Waves and Fields in Inhomogenous Media

Weng Cho Chew is the author of Waves and Fields in Inhomogenous Media, published by Wiley.

• PREFACE xviiACKNOWLEDGMENTS xxi1 PRELIMINARY BACKGROUND 11.1 Maxwell's Equations 11.1.1 Differential Representations 11.1.2 Integral Representations 31.1.3 Time Harmonic Forms 41.1.4 Constitutive Relations 51.1.5 Poynting Theorem and Lossless Conditions 61.1.6 Duality Principle 91.2 Scalar Wave Equations 91.2.1 Acoustic Wave Equation 101.2.2 Scalar Wave Equation from Electromagnetics 121.2.3 Cartesian Coordinates 121.2.4 Cylindrical Coordinates 141.2.5 Spherical Coordinates 161.3 Vector Wave Equations 171.3.1 Boundary Conditions 181.3.2 Reciprocity Theorem 201.3.3 Plane Wave in Homogeneous, Anisotropic Media 221.3.4 Green's Function 241.4 Huygens' Principle 291.4.1 Scalar Waves 291.4.2 Electromagnetic Waves 311.5 Uniqueness Theorem 321.5.1 Scalar Wave Equation 331.5.2 Vector Wave Equation 35Exercises for Chapter 1 37References for Chapter 1 41Further Readings for Chapter 1 422 PLANARLY LAYERED MEDIA 452.1 One-Dimensional Planar Inhomogeneity 452.1.1 Derivation of the Scalar Wave Equations 452.1.2 Reflection from a Half-Space 482.1.3 Reflection and Transmission in a Multilayered Medium 492.1.4 Ricatti Equation for Reflection Coefficients 532.1.5 Specific Inhomogeneous Profiles 562.2 Spectral Representations of Sources 572.2.1 A Line Source 582.2.2 A Point Source 632.2.3 Riemann Sheets and Branch Cuts 662.3 A Source on Top of a Layered Medium 702.3.1 Electric Dipole Fields 712.3.2 Magnetic Dipole Fields 742.3.3 The Transverse Field Components 752.4 A Source Embedded in a Layered Medium 762.5 Asymptotic Expansions of Integrals 792.5.1 Method of Stationary Phase 792.5.2 Method of Steepest Descent 822.5.3 Uniform Asymptotic Expansions 872.6 Dipole Over Layered Media—Asymptotic Expansions 932.6.1 Dipole Over Half-Space (VMD) 932.6.2 Dipole Over Half-Space (VED) 982.6.3 Dipole Over a Slab 1012.6.4 Example of Uniform Asymptotic Expansion —Transmitted Wave in a Half-Space 1062.6.5 Angular Spectrum Representation 1102.7 Singularities of the Sommerfeld Integrals 1112.7.1 Absence of Branch Points 1122.7.2 Bounds on the Locations of Singularities 1142.7.3 Numerical Integration of Sommerfeld Integrals 1182.8 WKB Method 1212.8.1 Derivation of the WKB Solution 1212.8.2 Asymptotic Matching 1242.9 Propagator Matrix 1282.9.1 Derivation of the State Equation 1292.9.2 Solution of the State Equation 1292.9.3 Reflection from a Three-Layer Medium 1302.9.4 Reflection from an Inhomogeneous Slab 1312.10 Waves in Anisotropic, Layered Media 1332.10.1 Derivation of the State Equation 1332.10.2 Solution of the State Equation 1352.10.3 Reflection from an Interface of Anisotropic Half Spaces 1362.10.4 Reflection from a Slab 1372.10.5 Geometrical Optics Series 138Exercises for Chapter 2 140References for Chapter 2 151Further Readings for Chapter 2 1553 CYLINDRICALLY AND SPHERICALLY LAYERED MEDIA 1613.1 Cylindrically Layered Media—Single Interface Case 1613.1.1 Vector Wave Equation in Cylindrical Coordinates 1623.1.2 Reflection and Transmission of an Outgoing Wave 1633.1.3 Reflection and Transmission of a Standing Wave 1653.2 Cylindrically Layered Media—Multi-Interface Case 1673.2.1 The Outgoing-Wave Case 1673.2.2 The Standing-Wave Case 1703.3 Source in a Cylindrically Layered Medium 1723.3.1 Discrete, Angular-Wave-Number Representation 1733.3.2 Continuum, Angular-Wave-Number Representation 1773.4 Propagator Matrix—Cylindrical Layers 1793.4.1 Isotropic, Layered Media 1793.4.2 Anisotropic, Layered Media 1823.5 Spherically Layered Media—Single Interface Case 1843.5.1 Vector Wave Equation in Spherical Coordinates 1853.5.2 Reflection and Transmission of an Outgoing Wave 1873.5.3 Reflection and Transmission of a Standing Wave 1893.6 Spherically Layered Media—Multi-Interface Case 1913.6.1 The Outgoing-Wave Case 1913.6.2 The Standing-Wave Case 1923.7 Source in a Spherically Layered Medium 1933.8 Propagator Matrix—Spherical Layers 197Exercises for Chapter 3 199References for Chapter 3 204Further Readings for Chapter 3 2064 TRANSIENTS 2114.1 Causality of Transient Response 2114.1.1 The Kramers-Kronig Relation 2124.1.2 Causality and Contour of Integration 2144.2 The Cagniard-de Hoop Method 2154.2.1 Line Source in Free-Space—Two-Dimensional Green's Function 2164.2.2 Point Source in Free-Space—Three-Dimensional Green's Function 2194.2.3 Line Source Over Half-Space—Transient Response 2214.2.4 Dipole Over Half Space—Transient Response 2244.3 Multi-interface Problems 2274.4 Direct Inversion 2284.5 Numerical Integration of Fourier Integrals 2314.5.1 Direct Field in a Lossy Medium—Two- Diemnsional Case 2324.5.2 Direct Field in a Lossy Medium—Three- Dimensional Case 2334.6 Finite-Difference Method 2354.6.1 The Finite-Difference Approximation 2364.6.2 Stability Analysis 2394.6.3 Grid-Dispersion Error 2424.6.4 The Yee Algorithm 2444.7 Absorbing Boundary Conditions 2464.7.1 Engquist-Majda Absorbing Boundary Condition 2464.7.2 Lindman Absorbing Boundary Condition 2494.7.3 Bayliss-Turkel Absorbing Boundary Condition 2504.7.4 Liao's Absorbing Boundary Condition 251Exercises for Chapter 4 256References for Chapter 4 262Further Readings for Chapter 4 2655 VARIATIONAL METHODS 2715.1 Review of Linear Vector Space 2715.1.1 Inner Product Spaces 2715.1.2 Linear Operators 2745.1.3 Basis Functions 2755.1.4 Parseval's Theorem 2785.1.5 Parseval's Theorem for Complex Vectors 2795.1.6 Solutions to Operator Equations—A Preview 2805.1.7 The Eigenvalue Problem 2845.2 Variational Expressions for Self-Adjoint Problems 2855.2.1 General Concepts 2855.2.2 Rayleigh-Ritz Procedure—Self-Adjoint Problems 2885.2.3 Applications to Scalar Wave Equations 2915.2.4 Applications to Vector Wave Equations 2935.3 Variational Expressions for Non-Self-Adjoint Problems 2955.3.1 General Concepts 2955.3.2 Rayleigh-Ritz Procedure—Non-Self-Adjoint Problems 2975.3.3 Applications to Scalar Wave Equations 2985.3.4 Applications to Vector Wave Equations 2995.4 Variational Expressions for Eigenvalue Problems 3015.4.1 General Concepts 3015.4.2 Applications to Scalar Wave Equations 3035.4.3 Applications to Electromagnetic Problems 3045.5 Essential and Natural Boundary Conditions 3085.5.1 The Scalar Wave Equation Case 3085.5.2 The Electromagnetic Case 312Exercises for Chapter 5 315References for Chapter 5 321Further Readings for Chapter 5 3236 MODE MATCHING METHOD 3276.1 Eigenmodes of a Planarly Layered Medium 3276.1.1 Orthogonality of Eigenmodes in a Layered Medium 3286.1.2 Guided Modes and Radiation Modes of a Layered Medium 3306.2 Eigenfunction Expansion of a Field 3356.2.1 Excitation of Modes due to a Line Source 3356.2.2 The Use of Vector Notation 3376.3 Reflection and Transmission at a Junction Discontinuity 3406.3.1 Derivation of Reflection and Transmission Operators 3416.3.2 The Continuum Limit Case 3436.4 A Numerical Method to Find the Eigenmodes 3466.5 The Cylindrically Layered Medium Case 3516.5.1 Eigenmodes of a Cylindrically Layered Medium 3516.5.2 Differential Equations of a Cylindrical Structure 3536.5.3 Numerical Solution of the Eigenmodes 3546.5.4 Eigenfunction Expansion of a Field 3566.5.5 Reflection from a Junction Discontinuity 3586.6 The Multiregion Problem 3606.6.1 The Three-Region Problem 3606.6.2 The iV-Region Problem 362Exercises for Chapter 6 365References for Chapter 6 370Further Readings for Chapter 6 3727 DYADIC GREEN'S FUNCTIONS 3757.1 Dyadic Green's Function in a Homogeneous Medium 3757.1.1 The Spatial Representation 3767.1.2 The Singularity of the Dyadic Green's Function 3787.1.3 The Spectral Representation 3817.1.4 Equivalence of Spectral and Spatial Representations 3847.2 Vector Wave Functions 3877.2.1 Derivation of Vector Wave Functions 3877.2.2 Orthogonality Relationships of Vector Wave Functions 3887.2.3 Vector Wave Functions for Unbounded Media 3937.3 Dyadic Green's Function Using Vector Wave Functions 3977.3.1 The Integral Representations 3977.3.2 Singularity Extraction 3997.4 Dyadic Green's Functions for Layered Media 4107.4.1 A General, Isotropic, Inhomogeneous Medium 4107.4.2 Planarly Layered Media 4117.4.3 Cylindrically Layered Media 4147.4.4 Spherically Layered Media 4167.4.5 Reciprocity Considerations 418Exercises for Chapter 7 421References for Chapter 7 424Further Readings for Chapter 7 4268 INTEGRAL EQUATIONS 4298.1 Surface Integral Equations 4308.1.1 Scalar Wave Equation 4308.1.2 Vector Wave Equation 4338.1.3 The Anisotropic, Inhomogeneous Medium Case 4378.1.4 Two-Dimensional Electromagnetic Case 4398.2 Solutions by the Method of Moments 4438.2.1 Scalar Wave Case 4438.2.2 The Electromagnetic Case 4468.2.3 Problem with Internal Resonances 4518.3 Extended-Boundary-Condition Method 4538.3.1 The Scalar Wave Case 4538.3.2 The Electromagnetic Wave Case 4578.4 The Transition and Scattering Matrices 4598.5 The Method of Rayleigh's Hypothesis 4608.6 Scattering by Many Scatterers 4638.6.1 Two-Scatterer Solution 4638.6.2 iV-Scatterer Solution—A Recursive Algorithm 4658.7 Scattering by Multilayered Scatterers 4698.7.1 One-Interface Problem 4698.7.2 Many-Interface Problems 4718.8 Surface Integral Equation with Finite-Element Method 4758-9 Volume Integral Equations 4798.9.1 Scalar Wave Case 4808.9.2 The Electromagnetic Wave Case 4818.9.3 Matrix Representation of the Integral Equation 4838.10 Approximate Solutions of the Scattering Problem 4848.10.1 Born Approximation 4858.10.2 Rytov Approximation 487Exercises for Chapter 8 490References for Chapter 8 501Further Readings for Chapter 8 5059 INVERSE SCATTERING PROBLEMS 5119.1 Linear Inverse Problems 5119.1.1 Back-Projection Tomography 5149.1.2 Radon Transforms 5169.1.3 Diffraction Tomography 5199.1.4 Finite-Source Effect 5229.1.5 Nonuniqueness of the Solution 5249.2 One-Dimensional Inverse Problems 5269.2.1 The Method of Characteristics 5269.2.2 Transformation to a Schrodinger-like Equation 5329.2.3 The GePfand-Levitan Integral Equation 5349.2.4 The Marchenko Integral Equation 5419.2.5 The GePfand-Levitan-Marchenko Integral Equation 5439.3 Higher-Dimensional Inverse Problems 5479.3.1 Distorted Born Iterative Method 5489.3.2 Born Iterative Method 5539.3.3 Operator Forms of the Scattering Equations 554Exercises for Chapter 9 557References for Chapter 9 563Further Readings for Chapter 9 566APPENDIX A Some Useful Mathematical Formulas 571A.I Useful Vector Identities 571A.2 Gradient, Divergence, Curl, and Laplacian in Rectangular, Cylindrical, Spherical, and General Orthogonal Curvilinear Coordinate Systems 571A.3 Useful Integral Identities 573A.4 Integral Transforms 574APPENDIX B Review of Tensors 577APPENDIX C Generalized Functions 583APPENDIX D Addition Theorems 591References for Appendices 597Further Readings for Appendices 598INDEX 599