▎ 摘　　要

We perform a perturbative calculation of the third order optical conductivities of doped graphene, using approximations valid around the Dirac points and neglecting effects due to scattering and electron-electron interactions. In this limit analytic formulas can be constructed for the conductivities. We discuss in detail the results for third harmonic generation, the Kerr effect and two-photon carrier injection, parametric frequency conversion, and two-color coherent current injection. We find a complicated dependence on the chemical potential and photon energies. The linear dispersion causes resonances over a wide range of photon energies, and it is possible to obtain large optical nonlinearities by tuning the chemical potential.