) «a )) (H*p) j J J J J J \j J J J J J /

/ r5 z)\ -»-5» + 2(1 Wsilj(Oj'CjjOj y*Hp

a \ "4=4° Zp» *4* -4= /A -4= -4- (b) 2(% W511j(Gj*0\)Cj2j»Hp = 2^1 W511j(Cj«Cj)Cj2|H*°p

w = [w5ii(S*li)CH*li)H - 4(S»'S)(6«A5ii)S + 6b 5ii (6-1)6

+ 2(S2»B5ii)S - 4 ( M 5 U ^ > 3D5H%

(%»%) (H-H)A511 * 2 (H«H)B51 iH* - 4(H=H)C5i 1

M in - Csr.H2* * 2t5nfi* - l 5U]-P (C-17)

Wq-s o : W = 1 Woso,- CP6?)'

j 3 ■

:= W08 0 (P*P)4 ' (C-18) - .241

The complete expression for the wave aberration expansion includ ing the effects of tilts and decenters in terras of image plane quanti ties then is

W = AW2o (P'P) + AWnC^-p) + Wo4o(p°p) 2 + [(Wi3ifi - 31) *p] (p*p)

+ [W220m (H°H) - 2(H-A220m) + B220m ](P°P)

+ "^"[^222 H2-2HA222 + B222] °P2

+ " [W311( M ) f i ' - 2 ( M 31i)fi + 2B311fi - ( S - t o s n

+ B311H* - C311]•p

+ W o s o C p - p )3 + [ ( ^151H - A i 51) o p ] ( p * p )2

+ [w24 0m (H’H) - 2(H«A24 0m) + b240M]CP"?)2

.. + j I(W242H2 - 2 E 242 + B242)-P2 ] (P-P)

+ ( [ W331m (H-H)H - 2(H-A331m)H + 2B33 1 ^ - (H-6)^331M

+ B33i^H* - C33iM ]-p)(p°p)

+ 3 3 3 H3 - 3H A 333 + 3HB333 - Cgsgj'p"

+ [W420M (H-H) (H-H) - 4(H-H)(H-A420M) + 4B420M (^^)

+ 2(H2-B420m) - 4 (H«C42om) + d420m K P <‘P)

-Continued 242

+ y[W422(H«H)H2 - 2(SH)HA422 + 3 ( M ) B 4 22 > 2 C^ ^ 22)H2

- C4 22^* + 3B4 22H2 - 3HC422 + D422],,p2

+ [WgnCH-H) (H*H)Fi - 4 { M ) C^ A 5ii)li+ 6B511 (M)fi -

» 2Cfl2.l5u)S - 4(6^511)# ■ ■■' ■'■■■ , .

+ 3D511H - (H=H) (H«H)A5n + 2 (H°H)1$5n 1i* - 4(H^)^511

- H2Clil - ^lilH2* + 2 ^ 1> - l5ll]»P

* WodoCp-p)4 - (c-19)

In an effort to simplify the notation for Appendix E, the following definitions will be made,

[ ] 131 * W131S - ^131 (C-20)

C ]220m = W220M§«i - 2d«t220M * B220M (021)

[^222 = W222S2 - 2^222 + ^222 (C-22)

C h n = ^ 3 1 1 # H ) % - 2 ( S a 3ii)h 4. 2B3iil - ( m h n

♦ tlnfi* - t a n ' ' CC-23)

C^llSl = #151 H - A151 (C-24)

[ ]240m = #240M ^°H - 2 M 240m + B240m (C-25)

["*"] 242 = #24 2 S2 - 2^242 + B242 (C-26) 243

[ ]33Ijyj W331M(M )H - 2 (5 ^ 3 iM)3 + 2B331^ - (M )A 331 m

- «3»h ( C " 2 7 ) r-N ^ E J 3 3 3 W 333H 3 - 3 % 3 3 ♦ i $ i 33 - ^ 3 3 ( C - 2 8 )

E 3420m

+ 2(H2«'B420m) - 4(H«C420m) ■+ D420m (C-29)

2 f ] 4 2 2 Wi+22(H»H)H2 - 2 (H°H)HAi+22 + 3(H”H)"B422 “ 2(H=A422)H2

- C422H* '+■ 3B422H “ 3H5422 * "6422 (C” 30)

C l s u = WsnCH-H) (H-H)H - 4 C M ) C M 5 U ) H - 6B 511 C H ° H ) H

+ 2 ( S 2« I 1 h ) S - 4 ^ ^ 511) % * 5D 511H

+' - 4Ct”S)^5U

h 2c1h - clnil2* + 2D5UH* - Esir (c-3i)

/ 244

The special cases which arise when the centerof curvature of a surface lies at a pupil or an aspheric is at a pupil are treated by expanding the pupil dependence of the spherical aberration contributions of these surfaces. This gives,

w040 : W “ W040 t(p -1- Ap)«(p + 2p) ] 2

= Wo4ofCPeP)2 * 4 (pop)(p-ap) + 2(p.p)(^p«Ap)

+ 4(p«%p) * 4 (p»^p)(^p»lp) + (lp»Ap)2]

= Wo4o t (PeP)2 + 4(p«p) (p«Ap) + 4(p-p) (I'polp)

+ 2(p2 «Ap2) + 4(p-Ap)(Ap«Ap) + (Ap»Ap)2] (C-32)

#060= . W = WoeotCP + ^p)'Cp + ^P>]3

= W0g0[(p»p)3 + 6 (p°p)2 (p°Ap) + 3(p•p)2 (Ap °Ap)

+ 12Cp->p) (p«2p)2 + 12 (p-p) (p«^p) (^p«2p)

+ 3(pop)(Ap°Ap)2 + 8(poAp)3 + 12(p«Ap)2 (Zp°Zp)

+ 6(pVAp)(Zp«Ip)2 + (Zp«Ip)3] .

= #060[(P0P)3 + 6(p=p)2 (p«^p) + 9 (p°p)2 (Ip °Xp)

+ 6(p°p)(p2»tp2) + 18(p«p)(p»lp)(Ap-Ap)

+ 9(p°p)(ApoAp)2 + 2 (p3 °Ap3)

+ 6(p2oAp2)(Ap-Ap) + 6(p«Ap)(Ap-Ap)2

+ (Ip'Ap)2]. . CC-33) 245

These terms result in the following modifications to the image plane perturbation vectors

^131 " A131 '* I4W040j^Pj (C-34)

A 151 = A151 * I 6W060 jAP j (£-35) j

B220M = B220m * 44Wo40jCApj«Apj) (C-36) 3

B240M = B240M + I9W060 j^AP j°AP (C-37) j

, - 2 ^222 " % 2 2 * I2W040iAP j 2 (£-38)

j : ■

B242 = B242 + ^6W060jAPj 2 1 (C-39) -* .

^311 ™ ^311 + I4W040j^APj°APj^APj (G-40) 3

^331M = ^331M ^ (C-41)

£333 " £333 * I^WggQ .Ap .3 (C-42) j

D420M = D420M + I9W060jCApj«Apj)2 (C-43)

D422 = D422 + I6W060 j CAPj °APj)APj2 (044) - - - 3 - . :

E511 = E511 + I6W060j^Apj',Apj)2Apj . (£-45) 3 246

The technique of obtaining the normalized forms which appear in

Chapter 6 is to replace 9 by 9 - and factor out the coefficient.

The resulting normalized form is then written in a form which is con venient for nodal interpretation.

The following expansions are useful for following the replacement of fiby9- tklm

a) It - a = ft - a (C-46)

b) (9 - a)e(H - a) = H cH - 2Hea + a°a (C-47)

c) (9 - a )2 = fi2 - 2fia + a2 (G-48)

d) [(9 - a) o (9 . a)] (9 - a) = [9c9)9 - 2(9*a)9

4- 2 (a°a)H ^ (HeH)a a ^ * - (a°a)a (C-49)

e) (9 - a) 3 = 93 _ 3g2I + 39a2 ,_ a^ (C-50)

£} [ (H ”> a) ® (H - a) ]2 = (9® H) (H®9) - 4 (H° H) (H° a)

. + 4 (H®H) (a® a) + 2 . ( 9 2 e a2) - 4 ( 9 ° a) (a®a)

+ (a-a)(a-a) (C-Sl)

g) [(9 - a) = (9 - a) 1(8 . t)2 = (9-9)9 2 - 2(9-9)9a „

* 3(9.9) a2 - 2C9.a)92 -

+ 3(a°a)92 - 3 (9.a)(a«a) + (a-a)a2. (C-52) 247

The normalized forms then are found to be,

*040: ' . * *040

*131; * = [(*131^ - ^13l)°P](PeP)

= *l3l(^13.1<’P) (P°P) (054)

*220m : W = [*220^^°^ ~ 2(Ho"^220^ ^ B220m 3 (PeP)

= [*220M(^220MeS220M) + B220M " *220M(a220M*^220M) 1 (P‘P)

* = *220m [ [H220M'’H220jyi^ + ^220^3 ^P °P3 (C-55)

*222* * ~ ~2 [*222^^ - 21^222 + ^2223 CP^

1 i4-2 _».2 ->2 -»»« = Y [*222^222 + “ 222 " *222a2223"P

2 2 * = J *222^222 + ^222] "P2 ■ (C-56)

*311: * = [*3il (M )ff “ 2(S-X31i)S- + 2B311^ - (S-S)t311 +

+ ^311^* ~ ^3113"P

s [*311 (^311 °^31111 + 2B3116 - 2W 311 (agn-agiijH + $31 lS*

- WgnagiiS* - ^311 + W31l(a3ii°a3ii)a3ii]

= : *31l[$3110^31l)^311 + b311^*311 + 2b31 iff

- (C3II ~ ^31l3-*3il) 1 eP

K - "aVi ftflan * ^n ifraill-P

+ *3ii[2b3ii^ - (C311 - b 31 1a*3i1)]<'p (C-57) Wi;ica;5i-p)(p-p)2 [see Wl3l] CG-59)

[W240M (^H> - 2(H‘A240M) - B240m]

W240M [(^240m*^240m) + b240M ] (P’P )2 (C-60)

[see W220m ]

y {[W242H2 - 2l S 242 + ^242]°P2 }(P ° P )

| W242{[6242 + ^242]-P2}(P-p) (C-61)

[see W222 1

{[W331m (6 . M - 2 ( M 3 3lM)ft + 2B331^ - (S.S)^331m

+ ^331^* - ^33 *?} (P’P)

W331m {[^331M + ^ 3 1 M)^331M]°P}(P‘P)

+ W331M^ ™ (c 331M “ b331Ma*33iM^ ”P ^ P “P^ ------(C-62)

[see tV311 ]

J [w333^3 - 3H2A333 + 3iS333 - ^ 333] "P3

j [#333^333 + 5® 3 33 - ^333^333 " ^333 + W3 3 3a33 31 ’P3

1 ->3 ->2 _>3 ->2. ■j W333 [H333 + 3H333b333 - (C333 - 3b 3 3 3a-3 3 3) ] 'P ( C - 63) 249

Wu20M: W';. = [W420M(S .6 )(d.S) ■- 4 (M )c5 -X f20M) + 4Blt 2 oMc M )

2 + 2 (H 2 »$420m) - .4 20m) + D4 20m] (p -p )

v = [^420MCH420M‘H420M) CH420M°H420M) + 4 B 420MCH»H)

“ 4W420M (a420M °a>420M) CH'^) + 2 (62«B420m) •

“ 2W42oM(S 2 -a 4 20M)

- 4(H«'?42om) + 4W420M(S -a 4 2 0M) ( l 4 2 0M-a 4 20M) + D4 20M

" W420MCa420M*a420M)Ca420M"a420M) 1(P"P)

= W 4 2 0 m I ( H 4 2 0 m oH 420m) (H 420 m °H 420m) + 2 (H 420M eb420M) '

• , + 4b4 2 0M(S °H) - 4 (H °c 4 2 0 m ) + d420M

+ 4 (Ha42oMeb4 2oM) - 2a4 20M,b420MK p*p)

2 2 2 W - = W420m [ (H42 om + 2b4 2oM) ”H420M] (P 'P )

+ W420M [4b420M CH°H) - 2H«(2c420M - 2b4 20Ma4 20M)

2 2 + (d420M - 2a4 2oM4 4 2 QM)](p°p) (C-64)

W4 2 2 : W = j [W422(S -5 )S 2 - 2(S-S)3A422 + 3(5*5)1^22 - 2 (6 -A 422) 6 2

3 v 2 - ^ 422^ * + 38422^ - 3l5?422 + ^4 22 ] °P^ 250

= J [^422 (^422°%22)^422 + 3-Cff*3)Sj22 " 3W422 (S»S) a422

3 3 - ^422^* + W422a422H* + 3B422H 2 - 3W422(a422°a422)f^2

35^422 + 3W422Ha422(a422°a422) + D422

W422(a422°a422)a422 J"P2

.3 .* Y W422,[H[t22H1+22 + 3H1+22bit22H422 - c422H42;

+ 3b422H 2 - 3H c 422 + 5 ll22 + 3H*a422b422

• n2 + 3Ha422bi+22 - 3a422a422b 4 22 - c422a422^'P

1 -> ->z3 _>3 >>2 -y* W = j ^422^ [H422 + 3H422b422 ” (c422 “ 22b422) ] H422^ °P

1 ~$~n -> ->* ->-2 + J W422{3b422H - 3H(C422 - 9422^422)

r“^2. ->* _ ^ “>"2 + [a4 2 2 ~ c422a422JJ"P . (C-65)

Finally, separate out the lower order terms generated by the higher order aberration and redefine the lower order perturbation vectors to account for these terms.

W131 + W331M:

W = W131[(it- aisi)'?] (P-P)

+ W331M{[,(^331M + bssi^HsSl^l’PHp-p)

+ w 331MU 2 b 33lN1H - (c 331m - b33lMa33iM )]”p}(p«p). 251

Define

W 1 3 1 e “ W 131 + 2W3 31M b 3'31M ' (C -66)

^ 1 3 1 + W 331m ( c331 m - ? 33iM a 33l M ) (C-67) al 3 1E Wl31E

Then,

W = . W 1 3 1 E ( ^ 1 3 1 E °P) (P*P) + W 3 3 1m U ( S 3 3 1 m + S’331M)^331M ] "PKP'P)

(C-68)

W22 0M + W42 0M :

W = ...... W220m [ (H220MA °H220M) * + b220MKp*P)

- ^ . 2. ^ 2 _y2 + lV420M [(H420M + 2b420M) °H420M ] (P*P)

2 + W42 0M [4b420M (ti,H) 2H- (2c420m " 2b42oMa42 0M)

^2 ^2 \ ^ + Cd420M - 2a420M °b420M)1(P*P).

Define

K 22 0xIE " W 22 0m + 4W42 0>Ib42 0M (C-69)

-x. -> ->2 -j. A 2 2 0 M + w420m (2c42om- 2b42oMa420M) a220^ = ' ” ' ' ' Cc-/0) 220ME'

B22 0ME = B220M + W42 0M Cd420M “ 2a420M *b420 ) (C-71)

b220^.jE b 220j^/^_ :. . - a22 0m-°a9 9n,^- . . 72) 220ME &22°ME a220ME 252

Then

W " W22 0N1E t (H22 0N1E ^220^) + b220NEKp'P)

+ W420m [ ( H420m + b420M)°^42 0m ]CP’P) (C-73)

j W222

W = — W222 (H222 + b 222)°P 2

+ y ^422"t [H422 + 3H4 22b4 22 ~ (c422 " 3a422b 422) ]Hi+22}°P2

+ 2" ^ 422(35422^2 - 2H(— C422 - - j b422a422)

r^ 2 ->3 ->* . ->2 + [Q422 - c422a422J(°P *

Define _»3' _>3 ^ 2 c422 = c422 - 35422^422 (C-74)

^222% = #222 + 3W4 22b422 (C-75)

-> 3 ->■ ->2. A 222 + T #4 22 (c4 22 ~ b 422a422) a2 22p = : (C - 76) #222p

_v2 ^.2 ^.2 ^3 _»* B222g = b222 + #422(a422 - c422a422)

(C-77)

^ 2 -v2 ->2 b222E - B222e/W222e " a222E • (C-78) 253

Then

IV = i W222p[ (S222F + ^222E) «P2]

+ y W4 2 2 [(H422 + 3H422b422 ■- c422)^422°P^]• (C-79)

The final forms for the nodal interpretation presented in

Chapters 3 and 6 .then are

Spherical Aberration:

IV = Wo40 ( ^ p )2 + W060 ( H ) 3 + IV080(P-P)4 (C-80)

Coma: a) IV31: W = Wi3ig(^i3ig-p)(p°p)

2 2 * + IV331mU (H33iM + b 33lM)H331.M]-p} (p-p) . (C-81)

b) W51 : W = WiSlfilSl'&iV-V)2 (C-82).

? 1 ->3 ' ->2 ->3 _>0 c) W33: W = -r W 333 [(H333 ■ + 3H3 3 3b 3 3 3 - C3 3 3) °p ] (C-83)

Medial Surface:

IV = AW2o(p°p)r + . IV220im[CH220»rF,H220MT:)t., ^ E u_n22 0iNIE,n 22 0ilEl ++ b2^ 222 0,vfnH 0 ^- p”P)

2 2 2 + IV420M UH420M + b420M) ^420;^] (P”P) (C-84)

Astigmatism: 1 _^2 _^2 IV = j IV222E [ (H2 2 2 e + b222E) ‘P2

3 v ^2 -4.3 v* + j W422 [ (^422. + 3Hu22b422 - ^422)^^422 "P2 1 (C-85.) Oblique Spherical Aberration: a) W2 4 om : W = W240M [C^240M oS240M) + b240M ](PeP)2 (C-8 6 ) b) #242: # - y#242[(H242 + ^242) °P2] • (C-87)

Finally, a brief summary of the nodal properties is included along with illustrations in Figs. C-l to C-3.

Spherical Aberration:

Unaffected by tilts and decenters

Coma:

a) W3 1 : Term linear in field is zero at aigig.

Term cubic in field is zero at three collinear points in

the field S = a33lM + ii) 3 3 lM , _ 8 = a3 3 i^ - iS33lM,

S = a33lM -

b) #5 1 : Zero at 8 = ai5i

c) #33: Zeroes at three points in the field." These three points

are found by using a vector generalization of standard

algebraic solution to cubic equations.

Medial Surface:

Term square in field is displaced to and defocused propor

tional to b2 2 0 MEv

Term quartic in field is displaced to 2 4 2 Oj^ and develops addi

tional zeroes at 8 = a4 20^ + 1 8 4 2 oM > ^ ~ a420M - i&420M - / 255

Astigmatism:

Term squared in field is zero at H = a2 2 2 g * ^ 2 2 2 5 and

it = a222g ~ i^22 2 g “ Term quartic in field is zero at four points in the field.

One of these points is H = ai+2 2 * the other three are found by

solving the vector generalization of the standard algebraic

solution to cubic equations.

Oblique Spherical Aberration:

a) W2£toM: This surface is displaced to a2 4 oME and defocused

proportional to b240{yjg-

b) *242" This term is zero at H * a2 42 + 162 4 2 and

it = a242 - ib242 • 256

Fig. C-l. Nodal Properties of the Comatic Aberrations in a Perturbed System.

(a) Wk3i : Linear plus field cubed coma. 257

W5i Coma

Fig. C-l .--Continued Nodal Properties of the Comatic Aberrations in a Perturbed System

(b) W51: Fifth order coma. 258

X 333

Fig. C-l.--Continued Nodal Properties of the Comatic Aberrations in a Perturbed System

Media 1 Surface

420

Fig. C- The Nodal Properties of the Medial Focal Surface and Astigmatism in a Perturbed System.

(a) U'K2 0xj: The medial focal surface. 260

22 As t igmatism

'222 E

222 E

Fig. C-2.--Continued The Nodal Properties of the Medial Focal Surface and Astigmatism in a Perturbed System.

(b) IV 22= Astigmatism with respect to the medial surface. 261

M Spher ica >240,

240

Fig. C 5. The Nodal Properties of Oblique Spherical Aberration in a Perturbed System.

(a) W2i+0jq: The medial surface for oblique spherical aberration. 262

f242 Ob 1 i que Spherica1

242

Fig. C-3.--Continued The Nodal Properties of Oblique Spherical Aberration in a Perturbed System.

(b) W242• Oblique spherical aberration relative to the medial surface for oblique spherical aberration. APPENDIX D

VECTOR GENERALIZATION OF THE ALGEBRAIC SOLUTION TO CUBIC EQUATIONS

Algebraic Solution:

Given

x3 + ax + b = 0, CD-I)

Tet

V i j V 4 B b Zb2 a3 \ 72K & - - H r * 2 \ 4 27/ j

•(D-2) Then the zeroes occur at

A + B A - B A + B A - B x = A + B, v d , - / -3 .

CD-3)

Vector Generalization:

0 , CD-4) let

R = lA W s

(D-5)

_s. S # 2 . ( S Y

263 264 and

_a> —^ R + S x = CD-6)

'v R - S x = . (D- 7) 2

Then the zeroes occur at

H = 2x , -x + i/3~x , -x - i/T'x. (D-8) APPENDIX E

DETERMINING THE ORIENTATION OF

ELLIPTICAL COMA

Elliptical coma, W 3 3 3 , maps the circular zone mappings for . linear plus field cubed coma to ellipses. It is important to be able to find the orientation of these ellipses. This is done by finding the absolute angle Y, for the orientation of the major axis of the ellipse.

To find the angle Y one must consider the various components of the transverse aberration for linear plus field cubed coma and elliptical coma. The transverse aberration is given by,

1 2 vto = Cp-p)([ lis i + C 2[([ ] 131 + [ 333iM)°p]p 3 3 3 + J [ 1333P2* (E-l)

= M 3 ip2e^® 31 + 2M 31P2 cos(931 - (j))e^

+ | M 3 3P2el( 3 6 3 3 ' ; (E-2)

Where

p": = pe1

[ 1131 + [ 1131 = M 3 i e ^ 031 (E-4)

A 333 = Msse13033 . (E-5)

These three components are illustrated in Fig. E-l. The first component gives the displacement of the circular zone and is independent of .

265 266

The second component traces out the circular zone and the third component maps the circle to an ellipse.

The orientation of the major axis of the ellipse is found when component three, when extended, passes through the center of the circle.

The condition, from Fig.* E-l, is

Y() = 3033 - 2. (E-6 ) I /

S'I / 36333 " Z*

Fig. E-l. Finding the Orientation of Elliptical Coma. 267

From the figure we can find,

B = 831 -

26 + 'jl = 180 (E-8)

i|j + a = 180 • (E-9)

Y + a = 831 (E-10) which gives

.Y + 2(831 - = 831 (E-ll)

Y = 2

Solving for

2 - 631 = 3833 - 2 (E-13)

=■ i [3633 + 631] • (E-14)

The major axis of the elliptical mapping of the circular zones, then, is oriented at the angle y where,

Y = | (3833 - 631). (E-15) APPENDIX F

EXTENDING THE EXPRESSIONS FOR ANALYZING PERTURBED SYSTEMS

TO FIFTH ORDER IN THE WAVE ABERRATION EXPANSION

This appendix contains the expressions for the wave aberration

expansion, transverse aberration, rms wavefront and rms spot size for

a perturbed optical including terms through fifth order in the wave aberrationexpansion. The notation is defined in Appendix C. The

expressions are obtained using the techniques presented in Chapter 4.

The wave aberration expansion can be written as

W = AW2o (pV p) + AWi i (H«"p) + W q ^ o Cp^ p )2

+ ([ 1131 °p ) (P °P) + [ + jt ]2 2 2 2 ° P ^

+ [ ]311°P

+ w 060 (P°P)3 + (t ]l51“P) (P°P)2

+ [ ]240[^(P*P)2 + y ([ ]Z2h2°t2) (p°p)

+ ([ 1 SSljvi’P) (P°P) + j l ] 3333eP 3

+ [ ]420m^p °p^ * ]2b22°Jp2

+ [ Isii'P

* W080(?-P)‘*- CF' 1:

268 269 From this, the expression for the transverse aberration is found to be,

n'u'e = VW = 2AW2 oP + ]H + 4Wq 4 o (p°p)p

+ 2([ ] i 3i °p)p + (p °p) [ ] i 31 + 2[ ]220^P

+ t 12222P* + I 1311+ 6W060 (P 0P^2P

+ 4 ([ ]151*p)(p ep)p + (p°p)2 [ ]151

+ '4[ ]240m ^ ° p-*P + Cl ]22420P2) p + Cp°p)[ ]2242P*

+ 2([ ] 331m °P^P + Cp""^ [ ]331M + ft ] 3333 Cp *)2

+ 2[ ]42 0mp + t 12422P* + [ Is il + 8W0 8o(P°P)3P » . . . (F-2)

[see Eq. (4-11) ] 270

The equation for calculating the actual rms wavefront error in a perturbed system is

“rms = ■C(ij2} 2 = {ttD^ZO + W04 0 + fo" W060 + *0 80^ '

+ [ l22 0M +. [ ]240m + t ]420IJ2

+ m [(*0*0 + J w 060 + y - W 08o) * [ ]240rJ 2

+ lioa [Wo6° + 2Wo8oJ2

+ 44100 [W080J 2

+ [([ ]2222 + -ft ]2242 + [ ]2422) 0 ([ ]2222

+ j l 12242 + [ 32422^]

+ 7 E(AWllH + ^ -*131 + ] 151 + [ 3 311 + §[ 1 331M

+ rtsll) °(AW 1.1H + §[ ] 131 + j l 3151 + [ 1311

+ | H 331m . + M 51 1)].

+ J 2 \_(j- h s i + f-t 1151 + [ ] 331M^ 0(t ] 131

■’•frilSl + M 3 3 1 M)]

+ iiio l["l l 5 V " ] 15j

♦'lis [^®333-rt33 3 - 3 } } % . CF-3)

[see Eq. 4-37] 271

The corresponding expression for calculating the actual rms spot size in a perturbed system is.

n'u'e o qi o rms •^2 £^ A W 2 + Y W + + Y #060 + I" #080^ + t 3220 M

l240M + [ H 2 0 M]2

Kt'#040 + f #0 60 + |^#080 )0 + [

+ & CW06 0 '|+ T ~ W 0 8o ] :

1225.. M 2 1 / -^ 2. -a. -2 ^ \ / _&, 2 2 + 2 ]222 +■[ 3.242 + [ 3422/°\I 3222 + [ 3242

+ M 42 2 )

-2b. 2 2 + I 1 1242°[ ]242

+ ^ ( a W h ^ + TjiSl + riisi + 1^3 311 + [^3 331m

+ 1^ 3511) <■ (a#ii^ + rii-ai+ riisi+ n 3

+ n 33iM+ ' t h n )

+ 1 ( H i s i t+ $[^3151JL J151 + t^3 33iM)°([ 3l31 •

. + 1 ^ 1 5 1 + n 331M)

loCriisrnisi]

(F-4) + 16 [ [ ]333°[ - [see Eq. (4-14)] LIST OF REFERENCES

Bom, M., and E . Wolf, Principles of Optics, Pergamon Press, London (1975) .

Buchdahl, H. A., Optical Aberration Coefficients, Oxford University Press, London (1954).

Buchroeder, R. A., "Tilted Component Optical Systems," PhD dissertation. Optical Sciences Center, University of Arizona (1976).

Burch, C. R., "On the Optical See-Saw Diagram," Mon. Not. R. Astron. Soc. 201:159 (1942).

Hopkins, G. W ., "Aberrational.Analysis of Optical Systems: a proximate ray trace approach," PhD dissertation. Optical Sciences Center, University of Arizona (1976).

Kingslake, R., Lens Design Fundamentals, Academic Press, New York (1.978).

Kreitzer, M. H., "Image Quality Criteria for Aberrated Systems," PhD dissertation. Optical Sciences Center, University of Arizona (1976). %■ Radowski, E. J., "Use of Orthonormalized Image Errors in Optical Design," Thesis, Institute of Optics, University of Rochester (1967).

Rimmer, M., "Optical Aberration Coefficients," Appendix IV of the Ordeals. II Program Manual, Tropel Inc., 52 West Ave., Fairport, New York (1965).

Robb, P. N., "Lens Design Using Optical Aberration Coefficients," pre sented at the OSA 1980 Spring Conference on Applied Optics (1980).

Shack, R. V., "The Use of Normalization in the Application of Simple Optical Systems, "Proc. SPIE, 54:155 (1974).

Smith, W. J., Modern Optical Engineering; the Design of Optical Systems, McGraw-Hill, New York (1966).

U . S. Department of Defense, Military Standardization Handbook; Optical Design, MlL-HNBK-141, Defense Supply Agency, Washington (1962) .

272 273

Unvala, H. A., "The Orthonormalization of Aberrations," Proceedings of the Conference on Lens Design with Large Computers (1967).

Wei ford W. T ., Aberrations of the Symmetrical Optical System, Academic Press, New York (1974).

Wiese, ii. E ., "Use of Physically Significant Merit Functions in Auto matic Lens Design," thesis. Optical Sciences Center, University of Arizona (1974). 5 634