Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics

TL;DR

This paper performs spectral analysis on the Born equation for electromagnetic scattering by arbitrarily shaped dielectrics, revealing a discrete spectrum and a shape-independent optical resonance mode at a critical permittivity.

Contribution

It introduces a spectral framework for the Born equation, demonstrating the discreteness of its spectrum and identifying a shape-independent resonance mode.

Findings

Born equation has a discrete spectrum

Spectral series converges, indicating stability

Identifies a shape-independent optical resonance mode at ε_r = -1

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$.