Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices

Abstract

In this work, we are concerned with the mathematical modeling of the electromagnetic (EM) scattering by arbitrarily shaped non-magnetic nanoparticles with high refractive indices. When illuminated by visible light, these particles can exhibit a very strong isotropic magnetic response, in addition to the electric resonance, resulting from the coupling of the incident wave with the circular displacement currents of the EM fields. We shall first introduce the mathematical concept of dielectric (subwavelength) resonances, then perform the asymptotic analysis in terms of the small particle size and the high contrast parameter. This enables us to derive the a priori estimates for the leading-order terms of dielectric resonances and the associated resonant modes, by making use of a Helmholtz decomposition for divergence-free vector fields. It turns out that these dielectric resonant fields are almost transverse electric in the quasi-static and high contrast regime, and hence present the feature of the magnetostatic fields. To address the existence of resonances, we apply the Gohberg-Sigal theory under a physical condition for the contrast, and show that there exist finitely many physically meaningful resonances in the fourth quadrant. The second part of this work is for the quantitative investigation of the enhancement of the scattered field and the cross sections when the dielectric resonance occurs. In doing so, we develop a novel multipole radiation framework that directly separates the electric and magnetic multipole moments and allows us to clearly see their orders of magnitude and blow-up rates. We show that at the dielectric subwavelength resonant frequencies, the nanoparticles with high refractive indices behave like the coupling of an electric dipole with a resonant magnetic dipole. By using the spherical multipole expansion, we further show how to explicitly calculate the quasi-static dielectric resonance and the approximate scattered field which helps validate our general results and formulas.