Scattering of surface plasmon polaritons in a graphene multilayer photonic crystal with inhomogeneous doping

The propagation of a surface plasmon polariton along a stack of doped graphene sheets is considered. This auxiliary problem is used to discuss: (i) the scattering of such a mode at an interface between the stack and the vacuum, and (ii) the scattering at an interface where there is a sudden change of the electronic doping. The formalism is then extended to the barrier problem. In this system rich physics is found for the plasmonic mode, showing total reflection, total transmission, Fabry-P\'erot oscillations, and coupling to photonic modes.