Difference Frequency Generation of Surface Plasmon-Polaritons in Landau Quantized Graphene

We develop a rigorous quantum-mechanical theory of the nonlinear optical process of difference frequency generation of surface plasmon-polaritons in Landau-quantized graphene. Although forbidden in the electric-dipole approximation, the second-order susceptibility is surprisingly high, equivalent to the bulk magnitude above ${10}^{\ensuremath{-}3}$ m/V. We consider the graphene monolayer as a nonlinear optical component of a monolithic photonic chip with integrated pump fields. The nonlinear power conversion efficiency of the order of tens of $\ensuremath{\mu}\mathrm{W}/{\mathrm{W}}^{2}$ is predicted from structures of $10--100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{m}$ size. We investigate a variety of waveguide configurations to identify the optimal geometry for maximum efficiency.