On the dynamic toroidal multipoles from localized electric current distributions

TL;DR

This paper critically examines the concept of dynamic toroidal multipoles, demonstrating that they lack independent physical significance and are merely higher-order terms in electric multipole expansions.

Contribution

The authors analytically show that toroidal multipoles do not have independent physical meaning and are just mathematical components of electric multipole expansions.

Findings

Toroidal multipoles do not have independent physical meaning.

Non-radiative components in split electric and toroidal parts cancel each other.

Toroidal multipoles are higher order terms in electric multipole expansion.

Abstract

We analyze the dynamic toroidal multipoles and prove that they do not have an independent physical meaning with respect to their interaction with electromagnetic waves. We analytically show how the split into electric and toroidal parts causes the appearance of non-radiative components in each of the two parts. These non-radiative components, which cancel each other when both parts are summed, preclude the separate determination of each part by means of measurements of the radiation from the source or of its coupling to external electromagnetic waves. In other words, there is no toroidal radiation or independent toroidal electromagnetic coupling. The formal meaning of the toroidal multipoles is clear in our derivations. They are the higher order terms of an expansion of the multipolar coefficients of electric parity with respect to the electromagnetic size of the source.