# Design and Research of Photonic Reservoir Computing for Optical Channel Equalization
## 2 Design of Photonic Reservoir Computing and Verification System  
_Photonic Reservoir Computing Structure_  
The PhRC architecture is divided into three layers: the input layer, the reservoir layer, and the readout layer. Each of these layers can be fabricated using complementary metal oxide semiconductor (CMOS) technology on SOI materials. Since the performance of the optical readout layer is similar to that of an optoelectronic readout layer equipped with ridge regression algorithms, in order to reduce the overall simulation time, all four PhRC structures in this paper adopt optoelectronic readout layers, with only the input layer and the reservoir layer implemented using photonic integrated devices. The rectangular reservoir layer structures with 12 nodes, 24 nodes, 36 nodes, and 48 nodes are shown in Figure 1a, b, c, and d, respectively. In the simulation, the input optical signal is a discrete time signal, and the state update equation between the input layer and the reservoir layer is as follows:  
\[x(t+1)=W^{in}u(t+1)+Wx(t) \tag{1}\]  
where \(x(\cdot)\) and \(u(\cdot)\) are vectors, representing the reservoir layer status and input layer status, respectively. \(W^{in}\) and \(W\) are matrices, representing the weights of input layer and output layer, respectively. \(t\) and \(t+1\) represent the current moment and the next moment, respectively. The state update equation between the reservoir layer and the readout layer is as follows:  
\[y(t+1)=f^{out}(W^{out}x(t+1)) \tag{2}\]  
where \(y(\cdot)\) is a vector, which represents the output signal of the output layer, \(W^{out}\) is a matrix, which represents the weight of the readout layer. \(f^{out}(\cdot)\) is the activation function. Taking the 12-node layout as an example, the overall PhRC structure is shown in Figure 2. In the input layer, the distorted optical signal output from the fiber enters the on-chip waveguide via a grating coupler (GC), and then it is equally divided into seven parts by a \(1\times 7\) MMI to enter the reservoir layer. The transmission relationship of the input layer is as follows:  
\[\left[E_{il\_out1}\quad E_{il\_out2}\quad\ldots\quad E_{il\_out2}\right]^{T}= \frac{1}{\sqrt{7}}10^{-0.1(L_{CC}+L_{MMI})}E_{input} \tag{3}\]  
Figure 1: (**a**) Twelve nodes, (**b**) 24 nodes, (**c**) 36 nodes and (**d**) 48 nodes in reservoir layer layouts. where \(E_{il\_out1}\), \(E_{il\_out2}\), \(\ldots\), \(E_{il\_out7}\) are electric fields of the seven optical signals output from the input layer. \(E_{input}\) is the electric field of the distorted signal. \(L_{GC}\) and \(L_{MMI}\) represent the insertion losses of GC and MMI, respectively, with the unit being dB. In the reservoir layer, DCs and waveguide delay lines represent nodes and delay elements, respectively. To minimize the use of waveguide crossing devices within the reservoir layer, both the input and output positions are placed at the outermost nodes. The number of input and output ports in the reservoir layer is seven each, and their transmission relationship is as follows:  
\[\left[E_{re\_out1}\quad E_{re\_out2}\quad\ldots\quad E_{re\_out7}\right]^{T}=S _{re}\left[E_{il\_out1}\quad E_{il\_out2}\quad\ldots\quad E_{il\_out7}\right]^{T} \tag{4}\]  
where \(E_{re\_out1}\), \(E_{re\_out2}\), \(\ldots\), \(E_{re\_out7}\) are the electric fields of the seven optical signals' output from the reservoir layer. Due to the large number of devices in the reservoir, representing them in mathematical form would be relatively complex. Therefore, the transmission matrix is denoted by \(S_{re}\). The length of the waveguide delay line in the reservoir layer is represented by a length coefficient, which is the ratio of the duration of signal transmission on the waveguide delay line to the time interval of symbols, as shown in the following equation:  
\[rid=\frac{\tau}{T_{symbol}} \tag{5}\]  
where \(\tau\) is delay time of the waveguide delay line. \(T_{symbol}\) represents one symbol period.