Visualizing and Optimizing Phase Matching in Nonlinear Guided-mode Resonators with the Green's Function Integral Method

Abstract: Efficient nonlinear frequency conversion in nanophotonics requires not only strong fundamental field but also precise phase matching among distributed nonlinear sources. Here, we develop the two-dimensional Green's function integral method (GFIM), which enables direct visualization and optimization of phase matching in nonlinear guided-mode resonators. Using GFIM phase analysis, we generalize the phase-matching factor (PMF) as a rigorous metric of spatial phase coherence in harmonic generation, revealing severe phase mismatch in conventional guide mode resonators. Guided by phase-matching profiles, we propose design strategies to improve the phase coherence, particularly by introducing a high-index waveguide layer that confines the fundamental field in the nonlinear material to regions where the harmonic Green's function varies slowly. This configuration achieves a PMF exceeding 0.91, approaching the ideal value of unity, and yields a record SHG efficiency of 26.7% at a low pump intensity of 2 kW/$\mathrm{cm}^2$. These results establish the GFIM-based phase-matching visualization as an effective strategy for compact, high-performance nonlinear photonic devices.