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Abstract
Nonlinear impairment in a high-speed orbital angular momentum (OAM) mode-division multiplexing (MDM) optical fiber communication system presents high complexity and strong stochasticity due to the massive optoelectronic devices. In this paper, we propose an Affinity Network (AffinityNet) nonlinear equalizer for an OAM-MDM intensity-modulation direct-detection (IM/DD) transmission with four OAM modes. The labeled training and testing signals from the OAM-MDM system can be regarded as “small sample” and “large target”, respectively. AffinityNet can be used to build an accurate nonlinear model using “small sample” based on few-shot learning and can predict the stochastic characteristic nonlinearity of OAM-MDM with a high level of generalization. As a result, the AffinityNet nonlinear equalizer can effectively compensate the stochastic nonlinearity in the OAM-MDM system, despite the large difference between the training and testing signals due to the stochastic nonlinear impairment. An experiment was conducted on a 400 Gbit/s transmission with four OAM modes using a pulse amplitude modulation-8 (PAM-8) signal over a 2 km ring-core fiber (RCF). Our experimental results show that the proposed nonlinear equalizer outperformed the conventional Volterra equalizer with improvements in receiver sensitivity of 1.7, 1.8, 3, and 3.3 dB for the four OAM modes at the 15% forward error correction (FEC) threshold, respectively. In addition, the proposed equalizer outperformed a convolutional neural network (CNN) equalizer with improvements in receiver sensitivity of 0.8, 0.5, 0.9, and 1.4 dB for the four OAM modes at the 15% FEC threshold. In the experiment, a complexity reduction of 37% and 83% of the AffinityNet equalizer is taken compared to the conventional Volterra equalizer and CNN equalizer, respectively. The proposed equalizer is a promising candidate for a high-speed OAM-MDM optical fiber communication system.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The current network of data centers has become an indispensable form of infrastructure for Internet and cloud computing and carries data traffic communications for service providers such as Google, Microsoft, and Alibaba [1]. Short-reach intensity-modulation direct-detection (IM-DD) optical communication systems are typically used for optical fiber interconnections of data centers [2]. Short-reach optical communication systems require high speed, low power consumption, and high levels of reliability in order to meet the data transmission requirements of data centers. In recent years, with the rapid growth of the Internet and related industries, the demand for transmission capacity of short-reach optical interconnect data centers has increased exponentially. As a result, there is an urgent need to expand the capacity of short-reach optical communication systems. Mode-division multiplexing (MDM) allows for the data transmission via multiple independent communication channels at the same time and frequency resources, which can be spatially separated from each other [3,4]. In conventional MDM optical systems, data can be loaded onto multiple orthogonal polarization modes, as the orthogonality between the modes guarantees independent transmission of each mode without crosstalk. The capacity of the system can be significantly increased with the number of orthogonal modes [5].
Orbital angular momentum (OAM) multiplexing is an emerging technology for MDM. In this technique, data are loaded onto mutually orthogonal vortex beams with different topological charges and transmitted through a weakly coupled RCF [6–9]. Since OAM modes with different orders are orthogonal to each other, the use of OAM-MDM can significantly expand the capability of the optical communication system. Hence, OAM-MDM technology has become an important area of research with the aim of breaking through the capacity limitations of short-reach optical communication systems in data centers [10,11].
Nonlinear impairment is an important factor that affects the transmission performance in OAM-MDM optical communication systems. Compared with conventional optical communication, many more photoelectric devices are used, which introduces severe nonlinear impairment to the OAM-MDM system. The nonlinear impairment generated by different optical devices, such as spatial light modulators (SLMs) [12], Mach-Zehnder modulators (MZMs) and photodetectors (PDs), are coupled with each other, creating an ultra-complicated nonlinear model for OAM-MDM. In addition, the mode coupling in the OAM-MDM system means that the nonlinear impairment has strong stochastic characteristic due to the random mode coupling. In conventional single-mode optical fiber (SMF) communication systems, fixed nonlinear models are usually established to mitigate nonlinear impairment, such as the Volterra series [13], digital predistortion (DPD) [14], and lookup table (LUT) [15]. However, nonlinear models of OAM-MDM systems cannot be accurately fitted using these methods due to its complexity and stochasticity. Therefore, it is important to develop a new nonlinear equalizer to compensate the stochastic nonlinear impairment of OAM-MDM optical fiber communication systems.
In recent years, machine-learning algorithms have been widely used to compensate the nonlinear impairment in optical fiber communication due to their powerful nonlinear fitting capability, such as the convolutional neural network (CNN) [16], transfer learning and waveform regression-based artificial neural network, etc. [17–20]. Machine-learning-based nonlinear equalizers also have the potential to compensate the severe nonlinear impairment in OAM-MDM optical fiber communication. However, these nonlinear equalizers also have a major drawback in this context: although they yield improved performance compared with traditional equalizers, they cannot overcome the strong stochastic nonlinearity of OAM-MDM system, meaning that they are unsuitable for high-speed OAM-MDM optical fiber communication systems.
In this paper, a nonlinear equalizer based on the Affinity Network Model (AffinityNet) is proposed and experimentally demonstrated in terms of the nonlinearity compensation of a high-speed OAM-MDM optical fiber communication system. AffinityNet is a data-efficient few-shot learning model that can learn features from a limited number of examples with supervised information, which improves the generalization capability [21]. The labeled training signal from the OAM-MDM system can therefore be regarded as “small sample”, and the testing signal can be regarded as “large target”. AffinityNet can build an accurate nonlinear model using “small sample” and can predict the stochastic characteristic nonlinearity of OAM-MDM with a high level of generalization. As a result, although there is a large difference between the training and testing signals due to the stochastic nonlinear impairment, the AffinityNet nonlinear equalizer can effectively compensate the stochastic nonlinearity in an OAM-MDM system as it is a nonlinear model that uses few-shot learning. Thus, the AffinityNet equalizer proposed in this manuscript is well-suited to high-speed OAM-MDM optical fiber communication systems.
2. Principle
Considering the original transmitted signal in the OAM-MDM system, the received signal can be expressed as
where the vector $y(n) = [{y_1},{y_2}, \ldots ,{y_n}]$ denotes the received PAM-8 symbol sequence; H denotes the channel response, consisting of mode coupling and stochastic nonlinear impairment; $x(n) = [{x_1},{x_2}, \ldots ,{x_n}],\textrm{ }{x_n} \in \{ - 7, - 5, - 3, - 1, + 1, + 3, + 5, + 7\}$ denotes the original PAM-8 symbol sequence; and $noise(n)$ denotes the additive noise generated in the high-speed OAM-MDM system. Each received symbol ${y_n}$ is input to the nonlinear equalizer to calculate the corresponding estimated symbol ${\bar{x}_n}$ for compensation of the distorted PAM-8 signal.Before the nonlinear compensation with the few-shot learning based AffinityNet equalizer, the received PAM-8 symbols are preprocessed, as shown in Fig. 1. The current symbol ${y_n}$ is wrapped with its L preceding and L succeeding symbols to form a feature vector $[{y_{n - L}}, \ldots ,{y_{n - 1}},{y_n},{y_{n + 1}}, \ldots ,{y_{n + L}}]$. The distorted time-series signal $y(n) = [{y_1},{y_2}, \ldots ,{y_n}]$ is transformed into a matrix composed of $(n - 2L)$ feature vectors with a length of $M\textrm{ }(M = 2L + 1)$. This feature matrix can be expressed as
The dataset $Y$ is then divided into two parts to form the training dataset ${Y_P} = [{Y_1},{Y_2}, \ldots ,{Y_p}]$ and the testing dataset ${Y_Q} = [{Y_1},{Y_2}, \ldots ,{Y_q}]$.
The schematic diagram of the AffinityNet equalizer is shown in Fig. 2. An AffinityNet equalizer consists of an input layer, a feature attention layer, a k-Nearest-Neighbor (kNN) attention pooling layer, two fully connected layers, and an output layer.
The training dataset ${Y_P}$ is applied to the AffinityNet equalizer to determine the parameters. Therefore, at first the training data is passed to the input layer in the form of a feature vector ${Y_i}\textrm{ }(i = 1,2, \ldots ,p)$ of the training data set ${Y_P}$.
Second, the input sample ${Y_i}$ undergoes element-wise multiplication with a feature attention vector w in the feature attention layer. This is expressed as
where ${\textrm{h}_i}$ is the output by the feature attention layer, and ${\odot}$ is the element-wise multiplication operator. In Eq. (3), the feature attention vector w is defined as $w = ({w_1},{w_2}, \cdots ,{w_M}),\sum\limits_{j = 1}^M {{w_j} = 1,{w_j} \ge 0}$. The feature vector is extracted through element-wise multiplication in the feature attention layer.Third, the kNN attention pooling layer consists of $k\textrm{ }(k < p)$ feature vectors with length M. ${\textrm{h}_j}\textrm{ }(j = 1,2, \ldots ,p,j \ne i)$ is defined as the k nearest neighbor feature vectors near the feature vector ${\textrm{h}_i}$. In this layer, the spatial distance $a({\cdot} , \cdot )$ between the feature vector ${\textrm{h}_i}$ and the neighboring feature vector ${\textrm{h}_j}$ can be calculated using the Gaussian attention kernel, which can be expressed as
The parameter $\sigma$ represents the smoothing factor, and $N(i)$ denotes the set of $k$ nearest neighbors of the sample ${\textrm{h}_i}$. In Eq.4, ${\left\|{{\textrm{h}_i} - {\textrm{h}_j}} \right\|^2}$ denotes the Euclidean distance between ${\textrm{h}_i}$ and ${\textrm{h}_j}$. However, the calculation of the Euclidean distance is very complex, involving a power operation, and the Manhattan operator is therefore used in place of the Euclidean distance to reduce the complexity of the AffinityNet equalizer [22]. Equation (4) can be rewritten as
The output of the kNN attention pooling layer is a new feature vector ${\textrm{h}_i}^{\prime}$ for the input sample ${Y_i}$, which is expressed as
After the calculation of the kNN attention pooling layer, the spatial distance between the feature vectors is shortened, meaning that all feature vectors are brought close to each other through the kNN cluster method.
Fourth, the two fully connected layers are then used to perform two regression calculations on ${\textrm{h}_i}^{\prime}$ with the weight vectors (${W_1},{W_2}$) and the bias vectors (${B_1}$, ${B_2}$), followed by a ReLU() activation function for each fully connected layer. The whole process can be expressed as
The dimensions of ${W_1}$, ${B_1}$, ${W_2}$ and ${B_2}$ are the same as the numbers of neurons in the fully connected layers, and $\textrm{h}_i^{^{\prime\prime}}$ is the new feature vector for the input sample ${Y_i}$.
Finally, the single neuron of the output layer is used to perform a linear calculation with the weight vector (${W_3}$) and the bias vector (${B_3}$), without an activation function. The estimated value ${\bar{x}_i}$ of the sample ${Y_i}$ output of the output layer can be expressed as
After obtaining the estimated value ${\bar{x}_i}$ for sample ${Y_i}$, the AffinityNet equalizer uses the Mean Squared Error (MSE) function to calculate the loss of the training process, and applies the Adam gradient descent function to update the model parameters ($w$, ${W_1}$, ${B_1}$, ${W_2}$, ${B_2}$, ${W_3}$, ${B_3}$) using backpropagation. The distortion symbols corresponding to the testing set ${Y_Q} = [{Y_1},{Y_2}, \ldots ,{Y_q}]$ can be compensated by using the forward propagation process of the AffinityNet equalizer after the training process.
Conventional equalizers fit the nonlinear model in a sample-by-sample manner, and only learn the features of the nonlinearity for the “average point” of one signal level. However, in a high-speed OAM-MDM system, the strongly stochastic nature of the nonlinearity results in a large gap in terms of the nonlinear impairment between the training and testing samples of the nonlinear equalizer. Hence, conventional equalizers find it hard to compensate the nonlinear impairment of the OAM-MDM due to their weak generalization capability, as shown in Fig. 3(a). In contrast, the AffinityNet equalizer, based on few-shot learning, use the kNN attention pooling layer to cluster the feature vectors of the training samples with the same label in the feature space, and can learn the features of the nonlinearity for “a class” of one signal level, rather than an “average point”. This improves the generalization capability significantly, as shown in Fig. 3(b). When the AffinityNet equalizer is applied to the testing samples, the kNN attention pooling layer clusters the testing samples and effectively compensates the nonlinearity of each class. The training sample for the OAM-MDM can be regarded as the “small samples”, and the nonlinearity feature of the “small samples” is extended to give strong generalization through clustering by the kNN attention pooling layer. As a result, the testing sample for the OAM-MDM system can be regarded as the “large target”, and the nonlinearity of the “large target” is compensated through the generalized feature of “one class” learned using the AffinityNet equalizer. In this way, the few-shot learning-based AffinityNet equalizer can effectively compensate the stochastic nonlinearity of the high-speed OAM MDM system.
Fig. 3. The Generalization ability of (a) a conventional equalizer; (b) the few-shot based AffinityNet equalizer.
3. Experimental
3.1. Experimental setup
To verify the effectiveness of the proposed AffinityNet nonlinear equalizer, an experiment was conducted on a 400 Gbit/s OAM-MDM IM/DD transmission with four OAM modes over a 2 km RCF, as shown in Fig. 4. To ensure that the equalizer learns the nonlinear characteristics of the channel, rather than the generated pseudo-random sequence, the random number seeds corresponding to the training data and the test data are different. This difference generates data sequences with different random characteristic, which guarantees that the proposed equalizer learns the nonlinear characteristics of the channel, rather than the random sequence.
Fig. 4. Experimental setup of the four modes OAM-MDM IM/DD optical fiber communication system (DSP: digital signal process; AWG: arbitrary waveform generator; EA: electric amplifier; MZM: Mach-Zehnder modulator; PC: polarization controller; EC: external cavity; EDFA: erbium-doped fiber amplifiers; OC: optical couplers; SMF: single-mode fiber; Col.: collimator; LP: linear polarizer; SLM: spatial light modulator; HWP: half-wave plate; QWP: quarter-wave plate; PBS/PBC: polarization beam combiner; BS/BC: beam splitter/combiner; RCF: ring-core fiber; VPP: vortex phase plate; BPF: band-pass filter; PD: photodetector; OSC: oscilloscope; BER: bit error rate).
At the transmitter, a pseudo-random sequence of length 132, with elements ranging from 1 to 20, is generated. The 132 numbers were utilized as random number seeds to generate random binary sequences with a length of 1000. These sequences were subsequently combined to form random sequences with a length of 217. The newly generated random sequence is mapped into a PAM-8 symbol sequence within the digital signal processing (DSP) block. The electrical signal was generated with a 35 GHz arbitrary waveform generator (AWG) at a sampling rate of 65 GSa/s. After passing the electrical signal through a super high frequency (SHF) electronic amplifier (EA), it was used to modulate an optical carrier with a wavelength of 1550.12 nm through a MZM to generate a double-side band optical signal. The optical signal generated in this way is then split into four branches using three optical couplers (OCs) and amplified using four erbium-doped fiber amplifiers (EDFAs). The four branches of the signal were delayed separately for decorrelation of the data pattern. There are two polarization control devices in the experiment. One is before the collimator. To meet the incident requirement of the spatial light modulator, four linear polarizers (LPs) were employed to ensure that the polarization directions of the signal light are aligned to that of the SLM. Furthermore, to maximize the power of the signal light after it passes through the linear polarizer, four polarization controllers (PCs) are utilized to adjust the polarization state of the light within the fiber. The other polarization devices are placed in front of the polarization beam combiners (PBCs). The linearly polarized signal light is divided into two orthogonal linear polarization beams using the PBC. The polarization direction is optimized by adjusting the half-wave plate (HWP) to ensure maximum intensity of the combined polarization beams. All signals were reflected by a 1920 × 1080 phase-only SLM for the conversion of OAM modes (l=<2, 3, 4, 5>). In this way, an aggregate capacity of 400 Gbit/s could be realized with a 33 GBaud PAM-8 signal per mode in the OAM-MDM IM/DD transmission. Two OAM modes were passed to a HWP for a 90° rotation. The polarization states are controlled to guarantee the maximum power of the light. The four branches were then combined using PBCs and a beam combiner (BC). Finally, the combined OAM modes were converted to circular polarization by a quarter-wave plate (QWP) and coupled to a 2 km RCF.
In Fig. 4, inset (i) shows the cross section of the RCF, and the intensity profiles of the four OAM modes (l=<2, 3, 4, 5>) are presented in insets (ii)–(v), respectively. At the receiver side, the multiplexed OAM modes is split into four beams through the BS, and these are converted into Gauss beams through the vortex phase plates (VPPs) with opposite topological charges, respectively. The four Gauss beams are coupled to the SMFs through the collimator (Col.), and then are converted into an electronic signal by employing four 50 GHz PDs. The electric waveform is recorded using a real-time oscilloscope at 100GSa/s, and is processed by the offline DSP block, which applies resampling, a lowpass filter, symbol synchronization, the AffinityNet equalizer and the BER calculation. The BER is calculated based on a bit-by-bit comparison to verify the performance of the proposed scheme. 300,000 bits are involved in the BER calculation.
3.2. Experimental results and analysis
Figure 5 illustrates the process of applying the AffinityNet equalizer to one testing sample, consisting of [1.4, 3.1, -6.5]. This sample is transformed to ${\textrm{h}_i}$ ([0.266, 1.798, -1.495]) by the feature attention layer with the feature attention vector $w = [0.19,\textrm{ }0.58,\textrm{ }0.23]$. The nearest neighbor feature vector ${\textrm{h}_j}$ ([0.152, 1.682, -1.564], [0.266, 1.856, -1.633] and [0.209, 1.566, -1.564]) are calculated based on the nearest neighbor testing samples ([0.8, 2.9, -6.8], [1.4, 3.2, -7.1] and [1.1, 2.7, -6.8]). In the kNN attention pooling layer, the feature vector ${\textrm{h}_i}$ is transformed to ${\textrm{h}_i}^{\prime}$ ([0.2407, 1.7156, -1.5269]). The spatial distance between the testing samples is reduced, and clustering is achieved. Finally, the estimated symbol is calculated by the fully connected layer using ${\textrm{h}_i}^{\prime}$. In this way, the large number of distorted symbols, which are regarded as the “large target”, can be compensated. The AffinityNet equalizer can overcome the strongly stochastic nonlinearity of the high-speed OAM-MDM IM/DD system by few-shot learning.
The performance of the AffinityNet nonlinear equalizer was also investigated, as shown in Figs. 6(a)–(d). The parameters used by the Volterra, CNN and AffinityNet equalizers are shown in Table 1. All three equalizers operated at one sample per symbol.
Fig. 6. Measured BER versus received optical power (ROP) for (a) l = 2, (b) l = 3, (c) l = 4, (d) l = 5 in the OAM MDM IM/DD transmission system over a 2 km RCF.
Table 1. Parameters used by the three equalizers
In Table 1, S denotes the order of the Volterra equalizer; D is denotes the number of convolution layers of the CNN; K is the length of the convolution kernel; ${N_{W1}}$ is the number of neurons in the first fully connected layer; and ${N_{W2}}$ is the number of neurons in the second fully connected layer. The values of the parameters were chosen to give the best equalization performance. Compared with a decision based on the minimum Euclidean distance (MED), the AffinityNet equalizer improved the receiver sensitivity by 2.3, 1.9, 4, and 3.5 dB for the four OAM modes (l=<2, 3, 4, 5>), respectively, at the 15% forward error correction (FEC) threshold. Compared with the Volterra equalizer, the AffinityNet equalizer improved the receiver sensitivity by 1.7, 1.8, 3, and 3.3 dB, respectively, for the four OAM modes at the 15% FEC threshold. Compared with the CNN equalizer, the AffinityNet equalizer improved the receiver sensitivity by 0.8, 0.5, 0.9, and 1.4 dB for the four OAM modes at the 15% FEC threshold. These results validate the effectiveness of the proposed AffinityNet equalizer in terms of compensating the stochastic nonlinear impairment in the OAM-MDM system.
Figure 7 presents eye diagrams for the received signal for the OAM mode l = 2 at ROP = 8 dBm. Figure 7(a) shows the eye diagram without a nonlinear equalizer, and Figs. 7(b)–(d) show eye diagrams for the received signal after processing with the Volterra, CNN and AffinityNet equalizers, respectively. The eye diagrams are improved slightly when the Volterra and CNN equalizers are used; however, the eye diagram for the AffinityNet equalizer is significantly clearer. These results demonstrate the good performance of AffinityNet when applied to a high-speed OAM-MDM IM/DD system, which is due to the use of few-shot learning.
Fig. 7. Measured result for 99Gbit/s PAM-8 eye diagram for OAM mode l = 2 with (a) no equalizer; (b) Volterra equalizer; (c) CNN equalizer; (d) AffinityNet equalizer.
The effects of the number of nearest neighbors k and the memory length M on the AffinityNet equalizer were also investigated. Figures 8 (a)-(d) shows the BER performance of the AffinityNet equalizer for the four OAM modes with different values of k when the ROP is set to 8 dBm. In general, the BER performance of AffinityNet equalizer improves as the number of nearest neighbors increases from 1 to 3 but deteriorates as this number increases further from 3 to 6, indicating that the best BER performance can be achieved when k is 3. Figures 9 (a) –(c) illustrate the effect of varying k on the performance of the AffinityNet equalizer. As shown in Fig. 9(a), the AffinityNet equalizer cannot cluster all samples belonging to the same class when $k < 3$, resulting in weakened generalization of the equalizer and meaning that it cannot overcome the stochastic nonlinear impairment in the OAM-MDM system when $k < 3$. Conversely, AffinityNet clusters samples from different class into one class when $k > 3$, meaning that its compensation ability is weakened. It can be seen from Fig. 9(b) that the proposed AffinityNet equalizer clusters samples belonging to the same class when $k = 3$, resulting in the best equalization performance with strong generalization ability.
Fig. 8. BER contour plots for (a) l = 2, (b) l = 3, (c)l = 4, (d)l = 5 versus the number of nearest neighbors k for various values of memory length M.
It can also be seen that there is no significant difference in the performance of the AffinityNet equalizer when M is set to different values. However, the complexity of AffinityNet equalizer is increases with the memory length. We therefore set the memory length M to 3 for the sample cost of the AffinityNet equalizer.
The complexity of the proposed equalizer was compared with that of the Volterra and CNN equalizers. The number of real multiplications for the Volterra equalizer can be expressed as [23]
The number of real multiplications for the CNN equalizer can be expressed as
The complexity of the AffinityNet equalizer can be expressed as
The parameters of the three equalizers were the same for all four OAM modes. The complexity of the three equalizers was calculated based on the parameters given in the Table 1. As shown in Table 2, compared with the Volterra and CNN equalizers, the AffinityNet equalizer has a complexity that is lower by 37% and 83%, respectively. Hence, the proposed equalizer has both good performance and low complexity.
Table 2. Complexity of three alternative equalizers considered in the experiment
4. Conclusion
A nonlinear equalizer based on few-shot learning-based AffinityNet has been proposed for OAM-MDM IM/DD optical fiber communication in this paper. Conventional nonlinear equalizers are difficult to compensate the nonlinear impairment in OAM-MDM transmission due to the large differences between the training and testing symbols induced by the stochastic nonlinearity. In the AffinityNet nonlinear equalizer, the training and the testing symbols can be considered as “small sample” and “large target”, respectively. The proposed AffinityNet equalizer can learn an accurate model of the OAM-MDM using few-shot learning, and can effectively compensate stochastic nonlinear impairment due to its high generalization ability. Our experimental results show that the proposed nonlinear equalizer outperforms the conventional Volterra equalizer with improvements in receiver sensitivity of 1.7, 1.8, 3, and 3.3 dB for the four OAM modes at the 15% FEC threshold, respectively. It also outperforms a CNN equalizer with improvements in receiver sensitivity of 0.8, 0.5, 0.9, and 1.4 dB for the four OAM modes at the 15% FEC threshold, respectively. The complexity of the AffinityNet equalizer is also greatly reduced compared with the other two equalizers, and it shows great potential for the compensation of stochastic nonlinearity impairment in OAM-MDM IM/DD transmission.
Funding
National Key Research and Development Program of China from Ministry of Science and Technology (2019YFA0706300); National Natural Science Foundation of China under Grants (62105026, 62205023); Fundamental Research Funds for the Central Universities; 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.
References
1. F. Han, M. Wang, Y. Cui, Q. Li, R. Liang, Y. Liu, and Y. Jiang, “Future Data Center Networking: From Low Latency to Deterministic Latency,” IEEE Network 36(1), 52–58 (2022). [CrossRef]
2. K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. Lau, “Digital Signal Processing for Short-Reach Optical Communications: A Review of Current Technologies and Future Trends,” J. Lightwave Technol. 36(2), 377–400 (2018). [CrossRef]
3. S. Randel, R. Ryf, A. Sierra, P. Winzer, A. Gnauck, C. Bolle, R. Essiambre, D. Peckham, A. McCurdy, and R. Lingle, “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011). [CrossRef]
4. R. Ryf, S. Randel, A. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. Burrows, R. Essiambre, P. Winzer, D. Peckham, A. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30(4), 521–531 (2012). [CrossRef]
5. J. Zhang, Y. Wen, H. Tan, J. Liu, L. Shen, M. Wang, J. Zhu, C. Guo, Y. Chen, Z. Li, and S. Yu, “80-Channel WDM-MDM Transmission over 50-km Ring-Core Fiber Using a Compact OAM DEMUX and Modular 4×4 MIMO Equalization,” Optical Fiber Communication Conference 2019 (OFC, 2019), W3F. 3.
6. L. Zhu, H. Yao, H. Chang, Q. Tian, Q. Zhang, and X. Xin, “Adaptive Optics for Orbital Angular Momentum-Based Internet of Underwater Things Applications,” IEEE Internet Things J. 9(23), 24281–24299 (2022). [CrossRef]
7. H. Chang, X. Yin, H. Yao, J. Wang, R. Gao, J. An, and L. Hanzo, “Low-Complexity Adaptive Optics Aided Orbital Angular Momentum Based Wireless Communications,” IEEE Trans. Veh. Technol. 70(8), 7812–7824 (2021). [CrossRef]
8. L. Zhu, X. Xin, H. Chang, X. Wang, Q. Tian, Q. Zhang, R. Gao, and B. Liu, “Security enhancement for adaptive optics aided longitudinal orbital angular momentum multiplexed underwater wireless communications,” Opt. Express 30(6), 9745–9772 (2022). [CrossRef]
9. H. Chang, X. Yin, H. Yao, J. Wang, R. Gao, X. Xin, and M. Guizani, “Adaptive Optics Compensation for Orbital Angular Momentum Optical Wireless Communications,” IEEE Trans. Wireless Commun. 21(12), 11151–11163 (2022). [CrossRef]
10. S. Zhou, X. Liu, R. Gao, Z. Jiang, H. Zhang, and X. Xin, “Adaptive Bayesian neural networks nonlinear equalizer in a 300-Gbit/s PAM8 transmission for IM/DD OAM mode division multiplexing,” Opt. Lett. 48(2), 464–467 (2023). [CrossRef]
11. F. Wang, R. Gao, S. Zhou, Z. Li, Y. Cui, H. Chang, F. Wang, D. Guo, C. Yu, X. Liu, Z. Dong, Q. Zhang, Q. Tian, F. Tian, Y. Wang, X. Huang, J. Yan, L. Jiang, and X. Xin, “Probabilistic neural network equalizer for nonlinear mitigation in OAM mode division multiplexed optical fiber communication,” Opt. Express 30(26), 47957–47969 (2022). [CrossRef]
12. A. Miniewicz, J. Mysliwiec, P. Pawlaczyk, and M. Zielinski, “Photorefractive-like all-optical switching in nematic-photoconducting polymer liquid crystal cell,” Mol. Cryst. Liq. Cryst. 489(1), 119–134 (2008). [CrossRef]
13. C. Chuang, W. Chang, C. Wei, C. Ho, and J. Chen, “Sparse Volterra Nonlinear Equalizer by Employing Pruning Algorithm for High-Speed PAM-4 850-nm VCSEL Optical Interconnect,” Optical Fiber ommunication Conference 2019 (OFC, 2019), M1F. 2.
14. Y. Bao, Z. Li, J. Li, X. Feng, B. Guan, and G. Li, “Nonlinearity mitigation for high-speed optical OFDM transmitters using digital pre-distortion,” Opt. Express 21(6), 7354–7361 (2013). [CrossRef]
15. J. Ke, Y. Gao, and J. Cartledge, “400 Gbit/s single-carrier and 1 Tbit/s three-carrier superchannel signals using dual polarization 16-QAM with look-up table correction and optical pulse shaping,” Opt. Express 22(1), 71–83 (2014). [CrossRef]
16. C. Chuang, L. Liu, C. Wei, J. Liu, and J. Chen, “Convolutional Neural Network based Nonlinear Classifier for 112-Gbps High Speed Optical Link,” Optical Fiber Communication Conference 2018 (OFC, 2018), pp. 1–3.
17. E. Liu, Z. Yu, Y. Song, K. Sun, H. Huang, F. Yin, Y. Zhou, and K. Xu, “TL-ANN Based Nonlinear Equalization for Multi-Users Radio Over Fiber System,” J. Lightwave Technol. 41(5), 1399–1405 (2023). [CrossRef]
18. F. Wang, Y. Wang, W. Li, B. Zhu, J. Ding, K. Wang, C. Liu, C. Wang, M. Kong, L. Zhao, W. Zhou, F. Zhao, and J. Yu, “Echo State Network Based Nonlinear Equalization for 4.6 km 135 GHz D-Band Wireless Transmission,” J. Lightwave Technol. 41(5), 1278–1285 (2023). [CrossRef]
19. X. Huang, D. Zhang, X. Hu, C. Ye, and K. Zhang, “Low-Complexity Recurrent Neural Network Based Equalizer With Embedded Parallelization for 100-Gbit/s/λ PON,” J. Lightwave Technol. 40(5), 1353–1359 (2022). [CrossRef]
20. P. Freire, D. Abode, J. Prilepsky, N. Costa, B. Spinnler, A. Napoli, and S. Turitsyn, “Transfer Learning for Neural Networks-Based Equalizers in Coherent Optical Systems,” J. Lightwave Technol. 39(21), 6733–6745 (2021). [CrossRef]
21. T. Ma and A. Zhang, “AffinityNet: Semi-Supervised Few-Shot Learning for Disease Type Prediction,” Proceedings of the AAAI Conference on Artificial Intelligence 33(01), 1069–1076 (2019). [CrossRef]
22. Y. Huang, W. Jin, B. Li, P. Ge, and Y. Wu, “Automatic Modulation Recognition of Radar Signals Based on Manhattan Distance-Based Features,” IEEE Access 7, 41193–41204 (2019). [CrossRef]
23. G. Yadav, C. Chuang, K. Feng, J. Yan, J. Chen, and Y. Chen, “Reducing computation complexity by using elastic net regularization based pruned Volterra equalization in a 80 Gbps PAM-4 signal for inter-data center interconnects,” Opt. Express 28(26), 38539–38552 (2020). [CrossRef]
