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

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.