| Summary: | 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 intraband 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 that 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.
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