# Numerical Simulation of an InP Photonic Integrated Cross-Connect for Deep Neural Networks on [PERSON] Shi
## 2 Photonic Deep Neural Network with Weight-SOAs  
The implementation of deep neural networks via a photonic approach takes advantage of the available parallelism of light beams. Figure 1 depicts the envisaged photonic deep neural network which uses wavelength division multiplexing (WDM) input signals, from the photonic neuron to the large-scale neural network. Here in particular, we realize multiple weighted additions, linear operations in an artificial neuron, via a broadcast-and-weight architecture, which are the most computational heavy elements in the neural network. The basic element of the neural network is an artificial neuron. Figure 1a depicts the basic neuron model with the output signal being \(y_{j}=f(\sum W_{ij}\ x_{i}+b_{j})\), where \(f\) is the activation function, \(x_{i}\) is the \(i\)th element of the input vector, \(W_{ij}\) is the weight factor for the input value \(x_{i}\) and \(b_{j}\) is the bias in the \(j\)th neuron, with the weighted addition given by \(\sum W_{ij}x_{i}\). The output of one full layer of M neurons can be expressed as a vector: \(y=f(W\cdot x+b)\), where \(x\) is the input vector with N elements,\(W\) is the \(M\times N\) weight matrix, \(b\), a bias vector with M elements, with matrix multiplication \(W\):\(x\). Figure 1b illustrates the corresponding photonic implementation with SOAs. In this instance, the input \(x\) is encoded onto several channels at different wavelengths and the individual input is weighted with the given gain/attenuation provided by an SOA. The weighted signals are then combined into a WDM signal and sent to the nonlinear function to provide a single wavelength neuron output. The nonlinear activation function can be realized in several ways, e.g., by employing the combination of a photodetector and a modulator [26], saturable absorbers [25], excitable lasers [28; 29], wavelength converters [30], and phase change materials [31]. In this simulation work, we use a photodetector and off-line processing for nonlinear function and we mainly focus on the operation of weighted addition for the matrix multiplication. Utilizing photodetectors at the output of the matrix multiplication, the detected summation of all the weighted signals results in  
\[V=(R\cdot Z_{0}\psi_{\pi})\cdot x\cdot\exp\left[h(I)\right]\]  
where \(R\) is the signal detection response, assumed to be constant for dense WDM signals, \(Z_{0}\) is the PD characteristic impedance and \(v_{\pi}\) the voltage at \(\pi\) phase shift. The vector \(h\) (\(I\)) has N elements, the \(i\)th element \(h(I_{i})\) is the gain integrated over the length of the SOA for weighting input \(x_{i}\), where the injection current is \(I_{i}\). The outputs are then sent to the nonlinear function which processes the signal and produces the outputs of the neuron. One full neural layer consists of linear matrix multiplications and nonlinear operations. The details of a neuron layer with four neurons, as used in this paper, are illustrated in Figure 1c, where the input WDM signal is selected by using a port selector that directs the desired input signal to this layer to be processed (see chip picture). The input signal is split and sent to the neurons (one neuron highlighted with a blue box). The AWG in the neuron de-multiplexes the input into individual channels, whose weight is assigned with different gains by using different weighted SOAs as shown in Figure 1b. The combined weighted signals from the four output ports pass through the activation function \(f_{i}\)  
Figure 1: Photonic deep neural network based on the broadcast-and-weight architecture. **(a)** The artificial neuron model. **(b)** The implementation via arrays of semiconductor optical amplifiers (SOAs). **(c)** One full layer of neurons by exploiting one wavelength division multiplexing (WDM) input, with a shaded photonic integrated circuit micrography at the back, to underline that part of the circuitry that is realized on chip. **(d)** Scheme of a three-layer photonic deep neural network.