Perturbation Theory of Optical Resonances of Deformed Dielectric Spheres

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.

Perturbation theory of nearly spherical dielectric optical resonators

“…Both equations ( 25) and ( 26) are characterized by the index l, so that for each value of l there will be a different set of solutions. To find these solutions, we write k 1 a = k 0 an 1 ≡ xn 1 and k 2 a = k 0 an 2 ≡ xn 2 in (25)(26), where the dimensionless wave number x is defined as x ≡ k 0 a. Then, we introduce the compact notation (the irrelevant prefactor x/i is introduced for later notational convenience),…”

“…As a first step towards a perturbation theory, we must verify that the system (69) reduces to the two equations (25)(26) for ε → 0, that is when the resonator is perfectly spherical. In this case, from (42) it follows that n = 0, and (65) become…”

“…However, such deformations manifest nonzero correlations, the effects of which are always disclosed by second-order perturbation theory [23,24]. Further details on the theory, omitted here due to lack of space, can be found in Julius Gohsrich's master thesis [25].…”