A generalized(Formula presented) (Formula presented)-space coherent averaging method for engineering lattices of spin–orbit beams

Spin–orbit beams, in which the orbital angular momentum degree of freedom is coupled to a two-level system such as polarization of light or spin in electrons and neutrons, have gained significant interest for their unique propagation properties and potential applications in imaging, material characterization, optical trapping, and quantum information processing. In this work we introduce a method for generating and engineering two-dimensional lattices of such spin–orbit beams based on coherent averaging in (Formula presented) (Formula presented) -space. By programming the angle, amplitude, and polarization of a set of input beams we obtain precise control over lattice geometry and period, as well as the orbital and radial degrees of freedom inside each unit cell. We explore both electromagnetic and matter wave implementations, and we experimentally demonstrate the generation and characterization of a micron-scale optical hexagonal lattice with well defined orbital and radial numbers in each unit cell. The described methods provide a robust and general method of generating and controlling structured waves such as optical skyrmions and matter wave implementations of orbit and spin–orbit beams.