  Topology-driven nonlinear light-matter effects open up new paradigms for both
topological photonics and nonlinear optics. Here, we propose to achieve
high-efficiency second-harmonic generation in a second-order photonic
topological insulator. Such system hosts highly localized topological corner
states with large quality factors for both fundamental and second harmonic
waves, which could be matched perfectly in frequency by simply tuning the
structural parameters. Through the nonlinear interaction of the doubly resonant
topological corner states, unprecedented frequency conversion efficiency is
predicted. In addition, the robustness of the nonlinear process against defects
is also demonstrated. Our work opens up new avenues toward topologically
protected nonlinear frequency conversion using high-order photonic topological
modes.
