Threshold Analysis for the E-Polarized Modes of Microsize Graphene Strip Grating Laser

We consider the threshold conditions for the E-polarized lasing modes of infinite flat grating of graphene strips lying on dielectric-slab substrate. The conductivity of graphene is modeled 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 use of Discrete Inverse Fourier Transform. Then, the problem reduces to a Fredholm 2nd-kind matrix equation for the unknown Floquet harmonic amplitudes. Therefore, the convergence of the resulting code is guaranteed by the Fredholm theorems. Since plasmon modes are absent in the case of E-polarization, we study the high-threshold modes of the substrate slightly perturbed by the presence of grating and ultra-low-threshold lattice modes of the entire grating as a periodic open cavity. Our results show that the lattice modes are mediated by the substrate guided waves.