Advanced Geometrical Optics (eBook)
XXIV, 460 Seiten
Springer Singapore (Verlag)
978-981-10-2299-9 (ISBN)
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.
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.
Preface 7
Acknowledgements 9
Contents 10
A New Light on Old Geometrical Optics (Raytracing Equations of Geometrical Optics) 23
1 Mathematical Background 25
1.1 Foundational Mathematical Tools and Units 25
1.2 Vector Notation 27
1.3 Coordinate Transformation Matrix 29
1.4 Basic Translation and Rotation Matrices 31
1.5 Specification of a Pose Matrix by Using Translation and Rotation Matrices 37
1.6 Inverse Matrix of a Transformation Matrix 38
1.7 Flat Boundary Surface 39
1.8 RPY Transformation Solutions 41
1.9 Equivalent Angle and Axis of Rotation 42
1.10 The First- and Second-Order Partial Derivatives of a Vector 44
1.11 Introduction to Optimization Methods 48
References 50
2 Skew-Ray Tracing of Geometrical Optics 51
2.1 Source Ray 51
2.2 Spherical Boundary Surfaces 54
2.2.1 Spherical Boundary Surface and Associated Unit Normal Vector 54
2.2.2 Incidence Point 56
2.2.3 Unit Directional Vectors of Reflected and Refracted Rays 59
2.3 Flat Boundary Surfaces 66
2.3.1 Flat Boundary Surface and Associated Unit Normal Vector 66
2.3.2 Incidence Point 68
2.3.3 Unit Directional Vectors of Reflected and Refracted Rays 69
2.4 General Aspherical Boundary Surfaces 77
2.4.1 Aspherical Boundary Surface and Associated Unit Normal Vector 77
2.4.2 Incidence Point 79
2.5 The Unit Normal Vector of a Boundary Surface for Given Incoming and Outgoing Rays 86
2.5.1 Unit Normal Vector of Refractive Boundary Surface 87
2.5.2 Unit Normal Vector of Reflective Boundary Surface 89
References 90
3 Geometrical Optical Model 92
3.1 Axis-Symmetrical Systems 92
3.1.1 Elements with Spherical Boundary Surfaces 97
3.1.2 Elements with Spherical and Flat Boundary Surfaces 98
3.1.3 Elements with Flat and Spherical Boundary Surfaces 99
3.1.4 Elements with Flat Boundary Surfaces 100
3.2 Non-axially Symmetrical Systems 108
3.3 Spot Diagram of Monochromatic Light 118
3.4 Point Spread Function 120
3.5 Modulation Transfer Function 125
3.6 Motion Measurement Systems 130
References 134
4 Raytracing Equations for Paraxial Optics 136
4.1 Raytracing Equations of Paraxial Optics for 3-D Optical Systems 136
4.1.1 Transfer Matrix 138
4.1.2 Reflection and Refraction Matrices for Flat Boundary Surface 139
4.1.3 Reflection and Refraction Matrices for Spherical Boundary Surface 140
4.2 Conventional 2 × 2 Raytracing Matrices for Paraxial Optics 144
4.2.1 Refracting Boundary Surfaces 145
4.2.2 Reflecting Boundary Surfaces 146
4.3 Conventional Raytracing Matrices for Paraxial Optics Derived from Geometry Relations 149
4.3.1 Transfer Matrix for Ray Propagating Along Straight-Line Path 150
4.3.2 Refraction Matrix at Refractive Flat Boundary Surface 152
4.3.3 Reflection Matrix at Flat Mirror 154
4.3.4 Refraction Matrix at Refractive Spherical Boundary Surface 156
4.3.5 Reflection Matrix at Spherical Mirror 159
References 163
5 Cardinal Points and Image Equations 164
5.1 Paraxial Optics 164
5.2 Cardinal Planes and Cardinal Points 166
5.2.1 Location of Focal Points 167
5.2.2 Location of Nodal Points 169
5.3 Thick and Thin Lenses 170
5.4 Curved Mirrors 172
5.5 Determination of Image Position Using Cardinal Points 174
5.6 Equation of Lateral Magnification 175
5.7 Equation of Longitudinal Magnification 176
5.8 Two-Element Systems 177
5.9 Optical Invariant 180
5.9.1 Optical Invariant and Lateral Magnification 181
5.9.2 Image Height for Object at Infinity 182
5.9.3 Data of Third Ray 183
5.9.4 Focal Length Determination 185
References 186
6 Ray Aberrations 187
6.1 Stops and Aperture 187
6.2 Ray Aberration Polynomial and Primary Aberrations 189
6.3 Spherical Aberration 191
6.4 Coma 193
6.5 Astigmatism 197
6.6 Field Curvature 199
6.7 Distortion 200
6.8 Chromatic Aberration 201
References 203
New Tools for Optical Analysis and Design (First-Order Derivative Matrices of a Ray and its OPL) 204
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
7.1 Jacobian Matrix of Ray 207
7.2 Jacobian Matrix {{/boldpartial {/bar{/bf{R}}}_{/bf i} } / {/boldpartial {/bar{/bf{R}}}_{/bf i - 1} }} for Flat Boundary Surface 208
7.2.1 Jacobian Matrix of Incidence Point 209
7.2.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray 210
7.2.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray 210
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
7.3 Jacobian Matrix {{/boldpartial} {/bar{/bf{R}}}_{/bf i} } / {/boldpartial {/bar{/bf{R}}}_{/bf i - /bf1} }} for Spherical Boundary Surface 214
7.3.1 Jacobian Matrix of Incidence Point 215
7.3.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray 216
7.3.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray 217
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
7.4 Jacobian Matrix {{/partial {/bar{/bf{R}}}_{/bf{i}} } / {/partial {/bar{/bf{X}}}_{/bf{i}} }} for Flat Boundary Surface 220
7.4.1 Jacobian Matrix of Incidence Point 221
7.4.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray 222
7.4.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray 222
7.4.4 Jacobian Matrix of {/bar{/bf{R}}}_{/bf{i}} with Respect to {/bar{/bf{X}}}_{/bf{i}} 223
7.5 Jacobian Matrix {{/boldpartial {/bar{/bf{R}}}_{{/bf i}} } / {/boldpartial {/bar{/bf{X}}}_{{/bf i}} }} for Spherical Boundary Surface 225
7.5.1 Jacobian Matrix of Incidence Point 226
7.5.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray 227
7.5.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray 227
7.5.4 Jacobian Matrix of {/bar{/bf{R}}_{/bf{i}}} with Respect to {/bar{/bf{X}}_{/bf{i}}} 228
7.6 Jacobian Matrix of an Arbitrary Ray with Respect to System Variable Vector 229
Appendix 1 232
Appendix 2 234
References 237
8 Jacobian Matrix of Boundary Variable Vector {/bar{/bf{X}}}_{/bf{i}} with Respect to System Variable Vector {/bar{/bf{X}}}_{/bf{sys}} 238
8.1 System Variable Vector 238
8.2 Jacobian Matrix {/bf{d}}/bar{/bf{X}}_{/bf{0}}/{/bf{d}}/bar{/bf{X}}_{/bf{sys}} of Source Ray 239
8.3 Jacobian Matrix {/bf{d}}/bar{/bf{X}}_{/bf{i}} /{/bf{d}}/bar{/bf{X}}_{/bf{sys}} of Flat Boundary Surface 240
8.4 Jacobian Matrix {/bf{d}}/bar{/bf{X}}_{/bf{i}} /{/bf{d}}/bar{/bf{X}}_{/bf{sys}} of Spherical Boundary Surface 245
Appendix 1 252
Appendix 2 255
Appendix 3 257
Appendix 4 260
References 262
9 Prism Analysis 263
9.1 Retro-reflectors 263
9.1.1 Corner-Cube Mirror 263
9.1.2 Solid Glass Corner-Cube 265
9.2 Dispersing Prisms 266
9.2.1 Triangular Prism 267
9.2.2 Pellin-Broca Prism and Dispersive Abbe Prism 268
9.2.3 Achromatic Prism and Direct Vision Prism 269
9.3 Right-Angle Prisms 271
9.4 Porro Prism 272
9.5 Dove Prism 273
9.6 Roofed Amici Prism 274
9.7 Erecting Prisms 275
9.7.1 Double Porro Prism 275
9.7.2 Porro-Abbe Prism 277
9.7.3 Abbe-Koenig Prism 278
9.7.4 Roofed Pechan Prism 279
9.8 Penta Prism 280
Appendix 1 281
References 282
10 Prism Design Based on Image Orientation 284
10.1 Reflector Matrix and Image Orientation Function 284
10.2 Minimum Number of Reflectors 291
10.2.1 Right-Handed Image Orientation Function 292
10.2.2 Left-Handed Image Orientation Function 294
10.3 Prism Design Based on Unit Vectors of Reflectors 294
10.4 Exact Analytical Solutions for Single Prism with Minimum Number of Reflectors 299
10.4.1 Right-Handed Image Orientation Function 301
10.4.2 Left-Handed Image Orientation Function 301
10.4.3 Solution for Right-Handed Image Orientation Function 302
10.4.4 Solution for Left-Handed Image Orientation Function 305
10.5 Prism Design for Given Image Orientation Using Screw Triangle Method 308
References 311
11 Determination of Prism Reflectors to Produce Required Image Orientation 312
11.1 Determination of Reflector Equations 312
11.2 Determination of Prism with n = 4 Boundary Surfaces to Produce Specified Right-Handed Image Orientation 315
11.3 Determination of Prism with n = 5 Boundary Surfaces to Produce Specified Left-Handed Image Orientation 319
Reference 324
12 Optically Stable Systems 325
12.1 Image Orientation Function of Optically Stable Systems 325
12.2 Design of Optically Stable Reflector Systems 328
12.2.1 Stable Systems Comprising Two Reflectors 328
12.2.2 Stable Systems Comprising Three Reflectors 329
12.2.3 Stable Systems Comprising More Than Three Reflectors 330
12.3 Design of Optically Stable Prism 332
Reference 334
13 Point Spread Function, Caustic Surfaces and Modulation Transfer Function 335
13.1 Infinitesimal Area on Image Plane 336
13.2 Derivation of Point Spread Function Using Irradiance Method 338
13.3 Derivation of Spot Diagram Using Irradiance Method 342
13.4 Caustic Surfaces 343
13.4.1 Caustic Surfaces Formed by Point Source 344
13.4.2 Caustic Surfaces Formed by Collimated Rays 346
13.5 MTF Theory for Any Arbitrary Direction of OBDF 349
13.6 Determination of MTF for Any Arbitrary Direction of OBDF Using Ray-Counting and Irradiance Methods 352
13.6.1 Ray-Counting Method 352
13.6.2 Irradiance Method 353
Appendix 1 360
Appendix 2 361
Appendix 3 362
Appendix 4 362
References 365
14 Optical Path Length and Its Jacobian Matrix 368
14.1 Jacobian Matrix of OPLi Between (i& !hx00A0
14.1.1 Jacobian Matrix of OPLi with Respect to Incoming Ray /bar{/hbox{R}}_{{{/rm i} - 1}} 369
14.1.2 Jacobian Matrix of OPLi with Respect to Boundary Variable Vector {/bar{/hbox{X}}}_{{/rm i}} 370
14.2 Jacobian Matrix of OPL Between Two Incidence Points 372
14.3 Computation of Wavefront Aberrations 377
14.4 Merit Function Based on Wavefront Aberration 383
References 384
A Bright Light for Geometrical Optics (Second-Order Derivative Matrices of a Ray and its OPL) 385
15 Wavefront Aberration and Wavefront Shape 387
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
15.1.1 Hessian Matrix of Incidence Point /bar{/bf{P}}_{/bf{i}} 389
15.1.2 Hessian Matrix of Unit Directional Vector /bar{{/cal {/boldell} }}_{{/rm i}} of Reflected Ray 389
15.1.3 Hessian Matrix of Unit Directional Vector /bar{{/cal {/boldell} }}_{{/rm i}} of Refracted Ray 389
15.2 Hessian Matrix {/boldpartial }^{2} {{/bar{/bf{R}}}}_{/bf{i}} /{/boldpartial } {{/bar{/bf{R}}}}_{{/bf{i} - 1}}^{ 2} for Spherical Boundary Surface 390
15.2.1 Hessian Matrix of Incidence Point /bar{/bf{P}}_{/bf{i}} 390
15.2.2 Hessian Matrix of Unit Directional Vector /bar{{/cal {/boldell} }}_{{/rm i}} of Reflected Ray 391
15.2.3 Hessian Matrix of Unit Directional Vector /bar{{/cal {/boldell} }}_{{/rm i}} of Refracted Ray 391
15.3 Hessian Matrix of /bar{/bf{R}}_{/bf{i}} with Respect to Variable Vector /bar{/bf{X}}_{/bf 0} of Source Ray 392
15.4 Hessian Matrix of /bf{OPL}_{/bf{i}} with Respect to Variable Vector /bar{/bf{X}}_{/bf 0} of Source Ray 394
15.5 Change of Wavefront Aberration Due to Translation of Point Source /bar{/bf{P}}_{/bf 0} 396
15.6 Wavefront Shape Along Ray Path 401
15.6.1 Tangent and Unit Normal Vectors of Wavefront Surface 403
15.6.2 First and Second Fundamental Forms of Wavefront Surface 404
15.6.3 Principal Curvatures of Wavefront 406
Appendix 1 413
Appendix 2 414
References 417
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
16.1 Hessian Matrix of a Ray with Respect to System Variable Vector 418
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
16.2.1 Hessian Matrix of Incidence Point /bar{/hbox{/bf P}}_{{/bf i}} 420
16.2.2 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/rm i}} of Reflected Ray 421
16.2.3 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/rm i}} of Refracted Ray 421
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
16.3.1 Hessian Matrix of Incidence Point /bar{/hbox{/bf P}}_{{/bf i}} 423
16.3.2 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Reflected Ray 424
16.3.3 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Refracted Ray 424
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
16.4.1 Hessian Matrix of Incidence Point /bar{/hbox{/bf P}}_{{/bf i}} 425
16.4.2 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Reflected Ray 426
16.4.3 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Refracted Ray 426
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
16.5.1 Hessian Matrix of Incidence Point /bar{/hbox{/bf P}}_{{/bf i}} 428
16.5.2 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Reflected Ray 428
16.5.3 Hessian Matrix of Unit Directional Vector /bar{{/boldell}}_{{/bf i}} of Refracted Ray 429
Appendix 1 430
Appendix 2 433
Reference 436
17 Hessian Matrix of Boundary Variable Vector /bar{/bf{X}}_{{/bf i}} with Respect to System Variable Vector {/bar{/bf{X}}}_{{/bf{sys}}} 437
17.1 Hessian Matrix {/bf{ /partial}}^{2} {/bar{/bf{X}}}_{0} /{/bf{/partial}}{/bar{/bf{X}}}_{/bf{sys}}^{2} of Source Ray 437
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
17.3 Design of Optical Systems Possessing Only Flat Boundary Surfaces 442
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
17.5 Design of Retro-reflectors 449
Appendix 1 453
Appendix 2 455
Appendix 3 457
Appendix 4 458
References 461
18 Hessian Matrix of Optical Path Length 462
18.1 Determination of Hessian Matrix of OPL 462
18.1.1 Hessian Matrix of OPLi with Respect to Incoming Ray {/bar{/bf{R}}}_{{{/bf{i - 1}}}} 464
18.1.2 Hessian Matrix of OPLi with Respect to {/bar{/bf{X}}}_{/bf{i}} and {/bar{/bf{R}}}_{{{/bf{i - 1}}}} 464
18.1.3 Hessian Matrix of OPLi with Respect to Boundary Variable Vector {/bar{/bf{X}}}_{/bf{i}} 464
18.2 System Analysis Based on Jacobian and Hessian Matrices of Wavefront Aberrations 465
18.3 System Design Based on Jacobian and Hessian Matrices of Wavefront Aberrations 467
Reference 468
VITA 469
| Erscheint lt. Verlag | 20.10.2016 |
|---|---|
| Reihe/Serie | Progress in Optical Science and Photonics | Progress in Optical Science and Photonics |
| Zusatzinfo | XXIV, 460 p. 222 illus., 193 illus. in color. |
| Verlagsort | Singapore |
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik |
| Naturwissenschaften ► Physik / Astronomie ► Optik | |
| Technik | |
| Schlagworte | Hessian matrices • Homogeneous coordinate notation • Jacobian Matrix • Modulation Transfer Function • Optically Stable Systems • Paraxial Optics • Point Spread Function • Prism design • Skew-Ray Tracing • Wavefront Aberrations |
| ISBN-10 | 981-10-2299-2 / 9811022992 |
| ISBN-13 | 978-981-10-2299-9 / 9789811022999 |
| Haben Sie eine Frage zum Produkt? |
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