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.
Author(s): Gang Bao, Peijun Li
Series: Applied Mathematical Sciences, 208
Publisher: Springer
Year: 2021
Language: English
Pages: 366
City: Singapore
Preface
Contents
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
Appendix Appendices
A.1 Vector and Integral Identities
A.2 Vector Spaces
A.3 Sobolev Spaces
A.4 Linear Operators
A.5 Variational Formulations
A.6 Ritz–Galerkin Methods for Variational Problems
References
Index
