Origin of the anapole condition as revealed by a simple expansion beyond the toroidal multipole

Shi-Qiang Li and Kenneth B. Crozier

Phys. Rev. B 97, 245423 – Published 25 June 2018

Abstract

Toroidal multipoles are a topic of increasing interest in the nanophotonics and metamaterials communities. In this paper, we separate out the toroidal multipole components of multipole expansions in polar coordinates (two- and three-dimensional) by expanding the Bessel or spherical Bessel functions. We discuss the formation of the lowest order of magnetic anapoles from the interaction between the magnetic toroidal dipole and the magnetic dipole. Our method also reveals that there are higher order current configurations other than the electric toroidal multipole that have the same radiation characteristics as the pure electric dipole. Furthermore, we find that the anapole condition requires that there is a perfect cancellation of all higher order current configurations.

Received 13 November 2017
Revised 27 May 2018

DOI:https://doi.org/10.1103/PhysRevB.97.245423

Physics Subject Headings (PhySH)

Metamaterials Scattering theory

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

Shi-Qiang Li^1,* and Kenneth B. Crozier^1,2,†

¹Department of Electrical and Electronic Engineering, University of Melbourne, Victoria 3010, Australia
²School of Physics, University of Melbourne, Victoria 3010, Australia

^*lishiqia@gmail.com
^†kenneth.crozier@unimelb.edu.au

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Vol. 97, Iss. 24 — 15 June 2018

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Images

Figure 1
Decomposition of the scattering cross section of a nanotube in the 2D multipole expansion. The incident plane wave has a wave vector along the $\hat{x}$ direction. TE wave case is shown as panel (a) and TM wave case is shown as panel (b). Inset of (a) shows a schematic of a cross section of the nanotube, whose inner r and outer R radii are 50 nm and 150 nm, respectively. The host medium is air with $n = 1$ and the tube has a refractive index of 3.5. Insets in panels (a) and (b) are mode profiles of various orders (amplitude of the electric field), with details explained in the text. The expanded toroidal terms and Cartesian dipole terms are plotted in (c) and (d) for TE and TM cases, respectively. These are partial scattering cross sections calculated based on the separation shown in Eq. (5).
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Figure 2
(a) Instantaneous electric ( $E_{z}$ , color map) and magnetic ( $H_{x}$ and $H_{y}$ , quiver plot) fields around nanotube for TE mode. It can be seen that two loops are formed in the magnetic field pattern. Fields are calculated for illumination wavelength, denoted by $T_{a}$ , in Fig. 1. Direction of toroidal moment is indicated in this panel by vector also denoted by $T_{a}$ . (b) Instantaneous magnetic ( $H_{z}$ , color map) and electric ( $E_{x}$ and $E_{y}$ , quiver plot) fields around nanotube for TM mode. Calculation is performed for illumination wavelength, denoted by $T_{b}$ , in Fig. 1. Vector also denoted by $T_{b}$ in this panel gives direction of toroidal moment.
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Figure 3
(a) Spherical multipole coefficients for the dipole moments of order 1 and degree 1. The dipole moments with (solid red line) and without (solid blue line) toroidal moment separation are calculated with Eq. (6a). The toroidal moment [found using Eq. (6b)] is plotted as the solid yellow curve. The curve with red diamond markers is the coefficient converted from Eq. (7) and that with yellow markers is the coefficient converted from Eq. (8). The insets are the field patterns of the electric field strength cut at the center of the nanosphere at two representative wavelengths (the pure toroidal condition and the anapole condition). The solid purple curve plots the next higher order term $[{(k r)}^{5}]$ . (b) The spherical multipole coefficients for the quadrupole moments of order 2 and degree 1. The quadrupole moments with (solid red line) and without (solid blue line) toroidal quadrupole moment separation are calculated with Eq. (6a). The toroidal quadrupole moment [found using Eq. (6b)] is plotted as the solid yellow curve. The solid purple curve plots the next higher order term $[{(k r)}^{5}]$ . The insets are the field patterns of the electric field strength cut at the center of the nanosphere at the toroidal condition and the anapole condition.
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