Structured Harmonic Generation via Geometric Phase Enabled Pump Shaping

Abstract: 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.