Preface Foreword Acknowledgment xvi xviii xix 1 Basic Equations 1 1.1 The Maxwell Equations 1 1.1.1 Boundary Conditions at Interfaces 4 1.1.2 Energy Conservation and Poynting s Theorem 9 1.2 Constitutive Relations 11 1.2.1 Isotropic Media with Dispersion 12 1.2.2 Examples 20 1.2.3 General Linear Media with Dispersion 28 1.2.4 Energy and Passivity 34 1.3 Time-Harmonic Fields and Fourier Transform 38 1.3.1 The Maxwell Equations 41 1.3.2 Constitutive Relations 42 1.3.3 Poynting s Theorem, Active, Passive, and Lossless Media 57 1.3.4 Sum Rules for the Constitutive Relations 65 1.3.5 Reciprocity 70 1.3.6 Special Type of Solutions 73 1.3.7 Ellipse of Polarization 74 1.4 Coherence and Degree of Polarization 83 1.4.1 Unpolarized Field 88 1.4.2 Completely Polarized Field 88 1.4.3 General Degree of Polarization 89 1.4.4 The Stokes Parameters 90 1.4.5 The Poincaré Sphere 93 Problems for Chapter 1 94 vii
viii 2 The Green Functions and Dyadics 99 2.1 The Green Functions in Isotropic Media 99 2.1.1 Potentials and Gauge Transformations 100 2.1.2 Canonical Problem in Homogeneous Space 103 2.1.3 Non-Radiating Sources 106 2.1.4 Generalizations 108 2.2 The Green Dyadics in Isotropic Media 112 2.2.1 Free-Space Green Dyadic 112 2.2.2 The Green Dyadic for the Electric Field in Free Space 113 2.2.3 Depolarizing Dyadic 115 2.3 The Green Dyadic in Anisotropic Media 117 2.4 The Green Dyadic in Biisotropic Media 118 2.5 Čerenkov Radiation 120 2.5.1 Energy Radiation 126 2.6 Time-Domain Problem 128 2.6.1 Potentials and Gauge Transformations 129 2.6.2 Canonical Problem in Free Space 130 2.6.3 Causality 132 Problems for Chapter 2 137 3 Integral Representation of Fields 141 3.1 Two Scalar Fields 141 3.1.1 Integral Representation of a Scalar Field 142 3.1.2 Integral Representation of a Scalar Field Alternative 144 3.2 Vector and Scalar Fields 150 3.2.1 Integral Representation of a Vector Field 151 3.2.2 Integral Representation of a Vector Field Alternative 153 3.3 Integral Representations of the Maxwell Equations 154 3.3.1 Elimination of Normal Component 156 3.4 Dyadic and Vector Fields 157 3.4.1 Integral Representation of the Electric Field Dyadic Version 158 3.4.2 Alternative Representation of the Electric Field Magnetic Case 161 3.5 Limit Values of the Scalar Integral Representations 162 3.5.1 Corners and Wedges 166 3.6 Limit Values of the Vector Integral Representations Vector Version 168 3.6.1 Maxwell Equations 170 3.6.2 Corners and Wedges 170 3.7 Limit Values of the Vector Integral Representations Dyadic Version 171
ix 3.8 Integral Representation for Biisotropic Materials 174 3.9 Integral Representations in the Time Domain 180 3.9.1 Surface Integral Representations of the Maxwell Equations 182 Problems for Chapter 3 183 4 Introductory Scattering Theory 185 4.1 The Far Zone 187 4.1.1 Volume Integral Formulation 188 4.1.2 Surface Integral Formulation 191 4.1.3 Translation of the Origin 199 4.2 Cross Sections 200 4.3 Scattering Dyadic (Matrix) 204 4.3.1 Spherical Coordinate Representation 210 4.3.2 Coherency Matrix 214 4.3.3 Mueller Matrix or Phase Matrix 217 4.3.4 Superposition 221 4.3.5 Translation of the Origin 222 4.3.6 Reciprocity of the Scattering Dyadic 222 4.4 Optical Theorem 227 4.4.1 Extinction 230 4.5 Plane Interface Case and Babinet s Principle 233 4.5.1 Babinet s Principle 239 Problems for Chapter 4 241 5 Scattering in the Time Domain 245 5.1 The Scattering Problem 246 5.1.1 The Incident Field 246 5.1.2 Scattering Problem Formulation 248 5.1.3 Scattered Field 250 5.1.4 Far Field Amplitude 251 5.1.5 Scattering Dyadic 254 5.2 Energy Balance in the Time Domain 255 5.3 Connection to the Time-Harmonic Results 255 5.4 Optical Theorem 256 5.5 Some Applications of the Optical Theorem 259 5.5.1 Several Scatterers 259 5.5.2 Layered Scatterers 261 Problems for Chapter 5 263
x 6 Approximations and Applications 265 6.1 Long Wavelength Approximation 265 6.1.1 Near Field Approximation 265 6.1.2 Far Field Amplitude 267 6.1.3 The Scattering Dyadic 272 6.1.4 Cross Sections 272 6.1.5 Internal Field 274 6.1.6 Polarizability Dyadics 279 6.2 Weak-Scatterer Approximation 287 6.2.1 Born Approximation 287 6.3 High-Frequency Approximation 289 6.3.1 Aperture Formulation 291 6.3.2 Reflection at a Metallic Surface 292 6.3.3 Physical Optics Approximation 294 6.3.4 Geometrical Optics Approximation 306 6.4 Sum Rule for the Extinction Cross Section 320 6.4.1 Additional Sum Rules 329 6.5 Scattering by Many Scatterers Multiple Scattering 335 6.5.1 Far Field Approximation 338 6.5.2 Single Scattering 341 Problems for Chapter 6 343 7 Spherical Vector Waves 349 7.1 Preparatory Discussions 350 7.2 Definition of Spherical Vector Waves 355 7.2.1 Expansions of the Fields 359 7.3 Orthogonality and Reciprocity Relations 360 7.3.1 Spherical Scalar Waves 363 7.3.2 Power Transport 364 7.4 Some Properties of the Spherical Vector Waves 365 7.4.1 Linear Independence 365 7.4.2 The Translation Matrices 366 7.5 Expansion of the Green Dyadic 367 7.5.1 The Green Function in Free Space 368 7.5.2 The Green Dyadic for the Electric Field in Free Space 369 7.5.3 Free-Space Green Dyadic 370 7.6 Null-Field Equations 371 7.7 Expansion of Sources 374 7.7.1 Expansion of a Plane Wave 374 7.7.2 Expansion of a Vertical Electric Dipole 375
xi 7.8 Far Field Amplitude and the Transition Matrix 376 7.8.1 Scattering Dyadic 379 7.8.2 Cross Sections 383 7.8.3 Generalized Optical Theorem 392 7.8.4 The Decrease of the Scattered Field 395 7.9 Dipole Moments of a Scatterer 397 Problems for Chapter 7 403 8 Scattering by Spherical Objects 407 8.1 Scattering by a Perfectly Conducting Sphere 407 8.1.1 Long Wavelength Approximation 414 8.1.2 High-Frequency Asymptotics 416 8.2 Scattering by a Dielectric Sphere 418 8.2.1 Internal Field 425 8.2.2 Long Wavelength Approximation 426 8.2.3 Resonances 431 8.2.4 Interference Structure 434 8.3 Scattering by Layered Spherical Objects 435 8.3.1 Resonance Frequencies in a Spherical Cavity 443 8.4 Scattering by an Anisotropic Sphere 447 8.4.1 Radial Expansion Functions 448 8.4.2 Transition Matrix 455 8.4.3 Non-Uniqueness of the Scattering Problem 458 8.5 Scattering by a Biisotropic Sphere 465 8.5.1 Spherical Vector Waves in a Biisotropic Material 465 8.5.2 The Transition Matrix for a Biisotropic Sphere 466 8.5.3 Long Wavelength Approximation 472 Problems for Chapter 8 475 9 The Null-Field Approach 479 9.1 The T-Matrix for a Single Homogeneous Scatterer 479 9.1.1 Perfectly Conducting Scatterer 480 9.1.2 Dielectric Scatterer 489 9.2 The T-Matrix for a Collection of Scatterers 496 9.2.1 Iterative Solution 503 9.2.2 Cross Sections 505 9.3 Obstacle above a Ground Plane 506 9.3.1 Formulation of the Problem 506 9.3.2 Integral Representation of the Solution 507 9.3.3 Transformation between Solutions 508
xii 9.3.4 Incident Electric Field 509 9.3.5 Utilizing the Surface Integral Representation 511 9.3.6 Expansion and Elimination of the Surface Fields 512 9.3.7 Decomposition of the Scattered Field 514 Problems for Chapter 9 517 10 Propagation in Stratified Media 519 10.1 Basic Equations 520 10.1.1 Decomposition of Dyadics 520 10.2 The Fundamental Equation 521 10.2.1 The Fourier Transform of the Fields 521 10.2.2 Decomposition of the Maxwell Equations 523 10.3 Wave Splitting 526 10.3.1 Power Flux Density 529 10.3.2 Wave Splitting and Projection Dyadics 531 10.4 Propagation of Fields the Propagator Dyadic 532 10.4.1 Reflection and Transmission Dyadics 533 10.4.2 Slab above Ground 535 10.4.3 Composition of Two Slabs 537 10.5 Propagator Dyadics Homogeneous Layers 538 10.5.1 Single Layer 538 10.5.2 Homogeneous Layer Distinct Eigenvalues (Projection Dyadics) 539 10.5.3 Homogeneous Layer Distinct Eigenvalues 541 10.5.4 Several Layers 543 10.6 Examples 544 10.6.1 Isotropic Media 544 10.6.2 Biisotropic Media 557 10.6.3 Anisotropic Media 559 10.7 Numerical Computations 562 10.7.1 Reflectivity and Transmissivity 562 10.7.2 Example Dielectric Slab with Uniaxial Layers 563 10.7.3 Example Bianisotropic Media 565 10.8 Asymptotic Analysis 566 10.9 The Green Dyadic 570 10.9.1 Particular Solution or Free-Space Solution 571 10.9.2 Homogeneous Solution in Free Space 575 10.9.3 General Solution 576 10.9.4 The Transmitted Field 580 Problems for Chapter 10 581
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
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
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