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## Chapter 2 Scattering problems for unbounded inhomogeneous layers

**Abstract**

The goal of this chapter is two-folds. It first introduces the scattering problem that motivates the main developments in this thesis, namely the scattering problem for perturbed infinite periodic layers. It second complements the results of the literature on non existence of guided modes using Rellich type identities. We show in particular that if absorption is present in the layer, then one can avoid the strict monotonicity assumption for the real part of the refractive index. This allows in particular considering the problem where the layer is embedded in the free space. We also sketch the simpler results for periodic problems and the cases where the layer lies on a substrate with Dirichlet boundary conditions.

**Introduction**

We examine in this chapter the well-posedness of the scattering problem from unbounded dielec-tric layers in the harmonic regime. The case of locally perturbed periodic layers, which is central in this thesis, can be seen as a particular case of this problem. The analysis for the latter can also rely on results for periodic layers but we were not able to obtain more general results than those for ”general” unbounded layers.

There exists an abundant literature on scattering problems from unbounded structures [6, 9, 37, 86, 34, 107, 106]. A first issue is to specify the radiation condition imposed at infinity and accordingly the solution space. The radiation condition can also be substituted with the use of adhoc weighted spaces [35, 9]. We adopt here the simplest form of radiation conditions based of the application of the Fourier transform in the longitudinal direction of the layer [86, 37]. This allows the introduction of a Dirichlet-to-Neumann mapping to bound the domain in the direction “orthogonal” to the layer and then to use a variational technique to study the scattering problem. In the case on unbounded layers, the variational domain is still unbounded in transverse directions and therefore the use of Fredholm alternative is not possible (as in the case of scattering problems from bounded domains). A by now widely used technique to get around this diﬃculty, is to derive a priori estimates on the solution using specific multipliers (and yielding the so-called Rellich identities). One then verifies an inf-sup condition for the variational formulation to conclude on the well-posedness of the problem. This is for instance the approach applied in [86] to study the scattering problem from dielectric layers. After recalling some results from the latter reference, we shall complement them by studying a specific case with absorption. The main reason of considering this case is to allow the index of refraction to be the same on both sides of the layer (this cannot be the case in [86]). We obtain a Rellich identity independently from strict monotonicity of the refractive index by assuming a definite positive imaginary part of the refractive index on a layer of positive thickness (see Fig 2.2).

In the case of periodic layers, the sign for the imaginary part does not need to hold on a the whole layer of positive thickness. This is due to the fact that the variational domain becomes bounded (using the periodicity condition). The well posedness of the problem can then be deduced from the use of a Fredholm alternative in the cases of quasi-periodic sources (or incident waves). When the incident wave is no longer quasi-periodic, the analysis can no longer be recast to the use of a Fredholm alternative (since the obtained bound on the solution is not explicit in terms of the quasi-periodicity parameter). One can rely for this case on the results for unbounded layers that need stronger assumptions of the absorption coeﬃcient or the strict monotonicity of the real part.

The case of locally perturbed periodic layers can be seen as a compact perturbation of the scattering problem from unbounded layers. A Fredholm alternative is used to infer results on the well-posedness for this problem.

Another case where the assumption on the monotonicity of the real part of the refractive index can be relaxed is the case of half-space scattering problems where a Dirichlet boundary condition is imposed on the half-space boundary. Although very similar problems are already studied in number of papers [86, 38], we shall provide here a proof of this case for the reader’s convenience. Including this case in our presentation is not essential but can be useful for the reader to get a hint on how things would simplify in this case. Let us also point out that similarly to this chapter, the results of Chapters 3-5 can be easily extended to the case of half space with Dirichlet (or Neumann boundary conditions, if one assumes that the direct problem is well posed) with minor and obvious adaptations.

The outline of the chapter is as follows. We first present the setting of the scattering problem for unbounded layers in Section (2.2). We introduce the Rayleigh radiation condition and estab-lish the variational formulation of the problem. We then recall some results from the literature on the well-posedness of the problem under some monotonicity conditions on the real part of the refractive index. We present in Section 2.3 the main result of this chapter on the well-posedness of the problem for the case where absorption is present is some layer of positive thickness. We establish the Rellich identity that allows the derivation of well-posedness results. We indicate in Section 2.4 how the assumptions can be simplified for the case of periodic layers and how one can simply address the case of locally perturbed layers. Section 2.5 is dedicated to the study of problems that can be seen as “compact perturbations” of the problem studied in the previous sections. We end up this chapter with Section 2.6 where the case of the scattering problem in half space with Dirichlet boundary conditions is studied.

### Setting of the problem

We consider the Helmholtz equation given by u + k2nu = g in Rd (2.1)

with a source term g 2 L2(Rd) and where the total field u satisfies some radiation condition (that will be specified later) with respect to the direction xd and where n denotes the index of refraction. We consider the case where the wavenumber k is positive and real valued. In the following, D denotes a domain such that n = 1 outside D and we shall assume that D h for some h > 0 where h := Rd 1 ] h; h[ (See Fig. 2.1). In a first step,

**A Rellich identity for the case with absorption**

We shall study in this section the well-posedness of the problem for the case where some absorp-tion in present in the media. More specifically, let us denote by De the domain such that Im n = 0 outside De. Then we shall make the following assumption (the assumption is illustrated in Fig. 2.2) Assumption 2.3.1. The index of refraction n 2 L1(Rd) with non negative imaginary part. Moreover, the domain De contains a thin infinite layer := Rd 1 ] ; [, > 0 and the refractive index n has positive definite imaginary part in De, i.e. 9 c0 > 0 such that Im n c0 in De.

#### Well-posedness of the problem

The uniqueness of solutions to the problem holds if the refractive index np has a positive imagi-nary part in a sub-domain O h0. This is made precise in the following theorem.

Theorem 2.4.4. Assume that the refractive index np 2 L1( h0) and that Im (np) > 0 in a sub-domain O h0. Then problem (2.69) is well-posed.

**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

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**