Lectures on Electromagnetic Field Theory, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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Lectures on Electromagnetic Field Theory Weng Cho CHEW1 Fall 2020, Purdue University 1Updated: December 3, 2020 Contents iii 8.3.1 Cold Collisionless Plasma Medium . . . . . . . . . . . . . . . . . . . . 83 8.3.2 Bound Electron Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.3.3 Damping or Dissipation Case . . . . . . . . . . . . . . . . . . . . . . . 85 8.3.4 Broad Applicability of Drude-Lorentz-Sommerfeld Model . . . . . . . 86 8.3.5 Frequency Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3.6 Plasmonic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9 Waves in Gyrotropic Media, Polarization 91 9.1 Gyrotropic Media and Faraday Rotation . . . . . . . . . . . . . . . . . . . . . 91 9.2 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.2.1 Arbitrary Polarization Case and Axial Ratio1 . . . . . . . . . . . . . . 96 9.3 Polarization and Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10 Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condi- tion, Energy Density 101 10.1 Spin Angular Momentum and Cylindrical Vector Beam . . . . . . . . . . . . 102 10.2 Momentum Density of Electromagnetic Field . . . . . . . . . . . . . . . . . . 103 10.3 Complex Poynting’s Theorem and Lossless Conditions . . . . . . . . . . . . . 104 10.3.1 Complex Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 104 10.3.2 Lossless Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 10.4 Energy Density in Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . 107 11 Transmission Lines 111 11.1 Transmission Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 11.1.1 Time-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 11.1.2 Frequency-Domain Analysis–the Power of Phasor Technique Again! . . 116 11.2 Lossy Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 12 More on Transmission Lines 121 12.1 Terminated Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.1.1 Shorted Terminations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 12.1.2 Open Terminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 12.2 Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 12.3 VSWR (Voltage Standing Wave Ratio) . . . . . . . . . . . . . . . . . . . . . . 128 13 Multi-Junction Transmission Lines, Duality Principle 133 13.1 Multi-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 133 13.1.1 Single-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . 135 13.1.2 Two-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . 136 13.1.3 Stray Capacitance and Inductance . . . . . . . . . . . . . . . . . . . . 139 13.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13.2.1 Unusual Swaps2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.3 Fictitious Magnetic Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1This section is mathematically complicated. It can be skipped on first reading. 2This section can be skipped on first reading. iv Electromagnetic Field Theory 14 Reflection, Transmission, and Interesting Physical Phenomena 145 14.1 Reflection and Transmission—Single Interface Case . . . . . . . . . . . . . . . 145 14.1.1 TE Polarization (Perpendicular or E Polarization)3 . . . . . . . . . . . 146 14.1.2 TM Polarization (Parallel or H Polarization)4 . . . . . . . . . . . . . . 148 14.2 Interesting Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 149 14.2.1 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 150 15 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines 155 15.1 Brewster’s Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 15.1.1 Surface Plasmon Polariton . . . . . . . . . . . . . . . . . . . . . . . . . 158 15.2 Homomorphism of Uniform Plane Waves and Transmission Lines Equations . 160 15.2.1 TE or TEz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 15.2.2 TM or TMz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 16 Waves in Layered Media 165 16.1 Waves in Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 16.1.1 Generalized Reflection Coefficient for Layered Media . . . . . . . . . . 166 16.1.2 Ray Series Interpretation of Generalized Reflection Coefficient . . . . 167 16.1.3 Guided Modes from Generalized Reflection Coefficients . . . . . . . . 168 16.2 Phase Velocity and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . 168 16.2.1 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 16.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 16.3 Wave Guidance in a Layered Media . . . . . . . . . . . . . . . . . . . . . . . . 173 16.3.1 Transverse Resonance Condition . . . . . . . . . . . . . . . . . . . . . 173 17 Dielectric Waveguides 175 17.1 Generalized Transverse Resonance Condition . . . . . . . . . . . . . . . . . . 175 17.2 Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 17.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 17.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 17.2.3 A Note on Cut-Off of Dielectric Waveguides . . . . . . . . . . . . . . . 184 18 Hollow Waveguides 185 18.1 General Information on Hollow Waveguides . . . . . . . . . . . . . . . . . . . 185 18.1.1 Absence of TEM Mode in a Hollow Waveguide . . . . . . . . . . . . . 186 18.1.2 TE Case (Ez = 0, Hz 6= 0, TEz case) . . . . . . . . . . . . . . . . . . . 187 18.1.3 TM Case (Ez 6= 0, Hz = 0, TMz Case) . . . . . . . . . . . . . . . . . . 189 18.2 Rectangular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 18.2.1 TE Modes (Hz 6= 0, H Modes or TEz Modes) . . . . . . . . . . . . . . 190 3These polarizations are also variously know as TEz , or the s and p polarizations, a descendent from the notations for acoustic waves where s and p stand for shear and pressure waves respectively. 4Also known as TMz polarization. Contents v 19 More on Hollow Waveguides 193 19.1 Rectangular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . 194 19.1.1 TM Modes (Ez 6= 0, E Modes or TMz Modes) . . . . . . . . . . . . . 194 19.1.2 Bouncing Wave Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 195 19.1.3 Field Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 19.2 Circular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 19.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 19.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 20 More on Waveguides and Transmission Lines 205 20.1 Circular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 20.1.1 An Application of Circular Waveguide . . . . . . . . . . . . . . . . . 206 20.2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface Plasmonic Modes 209 20.2.1 Quasi-TEM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 20.2.2 Hybrid Modes–Inhomogeneously-Filled Waveguides . . . . . . . . . . . 210 20.2.3 Guidance of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 20.3 Homomorphism of Waveguides and Transmission Lines . . . . . . . . . . . . . 212 20.3.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 20.3.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 20.3.3 Mode Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 21 Cavity Resonators 219 21.1 Transmission Line Model of a Resonator . . . . . . . . . . . . . . . . . . . . . 219 21.2 Cylindrical Waveguide Resonators . . . . . . . . . . . . . . . . . . . . . . . . 221 21.2.1 βz = 0 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 21.2.2 Lowest Mode of a Rectangular Cavity . . . . . . . . . . . . . . . . . . 224 21.3 Some Applications of Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 225 21.3.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 21.3.2 Electromagnetic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 227 21.3.3 Frequency Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 22 Quality Factor of Cavities, Mode Orthogonality 233 22.1 The Quality Factor of a Cavity–General Concept . . . . . . . . . . . . . . . . 233 22.1.1 Analogue with an LC Tank Circuit . . . . . . . . . . . . . . . . . . . . 234 22.1.2 Relation to Bandwidth and Pole Location . . . . . . . . . . . . . . . . 236 22.1.3 Wall Loss and Q for a Metallic Cavity . . . . . . . . . . . . . . . . . . 237 22.1.4 Example: The Q of TM110 Mode . . . . . . . . . . . . . . . . . . . . . 239 22.2 Mode Orthogonality and Matrix Eigenvalue Problem . . . . . . . . . . . . . . 240 22.2.1 Matrix Eigenvalue Problem (EVP) . . . . . . . . . . . . . . . . . . . . 240 22.2.2 Homomorphism with the Waveguide Mode Problem . . . . . . . . . . 241 22.2.3 Proof of Orthogonality of Waveguide Modes5 . . . . . . . . . . . . . . 242 5This may be skipped on first reading. viii Electromagnetic Field Theory 33 High Frequency Solutions, Gaussian Beams 361 33.1 Tangent Plane Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 362 33.2 Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 33.2.1 Generalized Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 365 33.3 Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 33.3.1 Derivation of the Paraxial/Parabolic Wave Equation . . . . . . . . . . 366 33.3.2 Finding a Closed Form Solution . . . . . . . . . . . . . . . . . . . . . 367 33.3.3 Other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 34 Scattering of Electromagnetic Field 371 34.1 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 34.1.1 Scattering by a Small Spherical Particle . . . . . . . . . . . . . . . . . 373 34.1.2 Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 375 34.1.3 Small Conductive Particle . . . . . . . . . . . . . . . . . . . . . . . . . 378 34.2 Mie Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 34.2.1 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 34.2.2 Mie Scattering by Spherical Harmonic Expansions . . . . . . . . . . . 381 34.2.3 Separation of Variables in Spherical Coordinates6 . . . . . . . . . . . . 381 35 Spectral Expansions of Source Fields—Sommerfeld Integrals 383 35.1 Spectral Representations of Sources . . . . . . . . . . . . . . . . . . . . . . . . 383 35.1.1 A Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 35.2 A Source on Top of a Layered Medium . . . . . . . . . . . . . . . . . . . . . . 389 35.2.1 Electric Dipole Fields–Spectral Expansion . . . . . . . . . . . . . . . . 389 35.3 Stationary Phase Method—Fermat’s Principle . . . . . . . . . . . . . . . . . . 392 35.4 Riemann Sheets and Branch Cuts7 . . . . . . . . . . . . . . . . . . . . . . . . 396 35.5 Some Remarks8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 36 Computational Electromagnetics, Numerical Methods 399 36.1 Computational Electromagnetics, Numerical Methods . . . . . . . . . . . . . 401 36.2 Examples of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 401 36.3 Examples of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 402 36.3.1 Volume Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 402 36.3.2 Surface Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 404 36.4 Function as a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 36.5 Operator as a Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 36.5.1 Domain and Range Spaces . . . . . . . . . . . . . . . . . . . . . . . . 406 36.6 Approximating Operator Equations with Matrix Equations . . . . . . . . . . 407 36.6.1 Subspace Projection Methods . . . . . . . . . . . . . . . . . . . . . . . 407 36.6.2 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 36.6.3 Matrix and Vector Representations . . . . . . . . . . . . . . . . . . . . 408 36.6.4 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 6May be skipped on first reading. 7This may be skipped on first reading. 8This may be skipped on first reading. Contents ix 36.6.5 Differential Equation Solvers versus Integral Equation Solvers . . . . . 410 36.7 Solving Matrix Equation by Optimization . . . . . . . . . . . . . . . . . . . . 410 36.7.1 Gradient of a Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 411 37 Finite Difference Method, Yee Algorithm 415 37.1 Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . . . . 415 37.1.1 The Finite-Difference Approximation . . . . . . . . . . . . . . . . . . . 416 37.1.2 Time Stepping or Time Marching . . . . . . . . . . . . . . . . . . . . . 418 37.1.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 37.1.4 Grid-Dispersion Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 37.2 The Yee Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 37.2.1 Finite-Difference Frequency Domain Method . . . . . . . . . . . . . . 427 37.3 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 428 38 Quantum Theory of Light 431 38.1 Historical Background on Quantum Theory . . . . . . . . . . . . . . . . . . . 431 38.2 Connecting Electromagnetic Oscillation to Simple Pendulum . . . . . . . . . 434 38.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 38.4 Schrödinger Equation (1925) . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 38.5 Some Quantum Interpretations–A Preview . . . . . . . . . . . . . . . . . . . . 443 38.5.1 Matrix or Operator Representations . . . . . . . . . . . . . . . . . . . 444 38.6 Bizarre Nature of the Photon Number States . . . . . . . . . . . . . . . . . . 445 39 Quantum Coherent State of Light 447 39.1 The Quantum Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 39.1.1 Quantum Harmonic Oscillator Revisited . . . . . . . . . . . . . . . . . 448 39.2 Some Words on Quantum Randomness and Quantum Observables . . . . . . 450 39.3 Derivation of the Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . 451 39.3.1 Time Evolution of a Quantum State . . . . . . . . . . . . . . . . . . . 453 39.4 More on the Creation and Annihilation Operator . . . . . . . . . . . . . . . . 454 39.4.1 Connecting Quantum Pendulum to Electromagnetic Oscillator9 . . . . 457 39.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 9May be skipped on first reading. x Electromagnetic Field Theory