Perturbed Green's functions in electromagnetic scattering

The solution of a problem involving the scattering of an incident electromagnetic field by an obstacle is easily solved when the obstacle has a boundary corresponding to a constant coordinate surface for which an analytic solution for the Green's function satisfying appropriate boundary conditions can be found. When the boundary does not satisfy this condition one generally has to solve an integral equation for the unknown boundary values of the field and/or its normal derivative at the surface. In the more restricted case when the boundary deviates by a small amount from a constant coordinate surface various perturbation solutions can be developed. We examine approximate solutions that can be constructed by perturbing the Green's function. In order to keep the discussion as simple as possible and yet illustrate the concepts involved we consider only problems that can be solved using a scalar Green's function. We start with the Green's function for the canonical problem of an obstacle with a constant coordinate surface and consider various perturbations of this Green's function and the resultant field solutions.