Identify the fundamental-harmonic modes responsible for modal phase matching

Identify the exact fundamental-harmonic spatial modes that interact to produce the observed second-harmonic TM21 mode through modal phase matching at a 249 μm poling period in the silicon-nitride waveguide.

Background

The paper reports an elevated second-harmonic baseline without an applied programmable illumination pattern, attributing this behavior to modal phase matching caused by near-zero wave-vector mismatch between particular spatial-mode combinations. In the reported measurement, the generated second harmonic was verified to occupy the TM21 mode, but the specific fundamental-harmonic modes participating in the interaction were not resolved. Determining these modes would clarify the underlying multimode nonlinear process and help guide dispersion engineering for modal phase matching.