Physics Research International

Volume 2014, Article ID 615605, 10 pages

http://dx.doi.org/10.1155/2014/615605

## Analytical Approximations of Whispering Gallery Modes in Anisotropic Ellipsoidal Resonators

^{1}Institute of Applied Physics, Friedrich-Schiller University Jena, Max-Wien-Platz, 07743 Jena, Germany^{2}Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany^{3}Institute for Optics, Information and Photonics, University of Erlangen-Nurmberg, Staudtstraße 7/B2, 91058 Erlangen, Germany

Received 1 May 2014; Accepted 12 November 2014; Published 10 December 2014

Academic Editor: Lorenzo Pavesi

Copyright © 2014 Marco Ornigotti and Andrea Aiello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numerical evolutions of whispering gallery modes of both isotropic and anisotropic spheroidal resonators are presented and are used to build analytical approximations of these modes. Such approximations are carried out mainly to have the possibility to have manageable analytic formulas for the eigenmodes and eigenfrequencies of anisotropic resonators. A qualitative analysis of ellipsoidal anisotropic modes in terms of superposition of spherical modes is also presented.

#### 1. Introduction

Thanks to their extremely high quality-factor (up to ), whispering gallery mode (WGM) resonators are nowadays very promising devices for applications in optoelectronics [1], bio-sensing [2, 3], nonlinear optics [4], and other fields of applied physics. Although the first analysis of such devices dates back to 1939 [5], this topic remained unconsidered for many years. Recently, however, the need of compact optical devices with great performances (together with the possibility to manufacture not only silica microspheres but WGM resonators of any material and shape [6, 7]) had contributed to make WGMs experience a new renaissance. Unfortunately, while the theory of WGMs in microspheres is well-established, and precise calculations of eigenmodes, radiative losses, field distributions, and other physical quantities have been carefully carried out [1], when approaching the study of anisotropic devices, analytical solutions cannot be found for geometries different from an ideal sphere [8] or cylinder. In this case, it is more convenient to solve the problem with the help of numerical codes. An analytical solution, on the other side, would be preferred, as it makes easier to catch the physics behind a phenomenon and it gives the possibility to easily predict its evolution. But how to validate an analytical approximation of a solution when a real analytical solution does not exist? To answer to this question, we propose in this paper to use a numerical finite element solver (COMSOL Multiphysics [9]) to firstly find an appropriate approximation of the problem in exam and then use that approximation to check the validity of a simple analytical model for the solution.

In this paper we study whispering gallery modes (WGMs) in isotropic and anisotropic dielectric ellipsoidal resonators. Whereas nonspherical WGMs have been already studied in the past, and their analytical expression was given in terms of spherical-like functions [10, 11] or superposition of spheroidal modes [12], to the best of our knowledge no papers have been written on analytical solution for the anisotropic case. We then intend to fill this gap by finding with COMSOL a suitable approximation for WGMs in these resonators and then presenting a correspondent robust and accurate analytical solution. The approximation we present is based on the observation that since WGMs are localized near the resonator surface, an effective approximation of an ellipsoidal resonator near its surface could be represented by a toroidal resonator of circular cross section, the radius of the torus being equal to the major axis of the resonator and the radius of the circular cross section being equal to the rim radius of the resonator at the considered surface. This result gives us the possibility to substitute, under certain conditions, the complete set of spheroidal wave functions [12] that characterize the ellipsoidal resonator with the complete set of spherical wave functions [8, 13]. This permits us to manage a simple and analytical set of eigenmodes that can be easily used for theoretical predictions of nonlinear or quantum optical effects in these resonators.

This paper is organized as follows: in Section 2 the model used to simulate with COMSOL such a resonator is presented, together with a brief recall of the weak form expression for Maxwell’s equations in a rotationally symmetric resonator that is used as a model for the calculations. Section 3 is then devoted to present the results of the COMSOL simulations we made both for the isotropic and for the anisotropic cases, with a direct comparison between the field distributions and the eigenvalues of both the ellipsoidal and the toroidal resonators. Finally, in Section 4 conclusions are presented.

#### 2. The Model

##### 2.1. Geometry of the Resonator

Let us consider an ellipsoid of revolution filled by a dielectric medium, rotationally invariant around the -axis and characterized by the in-plane major axis , that is, the radius of the circumference in the plane , and the out-of-plane minor axis , that is, the minor axis of the ellipse in the plane . Such an ellipsoid is depicted in Figure 1, and its cartesian equation is the following:
where the major axis is mm and the rim radius is mm; the value of the minor axis is then determined from these two numerical values with easy geometrical considerations and it turns out to be mm. According to [4], in fact, it is possible to obtain the value of in the following way. Let us firstly define as the angle between the vector describing the position of point on the surface of the ellipse and the major axis of the ellipse itself. Therefore, at , the rim radius of the ellipse is simply given by . This result is used here to infer, from the data presented in [4], the correct value of the minor axis . Following [4], we consider this resonator to be made of lithium niobate (LiNbO_{3}), an anisotropic crystal characterized by the following dielectric tensor:
where is the dielectric constant in the -plane and is the dielectric constant parallel to the -axis. However, in considering the resonator as isotropic, we assume that its dielectric tensor will be diagonal, with nonzero elements that are equal to .