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El. knyga: Advanced Geometrical Optics

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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.

Recenzijos

Professor Lin has greatly expanded the breadth and scope of using matrix representation in geometrical optics. Advanced Geometrical Optics could be used in teaching graduate and advanced-undergraduate students and is enthusiastically recommended for those interested in geometrical optics, optical design, and optimization. (R. Barry Johnson, Contemporary Physics, Vol. 58 (3), April, 2017)

This book is concerned with the study of geometrical optics using a matrix method. Since the author intends to make this book a useful reference for geometrical optics courses oriented to graduate and senior undergraduate students, many illustrative examples are presented. Overall, this is an interesting book modernizing the classical subject of geometrical optics. (Jichun Li, zbMATH, Vol. 1366.78001, 2017) 

Part I A New Light on Old Geometrical Optics (Raytracing Equations of Geometrical Optics)
1 Mathematical Background
3(26)
1.1 Foundational Mathematical Tools and Units
3(2)
1.2 Vector Notation
5(2)
1.3 Coordinate Transformation Matrix
7(2)
1.4 Basic Translation and Rotation Matrices
9(6)
1.5 Specification of a Pose Matrix by Using Translation and Rotation Matrices
15(1)
1.6 Inverse Matrix of a Transformation Matrix
16(1)
1.7 Flat Boundary Surface
17(2)
1.8 RPY Transformation Solutions
19(1)
1.9 Equivalent Angle and Axis of Rotation
20(2)
1.10 The First- and Second-Order Partial Derivatives of a Vector
22(4)
1.11 Introduction to Optimization Methods
26(3)
References
28(1)
2 Skew-Ray Tracing of Geometrical Optics
29(42)
2.1 Source Ray
29(3)
2.2 Spherical Boundary Surfaces
32(12)
2.2.1 Spherical Boundary Surface and Associated Unit Normal Vector
32(2)
2.2.2 Incidence Point
34(3)
2.2.3 Unit Directional Vectors of Reflected and Refracted Rays
37(7)
2.3 Flat Boundary Surfaces
44(11)
2.3.1 Flat Boundary Surface and Associated Unit Normal Vector
44(2)
2.3.2 Incidence Point
46(1)
2.3.3 Unit Directional Vectors of Reflected and Refracted Rays
47(8)
2.4 General Aspherical Boundary Surfaces
55(9)
2.4.1 Aspherical Boundary Surface and Associated Unit Normal Vector
55(2)
2.4.2 Incidence Point
57(7)
2.5 The Unit Normal Vector of a Boundary Surface for Given Incoming and Outgoing Rays
64(7)
2.5.1 Unit Normal Vector of Refractive Boundary Surface
65(2)
2.5.2 Unit Normal Vector of Reflective Boundary Surface
67(1)
References
68(3)
3 Geometrical Optical Model
71(44)
3.1 Axis-Symmetrical Systems
71(16)
3.1.1 Elements with Spherical Boundary Surfaces
76(1)
3.1.2 Elements with Spherical and Flat Boundary Surfaces
77(1)
3.1.3 Elements with Flat and Spherical Boundary Surfaces
78(1)
3.1.4 Elements with Flat Boundary Surfaces
79(8)
3.2 Non-axially Symmetrical Systems
87(10)
3.3 Spot Diagram of Monochromatic Light
97(2)
3.4 Point Spread Function
99(5)
3.5 Modulation Transfer Function
104(5)
3.6 Motion Measurement Systems
109(6)
References
113(2)
4 Ray tracing Equations for Paraxial Optics
115(28)
4.1 Raytracing Equations of Paraxial Optics for 3-D Optical Systems
115(8)
4.1.1 Transfer Matrix
117(1)
4.1.2 Reflection and Refraction Matrices for Flat Boundary Surface
118(1)
4.1.3 Reflection and Refraction Matrices for Spherical Boundary Surface
119(4)
4.2 Conventional 2 × 2 Raytracing Matrices for Paraxial Optics
123(5)
4.2.1 Refracting Boundary Surfaces
124(1)
4.2.2 Reflecting Boundary Surfaces
125(3)
4.3 Conventional Raytracing Matrices for Paraxial Optics Derived from Geometry Relations
128(15)
4.3.1 Transfer Matrix for Ray Propagating Along Straight-Line Path
129(2)
4.3.2 Refraction Matrix at Refractive Flat Boundary Surface
131(2)
4.3.3 Reflection Matrix at Flat Mirror
133(2)
4.3.4 Refraction Matrix at Refractive Spherical Boundary Surface
135(3)
4.3.5 Reflection Matrix at Spherical Mirror
138(4)
References
142(1)
5 Cardinal Points and Image Equations
143(24)
5.1 Paraxial Optics
143(2)
5.2 Cardinal Planes and Cardinal Points
145(4)
5.2.1 Location of Focal Points
146(2)
5.2.2 Location of Nodal Points
148(1)
5.3 Thick and Thin Lenses
149(2)
5.4 Curved Mirrors
151(2)
5.5 Determination of Image Position Using Cardinal Points
153(1)
5.6 Equation of Lateral Magnification
154(1)
5.7 Equation of Longitudinal Magnification
155(1)
5.8 Two-Element Systems
156(3)
5.9 Optical Invariant
159(8)
5.9.1 Optical Invariant and Lateral Magnification
160(1)
5.9.2 Image Height for Object at Infinity
161(1)
5.9.3 Data of Third Ray
162(2)
5.9.4 Focal Length Determination
164(1)
References
165(2)
6 Ray Aberrations
167(20)
6.1 Stops and Aperture
167(2)
6.2 Ray Aberration Polynomial and Primary Aberrations
169(2)
6.3 Spherical Aberration
171(2)
6.4 Coma
173(4)
6.5 Astigmatism
177(2)
6.6 Field Curvature
179(1)
6.7 Distortion
180(1)
6.8 Chromatic Aberration
181(6)
References
183(4)
Part II New Tools for Optical Analysis and Design (First-Order Derivative Matrices of a Ray and its OPL)
7 Jacobian Matrices of Ray Ri with Respect to Incoming Ray Ri--1 and Boundary Variable Vector Xi
187(32)
7.1 Jacobian Matrix of Ray
188(1)
7.2 Jacobian Matrix ∂Ri/∂Ri--1 for Flat Boundary Surface
189(6)
7.2.1 Jacobian Matrix of Incidence Point
190(1)
7.2.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray
191(1)
7.2.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray
191(1)
7.2.4 Jacobian Matrix of Ri with Respect to Ri---1 for Flat Boundary Surface
192(3)
7.3 Jacobian Matrix ∂Ri/∂Ri--1 for Spherical Boundary Surface
195(6)
7.3.1 Jacobian Matrix of Incidence Point
196(1)
7.3.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray
197(1)
7.3.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray
198(1)
7.3.4 Jacobian Matrix of Ri with Respect to Ri?1 for Spherical Boundary Surface
198(3)
7.4 Jacobian Matrix ∂Ri/∂Xi for Flat Boundary Surface
201(5)
7.4.1 Jacobian Matrix of Incidence Point
202(1)
7.4.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray
203(1)
7.4.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray
203(1)
7.4.4 Jacobian Matrix of Ri with Respect to Xi
204(2)
7.5 Jacobian Matrix ∂Ri/∂Xi for Spherical Boundary Surface
206(4)
7.5.1 Jacobian Matrix of Incidence Point
207(1)
7.5.2 Jacobian Matrix of Unit Directional Vector of Reflected Ray
208(1)
7.5.3 Jacobian Matrix of Unit Directional Vector of Refracted Ray
208(1)
7.5.4 Jacobian Matrix of R with Respect to Xi
209(1)
7.6 Jacobian Matrix of an Arbitrary Ray with Respect to System Variable Vector
210(9)
Appendix 1
213(2)
Appendix 2
215(3)
References
218(1)
8 Jacobian Matrix of Boundary Variable Vector Xi with Respect to System Variable Vector Xsys
219(26)
8.1 System Variable Vector
219(1)
8.2 Jacobian Matrix dX0/dXsys of Source Ray
220(1)
8.3 Jacobian Matrix dXi/dXsys of Flat Boundary Surface
221(5)
8.4 Jacobian Matrix dXi/dXsys of Spherical Boundary Surface
226(19)
Appendix 1
233(3)
Appendix 2
236(2)
Appendix 3
238(3)
Appendix 4
241(2)
References
243(2)
9 Prism Analysis
245(22)
9.1 Retro-reflectors
245(3)
9.1.1 Corner-Cube Mirror
245(2)
9.1.2 Solid Glass Corner-Cube
247(1)
9.2 Dispersing Prisms
248(5)
9.2.1 Triangular Prism
249(1)
9.2.2 Pellin-Broca Prism and Dispersive Abbe Prism
250(1)
9.2.3 Achromatic Prism and Direct Vision Prism
251(2)
9.3 Right-Angle Prisms
253(1)
9.4 Porro Prism
254(1)
9.5 Dove Prism
255(1)
9.6 Roofed Amici Prism
256(1)
9.7 Erecting Prisms
257(5)
9.7.1 Double Porro Prism
257(2)
9.7.2 Porro-Abbe Prism
259(1)
9.7.3 Abbe-Koenig Prism
260(1)
9.7.4 Roofed Pechan Prism
261(1)
9.8 Penta Prism
262(5)
Appendix 1
263(1)
References
264(3)
10 Prism Design Based on Image Orientation
267(28)
10.1 Reflector Matrix and Image Orientation Function
267(7)
10.2 Minimum Number of Reflectors
274(3)
10.2.1 Right-Handed Image Orientation Function
275(2)
10.2.2 Left-Handed Image Orientation Function
277(1)
10.3 Prism Design Based on Unit Vectors of Reflectors
277(5)
10.4 Exact Analytical Solutions for Single Prism with Minimum Number of Reflectors
282(9)
10.4.1 Right-Handed Image Orientation Function
284(1)
10.4.2 Left-Handed Image Orientation Function
284(1)
10.4.3 Solution for Right-Handed Image Orientation Function
285(3)
10.4.4 Solution for Left-Handed Image Orientation Function
288(3)
10.5 Prism Design for Given Image Orientation Using Screw Triangle Method
291(4)
References
294(1)
11 Determination of Prism Reflectors to Produce Required Image Orientation
295(14)
11.1 Determination of Reflector Equations
295(3)
11.2 Determination of Prism with n = 4 Boundary Surfaces to Produce Specified Right-Handed Image Orientation
298(4)
11.3 Determination of Prism with n = 5 Boundary Surfaces to Produce Specified Left-Handed Image Orientation
302(7)
Reference
307(2)
12 Optically Stable Systems
309(10)
12.1 Image Orientation Function of Optically Stable Systems
309(3)
12.2 Design of Optically Stable Reflector Systems
312(4)
12.2.1 Stable Systems Comprising Two Reflectors
312(1)
12.2.2 Stable Systems Comprising Three Reflectors
313(1)
12.2.3 Stable Systems Comprising More Than Three Reflectors
314(2)
12.3 Design of Optically Stable Prism
316(3)
Reference
318(1)
13 Point Spread Function, Caustic Surfaces and Modulation Transfer Function
319(34)
13.1 Infinitesimal Area on Image Plane
320(2)
13.2 Derivation of Point Spread Function Using Irradiance Method
322(4)
13.3 Derivation of Spot Diagram Using Irradiance Method
326(1)
13.4 Caustic Surfaces
327(6)
13.4.1 Caustic Surfaces Formed by Point Source
328(2)
13.4.2 Caustic Surfaces Formed by Collimated Rays
330(3)
13.5 MTF Theory for Any Arbitrary Direction of OBDF
333(3)
13.6 Determination of MTF for Any Arbitrary Direction of OBDF Using Ray-Counting and Irradiance Methods
336(17)
13.6.1 Ray-Counting Method
336(1)
13.6.2 Irradiance Method
337(7)
Appendix 1
344(1)
Appendix 2
345(1)
Appendix 3
346(1)
Appendix 4
346(3)
References
349(4)
14 Optical Path Length and Its Jacobian Matrix
353(20)
14.1 Jacobian Matrix of OPLi Between (i--1)th and ith Boundary Surfaces
353(4)
14.1.1 Jacobian Matrix of OPLi with Respect to Incoming Ray Ri--1
354(1)
14.1.2 Jacobian Matrix of OPLi with Respect to Boundary Variable Vector Xi
355(2)
14.2 Jacobian Matrix of OPL Between Two Incidence Points
357(5)
14.3 Computation of Wavefront Aberrations
362(6)
14.4 Merit Function Based on Wavefront Aberration
368(5)
References
369(4)
Part III A Bright Light for Geometrical Optics (Second-Order Derivative Matrices of a Ray and its OPL)
15 Wavefront Aberration and Wavefront Shape
373(32)
15.1 Hessian Matrix ∂2Ri/∂Ri2--1 for Flat Boundary Surface
374(2)
15.1.1 Hessian Matrix of Incidence Point Pi
375(1)
15.1.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
375(1)
15.1.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
375(1)
15.2 Hessian Matrix ∂2 Ri/∂Ri2--1 for Spherical Boundary Surface
376(2)
15.2.1 Hessian Matrix of Incidence Point Pi
376(1)
15.2.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
377(1)
15.2.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
377(1)
15.3 Hessian Matrix of Ri with Respect to Variable Vector X0 of Source Ray
378(2)
15.4 Hessian Matrix of OPLi with Respect to Variable Vector X0 of Source Ray
380(2)
15.5 Change of Wavefront Aberration Due to Translation of Point Source P0
382(5)
15.6 Wavefront Shape Along Ray Path
387(18)
15.6.1 Tangent and Unit Normal Vectors of Wavefront Surface
389(1)
15.6.2 First and Second Fundamental Forms of Wavefront Surface
390(2)
15.6.3 Principal Curvatures of Wavefront
392(7)
Appendix 1
399(1)
Appendix 2
400(3)
References
403(2)
16 Hessian Matrices of Ray Ri with Respect to Incoming Ray Ri--1 and Boundary Variable Vector Xi
405(20)
16.1 Hessian Matrix of a Ray with Respect to System Variable Vector
405(2)
16.2 Hessian Matrix ∂2Ri/∂X2i for Flat Boundary Surface
407(2)
16.2.1 Hessian Matrix of Incidence Point Pi
407(1)
16.2.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
408(1)
16.2.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
408(1)
16.3 Hessian Matrix ∂2Ri/∂Xi∂Ri--1 for Flat Boundary Surface----
409(3)
16.3.1 Hessian Matrix of Incidence Point Pi
410(1)
16.3.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
411(1)
16.3.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
411(1)
16.4 Hessian Matrix ∂2Ri/∂Xi2 for Spherical Boundary Surface
412(2)
16.4.1 Hessian Matrix of Incidence Point Pi
412(1)
16.4.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
413(1)
16.4.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
413(1)
16.5 Hessian Matrix ∂2Ri/∂Xi∂Ri--1 for Spherical Boundary Surface
414(11)
16.5.1 Hessian Matrix of Incidence Point Pi
415(1)
16.5.2 Hessian Matrix of Unit Directional Vector li of Reflected Ray
415(1)
16.5.3 Hessian Matrix of Unit Directional Vector li of Refracted Ray
416(1)
Appendix 1
417(3)
Appendix 2
420(3)
Reference
423(2)
17 Hessian Matrix of Boundary Variable Vector Xi with Respect to System Variable Vector Xsys
425(26)
17.1 Hessian Matrix ∂2X0/∂X2sys of Source Ray
425(1)
17.2 Hessian Matrix ∂2Xi/∂X2sys for Flat Boundary Surface
426(4)
17.3 Design of Optical Systems Possessing Only Flat Boundary Surfaces
430(3)
17.4 Hessian Matrix ∂2Xi/∂X2sys for Spherical Boundary Surface
433(4)
17.5 Design of Retro-reflectors
437(14)
Appendix 1
441(2)
Appendix 2
443(2)
Appendix 3
445(1)
Appendix 4
446(3)
References
449(2)
18 Hessian Matrix of Optical Path Length
451(8)
18.1 Determination of Hessian Matrix of OPL
451(3)
18.1.1 Hessian Matrix of OPLi with Respect to Incoming Ray Ri--1
453(1)
18.1.2 Hessian Matrix of OPLi with Respect to Xi and Ri--1
453(1)
18.1.3 Hessian Matrix of OPLi with Respect to Boundary Variable Vector Xi
453(1)
18.2 System Analysis Based on Jacobian and Hessian Matrices of Wavefront Aberrations
454(2)
18.3 System Design Based on Jacobian and Hessian Matrices of Wavefront Aberrations
456(3)
Reference
457(2)
VITA 459
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.
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