Geometrical Optics of Inhomogeneous Media

Geometrical Optics of Inhomogeneous Media

by Yury A. Kravtsov, Yury I. Orlov

Paperback(Softcover reprint of the original 1st ed. 1990)

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This monograph is concerned with the fundamentals of up-to-date geo­ metrical optics treated as an approximate method of wave theory. Geometrical optics has changed dramatically over the last two decades. Primarily, it has acquired a number of novel disciplines: space-time geo­ metrical optics, the quasi-isotropic approximation, the modern theory of caustics related to catastrophe theory, and perturbation techniques for rays, to name only a few. Another acquisition is the reliable boundaries of appli­ cability for geometrical optics, based upon the concept of the Fresnel volume for a ray. These recent additions to the field are the focus of dis­ cussion in the book. We did not attempt to separate study-oriented and illustrative material from that intended for professionals, but rather we spread it throughout the text to facilitate for the reader the mastering of this attractive, intuitively appealing and efficient ray method. In preparing the manuscript we used a set of lecture notes devised for All-Union Schools on Diffraction and Wave Propagation, published in Rus­ sian. Sections 2.1-4,6 and 10 result from joint efforts of both authors. The other material of the book we wrote separately. I contributed Sects. 2.5,9 and 3.17 and Chap.4; Yu.l. Orlov prepared the rest. Unfortunately, he could not take part in the preparation of the English edition, as he died in 1982 at the age of 41, on the verge of what would have been great achieve­ ments considering his strong and original talent.

Product Details

ISBN-13: 9783642840333
Publisher: Springer Berlin Heidelberg
Publication date: 12/30/2011
Series: Springer Series on Wave Phenomena , #6
Edition description: Softcover reprint of the original 1st ed. 1990
Pages: 312
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Introduction.- 2. The Scalar Wave Field.- 2.1 Equations of Geometrical Optics.- 2.1.1 Mathematical Background.- 2.1.2 Field Expansion in a Dimensionless Parameter.- 2.1.3 Field Expansion in Inverse Wave Numbers.- 2.1.4 Initial Conditions for the Eikonal and Amplitude Equations.- 2.1.5 Asymptotic Nature of the Ray Series.- 2.2 Rays and the Eikonal.- 2.2.1 The Method of Characteristics.- 2.2.2 Ray Equations and the Eikonal.- 2.2.3 Curvature and Torsion of Rays.- 2.2.4 Initial Conditions for Rays. Ray Coordinates.- 2.2.5 Ray Families and Phase Fronts.- 2.2.6 The Fermat Principle.- 2.2.7 Ray Equations in Curvilinear Coordinates.- 2.2.8 Other Types of Ray.- 2.3 Wave Amplitude.- 2.3.1 Formal Solution of the Transport Equation.- 2.3.2 Rays and the Direction of Energy Flow.- 2.3.3 Conservation of Energy Flux in a Ray Tube.- 2.3.4 The Field due to a Point Source in an Inhomogeneous Medium.- 2.3.5 The Resultant Field in the Ray-Optics Approximation.- 2.3.6 The Field Amplitudes of Higher-Order Approximations.- 2.3.7 Accounting for Weak Absorption.- 2.4 Caustics.- 2.4.1 Fundamental Properties.- 2.4.2 Wave-Field Focusing on Caustics.- 2.4.3 Types of Caustics.- 2.4.4 Structurally Stable and Unstable Caustics in Physical Problems.- 2.4.5 Other Types of Caustics.- 2.4.6 Singularities of Phase Fronts.- 2.4.7 Phase Shifts at Caustics.- 2.5 Reflection and Refraction of Waves at Interfaces.- 2.5.1 The Locality Principle in Wave Reflection.- 2.5.2 Relations for Rays and Eikonals.- 2.5.3 Reflection Formulas for Amplitude.- 2.5.4 Reflection from Weak Interfaces.- 2.5.5 The Geometrical Optics of Surface Waves.- 2.6 Reciprocity of Rays and Caustics.- 2.6.1 The Reciprocity Theorem.- 2.6.2 Reciprocity Relations for Rays and Caustics.- 2.7 Space-Time Geometrical Optics.- 2.7.1 The Wave Equation for Media with Temporal (Frequency) Dispersion.- 2.7.2 Necessary Conditions for the Geometrical-Optics Applied to Quasi-Monochromatic Wave Packets.- 2.7.3 Differential Form of the Constitutive Equation (2.7.2).- 2.7.4 Eikonal and Transport Equations.- 2.7.5 Space-Time Rays.- 2.7.6 Initial Conditions.- 2.7.7 Eikonal and Wave Amplitude.- 2.7.8 Space-Time Caustics.- 2.7.9 Propagation of Field Discontinuities in Nondispersive Media.- 2.8 Separation of Variables in the Eikonal Equation.- 2.8.1 The Complete Integral of the Eikonal Equation.- 2.8.2 Separation of Variables in Two Dimensions (Cartesian Coordinates).- 2.8.3 Separation of Variables in Two Dimensions (Curvilinear Orthogonal Coordinates).- 2.8.4 Separation of Variables in Three-Dimensional Space.- 2.8.5 Incomplete Separation of Variables.- 2.8.6 The Complete Integral of Eikonal and Ray Equations.- 2.9 Perturbation Techniques for Geometrical-Optics Equations.- 2.9.1 The Perturbation Method for the Eikonal.- 2.9.2 The Perturbation Method for Rays.- 2.9.3 Perturbations in Homogeneous Media.- 2.9.4 Perturbations in Nonhomogeneous Media.- 2.10 Applicability of Geometrical Optics.- 2.10.1 Existent Estimators of Method’s Errors.- 2.10.2 Fresnel Zones and Fresnel Volume of Rays in Inhomogeneous Media.- 2.10.3 The Physical Meaning of the Ray.- 2.10.4 Heuristic Criteria on Geometrical-Optics Applicability.- 2.10.5 Applicability Conditions for Space-Time Geometrical Optics.- 2.10.6 Heuristic Accuracy Estimates of Geometrical Optics.- 2.10.7 Estimating the Width of a Caustic Zone.- 2.10.8 Indistinguishability of Rays in the Caustic Zone.- 2.10.9 Observability of Caustics.- 2.10.10 Field Estimations Beyond the Validity Region of Geometrical Optics.- 2.10.11 Field-Focusing Indices at Caustics.- 2.10.12 Stability with Respect to Small Perturbations.- 2.10.13 Wave-Pattern Analysis in General.- 3. Applications of the Ray Methods.- 3.1 Waves in Homogeneous Media.- 3.1.1 Rays and the Eikonal.- 3.1.2 The Wave Amplitude.- 3.1.3 Caustics.- 3.1.4 The Plane Phase-Amplitude Screen.- 3.1.5 The Sinusoidal Phase Screen. An Illustrative Example.- 3.1.6 Applicability Conditions for Geometrical Optics.- 3.1.7 Geometrical Optics in Far and Near Antenna Fields. Wave Beam Propagation.- 3.1.8 On the Phase Center of an Antenna or a Scatterer.- 3.1.9 Field Near a Lens Focus.- 3.1.10 Field at the Focus of a Lens with Cylindrical (Spherical) Aberration.- 3.2 Reflection and Refraction at an Interface Between Homogeneous Media.- 3.2.1 Reflection Formulas.- 3.2.2 Divergence of Reflected and Refracted Rays.- 3.2.3 Effective Scattering Surface of a Body in the Geometrical-Optics Approximation.- 3.2.4 Reflection Far Field of a Directional Point Source.- 3.2.5 Caustics of Refracted and Reflected Rays.- 3.2.6 Examples of Catacaustics and Diacaustics.- 3.2.7 Applicability of Reflection Formulas.- 3.2.8 The Invalidity Domain in the Vicinity of a Tangent Ray.- 3.2.9 Wave Diffraction at a Surface of Variable Impedance.- 3.3 Rays and Caustics in Plane-Stratified Media.- 3.3.1 Ray Equations.- 3.3.2 Ray Tracing in a Plane-Stratified Medium.- 3.3.3 Equations of Caustics, and the Geometry of the Ray Family.- 3.3.4 Rays and Caustics due to a Point Source in an Inhomogeneous Medium.- 3.3.5 Rays and Caustics in a Linear Layer.- 3.3.6 Layers of Other Profiles.- 3.3.7 Plane Waves in a Parabolic Layer.- 3.3.8 A Point Source in a Parabolic Layer.- 3.4 Wave Fields in Plane-Stratified Media.- 3.4.1 The Field of an Arbitrary Wave.- 3.4.2 The Field of a Plane Wave.- 3.4.3 Fields of Point and Linear Sources.- 3.4.4 A Point Source in a Linear Layer.- 3.4.5 A Point Source in a Parabolic Layer.- 3.4.6 The Fresnel Volumes in Plane-Stratified Media.- 3.4.7 Validity Conditions of the Geometrical-Optics Approximation.- 3.5 Waves in Radially Inhomogeneous Media.- 3.5.1 Ray Equations for Spherically Stratified Media.- 3.5.2 The Eikonal Function for Spherically Stratified Media.- 3.5.3 Cylindrically Stratified Media.- 3.5.4 Ray Geometry.- 3.5.5 The Field due to a Point Source.- 3.5.6 The Field of a Plane Wave.- 3.5.7 Caustics.- 3.6 Tapered and Other Inhomogeneous Media.- 3.6.1 The Eikonal and Rays in a Tapered Medium.- 3.6.2 The Field of a Plane Wave.- 3.6.3 The Field due to a Linear Source.- 3.6.4 Ray Equations in a Two-Dimensional Medium with a Special Profile.- 3.6.5 The Field of a Point Source (Axially Symmetric Problem).- 3.6.6 A Plane Wave Incident on the Two-Dimensionally Inhomogeneous Medium.- 3.6.7 Weakly Inhomogeneous, Quasi-Stratified, and Random Media.- 3.7 Geometrical Optics of Waveguides and Resonators.- 3.7.1 Geometrical Optics of Waveguides.- 3.7.2 Ray Description of Modes in Uniform Waveguides.- 3.7.3 Adiabatic Modes of Smoothly Nonuniform Waveguides.- 3.7.4 Ionospheric Wave Channels. The Adiabatic Invariant Method.- 3.7.5 Underwater-Sound Ducts. Summing-Up Incoherent Wave Fields.- 3.7.6 Optical Fibers.- 3.7.7 Mode Conversion in Smoothly Nonuniform Waveguides.- 3.7.8 Normal Modes in Cavity Resonators.- 3.8 Wave Scattering at Localized Inhomogeneities.- 3.8.1 Effective Scattering Surface.- 3.8.2 Scattering by a Body in an Inhomogeneous Medium.- 3.8.3 Effective Scattering Surface of a Spherically Stratified Inhomogeneity.- 3.8.4 Effective Scattering Surface of a Perfectly Conducting Sphere in a Spherically Stratified Medium.- 3.8.5 Effective Scattering Surface of a Specific Two-Dimensionally Inhomogeneous Formation.- 3.8.6 Scattering of a Spherical Wave by a Localized Inhomogeneity.- 3.8.7 Scattering by Weak Localized Inhomogeneities.- 3.9 Pulse Propagation.- 3.9.1 General Relations for the Plasma (Guided) Dispersion Law.- 3.9.2 A Homogeneous Medium with an Arbitrary Dispersion Law.- 3.9.3 A Plane, Frequency-Modulated Pulse in a Homogeneous Medium.- 3.9.4 Dispersive Compression of FM Pulses in Homogeneous Media.- 3.9.5 Plane-Stratified Dispersive Media.- 3.9.6 Near and Far Fields of a Pulse.- 3.10 Numerical Methods in the Geometrical Optics of Inhomogeneous Media.- 3.10.1 The Ray-Tracing Analysis.- 3.10.2 Computing the Eikonal and Wave Amplitude.- 3.10.3 Problems of Numerical Analysis.- 3.11 Inverse Problems of Geometrical Optics.- 3.11.1 Reflection and Refraction at Interfaces.- 3.11.2 Inverse Problems for Given Models of the Inhomogeneous Medium.- 3.11.3 Multidimensional Inverse Problems.- 3.11.4 Nonstationary Inverse Problems.- 4. Vector Wave Fields.- 4.1 Transverse Electromagnetic Waves in Isotropic Media.- 4.1.1 Maxwell Equations for Monochromatic Waves.- 4.1.2 The Debye Expansion and the Iterative Equations.- 4.1.3 The Eikonal Equation.- 4.1.4 Transverse Nature of Zero Approximation Waves. Polarization Degeneracy.- 4.1.5 Consistency of the First-Approximation Equations.- 4.1.6 Conserving Energy Flow in a Ray Tube.- 4.1.7 Preserving the Polarization Ellipse.- 4.1.8 Rotation of Field Vectors (Rytov’s Law).- 4.1.9 Polarization of Transverse Waves.- 4.1.10 Longitudinal Components of the Field.- 4.1.11 Reflection of Transverse Electromagnetic Waves from Interfaces.- 4.1.12 Polarization Degeneracy in Problems of Quantum Mechanics and Theory of Elasticity.- 4.2 Independent Normal Waves in an Anisotropic Medium.- 4.2.1 Equation of the Eikonal.- 4.2.2 Independent Normal Mode.- 4.2.3 Ray Equations.- 4.2.4 Solving the Eikonal Equation.- 4.2.5 Definition of Mode Polarization Vectors.- 4.2.6 Consistency of Equations of the First Approximation.- 4.2.7 The Transfer Equation.- 4.2.8 Equation for the Argument of a Complex Amplitude.- 4.2.9 Rays and Energy Paths. The Fresnel Volumes in Anisotropic Media.- 4.2.10 An Account of Weak Absorption.- 4.2.11 Reflection and Refraction at the Boundaries of Anisotropic Media.- 4.2.12 Some Specific Results.- a) The Field of a Point Source in an Anisotropic Medium.- b) Waves in Plane-Stratified Anisotropic Media.- c) Separation of Variables in the Equation of Eikonal in the General Case.- d) Perturbation Theory.- 4.2.13 Divergence of First-Approximation Fields at Polarization Degeneracy.- 4.2.14 Other Vector Problems.- 4.3 Interaction of Normal Modes in Inhomogeneous Anisotropic Media.- 4.3.1 Waves in Weakly Anisotropic Media. The Quasi-Isotropic Approximation.- 4.3.2 Different Forms of the Equations of the Quasi-Isotropic Approximation.- 4.3.3 Solution Techniques for the Quasi-Isotropic Approximation.- 4.3.4 On the Error of the Quasi-Isotropic Approximation.- 4.3.5 Applications of the Quasi-Isotropic Approximation.- 4.3.6 The Quasi-Degenerate Approximation of Geometrical Optics.- 4.4 Equations of Geometrical Optics for Nonharmonic Electromagnetic Waves in the General Case of Inhomogeneous and Nonstationary Dispersive Media.- 4.4.1 The Maxwell Equations in Inhomogeneous and Nonstationary Media of Temporal and Spatial Dispersion.- 4.4.2 The Constitutive Equation in Differential Form.- 4.4.3 Equations for the Fields in Zeroth and First Approximations.- 4.4.4 The Eikonal Equation. Space-Time Rays.- 4.4.5 The Transfer Equation for Independent Normal Modes in an Anisotropic Medium.- 4.4.6 The Group Velocity Theorem.- 4.4.7 Integrating the Transfer Equation Along Space-Time Rays.- 4.4.8 Transverse Modes in an Isotropic Medium.- 4.4.9 Longitudinal Waves in an Isotropic Medium.- 4.4.10 Waves in Weakly Anisotropic Media.- 4.5 Constitutive Equations for Nonstationary and Inhomogeneous Dispersive Media. Existence of Adiabatic Invariance.- 4.5.1 Corrections to the Quasi-Stationary Permittivity Tensor.- 4.5.2 Physical Phenomena Due to the Deviation of ??? from Its Quasi-Stationary Value.- 4.5.3 Existence of the Adiabatic Invariant.- 4.5.4 Phenomenological Evaluation of the Anti-Hermitian Part of the Correction for the Quasi-Stationary Permittivity Tensor in Transparent Media.- 4.6 Wave Processes in Nonstationary Media.- 4.6.1 One-Dimensional Problem. General Relationships.- 4.6.2 Nonstationary Nondispersive Media.- 4.6.3 Nonstationary Dispersive Media.- 4.6.4 Evolution of Short Pulses.- 4.6.5 Reflection from Moving Interfaces.- 4.6.6 Perturbation Theory in Nonstationary Problems.- 5. Conclusion.- References.

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