Scattering by a Grating. I

Consideration is given to scattering of plane waves by a transmission grating upon whose elements the wave function vanishes. By an application of Green's theorem, the problem is formulated in terms of integral equations for a finite, or infinite, number of elements. The equations are put in a form more suitable for the study of interaction phenomena by the subtraction of a certain series. The modified equations then correspond, more or less, to the excitation of each element of the grating by the incident field, and two plane waves propagated in opposite directions along the grating. A solution is here attempted only for the infinite grating of identical elements. Attention is confined to the region of "Rayleigh wavelengths", where interaction is important, and the variation in spectral intensity for a grating of elements of arbitrary form is discussed in a semiquantitative manner; the existence of anomalies is inferred. An explicit solution is obtained for small scatterers of elliptical cross section, and the behavior of the spectral intensity is considered in some detail.