  Reciprocal spin-orbit coupling (SOC) via geometric phase with flat optics
provides a promising platform for shaping and controlling paraxial structured
light. Current devices, from the pioneering q-plates to the recent J-plates,
provide only spin-dependent wavefront modulation without amplitude control.
However, achieving control over all the spatial dimensions of paraxial SOC
states requires spin-dependent control of corresponding complex amplitude,
which remains challenging for flat optics. Here, to address this issue, we
present a new type of flat-optics elements termed structured geometric phase
gratings that is capable of conjugated complex-amplitude control for orthogonal
input circular polarizations. By using a microstructured liquid crystal
photoalignment technique, we engineered a series of flat-optics elements and
experimentally showed their excellent precision in arbitrary SOC control. This
principle unlocks the full-field control of paraxial structured light via flat
optics, providing a promising way to develop an information exchange and
processing units for general photonic SOC states, as well as extra-/intracavity
mode convertors for high-precision laser beam shaping.
