Fioralba Cakoni

Houssem Haddar
email: houssem.haddar@inria.fr

Thi-Phong Nguyen

New Interior Transmission Problem Applied to a Single Floquet-Bloch Mode Imaging of Local Perturbations in Periodic Media

This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior transmission problem, which until now was an open problem except for the special case when the local perturbation did not intersect the background inhomogeneities. The analysis of this new interior transmission problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.

Introduction

Nondestructive testing of period media is an important problem with grown interest since periodic material are part of many fascinating engineering structures with many technological use such as nanograss. In many situation the periodicity of the healthy periodic material is complicated or difficult to model mathematically, hence computing its Green's function is computationally expensive or even impossible. On the other hand, when looking for flows in such complex media, the option of reconstructing everything, i.e. both periodic structure and defects, may not be viable. The approach used in this paper provides a criteria to reconstruct the support of anomalies without explicitly know or reconstruct the background. The imaging method is based on the generalized linear sampling method which was first introduced in [START_REF] Audibert | Qualitative methods for heterogeneous media[END_REF], [START_REF] Audibert | A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements[END_REF]. This method falls in the class of qualitative approaches to inverse scattering. We refer the reader to [START_REF] Kirsch | The Factorization Method for Inverse Problems[END_REF] and [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] for a description of various aspects of such approaches. Qualitative methods have been applied to the imaging of many periodic structure, see [START_REF] Arens | Scattering by biperiodic layered media: The integral equation approach[END_REF], [START_REF] Arens | A complete factorization method for scattering by periodic structures[END_REF], [START_REF] Bourgeois | On the identification of defects in a periodic waveguide from far field data[END_REF], [START_REF] Elschner | Inverse scattering of elastic waves by periodic structures: uniqueness under the third or fourth kind boundary conditions[END_REF], [START_REF] Lechleiter | Factorization Method for Electromagnetic Inverse Scattering from Biperiodic Structures[END_REF], [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF], [START_REF] Lechleiter | Reconstruction of local perturbations in periodic surfaces[END_REF], [START_REF] Liem Nguyen | Spectral Methods for Direct and Inverse Scattering from Periodic Structures[END_REF] for a sample of work. In the case of our problem, we use an adapted version of so-called differential linear sampling method which process the measured data against the data coming from the healthy background. The idea of using differential measurements for sampling methods was first introduced in [START_REF] Audibert | Identifying defects in an unknown background using differential measurements[END_REF] where the response of the background was measured, and was adapted to the case of locally perturbed periodic layers in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF], [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF]. For the latter, the response of the periodic background does not need to be measured. It is replaced by the extraction of measurements associated with a single Floquet-Bloch mode to encode some differential behavior for the indicator functions. This extraction requires information only on the period size of the background. The justification of this method makes essential use of a non-standard interior transmission problem whose well-posedness was open, and this limited its use to the case when the defect does not intersect the inhomogeneous components of the background. In this paper we provide sufficient conditions for solvability of this non-standard interior transmission problem which allow us to design a differential imaging function for more general location of defects. Let us introduce the problem we consider here. More specifically, we are concerned with nondestructive testing of a penetrable infinite layer in R d , d = 2, 3 which is periodic with respect to d -1 first variables. Let L 1 , • • • , L d-1 , L j > 0, j = 1, • • • , d-1 denote the periods of each of these d-1 variables, respectively. The d -1 periodic refractive index of this periodic layer, denoted here by n p , from physical consideration is a bounded function, has positive real part Re (n p ) and nonnegative imaginary part Im (n p ) ≥ 0. Furthermore for simplicity we assume that this periodic layer is embedded in a homogeneous background with refractive index normalized to one, i.e. n p = 1 for |x d | > h for some fixed h > 0. This is what we refer to as the healthy material. We assume that one or finitely many cells of the layer are locally damaged. This means that in a compactly supported region ω (which can have multiple connected components) the refractive index differs from n p . Let us call n the refractive index of the damaged layer (which is not any longer periodic), i.e. n = n p only in ω. The goal is to determine the support of the damaged region ω by using the measured scattered field outside the layer due to appropriate incident fields (to become precise later). The challenging task however is to resolve ω without an explicit knowledge of n p (which in practice can have a complicated form) nor reconstructing it, but just using the fact that n p is d -1 periodic with known periods L 1 , • • • , L d-1 under some technical restriction which will be explained in the paper. More specifically, our analysis can be carried through for a M -period truncation of the infinite layer (containing the local defect), for M large enough, which is then extended as M L-periodic layer. As it is shown in [START_REF] Haddar | A volume integral method for solving scattering problems from locally perturbed infinite periodic layers[END_REF], this truncation is equivalent to approximating the problem in the Floquet-Bloch domain using uniform discretization of the Floquet-Bloch variable and a trapezoidal rule to approximate the discretized solution. However, let us remark that from the numerical point of view, our inversion algorithm does not require this M L-periodicity assumption. This is why it is indeed desirable to remove this technical assumption in our analysis. We also note that this assumption is not needed in [START_REF] Bourgeois | On the identification of defects in a periodic waveguide from far field data[END_REF] and [START_REF] Lechleiter | Reconstruction of local perturbations in periodic surfaces[END_REF] for the analysis of the inverse problem. The reason is that in these papers the local defect is reconstructed assuming that the periodic background is known, which is not the case in our paper, as mentioned above. The paper is configured as follows. In the next section we formulate the direct and inverse problem, define the measurements operator and recall some of its properties which are essential to our imaging method. Section 3 is devoted to introducing the near field operator corresponding to a single Floquet-Bloch mode that enables us to use a differential imaging approach. Most importantly here we study the properties of this operator which bring up the new interior transmission problem. Section 4 deals with the analysis of this new interior transmission problem. In the last section we build the differential imaging function and study its behavior for various positions of defective regions. In addition, here we provide some numerical example showing the viability of our inversion method.

Formulation of the Problem

In this section we give a rigorous formulation of the direct and inverse scattering problem we consider here. In order to motivate the new interior transmission problem which is our main concern, we recall the differential linear sampling method that was first introduced in [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] (see also [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]). This method recovers the support of local perturbations of a periodic layer without needing to compute the Green's function of the periodic layer. However to do so we must make some technical mathematical restrictions aimed to preserve some kind of periodicity for the damaged layer. In particular, we truncate the damaged infinite periodic layer by considering M periods (with M large enough to contain the defect) and extend it periodically, yielding to a M L := (M L 1 , • • • , M L d-1 )periodic layer. We call again n the refractive index of the M L-periodic extension of the truncated part. In this section we formulate rigorously this construction where we base our inversion algorithm. We remark that it is highly desirable to remove this mathematical artifact.

The Direct Scattering Problem

Here we adopt the notations from [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]. Recall that the parameter

L := (L 1 , • • • , L d-1 ) ∈ R d-1 , L j > 0, j = 1, • • • , d -1
:= (M 1 , • • • , M d-1 ) ∈ N d-1
refers to the number of periods in the truncated domain. A function defined in R d is called L periodic if it is periodic with period L with respect to the d -1 first variables. We consider in the following M L-periodic Helmholtz equation (vector multiplications is to be understood component wise, i.e.

M L = (M 1 L 1 , • • • , M d-1 L d-1
)). In this problem, the total field u

satisfies ∆u + k 2 nu = 0 in R d , d = 2, 3 u is M L-periodic (1)
where k > 0 is the wave number. We assume that the index of refraction n ∈ L ∞ (R d ) M

n = n p = 1 h -h ω O O c Ω 0 L Λ := O ∪ ω, D := Λ ∪ O c M L
Ω m = M - L , M + L × R with Ω m := -L 2 + mL, L 2 + mL × R, M - L := -M 2 + 1 2 L, M + L := M 2 + 1 2 L, and Z d-1 M := {m ∈ Z d-1 , -M 2 + 1 ≤ m ≤ M 2 , = 1, . . . , d -1}, where we use the notation a, b := [a 1 , b 1 ] × • • • × [a d-1 , b d-1
] and • denotes the floor function. We also shall use the notation

a := |a 1 • a 2 • • • a d-1 |.
Without loss of generality we assume that there is a local perturbation ω located in only one period, say Ω 0 (note the case when more periods are defective, the assumption holds true by grouping these cells as one cell with different period). This problem is treated in [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] under a strict assumption that the local perturbation does not intersect with the periodic background. In this work, we remove this assumption, and allow for the local perturbation to be located everywhere in Ω 0 . We call D p the support of n p -1 and D = D p ∪ ω, note that n = 1 outside D. For the justification of our inversion method (that relies on a unique continuation argument) we make the assumption that R d \ D is connected.

We consider down-to-up or up-to-down incident plane waves of the form

u i,± (x, j) = -i 2 β # (j) e iα # (j)•x±iβ # (j)x d (2)
where

α # (j) := 2π M L j and β # (j) := k 2 -|α # (j)| 2 , Im (β # (j)) ≥ 0, j ∈ Z d-1 and x = (x, x d ) ∈ R d-1 × R. (Note that α # (j) is a vector defined component-wise). Then the scattered field u s = u -u i satisfies ∆u s + k 2 nu s = -k 2 (n -1)u i in R d , u s is M L-periodic (3)
and we impose as a radiation condition the Rayleigh expansions:

   u s (x, x d ) = ∈Z d-1 u s + ( )e i(α # ( )•x+β # ( )(x d -h)) , ∀ x d > h, u s (x, x d ) = ∈Z d-1 u s -( )e i(α # ( )•x-β # ( )(x d +h)) , ∀ x d < -h, (4)
where the Rayleigh coefficients u s ± ( ) are given by

u s + ( ) := 1 | M - L , M + L | M - L ,M + L u s (x, h)e -iα # ( )•x dx , u s -( ) := 1 | M - L , M + L | M - L ,M + L u s (x, -h)e -iα # ( )•x dx . (5)
We shall use the notation

Θ h := M - L , M + L ×] -h, h[ Γ h M := M - L , M + L × {h}, Γ -h M := M - L , M + L × {-h}.
w ∈ H 1 # (Θ h ) satisfying ∆w + k 2 nw = k 2 (1 -n)f (6)
together with the Rayleigh radiation condition (4). Then we make the following assumption:

Assumption 1. The refractive index n and k > 0 are such that (6) with n and with n replaced by n p are both well-posed for all f ∈ L 2 (Ω h M ). We remark that the solution w ∈ H 1 # (Θ h ) of ( 6) can be extended to a function in Θ satisfying ∆w + k 2 nw = k 2 (1 -n)f , using the Rayleigh expansion (4). We denote by Φ(n p ; •) the fundamental solution to

         ∆Φ(n p ; •) + k 2 n p Φ(n p ; •) = -δ 0 , Φ(n p ; •) is M L -periodic,
and the Rayleigh radiation condition (4). [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] Then w has the representation as

w(x) = - D k 2 (n p -n)w + k 2 (1 -n)f (y)Φ(n p ; x -y) dy . (8)
For sufficient conditions that guarantee Assumption 1 we refer the reader to [START_REF] Haddar | A volume integral method for solving scattering problems from locally perturbed infinite periodic layers[END_REF], [START_REF] Kirsch | The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer[END_REF], [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] and the references therein.

The Inverse Problem

The inversion method is based on the so-called the Generalized Linear Sampling Method, which was first introduced in [3], [START_REF] Audibert | A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements[END_REF] (see also [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF]Chapter 2]), augmented with the idea of differential imaging introduced in [START_REF] Audibert | Identifying defects in an unknown background using differential measurements[END_REF] which was adapted to this problem in [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF].

As described above we have two choices of interrogating waves. If we use down-to-up (scaled) incident plane waves u i,+ (x; j) defined by (2), then our measurements (data for the inverse problem) are given by the Rayleigh sequences

u s + ( ; j), (j, ) ∈ Z d-1 × Z d-1 ,
whereas if we use up-to-down (scaled) incident plane waves u i,-(x; j) defined by (2) then our measurements are given the Rayleigh sequences

u s -( ; j), (j, ) ∈ Z d-1 × Z d-1 .
These measurements define the so-called near field (or data) operator which is used to derive the indicator function of the defect. More specifically, let us consider the (Herglotz) operators H + : 2 (Z d-1 ) → L 2 (D) and H -: 2 (Z d-1 ) → L 2 (D) defined by

H ± a := j∈Z d-1 a(j)u i,± (•; j) D , ∀ a = {a(j)} j∈Z d-1 ∈ 2 (Z d-1 ). (9)
Then H ± is compact and its adjoint (

H ± ) * : L 2 (D) → 2 (Z d-1
) is given by [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] (

H ± ) * ϕ := { ϕ ± (j)} j∈Z d-1 , where ϕ ± j := D ϕ(x)u i,± (•; j)(x) dx . ( 10
)
Let us denote by H ± inc (D) the closure of the range of H ± in L 2 (D). We then consider the (compact) operator G ± :

H ± inc (D) → 2 (Z d-1 ) defined by G ± (f ) := { w ± ( )} ∈Z d-1 , (11)
where { w ± ( )} ∈Z d-1 is the Rayleigh sequence of w ∈ H 1 # (Θ h ) the solution of [START_REF] Bourgeois | On the identification of defects in a periodic waveguide from far field data[END_REF]. We now define the sampling operators N ± :

2 (Z d-1 ) → 2 (Z d-1 ) by N ± (a) = G ± H ± (a). ( 12
)
By linearity of the operators G ± and H ± we also get an equivalent definition of N ± directly in terms of measurements as

[N ± (a)] = j∈Z d-1 a(j) u s ± ( ; j), ∈ Z d-1 . ( 13
)
The following properties of G ± and H ± are crucial to our inversion method. To state them, we must recall the standard interior transmission problem:

(u, v) ∈ L 2 (D)×L 2 (D) such that u -v ∈ H 2 (D) and              ∆u + k 2 nu = 0 in D, ∆v + k 2 v = 0 in D, u -v = ϕ on ∂D, ∂(u -v)/∂ν = ψ on ∂D, (14)
for given (ϕ, ψ) ∈ H 3/2 (∂D) × H 1/2 (∂D) where ν denotes the outward normal on ∂D. k is called a transmission eigenvalue if the homogeneous problem [START_REF] Kirsch | The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer[END_REF], i.e. with ϕ = 0 and ψ = 0, has non-trivial solutions. Up-to-date results on this problem can be found in [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF]Chapter 3] where in particular one finds sufficient solvability conditions. In the sequel we make the following assumption. If the boundary of D intersects the boundary of Ω 0 , then the previous interior transmission problem should be augmented with periodicity conditions on ∂D ∩ ∂Θ. Since this condition does not affect the assumptions on the solvability of the interior transmission problem (in H 2 (D) with periodic conditions on ∂D ∩ ∂Θ) nor requires any substantial modification of the arguments below (other than changing the solution space), we make the choice of simplifying this technicality and assume that ∂D ∩ ∂Ω 0 = ∅.

Assumption 2. ∂D ∩ ∂Ω 0 = ∅ and the refractive index n and the wave number k > 0 are such that (14) has a unique solution.

In particular, if Re (n -1) > 0 or -1 < Re (n -1) < 0 uniformly in a neighborhood of ∂D inside D the interior transmission problem [START_REF] Kirsch | The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer[END_REF] satisfies the Fredholm alternative, and the set of real standard transmission eigenvalues is discrete (possibly empty). Thus Assumption 2 holds as long as k > 0 is not a transmission eigenvalue. From now on, for z ∈ Θ h , we denote by Φ ± (•; z) := { Φ ± ( ; z)} ∈Z d-1 the Rayleigh sequences of Φ(n p , z) with n p = 1 define in [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] given by

Φ ± ( ; z) := i 2 M L β # ( ) e -i(α # ( )z-β # ( )|z d ∓h|) . ( 15
)
Lemma 2.1 (Lemma 3.3 in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]). The operator H ± is compact and injective. Let H ± inc (D) be the closure of the range of

H ± in L 2 (D). Then H ± inc (D) = H inc (D) := {v ∈ L 2 (D) : ∆v + k 2 v = 0 in D}. ( 16
)
Theorem 2.2 (Theorem 3.5 in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]). Assume that Assumptions 1 and 2 hold. Then the operator G ± : H inc (D) → 2 (Z d-1 ) defined by ( 11) is injective with dense range. Moreover Φ ± (•; z) belongs to R(G ± ) if and only if z ∈ D.

Another main ingredient is a symmetric factorization of an appropriate operator given in terms of N ± . To this end, for a generic operator F : H → H, where H is a Hilbert space, with adjoint F * we define

F := |Re (F)| + |Im (F)| ( 17
)
where Re (F) :

= 1 2 (F + F * ), Im (F) := 1 2i (F -F * ). Now if T : L 2 (D) → L 2 (D) is defined by Tf := k 2 (n -1)(f + w| D ) (18)
with w being the solution of ( 6), we have the following result under Assumptions 1 and 2.

Theorem 2.3 (Theorem 4.2 in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]). The following factorization holds

N ± = (H ± ) * T H ± , (19)
where T :

L 2 (D) → L 2 (D) is self-adjoint and coercive on H inc (D). Moreover, z ∈ D if and only if Φ ± (•; z) ∈ R (N ± ) 1/2 .
The above theorem provides a rigorous method to recover the support of D. However this is not satisfactory since the aim is find only the support of ω and trying to reconstruct everything may not be feasible due to possible complicated structure of the periodic media and even useless if ω ⊂ D p . Our goal is to derive an imaging method that resolves only ω without knowing or recovering D p . This leads us to introducing next the sampling operator for a single Floquet-Bloch mode whose analysis will bring up a new interior transmission problem. We end this section by introducing some more notations to be used in the sequel.

Definition 2.4. A function u is called quasi-periodic with parameter ξ = (ξ 1 , • • • , ξ d-1 ) and period L = (L 1 , • • • , L d-1
), with respect to the first d -1 variables (briefly denoted as ξ-quasi-periodic with period L) if:

u(x + jL, x d ) = e iξ•(jL) u(x, x d ), ∀j ∈ Z d-1 .
Let q be a fixed parameter in Z d-1 M , we denote by Φ q (x) the solution to

∆Φ q (x) + k 2 Φ q (x) = -δ 0 in Ω 0 (20)
and is α q quasi-periodic with period L with α q := 2πq/(M L).

Then Φ q (• -z) remains the solution to ∆Φ q (• -z) + k 2 Φ q (• -z) = -δ z in Ω 0 for all z ∈ R d . The Rayleigh coefficients Φ ± q (•; z) of Φ q (• -z) are given by Φ ± q (j; z) = i 2 L β # (q+M ) e -i(α # (q+M )z-β # (q+M )|z d ∓h|) if j = q + M , ∈ Z d-1 , 0 if j = q + M , ∈ Z d-1 . (21)
Furthermore we assume that each period of D p is composed by J ∈ N disconnected components and the defect ω as well as the components that contains or have non-empty intersection with ω are in one cell, which we denote by Ω 0 (otherwise we could rearrange the period). For convenience, we now introduce some additional notations. We denote by O the union of the components of D p ∩ Ω 0 that have nonempty intersection with ω, and by O c its complement in D p ∩ Ω 0 , i.e the union of all the components of D p ∩ Ω 0 that do not intersect ω.

O c p = m∈Z d-1 M O c + ν m , Λ p := m∈Z d-1 M Λ + ν m and D p := m∈Z d-1 M D + ν m (22) Remark that D p ≡ D p ∪ ∪ m∈Z d-1 M ω + ν m contains D and the L-periodic copies of ω \ D p . We remark that n = n p = 1 in D p \ D.

The Near Field Operator for a Single Floquet-Bloch Mode

Let us define the operator I q : 2 (Z d-1 ) → 2 (Z d-1 ) given by

(I q a)( ) = a(j), if = q + jM 0, else. (23)
The adjoint

I * q : 2 (Z d-1 ) → 2 (Z d-1
) is then (I * q b)(j) = b(q + jM ). The single Floquet-Bloch mode Herglotz operator H ± q : 2 (Z d-1 ) → L 2 (D) is defined by

H ± q a := H ± I q a = j a(j)u i,± (•; q + jM )| D ( 24
)
and the single Floquet-Bloch mode near field (or data) operator

N ± q : 2 (Z d-1 ) → 2 (Z d-1 ) is defined by N ± q a = I * q N ± I q a. (25)
We remark that H ± q a is an α q -quasi-periodic function with period L. The sequence N ± q a corresponds to the Fourier coefficients of the α q -quasi-periodic component of the scattered field in the decomposition (30). This operator is then somehow associated with α q -quasi-periodicity. One immediately sees from the factorization N ± = (H ± ) * T H ± that the following factorization holds.

N ± q = (H ± q ) * T H ± q . ( 26
)
For later use we also define the operator G ± q : R(H ± q ) → 2 (Z d-1 ) by

G ± q = (H ± q ) * T| R(H ± q ) (27)
where the operator T is defined by [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF].

Observing that ϕ(j; x)

:= e iα # (j)•x = e i 2π M L j•x , j ∈ Z d-1
is a Fourier basis of M L periodic function in L 2 (Θ), we have that any w ∈ L 2 (Θ) which is M L periodic, has the expansion

w(x) = j∈Z d-1 w(j, x d )ϕ(j; x), where w(j, x d ) := 1 M L Θ w(x)ϕ(j; x) dx .
(28) Spliting j by module M we can arrange the expansion of w as

w(x) = q∈Z d-1 M ∈Z d-1 w(q + M , x d )ϕ(q + M ; x) , (29)
where ϕ(q + M ; x) is α q -quasi-periodic with period L, here α q := 2π M L q. Letting

w q := ∈Z w(q + M , x d )ϕ(q + M ; x)
we have that w q is α q -quasi-periodic with period L. Thus any M L-periodic function

w ∈ L 2 (Θ) can be decomposed w = q∈Z M w q (30)
where w q is α q -quasi-periodic with period L. Moreover, by the orthogonality of the Fourier basis {ϕ(j; •)} j∈Z , we have that

w q ± (j) = 0 if j = q + M , ∈ Z and w ± (q + M ) = w q ± (q + M ) ( 31
)
where w q ± (j) the Rayleigh sequence of w q defined in [START_REF] Audibert | A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements[END_REF]. By definition of G ± q , we see that G ± q (f ) is a Rayleigh sequence of w ± (j) at all indices j = q + M , ∈ Z, where w is solution of [START_REF] Bourgeois | On the identification of defects in a periodic waveguide from far field data[END_REF]. Seeing also the line above that theses coefficients come from the Rayleigh sequence of w q where w q is one of the component of w using the decomposition (30), which is α q -quasi periodic. We now assume that f | Dp is α q -quasi-periodic. Then, using the decomposition (30) for w, and that fact that n p is periodic, f is α q -quasiperiodic and n -n p is compactly supported in one period Ω 0 , (6) becomes

∆w q + k 2 n p w q = k 2 (n p -n)w + k 2 (1 -n)f in Ω 0 . ( 32
)
Denoting by w := w -w q , the previous equation is equivalent to

∆w q + k 2 nw q = k 2 (n p -n) w + k 2 (1 -n)f in Ω 0 . (33)
Therefore, operator G ± q : R(H ± q ) → 2 (Z d-1 ) can be equivalently defined as

G ± q (f ) := I * q { w q ± ( )} ∈Z d-1 , (34)
where w q solution of (33) and w q + w is solution of ( 6). This definition is helpful for proving the following properties for H ± q and G ± q that are the counterpart results to Lemma 2.1 and Lemma 2.2, now needed for the operator N ± q .

Lemma 3.1. The operator H ± q is injective and

R(H ± q ) = H q inc (D) := {v ∈ L 2 (D), ∆v + k 2 v = 0 in D and v| Dp is α q -quasi-periodic}.
Proof. The proof of this lemma follows the lines of the proof of Lemma 5.1 in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] slightly modified to account for more general location of the defect. H ± q is injective since H ± is injective and I q is injective. We now prove that (H ± q ) * is injective on H q inc (D). Let ϕ ∈ H q inc (D) and assume (H ± q ) * (ϕ) = 0. We define

u(x) := 1 M D Φ q (x -y)ϕ(y) dy .
where Φ q (x) has expansion

Φ q (x) = i 2 L ∈Z d-1 1 β # (q + M ) e iα # (q+M )•x+iβ # (q+M )|x d | . ( 35
) with α # (q + M ) = 2π M L (q + M ) and β # (q + M ) = k 2 -|α # (q + M )| 2
. By definition of u and the expansion of Φ q we have that

u + (j) = 1 | M - L , M + L | M - L ,M + L 1 M D Φ q ((x, h) -y)ϕ(y) dy e -iα # (j)•x dx =      D ϕ(y) i 2β # (j) e iα # (j)-iβ # (j)(x d -h) = (H + ) * (ϕ) (j), if j = q + M 0 if j = q + M (
36) which implies that u ± (j) = 0 for all j = q + M and u ± (q + M ) = ((H ± ) * (ϕ)) (q + M ) = ((H ± q ) * (ϕ))( ) = 0. Therefore u has all Rayleigh coefficients equal 0, which implies that u = 0, for ± x d > h.

We now observe that for all y ∈ D, ∆Φ q (• -y)

+ k 2 Φ q (• -y) = 0 in the complement of D p . This implies that ∆u + k 2 u = 0 in R d \ D p .
Using a unique continuation argument we infer that u = 0 in Θ \ D p . Therefore, u ∈ H 2 0 ( D p ) by the regularity of volume potentials. We now consider two cases:

If ω ⊂ D p , then D p ≡ D p , i.e u ∈ H 2 0 (D p )
. Moreover, by definition, u verifies ∆u + k 2 u = -ϕ in D p and using the fact that ∆ϕ + k 2 ϕ = 0 in D p we finally have

-ϕ 2 L 2 (Dp) = Dp (∆u + k 2 u)ϕ dx = Dp u(∆ϕ + k 2 ϕ) dx = 0. ( 37
)
This proves that ϕ = 0 and yields the injectivity of (H ± q ) * on H q inc (D).

If ω ⊂ D p , let denote by ω c := ω \ D p then ω c = ∅. We then rewrite u(x) as

u(x) := 1 M ω c Φ q (x -y)ϕ(y) dy + 1 M Dp Φ q (x -y)ϕ(y) dy .
Since ϕ| Dp and Φ q (•) are α q -quasi-periodic functions with period L, then for x ∈ D p we have

Dp∩Ωm Φ q (x -y)ϕ(y) dy = Dp∩Ω 0 Φ q (x -y)ϕ(y) dy , ∀ m ∈ Z d-1 M . Therefore, for x ∈ D p ∩ Ω m . u(x) := 1 M ω c Φ q (x -y)ϕ(y) dy + Dp∩Ωm Φ q (x -y)ϕ(y) dy (38) We recall that ∆Φ q (• -x) + k 2 Φ q (• -x) = -δ x in Ω m and ∆Φ q (• -x) + k 2 Φ q (• -x) = 0 in ω c (by quasi-periodicity of Φ q (• -x)). Hence, from (38) we obtain that for m ∈ Z d-1 M , ∆u(x) + k 2 u(x) = -ϕ(x) in D p ∩ Ω m . ( 39
) Let us set for m ∈ Z d-1 M ϕ m (x) := e iαq•mL ϕ(x -mL) for x ∈ ω c + mL.
Then we have, using the α q -quasi-periodicity of Φ q (•) that for x ∈ ω c + mL

u(x) := 1 M ω c +mL Φ q (x -y)ϕ m (y) dy + 1 M Dp Φ q (x -y)ϕ(y) dy where Φ q (• -x) + k 2 Φ q (• -x) = -0 in D p and ∆Φ q (• -x) + k 2 Φ q (• -x) = -δ x in ω c + mL. We then get ∆u(x) + k 2 u(x) = -ϕ m in ω c + mL. (40)
Now define the function ϕ by

ϕ = ϕ in D p and ϕ = ϕ m in ω c + mL with m ∈ Z d-1 M .
Clearly, we have

∆ ϕ + k 2 ϕ = 0 in D p .
Since u ∈ H 2 0 ( D p ) we then have

Dp (∆u + k 2 u) ϕ = 0.
This implies according to (39) and (40) that

Dp |ϕ| 2 dx + M ω c |ϕ| 2 = 0,
which implies ϕ = 0 in D. This proves the injectivity of (H ± ) * on H q inc (D) and hence proves the Lemma.

The following theorem concerning the properties G ± q requires the solvability of a new interior transmission problem (to be formulated later), which up to this study was an open problem except for the case when ω ∩ D p = ∅ investigated in [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF]. Assumption 3. The refractive index n and k > 0 are such that the new interior transmission problem defined in Definition 3.3 has a unique solution.

Section 4 is dedicated to derive sufficient conditions for which Assumption 3 holds true. Theorem 3.2. Suppose that Assumptions 1, 2 and 3 hold. Then the operator G ± q :

H q inc (D) → 2 (Z d-1
) is injective with dense range.

Proof. To prove the injectivity of G q we assume that f ∈ H q inc (D) such that G q (f ) = 0. Let w be solution of ( 6) with data f . Since f | Dp is α q -quasi-periodic and n p -n is compactly supported of period Ω 0 we then have from (34) that

G ± q (f ) := I * q { w q ± ( )} ∈Z d-1 ,
where { w q ± ( )} ∈Z d-1 is the Rayleigh sequence of w q and w q is solution to

∆w q + k 2 nw q = k 2 (n p -n)(w -w q ) + k 2 (1 -n)f in Ω 0 (41)
In particular we have that ∆w q + k 2 w q = 0 in Θ \ D p (we recall notations in ( 22)). Using a similar unique continuation argument as at the beginning of the proof of Lemma 3.1 we deduce that

w q = 0 in Θ \ D p . Actually, if ω ⊂ D p then D p ≡ D p thus f | Dp is α q -quasi-periodic. However, if ω ⊂ D p , f | Dp is not α q -quasi-periodic.
To restore α q -quasi-periodicity, we introduce another function f by

f := f in Θ \ Λ p e iαqmL f | Λ in O + mL, ∀ m ∈ Z M . (42)
i.e we keep f the same outside Λ p and extend the values of f in Λ by α q -quasi-periodicity to Λ p . Since f | Dp is α q -quasi-periodic then the definition of

f implies that f = f also in Λ p ∩ D p . Therefore f = f in D and f = f in Θ \ D p (in other words f = f in D p \ D)
and f | Dp is α q -quasi-periodic with period L. We remark that in the special case when ω ⊂ D p , f ≡ f and D p ≡ D p . We now write (41) in terms of f (since f ≡ f in D)

∆w q + k 2 nw q = k 2 (n p -n)(w -w q ) + k 2 (1 -n) f in Ω 0 , (43)
hence it is enough to prove that f = 0 in D p . To this end, using the fact that

f = f in D (recall that D = Λ ∪ O c = D p ∩ Ω 0 = D ∩ Ω 0 ) then f verifies ∆ f + k 2 f = 0 in D p
and by the α q -quasi-periodicity of w q and f , it is sufficient to prove that f = 0 in the cell Ω 0 , i.e, proving that the following problem

   ∆w q + k 2 nw q = k 2 (n p -n)(w -w q ) + k 2 (1 -n) f in D, ∆ f + k 2 f = 0 in D, (44)
has trivial solution (w q , f )

∈ H 2 0 ( D) × L 2 ( D). Since in O c , n = n p and O c ∩ Λ = ∅, then (w q , f ) ∈ w q ∈ H 2 0 (O c ) × L 2 (O c ) verifies ∆w q + k 2 nw q = k 2 (1 -n)f in O c , ∆f + k 2 f = 0 in O c . ( 45
)
Assumption 2 implies that equation (45) has a trivial solution, and therefore

w q = f = 0 in O c .
It remains to prove that f = 0 in Λ. We can now rewrite (44) as a problem only in Λ

w q ∈ H 2 0 (Λ) and f ∈ L 2 (Λ)    ∆w q + k 2 nw q = k 2 (n p -n)(w -w q ) + k 2 (1 -n) f in Λ, ∆ f + k 2 f = 0 in Λ. (46)
To deal with this problem, we first express the quantity w -w q in terms of f using the property that f = 0 outside Λ. To this end, recalling that f = f in D, we can write [START_REF] Bourgeois | On the identification of defects in a periodic waveguide from far field data[END_REF] in terms of f as ∆w

+ k 2 n p w = k 2 (n p -n)w + k 2 (1 -n) f (47)
and then have

w(x) = - D k 2 (n p -n)w + k 2 (1 -n) f (y)Φ(n p ; x -y) dy . (48)
Using the facts that f = 0 and n = n p in O c p , i.e.

n p = n = 1 in Λ p \ D we have w(x) = - D\O c p k 2 (n p -n)w + k 2 (1 -n) f (y)Φ(n p ; x -y) dy . = - Λp k 2 (n p -n)w + k 2 (1 -n) f (y)Φ(n p ; x -y) dy . = -k 2 Λ (n p -n)w + (1 -n) f (y)Φ(n p ; x -y) dy -k 2 Λp\Λ (1 -n p ) f (y)Φ(n p ; x -y) dy . (49)
Moreover, since w q ∈ H 2 0 (Λ), then for all θ ∈ H 2 (Λ) satisfying ∆θ + k 2 n p θ = 0 we have

Λ ∆w q + k 2 n p w q θ dx = 0, ( 50
)
implying from equation (46), that

Λ (n p -n)w + (1 -n) f θ dx = 0. ( 51
)
Remark that for x / ∈ Λ, ∆Φ(n p ; x -y) + k 2 n p Φ(n p ; x -y) = 0 for all y ∈ Λ. Combined with f = 0 outside Λ p , we then conclude from (49) that

w(x) = - Λp\Λ k 2 (1 -n p ) f (y)Φ(n p ; x -y) dy for x / ∈ Λ. ( 52
)
Next we define

w(x) = - Λp\Λ k 2 (1 -n p ) f (y)Φ(n p ; x -y) dy , x ∈ Θ. ( 53
)
then w -w ∈ H 2 0 (Λ) and ∆ w + k 2 n p w = 0 in Λ. We now keep w and w q as above and let w := w q + w in Λ which obviously verifies

∆ w + k 2 n w = k 2 (1 -n) f in Λ.
By Assumption 1 we have w = w in Λ. This proves that w = w -w q in Λ. Moreover, using the α q quasi-periodicity of f in Λ p and the periodicity of n p we have that

Λ+mL k 2 (1 -n p ) f (y)Φ(n p ; x -y) dy = Λ k 2 (1 -n p ) f (y + mL)Φ(n p ; x -(y + mL)) dy = e iαqmL Λ k 2 (1 -n p ) f (y)Φ(n p ; x -mL -y) dy . Letting for y ∈ Λ Φ(x, y) := 0 =m∈Z M e iαqmL Φ(n p ; x -mL -y), (54)
we see that w defined by ( 53) is equivalent to

w(x) = - Λ k 2 (1 -n p ) f (y) Φ(x, y) dy . ( 55
)
This leads us to define the operator

S k : L 2 (Λ) → L 2 (Λ) S k : f → - Λ k 2 (1 -n p )f (y) Φ(x, y) dy . ( 56
)
From the smoothing property of the volume potential we have that the operator S k is compact and S k (f ) satisfies

∆ S k (f ) + k 2 n p S k (f ) = 0 in R d \ Λ.
From the above, we write w -w q in terms S k (f ) then reformulate (46) for w 0 := w q and f , we finally obtain

w 0 ∈ H 2 0 (Λ), f ∈ L 2 (Λ),      ∆w 0 + k 2 nw 0 = k 2 (1 -n)f + k 2 (n p -n) S k (f ) in Λ, ∆f + k 2 f = 0 in Λ. ( 57
)
This problem is the homogeneous version of the new interior transmission problem defined in Definition 3.3 below, where u := w 0 + f . Assumption 3 now implies that w 0 = f = 0 in Λ, which proves the injectivity of G q .

Definition 3.3 (The new interior transmission problem). Find (u,

f ) ∈ L 2 (Λ) × L 2 (Λ) such that u -f ∈ H 2 (Λ) and              ∆u + k 2 nu = k 2 (n p -n) S k (f ) in Λ, ∆f + k 2 f = 0 in Λ, u -f = ϕ on ∂Λ, ∂(u -f )/∂ν = ψ on ∂Λ, (58)
for given (ϕ, ψ) ∈ H 3/2 (∂Λ) × H 1/2 (∂Λ) where

S k : L 2 (Λ) → L 2 (Λ) : f → - Λ k 2 (1 -n p )f (y) 0 =m∈Z M e iαqmL Φ(n p ; x -mL -y) dy , (59)
Φ(n p ; •) is the M L-periodic fundamental solution given by [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF], and ν denotes the unit normal on ∂Λ outward to Λ. For sake of completeness, we end this section with proving a range statement for the operator G ± q which is used in the imaging algorithm. To this end, we recall Φ q (• -z) defined by (20), I q given by (23) with adjoint I * q . Theorem 3.5. Suppose that Assumptions 1, 2 and 3 hold. Then,

I * q Φ ± q (•; z) ∈ R(G ± q ) if and only if z ∈ D p .
Proof. We consider two cases:

Case 1: z ∈ D p = Λ p ∪ O c p . (i) If z ∈ O c p : Let (u, v) ∈ L 2 (D) × L 2 (D)
be the unique solution of ( 14) with ϕ := Φ q (• -z)| ∂D and ψ := ∂Φ q (• -z)/∂ν| ∂D and define

w = u -v in O c p Φ q in Θ \ O c p .
Then w ∈ H 2 loc (Θ) and verifies equation ( 6) with f = v in O c p and f = -Φ q in Θ \ O c p . Therefore G ± (f ) = Φ ± q (•; z). Furthermore u| O c p and v| O c p are α q -quasi-periodic (due to the periodicity of domain O c p and α q -quasi-periodicity of the data), hence f is also α q quasi-periodic. This implies f ∈ H q inc (D) and

G ± q (f ) = I * q G ± q (f ) = I * q Φ ± q (•; z). (ii) If z ∈ Λ p : We first consider that z ∈ Λ = Λ p ∩ Ω 0 . Let (u, v) ∈ L 2 (Λ p ) × L 2 (Λ p )
be the α q -quasi-periodic extension of (u Λ , v Λ ), where (u Λ := u, v Λ := f ) is the solution of the new interior transmission problem in Definition 3.3 with ϕ := Φ q (• -z)| ∂Λ and ψ =: ∂Φ q (• -z)/∂ν| ∂Λ . We then define

w q = u -v in Λ p Φ q in Θ \ Λ p .
Let f := v in Λ p and f := -Φ q in Θ\Λ p then f ∈ H q inc (D) and w q ∈ H 2 loc (Θ) satisfies the scattering problem (33) with data f . Furthermore, w defined such as w := w q + S k (f ) in Λ and w := w q in D \ Λ is solution to (6) with data f . Therefore G ± q (f ) = I * q Φ ± q (•; z). We next consider z ∈ Λ + mL with 0 = m ∈ Z d-1 M , and recall that Φ ± q (•; z) = e imL•αq Φ ± q (•; z -mL). If we take f ∈ H q inc (D) such that G ± q (f ) = I * q Φ ± q (•; z -mL), which is possible by the previous step since z -mL ∈ Λ, then

G ± q (e imL•αq f ) = I * q ( Φ ± q (•; z)).
Case 2: z / ∈ D p . If G ± q (v) = I * q Φ ± q (•; z), then using the same unique continuation argument as in the proof of Lemma 3.2 we obtain w q = Φ q in Θ \ D p where w q is defined by (30) with w being the solution of ( 6) with f = v. This gives a contradiction since w q is locally H 2 in Θ \ D p while Φ q (• -z) is not.

The Analysis of the New Interior Transmission Problem

This section is devoted to the study of the solvability of the new interior transmission problem in Definition 3.3. It provides sufficient conditions on n and k for which this problem is well-posed, i.e. such that Assumption 3 holds. As described in the previous section the solvability of the new interior transmission problem is fundamental to ensuring the properties needed for the imaging of the defect ω with a single Floquet-Bloch mode. Up to now the only case that could be handled was when ω ∩ D = ∅ [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] (see also [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]). Here we provide a general analysis that cover all possible cases. Our approach generalizes [START_REF] Kirsch | A note on Sylvester's proof of discreteness of interior transmission eigenvalues[END_REF] and [START_REF] Sylvester | Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators[END_REF]. We start with proving the following technical lemma: Lemma 4.1. Assume that n p > α > 0 on R d . Then there exists θ > 0 and C > 0 such that

S iκ (f ) L 2 (Λ) ≤ Ce -θκ f L 2 (Λ) , κ > 0 for all f ∈ L 2 (Λ).
Proof. Denoting w := S iκ (f ) and f the extension of f as α q -quasi-periodic in Λ p , we have that w

(x) = κ 2 Λp\Λ (1 -n p ) f (y)Φ(n p ; x -y) dy .
where Φ(n p ; •) denotes here the M L -periodic fundamental solution associated with k = iκ. An application of the Cauchy-Schwarz inequality an the Fubini theorem implies

w 2 L 2 (Λ) ≤ κ 4 |Λ p \ Λ| Λ Λp\Λ (n p -1) f (y)Φ(n p ; x -y) 2 dy dx = κ 4 |Λ p \ Λ| Λp\Λ (n p -1) f (y) 2 Λ Φ(n p ; x -y) 2 dx dy .
Next we let Σ = {z := x -y, x ∈ Λ, y ∈ Λ p \ Λ},

d max ∈ R : d max > sup{|z|, z ∈ Σ} and d := d(Λ, Λ p \ Λ), and remark that ∀x ∈ Λ, ∀y ∈ Λ p \ Λ, |x -y| > d, hence Σ ⊂ B := B(0, d max ) \ B(0, d) (60)
where B(0, d) is a ball of radii d and centered at the origin. Remark that d > 0 by Assumption 2. We then have

w 2 L 2 (Λ) ≤ κ 4 |Λ p \ Λ| Λp\Λ (n p -1) f (y) 2 dy B Φ(n p ; z) 2 dz . Since f = f in Λ, f is quasi-periodic and n p is periodic in Λ p , then Λp\Λ (n p -1) f (y) 2 dy = (|M |-1) Λ (n p -1) f (y) 2 dy ≤ (|M |-1) sup Λ |1-n p | f 2 L 2 (Λ) .
We now prove that, there exists θ > 0 and

C 0 > 0 such that B Φ(n p ; z) 2 dz ≤ C 0 e -θκ . (61)
We recall that Φ(n p ; z) is M L-periodic and satisfies

(∆ -κ 2 n p )Φ(n p ; •) = -δ 0 in Θ (62)
and Φ(n p ; •) ∈ L 2 (Θ) (or equivalently the Rayleigh radiation condition (4) with k = iκ). Consider the fundamental solution Ψ ∈ L 2 (R d ) satisfying

(∆ -κ 2 n p )Ψ = -δ 0 in R d .
Since n p is positive definite, then one can prove that e γκ|x| Ψ(x) ∈ L 2 (R d ) for γ > 0 sufficiently small (following the lines of the proof of Theorem 4.4 in [START_REF] Haddar | A volume integral method for solving scattering problems from locally perturbed infinite periodic layers[END_REF]). The function

u s := Φ(n p ; •) -Ψ satisfies ∆u s -κ 2 n p u s = 0 in Θ
and application of the Green formula and using the periodicity conditions of Φ(n p ; •) imply

Θ |∇u s | 2 +κ 2 n p |u s | 2 dx ≤ C Ψ H 1 2 (∂Θ) ∂u s ∂ν H -1 2 (∂Θ) + u s H 1 2 (∂Θ) ∂Ψ ∂ν H -1 2 (∂Θ)
for some constant C independent from κ. The decay property of Ψ implies that

Ψ H 1 2 (∂Θ) + ∂Ψ ∂ν H -1 2 (∂Θ) ≤ e -γ 1 κ
for κ sufficiently large and γ 1 > 0 a constant independent from κ. Therefore, by classical continuity properties for traces and normal traces and the fact that ∆u s = κ 2 n p u s , one conclude that for κ sufficiently large

Θ (|∇u s | 2 + |u s | 2 )dx ≤ Ce -γ 1 κ
for some constant C independent from κ. Since B is contained in the union of Θ and two M L periods, one to the left of Θ and the other to the right, using the M L-periodicity of u s we have that B (|∇u

s | 2 + |u s | 2 )dx ≤ 3Ce -γ 1 κ .
One then obtain the desired estimate (61) by writing Φ(n p ; •) = Ψ + u s and combining the previous estimate with the exponential decay of Ψ.

We now turn our attention to the analysis of the new interior transmission problem in Definition 3.3. For the given (ϕ, ψ) ∈ H 3/2 (Λ) × H 1/2 (Λ) in (58) we construct a lifting function u 0 ∈ H 2 (Λ) such that u 0 | ∂Λ = ϕ and ∂u 0 /∂ν| ∂Λ = ψ. Hence w := u-u 0 -f ∈ H 2 0 (Λ) and we let F := (∆u 0 +k 2 nu 0 )/k 2 ∈ L 2 (Λ). To further simplify notation, we set λ := -k 2 ∈ C, q := n -1, p := n -n p and v := -w/k 2 ∈ H 2 0 (Λ). In these notations, the problem we need to solve reads:

v ∈ H 2 0 (Λ) and f ∈ L 2 (Λ),    ∆v -λ(q + 1)v = qf + p S √ -λ (f ) + F in Λ, ∆f -λf = 0 in Λ, ( 63
)
for a given F ∈ L 2 (Λ), where S √ -λ (f ) is given by (59). We remark that (63) is a modification of the problem considered in [START_REF] Kirsch | A note on Sylvester's proof of discreteness of interior transmission eigenvalues[END_REF] (see also [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF]Section 3.1.3]). We write this problem in an equivalent variational form. To this end, let us denote

X := H 2 0 (Λ)×L 2 (Λ) with the norm (v, f ) 2 X := v 2 H 2 (Λ) + f 2 L 2 (Λ)
and the corresponding inner product •, • X . Then we define the sesquilinear form a k : X × X → C by

a λ (v, f ; φ, ψ) = Λ (∆φ -λφ)f dx + Λ ∆v -λ(q + 1)v ψ -qf + p S √ -λ f ψ dx , ( 64
)
for (v, f ) ∈ X and (φ, ψ) ∈ X and the bounded linear operator A λ : X → X defined by means of the Riesz's representation theorem

a λ (v, f ; φ, ψ) = A λ (v, f ), (φ, ψ) X . ( 65
)
Letting ∈ X be the Riesz's representative of the conjugate linear functional

, (φ, ψ) X = Λ F ψ dx , solving (63) is equivalent to solving A λ (v, f ) = or a λ (v, f ; φ, ψ) = Λ F ψ dx , ∀ (φ, ψ) ∈ X. ( 66
)
In a similar fashion, we define the sesquilinear form b

k : X × X → C by b λ (v, f ; φ, ψ) = Λ (∆φ -λφ)f dx + Λ ∆v -λv -qf ψ dx
with the associated bounded linear operator B λ :

X → X such that b λ (v, f ; φ, ψ) = B λ (v, f ), (φ, ψ) X , Lemma 4.2.
For any λ, µ ∈ C, the differences A λ -B µ and A λ -A µ are compact.

Proof. Let (v j , f j ) ∈ X be an arbitrary sequence converging weakly to (0, 0) in X. We must show that (A λ -B µ )(v j , f j ) converges (0, 0) strongly in X. Recall that

(A λ -B µ )(v, f ), (φ, ψ) X = a λ -b µ (v, f ; φ, ψ)
where

a λ -b µ (v j , f j ; φ, ψ) = (µ -λ) Λ f j φ + Λ µ -λ(q + 1) v j ψ - Λ p( S √ -λ f j -S √ -µ f j )ψ.
Let us take (φ, ψ) ∈ X such that (φ, ψ) X = 1 and define g j ∈ H 1 (Λ) such that ∆g j = f j and g j = 0 on ∂Λ. Hence g j 0 in H 1 (Λ) implying that g j → 0 in L 2 (Λ) by compact embedding of H 1 (Λ) in L 2 (Λ). Obviously,

Λ f j φ = Λ ∆g j φ = Λ g j ∆φ ≤ ∆φ L 2 (Λ) g j L 2 (Λ) ≤ g j L 2 (Λ) .
Similarly, since v j 0 in H 2 (Λ), then v j → 0 in L 2 (Λ), and

Λ µ-λ(q +1) v j ψ ≤ sup Λ |µ-λ(q +1)| v j L 2 (Λ) ψ j L 2 (Λ) ≤ sup Λ |µ-λ(q +1)| v j L 2 (Λ) and Λ p( S √ -λ f j -S √ -µ f j )ψ ≤ sup Λ |p| S √ -λ f j -S √ -µ f j L 2 (Λ) ψ j L 2 (Λ) ≤ sup Λ |p| ( S √ -λ -S √ -µ )f j L 2 (Λ) . Since ( S √ -λ -S √ -µ ) : L 2 (Λ) → L 2 (Λ) is a compact linear operator, then ( S √ -λ - S √ -µ )f j L 2 (Λ) converge to 0 strongly in L 2 (Λ). Therefore (A λ -B µ )(v j , f j ) X→X = sup (φ,ψ) X =1 |(a λ -b µ )(v j , f j ; φ, ψ)| ≤ c v j L 2 (Λ) + g j L 2 (Λ) + S λ f j -S µ f j L 2 (Λ) -→ 0.
The proof for A λ -A µ follows exactly the same lines.

At this point we need to assume a sign condition on q := n -1. To this end, let R be a neighborhood of ∂Λ in Λ and denote by

q min := inf Λ Re (q) > -1, q := inf R Re (q), q := sup R Re (q). ( 67
)
The following lemmas, which are proven first in [START_REF] Kirsch | A note on Sylvester's proof of discreteness of interior transmission eigenvalues[END_REF] for real refractive index and adapted to the case of complex refractive index in [7, Section 3.1.3]), play an important role in our analysis.

Lemma 4.3. Assume that q ∈ L ∞ (Λ) is such that q min + 1 > 0 and either q > 0 or q < 0. Then there exists c > 0 and d > 0 such that for all λ > 0 the following estimates holds

Λ\R |f | 2 dx ≤ ce -2d √ λ R |Re (q)||f | 2 dx ≤ ce -2d √ λ R |q||f | 2 dx ( 68
)
for all f ∈ L 2 (Λ) solution of ∆f -λf = 0.
Proof. See Lemma 2.3 in [START_REF] Kirsch | A note on Sylvester's proof of discreteness of interior transmission eigenvalues[END_REF] if q is real and Lemma 3.14 in [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] for q complex. Lemma 4.4. Assume that q ∈ L ∞ (Λ) is such that q min + 1 > 0 and either q > 0 or q < 0. Then for sufficiently large λ > 0, the operator B λ is an isomorphism form X onto itself.

Proof. See Theorem 2.7 in [START_REF] Kirsch | A note on Sylvester's proof of discreteness of interior transmission eigenvalues[END_REF] if q is real and Lemma 3.15 in [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] for q complex.

We now proceed with proving that the operator A λ : X → X is an isomorphism for sufficiently large λ. We adopt two different approaches for the cases q < 0 (Theorem 4.5) and q > 0 (Theorem 4.6).

Theorem 4.5. Assume that q ∈ L ∞ (Λ) is such that q min + 1 > 0 and q < 0. Then, for sufficiently large λ > 0, the operator A λ : X → X is an isomorphism.

Proof. It is sufficient to prove the injectivity of A λ since B λ is an isomorphism from Lemma 4.4 and and A λ -B λ is compact from Lemma 4.2. To this end, assume that

(v, f ) ∈ X is such that A λ (v, f ) = 0, i.e, a λ j (v, f ; φ, ψ) = 0 for all (φ, ψ) ∈ X. As such, (v, f ) satisfies    ∆v -λ(q + 1)v = qf + p S i √ λ (f ) ∆f -λf = 0 in Λ. ( 69
)
Multiplying the first equation of (69) with f we obtain

Λ q|f | 2 + Λ p S i √ λ (f )f = Λ ∆v -λ(q + 1)v f = Λ ∆v -λv f - Λ qvf .
First observe that Λ (∆v -λv)f = 0 since v ∈ H 2 0 (Λ) and ∆f -λf = 0 in Λ. Therefore

Λ q|f | 2 = - Λ p S i √ λ (f )f -λ Λ qvf . (70)
Multiplying now the first equation of (69) with v and integrating by parts we obtain

Λ |∇v| 2 + λ(q + 1)|v| 2 dx = - Λ qf v dx - Λ p S i √ λ (f )v dx . (71)
Now assume that q < 0 and taking the real part of the above we write

Λ |∇v| 2 + λ(Re (q) + 1)|v| 2 = Re - Λ qf v - Λ p S i √ λ (f )v . (72)
From ( 70) and (72) noting that λ is real, we deduce that

Λ |∇v| 2 + λ(Re (q) + 1)|v| 2 + 1 λ Λ -Re (q)|f | 2 dx (73) = Re Λ p S i √ λ (f )( 1 λ f -v) dx .
For large enough λ > 0, we set

ρ := max q ∞ ce -2d √ λ , ce -2d √ λ < 1. Then Lemma 4.3 implies (1 -ρ) R |Re (q)||f | 2 dx ≤ Λ -Re (q)|f | 2 dx ≤ (1 + ρ) R |Re (q)||f | 2 dx . (74)
The latter inequalities implies

Λ λ(Re (q) + 1)|v| 2 + 1 λ Λ -Re (q)|f | 2 dx (75) ≥ λ Λ (Re (q) + 1)|v| 2 + 1 -ρ λ R |Re (q)||f | 2 dx .
From Lemma 4.1 we infer the existence of c > 0 such that p S i

√ λ (f ) L 2 (Λ) ≤ ce -θ √ λ f L 2 (Λ) . We then have Λ Re p S i √ λ (f )( 1 λ f -v) dx ≤ ce -θ √ λ f L 2 (Λ) v L 2 (Λ) + 1 λ f L 2 (Λ)
Using Young's inequality then splitting the integral of |f | 2 into the domains R and Λ \ R we finally have

Λ Re p S i √ λ (f )( 1 λ f -v) dx ≤ 2c λ e -θ √ λ f 2 L 2 (Λ) + cλ 4 e -θ √ λ v 2 L 2 (Λ) ≤ 2c λ e -θ √ λ (1 + ρ) f 2 L 2 (R) + cλ 4 e -θ √ λ v 2 L 2 (Λ) ≤ 2c(1 + ρ) λq e -θ √ λ |Re (q)|f 2 L 2 (R) + cλ 4(q min + 1) e -θ √ λ Re (q) + 1v 2 L 2 (Λ) . (76)
For λ sufficiently large such that

2c(1 + ρ) λq e -θ √ λ ≤ 1 -ρ λ -ε and cλ 4(q min + 1) e -θ √ λ ≤ λ -ε,
with ε > 0 small enough we finally obtain from (73), ( 75) and (76)

ε Re (q) + 1v 2 L 2 (Λ) + ε |Re (q)|f 2 L 2 (R) ≤ 0.
This prove that v = 0 in Λ and f = 0 in R. Inequality (74) then implies f = 0 in Λ which ends the proof.

Theorem 4.6. Assume that q ∈ L ∞ (Λ) is such that q min + 1 > 0 and q > sup R (Im (q)) 2 ≥ 0. Then, for sufficiently large λ > 0, the operator A λ : X → X is an isomorphism.

Proof. Again similarly to the proof of Theorem 4.5 it is sufficient to prove that A λ is injection, which will do by contradiction. Assume to the contrary that there exists a sequence

λ j → ∞ and (v j , f j ) ∈ X with (v j , f j ) L 2 (Λ) = 1 and A λ j (v j , f j ) = 0. Then v j ∈ H 2 0 (Λ) and f j ∈ L 2 (Λ) satisfy ∆v j -λ j (q + 1)v j = qf j + p S i √ λ j (f j ) and ∆f j -λ j f j = 0. ( 77
)
For given ε > 0 small enough, from Lemma 4.1 and Lemma 4.3 and the fact that (v j , f j ) L 2 (Λ) = 1, we have that for λ j > 0 large enough

p S i √ λ j (f j ) L 2 (Λ) f j L 2 (Λ) + |q|f j 2 L 2 (Λ\R) (78)
+ λ j |q|f j 2 L 2 (Λ\R) |q|v j 2 L 2 (Λ\R) ≤ ε 2 . From (70), we observe that R Re (q)|f j | 2 dx = Re -λ j R qv j f j dx - Λ\R Re (q)|f j | 2 dx -Re Λ p S i √ λ (f j )v j dx + λ j Λ\R qv j f j dx
Estimating the last three terms from (78) and using Cauchy-Schwarz inequality we now obtain

R Re (q)|f j | 2 dx ≤ ε 2 + λ j R Re (q)|f j | 2 dx 1/2 R |q| 2 Re (q) |v j | 2 dx 1/2 . This implies that R Re (q)|f j | 2 dx 1/2 ≤ ε + λ j R |q| 2 Re (q) |v j | 2 dx 1/2 , (79)
(where we used for A > 0 and B > 0, 2 and taking the square root yields A ≤ ε + B). From (71), we see that (using again (v j , f j ) X = 1)

A 2 ≤ ε 2 + AB implies (A -B/2) 2 ≤ ε 2 + B 2 /4 ≤ (ε + B/2)
λ j R 1 + Re (q) |v j | 2 dx ≤ λ j Λ 1 + Re (q) |v j | 2 dx (80) ≤ R Re (q)|f j | 2 dx 1/2 R |q| 2 Re (q) |v j | 2 dx 1/2 + ε 2
since (1 + Re (q)) > 0 in Λ. Combining the last two inequalities yield

λ j R 1 + Re (q) |v j | 2 dx ≤ λ j R |q| 2 Re (q) |v j | 2 dx + ε R |q| 2 Re (q) |v j | 2 dx 1/2 + ε 2 . (81)
The assumption q > sup R (Im (q)) 2 ≥ 0 in particular implies inf R Re (q)-(Im (q)) 2 Re (q)

= δ 0 > 0, hence we have

λ j δ 0 R |v j | 2 dx ≤ ε R |q| 2 Re (q) |v j | 2 dx 1/2 + ε 2 .
Choosing λ j such that 4λ j δ 0 ≥ |q| 2 Re (q) L ∞ (R) we obtain that

λ j δ 0 R |v j | 2 dx ≤ 2ε λ j δ 0 R |v j | 2 dx 1/2 + ε 2 .
which implies

λ j δ 0 R |v j | 2 dx 1/2 ≤ 3ε
and proves that λ j δ 0

R |v j | 2 dx → 0 as j → ∞.
Coming back to (80) we deduce that

Λ (1 + Re (q))|v j | 2 dx → 0 as j → ∞
and from (79), since Re (q) is positive definite in R, we deduce that

R |f j | 2 dx → 0 as j → ∞. Lemma 4.3 then implies Λ |f j | 2 dx → 0 as j → ∞.
The last two zero limits contradict the fact that (v j , f j ) L 2 (Λ) = 1, which proves injectivity.

Recall that solving the new interior transmission problem formulated in Definition 3.3 is equivalent to solving

A k (v, f ) = in X (82)
(where we go back to the original eigen-parameter k ∈ C which is such that λ = -k 2 ). Let us fix λ 0 > 0 such that Theorem 4.5 or Theorem 4.6 holds, i.e. A λ 0 is an isomorphism and let k 0 = √ λ 0 i. Then we can rewrite (82) as

(v, f ) + A -1 k 0 (A k -A k 0 )(v, f ) = A -1 k 0 or (I -C k )(v, f ) = A -1 k . ( 83
)
where

C k := A -1 k 0 (A k -A k 0 ) is compact from Lemma 4.2.
Thus the Fredholm alternative applies to (82). In particular a solution of the new interior transmission problem exists provided k is not a new transmission eigenvalue defined in Definition 3.4. To show that the set of new transmission eigenvalues is discrete we appeal to Fredholm Analytic Theory (see e.g. [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]) since the mapping k → C k is analytic in C. Thus we have proven the following theorem: Theorem 4.7. Assume that the bounded function n satisfies inf Λ Re (n) > 0, Im (n) ≥ 0, and there exists a neighborhood R of ∂Λ inside Λ such that either sup R Re (n -1) < 0 or inf R Re (n -1) > sup R (Im (n)) 2 ≥ 0. Then the new interior transmission formulated in Definition 3.3 has a unique solution depending continuously on the data ϕ and ψ provided k ∈ C is not a new transmission eigenvalue defined in Definition 3.4. In particular the set of new transmission eigenvalues in C is discrete (possibly empty) with +∞ as the only possible accumulation point.

Note that Theorem 4.7 provides sufficient conditions under which Assumption 3 hold. It is highly desirable to show if and when real new transmission eigenvalues exist, because for such real wave numbers our imaging algorithm introduced in the next section fails.

A Differential Imaging Algorithm

We now apply all the results of Section 2 and Section 4 above to design an algorithm that provides us with the support of the defect ω without reconstructing D p or computing the Green's function of the periodic media. We follow the idea proposed in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] and build a differential imaging functional by comparing the application of the Generalized Linear Sampling algorithm to respectively the operators N ± and N ± q . The new results obtained in Theorem 4.7 allow us to justify this algorithm for general location of ω (possibly multi-component).

Description and analysis of the algorithm

Throughout this section we assume that Assumptions 1, 2 and 3 hold. For sake of simplicity of presentation we only state the results when the measurements operator N + is available. Exactly the same holds for the operator N -by changing everywhere the exponent + to -. For given φ and a in 2 (Z d-1 ) we define the functionals

J + α (φ, a) := α(N + a, a) + N + a -φ 2 , J + α,q (φ, a) := α(N + q, a, a) + N + q a -φ 2 (84)
with N + q, := I * q N + I q . Let a α,z , a α,z q and ãα,z q in (Z d-1 ) verify (i.e. are minimizing sequences)

J + α ( Φ + (•; z), a α,z ) ≤ inf a∈ 2 (Z d-1 ) J + α ( Φ + (•; z), a) + c(α) J + α ( Φ + q (•; z), a α,z q ) ≤ inf a∈ 2 (Z d-1 ) J + α ( Φ + q (•; z), a) + c(α) J + α,q (I * q Φ + q (•; z), ãα,z q ) ≤ inf a∈ 2 (Z d-1 ) J + α,q (I * q Φ + q (•; z), a) + c(α) (85)
with c(α) α → 0 as α → 0. Here Φ ± (•; z) are the Rayleigh coefficients of Φ(1, z) (i.e. Φ(n p ; z) defined by [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF] with with n p = 1) given by ( 15) and Φ ± q (•; z) are the Rayleigh coefficients of Φ q (• -z) given by (21).

The standard analysis of the generalized linear sampling method (see e.g. [7, Section 2.2]) making use of the factorization of N in Theorem 2.3 along with all the properties of the involved operators developed in Section 2.2 and Section 3 imply the following results (see also [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] and [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] for detailed proofs)

(N + a α,z q , a α,z q ) < ∞. Moreover, if z ∈ D p then H + a α,z q → v z in L 2 (D)
where (u z , v z ) is the solution of problem [START_REF] Kirsch | The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer[END_REF] with ϕ = Φ q (• -z) and ψ = ∂Φ q (• -z)/∂ν on ∂D.

(iii) z ∈ D p if and only if lim

α→0 (N + q, ãα,z q , ãα,z q ) < ∞. Moreover, if z ∈ D p then H + q ãα,z q → h z in L 2 (D)
where h z is defined by

h z = -Φ q (• -z) in Λ p v z in O c p if z ∈ O c p h z = v z in Λ p -Φ q (• -z) in O c p if z ∈ Λ p (86)
where (u z , v z ) is the solution of problem [START_REF] Kirsch | The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer[END_REF] with ϕ = Φ q (• -z) and ψ = ∂Φ q (• -z)/∂ν on ∂D and ( u z , v z ) is α q -quasi-periodic extension of the solution (u, f ) of the new interior transmission problem in Definition (3.3) with ϕ = Φ q (• -z) and ψ = ∂Φ q (• -z)/∂ν on ∂Λ.

Proof. The proof of the items (i) and (ii), we refer to [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF]. The proof of items (iii) is a direct application of Theorem A.4 in [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] in combination with Theorem 3.5.

We then consider the following imaging functional that characterizes Λ,

I + α (z) = (N + a α,z , a α,z ) 1 + (N + a α,z , a α,z ) D + (a α,z q , ãα,z q ) -1 (87)
where for a and b in 2 (Z d-1 ),

D + (a, b) := N + (a -I q b), (a -I q b) .
Before giving the main theorem for the characterization of the defect we need the following assumption.

Assumption 4. The refractive indexes n, n p and wave-number k are such that

             ∆u + k 2 nu = 0 in ω, ∆v + k 2 n p v = 0 in ω, u -v = 0 on ∂ω, ∂(u -v)/∂ν = 0 on ∂ω (88)
has only the trivial solution.

This assumption is satisfied if n -n p does not change sign in a neighborhood of the boundary of domain ω, or if either n or n p have non-zero imaginary part (see e.g [START_REF] Cakoni | Inverse scattering theory and transmission eigenvalues[END_REF]).

I + α (z) = 0. If z ∈ O c p , let (u z , v z ) ∈ L 2 (D) × L 2 (
D) be the unique solution of ( 14) with ϕ := Φ q (• -z)| ∂D and ψ = ∂Φ q (• -z)/∂ν| ∂D then from Lemma 5.1(ii), H + a α,z → v z . Note that in this case u z = 0 and v z := -Φ q (• -z) outside O c p . Now let h z ∈ L 2 ( D p ) defined by (86). Then from Lemma 5.1(iii), H + q ãα,z q → h z in L 2 (D). Furthermore, by definition of h z and the fact that v z = -Φ q (• -z) in Λ p , we have that v z coincide with h z in D. From the factorization of N + and the definition of H + q we have D + (a α,z q , ãα,z q ) = N + (a α,z q -I q ãα,z q ), a α,z q -I q ãα,z q = T (H + a α,z q -H + q ãα,z q ), H + a α,z q -H + q ãα,z q ≤ T H + a α,z q -H + q ãα,z q 2 L 2 (D) → 0, as α → 0.

Since from Lemma 5.1(i), (N + a α,z , a α,z ) remains bounded as α → 0, we can finally conclude from the above that lim α→0

I + α (z) = 0 if z ∈ O c p . Case 2: z ∈ D \ O c p ⊂ Λ p .
Then again by Lemma 5.1(i), (N + a α,z , a α,z ) remains bounded and by Lemma 5.1(iii), (N q, ãα,z q , ãα,z q ) remain bounded. Using the factorization of N + q, and the fact that N + q, = I * q N + I q we can write

(N + I q ãα,z q , I q ãα,z q ) → (T h z , h z ) < +∞,
where h z ∈ L 2 ( D p ) defined in (86). In this case, we need to consider the location of z in two sub-domain. First, if z ∈ ω \ D p (the part of defect outside periodic domain), then by Lemma 5.1(ii) (N + a α,z q , a α,z q ) → +∞ as α → 0. This implies, D + (a α,z q , ãα,z q ) ≥ (N + a α,z q , a α,z q ) -(N + I q ãα,z q , I q ãα,z q ) → +∞.

We then conclude that lim α→0

I + α (z) = 0 if z ∈ ω \ D p .
unique continuation implies that v z = -Φ q (• -z) in Λ \ {z}. But this is a contradiction, since by the interior regularity of solutions of Helmholtz equation v z is infinitely many times differentiable whereas Φ has a singularity at z. This proves that v z | Op is α q -quasi periodic with period L, and hence v z = h z in D \ O c p . We now have from the estimate

C H + a α,z q -H + q ãα,z q 2 L 2 (D) ≤ D + (a α,z q , ãα,z q ) ≤ T H + a α,z q -H + q ãα,z q 2 L 2 (D)
where C is the coercivity constant associated with T , that D + (a α,z q , ãα,z q ) is bounded and does not go to 0 as α → 0. Thus

lim α→0 I + α (z) = 0 if z ∈ O p .
This ends the proof of the theorem.

Theorem 5.2 shows that the functional I + α (z) provides an indicator function for D \ O c , i.e. the defect and the periodic components of the background that intersects ω. However, the proof of Theorem 5.2 indicates that although the values of I + α (z) are positive only in D \ O c , they are smaller for z ∈ O p compared to the values of I + α (z) for z ∈ ω \ D p . Therefore, if the defect ω has nonempty intersection with the periodic background then the reconstruction of O p (i.e. the components that have no nonempty intersection with the defect) are not displayed as clearly as the reconstruction of the part of defect outside D p . This is illustrated in the following numerical experiments. We recall that exactly the same can be shown for down-to-up incident field, by simply replacing the upper index + with -. It is also possible to handle the case with noisy data, and we refer the reader to [START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF] and [START_REF] Phong | Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers[END_REF] for more detailed discussion.

Numerical Experiments

We conclude by showing several numerical examples to test our differential imaging algorithm. We limit ourselves to examples in R 2 . The data is computed with both downto-up and up-to-down plane-waves by solving the forward scattering problem based on the spectral discretization scheme of the volume integral formulation of the problem presented in [START_REF] Haddar | A volume integral method for solving scattering problems from locally perturbed infinite periodic layers[END_REF]. Let us denote by

Z d-1 inc := {j = q + M , q ∈ Z d-1 M , ∈ Z d-1 and ∈ -N min , N max }
the set of indices for the incident waves (which is also the set of indices for measured Rayleigh coefficients). The values of all parameters used in our experiments will be indicated below. The discrete version of the operators N ± are given by the N inc × N inc matrixes N ± := u s ± ( ; j)

,j∈Z d-1 inc . ( 92
)
Random noise is added to the data. More specifically, in our computations we use

N ±,δ (j, ) := N ± (j, ) 1 + δA(j, ) , ∀(j, ) ∈ Z d-1 inc × Z d-1 inc ( 93
)
where A = (A(j, )) N inc ×N inc is a matrix of uniform complex random variables with real and imaginary parts in [-1, 1] 2 and δ > 0 is the noise level. In our examples we take δ = 1%.

For noisy data, one needs to redefine the functionals J + α and J + α,q as J +,δ α (φ, a) := α (N +,δ a, a) + δ N +,δ a 2 + N +,δ a -φ 2 , J +,δ α,q (φ, a) := α (N +,δ I q a, I q a) + δ N +,δ a 2 + N +,δ q a -φ 2

We then consider a α,z δ , a α,z q,δ and ãα,z q,δ in (Z d-1 ) as the minimizing sequence of, respectively, J +,δ α ( Φ + (•; z), a), J +,δ α ( Φ + q (•; z), a) and J +,δ α,q ( Φ + q (•; z), a).

The noisy indicator function takes the form

I +,δ α (z) = G +,δ (a α,z δ ) 1 + G +,δ (a α,z δ ) D +,δ (a α,z q,δ , ãα,z q,δ ) -1 (95)
where for a and b in 2 (Z d-1 ), D +,δ (a, b) := N +,δ (a -I q b), (a -I q b) and G +,δ (a) := (N +,δ a, a) + δ N +,δ a 2 .

Defining in a similar way the indicator function I -,δ (z) corresponding to up-to-down incident waves, we use the following indicator function in our numerical examples I δ (z) := I +,δ (z) + I -,δ (z).

In the following example, we consider the same periodic background D p , in which each cell consists of two circular components, namely the discs with radii r 1 , r 2 specified below. The physical parameters are set as k = 3.5π/3.14, n p = 2 inside the discs and n p = 1 otherwise. (96)

Letting λ := 2π/k be the wavelength, the geometrical parameters are set as the period L = πλ, the width of the layer h = 1.5λ, r 1 = 0.3λ, and r 2 = 0.4λ. (97)

Finally we choose the truncated model M = 3, N min = 5 and N max = 5 and q = 1 (98)

The reconstructions are displayed by plotting the indicator function I δ (z).

Example 1: In the first example, we consider the perturbation ω be a disc of radius r ω = 0.25λ with the refractive index n = 4 and located in the component of radii r 2 (see Figure 2-left). In this case we reconstruct the whole component which contains the defect and its L-periodic copies (see Figure 2-right). We remark that this is the unfortunate case when it is not possible to determine in which period is the defect located. This is in accordance with what Theorem 5.2 predicts. As already mention the case when the defect is entirely included in a component of the periodic background is ambiguous in the sense that the actual defective period can not be determined. We conclude by remarking that everything here can be adapted to the case when the background periodic layer is composed of inhomogeneities embedded in homogeneous media with constant refractive index n 0 = 1. The only difference is in the choice in the fundamental solution which in this case should correspond to a piecewise homogeneous media instead of n = 1.

  refers to the periodicity of the media with respect to the first d -1 variables and M

Figure 1 :

 1 Figure 1: Sketch of the geometry for the M L-periodic problem

  Furthermore, we denote by Λ := O ∪ ω and by D := Λ ∪ O c . Obviously, D = D ∩ Ω 0 . (see also Fig. 1 and note that if ω does not intersect with D p then O ≡ ∅, O c ≡ D p ∩ Ω 0 and Λ = ω). Let us denote by ν m := (mL, 0) ∈ R d , the translate vector from Ω 0 to Ω m , m ∈ Z d-1 . We consider the following L-periodic copies of the aforementioned regions

Definition 3 . 4 .

 34 Values of k ∈ C for which the homogeneous problem with ϕ = ψ = 0, are called new transmission eigenvalues.

Lemma 5 . 1 .

 51 (i) z ∈ D if and only if lim α→0 (N + a α,z , a α,z ) < ∞. Moreover, if z ∈ D then H + a α,z → v z in L 2 (D) where (u z , v z )is the solution of problem (14) with ϕ = Φ(1; z) and ψ = ∂Φ(1; z)/∂ν on ∂D. (ii) z ∈ D p if and only if lim

  α→0

Theorem 5 . 2 .

 52 Under Assumptions 1, 2,3 and 4, we have that z ∈ D \ O c p if and only if lim α→0 I + α (z) > 0. (Note that D \ O c p = ω ∪ O p contains the physical defect ω and O p := D p \ O c p the components of D p which have nonempty intersection with the defect). Proof. Case 1: z / ∈ D \ O c p . If z / ∈ D then from Lemma 5.1(i) we have that (N + a α,z , a α,z ) → +∞ as α → 0 and therefore lim

  α→0

Figure 2 :

 2 Figure 2: Left: The exact geometry for Example 1. Right: The reconstruction using z → I δ (z).

Figure 3 :Figure 4 :

 34 Figure 3: Left: The exact geometry for Example 2. Right: The reconstruction of the local perturbation using z → I δ (z).

Example 3 .

 3 Example 3 just show that when the defect has no intersection with the periodic background, the indicator function I δ (z) allows to reconstruct the true defect including its true location. Here the defect is a disc of r ω = 0.2λ with n = 3. More examples of this case can be found in[START_REF] Haddar | Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers[END_REF].

Figure 5 :Example 4 .

 54 Figure 5: Left: The exact geometry for Example 3. Right: The reconstruction using z → I δ (z).

Figure 6 :Example 5 .Figure 7 :

 657 Figure 6: Left: The exact geometry for Example 4. Right: The reconstruction using z → I δ (z).

Acknowledgements

The research of F. Cakoni is supported in part by AFOSR Grant FA9550-17-1-0147 and NSF Grant DMS-1602802. The research of T-P. Nguyen is supported in part by NSF Grant DMS-1602802.

Next, if z ∈ D p ∩ Λ p = O p , again by Lemma 5.1(ii) (N + a α,z q , a α,z q ) → (T v z , v z ) where v z ∈ L 2 (D) is defined in Lemma 5.1(ii). In this case v

p . On the other hand, h z | Op is α q -quasi-periodic with period L while v z | Op is not α q -quasi-periodic with period L (recall that (u z , v z ) is a solution of ( 14) defined in O c p with ϕ = Φ q (• -z) and ψ = ∂Φ q (• -z)/∂ν on ∂(D \ O c p ). Indeed, assume to the contrary that v z | Op is α q -quasi periodic with period L. For a fixed arbitrary 0 = m ∈ Z M , let us define

Next, we let

and observe that ( u z , v z ) verify

where we use the fact that from (89) the Cauchy data of u z and Φ q (• -z) + v z coincide on ∂O ∩ ω. From the definition of u z and (90) we have that (u z , u z ) is a solution to

Since n = n p = n in Λ \ ω, then u z and u z satisfy the same equation in Λ \ ω and share the same Cauchy data on ∂Λ \ ω, hence by the unique continuation u z = u z in Λ \ ω. Therefore, (u z , u z ) is a solution of (88) and Assumption 4 implies that u z = u z = 0 in ω, and consequently by definition v z + Φ q (• -z) = 0 in ω \ O (recall Λ = O ∪ ω). On the other hand, we have that v := v z + Φ q (• -z) satisfies ∆v + k 2 v = 0 in Λ \ {z}, hence


Madelyn Rojas Castro

Marina Travanca

Marta Avalos Fernandez

Ludivine Orriols

Director Master in Biostatistics, Researcher in Biostatistics, †, ‡ David Valentin Conesa

Emmanuel Lagarde
email: emmanuel.lagarde@u-bordeaux.fr

Epidemiology PhD Student Madelyn Y Rojas

Marina Travanca Biostatistician

Marta Avalos Researcher

In Biostatistics

Postdoc Epidemiology

Mavie-lab sports: a mhealth for injury prevention and risk management in sport

MAVIE-Lab Sports contains a decision support system (DSS) to self-management the potential risk of having an injury and to evaluate the adoption of preventive measures. The App allows participants, initially to compare the population risk between different sports. Then, to evaluate their personal injury risk for a range of sports.

Finally, to experiment the potential reduction or increase of risk due to behavioral changes or the adoption of protective devices, equipment or environments in the practice sportive.

The MAVIE-Lab algorithms were developed using detailed MAVIE cohort data, health, demographic, training practices information and the occurrence of injuries, there causes, consequences and severity. The model was constructed to predict the injury risk using probabilistic reasoning and graphical modelling through Bayesian Networks [3][4][5] . This approach combines qualitative and quantitative modelling, allowing the combination of MAVIE data evidences and prior experts information 6,7 about risk and protection factors and causal relations between them 8 . Keywords: eHealth, Injuries, Bayesian, Prediction, Machine Learning. Sub-theme: 2. Unintentional Injury. 2.9 Sport and recreation.


David Gloaguen
email: gloaguen@lamm.univ-nantes.fr

Manuel François

Ronald Guillen

Jean Royer

D Gioaguen

M Fran9ois

R Guillen\

Elastoplastic self-consistent modelisation of internal stresses in rolled Zr702a

Keywords: Zirconium, elastoplastic self-consistent model, internal stresses, X-ray diffraction, plastic anisotropy

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I. Introduction

Zirconium alloys are widely used in nuclear industry because they exhibit a high resistance to corrosion, a good thermal conductivity and a low neutron cross section. They have a hexagonal close packed structure at room temperature. These alloys present anisotropic thermal, elastic and plastic properties at mesoscopic (grain) and macroscopic levels. These properties and the crystallographic texture explain the appearance and the development of internal stresses when an elastoplastic deformation is introduced. Two types of internal stresses can be distinguished :

Thermal internal stresses, which may arise from thermal anisotropy. During the cooling process after a heat treatment, each grain is subjected to a contraction which depends on crystallographic direction. Elastic strain (and stress) must be present to maintain the compatibility of total strain between adjacent grains. Thermal stresses can modify the macroscopic elastoplastic response of the material, particularly for zirconium alloys [1].

Mechanical internal stresses, induced by the elastoplastic anisotropy and the requirement of compatibility.

The purpose of the present work is to make a complete study concerning the development and the evolution of internal strain introduced by tensile tests in rolled textured polycristalline Zr-a after a cooling process. Self consistent models (thermoelastic and elastoplastic) were used to simulate the thermal and the tensile response of samples tested along the rolling (RD) and the transverse (TD) directions of the sheet. In this study, the predicted residual stresses in a rolled Zr-a have been compared with experimental data provided by X-ray diffraction experiments in order to obtain more informations about the different factors responsible for the appearance of residual elastic mesostrains. This confrontation allows a better understanding and interpretation of X-ray diffraction and mechanical results. X-ray diffraction technique is one of the most used for the characterization of the mechanical state at mesoscopic level of the samples.

Experimental procedure

The investigated material is a commercially pure zirconium-a (Zr702). Its chemical composition is given in Table I. The metallographic examination of the annealed material revealed an average grain size of approximately 20 !J.m.

Mechanical testing

To analyse the influence of loading direction and texture on the mechanical response of the material, samples of dimensions 2xl.5x43.4 mm 3 were cut from the rolling (RD) and the transverse directions (TD) of the sheet. The specimens were submitted to uniaxial tensile tests at room temperature, at a constant strain rate of 3.8 10• 3 s• 1 with different total strains. Samples were 1 and 2 % deformed of in the two loading directions. The tensile loading curves are depicted in Figure 1. The experiments were carried out in two stages:

�� 500 - --------- ----� soo I 400 -----1 400 --experimental ----predicted 100 --experimental ----predicted �---�--�---�-- 0,5 1,5 0 0,5 1,5 Strain(%) Strain(%) �------------------- ---__ _ ] (a)
• Characterization of internal stresses due to thermal anisotropy in an undeformed sample after a thermal treatment at 650°C during 4.5 h.

• Determination of the evolution of internal stresses due to plastic anisotropy for samples deformed samples by uniaxial tensile both along rolling and transverse directions.

Measurements were carried out in the unloaded state. These experiments were carried out on a D500 SIEMENS goniometer with a Cr Ka radiation. An Q goniometric assembly with a scintillation detector was used. Two plane families were studied : { 10 T 4} at 28 = 156,7° and { 20 2 2} at 28 = 137,2°. Diffractograms were recorded for fifteen tilt angles \jf varying between -45 and 45° and for azimuth angles <1> = 0, 45 and 90°. The direction <1> = oo corresponds to the tensile direction. The X-ray beam output collimator has 0.5 mm diameter and a diffracted-beam slit limits the equatorial divergence to an angle of 0,6°. The sample shows a typical texture of rolled Zirconium. The basal poles are preferentially oriented at an angle of 32° from the normal direction towards the transverse direction, while the prism poles exhibit a weak maximum in the rolling direction. To avoid surface effect because of the mechanical history of the sheet or oxygen contamination during the thermal treatment, the specimens were electrochemically polished. Electro-polishing was carried out at 298 K under a potential difference of 30 V in a 10 % perchloric acid + 90 % acetic acid solution. Stress measurements by X-ray diffraction have been made at a depth of 100 !J.m. The experimental data from the mechanical tests and X-ray diffraction are presented and compared with the results from our model in the fourth section.

Ill. Self-Consistent Modeling

We consider that thermal contractions do not introduce plastic deformation in the material during the cooling process [1]. In this case, a thermoelastic formulation is sufficient to simulate the evolution of thermal stresses associated with cooling. An elastoplastic self-consistent scheme was then used to predict the mechanical behaviour of titanium polycristal under an uniaxial tension. The model used for our simulations was presented in a previous work [2]. More details can be found in references [3,4]. The self consistent model is based on the following assumptions :

• Each grain is considered as a spherical inclusion embedded in a Homogeneous Effective Medium (HEM) having the average properties of the aggregate.

•

The mesoscopic stress cr and the mesoscopic strain E in the inclusion are homogeneous.

•

The overall response of the material is determined from averaging the microscopic terms (cr, E) over a representative set of orientations.

•

The macroscopic stress :E and macroscopic strain E are considered as uniform. The properties of the medium are initially undetermined and must be solved by iterations. Each grain is treated as an inhomogeneity with elastic (c) and thermal (a) constants, slip and twin mechanisms characteristic of a single crystal.

•

The texture is known, the polycristal is represented by a weighted discrete distribution of orientations (Euler angles ( <p 1 <j>cp2 )). Lattice rotation (reorientation by slip and twinning) and texture changes are incorporated in the model.

Thermal residual stresses

The thermal strains due to cooling from annealing temperature to room temperature are calculated with the thermoelastic model. We suppose that the polycristal is stress-free at 923 K. The crystallographic orientation distribution function of the sample was calculated from the pole figures of the sample. A set of 1000 orientations was taken randomly and each orientation was given a volume fraction according to the texture function. The thermal and elastic properties of single crystal zirconium are listed in Table 2. The thermal strains due to a temperature variation from 923 to 298 K are calculated. The thermal and elastic coefficients depend on temperature. We have taken into account the variation of cii and a.ii with temperature [5,6] Table 2 .Thermal expansion coefficients (x 10 K ) [1] and elastic constants (GPa) [START_REF] Simmons | Single crystal elastic constants and calculated aggregate properties[END_REF] at smgle crystal level and at room temperature.

Mechanical residual stresses

We use an incremental elastoplastic self-consistent scheme to simulate the mechanical loading. At each strain increment, the non-linear behaviour is approximated by the tangent behaviour. The stress and strain rates, (J and t, of each grain are linearly linked by an elastoplastic tensor I. This tensor accounts for the plastic and elastic anisotropy of the grains and the hardening effects of the deformation systems during plasticity.

The tensor I has the form :

l=c-_L c . .R h •• (Rg •• c .. R h +Hg h )-1 •• Rg •• c (1)
h. g

where H is the hardening matrix and Rg is the Schmid tensor of a system g. The summation is performed over all the active systems h and g. At macroscopic level, the relation between the macroscopic stress rate and strain rate is given by a similar relation with the introduction of tensor L. a and t of each grain are linked to ! and E by an interaction tensor L • function of an Eshelby tensor taking into account the intrinsic elastic and plastic anisotropy of the HEM as well as the anisotropy induced by the plastic flow. It is a function of L, the shape and the orientation of each grain:

a-:t = L..[I -sesh -1 ]•• (t-E)= -L* •• (t-E) (2)
The overall elastoplastic tensor L is a weighted average of the mesoscopic tensor I : 3) is a non linear implicit equation because Sesh depends on the unknown L. The Eshelby tensor is calculated by an integral equation [START_REF] Tenckhoff | Deformation Mechanisms, Texture, and Anisotropy in Zirconium and Zircaloy, STP 966[END_REF] which takes into account the plastic anisotropy. Equation ( 3) is solved by iteration. Once L is known, by specifying an overall stress or strain, the model can give the corresponding stress or strain tensors for each orientation. Thus, we can describe the mechanical response of the polycrystal at mesoscopic scale. For hexagonal materials, like zirconium, because of the relatively strong crystallographic anisotropy, several modes may be active : slip and/or twinning systems. In such materials, the prismatic glide �0 I 1 Xn2o) is reported to be the main active deformation mode. First-order pyramidal slip �0 I 1 Xn23) and basal slip {0002} (1120) are generally presented as secondary slip modes [START_REF] Tenckhoff | Deformation Mechanisms, Texture, and Anisotropy in Zirconium and Zircaloy, STP 966[END_REF]. At room temperature, twinning has also been observed in zirconium samples [START_REF] Akhtar | [END_REF]10]. Twinning of the � 0 I 2} and � 121} types are expected in extension along the c axis whereas twinning of the � 122} and � 0 I 1} types are expected in compression along the c axis. In the model, critical resolved shear stresses (CRSS) and the hardening law for slip and twinning are considered as controlling parameters. As their values depend on many factors (oxygen content, grain size, temperature ... ), they are considered as unknown. Consequently, they are determined through a fitting of the predicted behaviour on the experimental tensile curve in RD.

L = <l .. [I+Sesh •• L-1 •• Air 1 > (3) Equation (

IV. Results And Comparaison Between Model And Experiment

IV.l. Mechanical response

The initial CRSS are taken as identical for all systems of a system type. The deformation systems introduced in the model are assumed to be : prismatic slip, first-order pyramidal slip and �0 T 2} twinning. In this work, we considered a linear hardening law, for simplicity, the coefficient Hgr is equal to Hgg for any deformation modes r: i g = H g Iy r , i.e., latent hardening is equal to the selfr hardening. The CRSS obtained are : 't = 120 MPa for prismatic mode, 't = 200 MPa for first-order pyramidal mode and t = 220 MPa for tensile twinning. The hardening coefficients were found to be 140 MPa for prismatic slip, 200 MPa for pyramidal slip and 240 MPa for twin mode. The CRSS for the prismatic systems is determined to reproduce the experimental yield stress in the RD. The CRSS values are adjusted to give a good description of the tensile test performed in the RD of the sheet. The hardening coefficients are chosen to be consistent with the slope of the experimental curve in the RD of the sample. The obtained parameters are then taken for the simulation of the tensile test in the TD. The experimental tests show several differences between the two directions : the yield stress is higher, the elastoplastic transition is faster and the curve shows a plateau at 0.6% strain in the transverse direction. The measured yield stresses are 205 MPa and 290 MPa (stress differential of 85 MPa) in the RD an TD respectively. The corresponding Young's modulus are 93 GPa and 96

GPa. These values are similar because of the low elastic anisotropy. With the model developed here, the predicted Young's modulus are 93 GPa and 95 GPa in agreement with the experimental ones. The stress differential can be attributed to thermal stresses [2]. The predicted yield stresses are 205 MPa and 275 MPa in the RD an TD respectively. When no thermal stresses are assumed, the model predicts a low stress differential ( <15 MPa) which could be attributed to texture. For the test along the RD, the elastoplastic transition starts with the activation of prismatic slip. At 0.7 % strain, pyramidal slip is activated and when the deformation reaches 1.3 % deformation, the polycrystal is fully plastic. This sequence explains correctly the experimental loading slope. At 1% deformation, the proportions of activated systems are: 83% for prismatic mode, 16.5 % for pyramidal slips and 0.5% for tensile twinning. Finally, we obtained a reasonable distribution in agreement with the experimental observations: prismatic glide is the main active mechanism, pyramidal slip is a secondary slip mode and twinning contributes very few to plastic deformation. When tension is simulated in the TD, the yield stress is higher and the plastic deformation begins by the activation of prismatic slip. At 0.4% strain, pyramidal systems become active and the material is fully plastic at about 0.9%. At 1%, the distribution of activated systems is : 75%, 24.2% and 0.8% for the prismatic, pyramidal and tensile twin respectively. The model fails to reproduce correctly the plateau in the curve. Turner & al [11] attributed this part to the relaxation of stresses by twinning. In our model, this phenomenon is incorporated, but we found that plastic deformation due to twinning is less than 1.0% and this result is coherent with experimental observations. Consequently, it seems clear that twinning cannot explain the plateau of the curve in the TD. If we take lower values of CRSS or hardening coefficients that doesn't modify this tendency. Higher dislocation annihilation with a solicitation along TD could be explain this phenomenon.

IV.2. X-ray diffraction results

Strains measurements have been made with two plane families : �0 I 4} and {2022} planes. The stress was obtained from the sin\11 relation. In the present section, it should be remembered that, due to the homogeneous macroscopic plastic flow, there is no macroscopic residual stress. The measured and simulated stresses are relative to the diffracting volume, i.e. to groups of crystallites with specific orientation. The mentioned stresses are thus pseudo-macroscopic stresses. X-ray analysis has been made with a removed thickness of 100 J.lm by electro polishing. Figure 2 presents pseudo-macroscopic stress after unloading as a function of total deformation (prestrain) in our samples using the �OI 4} and {2022} planes with <1> = oo and a solicitation along the RD. The two planes exhibit an opposite behaviour : �0 I 4} plane is in traction and {2022} plane is in compression with a stress value that increases with the amount of prestrain. At 2% strain, cr value reaches 30 MPa for � 0 I 4} plane and -100 MPa for {2022} plane. X -ray measurements show the existence of plastic anisotropy. For comparison, in cold rolled zircaloy-4 cladding tubes [12], the stresses obtained by XRD depend strongly on the analysed plane family (+168 MPa for the �OI4} plane and -193 MP a for the {W22} plane). This significant difference is also observed in the case of zirconium.

IV .3. Predicted mesoscopic strains

The next step will be to validate the assumptions on the deformation modes activated in our calculations. For that, the confrontation between stress measurements with diffraction techniques on these samples and simulation at mesoscopic level has been achieved. Simulations have been performed on a polycrystal with different combinations of plastic modes (pyramidal or basal slip as the main active deformation mode). The mesoscopic stresses were then averaged and projected on the <1>\lf direction to simulate the X-ray behaviour. Froin the slope of £<1>\Jf curves, we calculated an estimation of the pseudo-macroscopic residual stresses as a function of total prestrain. It was found that, only when prismatic glide was taken as the main deformation mode, the simulations could match the experimental results : traction for � 0 I 4} and compression for {w22} planes. Figure 2 presents this evolution. Stresses values are different for the two studied planes. Although the predictions are in qualitative agreement with experimental evidence, the residual stresses are overpredicted by the self-consistent model. It is a well-known feature of this model [13] which is too "stiff' because it doesn't take into account the fluctuations of moduli in the matrix and the inclusion due to the nonlinearity of the mechanical behaviour [14]. In our case, the model overestimates the observed residual stress by at least a factor two. In this study, the thermal residual stresses in a rolled a-Zr using a self-consistent model have been evaluated and compared with X-ray diffraction results. A good agreement between the two methods has been obtained. An elastoplastic self-consistent approach has been developed in order to modelise the mechanical response of rolled Zr under uniaxial loading. We showed that thermal stresses can explain the stress differential between RD and TD of the sample and that the elastoplastic transition can be explained by texture and internal stresses. The choice of prismatic slip as principal deformation mode can explain the opposite pseudo-macrostresses values for the two studied planes. These predicted results can be only obtained for a single set of slips modes and hardening parameters. This analysis shows that X-ray diffraction stress analysis could constitute an effective validation or identifi cation of a modelization.

� -

2 .

 2 (b) Figure!: Experimental and calculated stress-strain curves for a tensile test in the rolling (a) and transverse direction (b) of the sample 11.Stress measurements by X-ray diffraction technique

  .

1 - 1 ' 2 Figure2:

 112 Figure2: Evolution experimental (a) and calculated (b) of cr with total strain for <1> = 0°. •: {!022} plane, •: �0 I 4} plane. Axis loading is parallel to RD.V. Conclusions