Eigenmodes of spherical dielectric cavities: coupling of internal and external rays

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Abstract
Eigenmodes of spherical dielectric cavities are investigated within the framework of geometrical optics. A model for the coupling of internal and external rays is presented. Approximate expressions for the expansion coefficients of Mie theory are derived. Formulas for the modulus and the phase are given. The systematic behavior of the phase angle is investigated. All expressions are explicit equations and may be used to calculate the expansion coefficients as a function of order l or size parameter x. All features of the presented results are understandable in terms of light rays. A plane interface analog is presented. The so-called resonance regime (Λ/n<x<Λ), where n is the relative refractive index and Λ=l+1/2, is considered especially. An implicit equation for these resonance positions is rederived, and explicit relations for their strengths and widths are given. All findings are in agreement with results derived from Mie theory.