In the past few decades, artificial intelligence (AI) algorithms have been applied in various fields. Among them, neural network algorithms have become the common paradigm of modern AI and have achieved remarkable achievements in image recognition, natural language processing, speech recognition, and recommendation systems. However, the training process of these digital neural networks demands a large amount of time and energy. Therefore, optical computing solutions, with their multidimensional, high-speed, and low-energy advantages, have become a popular research area in AI applications. Extreme learning machine (ELM) is a machine learning paradigm where most connections in the model are established through randomly initialized nonlinear hidden nodes, and only a small part of the weights are adjusted during training after down-sampling. The advantage of this model is that it significantly reduces the training time as it replaces the time-consuming backpropagation with simple ridge regression. Nevertheless, in digital ELMs, the model performance heavily depends on the number of nodes in the hidden layer, which may lead to large memory consumption. To address this problem, we propose an optical extreme learning machine (optical ELM) by implementing random projections in the free-space propagation and investigating the effect of defocus on optical ELM. The optical aberrations, errors, and defects existing in the experimental process act as random components in the optical ELM, corresponding to the random transmission matrix in the digital ELM. This approach enables the passive realization of a large-scale hidden layer. We aim to design an easily deployable optical ELM that does not require complex processing of input and output data but achieves random projection through passive propagation in the optical domain. This method intends to simplify the system architecture while taking advantage of optical technology to achieve efficient parallel computing.

Fig. 1a shows the architecture of the digital ELM, defining the random projection process in the digital ELM as a randomly generated transmission matrix W, which describes the random linear mapping process between the input images and the output images. The matrix H is defined as the result of the input image matrix X multiplied by the transmission matrix W, with the ReLU function added as a nonlinear activation function. The calculation formula for H is given by H=ReLUXW. Subsequently, the parameters are trained using the ridge regression algorithm with β=(HTH+cI)-1HTT. Figs. 1(b) and 2 show the architecture and experimental setup of the optical ELM. A 532 nm wavelength laser is used, and its power is regulated by a Glan-Thompson polarizer and a half-wave plate. The input image is provided through a spatial light modulator (SLM) and propagates through free space. Optical aberrations, errors, and defects in the experimental process are defined as random transmission matrix W for the optical ELM. The matrix H is defined as the result of the input image matrix X multiplied by the transmission matrix W, using the nonlinear response function G of the camera as the nonlinear activation function. The calculation formula for H is given by H=GXW, and the parameters are trained using the ridge regression algorithm with β=(HTH+cI)-1HTT.

Figs. 3 and 4 show the simulation results for the digital ELM. First, under random projection, increasing both the input size and the number of hidden nodes remarkably improves ELM performance. Second, for high-resolution images, down-sampling in the optical domain is a more effective way to reduce computational burden. Figs. 5 and 6 illustrate the experimental results of the optical ELM. It shows that the utilization of the inherent random aberrations and errors of optical experiments and the nonlinear response of the camera in the framework is effective and can provide the necessary random mapping for optical ELM. Increasing the number of hidden nodes is related to the improvement of model performance. However, the propagation distance (PD) has a minimal impact on the model’s performance.

We present a framework for optical ELM and provide a detailed analysis and experimentation. The experimental results show that the optical ELM can achieve a certain classification accuracy under specific parameter settings. By using the inherent random aberrations and defects during free-space propagation and the nonlinear response of the camera, this approach replaces the time-consuming and energy-intensive random mapping process used in digital ELM, thus enhancing hardware efficiency. This study validates the effectiveness of optical transmission in passively processing large-scale image data. Compared with other complex systems, this design only requires a simple deployment of optical pathways while achieving the initial research goal: to design an easily deployable optical ELM that does not require complex processing of input data. However, there is still room for improvement in this experiment. Thus, Fig. 7 shows a potential improvement scheme by introducing a nonlinear scattering medium during free-space propagation, which can provide the model with more randomness and nonlinear effects, thereby enhancing model performance.