As an optical processor, a diffractive deep neural network (D²NN) utilizes engineered diffractive surfaces designed through machine learning to perform all-optical information processing, completing its tasks at the speed of light propagation through thin optical layers. With sufficient degrees of freedom, D²NNs can perform arbitrary complex-valued linear transformations using spatially coherent light. Similarly, D²NNs can also perform arbitrary linear intensity transformations with spatially incoherent illumination; however, under spatially incoherent light, these transformations are nonnegative, acting on diffraction-limited optical intensity patterns at the input field of view. Here, we expand the use of spatially incoherent D²NNs to complex-valued information processing for executing arbitrary complex-valued linear transformations using spatially incoherent light. Through simulations, we show that as the number of optimized diffractive features increases beyond a threshold dictated by the multiplication of the input and output space-bandwidth products, a spatially incoherent diffractive visual processor can approximate any complex-valued linear transformation and be used for all-optical image encryption using incoherent illumination. The findings are important for the all-optical processing of information under natural light using various forms of diffractive surface-based optical processors.

Large-scale linear operations are the cornerstone for performing complex computational tasks. Using optical computing to perform linear transformations offers potential advantages in terms of speed, parallelism, and scalability. Previously, the design of successive spatially engineered diffractive surfaces forming an optical network was demonstrated to perform statistical inference and compute an arbitrary complex-valued linear transformation using narrowband illumination. We report deep-learning-based design of a massively parallel broadband diffractive neural network for all-optically performing a large group of arbitrarily selected, complex-valued linear transformations between an input and output field of view, each with N_i and N_o pixels, respectively. This broadband diffractive processor is composed of N_w wavelength channels, each of which is uniquely assigned to a distinct target transformation; a large set of arbitrarily selected linear transformations can be individually performed through the same diffractive network at different illumination wavelengths, either simultaneously or sequentially (wavelength scanning). We demonstrate that such a broadband diffractive network, regardless of its material dispersion, can successfully approximate N_w unique complex-valued linear transforms with a negligible error when the number of diffractive neurons (N) in its design is ≥2N_wN_iN_o. We further report that the spectral multiplexing capability can be increased by increasing N; our numerical analyses confirm these conclusions for N_w > 180 and indicate that it can further increase to N_w ∼ 2000, depending on the upper bound of the approximation error. Massively parallel, wavelength-multiplexed diffractive networks will be useful for designing high-throughput intelligent machine-vision systems and hyperspectral processors that can perform statistical inference and analyze objects/scenes with unique spectral properties.

Optical computing provides unique opportunities in terms of parallelization, scalability, power efficiency, and computational speed and has attracted major interest for machine learning. Diffractive deep neural networks have been introduced earlier as an optical machine learning framework that uses task-specific diffractive surfaces designed by deep learning to all-optically perform inference, achieving promising performance for object classification and imaging. We demonstrate systematic improvements in diffractive optical neural networks, based on a differential measurement technique that mitigates the strict nonnegativity constraint of light intensity. In this differential detection scheme, each class is assigned to a separate pair of detectors, behind a diffractive optical network, and the class inference is made by maximizing the normalized signal difference between the photodetector pairs. Using this differential detection scheme, involving 10 photodetector pairs behind 5 diffractive layers with a total of 0.2 million neurons, we numerically achieved blind testing accuracies of 98.54%, 90.54%, and 48.51% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively. Moreover, by utilizing the inherent parallelization capability of optical systems, we reduced the cross-talk and optical signal coupling between the positive and negative detectors of each class by dividing the optical path into two jointly trained diffractive neural networks that work in parallel. We further made use of this parallelization approach and divided individual classes in a target dataset among multiple jointly trained diffractive neural networks. Using this class-specific differential detection in jointly optimized diffractive neural networks that operate in parallel, our simulations achieved blind testing accuracies of 98.52%, 91.48%, and 50.82% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively, coming close to the performance of some of the earlier generations of all-electronic deep neural networks, e.g., LeNet, which achieves classification accuracies of 98.77%, 90.27%, and 55.21% corresponding to the same datasets, respectively. In addition to these jointly optimized diffractive neural networks, we also independently optimized multiple diffractive networks and utilized them in a way that is similar to ensemble methods practiced in machine learning; using 3 independently optimized differential diffractive neural networks that optically project their light onto a common output/detector plane, we numerically achieved blind testing accuracies of 98.59%, 91.06%, and 51.44% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively. Through these systematic advances in designing diffractive neural networks, the reported classification accuracies set the state of the art for all-optical neural network design. The presented framework might be useful to bring optical neural network-based low power solutions for various machine learning applications and help us design new computational cameras that are task-specific.