  Remarkable optical and electrical properties of two-dimensional (2D)
materials, such as graphene and transition-metal dichalcogenide (TMDC)
monolayers, offer vast technological potential for novel and improved
optoelectronic nanodevices, many of which relying on nonlinear optical effects
in these 2D materials. This article introduces a highly effective numerical
method for efficient and accurate description of linear and nonlinear optical
effects in nanostructured 2D materials embedded in periodic photonic structures
containing regular three-dimensional (3D) optical materials, such as
diffraction gratings and periodic metamaterials. The proposed method builds
upon the rigorous coupled-wave analysis and incorporates the nonlinear optical
response of 2D materials by means of modified electromagnetic boundary
conditions. This allows one to reduce the mathematical framework of the
numerical method to an inhomogeneous scattering matrix formalism, which makes
it more accurate and efficient than previously used approaches. An overview of
linear and nonlinear optical properties of graphene and TMDC monolayers is
given and the various features of the corresponding optical spectra are
explored numerically and discussed. To illustrate the versatility of our
numerical method, we use it to investigate the linear and nonlinear
multiresonant optical response of 2D-3D heteromaterials for enhanced and
tunable second- and third-harmonic generation. In particular, by employing a
structured 2D material optically coupled to a patterned slab waveguide, we
study the interplay between geometric resonances associated to guiding modes of
periodically patterned slab waveguides and plasmon or exciton resonances of 2D
materials.
