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Third harmonic generation of terahertz radiation is expected to occur inmonolayer graphene due to the nonlinear relationship between the crystalmomentum and the current density. In this work, we calculate the terahertznonlinear response of graphene inside a parallel-plate waveguide including pumpdepletion, self-phase, and cross-phase modulation. To overcome the phasemismatching between the pump field and third-harmonic field at high inputfields due to self-phase and cross-phase modulation, we design a waveguide withtwo dielectric layers with different indices of refraction. We find that, bytuning the relative thicknesses of the two layers, we are able to improve phasematching, and thereby increase the power efficiency of the system by more thana factor of two at high powers. With this approach, we find that dispite theloss in this system, for an incident frequency of $2$ THz, we are able toachieve power efficiencies of $75 \%$ for graphene with low Fermi energies of$20$ meV and up to $35\%$ when the Fermi energy is $100$ meV.

Optimized nonlinear terahertz response of graphene in a parallel-plate waveguide