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Abstract
Orbital angular momentum (OAM) mode-division multiplexing (MDM) is a key technique to achieve ultra-high-capacity optical fiber communications. However, the high nonlinear impairment from optoelectronic devices, such as spatial light modulators, modulators, and photodiodes, is a long-standing challenge for OAM-MDM. In this paper, an equalizer based on a probabilistic neural network (PNN) is presented to mitigate the nonlinear impairment for an OAM-MDM fiber communication system with 32 GBaud Nyquist pulse amplitude modulation-8 (PAM8) intensity-modulation direct-detection (IM-DD) signals. PNN equalizer can calculate the distribution of the nonlinearity using Bayesian decision theory and thus mitigate the stochastic nonlinear impairment of the received signal. Experimental results show that compared with the convolutional neural network (CNN) equalizer, the PNN equalizer improves the receiver sensitivity by 0.6dB and 2dB for two OAM modes with l = + 3 and l = + 4 at the 20% FEC limit, respectively. Moreover, compared with Volterra or CNN equalizers, the PNN equalizer can reduce the computation complexity significantly, which has great potential to mitigate the nonlinear signal distortions in high-speed IM-DD OAM-MDM fiber communication systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Intensity-modulation direct-detection (IM-DD) schemes are preferred in short-reach applications due to their low system cost and power consumption [1]. In recent years, the demand for high capacity for IM-DD communication has exploded significantly in various industries, such as intra-data-center networks, the Internet of Things, cloud services, and other industries, which is difficult to achieve with conventional single-mode fiber (SMF) communication. Therefore, few-mode/multi-mode fiber (FMF/MMF) has attracted a great of attention due to significant data transmission ability with the mode multiplexing (MDM) technology. MDM can expand the optical communication capacity significantly by using linearly polarized (LP) modes or Orbital Angular Momentum (OAM) [2–5]. Especially for the OAM mode, the vortex wavefront phase, as a new degree of freedom, can provide an infinite number of orthogonal states in theory [6–8]. As a result, the IM-DD OAM-MDM transmission system has great potential to improve the transmission capacity, and some related transmission experiments have been reported by using the MMF [9,10]. However, mode coupling leads to serious crosstalk between the signals in the different modes of the MDM system [11]. Recently, ring-core fiber (RCF) has become an important carrier in the MDM system with low crosstalk between different modes [12,13]. In RCF, the effective refractive index difference ($\Delta {n_{eff}}$) between different mode groups is larger than that of conventional FMF/MMF, weakening the coupling between the adjacent modes of the RCF [14]. Therefore, an OAM-MDM system based on RCF transmission makes the large capacity IM-DD communication possible.
In an IM-DD OAM-MDM system, the nonlinear impairment caused by devices (e.g., modulator, electronic amplifier, and photodetector) is the most serious limitation on the performance of the system [15], which causes strong signal distortion. Especially, spatial light modulators (SLM) are always employed to generate the OAM lights in the OAM-MDM system. The SLM contains many nonlinear materials, such as liquid crystal, nonlinear polymer, and photorefractive material, which introduces serious nonlinear impairment [16]. In general, nonlinear impairment can be mitigated by adjusting the operating parameters of photoelectric devices, such as the output power of the electronic amplifier (EA) or the paranoid voltage of the modulator. However, these operations will also lead to a reduction in the signal-to-noise ratio (SNR), resulting in system deterioration [17]. Therefore, digital signal processing (DSP) becomes an alternative method to mitigate the nonlinear impairment of the communication system, such as the Volterra series [18], Digital Predistortion (DPD) [19], and Lookup Table (LUT) [20]. DSP algorithms mitigate the nonlinear impairment in the fiber optical communication systems by fitting a nonlinear mathematical model of the system. However, there are two difficulties in mitigating the device nonlinear impairment in OAM-MDM systems using traditional DSP algorithms. First, when many optoelectronic devices are working in the OAM-MDM system, nonlinearities from different devices can couple with each other, resulting in high complexity of the entire system nonlinearity. Second, the mode coupling in the OAM-MDM system also make a strong stochastic characteristic of the nonlinear impairment. Hence it is very difficult to mitigate the nonlinear impairment in the OAM-MDM system by using conventional DSP algorithms [16–21].
In recent years, machine learning has been regarded as a new direction to compensate nonlinear impairment in fiber communication [22]. Machine learning algorithms can learn the characteristics of the nonlinearity from a training signal and build an accurately fitting model to compensate the nonlinear impairment [23–28], including Deep Neural Network (DNN) [26], Convolutional Neural Network (CNN) [27], and Long Short-Term Memory (LSTM) Network [28]. These algorithms can fit complex nonlinear models accurately and compensate the nonlinear impairment effectively. However, most machine learning algorithms must learn the characteristics of the nonlinearity through a training data set. The strong stochastic nonlinearity in the OAM-MDM system leads to a huge difference between the training data and testing data, resulting in an inaccurate model. In addition, many machine learning algorithms involve high DSP complexity, which is not suitable for high-speed fiber communications.
In this paper, we propose an equalizer based on a probabilistic neural network (PNN) for the nonlinear mitigation in the OAM-MDM communication system with IM-DD transmission. This equalizer can calculate the distribution of the nonlinearity using the statistical properties of the received data [22], hence it can effectively fit a nonlinear model even to an OAM-MDM system with high stochasticity and complexity. An experiment is carried out to verify the effectiveness of the PNN equalizer, and the results demonstrate that the accuracy and complexity of the PNN equalizer are improved significantly compared with the conventional Volterra equalizer or CNN equalizer.
2. Principle
Nonlinear distortion caused by the intrinsic nonlinear characteristics of devices, such as the response of EA, the self-phase modulation of the spatial light modulator (SLM) [15], and the square-law detection and saturation effect of the photodetector (PD) [29], can seriously affect the communication performance. We propose a PNN nonlinear equalizer for the mitigation of nonlinear impairment in the IM-DD OAM-MDM system. In a high-speed IM-DD OAM-MDM system with pulse amplitude modulation-8 (PAM-8) modulation, different levels of the electrical signal can be regarded as different categories. The received symbols are divided into training samples and testing samples. The PNN equalizer calculates the Euclidean distance between the training sample and the testing sample, and the likelihood probability of the testing samplés belonging to each category is calculated by using a Gaussian probability density function (PDF). Then, the posterior probability is obtained by using the Bayesian formula. Finally, each received symbol is classified to its corresponding category by using Bayesian decision theory, mitigating the stochastic nonlinear impairment of the high-speed OAM-MDM system.
2.1. Principle of the PNN equalizer
The PNN nonlinear equalizer is illustrated in Fig. 1. It consists of an input layer, a pattern layer, a summation layer, and an output layer [30,31]. $[{Y_1},{Y_2}, \cdots ,{Y_p}]$ is the set of training samples, $[{Y_1},{Y_2}, \cdots ,{Y_q}]$ is the set of testing samples, and $C = \{ {c_1},{c_2}, \cdots ,{c_i}\}$ is the set of categories.
At first, each testing sample ${Y_q}$ is transmitted through the input layer to the pattern layer, and each training sample ${Y_p}$ is considered as a pattern layer neuron. Note that the number of neurons is equal to q in the input layer.
Second, the Euclidean distance between a testing sample and each pattern layer neuron is calculated at the pattern layer. Then, the probability density of each pattern layer neuron can be calculated through Gaussian PDF, which is expressed by
Third, in the summation layer, the summation layer neurons add up the probability density values belonging to the same category and calculate the likelihood probability of the testing sample
ni are the number of pattern neurons in each category, which have the same value for different categories. Then the posterior probability of each category can be calculated with the Bayesian formula in the summation layer P(ci) is the prior probability, which is equal to 1/(8 M) for PAM-8.Finally, the output layer outputs the classification decision based on Bayesian decision theory. The testing sample can be assigned to the category with the highest posterior probability
2.2 PNN equalizer for OAM mode division multiplexed transmission
Considering the original signal $x(n)$ which is supposed to be transmitted in the OAM-MDM system, the received signal $y(n)$ can be expressed as [32]
where H function denotes the channel response, which includes nonlinear distortion, and noise(n) denotes the additive noise generated in the system. The distorted time-series signal y(n)= [y1, y2, …, yn] is a vector containing n sequence PAM-8 symbols. Before the processing by the PNN equalizer, the samples need to be pre-processed. The received signal is transformed from cascade to parallel with a stride of one to form a dataset $Y = {\{ }[{y_1},{y_2}, \cdots ,{y_M}],[{y_2},{y_3}, \cdots ,{y_{M + 1}}], \cdots {\} = \{ }{Y_1}\textrm{, }{Y_2}\textrm{, }\ldots \textrm{, }{Y_m}{\} }$. The dataset Y consisting of vectors can be expressed asSince a distorted PAM-8 signal y(n) consists of eight electrical level values in $\{{\pm} 1, \pm 3, \pm 5, \pm 7\}$. ${Y_m}$ can be divided into 8M categories $C = \{ {c_1},{c_2}, \cdots ,{c_i}\}$. The process of the PNN equalizer for the high-speed OAM-MDM optical communication system with PAM-8 modulation is illustrated in Fig. 2. First, the testing sample is transmitted into the input layer. Then at the pattern layer, training samples are divided into 8M categories as different pattern layer neurons. The Euclidean distance and the probability density between the testing sample and each pattern layer neuron are calculated by using Eq. (1). After that, the summation layer calculates the posterior probability of each category by Eq. (2) and (3). Finally, the output layer chooses the optimized category from $C = \{ {c_1},{c_2}, \cdots ,{c_i}\}$ according to the maximum posterior probability.
The structure of the complete algorithm of the PNN equalizer is shown in Algorithm 1.
Algorithm 1. Structure of PNN equalizer
Xj represents the symbol after equalization, and yq represents the symbol after the clock recovery algorithm processing and before the nonlinear equalization processing.
In a high-speed OAM-MDM optical fiber communication system, the nonlinear impairment is highly stochastic due to the mode coupling. It is difficult to compensated for this stochasticity using a convectional nonlinear equalizer with the fixed parameter. However, the PNN equalizer can calculate the statistical properties of the distribution of the nonlinearity using Bayesian decision theory, hence the prior probability $p({c_i})$ and the likelihood probability $p({y_q}|{c_i})$ can be obtained from the training data. Since the nonlinearities of the training samples and the testing samples have the same distribution, the testing samples can be classified according to the posterior probability of the nonlinearity, rather than the minimum Euclidean distance, which can greatly mitigate the nonlinear impairment.
3. Experimental
3.1 Experimental setup
The experimental setup of the OAM-MDM fiber communication system with IM-DD is illustrated in Fig. 3. A 32 GBaud PAM-8 signal was transmitted over a 2.3km RCF, which is detected by using a receiver with an offline DSP. At the transmitter, a pseudo-random sequence with the length of 217 is generated and mapped into a PAM8 symbol sequence. The electrical signal is generated by using an arbitrary waveform generator (AWG) at the sampling rate of 64 GSa/s. After an EA, the electrical signal is utilized to modulate an optical carrier at the wavelength of 1550 nm through a Mach-Zehnder modulator (MZM) to generate a double-sideband optical signal. Then the generated optical signal is split into two branches by using an optical coupler (OC), and amplified by using an erbium-doped fiber amplifier (EDFA). One of them is delayed by using a 10m SMF for data mode decorrelation. A polarization controller (PC) follows to adjust the polarization direction of the signal light to guarantee the maximum power.
Through the collimators (Col.), two light beams are transmitted from the SMF to a SLM via a linear polarization (LP), respectively. Two Gauss light beams are converted into OAM beams of $l ={+} 3$ and $l ={+} 4$ through two vortex patterns of the SLMs. Then the two OAM beams are combined into one beam by employing a polarization beam splitter (PBS). Considering the transmission characteristics of the RCF, the multiplexed OAM beam is coupled into a 2.3 km RCF for transmission through a quarter-wave plate (QWP).
Insets (i) present the cross section of the RCF. The intensity profiles of two OAM modes (ii) $l ={+} 3$ and (iii) $l ={+} 4$ after 2.3km RCF are also presented. At the receiver side, the multiplexed OAM beam is split into two beams through a beam splitter (BS), and two QWPs are also used to convert the OAM beams into linearly polarized beams. Then the two beams are converted into Gauss beams through the vortex phase plate (VPP). The two Gauss beams are coupled into the SMFs through the collimators and converted into an electronic signal by employing two PDs. The electric waveform is recorded by using a real-time oscilloscope at 100GS/s. The offline DSP is composed of resampling, symbol synchronization, and the PNN equalizer. The Gardner clock recovery algorithm is used for symbol synchronization, and the equalizer operated at one sample per symbol. The bit error ratio (BER) is calculated by a bit-by-bit comparison to verify the performance of the proposed scheme.
3.2 Experimental results and analysis
An experiment was carried out by employing a 32 GBaud PAM-8 OAM-MDM optical fiber communication system with RCF transmission over 2.3 km to verify the effectiveness of the PNN nonlinear equalizer. Two OAM modes with $l ={+} 3$ and $l ={+} 4$ were transmitted by employing PAM-8 modulation. Figure 4 illustrates the process of PNN equalizer for one testing sample of [0.8085331, 6.1117324, 4.3859674]. In this experiment, the length of the training samples and testing samples are 25600 and 105472, respectively. The memory length is set as M = 3, hence in the pattern layer, the training samples are divided into 512 categories. The proposed PNN equalizer calculates the probability density based on the Euclidean distance between the testing sample and the training sample to derive the probability density distribution of the testing sample using Eq. (1). Insets in Fig. 4 show the Euclidean distances between the test sample and four categories of [-1, -7, 7], [1,7,5], [3, -1, 7], and [5, -7, -7]. The Euclidean distance calculated from the testing sample and the category of [1,7,5] is the smallest. Figure 4 shows the likelihood probability of the testing sample calculated by the summation layer of the PNN equalizer. The distribution of the likelihood probability reaches the peak at the category of [1,7,5], hence this testing sample is classified into the category of [1,7,5].
Fig. 4. Likelihood distribution of a test sample with M = 3. Insets show Euclidean distances of [-1, -7, 7], [1,7,5], [3, -1, 7], and [5, -7, -7].
The performance of the PNN nonlinear equalizer has been investigated, as shown in Fig. 5(a) and (b). Due to the stochastic nonlinear impairment of the OAM-MDM system, the BER calculated by minimum Euclidean distance (MED)-based decision cannot be lower than the 20% forward error correction (FEC) limit of 2.4 × 10−2 with low received optical power (ROP). With the processing of the PNN equalizer, the BERs of both OAM modes are lower than the 20% FEC threshold. This result validates the effectiveness of the proposed PNN equalizer for mitigating nonlinear impairment in OAM-MDM transmission systems.
Fig. 5. Measured BER versus ROP for (a) l=+3, (b) l=+4 in the IM-DD OAM-MDM transmission system over a 2.3 km RCF.
Then, the performance of PNN nonlinear equalizer was investigated in comparison with the conventional Volterra nonlinear equalizer and the CNN nonlinear equalizer. Figure 5(a) and (b) also show the BER performance for two OAM modes by employing Volterra equalization, CNN equalization, and PNN equalization, respectively. The memory lengths of the used three equalizers are shown in Table 1.
Table 1. The memory length of the equalizers
Due to the stochastic nonlinearity in the OAM-MDM system, the nonlinear model is difficult to fit with the Volterra series or the CNN algorithm accurately. Hence it can be seen that the proposed PNN nonlinear equalizer outperforms other two equalization methods. Compared with the Volterra nonlinear equalizer, the PNN nonlinear equalizer has about 1dB and 2.5dB benefit with OAM modes of l=+3 and l=+4 at the 20% FEC limit, respectively. Compared with the CNN nonlinear equalizer, the PNN nonlinear equalizer improves the BER by about 0.6dB and 2dB with OAM modes of l=+3 and l=+4 at the 20% FEC limit, respectively. Although the nonlinear impairments in the OAM-MDM system have complex stochastic properties due to mode coupling, the PNN equalizer can accurately calculate the posterior probabilities of the testing samples based on the probability distribution of the training samples and compensate the signals according to Bayesian decision theory because the test samples have the same probability distribution as the training samples. Therefore, compared with other nonlinear equalizer, the PNN equalizer can effectively mitigate the nonlinear impairment in an OAM-MDM system.
The performance of the Volterra, CNN, and PNN equalizers was also investigated in a 5 km standard single-mode fiber system (SSMF). Three equalizers also improve the performance of the SMF transmission system. However, due to the less optoelectronic devices used in the SMF system, the nonlinearity is not as strong as the OAM-MDM system. Therefore, in the short distance SSMF system, compared with the Volterra equalizer and the CNN equalizer, the receiver sensitivity improvement of the PNN equalizer is less than that in the IM-DD OAM-MDM system. As shown in Fig. 6, Compared with the Volterra and CNN equalizer, the PNN equalizer improves the receiver sensitivity by 1 dB and 0.8 dB at the 1E-3 BER limit, respectively.
In the PNN nonlinear equalizer, the smoothing parameter $\sigma$ is a key factor for the mitigation of nonlinear impairment. Figure 7(a) and (b) illustrate the performance on BER of two OAM modes with different $\sigma$ at ROP of 1 dBm. For both OAM modes, the optimal smoothing factor is 0.08: it achieves the best BER performance at different memory lengths. With smoothing parameter lower than 0.08, the probability density distribution for each category is concentrated. Hence when the testing sampling is on the border of one category, the probability density of the testing sampling for this category is very small. However, the probability density of the testing sample for the neighboring category is also very small, resulting in a mistaken classification by the PNN nonlinear equalizer, as shown in Fig. 7(c). In contrast, with smoothing parameter higher than 0.08, the probability density distribution of each category is dispersed. Hence when the testing sampling is on the border of one category, the probability density of the testing sample for this category is very large. However, the probability density of the testing sample for the neighboring category is also very large due to the wide PDF, also resulting in a mistaken classification by the PNN nonlinear equalizer, as shown in Fig. 7(d). When the smoothing parameter is 0.08, the PDF is optimized. Hence when the testing sample is on the border of one category, the probability density of the testing sample for this category is larger than that for the neighboring category, resulting in a low BER for the performance of the PNN nonlinear equalizer, as shown in Fig. 7(e).
Fig. 7. BER contour plots for (a) l=+3, (b) l=+4 in the IM-DD OAM-MDM transmission system over a 2.3 km RCF; The PDF for (c) $\sigma < 0.08$, (d) $\sigma > 0.08$, (e) $\sigma \textrm{ = }0.08$.
The influence of the memory length on the PNN nonlinear equalizer has been investigated. Figure 8(a) and (b) show the BER contour maps of two OAM modes with different memory lengths. With increasing memory length M, the BER decreases due to the accurate fitting of the nonlinear model of the OAM-MDM communication system by using the PNN nonlinear equalizer. Note that an increase of memory length will also increase computation complexity, hence there is a trade-off between BER performance and computing memory. Therefore, under a certain compensation performance requirement, an appropriate memory length should be chosen in exchange for a reduction in system complexity.
Fig. 8. BER contour plots for (a) l=+3, (b) l=+4 in the IM-DD OAM-MDM transmission system over a 2.3 km RCF with $\sigma \textrm{ = }0.08$.
The complexities of Volterra, CNN, and PNN nonlinear equalizers have been also investigated. The complexity of the Volterra equalizer, in terms of the number of real multiplications, is [33]
$S$ denotes the order of the Volterra equalizer. The total number of multiplications to implement the Volterra equalizer increases as the memory length increases.The complexity of the CNN is
The complexity of PNN is [34]
$p$ denotes the number of neurons in the pattern layer, which is equal to ni*8M.The complexities of three equalizers are shown in Fig. 9. The parameters of three equalizers are set corresponding to the BER performance shown in Fig. 5. S, D, ${K_l}$, ${C_1}$, ${C_2}$ were set as 3, 2, $0.8\ast M$, 64, 128, respectively. As shown in Fig. 9, the complexity of the CNN equalizer is always higher than that of the PNN equalizer. When the memory length is less than 87, the complexity of the PNN equalizer is higher than that of the Volterra equalizer. In contrast, when the memory length is greater than 87, the complexity of the PNN equalizer is lower than that of the Volterra equalizer.
Figure 10 illustrates the performance on BER of the Volterra-equalizer and the PNN equalizer with different memory lengths at l = 3 and ROP = -3. With increasing memory length, the BER of the OAM-MDM system with two equalizers gradually decreases. However, the performance of the Volterra equalizer is always lower than that of the PNN equalizer, which even unable to reach the 20% FEC threshold. Hence although the complexity of the Volterra equalizer is lower than that of the PNN equalizer when the memory length is less than 87, the PNN equalizer shows better performance improvement. Therefore, in the OAM-MDM system, the PNN equalizer is adopted to mitigate the nonlinear impairment.
4. Conclusion
A nonlinear equalizer for OAM-MDM optical fiber communication based on PNN machine-learning algorithm has been proposed in this paper. Due to the complexity and stochasticity of an OAM-MDM system, the nonlinear impairment is difficult to compensate with conventional nonlinear equalizers. The PNN equalizer can compensate the nonlinear impairment in the high-speed OAM-MDM system by calculating the posterior probability of the received signal using the probability distribution of the training samples. An experiment was carried out by employing a 32 GBaud PAM-8 OAM-MDM optical fiber communication system with RCF transmission over 2.3 km to verify the effectiveness of the proposed PNN equalizer. The experimental results demonstrate that the PNN nonlinear equalizer can mitigate the nonlinear impairment effectively. Compared with the Volterra equalizer, the PNN equalizer improves the receiver sensitivity by 1dB and 2.5dB with OAM modes l=+3 and l=+4 at the 20% FEC limit, respectively. Compared with the CNN equalizer, the PNN equalizer improves the receiver sensitivity by 0.6dB and 2dB in two OAM modes, respectively. Moreover, the complexity of the PNN equalizer is reduced greatly compared with that of the other two equalizers. The PNN nonlinear equalizer possesses both high performance and low complexity, providing a great potential to mitigate the nonlinear signal distortions in high-speed IM-DD OAM-MDM fiber communication systems.
Funding
National Key Research and Development Program of China from Ministry of Science and Technology (2019YFA0706300); National Natural Science Foundation of China for Excellent Young Scholars (62022016); National Natural Science Foundation of China (61835002, 61727817, 62021005, 62105026); Beijing Municipal Natural Science Foundation (4222075).
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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