Lorentz resonance in the homogenization of plasmonic crystals

We explain the Lorentz resonances in plasmonic crystals that consist of two-dimensional nano-dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a non-magnetic bulk dielectric. We identify local geometric resonances on the length scale of the small-scale period. From a materials perspective, the graphene surface exhibits a dispersive surface conductance captured by the Drude model. Together these phenomena conspire to generate Lorentz resonances at frequencies controlled by the surface geometry and the surface conductance. The Lorentz resonances found in the frequency response of the effective dielectric tensor of the bulk metamaterial are shown to be given by an explicit formula, in which material properties and geometric resonances are decoupled. This formula is rigorous and obtained directly from corrector fields describing local electrostatic fields inside the heterogeneous structure. Our analytical findings can serve as an efficient computational tool to describe the general frequency dependence of periodic optical devices. As a concrete example, we investigate two prototypical geometries composed of nanotubes and nanoribbons.