Retardation-induced phase singularities in coupled plasmonic oscillators

We show that two coupled, stacked plasmonic scatterers exhibit surprisingly complex excitation spectra as soon as retardation plays a role in the coupling. At a certain stacking distance and a certain frequency, the amplitude and phase exhibit sharp features, and the phase difference between the scatterers shows a singularity, which constitutes a vortex. Above this singularity, the antisymmetric oscillation that arises from plasmon hybridization ceases to exist and is replaced by a second, symmetric oscillation. We examine the distance-dependent behavior of these phenomena by an analytical coupled-dipole model as well as by numerical simulations of stacked split-ring resonators and show that the singularity can be explained as an antiresonance. Furthermore, we present a simple necessary and sufficient condition under which the oscillations have a phase difference of exactly zero or in case of arbitrary retardation and therefore exhibit the intrinsic symmetry of the system. We show that this condition implies that the spectral positions of (anti) symmetric oscillations do not coincide with the spectral positions of the (anti) symmetric mode. The vortices that are present at the phase singularities might be used to generate light with angular optical momentum in plasmonic metasurfaces.