  We have proposed and demonstrated a scalable and efficient scheme for
programmable unitary operations in orbital angular momentum (OAM) domain. Based
on matrix decomposition into diagonal and Fourier factors, arbitrary matrix
operators can be implemented only by diagonal matrices alternately acting on
orbital angular momentum domain and azimuthal angle domain, which are linked by
Fourier transform. With numerical simulations, unitary matrices with
dimensionality of 3*3 are designed and discussed for OAM domain. Meanwhile, the
parallelism of our proposed scheme is also presented with two 3*3 matrices.
Furthermore, as an alternative to verify our proposal, proof of principle
experiments have been performed on path domain with the same matrix
decomposition method, in which an average fidelity of 0.97 is evaluated through
80 experimental results with dimensionality of 3*3.
