The rapid development in fields such as artificial intelligence, autonomous driving, and big data has brought higher demands for computational tools due to the massive amount of data involved. In recent years, with the development of ultra-large-scale integrated circuits, the volume of electronic computers has been significantly reduced, and the data processing speed has been greatly improved. However, electronic devices are gradually constrained by physical limitations such as quantum effects, which slow down the improvement speed of low-power and miniaturized digital computing circuits. In addition, traditional analog signal processing requires processes such as analog-to-digital conversion, signal processing units, and digital-to-analog conversion. The inevitable conversion delays and high-power consumption of electronic devices make traditional analog computing incapable of large-scale information processing. Therefore, researchers are devoted to developing new computing systems to overcome the limitations of traditional electronic computing systems. One of the technologies that has attracted significant attention is the construction of all-optical systems for information transmission and processing using optical signals as carriers.

Optical information processing technology has attracted increasing attention due to its advantages such as ultra-high speed, large bandwidth, and low loss. People have attempted to introduce optical methods to improve the performance of information processing and have successfully designed various optical analog computing devices. Compared with electronic signal processing systems, optical signal processing can be categorized into digital computing and analog computing. The earliest digital computing of optical signals uses a liquid crystal spatial light modulator based on optoelectronic mixing for logical operations. Optical analog computing does not involve optoelectronic conversion and can directly manipulate optical signals in time and space domains. Due to the parallelism of physical processes such as light field promotion and interference in space, spatial optical analog computing offers advantages in information processing such as ultra-fast speed, high throughput, and low energy consumption. These attributes highlight its significant potential for applications in image processing, edge detection, and machine learning.

Traditional spatial optical analog operations often employ the Fourier optical 4f system, which involves components such as lenses and filters. However, in recent years, the rapid development of micro-nano optics and fabrication processes has made it possible to realize spatial analog computing using devices at sub-wavelength scales. This opens up possibilities for miniaturization, on-chip integration, and integration of optical computing systems. At present, research on spatial optical analog computing primarily focuses on achieving spatial differentiation, integration, and equation solving. The main design principles can be classified into three categories: effective medium theory, resonance principle, and non-resonance principle (Fig. 1). Spatial analog computing can be achieved by integrating metasurfaces and GRIN lenses into a 4f system. By designing the spatial distribution of transmission (or reflection) rates of the metasurface, researchers can obtain the spatial spectral transfer function for the desired mathematical operations. However, this method requires the introduction of spatial Fourier transform and inverse Fourier transform, resulting in larger device dimensions. Another approach involves constructing a multilayer flat-stack structure using multiple materials, where various spatial optical analog computations can be achieved by adjusting the refractive index and thickness parameters of each layer (Fig. 2). Devices designed based on the equivalent medium theory have complex structures and pose challenges in practical fabrication. In a resonant system, the excitation of resonance requires momentum matching, which leads to different responses of the resonant structure to wavevector components in different directions of the incident light. This allows the spatial response of the propagating optical field in the structure to conform to specific optical simulation operations at certain frequencies, without the need for Fourier transform (Fig. 3). In contrast to the limited spatial bandwidth of resonant-based analog devices, under specific conditions in non-resonant systems, researchers can obtain the spatial spectral transfer function required for spatial analog operations based on the spin Hall effect, Brewster effect, and PB phase (Fig. 4). Spatial optical analog computing enables high-speed, high-throughput, and low-power information processing. Spatial differentiation and convolution operations can be directly applied to image edge detection and are promising for applications in pattern recognition, machine vision, and other fields (Fig. 5). Finally, the existing challenges and research prospects of spatial optical analog computing are discussed.

We summarize the development of optical spatial analog computing and focus on the research progress and applications of optical spatial analog computing with metasurfaces in different theoretical models and systems. By incorporating artificial nanostructures to replace conventional large-scale optical components, metasurfaces enable the development of miniaturization and integration of spatial optical analog computing devices. Furthermore, we analyze the latest advances in spatial optical analog computing based on physical effects such as spin-orbit coupling and topology, which present novel avenues for achieving ultra-wideband and high-speed information processing. Lastly, we discuss the existing challenges and research prospects associated with optical spatial analog computing, shedding light on future directions for this field. With the development of information technology and the increasing demand for processing performance, optical information processing methods are gradually emerging. The design and implementation of various optical analog computing devices have become increasingly important for technological development and performance improvement. Spatial optical analog computing combined with metasurfaces has made unexpected progress, but it still suffers from some challenges such as the fabrication technology of metasurfaces, energy efficiency of optical analog computing, and reconfigurable computing. In the future, spatial analog computing combined with metasurfaces will unveil more innovative approaches, promoting the development of spatial analog computing from simple to complex. With continuous innovative advances in technology, combining various computing devices and achieving multi-functional optical computing chips may further boost fields such as high-throughput optical communication and optical imaging in the future.