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Abstract
As a key technique for achieving ultra-high capacity optical fiber communications, orbital angular momentum (OAM) mode-division multiplexing (MDM) is affected by severe nonlinear impairments, including modulation related nonlinearities, square-law nonlinearity and mode-coupling-induced nonlinearity. In this paper, an equalizer based on a hidden conditional random field (HCRF) is proposed for the nonlinear mitigation of OAM-MDM optical fiber communication systems with 20 GBaud three-dimensional carrierless amplitude and phase modulation-64 (3D-CAP-64) signals. The HCRF equalizer extracts the stochastic nonlinear feature of the OAM-MDM 3D-CAP-64 signals by estimating the conditional probabilities of the hidden variables, thereby enabling the signals to be classified into subclasses of constellation points. The nonlinear impairment can then be mitigated based on the statistical probability distribution of the hidden variables of the OAM-MDM transmission channel in the HCRF equalizer. Our experimental results show that compared with a convolutional neural network (CNN)-based equalizer, the proposed HCRF equalizer improves the receiver sensitivity by 2 dB and 1 dB for the two OAM modes used here, with l = + 2 and l = + 3, respectively, at the 7% forward error correction (FEC) threshold. When compared with a Volterra nonlinear equalizer (VNE) and CNN-based equalizer, the computational complexity of the proposed HCRF equalizer was found to be reduced by 30% and 41%, respectively. The bit error ratio (BER) performance and reduction in computational complexity indicate that the proposed HCRF equalizer has great potential to mitigate nonlinear distortions in high-speed OAM-MDM fiber communication systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In view of the growth in data traffic over recent years in diverse fields such as data centers, the Internet of Things, 4 K video, and cloud computing, there is a need for current short-reach infrastructure to expand rapidly to keep up with this trend. Intensity modulation and direct detection (IM/DD)-based optical fiber communication systems have been widely used due to their low cost, low power consumption, and small footprint size. In IM/DD-based systems, the modulation formats mainly consist of pulse amplitude modulation (PAM), discrete multi-tone (DMT), and carrierless amplitude and phase (CAP) modulation [1]. Of these, CAP stands out as it utilizes a digital filtering method to create multichannel signals without requiring additional expensive radio frequency sources or mixers for down conversion, making it a viable modulation format for IM/DD-based short reach scenarios [2–6]. In particular, multi-dimensional CAP modulation has been proposed based on the use of orthogonal shaping filters with additional dimensions to further expand transmission capacity [7–10].
Mode division multiplexing (MDM) technologies with few-mode fiber/multi-mode fiber (FMF/MMF) have been widely investigated in recent years. FMF/MMF-based MDM technology provides a high capacity by using multiple linearly polarization (LP) modes or orbital angular momentum (OAM) modes as independent data channels in a single optical fiber. OAM-MDM technology exploits the orthogonality of OAM modes to modulate signals in different OAM modes with different mode channels [11–16]. Theoretically, the transmission capacity of optical communication systems can be significantly boosted through the use of OAM-MDM [17,18]; however, the main restriction on OAM transmission in FMF or MMF is the crosstalk with mode coupling [19] and a special ring-core fiber (RCF) has been designed to suppress the mode group crosstalk with large effective refractive index differences between the OAM modes [18–21]. In general, an OAM-MDM IM/DD system with 3D-CAP modulation based on RCF has the potential to significantly enhance the transmission capacity of optical communication systems significantly.
In an OAM-MDM IM/DD system with 3D-CAP modulation, the transmission performance of the system is limited by the inter-mode/mode group (MG) crosstalk, which is unavoidable due to the imperfections in the fabrication of the fibers and multiplexers/demultiplexers. Modulation- and detection-induced nonlinearities also act as limitations that impact the performance of the OAM-MDM system [22,23]. Modulation-induced nonlinearities are arise from electro/optical devices such as electrical amplifiers (EAs) and electro/optical modulators [14], whereas detection-induced nonlinearities are caused by the square-law nonlinearity of photodiodes (PDs). Moreover, the nonlinear distortions of one OAM mode can also affect other modes, due to the mode coupling. In addition, the modal dispersion, including chromatic dispersion (CD) and intra-MG differential mode delay (DMD) combined with square-law detection, causes power fading and aggravates the memory characteristic of the nonlinear distortions [14]. The transmission performance of the OAM-MDM 3D-CAP signal will be degraded due to the complex nonlinearities mentioned above.
In general, Volterra nonlinear equalizers (VNEs) are always used to equalize the intrinsic nonlinear impairments in single mode fiber (SMF) systems [24,25]. However, in OAM-MDM systems, it is difficult for a VNE to compensate for the nonlinearity due to its stochasticity. Numerous equalization methods based on neural networks (NNs) and natural language processing (NLP) models have been reported, such as convolutional neural networks (CNNs) [26], bidirectional long short-term memory networks (Bi-LSTM) [27], bidirectional recurrent neural networks (Bi-RNNs) [28], and gated recurrent units (GRUs) [29], and their capability to mitigate the nonlinear impairment in various application scenarios has been demonstrated. However, it is difficult to handle the stochastic and complex nonlinearity in OAM-MDM systems with these algorithms due to their low equalization performance and excessive computational complexity.
In this paper, a nonlinear equalizer based on the hidden conditional random field (HCRF) approach is presented for a 3D-CAP-64 modulated OAM-MDM IM/DD system with RCF transmission. The proposed HCRF equalizer extracts the features of the signal sequence through the use of additional hidden variables, and mitigates the nonlinear impairment of the OAM-MDM based on the conditional probabilities of hidden variables. An OAM-MDM 3D-CAP-64 IM/DD optical fiber communication system is implemented and shown to achieve 240 Gb/s transmission via a 2.3 km RCF transmission, thus demonstrating that the proposed HCRF equalizer can reliably mitigate nonlinear impairments with low computational complexity.
2. Principle
2.1 Principle of 3D-CAP and filter design
The principle of operation of the 3D-CAP-64 modulation scheme in an OAM-MDM optical communication with RCF transmission is shown in Fig. 1. The bit data are initially converted to 3D-CAP-64 symbols following 3D constellation mapping. The 3D-CAP-64 symbols are then upsampled, filtered using three shaping filters $\{ {f_1},{f_2},{f_3}\}$, and summed up to generate 3D-CAP-64 signals. Multiple streams of 3D-CAP-64 signals are modulated on a light carrier and converted from Gaussian beams into OAM beams with different modes, and are then multiplexed for transmission through the RCF. At the receiver, the light is demultiplexed and converted into Gaussian beams, followed by the conversion of electrical signals through the PDs. These electrical signals from the PDs are then fed into matched filters $\{ {g_1},{g_2},{g_3}\}$ and separated into three streams corresponding to three dimensions. These streams are separately downsampled, equalized, and finally combined to obtain the bit data.
The Minimax optimization algorithm is applied to design the shaping filters ${\{{{f_1},{f_2},{f_3}} \}}$ [9,10], which can be expressed as
The taps of the shaping filters can be obtained with the Minimax algorithm. In this experiment, the tap number for each of the three filters was set to six and the corresponding taps values of the shaping filters $\{ {f_1},{f_2},{f_3}\}$ are shown in Fig. 2.
2.2 Principle of HCRF equalization for OAM-MDM 3D-CAP-64 signals
The principle of opration of the proposed HCRF equalizer for the OAM-MDM 3D-CAP-64 signal is illustrated below. The three dimensional signals after the matched filtering and downsampling can be expressed as
Fig. 3. (a) Preprocessing of 3D-CAP-64 signals; (b) subclass assignment for each dimension of 3D-CAP-64 signals.
Next, since each dimension of the 3D-CAP-64 signal can be considered as a PAM-4 signal, one dimension of the 3D-CAP-64 signal can be classified into four classes. In the HCRF equalizer, each class is therefore divided into ${n_h}$ subclasses, as shown in Fig. 3(b). The subclass set is $Q = \{ {q_1},{q_2},\ldots ,{q_N}\}$, where $N = 4{n_h}$.To achieve equalization of one dimension, a hidden variable sequence ${h^\ast } = [{h_1},{h_2},\ldots ,{h_T}]$ is defined that corresponds to the input sequence $x = [{x_1},{x_2},\ldots ,{x_T}]$ and ${h_i} = q \in Q$. The hidden variable ${h^\ast }$ can be obtained with the maximum conditional probability of the input sequence $x$, following [30], as
Fig. 4. The HCRF equalization for OAM-MDM 3D-CAP-64 signals. (a) definition of feature functions; (b) initialization; (c) forward recursion; (d) termination; (e) backward track.
The computational complexity of $P(h|x)$ in Eq. (3) is very high, since the normalization constant involves a sum of all of the possible hidden varibale sequences. Hence, a vector ${\boldsymbol F}(x,h,t)$ containing all the feature functions and a vector ${\boldsymbol \theta }$ containing all the weights are defined to simplify Eq. (3):
Equation (3) can then be rewritten as:
In Eq. (10), the unnormalized probability ${\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t)$ corresponding to the first input ${x_1}$ needs to be calculated for the first hidden variable ${h_1}$ when $t = 1$. When $t = 2,3,\ldots T$, for each possible transition from ${h_{t - 1}}$ to ${h_t}$, the unnormalized probability ${\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t)$ corresponding to the input ${x_t}$ is calculated and summed as $\sum {_1^{t - 1}} {\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t)$. The maximum value of ${\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t) + \sum {_1^{t - 1}} {\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t)$ for all possible transitions from ${h_{t - 1}}$ to ${h_t}$ is then calculated, sample by sample, to form a dynamic programming table. For the last hidden variable of ${h_T}$ where $t = T$, the maximum unnormalized probability $\max [\sum {_{t = 1}^T} {\boldsymbol \theta } \cdot {\boldsymbol F}(x,h,t)]$ is used to calculate ${h_T}$. Finally, the entire optimized hidden variable ${h^\ast } = [{h_1},{h_2},\ldots ,{h_T}]$ can be assigned through the Viterbi algorithm from the backtracked path of the dynamic programming table [31]. The steps of the process are described in detail below:
- (ii) Forward recursion: As shown in Fig. 4(c), when $t = 2,3,\ldots ,T$, for each $q \in Q$, $\pi [t,q]$ is recursively calculated with $\pi [t - 1,q^{\prime}]$ for all $q^{\prime} \in Q$ using Eq. (10).$$\pi [t,q] = {\max _{q^{\prime}}}[\pi [t - 1,q^{\prime}] + {\boldsymbol \theta } \cdot {\boldsymbol F}(x,{h_{t - 1}} = q^{\prime},{h_t} = q,t)]$$${\lambda [t,q]}$ is defined as the subclass ${q^{\prime}}$ corresponding to the $\pi [t,q]$.
- (iii) Termination: As shown in Fig. 4(d), when $t = T$, the last hidden variable ${h_T}$ is calculated as the maxmium of $\pi [T,q]$ for all $q \in Q$.
- (iv) Backward track: As shown in Fig. 4(e), $\pi [t,q]$ for all transitions corresponding to the input sequence x are combined to form a dynamic programming table and $\lambda [t,q]$ stores the subclass $q^{\prime}$ at $t - 1$ corresponding to $\pi [t,q]$. Then, based on the termination ${h_T}$, the optimized hidden variable ${h_t}$ can be obtained with the Viterbi algorithm through the $\lambda$ with the previously obtained ${h_{t + 1}}$ as:
The entire optimized hidden variable sequence $h^{*} = [{h_1},{h_2}, \ldots ,{h_T}]$ can be obtained applying the Viterbi algorithm backwards from ${h_T}$ to ${h_1}$.
Finally, the subclass sequences of three dimensions corresponding to the calculated hidden variable are separately demapped to their class sequences. According to the subclass assignment illustrated in Fig. 3(b), three class sequences are combined and mapped to the 3D-CAP-64 signal constellations.
When combined with optoelectronic devices, the random mode coupling in OAM-MDM typically generates strong nonlinear impairment with stochasticity, which is very difficult to compensate using conventional nonlinear equalizers. However, the proposed HCRF equalizer extracts the stochastic nonlinear features of the OAM-MDM 3D-CAP-64 signals by estimating the conditional probabilities of the hidden variables, thus enabling the signals to be classified into subclasses of constellation points. The nonlinear impairments can then be mitigated through the statistical probability distribution of the OAM-MDM transmission channel with the hidden variables in the HCRF equalizer.
2.3 Computational complexity analysis
In this section, we analyse the computational complexity of the proposed HCRF equalizer and two other commonly used equalizers, a VNE and a CNN-based equalizer. The number of real multiplications involved is used as a basis for an analysis and comparison of the computational complexity of different algorithms.
When processing one OAM-MDM 3D-CAP-64 symbol, according to Eqs. (10)–(15), the number of real multiplications for the HCRF nonlinear equalizer can be expressed as
where $M$ is the length of the memory, ${n_h}$ is the number of subclasses for each class and $N$ is the number of all subclasses which is equal to $4{n_h}$. ${N^2}$ refers the number of all possible transition paths between the current subclass and adjacent subclass, while the $3M + 1$ term represents the number of real multiplications for each possible subclass transition. It can be seen that the number of subclasses for each class ${n_h}$ is the most important term and that increasing ${n_h}$ will incur a sharper increase in computational complexity than increasing M.The number of real multiplications for the VNE can be expressed as
where $s$ denotes the order of the VNE and $d$ is the r-th order memory length [24].For the CNN-based equalizer, the number of real multiplications can be expressed as:
The computational complexity of each of three equalizers was evaluated and compared with the results of processing OAM-MDM 3D-CAP-64 signals, and the results are presented in the following section.
3. Experiments
3.1 Experimental setup
The experimental setup of the OAM-MDM RCF communication system with 3D-CAP-64 is illustrated in Fig. 5. A 20 Gbaud 3D-CAP-64 signal was transmitted over a 2.3 km RCF and detected by a receiver with an offline digital signal processing (DSP). A pseudo-random bit sequence (PRBS) was generated at the transmitter, and the bit data were converted into 3D-CAP-64 symbols following the 3D constellation mapping; these symbols were then filtered by using shaping filters and summed up to generate 3D-CAP-64 signals. The DSP at the transmitter side is shown in Fig. 5(b). The electrical 3D-CAP-64 signals were generated by using an arbitrary waveform generator (AWG, Keysight 8194A) at a sampling rate of 120 GSa/s, and were amplified by using an electrical amplifier (EA). The electrical 3D-CAP 64 signals were modulated onto an optical carrier with an external cavity laser (ECL) with a wavelength of 1550.12 nm using a Mach-Zehnder modulator (MZM). The modulated optical light was then split into two branches by using an optical coupler (OC), with one branch delayed with a 10 m SMF for data pattern decorrelation, and amplified by an erbium-doped fiber amplifier (EDFA). After being collimated and linear polarized, the two beams were converted into OAM beams with l = + 2 and l = + 3 by two spatial light modulators (labelled SLM1 and SLM2 in Fig. 5(a), respectively). The two OAM beams with orthogonal linear-polarizations were then multiplexed with low loss by using a half-wave plate (HWP) and a polarization beam combiner (PBC). The multiplexed OAM beams were converted into circular polarization states through a quarter-wave plate (QWP) and then coupled into the 2.3 km RCF.
Fig. 5. (a) Experimental setup of an OAM-MDM RCF communication system with 3D-CAP-64 modulation. DSP: digital signal processing; AWG: arbitrary waveform generator; MZM: Mach-Zehnder modulator; PC: polarization controller; ECL: external cavity laser; EDFA: erbium-doped fiber amplifiers; OC: optical coupler; SMF: single-mode fiber; Col.: collimator; LP: linear polarizer; SLM: spatial light modulator; HWP: half-wave plate; QWP: quarter-wave plate; PBC: polarization beam combiner; BS: beam splitter; MR: mirror; VPP: vortex phase plate; VOA: variable optical attenuator; PD: photodetector; OSC: oscilloscope, (b) DSP at the transmitter side, (c) offline DSP at the receiver side, (d) cross-sectional structure of the RCF, (e) intensity profiles of two OAM modes; (i) l = + 3 and (ii) l = + 2 after a 2.3 km RCF transmission.
The characteristics of the RCF employed here are summarized in Table 1. The cross section of the RCF is shown in Fig. 5(d). The values of the effective refractive index ${n_{eff}}$ of all guided modes of the employed RCF are shown in Fig. 6(a). The RCF was designed to minimize the effective refractive index difference $\Delta {n_{eff}}$ between the intra-MG modes to around $1 \times {10^{ - 5}}$, in order to suppress the frequency-selective fading of each MG channel and supports four azimuthal MGs from |l|=0 to |l|=3. The measured values of DMD for all supported modes of the employed RCF at 1550 nm are shown in Fig. 6(b). The relatively large values of DMD between high-order MGs indicate weak coupling among them, while the intra-MG DMD is almost constant for all MGs, meaning that intra-MG modes with similar characteristics can be considered as a single data channel. The crosstalk of -22.3 dB from |l| = 2 to |l| = 3 and crosstalk of -23.1 dB from |l| = 3 to |l| = 2 were measured using a vector network analyzer (VNA) in the 2.3 km RCF based system. In view of the relatively large $\Delta {n_{eff}}$, the two modes of |l|=2 and |l|=3 were chosen for the transmission experiment due to the low level of crosstalk.
Fig. 6. (a) Calculated values of ${n_{eff}}$ for all modes supported by the RCF at 1550 nm; (b) measured values of DMD for all modes supported by the RCF at 1550 nm
Table 1. Characteristics of the RCF
The intensity profiles of the two transmitted OAM modes l = + 2, + 3 are shown in Figs. 5(e) (ii) and (i), respectively. At the receiver side, the multiplexed OAM beams were split into two beams through a beam splitter (BS), and two QWPs were used to convert the OAM beams into linearly polarized beams. The two beams were then converted into Gaussian beams through two opposite vortex phase plates (VPPs). By changing the order of the VPPs and recording the power using a SMF-pigtailed power meter, the mode purities were measured and shown to be above 97% for both modes (l = + 2, + 3). The total insertion loss of the bulk-optics mode multiplexer was ∼8 dB, including the coupling loss into the ring core fiber, and the total insertion loss of the mode demultiplexer was ∼6 dB. The two Gaussian beams were coupled into SMFs through the collimators, and converted into electrical signals via two PDs. Two EDFAs and two variable optical attenuators (VOAs) were used to adjust the received optical powers (ROPs). Each ROPs was adjusted using a VOA before the beam was passed to the PD, and was measured with a SMF-pigtailed power meter. The electrical waveforms were captured and recorded with a real-time oscilloscope (OSC) at a sampling rate of 256 GSa/s. The offline DSP, as shown in Fig. 5(c), involved low-pass filtering (LPF), resampling, clock recovery, matched filtering, the HCRF equalizer, demapping, and bit error ratio (BER) calculation. The Gardner clock recovery algorithm was applied for the clock recovery while the BER was calculated with a bit-by-bit comparison to verify the performance of the proposed equalizer.
In the experiment, the length of the OAM-MDM 3D-CAP-64 signal sequence was 100000 for the proposed equalizer, of which 30% was used as a training dataset and 70% as a test dataset. The HCRF model was trained with the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimization technique [30] and L2 regularization was applied to avoid overfitting.
3.2 Experimental results and analysis
To verify the effectiveness of the proposed HCRF equaliser, an experiment was carried out on a 20 GBaud 3D-CAP-64 modulated OAM-MDM optical fiber communication system over 2.3 km RCF transmission. Two OAM modes (l = + 2 and l = + 3) were transmitted.
Figure 7 illustrates the process of the HCRF equalization for one dimension of the testing sequence of the OAM-MDM 3D-CAP-64 signals. In this experiment, the memory length $M$ was set to four, the number of subclasses for each class ${n_h}$ was four, and hence the number of subclasses N was therefore 16. The termination stage is shown in Fig. 7(a). $h_T^{}$ was set to ${q_1}$, since the value of $\pi [T,{q_1}]$ was a maximum. Figures 7(b) to (d) show the dynamic programming table at $t = T - 2,T - 1,T$ respectively, where different colors represent different values of $\pi [t,q]$. The horizontal and vertical coordinates are subclasses of $h_{t - 1}^{}$ and $h_t^{}$ according to Eq. (15). In Figs. 7(a)–(c), $\pi [t,q]$ denotes the maximum probability of the transition from $h_{t - 1}^{}$ to $h_t^{}$, and is marked in the darkest color with a red box. In Fig. 7(b), when $t = T$, ${h_{T - 1}}$ was set to ${q_8}$ because the maximum probability of the transition from $h_{T - 1}^{}$ to $h_T^{}$ was located at ${q_8}$. As shown in Fig. 7(c), when $t = T - 1$, ${h_{T - 2}}$ was set to ${q_4}$ because the maximum probability of the transition from $h_{T - 2}^{}$ to $h_{T - 1}^{}$ was located at ${q_4}$ and as shown in Fig. 7(d), when $t = T - 2$, ${h_{T - 3}}$ was set to ${q_4}$ because the maximum probability of the transition from $h_{T - 3}^{}$ to $h_{T - 2}^{}$ was located at ${q_4}$.The rest of the hidden variable sequence was obtained with the same backward track method, as can be seen in Fig. 7(e).
Fig. 7. Equalization process applied by the HCRF equalizer to a test sequence of OAM-MDM 3D-CAP-64 signals. dynamic programming table (a) at $t = T - 2$; (b) at $t = T - 1$; (c) at $t = T$; (d) termination; (e) backward track diagram.
The BER performance of the HCRF equalizer for modes l = + 2 and l = + 3 was evaluated as shown in Figs. 8(a) and (b), respectively. For each nonlinear equalizer, the BER performance was optimized based on the parameters settings and the complexity in terms of the number of real multiplications, as shown in Table 2. In addition, a feed-forward equalizer (FFE) was also implemented for comparison and to indicate the necessity of nonlinear equalization. Due to the nonlinear impairments of the OAM-MDM system, the BERs with the hard decision and with FFE could not achieve below the 7% forward error correction (FEC) limit of $3.8 \times {10^{ - 3}}$ in the ROP range of -2 to 8 dBm for l = + 2 and in the ROP range of 0 to 10 dBm for l = + 3. Hence it is necessary to use the nonlinear equalizer in the OAM-MDM system. The constellations of 3D-CAP-64 for ROP values of 8, 4, and 0 dBm for l = + 2 are shown in Figs. 9(a), (b), and (c) respectively. When processed with the HCRF equalizer, the BERs for both OAM modes were below the 7% FEC threshold when the ROP exceeded 4 dBm for the l = + 2 mode and when it was greater than 7 dBm for the l = + 3 mode, thereby demonstrating the effectiveness of the proposed HCRF equalizer.
Fig. 8. Measured BER versus ROP for (a) l = + 2 and (b) l = + 3 in the OAM-MDM 3D-CAP-64 RCF transmission system over a 2.3 km RCF with different equalizers; Measured BER versus ROP for (c) l = + 2 and (d) l = + 3 in the back-to-back OAM-MDM 3D-CAP-64 RCF transmission system with different equalizers.
Fig. 9. 3D-constellations for 3D-CAP-64 at ROP values of (a) 8 dBm; (b) 4 dBm; (c) 0 dBm for l = + 2.
Table 2. Parameter settings for the three equalizers shown in Fig.8
Performance comparisons with the conventional VNE and CNN-based equalizer are shown in Figs. 8(a) and (b). The proposed HCRF nonlinear equalizer outperformed the other two equalizers in terms of BER performance. Compared with the VNE, the the receiver sensitivity of the HCRF equalizer was improved by about 4 dB and 3 dB at the 7% FEC limit for two OAM modes (l = + 2 and l = + 3, respectively). Compared with the CNN-based equalizer, the receiver sensitivity of the HCRF equalizer was improved by about 2 dB and 1 dB at the 7% FEC limit for two OAM modes (l = + 2 and l = + 3, respectively). In contrast to the l = + 2 mode, the l = + 3 mode suffered relatively high performance degradation due to its relatively large modal dispersion and chromatic dispersion, resulting in a smaller improvement in receiver sensitivity. The BER performance for each of the three equalizers was evaluated for modes l = + 2 and l = + 3 using a back-to-back (BTB) configuration, which was implemented without RCF transmission between the OAM multiplexing and demultiplexing, as shown in Figs. 8(c) and (d), respectively. It can be seen that in the BTB transmission scenario, the HCRF equalizer achieved improvements in receiver sensitivity of 0.3 dB and 0.6 dB compared with the CNN-based equaliser and VNE, respectively.
For the HCRF nonlinear equalizer, the number of subclasses for each class ${n_h}$ and the length of memory M are two key parameters that affect the nonlinearity mitigation performance. Figures 10(a) and (b) illustrate the BER performance in the two OAM modes with different values of ${n_h}$ and M, at a ROP of 8 dBm. As shown in the contour plot, the effect of the two parameters ${n_h}$ and M on the BER performance was non-diagonal. When ${n_h}$ was increased, BER decreased gradually, and when M was increased, BER decreased. The complexity equation contains a squared term of ${n_h}$ and a primary term of M, meaning that the two parameters contribute differently to the complexity of the equalizer: the computational complexity increases significantly with the value of ${n_h}$ and more slowly with the value of M. Hence, in Fig. 8, ${n_h}$ was set as four and $M$ was set as five for the proposed equalizer in order to strike a balance between the BER performance and the computational complexity.
Fig. 10. BER contour plots for (a) l = + 2 and (b) l = + 3 with the HCRF equalizer in the OAM-MDM 3D-CAP-64 RCF transmission system for varying values of the model parameters.
The computational complexity of the three equalizers with similar BER performance was evaluated for the OAM mode l = + 2 with a ROP of 8 dBm as shown in Table 3.
Table 3. Complexity of the three equalizers with similar BER performance
When the number of real multiplications was used as an indicator of the computational complexity and the HCRF equaliser was compared with the conventional VNE and CNN-based equalisers, the complexity was found to be reduced by 30% and 41%, respectively, thus demonstrating that the proposed HCRF nonlinear equaliser has a relatively low computational complexity.
The maximum level of crosstalk tolerated by the proposed HCRF equalizer was evaluated. However, the parameters of the employed RCF is fixed. Hence, we fixed the power of one OAM mode and changed the power of the other OAM mode to adjust the crosstalk. First, mode l = + 2 was fixed with the power of 0 dBm, and the power of mode l = + 3 was adjusted from 0 to 10 dBm. Then, mode l = + 3 was fixed with the power of 0 dBm, and the power of mode l = + 2 was varied from 0 to 10 dBm. The BER performance of the two modes is shown in Fig. 11. For the parameter setting of the HCRF equalizer, M was set to 5 and ${n_h}$ was set to 5. It can be seen that the proposed HCRF equalizer could work properly with the mode power difference range from 0 to 10 dB. Therefore, the proposed equalizer is effective under a large crosstalk. In general, the proposed HCRF equalizer can compensate for the nonlinear impairments effectively through the statistical probability distribution of the hidden variables of the OAM-MDM transmission channel in the HCRF equalizer. Hence the proposed equalizer is expected to have good performance in other MDM systems, such as the LP mode-based MDM.
4. Conclusion
The use of an HCRF nonlinear equalizer for the impairment mitigation of OAM-MDM optical fiber communication was proposed in this work. The HCRF equalizer extracted the stochastic nonlinear feature of OAM-MDM 3D-CAP-64 signals by estimating the conditional probabilities of the hidden variables, which can classify the signal into subclasses of constellation points. Our results suggest that the nonlinear impairment could be mitigated through the statistical probabilistic distribution of the OAM-MDM transmission channel with the hidden variables in the HCRF equalizer. An experiment was conducted using a 20 Gbaud 3D-CAP-64 two-mode OAM-MDM optical fiber communication system over a 2.3 km RCF. The experimental results show that compared with a CNN-based equalizer, the HCRF equalizer improved the receiver sensitivity by 2 dB and 1 dB for OAM modes with l = + 2 and l = + 3 at the 7% FEC threshold, respectively. In addition, the HCRF equalizer reduced the computational complexity by 30% and 41% relative to the VNE and CNN-based equalizers, respectively. According to the BER performance and low computational complexity, the HCRF equalizer could provide an effective nonlinear mitigation method for high-speed IM/DD OAM-MDM optical fiber communication system.
Funding
National Natural Science Foundation of China (62022016); Science Fund for Creative Research Groups of China (62021005); National Key Research and Development Program of China from Ministry of Science and Technology (2019YFA0706300); Beijing Municipal Natural Science Foundation (4222075); Open Fund of IPOC (BUPT) (IPOC2022A04).
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. K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A.P.T. Lau, and C. Lu, “Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s short reach optical transmission systems,” Opt. Express 23(2), 1176–1189 (2015). [CrossRef]
2. Z. Tian, Y. Cai, L. Sun, H. Xu, C. Xiong, M. Gao, C. Jiang, X. Dong, B. Xia, G.N. Liu, and G. Shen, “Improving the spectral efficiency of visible light communications by a super-Nyquist multi-band CAP modulation,” Opt. Lett. 47(23), 6069–6072 (2022). [CrossRef]
3. B. Dirk, F. Geilhardt, R. Hülsermann, M. Kind, C. Lange, T. Monath, and E. Weis, “Opportunities for next-generation optical access,” IEEE Commun. Mag. 49(2), s16–s24 (2011). [CrossRef]
4. K. Jeung-Mo and S.K. Han, “A novel hybrid WDM/SCM-PON sharing wavelength for up-and down-link using reflective semiconductor optical amplifier,” IEEE Photonics Technol. Lett. 18(3), 502–504 (2006). [CrossRef]
5. W. Andreas OJ, B.E. Olsson, and P. Andrekson, “Single Cycle Subcarrier Modulation,” in Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optica Publishing Group, 2009), paper OTuE1.
6. N. Cvijetic, M. Cvijetic, M. Huang, E. Ip, Y. Huang, and T. Wang, “Terabit Optical Access Networks Based on WDM-OFDMA-PON,” J. Lightwave Technol. 30(4), 493–503 (2012). [CrossRef]
7. J. Zhang, B. Liu, Y. Mao, J. Ren, S. Han, S. Chen, X. Xu, X. Wu, X. Song, L. Jiang, and J. Zhao, “Enhanced 3D-CAP modulation with information-inserted time slots for seven-core fiber transmissions,” Opt. Express 28(17), 24991–24999 (2020). [CrossRef]
8. W. Xia, B. Liu, J. Ren, R. Ullah, X. Wu, Y. Mao, Y. Ma, S. Chen, Y. Wan, Q. Zhong, J. Zhao, Y. Bai, T. Yuan, and L. Zhao, “High-security 3D CAP modulation scheme based on a pyramid constellation design for 7-core fiber,” Opt. Express 31(4), 6659–6674 (2023). [CrossRef]
9. L. Sun, J. Du, and Z. He, “Multiband Three-Dimensional Carrierless Amplitude Phase Modulation for Short Reach Optical Communications,” J. Lightwave Technol. 34(13), 3103–3109 (2016). [CrossRef]
10. J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-Dimensional Probabilistically Shaped CAP Modulation Based on Constellation Design Using Regular Tetrahedron Cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020). [CrossRef]
11. 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]
12. 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]
13. Y. Zhu, K. Zou, Z. Zheng, and F. Zhang, “1 λ × 1.44 Tb/s free-space IM-DD transmission employing OAM multiplexing and PDM,” Opt. Express 24(4), 3967–3980 (2016). [CrossRef]
14. J. Zhang, J. Liu, Z. Lin, J. Liu, L. Shen, and S. Yu, “Nonlinearity-Aware Adaptive Bit and Power Loading DMT Transmission Over Low-Crosstalk Ring-Core Fiber with Mode Group Multiplexing,” J. Lightwave Technol. 38(21), 5875–5882 (2020). [CrossRef]
15. L. Zhu, C. Yang, D. Xie, and J. Wang, “Demonstration of km-scale orbital angular momentum multiplexing transmission using 4-level pulse-amplitude modulation signals,” Opt. Lett. 42(4), 763–766 (2017). [CrossRef]
16. L. Zhu, H. Yao, H. Chang, Q. Tian, Q. Zhang, X. Xin, and F. R. Yu, “Adaptive Optics for Orbital Angular Momentum-Based Internet of Underwater Things Applications,” IEEE Internet Things J. 9(23), 24281–24299 (2022). [CrossRef]
17. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
18. 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.
19. K. Ho and J. M. Kahn, “Mode Coupling and its Impact on Spatially Multiplexed Systems,” (2013). [CrossRef]
20. J. Zhang, G. Zhu, J. Liu, X. Wu, J. Zhu, C. Du, W. Luo, Y. Chen, and S. Yu, “Orbital-angular-momentum mode-group multiplexed transmission over a graded-index ring-core fiber based on receive diversity and maximal ratio combining,” Opt. Express 26(4), 4243–4257 (2018). [CrossRef]
21. C. Brunet, P. Vaity, B. Ung, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Design of a family of ring-core fiber for OAM,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optica Publishing Group, 2015), paper M3D.1.
22. J. Liu, G. Zhu, J. Zhang, Y. Wen, X. Wu, Y. Zhang, Y. Chen, X. Cai, Z. Li, Z. Hu, J. Zhu, and S. Yu, “Mode division multiplexing based on ring core optical fibers,” IEEE J. Quantum Electron. 54(5), 1–18 (2018). [CrossRef]
23. X. Jin, A. Gomez, K. Shi, B. C. Thomsen, F. Feng, G. S. D. Gordon, T. D. Wilkinson, Y. Jung, Q. Kang, P. Barua, J. Sahu, S. Alam, D. J. Richardson, D. C. O’Brien, and F. P. Payne, “Mode Coupling Effects in Ring-Core Fibers for Space-Division Multiplexing Systems,” J. Lightwave Technol. 34(14), 3365–3372 (2016). [CrossRef]
24. 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]
25. G. Saavedra, G. Liga, and P. Bayvel, “Volterra-Assisted Optical Phase Conjugation: A Hybrid Optical-Digital Scheme for Fiber Nonlinearity Compensation,” J. Lightwave Technol. 37(10), 2467–2479 (2019). [CrossRef]
26. C. Li, Y. Wang, J. Wang, H. Yao, X. Liu, R. Gao, L. Yang, H. Xu, Q. Zhang, P. Ma, and X. Xin, “Convolutional Neural Network-Aided DP-64 QAM Coherent Optical Communication Systems,” J. Lightwave Technol. 40(9), 2880–2889 (2022). [CrossRef]
27. P. J. Freire, Y. Osadchuk, B. Spinnler, A. Napoli, W. Schairer, N. Costa, J. E. Prilepsky, and S. K. Turitsyn, “Performance Versus Complexity Study of Neural Network Equalizers in Coherent Optical Systems,” J. Lightwave Technol. 39(19), 6085–6096 (2021). [CrossRef]
28. Y. Liu, V. Sanchez, P. Freire, J. Prilepsky, M. Koshkouei, and M. Higgins, “Attention-aided partial bidirectional RNN-based nonlinear equalizer in coherent optical systems,” Opt. Express 30(18), 32908–32923 (2022). [CrossRef]
29. C. Liu, C. Wang, W. Zhou, F. Wang, M. Kong, and J. Yu, “81-GHz W-band 60-Gbps 64-QAM wireless transmission based on a dual-GRU equalizer,” Opt. Express 30(2), 2364–2377 (2022). [CrossRef]
30. A. Quattoni, S. Wang, L. P. Morency, M. Collins, and T. Darrell, “Hidden conditional random fields,” IEEE Trans. Pattern Anal. Machine Intell. 29(10), 1848–1852 (2007). [CrossRef]
31. C. Suttons and A. McCallum, “An introduction to conditional random fields,” FNT in Machine Learning 4(4), 267–373 (2012). [CrossRef]
