Recognizing Optical Vortex Modes in Ultralow Illuminating Power With Convolutional Neural Network

Vortex beams' orbital angular momentum (OAM) has important application value in optical communication and other fields. Accurate measurement of their OAM topological charges is required for the application of vortex beams in the field of optical communication. As a type of convolutional neural network, DenseNet which combines the “Data Augmentation with Expansion” method, was introduced to measure the fractional topological charge of OAM images in the case of few samples. The OAM images used in this paper generally do not have high brightness, which simulates ultra-long-distance transmission situations in reality. The experiment shows that the classification accuracy of DenseNet network reaches 98.77%, with an average training time of about 1100 seconds, which is short. Our results showcase the potential of convolutional neural network approach to further study the OAM light in the application of free-space optical communication.

been proven depending on the degree of freedom of light being used.Recently, orbital angular momentum (OAM) of the light beams, has attracted increasing attention due to its promising potential application in optical communications.OAM beams could be used as a different physical dimension to generate a set of data carriers in space-division multiplexing (SDM) optical communication systems [1], [2], [3], which do not depend on the other properties of the light beam, such as the wavelength or polarization [4], [5], [6], [7], [8].Allen et al. have unveiled the role of OAM where optical vortex propagates in a paraxial beam.Vortex beams characterized by the factor exp(ilθ), e.g.Laguerre-Gaussian (LG) modes, can carry OAM of l per photon (here topological charge (TC) l is an integer number, represents the reduced Planck Constant) [9], [10], [11].In recent decades, with the development of the generation and manipulation of OAM, widespread applications rapidly emerged ranging from nanoscale and macroscopic optics to quantum optics [12], [13].Although the topological charge l refers to an integer in most OAM studies, a vortex beam with non-integer topological charge with discontinued phase step and broken annular intensity ring has attracted enormous attention especially in the fields of highdimension quantum and classical optical communications [14].In contrast to integer vortex beams, fractional vortex beams can mitigate the limitation of aperture size due to the imaging region of the detectors and effectively scale the communication capacity.
In recent years, deep learning (DL) has been used to identify optical vortex beams.In the case of integer topological charge, several deep neural networks have been presented and have good performance in distinguishing the OAM beams with different integer topological charge [15], [16].Moreover, to qualitatively distinguish the topological charge of the vortex beam, several neural network architectures have been suggested.Liu et al. have demonstrated a DL-assisted recognition approach of fractional OAM beams with the interval of Δl = 0.01 and performed a proof-of-principle experiment of OAM multiplexing technique [17].Cao et al. have proposed an improved residual neural network architecture to obtain the accurate recognition of the OAM beam in the atmospheric environment [18].Yang et al. investigated using convolutional neural networks to recognize OAM integer topological modes in the presence of atmospheric disturbances [19], [20], [21].In the case of turbid media, OAM images are transformed into speckle patterns [22], [23], [24].Huang et al. [25]  and identify the integer topological charges using a convolutional neural network (CNN).Lin et al. proposed the Diffractive Deep Neural Network (D 2 NN) [26], which uses an all-optical method to implement a deep neural network.Based on D 2 NN, many scholars have explored D 2 NN to recognize vortex light OAM modes.Zhao et al. used D 2 NN to classify the integer topological charge of OAM mode, which has the advantage of extremely fast classification speed [27].Zhan et al. used D 2 NN to identify the integer topological charge of OAM mode in oceanic turbulence, expecting that OAM mode recognition technology can be used for underwater wireless optical communication [28].
An important application for OAM mode recognition is freespace optical communication.In which, the introduction of OAM mode recognition technology can improve communication capacity.Compared with the integer topological charge, the OAM mode recognition technology of fractional topological charge can greatly improve communication capacity.Assume that in a free-space optical communication system, other factors remain fixed and it can accommodate n users to communicate simultaneously.When OAM technology is introduced and the topological charge is an integer, suppose the set of topological charge values is {2, 3, 4, 5, 6, 7, 8 }, then, 7n users can communicate simultaneously.If it is expanded to fractional topological charge and the topological charge interval Δl = 0.1, then the set of topological charge values is {2.0, 2.1, 2.2,..., 7.9, 8.0}.Theoretically, 61n users can communicate simultaneously at this time, which greatly improves communication capacity.
In free-space optical communication, when the beam is transmitted over a long distance, the brightness of the beam will gradually decrease.How to identify OAM mode under weak brightness is also an urgent problem that needs to be studied.At the same time, in real communication, there may not be so many samples.It is another urgent problem to identify OAM mode with few samples.
However, the current studies are mainly focused on identifying the topological charges of integer at higher luminance with more samples.The recognition of fractional topological charges at lower luminance and with few samples has been less studied.
In this work, we have proposed a method based on an excellent CNN model (DenseNet) to recognize the fractional vortex beams, especially when the power of the illuminating laser is weak enough to be even no more than 1 microwatt.The results show the applicability of the DenseNet design even for extremely low light illumination and reveal the potential of advanced neural networks in the diagnosis of complicated optical fields.

II. EXPERIMENTAL SETUP AND PRINCIPLE
We first briefly review the experimental method for collecting the OAM images with fractional TCs (as shown in Fig. 1) [29].In order to simulate the situation where the brightness of the laser beam is greatly reduced when the laser beam is transmitted over a long distance in free space, a very low-power laser source is used.In order to accurately obtain the OAM mode images, the experiment was conducted in a dark room, which is why the OAM images have dark background.A laser with a wavelength of 780 nm and a power of about 0.27 μW was used as the light source.A computer was used to transfer the prepared phase holograms to a spatial light modulator (SLM) screen, and a Laguerre-Gaussian laser beam was incident on the SLM, which was then shaped into corresponding vortex beams, each of which has a corresponding OAM mode.The vortex beam is transmitted for a certain distance in free space, and the transverse light intensity distribution of the vortex beam, that is, the OAM mode can be photographed with a CCD camera.By changing the phase holograms on SLM, OAM mode images can be obtained with different fractional topological charges.
The label of the dataset is the TC l in the ranges of [−2.0, −8.0] and [2.0, 8.0] with the interval of Δl = 0.1.After modulated by the SLM, the intensity patterns of 122 fractional OAM modes were recorded by a CCD camera.For each OAM mode, 10 images with different initial phase had been collected.The dataset of the fractional OAM beams was collected in very low illuminating conditions and have few images for DL training.Our aim in this paper is to develop an efficient CNN architecture for recognizing the factional OAM beams even with very dark background and few samples.
In the experiment 976 images were used as training sets and 244 images as test sets.In each category, there were 8 images in the training set while 2 images in the test set.The upper right corner of Fig. 1 shows images when the topological charges l in this data set are positive integers.It can be observed from the figure that the OAM images used in this paper generally do not have high brightness.This is because ultra-low laser power was used in this paper to simulate ultra-long-distance transmission situations in reality.
Fig. 2 shows a comparison of images when the topological charge is a fraction in the dataset.As can be seen from the images, it is sometimes difficult to determine the difference when the topological charge values differ by 0.1.
Used in this paper as shown in the lower left corner of Fig. 1, the DenseNet network is a convolutional neural network proposed by Gao Huang et al. in 2017.The architecture of DenseNet is shown in Fig. 3(a) [30].In the figure, BN-ReLU-Conv represents the combination of Batch Normalization (BN) [31], Rectified Linear Unit (ReLU) [32], [33], and Convolution.There are several Dense Blocks in Fig. 3(a).The specific internal structure of the Dense Block is shown in Fig. 3(b) [30].H L (.) in Fig. 3(b) is a composite transformation that represents a combination of BN-ReLU-Convolution.The subscript L represents the index of the layer, and x L represents the output of layer L. In the Dense Block, the Lth layer receives the feature maps of all previous layers, expressed as follows: In the formula, [x 0 , x 1 , . .., x L−1 ] represent the concatenation of x 0 , x 1 , . .., x L−1 .

III. EXPERIMENTAL RESULTS AND DISCUSSION
The 1220 OAM images obtained in this paper's experiment are divided into training and test sets and put into the DenseNet network for operation.For the training sets, two methods of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.data augmentation are used: The first method is "Random Data Augmentation," and the second method is "Data Augmentation with Expansion."For each original image in the training set, either the first or second data augmentation method is used.No data augmentation methods were used for the testing set.
The first method "Random Data Augmentation" and the experimental running results are described in detail below.For the training set, the following five methods are used in sequence in the Random Data Augmentation method: Flip left and right with a 50% probability; Flip up and down with a 50% probability; Perform an angular rotation with a 50% probability, and the rotation angle is a random number between [−30, 30]; Change the brightness by 10% with a 50% probability, which means the brightness is a random number between 90% and 110% of the original brightness; Transform the contrast by 10% with a 50% probability, which means the contrast is a random number between 90% and 110% of the original contrast.
Transforming an image with a 50% probability means there is a 50% chance of maintaining the original image.After five transformations, a small portion of the original images can be retained in the training set, (1/2) 5 = 32, meaning that 1/32 of the images have not undergone any transformation.At the same time, 1/32 of the images have undergone five transformations, some images have undergone one transformation, some images have undergone two transformations, some images have undergone three transformations, and some images have undergone four transformations.
The total number of training images remains unchanged at 976.Due to the fewer training images, the training time is much less than used in the "Data Augmentation with Expansion" method.The hardware environment for running this experiment was 2 GeForce RTX 3080 GPUs.The training parameters in the "Random Data Augmentation" method are: Batch Size = 8, Epochs = 80, Learning Rate = 0.001.
To evaluate the performance of the network, we use the notion of accuracy.The formula for accuracy is as follows: and (3)  The results of one run are shown in Fig. 4(a).From which, it can be seen that, at the beginning, the training accuracy and testing accuracy are relatively low.As the training progresses, both show an upward trend.Finally, the training accuracy stabilizes at 100%, while the maximum testing accuracy is only 50.0%, which is unstable and presents oscillation.Fig. 4(b) indicates similar trend in the training process.The final training accuracy is 100%, while the maximum testing accuracy is only 25.0%, which presents oscillation too.From these results of two runs of Random Data Augmentations, it can be seen that due to too little training data, the final testing accuracy is not ideal and far from meeting our expectations.
The results of the last five runs with the Random Data Augmentation method are shown in Table I.From the records in the table, it can be seen that when epoch = 80, the average Testing Accuracy of the Random Data Augmentation method is 0.4000, which means that the average classification accuracy is 40.0%.The average training speed of the Random Data Augmentation method is 135.2 samples per second.The average total training time is 574.8 seconds.
The average Testing Accuracy of the five runs was 40.0%, far below the results we had expected.From Fig. 5, it can be seen that the testing accuracy presents fluctuations and the results are unstable.For this data set, the amount of data is small and there are many categories, with only 10 images in each category.Experiments show that using "Random Data Augmentation" does not work well in this very few-shot situation.
The second method, "Data Augmentation with Expansion", and its experimental results are described in detail below.The "Data Augmentation with Expansion" method is essentially the same as the "Random Data Augmentation" method.With the only difference in that after the image undergoes data augmentation transformation, the "Data Augmentation with Expansion" method retains both the original image and the transformed image, thereby expanding the number of images.
The "Data Augmentation with Expansion" method significantly increases the amount of data by retaining both the original image and the transformed image.Due to five data transformations, each original image is expanded to six images.The expanded training set has 5856 images.No data augmentation methods were used for the testing set.The training parameters in the "Data Augmentation with Expansion" method are: Batch Size = 8, Epochs = 25, Learning Rate = 0.001.
The results of two runs with the "Data Augmentation with Expansion" method are shown in Fig. 5, where the solid blue Table II shows the records of the last five runs.From the records in the table, it can be seen that when epoch = 25: The average Testing Accuracy for DenseNet is 0.9877, which means, Fig. 6.Confusion matrix map of a run using the "data augmentation and expansion" method.The abscissa in the confusion diagram is the "Predicted Topological Charges", and the ordinate is the "True Topological Charges".Whether they are the "Predicted Topological Charges" or the "True Topological Charges", their negative topological charge labels start from -2.0 and end at -8.0, with a difference of -0.1; their positive topological charge labels start from 2.0 and end at 8.0, with a difference of 0.1.Both "Predicted Topological Charges" and "True Topological Charges" have 122 labels corresponding to 122 categories.Due to the large number of categories, this confusion matrix is relatively large.In order to facilitate intuitive observation, the larger the number in the confusion matrix map, the darker the color.
the average classification accuracy reached 98.77%.The average training speed of the DenseNet network is 134.1 samples per second.The average total training time of the DenseNet network is 1090.6 seconds.Fig. 6 is the confusion matrix map of a run using the "Data Augmentation with Expansion method".The abscissa in the confusion diagram is the "predicted label", and the ordinate is the "true label".In the confusion matrix map, the numbers on the diagonal represent the number of correct classifications, and the numbers off the diagonal represent the number of incorrect classifications.It can be seen from the confusion matrix map that the numbers on the diagonal are relatively large, indicating that the number of correctly categorized samples occupies the majority.Most of the numbers off the diagonal are 0, indicating that there are very few misclassified samples.If you run the "Data Augmentation with Expansion method" program multiple times to obtain a confusion matrix diagram, the numbers in the confusion matrix diagram will be different, but they all follow the above rules.
From the above results, it can be seen that using the DenseNet network with the "Data Augmentation with Expansion" method, the training time is relatively short.Using our experimental equipment, on this dataset, it only takes about 1100 seconds to complete the training.The average classification accuracy rate has reached 98.77%, which means it can be well qualified for the classification task of fractional topological charge of OAM images.

IV. CONCLUSION
In this paper, The DenseNet convolutional neural network is used to classify these OAM images with fractional TCs.Two methods of "Random Data Augmentation" and "Data Augmentation with Expansion" are explored for the identification of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
fractional topological charges in OAM images.For this dataset, there are only 10 pictures in each category due to the small amount of data.In this very few-shot case, we have shown that "Random Data Augmentation" does not work well.However, the "Data Augmentation with Expansion" method expands the number of images and is competent for the classification task of this dataset.With the fast training speed, the DenseNet network can be trained faster, even for a new task, and it can be put into use after the training is completed, even if the OAM images are quite different from this dataset.The OAM images used in this article have a dark background indicating a weak intensity.This simulates the situation of ultra-long-distance transmission in reality.Therefore, the method used in this article has practical value.In summary, this paper has proposed a DenseNet network that combines the "Data Augmentation with Expansion" method, which can be used to classify low-brightness OAM images with fractional topological charge in the case of few-shot, and has a certain application prospect in the field of vortex optical communication.It should be acknowledged that there are still shortcomings in the research of this article.Next, we will further explore to obtain OAM mode images at lower power.What's more we will further study changes of the model in performance when the power changes.

Fig. 1 .
Fig. 1.Schematic diagram of the experiment.A laser is used as the light source, which is modulated by a Spatial Light Modulator (SLM) to generate OAM beams with the desired fractional topological charge, and the luminance maps (i.e., the OAM mode images) of these OAM beams are shown in the upper-right dashed box.The CCD camera records these OAM mode images and transmits them to the computer.The lower left dashed box indicates the use of the DenseNet network to categorize the OAM mode images.The classification categories, i.e., topological charges, range from l = ±2.0,l = ±2.1,..., to l = ±8.0,for a total of 122 categories.Running the DenseNet network in a computer allows each OAM mode image entered to be categorized into one of these 122 categories.

Fig. 2 .
Fig. 2. Comparison of OAM images with of fractional topological charge values (random values within the range of [0,1] radians in phase).In the figure, l represents topological charge, θ represents phase.

Fig. 4 .
Fig. 4. Training accuracy diagram of two runs with the Random Data Augmentation method shows that (a) the Training Accuracy is 1.0000, and the Testing Accuracy is 0.5000; (b) the Training Accuracy is 1.0000, and the Testing Accuracy is 0.2500.

Fig. 5 .
Fig. 5. Training accuracy diagram of two runs with "Data Augmentation with Expansion method".The results show that (a) the Training Accuracy is 1.0000, and the Testing Accuracy is 0.9877; (b) the Training Accuracy is 1.0000, and the Testing Accuracy is 0.9959.TABLE II THE RESULTS OF THE LAST FIVE RUNS OF THE "DATA AUGMENTATION WITH EXPANSION" METHOD

TABLE I RESULTS
OF THE FIVE MOST RECENT RUNS USING THE RANDOM DATA AUGMENTATION METHOD represent the training accuracy curve, and the dotted red lines the testing accuracy curve.