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We discuss the scattering of graphene surface plasmon-polaritons (SPPs) at aninterface between two semi-infinite graphene sheets with different dopinglevels and/or different underlying dielectric substrates. We take into accountretardation effects and the emission of free radiation in the scatteringprocess. We derive approximate analytic expressions for the reflection and thetransmission coefficients of the SPPs as well as the same quantities for theemitted free radiation. We show that the scattering problem can be recast as aFredholm equation of the second kind. Such equation can then be solved by aseries expansion, with the first term of the series correspond to ourapproximated analytical solution for the reflection and transmissionamplitudes. We have found that almost no free radiation is emitted in thescattering process and that under typical experimental conditions theback-scattered SPP transports very little energy. This work provides atheoretical description of graphene plasmon scattering at an interface betweendistinct Fermi levels which could be relevant for the realization of plasmoniccircuitry elements such as plasmonic lenses or reflectors, and for controllingplasmon propagation by modulating the potential landscape of graphene.

Scattering of graphene plasmons at abrupt interfaces: an analytic and numeric study