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Preface |
7 |
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Contents |
9 |
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1 Introducing Reflection |
15 |
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1.1 The Electromagnetic {\varvec s} Wave |
15 |
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1.2 The Electromagnetic {\varvec p} Wave |
20 |
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1.3 Particle Waves |
26 |
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1.4 Acoustic Waves |
30 |
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1.5 Scattering and Reflection |
34 |
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1.6 A Look Ahead |
38 |
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References |
53 |
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Further Readings |
53 |
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2 Exact Results |
55 |
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2.1 Comparison Identities, and Conservation and Reciprocity Laws |
55 |
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2.2 General Expressions for {\varvec r}_{{\varvec s}} and {\varvec r}_{{\varvec p}} |
60 |
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2.3 Reflection at Grazing Incidence, and the Existence of a Principal Angle |
66 |
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2.4 Reflection by a Homogeneous Layer |
69 |
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2.5 The Tanh, Exp and Rayleigh Profiles |
75 |
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References |
86 |
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Further Readings |
87 |
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3 Reflection of Long Waves |
88 |
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3.1 Integral Equation and Perturbation Theory for the s Wave |
88 |
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3.2 The s Wave to Second Order in the Interface Thickness |
92 |
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3.3 Integral Invariants |
94 |
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3.4 \left| {{{\varvec r}}_{{{\varvec p}}} } \right|^{{{\bf 2}}} and {{\varvec r}}_{{{\varvec p}}} {{\varvec /r}}_{{{\varvec s}}} to Second Order |
97 |
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3.5 Reflection by a Thin Film Between Like Media |
101 |
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3.6 Six Profiles and Their Integral Invariants |
103 |
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References |
106 |
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Further Readings |
106 |
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4 Variational Theory |
107 |
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4.1 A Variational Expression for the Reflection Amplitude |
107 |
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4.2 Variational Estimate for rs in the Long Wave Case |
110 |
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4.3 Exact, Perturbation and Variational Results for the sech2 Profile |
111 |
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4.4 Variational Theory for the p Wave |
115 |
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4.5 Reflection by a Layer Between Like Media |
118 |
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4.6 The Hulth00E9n-Kohn Variational Method Applied to Reflection |
122 |
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4.7 Variational Estimates in the Short Wave Case |
124 |
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References |
126 |
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Further Readings |
126 |
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5 Equations for the Reflection Amplitudes |
127 |
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5.1 A First Order Non-linear Equation for an {\varvec s} Wave Reflection Coefficient |
127 |
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5.2 An Example: Reflection by the Linear Profile |
129 |
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5.3 Differential Equation for a {\varvec p} Wave Reflection Coefficient |
132 |
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5.4 Upper Bounds on {\varvec R}_{{\varvec s}} and on {\varvec R}_{{\varvec p}} |
134 |
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5.5 Long Wave Expansions |
136 |
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5.6 Differential Equations for the Reflection Amplitudes |
140 |
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5.7 Weak Reflection: The Rayleigh Approximation |
142 |
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5.8 Iteration of the Integral Equation for {\varvec r} |
143 |
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References |
145 |
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6 Reflection of Short Waves |
147 |
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6.1 Short Wave Limiting Forms for Some Solvable Profiles |
147 |
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6.2 Approximate High-Frequency Waveforms |
151 |
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6.3 Profiles of Finite Extent with Discontinuities in Slope at the Endpoints |
154 |
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6.4 Reflection Amplitude Estimates from a Comparison Identity |
157 |
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6.5 Perturbation Theory for Short Waves |
162 |
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6.6 Short Wave Results for {\varvec r}_{{\varvec p}} and {\varvec r}_{{\varvec p}} \varvec{/}{\varvec r}_{{\varvec s}} |
164 |
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6.7 A Single Turning Point: Total Reflection |
171 |
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6.8 Two Turning Points, and Tunneling |
178 |
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References |
184 |
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Further Readings |
185 |
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7 Simple Anisotropy |
186 |
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7.1 Anisotropy with Azimuthal Symmetry |
186 |
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7.2 Ellipsometry of a Thin Film on an Isotropic Substrate |
190 |
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7.3 Thin Film on an Anisotropic Substrate |
193 |
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7.4 General Results for Anisotropic Stratifications with Azimuthal Symmetry |
195 |
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7.5 Differential Equations for the Reflection Amplitudes |
196 |
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7.6 Reflection from the Ionosphere |
198 |
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References |
200 |
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Further Readings |
201 |
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8 Uniaxial Anisotropy |
202 |
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8.1 Propagation Within Homogeneous Anisotropic Media |
202 |
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8.2 Dielectric Tensor and Normal Modes in Uniaxial Crystals |
204 |
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8.3 Uniaxial Crystal Reflection and Transmission Amplitudes |
207 |
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8.4 Bounds and Zeros of the Reflection Amplitudes, the Polarizing Angle |
210 |
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8.5 External Reflection from an Immersed Crystal |
212 |
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8.6 Normal-Incidence Reflection and Transmission |
213 |
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8.7 Normal Incidence on a Uniaxial Plate |
216 |
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8.8 Isotropic Layer on a Uniaxial Substrate |
220 |
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8.9 Optical Properties of a Uniaxial Layer |
222 |
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References |
223 |
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9 Ellipsometry |
225 |
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9.1 Polarizer--Sample--Analyser |
225 |
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9.2 Polarizer--Compensator--Sample--Analyser |
227 |
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9.3 Polarizer--Sample--Compensator--Analyser |
228 |
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9.4 Polarizer--Modulator--Sample--Analyser |
229 |
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9.5 Polarizer--Sample--Modulator--Analyser |
231 |
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9.6 Ellipsometric Measurements: The Principal Angle |
231 |
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9.7 Transmission Ellipsometry |
232 |
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9.7.1 Polarizer--Sample--Analyser |
233 |
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9.7.2 Polarizer--Compensator--Sample--Analyser |
233 |
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9.7.3 Polarizer--Sample--Compensator--Analyser |
234 |
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9.7.4 Transmission Ellipsometry with a Polarization Modulator |
235 |
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9.8 Reflection and Transmission Ellipsometry of a Homogeneous Layer |
235 |
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9.9 Reflection Ellipsometry of Uniaxial Crystals |
238 |
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References |
240 |
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10 Absorption |
242 |
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10.1 Fresnel Reflection Formulae for an Absorbing Medium |
243 |
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10.2 General Results for Reflection by Absorbing Media |
249 |
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10.3 Dielectric Layer on an Absorbing Substrate |
250 |
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10.4 Absorbing Film on a Non-absorbing Substrate |
251 |
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10.5 Thin Inhomogeneous Absorbing Films |
254 |
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10.6 Attenuated Total Reflection, Surface Waves |
258 |
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10.7 Attenuated Total Reflection: Second Example |
265 |
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10.8 Reflection by a Diffuse Absorbing Interface: The Tanh Profile |
268 |
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10.9 Zero Reflection from Dielectric Layer on Absorbing Substrate |
271 |
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References |
271 |
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11 Inverse Problems |
273 |
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11.1 Reflection at a Sharp Boundary |
274 |
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11.2 Homogeneous Film Between Like Media |
277 |
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11.3 Inversion of Transmission Ellipsometric Data for a Homogeneous Nonabsorbing Layer |
279 |
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11.4 Inversion of Reflection Ellipsometric Data for a Homogeneous Nonabsorbing Layer |
280 |
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11.5 Synthesis of a Profile from r as a Function of Wavenumber |
281 |
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11.6 Inversion of the Rayleigh Approximation |
284 |
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11.7 Principal Angle of an Absorber |
286 |
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References |
287 |
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Further Readings |
288 |
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12 Matrix and Numerical Methods |
289 |
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12.1 Matrices Relating the Coefficients of Linearly Independent Solutions |
289 |
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12.2 Matrices Relating Fields and Their Derivatives |
293 |
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12.3 Multilayer Dielectric Mirrors at Normal Incidence |
298 |
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12.4 Reflection of Long Waves |
301 |
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12.5 Absorbing Stratified Media: Some General Results |
303 |
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12.6 High Transparency of an Absorbing Film in a Frustrated Total Reflection Configuration |
306 |
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12.7 Comparison of Numerical Approaches |
308 |
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12.8 Numerical Methods Based on the Layer Matrices |
309 |
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12.9 Variable Step Size, Profile Truncation, Total Reflection and Tunneling, Absorption, and Calculation of Wavefunctions |
314 |
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References |
317 |
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13 Periodically Stratified Media |
318 |
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13.1 Electromagnetic Waves in Stratified Media |
319 |
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13.2 Periodic Structures, Multilayer Dielectric Mirrors |
324 |
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13.3 Omnidirectional Reflection by Multilayer Dielectric Mirrors |
330 |
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13.3.1 Band Edges at Oblique Incidence for a General Stack |
332 |
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13.3.2 Refractive Indices for Which Omnidirectional Reflection Exists |
334 |
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13.4 Form Birefringence |
336 |
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13.5 Absorbing Periodically Stratified Media |
339 |
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13.5.1 Reflection of s-Polarized Plane Waves |
340 |
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13.5.2 Reflection of p-Polarized Plane Waves |
342 |
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13.5.3 Application to an Absorbing Quarter-Wave Stack |
344 |
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References |
345 |
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14 Rough or Structured Surfaces |
347 |
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14.1 Reflection from Rough Surfaces: The Rayleigh Criterion |
348 |
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14.2 Corrugated Surfaces, Diffraction Gratings |
349 |
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14.3 Scattering of Light by Liquid Surfaces |
355 |
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14.4 The Surface Integral Formulation of Scattering by Rough Surfaces |
359 |
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14.5 Absorbing and Rough Surfaces that Are Wet |
362 |
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14.6 Coherent Backscattering |
364 |
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References |
365 |
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Further Readings |
367 |
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15 Particle Waves |
369 |
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15.1 General Results |
369 |
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15.2 Some Exactly Solvable Profiles |
373 |
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15.3 Perturbation and Variational Theories |
380 |
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15.4 Long Waves, Integral Invariants |
382 |
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15.5 Riccati-Type Equations |
384 |
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15.6 Reflection of Short Waves |
386 |
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15.7 Absorption, the Optical Potential |
388 |
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15.8 Inversion of a Model Reflection Amplitude |
391 |
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15.9 Time Delay in the Reflection of Wavepackets |
393 |
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References |
396 |
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16 Neutron and X-ray Reflection |
397 |
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16.1 Common Features of X-ray and Neutron Optics |
398 |
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16.2 Reflection Near the Critical Angle |
399 |
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16.3 Reflection by Profiles Without Discontinuities |
403 |
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16.4 Reflection by Profiles with Discontinuities |
406 |
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16.5 Total Reflection: Extraction of the Phase in Lloyd's Mirror Experiments |
411 |
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16.6 Reflection of Neutrons by Periodic Stratifications |
416 |
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16.7 Neutron Reflection by Magnetic Materials |
420 |
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References |
422 |
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17 Acoustic Waves |
424 |
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17.1 General Relations for Stratified Media |
424 |
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17.1.1 General Results for the Reflection and Transmission Amplitudes |
425 |
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17.2 Matrix Methods |
430 |
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17.3 Low-Frequency Reflection and Transmission |
436 |
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17.4 High-Frequency Limiting Forms |
439 |
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17.5 Exact Solutions for the exp-lin and exp-exp Stratifications |
444 |
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17.6 An Upper Bound on the Acoustic Reflectivity |
446 |
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17.7 Profiles with Discontinuities in Density or Sound Speed |
449 |
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Appendix: Universal Properties of Acoustic Pulses and Beams |
453 |
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References |
456 |
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18 Chiral Isotropic Media |
457 |
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18.1 Constitutive Relations |
458 |
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18.2 Reflection and Transmission Amplitudes, Conservation Laws |
460 |
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18.2.1 Differential Reflectance, Ellipsometry |
464 |
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18.3 Wave Propagation in Chiral Media |
465 |
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18.3.1 Eigenstates of Curl |
467 |
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18.3.2 Boundary Conditions |
468 |
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18.4 Reflection from an Achiral--Chiral Interface |
469 |
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18.4.1 Wavefunctions |
470 |
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18.4.2 Reflection and Transmission Amplitudes |
470 |
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18.4.3 The Angles {\varvec{\uptheta}}_{{\bf B}} , {\varvec{\uptheta}}_{{\bf pp}}, {{\varvec and}}\,{\varvec{\uptheta}}_{{\bf pol}} |
474 |
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18.5 Optical Properties of a Chiral Layer |
475 |
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18.5.1 Normal Incidence |
476 |
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18.5.2 Optical Properties Near the Critical Angles |
478 |
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References |
479 |
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19 Pulses and Wavepackets |
480 |
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19.1 Reflection of Nearly Monochromatic Pulses: The Time Delay |
480 |
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19.2 Nonreflection of Wavepackets by a Subset of the sech2 Potentials |
484 |
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19.2.1 Construction of Reflectionless Wavepackets |
485 |
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19.3 Exact Solutions of Total and Partial Reflection of Wavepackets |
488 |
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Appendix: Universal Properties of Electromagnetic Pulses |
492 |
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Conservation Laws, Energy-Momentum Inequalities |
494 |
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Angular Momentum, Helicity |
496 |
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Lorentz Transformation of Pulses |
498 |
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References |
500 |
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Further Readings |
501 |
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20 Finite Beams |
502 |
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20.1 Universal Properties of Scalar and of Electromagnetic Beams |
502 |
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20.1.1 Bateman Integral Solution of the Wave Equation |
504 |
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20.1.2 Conservation Laws and Beam Invariants |
505 |
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20.1.3 Non-existence Theorems |
507 |
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20.1.4 Focal Plane Zeros |
508 |
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20.2 Reflection of Beams: The Lateral Beam Shift |
510 |
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20.3 Reflection of Gaussian Beams |
514 |
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20.3.1 Reflection at a Potential Spike (Delta Function) |
516 |
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20.3.2 Reflection at a Sharp Boundary Between Two Media |
517 |
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Appendix 1: Total Internal Reflection: The rs, rp Phases and Their Difference |
518 |
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Total Reflection by the Hyperbolic Tangent Profile |
522 |
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Appendix 2: Polarization of Electromagnetic Beams |
524 |
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Examples of Exactly and Approximately Linearly Polarized Beams |
526 |
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Approximately Circularly Polarized Beams |
528 |
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References |
529 |
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Further Readings |
530 |
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Appendix: Reflection and Transmission Formulae |
531 |
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Index |
536 |
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