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

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.