Two-dimensional materials constitute an exciting platform for nonlinear
optics with large nonlinearities that are tunable by gating. Hence,
gate-tunable harmonic generation and intensity-dependent refraction have been
observed in e.g. graphene and transition-metal dichalcogenides, whose
electronic structures are accurately modelled by the (massive) Dirac equation.
We exploit on the simplicity of this model and demonstrate here that arbitrary
nonlinear response functions follow from a simple iterative approach. The power
of this approach is illustrated by analytical expressions for harmonic
generation and intensity-dependent refraction, both computed up to ninth order
in the pump field. Moreover, the results allow for arbitrary band gaps and
gating potentials. As illustrative applications, we consider (i)
gate-dependence of third- and fifth-harmonic generation in gapped and gapless
graphene, (ii) intensity-dependent refractive index of graphene up to ninth
order, and (iii) intensity-dependence of high-harmonic generation.Comment: 6 pages, 5 figures. Supplemental material: 6 pages, 2 figure
