Designing robust diffractive neural networks with improved transverse shift tolerance

We propose a method for the design of diffractive neural networks (DNNs) for image classification, which takes into account the positioning errors (transverse shifts) of phase diffractive optical elements (DOEs) constituting the DNN. In this method, the error in solving the classification problem is represented by a functional depending on the phase functions of the DOEs and on random vectors describing the transverse shifts of the DOEs. 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. Explicit expressions are obtained for the derivatives of the error functional. It is shown that the calculation of the derivatives of this functional using the Monte Carlo method corresponds to the DNN training method, in which the DOEs have random transverse shifts. By 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.