On the scatttering of waves by an infinite grating

Using Green's function methods, we express the field of a grating of cylinders excited by a plane wave as certain sets of plane waves: a transmitted set, a reflected set, and essentially the sum of the two "inside" the grating. The transmitted set is given byψₒ + 2υG(θυ, θₒ)ψυ, where theψ's are the usual infinite number of plane wave (propagating and surface) modes;G(θυ, θₒ)is the "multiple scattered amplitude of a cylinder in the grating" for direction of incidenceθₒand observationθυ; and the C's are known constants. (For a propagating mode,Cυis proportional to the number of cylinders in the first Fresnel zone corresponding to the direction of modev.) We show (for cylinders symmetrical to the plane of the grating) thatG(θ,θₒ)= g(θ,θₒ) +(Συ - ∫ dv)Cυ[g(θ,θυ + g (θ,π - θυ)G(π-θυ,θₒ)], wheregis the scattering amplitude of an isolated cylinder. This inhomogeneous "sum-integral" equation forGis applied to the "Wood anomalies" of the analogous reflection grating; we derive a simple approximation indicating extrema in the intensity at wavelengths slightly longer than those having a grazing mode. These extrema suggest the use of gratings as microwave filters, polarizers, etc.