New Interior Transmission Problem Applied to a Single Floquet-Bloch Mode Imaging of Local Perturbations in Periodic Media
This work provides a theoretical foundation for a non-iterative imaging method in periodic structures, addressing a previously open problem for perturbations intersecting background inhomogeneities.
The paper solves an open interior transmission problem to justify a differential linear sampling method for imaging local perturbations in periodic media, without requiring the Green's function or reconstructing the background. Numerical examples confirm the theoretical indicator function.
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.