Flexible generation of vector beams based on the noncommutation of Pancharatnam–Berry phase elements

We examine the noncommutation of Pancharatnam–Berry phase elements in the generation of vector beams. Any desirable vector beams on the two high-order Poincaré spheres can be generated by incorporating the noncommutation and certain input polarization states. We show that the vector polarization states can be switched between two different high-order Poincaré spheres by simply exchanging the order of two Pancharatnam–Berry phase elements. More importantly, the point-to-point conversion of vector beams makes our scheme highly flexible and easy to operate.