Integral equations for scattering by a particle in a layer

In applications such as the alignment of layers in an integrated circuit or the application of a coating to a surface we may need to determine the light scattered by a dielectric or conducting particle of size comparable to the wavelength in a dielectric layer. We apply the method of the single integral equation to solve this problem and find that the minimum number of required unknown surface fields depends on the geometrical configuration of the system. This method involves the definition of auxiliary fields that coincide with physical fields in one region, obey the same equations everywhere, obey the radiation condition, and satisfy certain boundary conditions involving jumps that are the unknown surface functions in the integral equations. We apply this method to particles in layers. We find that for certain configurations the number of unknowns exceeds the number of boundaries. We also estimate the memory requirements for such a three-dimensional problem.