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.