MAXWELL EQUATIONS IN A PERIODIC STRUCTURE

Consider a diffraction of a beam of particles in R3 when the dielectric coefficient is a constant epsilon-1 above a surface S and a constant epsilon-2 below a surface S, and the magnetic permeability is constant throughout R3. S is assumed to be periodic in the x1 direction and of the form x1 = f1(s), x3 = f3(s), x2 arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in R3 having the form of refracted waves for x3 >> 1 and of transmitted waves for -x3 >> 1 if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the epsilon's, except for a discrete number, there exists a unique solution to the Maxwell equations.