  Third harmonic generation of terahertz radiation is expected to occur in
monolayer graphene due to the nonlinear relationship between the crystal
momentum and the current density. In this work, we calculate the terahertz
nonlinear response of graphene inside a parallel-plate waveguide including pump
depletion, self-phase, and cross-phase modulation. To overcome the phase
mismatching between the pump field and third-harmonic field at high input
fields due to self-phase and cross-phase modulation, we design a waveguide with
two dielectric layers with different indices of refraction. We find that, by
tuning the relative thicknesses of the two layers, we are able to improve phase
matching, and thereby increase the power efficiency of the system by more than
a factor of two at high powers. With this approach, we find that dispite the
loss in this system, for an incident frequency of $2$ THz, we are able to
achieve power efficiencies of $75 \%$ for graphene with low Fermi energies of
$20$ meV and up to $35\%$ when the Fermi energy is $100$ meV.
