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Abstract
Atmospheric turbulence has an adverse impact on orbital angular momentum (OAM) beam transmission, resulting in power fluctuations and mode crosstalk. These challenges are particularly pronounced in OAM multiplexing links. In this paper, we propose and demonstrate a novel network architecture that integrates convolutional layers and residual structures to address the issue of turbulence phase compensation. By harnessing the local feature learning capability of convolutional layers and the information-preserving function of residual structures, we aim to mitigate the adverse effects of network depth on information loss. By employing the proposed network, we compensate the turbulence phase directly using the received intensity distributions for free space multiplexed integer and fractional order OAM links, respectively. The obtained results show that the received optical power can be improved for more than 10 dB for integer order OAM multiplexed FSO links under weak to strong turbulence conditions, while 9 dB for fractional-order OAM multiplexed FSO links. Moreover, mode crosstalk can be reduced for about 10 dB under 4 OAM modes multiplexed links under turbulence strength D/r0=5. The proposed deep learning based atmospheric turbulence compensation method can predict phase screens rapidly and accurately, thus enhancing the dependability of future OAM multiplexing technology.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams carrying orbital angular momentum (OAM) that exploit the spatial structure physical dimension of electromagnetic waves have been intensively investigated in wide applications such as optical manipulation, optical tweezers, sensor, imaging, metrology, astronomy, and quantum entanglement [1–5]. OAM-carrying light beam is a helically phased beam comprising an azimuthal phase term exp(ilφ), carrying an OAM of lℏ per photon (ℏ: reduced Plank’s constant), where l is referred to topological charge and φ is the azimuthal angle [6]. Due to the intrinsic orthogonality and unbounded states of OAM modes, one can realize high-capacity and spectrally efficient twisted optical communications through efficiently OAM multiplexing. Consequently, OAM beam has shown great potential in optical communications, especially in free-space optical (FSO) communications [7].
However, in an FSO link, the time-varying atmospheric turbulence-induced random fluctuation in the refractive index of air can break the transverse amplitude and phase distribution of the transmitted OAM beam, resulting in the broadening of the beam OAM spectrum and introducing inter-mode crosstalk in the OAM optical communication system [8,9]. Thus, it is necessary to establish phase distortion compensation measures in the OAM-based FSO communication systems. Currently, adaptive optics suppression schemes can be divided into two categories: wavefront sensor (WFS)-based methods that rely on the measurement of distorted phase and WFS-free methods [10,11]. However, conventional WFSs cannot accurately measure the distorted phase of OAM beams due to their phase singularities. To address this issue, Ren et al. introduced a probe beam in 2014 to measure the aberration of the probe beam wavefront and obtain the corrected phase [12]. In 2016, Fu et al. proposed a pre-compensation scheme for OAM beams that combines a Gaussian probe beam with the GS algorithm [13]. The turbulence-induced phase distortion was probed using the Gaussian probe beam, and then the original and distorted images were analyzed using the GS algorithm to calculate the pre-compensation phase. This pre-compensation phase was added to the OAM beam at the transmitting end. The experimental results demonstrated that this scheme has the potential for application in OAM data transmission systems, as it can effectively compensate for single or composite OAM beams. The Gerchberg-Saxton (GS) algorithm is often used in phase retrieval algorithm-based WFS-free turbulence suppression schemes to correct the aberrated vortex beam. Although the GS algorithm performs well in phase recovery [14], it is computationally complex and has a long response time when faced with strong turbulence [15]. Recently, the stochastic parallel gradient descent algorithm (SPGD) has become a core solution for turbulence suppression due to its excellent phase correction capability and is widely used in OAM optical communication [16,17]. However, the SPGD method is sensitive to gain coefficients, which may cause the optimization curve to become trapped in a local optimal solution, increasing the instability of the turbulence compensation system. Therefore, in practical engineering, it is necessary to consider various factors and choose the most suitable method for correcting vortex beam distortion to better solve the problem of turbulence suppression.
In recent years, deep learning techniques have rapidly developed in image recognition and classification, garnering considerable attention in optics [18,19]. Traditional phase extraction methods have many problems, including the accurate extraction of turbulent phase features, which deep neural networks are adept at handling. Therefore, using deep learning techniques to extract turbulent phase features from aberrated light intensity has become key to solving these problems. In 2019, J. Liu et al. proposed a deep CNN model that learned the relationship between the input beam's intensity distribution and the turbulent phase, using the Gaussian Probe Beam (GPB) as the object of information extraction. The bit error rate (BER) of the OAM channel is reduced by almost two orders of magnitude [20]. In the same year, Q.W. Xu et al. used a neural network to extract the turbulent phase of a 20 m transmitted beam quickly and accurately through beacon light [21]. Although the turbulent phase displays a high degree of consistency in terms of phase amplitude distribution and image details, the neural network in the paper is limited in its ability to predict the turbulent phase and ideal output due to being trained on data with the same intensity magnitude of turbulence and only short transmission distances. A year later, W. Xiong et al. demonstrated the effectiveness of using a CNN to extract turbulence information from a standard and distorted vortex beam at a distance of 50 m and recover its distortion [22]. In 2021, X. Wang et al. applied this adaptive correction technique to a quantum key distribution (QKD) system, effectively improving the mode purity of OAM states [23]. Deep learning compensation methods have proven to be more accurate and faster in correcting distortion than AO systems. However, these methods still suffer from some problems, such as insufficient perception and representation of spot features, leading to gradient disappearance, explosion or overfitting, making it challenging to accurately complete the reconstruction task in real-time [24,25]. Moreover, Current research focuses on a single OAM mode for turbulence compensation, it is not clear how it applies to multiplexed vortex beams. Communication-based on OAM state multiplexing can increase system capacity, but turbulence interference can cause inter-mode crosstalk between OAM states at the receiver side [26]. Additionally, the composite fractional-order vortex beam has a stronger light intensity distribution when used as a carrier for information transmission, making it more favorable to information transmission. Therefore, methods that can quickly compensate for the distorted vortex beams under atmospheric turbulence, are still needed for both integer and fractional order OAM multiplexed FSO links.
In this paper, we propose a deep learning based phase compensation method for both integer and fractional order OAM multiplexed FSO links to enhance the performance under atmospheric turbulence conditions. We evaluate the accuracy and efficiency of the network and demonstrate that after training, the CNN model can learn to generate compensated phase screens by analyzing the intensity distribution of disturbed multiplexed OAM beams under atmospheric turbulence conditions. The received optical power can be improved for more than 10 dB for integer order OAM multiplexed FSO links under weak to strong turbulence conditions, while 9 dB for fractional-order OAM multiplexed FSO links. Additionally, mode crosstalk can be reduced for about 10 dB under 4 OAM modes multiplexed links. Furthermore, the performance of the proposed CNN model under a 10 OAM modes multiplexed FSO link is also evaluated, which can also improve the received power and reduce the mode crosstalk under weak to strong turbulence conditions.
2. Concept and network structure
To predict the turbulence phase for OAM multiplexed FSO links, we introduce a structured approach utilizing an encoder-decoder framework to predict atmospheric phase turbulence, aiming to enhance efficiency and sensitivity to local phase distributions, called Residual Net and Deconvolution Neural Network (RDNN) as shown in Fig. 1 [27–29]. In our RDNN, we introduce residual structures at every layer of the base network, complemented by an attention mechanism. Additionally, the network incorporates encoding and decoding structures.
Fig. 1. Network model structure for turbulence phase prediction and compensation in OAM multiplexed FSO links. BN: Batch Normalization; ReLU: Rectified Linear Unit; Conv: Convolution; Deconv: Deconvolution.
The encoder layer incorporates convolution and pooling layers, among others, to extract meaningful turbulence phase information from the disturbed beam. To achieve nonlinear downsampling and reduce the feature map resolution by 8 times compared to the original image, we employ a CNN followed by three maxpooling layers. Additionally, we enhance the extraction of phase information by integrating a Convolutional Block Attention Module [30] (CBAM) attention mechanism within the encoder layer structure. This attention mechanism improves the network's understanding of the input data.
In the decoder layer, we transform the output feature map from the encoding layer back to the original data form. This is accomplished by employing three deconvolution layers for 8-fold upsampling, ensuring consistency in the size of the original input image. We also incorporate a CBAM attention mechanism at the end of the decoding layer to further extract information from the disturbed beam.
Finally, to facilitate enhanced extraction of atmospheric turbulence phase features, we utilize a residual structure that connects the encoding and decoding layers in pairs. This structure promotes feature extraction within the network.
Figure 2 visualizes the employed upsampling and downsampling techniques utilized in this study. Part (a) depicts the downsampling operation, where the original input image undergoes feature encoding through convolution and pooling [31,32]. This process leads to the generation of encoded features characterized by reduced dimensionality. Part (b) corresponds to the upsampling operation, which involves decoding the encoded features through inverse pooling and deconvolution, commonly known as resolution amplification [33]. In our network architecture, we incorporate a total of four downsampling and four upsampling operations within the encoding phase. The upsampling procedure is achieved through the utilization of maxpooling and convolution methods.
Fig. 3. The diagram of the overall framework of the Channel-wise Attention Mechanism (CBAM) attention mechanism. (a) comprises two modules, namely, the Channel Attention Module and the Spatial Attention Module, and the intermediate feature maps are adaptively refined by our modules in each convolutional block of the deep network. (b)As illustrated, the channel sub-module utilizes both max-pooling outputs and average-pooling outputs with a shared network; (c) the spatial sub-module utilizes similar two outputs that are pooled along the channel axis and forward them to a convolution layer.
To enhance the capability of feature extraction in the encoder and upsampling in the decoder, we introduce the Convolutional Block Attention Module (CBAM) attention mechanism (Fig. 3). This mechanism combines the Channel Attention Module and Spatial Attention Module to process the incoming feature layers. The Channel Attention Module begins by performing global average pooling (AvgPool) and global max pooling on the incoming feature layers. These pooling operations are conducted along the height and width dimensions of the feature layers. The results of the average pooling and max pooling are then fed into a shared Multi-Layer Perceptron (MLP) for further processing. The shared MLP weights are applied to the pooled results, and the Sigmoid activation function is utilized to obtain the channel attention map. This map represents the weights assigned to each channel in the input feature layer, ranging from 0 to 1. Finally, the weights are multiplied with the input feature layer channel-wise to incorporate the channel attention information. The calculation process is demonstrated as follows:
In this process, the sigmoid function denoted by $\mathrm{\sigma} $ is utilized. The weights and biases involved in the MLP are represented as ${\textrm{W}_0} \in {{\mathbb R}^{\textrm{c} \times \frac{\textrm{c}}{\textrm{r}}}},{\textrm{b}_0} \in {{\mathbb R}^{\frac{\textrm{c}}{\textrm{r}}}},{\textrm{W}_1} \in {{\mathbb R}^{\frac{\textrm{c}}{\textrm{r}} \times \textrm{c}}},$ and ${\textrm{b}_1} \in {{\mathbb R}^\textrm{c}}$. The ReLU activation function is applied between the two layers of the MLP.
In contrast to the channel attention module, the spatial attention module serves the purpose of evaluating the significance of various regions within the input image. It complements the channel attention module by specifically attending to spatial information. The spatial attention module initially applies average pooling and max pooling operations along the channel dimension for each feature point in the input feature layer. The resulting pooled values are then stacked, generating a feature descriptor that comprises two 2D maps: $\textrm{F}_{\textrm{avg}}^\textrm{s} \in {{\mathbb R}^{1 \times \textrm{H} \times \textrm{W}}}$ and $\textrm{F}_{\textrm{max}}^\textrm{s} \in {{\mathbb R}^{1 \times \textrm{H} \times \textrm{W}}}$ . These maps represent the characteristics of average pooling and maximum pooling across the channels, respectively. Following this, a standard convolution layer with one channel is employed to convolve the stacked maps, and subsequently, the Sigmoid activation function is applied to obtain the 2D spatial attention map. This map determines the weight value assigned to each feature point in the input feature map, which ranges from 0 to 1. The specific calculation process is illustrated as follows:
Here, ${\mathrm{\mathbb{M}}_\textrm{c}}(\textrm{F})$ represents the 2D spatial attention map, $\mathrm{\sigma} $ denotes the Sigmoid function, $\textrm{Stack}$ represents the stacking of feature maps, and $\textrm{Conv}$ denotes the convolution operation.
3. Simulation results
Based on the aforementioned theory, numerical simulations are conducted for both integer and fractional order OAM multiplexed FSO links under weak to strong turbulence conditions. The parameters are configured as follows: beam waist radius ${w_0} = 1\textrm{ mm}$, wavelength $\lambda = 1.55\textrm{ um}$, propagation distance z = 100 m, and sampling point $\textrm{N = 800}$, respectively. Figure 4 displays the outcomes intensity distributions of three distinct integer-order OAM beams transmitted through a 100 m free-space link.
Fig. 4. Simulated intensity distributions of three distinct integer-order OAM beams transmitted through a 100 m free-space link. (a) single OAM transmission link (l = 3), (b) two OAM multiplexing link (l=±4), (c) four OAM multiplexing link (l=±2, ± 4).
The spots of the beam disperse gradually with the superposition of the topological charge numbers. The atmospheric turbulence affecting the transmitted beam can be represented as a turbulent phase, and extracting atmospheric turbulence information involves isolating the turbulent phase from the distorted beam. In this study, we conducted numerical simulations to analyze the intensity distribution of three different transmission methods, namely single OAM transmission link, two OAM multiplexing link, and four OAM multiplexing link, under atmospheric turbulence for transmission distances ranging from $90\textrm{m}$ to $100\textrm{m}$. A total of 40,000 training sets and 4000 test sets were used for each set, incorporating 10 turbulence intensities ranging from $C_n^2 = 1 \times {10^{ - 13}}$ to $C_n^2 = 1 \times {10^{ - 14}}$. The image parameters of the input network were set as follows: light intensity image pixels of 64 ${\times}$ 64, beam wavelength of 1.55 ${\mathrm{\mu} \mathrm{m}}$, beam waist of 0.05$\textrm{m}$, and turbulence inner and outer scales of ${l_0} = 0.0001\textrm{ }m$ and ${L_0} = 3\textrm{m}$, respectively. The network training parameters included 500 iterations (Epoch), a BatchSize of 256, and a learning rate of 0.001.
The rate of convergence and loss value are the most important indicators for assessing the robustness of a network model. Figure 5 displays the loss function curves, where the solid and dashed lines indicate the loss values of the validation and training sets, correspondingly. After 250 iterations, the loss values of the training set stabilize near the steady state. The loss curve exhibits rapid convergence within approximately 50 iterations, with the loss value dropping to 0.0018 at 100 iterations, and stabilizing at 0.0009 after 200 iterations, demonstrating that our model has strong prediction performance. The loss curve in the figure exhibits non-smoothness due to variations in the updates of the turbulence phase during the training process. Not every update ensures a closer approximation to the actual turbulence phase compared to the previous one, resulting in local fluctuations of the loss function value. However, the overall trend of the loss function is monotonically decreasing.
Fig. 5. Loss values of the training and validation sets plotted against the number of iterations in the algorithm. The learning rate is dynamically adjusted every 20 epochs, gradually decreasing towards the optimal solution as the number of iterations increases.
Table 1 presents the average loss values of each vortex beam after the convergence of the loss curve. Specifically, the vortex beam with topological charge 3 demonstrates an average loss value of 0.0009 after 400 iterations. In contrast, the remaining two composite vortex beams exhibit similar loss values, with an average of 0.0018. Regarding the validation set, the vortex beam with topological charge 3 has an average loss value of 0.0072.
Table 1. Loss values of vortex beams with varying topological charge numbers in the training and test sets.
To better illustrate the compensated phase, the phase distribution predicted by the network at varying iterations is depicted in Fig. 6 of this paper. As evident from Fig. 6, the predicted atmospheric turbulence phase screen converges towards the real phase screen with increasing number of iterations. The predicted turbulence phase and the actual turbulence phase exhibit significant consistency in terms of phase distribution and image details. This is evident to about phase distribution and image details.
Fig. 6. The phase distribution of atmospheric turbulence generated by the network at varying iterations for integer-order OAM is depicted in a series of images from left to right, showing the predicted phase maps for the validation set at the 1st, 9th, 99th, and 499th iterations. The image on the far right matches the actual phase map, with the colorbar representing the range of values for atmospheric turbulence phase distribution.
Moreover, the extraction effect remains consistent irrespective of the turbulence strength, validating the efficacy of the proposed network architecture in extracting phases from turbulence. The reduction effect does not significantly decrease with increasing turbulence strength, suggesting the broad applicability of the proposed method. Using the i7-1065G7 CPU, the computational time for predicting the turbulent phase is a mere 5 ms, which is significantly shorter than the time required for turbulence freezing. Thus, utilizing this network can enhance the performance of the integer-order vortex beam in mitigating turbulence effects.
Besides the perturbation of intensity distribution of light, atmospheric turbulence may also impact the power received. Therefore, the effect of turbulence compensation can be evaluated by comparing the power received before and after compensation. In this paper, random phase masks are generated to simulate the power received undulation with and without compensation for three different topological charge numbers of vortex beams at atmospheric turbulence intensities of ${D / {{r_0}}} =$1,5 and 10 respectively, with a transmit power of 0 dBm. For each condition, 50 beams are generated in the experiment and received at a distance of 100 m, as depicted in Fig. 7. It is observed that the received power decreases as the turbulence intensity increases, which presents a challenge to the reliability of OAM transmission. To maintain the communication performance, the distortion caused by turbulence requires correction.
Fig. 7. Comparison of received power after interference and phase compensation for vortex beams with different integer order topological charge numbers. (a) single OAM transmission link (l = 3), (b) two OAM multiplexing link (l=±4), (c) four OAM multiplexing link (l=±2, ± 4).
The predictive accuracy of the network model directly impacts the compensation effect. In this paper, turbulence intensities of three different levels are initially predicted and compensated on the beam. The average received power of the uncompensated third-order vortex beam is −12.95 dBm, and after compensation, the average received power increases to −1.77 dBm, a gain of 11.17 dB. After compensating for the OAM multiplexed link vortex beam with a topological charge number of ±4, the average received power per channel is −4.92 dBm, representing an increase of 10.54 dB. For the multiplexed vortex beam with ±2, ± 4 topological charge numbers after compensation, the average received power per channel is −9.8 dBm, a gain of 15 dB.
Based on the simulation results, it is evident that the received power decreases with increasing atmospheric turbulence when the OAM multiplexed-state propagates in the free-space link. The higher the number of OAM multiplexes, the lower the received power on the receiving screen. And compensation can mitigate the interference caused by atmospheric turbulence. This indicates that it is capable of effectively correcting turbulence, as long as valid information is provided by the phase screen. Furthermore, the network provides better compensation for multiplexed-state OAM compared to single-transmission OAM.
Figure 8 shows the spiral spectrum distribution of OAM multiplexed states transmitted at a distance of 100 m for different turbulence intensities. Prior to phase compensation, the pure states of the multiplexed vortex beams are disrupted, causing each OAM state to mix with different topological charge values and spread to other channels. Consequently, the orthogonality between OAM states is lost, resulting in crosstalk in adjacent modes. Moreover, the crosstalk between OAM states in adjacent modes increases with turbulence intensity. Specifically, mode crosstalk occurs between the topological charge numbers of −5 and 5, making it challenging to accurately determine the original topological charge number from the OAM spectrum distribution due to the turbulence-induced distortion and severe crosstalk between adjacent modes.
Fig. 8. The spectral distribution of two distinct integer-order OAM beams transmitted through a 100 m free-space link. (a) uncompensated two OAM multiplexing link (l=±4), (b) compensated two OAM multiplexing link (l=±4), (c)uncompensated four OAM multiplexing link (l=±2, ± 4), (d)compensated four OAM multiplexing link (l=±2, ± 4).
However, after compensation, the strict orthogonality between beams is restored, mitigating the OAM beam power migration caused by interference. Figure 8(b) and (d) show that the relative power of OAM states with topological charge numbers of −4 and +4, and −4, −2, + 2, + 4, respectively, is proportionally larger, which is consistent with the actual OAM multiplexed topological charge numbers, indicating the good compensation performance of the proposed network in this paper.
Once the phase information of the aberration is obtained, the original beam can be compensated by applying the reverse phase to the aberrated beam. Figure 9 illustrates the compensation effect of the disturbed beam as it passes through the network under different turbulence intensities. The figure clearly demonstrates that atmospheric turbulence disrupts the intensity distribution of the beam, with distortion increasing proportionally with turbulence intensity. At a turbulence intensity of ${D / {{r_0} = 1}}$, the vortex beam experiences minimal disturbance and no significant drift. At ${D / {{r_0} = 5}}$, the beam exhibits slight drift and distortion of the spot shape. At ${D / {{r_0} = 5}}$, the spot center is severely distorted, the spot drifts significantly, and the initial circular shape becomes irregular. This phenomenon is primarily attributed to the increased turbulence intensity, which amplifies the fluctuation of turbulence phase and intensifies its effect on the beam phase, resulting in more pronounced turbulence effects, including phase undulation, light intensity undulation, and spot drift.
Fig. 9. Intensity distributions of vortex beams with and without phase compensation are compared for integer-order vortex beams at various turbulence intensities, with a fixed value of $\triangle z$=100 m.
Figure 9 indicates that turbulence-induced disturbances have been successfully mitigated, and after compensation, the beam shows a uniform intensity distribution in the plot. The phase of low-intensity turbulence has been attenuated to a weaker level, and the phase of strong turbulence has been minimized as much as possible. During training of the network, Δz values ranging from 90 m to 100 m and values ranging from to are randomly chosen, allowing the same CNN model to be used for testing the efficacy of turbulence compensation under varying turbulence intensities and distances.
Currently, research on phase compensation of vortex beams primarily focuses on integer-order transmission, but compensation for fractional-order vortex beams is also crucial. This is because fractional-order vortex beams possess a strong encoding capability and exhibit a gap in the bright ring, which is expected to have potential applications in microparticle manipulation. In order to explore the application of our proposed network to fractional-order vortex beams, further numerical simulations of fractional-order vortex beams were carried out in this study. The parameters are set as follows: beam waist radius ${w_0} = 1$mm, wavelength $\lambda = 1.55\textrm{um}$, propagation distance z = 100 m, and topological charge numbers $l = 2.5,l ={\pm} 2.5,l ={\pm} 2.5, \pm 4.5$, respectively. The simulation results are shown in Fig. 10.
Fig. 10. Simulated intensity distributions of three distinct fractional-order OAM beams transmitted through a 100 m free-space link. (a) single OAM transmission link (l = 2.5), (b) two OAM multiplexing link (l=±2.5), (c) four OAM multiplexing link (l=±2.5, ± 4.5).
The phase of vortex light with integer order will coincide with z = 0 after one week of rotation. In contrast, vortex light with fractional-order rotating for one week will exhibit an excess phase of $2\pi u$ (where u is the fractional part of l). This results in a jump in the x-axis, creating a radial gap in the intensity of the light. The intensity distribution of the composite vortex beam, with superimposed fractional order, shows a distinct pattern of bright and dark spots. In this study, we simulate the intensity distribution images of three fractional-order beams transmitted over a distance of 90-100 m under atmospheric turbulence, with topological charge numbers of $l = 2.5,l ={\pm} 2.5,l ={\pm} 2.5, \pm 4.5$, respectively, while keeping the other parameters consistent with the above experiments. The curves of the loss values are depicted in Fig. 11. It shows that the loss curve converges rapidly, reaching a minimum value of 0.0009 after around 50 iterations and decreasing further after 200 iterations.
Fig. 11. Loss#values of the training and validation sets plotted against the iterations of the algorithm. We dynamically adjust the learning rate every 20 epochs, gradually decreasing it as the number of iterations increases, approaching the optimal solution.
Table 2 also reveals that the overall prediction performance of the fractional-order vortex beam is inferior to that of the integer-order vortex beam, as indicated by the higher average loss of 0.0095 compared to 0.0089 for the integer-order vortex beam.
Table 2. Loss values of vortex beams with varying topological charge numbers in the training and test sets.
The figure also demonstrates that the loss of multiplexing link is higher than that of single OAM transmission link. Likewise, Fig. 12 displays the predicted phase distribution by the network for different numbers of iterations during the training of fractional-order vortex beams.
Fig. 12. Predicted phase diagrams of atmospheric turbulence for fractional order OAM at different iterations by the network are shown. The series of five images, arranged from left to right, display the predicted phase maps for the validation dataset at 1, 9, 99, and 499 iterations, with the rightmost image representing the actual phase map. The colorbar indicates the distribution of atmospheric turbulence phases.
The compensation performance of integer-order vortex beams and fractional-order vortex beams is compared in this paper. Additionally, simulations are conducted to analyze the received power variations with and without compensation for three fractional-order OAM multiplexing links at atmospheric turbulence strengths of ${D / {{r_0} = }}$1,5 and 10 respectively, with a transmit power of 0 dBm, as shown in Fig. 13. The average received power after compensation for the vortex beam with a topological charge number of 2.5 is −2.1 dBm, an increase of 11.3 dB. For the vortex beam with a multiplexed topological charge number of ±2.5, the average received power after compensation is −5.7 dBm per channel, up by 9.3 dB. The average received power for the vortex beam with multiplexed topological charge numbers of ±2.5 and ±4.5 is −7.9 dBm, an increase of 9 dB. By comparing the above results, we can draw the conclusion that with an increase in the number of multiplexed modes, there is a slight decline in compensation performance (an average decrease of 0.8 dB). Moreover, fractional-order compensation results are slightly worse than integer-order (with an average decrease of 1.9 dB).
Fig. 13. The received power after interference was compared for three different fractional-order topological charge numbers of the vortex beam with the received power after phase compensation. (a) single OAM transmission link (l = 2.5), (b) two OAM multiplexing link (l=±2.5), (c) four OAM multiplexing link (l=±2.5, ± 4.5).
This paper presents a simulation of the compensation effect of a disturbed fractional-order beam passing through the network, as depicted in Fig. 14. The results indicate that the predicted turbulence phase screen can effectively compensate the distorted beam when the fractional-order beam is fed into the network. Specifically, the compensation effect is most pronounced when the turbulence intensity is ${D / {{r_0} = }}1$, resulting in the compensated beam being closest to the original beam. However, in the case of strong turbulence with ${D / {{r_0} = }}10$, there remains a noticeable gap between the compensated beam and the original beam.
Fig. 14. The intensity distribution of vortex beams with and without phase compensation for fractional-order vortex beams under varying turbulence intensities at a propagation distance of $\triangle z\textrm{ = }100\textrm{m}$.
Additionally, to investigate the accuracy of predicting the ten OAM multiplexing link (l = −14,−11,−8,−5,−2,1,4,7,10,13), a set of multiplexing link is simulated in this study using the same parameter settings as mentioned earlier. The intensity distribution of the transmitted beam after propagating 100 m is depicted in Fig. 15. The results indicate that the network is more effective in compensating for low intensity turbulence. As turbulence intensity increases, beam distortion becomes more severe, making it more challenging to recover the original light intensity distribution.
Fig. 15. (a) Simulated intensity distributions of OAM beams transmitted through a 100 m free-space link. (l = −14, −11, −8, −5, −2, 1, 4, 7, 10, 13). (b) The intensity distribution of vortex beams with and without phase compensation is examined for 10-mode vortex beams at ten OAM multiplexing link, with a propagation distance of $\triangle z = 100m$.
The loss curve is shown in Fig. 16. From the loss curve, it can be observed that there is a rapid convergence trend up to approximately 15 epochs, with a significant decrease in loss value. After 150 epochs, the loss curve stabilizes, with an average loss value of 0.0019 for the training set and 0.0144 for the test set. A comparison between vortex beams of integer order, fractional order, and ten-mode multiplexing reveals a higher loss value for the latter. This suggests that predicting vortex beams with a higher multiplexed topology charge number is more challenging.
Fig. 16. The loss values of the training and validation sets are plotted against the number of epochs in the algorithm.
The power spectrum of the 10 OAM multiplexing link, transmitted over a distance of 100 meters under different turbulence intensities, is shown in Fig. 17. It can be observed from the figure that with increasing turbulence intensity, the crosstalk among different OAM modes becomes more severe, rendering accurate distinction of the original multiplexed mode impossible, especially between the topological charge numbers $l ={-} 15$ and $l = 30$. After compensation, the average received power of the original multiplexed mode is −11.9 dBm, while the adjacent mode has an average received power of −28.5 dBm, and the topological charge distribution of the original beam can still be accurately distinguished from the compensated figure. The results demonstrate that the higher turbulence intensity leads to more severe crosstalk among OAM modes, making accurate distinction of the original multiplexed mode challenging, but compensation techniques can mitigate the crosstalk and restore the original topological charge distribution of the OAM beam.
Fig. 17. The received power of 10 OAM multiplexing link after interference is compared with that after phase compensation. (a) ${D / {{r_0} = }}1$, (b) ${D / {{r_0} = }}5$, (c) ${D / {{r_0} = }}10$.
As illustrated in Fig. 18, the network successfully compensates for the 10 OAM multiplexing link, effectively mitigating the inter-mode crosstalk among the OAM beams caused by interference. This observation highlights the strong applicability of the network proposed in this study.
Fig. 18. The optical angular momentum (OAM) spectral distribution of ten OAM multiplexing link with and without phase compensation was analyzed at varying turbulence intensities ($\triangle z = 100{\rm m}$).
4. Conclusion
In this paper, we present a CNN-based model for compensating atmospheric turbulence. Moreover, we successfully compensate for the OAM multiplexed free-space optical communication link. The network's impact on the multiplexed state of the OAM beam is analyzed across three dimensions: the distribution of received power before and after compensation, the crosstalk among OAM spectral modes, and the intensity distribution of compensation before and after turbulence. This research demonstrates the effectiveness of the proposed RDNN-based model in mitigating mode crosstalk among multiplexed OAM spectral modes. Following phase compensation, the beam's received power is enhanced by a minimum of 2.23 dBm during 100 m free-space link transmission under varying turbulence intensities, the received optical power can be improved for more than 8 dB for both integer and fractional order OAM multiplexed FSO links under weak to strong turbulence conditions, effectively mitigating the mode crosstalk. Moreover, the received power of the ten OAM multiplexing link improves from −20.6 dB to −12.4 dB. This study also reveals that wavefront correction method based on deep learning can reduce costs while maintaining high correction speed and accuracy, thereby positively impacting research and applications across diverse fields. Moreover, its capability to adapt to a range of complex communication environments, such as strong turbulence or underwater communication, renders it highly applicable in practical scenarios. Future research will concentrate on correction techniques for multiplexed links to better cater to the requirements of multi-user communication. Additionally, to enhance the phase correction effect, one can experimentally construct a genuine turbulent environment and acquire real training samples to further enhance the performance of the wavefront correction algorithm. Another significant application direction involves encoding the transmission of the phase-corrected vortex beam to enhance communication reliability.
While our experiments have primarily focused on OAM beams, there is significant potential for extending the application of our network to other mode bases, such as LG beams and Bessel beams. The underlying principles of wavefront correction and phase compensation, which form the basis of our neural network, are not inherently limited to specific beams. These principles can be adapted to various beam modes. The key lies in training the network with appropriate datasets that represent the specific characteristics and distortions associated with the target beam modes. Therefore, our neural network structure holds promise for predicting and compensating different beam types, broadening its applicability to a wide range of optical communication scenarios. This approach can be employed in diverse domains, including optical communication and optical sensing, to offer more dependable solutions for information transmission and data processing. When integrated with frequently employed optical devices, it becomes possible to achieve rapid and precise feedback for phase compensation.
Funding
National Natural Science Foundation of China (12104078); China Postdoctoral Science Foundation (2021M700561); Natural Science Foundation of Chongqing (cstc2021jcyj-bshX0223); Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000622).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. J. Winzer, “Modulation and Multiplexing in Optical Communications,” in Conference on Lasers and Electro-Optics 2009 (Optica Publishing Group, 2009), p. CTuL3.
2. Xiang Zhou and Jianjun Yu, “Multi-Level, Multi-Dimensional Coding for High-Speed and High-Spectral-Efficiency Optical Transmission,” J. Lightwave Technol. 27(16), 3641–3653 (2009). [CrossRef]
3. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nat 467(7313), 301–304 (2010). [CrossRef]
4. S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23(3), 3075–3087 (2015). [CrossRef]
5. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016). [CrossRef]
6. D. M. Fatkhiev, M. A. Butt, E. P. Grakhova, R. V. Kutluyarov, I. V. Stepanov, N. L. Kazanskiy, S. N. Khonina, V. S. Lyubopytov, and A. K. Sultanov, “Recent Advances in Generation and Detection of Orbital Angular Momentum Optical Beams—A Review,” Sensors 21(15), 4988 (2021). [CrossRef]
7. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
8. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef]
9. J. A. Anguita, J. Herreros, and I. B. Djordjevic, “Coherent Multimode OAM Superpositions for Multidimensional Modulation,” IEEE Photonics J. 6(2), 1–11 (2014). [CrossRef]
10. M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17(8), 1440–1453 (2000). [CrossRef]
11. Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. J. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015). [CrossRef]
12. Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014). [CrossRef]
13. S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Pre-correction of distorted Bessel–Gauss beams without wavefront detection,” Appl. Phys. B 123(12), 275 (2017). [CrossRef]
14. Y. Ren, H. Huang, J.-Y. Yang, Y. Yan, N. Ahmed, Y. Yue, A. E. Willner, K. Birnbaum, J. Choi, B. Erkmen, and S. Dolinar, “Correction of Phase Distortion of an OAM Mode using GS Algorithm based Phase Retrieval,” in Conference on Lasers and Electro-Optics 2012 (Optica Publishing Group, 2012), p. CF3I.4.
15. Y. Liang, X. Su, C. Cai, L. Wang, J. Liu, H. Wang, and J. Wang, “Adaptive turbulence compensation and fast auto-alignment link for free-space optical communications,” Opt. Express 29(24), 40514–40523 (2021). [CrossRef]
16. G. Xie, Y. Ren, H. Huang, M. P. J. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015). [CrossRef]
17. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998). [CrossRef]
18. H. Zhou, Z. Pan, M. I. Dedo, and Z. Guo, “High-efficiency and high-precision identification of transmitting orbital angular momentum modes in atmospheric turbulence based on an improved convolutional neural network,” J. Opt. 23(6), 065701 (2021). [CrossRef]
19. Z. Wang and Z. Guo, “Adaptive Demodulation Technique for Efficiently Detecting Orbital Angular Momentum (OAM) Modes Based on the Improved Convolutional Neural Network,” IEEE Access 7, 163633–163643 (2019). [CrossRef]
20. J. Liu, P. Wang, X. Zhang, Y. He, X. Zhou, H. Ye, Y. Li, S. Xu, S. Chen, and D. Fan, “Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication,” Opt. Express 27(12), 16671–16688 (2019). [CrossRef]
21. Q.-W. Xu, P.-P. Wang, Z.-J. Zeng, Z.-B. Huang, X.-X. Zhou, J.-M. Liu, Y. Li, S.-Q. Chen, and D.-Y. Fan, “Extracting atmospheric turbulence phase using deep convolutional neural network,” Acta Phys. Sin. 69(1), 014209 (2020). [CrossRef]
22. W. Xiong, P. Wang, M. Cheng, J. Liu, Y. He, X. Zhou, J. Xiao, Y. Li, S. Chen, and D. Fan, “Convolutional Neural Network Based Atmospheric Turbulence Compensation for Optical Orbital Angular Momentum Multiplexing,” J. Lightwave Technol. 38(7), 1712–1721 (2020). [CrossRef]
23. X. Wang, T. Wu, C. Dong, H. Zhu, Z. Zhu, and S. Zhao, “Integrating deep learning to achieve phase compensation for free-space orbital-angular-momentum-encoded quantum key distribution under atmospheric turbulence,” Photonics Res. 9(2), B9–B17 (2021). [CrossRef]
24. S. Jiang, H. Chi, X. Yu, S. Zheng, X. Jin, and X. Zhang, “Coherently demodulated orbital angular momentum shift keying system using a CNN-based image identifier as demodulator,” Opt. Commun. 435, 367–373 (2019). [CrossRef]
25. Z. Wang, M. I. Dedo, K. Guo, K. Zhou, F. Shen, Y. Sun, S. Liu, and Z. Guo, “Efficient Recognition of the Propagated Orbital Angular Momentum Modes in Turbulences With the Convolutional Neural Network,” IEEE Photonics J. 11(3), 1–14 (2019). [CrossRef]
26. Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014). [CrossRef]
27. V. Badrinarayanan, A. Kendall, and R. Cipolla, “SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(12), 2481–2495 (2017). [CrossRef]
28. S. Minaee, Y. Boykov, F. Porikli, A. Plaza, N. Kehtarnavaz, and D. Terzopoulos, “Image Segmentation Using Deep Learning: A Survey,” IEEE Trans. Pattern Anal. Mach. Intell. 44(7), 1 (2021). [CrossRef]
29. D. Teneketzis, “On the Structure of Optimal Real-Time Encoders and Decoders in Noisy Communication,” IEEE Trans. Inf. Theory 52(9), 4017–4035 (2006). [CrossRef]
30. S. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, “CBAM: Convolutional Block Attention Module,” in 15th European Conference on Computer Vision, V. Ferrari, M. Hebert, C. Sminchisescu, and Y. Weiss, eds., Lecture Notes in Computer Science (Springer International Publishing, 2018), Vol. 11211, pp. 3–19.
31. Dumitrescu and Boiangiu, “A Study of Image Upsampling and Downsampling Filters,” Computers 8(2), 30 (2019). [CrossRef]
32. T. Frajka and K. Zeger, “Downsampling dependent upsampling of images,” Signal Process. Image Commun. 19(3), 257–265 (2004). [CrossRef]
33. D. Ferstl, C. Reinbacher, R. Ranftl, M. Ruether, and H. Bischof, “Image Guided Depth Upsampling Using Anisotropic Total Generalized Variation,” in 2013 IEEE International Conference on Computer Vision (IEEE, 2013), pp. 993–1000.
