Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication

Junmin Liu; Peipei Wang; Xiaoke Zhang; Yanliang He; Xinxing Zhou; Huapeng Ye; Ying Li; Shixiang Xu; Shuqing Chen; Dianyuan Fan

doi:10.1364/OE.27.016671

1. Introduction

Vortex beam (VB) is a structured light which carries orbital angular momentum (OAM) [1]. Because of the phase singularity in the beam cross-section, VB presents a “doughnut” intensity distribution. And its helical phase wave-front can be characterized by an azimuthal phase factor of $\exp (i l θ),$ where $l$ is the topological charge (TC) and $θ$ is the azimuth angle. Theoretically, $l$ can take any integer value, and the VBs with different OAM modes are mutually orthogonal [2]. These features reward VBs with many remarkable optical properties, and show promising applications in optical trapping [3–5], optical microscopy and imaging [6–8], quantum information [9–11], optical storage [12], and optical communications [13–16], etc. Particularly in optical communication, VB can provide an additional degree of freedom in increasing communication capacity [17,18]. By multiplexing OAM modes, the bit-rate of 1.37 Tbit/s and the spectral efficiency of 25.6 bit/s/Hz are realized in free space optical communication [19]. And a 100 Tbit/s free-space data link is also achieved by three-dimensionally multiplexing OAM, polarization, and wavelength [20]. However, the spiral phase of VB is sensitive to the transmission environment and easy to be distorted. Therefore, the atmospheric turbulence (AT) resulting from the inhomogeneity of temperature and pressure in atmosphere will severely deteriorate the VBs transmitted in free space. The wave-front distortion, coherence destruction, and orthogonality destruction of the multiplexed VBs are the main effects of AT, which will directly increase the crosstalk among channels and reduce the communication performance [21–23].

Various methods have been proposed to improve the robustness of VBs against AT. The adaptive optics (AO) systems are the most common ways [24,25], and the frequently used algorithms are Gerchberg-Saxton (GS) and stochastic-parallel-gradient-descent (SPGD) algorithm [26]. However, both GS [27] and SPGD [28] algorithms require multiple iterations, which need a long processing time. Moreover, these algorithms have no learning and memory ability, and in the process of iterative calculations, the convergence is often stagnated due to the falling of local minima. Recently, the deep learning technology, which shows remarkable ability in image recognition and classification, has attracted extensive attentions in optics [29–31]. Utilizing the information capture and automatic classification capabilities of deep learning, the demodulation and demultiplexing technologies for OAM communications have been studied [32]. The turbulence correction in conjugate superposition OAM modes has been also realized, and this deep learning based compensation method even shows more accurate and faster correction ability than AO system [33]. However, the correction efficiency is dragged down because each superposition mode needs to be trained separately, and the turbulence corrections for non-conjugate modes are uninvolved, which severely block its practical application. Since the number of channels will determine the communication capacity density, eliminating non-conjugate OAM modes seems an enormous waste to multiplexing resources. Therefore, the methods with generalization ability in quickly compensating the turbulence distorted VBs are still urgently needed for practically applying the dimension of OAM modes.

In this paper, we propose and investigate a novel deep learning based AT compensation method for correcting the distorted VB and improving the performance of OAM multiplexing communication. To simultaneously compensate the turbulence distortion for multiple VBs, we introduce Gaussian probe beam (GPB) as information extraction object. After supervised training, the convolutional neural network (CNN) model can learn to produce compensation phase screen through the intensity distribution of GPB in randomly changed turbulent environments. After compensation, the mode purity of the distorted VB can be improved from 39.52% to 98.34% with the turbulence intensity of $C_{n}^{2}$ = 1 × 10⁻¹³. Constructing an OAM multiplexing communication link, the bit-error-rate (BER) of each OAM channel is reduced by almost two orders of magnitude after compensated by the trained CNN in moderate-strong AT. These results demonstrate that the CNN model we proposed can well predict the turbulent phase and shows good generalization ability in quickly and accurately compensating the distorted VBs, which makes the AT compensation great flexibility, and may provide an efficient solution for improving the performance of OAM based communication.

2. Theory of turbulence compensation based on CNN

2.1 The effects of AT on VB

AT always exist in free space, and affect the transmission quality of light beams. Particularly for VB, the helical phase wave-front makes it susceptible to AT and easy to be distorted. The researchers found that in a local homogeneous isotropic region, the statistical distribution of turbulence obeys the “2/3 law”, which can be expressed as [34]

D (r) = C_{n}^{2} r^{2 / 3}, l_{0} ≪ r ≪ L_{0},

where

D (r)

is the refractive index structure function,

r

represents the distance between two observation points,

l_{0}

and

L_{0}

are the inner and outer scale respectively.

C_{n}^{2}

is the atmospheric refractive index structure constant, representing the turbulence intensity. Based on this relationship, Kolmogorov proposed a power spectral density expression for refractive index fluctuations [35]

Φ_{n} (k) = 0.033 C_{n}^{2} k^{- 11 / 3},

where

k

is the wavenumber. However, the Kolmogorov spectral model only applies to the inertial region of

1 / L_{0} ≪ k ≪ 1 / l_{0}

. Thus, a Tatarskii power spectrum model that can extend the application range to the dissipation region is introduced. However, there are singularities at

k = 0

and

1 / L_{0} = 0 (L_{0} \to \infty)

. To resolve these problems, a Hill-Andrews spectrum model that can better simulate the authenticity of AT is proposed [35], which can be expressed as

\begin{array}{l} Φ_{n} (k_{x}, k_{y}) = 0.033 C_{n}^{2} [1 + 1.802 \sqrt{\frac{k_{x}^{2} + k_{y}^{2}}{k_{l}^{2}}} - 0.254 (^{\frac{k_{x}^{2} + k_{y}^{2}}{k_{l}^{2}}) - 7 / 12}] \\ \times \exp (\frac{k_{x}^{2} + k_{y}^{2}}{k_{l}^{2}}) (^{k_{x}^{2} + k_{y}^{2} + \frac{1}{L_{0}^{2}}) - 11 / 6}, \end{array}

where

Φ_{n} (k_{x}, k_{y})

represents the atmosphere refractive index power spectrum when the beam transmits in the

z

direction, and

k_{l} = 3.3 / l_{0}

.

Here, we use Hill-Andrews spectrum model to generate AT phase screen. The relationship between the refractive index power spectrum and phase spectrum is $Φ (k_{x}, k_{y}) = 2 π k_{0}^{2} Δ z Φ_{n} (k_{x}, k_{y})$ , $Δ z$ denotes the turbulence length. The variance of the phase spectrum can be written as

σ^{2} (k_{x}, k_{y}) = {(\frac{2 π}{N Δ L})}^{2} Φ (k_{x}, k_{y}),

where

Δ L

is the grid spacing. After fast Fourier Transform, the phase screen in the time domain can be obtained as

ϕ (x, y) = F F T [C \cdot σ (k_{x}, k_{y})],

where

ϕ (x, y)

is a

N \times N

phase screen,

F F T

is the Fourier Transform,

C

is an

N \times N

dimensional complex random number array with a mean of 0 and a variance of 1. When a beam

U (z, x, y)

propagates in AT over

Δ z

, the optical field at

z + Δ z

can be approximately described by

U (z + Δ z, x, y) \approx F F T^{- 1} [\exp (i A Δ z) \cdot F F T {\exp (i ϕ (x, y)) \times U (z, x, y)}] .

Here,

\exp (i A Δ z)

is the transfer function of Fresnel propagation,

A = (k_{x}^{2} + k_{y}^{2}) / (2 k)

,

F F T^{- 1}

represents the 2D Inverse Fourier Transform.

Figure 1 shows the effects of AT on the VBs with different TCs. The effect of AT on the beam is not only related to the $C_{n}^{2}$ but also to the source radius of the beam. To describe the strength variation of the AT more easily and intuitively, we set the beam waist radius to a fixed value. The turbulence intensities in Figs. 1(a) to1(d) are $C_{n}^{2} = 0$ , $C_{n}^{2} = 1 \times 10^{- 14},$ $C_{n}^{2} = 5 \times 10^{- 14},$ and $C_{n}^{2} = 1 \times 10^{- 13} .$ The AT phase screen is set with the size of $N \times N = 201 \times 201$ , the grid spacing of $ΔL= 0.0003 m,$ the inner scale of $l_{0} = 0.0001 m,$ and the outer scale of $L_{0} = 50 m$ . The intensity distributions and phase images of the VBs with TCs of 0, 1, and 3 through 20 m AT are presented in the right sides of each graph. From these figures, the phase fluctuation enhanced with the turbulence intensity, which directly affects the intensity and phase distributions of VBs. Under the influence of the same turbulence intensity, the distortion of VBs increases with the TC. More seriously, the distorted phase wave-front will lead to severe modes dispersion and crosstalk. Figure 2 shows the mode purity of the VB ( $l = 3$ ) experienced different turbulence intensities. As we can see that the larger the turbulence, the more severe the mode dispersion is, which results in a significant reduction in mode purity. As the turbulence intensity increases to $C_{n}^{2} = 1 \times 10^{- 13},$ the mode purity is only 39.52%.

Fig. 1 The intensity and phase distributions of VBs with different topological charges $(l = 0, 1, 3)$ under the influence of AT. (a) $C_{n}^{2} = 0$ (without turbulence); (b) $C_{n}^{2} = 1 \times 10^{- 14};$ (c) $C_{n}^{2} = 5 \times 10^{- 14};$ (d) $C_{n}^{2} = 1 \times 10^{- 13} .$

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Fig. 2 The mode purity of the VB $(l = 3)$ under the influence of AT. (a) $C_{n}^{2} = 1 \times 10^{- 14};$ (b) $C_{n}^{2} = 5 \times 10^{- 14};$ (c) $C_{n}^{2} = 1 \times 10^{- 13} .$

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2.2 AT compensation using deep learning

Deep learning, also known as artificial neural network, is one of the machine learning techniques based on data learning [36]. Its motivation lies in establishing a neural network of human brain-like analysis, and imitating the mechanism of human brain to process data. Different from other machine learning techniques, deep learning allows computational model contains multiple processing layers to learn data with multiple levels of abstraction, so that it can extract high-level abstract features. As long as enough the hidden units, the models can represent any complex but smooth functions. The neural network can learn a function $Y = f (X, θ)$ to approximate the real function $Y = f * (X),$ and the parameters $θ$ can be learned from the training data set $(X, Y)$ by the minimum loss function $L (f (X, θ), Y) .$

In this AT compensation task, a CNN is designed to learn the phase screen of AT by the intensity distribution of GPB. The training data set contains $N$ examples $(x_{i}, y_{i}),$ $i = 1 \cdot \cdot \cdot N,$ where $y_{i}$ is a gray image of AT phase screen produced by the Hill-Andrews model $(Δ z \in [15 m, 25 m],$ $C_{n}^{2} \in [1 \times 10^{- 14}, 1 \times 10^{- 13}]),$ and $x_{i}$ consists of two gray images (one is the intensity distribution of GPB influenced by AT in $y_{i},$ and the other is the intensity distribution of the GPB without turbulence).

The proposed CNN model is depicted in Fig. 3. The inputs of the network are the intensity distributions of GPBs, including the patterns with and without the influence of turbulence. The standard GPB without turbulence is shared, and only one light intensity distribution needs to be generated. The trained CNN model requires a better generalization ability, that is, it can process untrained data well. Also, the CNN model with strong generalization ability can effectively prevent overfitting. To achieve this objective, it is necessary to ensure that the training error and the model complexity penalty are small, wherein the model complexity penalty is affected by N and decreases with the increase of N. Therefore, 75600 GPB intensity patterns with specified turbulence range are randomly generated, among which 70000 of them are used as training data, 3000 are used as validation data, and the remaining 2600 are test data. The validation data is employed to check the training effect and observe whether the turbulence phase predicted by CNN is consistent with the ideal turbulence phase. If the effect is poor, we can stop the training and re-adjust the parameters or the model without waiting for the end of the training, which can save a lot of computing time. Besides, the validation data can be used to test the generalization ability of the model and whether overfitting has occurred. The size of each image is resized to $64 \times 64$ , and the images are normalized by the maximum gray value. The output of the network is the phase screen of AT with the size of $64 \times 64,$ which is also normalized into $[0, 1]$ by a proper normalization factor. The CNN architecture contains 15 learned layers, including 12 convolutional layers and 3 deconvolutional layers. The first convolutional layer uses $5 \times 5$ convolutional kernels, the other convolutional and deconvolutional layers use $3 \times 3$ convolutional kernels. The active function, Relu, is employed for every layer except the last one which uses the sigmoid as active function. Our loss functions are

l_{1} (f (x_{i}, θ), y_{i}) = {(y_{i} - f (x_{i}, θ))}^{2},

l_{2} (f (x_{i}, θ), y_{i}) = R e l u (y_{i} - f (x_{i}, θ) - a),

l_{3} (f (x_{i}, θ), y_{i}) = R e l u (f (x_{i}, θ) - y_{i} - a),

\begin{array}{l} L (f (X, θ), Y) = \sum_{i = 1}^{N} [l_{1} (f (x_{i}, θ), y_{i}) + l_{2} (f (x_{i}, θ), y_{i}) + l_{3} (f (x_{i}, θ), y_{i})] / N \\ = \sum_{i = 1}^{N} [{(y_{i} - f (x_{i}, θ))}^{2} + R e l u (y_{i} - f (x_{i}, θ) - a) + R e l u (f (x_{i}, θ) - y_{i} - a)] / N, \end{array}

where

a

is a constant factor,

R e l u (x) = \max (x, 0) .

The first item in Eq. (10) is used to make the predicted turbulence phase close to the true label

y_{i},

and the last two items are used to penalize the pixels which have more phase errors. Then the optimal parameters can be found by

Fig. 3 The architecture of the CNN model used for AT compensation. Conv: convolution; Deconv: deconvolution.

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θ_{o p t} = \min_{θ} (L (f (X, θ), Y) .

When the optimal parameters are explored, the trained CNN can predict the phase of AT $\hat{y}$ from the intensity distribution of GPB x:

\hat{y} = f (x, θ_{o p t}) .

After the predicted AT phase is obtained, the atmospheric compensation can be achieved by

y_{c o m} = y - G (\hat{y}),

where

G (\hat{y})

is a Gaussian filter used to smooth the predicted phase screen.The loss function is employed to estimate the degree of error between the predicted value and the real value, which is an important factor affecting the generalization ability of the CNN model. The graph of the loss function in the training

(Δ z \in [15 m, 25 m],

and

C_{n}^{2} \in [1 \times 10^{- 14}, 1 \times 10^{- 13}])

is shown in Fig. 4(a). As the number of iterations increase, the loss value becomes progressively smaller and tends to be stable. When the iterations increase to 16000, the loss value is already close to 0. To reflect the trend of loss curve more intuitively, we take five points A, B, C, D, E on the loss curve. The iterations and the loss value corresponding to these five points are given in Fig. 4(b). A loss value of 0.0143 is reached at the 13971th iteration, which demonstrates that the CNN model at this time has a good ability to extract effective information on turbulence. Figure 4(c) presents the phase images predicted by the CNN with the validation data at different training iterations and the corresponding actual phase screens. From the figures, we can see that when the iterations are sufficient, the predicted phase images are almost the same as the real phase screens, which indicates the CNN model we proposed has high accuracy.

Fig. 4 (a) Loss function curves. (b) The iteration numbers and loss values corresponding to A, B, C, D, and E, respectively. (c) AT phase diagrams.

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After extracted the distortion phase information, the phases turbulence can be compensated by loading a reverse phase to the distorted beams. Figure 5 illustrates the compensation effects of the trained CNN at different turbulence levels. As shown in the figures, the turbulence phase has been reduced to a weak level after compensation. It should be noted that the Gaussian beam is used as a probe beam to train the model, and the purpose is to extract the turbulence information from the GPB to compensate multiple VBs at the same time. Since the range of $Δ z$ is $[15 m, 25 m]$ and the range of $C_{n}^{2}$ is $[1 \times 10^{- 14}, 1 \times 10^{- 13}]$ in the training, the values of $Δ z$ and $C_{n}^{2}$ are randomly selected, which has a strong diversity. Therefore, when testing the turbulence compensation effects under different turbulence intensities and different distances, the same trained CNN model can be used.

Fig. 5 The phase compensation effect of the trained CNN under different AT intensities at the distance of $Δ z = 20 m .$ (a) $C_{n}^{2} =1 \times 10^{-14};$ (b) $C_{n}^{2} =5 \times 10^{-14};$ (c) $C_{n}^{2} =1 \times 10^{-13} .$

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The turbulence information extracted from the GPB by CNN is used for correcting VBs. AT will distort the intensity distribution of VBs, and the distortion increases with the turbulence intensity (see Fig. 6). However, after compensation, the deformed profiles are well corrected with a relatively uniform intensity distribution.

Fig. 6 The beam profiles of the VB $(l = 3)$ with and without compensation corresponding to different turbulence intensities, $Δ z = 20 m .$

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Except for the distortion of intensity distribution, AT also result in phase distortion. And the phase distortion of VB will cause the reduction of mode purity. Therefore, the turbulence correction ability can be measured by comparing the mode purity of the VBs before and after compensation. Figure 7 shows the mode purity of the VB $(l = 3)$ in different turbulence intensities at $Δ z = 20 m .$ It can be seen from the curves, the mode dispersion becomes more serious as the turbulence increases, which reduces the mode purity, and may lead to mode crosstalk in OAM communication. To guarantee the performance of communication, it is necessary to correct the optical carrier distorted by turbulence. And the prediction ability of the CNN model on the turbulence phase directly determines the correction effect. The low error prediction means a strong compensation ability in reducing mode dispersion and improving the OAM mode purity. Here, the ATs with five different intensities are compensated, and all the compensated mode purity are increased to over 97%. The specific values in Fig. 7(b) show that after compensation, the mode purity of the VB increases from 39.52% to 98.34% with the turbulence intensity of $C_{n}^{2} = 1 \times 10^{- 13} .$ This proves that the compensation ability of the CNN model is not limited to a certain turbulence intensity range. As long as the GPBs provide effective information, the CNN can correct all the turbulence well.

Fig. 7 The mode purity of the VB $(l = 3)$ under the influence of different AT intensities with and without CNN compensation, $Δ z = 20 m .$ (a) The mode purity curves as the function of turbulence intensity $C_{n}^{2} .$ (b) The mode purity before and after compensation for different turbulence intensities.

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Except for the strength of AT, the propagation distance will also affect the purity of OAM modes. Figure 8(a) presents the mode purity of the VB $(l = 3)$ with the turbulence intensity of $C_{n}^{2} =5 \times 10^{-14}$ and the propagation distance is increased from 15 m to 25 m. As shown in the figure, the influence of turbulence increases with the transmission distance. From Fig. 8(b), under the turbulence intensity of $C_{n}^{2} =5 \times 10^{-14},$ the mode purity of the VB at the propagation distance of 15 m and 25 m are 92.33% and 70.01% respectively, but both can be improved to 98.66% and 98.82% after compensation. These illustrate that the trained CNN model can accurately predict the AT phase screen and significantly improve the mode purity of VB by compensation. The time required for predicting turbulence is about 9 ms per image by using I5-6500 CPU, and the processing time can be further reduced with the acceleration of CPU.

Fig. 8 The mode purity of the VB $(l = 3)$ under the AT intensity of $C_{n}^{2} =5 \times 10^{-14}$ at different distance $Δ z .$ (a) The mode purity curves as the function of distance $Δ z .$ (b) The mode purity before and after compensation for different distances.

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3. OAM multiplexing communication link and results

The system diagram of the OAM multiplexing free space optical (FSO) communication link with atmospheric compensation is shown in Fig. 9. A laser beam (beam1) with the working wavelength of 1550 nm is modulated to carry 16 QAM-OFDM signals and split into two sub-beams which delayed with fibers for decorrelation. For each path, the signal beam is linearly polarized in the X-direction and projected onto a fork grating to load spatially helical phase wave-front $(l = 1, 3) .$ The multiplexed VBs are then combined with the Y-polarized GPB (beam2) by a polarizing beam splitter (PBS), and transmitted together through AT with the equivalent phase screen produced by the Hill-Andrews model. At the receiver, the spatial light modulator (SLM) is only sensitive to X-polarization and used to load the compensation phase. Thus, both the GPB and VBs will be distorted by the AT, but only the VBs are compensated.

Fig. 9 The system diagram of OAM multiplexing FSO communication link with AT compensation. PBS: polarization beam splitter; AT: AT; MR: mirror; SLM: spatial light modulator.

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After propagation, the GPB is separated from the multiplexed VBs by a PBS. And a CCD is employed to extract the intensity distribution of GPB, which is used as the input of CNN model to predict AT phase screen. Then the atmospheric compensation can be achieved by loading a reversed phase of the predicted phase screen on the SLM.

To make the simulation system more practical, the parameters of the beam1 generated from Laser 1 are set with a waist radius of $w = 0.00 2 m,$ and the working wavelength of $λ = 1550 n m .$ The beam 2 generated from Laser 2 are set with a waist radius of $w = 0.00 4 m,$ and the working wavelength of $λ = 1550 n m .$ The turbulence length of $Δ z \in [15 m, 25 m],$ and the structure constant of the index of refraction is $C_{n}^{2} \in [1 \times 10^{- 14}, 1 \times 10^{- 13}] .$ According to the system scaling theory, an experimental link can be used to emulate a practical outdoor long-range atmospheric turbulence transmission [37]. Considering the aperture size of the sender as $D_{r e}$ = 1cm, the transmission distance is equivalent to the out-door turbulence transmission distance of $Z \in [38.5 m, 62.5 m] .$

Since the signals carried by VB interfered with any noise will induce bit-errors, we employ Gaussian white noise to disturb signals and use the signal-to-noise ratio (SNR) to represent the noise level. Figure 10 shows the BERs of the OAM channels of $l = 3$ and $l =1$ with and without turbulence compensation at different SNRs. As we can see from the figures, the BERs decrease dramatically after compensation. When the SNR reaches 20 dB, the BERs of all channels are less than $10^{-3}$ , which indicates that the compensation method we proposed can well improve the communication performance of OAM multiplexing.

Fig. 10 The BER curves as the function of SNR for different turbulence intensities $C_{n}^{2}$ with and without compensation at the distance of $Δ z = 20 m .$ (a) $l = 3.$ (b) $l = 1 .$

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To further study the influence of turbulence to BER, we take the SNR as a fixed value (SNR = 20 dB) and analyze the BER under different turbulence intensities. Figure 11(a) shows the BER curves as the function of turbulence intensity $C_{n}^{2}$ with and without atmospheric compensation at the distance of $Δ z = 20 m .$ The demodulated constellations of the OAM channel of $l = 3$ at $C_{n}^{2} = 1 \times 10^{- 14},$ $C_{n}^{2} = 5 \times 10^{- 14},$ and $C_{n}^{2} = 1 \times 10^{- 13}$ are shown in Fig. 11(b), and its convergence can be measured by the vector error magnitude (EVM). As illustrated in the figures, the BER of each OAM channel at $C_{n}^{2} =2 .5 \times 10^{- 14}$ almost exceeds $10^{- 3}$ and increases with the $C_{n}^{2}$ without atmospheric compensation, which is because of the intermodal crosstalk induced by the turbulence. However, after compensation, the BERs of each OAM channel under different turbulence intensities are mostly less than $10^{-3}$ and keep stable. Besides, the distribution and convergence of the constellation can also directly reflect the turbulence compensation effect of the CNN model. As shown in Fig. 11(b), the constellation of the channel $l = 3$ is obviously converged after compensation, and the EVM values reduced from 0.3239 to 0.1558 with $C_{n}^{2} = 1 \times 10^{- 13},$ which indicate that the performance of the transmitted 16 QAM-OFDM signals have been greatly improved.

Fig. 11 (a) The BER curves as the function of turbulence intensity $C_{n}^{2}$ for each OAM channel with and without compensation at the distance of $Δ z = 20 m .$ (b) The constellations of the channel $l = 3$ at $C_{n}^{2} = 1 \times 10^{- 14},$ $C_{n}^{2} = 5 \times 10^{- 14},$ and $C_{n}^{2} = 1 \times 10^{- 13}$ without (top) and with (bottom) AT compensation.

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AT also limits the communication distance, making most of high-capacity FSO communications with OAM multiplexing only available in meter level. Here, we analyze the BER of each channel at different transmission distances. Figure 12(a) depicts the BER curves as the function of transmission distance (the length of turbulence) $Δ z$ for each OAM channel at $C_{n}^{2} = 5 \times 10^{- 14}$ and SNR = 20 dB. Figure 12(b) shows the demodulated constellations of each OAM channel at $Δ z = 15 m, 20 m, 25 m .$ As shown in the figures, before compensation, the BER of each channel increases with the transmission distance and remains above $10^{-2}$ from $Δ z = 15 m$ to $Δ z = 25 m .$ And the constellations become more scattered and blurred as the distance increases. These demonstrate that as the transmission distance increases, the influence of turbulence becomes larger, and the signals are deteriorated, which make it difficult to realize long-distance transmission of signals. However, the BERs are reduced by nearly two orders of magnitude, and the constellation converges well after compensation. For example, the EVM value of the compensated constellation decreases from 0.2802 to 0.1506 at $Δ z = 25 m .$

Fig. 12 (a) The BER curves as the function of transmission distance of $Δ z$ for each OAM channel with and without AT compensation at $C_{n}^{2} = 5 \times 10^{- 14} .$ (b) The constellations of the channel $l = 3$ at $Δ z = 15 m, 20 m, 25 m$ without (top) and with (bottom) atmospheric compensation.

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To verify the generalization ability of the CNN model, we use the previously trained model to perform the turbulence compensation for different multiplexed VBs and analyze its BER performance. Table 1 lists the BER of the OAM channel $l = 3$ at $C_{n}^{2} = 5 \times 10^{- 14},$ $Δ z = 20 m$ when the multiplexed OAM modes are $l = 2, 3,$ $l = 4, 3,$ $l = 5, 3$ respectively (SNR = 20 dB). The demodulated constellations of the OAM channel $l = 3$ corresponding to Table 1 are shown in Fig. 13. As presented in the list, the BER of the channel $l = 3$ of the $l = 5, 3$ multiplexing modes is less than $l = 2, 3$ and $l = 4, 3$ because the mode crosstalk power of $l = 5$ versus $l = 3$ is relatively small. After turbulence compensation, the BERs of $l = 3$ channel of the three multiplexing modes are all reduced to below $10^{-3} .$ And the compensated constellations are well converged, indicating that the turbulence compensation greatly mitigates the influence of mode crosstalk, and the trained CNN can flexibly and effectively compensate multiple OAM multiplexing modes.

Table 1. The BER of the OAM channel when the other multiplexed OAM mode is different

View Table | View all tables in this article

Fig. 13 The constellations of the channel $l = 3$ with multiplexed OAM modes $l = 2, 3,$ $l = 4, 3,$ $l = 5, 3$ with and without AT compensation.

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Considering the actual experimental conditions, we increase the transmission distance to 1000 m. Figure 14 shows the BER curves as the function of turbulence intensity $C_{n}^{2}$ at the transmission distance of $Δ z = 1000$ m and the corresponding demodulated constellations of OAM channel $l = 3.$ Comparing the distance of $Δ z = 20$ m, the BERs have significantly increased, and the convergence of constellations is reduced. The constellations of $C_{n}^{2} = 5 \times 10^{- 15}$ and $C_{n}^{2} = 1 \times 10^{- 14}$ are greatly blurred, and the EVM values increase to 0.4172 and 0.5550, respectively. This means that the greater the distance is, the more severe the OAM modes crosstalk caused by turbulence, resulting in poor signal quality. After compensated by the CNN model, the BERs are also reduced nearly by two magnitude orders. Meanwhile, the constellations converge, and its EVM values of $C_{n}^{2} = 5 \times 10^{- 15}$ and $C_{n}^{2} = 1 \times 10^{- 14}$ are decreased to 0.1611 and 0.1871, respectively. These demonstrate that the CNN model also shows good compensation ability for long distances transmission of VBs.

Fig. 14 (a) The BER curves as the function of turbulence intensity $C_{n}^{2}$ for each OAM channel with and without compensation at the transmission distance of $Δ z = 1000 m .$ (b) The constellations of the channel $l = 3$ at $C_{n}^{2} = 1 \times 10^{- 15},$ $C_{n}^{2} = 5 \times 10^{- 15},$ and $C_{n}^{2} = 1 \times 10^{- 14}$ without (top) and with (bottom) AT compensation.

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4. Discussion

The working mechanism of the proposed CNN model can be simply described as a mathematical problem. For the intensity distribution of standard GPB $E_{1}$ not affected by turbulence, the intensity distribution of GPB $E_{2}$ affected by turbulence and the ideal output turbulent phase P, there is a relationship of $E_{2} = f (P, E_{1}),$ where $f$ represents the light field conversion function of $E_{1}$ which is affected by P. After transformation, $P = g (E_{1}, E_{2})$ can be obtained, where $g$ is the transformed function of $f,$ and the CNN model can learn the $g$ function through a large number of $(E_{1}, E_{2})$ data pairs and the corresponding P. Once the expression is learned by the CNN model, it can directly output $P$ based on the input of $(E_{1}, E_{2}) .$ Since $E_{1}$ is fixed, the mapping relationship between $E_{2}$ and $P$ is the most key problem. Therefore, the fluctuation amplitude of the output $P$ increases with the turbulence on $E_{2},$ which means that whether the turbulence is strong or weak, the distance is long or short, the turbulent phase can be well predicted by the CNN model. Using CNN to train the weak turbulence $C_{n}^{2} \in [1 \times 10^{- 15}, 1 \times 10^{- 14}]$ and strong turbulence $C_{n}^{2} \in [1 \times 10^{- 13}, 1 \times 10^{- 12}]$ (see Fig. 15), the training loss function and the mean test loss value are reduced to below 0.05. The reason for the reduction of the prediction effect in strong turbulence is that to make a stark contrast between strong and weak turbulence, the normalization factors are unified. Some pixels of the saved turbulence phase will overflow the gray value range, which results in a reduction in effective information. From the perspective of algorithm, the reason why the information extraction ability of CNN is not affected by the global change of the objects is that CNN can automatically extract the intrinsic features of the image through a multi-layer structure, in which the convolution operation and subsampling make the features of objects locally related. These enable the CNN-based turbulence phase prediction applies to various turbulent environments.

Fig. 15 Loss function curve and mean test loss value for the weak AT of $C_{n}^{2} \in [1 \times 10^{- 15}, 1 \times 10^{- 14}]$ and strong AT of $C_{n}^{2} \in [1 \times 10^{- 13}, 1 \times 10^{- 12}],$ $Δ z \in [15 m, 25 m] .$

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In the process of constructing the CNN structure, considering our goal is not simple classification but predicting the whole turbulent phase, which requires a good capture of all the effective information of input. Thus, we chose a 15-layer structure, and the number of layers can be changed according to the actual needs. It is well known that the more complex the structure is, the more training data is needed to ensure a small mode complexity penalty, and make a relatively small test error. Table 2 lists the mean test loss corresponding to three different numbers of training data with $C_{n}^{2} = 5 \times 10^{- 15}$ and $Δ z = 1000$ m, and the test data is 3000. From the comparison results, the mean test loss decreases with increasing training data. Therefore, to ensure the predictive ability of the CNN model, 70000 training data are adopted. If better prediction performance is required, the training data or iterations need to be further increased.

Table 2. The mean test loss for different numbers of training data

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In the OAM multiplexed FSO communication, the criteria for evaluating the compensation capability of the CNN model are not only the loss function which measures the difference between the predicted phase and the ideal phase, but also the BER and constellation. In the free-space transmission process, the wave-front distortion and mode dispersion caused by turbulence will seriously destroy the orthogonality among channels and increase the mode crosstalk, resulting in poor communication performance. However, the CNN model can well predict the fluctuation amplitude of each point of the turbulent phase. After inverse compensation, the amplitude of each point of the phase can be reduced to a weak level. Therefore, the phase perturbation of the compensated optical carrier is extremely small, the orthogonality between the channels is effectively restored, and the mode crosstalk is inhibited, which directly reflected in the decrease of BER and the convergence of constellation. It should be noted that these are static processing. The equivalent phase screen is used to simulate the turbulence influence of VB in practice, and CNN is used to compensate the turbulence. It seems that the CNN model is does not apply to the actual experimental environment where the turbulence is time-varying. However, the AT can be “frozen” in an extremely short period, and researchers have found that the frozen time of turbulence is less than 10 ms [38,39]. Our turbulence prediction time is about 9 ms which can be further reduced by improving computer performance. The calculation speed of the CNN model is mainly determined by the calculation amount of the neural network, namely Floating-point Operations (FLOPs), and the computing power of the computer. The FLOPs of the CNN model can be expressed by the time complexity as $F \sim O (\sum_{i = 1}^{D} M_{i}^{2} \cdot K_{i}^{2} \cdot C_{i - 1} \cdot C_{i}),$ where $O$ represents the functional relationship between the computation amount of the neural network and the scale of the problem to be processed, $D$ is the number of convolutional layers of the CNN, $M$ is the size of the feature map output of each convolution kernel in the i-th convolution layer, $K$ is the size of each convolution kernel, $C_{i - 1}$ and $C_{i}$ represent the number of input and output channels for each convolution kernel respectively. When the CNN structure is determined, its FLOPs is a fixed value, and the calculation speed is mainly affected by the Floating-point Operations Per Second (FLOPS) of the computer. Here we run the CNN model with the I5-6500 CPU, the FLOPS of an entry-level GPU is about eight times faster, and a more advanced GPU GTX1080ti is about thirty times faster. Therefore, using a high-performance GPU is expected to shorten the turbulence prediction time of CNN to 0.3 ms. Thus, the CNN-based turbulence compensation method we proposed may provide a good solution for dynamic turbulence compensation. As for the devices of optical intensity collection and phase compensation that determines the entire compensation time, the commonly used SLM and CCD whose switching frequency are about 50 Hz. To achieve the real-time application, the SLM and CCD need to be replaced by other devices with high processing speed, such as digital micro-mirror device and complementary metal oxide semiconductor (CMOS) with operating frequency up to 1000 Hz.

Considering some devices may have time lags in a practical turbulence compensation system, which will cause the compensation time exceeding the turbulent freezing time, and the turbulence has changed. We use the turbulence phase predicted by the CNN model to compensate the turbulence, which has changed by different degrees in a reasonable time-scale. Figure 16 shows the compensation effect after turbulence has changed to varying degrees. The leftmost side is the original turbulent phase, the second to seventh columns are the phase distributions after turbulence has changed by 5.2%, 9.73%, 14.33%, 19.66%, 26.01%, and 33.26% respectively. The corresponding compensated turbulence phase is shown in the second row, where the used compensation phase is the turbulence phase predicted by CNN for the original turbulence. If the phase predicted by CNN has a high fitting degree with the original turbulence, the change in turbulence is equivalent to increasing the error of the predicted phase, resulting in that the turbulence after compensation cannot be reduced to very low intensity. And the greater the turbulence change is, the lower the compensation ability. Therefore, the compensation time should be minimized in the actual real-time turbulence compensation system.

Fig. 16 The compensation effect after turbulence has changed to varying degrees.

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5. Conclusion

In conclusion, we have proposed an AT compensation scheme for VBs based on the CNN model. After phase compensation, the mode purity of the VB is higher than 90% under different AT intensities. Especially in the strong turbulence intensity, the mode purity is improved from 39.52% to 98.34% with the turbulence intensity of $C_{n}^{2} =1 \times 10^{-13}$ . And an OAM multiplexing free-space optical communication link is also constructed to evaluate the performance of the AT compensation. The numerical simulation results show that the BER of each OAM channel decreases almost two orders of magnitude under different turbulence intensities after compensation. And the proposed CNN model spends only 9 ms to predict the phase screen on the CPU (I5-6500), which may provide a reference for real-time compensation of turbulence.

Funding

The National Natural Science Foundation of China (61805149, 61575127); Guangdong Natural Science Foundation (2016A030310065); The Program of Fundamental Research of Shenzhen Science and Technology Plan (JCYJ20180507182035270); The Natural Science Foundation Guangdong Education Department (2015KTSCX124, 2015KQNCX146); Science and Technology Project of Shenzhen (ZDSYS201707271014468), and The Educational Commission of Guangdong Province (2016KCXTD006).

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Multiplexed OAM modes	BER (without compensation)	BER (with compensation)
l₁ = 2, l₂ = 3	0.0366	9.4551 × 10⁻⁴
l₁ = 4, l₂ = 3	0.0318	3.2051 × 10⁻⁴
l₁ = 5, l₂ = 3	0.0243	3.2053 × 10⁻⁴

The number of training data	Mean test loss
10000	0.00548
40000	0.00344
70000	0.00291

Multiplexed OAM modes	BER (without compensation)	BER (with compensation)
l₁ = 2, l₂ = 3	0.0366	9.4551 × 10⁻⁴
l₁ = 4, l₂ = 3	0.0318	3.2051 × 10⁻⁴
l₁ = 5, l₂ = 3	0.0243	3.2053 × 10⁻⁴

The number of training data	Mean test loss
10000	0.00548
40000	0.00344
70000	0.00291

Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication

Abstract

1. Introduction

2. Theory of turbulence compensation based on CNN

2.1 The effects of AT on VB

2.2 AT compensation using deep learning

3. OAM multiplexing communication link and results

4. Discussion

5. Conclusion

Funding

References

Cited By

Figures (16)

Tables (2)

Equations (13)

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