Coherence Awareness in Diffractive Neural Networks

TL;DR

This work addresses how partial spatial and temporal coherence affects all-optical, diffractive neural networks, challenging the assumption that either fully coherent or fully incoherent illumination suffices. It introduces a coherence-aware training framework that propagates fields from each source point through the network and supports linear and nonlinear layers, guided by the van Cittert–Zernike theorem. The authors demonstrate that networks can be trained for any prescribed coherence and show performance trends on MNIST and BloodMNIST phase-object datasets, including coherence-blind variants that resist illumination changes. This framework paves the way for deploying all-optical neural networks under natural lighting, with potential impact across imaging, sensing, and autonomous systems by enabling coherence-aware or coherence-robust optical computation.

Abstract

Diffractive neural networks hold great promise for applications requiring intensive computational processing. Considerable attention has focused on diffractive networks for either spatially coherent or spatially incoherent illumination. Here we illustrate that, as opposed to imaging systems, in diffractive networks the degree of spatial coherence has a dramatic effect. In particular, we show that when the spatial coherence length on the object is comparable to the minimal feature size preserved by the optical system, neither the incoherent nor the coherent extremes serve as acceptable approximations. Importantly, this situation is inherent to many settings involving active illumination, including reflected light microscopy, autonomous vehicles and smartphones. Following this observation, we propose a general framework for training diffractive networks for any specified degree of spatial and temporal coherence, supporting all types of linear and nonlinear layers. Using our method, we numerically optimize networks for image classification, and thoroughly investigate their performance dependence on the illumination coherence properties. We further introduce the concept of coherence-blind networks, which have enhanced resilience to changes in illumination conditions. Our findings serve as a steppingstone toward adopting all-optical neural networks in real-world applications, leveraging nothing but natural light.

Figures (18)

• Figure 1: The influence of the degree of spatial coherence on the output of non-imaging optical systems. (a) A non-imaging optical system. (b) The left pane depicts intensity patterns resulting from the optical system illustrated in (a), for a diffractive element whose pixels are drawn independently at random in the range $[0,2\pi]$. The object is a digit of dimensions 0.84 mm $\times$ 0.84 mm. The intensity patterns are calculated for 35 different values of $d$ which translate to different coherence lengths between 800 $\mu$m and 2 $\mu$m. All the results are for the case where $a=D$ and $d_i$ is the same across experiments. The plot on the right shows the centered cosine similarity between the intensity pattern at the output plane and the intensity pattern corresponding to fully incoherent (red) and fully coherent (blue) illumination. A gray rectangle marks the area where the centered cosine similarity is below 0.9. The dashed line indicates where $d \approx d_i$. The x-axis is at log-scale. The results are averaged over 5 different digits, where for each digit we used 5 different random phase masks. (c) The same as in (b) only for an imaging system, i.e. where the diffractive element is a lens with quadratic phase that preforms unit imaging up to rotation. (d) Several scenarios where $d \approx d_i$ and $a \approx D$, in which the system inherently does not operate in the incoherent regime.
• Figure 2: Optical configuration for coherence-aware networks. A planar, incoherent, uniformly bright source of area $A_S$ is placed a distance $d$ from the object plane. The parameters $A_S$ and $d$ control the degree of spatial coherence at the object plane. As opposed to networks for fully coherent and fully incoherent light, here computing the intensity at the output plane necessitates propagating fields from the source plane rather than starting at the object plane. The figure presents an example of a classification network comprising power-preserving linear and nonlinear diffractive layers. The output plane contains an array of detectors, one for each object class. In this setting, the network may be optimized to concentrate light on the detector corresponding to the predicted class.
• Figure 3: The proposed method for simulating a network with partially spatial coherent light. A spatial incoherent square light source is located at a distance $d$ in front of an object plane. Each pixel (radiator) in the light source is separately propagated and multiplied by the object's amplitude. Each multiplication result is separately propagated through the diffractive network. The intensity is calculated in the output plane for each of the light source's pixels and summed for all of them. The resulting intensity in the output plane is summed in pre-determined areas, marked by yellow squares. The area with the highest intensity is chosen as the correct classification, marked with a cyan square.
• Figure 4: Versatility of the proposed framework. The left-hand side of the figure depicts diffractive networks with a single phase mask, both without (top) and with (bottom) a nonlinear element placed right after it. The right-hand side of the figure depicts diffractive networks with two linear phase masks. The networks are trained for classification of handwritten digits and fashion items. (a),(d) Examples of items for classification in the object plane. (b),(e) The learned phase masks of the linear and nonlinear diffractive networks. (c),(f) Illustrations of the output intensity patterns indicating the network's classification predictions. The cyan square corresponds to the class of the ground-truth digit. In all cases except for the top pane in (c), the networks' predictions are correct. In the top pane of (c), we show an example of an incorrect prediction of a linear network, indicated by a red square.
• Figure 5: Effect of spatial and temporal coherence on the performance of diffractive networks. The top part of the figure shows the accuracy rates for different levels of spatial and temporal coherence, for nonblind (a) and blind (c) settings. The horizontal axis measures spatial coherence via coherence length. The vertical axis measures temporal coherence and has two scales, one for coherence time (left) and another for the associated wavelength bandwidth around the central wavelength of 550 nm (right). Both axes are on a logarithmic scale. The different colors on the graph indicate the resulting accuracy. The edges of each graph indicate different spatially and temporally coherent states, from the temporally and spatially incoherent B to the temporally and spatially coherent C. The bottom part of the figure displays the intensity patterns at the output plane for the nonblind (b) and blind (d) settings. The images are shown for the same input image from the test set (a digit 7), for the edge cases marked by A, B, C, and D in (a) and (b).