本书系统地阐述周期性结构中的正、反散射理论，它涵盖了周期结构中麦克斯韦方程组正、反散射问题的几乎所有主题，包括数学模型、物理原理、数学分析以及计算方法。该书首先介绍了电磁场和光栅的基本理论。对于正散射问题，本书详细介绍变分方法来研究解的适定性以及自适应有限元的数值计算方法。对于反散射问题，本书讨论了解的唯一性、稳定性以及光栅界面重构的数值方法。此外，本书也介绍了此领域中的重要最新发展，例如近场成像、时域散射问题、非线性光学和最优设计问题等。该书可以为攻读硕士和博士学位的研究生提供初步的材料，以介绍周期性结构中麦克斯韦方程的正、反散射理论。它还为应用和计算数学的研究人员以及不同相关学科，例如电磁学和光学的工程人员提供了最新的深刻的数学结果和具有挑战性的问题。

1 Maxwell's Equations
1.1 Electromagnetic Waves
1.2 Jump and Boundary Conditions
1.3 Two Fundamental Polarizations
References
2 Diffraction Grating Theory
2.1 Perfectly Conducting Gratings
2.2 Dielectric Gratings
2.3 Biperiodic Gratings
2.3.1 Perfect Electric Conductors
2.3.2 Dielectric Media
References
3 Variational Formulations
3.1 The Dirichlet Problem
3.2 The Transmission Problem
3.3 Biperiodic Structures
3.3.1 Function Spaces
3.3.2 The Transparent Boundary Condition
3.3.3 The Variational Problem
References
4 Finite Element Methods
4.1 The Finite Element Method
4.1.1 Finite Element Analysis for TE Polarization
4.1.2 Finite Element Analysis for TM Polarization
4.2 Adaptive Finite Element PML Method
4.2.1 The PML Formulation
4.2.2 Transparent Boundary Condition for the PML Problem
4.2.3 Error Estimate of the PML Solution
4.2.4 The Discrete Problem
4.2.5 Error Representation Formula
4.2.6 A Posteriori Error Analysis
4.2.7 Numerical Results
4.3 Adaptive Finite Element DtN Method
4.3.1 The Discrete Problem
4.3.2 A Posteriori Error Analysis
4.3.3 TM Polarization
4.3.4 Numerical Results
4.4 Adaptive Finite Element PML Method for Biperiodic Structures
4.4.1 The PML Formulation
4.4.2 Transparent Boundary Condition for the PML Problem
4.4.3 Convergence of the PML Solution
4.4.4 The Discrete Problem
4.4.5 A Posteriori Error Analysis
4.4.6 Numerical Results
References
5 Inverse Diffraction Grating
5.1 Uniqueness Theorems
5.1.1 The Helmholtz Equation
5.1.2 Maxwell's Equations
5.2 Local Stability
5.2.1 The Helmholtz Equation
5.2.2 Maxwell's Equations
5.3 Numerical Methods
References
6 Near-Field Imaging
6.1 Near-Field Data
6.1.1 The Variational Problem
6.1.2 An Analytic Solution
6.1.3 Convergence of the Power Series
6.1.4 The Reconstruction Formula
6.1.5 Error Estimates
6.1.6 Numerical Results
6.2 Far-Field Data
6.2.1 The Reduced Problem
6.2.2 Transformed Field Expansion
6.2.3 The Reconstruction Formula
6.2.4 A Nonlinear Correction Scheme
6.2.5 Numerical Results
6.3 Maxwell's Equations
6.3.1 The Reduced Model Problem
6.3.2 Transformed Field Expansion
6.3.3 The Zeroth Order Term
6.3.4 The First Order Term
6.3.5 The Reconstruction Formula
6.3.6 Numerical Results
References
7 Related Topics
7.1 Method of Boundary Integral Equations
7.1.1 Model Problems
7.1.2 Quasi-periodic Green's Function
7.1.3 Boundary Integral Operators
7.1.4 Boundary Integral Equations
7.1.5 Integral Formulas for Rayleigh's Coefficients
7.2 Time-Domain Problems
7.2.1 Problem Formulation
7.2.2 Time-Domain Transparent Boundary Condition
7.2.3 The Reduced Problem
7.2.4 A Priori Estimates
7.3 Nonlinear Gratings
7.3.1 SHG Model
7.3.2 TE-TE Polarization
7.3.3 TM-TE Polarization
7.4 Optimal Design Problems
7.4.1 The Model Problem
7.4.2 The Optimal Design Problem
7.4.3 Homogenization of the Design Problem
7.4.4 The Relaxed Problem
References
Appendices
Index
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