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Physics - Classical Continuum Physics | Caustics, Catastrophes and Wave Fields

Caustics, Catastrophes and Wave Fields

Kravtsov, Yu.A., Orlov, Yu.I.

Translated by Edelev, M.G.

2nd ed. 1993, XII, 216 pp. 60 figs.

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  • About this book

Caustics, Catastrophes and Wave Fields in a sense continues the treatment of the earlier volume 6 "Geometrical Optics of Inhomogeneous Media" in the present book series, by analysing caustics and their fields on the basis of modern catastrophe theory. This volume covers the key generalisations of geometrical optics related to caustic asymptotic expansions: The Lewis-Kravtsov method of standard functions, Maslov's method of canonical operators, Orlov's method of interference integrals, as well as their modifications for penumbra, space-time, random and other types of caustics. All the methods are amply illustrated by worked problems concerning relevant wave-field applications.

Content Level » Research

Keywords » Oscillation - Phase - Wave - acoustics - asymptotics - caustics - geometrical optics - interference - optics - uniform caustics

Related subjects » Classical Continuum Physics - Optics & Lasers

Table of contents 

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.

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