Theory of Reflection (eBook)

John Lekner is Professor of Theoretical Physics at Victoria University of Wellington. He has a PhD from the University of Chicago, and has taught at Cambridge, where he was a Fellow of Emmanuel College. He has worked mainly in statistical physics, quantum mechanics and electrodynamics. He has published more than 140 papers, of which about 40 are reflection-related.

Preface 7
Contents 9
1 Introducing Reflection 15
1.1 The Electromagnetic {/varvec s} Wave 15
1.2 The Electromagnetic {/varvec p} Wave 20
1.3 Particle Waves 26
1.4 Acoustic Waves 30
1.5 Scattering and Reflection 34
1.6 A Look Ahead 38
References 53
Further Readings 53
2 Exact Results 55
2.1 Comparison Identities, and Conservation and Reciprocity Laws 55
2.2 General Expressions for {/varvec r}_{{/varvec s}} and {/varvec r}_{{/varvec p}} 60
2.3 Reflection at Grazing Incidence, and the Existence of a Principal Angle 66
2.4 Reflection by a Homogeneous Layer 69
2.5 The Tanh, Exp and Rayleigh Profiles 75
References 86
Further Readings 87
3 Reflection of Long Waves 88
3.1 Integral Equation and Perturbation Theory for the s Wave 88
3.2 The s Wave to Second Order in the Interface Thickness 92
3.3 Integral Invariants 94
3.4 /left| {{{/varvec r}}_{{{/varvec p}}} } /right|^{{{/bf 2}}} and {{/varvec r}}_{{{/varvec p}}} {{/varvec /r}}_{{{/varvec s}}} to Second Order 97
3.5 Reflection by a Thin Film Between Like Media 101
3.6 Six Profiles and Their Integral Invariants 103
References 106
Further Readings 106
4 Variational Theory 107
4.1 A Variational Expression for the Reflection Amplitude 107
4.2 Variational Estimate for rs in the Long Wave Case 110
4.3 Exact, Perturbation and Variational Results for the sech2 Profile 111
4.4 Variational Theory for the p Wave 115
4.5 Reflection by a Layer Between Like Media 118
4.6 The Hulth00E9n-Kohn Variational Method Applied to Reflection 122
4.7 Variational Estimates in the Short Wave Case 124
References 126
Further Readings 126
5 Equations for the Reflection Amplitudes 127
5.1 A First Order Non-linear Equation for an {/varvec s} Wave Reflection Coefficient 127
5.2 An Example: Reflection by the Linear Profile 129
5.3 Differential Equation for a {/varvec p} Wave Reflection Coefficient 132
5.4 Upper Bounds on {/varvec R}_{{/varvec s}} and on {/varvec R}_{{/varvec p}} 134
5.5 Long Wave Expansions 136
5.6 Differential Equations for the Reflection Amplitudes 140
5.7 Weak Reflection: The Rayleigh Approximation 142
5.8 Iteration of the Integral Equation for {/varvec r} 143
References 145
6 Reflection of Short Waves 147
6.1 Short Wave Limiting Forms for Some Solvable Profiles 147
6.2 Approximate High-Frequency Waveforms 151
6.3 Profiles of Finite Extent with Discontinuities in Slope at the Endpoints 154
6.4 Reflection Amplitude Estimates from a Comparison Identity 157
6.5 Perturbation Theory for Short Waves 162
6.6 Short Wave Results for {/varvec r}_{{/varvec p}} and {/varvec r}_{{/varvec p}} /varvec{/}{/varvec r}_{{/varvec s}} 164
6.7 A Single Turning Point: Total Reflection 171
6.8 Two Turning Points, and Tunneling 178
References 184
Further Readings 185
7 Simple Anisotropy 186
7.1 Anisotropy with Azimuthal Symmetry 186
7.2 Ellipsometry of a Thin Film on an Isotropic Substrate 190
7.3 Thin Film on an Anisotropic Substrate 193
7.4 General Results for Anisotropic Stratifications with Azimuthal Symmetry 195
7.5 Differential Equations for the Reflection Amplitudes 196
7.6 Reflection from the Ionosphere 198
References 200
Further Readings 201
8 Uniaxial Anisotropy 202
8.1 Propagation Within Homogeneous Anisotropic Media 202
8.2 Dielectric Tensor and Normal Modes in Uniaxial Crystals 204
8.3 Uniaxial Crystal Reflection and Transmission Amplitudes 207
8.4 Bounds and Zeros of the Reflection Amplitudes, the Polarizing Angle 210
8.5 External Reflection from an Immersed Crystal 212
8.6 Normal-Incidence Reflection and Transmission 213
8.7 Normal Incidence on a Uniaxial Plate 216
8.8 Isotropic Layer on a Uniaxial Substrate 220
8.9 Optical Properties of a Uniaxial Layer 222
References 223
9 Ellipsometry 225
9.1 Polarizer--Sample--Analyser 225
9.2 Polarizer--Compensator--Sample--Analyser 227
9.3 Polarizer--Sample--Compensator--Analyser 228
9.4 Polarizer--Modulator--Sample--Analyser 229
9.5 Polarizer--Sample--Modulator--Analyser 231
9.6 Ellipsometric Measurements: The Principal Angle 231
9.7 Transmission Ellipsometry 232
9.7.1 Polarizer--Sample--Analyser 233
9.7.2 Polarizer--Compensator--Sample--Analyser 233
9.7.3 Polarizer--Sample--Compensator--Analyser 234
9.7.4 Transmission Ellipsometry with a Polarization Modulator 235
9.8 Reflection and Transmission Ellipsometry of a Homogeneous Layer 235
9.9 Reflection Ellipsometry of Uniaxial Crystals 238
References 240
10 Absorption 242
10.1 Fresnel Reflection Formulae for an Absorbing Medium 243
10.2 General Results for Reflection by Absorbing Media 249
10.3 Dielectric Layer on an Absorbing Substrate 250
10.4 Absorbing Film on a Non-absorbing Substrate 251
10.5 Thin Inhomogeneous Absorbing Films 254
10.6 Attenuated Total Reflection, Surface Waves 258
10.7 Attenuated Total Reflection: Second Example 265
10.8 Reflection by a Diffuse Absorbing Interface: The Tanh Profile 268
10.9 Zero Reflection from Dielectric Layer on Absorbing Substrate 271
References 271
11 Inverse Problems 273
11.1 Reflection at a Sharp Boundary 274
11.2 Homogeneous Film Between Like Media 277
11.3 Inversion of Transmission Ellipsometric Data for a Homogeneous Nonabsorbing Layer 279
11.4 Inversion of Reflection Ellipsometric Data for a Homogeneous Nonabsorbing Layer 280
11.5 Synthesis of a Profile from r as a Function of Wavenumber 281
11.6 Inversion of the Rayleigh Approximation 284
11.7 Principal Angle of an Absorber 286
References 287
Further Readings 288
12 Matrix and Numerical Methods 289
12.1 Matrices Relating the Coefficients of Linearly Independent Solutions 289
12.2 Matrices Relating Fields and Their Derivatives 293
12.3 Multilayer Dielectric Mirrors at Normal Incidence 298
12.4 Reflection of Long Waves 301
12.5 Absorbing Stratified Media: Some General Results 303
12.6 High Transparency of an Absorbing Film in a Frustrated Total Reflection Configuration 306
12.7 Comparison of Numerical Approaches 308
12.8 Numerical Methods Based on the Layer Matrices 309
12.9 Variable Step Size, Profile Truncation, Total Reflection and Tunneling, Absorption, and Calculation of Wavefunctions 314
References 317
13 Periodically Stratified Media 318
13.1 Electromagnetic Waves in Stratified Media 319
13.2 Periodic Structures, Multilayer Dielectric Mirrors 324
13.3 Omnidirectional Reflection by Multilayer Dielectric Mirrors 330
13.3.1 Band Edges at Oblique Incidence for a General Stack 332
13.3.2 Refractive Indices for Which Omnidirectional Reflection Exists 334
13.4 Form Birefringence 336
13.5 Absorbing Periodically Stratified Media 339
13.5.1 Reflection of s-Polarized Plane Waves 340
13.5.2 Reflection of p-Polarized Plane Waves 342
13.5.3 Application to an Absorbing Quarter-Wave Stack 344
References 345
14 Rough or Structured Surfaces 347
14.1 Reflection from Rough Surfaces: The Rayleigh Criterion 348
14.2 Corrugated Surfaces, Diffraction Gratings 349
14.3 Scattering of Light by Liquid Surfaces 355
14.4 The Surface Integral Formulation of Scattering by Rough Surfaces 359
14.5 Absorbing and Rough Surfaces that Are Wet 362
14.6 Coherent Backscattering 364
References 365
Further Readings 367
15 Particle Waves 369
15.1 General Results 369
15.2 Some Exactly Solvable Profiles 373
15.3 Perturbation and Variational Theories 380
15.4 Long Waves, Integral Invariants 382
15.5 Riccati-Type Equations the Rayleigh Approximation
15.6 Reflection of Short Waves 386
15.7 Absorption, the Optical Potential 388
15.8 Inversion of a Model Reflection Amplitude 391
15.9 Time Delay in the Reflection of Wavepackets 393
References 396
16 Neutron and X-ray Reflection 397
16.1 Common Features of X-ray and Neutron Optics 398
16.2 Reflection Near the Critical Angle 399
16.3 Reflection by Profiles Without Discontinuities 403
16.4 Reflection by Profiles with Discontinuities 406
16.5 Total Reflection: Extraction of the Phase in Lloyd's Mirror Experiments 411
16.6 Reflection of Neutrons by Periodic Stratifications 416
16.7 Neutron Reflection by Magnetic Materials 420
References 422
17 Acoustic Waves 424
17.1 General Relations for Stratified Media 424
17.1.1 General Results for the Reflection and Transmission Amplitudes 425
17.2 Matrix Methods 430
17.3 Low-Frequency Reflection and Transmission 436
17.4 High-Frequency Limiting Forms 439
17.5 Exact Solutions for the exp-lin and exp-exp Stratifications 444
17.6 An Upper Bound on the Acoustic Reflectivity 446
17.7 Profiles with Discontinuities in Density or Sound Speed 449
Appendix: Universal Properties of Acoustic Pulses and Beams 453
References 456
18 Chiral Isotropic Media 457
18.1 Constitutive Relations 458
18.2 Reflection and Transmission Amplitudes, Conservation Laws 460
18.2.1 Differential Reflectance, Ellipsometry 464
18.3 Wave Propagation in Chiral Media 465
18.3.1 Eigenstates of Curl 467
18.3.2 Boundary Conditions 468
18.4 Reflection from an Achiral--Chiral Interface 469
18.4.1 Wavefunctions 470
18.4.2 Reflection and Transmission Amplitudes 470
18.4.3 The Angles {/varvec{/uptheta}}_{{/bf B}} , {/varvec{/uptheta}}_{{/bf pp}}, {{/varvec and}}/,{/varvec{/uptheta}}_{{/bf pol}} 474
18.5 Optical Properties of a Chiral Layer 475
18.5.1 Normal Incidence 476
18.5.2 Optical Properties Near the Critical Angles 478
References 479
19 Pulses and Wavepackets 480
19.1 Reflection of Nearly Monochromatic Pulses: The Time Delay 480
19.2 Nonreflection of Wavepackets by a Subset of the sech2 Potentials 484
19.2.1 Construction of Reflectionless Wavepackets 485
19.3 Exact Solutions of Total and Partial Reflection of Wavepackets 488
Appendix: Universal Properties of Electromagnetic Pulses 492
Conservation Laws, Energy-Momentum Inequalities 494
Angular Momentum, Helicity 496
Lorentz Transformation of Pulses 498
References 500
Further Readings 501
20 Finite Beams 502
20.1 Universal Properties of Scalar and of Electromagnetic Beams 502
20.1.1 Bateman Integral Solution of the Wave Equation 504
20.1.2 Conservation Laws and Beam Invariants 505
20.1.3 Non-existence Theorems 507
20.1.4 Focal Plane Zeros 508
20.2 Reflection of Beams: The Lateral Beam Shift 510
20.3 Reflection of Gaussian Beams 514
20.3.1 Reflection at a Potential Spike (Delta Function) 516
20.3.2 Reflection at a Sharp Boundary Between Two Media 517
Appendix 1: Total Internal Reflection: The rs, rp Phases and Their Difference 518
Total Reflection by the Hyperbolic Tangent Profile 522
Appendix 2: Polarization of Electromagnetic Beams 524
Examples of Exactly and Approximately Linearly Polarized Beams 526
Approximately Circularly Polarized Beams 528
References 529
Further Readings 530
Appendix: Reflection and Transmission Formulae 531
Index 536