Designing robust diffractive neural networks with improved transverse shift tolerance

A wide range of practically important problems is nowadays efficiently solved using artificial neural networks. This gave momentum to intensive development of their optical implementations, among which, the so-called diffractive neural networks (DNNs) constituted by a set of phase diffractive optical elements (DOEs) attract considerable research interest. In the practical implementation of DNNs, one of the standing problems is the requirement for high positioning accuracy of the DOEs. In this work, we address this problem and propose a method for the design of DNNs for image classification, which takes into account the positioning errors (transverse shifts) of the DNN elements. In the method, the error of solving the classification problem is represented by a functional depending on the phase functions of the DOEs and on random vectors describing their transverse shifts. The mathematical expectation of this functional is used as an error functional in the gradient method for calculating the DNN taking into account the transverse shifts of the DOEs. It is shown that the calculation of the derivatives of this functional corresponds to the DNN training method, in which the DOEs have random transverse shifts. Using the proposed gradient method, DNNs are designed that are robust to transverse shifts of the DOEs and enable solving the problem of classifying handwritten digits at a visible wavelength. Numerical simulations demonstrate good performance of the designed DNNs at transverse shifts of up to 17 wavelengths.