Inverse diffraction by a doubly periodic structure

Let us consider the scattering of electromagnetic waves by a doubly periodic structure. Above the structure, the medium is assumed to be homogeneous with a fixed real dielectric coefficient. The medium is a perfect conductor below the structure. For a given incident plane wave, the tangential electric field is measured away from the structure. An inverse problem arises: To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of an eigenvalue problem.