Degenerate four-wave mixing in nonlinear resonators comprising two-dimensional materials: A coupled-mode theory approach

Abstract

Two-dimensional (2D) or sheet materials have been recently recognized as fascinating materials for nonlinear photonics. Here, we develop a rigorous mathematical framework based on perturbation theory and temporal coupled-mode theory capable of analyzing third-order, ${\ensuremath{\chi}}^{(3)}$, multichannel nonlinear processes in resonant systems comprising 2D materials. The framework is applied to model degenerate four-wave mixing in a guided-wave graphene plasmon-polariton resonant structure, consisting of a standing-wave resonator directly coupled to access waveguides. The results obtained with the proposed framework are compared with full-wave finite-element simulations revealing excellent agreement. Aside from being accurate and efficient, our framework allows for selectively incorporating different nonlinear phenomena, identifying their unique impact on the nonlinear response and providing valuable physical insight. We are, thus, able to specify the optimal operating point leading to maximum conversion efficiency for the generated wave in a multiparameter space. In addition, we identify unstable operating regimes exhibiting optical bistability or limit cycles, thoroughly characterizing the component performance. Our framework enables the study of diverse multichannel phenomena (frequency generation, frequency mixing, and parametric amplification) in the thriving field of 2D material photonics, thus allowing for assessing the potential of these exciting materials for practical nonlinear applications.