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## Scattering problems for unbounded inhomogeneous layers

### A volume integral method for solving the scattering problem

#### The spectral volumetric method for solving the direct scattering problem

**Table of contents :**

**1 Introduction **

**2 Scattering problems for unbounded inhomogeneous layers**

2.1 Introduction

2.2 Setting of the problem

2.2.1 Construction of the DtN operators

2.2.2 Variational formulation of the problem

2.3 A Rellich identity for the case with absorption

2.4 Scattering problem from periodic layers

2.4.1 The DtN operator for quasi-periodic problems

2.4.2 Well-posedness of the problem

2.5 The case of infinite layers with local perturbations

2.6 On the case of the scattering problem for half space

2.6.1 Variational formulation of the problem

2.6.2 The Rellich identity for the case without absorption

**3 A volume integral method for solving the scattering problem **

3.1 Introduction

3.2 Setting of the problem

3.2.1 Introduction of the problem and notation

3.2.2 Formulation of the problem using the Floquet-Bloch transform

3.3 Volume integral formulation of the quasi-periodic problem

3.3.1 Setting of the volume integral equation

3.3.2 Periodization of the integral equation

3.3.3 Spectral approximation of problem (3.19)

3.4 Discretization of the problem and convergence analysis

3.4.1 Discretization and convergence in the Floquet-Bloch variable

3.4.2 Discretization in the spatial variable

3.5 Numerical Algorithm and Experiments

3.5.1 Numerical examples for real wave numbers

**4 Inverse scattering problem from locally perturbed periodic media **

4.1 Introduction

4.2 Setting of the direct scattering problem

4.3 Setting of the inverse problem

4.3.1 Definition of the sampling operator

4.3.2 Some useful properties for sampling methods

4.4 Application to Sampling methods

4.4.1 The Linear Sampling Method (LSM)

4.4.2 The Factorization Method

4.4.3 The Generalized Linear Sampling Method (GLSM)

4.4.4 Reconstruction of the periodic domain Dp from N

4.4.5 Validating Numerical Experiments

4.5 On the Use of Differential measurements

4.5.1 Theory of the Differential LSM

4.5.2 A numerical example

4.6 Sampling methods for a single Floquet-Bloch mode

4.6.1 Near field operator for a fixed Floquet-Bloch mode

4.6.2 Some properties of the operators H q , Nq and Gq

4.6.3 A new differential imaging functional

4.6.4 Numerical validating examples

**5 The case of TM-mode **

5.1 Introduction

5.2 The spectral volumetric method for solving the direct scattering problem

5.2.1 Volume integral formulation of the quasi-periodic problem

5.2.2 Discretization of the locally perturbed periodic problem

5.3 The sampling methods for the TM-mode

5.3.1 Setting of the inverse problem and definition of the sampling operator

5.3.2 Some key properties of the introduced operators

5.3.3 Sampling methods for a single Floquet-Bloch mode

**6 Surface potential formulation of a scattering problem **

6.1 Introduction

6.2 Setting of the problem

6.3 Reformulation of the problem via surface potentials

6.3.1 Surface potentials and some classical results

6.3.2 The system of surface integral equations

6.4 The Fredholm property of the surface integral operator Z(ki; ke)

6.4.1 The existence and uniqueness of solutions to the scattering problem

**7 Electromagnetic scattering with sign-changing coefficients **

7.1 Introduction

7.2 Periodic Electromagnetic Scattering

7.3 T-Coercivity Framework

7.4 Fredholm Alternative

**8 Conclusion and Perspectives **

**A Abstract theoretical foundations of the sampling methods **

A.1 Main theorem for the F] method

A.2 Theoretical foundations of GLSM

**Bibliography**