Vortex beams carry orbital angular momentum and have a phase factor exp( ilθ ), where l is the topological charge number and is direction angle. Theoretically, l can take any integer value, and different orbital angular momentum modes are mutually orthogonal. Therefore, in optical communication, the orbital angular momentum can be used for information transmission and exchange or multiplexed to improve communication capacity. However, vortex beams are affected by turbulence when transmitting in atmospheric turbulence, which results in the distortion of their spiral phase and causes inter-mode crosstalk and reduced communication reliability. Many studies focus on compensating the phase distortion of vortex beams, with adaptive optics systems commonly used. However, such methods require multiple iterations, converge slowly, and easily fall into local minima. In recent years, convolutional neural networks have attracted extensive attention in various fields due to their powerful image processing capabilities. Therefore, in this paper, convolutional neural networks are used to extract atmospheric turbulence information from distorted light intensity distribution and recover its distortion. This deep learning-based compensation method has even more accurate and faster correction capability than adaptive optics systems. In view of this, convolutional neural networks are employed herein for the phase prediction of atmospheric turbulence to achieve the phase compensation of Laguerre-Gaussian (LG) beams and improve modal detection accuracy and communication reliability.
In this paper, a novel convolutional neural network, i.e., deep phase estimation network, is constructed to achieve the prediction of turbulent phases. With this proposed deep network, a mapping between light intensity and turbulent phase caused by atmospheric turbulence is established. Here two strategies are used for learning and predicting the turbulent phase respectively: one uses a Gaussian beam as the probe beam, and the other makes a direct prediction with an LG beam carrying information without a probe beam. In the target plane, the turbulent phase is predicted by intensity, and the field is corrected by the predicted phase. The inputs of the networks of the two schemes are a Gaussian beam and an LG beam, respectively, and the output is the corresponding predicted phase of atmospheric turbulence. The deep phase estimation network performs feature extraction of the input light intensity profile by under-sampling through the encoder, learns the atmospheric turbulence feature parameters by up-sampling through the decoder to achieve the reconstruction of the equivalent atmospheric turbulence phase screen, and finally outputs the results. By learning and training a large number of samples, the network structure proposed in this paper can achieve good prediction results at a transmission distance of 500 m. In addition, five sets of intensity profiles with different turbulence intensities are set for testing and verifying the network to prove that the network has strong generalization ability. Then, the compensation is achieved by loading the reverse phase of the predicted phase on the distorted beam to exert the correction effect.
In this paper, we construct a deep phase estimation network consisting of 15 convolutional layers, 3 deconvolutional layers, and 3 skip connections (Fig. 6) by using an encoder-decoder architecture, which can achieve phase prediction at long transmission distances. At a transmission distance of 500 m, after the network is trained with the distorted beam at different turbulence intensities, it can predict the turbulence phase screen with a high agreement with the simulation results of tests at five different turbulence intensities (Fig. 7). The prediction results are evaluated by calculating the mean square error between them and the simulation results, and it is found that the network can effectively extract turbulence information and has strong generalization ability (Table 2). The beam phase correction is achieved by loading the predicted phase reverse to the distorted beam, and the intensity profile (Fig. 8) and phase (Fig. 10) are corrected to a large extent. The mode purity of the corrected beam is greatly improved, and the mean square error of the intensity image is significantly reduced (Table 3).
The results show that the deep phase estimation network created in this paper can achieve phase prediction more accurately, and it is trained to automatically learn the mapping relationship between the input sample light intensity distribution and the turbulent phase and output the predicted phase. The phase compensation of the vortex beam is achieved based on the predicted phase. The compensation effects of two schemes using a Gaussian probe beam and not using a probe beam are investigated separately, and both of them are effective in correcting the distorted phase. They can predict the turbulent phase accurately under tests at five different turbulent intensities. After compensation, the mode purity of the beam reaches more than 95%, and the mean square errors of the compensated light intensity image and the source plane are both significantly reduced. Even in the case of
