Perturbation Theory of Optical Resonances of Deformed Dielectric Spheres

TL;DR

This paper develops a perturbation theory to analyze how optical resonances in dielectric spheres change when the shape is deformed, providing a systematic way to predict resonance shifts and leakage.

Contribution

A novel perturbation approach for calculating shifts in optical resonances of deformed dielectric spheres up to second order.

Findings

The method accurately predicts resonance shifts for small deformations.

Resonance eigenvalues are obtained via finite-dimensional linear eigenvalue equations.

Applicability criteria for the perturbation theory are derived and validated.

Abstract

Light injected into a spherical dielectric body may be confined very efficiently via the mechanism of total internal reflection. The frequencies that are most confined are called resonances. If the shape of the body deviates from the perfect spherical form the resonances change accordingly. In this thesis, a perturbation theory for the optical resonances of such a deformed sphere is developed. The optical resonances of such an open system are characterized by complex eigenvalues, where the real part relates to the frequency of the resonant light and the imaginary part to the energy leakage out of the system. As unperturbed and analytically solvable problem serves the homogeneous dielectric sphere, and the corrections to its eigenvalues are determined up to and including second order for any polarization of light. For each order, the corrections of the optical resonances are determined by a finite-dimensional linear eigenvalue equation, similar to degenerate time-independent perturbation theory in quantum mechanics. Furthermore, geometrically intuitive applicability criteria are derived. To check the validity of the presented method, it is applied and compared to an analytically solvable problem.