Generalized nonlinear Snell's law at χ (2) modulated nonlinear metasurfaces: anomalous nonlinear refraction and reflection.

The generalized nonlinear Snell's law at the χ(2) modulation surface is deduced from the universal model of χ(2) processes in nonlinear photonic crystals (NPhCs). Based on the generalized nonlinear Snell's law, the anomalous refraction and reflection geometries are predicted and observed at crystal's inner boundary on nonlinear metasurfaces formed by one-dimensional NPhCs. The emitted second harmonic is observed, which obeys the law and appears to turn into multiple orders compared to that in a bulk crystal. Furthermore, the analysis shows a potential way to achieve nonlinear over-reflection by using submicron periodic χ(2) modulation in metasurfaces.