Which plasmonic nano-antenna is best for free-space out-coupling of single photon emission from a hyperbolic metamaterial resonator?

Recently, hyperbolic metamaterials (HMMs) have shown large emission-rate/Purcell enhancement for emitters coupled to them. However, due to the large momentum ($ k $) mismatch between the high-k hyperbolic modes of HMMs and free-space modes, the far-field out-coupling of the emission is limited and requires an antenna. In this work, we present an in-depth theoretical study of the performance of some commonly known plasmonic antennas-cylindrical, cuboid, crossed, and bow-tie-when coupled to a HMM. Of all these antennas, the cylindrical antenna was observed to be the best for out-coupling to the high-$ k $ HMM modes with the Purcell factor and collection efficiencies reaching 1000 and 0.5, respectively. The hyperbolic HMM modes are observed to be efficiently coupled to the resonance modes of the cylindrical antenna, with the antenna modes getting effectively out-coupled into free space. These values are expected to result in two to three orders of fluorescence enhancement from a solid-state single photon source. For other antennas, the greater momentum mismatch between the hyperbolic HMM modes and the antenna modes resulted in relatively much weaker free-space out-coupling.