Generalized linear sampling methods for locally perturbed periodic layers

Résumé

We are interested in this PhD thesis in studying the inverse scattering problem for the reconstruction of defects in unbounded periodic structures, without a knowledge of the periodic background, and without assuming that the defect is also periodic. We first study the inverse problem using quasi-periodic incident waves. More precisely we employ the GLSM to solve the inverse problem for a single Floquet-Bloch mode. We then consider the case where non-quasi periodic incident sources are applied, and we employ the GLSM to define an indicator function for the full domain. Based on these two results, we apply the differential imaging to construct an indicator function allowing us to directly construct the defect. We give some numerical examples for the reconstruction obtained using the indicator function associated with the single Floquet-Bloch mode. We finally give a study for the well-posedness of the interior transmission problem by considering the cases when the perturbation is included in the periodic background. The study is based on the application of the Floquet-Bloch transform, a discretization with respect to the Floquet-Bloch variable and a convergence analysis to construct a solution for the Interior Transmission Problem.

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