An inverse scattering problem for a layered periodic structure

This article is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable periodic structure in a periodic layered medium. Having established the well-posedness of the direct problem by the integral equation method, we show that the measurements, corresponding to a countably infinite number of incident quasi-periodic waves above the structure, uniquely determine both the penetrable periodic interface and the impenetrable periodic structure together with its physical property. To this end, an important role is played by a priori estimates of the solution and a mixed reciprocity relation for the direct problem.