An inverse problem in periodic diffractive optics: global uniqueness with a single wavenumber

We consider the problem of recovering a perfectly reflecting two-dimensional diffraction grating from the knowledge of one wavenumber, one incident direction and the total field measured above the grating. We prove a global uniqueness result within the class of polygonal grating profiles. The proof relies on the analyticity of solutions to the Helmholtz equation and the Rayleigh expansion of the scattered field.