  The existence of large nonlinear optical coefficients is one of the
preconditions for using nonlinear optical materials in nonlinear optical
devices. For a crystal, such large coefficients can be achieved by matching
photon energies with resonant energies between different bands, and so the
details of the crystal band structure play an important role. Here we
demonstrate that large third-order nonlinearities can also be generally
obtained by a different strategy: As any of the incident frequencies or the sum
of any two or three frequencies approaches zero, the doped or excited
populations of electronic states lead to divergent contributions in the induced
current density. We refer to these as intraband divergences, by analogy with
the behavior of Drude conductivity in linear response. Physically, such
resonant processes can be associated with a combination of inraband and
interband optical transitions. Current-induced second order nonlinearity,
coherent current injection, and jerk currents are all related to such
divergences, and we find similar divergences in degenerate four wave mixing and
cross-phase modulation under certain conditions. These divergences are limited
by intraband relaxation parameters, and lead to a large optical response from a
high quality sample; we find they are very robust with respect to variations in
the details of the band structure. To clearly track all of these effects, we
analyze gapped graphene, describing the electrons as massive Dirac fermions;
under the relaxation time approximation, we derive analytic expressions for the
third order conductivities, and identify the divergences that arise in
describing the associated nonlinear phenomena.
