Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Translated by authors With 259 Figures Springer
Contents 1 Basic Electromagnetic Theory 1 1.1 Maxwell's Equations 1 1.1.1 Basic Maxwell Equations 2 1.1.2 Maxwell's equations in Material Media 5 1.1.3 Complex Maxwell Equations 12 1.1.4 Complex Permittivity and Permeability 14 1.1.5 Complex Maxwell equations in Anisotropic Media.. 16 1.1.6 Equivalent Magnetic Charge and Current 17 1.2 Boundary Conditions 18 1.2.1 General Boundary Conditions 18 1.2.2 The Short-Circuit Surface 19 1.2.3 The Open-Circuit Surface 21 1.2.4 She Impedance Surface 21 1.3 Wave Equations 22 1.3.1 Time-Domain Wave Equations 22 1.3.2 Solution to the Homogeneous Wave Equations... 24 1.3.3 Frequency-Domain Wave Equations 27 1.4 Poynting's Theorem 28 1.4.1 Time-Domain Poynting Theorem 28 1.4.2 Frequency-Domain Poynting Theorem 30 1.4.3 Poynting's Theorem for Dispersive Media 33 1.5 Scalar and Vector Potentials 37 1.5.1 Retarding Potentials: d'aiembert's Equations 38 1.5.2 Solution of d'aiembert's Equations 40 1.5.3 Complex d'alembert Equations 42 1.6 Hertz Vectors 43 1.6.1 Instantaneous Hertz Vectors 43 1.6.2 Complex Hertz Vectors 45 1.7 Duality 46 1.8 Reciprocity 47 Problems 49
VIII Contents 2 Introduction to Waves 51 2.1 Sinusoidal Uniform Plane Waves 51 2.1.1 Uniform Plane Waves in Lossless Simple Media... 52 2.1.2 Plane Waves in Lossy Media: Damped Waves 57 2.2 Polarization of Plane Waves 61 2.2.1 Combination of Two Mutually Perpendicular Linearly Polarized Waves 61 2.2.2 Combination of Two Opposite Circularly Polarized Waves 66 2.2.3 The Jones Matrix 67 2.2.4 Stokes Parameters and the Poincare Sphere 67 2.2.5 The Degree of Polarization 69 2.3 Reflection and Refraction of Plane Waves 70 2.3.1 SnelPs Law 70 2.3.2 Fresnel's Law, Reflection and Refraction Coefficients. 72 2.3.3 Reflection at a Perfectly Conductive Plane 76 2.3.4 The Brewster Angle 79 2.3.5 Total Reflection and the Critical Angle 81 2.3.6 Decaying Fields and Slow Waves 83 2.3.7 The Goos-Hänchen Shift 86 2.3.8 Reflection Coefficients at Dielectric Interfaces 87 2.3.9 Reflection and Transmission of Plane Waves at Interfaces Between Lossless and Lossy Media... 89 2.3.10 Transformation of Impedance and Reflection Coefficients 91 2.4 Transmission-Line Simulation of Electromagnetic Waves... 93 2.4.1 The Telegraph Equations and Their Solutions 94 2.4.2 The Reflection Coefficient, Standing Wave Ratio, and Impedance in a Lossless Line 98 2.4.3 States of a Transmission Line 103 2.4.4 Transmission»Line Charts 105 2.4.5 The Equivalent Transmission Line of Guided-Wave Systems 110 2.5 Network Simulation of Electromagnetic Waves Ill 2.5.1 Network Matrix and Parameters of a Linear Multi-port Network 112 2.5.2 The Network Matrices of the Reciprocal, Lossless, Source-Free Multi-port Networks 118 2.5.3 Two-Port Networks 122 2.5.4 The Network Parameters of Some Basic Circuit Elements 130 2.6 Reflection and Transmission of Waves at Multi-layer Media and Impedance Transducers 136
Contents IX 2.6.1 Single Dielectric Layer, The Л/4 Impedance Transducer 136 2.6.2 The Double Dielectric Layer: Double-Section Impedance Transducers 141 2.6.3 The Design of a Multiple Dielectric Layer or Multi-section Impedance Transducer 142 2.6.4 The Small-Reflection Approach 147 2.6.5 A Multi-layer Coating with an Alternating Index... 153 Problems 155 3 Time-Varying Boundary-Value Problems 159 3.1 Uniqueness Theorem for Time-Varying-Field Problems... 159 3.1.1 Uniqueness Theorem for the Boundary-Value Problems of Helmholtz's Equations 160 3.1.2 Uniqueness Theorem for the Boundary-Value Problems with Complicated Boundaries 162 3.2 Solution of Vector Helmholtz Equations in Orthogonal Curvilinear Coordinates 165 3.2.1 Orthogonal Curvilinear Coordinate Systems 165 3.2.2 Method of Borgnis'Potentials 167 3.2.3 Method of Hertz Vectors 173 3.2.4 Method of Longitudinal Components 174 3.3 Boundary Conditions of Helmholtz's Equations 177 3.4 Separation of Variables 178 3.5 Electromagnetic Waves in Cylindrical Systems 180 3.6 Solution of Helmholtz's Equations in Rectangular Coordinates 184 3.6.1 Set 2 as и 3 184 3.6.2 Set x от у as щ 187 3.7 Solution of Helmholtz's Equations in Circular Cylindrical Coordinates 188 3.8 Solution of Helmholtz's Equations in Spherical Coordinates. 193 3.9 Vector Eigenfunctions and Normal Modes 198 3.9.1 Eigenvalue Problems and Orthogonal Expansions... 198 3.9.2 Eigenvalues for the Boundary-Value Problems of the Vector Helmholtz Equations 201 3.9.3 Two-Dimensional Eigenvalues in Cylindrical Systems. 203 3.9.4 Vector Eigenfunctions and Normal Mode Expansion. 203 3.10 Approximate Solution of Helmholtz's Equations 206 3.10.1 Variational Principle of Eigenvalues 206 3.10.2 Approximate Field-Matching Conditions 208 Problems 212
X Contents 4 Metallic Waveguides and Resonant Cavities 213 4.1 General Characteristics of Metallic Waveguides 214 4.1.1 Ideal-Waveguide Model 214 4.1.2 Propagation Characteristics 215 4.1.3 Dispersion Relations 216 4.1.4 Wave Impedance 217 4.1.5 Power Flow 218 4.1.6 Attenuation 219 4.2 General Characteristics of Resonant Cavities 221 4.2.1 Modes and Natural Frequencies of the Resonant Cavity 221 4.2.2 Losses in a Resonant Cavity: the Q Factor 222 4.3 Waveguides and Cavities in Rectangular Coordinates 223 4.3.1 Rectangular Waveguides 223 4.3.2 Parallel-Plate Transmission Lines 231 4.3.3 Rectangular Resonant Cavities 234 4.4 Waveguides and Cavities in Circular Cylindrical Coordinates 238 4.4.1 Sectorial Cavities 238 4.4.2 Sectorial Waveguides 241 4.4.3 Coaxial Lines and Coaxial Cavities 242 4.4.4 Circular Waveguides and Circular Cylindrical Cavities 247 4.4.5 Cylindrical Horn Waveguides and Inclined-Plate Lines 255 4.4.6 Radial Transmission Lines and Radial Line Cavities 257 4.5 Waveguides and Cavities in Spherical Coordinates 260 4.5.1 Spherical Cavities 261 4.5.2 Biconical Lines and Biconical Cavities 263 4.6 Reentrant Cavities 267 4.6.1 Exact Solution for the Reentrant Cavity 269 4.6.2 Approximate Solution for the Reentrant Cavity... 271 4.7 Fabry-Perot Cavities 276 4.8 Principle of Perturbation 278 4.8.1 Cavity Wall Perturbations 278 4.8.2 Material Perturbation of a Cavity 281 4.8.3 Cutoff Frequency Perturbation of a Waveguide... 284 4.8.4 Propagation Constant Perturbation of a Waveguide. 285 Problems 287 5 Dielectric Waveguides and Resonators 289 5.1 Metallic Waveguide with Different Filling Media 291 5.1.1 The Possible Modes 291 5.1.2 LSE and LSM Modes 294 5.2 Symmetrical Planar Dielectric Waveguides 299
Contents XI 5.2.1 TM Modes 300 5.2.2 ТЕ Modes 302 5.2.3 Cutoff Condition, Guided Modes, and Radiation Modes 304 5.2.4 Dispersion Characteristics of Guided Modes 305 5.2.5 Radiation Modes 306 5.2.6 Fields in Symmetrical Planar Dielectric Waveguides. 307 5.2.7 The Lowest Modes in Symmetrical Planar Dielectric Waveguides 307 5.2.8 Dielectric Coated Conducting Plane 310 5.3 Asymmetrical Planar Dielectric Waveguides 310 5.3.1 TM Modes 311 5.3.2 ТЕ Modes 313 5.3.3 Dispersion Characteristics of Asymmetrical Planar Dielectric Waveguide 314 5.3.4 Fields in Asymmetrical Planar Dielectric Waveguides. 315 5.4 Rectangular Dielectric Waveguides 317 5.5 Circular Dielectric Waveguides and Optical Fibers 322 5.5.1 General Solutions of Circular Dielectric Waveguides. 322 5.5.2 Nonmagnetic Circular Dielectric Waveguides 334 5.5.3 Weakly Guiding Optical Fibers 342 5.5.4 Linearly Polarized Modes in Weakly Guiding Fibers. 344 5.5.5 Dominant Modes in Circular Dielectric Waveguides.. 347 5.5.6 Low-Attenuation Optical Fibers 349 5.6 Dielectric-Coated Conducting Cylinders 350 5.7 Dielectric Resonators 352 5.7.1 Perfect-Magnetic-Wall Approach 353 5.7.2 Cutoff-Waveguide Approach 356 5.7.3 Cutoff-Waveguide, Cutoff-Radial-Line Approach... 358 5.7.4 Dielectric Resonators in Microwave Circuits 360 Problems 362 6 Periodic Structures and the Coupling of Modes 365 6.1 Characteristics of Slow Waves 366 6.1.1 Dispersion Characteristics 366 6.1.2 Interaction Impedance 367 6.2 A Corrugated Conducting Surface as a Uniform System... 368 6.2.1 Unbounded Structure 368 6.2.2 Bounded Structure 370 6.3 A Disk-Loaded Waveguide as a Uniform System 371 6.4 Periodic Systems 374 6.4.1 Floquet's Theorem and Space Harmonics 374 6.4.2 The uj-ß Diagram of Periodic Systems 378 6.4.3 The Band-Pass Character of Periodic Systems... 378 6.4.4 Fields in Periodic Systems 381
6.4.5 Two Theorems on Lossless Periodic Systems 383 6.4.6 The Interaction Impedance for Periodic Systems... 384 6.5 Corrugated Conducting Plane as a Periodic System 384 6.6 Disk-Loaded Waveguide as a Periodic System 388 6.7 The Helix 392 6.7.1 The Sheath Helix 393 6.7.2 The Tape Helix 402 6.8 Coupling of Modes 410 6.8.1 Coupling of Modes in Space 410 6.8.2 General Solutions for Codirectional Coupling 414 6.8.3 Waveguide Couplers and Switches 416 6.8.4 Coupling Coefficient of Dielectric Waveguides 418 6.9 Distributed Feedback (DFB) Structures 420 6.9.1 Principle of DFB Structures 420 6.9.2 DFB Transmission Resonator 424 6.9.3 A Multiple Layer as a DFB Transmission Resonator. 426 6.9.4 The Quarter-Wave-Shifted DFB Resonator 427 Problems 430 Electromagnetic Waves in Dispersive Media 433 7.1 Classical Theory of Dispersion in Material Media 434 7.1.1 Complex Susceptibility and Complex Permittivity.. 434 7.1.2 Kramers-Kronig Relations 437 7.1.3 Complex Index of Refraction. 437 7.1.4 Normal and Anomalous Dispersion 439 7.2 Complex Index for Metals 440 7.3 Behavior at Low Frequencies, Electric Conductivity 441 7.4 Behavior at High Frequencies, Plasma Frequency 442 7.5 Complex propagation coefficient and Phase Velocity 443 7.6 Group Velocity 445 7.7 Signal Velocity 449 7.8 Velocity of Energy Flow 450 Problems 452 Electromagnetic Waves in Anisotropic Media 453 8.1 Anisotropic Media and Their Constitutional Relations... 453 8.1.1 Constitutional Equations for Anisotropic Media... 453 8.1.2 Symmetrical Properties of the Constitutional Tensors 454 8.2 Governing Equations for Fields and Waves in Anisotropic Media 457 8.2.1 Maxwell Equations and Wave Equations in Anisotropic Media 457 8.2.2 Wave Vector and Poynting's Vector in Anisotropic Media 458 8.2.3 kdb Coordinate System 460
Contents XIII 8.3 Reciprocal Dielectric Crystals 463 8.3.1 Isotropic Crystals 464 8.3.2 Uniaxial Crystals 464 8.3.3 Biaxial Crystals 464 8.4 Electromagnetic Waves in Uniaxial Crystals 464 8.4.1 General Expressions 465 8.4.2 Plane Waves Propagating in the Direction of the Optical Axis 468 8.4.3 Plane Waves Propagating in the Direction Perpendicular to the Optical Axis 468 8.4.4 Plane Waves Propagating in an Arbitrary Direction. 470 8.5 General Formalisms of EM Waves in Reciprocal Media... 472 8.5.1 Index Ellipsoid 472 8.5.2 Dispersion Equations for the Plane Waves in Reciprocal Media 476 8.5.3 Normal and Effective-Index Surfaces 480 8.5.4 Phase Velocity and Group Velocity of the Plane Waves in Reciprocal Crystals 484 8.6 Waves in Electron Beams 485 8.6.1 Permittivity Tensor for an Electron Beam 485 8.6.2 Space Charge Waves 488 8.7 Nonreciprocal Media 492 8.7.1 Stationary Plasma in a Finite Magnetic Field 493 8.7.2 Ferrite in a Finite Magnetic Field, Gyromagnetic Media 496 8.8 Electromagnetic Waves in Nonreciprocal Media 506 8.8.1 Plane Waves in a Stationary Plasma in a Finite Magnetic Field 507 8.8.2 Plane Waves in Saturated-Magnetized Ferrites... 510 8.9 Magnetostatic Waves 519 8.9.1 Magnetostatic Wave Equations 520 8.9.2 Magnetostatic Wave Modes 523 Problems 533 9 Gaussian Beams 535 9.1 Fundamental Gaussian Beams 535 9.2 Characteristics of Gaussian Beams 538 9.2.1 Condition of Paraxial Approximation 538 9.2.2 Beam Radius, Curvature Radius of Phase Front, and Half Far-Field Divergence Angle 539 9.2.3 Phase Velocity 540 9.2.4 Electric and Magnetic Fields in Gaussian Beams... 541 9.2.5 Energy Density and Power Flow 542 9.3 Transformation of Gaussian Beams 543 9.3.1 The q Parameter and Its Transformation 543
XIV Contents 9.3.2 ABCD Law and Its Applications 547 9.3.3 Transformation Through a Non-thin Lens 549 9.4 Elliptic Gaussian Beams 550 9.5 Higher-Order Modes of Gaussian Beams 553 9.5.1 Hermite-Gaussian Beams 554 9.5.2 Laguerre-Gaussian Beams 558 9.6 Gaussian Beams in Quadratic Index Media 560 9.6.1 The General Solution 562 9.6.2 Propagation in Media with a Real Quadratic Index Profile 564 9.6.3 Propagation in Medium with an Imaginary Quadratic Index Profile 565 9.6.4 Steady-State Hermite-Gaussian Beams in Media with a Quadratic Index Profile 567 9.7 Optical Resonators with Curved Mirrors 569 9.8 Gaussian Beams in Anisotropic Media 572 Problems 577 10 Scalar Diffraction Theory 579 10.1 Kirchhoff's Diffraction Theory 579 10.1.1 Kirchhoff Integral Theorem 579 10.1.2 Fresnel-Kirchhoff Diffraction Formula 581 10.1.3 Rayleigh-Sommerfeld Diffraction Formula 583 10.2 Fraunhofer and Fresnel Diffraction 585 10.2.1 Diffraction Formulas for Spherical Waves 585 10.2.2 Fraunhofer Diffraction at Circular Apertures 587 10.2.3 Fresnel Diffraction at Circular Apertures 590 10.3 Diffraction of Gaussian Beams 592 10.3.1 Fraunhofer Diffraction of Gaussian Beams 592 10.3.2 Fresnel Diffraction of Gaussian Beams 595 10.4 Diffraction of Plane Waves in Anisotropic Media 597 10.4.1 Fraunhofer Diffraction at Square Apertures 598 10.4.2 Fraunhofer Diffraction at Circular Apertures 603 10.4.3 Fresnel Diffraction at Circular Apertures 606 10.5 Refraction of Gaussian Beams in Anisotropic Media 610 10.6 Eigenwave Expansions of Electromagnetic Fields 615 10.6.1 Eigenmode Expansion in a Rectangular Coordinate System 616 10.6.2 Eigenmode Expansion in a Cylindrical Coordinate System 617 10.6.3 Eigenmode Expansion in Inhomogeneous Media... 620 10.6.4 Eigenmode Expansion in Anisotropic Media 622 10.6.5 Eigenmode Expansion in Inhomogeneous and Anisotropic Media 623
Contents XV 10.6.6 Reflection and Refraction of Gaussian Beams on Medium Surfaces 625 Problems 628 A SI Units and Gaussian Units 631 A.l Conversion of Amounts 631 A.2 Conversion of Formulas 632 В Vector Analysis 633 B.l Vector Differential Operations 633 B.l.l General Orthogonal Coordinates 633 B.l.2 General Cylindrical Coordinates 634 B.1.3 Rectangular Coordinates 635 B.l.4 Circular Cylindrical Coordinates 635 B.l.5 Spherical Coordinates 636 B.2 Vector Formulas 636 B.2.1 Vector Algebric Formulas 636 B.2.2 Vector Differential Formulas 637 B.2.3 Vector Integral Formulas 637 B.2.4 Differential Formulas for the Position Vector 638 С Bessel Functions 639 C.l Power Series Representations 639 C.2 Integral Representations 640 C.3 Approximate Expressions 640 C.3.1 Leading Terms of Power Series C.3.2 (Small-Argument Approximation) 640 Leading Terms of Asymptotic Series (Large-Argument Approximation) 640 C.4 Formulas for Bessel Functions 640 C.4.1 Recurrence Formulas 640 C.4.2 Derivatives 641 C.4.3 Integrals 641 C.4.4 Wronskian 641 C.5 Spherical Bessel Functions 642 C.5.1 Bessel Functions of Order n + 1/2 642 C.5.2 Spherical Bessel Functions 642 C.5.3 Spherical Bessel Functions Used by Schelkunoff... 642 D Legendre Functions 643 D.l Legendre Polynomials 643 D.2 Associate Legendre Polynomials 643 D.3 Formulas for Legendre Polynomials 644 D.3.1 Recurrence Formulas 644 D.3.2 Derivatives 644
XVI Contents D.3.3 Integrals 644 E Matrices and Tensors 645 E.l Matrix 645 E.2 Matrix Algebra 646 E.2.1 Definitions 646 E.2.2 Matrix Algebraic Formulas 646 E.3 Matrix Functions 647 E.4 Special Matrices 648 E.5 Tensors and Vectors 649 Physical Constants and Smith Chart 651 Bibliography 653 Index 659