Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics

Abstract

Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling the interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact and Hilbert--Schmidt operator polynomials are constructed from the Green operator of electromagnetic scattering on scatterers with smooth boundaries. As a consequence, it is shown that the strongly singular Born equation has a discrete spectrum, and that the spectral series $ \sum_\lambda|\lambda|^2|1+2\lambda|^4$ is convergent, counting multiplicities of the eigenvalues $ \lambda$. This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity $ \epsilon_r=-1$.