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

TL;DR

This paper introduces a new interior transmission problem to justify a differential linear sampling method for imaging local perturbations in periodic media, avoiding the need for Green's functions and handling complex background inhomogeneities.

Contribution

It develops the analysis of a novel interior transmission problem crucial for the imaging method, extending applicability to perturbations intersecting background inhomogeneities.

Findings

The new interior transmission problem is well-posed.

The imaging method accurately reconstructs perturbations.

Numerical examples validate the theoretical results.

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 \mfied{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.