Effective-medium description of a metasurface composed of a periodic array of nanoantennas coupled to a metallic film

We compute the reflectance properties of a metasurface that consists of a doubly periodic array of patch nanoantennas strongly coupled to a metallic film. Each plasmonic patch antenna can be accurately modeled as a polarizable, radiating, magnetic dipole. By accounting for interactions amongst the dipoles, an equivalent surface polarizability can be obtained, from which the effective surface impedance, reflectivity, and other homogenized quantities of interest can be obtained. When the metasurface is extremely close to the metal film, the interaction between constituent dipoles is dominated by surface plasmon mediation. We calculate analytically the dipole interaction constant by explicitly evaluating the infinite sum of fields from all the dipoles in the lattice. While a single film-coupled nanoparticle exhibits anomalous loss due to coupling to surface plasmons, we find that for the lattice of dipoles, the radiation reaction force due to the coupling to the surface plasmon modes is exactly canceled by the interaction constant; the lattice thereby conserves energy in the limit of zero Ohmic loss. When Ohmic losses are present, absorption to surface plasmons reemerges and can be compared with the losses to radiation and Ohmic absorption in the metasurface.