Nonlinear graphene plasmonics: Amplitude equation for surface plasmons

Using perturbation expansion of Maxwell equations, an amplitude equation is derived for nonlinear transverse magnetic (TM) and transverse electric (TE) surface plasmon waves supported by graphene. The equation describes the interplay between in-plane beam diffraction and nonlinearity due to light intensity induced corrections to graphene conductivity and susceptibility of dielectrics. For strongly localized TM plasmons, graphene is found to bring the superior contribution to the overall nonlinearity. In contrast, nonlinear response of the substrate and cladding dielectrics can become dominant for weakly localized TE plasmons.