Preface

Chapter 1. Basic Statements

1.1. The Formulation of Boundary Value and Initial Boundary Value Problems in the Theory of Diffraction Gratings

1.1.1. Main Equations

1.1.2. Domains of Analysis, Boundary and Initial Conditions

1.1.3. Time Domain: Initial Boundary Value Problems

1.1.4. Frequency Domain: Boundary Value Problems

1.2. The General Physical Picture: Main Definitions and Consequences from Conservation Laws and Reciprocity Theorems

1.2.1. The Diffraction Problems for Plane Waves

1.2.2. The Simplest Physical Consequences from the Poynting Theorem and the Lorentz Lemma

1.3. The Spectral Theory of Gratings

1.3.1. Introduction

1.3.2. The Grating as an Open Periodic Resonator

1.3.3. The Grating as an Open Periodic Waveguide

1.3.4. Some Physical Results of Spectral Theory

Chapter 2. Analytic Regularization Methods

2.1. General Description and Classification of the Analytic Regularization Methods: History, Provenance and Survey

2.2. The Riemann-Hilbert Problem Method and its Generalization

2.2.1. Classical Dual Series Equations and the Riemann-Hilbert Problem

2.2.2. Classical Dual Series Equations with "Matrix Perturbation"

2.2.3. Dual Series Equations with the Non Unit Coefficient of Conjugation

2.2.4. The System of Dual Series Equations and Riemann-Hilbert Vector Problem

2.3. Inversion of Convolution-Type Matrix Operators in Equation Systems of the Mode Matching Technique

2.3.1. Knife Gratings: Systems of First Kind Equations and Analytic Regularization of the Problem

2.3.2. Matrix Scheme of Analytic Regularization Procedure

2.4. Electromagnetic Wave Diffraction by Gratings in Presence of a Chiral Isotropic Medium

2.4.1. Field Presentation in Chiral Medium

2.4.2. Formulation of the Problem

2.4.3. The Systems of Dual Series Equations

2.4.4. An Algebraic System of the Second Kind

2.4.5. Numerical Analysis for Grating and Chiral Half-Space

2.4.6. Strip Grating with Layered Medium

2.4.7. Electromagnetic Properties of a Strip Grating with Layered Medium in the Resonant Frequency Range

2.5. Resonant Scattering of Electromagnetic Waves by Gratings and Interfaces between Anisotropic Media and Metamaterials

2.5.1. Resonant Wave Scattering by a Strip Grating Loaded with a Metamaterial Layer

2.5.2. The Plane Wave Diffraction from a Strip Grating with Anisotropic Medium

2.6. Diffraction of Quasi-Periodic Waves by Obstacles with Cylindrical Periodical Wavy Surfaces

2.6.1. The Dirichlet Diffraction Problem

2.6.2. Reduction of the Dirichlet BVP to the Integral Equations

2.6.3. Investigation of the Differential Properties of the Integral Equation Kernel

2.6.4. Additive Splitting of the Integral Equation Kernel into a Sum of Main Singular Part and Some More Smooth Function

2.6.5. Reduction of the Integral Equation to an Infinite System of Linear Algebraic Equations of the First Kind

2.6.6. Construction of an Infinite System of Linear Algebraic Equations of the Second Kind

2.6.7. The Neumann Diffraction Problem

Chapter 3. C-Method: From the Beginnings to Recent Advances

3.1. Introduction

3.2. Classical C-Method

3.2.1. Modal Equations in Cartesian Coordinates and Quasi-Periodic Green Function

3.2.2. New Coordinate System

3.2.3. Modal Equation in the Translation Coordinate System

3.3. Diffraction of a Plane Wave by a Modulated Surface Grating

3.3.1. Posing the Problem

3.3.2. Tangential Component of a Vector Field at a Coordinate Surface

3.3.3. Boundary Conditions

3.4. Adaptive Spatial Resolution

3.4.1. Trapezoidal Grating

3.4.2. Lamellar Grating and Adaptive Spatial Resolution

3.5. Curved Strip Gratings

3.6. Sev