Anapoles and Flying Doughnuts

Toroidal excitations can exist both in matter, as represented by the toroidal multipoles, as well as in free-space in the form of “Flying Doughnuts” [1]. Toroidal multipoles provide significant contributions to the electromagnetic response of matter and together with the conventional electric and magnetic multipoles can lead to non-radiating configurations, termed anapoles. Indeed, anapoles can be formed by a superposition of an electric dipole and a toroidal dipole, which, owing to their identical radiation properties, allow for complete cancellation of radiated electromagnetic fields outside the source. However, although anapoles do not radiate electromagnetic fields, they do act as sources of vector potential (which cannot be eliminated by a change of gauge). On the other hand, Flying Doughnuts are few-cycle electromagnetic pulses with non-trivial spatiotemporal coupling and toroidal configuration of electromagnetic fields that propagate in free-space at the speed of light. They are exact solutions to Maxwell's equations and exhibit strong longitudinal field components along the propagation direction. The spatial and temporal dependence of the Flying Doughnut pulse cannot be separated from one another, which results in a spatially varying frequency spectrum. In particular, the Flying Doughnut pulse exhibits a frequency spectrum that varies across the wavefront with shorter wavelengths localized closer to the center of the pulse and longer wavelengths dominating the outer regions of the pulse. Importantly, this spatial variation remains invariant upon propagation of the pulse, as well as focusing and defocusing, indicating that the Flying Doughnut pulse is isodiffracting. This spatiotemporal coupling in combination with the doughnut-like arrangement of electromagnetic fields, leads to a complex topological structure in the form of spectrally broadband vortices with multiple singularities in both the electric and magnetic fields. Flying Doughnut pulses can interact with matter in unique ways, which result in non-trivial field transformations upon reflection from perfectly conducting and dielectric interfaces. Moreover, the interactions of Flying Doughnut pulses with spherical dielectric particles can lead to the excitation of toroidal resonances and non-radiating configurations (anapoles).