  In this paper, we propose a novel numerical method for modeling
nanostructures containing dispersive and nonlinear two-dimensional (2D)
materials, by incorporating a nonlinear generalized source (GS) into the
finite-difference time-domain (FDTD) method. Starting from the expressions of
nonlinear currents characterizing nonlinear processes in 2D materials, such as
second- and third-harmonic generation, we prove that the nonlinear response of
such nanostructures can be rigorously determined using two linear simulations.
In the first simulation, one computes the linear response of the system upon
its excitation by a pulsed incoming wave, whereas in the second one the system
is excited by a nonlinear generalized source, which is determined by the linear
near-field calculated in the first linear simulation. This new method is
particularly suitable for the analysis of dispersive and nonlinear 2D
materials, such as graphene and transition-metal dichalcogenides, chiefly
because, unlike the case of most alternative approaches, it does not require
the thickness of the 2D material. In order to investigate the accuracy of the
proposed GS-FDTD method and illustrate its versatility, the linear and
nonlinear response of graphene gratings have been calculated and compared to
results obtained using alternative methods. Importantly, the proposed GS-FDTD
can be extended to 3D bulk nonlinearities, rendering it a powerful tool for the
design and analysis of more complicated nanodevices.
