Structured Harmonic Generation via Geometric Phase Enabled Pump Shaping

Nonlinear optics is crucial for shaping the spatial structure of shortwave light and its interactions with matter, but achieving this through simple harmonic generation with a single pump is challenging. This study demonstrates nonlinear spin-orbit conversion using spin-dependent pump shaping via geometric phase, allowing the direct creation of desired structured harmonic waves from a Gaussian pump beam. By using the liquid-crystal flat optical elements fabricated with photoalignment, we experimentally produce higher-order cylindrically vectorial modes in second harmonic fields. We examine the vectorial spatial wavefunctions, their propagation invariance, and nonlinear spin-orbit conversion. Our results provide an efficient method for full structuring nonlinear light in broader harmonic systems, with significant applications in laser micromachining and high-energy physics.