Directivity and Efficiency of the THz and FIR Smith-Purcell Radiation Phased Array Antenna Made of Graphene Nanotubes

We consider radiation characteristics of a phased array meta-antenna composed of finite number of graphene nanotubes with low-permittivity dielectric cores, driven by a modulated beam of charged particles. Monolayer graphene's properties are accounted for using the quantum Kubo formalism for its conductivity and resistive-sheet boundary conditions. By inverting analytically the single-tube part of the full problem, we cast it into a well-conditioned system of linear equations. This semi-analytical technique possesses guaranteed convergence and has high computational efficiency. Numerical results display the resonances on the plasmon modes of graphene tubes and the lattice modes of the grating near the Rayleigh Anomaly frequencies of similar infinite grating. We clarify the role of the number of nanotubes and their chemical potential. We show that the radiation efficiency is generally high, close to 100%, at all frequencies except the band around the frequency of the so-called invisibility effect, where the scattering has a deep minimum. The directivity is proportional to the number of grating elements and reaches maxima at the red side of the Rayleigh Anomalies.