Microsize graphene strip grating on gain substrate laser: comparison of the H- and E-polarized mode threshold conditions

We investigate the H- and E-polarized natural modes of infinite flat grating of graphene strips lying on dielectric-slab substrate. The conductivity of graphene is modelled using the quantum-theory 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 each field problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics that we analytically regularize based on the inversion of its static part. 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 and that those of the H-polarized plasmon modes can be controlled by graphene's chemical potential within the factor of three.