Basic Equations of the Lasing Eigenvalue Problem for Graphene Strips-on-Substrate Grating, H-Polarization

We consider the H-polarized lasing modes of an infinite flat grating of graphene monolayer strips, patterned on top of a dielectric-slab substrate. The electron conductivity of graphene is modeled using the Kubo formulas. The material of the substrate is assumed to possess the gain, which compensates for the radiation and ohmic losses, so that the modes are at the threshold of lasing. The eigenvalues of this problem are the pairs of real-valued numbers: the frequency and the threshold value of the gain index, specific for each mode. To find them, we reduce this problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics. Then we perform analytical regularization of this equation, based on the inversion of its static part. This yields a Fredholm second-kind infinite matrix equation and the associated determinantal equation for the eigenvalues, convergence of which to the exact values is guaranteed with larger truncation numbers. We can see that both the frequency and the threshold gain values of the lasing modes can be controlled by the variation of the graphene parameters.