Contents. 1 Basic Equations 1. Acknowledgment. 1.1 The Maxwell Equations Constitutive Relations 11

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1 Preface Foreword Acknowledgment xvi xviii xix 1 Basic Equations The Maxwell Equations Boundary Conditions at Interfaces Energy Conservation and Poynting s Theorem Constitutive Relations Isotropic Media with Dispersion Examples General Linear Media with Dispersion Energy and Passivity Time-Harmonic Fields and Fourier Transform The Maxwell Equations Constitutive Relations Poynting s Theorem, Active, Passive, and Lossless Media Sum Rules for the Constitutive Relations Reciprocity Special Type of Solutions Ellipse of Polarization Coherence and Degree of Polarization Unpolarized Field Completely Polarized Field General Degree of Polarization The Stokes Parameters The Poincaré Sphere 93 Problems for Chapter 1 94 vii

2 viii 2 The Green Functions and Dyadics The Green Functions in Isotropic Media Potentials and Gauge Transformations Canonical Problem in Homogeneous Space Non-Radiating Sources Generalizations The Green Dyadics in Isotropic Media Free-Space Green Dyadic The Green Dyadic for the Electric Field in Free Space Depolarizing Dyadic The Green Dyadic in Anisotropic Media The Green Dyadic in Biisotropic Media Čerenkov Radiation Energy Radiation Time-Domain Problem Potentials and Gauge Transformations Canonical Problem in Free Space Causality 132 Problems for Chapter Integral Representation of Fields Two Scalar Fields Integral Representation of a Scalar Field Integral Representation of a Scalar Field Alternative Vector and Scalar Fields Integral Representation of a Vector Field Integral Representation of a Vector Field Alternative Integral Representations of the Maxwell Equations Elimination of Normal Component Dyadic and Vector Fields Integral Representation of the Electric Field Dyadic Version Alternative Representation of the Electric Field Magnetic Case Limit Values of the Scalar Integral Representations Corners and Wedges Limit Values of the Vector Integral Representations Vector Version Maxwell Equations Corners and Wedges Limit Values of the Vector Integral Representations Dyadic Version 171

3 ix 3.8 Integral Representation for Biisotropic Materials Integral Representations in the Time Domain Surface Integral Representations of the Maxwell Equations 182 Problems for Chapter Introductory Scattering Theory The Far Zone Volume Integral Formulation Surface Integral Formulation Translation of the Origin Cross Sections Scattering Dyadic (Matrix) Spherical Coordinate Representation Coherency Matrix Mueller Matrix or Phase Matrix Superposition Translation of the Origin Reciprocity of the Scattering Dyadic Optical Theorem Extinction Plane Interface Case and Babinet s Principle Babinet s Principle 239 Problems for Chapter Scattering in the Time Domain The Scattering Problem The Incident Field Scattering Problem Formulation Scattered Field Far Field Amplitude Scattering Dyadic Energy Balance in the Time Domain Connection to the Time-Harmonic Results Optical Theorem Some Applications of the Optical Theorem Several Scatterers Layered Scatterers 261 Problems for Chapter 5 263

4 x 6 Approximations and Applications Long Wavelength Approximation Near Field Approximation Far Field Amplitude The Scattering Dyadic Cross Sections Internal Field Polarizability Dyadics Weak-Scatterer Approximation Born Approximation High-Frequency Approximation Aperture Formulation Reflection at a Metallic Surface Physical Optics Approximation Geometrical Optics Approximation Sum Rule for the Extinction Cross Section Additional Sum Rules Scattering by Many Scatterers Multiple Scattering Far Field Approximation Single Scattering 341 Problems for Chapter Spherical Vector Waves Preparatory Discussions Definition of Spherical Vector Waves Expansions of the Fields Orthogonality and Reciprocity Relations Spherical Scalar Waves Power Transport Some Properties of the Spherical Vector Waves Linear Independence The Translation Matrices Expansion of the Green Dyadic The Green Function in Free Space The Green Dyadic for the Electric Field in Free Space Free-Space Green Dyadic Null-Field Equations Expansion of Sources Expansion of a Plane Wave Expansion of a Vertical Electric Dipole 375

5 xi 7.8 Far Field Amplitude and the Transition Matrix Scattering Dyadic Cross Sections Generalized Optical Theorem The Decrease of the Scattered Field Dipole Moments of a Scatterer 397 Problems for Chapter Scattering by Spherical Objects Scattering by a Perfectly Conducting Sphere Long Wavelength Approximation High-Frequency Asymptotics Scattering by a Dielectric Sphere Internal Field Long Wavelength Approximation Resonances Interference Structure Scattering by Layered Spherical Objects Resonance Frequencies in a Spherical Cavity Scattering by an Anisotropic Sphere Radial Expansion Functions Transition Matrix Non-Uniqueness of the Scattering Problem Scattering by a Biisotropic Sphere Spherical Vector Waves in a Biisotropic Material The Transition Matrix for a Biisotropic Sphere Long Wavelength Approximation 472 Problems for Chapter The Null-Field Approach The T-Matrix for a Single Homogeneous Scatterer Perfectly Conducting Scatterer Dielectric Scatterer The T-Matrix for a Collection of Scatterers Iterative Solution Cross Sections Obstacle above a Ground Plane Formulation of the Problem Integral Representation of the Solution Transformation between Solutions 508

6 xii Incident Electric Field Utilizing the Surface Integral Representation Expansion and Elimination of the Surface Fields Decomposition of the Scattered Field 514 Problems for Chapter Propagation in Stratified Media Basic Equations Decomposition of Dyadics The Fundamental Equation The Fourier Transform of the Fields Decomposition of the Maxwell Equations Wave Splitting Power Flux Density Wave Splitting and Projection Dyadics Propagation of Fields the Propagator Dyadic Reflection and Transmission Dyadics Slab above Ground Composition of Two Slabs Propagator Dyadics Homogeneous Layers Single Layer Homogeneous Layer Distinct Eigenvalues (Projection Dyadics) Homogeneous Layer Distinct Eigenvalues Several Layers Examples Isotropic Media Biisotropic Media Anisotropic Media Numerical Computations Reflectivity and Transmissivity Example Dielectric Slab with Uniaxial Layers Example Bianisotropic Media Asymptotic Analysis The Green Dyadic Particular Solution or Free-Space Solution Homogeneous Solution in Free Space General Solution The Transmitted Field 580 Problems for Chapter

7 xiii APPENDIX A Vectors and Linear Transformations 583 A.1 Vectors 583 A.2 Linear Transformations, Matrices, and Dyadics 584 A.2.1 Projections 588 A.3 Rotation of Coordinate System 588 A.3.1 Euler Angles 591 A.3.2 Quaternions 592 APPENDIX B Bessel Functions 599 B.1 Bessel and Hankel Functions 599 B.1.1 Useful Integrals 604 B.2 Modified Bessel Functions 605 B.3 Spherical Bessel and Hankel Functions 607 B.3.1 Integral Representations 612 B.3.2 Modulus of a Spherical Hankel Function 615 B.3.3 Related Functions 617 APPENDIX C Spherical Harmonics 621 C.1 Legendre Polynomials 621 C.1.1 Combinations of Legendre Polynomials 623 C.2 Associated Legendre Functions 625 C.3 Spherical Harmonics 627 C.4 Vector Spherical Harmonics 630 C.5 Addition Theorem for the Legendre Polynomials 635 C.6 Transformation Formulas 637 APPENDIX D The Fourier and Other Transforms 639 D.1 The Fourier Transform 639 D.1.1 Paley Wiener Theorem 640 D.1.2 The Poisson Summation Formula 640 D.2 Hilbert Transform and Plemelj s Formulas 641 D.2.1 Integral Identities 643 D.3 Meĭman s Theorem 645 D.3.1 Zeros in the Upper Complex Half-Plane 647 D.4 Positive-Definite Functions 649 D.5 Herglotz Functions 651 D.6 The Watson Transformation 655 D.7 Zeros and Poles of an Analytic Function 656

8 xiv APPENDIX E Relativity 659 E.1 Lorentz Transformation 659 E.2 Transformation of the Electromagnetic Fields 659 E.3 Boundary Conditions at a Moving Interface 660 APPENDIX F Some Useful Mathematical Results 663 F.1 Cayley Hamilton Theorem 663 F.2 Projection Dyadics 664 F.2.1 Distinct Eigenvalues 664 F.2.2 Diagonalizable Case 667 F.2.3 Baker Campbell Hausdorff Formula 669 F.3 Hermitian Forms 669 F.3.1 Positive-Definite Dyadics and Positive-Definite Matrices 670 F.4 Möbius Transform 670 F.5 Solid Angle 672 F.6 Helmholtz Theorem 672 F.6.1 Uniqueness of the Decomposition 674 F.7 The Translation Matrices 675 F.7.1 Wigner 3-j Symbol 676 F.8 Volterra Equations 679 F.9 Vectors and Linear Operators in Hilbert Spaces 680 F.9.1 Function Spaces 681 APPENDIX G Asymptotic Evaluation of Integrals 683 G.1 One-Dimensional Case 683 G.2 Multi-Dimensional Case 684 G.3 Computation of an Integral 687 APPENDIX H The Nabla Operator in Curvilinear Coordinate Systems 689 H.1 Cartesian Coordinate System 689 H.2 Circular Cylindrical (Polar) Coordinate System 689 H.3 Spherical Coordinate System 690 APPENDIX I Notation 693 I.1 Sets 693 I.2 Volumes and Surfaces 693

9 xv I.3 Vectors and Transformations 693 I.4 Symbols and Functions 695 I.5 Real and Imaginary Parts of Numbers and Dyadics 696 I.6 Curvilinear Coordinates 697 APPENDIX J Units and Constants 699 Bibliography 701 Answers to Problems 717 Index 727