Results and Discussions Fig. 6 shows the training results and confusion matrix of 0.01-order fractional POVB after a MI under the ideal environment. In Fig. 6 (a), the solid line is the training accuracy curve, indicating that the training accuracy reaches 100% after three epochs. The dashed line is the verification accuracy curve, indicating that the verification accuracy reaches 100% after eight epochs. In Fig. 6 (b), the results of the confusion matrix show that the test accuracy is 100%. Fig. 8 (a) and Fig. 8 (b) show the training results and confusion matrix of 0.01-order fractional POVB after an MI under a sector-shaped opaque obstacle of 90°. In Fig. 8 (a), the solid line is the training accuracy curve, indicating that the training accuracy reaches 100% after three epochs. The dashed line is the verification accuracy curve, indicating that the verification accuracy reaches 100% after eight epochs. In Fig. 8 (b), the results of the confusion matrix show that the test accuracy is 100%. Fig. 8 (c) and Fig. 8 (d) show the training results and confusion matrix of 0.01-order fractional POVB after an MI under a sector-shaped opaque obstacle of 180°. In Fig. 8 (c), the solid line is the training accuracy curve, indicating that the training accuracy reaches 100% after five epochs. The dashed line is the verification accuracy curve, indicating that the verification accuracy reaches 100% after eight epochs. In Fig. 8 (d), the results of the confusion matrix show that the test accuracy is 99.5%. According to the above results, we can prove that our method is feasible and efficient.

Perfect optical vortex beams (POVBs) are widely applied in particle manipulation, optical communication, and laser material processing for the constant spot size under different topological charges (TCs). Compared with the integer-order POVB, the fractional POVB which is a dark hollow beam with an opening in the angular intensity distribution is more flexible in particle manipulation and beam shaping. In addition, the fractional POVB carries the information with fractional TC orders and has a greater communication capacity. In order to realize the above applications of the fractional POVB, the accurate recognition of the orbital angular momentum (OAM) mode is of great significance. In this paper, a method combining convolutional neural network (CNN) and multiaperture interferometer (MI) is proposed to recognize the modes of 0.01-order fractional POVB. Experimental results show that the recognition accuracy of 0.01-order fractional POVB reaches 100% under an ideal environment. Under the condition of a sector-shaped opaque obstacle of 90° and 180°, the recognition accuracy of 0.01-order fractional POVB reaches 100% and 99.5%, respectively. This study provides a new method for recognizing 0.01-order fractional POVB, which is of great significance for the application and promotion of this beam.

Our method for fractional POVB recognition combines an MI and a CNN. First, the POVB to be detected is sent to the MI, and interference patterns are collected at the output of the interferometer. In this work, the MI is a seven-aperture plate that is realized through a spatial light modulator (SLM). The aperture radius r0 equals 0.25 mm. The interference patterns have a one-to-one correspondence to the TC of the input beam. Secondly, a CNN model is trained with the interference patterns of 0.01-order fractional POVB. The network structure is shown in Fig. 3, and it is a six-layer network consisting of four convolutional blocks and two fully connected layers. The full dataset of the CNN model contains 4000 intensity images, which are labeled by 10 different TCs from l=8.01 to l=8.10. The intensity images of POVB are collected by a CCD. The dataset is divided into the training set, validation set, and test set according to the ratio of 7∶2∶1. The training set and validation set are put into the designed model in this experiment for training, while the test set is not placed into the model training but is used to test the robustness of the model. Finally, the trained model is tested by the test set. The sector-shaped opaque obstacle in a non-ideal environment is simulated by SLM. The number of collected datasets and the experimental procedure in the non-ideal environment case are the same as those in the ideal environment.

In this paper, a method combing CNN with MI is proposed to accurately classify 0.01-order fractional POVB under ideal and non-ideal environments. This method utilizes the one-to-one relationship between the TC of the input beam and the intensity pattern of the interferometer and the classification ability of CNN to accurately classify the 0.01-order fractional POVB. The experimental results show that in the ideal environment, the recognition accuracy of this method for 0.01-order fractional POVB reaches 100%. For the non-ideal environments with a sector-shaped opaque obstacle of 90° and 180°, the recognition accuracy of this method for 0.01-order fractional POVB is 100% and 99.5%, respectively. The proposed method provides a new scheme for the recognition of fractional POVB. We hope that it can be helpful in the applications of fractional optical vortices.