Maxwell’s equations in a periodic structure

Abstract

Consider a diffraction of a beam of particles in R 3 {\mathbb {R}^3} when the dielectric coefficient is a constant ε 1 {\varepsilon _1} above a surface S S and a constant ε 2 {\varepsilon _2} below a surface S S , and the magnetic permeability is constant throughout R 3 {\mathbb {R}^3} . S S is assumed to be periodic in the x 1 {x_1} direction and of the form x 1 = f 1 ( s ) , x 3 = f 3 ( s ) , x 2 {x_1} = {f_1}(s),\,{x_3} = {f_3}(s),\,{x_2} arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in R 3 {\mathbb {R}^3} having the form of refracted waves for x 3 ≪ 1 {x_3} \ll 1 and of transmitted waves for − x 3 ≫ 1 - {x_3} \gg 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 ε \varepsilon ’s, except for a discrete number, there exists a unique solution to the Maxwell equations.