# Integrated Photonic Neural Networks for Equalizing Optical Communication Signals: A Review
## 2 Fundamentals of Photonic Neural Networks Working as Equalizers  
Photonic neural networks are part of a research field that explores the use of photonics to build artificial neural networks on integrated circuits [82]. Unlike classical neural networks, which use conventional electronics for data processing, photonic neural networks employ optical components, such as lasers, optical fibers, waveguides, and photonic integrated circuits to perform computations and transmit information [83]. Classical artificial neural networks operating in the electronic domain were inspired by biological neural connections and consist of learning models employed primarily to determine a prediction model based on a large amount of input data [84]. Specifically, each neuron interconnected with all others in the network structure receives input information from other neurons or external stimuli, processes this information, and provides a transformed output signal to other neurons or external outputs [85]. Thus, classical neural networks can efficiently recognize and learn interconnected patterns between previously unseen input data sets and the related target values [85]. After the training stage, these neural networks can be applied to new, independent input data to predict the system dynamics with high accuracy [86]. Photonic neural networks aim to perform these same steps but using light. The most basic structure of a photonic neural network that was recently employed as an equalizer is called a perceptron, which consists of a photonic neural network with only one layer, in which the information flow in its structure occurs in a single direction, from its input to the output, without any feedback connection. An integrated version of a photonic perceptron was realized in 2022 by [PERSON] et al. [76], composed of delay lines and phase modulators, and a sketch of it is schematically illustrated in Figure 1.  
In the device depicted in Figure 1, an optical input signal \(u(t)\) is split into four copies \((u_{k}(t),k=1,\ldots,4)\), which propagate in the four waveguides, which are spiral-wound to act as delay lines and thus delay copies of the input signal. Thermal heaters (yellow lines) are placed above the waveguides to heat them and thus transmit a specific phase \(\phi_{k}\) to the delayed input signals (phase shifters). Consequently, the phase encoded through the weights, \(w_{k}=exp(i\phi_{k})\), is assigned to each delayed copy. Finally, these delayed copies are weighted and summed coherently by a combiner, and the output signal is collected by a fiber and detected by a photodetector. The output of the photonic neural network \(y(t)\) is then mathematically described by [76]  
\[y(t)=\left|\sum_{k=1}^{4}u(t-k\Delta t)w_{k}\right|^{2}. \tag{1}\]  
where \(\Delta t\) represents the time delay of each delay line. There are also photonic neural networks that can be used in a framework called reservoir computing that can be used as equalizers, which consist of a simplification of a classical recurrent neural network [79]. It is worth mentioning that recurrent neural networks are distinguished by their well-known memory capability, as they receive information from previous inputs to adjust the current input and output values [87]. Specifically, photonic reservoir computing comprises three parts: the input layer, the reservoir (hidden layer), and the output layer. The weights in the input and reservoir are randomly selected and kept fixed, and the connections in the reservoir are recurrent, as illustrated in Figure 2. Furthermore, only the weights in the output layer are adjusted, which can be achieved through a simple linear regression, making the training process fast and simple.