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

Abstract

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.

Electrostatic images and T-matrices for simple geometries

“…For higher n the harmonics can be generated by repeated application of the operator r∂ r . The toroidal harmonics follow a recurrence relation involving r∂ r , and this recurrence is also satisfied by coefficients c m nk , s m nk relating toroidal and spherical harmonics (see (7.48) and appendix of [25]). We can then deduce the differential Figure 7.2: Representation of the source charge configurations for the low order ring toroidal harmonics.…”

“…• Chapter 7: Matt Majic, "Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics", Journal of Quantitative Spectroscopy & Radiative Transfer, 2019 [25].…”

“…The harmonics for n = 1 may be generated by application of the operators ∂ z = ∂ /∂ z and r∂ r (see appendix of [25] for details), so we apply these to the integral expression (7.12), first r∂ r :…”

“…The integrals can be evaluated analytically (see appendix of [25]), and the matrix elements are then…”

“…and present the following linear relationships between spherical and toroidal harmonics (see appendix of [25]):…”