Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers

Abstract

International audienceThis paper is dedicated to the design and analysis of sampling methods to reconstruct the shape of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency. We first introduce the model problem that corresponds with the semi-discretized version of the continous model with respect to the Floquet-Bloch variable. We then present the inverse problem setting where (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and analyze two possible factorizations of this operator. We then establish sampling methods to identify the defect and the periodic background geometry from this operator measurement. We also show how one can recover the geometry of the background independently from the defect. We then introduce and analyze the single Floquet-Bloch mode measurement operators and show how one can exploit them to built an indicator function of the defect independently from the background geometry. Numerical validating results are provided for simple and complex backgrounds

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

“…Here we adopt the notations in [11]. Recall that the parameter L := (L 1 , · · · , L d−1 ) ∈ R d−1 , L j > 0, j = 1, · · · , d − 1 refers to the periodicity of the media with respect to the first d − 1 variables and M := (M 1 , · · · , M d−1 ) ∈ N d−1 refers to the number of periods in the truncated domain.…”

“…(ii) z ∈ D p if and only if lim Proof. The proof of the items (i) and (ii), we refer to [11]. The proof of items (iii) is a direct application of Theorem A.4 in [11] in combination with Theorem 3.5.…”

“…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 [11] and [18] for more detailed discussion.…”

“…Here the defect is a disc of r ω = 0.2λ with n = 3. More examples of this case can be found in [11]. Example 4.…”

Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements

“…Up to our knowledge, the sampling methods for locally perturbed infinite periodic layers and far field settings have not been treated in the literature. We refer to [13,10] where differential sampling methods have been designed to reconstruct defects in periodic backgrounds without knowledge of the background using propagative and evanescent modes. However, the method applied in [13,10] has been only justified for the case of periodic defects with periodicity length that is the multiple of the background periodicity.…”