Vectorial freeform focal line synthesis: decoding the Poincaré sphere with tailored light trajectories.

Coordinated manipulation of light field propagation is critically important and profoundly challenging for both fundamental research in optics and the expansion of its applications. Achieving independent spatial control over the properties of light-amplitude, phase, and polarization-lies at the core of this challenge. Here, we propose a vectorial synthesis scheme based on structured focal lines generated by a freeform phase mask to simultaneously tailor their spatial distributions. By strategically engineering the parameters of a freeform phase mask, we construct a pair of structured focal lines carrying customized phase profiles and mutually orthogonal polarizations. The coherent superposition of these two orthogonally polarized focal lines, serving as the basis beams, enables the generation of vector vortex beams with arbitrarily designable polarization states on the Poincaré sphere. Both numerical simulations and experimental demonstrations confirm that our method facilitates independent and effective manipulation of the amplitude, phase, and polarization along the prescribed two-dimensional or three-dimensional focal trajectories of these so-called freeform vector vortex beams (FVVBs). This work offers a flexible new approach for the regulation of multidimensional light fields.