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

Abstract

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 didn't 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.

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

“…The calculation (10) implicates that v s vanishes on {Im n p > 0} and the theorem of unique continuation gives v s = 0 on the connected set D c including {Im n p > 0} \ D. Since we assumed to have no transmission eigenfunctions, the scattered field v s has to vanish everywhere and in consequence v = 0 holds as claimed.…”

“…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.…”

Analysis of sampling methods for imaging a periodic layer and its defects

“…We set u N := w − w N , then u N ∈ H 2 loc (Ω R ) satisfies (6) with the right hand side of the first equation is replaced by…”

“…For the justification of the method we assume that the local perturbation does not intersect the periodic background. The case where this intersection is not empty requires the study of an interior transmission problem that has a non standard structure similar to the one considered in [20]. For the sake of conciseness we leave this to future investigations.…”

Near-field imaging of locally perturbed periodic surfaces

“…In the paper [Zha18], a high order numerical method has been proposed based on the Floquet-Bloch transform and this method is used in this paper to produce the measured data. For a fast imaging method to reconstruct the local perturbations in periodic media with the help of the Bloch transform, we refer to [CHN18].…”