Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics

• Published in: Journal of quantitative spectroscopy & radiative transfer 2019-09, Vol.235, p.287-299
• Main Author: Majic, Matt
• Format: Article
• Language: English
• Subjects: Analytic; Rayleigh scattering; T-matrix; Toroidal harmonics; Torus
• ISSN: 0022-4073; 1879-1352
• DOI: 10.1016/j.jqsrt.2019.07.010

•”Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics”.•New analytic expressions for solving the extended boundary condition method for a dielectric torus in the static limit•Presentation of series relationships between toroidal and spherical harmonics in English (they have been presented before but in a Russian paper which is hard to access). New expressions for the series coefficients are also derived.•Analytic expressions for the long wavelength limit of the conventional electromagnetic T-matrix for a torus.•New method of calculating the plasmon resonance conditions.•New simple analytic approximations for the T-matrix and polarizability of a thin dielectric torus. Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived.