Low frequency solution for electromagnetic waves through a doubly-periodic magnetodielectric layer

The problem of electromagnetic wave scattering by a doubly-periodic magnetodielectric layer is solved by a new method based on the rigorous integro-differential equations. The Galerkin method is applied to reduce these equations to a set of second-order differential equations with constant coefficients in functionals. A closed-form approximate solution is given to the problem of scattering of a low-frequency electromagnetic wave through the layer with arbitrary distribution of dielectric permittivity and magnetic permeability over the period. Based on such "one-mode" approximation we derive simple expressions for the scattering coefficients