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 by
\psi_{o} + 2\SigmaC_{\upsilon}G(\theta_{\upsilon}, \theta_{o})\psi_{\upsilon}
, where the
\psi
's are the usual infinite number of plane wave (propagating and surface) modes;
G(\theta_{\upsilon}, \theta_{o})
is the "multiple scattered amplitude of a cylinder in the grating" for direction of incidence
\theta_{o}
and observation
\theta_{\upsilon}
; and the C's are known constants. (For a propagating mode,
C_{\upsilon}
is proportional to the number of cylinders in the first Fresnel zone corresponding to the direction of mode
v
.) We show (for cylinders symmetrical to the plane of the grating) that
G(\theta,\theta_{o})= g(\theta,\theta_{o}) +(\Sigma_{\upsilon} - \int dv)C_{\upsilon}[g(\theta,\theta_{\upsilon} + g (\theta,\pi - \theta_{\upsilon})G(\pi-\theta_{\upsilon},\theta_{o})]
, where
g
is the scattering amplitude of an isolated cylinder. This inhomogeneous "sum-integral" equation for
G
is 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.