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This book addresses recent developments in mathematical analysis and computational methods for solving direct and inverse problems for Maxwell’s equations in periodic structures. The fundamental importance of the fields is clear, since they are related to technology with significant applications in optics and electromagnetics. The book provides both introductory materials and in-depth discussion to the areas in diffractive optics that offer rich and challenging mathematical problems. It is also intended to convey up-to-date results to students and researchers in applied and computational mathematics,and engineering disciplines as well.
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Contents
1 Maxwell’s Equations 1
1.1 Electromagnetic Waves 1
1.2 Jump and Boundary Conditions 6
1.3 Two Fundamental Polarizations 9
References 12
2 Diffraction Grating Theory 13
2.1 Perfectly Conducting Gratings 14
2.2 Dielectric Gratings 22
2.3 Biperiodic Gratings 32
2.3.1 Perfect Electric Conductors 33
2.3.2 Dielectric Media 38
References 42
3 Variational Formulations 45
3.1 The Dirichlet Problem 46
3.2 The Transmission Problem 53
3.3 Biperiodic Structures 59
3.3.1 Function Spaces 60
3.3.2 The Transparent Boundary Condition 68
3.3.3 The Variational Problem 76
References 84
4 Finite Element Methods 87
4.1 The Finite Element Method 89
4.1.1 Finite Element Analysis for TE Polarization 90
4.1.2 Finite Element Analysis for TM Polarization 94
4.2 Adaptive Finite Element PML Method 98
4.2.1 The PML Formulation 99
4.2.2 Transparent Boundary Condition for the PML Problem 102
4.2.3 Error Estimate of the PML Solution 105
4.2.4 The Discrete Problem 108
4.2.5 Error Representation Formula 109
4.2.6 A Posteriori Error Analysis 111
4.2.7 Numerical Results 114
4.3 Adaptive Finite Element DtN Method 118
4.3.1 The Discrete Problem 120
4.3.2 A Posteriori Error Analysis 122
4.3.3 TM Polarization 125
4.3.4 Numerical Results 126
4.4 Adaptive Finite Element PML Method for Biperiodic Structures 130
4.4.1 The PML Formulation 132
4.4.2 Transparent Boundary Condition for the PML Problem 135
4.4.3 Convergence of the PML Solution 140
4.4.4 The Discrete Problem 145
4.4.5 A Posteriori Error Analysis 148
4.4.6 Numerical Results 153
References 158
5 Inverse Diffraction Grating 163
5.1 Uniqueness Theorems 164
5.1.1 The Helmholtz Equation 165
5.1.2 Maxwell’s Equations 170
5.2 Local Stability 175
5.2.1 The Helmholtz Equation 176
5.2.2 Maxwell’s Equations 182
5.3 Numerical Methods 193
References 200
6 Near-Field Imaging 205
6.1 Near-Field Data 208
6.1.1 The Variational Problem 210
6.1.2 An Analytic Solution 215
6.1.3 Convergence of the Power Series 219
6.1.4 The Reconstruction Formula 224
6.1.5 Error Estimates 228
6.1.6 Numerical Results 232
6.2 Far-Field Data 233
6.2.1 The Reduced Problem 236
6.2.2 Transformed Field Expansion 238
6.2.3 The Reconstruction Formula 242
6.2.4 A Nonlinear Correction Scheme 243
6.2.5 Numerical Results 244
6.3 Maxwell’s Equations 245
6.3.1 The Reduced Model Problem 248
6.3.2 Transformed Field Expansion 249
6.3.3 The Zeroth Order Term 253
6.3.4 The First Order Term 254
6.3.5 The Reconstruction Formula 256
6.3.6 Numerical Results 258
References 261
7 Related Topics 267
7.1 Method of Boundary Integral Equations 267
7.1.1 Model Problems 268
7.1.2 Quasi-periodic Green’s Function 270
7.1.3 Boundary Integral Operators 273
7.1.4 Boundary Integral Equations 277
7.1.5 Integral Formulas for Rayleigh’s Coefficients 280
7.2 Time-Domain Problems 282
7.2.1 Problem Formulation 282
7.2.2 Time-Domain Transparent Boundary Condition 286
7.2.3 The Reduced Problem 291
7.2.4 A Priori Estimates 297
7.3 Nonlinear Gratings 302
7.3.1 SHG Model 303
7.3.2 TE-TE Polarization 305
7.3.3 TM-TE Polarization 310
7.4 Optimal Design Problems 315
7.4.1 The Model Problem 316
7.4.2 The Optimal Design Problem 318
7.4.3 Homogenization of the Design Problem 320
7.4.4 The Relaxed Problem 323
References 326
Appendices 331
Tndex 351
Book list of the Series in Information and Computational Science 357
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