## 2 The Parallel Reservoir Computing
### Proposed Scheme of Optoelectronic Reservoir Computing  
The internal dynamics of the optoelectronic reservoir are enriched by the proposed parallel optoelectronic RC scheme using optical polarization multiplexing methodology, supported by a novel double-loop scheme with a dual-polarization Mach-Zehnder modulator (DPol-MZM). The structure is shown in Figure 2.  
A continuous-wave laser is used as the input of the DPol-MZM. The optical carrier passes through the optical polarization controller (PC) into two orthogonal beams. It enters  
Figure 1: Schematic diagram of RC. \(W_{in}\) is the input connection weight matrix, \(W_{res}\) is the internal connection weight matrix, \(W_{out}\) is the output connection weight matrix, \(\tau\) is the delay time, \(x(n)\) is the state function, and NL is the nonlinear node.  
Figure 2: DPol-MZM-based double-loop reservoir structure diagram. LD: laser diode; PC: polarization controller; DPol-MZM: dual-polarization Mach–Zehnder modulator; OA: optical attenuator; PD: photodetector; AMP: amplifier; BPF: bandpass filter.  
into two arms of the DPol-MZM (e.g., Fujitsu FTM7980), and then the intensities of these two optical carriers are modulated by the voltage related to the input signal. The state function \(x(n)\) is collected and used as the output of RC. Then, it is injected into two feedback loops with different delays and different optical attenuation coefficients. The up and down loops have various delays corresponding to \(\tau_{1}\) and \(\tau_{2}\) in the figure. The virtual node interval \(\theta\) is equal, so the number of virtual nodes \(N\) between the two loops is different. By setting the delays in the two loops differently, the virtual nodes of the two loops are non-symmetrically coupled, and the internal dynamics of the reservoir are enriched. Setting different light attenuation coefficients also enriches the internal dynamics. Next, the optical signals of these two loops are converted into electrical signals by photodetectors (PDs), and they are combined with the pre-processed input signals to feedback to the two arms of the DPol-MZM, forming a parallel optoelectronic RC scheme with two delay feedback loops. The combiner is actually the coupler in the implementation of the scheme. Here, the polarization is not controlled in the loop, and the optical polarization orthogonality is controlled inside the DPol-MZM.  
A delay differential equation model is used to analyze the system. The dynamic equation is shown as follows:  
\[\frac{dx(n)}{d\ t}=\frac{1}{\tau_{L}}\left\{\begin{array}{l}-x(n)-\frac{1}{ \tau_{H}}y(n)+\\ \beta\left[\begin{array}{l}\cos^{2}(\pi\frac{a_{1}x(n-\tau_{1})+\tau_{2}y_{ \pi}*u(n)+\rho_{1}}{2\
u_{\pi}})+\\ \cos^{2}(\pi\frac{a_{2}x(n-\tau_{2})+\tau_{3}*W_{in}*u(n)+\rho_{2}}{2\
u_{\pi} })\end{array}\right]\end{array}\right\} \tag{2}\]  
\[\frac{dy(n)}{d\ t}=x(n) \tag{3}\]  
where \(x(n)\) is the state function, \(\tau_{H}\) is the time constant of the high-pass filtering effect of the loop, \(\tau_{L}\) is the time constant of the low-pass filtering effect of the loop, and \(\beta\) is the normalized feedback coefficient. The function of \(a_{1,2}\) is equivalent to the internal connection weight matrix of the neural network, \(\tau_{1,2}\) is the two-loop delays, \(\gamma\) is the scaling parameter for the input signal, \(W_{in}\) is the input connection weight matrix, \(u(n)\) is the input signal, \(\varphi_{1,2}\) is the bias voltage of the two arms of DPol-MZM, and \(V_{\pi}\) is the half-wave voltage of the modulator. The typical values of the fixed parameters in the system are presented in Table 1. Additional parameters are optimized in the numerical simulations to optimize the nonlinear channel equalization performance.