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Conditions for a pure toroidal dipole source

Recently, the physical significance of dynamic toroidal multipoles in the context of electrodynamics has been put under discussion. Indeed, the latter can be shown to arise simply from a Taylor series of the exact source (Cartesian) multipole moments. The split into elementary and toroidal parts was demonstrated to lead to an unphysical result were forbidden components of the momentum transform of the current could radiate into free space. In this contribution, we elaborate the conditions that a current distribution must necessarily satisfy to be considered a ‘pure’ toroidal dipole source. We demonstrate for the first time that symmetry prevents such current distribution to radiate as an elementary electric dipole moment, without leading to an unphysical result. Thus, while both elementary electric dipole and toroidal dipoles are indistinguishable outside the source, they display topologically distinct characteristics within the smallest spherical surface enclosing the source itself and have different physical origin. Based on our results, a pure ‘toroidal’ source can be designed. We believe the outcome of our investigations will help clarify further the formal meaning of the toroidal multipoles.


INTRODUCTION
In modern physics, multipole expansions are commonly used tools to understand a broad range of phenomena in optics, nuclear physics and condensed matter [1,2]. It is nowadays well-known that the so-called 'Cartesian' or multipole expansions of the source, which arise in a Taylor series of the vector potential [3,4], require the introduction of dynamic toroidal moments [4]. The most well-known constituent of the toroidal family is the toroidal dipole, first identified [5] and observed as a static current distribution. It was later shown to naturally give rise to a different form of long-range order in ferroics violating both space and time inversion symmetries [6], and predicted to have a strong impact in the electronic properties of several molecular structures [2]. Nowadays, dynamic toroidal moments in electrodynamics are a subject of renewed scientific and technological interest in the growing field of all-dielectric nanophotonics [7][8][9][10][11][12] due to their role in the formation of the nonradiating sources known as anapoles [13].
Their physical relevance, as well as their formal meaning in electrodynamics, is still however a matter of on-going discussion [14,15]. Corbaton et.al. rigorously showed in [15] how the toroidal moments appear as high order terms of a Taylor series of the exact source multipoles. Taking as a case example the electric and electric toroidal dipole, the split into elementary and toroidal terms leads to an unphysical result if considered separately, i.e. the toroidal dipole cannot, in fact, be regarded differently from the elementary electric dipole moment.
In this work, we show that such a restriction can in fact be circumvented under a few special conditions, which effectively allow a 'pure' naturally broadband electric toroidal dipole to arise in the absence of the elementary electric dipole. With our analysis, we demonstrate that, while indistinguishable for an observer outside the source, the physical

GENERAL THEORETICAL CONSIDERATIONS
We start with the dipolar vector of electric parity in the exact form [15]: The Bessel functions within the integral act as 'filters' for the momentum components of the current, such that only those with total momentum / c Z p contribute to radiation. A Taylor series of Eq.(1) leads to the well-known expressions of the elementary dipole 1 Z p and toroidal dipole 1 Z t moments: In what follows, it will be convenient to recall the expression for the momentum transform of a localized current distribution: comparing Eqs.  [15] concluded that, since each term in the expansion contains nonradiative terms, the elementary dipole and the toroidal dipole moment couple identically to electromagnetic radiation, and, as such, there is no need for separating their contributions outside the source.
Here, we focus however on the way that the elementary (electric) and toroidal terms are excited within the source itself. Within the smallest spherical volume enclosing the source, there is no restriction as to which components ( ) Z J p can contribute to the electromagnetic fields, and therefore the previous arguments do not apply.

RESULTS AND DISCUSSION
First, we focus on a simple way to obtain a pure 'toroidal' source, i.e. with 1 validating this condition would have Z J p 0 0 , therefore allowing the toroidal part to radiate by itself. This idea was discussed in Ref. [15] and discarded as trivial, since in the far field pure toroidal radiation cannot be distinguished from that of the elementary dipole. However, a current distribution validating such condition has a very well-defined topology, very different from that of an elementary electric dipole. Such a current must, in fact, not only be divergenceless