Dielectric scattering resonances for high-refractive resonators with cubic nonlinearity

TL;DR

This paper develops a rigorous mathematical framework to analyze nonlinear dielectric resonances in high-refractive index resonators with Kerr nonlinearity, revealing bifurcations, symmetry-breaking, and asymptotic behaviors in different dimensions.

Contribution

It provides the first rigorous analysis of nonlinear dielectric resonances in high-index resonators, including existence, asymptotics, and bifurcation phenomena in 2D and 3D settings.

Findings

Existence of nonlinear dielectric resonances bifurcating from linear ones.

Symmetry-breaking bifurcation in 3D dimer resonators at critical amplitude.

No symmetry-breaking bifurcation in 2D due to logarithmic singularity.

Abstract

This work establishes a rigorous mathematical framework for the analysis of nonlinear dielectric resonances in wave scattering by high-index resonators with Kerr-type nonlinearities. We consider both two- and three-dimensional settings and prove the existence of nonlinear dielectric resonances in the subwavelength regime, bifurcating from the zero solution at the corresponding linear resonances. Furthermore, we derive asymptotic expansions for the nonlinear resonances and states in terms of the high contrast parameter $\tau$ and the normalization constant. For a symmetric dimer of resonators, these small-amplitude nonlinear resonant states exhibit either symmetric or antisymmetric profiles. In three dimensions, under conditions valid in the dilute regime, we prove that as the field amplitude increases, mode hybridization induces a symmetry-breaking bifurcation along the principal symmetric solution branch at a critical amplitude. This bifurcation gives rise to two asymmetric resonant states, each localized on one of the particles in the dimer. Remarkably, in two dimensions, we show that no such symmetry-breaking bifurcation exists along the principal solution branches, owing to the distinct scaling behavior of the principal nonlinear subwavelength resonance arising from the logarithmic singularity.