Emission Frequencies and Thresholds for Microsize Graphene Strip Grating Laser on Gain Substrate

We investigate the H-polarized lasing modes of infinite flat grating of graphene strips lying on dielectric-slab substrate. The conductivity of graphene is modelled using the Kubo formulas, while the substrate material is assumed to have gain that offsets the radiation and ohmic losses and allows the modes to reach the lasing threshold. We look for the frequency and the gain index threshold value, specific to each mode, as eigenvalues. To find them, we reduce the field problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics, which we analytically regularize based on the inversion of its static component. This results in a Fredholm second-kind infinite determinantal equation for eigenvalues. Increasing the matrix truncation number guarantees the convergence of these quantities to their exact values. Our findings reveal that the lowest thresholds are associated with the lattice modes (LMs) and that those of the plasmon modes can be controlled by graphene’s chemical potential.