Keywords: 
• geometric phase /
• liquid crystal /
• flat optics /
• light field shaping

Abstract:   Significance   Flat optics elements based on geometric phase, owing to their low cost, integrability, and versatility, have been widely used in shaping of light's spatial structure. Notably, current SOC (spin-orbit coupling) devices, such as the best-known q-plates, provide only spatial phase modulation with SoP (state of polarization)-switchable behavior. The absence of amplitude control prevents research scholars from accessing light's full spatial degrees of freedom, thus limiting their application in corresponding studies. This team demonstrates a series of novel flat optics elements with liquid-crystal geometric phase, which unlocks the full-field control of paraxial structured light, providing a powerful toolbox for relevant experimental studies and especially for high-dimensional classical/quantum information.  Progress  To control a paraxial SOC state in all its spatial degrees of freedom, spin-dependent complex amplitude modulation provides an essential alternative. But up to now, it has remained elusive with flat optics. This paper fills this gap by putting forward a new type of geometric phase element termed structured geometric-phase grating (SGPG), featuring a spatially-varying grating cycle, depth and orientation (Fig.1). In addition, the joint team also demonstrated the vector wavefront control technology based on the geometric phase of liquid crystals, and developed a series of liquid crystal geometric phase elements (e.g., mode convertor (Fig.2(a)) and high-order spatial mode generator (Fig.2(b)) with the full dimensional control ability.  Conclusions and Prospects  Such a crucial advance, compared with the present geometric phase elements, unlocks the control of paraxial structured light in all spatial dimensions, and paves the way for arbitrary SOC conversion via flat optics. This capability makes it a key extra-/intracavity component to build a structured laser that has greater tunability in beam structure, compared with reported systems based on q-plate and metasurface. For quantum optics, the proposed reciprocal SOC interface allows to implement a Bell measurement for arbitrary SOC states, which is the basis for the teleportation scheme for SOC photon pairs. Moreover, owing to the capability of full-field spatial mode control, the device also paves the way for quantum control of high-dimension photonic skyrmions. Beyond single-beam vector mode control, this principle can further realize multiple vector mode control through the addition of a Dammann grating structure. This represents a promising way to develop information exchange and processing units working for photonic SOC states, that is, vector-mode multiplexers and demultiplexers.