## 1 Introduction  
Vortex beams with spiral phase structures have been used extensively in information transmission, radar imaging and rotational target detection, since [PERSON] first investigated the Laguerre-Gaussian vortex beam and its orbital angular momentum (OAM) in 1992 [1]. Theoretically, there are infinite kinds of eigenstates; different eigenstates are orthogonal to each other, which is quite important in terms of improving communication capacity and imaging resolution in remote sensing. Multiplexing different OAM beams can effectively avoid the crosstalk between different modes in a channel, providing a new communication dimension that is no longer limited to amplitude, phase, frequency and polarization, thereby greatly improving the communication capacity [2, 3]. In free-space OAM communication systems, the receiver needs to demodulate the OAM beam to recover the information sequence. Traditional OAM demodulation techniques, such as spatial light modulators, the diffraction method, the cylindrical lens method, plane wave interferometry and spherical wave interferometry, are based on optical hardware and have been researched extensively [4, 5, 6, 7]. The OAM beam is pre-processed by optical hardware to obtain optical pattern features that can be distinguished by the naked eye [8]. However, on account of the high cost and the limited processing capability of optical hardware, high-performance transmission cannot be guaranteed with a cost-effective vortex optical communication system. In recent years, with the rapid increase in computing power, OAM mode recognition based on deep learning has attracted growing attention. Some researchers have studied OAM mode recognition based on neural networks. [PERSON] et al. proposed a self-organizing competitive neural network (SOM) based OAM mode recognition which verified the feasibility of machine learning in vortex optical communication systems for the first time [9]. They built a long-distance vortex optical communication system on the sea between the Canary Islands, achieving a recognition accuracy of 91.67%, which verified the possibility of the use of vortex beams for long-distance information transmission [10]. Deep neural network (DNN)-based recognition of different OAM modes is proposed in [11]. Convolutional neural networks (CNNs) are proposed in OAM mode recognition by [PERSON] et al. [12]; that method achieved a mode recognition accuracy of up to 99%, which is far superior to the traditional methods. In addition, the CNN-based mode identification method is robust in terms of the influence of turbulence intensity, data size, sensor noise and pixels. Such research has paved the way for the application of CNN in OAM mode detection. Subsequently, many achievements in OAM mode recognition have been realized based on CNNs [13; 14]. In 2017, [PERSON] et al. compared the performance of a K nearest neighbor neural network, a plain Bayesian classifier, a Back Projection artificial neural network and a CNN as OAM mode classifiers under different turbulence conditions. They observed that CNN yielded the best results [15]. The same authors improved the original LeNet-5 network and proposed a decoder scheme that could simultaneously implement OAM mode and turbulence intensity recognition [16]. OAM mode recognition technology combined with turbo channel coding is proposed in [17]; this approach effectively improved the recognition accuracy and reliability of communication transmission. Similarly, many scholars have worked on niche applications of OAM mode detection. In 2018, [PERSON] et al. applied a CNN to learn a received OAM light intensity map under different tilt angles by adding a view-pooling layer. They also used a hybrid data collection technique to improve the performance [18]. Misaligned hyperfine OAM mode recognition was carried out in [19]. Machine learning based the recognition of fractional optical vortex modes in atmospheric environment was studied by [PERSON] et al. [20].