Lectures on Electromagnetic Field Theory, Lecture notes of Quantum Mechanics

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Lectures on Electromagnetic Field Theory Weng Cho CHEW1 Spring 2020, Purdue University 1Updated: April 24, 2020 Contents iii 9 Waves in Gyrotropic Media, Polarization 81 9.1 Gyrotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9.2 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.2.1 Arbitrary Polarization Case and Axial Ratio . . . . . . . . . . . . . . 86 9.3 Polarization and Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 10 Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condi- tion, Energy Density 91 10.1 Spin Angular Momentum and Cylindrical Vector Beam . . . . . . . . . . . . 91 10.2 Complex Poynting’s Theorem and Lossless Conditions . . . . . . . . . . . . . 93 10.2.1 Complex Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 93 10.2.2 Lossless Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 10.3 Energy Density in Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . 95 11 Transmission Lines 99 11.1 Transmission Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.1.1 Time-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.1.2 Frequency-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . 103 11.2 Lossy Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 12 More on Transmission Lines 109 12.1 Terminated Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 109 12.1.1 Shorted Terminations . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 12.1.2 Open terminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 12.2 Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 12.3 VSWR (Voltage Standing Wave Ratio) . . . . . . . . . . . . . . . . . . . . . . 116 13 Multi-Junction Transmission Lines, Duality Principle 121 13.1 Multi-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 121 13.1.1 Single-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . 122 13.1.2 Two-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . 124 13.1.3 Stray Capacitance and Inductance . . . . . . . . . . . . . . . . . . . . 127 13.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 13.2.1 Unusual Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 13.2.2 Fictitious Magnetic Currents . . . . . . . . . . . . . . . . . . . . . . . 129 14 Reflection, Transmission, and Interesting Physical Phenomena 131 14.1 Reflection and Transmission—Single Interface Case . . . . . . . . . . . . . . . 131 14.1.1 TE Polarization (Perpendicular or E Polarization)1 . . . . . . . . . . . 132 14.1.2 TM Polarization (Parallel or H Polarization) . . . . . . . . . . . . . . 134 14.2 Interesting Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 135 14.2.1 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 135 1These polarizations are also variously know as the s and p polarizations, a descendent from the notations for acoustic waves where s and p stand for shear and pressure waves respectively. iv Electromagnetic Field Theory 15 More on Interesting Physical Phenomena 141 15.1 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, Trans- mission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 15.1.1 Brewster Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 15.1.2 Surface Plasmon Polariton . . . . . . . . . . . . . . . . . . . . . . . . . 144 15.2 Homomorphism of Uniform Plane Waves and Transmission Lines Equations . 146 15.2.1 TE or TEz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 15.2.2 TM or TMz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 16 Waves in Layered Media 149 16.1 Waves in Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 16.1.1 Generalized Reflection Coefficient for Layered Media . . . . . . . . . . 150 16.2 Phase Velocity and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . 151 16.2.1 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 16.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 16.3 Wave Guidance in a Layered Media . . . . . . . . . . . . . . . . . . . . . . . . 155 16.3.1 Transverse Resonance Condition . . . . . . . . . . . . . . . . . . . . . 155 17 Dielectric Waveguides 157 17.1 Generalized Transverse Resonance Condition . . . . . . . . . . . . . . . . . . 157 17.2 Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 17.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 17.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 17.2.3 A Note on Cut-Off of Dielectric Waveguides . . . . . . . . . . . . . . . 165 18 Hollow Waveguides 167 18.1 Hollow Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 18.1.1 Absence of TEM Mode in a Hollow Waveguide . . . . . . . . . . . . . 168 18.1.2 TE Case (Ez = 0, Hz 6= 0) . . . . . . . . . . . . . . . . . . . . . . . . 169 18.1.3 TM Case (Ez 6= 0, Hz = 0) . . . . . . . . . . . . . . . . . . . . . . . . 171 18.2 Rectangular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 18.2.1 TE Modes (H Mode or Hz 6= 0 Mode) . . . . . . . . . . . . . . . . . . 172 19 More on Hollow Waveguides 175 19.1 Rectangular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . 175 19.1.1 TM Modes (E Modes or Ez 6= 0 Modes) . . . . . . . . . . . . . . . . . 175 19.1.2 Bouncing Wave Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 176 19.1.3 Field Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 19.2 Circular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 19.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 19.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Contents v 20 More on Waveguides and Transmission Lines 185 20.1 Circular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 20.1.1 An Application of Circular Waveguide . . . . . . . . . . . . . . . . . . 186 20.2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface Plasmonic Modes 189 20.2.1 Quasi-TEM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 20.2.2 Hybrid Modes–Inhomogeneously-Filled Waveguides . . . . . . . . . . . 190 20.2.3 Guidance of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 20.3 Homomorphism of Waveguides and Transmission Lines . . . . . . . . . . . . . 192 20.3.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 20.3.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 20.3.3 Mode Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 21 Resonators 197 21.1 Cavity Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 21.1.1 Transmission Line Model . . . . . . . . . . . . . . . . . . . . . . . . . 197 21.1.2 Cylindrical Waveguide Resonators . . . . . . . . . . . . . . . . . . . . 199 21.2 Some Applications of Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 203 21.2.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 21.2.2 Electromagnetic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 205 21.2.3 Frequency Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 22 Quality Factor of Cavities, Mode Orthogonality 209 22.1 The Quality Factor of a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 209 22.1.1 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 22.1.2 Relation to the Pole Location . . . . . . . . . . . . . . . . . . . . . . . 210 22.1.3 Some Formulas for Q for a Metallic Cavity . . . . . . . . . . . . . . . 212 22.1.4 Example: The Q of TM110 Mode . . . . . . . . . . . . . . . . . . . . . 213 22.2 Mode Orthogonality and Matrix Eigenvalue Problem . . . . . . . . . . . . . . 214 22.2.1 Matrix Eigenvalue Problem (EVP) . . . . . . . . . . . . . . . . . . . . 214 22.2.2 Homomorphism with the Waveguide Mode Problem . . . . . . . . . . 215 22.2.3 Proof of Orthogonality of Waveguide Modes . . . . . . . . . . . . . . . 216 23 Scalar and Vector Potentials 219 23.1 Scalar and Vector Potentials for Time-Harmonic Fields . . . . . . . . . . . . . 219 23.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 23.1.2 Scalar and Vector Potentials for Statics, A Review . . . . . . . . . . . 219 23.1.3 Scalar and Vector Potentials for Electrodynamics . . . . . . . . . . . . 220 23.1.4 More on Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . 222 23.2 When is Static Electromagnetic Theory Valid? . . . . . . . . . . . . . . . . . 223 23.2.1 Quasi-Static Electromagnetic Theory . . . . . . . . . . . . . . . . . . . 228 24 Circuit Theory Revisited 229 24.1 Circuit Theory Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 24.1.1 Kirchhoff Current Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 24.1.2 Kirchhoff Voltage Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 viii Electromagnetic Field Theory 34.2.1 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 34.2.2 Mie Scattering by Spherical Harmonic Expansions . . . . . . . . . . . 341 34.2.3 Separation of Variables in Spherical Coordinates . . . . . . . . . . . . 341 35 Spectral Expansions of Source Fields 343 35.1 Spectral Representations of Sources . . . . . . . . . . . . . . . . . . . . . . . . 343 35.1.1 A Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 35.2 A Source on Top of a Layered Medium . . . . . . . . . . . . . . . . . . . . . . 348 35.2.1 Electric Dipole Fields–Spectral Expansion . . . . . . . . . . . . . . . . 349 35.3 Stationary Phase Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 35.4 Riemann Sheets and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 355 35.5 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 36 Computational Electromagnetics, Numerical Methods 357 36.1 Computational Electromagnetics and Numerical Methods . . . . . . . . . . . 358 36.1.1 Examples of Differential Equations . . . . . . . . . . . . . . . . . . . . 358 36.1.2 Examples of Integral Equations . . . . . . . . . . . . . . . . . . . . . . 359 36.1.3 Surface Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . 359 36.1.4 Function as a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 36.1.5 Operator as a Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 36.1.6 Approximating Operator Equations with Matrix Equations . . . . . . 364 36.2 Subspace Projection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 36.2.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 36.2.2 Differential Equation Solvers versus Integral Equation Solvers . . . . . 366 36.3 Solving Matrix Equation by Optimization . . . . . . . . . . . . . . . . . . . . 367 36.3.1 Gradient of a Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 368 37 Finite Difference Method, Yee Algorithm 371 37.1 Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . . . . 371 37.1.1 The Finite-Difference Approximation . . . . . . . . . . . . . . . . . . . 372 37.1.2 Time Stepping or Time Marching . . . . . . . . . . . . . . . . . . . . . 374 37.1.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 37.1.4 Grid-Dispersion Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 37.2 The Yee Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 37.2.1 Finite-Difference Frequency Domain Method . . . . . . . . . . . . . . 383 37.3 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 383 38 Quantum Theory of Light 387 38.1 Historical Background on Quantum Theory . . . . . . . . . . . . . . . . . . . 387 38.2 Connecting Electromagnetic Oscillation to Simple Pendulum . . . . . . . . . 390 38.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 38.4 Schrödinger Equation (1925) . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 38.5 Some Quantum Interpretations–A Preview . . . . . . . . . . . . . . . . . . . . 399 38.5.1 Matrix or Operator Representations . . . . . . . . . . . . . . . . . . . 400 38.6 Bizarre Nature of the Photon Number States . . . . . . . . . . . . . . . . . . 401 Contents ix 39 Quantum Coherent State of Light 403 39.1 The Quantum Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 39.1.1 Quantum Harmonic Oscillator Revisited . . . . . . . . . . . . . . . . . 404 39.2 Some Words on Quantum Randomness and Quantum Observables . . . . . . 406 39.3 Derivation of the Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . 407 39.3.1 Time Evolution of a Quantum State . . . . . . . . . . . . . . . . . . . 409 39.4 More on the Creation and Annihilation Operator . . . . . . . . . . . . . . . . 410 39.4.1 Connecting Quantum Pendulum to Electromagnetic Oscillator . . . . 412 39.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 x Electromagnetic Field Theory