Maxwell’s equations in a periodic structure

Consider a diffraction of a beam of particles in when the dielectric coefficient is a constant above a surface and a constant below a surface , and the magnetic permeability is constant throughout . is assumed to be periodic in the direction and of the form arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in having the form of refracted waves for and of transmitted waves for if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the 's, except for a discrete number, there exists a unique solution to the Maxwell equations.