Yu. A. Kravtsov Yu.I. Orlov

Caustics, Catastrophes and Wave Fields Second Edition

With 60 Figures

Springer

Contents

1 Introduction 1 1.1 Caustic Fields in Physical Problems 1 1.2 The Geometrical Aspect of the Caustic Problem 4 1.3 The Wave Aspect of the Caustic Problem 5

2 Rays and Caustics 8 2.1 Equations of Geometrical Optics 8

2.1.1 The Scalar Problem 8 2.1.2 Electromagnetic Waves in an Isotropic Medium 11 2.1.3 Electromagnetic Waves in an Anisotropic Medium . . . 12

2.2 The Role of Rays in the Method of Geometrical Optics . . . . 13 2.2.1 The Locality Principle 13 2.2.2 Rays as Energy and Phase Trajectories 13 2.2.3 Fresnel Volume of a Ray: The Physical Content of the

Ray Concept 14 2.2.4 Heuristic Criteria of Applicability for Ray Theory . . . . 16 2.2.5 Distinguishability of Rays 17

2.3 Physical Characteristics of Caustics 17 2.3.1 Caustics as Envelopes of Ray Families 17 2.3.2 Caustic Phase Shift 18 2.3.3 Caustic Zone and Caustic Volume 19 2.3.4 Ray Estimates of Fields at Caustics and in Focal Spots 23 2.3.5 Indistinguishability of Rays in a Caustic Zone 24 2.3.6 Reality of Caustics 25 2.3.7 A Remark on Multipath Propagation 26

2.4 Complex Rays 27 2.4.1 Main Properties of Complex Rays 27 2.4.2 Reflection of a Plane Wave from a Linear Slab 29 2.4.3 Nonlocal Nature of Complex Rays 30 2.4.4 Domain of Localization of Complex Rays 32

3 Caustics as Catastrophes 34 3.1 Mappings Induced by Rays 34

3.1.1 The Ray Surface and Lagrange's Manifold 34 3.1.2 Classification of Structurally Stable Caustics 36

3.2 Classification of Typical Caustics 39

Contents

3.2.1 Generating Function: Codimension and Corank 39 3.2.2 Caustic Surfaces of Low Codimension 40 3.2.3 Caustics of High Codimension 44 3.2.4 Subordinance Relations 47

Typical Integrals of Catastrophe Theory 48 4.1 Standard Caustic Integrals 48

4.1.1 Use of Generating Functions as Phase Functions . . . . 48 4.1.2 Reducing Integrals to Normal Form 51 4.1.3 Multiplicity of Standard Integrals 53

4.2 The Airy Integral 54 4.2.1 Basic Properties 54 4.2.2 The Airy Differential Equation 57 4.2.3 An Example of Airy-Integral Solution to the Wave

Problem 57 4.2.4 The Airy Integral as a Standard Function

for the One-Dimensional Wave Equation 58 4.2.5 Applicability Conditions of the Uniform Airy

Asymptotic in One-Dimensional Problems 59 4.3. The Pearcey Integral 60

4.3.1 Properties 60 4.3.2 Focusing in the Presence of Cylindrical Aberration . . . 61 4.3.3 Caustic Indices and Field Structure 63

4.4 Other Typical Integrals 64 4.4.1 Generalized Airy Functions 64 4.4.2 Fresnel Criteria for Transition to Subasymptotics . . . . 66 4.4.3 Field Structure in Different Areas of the External

Variable Domain 67 4.4.4 Integrals of the Dm+l Series 68 4.4.5 Caustics with a Large Number of Rays 69 4.4.6 Calculation of Standard Integrals 71

Uniform Caustic Asymptotics Derived with Standard Integrals . . . . 73 5.1 Uniform Airy Asymptotic of a Scalar Field 73

5.1.1 Heuristic Foundation of the Method of Standard Integrals 73

5.1.2 Guessing at a Form of Solution 74 5.1.3 Equations for Unknown Functions 75 5.1.4 Relation of the Airy Asymptotic to the Ray Fields . . . 77 5.1.5 Field in the Caustic Shadow 79 5.1.6 Local Field Asymptotic near a Caustic 80 5.1.7 Interpolation Formula for a Caustic Field 85 5.1.8 Estimating the Coefficient of the Airy Function

Derivative 85 5.1.9 The Geometric Backbone and Wave "Flesh" 86 5.1.10 Uniform Airy Asymptotic of an EM Field 87

Contents IX

5.1.11 Local Asymptotic of an EM Field 89 5.1.12 One-Dimensional Problem 90 5.1.13 Applicability Conditions for the Airy Asymptotic . . . . 91

5.2 Uniform Caustic Asymptotics Based on General Standard Integrals 92 5.2.1 Structure of a Solution 92 5.2.2 Equations for Phase and Amplitude Functions 93 5.2.3 Relation to Geometrical Optics 94 5.2.4 General Scheme to Compute Caustic Fields 97 5.2.5 Uniform Caustic Asymptotic of an EM Field 98 5.2.6 The Ray Skeleton and Uniform Caustic Asymptotics . 99 5.2.7 Some Specific Situations 99 5.2.8 Local Asymptotics 101

5.3 Illustrative Examples 103 5.3.1 The Circular Caustic 103 5.3.2 Point Source in a Linear Slab 106 5.3.3 Swallowtail Caustics in a Linear Layer Bordering

upon a Homogeneous Halfspace 108 5.3.4 Butterfly in a Parabolic Plasma Layer I l l 5.3.5 Elliptic Umbilic Formed by an Antenna

in a Plasma Layer I l l 5.3.6 Elliptic Umbilics in Underwater Acoustics 112 5.3.7 How Far Can We Advance in Constructing Caustic

Asymptotics? 113 5.3.8 Do Swallowtails Exist in Two Dimensions? 114

6 Maslov's Method of the Canonical Operator 116 6.1 Principal Relationships 116

6.1.1 The Wave Equation in the Coordinate-Momentum Representation 116

6.1.2 Asymptotic Solution of the Wave Equation 117 6.1.3 Elimination of Field Divergence at Caustics 119 6.1.4 The Canonical Operator 120 6.1.5 Remarks on Applicability Conditions 121

6.2 Specific Problems 122 6.2.1 Plane Wave in a Linear Layer 122 6.2.2 Diffraction on a Phase Screen 124 6.2.3 Asymptotic Solution of the Parabolic Equation 126 6.2.4 Miscellaneous Problems 127

6.3. Generalization by Using Fractional Transformations 128 6.3.1 Fractional Fourier Transformation 128 6.3.2 Fractional Representation for Two-Dimensional

Propagation 129 6.3.3 Construction of the Overall Field 131 6.3.4 Advantages of the Alonso-Forbes Representation 134

X Contents

7 Method of Interference Integrals 135 7.1 Ray Type Integrals 135

7.1.1 Wide and Narrow Sense Interpretations 135 7.1.2 Eiconals and Amplitudes of Partial Waves 136 7.1.3 Virtual Rays 140 7.1.4 Specific Problems 141

7.2 Caustic Integrals 143 7.2.1 Airy Function Based Integrals 143 7.2.2 Use of Miscellaneous Special Functions 144 7.2.3 Specific Problems 144

7.3 Additional Topics and Generalizations 146 7.3.1 Comparison with Maslov's Method 146 7.3.2 Implementation of Interference-Integral Algorithms . . . 146 7.3.3 Applicability Limits 147 7.3.4 Some Generalizations 147

8 Penumbra Caustics 148 8.1 Broken Penumbra Caustics 148

8.1.1 Broken Caustics in Diffraction at Screens 148 8.1.2 A Uniform Asymptotic 150 8.1.3 Particular Cases 151 8.1.4 A Uniform Asymptotic for an EM Field . 152 8.1.5 Broken Caustics of Higher Dimension 152 8.1.6 Broken Caustics at Discontinuities of Phase-Front

Curvature and Jumps of Refractive Index 153 8.2 Penumbra Caustics of Diffraction Rays 154

8.2.1 Generation of Caustics 154 8.2.2 Asymptotic Solution 156 8.2.3 Properties of the Asymptotic Solution 156 8.2.4 Some Generalizations 157

8.3 Penumbra Caustics and Edge Catastrophes 157 8.3.1 Simple Edge Catastrophes 157 8.3.2 Typical Integrals of Edge Catastrophe Theory 158 8.3.3. Corner Catastrophes 159

9 Modifications and Generalizations of Standard Integrals and Functions 160 9.1 Nonpolynomial Phase Standard Integrals 160

9.1.1 Standard Integrals with Arbitrary Phase Functions . . . 160 9.1.2 Uniform Asymptotics Based on Standard Integrals

with Arbitrary Phase Functions 160 9.1.3 Bessel Function Based Uniform Asymptotics

near Simple Caustics 161 9.1.4 Contour Standard Integrals 163

Contents XI

9.2 Structurally Unstable Caustics 163 9.2.1 Structurally Stable and Unstable Objects 163 9.2.2 Uniform Asymptotics for Axially Symmetric Caustics . 164 9.2.3 A Uniform Asymptotic for an Axial Caustic 166 9.2.4 Applicability of Axial Caustic Asymptotics

in the Presence of Aberrations 167 9.3 Standard Integrals with Amplitude Correction 168

9.3.1 Integrals of Weighted Rapidly Oscillating Functions . . 168 9.3.2 Uniform Penumbral Asymptotics near a Fuzzy

Light-Shadow Boundary 168 9.3.3 Broken Caustics near Diffused Shadow 170

9.4 Reflection from a Barrier and Oscillations in a Potential Well 171 9.4.1 Weber Equation and Functions 171 9.4.2 Asymptotic Solution to One-Dimensional Reflection

from a Barrier 172 9.4.3 Penetration of a Plane Wave Through a Barrier 174 9.4.4 Asymptotic Representation of the Field for a Barrier

with Variable Parameters 176 9.4.5 Waveguiding Caustics 178 9.4.6 Caustics Confining "Bouncing Ball" Oscillations 181 9.4.7 Applicability of the Weber Asymptotic 182

9.5 Standard Functions Induced by Ordinary Differential Equations 184 9.5.1 Using Second-Order Differential Equations

as Standards 184 9.5.2 Uniform Asymptotics of 3-D Wave Problems

Developed with 1-D Standard Functions 185 9.5.3 Caustics for an Ellipsoid Cavity 186 9.5.4 Extension of EM Oscillations 188 9.5.5 Multibarrier Problems: Coupled Oscillations 188 9.5.6 Caustics with Arbitrary Order of Ray Contact 188 9.5.7 Standard Equations of Order Higher than Two 189 9.5.8 Interpolation Formulas for Oscillating Integrals 189

10 Caustics Revisited 190 10.1 Caustics in Dispersive Media 190

10.1.1 Space-Time Caustics 190 10.1.2 A Uniform Field Asymptotic for Space-Time Caustics 192 10.1.3 Caustics with Anomalous Phase Shift 193 10.1.4 Broken Space-Time Caustics 193 10.1.5 Space-Time Lenses 193 10.1.6 Uniform Asymptotics in Media with Spatial Dispersion 194

10.2 Caustics in Anisotropic Media 194 10.2.1 Description of Caustic Fields 194 10.2.2 Exceptional Directions of Radiative Transfer 195

XII Contents

10.2.3 Focusing of Waves at the Interface of Anisotropic and Isotropic Media 196

10.2.4 Caustics with Anomalous Phase Shift 196 10.3 Complex Caustics 197 10.4 Random Caustics 198 10.5 Caustics in Quantum Mechanical Problems 200 10.6 Concluding Remarks 201

References 202

List of Symbols 214

Subject Index 215