Analysis of the spectrum of the single integral equation for scattering from dielectric objects

Surface integral equations are widely used to analyze electromagnetic scattering from a dielectric object residing in free space. Methods for analyzing scattering from dielectric bodies have largely relied on either the PMCHWT or the Muller formulations. It is well known that the PMCHWT formulation result in a first kind Fredholm integral equation, is not well-conditioned and leads to slow convergence. The Muller formulation, on the other hand, results in a second-kind integral equation and is well-conditioned as the hyper-singular terms nearly cancel for low-contrast ratios. However, the Muller formulation is not very accurate for very high-contrast materials. Alternatively, it has been shown that scattering from a dielectric body can be computed using a single unknown and a set of cascaded equations, viz., the single integral equation (SIE) [1]. Existing literature [2] has reported that the condition number of the impedance matrix is excellent. However, no work has been attempted to analyze the convergence and uniqueness of the SIE operator. In this thesis, a detailed analysis is carried out to reveal the underlying mathematical properties of the operator. It will be shown that this operator does not produce unique solutions at internal resonance frequencies. Nonetheless, we suggest a method to overcome spurious resonances and propose an integral equation that is accurate for arbitrary material contrast ratios, while still preserving its well-conditioned nature.