Eigenvalue Analysis of the 1D Internal Scattering Operator for a Homogeneous Medium

The electromagnetic scattering problem for dielectric objects involves a linear integral equation that relates the total field to the scattered field inside the object. Numerical solution methods require discretization by choosing a suitable set of basis functions. An optimal choice of the expansion functions not only achieves numerical efficiency but also allows the correct subspaces to be spanned. We resort to the spectral decomposition of the pertinent operator to investigate such optimal functions. A one-dimensional (1D) geometry (i.e., scattering by a homogenous dielectric slab) is considered because it allows us to derive some analytical results and simple closed-form solutions to be used in the numerical verifications. Then, the spectral decomposition is performed numerically. The analysis of the eigenvalues and the eigenfunctions allows for predicting the required number once the maximum slab permittivity is given. In turn, the corresponding maximum number of Fourier harmonics can also be established, as they provide the basis to expand the eigenfunctions.