Lin Advanced Geometrical Optics

1. Auflage 2017
ISBN: 978-981-10-2299-9
Verlag: Springer Nature Singapore
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 4, 470 Seiten

Reihe: Progress in Optical Science and Photonics

This book computes the first- and second-order derivative matrices of skew ray and optical path length, while also providing an important mathematical tool for automatic optical design. This book consists of three parts. Part One reviews the basic theories of skew-ray tracing, paraxial optics and primary aberrations - essential reading that lays the foundation for the modeling work presented in the rest of this book. Part Two derives the Jacobian matrices of a ray and its optical path length. Although this issue is also addressed in other publications, they generally fail to consider all of the variables of a non-axially symmetrical system. The modeling work thus provides a more robust framework for the analysis and design of non-axially symmetrical systems such as prisms and head-up displays. Lastly, Part Three proposes a computational scheme for deriving the Hessian matrices of a ray and its optical path length, offering an effective means of determining an appropriate search direction when tuning the system variables in the system design process.


Dr. PD Lin is a distinguished Professor of Mechanical Engineering Department at National Cheng Kung University, Taiwan, where he has been since 1989. He earned his BS and MS from that university in 1979 and 1984, respectively. He received his Ph.D. in Mechanical Engineering from Northwestern University, USA, in 1989. He has served as an associate editor of Journal of the Chinese Society of Mechanical Engineers since 2000. He has published over 80 papers and supervised over 60 MS and 11 Ph.D. students. His research interests include geometrical optics and error analysis in multi-axis machines. In geometrical optics, he employs homogeneous coordinate notation to compute the first- and second-order derivative matrices of various optical quantities. It is one of the important mathematical tools for automatic optical design.

Autoren/Hrsg.

Lin, Psang Dain

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgements;9
3;Contents;10
4;A New Light on Old Geometrical Optics (Raytracing Equations of Geometrical Optics);23
5;1 Mathematical Background;25
5.1;1.1 Foundational Mathematical Tools and Units;25
5.2;1.2 Vector Notation;27
5.3;1.3 Coordinate Transformation Matrix;29
5.4;1.4 Basic Translation and Rotation Matrices;31
5.5;1.5 Specification of a Pose Matrix by Using Translation and Rotation Matrices;37
5.6;1.6 Inverse Matrix of a Transformation Matrix;38
5.7;1.7 Flat Boundary Surface;39
5.8;1.8 RPY Transformation Solutions;41
5.9;1.9 Equivalent Angle and Axis of Rotation;42
5.10;1.10 The First- and Second-Order Partial Derivatives of a Vector;44
5.11;1.11 Introduction to Optimization Methods;48
5.12;References;50
6;2 Skew-Ray Tracing of Geometrical Optics;51
6.1;2.1 Source Ray;51
6.2;2.2 Spherical Boundary Surfaces;54
6.2.1;2.2.1 Spherical Boundary Surface and Associated Unit Normal Vector;54
6.2.2;2.2.2 Incidence Point;56
6.2.3;2.2.3 Unit Directional Vectors of Reflected and Refracted Rays;59
6.3;2.3 Flat Boundary Surfaces;66
6.3.1;2.3.1 Flat Boundary Surface and Associated Unit Normal Vector;66
6.3.2;2.3.2 Incidence Point;68
6.3.3;2.3.3 Unit Directional Vectors of Reflected and Refracted Rays;69
6.4;2.4 General Aspherical Boundary Surfaces;77
6.4.1;2.4.1 Aspherical Boundary Surface and Associated Unit Normal Vector;77
6.4.2;2.4.2 Incidence Point;79
6.5;2.5 The Unit Normal Vector of a Boundary Surface for Given Incoming and Outgoing Rays;86
6.5.1;2.5.1 Unit Normal Vector of Refractive Boundary Surface;87
6.5.2;2.5.2 Unit Normal Vector of Reflective Boundary Surface;89
6.6;References;90
7;3 Geometrical Optical Model;92
7.1;3.1 Axis-Symmetrical Systems;92
7.1.1;3.1.1 Elements with Spherical Boundary Surfaces;97
7.1.2;3.1.2 Elements with Spherical and Flat Boundary Surfaces;98
7.1.3;3.1.3 Elements with Flat and Spherical Boundary Surfaces;99
7.1.4;3.1.4 Elements with Flat Boundary Surfaces;100
7.2;3.2 Non-axially Symmetrical Systems;108
7.3;3.3 Spot Diagram of Monochromatic Light;118
7.4;3.4 Point Spread Function;120
7.5;3.5 Modulation Transfer Function;125
7.6;3.6 Motion Measurement Systems;130
7.7;References;134
8;4 Raytracing Equations for Paraxial Optics;136
8.1;4.1 Raytracing Equations of Paraxial Optics for 3-D Optical Systems;136
8.1.1;4.1.1 Transfer Matrix;138
8.1.2;4.1.2 Reflection and Refraction Matrices for Flat Boundary Surface;139
8.1.3;4.1.3 Reflection and Refraction Matrices for Spherical Boundary Surface;140
8.2;4.2 Conventional 2 × 2 Raytracing Matrices for Paraxial Optics;144
8.2.1;4.2.1 Refracting Boundary Surfaces;145
8.2.2;4.2.2 Reflecting Boundary Surfaces;146
8.3;4.3 Conventional Raytracing Matrices for Paraxial Optics Derived from Geometry Relations;149
8.3.1;4.3.1 Transfer Matrix for Ray Propagating Along Straight-Line Path;150
8.3.2;4.3.2 Refraction Matrix at Refractive Flat Boundary Surface;152
8.3.3;4.3.3 Reflection Matrix at Flat Mirror;154
8.3.4;4.3.4 Refraction Matrix at Refractive Spherical Boundary Surface;156
8.3.5;4.3.5 Reflection Matrix at Spherical Mirror;159
8.4;References;163
9;5 Cardinal Points and Image Equations;164
9.1;5.1 Paraxial Optics;164
9.2;5.2 Cardinal Planes and Cardinal Points;166
9.2.1;5.2.1 Location of Focal Points;167
9.2.2;5.2.2 Location of Nodal Points;169
9.3;5.3 Thick and Thin Lenses;170
9.4;5.4 Curved Mirrors;172
9.5;5.5 Determination of Image Position Using Cardinal Points;174
9.6;5.6 Equation of Lateral Magnification;175
9.7;5.7 Equation of Longitudinal Magnification;176
9.8;5.8 Two-Element Systems;177
9.9;5.9 Optical Invariant;180
9.9.1;5.9.1 Optical Invariant and Lateral Magnification;181
9.9.2;5.9.2 Image Height for Object at Infinity;182
9.9.3;5.9.3 Data of Third Ray;183
9.9.4;5.9.4 Focal Length Determination;185
9.10;References;186
10;6 Ray Aberrations;187
10.1;6.1 Stops and Aperture;187
10.2;6.2 Ray Aberration Polynomial and Primary Aberrations;189
10.3;6.3 Spherical Aberration;191
10.4;6.4 Coma;193
10.5;6.5 Astigmatism;197
10.6;6.6 Field Curvature;199
10.7;6.7 Distortion;200
10.8;6.8 Chromatic Aberration;201
10.9;References;203
11;New Tools for Optical Analysis and Design (First-Order Derivative Matrices of a Ray and its OPL);204
12;7 Jacobian Matrices of Ray {\bar{\bf{R}}}_{\bf{i}} with Respect to Incoming Ray {\bar{\bf{R}}}_{{{\bf{i - 1}}}} and Boundary Variable Vector {\bar{\bf{X}}}_{\bf{i}};206
12.1;7.1 Jacobian Matrix of Ray;207
12.2;7.2 Jacobian Matrix {{\boldpartial {\bar{\bf{R}}}_{\bf i} } / {\boldpartial {\bar{\bf{R}}}_{\bf i - 1} }} for Flat Boundary Surface;208
12.2.1;7.2.1 Jacobian Matrix of Incidence Point;209
12.2.2;7.2.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray;210
12.2.3;7.2.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray;210
12.2.4;7.2.4 Jacobian Matrix of {\bar{\bf{R}}}_{\bf{i}} with Respect to {\bar{\bf{R}}}_{{{\bf{i}} - \bf1}} for Flat Boundary Surface;211
12.3;7.3 Jacobian Matrix {{\boldpartial} {\bar{\bf{R}}}_{\bf i} } / {\boldpartial {\bar{\bf{R}}}_{\bf i - \bf1} }} for Spherical Boundary Surface;214
12.3.1;7.3.1 Jacobian Matrix of Incidence Point;215
12.3.2;7.3.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray;216
12.3.3;7.3.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray;217
12.3.4;7.3.4 Jacobian Matrix of {\bar{\bf{R}}}_{\bf{i}} with Respect to {\bar{\bf{R}}}_{{{\bf{i}} -\bf 1}} for Spherical Boundary Surface;217
12.4;7.4 Jacobian Matrix {{\partial {\bar{\bf{R}}}_{\bf{i}} } / {\partial {\bar{\bf{X}}}_{\bf{i}} }} for Flat Boundary Surface;220
12.4.1;7.4.1 Jacobian Matrix of Incidence Point;221
12.4.2;7.4.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray;222
12.4.3;7.4.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray;222
12.4.4;7.4.4 Jacobian Matrix of {\bar{\bf{R}}}_{\bf{i}} with Respect to {\bar{\bf{X}}}_{\bf{i}};223
12.5;7.5 Jacobian Matrix {{\boldpartial {\bar{\bf{R}}}_{{\bf i}} } / {\boldpartial {\bar{\bf{X}}}_{{\bf i}} }} for Spherical Boundary Surface;225
12.5.1;7.5.1 Jacobian Matrix of Incidence Point;226
12.5.2;7.5.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray;227
12.5.3;7.5.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray;227
12.5.4;7.5.4 Jacobian Matrix of {\bar{\bf{R}}_{\bf{i}}} with Respect to {\bar{\bf{X}}_{\bf{i}}};228
12.6;7.6 Jacobian Matrix of an Arbitrary Ray with Respect to System Variable Vector;229
12.7;Appendix 1;232
12.8;Appendix 2;234
12.9;References;237
13;8 Jacobian Matrix of Boundary Variable Vector {\bar{\bf{X}}}_{\bf{i}} with Respect to System Variable Vector {\bar{\bf{X}}}_{\bf{sys}};238
13.1;8.1 System Variable Vector;238
13.2;8.2 Jacobian Matrix {\bf{d}}\bar{\bf{X}}_{\bf{0}}/{\bf{d}}\bar{\bf{X}}_{\bf{sys}} of Source Ray;239
13.3;8.3 Jacobian Matrix {\bf{d}}\bar{\bf{X}}_{\bf{i}} /{\bf{d}}\bar{\bf{X}}_{\bf{sys}} of Flat Boundary Surface;240
13.4;8.4 Jacobian Matrix {\bf{d}}\bar{\bf{X}}_{\bf{i}} /{\bf{d}}\bar{\bf{X}}_{\bf{sys}} of Spherical Boundary Surface;245
13.5;Appendix 1;252
13.6;Appendix 2;255
13.7;Appendix 3;257
13.8;Appendix 4;260
13.9;References;262
14;9 Prism Analysis;263
14.1;9.1 Retro-reflectors;263
14.1.1;9.1.1 Corner-Cube Mirror;263
14.1.2;9.1.2 Solid Glass Corner-Cube;265
14.2;9.2 Dispersing Prisms;266
14.2.1;9.2.1 Triangular Prism;267
14.2.2;9.2.2 Pellin-Broca Prism and Dispersive Abbe Prism;268
14.2.3;9.2.3 Achromatic Prism and Direct Vision Prism;269
14.3;9.3 Right-Angle Prisms;271
14.4;9.4 Porro Prism;272
14.5;9.5 Dove Prism;273
14.6;9.6 Roofed Amici Prism;274
14.7;9.7 Erecting Prisms;275
14.7.1;9.7.1 Double Porro Prism;275
14.7.2;9.7.2 Porro-Abbe Prism;277
14.7.3;9.7.3 Abbe-Koenig Prism;278
14.7.4;9.7.4 Roofed Pechan Prism;279
14.8;9.8 Penta Prism;280
14.9;Appendix 1;281
14.10;References;282
15;10 Prism Design Based on Image Orientation;284
15.1;10.1 Reflector Matrix and Image Orientation Function;284
15.2;10.2 Minimum Number of Reflectors;291
15.2.1;10.2.1 Right-Handed Image Orientation Function;292
15.2.2;10.2.2 Left-Handed Image Orientation Function;294
15.3;10.3 Prism Design Based on Unit Vectors of Reflectors;294
15.4;10.4 Exact Analytical Solutions for Single Prism with Minimum Number of Reflectors;299
15.4.1;10.4.1 Right-Handed Image Orientation Function;301
15.4.2;10.4.2 Left-Handed Image Orientation Function;301
15.4.3;10.4.3 Solution for Right-Handed Image Orientation Function;302
15.4.4;10.4.4 Solution for Left-Handed Image Orientation Function;305
15.5;10.5 Prism Design for Given Image Orientation Using Screw Triangle Method;308
15.6;References;311
16;11 Determination of Prism Reflectors to Produce Required Image Orientation;312
16.1;11.1 Determination of Reflector Equations;312
16.2;11.2 Determination of Prism with n = 4 Boundary Surfaces to Produce Specified Right-Handed Image Orientation;315
16.3;11.3 Determination of Prism with n = 5 Boundary Surfaces to Produce Specified Left-Handed Image Orientation;319
16.4;Reference;324
17;12 Optically Stable Systems;325
17.1;12.1 Image Orientation Function of Optically Stable Systems;325
17.2;12.2 Design of Optically Stable Reflector Systems;328
17.2.1;12.2.1 Stable Systems Comprising Two Reflectors;328
17.2.2;12.2.2 Stable Systems Comprising Three Reflectors;329
17.2.3;12.2.3 Stable Systems Comprising More Than Three Reflectors;330
17.3;12.3 Design of Optically Stable Prism;332
17.4;Reference;334
18;13 Point Spread Function, Caustic Surfaces and Modulation Transfer Function;335
18.1;13.1 Infinitesimal Area on Image Plane;336
18.2;13.2 Derivation of Point Spread Function Using Irradiance Method;338
18.3;13.3 Derivation of Spot Diagram Using Irradiance Method;342
18.4;13.4 Caustic Surfaces;343
18.4.1;13.4.1 Caustic Surfaces Formed by Point Source;344
18.4.2;13.4.2 Caustic Surfaces Formed by Collimated Rays;346
18.5;13.5 MTF Theory for Any Arbitrary Direction of OBDF;349
18.6;13.6 Determination of MTF for Any Arbitrary Direction of OBDF Using Ray-Counting and Irradiance Methods;352
18.6.1;13.6.1 Ray-Counting Method;352
18.6.2;13.6.2 Irradiance Method;353
18.7;Appendix 1;360
18.8;Appendix 2;361
18.9;Appendix 3;362
18.10;Appendix 4;362
18.11;References;365
19;14 Optical Path Length and Its Jacobian Matrix;368
19.1;14.1 Jacobian Matrix of OPLi Between (i&!hx00A0;?&!hx00A0;1)th and ith Boundary Surfaces;368
19.1.1;14.1.1 Jacobian Matrix of OPLi with Respect to Incoming Ray \bar{\hbox{R}}_{{{\rm i} - 1}};369
19.1.2;14.1.2 Jacobian Matrix of OPLi with Respect to Boundary Variable Vector {\bar{\hbox{X}}}_{{\rm i}};370
19.2;14.2 Jacobian Matrix of OPL Between Two Incidence Points;372
19.3;14.3 Computation of Wavefront Aberrations;377
19.4;14.4 Merit Function Based on Wavefront Aberration;383
19.5;References;384
20;A Bright Light for Geometrical Optics (Second-Order Derivative Matrices of a Ray and its OPL);385
21;15 Wavefront Aberration and Wavefront Shape;387
21.1;15.1 Hessian Matrix { \boldpartial }^{\bf 2} \bar{\bf{R}}_{\bf{i}} /{\boldpartial } \bar{\bf{R}}_{{\bf{i} - 1}}^{ \bf 2} for Flat Boundary Surface;388
21.1.1;15.1.1 Hessian Matrix of Incidence Point \bar{\bf{P}}_{\bf{i}};389
21.1.2;15.1.2 Hessian Matrix of Unit Directional Vector \bar{{\cal {\boldell} }}_{{\rm i}} of Reflected Ray;389
21.1.3;15.1.3 Hessian Matrix of Unit Directional Vector \bar{{\cal {\boldell} }}_{{\rm i}} of Refracted Ray;389
21.2;15.2 Hessian Matrix {\boldpartial }^{2} {{\bar{\bf{R}}}}_{\bf{i}} /{\boldpartial } {{\bar{\bf{R}}}}_{{\bf{i} - 1}}^{ 2} for Spherical Boundary Surface;390
21.2.1;15.2.1 Hessian Matrix of Incidence Point \bar{\bf{P}}_{\bf{i}};390
21.2.2;15.2.2 Hessian Matrix of Unit Directional Vector \bar{{\cal {\boldell} }}_{{\rm i}} of Reflected Ray;391
21.2.3;15.2.3 Hessian Matrix of Unit Directional Vector \bar{{\cal {\boldell} }}_{{\rm i}} of Refracted Ray;391
21.3;15.3 Hessian Matrix of \bar{\bf{R}}_{\bf{i}} with Respect to Variable Vector \bar{\bf{X}}_{\bf 0} of Source Ray;392
21.4;15.4 Hessian Matrix of \bf{OPL}_{\bf{i}} with Respect to Variable Vector \bar{\bf{X}}_{\bf 0} of Source Ray;394
21.5;15.5 Change of Wavefront Aberration Due to Translation of Point Source \bar{\bf{P}}_{\bf 0};396
21.6;15.6 Wavefront Shape Along Ray Path;401
21.6.1;15.6.1 Tangent and Unit Normal Vectors of Wavefront Surface;403
21.6.2;15.6.2 First and Second Fundamental Forms of Wavefront Surface;404
21.6.3;15.6.3 Principal Curvatures of Wavefront;406
21.7;Appendix 1;413
21.8;Appendix 2;414
21.9;References;417
22;16 Hessian Matrices of Ray {\bar{{\bf R}}}_{{\bf i}} with Respect to Incoming Ray {\bar{{\bf R}}}_{{{{\bf i - 1}}}} and Boundary Variable Vector {\bar{{\bf X}}}_{{\bf i}};418
22.1;16.1 Hessian Matrix of a Ray with Respect to System Variable Vector;418
22.2;16.2 Hessian Matrix \boldpartial^{2} {\bar{\hbox{\bf R}}}_{{\rm i}} /\boldpartial {\bar{\hbox{\bf X}}}_{{\bf i}}^{\bf 2} for Flat Boundary Surface;420
22.2.1;16.2.1 Hessian Matrix of Incidence Point \bar{\hbox{\bf P}}_{{\bf i}};420
22.2.2;16.2.2 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\rm i}} of Reflected Ray;421
22.2.3;16.2.3 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\rm i}} of Refracted Ray;421
22.3;16.3 Hessian Matrix \boldpartial^{\bf2} {\bar{\hbox{\bf R}}}_{{\bf i}} /\boldpartial {\bar{\hbox{\bf X}}}_{{\bf i}} \boldpartial {\bar{\hbox{\bf R}}}_{{{{\bf i}} - \bf1}} for Flat Boundary Surface;422
22.3.1;16.3.1 Hessian Matrix of Incidence Point \bar{\hbox{\bf P}}_{{\bf i}};423
22.3.2;16.3.2 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Reflected Ray;424
22.3.3;16.3.3 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Refracted Ray;424
22.4;16.4 Hessian Matrix \boldpartial^{\bf 2} {\bar{\hbox{\bf R}}}_{{\bf i}} /\boldpartial {\bar{\hbox{\bf X}}}_{{\bf i}}^{\bf 2} for Spherical Boundary Surface;425
22.4.1;16.4.1 Hessian Matrix of Incidence Point \bar{\hbox{\bf P}}_{{\bf i}};425
22.4.2;16.4.2 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Reflected Ray;426
22.4.3;16.4.3 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Refracted Ray;426
22.5;16.5 Hessian Matrix \boldpartial^{\bf 2} {\bar{\hbox{\bf R}}}_{{\bf i}} /\boldpartial {\bar{\hbox{\bf X}}}_{{\bf i}} \boldpartial {\bar{\hbox{\bf R}}}_{{{{\bf i}} -\bf 1}} for Spherical Boundary Surface;427
22.5.1;16.5.1 Hessian Matrix of Incidence Point \bar{\hbox{\bf P}}_{{\bf i}};428
22.5.2;16.5.2 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Reflected Ray;428
22.5.3;16.5.3 Hessian Matrix of Unit Directional Vector \bar{{\boldell}}_{{\bf i}} of Refracted Ray;429
22.6;Appendix 1;430
22.7;Appendix 2;433
22.8;Reference;436
23;17 Hessian Matrix of Boundary Variable Vector \bar{\bf{X}}_{{\bf i}} with Respect to System Variable Vector {\bar{\bf{X}}}_{{\bf{sys}}};437
23.1;17.1 Hessian Matrix {\bf{ \partial}}^{2} {\bar{\bf{X}}}_{0} /{\bf{\partial}}{\bar{\bf{X}}}_{\bf{sys}}^{2} of Source Ray;437
23.2;17.2 Hessian Matrix {\bf{\partial}}^{\bf{2}} {\bar{\bf{X}}}_{\bf{i}} /{\bf{\partial}}{\bar{\bf{X}}}_{\bf{sys}}^{\bf{2}} for Flat Boundary Surface;438
23.3;17.3 Design of Optical Systems Possessing Only Flat Boundary Surfaces;442
23.4;17.4 Hessian Matrix {\bf{\partial}}^{\bf{2}} {\bar{\bf{X}}}_{\bf{i}} /{\bf{\partial}}{\bar{\bf{X}}}_{\bf{sys}}^{\bf{2}} for Spherical Boundary Surface;445
23.5;17.5 Design of Retro-reflectors;449
23.6;Appendix 1;453
23.7;Appendix 2;455
23.8;Appendix 3;457
23.9;Appendix 4;458
23.10;References;461
24;18 Hessian Matrix of Optical Path Length;462
24.1;18.1 Determination of Hessian Matrix of OPL;462
24.1.1;18.1.1 Hessian Matrix of OPLi with Respect to Incoming Ray {\bar{\bf{R}}}_{{{\bf{i - 1}}}};464
24.1.2;18.1.2 Hessian Matrix of OPLi with Respect to {\bar{\bf{X}}}_{\bf{i}} and {\bar{\bf{R}}}_{{{\bf{i - 1}}}};464
24.1.3;18.1.3 Hessian Matrix of OPLi with Respect to Boundary Variable Vector {\bar{\bf{X}}}_{\bf{i}};464
24.2;18.2 System Analysis Based on Jacobian and Hessian Matrices of Wavefront Aberrations;465
24.3;18.3 System Design Based on Jacobian and Hessian Matrices of Wavefront Aberrations;467
24.4;Reference;468
25;VITA;469