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.
Highlights
In this work we aim at determining frequencies and linewidths of electromagnetic resonances of nearly spherical dielectric cavities of arbitrary size and deformation
Eqs. (37) and (45) imply that the physical conditions for the applicability of the perturbation theory require that the magnitude of the deformation function h(θ, φ) and its gradient ∇h(θ, φ), both evaluated on the unit sphere S, must be O(ε), namely, max{|h(θ, φ)|}S ≈ max{|∇h(θ, φ)|}S 1. (49)
We have developed a boundary-condition perturbation theory to determine the electromagnetic resonances of nearly spherical dielectric resonators
Summary
In this work we aim at determining frequencies and linewidths of electromagnetic resonances of nearly spherical dielectric cavities of arbitrary size and (small) deformation. Level of approximation, the frequencies and the linewidths of the electromagnetic resonances of nearly spherical dielectric cavities [6] In principle, determining such resonances is a conceptually simple boundary-value problem: One must solve Maxwell’s equations for the fields inside (medium 1) and outside (medium 2) the cavity and match these fields at the interface between the two media. II we establish the notation and we review the classical Mie solution [27] for the scattering of electromagnetic waves by dielectric spheres This is functional to the perturbation theory to be developed because the Mie solution will be taken as the zeroth-order approximation.
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