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This paper considers the imaging of local perturbations of an infinitepenetrable periodic layer. A cell of this periodic layer consists of severalbounded inhomogeneities situated in a known homogeneous media. We use \mfied{adifferential linear sampling method} to reconstruct the support ofperturbations without using the Green's function of the periodic layer norreconstruct the periodic background inhomogeneities. The justification of thisimaging method relies on the well-posedeness of a nonstandard interiortransmission problem, which until now was an open problem except for thespecial case when the local perturbation didn't intersect the backgroundinhomogeneities. The analysis of this new interior transmission problem is themain focus of this paper. We then complete the justification of our inversionmethod and present some numerical examples that confirm the theoreticalbehavior of the differential indicator function determining the reconstructableregions in the periodic layer.

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