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Tables of Contents for Caustics, Catastrophes and Wave Fields
Chapter/Section Title
Page #
Page Count
1 Introduction
1
7
1.1 Caustic Fields in Physical Problems
1
3
1.2 The Geometrical Aspect of the Caustic Problem
4
1
1.3 The Wave Aspect of the Caustic Problem
5
3
2 Rays and Caustics
8
26
2.1 Equations of Geometrical Optics
8
5
2.1.1 The Scalar Problem
8
3
2.1.2 Electromagnetic Waves in an Isotropic Medium
11
1
2.1.3 Electromagnetic Waves in an Anisotropic Medium
12
1
2.2 The Role of Rays in the Method of Geometrical Optics
13
4
2.2.1 The Locality Principle
13
1
2.2.2 Rays as Energy and Phase Trajectories
13
1
2.2.3 Fresnel Volume of a Ray: The Physical Content of the Ray Concept
14
2
2.2.4 Heuristic Criteria of Applicability for Ray Theory
16
1
2.2.5 Distinguishability of Rays
17
1
2.3 Physical Characteristics of Caustics
17
10
2.3.1 Caustics as Envelopes of Ray Families
17
1
2.3.2 Caustic Phase Shift
18
1
2.3.3 Caustic Zone and Caustic Volume
19
4
2.3.4 Ray Estimates of Fields at Caustics and in Focal Spots
23
1
2.3.5 Indistinguishability of Rays in a Caustic Zone
24
1
2.3.6 Reality of Caustics
25
1
2.3.7 A Remark on Multipath Propagation
26
1
2.4 Complex Rays
27
7
2.4.1 Main Properties of Complex Rays
27
2
2.4.2 Reflection of a Plane Wave from a Linear Slab
29
1
2.4.3 Nonlocal Nature of Complex Rays
30
2
2.4.4 Domain of Localization of Complex Rays
32
2
3 Caustics as Catastrophes
34
14
3.1 Mappings Induced by Rays
34
5
3.1.1 The Ray Surface and Lagrange's Manifold
34
2
3.1.2 Classification of Structurally Stable Caustics
36
3
3.2 Classification of Typical Caustics
39
9
3.2.1 Generating Function: Codimension and Corank
39
1
3.2.2 Caustic Surfaces of Low Codimension
40
4
3.2.3 Caustic of High Codimension
44
3
3.2.4 Subordinance Relations
47
1
4 Typical Integrals of Catastrophe Theory
48
25
4.1 Standard Caustic Integrals
48
6
4.1.1 Use of Generating Functions as Phase Functions
48
3
4.1.2 Reducing Integrals to Normal Form
51
2
4.1.3 Multiplicity of Standard Integrals
53
1
4.2 The Airy Integral
54
6
4.2.1 Basic Properties
54
3
4.2.2 The Airy Differential Equation
57
1
4.2.3 An Example of Airy-Integral Solution to the Wave Problem
57
1
4.2.4 The Airy Integral as a Standard Function for the One-Dimensional Wave Equation
58
1
4.2.5 Applicability Conditions of the Uniform Airy Asymptotic in One-Dimensional Problems
59
1
4.3 The Pearcey Integral
60
4
4.3.1 Properties
60
1
4.3.2 Focusing in the Presence of Cylindrical Aberration
61
2
4.3.3 Caustic Indices and Field Structure
63
1
4.4 Other Typical Integrals
64
9
4.4.1 Generalized Airy Functions
64
2
4.4.2 Fresnel Criteria for Transition to Subasymptotics
66
1
4.4.3 Field Structure in Different Areas of the External Variable Domain
67
1
4.4.4 Integrals of the D(m + 1) Series
68
1
4.4.5 Caustics with a Large Number of Rays
69
2
4.4.6 Calculation of Standard Integrals
71
2
5 Uniform Caustic Asymptotics Derived with Standard Integrals
73
43
5.1 Uniform Airy Asymptotic of a Scalar Field
73
19
5.1.1 Heuristic Foundation of the Method of Standard Integrals
73
1
5.1.2 Guessing at a Form of Solution
74
1
5.1.3 Equations for Unknown Functions
75
2
5.1.4 Relation of the Airy Asymptotic to the Ray Fields
77
2
5.1.5 Field in the Caustic Shadow
79
1
5.1.6 Local Field Asymptotic near a Caustic
80
5
5.1.7 Interpolation Formula for a Caustic Field
85
1
5.1.8 Estimating the Coefficient of the Airy Function Derivative
85
1
5.1.9 The Geometric Backbone and Wave "Flesh"
86
1
5.1.10 Uniform Airy Asymptotic of an EM Field
87
2
5.1.11 Local Asymptotic of an EM Field
89
1
5.1.12 One-Dimensional Problem
90
1
5.1.13 Applicability Conditions for the Airy Asymptotic
91
1
5.2 Uniform Caustic Asymptotics Based on General Standard Integrals
92
11
5.2.1 Structure of a Solution
92
1
5.2.2 Equations for Phase and Amplitude Functions
93
1
5.2.3 Relation to Geometrical Optics
94
3
5.2.4 General Scheme to Compute Caustic Fields
97
1
5.2.5 Uniform Caustic Asymptotic of an EM Field
98
1
5.2.6 The Ray Skeleton and Uniform Caustic Asymptotics
99
1
5.2.7 Some Specific Situations
99
2
5.2.8 Local Asymptotics
101
2
5.3 Illustrative Examples
103
13
5.3.1 The Circular Caustic
103
3
5.3.2 Point Source in a Linear Slab
106
2
5.3.3 Swallowtail Caustics in a Linear Layer Bordering upon a Homogeneous Halfspace
108
3
5.3.4 Butterfly in a Parabolic Plasma Layer
111
1
5.3.5 Elliptic Umbilic Formed by an Antenna in a Plasma Layer
111
1
5.3.6 Elliptic Umbilics in Underwater Acoustics
112
1
5.3.7 How Far Can We Advance in Constructing Caustic Asymptotics?
113
1
5.3.8 Do Swallowtails Exist in Two Dimensions?
114
2
6 Maslov's Method of the Canonical Operator
116
19
6.1 Principal Relationships
116
6
6.1.1 The Wave Equation in the Coordinate-Momentum Representation
116
1
6.1.2 Asymptotic Solution of the Wave Equation
117
2
6.1.3 Elimination of Field Divergence at Caustics
119
1
6.1.4 The Canonical Operator
120
1
6.1.5 Remarks on Applicability Conditions
121
1
6.2 Specific Problems
122
6
6.2.1 Plane Wave in a Linear Layer
122
2
6.2.2 Diffraction on a Phase Screen
124
2
6.2.3 Asymptotic Solution of the Parabolic Equation
126
1
6.2.4 Miscellaneous Problems
127
1
6.3 Generalization by Using Fractional Transformations
128
7
6.3.1 Fractional Fourier Transformation
128
1
6.3.2 Fractional Representation for Two-Dimensional Propagation
129
2
6.3.3 Construction of the Overall Field
131
3
6.3.4 Advantages of the Alonso-Forbes Representation
134
1
7 Method of Interference Integrals
135
13
7.1 Ray Type Integrals
135
8
7.1.1 Wide and Narrow Sense Interpretations
135
1
7.1.2 Eiconals and Amplitudes of Partial Waves
136
4
7.1.3 Virtual Rays
140
1
7.1.4 Specific Problems
141
2
7.2 Caustic Integrals
143
3
7.2.1 Airy Function Based Integrals
143
1
7.2.2 Use of Miscellaneous Special Functions
144
1
7.2.3 Specific Problems
144
2
7.3 Additional Topics and Generalizations
146
2
7.3.1 Comparison with Maslov's Method
146
1
7.3.2 Implementation of Interference-Integral Algorithms
146
1
7.3.3 Applicability Limits
147
1
7.3.4 Some Generalizations
147
1
8 Penumbra Caustics
148
12
8.1 Broken Penumbra Caustics
148
6
8.1.1 Broken Caustics in Diffraction at Screens
148
2
8.1.2 A Uniform Asymptotic
150
1
8.1.3 Particular Cases
151
1
8.1.4 A Uniform Asymptotic for an EM Field
152
1
8.1.5 Broken Caustics of Higher Dimension
152
1
8.1.6 Broken Caustics at Discontinuities of Phase-Front Curvature and Jumps of Refractive Index
153
1
8.2 Penumbra Caustics of Diffraction Rays
154
3
8.2.1 Generation of Caustics
154
2
8.2.2 Asymptotic Solution
156
1
8.2.3 Properties of the Asymptotic Solution
156
1
8.2.4 Some Generalizations
157
1
8.3 Penumbra Caustics and Edge Catastrophes
157
3
8.3.1 Simple Edge Catastrophes
157
1
8.3.2 Typical Integrals of Edge Catastrophe Theory
158
1
8.3.3 Corner Catastrophes
159
1
9 Modifications and Generalizations of Standard Integrals and Functions
160
30
9.1 Nonpolynomial Phase Standard Integrals
160
3
9.1.1 Standard Integrals with Arbitrary Phase Functions
160
1
9.1.2 Uniform Asymptotics Based on Standard Integrals with Arbitrary Phase Functions
160
1
9.1.3 Bessel Function Based Uniform Asymptotics near Simple Caustics
161
2
9.1.4 Contour Standard Integrals
163
1
9.2 Structurally Unstable Caustics
163
5
9.2.1 Structurally Stable and Unstable Objects
163
1
9.2.2 Uniform Asymptotics for Axially Symmetric Caustics
164
2
9.2.3 A Uniform Asymptotic for an Axial Caustic
166
1
9.2.4 Applicability of Axial Caustic Asymptotics in the Presence of Aberrations
167
1
9.3 Standard Integrals with Amplitude Correction
168
3
9.3.1 Integrals of Weighted Rapidly Oscillating Functions
168
1
9.3.2 Uniform Penumbral Asymptotics near a Fuzzy Light-Shadow Boundary
168
2
9.3.3 Broken Caustics near Diffused Shadow
170
1
9.4 Reflection from a Barrier and Oscillations in a Potential Well
171
13
9.4.1 Weber Equation and Functions
171
1
9.4.2 Asymptotic Solution to One-Dimensional Reflection from a Barrier
172
2
9.4.3 Penetration of a Plane Wave Through a Barrier
174
2
9.4.4 Asymptotic Representation of the Field for a Barrier with Variable Parameters
176
2
9.4.5 Waveguiding Caustics
178
3
9.4.6 Caustics Confining "Bouncing Ball" Oscillations
181
1
9.4.7 Applicability of the Weber Asymptotic
182
2
9.5 Standard Functions Induced by Ordinary Differential Equations
184
6
9.5.1 Using Second-Order Differential Equations as Standards
184
1
9.5.2 Uniform Asymptotics of 3-D Wave Problems Developed with 1-D Standard Functions
185
1
9.5.3 Caustics for an Ellipsoid Cavity
186
2
9.5.4 Extension of EM Oscillations
188
1
9.5.5 Multibarrier Problems: Coupled Oscillations
188
1
9.5.6 Caustics with Arbitrary Order of Ray Contact
188
1
9.5.7 Standard Equations of Order Higher than Two
189
1
9.5.8 Interpolation Formulas for Oscillating Integrals
189
1
10 Caustics Revisited
190
12
10.1 Caustics in Dispersive Media
190
4
10.1.1 Space-Time Caustics
190
2
10.1.2 A Uniform Field Asymptotic for Space-Time Caustics
192
1
10.1.3 Caustics with Anomalous Phase Shift
193
1
10.1.4 Broken Space-Time Caustics
193
1
10.1.5 Space-Time Lenses
193
1
10.1.6 Uniform Asymptotics in Media with Spatial Dispersion
194
1
10.2 Caustics in Anisotropic Media
194
3
10.2.1 Description of Caustic Fields
194
1
10.2.2 Exceptional Directions of Radiative Transfer
195
1
10.2.3 Focusing of Waves at the Interface of Anisotropic and Isotropic Media
196
1
10.2.4 Caustics with Anomalous Phase Shift
196
1
10.3 Complex Caustics
197
1
10.4 Random Caustics
198
2
10.5 Caustics in Quantum Mechanical Problems
200
1
10.6 Concluding Remarks
201
1
References
202
12
List of Symbols
214
1
Subject Index
215