PLANE WAVE SPECTRA IN GRATING THEORY: IV. SCATTERING BY A FINITE GRATING OF IDENTICAL CYLINDERS

Abstract

Scattering of time-harmonic plane waves by a finite grating of identical elements is examined by mathematical decomposition of an infinite grating. An equation is derived which relates the scattering function of the finite grating to that of a single element in the infinite grating. This is manipulated to make possible a series solution involving scattering functions for semi-infinite gratings. The first term is equal to the scattering function obtained by assuming that each cylinder of the finite grating scatters as an element of an infinite grating, plus end-effect corrections which involve scattering functions for semi-infinite gratings. Other terms tend to zero as the width of the grating tends to infinity. These higher-order terms are of most importance when circumstances are favorable for the existence of grating anomalies.The qualitative form of the scattered far field is examined. It is suggested that the "single anomaly", observed experimentally, is actually the superposition of two effects. One is dependent on the angle of incidence (α), the other on the angle of observation (θ). This latter effect, although always present, usually occurs in directions of low field intensity and apparently has not been observed. When α is such that a single anomaly is to be expected, the θ-dependent effect occurs in directions of high intensity. Similarly, it is argued that the double anomaly is a superposition of four effects.