Toroidal eigenmodes in all-dielectric metamolecules

We present a thorough investigation of the electromagnetic resonant modes supported by systems of polaritonic rods placed at the vertices of canonical polygons. The study is conducted with rigorous finite-element eigenvalue simulations. To provide physical insight, the simulations are complemented with coupled mode theory (the analog of LCAO in molecular and solid state physics) and a lumped wire model capturing the coupling-caused reorganizations of the currents in each rod. The systems of rods, which form all-dielectric cyclic metamolecules, are found to support the unconventional toroidal dipole mode, consisting of the magnetic dipole mode in each rod. Besides the toroidal modes, the spectrally adjacent collective modes are identified. The evolution of all resonant frequencies with rod separation is examined. They are found to oscillate about the single-rod magnetic dipole resonance, a feature attributed to the leaky nature of the constituent modes. Importantly, we observe that ensembles of an odd number of rods produce larger frequency separation between the toroidal mode and its neighbor than the ones with an even number of rods. This increased spectral isolation, along with the low quality factor exhibited by the toroidal mode, favors the coupling of the commonly silent toroidal dipole to the outside world, rendering the proposed structure a prime candidate for controlling the observation of toroidal excitations and their interaction with the usually present electric dipole.