On the source type of integral equations for scattering by an object in a layered medium

This paper is concerned with the source type of integral equation to represent the scattering by an inhomogeneous object in a homogeneous layer of a planar layered medium in the frequency domain. By decomposing the scattered field into a particular and a general constituent the structure of the integral operator of the integral equation is identified. The particular constituent represents the scattered field inside the layer, that embodies the contrasting object, due to the presence of virtual contrast sources inside the inhomogeneous object, while the general constituent represents the interaction with the other layers due to the presence of source distributions on each side of the layer that embodies the contrasting object. The part due to the particular constituent has a convolution structure in all spatial directions. The part due to the general constituent consists of two terms, one has again a convolution structure with respect to all spatial coordinates, while the other has a convolution structure with respect to the horizontal coordinates and a correlation structure in the vertical coordinates. These properties facilitate a fast and efficient computation of the integral operator with the help of fast Fourier transforms. For the full 3D electromagnetic case, where the inhomogeneous object exhibits both an electric and magnetic contrast, the source type of integral equations are derived and their structure is discussed.