1. Introduction

The increasing popularity of AI-centric applications such as ChatGPT has imposed high-capacity requirements on IM/DD transmission for short-reach optical interconnections [1,2]. As there is a need to break through the Shannon limit on the capacity of a single-mode optical fiber (SMF), orbital angular momentum (OAM) mode division multiplexing (MDM) has been developed as an innovative multiplexing technology for enhancing capacity [3–10]. OAM-MDM IM/DD transmission has achieved ultra-high-capacity transmission over different distances by leveraging a ring-core fiber (RCF) with weak inter-MG coupling; for example, researchers have used a 20-Gbaud 3D CAP-64 architecture for OAM-MDM transmission over 2.3 km [3], a three-mode multiplexed PAM8 transmission for IM/DD OAM-MDM over 2.6 km [4], and a 4.32 Tbit/s PAM8 transmission for IM/DD OAM-MDM over 2 km [5].

However, in short-reach IM/DD OAM-MDM systems, significant nonlinear impairments arise from the optoelectronic devices used, such as Mach-Zehnder modulators (MZMs), spatial light modulators (SLMs) and photodiodes (PDs). Furthermore, the nonlinear distortion of each of the different OAM modes affects the other modes, due to the mode coupling in OAM-MDM transmission. Hence, OAM-MDM shows strong complex nonlinear impairment, and this is particularly prominent for multiple OAM mode transmission systems. For conventional block-structured OAM-MDM systems, AI-based compensation methods have achieved remarkable results following the optimization of independent blocks, such as probabilistic shaping (PS) [11], linear equalization [12] and nonlinear equalization [3–5]. However, although the methods described above can effectively improve the performance of OAM-MDM systems, it is still uncertain whether individually optimized blocks can lead to global optimization of an IM/DD OAM-MDM system due to the presence of mode coupling and complex nonlinear impairment [13]. Hence, the development of optimal modulation and equalization schemes for OAM-MDM systems remains a persistent challenge.

End-to-end (E2E) learning is an emerging technology based on an autoencoder (AE) that provides intelligent E2E optimization according to the impairment of a communication system [14–18]. An E2E strategy can achieve global optimization of joint modulation and equalization, and can compensate for signal impairment by treating the entire communication system as an AE, which offers great performance improvements for optical communications. Nan Chi et al. proposed a novel joint geometric shaping (GS) and post-equalization scheme for a fiber-THz integrated communication system [17]. Jianyang Shi et al. presented an E2E learning strategy to achieve joint pre-equalization and post-equalization for a visible light communication system [18]. Accurate estimation of the OAM-MDM channel is crucial for E2E PS or equalization. In conventional optical communication systems, deep learning strategies are usually adopted to provide accurate channel models, such as conditional generative adversarial networks (CGANs) [19,20], heterogeneous neural network networks [21,22] or a three-subnetwork architecture [23]. However, the channel model of the input signal exhibits different characteristics at different frequencies, due to the complex nonlinear impairments to OAM-MDM transmission. For example, the nonlinear impairment caused by optoelectronic devices is mainly seen at high frequencies [24], while the linear impairment caused by mode coupling is mainly observed at low frequency [25]. Channel modelling strategies based on conventional machine learning always build a mixed model of the transmission channel for different frequencies, resulting an inaccurate channel model for an OAM-MDM IM/DD system.

In this paper, we propose a scheme based on a frequency domain feature decoupling network (FDFDnet) and E2E learning for joint PS and equalization of an OAM-MDM IM/DD system. Our FDFDnet-based E2E learning scheme can accurately build a complex nonlinear model of an OAM-MDM system by separating the signal into different frequency domain features. Secondly, a FDFDnet-based E2E strategy for joint PS and equalization based on the CAP-32 modulation format is presented to compensate the signal impairment. An experiment is carried out to verify the effectiveness of the proposed method in an OAM-MDM IM/DD transmission with three modes; the results demonstrate that the proposed method can build an accurate channel model that can significantly mitigate the nonlinear impairments of the OAM-MDM IM/DD transmission. Thus, the proposed scheme is a promising contender for OAM-MDM transmission in ultra-high-capacity inter-data center interconnects.

2. End-to-end deep learning of joint probabilistic shaping and equalization for an OAM-MDM system using FDFDnet emulator

For a conventional block-structured OAM-MDM system, AI-based methods are commonly implemented by optimizing independent blocks. However, these individually optimized blocks cannot lead to the global optimization of the OAM-MDM system, and as a result, the development of optimal modulation and equalization schemes for OAM-MDM systems remains a challenge. In order to achieve an accurate channel response for an OAM-MDM system, a novel E2E learning strategy for joint PS and equalization based on an FDFDnet emulator is proposed for OAM-MDM transmission with CAP-32 modulation. The E2E learning scheme proposed in this paper is shown in Fig. 1.

 

Fig. 1. Diagram showing the proposed E2E learning strategy based on a FDFDnet emulator for an OAM-MDM system: (a) actual three-mode OAM-MDM system; (b) proposed OAM-MDM system based on AE.

Download Full Size | PDF

Figure 1(a) depicts a three-mode OAM-MDM system consisting of a transmitter, channel and receiver. In the proposed E2E learning scheme for joint PS and equalization, the whole OAM-MDM system is considered as an AE, with the transmitter regarded as the AE encoder and the receiver as the AE decoder. The channel is simulated with our FDFDnet emulator. The FDFDnet emulator separates the signal into different frequency domain features rather than building a mixed model for the transmission channel for different frequencies, meaning that it can accurately model the channel response of the OAM-MDM system even though the received signal is subject to mode coupling and complex nonlinear impairments. With the trained FDFDnet emulator, gradients can be processed from the AE decoder to the AE encoder, thereby achieving joint training of the transmitter and receiver. In this way, the AE encoder achieves the optimal probability distribution and the AE decoder achieves equalization. As shown in Fig. 1(b), the proposed optimal modulation and equalization scheme based on the FDFDnet emulator is realized through E2E learning, which can bring great performance improvements to an OAM-MDM system.

2.1 FDFDnet-based channel modelling strategy for an OAM-MDM system

In an IM/DD OAM-MDM system, the transmitted CAP-32 symbol sequence can be represented as $x(n) = [{{x_1},{x_2}, \cdot{\cdot} \cdot {x_n}} ],{x_n} \in ( - 1,1)$. The CAP-32 symbol sequence received by the OAM-MDM system is distorted due to the mode coupling and complex nonlinear impairment, and can be represented as $y(n) = [{{y_1},{y_2}, \cdot{\cdot} \cdot {y_n}} ]$.

Before simulating the channel response in the OAM-MDM, the transmitted and received CAP-32 signals are pre-processed as shown in Fig. 2. The current symbol ${x_i}(i = 1, \ldots n)$ is wrapped with its L preceding and L succeeding symbols to form a feature vector ${X_i} = [{{x_{i - L}}, \cdot{\cdot} \cdot ,{x_{i - 1}},{x_i}, \cdot{\cdot} \cdot ,{x_{i + L}}} ]$. The transmitted time-series signal $x(n) = [{{x_1},{x_2}, \cdot{\cdot} \cdot {x_n}} ]$ is transformed into a matrix composed of $(n - 2L)$ feature vectors with a length of $({2L + 1} )$. The feature vector matrix can be expressed as

 

Fig. 2. Data preprocessing.

Download Full Size | PDF

The received signal ${y_i}(i = 1,2, \ldots n)$ corresponding to the feature vector ${X_i}(i = 1, \ldots ,n - 2L)$ is taken as real data, and can be expressed as

The feature vector X and real data Y are combined to form the dataset $D = \{{X,Y} \}$. The dataset D is then divided into two parts to form the training dataset ${D_T} = \{{{X_T},{Y_T}} \}$ and the testing dataset ${D_P} = \{{{X_P},{Y_P}} \}$, where ${X_T} = [{{X_1},{X_2} \cdots {X_T}} ]$, ${Y_T} = [{{Y_1},{Y_2} \cdots {Y_T}} ]$, ${X_P} = [{{X_1},{X_2} \cdots {X_P}} ]$ and\[{Y_P} = [{{Y_1},{Y_2} \cdots {Y_P}} ]\]$({P + T = n - 2L} )$.

The structure of the FDFDnet emulator is depicted in Fig. 3(a). It consists of three blocks: an input layer, two fully connected layers, and an output layer. Each block represents a frequency domain of the OAM-MDM signal (Block 1, 2, 3 represent the low, middle, and high frequencies, respectively). The first two blocks also include a frequency low cut (FLC) pooling layer. The training dataset ${D_T}$ is applied to the FDFDnet emulator to determine all of the parameters. The training data is first passed to the input layer of the Block 1 in the form of a feature vector ${X_i}({i = 1,2, \ldots T} )$ of the training data set ${D_T}$.

 

Fig. 3. Schematic diagram of the FDFDnet emulator: (a) structure of Block 1; (b) structure of Block 2; (c) structure of Block 3.

Download Full Size | PDF

In the second step, in the Block 1, the input vectors ${X_i}$ with the length of $(2L + 1)$ are converted into the frequency domain with a fast Fourier transform (FFT). The low-frequency part is separated in the FLC pooling layer, and is expressed as

In the third step, in Block 1, the two fully connected layers are used to perform two regression calculations on ${F_{low}}$ with the weight vectors $({{W_1},{W_2}} )$ and the bias vectors $({{b_1},{b_2}} )$, and this is followed by a $ReLU ({} )$ activation function for each fully connected layer. The output layer is used to perform a linear calculation to generate two output vectors with the weight vector $({{W_3}} )$ and the bias vector $({{b_3}} )$. The whole process can be expressed as

The dimensions of ${W_1}$, ${b_1}$, ${W_2}$, ${b_2}$, ${W_3}$ and ${b_3}$ are the same as the number of neurons in the fully connected layers. ${Y_1}$ is the output vector of Block 1, and has a length of $({2L + 2} )$. ${\widehat y_1}$ is the reconstructed vector of ${F_{low}}$, with length $({2L + 1} )$. ${y_1}$ is the emulated signal based on the low-frequency feature vector. ${\widehat y_1}$ is calculated to avoid repeated training of signal that have been learned in subsequent blocks, while the ${y_1}$ is added to the output of the subsequent blocks to form the final emulated signal.

In the fourth step, the input of Block 2 is obtained by subtracting the reconstructed vector ${\widehat y_1}$ from the input vector ${X_i}$ of Block 1, which can be expressed as

In the fifth step, in Block 2, the input vector ${x_2}$ is taken using the same calculation process as for Block 1. Block 2 can then be used to extract the medium-frequency signals, which can be expressed as

In the sixth step, the input to Block 3 is obtained by subtracting the reconstructed vector ${\widehat y_2}$ from the input vector ${x_2}$ of Block2, which can be expressed as

The two fully connected layers are then used to perform two regression calculations on ${x_3}$ with the weight vectors $({{W_7},{W_8}} )$ and the bias vectors $({{b_7},{b_8}} )$, and this is followed by a $ReLU ({} )$ activation function for each fully connected layer. The output layer is then used to perform a linear calculation with the weight vector $({{W_9}} )$ and the bias vector $({{b_9}} )$, without an activation function. The whole process can be expressed as

Finally, the emulated signal $\hat{Y}$ of the input sample ${X_i}$ output from the FDFDnet emulator can be expressed as

After obtaining the estimated value $\hat{Y}$, FDFDnet applies the mean squared error (MSE) function to calculate the loss throughout the training process. The Adam gradient descent function is employed to iteratively train the model parameters $({{W_1},{b_1},{W_2},{b_2},{W_3},b{}_3,{W_4},{b_4},{W_5},{b_5},{W_6},{b_6},{W_7}}$,${{b_7},{W_8},{b_8},{W_9},{b_9}} )$ through the use of backpropagation. The distortion symbols corresponding to the testing set ${X_P} = [{{X_1},{X_2} \cdots {X_P}} ]$ can be simulated by using the forward propagation process of the FDFDnet emulator after the training process.

The combination of mode coupling in OAM-MDM with nonlinear impairment from optoelectronic devices typically results in strong complex nonlinear damage, which poses a challenge for traditional channel emulators based on a mixed model for different frequencies. However, since the FDFDnet emulator separates the signal into different frequency domain features rather than building a mixed model for the transmission channel for different frequencies, it can accurately model the channel response of the OAM-MDM system even though the received signal is subject to mode coupling and complex nonlinear impairment. Our FDFDnet emulator can obtain an accurate model from the different frequency domain features, thus forming an effective OAM-MDM transmission channel emulator for strong complex nonlinearity.

2.2 Joint probabilistic shaping and equalization for an OAM-MDM system based on AE

In this section, an AE-based scheme for joint optimization of PS and equalization for an OAM-MDM system with CAP-32 modulation is introduced, as depicted in Fig. 4. The system is composed of an AE encoder, a mapper, CAP modulation, a FDFDnet emulator, CAP demodulation, and an AE decoder. Both the encoder and decoder are neural networks (NNs) composed of fully connected layers.

 

Fig. 4. Diagram showing our AE scheme for PS and equalization in an OAM-MDM system. (a) AE Encoder, (b) Mapper, (c) FDFDnet emulator, (d) AE Decoder.

Download Full Size | PDF

At the transmitter, the AE encoder first generates a vector of probabilities ${P_M} = ({{p_1}, \ldots ,{p_M}} )$ to denote the probability for each constellation point ${C_M} = ({{c_1}, \ldots {c_M}} )$ and $M = {2^m}$ represents the modulation order, as shown in Fig. 4(a).

Next, as shown in Fig. 4(b), the constellation point probability ${P_M}$ and the batch size S are input to a sampler. In order to find the corresponding symbol number ${Q_M} = ({{q_1}, \ldots {q_M}} )$ for each constellation point, algorithm 2 from [26] is applied. It is implemented by minimizing the information entropy according to the constellation point probability ${P_M}$ and batch size S. The symbol number ${Q_M}$ are then converted to the binary representation to get the bit vector ${b_k} = ({{b_{1,k}}, \ldots {b_{m,k}}} ),k = 1, \ldots S$; these are encoded to the corresponding one-hot vector ${V_S} = ({v_1}, \ldots {v_S})$ of size $M \times S$, which is not zero at the index corresponding to the binary-to-integer conversion of the bit vector. Then, to consider the non-uniform constellation point probability, the constellation points ${C_M} = \{{{c_1}, \ldots {c_M}} \}$ are normalized based on the probability ${P_M}$, as follows:

Next, the symbol sequence $K = \{{{k_1}, \ldots {k_S}} \}$ is upsampled and filtered using two shaping filters for the CAP modulation, and the CAP signal $X = \{{{x_1}, \ldots {x_S}} \}$ is generated.

At the channel, as shown in Fig. 4(c), the CAP signal $X = \{{{x_1}, \ldots {x_S}} \}$ is then fed into the FDFDnet emulator to generate a received signal $Y = \{{{Y_1}, \ldots {Y_S}} \}$, which contains complex nonlinear impairment.

At the receiver, the impaired signal $Y = \{{{Y_1}, \ldots {Y_S}} \}$ is filtered using two shaping filters and then downsampled to generate the CAP demodulated signal $E = \{{{E_1}, \ldots {E_S}} \}$. Finally, as shown in Fig. 4(d), the decoder maps the CAP demodulated signal $E = \{{{E_1}, \ldots {E_S}} \}$ to a probability vector ${R_j}(j = 1, \ldots S)$.

In order to train E2E AE model accurately, a modified loss function based on the generalized mutual information (GMI) is used, which can be expressed as [15]

3. Experimental

3.1 Experimental setup

As shown in Fig. 5, to verify the accuracy of the proposed FDFDnet emulator, an experiment was carried out on a 300 Gbit/s OAM-MDM IM/DD transmission with three OAM modes over a 10 km RCF. At the transmitter, a pseudo-random bit sequence with the length of ${2^{18}}$ was generated and mapped onto a CAP-32 symbol sequence. The electrical signal was passed through the arbitrary waveform generator (AWG) and electrical amplification (EA) and then modulated by the MZM onto a double-sideband optical signal generated by a laser with a wavelength of 1550.12 nm. The optical signal was then divided into three branches using three optical couplers (OCs), and amplified using three erbium-doped fiber amplifiers (EDFAs). The three branches of the optical signal were delayed separately using a 10 m SMF for decorrelation of the data modes, and were simultaneously transmitted into space through a polarisation controller (PC), collimator (Col.), and linear polarizer (LP). Three SLMs were then used to modulate the three optical signals into three OAM modes ($l = 2,3,4$). In this way, 300Gbit/s transmission could be realized with a 20 GBaud CAP-32 signal per OAM mode in the OAM-MDM IM/DD transmission. The three optical signals in the OAM modes were then combined using polarization beam combiners (PBC) and a beam combiner (BC). Finally, the combined optical signal was converted to circular polarization via a quarter-wave plate (QWP) and transmitted through a 10 km RCF.

 

Fig. 5. Experimental setup for the E2E OAM-MDM system.

Download Full Size | PDF

In Fig. 5, inset (i) shows the cross section of the RCF, and insets (ii)–(iv) show the intensity profiles of the three OAM modes ($l = 2,3,4$), respectively. At the receiver, the multiplexed OAM mode was divided into three beams by two BSs, and these were then converted into Gaussian beams via three vortex phase plates (VPPs). The three Gauss beams were coupled to the SMFs through the Col., and converted into electronic signal via three PDs. The electrical signal was recorded using a real-time oscilloscope at 256GSa/s, and processed by offline DSP including resampling, synchronisation, CAP demodulation, AE-optimized equalization and BER calculation. Each set of 11 adjacent symbols of the electrical signal was input to the FDFDnet emulator as a feature vector ${X_i}$. The corresponding symbol after synchronization was used as real data ${Y_i}$ to sent to the FDFDnet emulator. The feature vector ${X_i}$ and real data ${Y_i}$ were used as the dataset for training the emulator. In the experiment, L was set to five, and the batch size and learning rate of emulator were set to 512 and 1e-3, respectively, for the three OAM modes.

In order to verify the performance of our FDFDnet emulator, we compared it with a traditional CGAN emulator. In addition, as the number of blocks in FDFDnet represent the number of frequency domains considered, we tested the impact of the number of blocks by using only two blocks, to simulate low and high frequencies, in a model called the FDFDnet-2 emulator. As the number of blocks increases, it can lead to a significant increase in the complexity of the emulator. The same dataset was used for training the FDFDnet-2 and CGAN emulators. The trained emulators were then integrated with the AE to obtain the optimal probability distribution and equalization for the OAM-MDM. The pseudo-random bit sequence was mapped onto a CAP-32 signal sequence using a distribution matcher according to the learned optimal probability distribution [26]. The CAP-32 signal was transmitted via AWG and modulated by an optical carrier with the MZM. The electric signal was then recorded and processed vis offline DSP, including resampling, synchronization, CAP demodulation, AE-optimized equalization and BER calculation. The values of the BER for the FDFDnet, FDFDnet-2 and CGAN emulators were compared to verify the effectiveness of the proposed scheme.

3.2 Experimental results and analysis

3.2.1 Performance of the FDFDnet emulator for the OAM-MDM system

In these experiments, the OAM-MDM system channels were simulated using the FDFDnet, FDFDnet-2 and CGAN emulators. The accuracy of each emulator was verified by comparing its output with the real OAM-MDM system channel signal. Figures 6(a), (b) and (c) illustrate the frequency domain emulated results trained with different emulators when the voltage peak-to-peak (Vpp) is 350 mV and received optical power (ROP) is 0 dBm. Compared with the FDFDnet-2 and CGAN emulator, the results from the FDFDnet emulator show higher consistency with the real OAM-MDM channel output. The fading trends of FDFDnet and FDFDnet-2 at high frequencies are more similar to the real channel output signal than that of the CGAN emulator.

 

Fig. 6. Channel output waveforms in the frequency domains based on the OAM-MDM system channel, CGAN emulator and FDFD emulator for (a) $l = 2$, (b) $l = 3$, (c) $l = 4$.

Download Full Size | PDF

To quantitatively represent the channel modelling effect of the real OAM-MDM system using the FDFDnet, FDFDnet-2 and CGAN emulators, we compared the normalized MSE (NMSE) and BER performance. Figures 7(a), (b) and (c) show the NMSE and BER performance for the signals generated by the FDFDnet, FDFDnet-2 and CGAN emulators, respectively, at different ROPs and for different OAM modes ($l = 2,3,4$). Compared with the traditional CGAN emulator, the maximum improvements in the modelling accuracy of the FDFDnet and FDFDnet-2 emulator are 30.8%, 26.3%, 31%, 11.5%, 10.1% and 9.6% at an ROP of −3dBm for the three modes, respectively. The average BER errors between the signal generated by the FDFDnet emulator and the real received signal were only 8%, 7.6% and 8.9% for different OAM modes ($l = 2,3,4$), respectively. The average BER errors between the signal generated by the FDFDnet-2 emulator and the real received signal were 15.3%, 14.5% and 15.8%. These were much lower than the values of 20.5%, 21.9% and 19.6% found for the CGAN emulator. This is because the FDFDnet-2 emulator only divides the complex nonlinear impairment into two parts to train the transmission channel model, while the FDFDnet emulator divides the complex nonlinear impairment into three parts, thereby achieving high accurate modeling of the transmission channel. This indicates that the proposed FDFDnet emulator is feasible for use in modelling an OAM-MDM system channel with complex nonlinear impairment.

 

Fig. 7. NMSE and BER value for the signal amplitude versus ROP for (a) $l = 2$, (b) $l = 3$, (c) $l = 4$ in an OAM MDM IM/DD transmission system over a 10 km RCF.

Download Full Size | PDF

The complexity of the proposed FDFDnet emulator was also compared with that of the CGAN emulator. The number of real multiplications carried out by the CGAN emulator can be expressed by [19]

The complexity of the FDFDnet emulator can be expressed as [27]

The complexity of the two emulators with similar NMSE performance can then be calculated based on the above parameters for different values of the ROP. As shown in Tables 1, 2 and 3, compared with the conventional CGAN emulator, the maximal reductions in complexity achieved by the FDFDnet emulator are 59.5%, 50.4%, and 52.8%, respectively, at an ROP of −3dBm, for the three modes. This indicates that the proposed FDFDnet emulator has both high performance and low complexity.

Table 1. Experimental results for the complexity of the CGAN and FDFDnet emulators for l = 2

View Table | View all tables in this article

Table 2. Experimental results for the complexity of the CGAN and FDFD emulators for l = 3

View Table | View all tables in this article

Table 3. Experimental results for the complexity of the CGAN and FDFD emulators for l = 4

View Table | View all tables in this article

3.2.2 Performance validation of AE-based joint PS and equalization for OAM-MDM system

In these experiments, the AE-based joint PS and equalization scheme was applied to the OAM-MDM system with the parameters shown in Table 4. We present the BER results based on the AE implementation of the FDFDnet, FDFDnet-2 and CGAN emulators with different values of Vpp for the three OAM modes in Figs. 8, 9 and 10. As can be seen from Figs. 8(a), 9(a) and 10(a), with an increase in the Vpp, the BER values for the AE-based joint PS and equalization scheme for the FDFDnet, FDFDnet-2 and CGAN emulators all decreases at first and then increases. Figures 8(b), 9(b) and 10(b) show that the entropy of the PS for the FDFDnet, FDFDnet-2 and CGAN emulators also increases at first and then decreases as Vpp increases. This is because with an increase in Vpp, the nonlinear effect of the system first decreases and then increases.

 

Fig. 8. Measured values of BER versus Vpp for $l = 2$ in an IM-DD OAM-MDM transmission system over a 10 km RCF.

Download Full Size | PDF

 

Fig. 9. Measured values of BER versus Vpp for $l = 3$ in an IM-DD OAM-MDM transmission system over a 10 km RCF.

Download Full Size | PDF

 

Fig. 10. Measured values of BER versus Vpp for $l = 4$ in an IM-DD OAM-MDM transmission system over a 10 km RCF.

Download Full Size | PDF

Table 4. Parameters of the AE

View Table | View all tables in this article

The AE-based joint PS and equalization scheme for the FDFDnet emulator yields better BER performance than for the FDFDnet-2 and CGAN emulators for various values of Vpps. Furthermore, as shown in Fig. 11, the values of entropy achieved by the AE for the FDFDnet emulator are 4.4, 4.42, 4.39, 4.7, 4.65, 4.72, 4.59, 4.62, 4.56, which are lower than those of 4.47, 4.52, 4.46, 4.73, 4.71, 4.76, 4.64, 4.68, 4.66 obtained for the FDFDnet-2 emulator and also lower than those of 4.5, 4.56, 4.48, 4.75, 4.74, 4.77, 4.65, 4.7, 4.68 obtained for the CGAN emulator at the same Vpp for three OAM modes. This is because the FDFDnet-2 emulator and CGAN emulator build a hybrid model for complex nonlinear impairment at different frequencies in the OAM-MDM system. However, the FDFDnet emulator can build complex nonlinear model by separating signal into different frequency domain features, and can therefore simulate the OAM-MDM transmission channel model accurately.

 

Fig. 11. Entropy of the trained PS-based FDFDnet, FDFDnet-2 and CGAN emulators for CAP-32 versus VPP for (a) $l = 2$, (b) $l = 3$,(c) in an IM-DD OAM-MDM transmission system over a 10 km RCF.

Download Full Size | PDF

Figure 12 illustrates the BER performance for different OAM modes ($l = 2,3,4$) versus ROP for the proposed FDFDnet, FDFDnet-2 and CGAN emulators when the Vpp is 300 mV. The measured ROP ranges from −7dBm to 3dBm. It is clear that the proposed FDFDnet-based joint PS and equalization scheme outperforms the other methods. Compared with the CAP-32 baseline scheme, the FDFDnet-based joint PS and equalization scheme achieves improvements in the receiver sensitivity of 5.5 dB, 5.1 dB, 5.3 dB for different OAM modes ($l = 2,3,4$) at the 20% FEC limit, respectively. Compared with the FDFDnet-2-based joint PS and equalization scheme, the FDFDnet-based joint PS and equalization scheme achieves improvements in receiver sensitivity of 2 dB, 2.5 dB, 2.9 dB for different OAM modes ($l = 2,3,4$) at the 20% FEC limit, respectively. Compared with the CGAN-based scheme, the FDFDnet-based scheme yields improvements in receiver sensitivity of 3 dB, 2.5 dB, 2.2 dB for different OAM modes ($l = 2,3,4$) at the 20% FEC limit, respectively. These results validate the effectiveness of the proposed FDFDnet-based joint PS and equalization scheme for mitigating nonlinear impairment in OAM-MDM transmission systems.

 

Fig. 12. Measured values of BER versus ROP for (a) $l = 2$, (b) $l = 3$,(c) $l = 4$ in an IM-DD OAM-MDM transmission system over a 10 km RCF.

Download Full Size | PDF

4. Conclusion

This paper has proposed an E2E learning scheme with joint PS and equalization based on an FDFDnet emulator for an OAM-MDM IM/DD system. As the channel model of the damaged signal exhibits different characteristics at different frequencies, it is difficult for traditional channel emulators to build a mixed model for the transmission channel for the different frequencies of OAM-MDM transmission. However, our FDFDnet emulator can accurately build a complex nonlinear model of an OAM-MDM system by separating the signal into different frequency domain features. Our experimental results show that the proposed FDFDnet emulator outperforms the conventional CGAN emulator, with improvements in modelling accuracy of 30.8%, 26.3% and 31% for OAM modes $l = 2,3,4$, respectively. Furthermore, an AE-based joint PS and equalization scheme for the FDFDnet emulator has been presented. In terms of the receiver sensitivity, the proposed FDFDnet emulator with E2E scheme outperforms the CGAN emulator by 3, 2.5, 2.2 dBm, the FDFDnet-2 emulator by 2, 2.5, 2.9 dBm, and the real channel by 5.5, 5.1, and 5.3 dBm for the three OAM modes, respectively. These experimental results demonstrate that the proposed AE-based joint PS and equalization scheme with the FDFDnet emulator is a promising candidate for OAM-MDM optical fiber communication.