Theory of Anisotropic Whispering Gallery Resonators

TL;DR

This paper provides an analytic solution for uniaxial spherical whispering gallery resonators, including mode calculations and asymptotic formulas, facilitating future quantum theory developments.

Contribution

It introduces an analytic approach to uniaxial spherical resonators using Debye potentials, with detailed boundary condition treatment and asymptotic analysis.

Findings

Analytic solution for uniaxial spherical resonator modes

Asymptotic expansion of the characteristic equation

Precision formula for mode calculation

Abstract

An analytic solution for an uniaxial spherical resonator is presented using the method of Debye potentials. This serves as a starting point for the calculation of whispering gallery modes (WGM) in such a resonator. Suitable approximations for the radial functions are discussed in order to best characterize WGMs. The characteristic equation and its asymptotic expansion for the anisotropic case is also discussed, and an analytic formula with a precision of the order $O[\nu^{-1}]$ is also given. Our careful treatment of both boundary conditions and asymptotic expansions makes the present work a particularly suitable platform for a quantum theory of whispering gallery resonators.