WAVES AND FIELDS IN INHOMOGENEOUS MEDIA
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1 WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society, Sponsor IEEE PRESS The Institute of Electrical and Electronics Engineers, Inc., New York
2 PREFACE xvii ACKNOWLEDGMENTS xxi 1 PRELIMINARY BACKGROUND Maxwell's Equations Differential Representations Integral Representations Time Harmonic Forms Constitutive Relations Poynting Theorem and Lossless Conditions Duality Principle Scalar Wave Equations Acoustic Wave Equation Scalar Wave Equation from Electromagnetics Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates Vector Wave Equations Boundary Conditions Reciprocity Theorem Plane Wave in Homogeneous, Anisotropic Media Green's Function Huygens' Principle Scalar Waves Electromagnetic Waves Uniqueness Theorem Scalar Wave Equation Vector Wave Equation 35 Exercises for Chapter 1 37 References for Chapter 1 41 Further Readings for Chapter 1 42
3 viii CONTENTS 2 PLANARLY LAYERED MEDIA One-Dimensional Planar Inhomogeneity Derivation of the Scalar Wave Equations Reflection from a Half-Space Reflection and Transmission in a Multilayered Medium Ricatti Equation for Reflection Coefficients Specific Inhomogeneous Profiles Spectral Representations of Sources A Line Source A Point Source Riemann Sheets and Branch Cuts A Source on Top of a Layered Medium Electric Dipole Fields Magnetic Dipole Fields The Transverse Field Components A Source Embedded in a Layered Medium Asymptotic Expansions of Integrals Method of Stationary Phase Method of Steepest Descent Uniform Asymptotic Expansions Dipole Over Layered Media Asymptotic Expansions Dipole Over Half-Space (VMD) Dipole Over Half-Space (VED) Dipole Over a Slab Example of Uniform Asymptotic Expansion Transmitted Wave in a Half-Space Angular Spectrum Representation Singularities of the Sommerfeld Integrals Absence of Branch Points Bounds on the Locations of Singularities Numerical Integration of Sommerfeld Integrals WKB Method Derivation of the WKB Solution Asymptotic Matching 124
4 ix 2.9 Propagator Matrix Derivation of the State Equation Solution of the State Equation Reflection from a Three-Layer Medium Reflection from an Inhomogeneous Slab Waves in Anisotropic, Layered Media Derivation of the State Equation Solution of the State Equation Reflection from an Interface of Anisotropic Half Spaces Reflection from a Slab Geometrical Optics Series 138 Exercises for Chapter References for Chapter Further Readings for Chapter CYLINDRICALLY AND SPHERICALLY LAYERED MEDIA Cylindrically Layered Media Single Interface Case Vector Wave Equation in Cylindrical Coordinates Reflection and Transmission of an Outgoing Wave Reflection and Transmission of a Standing Wave Cylindrically Layered Media Multi-Interface Case The Outgoing-Wave Case The Standing-Wave Case Source in a Cylindrically Layered Medium Discrete, Angular-Wave-Number Representation Continuum, Angular-Wave-Number Representation Propagator Matrix Cylindrical Layers Isotropic, Layered Media Anisotropic, Layered Media Spherically Layered Media Single Interface Case Vector Wave Equation in Spherical Coordinates Reflection and Transmission of an Outgoing Wave Reflection and Transmission of a Standing Wave Spherically Layered Media Multi-Interface Case 191
5 3.6.1 The Outgoing-Wave Case The Standing-Wave Case Source in a Spherically Layered Medium Propagator Matrix Spherical Layers 197 Exercises for Chapter References for Chapter Further Readings for Chapter TRANSIENTS Causality of Transient Response The Kramers-Kronig Relation Causality and Contour of Integration The Cagniard-de Hoop Method Line Source in Free-Space Two-Dimensional Green's Function Point Source in Free-Space Three-Dimensional Green's Function Line Source Over Half-Space Transient Response Dipole Over Half Space Transient Response Multi-interface Problems Direct Inversion Numerical Integration of Fourier Integrals Direct Field in a Lossy Medium Two- Diemnsional Case Direct Field in a Lossy Medium Three- Dimensional Case Finite-Difference Method The Finite-Difference Approximation Stability Analysis Grid-Dispersion Error The Yee Algorithm Absorbing Boundary Conditions Engquist-Majda Absorbing Boundary Condition Lindman Absorbing Boundary Condition Bayliss-Turkel Absorbing Boundary Condition Liao's Absorbing Boundary Condition 251
6 xi Exercises for Chapter References for Chapter Further Readings for Chapter VARIATIONAL METHODS Review of Linear Vector Space Inner Product Spaces Linear Operators Basis Functions Parseval's Theorem Parseval's Theorem for Complex Vectors Solutions to Operator Equations A Preview The Eigenvalue Problem Variational Expressions for Self-Adjoint Problems General Concepts Rayleigh-Ritz Procedure Self-Adjoint Problems Applications to Scalar Wave Equations Applications to Vector Wave Equations Variational Expressions for Non-Self-Adjoint Problems General Concepts Rayleigh-Ritz Procedure Non-Self-Adjoint Problems Applications to Scalar Wave Equations Applications to Vector Wave Equations Variational Expressions for Eigenvalue Problems General Concepts Applications to Scalar Wave Equations Applications to Electromagnetic Problems Essential and Natural Boundary Conditions The Scalar Wave Equation Case The Electromagnetic Case 312 Exercises for Chapter References for Chapter Further Readings for Chapter 5 323
7 xii CONTENTS 6 MODE MATCHING METHOD Eigenmodes of a Planarly Layered Medium Orthogonality of Eigenmodes in a Layered Medium Guided Modes and Radiation Modes of a Layered Medium Eigenfunction Expansion of a Field Excitation of Modes due to a Line Source The Use of Vector Notation Reflection and Transmission at a Junction Discontinuity Derivation of Reflection and Transmission Operators The Continuum Limit Case A Numerical Method to Find the Eigenmodes The Cylindrically Layered Medium Case Eigenmodes of a Cylindrically Layered Medium Differential Equations of a Cylindrical Structure Numerical Solution of the Eigenmodes Eigenfunction Expansion of a Field Reflection from a Junction Discontinuity The Multiregion Problem The Three-Region Problem The iv-region Problem 362 Exercises for Chapter References for Chapter Further Readings for Chapter DYADIC GREEN'S FUNCTIONS Dyadic Green's Function in a Homogeneous Medium The Spatial Representation The Singularity of the Dyadic Green's Function The Spectral Representation Equivalence of Spectral and Spatial Representations Vector Wave Functions Derivation of Vector Wave Functions Orthogonality Relationships of Vector Wave Functions 388
8 xiii Vector Wave Functions for Unbounded Media Dyadic Green's Function Using Vector Wave Functions The Integral Representations Singularity Extraction Dyadic Green's Functions for Layered Media A General, Isotropic, Inhomogeneous Medium Planarly Layered Media Cylindrically Layered Media Spherically Layered Media Reciprocity Considerations 418 Exercises for Chapter References for Chapter Further Readings for Chapter INTEGRAL EQUATIONS Surface Integral Equations Scalar Wave Equation Vector Wave Equation The Anisotropic, Inhomogeneous Medium Case Two-Dimensional Electromagnetic Case Solutions by the Method of Moments Scalar Wave Case The Electromagnetic Case Problem with Internal Resonances Extended-Boundary-Condition Method The Scalar Wave Case The Electromagnetic Wave Case The Transition and Scattering Matrices The Method of Rayleigh's Hypothesis Scattering by Many Scatterers Two-Scatterer Solution iv-scatterer Solution A Recursive Algorithm Scattering by Multilayered Scatterers One-Interface Problem Many-Interface Problems 471
9 xiv CONTENTS 8.8 Surface Integral Equation with Finite-Element Method Volume Integral Equations Scalar Wave Case The Electromagnetic Wave Case Matrix Representation of the Integral Equation Approximate Solutions of the Scattering Problem Born Approximation Rytov Approximation 487 Exercises for Chapter References for Chapter Further Readings for Chapter INVERSE SCATTERING PROBLEMS Linear Inverse Problems Back-Projection Tomography Radon Transforms Diffraction Tomography Finite-Source Effect Nonuniqueness of the Solution One-Dimensional Inverse Problems The Method of Characteristics Transformation to a Schrodinger-like Equation The Gel'fand-Levitan Integral Equation The Marchenko Integral Equation The Gel'fand-Levitan-Marchenko Integral Equation Higher-Dimensional Inverse Problems Distorted Born Iterative Method Born Iterative Method Operator Forms of the Scattering Equations 554 Exercises for Chapter References for Chapter Further Readings for Chapter APPENDIX A Some Useful Mathematical Formulas 571 A.l Useful Vector Identities 571
10 xv A.2 Gradient, Divergence, Curl, and Laplacian in Rectangular, Cylindrical, Spherical, and General Orthogonal Curvilinear Coordinate Systems 571 A.3 Useful Integral Identities 573 A.4 Integral Transforms 574 APPENDIX B Review of Tensors 577 APPENDIX C Generalized Functions 583 APPENDIX D Addition Theorems 591 References for Appendices 597 Further Readings for Appendices 598 INDEX 599
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