Spheroidal harmonic expansions for the solution of Laplace's equation for a point source near a sphere

Matt R. A. Majić, Baptiste Auguié, and Eric C. Le Ru
Phys. Rev. E 95, 033307 – Published 15 March 2017

Abstract

We propose a powerful approach to solve Laplace's equation for point sources near a spherical object. The central new idea is to use prolate spheroidal solid harmonics, which are separable solutions of Laplace's equation in spheroidal coordinates, instead of the more natural spherical solid harmonics. Using electrostatics as an example, we motivate this choice and show that the resulting series expansions converge much faster. This improvement is discussed in terms of the singularity of the solution and its analytic continuation. The benefits of this approach are further illustrated for a specific example: the calculation of modified decay rates of light emitters close to nanostructures in the quasistatic approximation. We expect the general approach to be applicable with similar benefits to the solution of Laplace's equation for other geometries and to other equations of mathematical physics.

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  • Received 4 October 2016

DOI:https://doi.org/10.1103/PhysRevE.95.033307

©2017 American Physical Society

Physics Subject Headings (PhySH)

General Physics

Authors & Affiliations

Matt R. A. Majić, Baptiste Auguié, and Eric C. Le Ru*

  • The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand

  • *eric.leru@vuw.ac.nz

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Issue

Vol. 95, Iss. 3 — March 2017

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