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Abstract
Spatial modes of light are susceptible to distortion, particularly by the presence of turbulence in atmospheric free-space links. The scattering of one mode to another disrupts the orthogonality among distinct orbital angular momentum (OAM) modes, leading to modal crosstalk between multiple channels. To enhance the performance of OAM-multiplexed free-space optical (FSO) communication, a convolutional neural network (CNN)-based turbulent OAM approach is proposed for compensating turbulence, with a specific focus on predicting the OAM of turbulence itself. An operator approach is utilized to extract the OAM component of atmospheric turbulence and the CNN is trained to predict the turbulent OAM coefficients. By employing the proposed network, the received power of the OAM-based FSO link can be improved by more than 10 dB under weak to strong turbulence conditions. Compared to Zernike modes, the turbulent OAM modes characterize most of the turbulence information using only a small number of orders. After compensation, when the strong turbulence strength D/r0 = 4, the received power of the transmitted beams with turbulent OAM improves by 4 dB over that with Zernike. Additionally, the crosstalk of multiplexed channels with turbulent OAM is reduced by 10 dB over that with Zernike under varying turbulence conditions.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multiplexing technologies, in conjunction with free-space optical (FSO) communications, are promising solutions to address the impending data crunch [1–4], such as utilizing spatial modes of light. One common strategy is to employ structured light carrying orbital angular momentum (OAM), which can be expressed with a phase factor $\exp (il\theta )$ [5–10]. The orthogonality of OAM modes enables OAM multiplexing communication systems to significantly enhance spectrum efficiency and communication capacity. Therefore, OAM has become the prominent mode of choice for communication studies and attracting considerable attention [11–13].
However, spatial modes of light are susceptible to distortion, particularly due to the presence of turbulence in atmospheric free-space links. Atmospheric turbulence induces random phase and intensity variations in the optical beam as it propagates through the atmosphere, leading to unintended mode crosstalk [14,15]. Consequently, the scattering of one mode to its neighbors disrupts the orthogonality of different OAM modes [16–17], hindering the efficient multiplexing and demultiplexing of OAM modes and affecting the overall communication performance.
To mitigate the adverse effects of atmospheric turbulence, adaptive optics (AO) technology has emerged as a robust and prominent approach for turbulence compensation in FSO communication systems [18–27]. AO techniques utilize phase retrieval algorithms or wavefront sensors (WFS) to accurately characterize the wavefront distortions, with the latter often requiring increased hardware complexity at the receiver [18,19]. Consequently, phase retrieval algorithms, such as the Gerchberg-Saxton (GS) algorithm [20–22] and Zernike-based deep learning algorithm [23–26], are commonly employed to address turbulence effects. However, these algorithms face challenges when dealing with strong turbulence and limited applicability to OAM multiplexing systems, making their approach primarily focused on phase recovery appear somewhat one-sided. Recent theoretical and experimental finding indicates that atmospheric turbulence distortions are independent of the original OAM mode, suggesting that the turbulence itself can be considered as imparting or extracting OAM from the optical beam [27]. Since the total OAM is conserved in a closed system, the OAM spreading observed when a beam passes through turbulence is a result of turbulent OAM coupling with the OAM of the beam. Therefore, precise determination of the OAM component in atmospheric turbulence can effectively compensate for the mode crosstalk in OAM beams, thereby enhancing the communication performance of OAM- multiplexed free-space optical links.
In this paper, a convolutional neural network (CNN)-based turbulent OAM approach is proposed for compensating turbulence in OAM-FSO links, with a specific focus on predicting the OAM of turbulence itself. An operator approach is utilized to extract the OAM component of atmospheric turbulence and the CNN is trained to predict the turbulent OAM coefficients. Under the condition that atmospheric turbulence remains constant during beam propagation, compensating any OAM beam in the FSO link proves to be effective. This universality is confirmed through long-path simulations comprising weak, moderate and strong turbulence. By employing the proposed network, the compensated OAM beams show significant improvement in phase and intensity distortions. The received power of the OAM-based FSO link can be improved by more than 10 dB under weak to strong turbulence conditions. Compared to Zernike modes, turbulent OAM modes effectively characterize most of the turbulence information using only a small number of orders. After compensation, when the strong turbulence strength D/r0 = 4, the received power of the transmitted beams with turbulent OAM improves by 4 dB over that with Zernike. Additionally, the crosstalk of multiplexed channels with turbulent OAM is reduced by 10 dB over that with Zernike under varying turbulence conditions.
2. Concept and theoretical framework
2.1 Turbulent OAM mode decomposition
The concept of the proposed turbulent OAM approach based on CNN is shown in Fig. 1. At the transmitter side, light beams can carry OAM associated with the spatial profile of the phase, which can be expressed by
where Ar is the amplitude distribution, ϕ is the azimuthal position, l is the topological charge of OAM mode. Given the confirmed existence of angular momentum in the turbulence, a cylinder of the atmosphere turbulence is visualized, as seen in Fig. 1(a). The expansion of turbulent OAM is linked to turbulence described by the T, which can be represented as a linear combination of orthogonal OAM factors where Ck, Δlk are coefficient and topological charge of kth turbulent OAM mode, respectively.
Fig. 1. Concept of the proposed CNN-based turbulent OAM approach. (a) A cylinder of the atmospheric turbulence is visualized. (b) The schematic diagram of the CNN framework. (c) The phase is reconstructed to inverse compensate for the distorted OAM-spreading beam.
When two OAM light fields are multiplied, it results in the convolution of their OAM spectral components [28]. In practical terms, this implies that the resulting output light field will encompass contributions from various OAM modes originating from this convolution process. Hence, when the turbulence containing OAM is considered as an amplitude-independent OAM superposition state, it becomes evident that the outcomes of the OAM spectral expansion remain consistent regardless of the specific individual OAM modes involved. After propagating the OAM beam El through the turbulence, its OAM components become
Assuming the OAM in atmospheric turbulence is symmetric with respect to the topological charge symbols, the mode spreading composition of OAM beam after the channel is arranged in a sequential manner
Here, E′ are column vectors and each column element corresponds to the OAM-spreading mode at a specific Ck. The expansion coefficients Ck are determined with
where E′-1 is the Moore-Penrose matrix. The turbulent OAM coefficients Ck can be extracted from the received optical field E by a method based on singular value decomposition [29]. Additionally, the intensity distribution I of captured images can also be expressed as followsConsequently, if a mapping relationship can be established between turbulent OAM coefficients and the captured optical intensity, it becomes feasible to accurately predict the OAM components of atmospheric turbulence and mitigate the adverse effects on the OAM beams in OAM-based FSO links.
2.2 Turbulent OAM mode prediction
Due to the highly dynamic nature of atmospheric turbulence, developing an explicit model capable of accurately matching the mapping poses a formidable challenge. Fortunately, deep learning provides a potent solution by leveraging extensive datasets for learning. In this work, CNN is employed to accurately extract turbulent OAM coefficient information from distorted intensity distributions and subsequently restore the distortion patterns.
As illustrated in Fig. 1 (b), the CNN is constructed based on the ResNet architecture, with the primary objective of predicting turbulent OAM coefficients using the distorted intensity of received beams [30]. The input intensity image is sized at 224 × 224 pixels. A sequence of residual blocks and max pooling layers are applied to extract the hierarchical high-dimensional features. These extracted features are then integrated through fully-connected layers to make predictions. Each residual block encompasses multiple layers, comprising rectified linear unit (ReLU) nonlinear activations and convolutions (Conv) with skip connections. As the loss function, the mean square error (MSE) between the estimated values and the actual turbulent OAM coefficients is chosen. The network is subsequently trained using a stochastic gradient descent optimizer to minimize the mean loss. The output of network is a vector of turbulent OAM coefficients, sized at 1 × N. Subsequently, the phase corresponding to the input intensity is reconstructed to inversely compensate for the distorted OAM-spreading beam, as depicted in Fig. 1(c).
In the previous work, deep learning algorithms commonly employed for extracting Zernike coefficients to characterize atmospheric turbulence phase [31,32]. These Zernike coefficients are given by a linear combination of Zernike polynomials
where ΦTurb is the atmospheric turbulence phase, Zj is the Zernike polynomial and αj is the Zernike coefficient. Nonetheless, Zernike-based deep learning approaches come with inherent drawbacks and limitations. In cases of moderately strong turbulence, it is often necessary to utilize higher-order Zernike polynomials, often involving tens of orders or even more [33]. When utilizing deep learning to extract mode coefficients from arbitrary light fields, it necessitates the coverage of all potential combinations of coefficients [34]. This entails a rapid expansion of the training dataset D with each additional mode where s indicates the step size to cover each possible case of mode combination, N indicates the number of mode orders adopted. Furthermore, it's important to highlight that in atmospheric turbulence, there can be substantial phase variations during the propagation of the beam. This indicates that Zernike polynomials may not be effective in capturing significant global phase changes.On the contrary, the turbulent OAM approach is explicitly tailored to address variations in OAM across the atmospheric transmission channel. In the context of free-space transmission, turbulent OAM modes primarily couple with neighboring orders of the transmitted modes, with minimal coupling to modes with larger order spacing. Consequently, adopting the turbulent OAM approach with a reduced number of modes holds promise for significantly reducing the required training data. This leads to shorter model training duration and reducing computational expense. In addition, the turbulent OAM approach provides a transmission matrix for OAM-based FSO communication. This not only facilitates the inverse function of crosstalk mitigation at the transmitter, but also enables the reconstruction of modal wavefront based on turbulent OAM at the receiver.
3. Simulation and results
In this paper, a set of OAM beams with a wavelength of 1550 nm is utilized and the beam size at the transmitter is fixed at 5 cm. The atmospheric turbulence effect is modeled using the Kolmogorov spectrum statistics model. Typically, turbulence strength is quantified using the ratio D/r0, where D represents the beam diameter and r0 denotes the Fried parameter. In this study, three different turbulence conditions are explored, namely D/r0 values of 1, 2, and 4, representing weak, moderate, and strong turbulence, respectively. Typically, higher-order OAM modes experience more pronounced distortion during transmission. Consequently, CNN requires either more training data or more complex network structures to effectively handle these higher-order OAM modes. In contrast, simpler OAM modes tend to yield higher prediction accuracy for the CNN due to their lower complexity and ease of information capture. To enhance prediction accuracy, it is feasible to increase the input channels of the CNN by incorporating multiple modes. However, this approach should be carefully considered, taking into account computational costs. Here, the intensity patterns of the distorted OAM mode (OAM0) after traversing a 1 km free space are employed as training samples for the CNN. In Fig. 2(a), it becomes evident that under weak turbulence condition, there is an increase in correlation between the reconstructed phase derived from N turbulent OAM terms. It is apparent that a mere five turbulent OAM modes suffice to effectively characterize the phase of the OAM-spreading beam. Furthermore, higher-order turbulent OAM modes beyond the fifth order contribute only marginally to the distortion of the beams. Consequently, only five OAM modes (OAM-2, OAM-1, OAM0, OAM + 1, OAM + 2) in atmospheric turbulence are considered in this paper.
Fig. 2. (a) The correlation between the reconstructed phase from N turbulent OAM terms. Insets show reconstructed phase screens produced from 5 and 15 terms. (b) The average values of relative coefficient errors calculated under varying turbulence conditions and the relative compensation errors for various OAM beams.
The CNN is trained on a desktop computer equipped with an Intel Core i7-12700 K CPU and NVIDIA GeForce RTX 3090 GPU, using the PyTorch framework. A significant number of appropriate training data is utilized to ensure that the CNN can extract accurate turbulent OAM coefficients from a given intensity image. The training dataset is divided into three categories based on the turbulence strengths (D/r0 = 1, 2, 4). Each kind has 80000 pairs of distorted images and their corresponding coefficient vectors. To quantitatively evaluate the prediction accuracy of the trained CNN, two key parameters are calculated: the relative coefficients error Ecoeff, which measures the disparity between the theoretical output vector C and the experimental prediction vector Cp, and the relative compensation error Ecomp, which is applied to various OAM beams using C and Cp. Here, these relative errors are defined as Ecoeff = | |C|2 - |CP|2 | and Ecomp = | ρ - ρp |. The histogram in Fig. 2(b) showcases the average values of these coefficient errors and compensation errors computed under varying turbulence conditions. Notably, it can be observed that both relative errors exhibit a gradual increase as turbulence intensity rises. Conversely, the compensation error tends to decrease with higher OAM orders when dealing with moderate to strong turbulence. This trend can be attributed to the fact that heightened turbulence intensity results in more pronounced beam distortion, influencing the CNN training process. Additionally, it exerts a greater impact on higher-order OAM beams, leading to a somewhat diminished compensation effect and consequently reducing the compensation error. Despite these variations, the achieved level of accuracy remains satisfactory. To further enhance prediction accuracy, additional learning samples or iterations can be integrated into the process. In the realm of OAM optical communication, the average error of less than 5% is regarded as highly accurate [35]. Thanks to the trained CNN's efficiency, turbulent OAM coefficients can be predicted in a single iteration, with each prediction requiring only 90 milliseconds. This timeframe is notably faster than the rapid atmospheric turbulence variations occurring within 100 milliseconds [36]. Therefore, the CNN effectively maintains consistent atmospheric turbulence conditions during beam propagation.
Firstly, different OAM modes (OAM0, OAM + 1, OAM + 2, OAM + 3) under the same turbulence condition (D/r0 = 1) are investigated. Among these, OAM0 is part of the training data, while OAM + 1, OAM + 2 and OAM + 3 are not included in the training set. The compensation results of mode purity are presented in Fig. 3. Notably, the mode purity of OAM0 beams significantly improves following the compensation of the reconstructed distorted phase. This observation underscores the effectiveness of the proposed approach in compensating trained OAM beams. Furthermore, even the more severely affected OAM + 1, OAM + 2 and OAM + 3 beams, owing to their higher mode, are accurately compensated. This demonstrates the firm universality of turbulent OAM modes in compensating for various transmitted OAM beams within the FSO link. This effectiveness arises from the fact that atmospheric turbulence primarily impacts the OAM-spreading induced by turbulence rather than the beams themselves. Consequently, the turbulent OAM approach based on CNN circumvents the need for additional training complexity, particularly in OAM multiplexing systems.
Fig. 3. The mode purity of the OAM beams before and after compensation under turbulence strength of D/r0 = 1. (a) OAM0, (b) OAM + 1, (c) OAM + 2, (d) OAM + 3.
To assess the compensation effect on communication performance, the received optical power of the same received OAM mode under varying levels of turbulence strength is investigated. Figure 4 displays the distribution and corresponding cumulative probability of the received power of the received OAM + 3 mode over 50 independent turbulence realizations under weak to strong turbulence (D/r0 = 1, 2, 4). As turbulence strengths increase, the average received power in FSO links decreases without compensation, while it significantly improves with compensation. For each random turbulence realization, the received power of the compensated OAM beam exceeds that without compensation, confirming the effectiveness of turbulent OAM approach to different turbulence compensation conditions. The fluctuation range of the received power with compensation is lower compared to that without compensation, as depicted in Figs. 4(a), (b), and (c). This suggests that the received power distribution can be improved and fluctuation reduced with the use of the turbulent OAM. Under weak turbulence (D/r0 = 1), the lowest received power with compensation is about 12 dB higher than without compensation. However, due to the wider range of main OAM modes possessed by moderate and strong turbulence, the improvement of received power and fluctuation tends to decrease as the turbulence strength increases. This trend can be effectively addressed by increasing the turbulent OAM decomposition modes.
Fig. 4. The received power and corresponding cumulative probability of the OAM3 beams before and after compensation under different turbulence strengths (a, d) D/r0 = 1, (b, e) D/r0 = 2, (c, f) D/r0 = 4.
Next, the intensity and phase distribution, as well as received power, of distorted OAM beams before and after compensation under different turbulence are shown in Fig. 5. Turbulence significantly affects the intensity distribution of OAM beams, causing increased distortion and energy diffusion as turbulence strength rises. Particularly at a turbulence strength of 4, it becomes challenging to distinguish the transmitted modes from the captured intensity distributions. This could have a substantial impact on the communication performance of FSO links. However, the proposed approach effectively corrects the intensity profiles of distorted beams, producing a more uniform annular intensity distribution akin to far-field light intensity without turbulence effects. Figure 5(b) demonstrates that the reconstructed phase distribution is nearly identical to the distorted phase of the received beams. This suggests that the turbulent OAM, used for turbulence characterization, can also directly characterize the phase of distorted OAM beams, eliminating the need for a WFS. Moreover, angular phase fluctuations introduced during transmission can be effectively restored, resulting in a relatively smooth phase distribution following reverse compensation. Conversely, restoring such angular phase fluctuations for Zernike modes presents a challenge, as sudden spiral-phase changes on the unit circle make it difficult to reconstruct. Therefore, it is a unique advantage of the turbulent OAM approach. As shown in Fig. 5(c), after compensation, the energy of distorted beams diffusing into other orders significantly decreases while the received power of the third-order increases notably. Under turbulence with D/r0 values of 1, 2, and 4, the received power of the transmitted beams improves by 9 dB, 11 dB, and 8 dB, respectively.
Fig. 5. (a) The intensity distribution of the OAM + 3 beams before and after compensation under turbulence strengths D/r0 = 1, 2 and 4, respectively. (b) The phase distribution of the OAM + 3 beams before and after compensation under turbulence strengths D/r0 = 1, 2 and 4, respectively. (c) The received power of the OAM + 3 beams before and after compensation under turbulence strengths D/r0 = 1, 2 and 4, respectively.
The performance of Zernike modes and turbulent OAM modes is compared. Turbulent OAM modes not only utilize OAM coefficients to directly characterize the phase of the distorted beam but also represent turbulent information as a superposition of turbulent OAM factors. In contrast, Zernike modes can only fit turbulence phase using Zernike polynomials. For comparison, the compensated phase screen is obtained by simulating the transmission of turbulence information characterized by 5 turbulent OAM coefficients (5-OAM), the first 6 Zernike coefficients (6-Zernike) and the first 21 Zernike coefficients (21-Zernike). Figure 6 displays the cumulative probability of received power before and after compensation for received beams (OAM0 and OAM + 2) under different turbulence conditions. The results demonstrate that Zernike polynomials with a small order, such as 5-OAM, do not produce a good compensating effect. Only when characterizing the weak and moderate turbulence information well, can 21-Zernike achieve improvement in received power comparable to the turbulent OAM compensation for OAM beams, as shown in Figs. 6(a), (b), (d) and (e). However, in strong turbulence (D/r0 = 4), the ability of 21-Zernike to represent turbulence information begins to be inferior to that of 5-OAM, shown in Figs. 6(c) and (f). Compared to Zernike modes, the received power of the transmitted beams with turbulent OAM improves by 4 dB. This phenomenon can be attributed to the fact that decomposing strong turbulence into Zernike modes requires a significantly higher number of high-order modes compared to OAM modes.
Fig. 6. (a, b, c) The comparison using the turbulent OAM coefficients and Zernike coefficients before and after compensation for the cumulative probability of OAM0 beams under turbulence strengths D/r0 = 1, 2 and 4, respectively. (d, e, f) The comparison using the turbulent OAM coefficients and Zernike coefficients before and after compensation for the cumulative probability of OAM + 2 beams under turbulence strengths D/r0 = 1, 2 and 4, respectively.
Moreover, within an OAM-based FSO link, orthogonal OAM modes are transmitted. Consequently, the turbulent OAM approach directly compensates for the transmitted OAM mode using turbulent OAM, resulting in a significant enhancement of the received power. It is worth highlighting that employing lower-order turbulent OAM, as opposed to Zernike method, not only considerably diminishes the necessary training data volume but also enables phase measurement of the OAM beam without relying on a WFS.
To investigate the benefits of the proposed turbulent OAM approach in OAM multiplexing communication systems, two turbulence characterization methods are compared to quantify the mode crosstalk before and after compensation. Figure 7 displays the results of crosstalk before and after compensation for multiplexed OAM channels (OAM + 1 and OAM + 3) under different turbulence. The power of crosstalk before compensation increases with increasing turbulence strength. However, after turbulent OAM compensation, the crosstalk in the $\textrm{OA}{\textrm{M}_3}$ channel becomes lower across different turbulence strengths. For D/r0 = 1, the crosstalk reduces from -13 dB to -27 dB. For D/r0 = 2, the crosstalk reduces from -6 dB to -21 dB. For D/r0 = 4, the crosstalk reduces from -3 dB to -14 dB. Compared to Zernike modes, the crosstalk of multiplexed channels with turbulent OAM is reduced by approximately 10 dB under varying turbulence conditions. The results clearly demonstrate that the turbulent OAM scheme excels in reducing crosstalk when compared to the Zernike mode. This superiority can be attributed to the fact that decomposing turbulence into OAM modes and applying inverse compensation is essentially a direct method of mitigating crosstalk between OAM modes during transmission. Therefore, it becomes evident that turbulent OAM stands out as the more fitting choice for OAM multiplexing communication systems.
Fig. 7. The comparison using the turbulent OAM coefficients and Zernike coefficients before and after compensation for the crosstalk power of the OAM + 3 channel before and after compensation under turbulence strengths D/r0 = 1, 2 and 4, respectively.
4. Conclusion
In conclusion, a CNN-based turbulent OAM approach is proposed and demonstrated for turbulence compensation, aimed at enhancing the performance of OAM-based FSO communication. An operator approach is utilized to extract the OAM component of atmospheric turbulence and the CNN is trained to predict the turbulent OAM coefficients. It reveals that different OAM beams undergo the same mode spreading according to the OAM components in atmospheric turbulence. Under the condition that atmospheric turbulence remains constant during beam propagation, compensating any OAM beam in the FSO link proves to be effective. Through long-path simulations comprising weak, moderate, and strong turbulence, the results show that after compensation, the received power of the OAM + 3 channel with turbulent OAM is increased by 9 dB, 11 dB, and 8 dB under turbulence conditions of D/r0 = 1, 2, and 4, respectively. Compared to Zernike modes, turbulent OAM modes effectively characterize most of the turbulence information using only a small number of orders. Notably, the received power of the transmitted beams with turbulent OAM improves by 4 dB at strong turbulence strength D/r0 = 4. Additionally, the crosstalk of multiplexed channels with turbulent OAM is reduced by 10 dB over that with Zernike under varying turbulence conditions. The results obtained demonstrate the potential of the turbulent OAM approach to enhance the system reliability of OAM-based FSO links, particularly when dealing with minimal mode orders in turbulent environments. Furthermore, under strong turbulence conditions, theoretically increasing the mode orders can achieve more effective compensation. Our approach offers a promising solution for OAM-multiplexed FSO communication and can be easily extended to other turbulent applications, such as remote sensing imaging and optical ranging.
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); Chongqing Graduate Student Research Innovation Project (CYS23447).
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.
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