Vectorial engineering of second-harmonic generation in silicon-based waveguides integrated with 2D materials

Integrating 2D materials onto on-chip photonic devices holds significant potential for nonlinear frequency conversion across various applications. The lack of inversion symmetry in monolayers of transition metal dichalcogenides (TMDs), e.g., MoS$_2$, is particularly attractive for enabling nonlinear phenomena based on $\chi^{(2)}$ in silicon-based photonic devices incorporated with these materials, which has been previously demonstrated. However, reports have largely overlooked the need to consider, in the nonlinear modal interaction, both the tensorial nature of the TMD's second-order susceptibility and the full vectorial nature of the electromagnetic fields. In this work, we investigate second-harmonic generation (SHG) in silicon nitride (SiN) waveguides integrated with a monolayer of MoS$_2$. We experimentally observed an enhancement in SHG in MoS$_2$-loaded waveguides compared to those without the monolayer. Notably, this enhancement occurred even when the dominant electric field component of the pump and/or signal mode was orthogonal to the TMD plane, highlighting co- and cross-polarized SHG interactions. This phenomenon cannot be predicted by the traditionally used scalar models. By taking into account the full vectorial and tensorial natures of the problem, we then designed a waveguide in which a TE pump mode is phase-matched to a TM second-harmonic mode. With a single 110-$\mu$m-long MoS$_2$ flake, we experimentally achieved $14\times$ frequency conversion enhancement relative to the non-phase-matched case and $220\times$ enhancement relative to free-space (normal-incidence) excitation. Our work, thus, introduces fundamental guidelines for the design and optimization of nonlinear silicon-photonic devices based on 2D-material hybrid integration. These guidelines are material independent and may lead to significant further conversion efficiency enhancement.