Semiclassical theory to optical resonant modes of a transparent dielectric spheroidal cavity

Abstract

We study the resonant scattering of light by a transparent dielectric spheroid. We try to understand the features of the resonant modes of a spheroidal optical cavity. In this way, we use an analogy between optics and quantum mechanics. Through this analogy it is possible to interpret resonances as quasibound states of light. Using semiclassical methods such as the WKB method and a uniform asymptotic expansion for spheroidal radial functions, we developed algorithms that permit us to calculate the resonance position as well as the resonance width.

Geometrical theory of whispering-gallery modes

“…These conditions plus integrals (14) completely coincide with those obtained by Bykov [26], [27], [28] if we transform ellipsoidal to spheroidal coordinates, and have clear geometrical interpretation. The integral for S 1 corresponds to the difference in lengths of the two geodesic curves on η r between two points P 1 = (ξ r , η r , φ 1 ) and P 2 = (ξ r , η r , φ 2 ).…”

“…Unfortunately they lead to extremely bulky infinite sets of equations which can be solved numerically only in simplest cases and the convergence is not proved. Exact characteristic equation for the eigenfrequencies in dielectric spheroid was suggested [14] without provement that if real could significantly ease the task of finding eigenfrequencies. However, we can not confirm this claim as this characteristic equation contradicts limiting cases with the known solutions i.e.…”

<title>Whispering gallery modes in axisymmetric resonators</title>

“…To find the circular integrals of phases kS (14) we should take into account the properties of phase evolutions on caustic and reflective boundary. Every touching of caustic adds π/2 (see for example 8 ) and reflection adds π.…”

Spheroidal microresonators for the optoelectronics

“…T h us for S 1 we h a ve one caustic shift of =2 a t r and one re ection from the equivalent boundary surface s (at the distance from the real surface), for S 2 { t wo t i m e s =2 due to caustic shifts at r and nothing for S 3 . k S 1 = 2 k s Z ; r @S 1 @ d = 2 (q ; 1=4) k S 2 = 2 k r Z ; r @S 2 @ d = 2 (p + 1 =2) k S 3 = k 2 Z 0 @S 3 @ d = 2 jmj (11) where q = 1 2 3::: { is the order of the mode, showing the number of the zero of the radial function on the surface, and p = l ; j mj = 0 1 2:::. These conditions plus integrals (9) completely coincide with those obtained by B y k ov 22, 24 if we transform ellipsoidal to spheroidal coordinates.…”

“…11,13,15,23 To approximate radial function (17) we proceed in the same way starting from elimination of the rst derivative:…”