Diffraction of light by a rectangular periodic relief of a coated metal surface

A theoretical analysis is made of the influence of the coating on the optical properties of a metal with a periodic relief of its reflecting surface. A system of equations is derived for the diffraction of light by a rectangular periodic relief of a metal surface with a three-layer coating. These equations apply to arbitrary depths of relief, any layer thicknesses in the coating, and any relief period. Analytic expressions for the diffraction efficiency are obtained and analyzed in some specific cases of interest.