Maxwell’s equations in a periodic structure

Abstract:

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