Generalized Tricomi–Gaussian Beams as a Structured Light Framework Enabled by Liquid‐Crystal Geometric‐Phase Polymer

ABSTRACT Tailoring the high‐dimensional degrees of freedom of structured light is central to modern photonics. However, existing descriptions often treat beam morphology, propagation dynamics, and topology separately, limiting a unified view of structured‐light evolution. Here, we introduce Tricomi–Gaussian beams (TGBs) as a generalized angular‐spectrum framework that systematically connects symmetric, asymmetric, and off‐axis structured beams within a common parameter space. By engineering the angular‐spectrum distribution, TGBs generalize a broad class of structured fields, reducing to conventional modes such as Bessel‐Gaussian beams under specific parameter limits while enabling symmetry‐broken structures with complex phase. Leveraging liquid‐crystal (LC) geometric‐phase optics, we experimentally realize the TGB family through continuous, high‐fidelity phase encoding in planar LC polymer films, while introducing spin‐dependent control through the geometric phase. We demonstrate that angular‐spectrum asymmetry governs the transverse phase distribution, giving rise to spin‐dependent rotational propagation dynamics. Meanwhile, the azimuthal phase term establishes well‐defined orbital angular momentum (OAM) states with robust phase singularities. Furthermore, despite turbulence‐induced intensity distortions, the phase singularities remain identifiable, highlighting the robustness of OAM states. This work establishes TGBs as a generalized angular‐spectrum framework linking beam symmetry, propagation dynamics, and topology, while demonstrating LC geometric‐phase polymer as a versatile platform for multidimensional wavefront engineering.