Perturbation theory of optical resonances of deformed dielectric spheres

TL;DR

This paper develops a perturbation theory to analyze how small deformations of dielectric spheres affect their optical resonances, providing analytical formulas for frequency shifts and linewidth modifications.

Contribution

It introduces a second-order perturbation approach using the Kapur-Peierls formalism to accurately predict resonance changes in deformed dielectric spheres.

Findings

Derived formulas for first- and second-order frequency shifts.

Validated the theory by comparing with existing results.

Applicable to deformed liquid drops and solid spheres.

Abstract

We analyze the optical resonances of a dielectric sphere whose surface has been slightly deformed in an arbitrary way. Setting up a perturbation series up to second order, we derive both the frequency shifts and modified linewidths. Our theory is applicable, for example, to freely levitated liquid drops or solid spheres, which are deformed by thermal surface vibrations, centrifugal forces or arbitrary surface waves. A dielectric sphere is effectively an open system whose description requires the introduction of non-Hermitian operators characterized by complex eigenvalues and not normalizable eigenfunctions. We avoid these difficulties using the Kapur-Peierls formalism which enables us to extend the popular Rayleigh-Schr\"{o}dinger perturbation theory to the case of electromagnetic Debye's potentials describing the light fields inside and outside the near-spherical dielectric object. We find analytical formulas, valid within certain limits, for the deformation-induced first- and second-order corrections to the central frequency and bandwidth of a resonance. As an application of our method, we compare our results with preexisting ones finding full agreement.