Lectures on Electromagnetic Field Theory
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Lectures on Electromagnetic Field Theory
02-May-2020 This set of lecture notes is from my teaching of ECE 604 Electromagnetic Field Theory
Lectures on
Electromagnetic Field Theory
Weng Cho CHEW
1Fall 2019, Purdue University
1Updated: December 4, 2019
Contents
Prefacexi
Acknowledgementsxii
1 Introduction, Maxwell's Equations1
1.1 Importance of Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 A Brief History of Electromagnetics . . . . . . . . . . . . . . . . . . . 3
1.2 Maxwell's Equations in Integral Form . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Static Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Coulomb's Law (Statics) . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Electric FieldE(Statics) . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Gauss's Law (Statics) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Derivation of Gauss's Law from Coulomb's Law (Statics) . . . . . . . 9
2 Maxwell's Equations, Dierential Operator Form15
2.1 Gauss's Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Gauss's Law in Dierential Operator Form . . . . . . . . . . . . . . . 18
2.1.2 Physical Meaning of Divergence Operator . . . . . . . . . . . . . . . . 19
2.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Stokes's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Faraday's Law in Dierential Operator Form . . . . . . . . . . . . . . 22
2.2.2 Physical Meaning of Curl Operator . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Maxwell's Equations in Dierential Operator Form . . . . . . . . . . . . . . . 24
3 Constitutive Relations, Wave Equation, Electrostatics, and Static Green's
Function25
3.1 Simple Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Emergence of Wave Phenomenon, Triumph of Maxwell's Equations . . . . . 26
3.3 Static Electromagnetics{Revisted . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Poisson's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.3 Static Green's Function . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.4 Laplace's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
i iiElectromagnetic Field Theory4 Magnetostatics, Boundary Conditions, and Jump Conditions35
4.1 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 More on Coulomb's Gauge . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Boundary Conditions{1D Poisson's Equation . . . . . . . . . . . . . . . . . . 37
4.3 Boundary Conditions{Maxwell's Equations . . . . . . . . . . . . . . . . . . . 39
4.3.1 Faraday's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.2 Gauss's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.3 Ampere's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.4 Gauss's Law for Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . 44
5 Biot-Savart law, Conductive Media Interface, Instantaneous Poynting's
Theorem45
5.1 Derivation of Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Boundary Conditions{Conductive Media Case . . . . . . . . . . . . . . . . . . 47
5.2.1 Electric Field Inside a Conductor . . . . . . . . . . . . . . . . . . . . . 47
5.2.2 Magnetic Field Inside a Conductor . . . . . . . . . . . . . . . . . . . . 49
5.3 Instantaneous Poynting's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Time-Harmonic Fields, Complex Power55
6.1 Time-Harmonic Fields|Linear Systems . . . . . . . . . . . . . . . . . . . . . 55
6.2 Fourier Transform Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Complex Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 More on Constitute Relations, Uniform Plane Wave63
7.1 More on Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.1.1 Isotropic Frequency Dispersive Media . . . . . . . . . . . . . . . . . . 63
7.1.2 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.1.3 Bi-anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.1.4 Inhomogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.1.5 Uniaxial and Biaxial Media . . . . . . . . . . . . . . . . . . . . . . . . 66
7.1.6 Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.2 Wave Phenomenon in the Frequency Domain . . . . . . . . . . . . . . . . . . 67
7.3 Uniform Plane Waves in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8 Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 73
8.1 Plane Waves in Lossy Conductive Media . . . . . . . . . . . . . . . . . . . . . 73
8.2 Lorentz Force Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3 Drude-Lorentz-Sommerfeld Model . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3.1 Frequency Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . 80
8.3.2 Plasmonic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9 Waves in Gyrotropic Media, Polarization83
9.1 Gyrotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.2.1 Arbitrary Polarization Case and Axial Ratio . . . . . . . . . . . . . . 89
9.3 Polarization and Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Contentsiii
10 Spin Angular Momentum, Complex Poynting's Theorem, Lossless Condi-
tion, Energy Density9310.1 Spin Angular Momentum and Cylindrical Vector Beam . . . . . . . . . . . . 93
10.2 Complex Poynting's Theorem and Lossless Conditions . . . . . . . . . . . . . 95
10.2.1 Complex Poynting's Theorem . . . . . . . . . . . . . . . . . . . . . . . 95
10.2.2 Lossless Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.3 Energy Density in Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . 97
11 Transmission Lines101
11.1 Transmission Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.1.1 Time-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.1.2 Frequency-Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . 105
11.2 Lossy Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
12 More on Transmission Lines109
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 Principle121
13.1 Multi-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 121
13.1.1 Single-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . 121
13.1.2 Two-Junction Transmission Lines . . . . . . . . . . . . . . . . . . . . . 122
13.1.3 Stray Capacitance and Inductance . . . . . . . . . . . . . . . . . . . . 126
13.2 Duality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
13.2.1 Unusual Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
13.2.2 Fictitious Magnetic Currents . . . . . . . . . . . . . . . . . . . . . . . 129
14 Re ection and Transmission, Interesting Physical Phenomena13314.1 Re
ection and Transmission|Single Interface Case . . . . . . . . . . . . . . . 13314.1.1 TE Polarization (Perpendicular or E Polarization)
1. . . . . . . . . . . 134
14.1.2 TM Polarization (Parallel or H Polarization) . . . . . . . . . . . . . . 136
14.2 Interesting Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 136
14.2.1 Total Internal Re
ection . . . . . . . . . . . . . . . . . . . . . . . . . . 13715 Interesting Physical Phenomena143
15.1 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, Trans-
mission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14315.1.1 Brewster Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.1.2 Surface Plasmon Polariton . . . . . . . . . . . . . . . . . . . . . . . . . 146
15.2 Homomorphism of Uniform Plane Waves and Transmission Lines Equations . 1481
These polarizations are also variously know as thesandppolarizations, a descendent from the notations
for acoustic waves wheresandpstand for shear and pressure waves respectively. ivElectromagnetic Field Theory15.2.1 TE or TE
zWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14815.2.2 TM or TM
zWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15016 Waves in Layered Media151
16.1 Waves in Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.1.1 Generalized Re
ection Coecient for Layered Media . . . . . . . . . . 15116.2 Phase Velocity and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . 154
16.2.1 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
16.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
16.3 Wave Guidance in a Layered Media . . . . . . . . . . . . . . . . . . . . . . . . 158
16.3.1 Transverse Resonance Condition . . . . . . . . . . . . . . . . . . . . . 158
17 Dielectric Waveguides161
17.1 Generalized Transverse Resonance Condition . . . . . . . . . . . . . . . . . . 161
17.2 Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
17.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
17.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
17.2.3 A Note on Cut-O of Dielectric Waveguides . . . . . . . . . . . . . . . 169
18 Hollow Waveguides171
18.1 Hollow Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
18.1.1 Absence of TEM Mode in a Hollow Waveguide . . . . . . . . . . . . . 172
18.1.2 TE Case (Ez= 0,Hz6= 0) . . . . . . . . . . . . . . . . . . . . . . . . 173
18.1.3 TM Case (
E z6= 0,Hz= 0) . . . . . . . . . . . . . . . . . . . . . . . . 17518.2 Rectangular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
18.2.1 TE Modes (H Mode orHz6= 0 Mode) . . . . . . . . . . . . . . . . . . 176
19 More on Hollow Waveguides179
19.1 Rectangular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . 179
19.1.1 TM Modes (E Modes orEz6= 0 Modes) . . . . . . . . . . . . . . . . . 179
19.1.2 Bouncing Wave Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 180
19.1.3 Field Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
19.2 Circular Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
19.2.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
19.2.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
20 More on Waveguides and Transmission Lines189
20.1 Circular Waveguides, Contd. . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
20.1.1 An Application of Circular Waveguide . . . . . . . . . . . . . . . . . . 189
20.2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface Plasmonic Modes 194
20.2.1 Quasi-TEM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
20.2.2 Hybrid Modes{Inhomogeneously-Filled Waveguides . . . . . . . . . . . 195
20.2.3 Guidance of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
20.3 Homomorphism of Waveguides and Transmission Lines . . . . . . . . . . . . . 196
20.3.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
20.3.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Contentsv
20.3.3 Mode Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
21 Resonators 203
21.1 Cavity Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
21.1.1 Transmission Line Model . . . . . . . . . . . . . . . . . . . . . . . . . 203
21.1.2 Cylindrical Waveguide Resonators . . . . . . . . . . . . . . . . . . . . 205
21.2 Some Applications of Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 208
21.2.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
21.2.2 Electromagnetic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 210
21.2.3 Frequency Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
22 Quality Factor of Cavities, Mode Orthogonality215
22.1 The Quality Factor of a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 215
22.1.1 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
22.1.2 Relation to the Pole Location . . . . . . . . . . . . . . . . . . . . . . . 216
22.1.3 Some Formulas forQfor a Metallic Cavity . . . . . . . . . . . . . . . 218
22.1.4 Example: TheQof TM110Mode . . . . . . . . . . . . . . . . . . . . . 219
22.2 Mode Orthogonality and Matrix Eigenvalue Problem . . . . . . . . . . . . . . 220
22.2.1 Matrix Eigenvalue Problem (EVP) . . . . . . . . . . . . . . . . . . . . 220
22.2.2 Homomorphism with the Waveguide Mode Problem . . . . . . . . . . 221
22.2.3 Proof of Orthogonality of Waveguide Modes . . . . . . . . . . . . . . . 222
23 Scalar and Vector Potentials225
23.1 Scalar and Vector Potentials for Time-Harmonic Fields . . . . . . . . . . . . . 225
23.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
23.1.2 Scalar and Vector Potentials for Statics, A Review . . . . . . . . . . . 225
23.1.3 Scalar and Vector Potentials for Electrodynamics . . . . . . . . . . . . 226
23.1.4 More on Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . 228
23.2 When is Static Electromagnetic Theory Valid? . . . . . . . . . . . . . . . . . 229
23.2.1 Quasi-Static Electromagnetic Theory . . . . . . . . . . . . . . . . . . . 234
24 Circuit Theory Revisited237
24.1 Circuit Theory Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
24.1.1 Kirchho Current Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
24.1.2 Kirchho Voltage Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
24.1.3 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
24.1.4 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
24.1.5 Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
24.2 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
24.2.1 Energy Storage Method for Inductor and Capacitor . . . . . . . . . . 244
24.2.2 Finding Closed-Form Formulas for Inductance and Capacitance . . . . 244
24.3 Importance of Circuit Theory in IC Design . . . . . . . . . . . . . . . . . . . 246
24.3.1 Decoupling Capacitors and Spiral Inductors . . . . . . . . . . . . . . . 249
viElectromagnetic Field Theory25 Radiation by a Hertzian Dipole251
25.1 Radiation by a Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 251
25.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
25.1.2 Approximation by a Point Source . . . . . . . . . . . . . . . . . . . . . 252
25.1.3 Case I. Near Field,r1 . . . . . . . . . . . . . . . . . . . . . . . . 254
25.1.4 Case II. Far Field (Radiation Field),r1 . . . . . . . . . . . . . . 255
25.1.5 Radiation, Power, and Directive Gain Patterns . . . . . . . . . . . . . 255
25.1.6 Radiation Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
26 Radiation Fields, Far Fields261
26.1 Radiation Fields or Far-Field Approximation . . . . . . . . . . . . . . . . . . 261
26.1.1 Far-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 262
26.1.2 Locally Plane Wave Approximation . . . . . . . . . . . . . . . . . . . 263
26.1.3 Directive Gain Pattern Revisited . . . . . . . . . . . . . . . . . . . . . 265
27 Array Antennas, Fresnel Zone, Rayleigh Distance269
27.1 Linear Array of Dipole Antennas . . . . . . . . . . . . . . . . . . . . . . . . . 269
27.1.1 Far-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 270
27.1.2 Radiation Pattern of an Array . . . . . . . . . . . . . . . . . . . . . . 270
27.2 When is Far-Field Approximation Valid? . . . . . . . . . . . . . . . . . . . . . 273
27.2.1 Rayleigh Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
27.2.2 Near Zone, Fresnel Zone, and Far Zone . . . . . . . . . . . . . . . . . 276
28 Dierent Types of Antennas{Heuristics277
28.1 Types of Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
28.1.1 Resonance Tunneling in Antenna . . . . . . . . . . . . . . . . . . . . . 277
28.1.2 Horn Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
28.1.3 Quasi-Optical Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 283
28.1.4 Small Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
29 Uniqueness Theorem291
29.1 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
29.1.1 Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
29.1.2 General Anisotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . 294
29.1.3 Hind Sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
29.1.4 Connection to Poles of a Linear System . . . . . . . . . . . . . . . . . 296
29.1.5 Radiation from Antenna Sources . . . . . . . . . . . . . . . . . . . . . 297
30 Reciprocity Theorem299
30.1 Reciprocity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
30.1.1 Conditions for Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . 302
30.1.2 Application to a Two-Port Network . . . . . . . . . . . . . . . . . . . 303
30.1.3 Voltage Sources in Electromagnetics . . . . . . . . . . . . . . . . . . . 304
30.1.4 Hind Sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
30.1.5 Transmit and Receive Patterns of an Antennna . . . . . . . . . . . . . 306
Contentsvii
31 Equivalence Theorem, Huygens' Principle309
31.1 Equivalence Theorem or Equivalence Principle . . . . . . . . . . . . . . . . . 309
31.1.1 Inside-Out Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
31.1.2 Outside-in Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
31.1.3 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
31.1.4 Electric Current on a PEC . . . . . . . . . . . . . . . . . . . . . . . . 311
31.1.5 Magnetic Current on a PMC . . . . . . . . . . . . . . . . . . . . . . . 312
31.2 Huygens' Principle and Green's Theorem . . . . . . . . . . . . . . . . . . . . 312
31.2.1 Scalar Waves Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
31.2.2 Electromagnetic Waves Case . . . . . . . . . . . . . . . . . . . . . . . 315
32 Shielding, Image Theory319
32.1 Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
32.1.1 A Note on Electrostatic Shielding . . . . . . . . . . . . . . . . . . . . . 319
32.1.2 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
32.2 Image Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
32.2.1 Electric Charges and Electric Dipoles . . . . . . . . . . . . . . . . . . 322
32.2.2 Magnetic Charges and Magnetic Dipoles . . . . . . . . . . . . . . . . . 325
32.2.3 Perfect Magnetic Conductor (PMC) Surfaces . . . . . . . . . . . . . . 327
32.2.4 Multiple Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
32.2.5 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
33 High Frequency Solutions, Gaussian Beams331
33.1 High Frequency Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
33.1.1 Tangent Plane Approximations . . . . . . . . . . . . . . . . . . . . . . 331
33.1.2 Fermat's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
33.1.3 Generalized Snell's Law . . . . . . . . . . . . . . . . . . . . . . . . . . 334
33.2 Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
33.2.1 Derivation of the Paraxial/Parabolic Wave Equation . . . . . . . . . . 335
33.2.2 Finding a Closed Form Solution . . . . . . . . . . . . . . . . . . . . . 336
33.2.3 Other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
34 Rayleigh Scattering, Mie Scattering339
34.1 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
34.1.1 Scattering by a Small Spherical Particle . . . . . . . . . . . . . . . . . 341
34.1.2 Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 342
34.1.3 Small Conductive Particle . . . . . . . . . . . . . . . . . . . . . . . . . 344
34.2 Mie Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
34.2.1 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
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