  Numerically solving the semiconductor Bloch equations within a
phenomenological relaxation time approximation, we extract both the linear and
nonlinear optical conductivities of doped graphene and gapped graphene under
excitation by a laser pulse. We discuss in detail the dependence of second
harmonic generation, third harmonic generation, and the Kerr effects on the
doping level, the gap, and the electric field amplitude. The numerical results
for weak electric fields agree with those calculated from available analytic
perturbation formulas. For strong electric fields when saturation effects are
important, all the effective third order nonlinear response coefficients show a
strong field dependence.
