  Composite and janus type metallo-dielectric nanoparticles are increasingly
considered as a means to control the spatial and temporal behavior of
electromagnetic fields in diverse applications such as coupling to quantum
emitters, achieve invisibility cloaks, and obtain quantum correlations between
qubits. We investigate the surface modes of a toroidal nano-structure and
obtain the canonical plasmon dispersion relations and resonance modes for
arbitrarily layered nanorings. Unlike particle plasmon eigenmodes in other
geometries, the amplitudes of the eigenmodes of tori exhibit a distinct forward
and backward coupling. We present the plasmon dispersion relations for several
relevant toroidal configurations in the quasistatic limit and obtain the
dominant retarded dispersion relations of a single ring for comparison, discuss
mode complementarity and hybridization, and introduce two new types of toroidal
particles in the form of janus nanorings. The resonance frequencies for the
first few dominant modes of a ring composed of plasmon supporting materials
such as gold, silver, and aluminum are provided and compared to those for a
silicon ring. A generalized Green's function is obtained for multilayer tori
allowing for calculation of the scattering response to interacting fields.
Employing the Green's function, the scalar electric potential distribution
corresponding to individual poloidal and toroidal modes in response to an
arbitrarily polarized external field and the field of electrons is obtained.
The results are applied to obtain the local density of states and decay rate of
a dipole near the center of the torus.
