1. Introduction

As a result of the exponential growth in demand for optical data transmission, use of traditional information carriers such as the amplitude [1–3] and phase [4–6] can no longer provide sufficient channel capacity. To overcome this difficulty, a great deal of research attention has been devoted to searching for new degrees of freedom to carry the required information. Among the novel types of light field under consideration, spatially structured light [7–12]has recently been demonstrated as a new degree of freedom that is suitable for carrying information. For example, orbital angular momentum (OAM) modes with different topological charges that are orthogonal to each other can theoretically provide an unlimited number of channels using the OAM multiplexing technique. Because of this advantage, OAM light is regarded as a potential information carrier and has recently been investigated intensively. Additionally, OAM light has already been applied in fields including optical links [13–15], imaging [16,17] and optical communication [18,19], and other integrated OAM emitters and receivers have also been studied heavily in recent years [20–23].

Specifically, for optical communication applications, the OAM of light is considered to be a promising degree of freedom for data multiplexing in both free space [24] and optical fibers [25,26]. For long-distance information transmission, optical fibers are critically important. However, because of the dispersion and scattering introduced by these optical fibers, the wavefront of an OAM beam will be scrambled during propagation; this in turn seriously limits detection of the beam, the mode numbers supported by the beam, and the transmission distance. Consequently, intensive efforts have been devoted to increasing the number of OAM modes supported by these fibers. Bozinovic et al. proposed a specially designed few-mode vortex fiber [25] that supports up to four low-order OAM modes at a single wavelength. Similarly, Brunet et al. proposed an optical fiber [26] with an air core and an annular index profile that supports up to nine modes. In addition to these specially designed fibers, use of commercial multi-mode fibers (MMFs) has also been shown to be a promising approach to increase the communication capacity through use of multiplexing techniques [18,27,28]. However, although OAM-based data transmission has been implemented on both the specially designed optical fibers and conventional MMFs, neither method has been able to support large numbers of OAM modes to date. Moreover, in all the approaches described above, specially designed optical fibers or self-made devices were required that further limited their application in real life scenarios.

To overcome the limitations described above, computational methods such as the speckle-correlation scattering matrix (SSM) approach [28] have been proposed for OAM mode transmission in conventional fibers. These methods calculate the transmission matrix (TM) of a given scattering medium [24] or MMF [28,29], and then use the SSM to achieve OAM-multiplexed data transmission through a commercial MMF. However, although this approach uses only conventional devices, the method is strongly reliant on the correlation coefficient of the speckle patterns, and because the speckle patterns produced by neighboring OAM modes are very similar, use of fractional topological charges, for example, remains challenging. Furthermore, the bit error rate (BER) increases when using high-order OAM modes because the phase distribution near the singularity point becomes more complex. In addition, the scattering introduced by fiber imperfections becomes increasingly severe for longer fibers. The longest transmission distance reported to date when using the TM method is only 35 m [28] and the number of OAM modes supported was only 12. More importantly, because external environmental fluctuations can change the TM of a long optical MMF dramatically, use of such a technique remains challenging in real applications, in which environmental fluctuations are unavoidable.

In this work, we demonstrate an artificial intelligence (AI)-assisted OAM-multiplexing data transmission system based on a 1-km-long conventional commercial MMF. The AI network was used to predict the OAM topological charges of an incident beam from a recorded speckle pattern in the presence of external fluctuations, thus enabling robust retrieval of the encoded information. We demonstrate that our system can support up to 100 OAM modes with topological charge intervals as small as 0.01 in parallel, which is the largest number of modes supported to date, to the best of our knowledge. In addition, the work shows that a single large network can be replaced with multiple small parallel networks, allowing the same or better performance to be achieved with far fewer training data, which is beneficial in saving computational capacity. Finally, we demonstrate high-fidelity image transmission via 16-bit multiplexed fractional OAM channels through a 1-km-long commercial MMF. In addition, a multiple parallel network architecture that enables 16-bit data transmission by running four 14-bit networks in parallel has also been studied.

2. Materials and methods

2.1 Experimental set-up and characterization

The experimental set-up is shown in Fig. 1(a). A He-Ne laser (633 nm; Thorlabs: HNL050L) is used to generate a linearly polarized monochromatic incident beam. The laser beam first passes through a half-wave plate for polarization control, and is then expanded using a lens pair (L1 and L2, with focal lengths of 30 mm and 180 mm, respectively). The beam is then directed to the spatial light modulator (SLM) using a nonpolarizing beam splitter cube (Thorlabs: CCM1-BS013) and modulated using spiral holograms [30,31] loaded onto the SLM. The modulated beam is then directed to the MMF by the same cube and coupled into a commercial gradient index MMF (105/125 ${\mathrm{\mu} \mathrm{m}}$, 1 km length) through a focusing lens with a focal length of 50 mm. A complementary metal-oxide-semiconductor (CMOS) camera (Thorlabs: CS505MU) is used to capture the speckle patterns that emerge from the distal end of the fiber. A convolutional neural network (CNN) [32–37]is then used to identify the OAM topological charges of the incident beam from a received speckle pattern (a description of the details of CNN is given in the Methods section). We subsequently collected the speckle dataset and trained the CNN to identify the OAM superposition modes and thus recover the transmitted data. Multimode fiber is sensitive to environmental fluctuation, especially when the fiber is long as external environment changes the transfer matrix of a MMF (It can be found in Fig. 1(b) of the manuscript and Fig. S1 of Supplement 1). During the experiments, the fiber is placed on an optical table, but no special attention was paid to improving either the mechanical stability of the fiber or the temperature of the room. The temperature varied from 21.6°C to 23.6°C during the 48 h measurement period. As mentioned previously, because the TM of a MMF can change rapidly with environmental fluctuations, the same incident topological charge may output many different speckle patterns with completely different appearances, as shown in Fig. 1(b). The details of how these speckle patterns vary with time and the incident topological charge are discussed in the Supplement 1. It is worth pointing out that, a multimode fiber used is long and very sensitive to environment fluctuations. When larger temperature fluctuation occurs, if there are enough training data, it would be expected that the network can still generalize to it (dataset size may increase dramatically).

 

Fig. 1. Experimental setup and speckle patterns. (a) Illustration of experimental setup (b) Demonstration of recorded speckle pattern variation with time under same incident topological charge (Details are discussed in Supplement 1)

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2.2 Methods

2.2.1 Network configuration

In our application, because the speckle patterns are much more complex than natural images, we prefer to use a deeper network to ensure that the CNN has sufficient capacity to distinguish the speckle patterns. Therefore, we adopted a canonical-type deep learning model to perform the image handling. A CNN with multiple layers can capture high-level image features, and a deeper network has a more powerful ability to extract the abstract features.

The CNN architecture is shown in Fig. 2. We used a deep residual network [38] to construct a deeper network, which can help to avoid problems such as gradients that vanish and explode [39] during the backpropagation process of CNN training. This helped us to build a deeper and more powerful network that would be robust with respect to environmental fluctuations.

 

Fig. 2. Architecture of the CNN. (a) The input image size is 224 × 224 pixels. The numbers and sizes of the feature maps extracted from the hidden layers are indicated by the boxes. The conv1 box contains a 7 × 7 convolutional layer with two strides; a batch normalization layer; a rectified linear unit (ReLU) activation function layer; and a max pooling layer with two strides. (b), (c), (d), and (e) represent the “bottleneck” building blocks used in conv2_X, conv3_X, conv4_X, and conv5_X, respectively, where the number x denotes the block number. The deep residual function of the CNN is mainly dependent on the “bottleneck” building block; this has been discussed in detail by He et al. [38]. The number of convolutional kernels is denoted in each block. Down-sampling is performed by using conv3_1, conv4_1, and conv5_1 with a stride of two.

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2.2.2 Training

All speckle patterns recorded by the camera were used to form the dataset, which was split into a training set, a validation set, and a test set at random, with fractions of 80%, 10%, and 10%, respectively. The training set was used to train the CNN and upgrade the parameters in each layer of the network model, the validation set was used to select the best model during the current process, and the test set was used to evaluate the performance of the selected model of the CNN. The ultimate goal is to obtain a well-trained CNN that can predict the OAM-multiplexed modes and also recover the information from the speckle patterns provided.

To train the CNN for the OAM-multiplexed mode classification task, the speckle patterns were first down-sampled to 224 × 224 to match the CNN’s input requirements. During the training process, we also minimized the cross-entropy loss function, as follows:

In this paper, the training of the CNN was performed on Python with hardware that included a NVidia Tesla V100-DGXS-32GB graphics processing unit (GPU), 250 GB of random access memory (RAM), and an Intel Xeon E5-2698 v4 central processing unit (CPU). After training, the CNN was considered to be established and was easily able to predict the incident OAM-multiplexed modes on the basis of the speckle patterns. In addition, because the trained network is only a few MB in size, it can be implemented easily using a conventional personal computer (PC).

3. Results of topological charge recognition

3.1 Integer topological charge recognition

As a proof-of-concept experiment, we first demonstrate the ability of a trained neural network to recognize individual OAM modes with either integer or fractional topological charges. In this study, the topological charge ranges from 1 to 50 at integer intervals (It would be problematic if the topological charge keep increasing, since the beam size maybe larger than the core size of the fiber, a lens with higher N.A. is needed in this case), with each mode including 2000 recorded images. In total, 100000 speckle images were used as the dataset. This dataset was then randomly divided into training, validation, and testing sets with fractions of 80%, 10%, and 10%, respectively. The speckle patterns were down-sampled to 224×224 pixels to match the CNN’s input requirements (details of the training and configuration of the network are presented in the Methods section). After training, the network is used to recognize incident topological charges from a recorded speckle pattern that has not been used during training. The results are presented in Fig. 3. To give a visualization of the trained CNN on mode recognition, a t-Distributed Stochastic Neighbor Embedding (t-SNE) [40] method has been implemented. Where t-SNE is a nonlinear method to project high-dimensional feature into a low-dimensional space such that the neighborhood probabilistic distribution of the higher-dimensional data is preserved in the low-dimensional vector space. The visualization of high-level feature vectors extracted by the CNN during the training process are shown in Fig. 3(a). It can be observed in Fig. 3(a) that, at the first few epochs of training (left), the recognition accuracy is only 21%. The dots with different colors (speckle patterns generated by different incident wavelengths) were not regrouped into different clusters and most of them are randomly distributed. It indicates that a network before well trained can hardly recognize the input speckle patterns to the corresponding incident wavelengths correctly. After more training epochs, the recognition accuracy increases, which is consistent with Fig. 3(a) (middle, and right). At the end, all the dots with the same color regroup to clusters and separate with dots of other colors, showing that the network can learn the high dimensional features of the speckle patterns and generalize during the training. Figure 3(b) shows that the training and testing accuracy reaches 100% after 25 epochs, thus indicating that an appropriately trained network can recognize the OAM mode accurately from a received speckle pattern. To provide a further demonstration of the network performance, a confusion matrix is shown in Fig. 3(c) from which all the tested OAM modes have been recognized correctly, thus illustrating the ability of the trained CNN to extract the general features of the vortex beams from the speckle patterns. Note that only the speckle patterns in the testing set that had not been used during network training were used to evaluate the performance of the trained CNN. This performance indicates that the network is suitably generalized to allow it to recognize the possible speckle patterns that can be generated by the same incident topological charge, thus confirming the potential of the proposed AI-assisted method of data transmission based on OAM.

 

Fig. 3. Identification of OAM modes with integer topological charges. (a) t-SNE visualization of speckle patterns in the testing set during the training process. (b) Training and testing accuracy characteristics of the CNN for speckle patterns generated with specific OAM modes. (c) Confusion matrix showing the excellent performance of the CNN in discriminating OAM modes ranging from 1 to 50.

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3.2 Fractional topological charge recognition

As the topological charge values increase, the complexity of the phase distribution around the singularity grows, which then leads to heavier diffraction effects and difficulties in focusing and coupling with the fiber. One important approach that can be used to expand the system capacity is to use more OAM states but with smaller topological charge differences, which means use of OAM beams with fractional topological charges. This approach would require the AI network to provide much higher resolution for recognition of the topological charges. In this section, we demonstrate that an appropriately trained network can also recognize fractional topological charges with intervals as small as 0.01. Here, 100 groups of speckle patterns with different OAM modes with intervals of 0.01 are collected as a dataset, where each pattern contains 2000 images. To demonstrate the generalization ability of the network, we selected discontinuous topological charge values here. Ten central topological charges containing both positive and negative topological charges, comprising ${l_n}$ = −50, −40, −30, −20, −10, 10, 20, 30, 40, 50, are used in this study. Speckle patterns generated by beams carrying 10 values close to each central topological charge are collected as a dataset (e.g., −49.96, −49.97, …, −50.05 for the central charge at $l = -50$), and the dynamic range of the network is as shown in Fig. 4(a). 100 groups of speckle patterns are collected in total. The datasets are then split and the networks are trained in the manner described in the previous section. As shown in Fig. 4(b), the test accuracy reaches 98.78% in this case. This result indicates that the system can also use fractional topological charges as information carriers, which in turn is beneficial for broadening of the system bandwidth.

 

Fig. 4. Identification of OAM modes with fractional topological charges. (a) Dynamic range of the network. Ten central topological charges with intervals of 10 are used in this study, and each group contains 10 topological charges with values that are close to each central topological charge at intervals of 0.01. (b) Confusion matrix of the test results, where the recognition accuracy reaches 98.78%.

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Note that although the training accuracy still reaches almost 100%, unlike previous cases in which all topological charges could be recognized correctly, when the size of the dataset increases (i.e., the number of groups increases), there are rare cases in which the topological charges from the test set cannot be recognized correctly, particularly for large topological charge values, as illustrated in Fig. 4(b). A part of the reason is surly because beam carries higher topological charges have more complex wave front. This effect would become increasingly severe when the number of topological charges used in the network increased further, and would thus limit the band expansion ability of the system. Two different approaches can be used to overcome this problem. The first is to use a more complex network, which means use of a network with more layers or with a more complex architecture as such a network may provide better feature detection abilities. The second approach is to increase the number of images used for the training process to enable the network to determine more statistical features of the speckles and thus be better generalized for all possible speckle patterns. There are clear drawbacks for the two methods proposed above, in that more powerful hardware and much longer training times would be required. In the following section, we will discuss how the adoption of a parallel network configuration can resolve the problem.

3.3 Bandwidth expansion with parallel networks

For the previous network that supported 100 fractional topological charges, 2,000 figures were used for training of each topological charge, which led to use of 200,000 figures in total. However, when using a system with a large bandwidth (in our case, with many different topological charge values), it is always desirable to use fewer data for the training of each topological charge. However, because the network uses statistical features to sort the input speckle pattern into pre-defined groups, the network would then require a specific number of images to generalize to the entire dataset. If insufficient numbers of images are available for the training, the trained network would then offer a poorer performance.

Here, we propose a new approach involving use of multiple parallel networks in which it is only necessary for each network to classify 10 values around a central topological charge; This approach has a much smaller Vapnik–Chervonenkis dimension [41] when compared with the previous case, thus allowing us to train each topological charge using fewer data because each network is trained to complete a simpler mission (with a smaller bandwidth). We thus ensure that these multiple smaller networks provide the same performance as an appropriately trained larger network. More specifically, we initially divide the dataset into a few subsets; each subset is then used to train an individual network, as shown in Fig. 5(a). All trained networks are then used in parallel to form a system, as illustrated in Fig. 5(b). In this way, the system contains multiple networks and each of these networks covers different topological charges. In general, when a speckle pattern is sent to a single trained network, the CNN outputs a vector in which each element is the probability that the topological charge of the incident beam matches the predicted topological charge (where the sum of all elements is equal to ‘1’). For a non-parallel configuration, the network then automatically selects the best match as the output. In contrast, in the proposed parallel network configuration, we asked each network to output a predicted topological charge only if the probability exceeds 98%; otherwise, the network will then conclude that the topological charge of the incident beam is not included in the specific bandwidth covered by this network (although, of course, it may appear in the other parallel networks). Because the speckle pattern is sent to all the networks simultaneously, the topological charge of the incident beam is guaranteed to be identified accurately, as long as the value in question has been trained by one of the parallel networks. In this way, the system bandwidth can be extended by a factor of N if N networks are used in parallel in the system.

 

Fig. 5. Parallel networks to expand the topological charge bandwidth without decreasing performance. (a) Illustration of training process of a parallel network system (b) Schematic showing how the parallel networks recognize an input speckle pattern that corresponds to a specific output. A given speckle pattern was sent to all trained networks in parallel and a corresponding value was output. (c) Performance of parallel networks in expanding the topological charge bandwidth, where each red box contains 10 topological charges with intervals of 0.01 (the axis is not to scale here).

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Here we present an experiment to verify the feasibility of the proposed approach. In this experiment, the same dataset that was used for Fig. 4 was applied. First, we select 100,000 figures (1,000 for each topological charge) at random to act as a new dataset. The size of this dataset is thus only half the size of that in the previous case. This dataset is then divided into 10 groups to train 10 different networks. The training process is the same as the method described in the previous section.

The test results are shown in Fig. 5(c); the confusion matrix shows that the recognition accuracy of speckle patterns in test set reaches 99.63%, including the large topological charge values. When compared with the previous case, because the task assigned to each individual network is easier, the performance of each network remains excellent, although fewer data are used overall. When combined, the parallel networks provide a system that can recognize all the incident topological charges accurately. In addition, the proposed approach avoids the need for huge numbers of dataset and thus saves the time consumed by dataset acquisition and reduces the hardware requirements. The results indicate that use of parallel networks to expand the bandwidth (i.e., the OAM mode numbers) of the system without requiring huge amounts of additional resources (typically time and hardware) is feasible. Use of parallel networks is thus a promising way to achieve a broader bandwidth without diminishing the system performance.

With the above approach, the bandwidth of the system is determined by the resolution of the topological charge, the number of classes for each parallel network and the threshold at which the parallel network predicts the topological charge. When the topological charge interval is larger, it is easier to achieve a broader bandwidth with high performance. When the number of classes for each parallel network is very big, the performance may be less satisfactory. When the threshold of the parallel network system is set to be very small, the accuracy of the system will be poor.

4. Image transmission with proposed system

In this section, we demonstrate that the proposed system can perform data transmission tasks using OAM multiplexing techniques, which represents a further step toward application of the system to data transmission. It has been demonstrated that both proposed configurations can perform the data transmission task successfully. More specifically, in the first case, a network with a 16-bit depth is trained, while four networks with 14-bit depths are trained in the second case. Comparison of the performances and complexities of the two configurations thus becomes possible.

4.1 Principle of OAM-multiplexing

We here first give some details of OAM-multiplexing. The electric field of a vortex beam can be described as follows:

 

Fig. 6. Illustration of OAM-multiplexing information transmission system. Information is encoded into an OAM superposition state. The superposed vortex beam is then coupled into a commercial MMF and propagates through the fiber. The speckle pattern generated at the distal end of the fiber is recorded using the camera. A CNN is then used to convert the received speckle pattern into the initial data.

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4.2 16-bit image transmission with single network

To provide a further demonstration of the data transmission ability of the proposed method, a 200×200 16-bit colored image of a dog has been transmitted using our system. A 16-bit image is used in this case because it is the most common way to encode images. Images coded using higher numbers of bits can also be transmitted by the system by using more OAM channels to perform the multiplexing. In the experiment, each colored pixel is encoded into a weighted matrix composed of three groups of 16 binary bits that correspond to the three primary colors (i.e., red, green, and blue) using 16 different OAM modes ($l = \textrm{from}\; 1.31\; \textrm{to}\; 1.46$) with adjacent intervals of $\Delta l = 0.01$.

In the encoding process, every bit takes a binary value of 1 or 0, as defined by the value ${c_n}$ given in Eq. (3). As shown in Fig. 7, an OAM superposition state containing 16 different topological charge values that correspond to the intensity value of a pixel is obtained. For example, a 16-bit OAM super-state that is encoded as “0100000111000100” represents the pixel intensity value of 16836. In this way, any color pixels can be encoded using the multiplexing OAM modes. All $200 \times 200 \times 3 = 120000$ data were sent to the system sequentially, and the speckle patterns were sent to the trained CNN to predict the superposed OAM modes and then convert them into the intensity value of the corresponding pixel.

 

Fig. 7. Details of transmission of OAM-encoded data of a color image through a commercial 1000-m-long MMF using the AI-assisted OAM multiplexing system. Each color pixel of the dog image was encoded into a weighted matrix of three groups of 16 binary bits by multiplexing 16 different fractional OAM modes. The trained CNN can then decode the information from the sequential speckle patterns. The red/green/blue (R/G/B) channels are all encoded in the same way and are transmitted in sequence.

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As shown in Fig. 7, the dog image has been transmitted through the system successfully, and a bit error rate (BER) of 0.02% has been achieved, where the BER is defined as the ratio of the number of incorrect pixels to the total number of pixels. It should be noted that, unlike the case shown in Fig. 4, the speckle patterns in this case are generated by multiplexed OAM modes rather than by a single mode; therefore, the phase distribution of the field is more complex here. However, after training, the system can still provide a satisfactory performance. In addition, during the experiment, no specific care has been taken with environmental control or the mechanical stability of the system. The performance of the system under these experimental conditions indicates that the trained CNN can extract the intrinsic features of the given speckles successfully and reject the environmental fluctuations. This result further confirms that AI-assisted OAM multiplexing data transmission represents a promising method to transmit data through commercial MMFs under real-life conditions because environmental fluctuations are unavoidable under such conditions.

4.3 16-bit image transmission with four parallel 14-bit networks

In this section, we demonstrate that the parallel network configuration can also perform the proposed data transmission task, and that the total number of figures required for training can be reduced when using this configuration. More specifically, four parallel 14-bit networks are trained to transmit the same 16-bit image shown in Fig. 7. In this case, each 14-bit network presents ${2^{14}}$ data levels, and the intensity values presented by these networks range from 0 to 16383, from 16384 to 32767, from 32768 to 49151, and from 49152 to 65535. A 16-bit image (with values ranging from 0–65535) can thus be encoded fully into four 14-bit networks. To train this network, we multiplexed 14 different high-order OAM modes with the adjacent interval $\Delta l = 0.01$ using the method described in the previous section. In addition, the number of figures used in the dataset has been reduced to 1,000 images per class. The total number of data required is only half that required for the previous case. When combined with the four trained networks, the 16-bit image transmission system is ready for testing. The results are presented in Fig. 8. Figure 8(a) shows that both configurations provide satisfactory performances when the number of figures used for training is more than 1,000. However, when the dataset is reduced in size to 500 images per class, the recognition accuracy of the single 16-bit network then drops to 78.17%. In the meantime, when the four 14-bit parallel network configuration is used, the global recognition accuracy of the system reaches 96.26%. The image quality comparison after transmission shown in Fig. 8(b) also confirms that the four 14-bit parallel network configuration can provide much better performance than the single network. These results indicate that use of the parallel network configuration can increase the network performance and simultaneously reduce the quantity of data required for training, which is impossible when using the solo network configuration. This newly proposed parallel network configuration has great potential for simplification of the application of deep learning techniques under conditions where huge datasets are required.

 

Fig. 8. Comparison of solo networks and parallel networks. (a) Performance of solo networks and parallel networks trained with different number of speckle patterns for each topological charge. (b) Image transmission via the 4×14-bit parallel network configuration (top) and via the single 16-bit network (bottom). Although the 4×14-bit parallel networks are trained with fewer data, they provide the better performance when compared with the single 16-bit network.

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5. Conclusion

In this paper, we have demonstrated an AI-assisted OAM-multiplexing data transmission system based on MMF speckle patterns. Appropriate training of a network enables a system with a single network to support transmission of up to 100 fractional OAM modes with intervals as small as 0.01 through a 1-km-long commercial MMF. This is, to the best of our knowledge, the largest reported number of OAM modes to be transmitted through such a long fiber. Furthermore, although no special care was taken with environmental control during the experiments, including maintaining the temperature, humidity, and mechanical stability of the fiber, the system was still able to provide excellent performance in the topological charge recognition task. This indicates that an appropriately trained network can reject noise perturbations effectively, which is an essential characteristic for implementation of the proposed system in real-life measurement applications. Additionally, a configuration using multiple parallel networks has also been proposed, and it has been demonstrated that the system bandwidth can be extended using this parallel network configuration rather than through use of more complex network structures. Such an arrangement can help save on both the computational capacity and the hardware requirements. Finally, we performed image transmission using both the single network and multiple network configurations, which demonstrated that the parallel network configuration can fulfill the tasks of a huge network while using far fewer data for the training stage; this is consistent with our conclusions. This work raises the possibility that network bandwidths can be extended without introducing complex architectures and exaggerated datasets.