Gouy-phase engineered space-time singularities in second harmonic generation

Space-time singularities represent points where the local structure of a wave field becomes locally undefined, and are central to phenomena ranging from fluid dynamics to cosmology. In optics, spatiotemporal optical vortices (STOVs) provide a versatile setting in which to study such singularities, but their controlled manipulation during nonlinear frequency conversion has remained elusive. Here we show that the evolution of STOV singularities in second-harmonic generation can be continuously engineered through spatial Gouy-phase control. In a 4f system, translating a thin nonlinear crystal about the Fourier plane imposes a tunable Gouy phase that governs the orientation, splitting, and trajectories of singularities in the second-harmonic field. A Hermite-Gaussian modal description identifies Gouy-phase-dependent mechanism underlying these dynamics, in agreement with spatiotemporal interferometry measurements. Beyond integer-order inputs, fractional-order inputs (l=0.5) result in a distinct regime in which Gouy-phase bias drives topological reconfiguration and directional energy redistribution. Our results establish the Gouy phase as a simple yet powerful control knob for space-time singularities, opening new opportunities for structured-light engineering in nonlinear and ultrafast photonics.