Artificial neural networks have achieved remarkable success in image recognition, natural language processing, robotics, medical diagnosis, and many other intelligent applications. However, the continuous expansion of model scale and data volume has imposed increasingly stringent requirements on computing power and energy efficiency. Conventional electronic computing architectures are gradually constrained by electronic interconnection bottlenecks, memory access latency, and power dissipation. Optical diffractive neural networks provide a promising alternative computing paradigm because optical waves can naturally perform parallel propagation, interference, and spatial transformation at the speed of light. In particular, metasurfaces composed of subwavelength artificial nanostructures offer high degrees of freedom for manipulating the amplitude, phase, polarization, and propagation direction of light within an ultrathin platform. Combining metasurface-enabled wavefront engineering with diffractive optical neural networks is therefore of great significance for developing compact, low-power, and highly parallel optical intelligent computing systems. This study aims to construct a single-layer optical diffractive neural network based on a Pancharatnam-Berry phase metasurface, simplify the physical architecture of conventional multilayer diffractive neural networks, and develop a dynamic adaptive optimization algorithm suitable for training such optical systems without relying on explicit gradient calculation.

A physical model of an amorphous silicon metasurface was established and verified by finite-difference time-domain simulation. The optical constants of amorphous silicon near the C-band were calibrated, and the geometric parameters of rectangular nanopillars were optimized to obtain efficient circular polarization conversion and full phase modulation at 1550 nm. By rotating each nanopillar, a Pancharatnam-Berry geometric phase was introduced to realize continuous phase modulation over the range from 0 to 2π. A metasurface modulation unit consisting of a 10×10 nanopillar array was then constructed as a large pixel in the diffractive neural network, and its phase response and edge effects were analyzed. The optical neural network model was described using scalar diffraction theory and the angular spectrum method. Input images were encoded into optical fields, propagated through free space, modulated by the metasurface phase layer, and detected at the output plane through optical intensity distributions. A multilayer diffractive neural network was first simulated as a reference to evaluate the effects of modulation-layer number, interlayer distance, training data scale, amplitude modulation, and nonlinear activation. Then, a single-layer diffractive neural network was designed by optimizing the Fresnel number, enabling a compact architecture with one modulation layer to provide effective feature aggregation and many-to-one mapping. To solve the training problem of optical diffractive neural networks, a dynamic adaptive optimization algorithm based on zeroth-order optimization was proposed. Instead of calculating gradients directly through backpropagation, the algorithm introduces random perturbations into the parameter space, compares the loss function before and after perturbation, and estimates the update direction through sampling and finite differences. This method is independent of explicit analytical gradients and is suitable for physical optical systems, black-box models, hardware-in-the-loop training, and online adaptive calibration. The algorithm was applied to the single-layer diffractive neural network for MNIST handwritten digit classification under phase-only and amplitude-phase modulation conditions.

Simulation results show that the optimized amorphous silicon nanopillar has strong Pancharatnam-Berry phase modulation capability near 1550 nm. When the nanopillar length is about 500 nm and the width is about 265 nm, high cross-circular polarization transmittance and low co-circular polarization transmittance are obtained, indicating effective polarization conversion. The additional phase varies linearly with the nanopillar rotation angle and covers the full 0 to 2π range. The 10×10 nanopillar array also maintains a stable phase response in the central region, confirming its feasibility as a modulation pixel for diffractive neural networks. For the multilayer diffractive neural network, increasing the number of modulation layers significantly improves the MNIST classification accuracy. With five modulation layers, optimized interlayer distance, and large-scale training data, the phase-only network achieves an accuracy of 92.65%. After amplitude modulation is introduced, the accuracy increases to 93.40%, showing that amplitude control can improve performance but brings limited gain and higher hardware complexity. With nonlinear activation, the accuracy further rises to 96.98%, indicating the importance of optical nonlinearity for enhancing network expressiveness, although practical physical implementation remains difficult. For the single-layer diffractive neural network, the phase-only structure achieves an accuracy of 88.81%, and the amplitude-phase structure achieves 89.25%. After using the proposed dynamic adaptive algorithm, the phase-only single-layer network reaches 91.82% accuracy on the MNIST test set, corresponding to 9182 correctly classified images among 10000 samples. Compared with the single-layer network without the algorithm, the accuracy is improved by 3.01 percentage points. The amplitude-phase mode with the dynamic adaptive algorithm achieves 91.51%, slightly lower than the phase-only case. These results show that the proposed algorithm is especially effective for phase-only single-layer diffractive neural networks and can significantly improve classification accuracy without increasing optical layers. The wavelength-dependent performance was also analyzed. When the same network parameters are used, the recognition accuracy reaches its maximum at 1550 nm and decreases as the wavelength deviates from this value. This is mainly caused by the wavelength dependence of the metasurface phase response, polarization conversion efficiency, diffraction phase factor, and Fresnel number. When different networks are trained at wavelengths from 1545 nm to 1555 nm, the accuracy remains above 90% near the design wavelength, showing reasonable wavelength tolerance. In addition, the equivalent computing power of the proposed single-layer network is estimated to be 23.09 KFLOPS, and the system-level computational energy efficiency is 0.43 KFLOPS/W. The current efficiency is limited mainly by the refresh time, interface transmission, detection, and electronic processing overheads of commercial devices rather than by optical propagation itself.

A metasurface-based optical diffractive neural network architecture is proposed and investigated. The results verify that amorphous silicon nanopillars can provide efficient Pancharatnam-Berry phase modulation at 1550 nm and can serve as physical modulation units for diffractive optical computing. By optimizing the Fresnel number, the conventional multilayer diffractive neural network is simplified into a single-layer structure, reducing alignment difficulty, propagation loss, and fabrication complexity. The proposed dynamic adaptive optimization algorithm improves the accuracy of the phase-only single-layer diffractive neural network from 88.81% to 91.82% on MNIST, demonstrating its effectiveness for gradient-free optimization in optical neural systems. This work provides a feasible route toward compact, low-power, and adaptive metasurface optical computing architectures. Future research may focus on all-optical nonlinear activation, high-speed modulation and detection, multidimensional multiplexing, and large-scale integrated metasurface platforms.