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Abstract
Orbital angular momentum (OAM) mode division multiplexing (MDM) has emerged as a new multiplexing technology that can significantly increase transmission capacity. In addition, probabilistic shaping (PS) is a well-established technique that can increase the transmission capacity of an optical fiber to close to the Shannon limit. However, both the mode coupling and the nonlinear impairment lead to a considerable gap between the OAM-MDM channel and the conventional additive white Gaussian noise (AWGN) channel, meaning that existing PS technology is not suitable for an OAM-MDM intensity-modulation direct-detection (IM-DD) system. In this paper, we propose a Bayesian generative adversarial network (BGAN) emulator based on an end-to-end (E2E) learning strategy with probabilistic shaping (PS) for an OAM-MDM IM/DD transmission with two modes. The weights and biases of the BGAN emulator are treated as a probability distribution, which can be accurately matched to the stochastic nonlinear model of OAM-MDM. Furthermore, a BGAN emulator based on an E2E learning strategy is proposed to find the optimal probability distribution of PS for an OAM-MDM IM/DD system. An experiment was conducted on a 200 Gbit/s two OAM modes carrierless amplitude phase-32(CAP-32) signal over a 5 km ring-core fiber transmission, and the results showed that the proposed BGAN emulator outperformed a conventional CGAN emulator, with improvements in modelling accuracy of 29.3% and 26.3% for the two OAM modes, respectively. Moreover, the generalized mutual information (GMI) of the proposed E2E learning strategy outperformed the conventional MB distribution and the CGAN emulator by 0.31 and 0.33 bits/symbol and 0.16 and 0.2 bits/symbol for the two OAM modes, respectively. Our experimental results demonstrate that the proposed E2E learning strategy with the BGAN emulator is a promising candidate for OAM-MDM IM/DD optic fiber communication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the explosive growth of global Internet traffic has imposed high-capacity requirements on IM/DD transmission for short-reach optical interconnections [1,2]. In order to overcome the capacity bottleneck of single-mode fiber (SMF), orbital angular momentum (OAM) mode division multiplexing (MDM) was developed as a new multiplexing technology [3–7]. Typical schemes for OAM-MDM IM/DD transmission are based on the use of ring-core fiber (RCF), and include a 400-Gbit/s OAM-MDM transmission scheme using PAM modulation, a 186.4-Gbit/s OAM-MDM transmission scheme with DMT modulation, and a 2.5-Gbaud 3D CAP-8 architecture for OAM-MDM transmission [8–10]. RCF-based OAM-MDM IM/DD communication systems have achieved ultra-high capacity transmission when used in optical interconnections.
Probabilistic shaping (PS) is a well-established technique in optical communication that can raise the transmission capacity to close to the Shannon limit without increasing the transmission power and complexity of the system [11]. However, the probability distribution in conventional PS technology always obeys a Maxwell-Boltzmann (MB) distribution according to the additive white Gaussian noise (AWGN) channel with a SMF. The mode coupling between different OAM modes significantly affects the channel model of the OAM-MDM, and in short-reach OAM-MDM IM/DD systems, nonlinear impairments are significantly increased due to the insufficient bandwidth of electro-optical devices such as electrical amplifiers (EAs), electro-optic modulators, and spatial light modulators (SLMs). In particular, the nonlinear effects of the SLM may be generated through several physical phenomena, such as self-phase modulation and memory effects, making it difficult to create an accurate model. Both the mode coupling and the nonlinear impairment lead to a considerable gap between the OAM-MDM channel and the conventional AWGN channel. Existing PS technology is therefore unsuitable for an OAM-MDM IM/DD system.
The end-to-end (E2E) learning strategy is a machine learning technique based on an autoencoder (AE) that provides a way of automatically tailoring signal modulation to arbitrary communication links [12]. Stark et al. were the first to present an E2E solution to achieve the optimal probability distribution of the AWGN channel [13]. Aref et al. proposed a novel AE-based learning of PS for coded-modulation systems [14]. An accurate channel model of a real optical fiber communication system is difficult to create, and deep-learning-based E2E frameworks are usually adopted to give an accurate channel model, such as CGANs [15,16], DNNs [17] or heterogeneous neural networks [18,19]. However, the channel model of the test signal is greatly changed due to the random mode coupling in time-variant OAM-MDM transmission. Channel modelling strategies based on conventional machine learning assign weight coefficients with fixed values based on the training signal, and as a result, it is difficult to use these strategies to build a stochastic channel model for an OAM-MDM IM/DD system.
In this paper, we propose a scheme based on a Bayesian generative adversarial network (BGAN) and E2E learning for the PS in an OAM-MDM IM/DD system. Our BGAN-based E2E learning scheme can accurately fit a stochastic nonlinear model of an OAM-MDM system by treating the weights and biases as probability distribution. Secondly, we use a BGAN-based E2E strategy to learn the optimal probability distribution based on the carrierless amplitude phase (CAP) modulation format. An experiment is carried out to verify the effectiveness of the proposed method, and the results demonstrate that the proposed strategy can be used to build an accurate channel model of the OAM-MDM, and significantly improves the performance of the PS in a time-variant OAM-MDM IM/DD system compared with the conventional MB distribution.
2. End-to-end deep learning of probabilistic shaping for OAM-MDM system using BGAN emulator
In traditional SMF optic communication systems, PS is commonly implemented using an MB distribution, which is optimal for the AWGN channel. However, in an OAM-MDM system, there is a significant gap between the OAM-MDM channel and the AWGN channel. As a result, existing PS techniques are not suitable for OAM-MDM system. In order to achieve an accurate channel response for an OAM-MDM system, a novel E2E learning strategy based on a BGAN emulator is proposed for OAM-MDM transmission with CAP-32 modulation. The E2E learning scheme considered in this paper is shown in Fig. 1.
Fig. 1. Diagram showing the proposed E2E learning strategy based on a BGAN emulator for an OAM-MDM system: (a) actual OAM-MDM system; (b) proposed OAM-MDM system based on AE.
Figure 1(a) shows an OAM-MDM system with CAP-32 modulation. In the proposed E2E learning scheme, the whole OAM-MDM system is implemented as an AE, including the transmitter, a channel based on an BGAN emulator, and the receiver, which enables joint training of the transmitter and receiver. The AE encoder achieves an optimal probability distribution ${P_M}$. The network parameters of the BGAN emulator are treated as a probability distribution rather than being fixed constants, meaning that the BGAN emulator can accurately model the channel response of the OAM-MDM system even though the received signal is subject to a strong stochastic nonlinear impairment. Then, as shown in Fig. 1(b), PS based on the BGAN emulator is achieved through E2E learning, thereby enhancing the performance of the OAM-MDM system.
2.1. BGAN-based channel modelling strategy for an OAM-MDM system
For a given transmitted signal originally in the OAM-MDM system, the received signal can be mathematically represented as
where n denotes the order of the CAP-32 symbol. $x(n) = [{{x_1},{x_2}, \cdot{\cdot} \cdot {x_n}} ],{x_n} \in ( - 1,1)$ denotes the normalised transmitted CAP-32 symbol sequence that has been upsampled with a factor of $m$. Hence, each symbol ${x_i}(i = 1,2, \ldots n)$ consists of m samples, and can be expressed as ${x_i} = [x_i^1,x_i^2 \cdots x_i^m]$. $noise(n)$ denotes the additive noise generated in the high-speed OAM-MDM system. $y(n) = [{{y_1},{y_2}, \cdot{\cdot} \cdot {y_n}} ]$ represents the received CAP-32 symbol sequence, where each symbol ${y_i}(i = 1,2, \ldots n)$ is also composed of m samples, that is, ${y_i} = [y_i^1,y_i^2 \cdots y_i^m]$. $H$ represents the complex channel response of the OAM-MDM system. Each transmitted symbol $x(n)$ will be input to the BGAN emulator to simulate the distorted symbol $\widehat y(n)$.Before emulating the channel response in the OAM-MDM, the transmitted and received CAP-32 signals are preprocessed as shown in Fig. 2. The current symbol ${x_i}$ is wrapped with its l preceding and l succeeding symbols to form a condition vector ${X_i} = [{{x_{i - l}} \cdot{\cdot} \cdot ,{x_{i - l}} \cdot{\cdot} \cdot {x_{i - 1}},{x_i},{x_{i + 1}} \cdot{\cdot} \cdot {x_{i + l}}} ]$. The transmitted time-series signal $x(n) = [{{x_1},{x_2}, \cdot{\cdot} \cdot {x_n}} ]$ is transformed into a matrix composed of $(n - 2l)$ condition vectors with a length of $(2l + 1)m$. The condition vector matrix can be expressed as
The received signal ${y_i}$ corresponding to the condition vector ${X_i}(i = (n - 2l))$ is taken as real data, and can be expressed as
The condition vector X and real data Y are combined to form the dataset $Q = \{{X,Y} \}$, which is then divided into two parts to give the training dataset ${Q_T} = \{{{X_T},{Y_T}} \}$ and the testing dataset ${Q_P} = \{{{X_P},{Y_P}} \}$, 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 ${Y_P} = [{{Y_1},{Y_2} \cdots {Y_P}} ]$.
The structure of the BGAN emulator is shown in Fig. 3(a). It consists of a generator and a discriminator, both of which comprise an input layer, a hidden layer, and an output layer. The generator models the channel of the OAM-MDM system to generate the emulated signal, while the discriminator compares the emulated signal with the real data. The BGAN emulator is trained using the training dataset ${Q_T}$ to calculate the probability distribution of the parameters (weights and bias). The probability distributions of the generator and discriminator parameters are updated iteratively through adversarial training until the emulated signal matches the real data.
Fig. 3. Schematic diagram of the BGAN emulator: (a) structure of the BGAN emulator; (b) training of the model parameters.
First, in the input layer, a set of standard normal distribution noise vectors z is randomly generated. The noise vectors enable the generator to generate diverse data to match the noise distribution in the channel. The condition vector ${X_i}$ and the noise vector z are then concatenated to form the input vector $({{X_i} + z} )$ for the generator.
Next, the hidden layer is formed from two fully connected layers. All the weights and biases for the generator are initialized to follow a standard normal distribution with mean 0 and standard deviation 1, as shown in Eq. (4):
In the next step, the real data ${Y_i}$ and the corresponding emulated signal ${\widehat y_i}$ are concatenated with the condition vector ${X_i}$, to serve as the input vector for the discriminator.
The hidden layer is also formed using two fully connected layers. All the weights and biases for the discriminator are initialized to follow a standard normal distribution with mean 0 and standard deviation 1, as shown in Eq. (6):
The likelihood probabilities of the parameter for the discriminator are then calculated using the hidden layer and output layer, and can be expressed as
The output P of the discriminator is a probability score indicating the “realness” of the emulated signal, which can be expressed as
Since the parameters of the BGAN emulator are determined according to the stochastic nonlinearity for the OAM-MDM, the purpose of training the BGAN emulator is to find the posterior distribution of the model parameters $p(\theta |{Q_T}),\theta = \{{{\theta_g},{\theta_d}} \}$ through the Bayesian formula, which can be expressed as
where $p({Q_T})$ denotes the marginal likelihood. The prior and likelihood probabilities can be obtained using Eqs. (4)–8. However, $p({Q_T})$ represents all possible values of parameter $\theta$, which is difficult to obtain. Hence, the variation inference is used as an analytical approximation technique to learn the posterior distribution $p(\theta |{Q_T})$ based on the prior and likelihood probabilities of the BGAN emulator parameters [20]. As shown in Fig. 3(b), variation inference can approximate the posterior distributions $p({\theta _g}|{Q_T})$ and $p({\theta _d}|{Q_T})$ by using the Gaussian distributions ${q_{{\omega _g}}}({{\theta_g}} )$ and ${q_{{\omega _d}}}({{\theta_d}} )$ with variation parameters ${\omega _g} = ({\mu ,\sigma } )$ and ${\omega _d} = ({\mu^{\prime},\sigma^{\prime}} )$. First, ${q_{{\omega _g}}}({{\theta_g}} )$ and ${q_{{\omega _d}}}({{\theta_d}} )$ are initialized as standard normal distributions, and variation learning is then used to optimise the parameters ${\omega _g} = ({\mu ,\sigma } )$ and ${\omega _d} = ({\mu^{\prime},\sigma^{\prime}} )$ for ${q_{{\omega _g}}}({{\theta_g}} )$ and ${q_{{\omega _d}}}({{\theta_d}} )$ by minimising the Kullback-Leibler divergence with respect to the posterior distributions $p({\theta _g}|{Q_T})$ and $p({\theta _d}|{Q_T})$ [21], as follows:After the variation learning, ${q_{{\omega _g}^ \ast }}({{\theta_g}} )\approx p({\theta _g}|{Q_T})$ and ${q_{{\omega _d}^ \ast }}({{\theta_d}} )\approx p({\theta _d}|{Q_T})$, indicating that the distribution of parameters of the BGAN emulator is optimised and matches the probability distribution of the stochastic nonlinearity for the OAM-MDM.
The combination of stochastic mode coupling in OAM-MDM with optoelectronic devices typically results in strong stochastic nonlinear damage, which poses a challenge for traditional channel emulators with fixed parameters. However, the parameters of the BGAN emulator follow a probability distribution rather than having fixed values. By leveraging the variational inference and updating the posterior probability distribution of the parameters using prior and likelihood probabilities, the stochastic nonlinear features of the OAM-MDM CAP-32 signal can be accurately extracted from the probability distribution of the parameters. Hence, our BGAN emulator can obtain an accurate model from the posterior probability distribution of each parameter, thus forming an effective OAM-MDM transmission channel emulator for stochastic nonlinearity.
2.2. Probabilistic shaping for OAM-MDM system based on AE
After obtaining the BGAN emulator, the gradient can process from the AE decoder to the AE encoder. In this way, the proposed E2E AE scheme with PS for an OAM-MDM system with CAP-32 modulation is implemented.
Our E2E AE scheme based on PS for an OAM-MDM system with CAP-32 modulation is depicted in Fig. 4. It consists of an encoder, a mapper, CAP modulation, a BGAN emulator, CAP demodulation, and a decoder. The encoder is composed of two fully connected layers, while the decoder is composed of three fully connected layers.
The constant t is first fed into the AE encoder, as shown in Fig. 4(a). The probability ${P_M}$ of the constellation point is calculated by the two fully connected layers of the encoder, and can be expressed as
Next, as shown in Fig. 4(b), in order to generate a symbol sequence with probability ${P_M}$, we apply Algorithm 2 from [22] with batch size N. We convert the symbol sequence to their one-hot representations to get one-hot vector matrix V with size $N \times M$ [14]. The one-hot vector matrix can be expressed as
Then, to keep the constellation power constant, the constellation point ${C_M} = \{{{c_1}, \ldots {c_M}} \}$ is normalised based on the probability ${P_M} = \{{{p_1}, \ldots ,{p_M}} \}$, which can be expressed as
where ${c_m}$($m = 1, \ldots M$) represents the constellation point, ${\hat{c}_m}$ represents the normalised constellation point, and ${\hat{C}_M} = \{{{{\hat{c}}_1}, \ldots {{\hat{c}}_M}} \}$. The one-hot vector matrix V is then multiplied by ${\hat{C}_M}$ to obtain the shaped symbol sequence $K$:Next, as shown in Fig. 4(c), the shaped symbol sequence $K = \{{{k_1}, \ldots {k_N}} \}$ is upsampled and filtered using two shaping filters ${f_1}$, ${f_2}$. It is then summed to generate a CAP signal $X = \{{{x_1}, \ldots {x_N}} \}$ to complete the CAP modulation.
As shown in Fig. 4(d), the CAP signal $X = \{{{x_1}, \ldots {x_N}} \}$ is then fed into the BGAN emulator to generate a received signal $Y = \{{{Y_1}, \ldots {Y_N}} \}$, which contains stochastic nonlinear impairment in the OAM-MDM channel.
At the receiver, the received signal $Y = \{{{Y_1}, \ldots {Y_N}} \}$ is sent to the matched filters ${g_1}$, ${g_2}$ and then downsampled to generate the CAP demodulated signal $E = \{{{E_1}, \ldots {E_N}} \}$, as shown in Fig. 4(e).
Finally, as shown in Fig. 4(f), the decoder maps the CAP demodulated signal $E = \{{{E_1}, \ldots {E_N}} \}$ to a probability vector ${R_j}(j = 1, \ldots N)$, which can be expressed as
When the probability vector $R = {\{{{R_1}, \ldots ,{R_N}} \}^T}$ has been obtained, the loss function is calculated based on R, V and ${P_M}$, and can be expressed as [13]
where $H({{P_M}} )$ is the entropy of ${P_M}$. The E2E AE uses L to calculate the loss for the training process, and the Adam gradient descent function is applied to update the model parameters $({W_1}^{\prime \prime },{b_1}^{\prime \prime },{W_2}^{\prime \prime },{b_2}^{\prime \prime },{W_1}^{\prime \prime \prime },{b_1}^{\prime \prime \prime },{W_2}^{\prime \prime \prime },{b_2}^{\prime \prime \prime },{W_3}^{\prime \prime \prime },{b_3}^{\prime \prime \prime })$ using backpropagation. The optimal probability distribution ${P_M}$ is found when the loss function L converges. In this way, the AE can obtain optimal PS for the OAM-MDM system through the use of a BGAN emulator.3. Experimental
3.1. Experimental setup
In order to verify the effectiveness of the proposed BGAN emulator, an experiment was carried out on a 200 Gbit/s OAM-MDM IM/DD transmission with two OAM modes over a 5 km RCF, as shown in Fig. 5.
At the transmitter, a pseudo-random bit sequence with the length of ${2^{18}}$ was generated. The bit data were converted into CAP-32 symbols following the constellation mapping, and then filtered using shaping filters and summed to generate CAP-32 signals. The electrical signal was generated using an arbitrary waveform generator (AWG). Every 11 adjacent symbols were input to the BGAN emulator as the condition vector ${X_i}$. In order to ensure that the emulator learns the nonlinear characteristics of the channel, rather than the generated pseudo-random sequence, the different random number seeds are utilized for the training data and the test data. This difference generates two data sequences with different characteristic, which guarantees that the proposed emulator learns the nonlinear characteristics of the channel, rather than the random sequence. Following electrical amplification (EA), the electric signal was utilised to modulate an optical carrier with a wavelength of $1550.12$ nm through a Mach-Zehnder Modulator (MZM), generating a double-sideband optical signal. Subsequently, an optical coupler (OC) was employed to split the generated optical signal into two branches, which were then amplified using an erbium-doped fiber amplifier (EDFA). One of these branches was delayed for data mode decorrelation using a 10 m SMF. Two polarisation controllers (PCs) were used to adjust the direction of polarisation of the signal light. The two light beams were transmitted to an SLM through the collimators (Col.), and the linear polarizers (LP). The two Gaussian light beams were converted into OAM beams with $l = 2$ and $l = 3$ through the two vortex patterns of the SLMs. These two OAM beams were then combined into a single beam by employing a polarisation beam splitter (PBS). In view of the transmission characteristics of the RCF, the multiplexed OAM beam was coupled to a 5 km RCF for transmission through a quarter-wave plate (QWP).
In Fig. 5, the insets (i) show the cross section of the RCF. The intensity profiles of the two OAM modes $l = 2$ and $l = 3$ after 5 km RCF transmission are shown in (ii) and (iii), respectively. At the receiving side, the multiplexed OAM beams were split into two using a beam splitter (BS), and were converted into linearly polarised beams using two QWPs. The two beams were then passed through the vortex phase plate (VPP), which converted them into Gauss beams. These two Gauss beams were coupled to SMFs via collimators and converted into electronic signals via two PDs. The electronic waveform was recorded using a real-time oscilloscope. We then sampled the electronic signal using the real-time oscilloscope to obtain the corresponding digital signal, which underwent sequential resampling, low-pass filtering and synchronisation before being sent to the BGAN emulator as the real data ${Y_i}$. In order to compare the performance between the BGAN and CGAN emulators, the digital signal was also sent to the conventional CGAN emulator via the same process.
The condition vector ${X_i}$ and real data ${Y_i}$ were used as a dataset for training the BGAN and CGAN emulators. The trained emulators were then integrated with the AE to obtain the optimal probability distribution for the OAM-MDM system. The original input bit sequence was mapped using a distribution matcher to produce a symbol sequence with the learned optimal probability distribution [23]. The symbol sequence was modulated to generate a CAP electrical signal using the AWG, while the electrical signal was modulated by an optical carrier with the MZM and transmitted to the OAM-MDM system, as shown in Fig. 5. The GMI and BER for the BGAN and CGAN emulators were compared to verify the performance of the proposed scheme.
3.2. Experimental results and analysis
3.2.1 Performance of the BGAN emulator for the OAM-MDM system
Figures 6(a) and (b) show the signal waveform in the time and frequency domains after transmission via the real OAM-MDM system channel in different OAM modes, the CGAN emulator, and the BGAN emulator at a received optical power (ROP) of 1dBm. Compared with the conventional CGAN emulator, the output signal waveform of the BGAN emulator shows higher consistency with the real channel output in both the time and frequency domains. In order to quantitatively represent the channel modelling effect of the OAM-MDM system using the BGAN and CGAN emulators, we adopt the normalised MSE (NMSE) of the power, following the method in [24]. This quantity can be expressed as
where k is the data length, y is the real OAM-MDM system channel output data after synchronisation, and $\hat{y}$ is the data generated by the emulator. Figures 7(a) and (b) show the NMSE for the signal generated by the CGAN and BGAN emulators at different ROPs in different OAM modes $\left( {l = \langle 2,3\rangle } \right)$. Compared with the conventional CGAN emulator, the maximum improvements in modelling accuracy of the BGAN emulator are 29.3% and 26.3% at an ROP of −3dBm for the two modes, respectively. This result verifies the effectiveness of the proposed BGAN emulator in terms of characterising the stochastic nonlinear impairment in an OAM-MDM system based on a probability distribution for the parameters.Fig. 6. Channel output waveforms in the time and frequency domains based on OAM-MDM system channel, CGAN emulator and BGAN emulator for (a) $l = 2$, (b) $l = 3$.
Fig. 7. NMSEs of signal amplitude versus ROP for (a) $l = 2$, (b) $l = 3$ in an OAM MDM IM/DD transmission system over a 5 km RCF.
To further validate the effectiveness of the BGAN emulator for the OAM-MDM system channel, we experimentally compared the BER performance of the BGAN and CGAN emulators under different ROPs of the CAP-32 transmitted signal, as shown in Fig. 8. The average BER errors between the signal generated by the BGAN emulator and the real received signal were only 9% and 8.9% for $l = 2$ and $l = 3$, respectively; these were much lower than the values of 23.5% and 21.9% found for the error between the signal generated by the CGAN emulator and the real received signal. This indicates that the proposed BGAN emulator is feasible for use in modelling the OAM-MDM system channel with stochastic nonlinear impairment.
Fig. 8. BER for CAP32 versus ROP for (a) $l = 2$, (b) $l = 3$ in an OAM-MDM IM/DD transmission system over a 5 km RCF.
To further validate the robustness of the BGAN emulator for the OAM-MDM system channel, we experimentally compared the NMSE performance of the fixed BGAN and retrained BGAN emulator under different voltage peak-to-peak (VPPs). In the fixed BGAN, the BGAN emulator is trained only under the VPPs of 350 mV, and tested under different VPPs transmitted signal with the range from 200 mV to 450 mV. In the retrained BGAN, the BGAN emulator is trained under the VPP from 200 mV to 450 mV, and tested under different VPPs transmitted signal with the same range. As shown in Fig. 9, the average NMSE errors between the fixed BGAN and retrained BGAN emulator were only 6.1% and 7.2% for $l = 2$ and $l = 3$, respectively. This indicates that the fixed BGAN emulator has good robustness and can adapt to different signal condition.
Fig. 9. The performance of the fixed trained BGAN emulator under different Vpps for (a) $l = 2$, (b) $l = 3$ in an OAM-MDM IM/DD transmission system over a 5 km RCF.
The complexity of both the proposed BGAN emulator and the CGAN emulator was also analysed. In terms of the number of real multiplications, the complexity of the BGAN emulator can be expressed as [25]:
where ${d_1}$ denotes the length of the condition vector, and ${d_2}$ denotes the length of the noise vector. ${h_1}$ denotes the number of neurons in the first hidden layer, ${h_2}$ is the number of neurons in the second hidden layer, and ${d_3}$ is the length of the generated data.The number of real multiplications for the CGAN emulator can be expressed as:
where ${k_1}$ denotes the number of neurons for the first hidden layer, ${k_2}$ denotes the number of neurons for the second hidden layer. ${d_1}$, ${d_2}$, and ${d_3}$ are the same as BGAN emulator.The complexity of the proposed BGAN emulator and the CGAN emulator was evaluated with the two OAM modes at different values of the ROP. The complexity of the two emulators with similar NMSE performance is presented in Tables 1 and 2. Compared with the conventional CGAN emulator, the maximal reductions in complexity achieved by the BGAN emulator are 36.3% and 33.7%, respectively, at an ROP of −3dBm for the two modes. We can conclude that the proposed BGAN emulator has both high performance and low complexity.
Table 1. Experimental results for the complexity of the CGAN and BGAN emulators for l = 2
Table 2. Experimental results for the complexity of the CGAN and BGAN emulators for l = 3
3.2.2 Performance validation of AE-based PS for OAM-MDM system
In this section, we report the use of an AE to perform PS for the OAM-MDM system. The parameters of the AE are shown in Table 3. The PS results based on the AE implementation of the BGAN and CGAN emulators with different input optical powers (IOPs) for the two OAM modes are shown in Figs. 10 and 11. With an increase in the IOP, the entropy of the AE-based PS for both the CGAN and BGAN emulators first increases and then decreases. This is because with an increase in IOP, the signal-to-noise ratio (SNR) of the signal increases at first, and the nonlinear effect of the system decreases. As the IOP continues to increase, the nonlinear effect increases, which results in a higher probability of the central constellation points being obtained based on the AE. Furthermore, as shown in Fig. 10, the values of entropy achieved by the AE for the BGAN emulator are 4.4, 4.42, 4.7, 4.65, 4.59, and 4.56, which are lower than those of 4.47, 4.52, 4.73, 4.71, 4.64, and 4.66 obtained for CGAN emulator at the same IOPs for two OAM modes. This is because it is hard for the conventional CGAN emulator with fixed parameters to simulate the stochastic nonlinear impairment caused by mode coupling in the OAM-MDM system. However, the BGAN emulator can model the probability distribution of the stochastic nonlinear impairment statistically, and can therefore simulate the OAM-MDM transmission channel model accurately. The BGAN emulator captures more complex channel impairments and therefore the values of entropy achieved by the AE for the BGAN emulator are lower than CGAN emulator at the same IOP.
Fig. 10. Learned PS for the (a)-(c), (g)-(i) BGAN and (d)-(f), (j)-(l) CGAN emulators versus IOP for (a)-(f) $l = 2$, (g)-(l) $l = 3$ in the IM-DD OAM-MDM transmission system over a 5 km RCF.
Fig. 11. Entropy of learned PS-based BGAN and CGAN emulators for CAP-32 versus IOP for (a) $l = 2$, (b) $l = 3$ in an IM-DD OAM-MDM transmission system over a 5 km RCF.
Table 3. Parameters of the AE
Figure 12 shows the measured GMI versus ROP for different OAM modes in the OAM-MDM IM-DD transmission system over a 5 km RCF. Compared with OAM-MDM transmission without PS, the GMIs of the AE-based PS with the BGAN emulator are increased by a maximum of 0.5 and 0.55 bits/symbol at an ROP of −3dBm for OAM modes $l = 2$ and $l = 3$, respectively. Compared with the MB distribution with the same entropy, the GMIs for the AE-based PS with BGAN emulator are increased by a maximum of 0.31 and 0.33 bits/symbol at an ROP of −3dBm for the two OAM modes, respectively. Compared with AE-based PS with the CGAN emulator, the GMIs for the AE-based PS with the BGAN emulator are increased by a maximum of 0.16 and 0.2 bits/symbol at ROP of −3dBm for the two OAM modes. This result verifies that the proposed AE-based BGAN emulator can achieve an optimal PS that matches the stochastic nonlinear impairment in the OAM-MDM system.
Fig. 12. Measured GMI versus ROP for (a) $l = 2$, (b) $l = 3$ in the IM-DD OAM-MDM transmission system over a 5 km RCF.
4. Conclusion
This paper has proposed an E2E learning scheme with PS based on a BGAN emulator for an OAM-MDM IM/DD system. Due to the random mode coupling in time-varying OAM-MDM transmission, it is difficult for traditional channel emulators to capture the stochastic nonlinear impairment with fixed weight coefficients in OAM-MDM transmission. However, the weights and biases of the BGAN emulator are regarded here as a probability distribution, and can accurately match the stochastic nonlinear model of OAM-MDM. Our experimental results show that the proposed BGAN emulator outperforms the conventional CGAN emulator, with improvements in modelling accuracy of 29.3% and 26.3% for OAM modes $l = 2$ and $l = 3$, respectively. Furthermore, an AE-based PS scheme for the BGAN emulator is presented. In terms of the GMI, the proposed BGAN emulator with AE-based PS outperforms the model without PS, the conventional MB distribution, and the CGAN emulator, by 0.5 and 0.55, 0.31 and 0.33, 0.16 and 0.2 bits/symbol for the two OAM modes, respectively. These experimental results demonstrate that the proposed AE-based PS scheme with the BGAN emulator is a promising candidate for OAM-MDM IM/DD optic fiber communication.
Funding
National Science Fund for Distinguished Young Scholars (62022016); National Key Research and Development Program of China (2019YFA0706300); National Natural Science Foundation of China (62105026, 62205023); Fundamental Research Funds for the Central Universities, Beijing Municipal Natural Science Foundation (4222075); BIT Research and Innovation Promoting Project (2023YCXY028).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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