Numerical simulation of the whispering gallery modes in prolate spheroids

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

“…We mention Initial Value Problems [4], Boundary Value Problems [5,6,11,12] and Sturm-Liouville problems [7][8][9][10]. In all these cases the proposed formulae show good stability properties and the developed codes turn out to be competitive with the existing software, in particular when the problem to be solved is stiff and the solution has a slope different from that of its derivatives.…”

“…The mainly used schemes are high order central finite differences, while other additional formulae are considered depending on the Sturm-Liouville classification of the endpoints of the interval. The proposed approach appears efficient and robust to solve with constant stepsize a particular singular Sturm-Liouville problem arising from the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform (see [9]), important for numerous applications in signal and image processing, and also for the solution of a two-parameter singular Sturm-Liouville problem derived from the numerical simulation of the so called 'whispering gallery' modes (WGMs) occurring inside a prolate spheroidal cavity, see [10].…”

Variable-step finite difference schemes for the solution of Sturm–Liouville problems

“…In the last few years High Order Finite Difference Schemes have been largely used to solve different problems concerning with differential equations of order greater than 1. We mention Initial Value Problems [4], Boundary Value Problems [5,6,11,12] and Sturm-Liouville problems [8,9,10]. In all these cases the proposed formulae show good stability properties and the developed codes turn out to be competitive with the existing software, in particular when the problem to be solved is stiff and the solution has a slope different from that of its derivatives.…”

Extension and Optimization of the Axisymmetric 2.5-D Eigensolver: Toward Far-Field Calculations in Stratified Backgrounds

“…Since the fields of a structure can always be expanded by eigenfunctions, eigenvalue problems are of importance. Among all the axisymmetric structures, only dipoles [16], [17], spheres [18]- [20], and spheroids [21]- [23] possess analytical eigensolutions, while other structures, such as disks and rings, have to be handled by numerical methods, such as the finite-difference time-domain (FDTD) method [24] and finite-element method (FEM) [25]. For the nanoparticle antennas, a 3-D simulation is appropriate because the scatter is relatively small.…”