Tatiana Levitina

Numerical simulation of the whispering gallery modes in prolate spheroids

Abstract In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.

Calculations of the morphology dependent resonances

∗Dipartimento di Matematica, Universita di Bari, Via E. Orabona 4, I-70125 Bari, Italy †Institut Computational Mathematics, TU Braunschweig, Pockelsstrasse 14, D-38106 Braunschweig, Germany ∗∗Dipartimento di Matematica e Fisica ‘E. De Giorgi’ , Universita del Salento, Via per Arnesano, I-73047 Lecce, Italy ‡Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria

Sampling formula for convolution with a prolate

Abstract Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This permits interpolation of such a convolution by the Walter and Shen sampling formula in terms of prolates, although the Fourier transform of the convolution is not necessarily even continuous and the concentration interval is twice as large as that of a prolate. Rigorous error estimates are given as dependent on the truncation limits. The accuracy achieved is tested by numerical examples.

Filter diagonalization: Filtering and postprocessing with prolates

A detailed account is given of a recent modification of the Filter Diagonalization technique that serves to analyze a signal spectrum within a selected energy range. Our approach employs for filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which are superior to other filters due to their special properties. In particular, prolates are simultaneously band-limited and highly concentrated at a finite time-interval, producing filters with optimal accuracy. In addition both features are acquired by the convolution of a band-limited function with a prolate, that permits the latter to be interpolated via the Walter and Shen sampling formula, which essentially simplifies the supplementary computations. Rigorous filtering error estimates are obtained. Test calculations illustrate the facilities of the presented modification.

Whispering gallery modes in oblate spheroidal cavities: Calculations with a variable stepsize

We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.