Fig. 1. Schematic diagrams of the three types of the OAM-encoded D2NN. The OAM beams illuminating the digits are multiplexed by 10 OAM modes ranging from −5 to +5 in equal proportions. The red numbers represent the topological charges of the OAM modes, while the black numbers in brackets correspond to the assumed digits associated with the OAM modes. The digit inputs are illuminated by the multiplexed OAM beams, and the predicted OAM beams are obtained in the output plane after modulation by the OAM-encoded D2NNs. The right side of the output plane shows the OAM spectra of the OAM beams. Three different configurations of OAM-encoded D2NNs have been described below: (a) single detector OAM-encoded D2NN for single-task classification, (b) single detector OAM-encoded D2NN for multitask classification, and (c) multidetector OAM-encoded D2NN for multitask classification.

Fig. 2. (a) The amplitude and phase distributions of the OAM beams are shown for the input plane, the diffractive layers, and the output plane. The input image is a handwritten digit “1” encoded as an OAM beam with +2 mode. (b) Schematic of the modulation of the light field by the single-detector OAM-encoded D2NN. (c) The OAM spectrum of the output OAM beams. The red plot corresponding to the OAM mode with the highest normalized intensity indicates the inferred category of the input digit. (d) The loss and accuracy functions for both the training and test sets. Three simulations were conducted for each set, and the corresponding results are represented by the three dashed lines. The solid lines represent the average results of the three function curves depicted by the dashed lines. (e) A confusion matrix summarizes the numerical classification results in the test set. The matrix provides a comprehensive overview of the performance of the single-detector OAM-encoded D2NN in recognizing the handwritten digits from the MNIST data set.

Fig. 3. (a) The amplitude and phase distribution of the OAM beams in the input plane, diffractive layers, and output plane. The input handwritten digits are “7” and “0,” which correspond to the multiplexed OAM beams that produce “-3” and “+1” OAM modes. (b) Schematic of the light field modulation by single-detector OAM-encoded D2NN for multitask classification. The OAM beam encodes two handwritten digits as the input. After undergoing OAM-encoded D2NN modulation, it produces a new OAM beam corresponding to two modes at the same spatial location. (c) The OAM spectrum of the output OAM beams. The two OAM modes detected by the detector with the highest normalized intensity represent the assumed categories of the input digits, and their classes are indicated by the red bars. (d) Loss function and accuracy during training and testing. Solid lines indicate the average result of the three-function curve represented by the dashed line. (e) The confusion matrix summarizes the numerical classification result in the test set.

Fig. 4. (a) From top to bottom, the multidetector OAM-encoded D2NN provides recognition for two digits, three digits, and four-digits, respectively. The amplitude and phase distribution of the OAM beams in the input plane, diffractive layers, and output plane. (b) Schematic of the light field modulation by four-detector OAM-encoded D2NN for multitask classification. Each input OAM beam at different positions encodes only one digit and generates the corresponding OAM mode of that digit at the output, which is detected by a detector at a fixed position. (c) The OAM spectrum of the output OAM beams. The two blue OAM spectra correspond to the OAM beams generated by the two-detector OAM-encoded D2NN, from top to bottom, respectively. The green OAM spectrum in the first row corresponds to the separate OAM beam in the first row of the three-detector OAM-encoded D2NN, and the green OAM spectra in the second and third rows correspond to the two OAM beams from left to right in the second row, respectively. The four red OAM spectra are arranged in a sequential relationship from left to right and from top to bottom.

Fig. 5. (a) The loss function and accuracy function of the two-detector, three-detector, and four-detector OAM-encoded D2NNs in training and testing are arranged from left to right. The solid line represents the average result of the function curves for the three simulations, which is represented by the dashed line. Their average accuracy in the test set is 70.94%, 52.41%, and 40.13%, respectively. (b) Confusion matrices of the three multidetector OAM-encoded D2NNs, summarizing the numerical classification results of the test set. Due to the large number of pixel points in the confusion matrices of the three-detector and four-detector OAM-encoded D2NNs, the confusion matrices are reduced and localized zoomed-in images are inserted.

Fig. 6. The different colored curves represent different diffractive networks, as illustrated in the square diagram located in the lower left corner. (a) The deviation of the pixel size and the layer spacing. The horizontal coordinate represents the error range from 0.8 times the pixel size and the corresponding layer spacing to 1.2 times the pixel size and the corresponding layer spacing. (b) The analysis of the deviation of the object misalignment in horizontal and vertical directions. (c) The analysis of the deviation of the misalignment layer. The left image represents a random misalignment error of 5% for each layer, while the right image represents a random misalignment error of 10% for each layer.

Fig. 7. (a) The left figure shows the geometrical model of the five layer D2NN with the pixel size of 50×50, and the right figure shows the mask model of the number “9” illuminated by the OAM beam. (b) The simulation of the incident OAM beam. (c) The simulation of the output plane by a one-layer D2NN with the pixel size of 30×30. (b), (c) The figures from left to right are amplitude distribution simulated with Python, amplitude distribution simulated with COMSOL Multiphysics software, phase distribution simulated with Python, and phase distribution simulated with COMSOL Multiphysics software.