On-Demand Longitudinal Transformations of Light Beams via an Amplitude-Phase-Decoupled Pancharatnam-Berry Phase Element

To overcome the inherent phase-only modulation limitation of conventional Pancharatnam-Berry (PB) phase elements, we propose a checkerboard-encoded design that enables simultaneous amplitude and phase modulation through PB phase engineering. This single-element platform directly generates polarization-rotating beams with continuously varying polarization angles along the propagation axis, eliminating traditional dynamic phase requirements. By strategically combining the proposed element with a liquid crystal q-plate and waveplates, we demonstrate on-demand longitudinal transformations of polarization states and higher-order Poincaré sphere beams. Significantly, we achieve, for the first time to our knowledge, the controllable propagation-transformed optical skyrmions over multiple cycles, whose topological textures periodically evolve with propagation distance. This amplitude-phase decoupling mechanism and the compatible solution for various longitudinal transformations significantly advance the development of a universal theoretical framework for longitudinal manipulation of light beams.