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.
Highlights
After the introduction of toroidal multipoles in the static case [1], the dynamic toroidal multipoles were presented as a new independent multipole family that had been previously ignored [2, 3]
We find that the two parts cannot be separately determined by measuring the electromagnetic fields produced by the source outside the source region
Only the two transverse degrees of freedom are needed to describe the electromagnetic fields produced by the sources outside the source region, where the field produced by the longitudinal degrees of freedom is identically zero
Summary
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 analyze the original split [2, 3] in the general multipolar case, and establish that both electric and toroidal parts contain non-radiative components, which appear due to the separation of terms of different order in the size of the source. These non-radiative components, which cancel each other when both parts are summed, preclude the separate determination of each part by measuring the radiation from the source outside the source region, or by measuring the coupling between the source and externally incident electromagnetic waves. We use the obtained insights to clarify some statements that are often found in the literature
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