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

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 nonspherical 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 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 (T 21 , T 12 , 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.

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

“…The harmonics for n = 1 may be generated by application of the operators ∂ z = ∂/∂z and r∂ r (see appendix B 1 for details), so we apply these to the integral expression (14), first r∂ r :…”

“…Spherical harmonics are far easier to compute, better known and more applicable than toroidal harmonics, so it is of interest to re-express this T -matrix on a basis of spherical harmonics -in particular, this allows computation of the long wavelength limit of the Tmatrix for electromagnetic or acoustic scattering, which is usually expressed on a basis of spherical wavefunctions. For the spheroid, the low frequency scattering problem has been re-expressed with spherical harmonics and the electromagnetic T -matrix has been obtained to third order in size parameter [13][14][15] by applying the series relationships between spherical and spheroidal harmonics. But the series relationships between spherical and toroidal harmonics are more complex and virtually unknown in the literature, exept for the low degrees: toroidal harmonics of degree zero, corresponding to the potential of rings of sinusoidal charge distributions, are known as series of spherical harmonics, and spherical harmonics corresponding to point charges and dipoles are known as series of toroidal harmonics.…”

Electrostatic images and T-matrices for simple geometries

“…• Chapter 5: Matt Majic, Eric C. Le Ru, "Quasistatic limit of the electric-magnetic coupling blocks of the T-matrix for spheroids", Journal of Quantitative Spectroscopy & Radiative Transfer, 2018 [24].…”

“…Spherical harmonics are far easier to compute, better known and more applicable than toroidal harmonics, so it is of interest to obtain the T-matrix on a basis of spherical harmonics -in particular, this allows computation of the long wavelength limit of the T-matrix for electromagnetic or acoustic scattering, which is usually expressed on a basis of spherical wavefunctions. For the spheroid, the low frequency scattering problem has been re-expressed with spherical harmonics and the electromagnetic T-matrix has been obtained to third order in size parameter [80,24,27] by applying the series relationships between spherical and spheroidal harmonics. But the series relationships between spherical and toroidal harmonics are more complex and virtually unknown in the literature, except for the low degrees.…”