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

We revisit the differential sampling method introduced in (Haddar and Nguyen 2017 Comput. Math. Appl. 74 2831-55) for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet-Bloch variable. The corner stone of the analysis is the justification of the generalized linear sampling method in this setting for a single Floquet-Bloch mode.