1. Introduction

In recent years, various multiplexing techniques have been widely used in optical fiber communication systems in order to exceed the Shannon limit on the capacity of a single mode fiber (SMF), including the wavelength, amplitude, phase, and polarization multiplexing. Orbital angular momentum (OAM) is an approach that can increase the channel capacity in an optical fiber communication system and is attracting growing interest [1]. OAM provides an additional degree of freedom in multiplexing, which depends on a phase term with infinite integer values of topological charge index l [2,3]. OAM transmission has been experimented over different distances [4–8]. It now can be achieved with multi-mode fiber (MMF) or few-mode fiber (FMF) over 8.8km [4]. One of the main restrictions on OAM transmission in FMF or MMF is crosstalk. A common approach for the compensation of the crosstalk is the multi-input multi-output (MIMO) digital signal processing (DSP). However, it is impractical to apply MIMO in real-time FMF or MMF transmission due to the high complexity of MIMO DSP. For instance, a 10-mode channel over FMF transmission would require a $\textrm{20} \times \textrm{20}$ MIMO DSP [9]. Compared with conventional fibers, ring core fiber (RCF) is a special type of optical fiber with large effective refractive index $({\Delta {\textrm{n}_{eff}}} )$ difference between mode groups (MGs), which can separate the eigenmodes effectively [10]. RCF has attracted a great deal of attention due to the dramatic reduction in complexity: an eight-mode OAM mode-division multiplexed transmission only needs $\textrm{4} \times \textrm{4}$ MIMO DSP for RCF transmission over 100km [11].

Although MIMO DSP could suppress crosstalk effectively, nonlinearity is still a key problem limiting the spectral efficiency. The nonlinear effect in communication system can be divided into two categories: optical fiber nonlinearity and device nonlinearity. Optical fiber nonlinearity, which include self-phase modulation (SPM), intra or inter cross-phase modulation (IXPM), and four wave mixing (FWM), often causes serious signals distortion over long distance transmission [12]. Digital backpropagation (DBP) algorithm and perturbation method are traditional approaches to compensate the nonlinearity of optical fiber, which are based on the analysis of the nonlinear Schrodinger equation (NLSE) of field propagation in optical fiber [13]. Besides, multiple opto-electronic devices are required to modulate signals for OAM mode-division multiplexed optical fiber communication, such as Mach-Zehnder modulator (MZM), photodiode (PD) and spatial light modulator (SLM), et al. [14]. Device nonlinearity is the inherent characteristic of these opto-electronic devices, which will cause serious deterioration on the signal [15]. Traditional nonlinearity compensation approaches, such as Volterra equalizer, usually compensate the nonlinearity through the Volterra series to fit nonlinear model of devices [16]. However, it is very hard to build an accurate theoretical model for multiple opto-electronic devices by using Volterra series. More importantly, some special opto-electronic devices are still lack the related theoretical model. For example, the nonlinear effect of the SLM may be generated through several physical phenomena, such as self-phase modulation and memory effect, which is difficulty to determine an accurate model [17,18]. Hence, few reports has been published for an accurate and comprehensive model of SLM nonlinearities. As a result, it is very hard to compensate the nonlinear impairment of the special opto-electronic devices in the OAM mode division multiplexed optical fiber communication through the conventional approaches, due to the uncertainty of the nonlinear model. Meanwhile, the computational complexity of conventional nonlinear equalizer, such as Volterra equalizer, is very high due to the complex multiplication operation.

Machine learning algorithms of classification could classify the test data by using the mathematics method based on the attribute from the training data, which is very feasible to build an accurate nonlinear model for the complex and uncertain opto-electronic devices. Machine learning algorithms for classification have all shown good performances and tolerance on nonlinearity mitigation. For instance, k-nearest neighbors (KNN) algorithm for mitigating nonlinearity in 16/64-QAM coherent optical communication system has been demonstrated [19]. Besides, the support vector machines (SVM) for the compensation of nonlinear equalization in singe-carrier and multi-carrier coherent optical orthogonal frequency-division multiplexing (OFDM) have also been explored [20–22]. Moreover, the neural network is a powerful tool to build the nonlinear model for optical fiber communication noise, showing good performances on nonlinearity mitigation in many researches, such as deep neural network (DNN), artificial neural network (ANN) and long short-term memory (LSTM) neural network [23–25]. Especially, the naïve Bayes algorithm, which is a statistical classification algorithm that obtains the posterior probability of a data set through the prior and likelihood function, is ideal to mitigate the nonlinear impairments in OAM optical fiber communication with high accuracy and low complexity [26].

In this manuscript, a data-defined naïve Bayes (DNB)-based decision scheme for nonlinear mitigation is presented for an OAM mode-division multiplexed optical fiber communication system. An experiment is carried out to verify the effectiveness of the proposed scheme, and the results demonstrate that the accuracy and calculation complexity of the DNB algorithm is significantly improved compared with other machine learning algorithms.

2. Principle

The nonlinearity of the opto-electronic device is a key problem in OAM mode division multiplexed optical fiber communication system. For example, the spatial light modulator (SLM) is a key opto-electronic devices in the system to generate the OAM beam, which consists of densely packed liquid crystal cells arranged in two-dimensional pixel arrays [27]. Many physical phenomena in the SLM may generate the nonlinear noise on the OAM beam, including self-phase modulation and memory effect. The liquid crystal cells would be rotated at different deviation angles and directions by loading the driven voltage, and then the phase modulation of the beam is carried out. However, the self-phase modulation is an inherent characteristics of liquid crystal cells whose reflective index can be present as $\textrm{n}(I )= {n_0} + \alpha I$, where ${n_0}$ is the constant refractive index of liquid crystal cell, I is the modulated light intensity through the SLM, and $\alpha$ is the nonlinear coefficient, which is dependent on the characteristic of liquid crystal cell and light frequency [17]. The nonlinear distort generated by the self-phase modulation of the liquid crystal cells would increase the nonlinear noise of the modulated OAM beam, leading to the deterioration of the orthogonality for OAM mode with different topological charge. As a result, the nonlinear impairment is accelerated through the self-phase modulation of the SLM. Besides, liquid crystal cells have strong memory effect, which is attributed to electric field induced charge transfer from liquid crystal molecules [28]. This effect would increase the nonlinear noise of the driven liquid crystal cells, which also deteriorate the OAM beam. Conventional approaches, such as Volterra equalizer and polynomial equalizer, usually compensate the nonlinearity through the nonlinear modeling of each device. However, few reports have been published for the accurate and comprehensive theoretical model for SLM nonlinearity due to the complex interplay between the liquid crystal cell and OAM beam. Therefore, the accuracy nonlinearity fitting for the OAM mode division multiplexed optical fiber communication system cannot be achieved with the Volterra polynomial due to the complexity of the nonlinear impairments. As a result, conventional approaches are hard to compensate the nonlinearity of OAM mode division multiplexed transmission.

Instead, the use of the DNB algorithm with a likelihood probability distribution is an alternative method to compensate the complex and uncertain nonlinear impairments in OAM mode-division multiplexed transmission. Although nonlinear impairments of some devices in this type of transmission is uncertain, an accurate nonlinear model can be built through the statistical probability of the nonlinear noise for each OAM mode. Therefore, the DNB algorithm based on the likelihood probability distribution of the nonlinear noise could mitigate the uncertain nonlinear impairments in OAM mode-division multiplexed transmission effectively.

2.1 Principle of the DNB based decision

The naïve Bayes algorithm can be described as a statistical classification algorithm that gives the posterior probability of a dataset based on a prior probability and likelihood probability. Consider a finite set $B = \{{X_1}\textrm{,}\ldots \textrm{, }{X_i},..\textrm{, }{X_n}\} $ of discrete random variables. Assuming that each variable ${\textrm{X}_i}$ is attributed to a class $C = \{{C_1}\textrm{, }\ldots \textrm{, }{C_i},\ldots \textrm{, }{C_l}\}_i$, the likelihood probability, $P({ B |{C_i}} )$ that describes the probability of each instance $B = \{{X_1}\textrm{,}\ldots \textrm{, }{X_i},..\textrm{, }{X_n}\}$ is attributed to a class ${C_i}$. And the variables ${\textrm{X}_i}$ are features of instance B. $P({ B |{C_i}} )$ can be given as [29]

Equation (2) shows that the probability of class ${C_i}$ can be predicted based on the prior probability and the likelihood probability [30]. From Eq. (2), it can be seen that both the posterior probability and the likelihood probability contain the probability distribution information for set B.

In an optical fiber communication system, the received signals always contain the features of the transmission channel. The DNB algorithm is therefore suitable for the mitigation of uncertain nonlinear impairments over OAM mode-division multiplexed transmission, since the distribution information of the uncertain nonlinear noise can be obtained based on the likelihood probability. The transmission signal can be described as set B, while the constellation point can be described as class ${C_i}$. The prior probability $({P({{C_i}} )} )$ and likelihood probability $({P({ B |{C_i}} )} )$ can be obtained from the training data, which contains the probability distribution of the uncertain nonlinear noise.

In Eq. (2), $\sum\limits_{i = 1}^n {P({{C_i}} )} P({ B |{C_i}} )\textrm{ = }P(B )$ can be ignored, since it is the same for all constellation points [31]. The posterior probability for each constellation point can then be computed based on the prior probability and likelihood probability, which can be simplified as [29], where a is the number of instance B:

For the received OAM signal, the posterior probability of each item of signal data ${f_i}(B )$ for each constellation point can be obtained by submitting the value of the I/Q axis into Eq. (3), and the decision process of the received OAM signal can then be executed directly based on the maximum posterior probability of each constellation point. The posterior probability also contains the distribution information of the uncertain nonlinear noise due to the likelihood probability according to Eq. (3). Hence, the DNB algorithm can effectively mitigate uncertain nonlinear impairments based on the posterior probability.

2.2 DNB based decision for OAM mode division multiplexed transmission

The proposed DNB-based decision scheme is divided into two stages, training and execution. In the training stage, DNB learns the prior and likelihood from the training data and compute the posterior probability of each class. A schematic diagram for DNB-based decision scheme is shown in Fig. 1. A known dataset of constellation diagram with four classes is shown in Fig. 1(a). The first step of the training progress is the calculation of the discrete probability density. One class is taken as an example, as shown in Fig. 1(b). One constellation diagram is divided into $\textrm{4} \times \textrm{4}$ spaces along the x- and y-axes. Then, the discrete probability density value of each space is calculated in the second step, as shown in Fig. 1(b). Finally, the continuous probability distribution is obtained by fitting the discrete probability density value, as shown in Fig. 1(c). The other three classes of the constellation diagram are fitted to get the corresponding continuous probability distributions in the same way. Therefore, four continuous probability distributions for the constellation diagram are obtained, as shown in Fig. 1(d). In the execution stage, one test data is received with the value of x- and y-axes, as presented in Fig. 1(e). The probability densities for four classes are then calculated from the learning distributions, respectively, as shown in Fig. 1(f) and 1(g). Finally, the constellation point of the received data is decided by the comparison of four classes with the highest probability, as shown in Fig. 1(h).

 

Fig. 1. DNB based decision schematic diagram.

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In the training stage, the OAM optical fiber system transfers a set of known data with a 32GBaud QPSK signal. The received signals are labelled with four classes, $C = \{{C_1}\textrm{, }{C_\textrm{2}}\textrm{, }{C_\textrm{3}},{C_\textrm{4}}\}_{i}$, corresponding to the four constellation points of QPSK. The prior probability ($P({{C_i}} )$) and likelihood probability ($P({ B |{C_i}} )$) are then calculated based on these training data using the DNB algorithm. For the likelihood probability ($P({ B |{C_i}} )$), most of the conventional probability distributions, such as the normal, Poisson and Rayleigh distributions, are unsuitable for the description of OAM mode-division multiplexed transmission due to uncertain nonlinearity of the opto-electronic device. Hence the likelihood probability distributions for each class, which indicate the probability function ${g_i}({x,y} )$, should be obtained from known training datasets utilizing the nonparametric kernel density estimation. The probability distribution for each class corresponding to each constellation point is calculated by interpolating a kernel function into a sample density at each coordinate $({x,y} )$, which is given by

At the execution stage, the coordinate $({x,y} )$ for each received test OAM signal is submitted to four ${g_i}({x\textrm{, }y} )$ probability functions to calculate four likelihood probabilities $P({ B |{C_i}} )$ corresponding to the four constellation points of QPSK modulation. Each posterior probability $P({ {{C_i}} |B} )$ for the four constellation points could then be obtained by multiplying the likelihood probability and prior probability, as shown in Eq. (4). Finally, the received OAM signal is classified as one constellation point based on the maximum value of the posterior probability, $Max\{ P({ {{C_i}} |B} )\} ,i \in 1,2,3,4$.

3. Experimental results

3.1 Experimental setup

The experimental setup for 32Gbaud OAM mode division multiplexed optical fiber communication system over 10km RCF transmission is represented as Fig. 2. At transmitter, data passed digital signal process (DSP) block is encoded onto waveform generated by arbitrary waveform generator (AWG) with a sample rate of 64GSa/s. The optical carrier is generated by using an external cavity (EC) laser with a wavelength of 1550.12nm [34], followed by the modulation of the Mach-Zehnder modulator (MZM). The modulated electrical data sequence is pseudo-random binary sequence (PRBS) with pattern length of 2¹⁸. Signal is amplified by using erbium-doped fiber amplifier (EDFA) for improving transmission distance. Then it is split into two components by using an optical coupler. One of the components is delayed through a 10m single mode fiber (SMF) to reduce correlation. Signal beams are coupled from fiber to space through polarization controllers (PC), collimators (Col.), and linear polarization (LP) [35]. For modulating OAM modes with $\textrm{l} ={+} 4$ and $\textrm{l} ={+} 5$, two spatial light modulators (SLM) are used. Insets (i) and (ii) present the amplitudes of OAM mode with topological charge $\textrm{l} ={+} 4$ and $\textrm{l} ={+} 5$. Then two OAM beams are multiplexed with a beam splitter (BS1).

 

Fig. 2. Experimental setup for 32Gband OAM mode division multiplexed optical fiber communication system over 10 km RCF transmission.

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Then the multiplexed OAM beams are split into two beams by using BS2. One beam transmits through the BS2 with the OAM modes of $\textrm{l} ={+} 4$ and $\textrm{l} ={+} 5$, while the other beam is reflected with BS2, three reflected mirrors, and BS3, respectively. As a result, this beam contains two OAM modes with $\textrm{l} = \textrm{ - }4$ and $\textrm{l} = \textrm{ - }5$ due to five times reflections. Thus after the combination of two beams by using the BS3, the multiplexed OAM beams contain four modes of $\textrm{l} ={\pm} 4$ and $\textrm{l} ={\pm} 5$. Then the multiplexed OAM beams is converted into circular polarization from linear polarization by a quarter wave plate (QWP1). These four circular polarization OAM modes are again split into two linear polarizations (horizontal and vertically polarization) by using PBS1, and combined again with the PBS2 and two reflected mirrors. QWP2 is used to convert the linear polarization back into the circular polarization (CP), As a result, the multiplexed OAM beams contain eight modes (<$\textrm{ + }4$, left CP>, <$\textrm{ + }4$, right CP>,<$\textrm{ - }4$, left CP>, <$\textrm{ - }4$, right CP>; <$\textrm{ + 5}$, left CP>, <$\textrm{ + 5}$, right CP>,<$\textrm{ - 5}$, left CP>, <$\textrm{ - 5}$, right CP>) in two mode groups. The eight OAM modes are multiplexed and coupled into the 10km RCF. Insets (iii) presents the cross section of RCF.

At receiver, all OAM modes are converted into linear polarization and split into two beams by using BS4. Two beams are passed through vortex phase plates (VPP) with opposite topological charge. After propagation of the VPP with opposite topological charge, the OAM beam with corresponding mode can be converted to Gaussian beams, which is coupled into the SMF with the collimator. The optical signals are received by using the coherent detection and processed by using the off-line DSP, which includes frequency offset estimation, $\textrm{4} \times \textrm{4}$ offline MIMO, clock recovery, phase recovery and DNB based decision [36]. Two uncorrelated datasets with the maximum normalized cross-correlation of 0.8% are transmitted for training and testing in order to keep the data independence.

3.2 Experimental results and analysis

An experiment is carried out using a 32Gband eight-mode OAM optical fiber communication system with RCF transmission over 10 km to verify the effectiveness of the proposed method. Two OAM mode groups containing eight OAM modes with |l|=4, |l|=5 and < left, right > circular polarization are transmitted using QPSK modulation. Figure 3 illustrates the training process for the experimental data under an OSNR of 23 dB. The proposed DNB algorithm trains a probability density for each class corresponding to the constellation point to obtain the likelihood probability using Eq. (6). The length of training dataset, validation dataset, and test dataset are 200, 10000, and 19807 constellation points, respectively. Two uncorrelated datasets with the maximum normalized cross-correlation of 0.8% are transmitted for training and testing in order to keep the data independence. In the training stage, the bandwidths are optimized as [0.0577, 0.0570, 0.0584, 0.0519], and the constellation diagram are divided into $\textrm{256} \times \textrm{256}$ spaces with the scale form along the x- and y-axes. Figure 3(a) shows the constellation diagram, while Fig. 3(b) shows the discrete probability density of four constellation points for each space using the Gaussian kernel function. The independent discrete probability density of the four constellation points along the x- and y-axes are shown in Fig. 3(c). All probability densities for the different constellation points are distributed with an approximately normal distribution due to the amplified spontaneous emission noise in the RCF channel. However, the normal distribution of each constellation point is deformed differently under the uncertain nonlinear impairment. For example, the distributions of the constellation points at (+0.5, +0.5) and (−0.5, −0.5) in Fig. 3(c) are deformed much more severely than those of the other two constellation points. A two-dimensional discrete probability density at each small space is then fitted to obtain a continuous probability distribution, as shown in Fig. 3(d). Figure 3(e) shows the contours (for the same probability density) of the four classes for four constellation points. The contour lines at the centre of each class are denser, indicating a higher probability of constellation coordinate points in the centre. Conversely, the contour lines at the edge are sparser, indicating a lower probability. The probability density corresponding to the two constellation points of (+0.5, +0.5) and (+0.5, −0.5) are distributed widely compared with the other two constellation points, indicating a strong nonlinear impairment. Based on the training process mentioned above, the probability densities of all training data with the OSNR from 14 dB to 23 dB are calculated to obtain the corresponding likelihood probability at different OSNR.

 

Fig. 3. Distributions in a training progress of one of experimental data constellation diagrams under the OSNR of 23 dB.

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Before the comparison between DNB algorithm and other machine learning algorithms, we firstly compensate the nonlinear noise using the conventional Volterra equalizer followed by the hard-decision. Simultaneously, the DNB based decision algorithm is also tested for all of the received OAM signals. The BER of the OAM signals by using the Volterra and DNB approaches are shown in Fig. 4. However, the BER by using the DNB algorithm is much lower than that by using the Volterra equalizer at each OSNR, which confirm the effectiveness of proposed DNB based decision algorithm for the mitigation of nonlinear impairments in OAM mode division multiplexed transmission. The nonlinear model of some opto-electronic devices, such as SLM, is difficult to fit precisely by using Volterra series. Hence the conventional Volterra equalizer cannot compensate nonlinear impairments in OAM mode division multiplexed transmission effectively. On the contrary, through the likelihood probability of the training data, the DNB algorithm could learn the probability distribution of the nonlinear noise, which mitigates the nonlinear impairment in OAM mode division multiplexed transmission effectively with the posterior probability.

 

Fig. 4. Performances on BER of 8 OAM modes.

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To evaluate the performance of different machine learning algorithms in terms of mitigating the nonlinear impairment, we test two other machine learning decision algorithms, KNN and SVM, which are widely used in the DSP of optical communication systems. Figure 4 shows the BER for eight OAM modes using the DNB, KNN and SVM algorithms. A comparison with different decision algorithms shows that the DNB outperformed the KNN algorithm in terms of BER by a minimum of 11% at $\textrm{l} ={+} \textrm{5}$ for OSNR=18 with right polarization, and a maximum of 66% at $\textrm{l} = \textrm{ - }4$ for OSNR=23 dB with right polarization using $\textrm{k} = \textrm{5}$ and Euclidean distance nearest neighbours. The BER for DNB exceeded that of the SVM algorithm by a minimum of 6% at $\textrm{l} ={+} 4$ for OSNR=14 dB with right polarization, and a maximum of 35% at $\textrm{l} = \textrm{ - }4$ for OSNR=23 dB with right polarization, using an extension of the original nonlinear binary classification in SVM.

Table 1 gives the computational complexity of DNB, KNN and SVM algorithm in this experiment, including the training complexity and running complexity. For training complexity, the complexity of DNB algorithms is $O\{ [\textrm{d} + \frac{1}{2}d(d - 1)]n\}$, where $\textrm{d}$ is dimensions ($\textrm{d = 2}$ for QPSK modulation), and $\textrm{n}$ is the number of training sample [37]. As an unsupervised classification algorithm, the training complexity of KNN is zero. The training complexity of SVM is $O[\frac{1}{2}\textrm{d}c(c - 1){n^2}]$ by using an extension of binary classification of SVM [38], where c and $\textrm{d}$ are constants for QPSK modulation. Hence the training complexity of SVM is much higher than that of the DNB because the training complexity of SVM is accumulated with the square operation (${n^2}$) compared with the DNB ($n$), especially with a large training sample.

Table 1. Computational complexity of DNB, KNN and SVM

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The running complexity is more likely to cause computational redundancy and transmission delay. For the DNB, the complexity is a fixed value $O(cd)$, that is independent of sample number since c and $\textrm{d}$ are constants for QPSK modulation. However, the running complexity of KNN is proportional to training sample and test sample, which is very high with a large sample. For the SVM, the running complexity of SVM is proportional to the number of support vector $\textrm{s}$, which is very large with the complex of the fuzy boundaries in the sample datasets. Therefore, compared with $O(mnk)$ of KNN (where m is the number of running sample, k is the number of k nearest points.) and $O(\frac{1}{2}\textrm{sd}c(c - 1))$ of SVM (where s is the number of support vector which indicates the number of samples that determine the boundary), the running complexity of $O(cd)$ for DNB is decreases significantly, as shown in Table 1 [37–39].

To verify the complexity of three machine learning algorithms, the computational time of three algorithms is recorded. The results are similar for each mode group at each OSNR. The computational time is represented as Fig. 5 with mode group $\textrm{l} = |\textrm{4} |$ at OSNR=14, 19, 23dB respectively. It can be seen that the computational time of DNB is significantly reduced compared with that of other two algorithms, which confirm the complexity comparison in Table 1.

 

Fig. 5. Computational time with mode group $l = |4 |$ at different OSNR.

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In this experiment, the Gaussian function is applied as the kernel function. According to [31], the values of the bandwidth and band border have an impact on the estimation results, which should be optimized through cross-validation in order to avoid under/over-fitting. To illustrate the effectiveness of the bandwidth and band border, BERs for different bandwidths and band borders for a dataset with OSNR=23dB are presented in Fig. 6. And space number N represents bandwidth which there is a monotone transformation given by Eq. (5), b represents the band border length calculated by the maximum minus the minimum in the training dataset. The performance in terms of the BER is optimized based on suitable values of the bandwidth and band border. For other values, the estimations would be under/over-fitted, which would reduce the BER performance.

 

Fig. 6. BERs under different bandwidth and band borders of a data set at OSNR=23 dB.

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4. Conclusion

A decision scheme for nonlinear mitigation of OAM optical fiber communication is proposed based on the machine learning of DNB algorithm in this manuscript. Due to the complex nonlinear model of the opto-electronic devices in the OAM mode division multiplexed transmission, the nonlinear impairment possesses a stochastic characteristic, which is difficult to compensate for conventional nonlinear compensation algorithm. The statistical algorithm in RCF provides an alternative method. The DNB algorithm could learn the likelihood probability of nonlinear noise, and mitigate the strong nonlinearity of the OAM communication system with posterior probability due to the uncertain device nonlinearity. An experiment for 32Gband 8 modes OAM optical fiber communication system over 10km RCF transmission is carried out to verify the effectiveness of the proposed method. The experimental results demonstrate that the accuracy and computational complexity of the DNB algorithm is improved significantly compared with classification algorithms KNN and SVM. The performances of DNB algorithm are improved by a maximum of 35% on BER and 0.4273dB on Q factor than SVM algorithm, and by a maximum of 66% on BER and 1.4461dB on Q factor than KNN algorithm. Moreover, the complexity of DNB decision algorithm is much lower compared with that of other two algorithms. The DNB decision algorithm employs a posterior probability distribution to mitigate the strong nonlinear impairments with the complexity of opto-electronic devices in OAM mode division multiplexed transmission, which possess high effectiveness and low complexity.