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A plane wave is incident (from free space) obliquely upon an infinitely large dielectric slab with thickness a thickness of d. a permittivity of £;. and permeability of y;. Find the reflected and transmitted fields and the field inside the dielectric slab for both perpendicular and parallel polarization. Consider specifically four cases: a) both & and pu, are positive, b) both & and py, are negative, ¢) gg is positive while yi, is negative d) 4 is negative while p14 is positive. Theory and Computation of Electromagnetic Fields THEORY AND COMPUTATION OF ELECTROMAGNETIC FIELDS THEORY AND COMPUTATION OF ELECTROMAGNETIC FIELDS Second Edition JIAN-MING JIN Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Copyright © 2015 by John Wiley & Sons, Inc. 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Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 1 2015 http://www.copyright.com http://www.wiley.com/go/permissions http://www.wiley.com CONTENTS Preface xv Acknowledgments xxi PART I Electromagnetic Field Theory 1 1 Basic Electromagnetic Theory 3 1.1 Review of Vector Analysis, 3 1.1.1 Vector Operations and Integral Theorems, 4 1.1.2 Symbolic Vector Method, 6 1.1.3 Helmholtz Decomposition Theorem, 9 1.1.4 Green’s Theorems, 9 1.2 Maxwell’s Equations in Terms of Total Charges and Currents, 11 1.2.1 Maxwell’s Equations in Integral Form, 12 1.2.2 Maxwell’s Equations in Differential Form, 17 1.2.3 Current Continuity Equation, 17 1.2.4 The Lorentz Force Law, 18 1.3 Constitutive Relations, 18 1.3.1 Electric Polarization, 19 1.3.2 Magnetization, 21 1.3.3 Electric Conduction, 22 1.3.4 Classification of Media, 23 1.4 Maxwell’s Equations in Terms of Free Charges and Currents, 25 vi CONTENTS 1.5 Boundary Conditions, 27 1.6 Energy, Power, and Poynting’s Theorem, 31 1.7 Time-Harmonic Fields, 33 1.7.1 Time-Harmonic Fields, 33 1.7.2 Fourier Transforms, 35 1.7.3 Complex Power, 37 1.7.4 Complex Permittivity and Permeability, 42 References, 46 Problems, 46 2 Electromagnetic Radiation in Free Space 53 2.1 Scalar and Vector Potentials, 53 2.1.1 Static Fields, 54 2.1.2 Time-Harmonic Fields and the Lorenz Gauge Condition, 58 2.2 Solution of Vector Potentials in Free Space, 61 2.2.1 Delta Function and Green’s Function, 61 2.2.2 Green’s Function in Free Space, 62 2.2.3 Field–Source Relations in Free Space, 63 2.2.4 Why Use Auxiliary Potential Functions, 64 2.2.5 Free-Space Dyadic Green’s Functions, 66 2.3 Electromagnetic Radiation in Free Space, 69 2.3.1 Infinitesimal Electric Dipole, 69 2.3.2 Finite Electric Dipole, 72 2.3.3 Far-Field Approximation and the Sommerfeld Radiation Condition, 73 2.3.4 Circular Current Loop and Magnetic Dipole, 76 2.4 Radiation by Surface Currents and Phased Arrays, 78 2.4.1 Radiation by a Surface Current, 78 2.4.2 Radiation by a Phased Array, 81 References, 84 Problems, 85 3 Electromagnetic Theorems and Principles 89 3.1 Uniqueness Theorem, 90 3.2 Image Theory, 94 3.2.1 Basic Image Theory, 94 3.2.2 Half-Space Field–Source Relations, 99 3.3 Reciprocity Theorems, 101 3.3.1 General Reciprocity Theorem, 101 3.3.2 Lorentz Reciprocity Theorem, 102 3.3.3 Rayleigh–Carson Reciprocity Theorem, 103 3.4 Equivalence Principles, 107 3.4.1 Surface Equivalence Principle, 107 3.4.2 Application to Scattering by a Conducting Object, 109 3.4.3 Application to Scattering by a Dielectric Object, 114 3.4.4 Volume Equivalence Principle, 116 CONTENTS vii 3.5 Duality Principle, 120 3.6 Aperture Radiation and Scattering, 121 3.6.1 Equivalent Problems, 121 3.6.2 Babinet’s Principle, 124 3.6.3 Complementary Antennas, 127 References, 128 Problems, 129 4 Transmission Lines and Plane Waves 135 4.1 Transmission Line Theory, 135 4.1.1 Governing Differential Equations and General Solutions, 135 4.1.2 Reflection and Transmission, 138 4.1.3 Green’s Function and Eigenfunction Expansion, 140 4.2 Wave Equations and General Solutions, 144 4.2.1 Wave Equations and Solution by Separation of Variables, 144 4.2.2 Characteristics of a Plane Wave, 146 4.2.3 Wave Velocities and Attenuation, 147 4.2.4 Linear, Circular, and Elliptical Polarizations, 151 4.2.5 Wave Propagation in Metamaterials, 154 4.3 Plane Waves Generated by a Current Sheet, 156 4.4 Reflection and Transmission, 159 4.4.1 Reflection and Transmission at Normal Incidence, 159 4.4.2 Reflection and Transmission at Oblique Incidence, 161 4.4.3 Total Transmission and Total Reflection, 164 4.4.4 Transmission into a Left-Handed Medium, 168 4.4.5 Plane Waves Versus Transmission Lines, 170 4.5 Plane Waves in Anisotropic and Bi-Isotropic Media, 174 4.5.1 Plane Waves in Uniaxial Media, 174 4.5.2 Plane Waves in Gyrotropic Media, 179 4.5.3 Plane Waves in Chiral Media, 183 References, 190 Problems, 191 5 Fields and Waves in Rectangular Coordinates 199 5.1 Uniform Waveguides, 199 5.1.1 General Analysis, 200 5.1.2 General Characteristics, 204 5.1.3 Uniform Rectangular Waveguide, 208 5.1.4 Losses in Waveguides and Attenuation Constant, 215 5.2 Uniform Cavities, 220 5.2.1 General Theory, 221 5.2.2 Rectangular Cavity, 223 5.2.3 Material and Geometry Perturbations, 226 5.3 Partially Filled Waveguides and Dielectric Slab Waveguides, 229 5.3.1 General Theory, 229 5.3.2 Partially Filled Rectangular Waveguide, 231 5.3.3 Dielectric Slab Waveguide on a Ground Plane, 236 viii CONTENTS 5.4 Field Excitation in Waveguides, 241 5.4.1 Excitation by Planar Surface Currents, 242 5.4.2 Excitation by General Volumetric Currents, 243 5.5 Fields in Planar Layered Media, 245 5.5.1 Spectral Green’s Function and Sommerfeld Identity, 245 5.5.2 Vertical Electric Dipole above a Layered Medium, 247 5.5.3 Horizontal Electric Dipole above a Layered Medium, 249 5.5.4 Dipoles on a Grounded Dielectric Slab, 251 References, 257 Problems, 257 6 Fields and Waves in Cylindrical Coordinates 261 6.1 Solution of Wave Equation, 261 6.1.1 Solution by Separation of Variables, 262 6.1.2 Cylindrical Wave Functions, 263 6.2 Circular and Coaxial Waveguides and Cavities, 266 6.2.1 Circular Waveguide, 267 6.2.2 Coaxial Waveguide, 273 6.2.3 Cylindrical Cavity, 276 6.3 Circular Dielectric Waveguide, 279 6.3.1 Analysis of Hybrid Modes, 279 6.3.2 Characteristics of Hybrid Modes, 283 6.4 Wave Transformation and Scattering Analysis, 287 6.4.1 Wave Transformation, 288 6.4.2 Scattering by a Circular Conducting Cylinder, 289 6.4.3 Scattering by a Circular Dielectric Cylinder, 293 6.4.4 Scattering by a Circular Multilayer Dielectric Cylinder, 296 6.5 Radiation by Infinitely Long Currents, 300 6.5.1 Line Current Radiation in Free Space, 300 6.5.2 Radiation by a Cylindrical Surface Current, 304 6.5.3 Radiation in the Presence of a Circular Conducting Cylinder, 306 6.5.4 Radiation in the Presence of a Conducting Wedge, 309 6.5.5 Radiation by a Finite Current, 312 References, 319 Problems, 320 7 Fields and Waves in Spherical Coordinates 325 7.1 Solution of Wave Equation, 325 7.1.1 Solution by Separation of Variables, 325 7.1.2 Spherical Wave Functions, 328 7.1.3 TEr and TMr Modes, 329 7.2 Spherical Cavity, 331 7.3 Biconical Antenna, 335 7.3.1 Infinitely Long Model, 335 7.3.2 Finite Biconical Antenna, 339 7.4 Wave Transformation and Scattering Analysis, 341 7.4.1 Wave Transformation, 342 CONTENTS ix 7.4.2 Expansion of a Plane Wave, 344 7.4.3 Scattering by a Conducting Sphere, 347 7.4.4 Scattering by a Dielectric Sphere, 352 7.4.5 Scattering by a Multilayer Dielectric Sphere, 357 7.5 Addition Theorem and Radiation Analysis, 360 7.5.1 Addition Theorem for Spherical Wave Functions, 360 7.5.2 Radiation of a Spherical Surface Current, 362 7.5.3 Radiation in the Presence of a Sphere, 368 7.5.4 Radiation in the Presence of a Conducting Cone, 370 References, 377 Problems, 377 PART II Electromagnetic Field Computation 383 8 The Finite Difference Method 385 8.1 Finite Differencing Formulas, 385 8.2 One-Dimensional Analysis, 387 8.2.1 Solution of the Diffusion Equation, 387 8.2.2 Solution of the Wave Equation, 389 8.2.3 Stability Analysis, 390 8.2.4 Numerical Dispersion Analysis, 392 8.3 Two-Dimensional Analysis, 393 8.3.1 Analysis in the Time Domain, 393 8.3.2 Analysis in the Frequency Domain, 395 8.4 Yee’s FDTD Scheme, 397 8.4.1 Two-Dimensional Analysis, 397 8.4.2 Three-Dimensional Analysis, 399 8.5 Absorbing Boundary Conditions, 402 8.5.1 One-Dimensional ABC, 403 8.5.2 Two-Dimensional ABCs, 404 8.5.3 Perfectly Matched Layers, 406 8.6 Modeling of Dispersive Media, 417 8.6.1 Recursive Convolution Approach, 417 8.6.2 Auxiliary Differential Equation Approach, 420 8.7 Wave Excitation and Far-Field Calculation, 422 8.7.1 Modeling of Wave Excitation, 422 8.7.2 Near-to-Far-Field Transformation, 426 8.8 Summary, 427 References, 428 Problems, 429 9 The Finite Element Method 433 9.1 Introduction to the Finite Element Method, 434 9.1.1 The General Principle, 434 9.1.2 One-Dimensional Example, 435 x CONTENTS 9.2 Finite Element Analysis of Scalar Fields, 439 9.2.1 The Boundary-Value Problem, 439 9.2.2 Finite Element Formulation, 440 9.2.3 Application Examples, 446 9.3 Finite Element Analysis of Vector Fields, 450 9.3.1 The Boundary-Value Problem, 450 9.3.2 Finite Element Formulation, 452 9.3.3 Application Examples, 456 9.4 Finite Element Analysis in the Time Domain, 465 9.4.1 The Boundary-Value Problem, 465 9.4.2 Finite Element Formulation, 466 9.4.3 Application Examples, 470 9.5 Discontinuous Galerkin Time-Domain Method, 472 9.5.1 Basic Idea, 473 9.5.2 Central-Flux DGTD Method, 475 9.5.3 Upwind-Flux DGTD Method, 478 9.5.4 Application Example, 481 9.6 Absorbing Boundary Conditions, 483 9.6.1 Two-Dimensional ABCs, 483 9.6.2 Three-Dimensional ABCs, 486 9.6.3 Perfectly Matched Layers, 488 9.7 Some Numerical Aspects, 494 9.7.1 Mesh Generation, 494 9.7.2 Matrix Solvers, 494 9.7.3 Higher-Order Elements, 495 9.7.4 Curvilinear Elements, 496 9.7.5 Adaptive Finite Element Analysis, 496 9.8 Summary, 497 References, 497 Problems, 499 10 The Method of Moments 505 10.1 Introduction to the Method of Moments, 506 10.2 Two-Dimensional Analysis, 510 10.2.1 Formulation of Integral Equations, 510 10.2.2