Uniqueness in determining rectangular grating profiles with a single incoming wave (Part I): TE polarization case

Abstract

We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a rectangular grating profile together with the refractive index beneath it can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. Our approach relies on the expansion of solutions to the Helmholtz equation and the corner singularity analysis of solutions to the inhomogeneous Laplace equation with a piecewise continuous source term in a sector. This paper also contributes to corner scattering theory for the Helmholtz equation in a special non-convex domain.