A large aperture telescope is needed to achieve long distance observations. The size of a single aperture telescope is limited by processing costs and other factors, and the segmented mirror technology is expected to break through the single aperture telescope limit. The key to the realization of segmented mirror technology is fine co-phasing. Currently, the most widely used technique for co-phasing detection is the broadband and narrowband Shack-Hartmann (S-H) method. The broadband S-H detection range is large, but the accuracy is low (30 nm), whereas the narrowband S-H method has a high detection accuracy of 6 nm; however, there is 2π ambiguity effect and its detection range is λ/2. The conventional cross-correlation algorithm uses two wavelengths to detect the co-phasing error, which effectively solves the 2π ambiguity effect in single wavelength detection and simultaneously improves the detection range. In this study, to address the slow detection speed and low accuracy of the current two-wavelength co-phasing detection method using the cross-correlation algorithm in the detection of large-range co-phasing errors, a two-wavelength co-phasing algorithm based on convolutional neural networks is proposed to achieve fast and accurate co-phasing detection in large-range co-phasing errors. First, the circular diffraction image splicing at the two wavelengths is used as the training data for the convolutional neural network. After training, the circular diffraction splicing image containing the piston error information is input into the trained model, and the piston error value is detected directly. The robustness of the convolutional network based on convolutional networks under different error situations is also analyzed.

Based on the principle of circular diffraction, the circular diffraction pattern with the piston error information is first obtained through software simulation, and the circular diffraction patterns corresponding to the piston error at the two wavelengths are used to splice and obtain the data set for training the network. The convolutional neural network is then constructed, and the model of the circular aperture diffraction pattern and piston error is trained using the established data set. Finally, after the convolutional neural network is trained, the circular diffraction pattern at the corresponding wavelength is collected by inserting a circular aperture mask between the sub-mirrors of the segmented mirror system, and the obtained circular diffraction pattern is used as the input of the neural network. The piston error between the two sub-mirrors is directly obtained using the trained convolutional neural network model. The robustness of the convolutional neural network is also analyzed for different error situations.

The convolutional neural network model is trained with 99.85% accuracy in the validation set and 99.9% accuracy in the test set, with a residual root-mean-square error (RMSE) of 36.7 nm (Fig. 6). The robustness of the convolutional neural network model under multiple error cases is discussed. When only the eccentricity error (R₂) is present, the residual RMSE of the convolutional neural network is less than 40 nm at R₂≤0.1 (Fig. 7). When only the noisy signal-noise-ratio (S_SNR) is present, the residual RMSE of the convolutional neural network is less than 40 nm for S_SNR≥40 (Fig. 8). When both errors are present, the residual RMSE of the convolutional neural network is less than 40 nm for S_SNR>40 and the eccentricity error R₂<0.1 (Fig. 9). It is also demonstrated that the number of prediction data samples has no significant effect on the prediction results of the convolutional neural network model. Finally, comparing the convolutional neural network-based detection method with the traditional cross-correlation algorithm (Fig. 10 and Table 2), the convolutional neural network takes 15.88 s to predict 1500 sets of piston error images successively under the same conditions. Only two images are predicted incorrectly, compared with the lower performance of the traditional cross-correlation algorithm.

Based on the principle of circular diffraction, this study proposes the use of a convolutional neural network for the co-phasing detection method to solve the problem of slow operation speed based on the cross-correlation algorithm in the current two-wavelength detection co-phase error method and to achieve faster and more accurate co-phasing detection of the segmented mirror. The robustness of the convolutional neural network under several error situations is also demonstrated. The detection method based on the convolutional neural network is compared with the traditional cross-correlation algorithm. The simulation analysis shows that the two-wavelength detection method based on the convolutional neural network can achieve the requirements of co-phasing detection with a large range, high accuracy, and fast detection speed. The study provides an experimental reference for the future application of the co-phasing detection method in engineering experiments.