Analytical solution for the diffraction of an electromagnetic wave by a graphene grating

An analytical method for diffraction of a plane electromagnetic wave at a periodically modulated graphene sheet is presented. Both interface corrugation and a periodic change in the optical conductivity are considered. Explicit expressions for reflection, transmission, absorption and transformation coefficients in arbitrary diffraction orders are presented. The dispersion relation and decay rates for graphene plasmons of the grating are found. Simple analytical expressions for the value of the bandgap in the vicinity of the first Brillouin zone edge are derived. The optimal amplitude and wavelength, guaranteeing the best matching of the incident light with graphene plasmons are found for the conductivity grating. The analytical results are in a good agreement with first-principles numeric simulations.