Universal approximation capabilities of coherent diffractive systems

Coherent optical computing systems are a promising avenue for increasing computation speed and solving energy requirements for machine learning applications. These systems utilize the diffraction process of coherent waves to perform calculations in the optical domain. While the diffraction process is linear in complex space $\mathbb{C}$, it empirically has been shown, that these systems are able to outperform standard linear matrix multiplications in $\mathbb{R}$, because photo-sensors project from complex space to real space. Here we give theoretical insights, of why this is the case and show, that a system consisting of multiple phase-plates, two output photo-detectors, and the appropriate input encoding is theoretically able to learn any one-dimensional function. We furthermore use these theoretical insights to show that encoding the input information solely in the intensity of the diffractive system is never enough to make the system a universal function aproximator. These results are useful to understand the capabilities of diffractive optical systems and improve their training.