Perturbation theory of nearly spherical dielectric optical resonators

TL;DR

This paper develops a boundary-condition perturbation theory to accurately predict the shifted and broadened optical resonances of nearly spherical dielectric resonators with small shape deviations.

Contribution

It extends Rayleigh's acoustic membrane perturbation theory to electromagnetic whispering-gallery modes in nearly spherical dielectric cavities, providing explicit formulas up to second order.

Findings

Explicit formulas for resonance frequency shifts and linewidths

Applicability conditions for perturbation theory

Second-order accuracy in shape deviations

Abstract

Dielectric spheres of various sizes may sustain electromagnetic whispering-gallery modes resonating at optical frequencies with very narrow linewidths. Arbitrary small deviations from the spherical shape typically shift and broaden such resonances. Our goal is to determine these shifted and broadened resonances. A boundary-condition perturbation theory for the acoustic vibrations of nearly circular membranes was developed by Rayleigh more than a century ago. We extend this theory to describe the electromagnetic excitations of nearly spherical dielectric cavities. This approach permits us to avoid dealing with decaying quasinormal modes. We explicitly find the frequencies and the linewidths of the optical resonances for arbitrarily deformed nearly spherical dielectric cavities, as power series expansions by a small parameter, up to and including second-order terms. We thoroughly discuss the physical conditions for the applicability of perturbation theory.