  We discuss the scattering of graphene surface plasmon-polaritons (SPPs) at an
interface between two semi-infinite graphene sheets with different doping
levels and/or different underlying dielectric substrates. We take into account
retardation effects and the emission of free radiation in the scattering
process. We derive approximate analytic expressions for the reflection and the
transmission coefficients of the SPPs as well as the same quantities for the
emitted free radiation. We show that the scattering problem can be recast as a
Fredholm equation of the second kind. Such equation can then be solved by a
series expansion, with the first term of the series correspond to our
approximated analytical solution for the reflection and transmission
amplitudes. We have found that almost no free radiation is emitted in the
scattering process and that under typical experimental conditions the
back-scattered SPP transports very little energy. This work provides a
theoretical description of graphene plasmon scattering at an interface between
distinct Fermi levels which could be relevant for the realization of plasmonic
circuitry elements such as plasmonic lenses or reflectors, and for controlling
plasmon propagation by modulating the potential landscape of graphene.
