  All-dielectric nanophotonics attracts ever increasing attention nowadays due
to the possibility to control and configure light scattering on high-index
semiconductor nanoparticles. It opens a room of opportunities for the designing
novel types of nanoscale elements and devices, and paves a way to advanced
technologies of light energy manipulation. One of the exciting and promising
prospects is associated with the utilizing so called toroidal moment being the
result of poloidal currents excitation, and anapole states corresponding to the
interference of dipole and toroidal electric moments. Here, we present and
investigate in details via the direct Cartesian multipole decomposition higher
order toroidal moments of both types (up to the electric octupole toroidal
moment) allowing to obtain new near- and far-field configurations. Poloidal
currents can be associated with vortex-like distributions of the displacement
currents inside nanoparticles revealing the physical meaning of the high-order
toroidal moments and the convenience of the Cartesian multipoles as an
auxiliary tool for analysis. We demonstrate high-order nonradiating anapole
states (vanishing contribution to the far-field zone) accompanied by the
excitation of intense near-fields. We believe our results to be of high
importance for both fundamental understanding of light scattering by high-index
particles, and a variety of nanophotonics applications and light governing on
nanoscale.
