The Rayleigh Hypothesis

Abstract

Frequently alluded in the foregoing chapters, we will now deal in more detail with the problem of Rayleigh’s hypothesis. In 1907, Lord Rayleigh published a paper on the dynamic theory of gratings, as mentioned earlier in Chap. 1. In this paper he presented a rigorous approach for solving plane wave scattering on periodic surfaces in Cartesian coordinates (see Fig. 6.1). Those gratings are of importance in many different fields of physics and in engineering. They are used as dispersive elements in grating spectrographs, for example. In his paper, Rayleigh used a series expansion of the scattered wave in terms of outgoing plane waves only, i.e., in terms of waves which move only away from the grating. He determined the unknown expansion coefficients later by application of the boundary conditions at the periodic surface appropriately, as discussed in Chap. 1. For the special case of a perpendicularly incident plane wave on a sinusoidal but perfectly conducting surface, he derived an equation system which is at first independent of the groove depth. But Rayleigh approximated this system later to allow for an iterative solution for shallow grooves.