Multipole Radiation of Modified Dipoles

Abstract

The multipole expansion decomposes the electromagnetic response of a medium to three families of elementary sources: electric (current), magnetic (transversal current oscillations) and toroidal (radial current oscillations) multipole moments. The latter had not been treated as a separate multipole family until quite recently [1] . The reason for this neglect origins from the impossibility to distinguish electric and toroidal multipoles far-field radiation patterns. This ambiguity is still giving rise to multiple discussions on whether toroidal multipoles should be separated from the electric multipoles. In this study, we address this issue by showing that electric and toroidal dipoles radiation patterns change differently when dipoles are being geometrically modified. Multipoles intensities and radiating fields were calculated by equations presented in [2] . Electric (red) and toroidal (blue) dipoles presented in figure 1 are assumed to be made of infinitely thin wires. Their radius vectors are r el and r tor described by equations (1) and (2) correspondingly. Assuming the uniform current flow I along the wires, we define a current element as ${\\mathbf{j}}dV = Id{\\mathbf{r}}$ . Then, we define the current in multipole moments equations as $d{\\mathbf{r}} = \\frac{{dr}}{{d\\varphi }}d\\varphi $ . We started by adjusting dipole dimensions so that they form the non-radiating system. The dipoles were then modified, destroying the anapole condition. First, we bent the electric dipole in accordance with the equation (1) as shown in figure 1a . The dipole was positioned so that its center of mass ( $\\left\\{ {0,{y_{cm}},0} \\right\\}$ ) matched the origin. Second approach was to partly flatten the toroidal dipole ( equation (2) ) as presented in figure 1b .