Universal linear optical operations on discrete phase-coherent spatial modes with a fixed and non-cascaded setup

Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary linear operations for N-dimensional phase-coherent spatial modes using meticulously designed phase gratings. As implemented on spatial light modulators (SLMs), the unitary transformation matrix has been realized with dimensionalities ranging from 7 to 24 and the corresponding fidelities are from 95.1% to 82.1%. For the non-unitary operators, a matrix is presented for the tomography of a 4-level quantum system with a fidelity of 94.9%. Thus, the linear operator has been successfully implemented with much higher dimensionality than that in previous reports. It should be mentioned that our approach is not limited to SLMs and can be easily applied on other devices. Thus, our proposal could provide another option to perform linear operations with a simple, fixed, error-tolerant and scalable scheme.