First Approach to the Green Functions: The Rayleigh Method

Abstract

In Sect. 1.3 we have considered a solution method for the scattering problems which was already used by Rayleigh to solve plane wave scattering on periodic gratings. Starting point was the approximation (1.21) of the scattered field by a finite series expansion in terms of any appropriate expansion functions. This approximation was assumed to hold everywhere in the outer region Γ+. The unknown expansion coefficients in this approximation have been determined afterwards by use of the additional boundary conditions at the scatterer surface ∂Γ (see (1.29), for example, if the outer Dirichlet problem is considered). Hereby it was assumed that the primary incident field is the known quantity. But if we look closer on (1.21) and (1.29) we can realize that there are two different sets of expansion functions in use. In (1.21), we have the expansion functions | ϕ i,τ (k 0, x)〉 defined everywhere in the outer region Γ+. On the other hand, concerning equation (1.29) we used the expansion functions | ϕ i,τ (k 0, x)〉∂Γ defined exclusively at the scatterer surface ∂Γ. The expansion coefficients resulting from the corresponding continuity conditions at the scatterer surface should have their meaning only for the approximation of the scattered field at this surface, as one might expect. Therefore, we have to clarify whether these expansion coefficients can be used also in approximation (1.21) or not. Before going into the details of deriving the Green function related to the outer Dirichlet problem of the Helmholtz and vector-wave equation we will clarify this aspect first.