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Toroidal dipole, first described by Ia. B. Zeldovich [Sov. Phys. JETP 33,1184 (1957)], is a distinct electromagnetic excitation that differs both fromthe electric and the magnetic dipoles. It has a number of intriguingproperties: static toroidal nuclear dipole is responsible for parity violationin atomic spectra; interactions between static toroidal dipole and oscillatingmagnetic dipole are claimed to violate Newton's Third Law while non-stationarycharge-current configurations involving toroidal multipoles have been predictedto produce vector potential in the absence of electromagnetic fields. Existenceof the toroidal response in metamaterials was recently demonstrated and is nowa growing field of research. However, no direct analytical link has yet beenestablished between the transmission and reflection of macroscopicelectromagnetic media and toroidal dipole excitations. To address thisessential gap in electromagnetic theory we have developed an analyticalapproach linking microscopic and macroscopic electromagnetic response of ametamaterial and showed, using a case study, the key role of the toroidaldipole in shaping the electromagnetic properties of the metamaterial.

Macroscopic electromagnetic response of metamaterials with toroidal resonances