Quasistatic limit of the electric-magnetic coupling blocks of the $T$-matrix for spheroids

TL;DR

This paper derives analytic expressions for the long-wavelength limit of all coupling blocks of the electromagnetic $T$-matrix for spheroids, extending previous electrostatic results to include magnetic interactions.

Contribution

It provides the first comprehensive analytic formulas for the electric-magnetic and magnetic-magnetic $T$-matrix blocks in the quasistatic limit for spheroidal particles.

Findings

Analytic expressions for $T^{21}$, $T^{12}$, $T^{11}$ blocks derived

Expressions expressed as finite sums using spheroidal harmonics

Includes formulas for auxiliary matrices in boundary condition method

Abstract

The $T$-matrix formally describes the solution of any electromagnetic scattering problem by a given particle in a given medium at a given wavelength. As such it is commonly used in a number of contexts, for example to predict the orientation-averaged optical properties of non-spherical particles. The $T$-matrix for electromagnetic scattering can be divided into four blocks corresponding physically to coupling between either magnetic or electric multipolar fields. Analytic expressions were recently derived for the electrostatic limit of the electric-electric $T$-matrix block $\mathbf T^{22}$, of prolate spheroids. In such an electrostatic approximation, all the other blocks were zero. We here analyse the long-wavelength limit for the other blocks ($\mathbf T^{21}$, $\mathbf T^{12}$, $\mathbf T^{11}$) corresponding to electric-magnetic, magnetic-electric, and magnetic-magnetic coupling respectively. Analytic expressions (finite sums) are obtained in the case of spheroidal particles by expressing the fields with solutions to Laplace's equation, expanding the fields in terms of spheroidal harmonics and applying the boundary conditions. Similar expressions are also presented for the auxiliary matrices in the extended boundary condition method, often used in conjunction with the $T$-matrix formalism.