Caustics, Catastrophes and Wave Fields

1 Introduction.- 1.1 Caustic Fields in Physical Problems.- 1.2 The Geometrical Aspect of the Caustic Problem.- 1.3 The Wave Aspect of the Caustic Problem.- 2 Rays and Caustics.- 2.1 Equations of Geometrical Optics.- 2.1.1 The Scalar Problem.- 2.1.2 Electromagnetic Waves in an Isotropic Medium.- 2.1.3 Electromagnetic Waves in an Anisotropic Medium.- 2.2 The Role of Rays in the Method of Geometrical Optics.- 2.2.1 The Locality Principle.- 2.2.2 Rays as Energy and Phase Trajectories.- 2.2.3 Fresnel Volume of a Ray: The Physical Content of the Ray Concept.- 2.2.4 Heuristic Criteria of Applicability for Ray Theory.- 2.2.5 Distinguishability of Rays.- 2.3 Physical Characteristics of Caustics.- 2.3.1 Caustics as Envelopes of Ray Families.- 2.3.2 Caustic Phase Shift.- 2.3.3 Caustic Zone and Caustic Volume.- 2.3.4 Ray Estimates of Fields at Caustics and in Focal Spots.- 2.3.5 Indistinguishability of Rays in a Caustic Zone.- 2.3.6 Reality of Caustics.- 2.3.7 A Remark on Multipath Propagation.- 2.4 Complex Rays.- 2.4.1 Main Properties of Complex Rays.- 2.4.2 Reflection of a Plane Wave from a Linear Slab.- 2.4.3 Nonlocal Nature of Complex Rays.- 2.4.4 Domain of Localization of Complex Rays.- 3 Caustics as Catastrophes.- 3.1 Mappings Induced by Rays.- 3.1.1 The Ray Surface and Lagrange’s Manifold.- 3.1.2 Classification of Structurally Stable Caustics.- 3.2 Classification of Typical Caustics.- 3.2.1 Generating Function: Codimension and Corank.- 3.2.2 Caustic Surfaces of Low Codimension.- 3.2.3 Caustics of High Codimension.- 3.2.4 Subordinance Relations.- 4 Typical Integrals of Catastrophe Theory.- 4.1 Standard Caustic Integrals.- 4.1.1 Use of Generating Functions as Phase Functions.- 4.1.2 Reducing Integrals to Normal Form.- 4.1.3 Multiplicity of Standard Integrals.- 4.2 The Airy Integral.- 4.2.1 Basic Properties.- 4.2.2 The Airy Differential Equation.- 4.2.3 An Example of Airy-Integral Solution to the Wave Problem.- 4.2.4 The Airy Integral as a Standard Function for the One-Dimensional Wave Equation.- 4.2.5 Applicability Conditions of the Uniform Airy Asymptotic in One-Dimensional Problems.- 4.3. The Pearcey Integral.- 4.3.1 Properties.- 4.3.2 Focusing in the Presence of Cylindrical Aberration.- 4.3.3 Caustic Indices and Field Structure.- 4.4 Other Typical Integrals.- 4.4.1 Generalized Airy Functions.- 4.4.2 Fresnel Criteria for Transition to Subasymptotics.- 4.4.3 Field Structure in Different Areas of the External Variable Domain.- 4.4.4 Integrals of the Dm+1 Series.- 4.4.5 Caustics with a Large Number of Rays.- 4.4.6 Calculation of Standard Integrals.- 5 Uniform Caustic Asymptotics Derived with Standard Integrals.- 5.1 Uniform Airy Asymptotic of a Scalar Field.- 5.1.1 Heuristic Foundation of the Method of Standard Integrals.- 5.1.2 Guessing at a Form of Solution.- 5.1.3 Equations for Unknown Functions.- 5.1.4 Relation of the Airy Asymptotic to the Ray Fields.- 5.1.5 Field in the Caustic Shadow.- 5.1.6 Local Field Asymptotic near a Caustic.- 5.1.7 Interpolation Formula for a Caustic Field.- 5.1.8 Estimating the Coefficient of the Airy Function Derivative.- 5.1.9 The Geometric Backbone and Wave “Flesh”.- 5.1.10 Uniform Airy Asymptotic of an EM Field.- 5.1.11 Local Asymptotic of an EM Field.- 5.1.12 One-Dimensional Problem.- 5.1.13 Applicability Conditions for the Airy Asymptotic.- 5.2 Uniform Caustic Asymptotics Based on General Standard Integrals.- 5.2.1 Structure of a Solution.- 5.2.2 Equations for Phase and Amplitude Functions.- 5.2.3 Relation to Geometrical Optics.- 5.2.4 General Scheme to Compute Caustic Fields.- 5.2.5 Uniform Caustic Asymptotic of an EM Field.- 5.2.6 The Ray Skeleton and Uniform Caustic Asymptotics.- 5.2.7 Some Specific Situations.- 5.2.8 Local Asymptotics.- 5.3 Illustrative Examples.- 5.3.1 The Circular Caustic.- 5.3.2 Point Source in a Linear Slab.- 5.3.3 Swallowtail Caustics in a Linear Layer Bordering upon a Homogeneous Halfspace.- 5.3.4 Butterfly in a Parabolic Plasma Layer.- 5.3.5 Elliptic Umbilic Formed by an Antenna in a Plasma Layer.- 5.3.6 Elliptic Umbilics in Underwater Acoustics.- 5.3.7 How Far Can We Advance in Constructing Caustic Asymptotics?.- 5.3.8 Do Swallowtails Exist in Two Dimensions?.- 6 Maslov’s Method of the Canonical Operator.- 6.1 Principal Relationships.- 6.1.1 The Wave Equation in the Coordinate-Momentum Representation.- 6.1.2 Asymptotic Solution of the Wave Equation.- 6.1.3 Elimination of Field Divergence at Caustics.- 6.1.4 The Canonical Operator.- 6.1.5 Remarks on Applicability Conditions.- 6.2 Specific Problems.- 6.2.1 Plane Wave in a Linear Layer.- 6.2.2 Diffraction on a Phase Screen.- 6.2.3 Asymptotic Solution of the Parabolic Equation.- 6.2.4 Miscellaneous Problems.- 6.3. Generalization by Using Fractional Transformations.- 6.3.1 Fractional Fourier Transformation.- 6.3.2 Fractional Representation for Two-Dimensional Propagation.- 6.3.3 Construction of the Overall Field.- 6.3.4 Advantages of the Alonso-Forbes Representation.- 7 Method of Interference Integrals.- 7.1 Ray Type Integrals.- 7.1.1 Wide and Narrow Sense Interpretations.- 7.1.2 Eiconals and Amplitudes of Partial Waves.- 7.1.3 Virtual Rays.- 7.1.4 Specific Problems.- 7.2 Caustic Integrals.- 7.2.1 Airy Function Based Integrals.- 7.2.2 Use of Miscellaneous Special Functions.- 7.2.3 Specific Problems.- 7.3 Additional Topics and Generalizations.- 7.3.1 Comparison with Maslov’s Method.- 7.3.2 Implementation of Interference-Integral Algorithms.- 7.3.3 Applicability Limits.- 7.3.4 Some Generalizations.- 8 Penumbra Caustics.- 8.1 Broken Penumbra Caustics.- 8.1.1 Broken Caustics in Diffraction at Screens.- 8.1.2 A Uniform Asymptotic.- 8.1.3 Particular Cases.- 8.1.4 A Uniform Asymptotic for an EM Field.- 8.1.5 Broken Caustics of Higher Dimension.- 8.1.6 Broken Caustics at Discontinuities of Phase-Front Curvature and Jumps of Refractive Index.- 8.2 Penumbra Caustics of Diffraction Rays.- 8.2.1 Generation of Caustics.- 8.2.2 Asymptotic Solution.- 8.2.3 Properties of the Asymptotic Solution.- 8.2.4 Some Generalizations.- 8.3 Penumbra Caustics and Edge Catastrophes.- 8.3.1 Simple Edge Catastrophes.- 8.3.2 Typical Integrals of Edge Catastrophe Theory.- 8.3.3. Corner Catastrophes.- 9 Modifications and Generalizations of Standard Integrals and Functions.- 9.1 Nonpolynomial Phase Standard Integrals.- 9.1.1 Standard Integrals with Arbitrary Phase Functions.- 9.1.2 Uniform Asymptotics Based on Standard Integrals with Arbitrary Phase Functions.- 9.1.3 Bessel Function Based Uniform Asymptotics near Simple Caustics.- 9.1.4 Contour Standard Integrals.- 9.2 Structurally Unstable Caustics.- 9.2.1 Structurally Stable and Unstable Objects.- 9.2.2 Uniform Asymptotics for Axially Symmetric Caustics.- 9.2.3 A Uniform Asymptotic for an Axial Caustic.- 9.2.4 Applicability of Axial Caustic Asymptotics in the Presence of Aberrations.- 9.3 Standard Integrals with Amplitude Correction.- 9.3.1 Integrals of Weighted Rapidly Oscillating Functions.- 9.3.2 Uniform Penumbral Asymptotics near a Fuzzy Light-Shadow Boundary.- 9.3.3 Broken Caustics near Diffused Shadow.- 9.4 Reflection from a Barrier and Oscillations in a Potential Well.- 9.4.1 Weber Equation and Functions.- 9.4.2 Asymptotic Solution to One-Dimensional Reflection from a Barrier.- 9.4.3 Penetration of a Plane Wave Through a Barrier.- 9.4.4 Asymptotic Representation of the Field for a Barrier with Variable Parameters.- 9.4.5 Waveguiding Caustics.- 9.4.6 Caustics Confining “Bouncing Ball” Oscillations.- 9.4.7 Applicability of the Weber Asymptotic.- 9.5 Standard Functions Induced by Ordinary Differential Equations.- 9.5.1 Using Second-Order Differential Equations as Standards.- 9.5.2 Uniform Asymptotics of 3-D Wave Problems Developed with 1-D Standard Functions.- 9.5.3 Caustics for an Ellipsoid Cavity.- 9.5.4 Extension of EM Oscillations.- 9.5.5 Multibarrier Problems: Coupled Oscillations.- 9.5.6 Caustics with Arbitrary Order of Ray Contact.- 9.5.7 Standard Equations of Order Higher than Two.- 9.5.8 Interpolation Formulas for Oscillating Integrals.- 10 Caustics Revisited.- 10.1 Caustics in Dispersive Media.- 10.1.1 Space-Time Caustics.- 10.1.2 A Uniform Field Asymptotic for Space-Time Caustics.- 10.1.3 Caustics with Anomalous Phase Shift.- 10.1.4 Broken Space-Time Caustics.- 10.1.5 Space-Time Lenses.- 10.1.6 Uniform Asymptotics in Media with Spatial Dispersion.- 10.2 Caustics in Anisotropic Media.- 10.2.1 Description of Caustic Fields.- 10.2.2 Exceptional Directions of Radiative Transfer.- 10.2.3 Focusing of Waves at the Interface of Anisotropic and Isotropic Media.- 10.2.4 Caustics with Anomalous Phase Shift.- 10.3 Complex Caustics.- 10.4 Random Caustics.- 10.5 Caustics in Quantum Mechanical Problems.- 10.6 Concluding Remarks.- References.- List of Symbols.