Avant d'entrer dans le vif du sujet, j'aimerais exprimer ma gratitude à tous ceux qui ont soutenu directement ou indirectement la réalisation de ma thèse.

Tout d'abord, je tiens à remercier ma famille pour m'avoir soutenu et accompagnée dans cette aventure difficile. En particulier, je remercie ma mère pour tous les sacrifices qu'elle a eue à faire pour que je ne manque de rien et pour toute la force de caractère qu'elle a su me transmettre. C'est en grande partie grâce à toi que je suis arrivée jusqu'ici, car tu n'as jamais eu à douter de mes choix ou de mon potentiel.

Ensuite, je tiens à remercier sincèrement mon tuteur de thèse, Benoit Charbonnier, qui m'a accompagné tout au long de mon parcours, dans les bons et surtout dans les mauvais moments. Tu as toujours su me pousser vers le haut sans aucun jugement et toujours avec bienveillance. Je me souviendrai toujours de nos désaccords, de nos débats et des découvertes scientifiques qui ont nourri mon quotidien. Grâce à toi, j'ai toujours eu la conviction que je pouvais réussir ma thèse malgré les difficultés, surtout à la fin. Enfin, je suis reconnaissante d'avoir fait ta connaissance et d'avoir pu être encadrée par une personne qui a vraiment eu la volonté de me voir réussir.

Je tiens également à remercier mon co-tuteur de thèse, Karim Hassan, pour son encadrement, ses conseils avisés et surtout sa disponibilité. Avec Benoit, vous avez toujours su me fournir les ressources nécessaires pour mener à bien ma thèse. Dans la même optique, j'adresse un grand merci à tous mes collègues du laboratoire LIPS, et plus particulièrement à Cécilia, qui m'a apporté son aide dans l'implémentation d'un nouveau processus de fabrication des lasers. Je n'oublie pas mon directeur de thèse Maxime Jacquot et toutes les personnes qui ont contribué indirectement à ma thèse, notamment Hélène Duprez dont le travail m'a permis de débuter cette thèse.

Enfin, je tiens à remercier mon ancienne collègue et aujourd'hui amie, Raouia Rhazi, tout simplement pour ton soutien moral, tes conseils et tout le temps que nous avons passé ensemble au fil des années. Merci de m'avoir accompagné, écouté et soutenu lorsque j'avais des doutes.

Merci à vous tous !

Table of contents

Motivation and Objectives

As the 21st century has unfolded, significant advances have been made, especially in the area of artificial intelligence (AI). As we delve deeper into understanding the intricacies of the brain, researchers have realized the limitations of conventional computing architectures in simulating the brain's parallel and distributed processing capabilities. These limitations have paved the way for novel computational models, such as spiking neural networks, that aim to capture the essence of the brain's information-processing mechanisms.

In hardware, photonic technology has enabled the development of novel spiking neural networks, which promise to provide faster and more energy-efficient computing solutions. Moreover, photonic platforms offer high-density integration enabling more complex and scalable neural network architectures. Thus, combining both technologies could result in unprecedented efficiency when tackling computationally demanding tasks.

In the context of this evolving AI era, the primary objective of this research is to explore the potential of photonic platforms to enable high-density spiking neural networks. By using an ultra-short pulsed semiconductor laser as the core component of our photonic network, we hope to build a scalable device that takes advantage of light-based information processing without compromising energy efficiency. I have organized this thesis as follows. The first Chapter will introduce the fundamentals of spiking neural networks and describe the current state of photonic networks with an emphasis on laser-based systems. Then, Chapter 2 will discuss the design of a spiking semiconductor laser and its implementation on the CEA hybrid technology platform. In Chapter 3, I will present our approach to training a large-scale photonic neural network. Finally, Chapter 4, provides conclusions and proposes a roadmap for future research. In this chapter, we will delve into the state-of-the-art of Spiking Neural Networks (SNNs) and explore their implementation in photonics platforms. The chapter begins by introducing the basic concepts and principles of SNNs in section 1.1. Additionally, popular learning algorithms used in the training of SNNs will be discussed, such as Spike-Timing-Dependent Plasticity (STDP) and backpropagation. Next, we will discuss the latest advancements in neuromorphic photonics and examine the challenges and opportunities in realizing efficient photonic SNNs in section 1.2. Examples of spiking neuron and synaptic functions implemented using lasers, modulators, and photodetectors will be presented. Neurons and glia are the primary cells that make up our brain's central nervous system (CNS) [1]. Each contributes to the process of information in its way, making them both essential for a healthy nervous system. However, since neurons are the more functionally significant of the two [1], our main interest will lie in understanding how they actually work to process information and how they have influenced bio-inspired Spiking Neural Networks (SNNs) throughout history.

Neuron overview

Neurons are excitable and conductive cells [1], specialized in processing information contained in a nerve impulse. Our brain is believed to contain more than a hundred billion interconnected neurons (10 11 ) [1], each of which can connect with more than 10 4 [2] other neurons at once, resulting in a high degree of connectivity. A famous drawing by the neuroscientist Ramon y Cajal illustrates their rich diversity (Fig 1 .1.a) [2]. Indeed, various classes of neurons have been distinguished according to their shape, location, and function in our CNS. Usually, they have a kind of tree-like shape with a cell body (or the soma), dendrites, an axon, and synapses, as shown in Dendrites (neuron's input) are the pathways for information from other neurons to reach the cell body, where it can be integrated and processed. The cell body then determines whether to transmit the information through a nerve impulse (electrical signal) [2]. A long singlebranched extension, called an axon, carries the outgoing signal (if any) to the neuron's output (axon terminals). Finally, synapses found at the ends of neuron axons facilitate the communication of signals from one neuron to another [1], [2].

Spike generation

All information a neuron receives is processed in its cell body [2]. A plasma membrane surrounds a neuron's cell body and separates its intracellular and extracellular environments. Several specific ion channels are found in the plasma membrane, enabling the passage of both positively (Na + ,Ca + , K + ) and negatively (Cl -) charged ionic species [4]. Comparatively, sodium ions have a higher concentration outside of the cell than potassium ions and vice versa (cf Fig 1.2.a). This unequal distribution of charged ions creates a potential difference (known as the resting potential) between the two media that is generally close to -70 mV [2]. The membrane is then said to be 'polarized'. From this state, if the neuron's cell is strongly stimulated, its membrane potential becomes more positive (i.e. depolarizes), resulting in the emission of an electrical pulse, also known as an action potential or a 'spike' [4] which is allor-none phenomenon [2]. is generally characterized by three phases lasting about 1-2 ms [2]. Membrane depolarization marks the first phase. In fact, stimulation allows certain positive ions to pass through the membrane more readily, such as Na + , which increases the membrane potential until it reaches its excitation threshold and releases spike [4]. Since the spike is a very short event, the membrane potential drops almost immediately after firing (repolarization phase, potassium ions move out of the cell). During this phase, the neuron can not fire another spike as it's in an 'absolute refractory' period [2] which marks its insensitivity to any external stimulation. Finally, before returning to its resting state, the membrane goes through a period of transient hyperpolarization where its potential is lower than its resting potential (around -80 mV). The neurons at this stage exhibit 'relative refractoriness' [2], which means they are capable of generating new spikes if sufficiently stimulated.

The synapse: How do neurons communicate?

The transmission of information between neuronal cells is done through tiny junctions known as synapses. In biology, synapses can connect two neurons, but eventually connect a neuron with an organ, and are classified as chemical or electrical [4]. Here we will only focus on the most common type found in our brain, which are chemical synapses. This type of synapse uses mediators called neurotransmitters to communicate. 3.a illustrates a chemical junction between an excitatory neuron (pre-synaptic) and a target neuron (post-synaptic). In fact, through the exocytosis process [6], a pre-synaptic action potential or spike signal is transmitted to a post-synaptic axon terminal using neurotransmitters, which are charged ions in synaptic vesicles. When ions are released, they diffuse into the post-synaptic neuron's extracellular fluid through a small gap (the synaptic cleft [4]), where they bind to specific receptors or gates. In response to these neurotransmitters, either the post-synaptic membrane is depolarized 1 and an Excitatory PostSynaptic Potential (EPSP) is generated or it is hyperpolarized 2 , resulting in an Inhibitory PostSynaptic Potential (IPSP) [4], [7], as illustrated in Fig 1 .3.b. Therefore, a neuron can generate an action potential only through the integration of several EPSPs arriving in a short period, whereas several IPSPs would make it inactive. Note that, only the timing and/or the frequency of spikes matters when it comes to neural processing [2]. An important property of chemical synapses is their plasticity, as they can modulate the signal strength transmitted between two neurons, thus strengthening or weakening their connectivity. As we will discuss later, neuronal plasticity is an essential component of learning and memory in the brain, which are essential for the configuration of large-scale neuronal circuits.

1.1.2 ANN to SNN: Third generation of neural network

1.1.2.

Historical background

It has been shown previously that a biological neural network contains multiple layers of neurons that are interconnected by synapses. The network is organized so that information is processed asynchronously and in a parallel manner [9]: a neuron can stimulate (or inhibit) many other downstream neurons in an independent manner. Information contained in spike events is then passed from layer to layer until the final layer produces the result. With this process scheme, our brain can perform more than 10 18 operations/s using only a few watts [10].

Models of Artificial Neural Networks (ANNs) are mathematical representations of how the brain processes information. The idea behind this is to build intelligent machines that 1 When the triggered ion channels are permeable to Na + ions, positive charge builds up inside the cell 2 Open ion channels allow Cl -ions to pass through, making the membrane more negative. [8]. Synaptic vesicles contain positive or negative charges or neurotransmitters. A post-synaptic neuron's terminal contains specific receptors on which neurotransmitters bind after a spike has occurred. (b) Characterization of EPSP (depolarization) and IPSP (hyperpolarization), induced by the stimulation of receptorgated ion channels permeable to Na + (left) and Cl -+ (right), respectively [7].

are capable of competing with the high capability of the human brain [9]. An ANN consists of neurons that are independent computational units, while synapses support the network connectivity. Different generations of neural networks have been proposed over the years, each distinguished by how the computing unit is modelled and the way information is encoded [11], [12].

First generation of ANN

In the early 1940s, McCulloch and Pitts developed the Threshold Logic Unit (TLU) a precursor to ANNs [12]. This unit, which will later be known as a perceptron, consists of multiple inputs, an adder, and a computational/activation function as depicted in Fig 1. 4.a. The neuron's inputs, representing a set of weighted incoming signals, are summed and an activation function (similar to a Heaviside function) is applied to the result. Therefore, when this weighted sum crosses a fixed threshold, the neuron's output is equal to 1, otherwise it is 0 [12], [13]. In other words, this type of neuron can only receive or produce digital values. Eventually, the sum can be biased to influence the neuron's final decision. Regarding biology, this would be like translating the presence of a spike (the neuron's action potential) by 1 and its absence by 0 without considering its firing time, which is then far from plausible. Additionally, due to its unimodal binary activation function, this neuron generation was strongly restricted to linear classification tasks (all Boolean functions except XOR), limiting its application.

Second generation of ANN

ANNs of the second generation were first developed to address complex nonlinear problems such as XOR [11]. Compared to the first generation, its neuron model performs continuous activation to the weighted sum inputs, for example using the sigmoid (see Fig 1.4.b), hyperbolic tangent, or more recently the Rectified Linear Unit (ReLU) functions [14]. As a result, both inputs and outputs can be real numbers comprising between 0 and 1. This is the concept behind rate coding, where the firing rate of neurons is used to process information. Feed-forward networks (discussed below) and MultiLayer Perceptrons (MLP) are typical second-generation neural networks [13]. In addition, the second generation of ANNs, together with the popular Back-Propagation (BP) learning algorithm, have proven to be highly effective in solving com-CHAPTER 1. INTRODUCTION AND STATE OF ART plex tasks (such as the XOR problem), classification tasks, and so on. However, although they are more powerful and efficient than the first generation of ANNs, they do have limitations to their computational power as they do not exploit the temporal aspect of spike signals [10], [11].

Third generation of ANN

Neuromorphic systems can be built more efficiently using the third generation of ANN, known as spiking neural networks (SNNs). Based on the concept of spiking neurons, these networks are the most accurate when it comes to representing biological reality in a model [13]. SNNs introduce time using event-based computations [2], where spikes are used to transmit information similarly to biological neurons, a concept that does not exist in the first two generations. Also, information can be transmitted either in the frequency of spikes (rate coding) or in the timing of spikes (temporal coding) [13], but not in their amplitudes. As shown in Fig 1.4.c, spiking neurons operate with a series of spikes that are generated whenever their internal state (modeled by its membrane potential) exceeds a certain value known as the threshold. This leads to an activation function being binary like the one of the first generation neurons, but without the loss of information that usually comes with binary coding, thanks to its spatiotemporal property [2]. Moreover, instead of transmitting information synchronously (as in the two first generations), spiking neurons are only active when they generate or receive spikes. This asynchronous property, which led to saving energy consumption [10], is one of the various advantages of using spike processing for hardware implementations. However, in practice, SNNs are computationally demanding and are limited regarding training options.

Spiking neuron models

Simulations of spiking neurons can be performed using a variety of mathematical models. These models use differential equations to describe some key properties of biological neurons observed experimentally, including Regular Spikes (RS), Intrinsic Bursts (IB), or CHattering (CH) [15]. Current neuron models are distinguished by both their biological realism and their computational complexity [15], [16]. Indeed, as models become more accurate and complex (with too many variables), it becomes difficult to implement them on traditional computing machines without being limited by available memory or processing power [16]. A spectrum of common neuron models is shown in Fig 1 .5, highlighting conductance-based models and integrate-and-fire models.

Conductance-based models

Conductance-based models are the closest possible biophysical simulation of neurons, representing Ion channels by conductances. One of those models is the Hodgkin-Huxley (HH) model [17]. HH model allows to describe with high accuracy the membrane potential, the opening and closing of N a + and K + channels as well as a leakage current L [15], [18]. An equivalent circuit of an HH neuron's membrane can be seen in Fig 1. 6. By considering equation 1.1, we can model the effect of a synaptic current passing through the neuron's membrane. and its electrical circuit equivalent (right), [19].

C m dV m dt = I(t) -G N a (V m -E N a ) -G K (V m -E K ) -G L (V m -E L ) (1.1)
With C m representing the neuron's membrane capacitance and V m its electrical potential.

G N a , G K , G L are respectively the conductances parameters of the sodium, potassium, and leaky channel and E N a ,E K , E L are the potentials associated with each channel. By solving this equation, this model can reproduce almost all the physiological properties [15]; however, It does require a large amount of computational power due to its complexity. For this reason, models like Fitzhugh-Nagumo (FN) [20] or Hindmarsh-Rose (HR) [21] have been proposed as alternatives to this model, admitting fewer variables, which simplify the computational settings and are less time-consuming to implement.

Threshold models

As we shift to the left in the spectrum, one finds neuron models that admit fewer variables and are easier to implement. Lapicque's (1907) leaky Integrate-and-Fire (LIF) neuron [2], [15] is one of the most popular models in this category. As this model will be further developed later in the document, key features are outlined here.

The LIF neuron can be modeled electrically as an RC circuit characterized by its integration time τ m = R m C m (see Fig 1.7). As for the HH model, the plasma membrane is characterized by a capacitor C, which is charged by a synaptic current regulated by sodium-potassium ion channels. These channels are modeled by a single leaky resistor R. The temporal evolution of the membrane potential V m (t) can thus be expressed as [18]: A typical LIF neuron's membrane potential evolution over time [19]. The neuron integrates several spikes, resets its potential to zero after a spike is released, and enters a refractory period before returning to its resting potential [22].

C m dV m dt = - 1 R m (V m -E L ) + I syn (t) (1.2)
Where R m is the leakage resistance and I syn (t) is the synaptic current flowing through the membrane. I syn (t) can also evolve according to its own dynamics, as we'll see later.

Here, whenever V m (t) crosses a threshold potential V th , a spike is released and the membrane returns to its resting potential, V rest = E L . In this way, neuron spiking activity can be regulated. This model can also simulate neuron insensitivity to external stimuli after a spike generation (a.k.a the relative refractory period) for greater realism [23]. Note that, the LIF neuron model is commonly used in two other versions in the literature: the Integrate-and-Fire (IF) model and the Spike Response Model (SRM). The IF model is simpler to implement since the leakage terms R m and E L are removed and the input spikes are directly integrated. On the other hand, the SRM model can be derived from the LIF model by integrating equation 1.2. [2]. This model is used to describe more accurately the membrane post-synaptic potential and the synaptic conductance, which makes it more complex to implement. In contrast, other models such as Izhikevich [15] and the Adaptive Leaky-Integrate-and-Fire (ALIF) [24] model propose some adjustments to the LIF model for improving biological accuracy while maintaining a lower computational cost. In summary, threshold models, as demonstrated in Izhikevitch's study [25], can provide a better compromise between biological realism and computational complexity, which is why they are widely used today in neural computing.

Information encoding

Implementation of SNNs involves converting raw spatio-temporal data into spikes, which can be further processed [2]. In biological neural networks, the timing of individual spikes or their numbers is the basic way to encode information as we explained earlier. Based on these observations, two encoding schemes were distinguished: the rate coding and the temporal coding [26].

Rate coding

Information is conveyed by using the frequency or the rate of spikes generated over a certain period. In general, there are three types of rate encoding (density coding, population coding) as detailed here [27], but the most used one is the count rate coding. Here, the neuron will fire at a low rate if it is weakly stimulated, otherwise, it will fire at a high rate. In this coding scheme, the timing of spikes does not matter and can even be random [27]. Rate coding is commonly used in standard ANNs and even in SNNs as it shows strong robustness to noise [2]. However, the effectiveness of this method depends on whether neurons can generate enough spikes per simulation quickly or after a long time [2], [27]. In addition, as each spike consumes energy, this method can be both energy-intensive and quite slow.

Temporal coding

This method only considers the exact spike timing, not their rates. As for rate coding, there are several forms of temporal coding (rank, synchronization, etc. see [27]), but the most basic and most popular approach is the time-to-first-spike (TTFS) or "latency" coding shown in Fig

1.8.b.
Here, information is encoded in the time delay between the onset of the stimulus and the first spike of the neuron. In this sense, a high latency means that the stimulus is weak and vice versa. So, compared to rate coding, only one spike per neuron is needed to process the data, reducing both the computation time and energy consumption of the computing platforms.

Encoding schemes like this have shown great capability in supervised training methods like [28] or [29]. Nonetheless, many neuroscientists believe the brain encodes and transmits data using both time and rate schemes [2], [26]. However, in this thesis, we will only use temporal coding due to the reasons mentioned above and for simplicity. Another important factor to consider when implementing a neuromorphic system is the neural network model. A neural network can be recurrent, feed-forward, or hybrid in topology [30]. Throughout this thesis, we will only consider feedforward architectures due to their simplistic implementation for classification tasks. Feed-forward neural network means that information only travels in one direction i.e. from left to right. As shown in Fig 1 .9, this type of neural network consists of an input layer (encoding), an output layer (readout), and at least one hidden layer (learning), all fully connected by synapses (represented with straight lines).

In the first layer of neurons, spike trains are used to encode (rate or time) and sparsely transmit input data to the rest of the network. Then, computations occur mostly in the hidden layer, i.e. each spiking neuron accumulates multiple incoming weighted spikes (referred to as fan-in) and fires an output spike (that is fanned out to downstream neurons) if its threshold is reached.

In contrast, neurons in an ANN perform Multiply-ACcumulate (MAC) operations [31], [32], a metric that provides information about the performance of an AI processor. Finally, at the end of the network, results are given by the final layer, where each neuron belongs to a certain class.

Figure 1.9: Schematic of a feed-forward spiking neural network. if the network has more than one hidden layer, then it is called deep-spiking neural network.

SNN learning approaches: State-of-the-art

SNNs learning is inspired by the phenomenon of neuroplasticity in our brain, or in the synapses [6], [33]. To be more precise, learning refers to the modification (strengthening or weakening) of synaptic connections within a network of neurons, which allows the latter to acquire and memorize information [6]. Thus, in the context of SNNs, it consists of adapting weights in the network (weights and biases) that have a direct influence on the network's global response. It is therefore necessary to repeat the learning process until either the network output matches the target outputs or the network stabilizes. The first case involves supervised learning, while the second refers to unsupervised learning. Generally, the learning phase is followed by the inference phase, in which new data is sent through the trained model to make conclusions about a specific application [7]. While ANNs can be trained efficiently using supervised learning techniques such as gradient descent methods [34], despite their recent popularity, SNNs lack a reliable training approach. It is only recently that robust training approaches have been CHAPTER 1. INTRODUCTION AND STATE OF ART developed for spike time-based SNNs. However, rate-based SNNs can be trained in the same way as ANNs. Some notable learning methods and strategies found in the literature are discussed here, classified as supervised or unsupervised.

Unsupervised learning

The concept of unsupervised learning describes machine learning in which no labels or categories are provided by the user. Typically, this method involves a bio-derived and local rule known as Spike-Time-Dependent Plasticity (STDP) [35], [36], which describes how the synaptic connection between two neurons changes depending on the timing of their spikes. Fig

1.
10 shows the temporal evolution of the STDP rule in the case of synaptic connections w ij between presynaptic (j) and postsynaptic (i) neurons. In the figure, two learning windows are shown, one representing long-term potentiation (LTP) and one representing long-term depression (LTD). LTP is the process by which the synaptic connection between neuron j and neuron i is strengthened when neuron j spikes before neuron i (the neuron j fires after having received a spike from neuron i). In contrast, if the opposite occurs, then the synaptic connection w ij is weakened, as there is no link between the firing of the neuron j and the reception of the presynaptic spike. In this case, we speak of long-term depression (LTD). The relative variation of the synaptic weight w ij can thus be translated by the following equations [18]: Figure 1.10: STDP learning windows. A change in synaptic weight is determined by the correlation between pre-and post-synaptic spikes [37].

∆w = M m=1 N n=1 W (t n i -t m j ) (1.3) With W (∆t) =    A + e -∆t τ + , if ∆t > 0 -A -e ∆t τ -, if ∆t < 0 (1.4)
Where ∆w is the total change in synaptic weight w ij , t n i is the n-th spike of neuron i, t m j is the m-th spike of neuron j and W(∆t) is a function of the STDP rule determined by equation 1.4. ∆t is the time delay between the events t pre i and t post j , A + ,A -, are the amplitudes of the LTP and LTD curves, τ + and τ -are the associated time constants. This form of STDP (also CHAPTER 1. INTRODUCTION AND STATE OF ART known as the "Vanilla" STDP) is advantageously used to train SNNs for real-time classification and features recognition tasks since it uses local information [38], [39], [40]. Moreover, as STDP is an unsupervised method, it does not require heavy human intervention and can be easily implemented on a neuromorphic chip. However, though it is simple to implement, STDP relies on statistical classification, which means that it requires massive data and resources to solve a problem. This can lead to inaccurate results, and convergence isn't always guaranteed, especially when the number of hidden layers increases [START_REF] Vigneron | A critical survey of STDP in Spiking Neural Networks for Pattern Recognition[END_REF]. The solution proposed by Mozafari et al. [START_REF] Mozafari | First-spike based visual categorization using reward-modulated STDP[END_REF] to increase STDP's accuracy and reliability is to incorporate supervision into the training by rewarding the system when it makes progress toward the desired output. This method allows to implement a reinforcement learning mechanism and it is referred to as Reinforcement STDP (R-STDP). Tavanaei et al [START_REF] Tavanaei | Acquisition of Visual Features Through Probabilistic Spike-Timing-Dependent Plasticity[END_REF] introduced another method in which the LTP synaptic weights are updated with an exponential function related to the magnitude of the current weight [START_REF] Vigneron | A critical survey of STDP in Spiking Neural Networks for Pattern Recognition[END_REF]. This method known as Probabilistic STDP (P-STDP) has shown great results regardless of the complexity of the neuron model used [START_REF] Vigneron | A critical survey of STDP in Spiking Neural Networks for Pattern Recognition[END_REF]. While STDP's performance has improved significantly when training with these strategies, it still struggles to match the state-of-the-art ANNs trained with gradient descent methods.

Supervised learning

Gradient-based optimization methods such as error backpropagation are the basis for supervised learning in SNN. This approach is illustrated in Fig 1 .11. It consists of adjusting the synaptic weights and biases of a deep neural network according to the error gradient computed between the desired (known) output and the neural network result. Thus, the model can predict a given output with high accuracy by minimizing the network error through multiple iterations. The backpropagation error method is less bio-plausible than the STDP method, but it has been proven to be highly effective for training deep ANNs, as reported in [START_REF] Zhu | Classification of MNIST Handwritten Digit Database using Neural Network[END_REF], [START_REF] Lee | Deeply-Supervised Nets[END_REF]. However, since this method is incompatible with the nature of spiking neurons, i.e. they are non-differentiable, SNNs did not immediately follow this trend. To overcome these limitations, a lot of work has been proposed specifically to train SNNs with the backpropagation method, to exploit SNN's full potential, particularly regarding energy efficiency and robustness to noise. These works can be divided into the two following categories:

ANN-to-SNN Conversion

Conversion-based algorithms can indirectly address the training problems of SNNs. There have been several works on this approach, including [START_REF] Hunsberger | Spiking Deep Networks with LIF Neurons[END_REF], [START_REF] Diehl | Fast-classifying, highaccuracy spiking deep networks through weight and threshold balancing[END_REF], [START_REF] Sengupta | Going Deeper in Spiking Neural Networks: VGG and Residual Architectures[END_REF], [START_REF] Han | RMP-SNN: Residual Membrane Potential Neuron for Enabling Deeper High-Accuracy and Low-Latency Spiking Neural Network[END_REF], and [START_REF] Patel | A Spiking Neural Network for Image Segmentation[END_REF]. In these studies, traditional ANNs with a pre-trained equivalent topology are typically used to train SNNs. Therefore, this technique involves two main phases: inference and learning (see Fig

1.12).

In the learning phase, an ANN's synaptic weights are optimized using the error backpropagation method. ANNs typically use neuron models that do not emit pulses, such as the standard rectified linear unit ReLU or an approximate model of the LIF introduced by [START_REF] Hunsberger | Spiking Deep Networks with LIF Neurons[END_REF], called "soft- Figure 1.12: Overview of conversion-based algorithm, taken from [START_REF] Abderrahmane | Hardware design of spiking neural networks for energy efficient brain-inspired computing[END_REF] LIF". Then, ANN's neuron model is converted into an LIF or IF neuron to form an SNN of the same network typology. The input and output data are now expressed in spike trains instead of analog values. Rate coding is most often used to convert input data into spikes using this conversion method as it is straightforward to convert an analog value to a spike rate or density. Following the learning step, optimized synaptic weights are used to initialize the SNN network in the inference step. In some techniques, such as in [START_REF] Diehl | Fast-classifying, highaccuracy spiking deep networks through weight and threshold balancing[END_REF] or in [START_REF] Rueckauer | Conversion of Continuous-Valued Deep Networks to Efficient Event-Driven Networks for Image Classification[END_REF], the weights are normalized before adding them to the SNN network so that the firing rates of the neurons are not too different. Instead, [START_REF] Sengupta | Going Deeper in Spiking Neural Networks: VGG and Residual Architectures[END_REF] introduces the Spike-Norm algorithm, which consists of normalizing the threshold of the spiking neuron in order to reduce the degradation of accuracy that usually happens during the conversion process. ANN-to-SNN conversion has the advantage of leveraging backpropagation methods and deep learning advances while demonstrating comparable performance to conventional ANNs. However, it suffers from high latency and high energy consumption [START_REF] Abderrahmane | Hardware design of spiking neural networks for energy efficient brain-inspired computing[END_REF] related to the conversion delay and the use of rate coding, respectively. So, even though some works have addressed some of these issues, such as the work of [START_REF] Rueckauer | Conversion of Continuous-Valued Deep Networks to Efficient Event-Driven Networks for Image Classification[END_REF], this method is less suitable for processing event-based data and harnessing the true potential of spike processing.

CHAPTER 1. INTRODUCTION AND STATE OF ART

Direct training algorithm

As a reminder, we consider spike time-based SNNs for information processing. However, spikes are discrete events in time, making their activation function discontinuous and therefore non-derivable [18]. As a result, gradient descent techniques must be approximated before being applied. A comparison of some direct learning methods that try to overcome this spike processing's inherent problem is presented in SpikeProp [START_REF] Verleysen | Proceedings / 8th European Symposium on Artificial Neural Networks[END_REF] was the first supervised algorithm that successfully trained multi-layer SNNs using the SRM neuron model. It is a time-based version of backpropagation learning that uses a linear gradient approximation called "surrogate gradient" [START_REF] Neftci | Surrogate Gradient Learning in Spiking Neural Networks: Bringing the Power of Gradient-Based Optimization to Spiking Neural Networks[END_REF], so that the activation function of the spiking neuron, comparable to the Heaviside step function, can be differentiated. As a result, gradient descent can be applied to minimize the error between the labeled spike times and the SNN output spike times. Several variants of this algorithm, such as the work of [START_REF] Kheradpisheh | S4NN: Temporal backpropagation for spiking neural networks with one spike per neuron[END_REF], [START_REF] Wu | Spatio-Temporal Backpropagation for Training High-Performance Spiking Neural Networks[END_REF], [START_REF] Zenke | SuperSpike: Supervised Learning in Multilayer Spiking Neural Networks[END_REF], [START_REF] Shrestha | SLAYER: Spike Layer Error Reassignment in Time[END_REF] also use the surrogate gradient strategy.

Another interesting way to train SNNs is to smooth the spiking neuron model, to make it continuous and thus differentiable. The Spike-Timing-Dependent Back-Propagation (STDBP) [START_REF] Zhang | Spike-Timing-Dependent Back Propagation in Deep Spiking Neural Networks[END_REF] method, uses a linear approximation of the SRM neuron model (Rel-PSP) to solve notable backpropagation problems in the spike domain, such as dead neurons and gradient exploding. With this neuron model, they achieved state-of-the-art performance on the MNIST (Modified National Institute of Standards and Technology) handwritten digits dataset. Parametric Leaky Integrate-and-Fire model (PLIF) neurons were introduced by [START_REF] Fang | Incorporating Learnable Membrane Time Constant to Enhance Learning of Spiking Neural Networks[END_REF] to speed up learning by optimizing weights along with time constants associated with each neuron. In comparison to most other models cited here, the PLIF strategy has shown a higher level of accuracy when classified the MNIST dataset, so far.

CHAPTER 1. INTRODUCTION AND STATE OF ART

Other significant works including the Remote Supervised Method (ReSuMe) [START_REF] Neural | ReSuMe -New Supervised Learning Method[END_REF], Back-Propagation Spike Timing Dependent Plasticity (BP-STDP) [START_REF] Tavanaei | BP-STDP: Approximating backpropagation using spike timing dependent plasticity[END_REF], and Supervised Spike Timing Dependent Plasticity (SSTDP) [START_REF] Liu | SSTDP: Supervised Spike Timing Dependent Plasticity for Efficient Spiking Neural Network Training[END_REF] proposes to combine the bio-plausibility of the STDP approach with the reliability and efficiency of the backpropagation technique. For instance, in SSTDP, weights are locally updated through STDP thanks to the temporal correction given by global backpropagation, which leads to fewer time steps and performance similar to the stateof-the-art ANNs. Hence, this hybrid approach can provide an algorithm that is bio-plausible as well as efficient.

A similar effort is made in the work of [START_REF] Wunderlich | Event-based backpropagation can compute exact gradients for spiking neural networks[END_REF] to improve SNNs performance with an algorithm called EventProp. In this work, backpropagation gradients can be computed without any approximation in an event-driven manner. As part of the learning process, an adjoint spiking neural network [START_REF] Lecun | A Theoretical Framework for Back-Propagation[END_REF] is used along with partial derivative jumps of the LIF model to backpropagate the exact error through the various layers. Furthermore, because it entirely leverages spike-based information processing, this novel and promising method is intended to be less energy and memory-intensive than the other works listed above. This thesis will further develop the working principles of this algorithm since it is our method of choice for training our SNN model.

SNNs implementation in photonics: state-of-the-art 1.2.1 Related works: neuromorphic photonics network

Integrating spike-based computing with neuromorphic photonic platforms offers many advantages, including, high operational speeds, high communication bandwidth, and massive parallelism, which address many of the limitations of electrical hardware [START_REF] Shastri | Photonics for artificial intelligence and neuromorphic computing[END_REF], [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF]. Recently, photonic circuits have been used to replicate spiking neural networks on different integration platforms such as CMOS, Silicon/III-V, etc [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. For building a neural network, it's important to study each key component separately, namely: the neuron, which applies a non-linear function, and synaptic connectivity, which is responsible for reconfigurability and memory in the network. Table 1.2 shows biological neural network building blocks and their equivalent photonic devices. Note that, in the literature, photonics devices that exhibit spiking behavior are often called 'spike processors' [START_REF] Tait | Photonic Neuromorphic Signal Processing and Computing[END_REF]. Another key element for any neural network is a robust training scheme that can handle noise and inaccuracies in the learning parameters. However, training a network with optical or photonic hardware is challenging because of their high sensitivity and the complexity of implementing training that implies gradient calculation, which is memory and time-intensive [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF], [START_REF] Xu | A survey of approaches for implementing optical neural networks[END_REF]. So most demonstrations are off-line, while the on-chip network is only used for inference phases [START_REF] Xu | A survey of approaches for implementing optical neural networks[END_REF]. There are only a few studies that demonstrate on-line photonic-based learning methods, mostly based on the STDP rule.

Therefore, this section will discuss recent progress made in photonic SNN [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF] integrated platforms, including photonic synapses and photonic neuron implementation. We will then briefly discuss integrated neural networks and recent advancements in photonic training schemes.

Photonic synaptic functions

A synaptic network is equivalent to a matrix multiplier as it mathematically performs parallel dot-product operations between the input vector (neuron's fan-in) and weights matrix before being processed by neurons. Many studies have been conducted in the field of photonics with the aim of developing on-chip optical matrix multipliers [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. For this purpose, reconfigurable devices and materials like Phase change materials (PCMs), Micro-Ring Resonators (MRRs), Mach-Zehnder Interferometer (MZI) meshes, and SOAs have been used. Table 1.4 summarizes some of the key characteristics of these devices, which are also called photonic synapses [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. The following subsections will focus on the usual photonic structures that are capable of emulating synaptic function, which include PCMs, MRRs, and MZIs. An effective way to realize a reconfigurable photonic device is to manipulate the refractive index (real part = optical phase or imaginary part = loss) of the medium in which the light propagates. This can be achieved using a phase change material (PCM), which is frequently used in optical switching applications [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. Indeed, this material has the unique ability to CHAPTER 1. INTRODUCTION AND STATE OF ART change from a crystalline state to an amorphous state (vice-versa) due to thermal effects induced optically or electrically [START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF]. Thus, PCMs are highly transmissive in their amorphous state, but absorb more light as they crystallize. Moreover, several intermediate levels can be reached during the transition from one state to another, allowing to implement optical weighting operations with arbitrary values [START_REF] Xu | A survey of approaches for implementing optical neural networks[END_REF]. Apart from their high compactness (table 1.4), another notable advantage of PCMs is the stability of their phase in time. This allows the use of non-volatile attenuators (weights) that do not require a constant power supply during the inference phase. Hence, large-scale, ultra-fast, high-integration-density synaptic networks with low energy consumption are plausible with this device. The reference [START_REF] Cheng | On-chip photonic synapse[END_REF] presents an all-optical silicon nitride (Si 3 N 4 ) photonic synapse that integrates a PCM with a waveguide. An overview of their on-chip photonic synapse (red box on the left) can be seen in Fig 1 .13.a, which consists of a tapered waveguide on which discrete chalcogenide (GST) PCM structures are incorporated. A SEM image of this region is shown in Fig 1 .14.a-(right). The authors use optical pulses with fixed time and energy to modulate the states of the PCM islands, and thus the synaptic weight between pre-and post-neuron synapses. This modulation will induce a variation in the absorption of the PCM cells, thus CHAPTER 1. INTRODUCTION AND STATE OF ART affecting the transmission strength of the optical electric field. They also demonstrated that synaptic weights vary as a function of the number of optical pulses sent through the waveguide, each with 243 pJ of energy and 50 ns of duration. Moreover, switching between arbitrary weights (known as "0, ""1, ""2, ""3" in the experiment) can be achieved just by sending a variety of known pulses (1000, 100, 50, and 1, respectively) [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF] to PCM elements. In addition, the authors have implemented an effective all-optical STDP rule by exploiting the reconfigurability of this photonic device.

Photonic synapses

An example of a large-scale photonic synaptic network based on PCM devices is shown in Fig 1 .13.b. This prototype proposed in the work of [START_REF] Feldmann | Parallel convolutional processing using an integrated photonic tensor core[END_REF] illustrates a photonic integrated PCM crossbar, which is a simple way to construct a matrix multiplier. This rectangular crossbar performs parallel MAC operations in a convolutional neural network (CNN) through N inputs and M outputs coupling elements. For instance, they achieved 10 12 MAC/s with 95.3% accuracy for this prototype, versus 96.1% for a numerically equivalent CNN model.

Micro-Ring Resonators (MRRs) bank

The microring resonators (MRRs) bank is another option for achieving a photonic reconfigurable weighting unit. MRRs have the advantage of being compact, easily tunable, and highly compatible with wavelength division multiplexing (WDM) technology (an essential building block for scalable photonic neural networks) [START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF], making them natural candidates for implementing synaptic networks capable of emulating a dense matrix multiplier compatible with on-chip integration.

An MRR weight bank with four channels was demonstrated in [START_REF] Tait | Multi-channel control for microring weight banks[END_REF]. The unit, as illustrated in Fig 1 .14.a, consists of multiple parallel-coupled microrings with increasing radii and two output ports, referred to as "through" and "drop" ports. The different radii allow the targeted wavelength to be different for each ring, each wavelength representing one element of the input data vector. To perform analog weighting operations, each MRR attenuates more or less the transmission of a specific WDM signal by tuning on and off resonance using thermal effects. This can be achieved with metal heaters at the top of the rings, as can be seen in Fig 1.14.b. With this design, the weights between the two ports can vary between -1 and +1, considering both the excitatory and inhibitory neuronal connections. Thus, by assembling this unit into banks, one can implement a linear weighted sum (or a matrix multiplier) in the form of a "broadcast and weight" (B&W) scheme [START_REF] Tait | Broadcast and Weight: An Integrated Network For Scalable Photonic Spike Processing[END_REF], using photodetectors (PDs) for summing operations.

Fig 1 .14.c shows a schematic representation of an on-chip optical matrix-vector multiplier [START_REF] Yang | On-chip optical matrix-vector multiplier for parallel computation[END_REF], consisting of a laser-modulator array, an MRR matrix (or banks), and a PD array. An integrated power splitter and wavelength multiplexer work together to split and converge light, in a B&W fashion. This configuration can perform the matrix-vector multiplication of A•B=C, where A is the transmissivity of the M × N MRRs matrix, B is a vector containing the output signals of N × 1 laser-modulator array, and C is the resulting vector representing the output signal intensity measured by the M × 1 PD array. The authors reported an operation rate An example of a synaptic dot product implementation with N × M MRRs units [START_REF] Yang | On-chip optical matrix-vector multiplier for parallel computation[END_REF]. Note that, the multiplier coefficients are all positive, whereas they can be either positive or negative in designs (a) and (b). MD: modulator, PD: photodetector of 8 × 10 7 MAC/s for parallel matrix multiplication using the on-chip microring modulator matrix, which corresponds to a processing speed of 20 Mb/s.

Despite MRRs' potential to implement a high-performance photonic matrix multiplexer, their application is limited due to their high sensitivity to thermal fluctuations and vibrations that require considerable control in order to prevent accuracy and stability losses. To overcome this problem, calibration techniques such as feedforward control [START_REF] Tait | Multi-channel control for microring weight banks[END_REF] or feedback control [START_REF] Jayatilleka | Wavelength tuning and stabilization of microringbased filters using silicon in-resonator photoconductive heaters[END_REF] have been studied -but it still makes up a real challenge to implementing the B&W approach.

Mach-Zehnder Interferometer (MZI) meshes

In the previous example, a change in the amplitude of transmission and incoherent power summation is used. As an alternative, coherent interference and changes in the optical phase (refractive index) can also be used for high-efficiency matrix multipliers. This can be achieved with a mesh of several Mach-Zehnder interferometers (MZIs), which is also CMOS-compatible.

An example of such a unit is introduced by [START_REF] Shen | Deep Learning with Coherent Nanophotonic Circuits[END_REF] and consists of integrated photonic bidirectional couplers, as depicted in the red rectangle of Fig 1 .15.a. Phase shifters, which can be located on any two straight arms of the MZI, control the division ratio and the relative phase between the two input modes, respectively, thereby allowing the MZI to be reconfigured and the weights to be tuned. Typically, this configuration is used as a unitary 2 × 2 matrix, which, arranged in a mesh with multiple other devices of the same type, can build an optical matrix multiplier, or optical interferometric unit (OIU). This unit is based on the singular value decomposition of a non-unitary weight matrix (of real values) M = U ΣV * , where U and V * are unitary matrices that can be implemented with MZIs and Σ, a diagonal matrix, usually implemented with optical attenuators or optical amplification materials. This unit's performance at a system level is detailed further in section 1.2.4.

When implementing an OIU, various mesh topologies exist, but the most commonly used are rectangular, triangular, and diamond [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF], [START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF], as illustrated in Fig 1 .15.b-d. Rectangular meshes offer more compactness, while triangular meshes support self-configuration mechanisms and finally, diamond topologies can provide more degrees of freedom for optimizing weight matrices during backpropagation learning [START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF]. However, photonic neural networks based on MZI units must deal with the large footprint of MZI and phase noise accumulation of all-optical configurations that can significantly limit their scalability [START_REF] Shen | Deep Learning with Coherent Nanophotonic Circuits[END_REF].

Photonic spiking neurons

A prerequisite for emulating a spiking neuron is to replicate its ability to exhibit excitable behavior. Excitability, which is the basis for all-or-none responses in neurons, is characterized by the following key properties [START_REF] Tait | Photonic Neuromorphic Signal Processing and Computing[END_REF], [START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF]: 1. temporal integration of a weighted sum of incoming spikes, 2. release of a spike when the integrated signals exceed a certain level, 3. immediately after a spike excursion, return of the neuron to its rest potential and exhibition of a refractory period. Considering these properties is therefore crucial for implementing a spike processor that is computationally efficient and robust to noise. Additionally, spike processors must also be capable of stimulating or inhibiting other neurons to ensure successful communication between layers of the network. This is the idea behind cascadability, which is another important requirement of neurons, where their output signal must be strong enough to drive other neurons [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF]. Numerous studies have shown that excitable photonic devices can exhibit spiking neural properties, but with faster operating speeds than their biological or electronics counterparts [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF], [START_REF] Xiang | A review: Photonics devices, architectures, and algorithms for optical neural computing[END_REF]. Furthermore, depending on their operating mechanisms, they can be classified as all-optical or opto-electronic (O/E/O), as we will explain in the following. Table 1.4 summarizes the current state of the art of spiking photonic neurons with electrical or optical injection.

Optoelectronic spiking neurons

The concept of optoelectronic or O/E/O (optical to electrical to optical) link was first introduced in [START_REF] Nahmias | An integrated analog O/E/O link for multi-channel laser neurons[END_REF] to enable integrated laser neurons to receive multiple input signals simultaneously. Typically, it consists of two sub-circuits, one for accumulating presynaptic optical inputs (O/E), and one for implementing the nonlinear functionality (E/O) [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF]. The O/E sub-circuit uses generally photodetectors to convert optical input signals into electrical signals, which in turn drives the E/O sub-circuit. As explained in ref [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF], nonlinear models such as lasers, modulators, or electronic circuits can be used to implement the spiking E/O sub-circuit, representing the photonic neuron. However, since we're only interested in lasers, we narrowed down this section to E/O sub-circuits based mainly on semiconductor lasers whose excitability is controlled by electrical injection. Readers are referred to [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF] or [START_REF] Shastri | Photonics for artificial intelligence and neuromorphic computing[END_REF] for an in-depth review of photonic continuous-time neurons. This class of neurons was first fully studied in 2013 by Nahmias et al [START_REF] Nahmias | A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing[END_REF], who demonstrated the isomorphism between an LIF neuron and a two-section semiconductor laser model. Indeed, it has been shown that semiconductor lasers with two sections (gain vs. saturable absorber) can demonstrate bistable or self-pulsing dynamics when excited above their thresholds [START_REF] Dubbeldam | Self-pulsations of lasers with saturable absorber: Dynamics and bifurcations[END_REF][START_REF] Ueno | Conditions for self-sustained pulsation and bistability in semiconductor lasers[END_REF][START_REF] Harder | Bistability and pulsations in semiconductor lasers with inhomogeneous current injection[END_REF][START_REF] Barbay | Excitability in a semiconductor laser with saturable absorber[END_REF]. In the case of spiking neuron implementation, excitability can be observed close to a regime of self-pulsing, which is also known as Q-switching [START_REF] Barbay | Excitability in a semiconductor laser with saturable absorber[END_REF]. Self-pulsing dynamics occurred at the relaxation oscillation frequency of the device and could be tuned by changing the laser bias. This non-linear mechanism will be further discussed in section 2.1.2 of Chapter 2. Here, we detail the functionality of two notable O/E/O neurons based on the work of [START_REF] Nahmias | A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing[END_REF] that have inspired many others, including this thesis.

CHAPTER 1. INTRODUCTION AND STATE OF ART

Two-section lasers

Photonic neurons can be implemented with two-section excitable lasers since they exhibit the same excitability properties as the LIF neuron model [START_REF] Nahmias | A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing[END_REF], [START_REF] Shastri | SIMPEL: Circuit model REFERENCES TO CHAPTER 1 for photonic spike processing laser neurons[END_REF]. For instance, the work of [START_REF] Peng | Temporal Information Processing With an Integrated Laser Neuron[END_REF] has demonstrated the dynamical LIF features of a distributed feedback (DFB) laser photonic neuron, both theoretically and experimentally. .a shows the proposed neuron, integrated in a hybrid III-V/silicon platform. The optical cavity consists of optically coupled absorbing (small section) and gain medium (large section) that are electrically isolated through an etching. To ensure a single-mode operation within the cavity, a DFB grating is used as a mirror. Note that, this type of laser emits light in the planar direction in contrast to vertical-cavity surface-emitting lasers (VCSEL), where light is emitted in the upward or downward [START_REF] Shastri | SIMPEL: Circuit model REFERENCES TO CHAPTER 1 for photonic spike processing laser neurons[END_REF]. The laser neuron is also connected to a pair of Balanced PhotoDetectors (BPDs), which modulate the carrier density of the gain medium with a photocurrent. Here, the photodetectors PD1 and PD2 function as inhibitory and excitatory CHAPTER 1. INTRODUCTION AND STATE OF ART synapses, so that the generated photocurrent can either stimulate or inhibit the laser neuron. Additionally, two constant bias currents are used to pump the laser, one of which provides gain and the other adjusts the excitability threshold. Therefore, under favorable pumping conditions, the interaction between the two media can lead to nonlinearities, favoring the transition of the laser to an excitable regime.

The spiking dynamics of this laser have been investigated theoretically using the wellknown Yamada model [START_REF] Yamada | A theoretical analysis of self-sustained pulsation phenomena in narrowstripe semiconductor lasers[END_REF] where spike generation depends on the recovery time of the gain medium. Furthermore, the authors simulated a network of laser neurons that was able to perform the XOR classification with 100% accuracy despite timing jitter.

Graphene-based excitable Lasers

Graphene naturally exhibits strong first-order absorption and Kerr effects [START_REF] Tamura | Inducing optical self-pulsation by electrically tuning graphene on a silicon microring[END_REF], and can be easily integrated into silicon photonic platforms as it is CMOS compatible. So, instead of the conventional saturable absorber (SA) semiconductor, authors in ref [START_REF] Shastri | Spike processing with a graphene excitable laser[END_REF], have proposed an integrated photonic neuron that used graphene as SA. This Graphene Excitable Laser (GEL) is analogous to a fiber-based graphene prototype (as highlighted in [START_REF] Shastri | Spike processing with a graphene excitable laser[END_REF]), whose excitability properties have been compared and demonstrated theoretically and then experimentally.

The integrated device has an electrically pumped multiple quantum well (MQW) gain medium, two graphene sheets (SA section), and a DFB grating for the cavity feedback, as illustrated in Fig 1.17.a. The device's behavior is simulated using the Yamada model as well as the fiber-based GEL. As shown in Figs 1.17.b-d, the simulated (pink) response of the integrated GEL's model presenting spiking dynamics has been compared to experimental results measured (blue) for the analogous fiber-based GEL device. The integrated GEL device has demonstrated spiking neurons' characteristics, such as excitability threshold, leaky integration, as well as absolute and relative refractory periods. Note that, even if both devices seem to display similar behaviors, the integrated device can operate on a faster time scale (ns vs µs, see Fig 1

All-optical spiking neurons

All-optical neurons describe nonlinear systems whose excitability arises from optical perturbations, which can be either coherent or incoherent light [START_REF] Prucnal | Recent progress in semiconductor excitable lasers for photonic spike processing[END_REF]. As a result, electro-optical conversion is not required when it comes to intra-neuron communication [START_REF] Sunny | A Survey on Silicon Photonics for Deep Learning[END_REF]. Moreover, compared to O/E/O spiking neurons, this category comes in a wide variety of configurations, ranging from excitable lasers to nonlinear passive cavities to photonic crystal nanocavities. For a long time, all-optical spiking neuron implementation has been limited by exotic manufacturing processes or a lack of appropriate devices [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF], [START_REF] Xu | A survey of approaches for implementing optical neural networks[END_REF]. However, recent advances in photonic device design as well as the development of a new photonic platform, such as silicon photonic platforms, have enabled the realization of all-optical neurons. Some of these neurons are reviewed below.

VCSEL-SA neurons

A VCSEL is a natural candidate for implementing a photonic neuron with attractive advantages. They can offer high network scalability due to their compactness in addition to their low power consumption, low cost, and easy integration [START_REF] Koyama | Recent Advances of VCSEL Photonics[END_REF]. One common way to achieve In ref [START_REF] Fu | Multilayer Photonic Spiking Neural Networks: Generalized Supervised Learning Algorithm and Network Optimization[END_REF], a VSCEL with an intracavity saturable absorber (VCSEL-SA) is employed as a primitive LIF neuron in a simulated multilayer photonic SNN. As can be seen in Fig 1 .18.a, the VCSEL-SA cavity consists of an amplifying medium (Gain) and a SA region, both composed of III-V materials. Two DFB mirrors are used to provide intra-cavity feedback. The laser neuron operates with optical injection, which means that input optical signals can cause the laser to fire a spike if its threshold is exceeded. After that, the recovery mechanism takes place inside the cavity, where the laser can hardly fire another spike (see Fig 1.18.b). This behaviour reflects the LIF neuron's fundamental properties, such as leaky integration, spike generation, and refractory period. This device was further used as an LIF unit embedded in a network to solve the XOR problem numerically, with promising results on both the Iris (96% of accuracy) and Wisconsin breast cancer (96.1% of accuracy) datasets.

Numerical studies on VCSEL-SA have also demonstrated other important neuronal properties, including cascadability [START_REF] Zhang | Spike encoding and storage properties in mutually coupled vertical-cavity surface-emitting lasers subject to optical pulse injection[END_REF] and inhibition [START_REF] Zhang | All-optical inhibitory dynamics in photonic neuron based on polarization mode competition in a VCSEL with an embedded saturable absorber[END_REF]. Furthermore, VCSELs-SA have shown great potential in encoding spikes, storing spikes, or sound detection [START_REF] Nahmias | A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing[END_REF], [START_REF] Zhang | Spike encoding and storage properties in mutually coupled vertical-cavity surface-emitting lasers subject to optical pulse injection[END_REF], [START_REF] Zhang | Spike encoding and storage properties in mutually coupled vertical-cavity surface-emitting lasers subject to optical pulse injection[END_REF]. However, as an alternative, some authors have shown numerically and experimentally that conventional VCSELs can also mimic neural non-linearity through the polarization switching effect or optical injection, as in ref [START_REF] Hurtado | Optical neuron using polarisation switching in a 1550nm-VCSEL[END_REF], [START_REF] Xiang | Emulation of Spiking Response and Spiking Frequency Property in VCSEL-Based Photonic Neuron[END_REF], [START_REF] Xiang | Cascadable Neuron-Like Spiking Dynamics in Coupled VCSELs Subject to Orthogonally Polarized Optical Pulse Injection[END_REF], but this requires the intervention of several third parties and specifically a second laser off-line.

Micropillar neurons

Another two-section gain and SA-laser neuron model with the same structure of a VCSEL was demonstrated in refs [START_REF] Selmi | Spike latency and response properties of an excitable micropillar laser[END_REF], [START_REF] Selmi | Relative Refractory Period in an Excitable Semiconductor Laser[END_REF] and [START_REF] Selmi | Temporal summation in a neuromimetic micropillar laser[END_REF] that consist of two and one InGaAs/AlGaAs QWs, respectively. Nonlinearity in this device is introduced by pumping the active region optically while having no pump in the passive region, acting thus as a saturable absorber (SA). The laser is therefore biased below its selfpulsing threshold and optical perturbations of varying amplitudes are sent onto the cavity to demonstrate excitability. As illustrated in Fig 1 .19.b, the laser shows a step-like behavior with an excitable threshold depending on the bias pump level (P). Here, the further the bias pump is from the self-pulsating threshold, the less likely the excitable regime is to be established. The authors in ref [START_REF] Selmi | Relative Refractory Period in an Excitable Semiconductor Laser[END_REF] have also shown that the micropillar laser can exhibit absolute and relative refractory periods thanks to its cavity gain recovery mechanism. In Ref [START_REF] Selmi | Temporal summation in a neuromimetic micropillar laser[END_REF], the temporal summation functionality (or the leaky integration) is experimentally demonstrated using this all-optical neuron. Finally, even though the micropillar laser has shown spiking properties comparable to neurons, it is not capable of supporting inhibitory functions, which can limit the expressiveness of photonic SNN.

PCM-based neurons

PCM cells can also serve in the design of photonic neurons when integrated with microring resonators (MRRs) as in [START_REF] Feldmann | Alloptical spiking neurosynaptic networks with self-learning capabilities[END_REF] and [START_REF] Chakraborty | A Photonic In-Memory Computing primitive for Spiking Neural Networks using Phase-Change Materials[END_REF]. Fig 1.20.a shows an SEM image of the all-optical PCMbased photonic neuron that can mimic the spiking properties of the IF neuron [START_REF] Feldmann | Alloptical spiking neurosynaptic networks with self-learning capabilities[END_REF]. As for the photonic synapse, the changing state of the PCM cell is used to influence the response of the MRR unit.

To generate an optical pulse, the ring uses an 'output' waveguide, which allows a probe or a bias input signal to be coupled to the MRR. Whenever the PCM is in its crystalline phase (thus very absorbent), the probe signal resonates with the ring, resulting in no output pulse. However, if the input signal is strong enough to turn the PCM amorphous, an output pulse will be generated. In this case, the input signal is the sum of the weighted input spikes from the pre-synaptic neurons-like, combined using a WDM multiplexer (MUX). In Fig 1 .20.b, the normalized MRR transmission is shown as a function of the input pulse energy accumulated in the 'input' waveguide at a fixed wavelength. The curve is similar to the activation function of an IF neuron (similar to the ReLU function), with the neuron emitting an output pulse above a certain threshold, here, 430 pJ. In addition, the PCM-based neuron model may also be capable of performing efficient excitatory and inhibitory functions in an all-optical large-scale photonic SNN with low consumption, as proposed in Ref [START_REF] Selmi | Temporal summation in a neuromimetic micropillar laser[END_REF].

It should be noted that this configuration is known as an all-optical non-coherent scheme since the output wavelength can differ from the input one, as we will see in section 1.2.4. In other words, this kind of neuron can be implemented with WDM technology, but its reliability may be compromised by inter-channel crosstalk. Furthermore, another notable drawback is its low cascadability due to the need for extended lasers for pulse generation.

Comparison of photonic spiking neurons

This section presents the current state of the art of photonic spiking neurons. Table 1.4 compares the performance of these neurons regarding CMOS compatibility, energy consumption, processing speed, and footprint. Note that, energy consumption is closely related to pump power whereas the processing speed is tied to the refractory period.

In general, all-optical neurons tend to be more energy efficient and faster than O/E/O neurons while being CMOS compatible, especially for passive cavities (Microresonator, MRR, and Nanobeam). Indeed, O/E/O neurons suffer from optoelectric (i.e., O/E and E/O) and wavelength conversion stages, including demultiplexing [START_REF] Shastri | Photonics for artificial intelligence and neuromorphic computing[END_REF], although their integration is easier. However, since all-optical neurons typically produce weaker output signals than their inputs, CHAPTER 1. INTRODUCTION AND STATE OF ART cascading them requires optical amplifiers [START_REF] Tamura | Inducing optical self-pulsation by electrically tuning graphene on a silicon microring[END_REF], which can limit their reliability and consequently scalability. Also, their high sensitivity to noise and fabrication-process variations [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF] is another concern for their scalability.

Finally, passive devices and integrated VCSELs are among the most compact neurons and are therefore highly scalable devices, allowing them to support a large number of input neurons (i.e. fan-in) for large-scale photonic SNN.

Photonic neural network implementation

Numerous researches have been conducted to realize high-performance and scalable photonic versions of neural networks integrated on a chip. O/E/O and all-optical links have allowed various photonic devices to achieve nonlinear activation functions that feature high processing speed, high bandwidth, and low power consumption [START_REF] Sunny | A Survey on Silicon Photonics for Deep Learning[END_REF]. This leads to the definition of what is called a processing network node (PNN) [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF], a simulation of a spiking neuron, which combines weight summation and firing functions into the same unit. So, an integrated neural network can be formed by interconnecting several of these computational nodes with a protocol that would enable high-density connectivity. In photonics, such a network can be directly implemented by employing WDM technology, which offers high optical bandwidth as well as high parallel processing. However, this implementation is only supported by neurons that admit the incoherent injection schemes. The following sub-sections give a review of large-scale SNNs implemented with the incoherent and coherent approach for the PNN model.

Incoherent approach

The incoherent approach involves PNN models that can operate at multiple wavelengths whether they are used in the O/E/O or all-optical link. Incoherent PNNs can leverage WDM technology to allow for weighted summation and reconfigurability schemes in a photonic network. The broadcast-and-weight (B&W) is a typical architecture that uses WDM to easily interconnect a scalable number of PNN on a photonic chip. Synaptic connections between nodes can be reconfigured efficiently with this protocol using multi-wavelength operations, as illustrated in to a network of N computational nodes linked to N filters, resulting in N 2 connections [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF].

Each filter represents a bank of tunable MRRs, which modulates the transmission of a specific wavelength, giving a positive or negative analog weight before being summed by photodetectors (BPDs). The corresponding processing node converts each electrical signal into an optical signal (O/E converter) and the output is finally fan-out to other nodes. Also, by using the BW framework, network topologies such as recurrent and feedforward can be implemented. This architecture was initially intended to be implemented using excitable lasers in the O/E/O link [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF], [START_REF] Tait | Broadcast and Weight: An Integrated Network For Scalable Photonic Spike Processing[END_REF], as shown in Fig 1 .21.b. However, this protocol has been recently extended to time-continuous neurons, as MZ modulators [START_REF] Tait | Demonstration of a silicon photonic neural network[END_REF] and derived to be used in an all-optical link as we will see below.

PCM-based all-optical SNN

The B&W protocol has been adapted in the work of [START_REF] Feldmann | Alloptical spiking neurosynaptic networks with self-learning capabilities[END_REF] to experimentally demonstrate an on-chip all-optical SNN. As previously explained, PCM cells can implement both synaptic and spiking neuronal functions in a silicon platform, enabling such a network to be realized. The proposed all-optical neural network architecture is illustrated in Fig 1 .22.a, consisting of an input layer, N hidden layers, and a single output layer. In addition to neural and synaptic functions, each of these layers is made up of a "collector" and a "distributor". The Fig1.22.b depicts a photonic representation of a single layer from the network built with a bank of MRRs sharing a bus waveguide. Thus, using a WDM scheme, the optical signals from the previous layers are combined into the single bus waveguide, forming the collector unit. Then, the distributor demultiplexes the resulting signal with MRRs and sends it to the neurons within this layer. Weighting and threshold operations are achieved using integrated PCM elements (see section), while summing operations are enabled by the WDM multiplexer, allowing for alloptical information processing. This all-optical device is one of the first neural circuits to be experimentally realized with adaptive and learning capabilities. The all-optical dimension of this circuit could allow it to process large amounts of information faster than biological neural networks with relatively low power. However, as discussed above, the performance of SNNs based on incoherent photonic neurons may be limited by crosstalk and wavelength conversion. As well, in this case, processing speed is also a notable concern due to the slow switching operation in PCM devices. In addition, several other optical elements such as amplifiers, lasers or modulators still need to be integrated on the same chip, which limits the scalability of such a network.

Coherent approach

A coherent system requires that the input wavelength be the same as the output wavelength, i.e. it operates with a single wavelength. Coherent photonic neural networks can only be implemented with all-optical links. As a result, this approach cannot benefit from WDM technology as the optical signals are hardly distinguishable from each other [START_REF] Shastri | Principles of Neuromorphic Photonics[END_REF]. Instead, processing nodes from this system manipulate other aspects of light, such as optical modes or light polarization, for network reconfigurability [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. Ref [START_REF] Shen | Deep Learning with Coherent Nanophotonic Circuits[END_REF] proposes an integrated coherent all-optical neural network whose scalable struc- It is pertinent to note that the proposed network relies on MZI units, which are in fact non-spiking neurons. Nevertheless, we consider this work as a case study to understand the concept of a coherent all-optical approach and how it differs from the non-coherent one. Indeed, as depicted in the figure, the network is implemented using several layers of integrated coherent PNN, each consisting of a collection of MZI meshes forming an optical interference unit (OIU) and a nonlinear optical unit (ONU). As we have seen in section 1.2.2.3, OIU is used to build a photonic matrix multiplier that couples light from one layer to another. In contrast, ONU implements nonlinear activation functions through saturable absorption, represented by the digital function f N L .

For demonstration purposes, the authors conducted a vowel recognition experiment with this architecture. A fabricated OIU structure is used to experimentally perform both optical matrix multiplication (red meshes) and attenuation (blue meshes) (see Fig 1 .23.b). Additionally, digital electronics is employed to implement off-chip nonlinear functions of the saturable absorber. Here, the input data are first pre-processed off-chip, and then encoded into electrical field amplitudes that propagate through the integrated OIU. Thus, the overall coherent circuit is trained off-line using backpropagation and its performance is evaluated on the set of vowels (A, B, C, and D). Fig 1 .23.c shows that the circuit was able to correctly classify vowels with 76.7 % accuracy compared to the 91.7% correctness achieved by the same architecture implemented on a conventional 64-bit computer.

Although wavelength operation is limited, coherent approaches can enable high-speed and energy-efficient photonic neural networks [START_REF] Shen | Deep Learning with Coherent Nanophotonic Circuits[END_REF]. Indeed, it is less subjected to electrical parasites as in O/E/O systems with a B&W scheme and multi-wavelength operation that can be bottleneck [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. However, there are some real challenges in synchronizing information as well as computational resolution resulting from phase noise accumulation that usually occurs in coherent systems [START_REF] Shastri | Photonics for artificial intelligence and neuromorphic computing[END_REF], [START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]. MZs have much larger footprints than tunable MRRs, which reduces the density of interconnection in this type of system, hence cascadability.

Chapter summary

This chapter provides a comprehensive overview of the current state-of-the-art in the field of spiking neural networks (SNNs) and their implementation in photonics.

First, we introduced the concept of SNNs and their significance in artificial intelligence (AI). The description of SNNs was mainly based on their two basic components: neurons and synapses, with emphasis on their similarities to biological neurons. Unlike traditional artificial neural networks that use continuous-valued activations, SNNs employ discrete, asynchronous spikes to transmit information between neurons. This spike-based communication allows for more efficient and realistic modeling of neural behavior.

In the second part, we reviewed how SNNs have been implemented in photonics to date, both at the device and circuit level. As we have seen, photonics-based neuromorphic systems offer a promising alternative to traditional electronic systems, with the potential for high speed, high bandwidth, and low power consumption. However, SNN learning in photonics still poses a considerable challenge in photonics due to the lack of efficient algorithms and appropriate hardware architectures that can handle a large amount of data. In this chapter, we present the design details of our laser photonic neuron based on an edge-emitting semiconductor laser with an intra-cavity SA. In the beginning, pulse generation mechanisms in semiconductor lasers will be reviewed in section 2.1, with an emphasis on passive Q-Switching (QS) and Mode-Locking (ML) mechanisms. Then, we will focus on the main design criteria of our laser system in section 2.2. The laser fabrication process on the CEA Leti's III-V/silicon platform will also be addressed. Finally, in section 2.3, the travelling wave model will be used to verify the spiking dynamics of our laser through numerical studies, and to investigate the influence of the SA parameters on its dynamics.

Short pulse generation in semiconductor laser

The inherent characteristics of semiconductor lasers, such as fast carrier decay as well as wide gain, make them attractive for ultra-short pulse generation. Two techniques are generally used for this purpose: Q-switching (QS) and mode-locking (ML). It has been shown in the previous chapter that integrating a saturable absorber (SA) into the laser's cavity, while using a single active material, can result in a more cost-effective and efficient implementation of the Q-switching technique. In this case, the operation can be classified as "passive" since no additional resources or energy are required, otherwise, we call it "active". An operation involving Q-switching is commonly described as a self-pulsing regime, which is the mechanism we are the most interested in. However, this mechanism can be altered when the laser is operated in multiple modes, resulting in an instability that implies Mode-Locking and is known as Q-Switched Mode-Locked or QS instability [1]. Here, after describing the mechanism of saturable absorption in a semiconductor, the following sections will describe the principles of each of these three mechanisms.

Saturable absorption in semiconductor

A saturable absorber (SA) is a device or material that exhibits a non-linear optical response to incident light, allowing for control over the intensity and transmission of light. In semiconductors, this property is mostly due to the Pauli exclusion principle [2], which states that two electrons cannot occupy the same energy state. This mechanism is illustrated in Fig 2 .1.a in the case of a bulk semiconductor with direct energy bands. At low incident light intensity (or low excitation) linear absorption dominates in the material i.e. photons with energies higher than the band gap are absorbed, resulting in electron-hole pairs that fill active states. However, as the intensity of incident light increases, the nonlinear absorption process becomes more evident since the final states are partially CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING occupied (Pauli effect) [2], resulting in a rapid decrease in absorption strength. The transmittance of the material is then at its highest and the absorber is said to have "bleached". This is followed by the recovery of the absorption through subsequent carrier redistribution and thermalization on the nanosecond time scale [4].

Note that, in the case of quantum well structure, although the absorption mechanism is not fundamentally different from the bulk structure, absorption can be more prominent due to the so-called Quantum-Confined Stark Effect (QCSE). This phenomenon occurs when a reverse bias is applied to the semiconductor material. The bias creates an electric field that will tilt the band structure (see Fig 2 .1.b) causing the energy bands to shift and the absorption to increase. As a result, generated carriers are swept out faster and absorption recovery time is shortened. According to the work of Karin et al [5], the recovery time in quantum well (QW) material can be related to the reverse bias V bias through the following function:

τ s = 2πm * L 2 w K B T exp( E b -L w e(V bias -V bi )/2d K B T ) (2.1)
where m * is the effective mass, L w is the QW width, E b is the confinement potential, V bi is the built-in potential, d is the width of the depletion region, K B is the Boltzmann constant and T, the temperature. According to the findings of [5], the absorption recovery time of a QW material decreased from 40 ps to a few ps under reverse biasing ranging from 0-4 V. Also, this technique has been widely used to create an absorbing medium for multi-contact semiconductor lasers, as reported in [6], [7], [8]. The application of a reverse bias to the absorbing section of a multi-contact laser is a commonly used method to reduce its recovery time. This is the method we will use in this thesis.

Passive Q-switching in semiconductor laser

Q-switching (QS) is a mode of operation of a laser, in which a cavity stores optical energy for a time, and then releases it in the form of a short, high-power pulse [1]. In general, pure QS involves only one axial mode for the laser [9]. Otherwise (in multi-mode operation), the QS efficiency is severely reduced and affected by intensity noise associated with modes that are not in phase, leading to instability related to Mode-Locking operation [9]. This phenomenon is described in more detail in Section 2.1.3.

Most QS semiconductor lasers consist of a gain section and at least a saturable absorber section that are electrically separated. With this configuration, a forward bias can be applied to the gain section for light amplification, while a reverse bias or unpumped absorber section provides absorption. Due to the absorber's nonlinear response, the cavity's Q-factor (defined as the ratio of the energy stored in a cavity to the energy lost per cycle) can be modulated. The conceptual mechanism of QS is shown in Fig 2 .2. When pumping starts, the laser is turned off since the absorber maintains a high loss state or low Q-factor in the cavity. Electric pumping increases the population inversion in the gain section, while absorption in the SA keeps the photon population low. However, the Pauli effect restricts the SA's ability to absorb indefinitely, leading to a state of saturation called "bleaching". Once the SA has bleached, the switch from high loss (low Q-factor) to low loss (high Q-factor) starts. The accumulated carriers deplete the gain through stimulated emission, which is transmitted rapidly as a short, intense light pulse. This process is then repeated as the gain and absorber reset, with the gain again lower than the cavity losses. As a result, a self-pulsing regime is achieved with a repetition rate that is limited by the relaxation oscillation frequency [10]) which for a semiconductor is about ten GHz [11].

For high-quality QS generation, specific parameters associated with the laser model namely, the two ratios da/dn (differential loss), and dg/dn (differential gain) as well as their respective recovery times, τ a and τ g , must satisfy the following conditions [11], [12]:

da dn > dg dn and τ a < τ g
Using the condition on the left, we ensure the SA reaches saturation before the gain, thus providing high-energy pulses. This condition is generally verified by semiconductors as they exhibit a sub-linear relationship between gain, absorption, and injected carriers, as we shall discuss later in this chapter. The second requirement (on the right) makes sure no unwanted oscillations occur during pumping. This can be satisfied by using non-uniform pumping conditions in the multi-contact laser. It should be mentioned that some values of these ratios could result in bistability between the fundamental ML and non-lasing regimes rather than QS operation for low drive currents [13], [14], [15]. However, this thesis will not address this particular regime.

Passive Mode-locking in semiconductor laser

Principle of operation

Another mechanism that could be involved in a multi-section cavity is the mode-locking (ML) mechanism. In contrast to the QS operation, this mechanism produces shorter pulses at much higher repetition rates but with less energy. ML works by forcing the longitudinal modes with CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING different frequencies in the laser cavity to have a fixed phase relationship [10]. As shown in Fig 2 .3.a, modes that are oscillating in phase are subjected to constructive interference, resulting in periodic ultra-short pulse trains at a repetition rate proportional to the inverse of the cavity round trip time defined as:

T rep = 2n g L c (2.2)
Where c is the speed of light, n g is the group refractive index, and L is the cavity length. In general, ML is also characterized by a time-bandwidth product that defines its minimum pulse duration as inversely proportional to its finite gain bandwidth. Therefore, the wider the gain bandwidth, the shorter the output pulse duration. Considering the high bandwidth available in semiconductors, high-quality pulses (nearly transform-limited) with time widths in the order of femtoseconds can be achieved. In a typical two-section semiconductor laser, passive ML can be considered primarily as a self-starting process that involves modulating cavity losses periodically at the laser cavity's round-trip time. Fig 2 .3.b illustrates the dynamics of time-varying gain and absorption loss in a passive ML laser. At first, unsaturated losses (blue curve) are greater than the gain, and spontaneous emission prevails in the cavity. As a result, longitudinal modes generated from noise have a random phase relationship with each other. A few round trips later, as the gain (red curve) reaches saturation, the random phase relation between modes becomes closer together, leading to noise peaks with higher intensity. This in turn saturates the absorber and enables a steady-state phase relation between modes and ML pulses are formed. The recovery mechanisms of these effects create a net gain window once per round trip that will amplify the fluctuations having the highest intensity and contribute to shortening the pulse as their leading and trailing parts will experience losses. The steady state is established when this effect is well balanced with pulse-broadening effects such as Self-Phase Modulation, detuning, and dispersion. These broadening effects are well referenced in [10]. Q-Switched Mode-Locked or QS instability means that a low-frequency QS envelope modulates the high-frequency ML pulses. This regime is typically observed when the laser operates slightly above its threshold, which corresponds to small pumping conditions. Due to insufficient available gain, the absorber cannot be fully saturated, resulting in the creation of net gain windows with amplitudes that change as the pump progressively increases [10]. This results in bunches of pulses of different amplitudes modulated by the Q-switching frequency with a temporality that lasts over several cavity rounds trip.

Q-switching instability is generally undesirable in applications where stability and reproducible properties are required. While a pure QS operation is clearly ideal for photonic neuromorphic applications, this regime still remains attractive due to the high pulse energy concentrated in a bunch of pulses. This is especially true if we take into consideration that our laser is unlikely to be pure single mode.

Design of the laser photonic neuron

This section describes step by step how we designed our spiking laser neuron. The neuron is an edge-emitting semiconductor laser optimized for pulsed operation through modification of the CEA Leti III-V/Silicon integrated continuous wave (CW) laser's heterostructure. The III-V/silicon platform offers tremendous flexibility in terms of design, allowing a wide range of active and passive photonic devices to be implemented in a single system. Moreover, edgeemitting lasers are naturally suitable for waveguide coupling with low losses, making them an attractive choice for scaling up all-optical or optoelectric processing network nodes (PNN). In order to provide a context for what follows, a brief review of the basic structure of the hybrid III-V/Si laser will be provided. Afterward, we will develop the building blocks of our laser neuron based on this heterostructure before describing its fabrication procedure.

CEA Leti's technology platform for continuous-wave lasers integration

2.2.1.1 Hybrid III-V/silicon laser hetero-structure Molecular bonding is the key technology used for hybrid laser integration on CEA Leti's III-V/silicon platform. Indeed, this technique involves direct bonding of III-V materials onto a low-loss Silicon-on-Insulator (SOI) substrate either at the die or wafer level [17]. The resulting hybrid structure is then able to benefit from the advantages of both III-V and silicon technologies, allowing for a reliable, highly efficient, and cost-effective laser device. In addition, this hybrid integration can be processed using standard CMOS technology, allowing for massive production at a low cost. In active waveguides, separated by a SiO 2 gap layer (typically 100 nm). The active waveguide at the top usually consists of multiple quantum wells (MQWs) structures surrounded by pand n-doped InP layers. This region is electrically pumped through p-and n-ohmic contacts to provide optical gain for light travelling through the cavity. At the bottom of this structure, optical functions such as rib waveguides, grating couplers, or Bragg reflectors are fabricated in an SOI waveguide. The rib waveguides are widened below the edges of the active waveguide to serve as mode transformers (or Tapers). This ensures that the supermodes formed by the hybrid structure are efficiently coupled with the silicon waveguide while still experiencing high optical gain in the center region of the III-V waveguide, as illustrated in Fig 2 .4 c. These Tapers generally achieve coupling efficiency higher than 90% [18], [19]. This enables light to be efficiently reflected by integrated mirrors, or collected by fiber-coupled surface grating couplers. Hence, optimizing mode transformers is crucial to achieve maximum efficiency in this type of hybrid laser architecture.

Continuous wave (CW) demonstration

An optical microscope image of an electrically pumped hybrid distributed feedback (DFB) laser is shown in

The laser spiking Neuron: overall design

Using the CEA Leti's laser prototype, we modified its heterogeneous amplifier region to build our laser photonic neuron. Thus, the fabrication process and amplifier design are quite similar, except for a few points that we will discuss below. The overall excitable device based on a 500-nm-thick SOI substrate is illustrated schematically in is used to separate the gain p-contact from the absorber p-contact, both physically and electrically. Therefore, all regions share the same n-contact. As the length of the SA is a critical parameter for efficient spike generation, we varied it on our lithographic mask for a case study. The absorber length is defined as a percentage fraction (starting from 2.5% to 10%) of the active section length, which can be either 400, 600, or 800 -µm-long.

A detailed discussion of the SA's design can be found in section 2.2.

• Mode transformers

The active SOI waveguides are bound using a 100-nm-thick SiO2 layer. As with the standard CW hybrid laser design, two silicon mode transformers are integrated to achieve adiabatic coupling between the two structures. Our design of the experiment (DOE) includes 25 different adiabatic silicon tapers with robust design, which can exhibit coupling efficiencies higher than 94%.

• Optical cavity

Two types of optical cavities are considered in this thesis: Fabry-Perot (FP) cavities and distributed feedback (DFB) cavities. Typically, the former is defined by either Sagnac loops or DBR (see Fig 2 .6.c) placed at both sides of the structure in order to form a resonant cavity. The DBR reflectors used for our lasers have a 200 nm grating period, 50% duty cycle and 10 nm etch depth, resulting in modal reflectivities up to 98% for the longer mirror (750 µm) and 70% for the shorter mirror (150 µm). The mirrors are partially etched into the underlying 500 nm-thick SOI waveguide, resulting in a low index contrast between the silicon and its native oxide, thereby narrowing its spectral selectivity. In other words, these DBR reflectors allow lasers to operate in a single longitudinal lasing mode, owing to their filtering capability.

Sagnac loops working as reflecting mirrors are also used to form single-mode FP cavities. These mirrors are based on 2x2 directional couplers and waveguide Sagnac interferometers. Overall, FP lasers with Sagnac loops have been designed to have power reflectivities of 35% for the front mirror and nearly 99% for the back mirror.

Another cavity type is DFB lasers, in which the cavity is fully defined under the III-V region (see Fig 2 .6.c). As for the DBR reflectors, the mirrors are partially etched along the rib layer of the SOI waveguide, which enables shorter cavities as opposed to FP cavities. Here, DFB structures are characterized by a quarter wavelength-shifted defect at the center of the cavity to promote single-mode operation at 1310 nm. Other typical parameters include an etch depth of 50 nm and a κ r L g product ranging from 1.5 to 3 for a grating length Lg varying between 400 and 800 um, κ r being the grating coupling strength constant.

Since the mirrors and adiabatic transitions have already been developed and optimized in previous PhD studies [18], [19], we decided to use them in this study. For this reason, we will not go into details about their design for the sake of simplification.

The next section will focus on describing the design of the SA medium which is the key part of our spiking laser.

Saturable Absorber's design 2.2.3.1 Preliminary Considerations

Two kinds of cavity configurations can be used for spike generation in semiconductor lasers. The first option implies placing the SA in the middle of the cavity, between two gain mediums as illustrated in Fig 2 .7.a. In this symmetric configuration, the main frequency is twice the fundamental repetition rate due to two counter-propagating pulses travelling through it, referred to as the colliding pulse mode (CPM). As the two pulses interact and interfere in the absorber section, saturation can occur more easily and efficiently with minimal pulse broadening. As an alternative, Fig 2 .7.b shows a two-section laser based on a self-colliding pulse mode (SCPM). Here, the SA is placed next to the high-reflectivity cavity mirror so that the pulse collides with itself. In this setup, the optical modes are separated by the fundamental cavity's repetition rate, c/2n g L. For a deeper understanding of these phenomena, please refer to [11], [20].

Generally, both configurations are used to produce high-quality passive ML in semiconductor lasers. However, the SCPM configuration has been shown to have similar and even better performance compared to the CPM arrangement with a shorter cavity, [11]. Nevertheless, in this thesis, we have chosen to implement our laser photonic neuron in the CPM configuration in order to keep the III-V to silicon tapers unchanged compared to the CW architecture. This shall give us a better starting point for this initial trial. Additionally, although the laser neuron will never be used in pure CW-ML, this CPM effect will be used to identify the laser's spiking dynamics regimes both numerically (see section 2.3) and experimentally. In both cases, the gain section(s) is (are) forward-biased while the absorber is reverse-biased.

Method for creating the saturable absorber

Designing and fabricating spiking lasers requires the separation of the gain and absorber sections electrically. Electrical isolation is essential to avoid current leakage between the two sections, which would reduce the effectiveness of the reverse bias applied to the absorber as well as increase thermal instability. Several techniques have been used to achieve highly resistive regions in semiconductor devices, including partial removal of highly p-doped layers [21], [22] or the ion implantation procedure [18], [19], [23], [24], [25]. Following our studies, in order to achieve a sufficiently high degree of isolation, we have chosen to implement both methods, thus ensuring the reliable operation of the spiking laser. To do so, small isolation cavities (see Fig 2.8) are first etched along the active waveguide and then implanted with hydrogen ions (H + ). To evaluate the effectiveness of these two techniques, we first studied the impact of those small cavities on the fundamental optical mode supported by the active waveguide. For the sake of simplicity, we simulated a single isolation cavity which is characterized by its adjustable length (L isol ) and depth (d isol ). The goal of this optical simulation is to find values for these two parameters that would provide a first level of isolation while maintaining a stable 2.2 shows the refractive indices for layers used in optical simulations. Here, we treated the MQW/n-InP cladding structure as an equivalent layer with an average refractive index, n MQW/n-clad . The following weighted geometric average relation [26] is used to calculate these indices:

n M QW/n-clad = m i=1 t i n 2 i m i=1 t i (2.3)
with t i and n i being the thickness and refractive index of each layer in the MQW/n-InP cladding structure, respectively. The 3D finite-difference time-domain (FDTD) method (by Lumerical software) is used for propagation analysis of the fundamental mode through the considered III-V waveguide. In these simulations, the length and depth of the isolation cavity vary from 1 µm to 10 µm and from 200 nm to 2000 nm, respectively. As part of the FDTD analysis, we extracted the full scattering matrix parameters of the structure to calculate both transmission and reflection coefficients. You can see the simulation results in Results indicate that the fundamental optical mode propagation is significantly altered near the MQW region. As the etch depth increases beyond 1.5 µm (the cavities are etched down to the top surface of the first layer of SCH), a noticeable loss in transmission is caused, as well as a significant increase in back-reflection, which certainly impacts the laser cavity. This change in transmission and reflection coefficients is exactly what we want to prevent. Therefore, we have limited the etch depth of the isolation cavity to 1 µm or less, to ensure that transmission of optical modes remains as expected. We also notice that for this cavity, the length parameter has a very limited impact compared to the etch depth. With these parameters, the current leakage between the gain sections and the SA can be limited to less than 2 mA per Volt of bias difference. The laser sections shall be biased around their threshold e.g. around +1V while the SA section may have to be strongly reverse biased e.g. around -3V. As a result, in our G-SA-G configuration, this would lead to 16 mA of leakage current. This is considered too high for the optimal behaviour of the laser neuron, hence we decided to explore the possibility of combining this isolation with H + injection. Thus, using the same setup, we simulated the effect that an H + implant would have on the isolation cavity's resistivity. The simulations were performed this time using the COMSOL Multiphysics commercial software. We used the AC/DC module from the software to evaluate the leakage current between two separated sections, representing the gain and the SA mediums respectively. Electrical resistivities of doped layers used for these simulations are collected from various experimental studies and are given in Table 2.2. The H + implant is implemented through a high-resistive region having the same geometrical parameters as the isolation cavity area. The electrical resistivity of this region is assumed to be ρ isol = 2.8 × 10 9 Ω.m. We deduced this resistivity from the work of [27], who experimentally studied the resistivity of p-type InP samples bombarded with H + protons. Fig 2.9.c,d depicted the leakage currents measured at a DC bias voltage of 1 V between the gain and SA contacts, for various parameters of the isolation cavity. It should be noted that Fig 2.9.d plots the dependence of the leakage current on the isolation cavity length for a fixed value of d isol = 0.4 µm. The red point on the curve, representing the case of a 10 µm-long implanted cavity, highlights the action of the implant. Thus, two important observations can be drawn from these two figures. First, without a proton implantation region (cf Fig 2.9.c), highly conductive layers such as p++ and p+ should at least be removed to ensure a first level of electrical isolation. The lowest leakage current (2 mA) is achieved with a cavity length of L isol = 10 µm, although it may be quite significant, particularly at high applied voltage. The second observation that we made is that protons implantation would lead to a more reliable isolation, as evidenced by the decrease in the leakage current from 2 mA to 0.03 pA for L isol = 10 µm and d isol = 0.4 µm.

To summarize this subsection, a two-step process will be followed to implement electrical isolation in our device. First, the active waveguide will be physically etched to separate the absorber contact from the gain contacts. In order to meet the requirements of our simulations, we chose a cavity with an area of 10µm × 6µm and an etch depth of 650 nm ± 100 nm (which include: 200 nm of InGaAs, 50 nm of InGaAsP and 400 nm of InP cladding). Second, H + protons will be implanted in the etched areas to guarantee effective electrical isolation between the two sections. However, in reality, the efficiency of the induced isolation can vary depending on the ion dose, mass, energy, and annealing temperature of the substrate. The following section describes how these parameters were determined.

Simulation of ion implantation

Ion implantation is a well-known process that has proven to be a reliable and effective way of modifying the electrical properties of materials for various applications. For instance, semiconductor lasers are commonly designed with regions of high electrical resistance around the active waveguide by using ion implantation, [7] [18], [19], [24]. Indeed, it allows the pump current to be confined to the middle of the structure, where the optical mode is at its maximum, maximizing pump current efficiency. The isolation works by reducing the concentration of free-charged carriers in semiconductors either through deep-level state creation within the bandgap or damage-induced compensation, [28], [29].

As mentioned previously, a number of parameters determine the effectiveness of ion implantation, such as ion energy, dose, mass, and substrate temperature. Here, the implant energy determines the range of penetration of the ions into the material, while the dose refers to the number of ions that are implanted per unit area. The ion's mass also influences the range of penetration induced, with lighter ones, such as H + ions, penetrating more deeply than heav- ier ones like O + or N + into the material. Temperature, on the other hand, can eliminate or strengthen the amount of damage that is caused to the material [30]. Hence, by optimizing these parameters, the implanted layer can be damaged in a controlled manner, thus preserving the quantum well layers from being altered. This is particularly critical because any damage to the quantum well layers could result in a change to its bandgap structure, which would in turn affect the device's performance.

Therefore, to determine the distribution of implanted ions in our structure, we used the Transport of Ions in Matter (TRIM) program from the Stopping and Range of Ions in Matter (SRIM) software package. The TRIM software uses the Monte Carlo method to predict the uniformity of ion-induced damage in a target material. For our application, it is necessary for the penetration depth to be large enough to create highly resistive regions while still not damaging MQW layers. To this end, we assessed the distribution profiles of H + protons at different energy values to identify the optimal parameters for the implant. As you can see from the plot in Fig 2 .10.b, the higher the energy, the further the ions can penetrate the material. At 230 keV, the average penetration depth corresponding to the peak concentration (known as the projected range), is evaluated at 2.3 µm, which is in direct interaction with the SHC upper layer. This projected range is observed to move far away from the MQW region at lower energies. Therefore, energies that are higher than 170 keV seem to be already critical for our design. However, our expectation is that even in the worst-case scenario, the SCH would act as a buffer layer, ensuring that the main MQW layers would remain unaffected. In addition, we also chose to conduct the implantation step at the very last stage of the laser fabrication process, specifically after the electrical contact deposition steps. In this way, the implant will not be exposed to high annealing temperatures (such as for instance during the back end of line (BEOL) fabrication steps) that could deteriorate its insulating effect. Otherwise, although the exact dose to achieve this step has not been studied here, our approach is to use the simulated dose value of 2 × 10 14 atoms/cm 2 , which is already used to create resistive zones in the CEA Leti CW laser, with a maximum energy of 170 keV.

Laser neuron fabrication

This section describes the fabrication process of a multisection laser using the CEA hybrid technology platform. Note that, as the fabrication procedure of the spiking lasers was not the priority of this thesis, we will limit our discussion to relevant details. We followed the standard fabrication sequences for traditional hybrid III-V/si laser, including silicon waveguide patterning, III-V wafer bonding, III-V waveguide patterning, and III-V contact metallization, in addition to extra steps for isolation cavities. In our case, silicon patterning is realized on the 200-mm -platform of the CEA Leti and then some of the lasers are dispatched to Leti's 100-mm platform, which uses different recipes for the metallization steps. Please note that the following description gives only the main steps in the fabrication of the lasers on the 200mm platform. The general layout of a spiking laser mask layout is shown in Fig 2 .11 and its corresponding process flow is schematically depicted in Fig 2 .12.

Silicon waveguide patterning

First, passive devices including rib waveguides, tapers, and gratings are etched onto a 200-mm SOI wafer. The SOI consists of a 300-or 500-nm-thick silicon layer on a 1000-nm buried oxide (BOX) layer mounted on a 725-µm-thick silicon substrate (cf Fig 2 .12.1). Passive silicon devices are created using traditional ultraviolet (UV) photolithography and selective reactive ion etching (RIE). Typically, these two techniques involve exposing a photoresist to UV light through a mask to create patterns, which are then transferred to the underlying silicon substrate through the RIE process. The final steps require stripping the remaining resist and cleaning the substrate. In some cases (waveguide or grating coupler etching levels), a hard mask can be used in addition to a photoresist to achieve higher selectivity and to provide deep etching capability. Thus, following silicon patterning, the entire structure is encapsulated with 650 nm of SiO2 in our case, then planarized using chemical-mechanical polishing (CMP) to achieve a 100 nm gap between the underlying silicon and the top SiO2 layer. Furthermore, CMP facilitates the subsequent step of molecular bonding by creating a smooth surface.

Wafer bonding

After the SOI patterning steps, molecular bonding (cf Fig 2 .12.2 can be performed between the III-V and SOI wafers by pre-activating both surfaces with oxygen plasma. Subsequently, high-temperature annealing is used to strengthen the bond and make it more resistant to temperature changes. Fig 2 .12.b shows a picture of the bonding of a 3-inch III-V wafer on a 200 mm SOI wafer. Following that, the InP substrate is etched away with HCl solution, and the resulting wafer is downsized to 3 inches.

Isolation cavities etching

After bonding, the wafer is prepared for the first level of electrical isolation through the etching of isolation cavities within the III-V waveguide. The etching sequence of this level is detailed in Fig 2 .12.3. A 100-nm thick SiN hard mask and a 3µm thick resist are used to define 650nm deep, 5-wide and 10-µm long cavities in the InP-p-contact layer. The next steps involve photolithography, hard mask etching, resist stripping, and finally dry etching of the exposed areas. Considering that the hard mask will serve in the subsequent process of defining the III-V ridge waveguide, it is not removed here.

III-V waveguide patterning

In order to define III-V (or gain) ridge waveguides, a 500-nm-thick SiN hard mask is first deposited on the substrate. In total, the SiN thickness is 600 nm where the isolation cavities are not present and 500 nm where they are. This particular step implies considering three different cases, which are depicted in Fig 2 .12.4.a-c.

The first case describes the shaping of the III-V ridge waveguide in the areas where no isolation cavities are present, as depicted in Fig 2 .12.4.a. Here, after another photolithography process, hard mask etching, and the removal of the resist, the 2-µm-thick InP top cladding layer is dry etched down to the beginning of the InGaAsP p-layer (SCH layer) using RIE. This sequence is then completed by wet and dry etching methods to structure the remaining layers of the MQW and to define the mesa level that separates the different laser sources on the wafer. Note that both of these levels rely on SiN spacers [17], which results in the MQW layer being 2 µm larger than the InP top cladding. The resulting structure is generally described as shallow-ridge architecture.

However, two singular cases appear where the isolation trenches are etched. Figure 2.12.4.b shows one of these cases, which corresponds to the etching of very narrow III-V ridge width (i.e. of 1.3 µm) compared to that of the isolation cavities. Indeed, due to this large difference in width, areas of vertical InP layers appear owing to the remains of SiN still present even after the hard mask etching process. In addition, due to the use of a spacer in the etching of the MQWs, these blocks of InP can also lead to excessive broadening of the MQWs. Figure 2.13.a shows a scanning electron microscope (SEM) image that confirms this trend. On the other hand, when the width of the III-V ridge waveguide is comparable to the width of the isolation cavity, SiN blocks produce cuvettes within the isolation cavities (see Fig 2 .12.4.c). This phenomenon can also be seen in the SEM image in Fig 2 .13.b. It should be noted that the impact of creating such features on laser performance is as yet unknown. The devices are then ready for metallization of the III-V contact using a multi-level BEOL.

III-V contacts metallization and H+ implantation

Before starting the multi-level BEOL process, the devices are first encapsulated with SiO2 and planarized. Afterward, III-V contacts are defined through photolithography and RIE steps, and Ti(7 nm) TiN(50nm) contacts are deposited on both n-InP and p-InGaAsP layers. As part of the BEOL, a second encapsulation is used to pattern two levels of plug vias, which are then filled with tungsten (W). Different sizes of vias are used here to connect the two levels of AlCu metals. A multi-level BEOL has been shown to reduce series resistance by a factor of two compared with a traditional one-level metallization. This can result in better performance and lower power consumption of the device. A more detailed description of this multi-level process design and performance is found in [17]. Fig 2 .12.5 shows a schematic view of the hybrid laser with the multi-level planar BEOL.

As the final step in laser fabrication, the second level of electrical isolation is added by implanting H+ protons into the isolation cavities. Although the sequence of this process is not yet clearly defined at the time this manuscript is written, the idea is to re-open the III-V contacts in order to implant the isolation cavities before they are re-encapsulated with oxide. Also, as a reminder, this step is performed at the end of BEOL to avoid exposing the H+ implant to high annealing temperatures that would reduce its insulating properties.

Modelling of integrated semiconductor spiking lasers

The numerical analysis of a multi-section laser plays a crucial role in this thesis, not only in the understanding of its underlying dynamics but also as a foundation for simulations at a larger scale. Moreover, this analysis can also serve as a reference for future experimental demonstrations. In our case, we used the Travelling Wave equation (TW) model as a numerical method to efficiently resolve the multi-sectional structure of the considered device. Indeed, the TW model, which is based on the Maxwell-Bloch equations is a more advanced version of the wellknown Rate Equation (RE) model. Additionally, it allows for taking into account higher-order transport mechanisms, as well as non-ideal effects such as gain saturation, thermal detuning, and gain dispersion which are ignored in the RE. Thus, the TW approach could provide more reliable results, especially when studying the dynamic behaviour of a device.

In the following sections, we discuss the basic TW model used to simulate and investigate the spiking dynamic of our laser neuron model. Particular attention is paid to the influence of the saturable absorber design parameters on its spiking regime. Besides, a stability analysis of our model's operating regime is conducted and discussed.

Travelling Wave model for three section laser

Let us consider a three-section Fabry-Perot (FP) edge-emitting semiconductor laser, as illustrated in Fig 2 .14. The laser consists of active and passive sections divided into multiple segments of optical length ∆z = ∆tv g , where ∆t represents the discretized time steps whereas Each section is discretized into segments of length ∆z. E ± (z, t) corresponds to the slowly varying complex amplitudes of the two counter-propagating optical fields travelling along the laser cavity. S a1/2 : gain section. S s : SA section. l a1/2 : length of the corresponding gain section. l s : length of the SA section. v g is the group velocity of light. Using the Traveling Wave (TW) model, we can determine the spatio-temporal evolution of the slowly varying forward and backward optical fields E + (z, t) and E -(z, t), propagating along the cavity z-axis, respectively. This results in the following set of coupled partial differential equations (PDEs) [31], [32]:

n g c 0 ∂ t E ± ± ∂ z E ± = -iβE ± -iκE ∓ + F ± sp (2.4)
with the boundary conditions established at the laser facets, z = 0 and z = L

E + (0, t) = r 0 E -(0, t), E -(L, t) = r L E + (L, t) (2.5)
In this system of equations, c 0 is the speed of light in vacuum, n g is the group refractive index, κ is the field coupling strength which is non-zero only in DBR sections and F ± sp are Langevin noise sources that describe spontaneous emission processes in active regions [33]. r 0,L are the complex field reflectivity coefficients at the laser facets. The local photon density is determined by normalizing the fields E(z, t) ± as:

|E(z, t)| 2 ≃ 1 E ph ν g σ (|E + (z, t)| 2 + |E -(z, t)| 2 )
With E ph = hc 0 /λ 0 , the photon energy, λ 0 , the operating wavelength ν g = c/n g , the group velocity, and σ the cross-section area of the active region.

In Eqs (2.4), the propagation factor β is expressed using two different models depending on whether the section is passive or active : CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING

β(z, t) = δ 0 -i α 2 if z ∈ S P δ 0 + ∆n(N ) + ∆n T (I) + i(g(N )-α) 2 -iD otherwise z ∈ S a1/2,s
In passive sections (S P ), such as waveguides or insulating sections where charge carriers are not present, β is a constant that only depends on δ 0 , the static detuning from the main frequency and α the internal optical losses.

In the case of the two gain (S a1/2 ) and SA (S s ) sections, the propagation parameter β is a more complex function that first expresses the dependency of the injected carrier density N on the optical peak gain g(N ) (at a fixed wavelength), defined here with the logarithm model (see [31]):

g(N ) = g(N, E ± ) = g ′ N tr ln( N N tr ) 1 1 + εΓ|E| 2 (2.6)
and the resulting change in the refractive index ∆n(N ) due to carrier injection, which can be described with the square root relation:

∆n(N ) = n ′ Γg ′ N tr k 0 N N tr (2.7)
With g ′ = ∂ N g(N tr ) and n ′ = ∂ N ∆n(N tr ), being the differential gain and index, both evaluated at the transparency carrier density N tr . Γ is the confinement factor of the fundamental mode, ε is the gain compression factor that is added to take into account the saturation of the gain in the presence of intense optical field, and k 0 = 2π/λ 0 .

The propagation factor β in the active region can also include thermal effects such as selfand cross-heating phenomena through the term ∆n T (I), which assumes a linear relationship between the refractive index and an applied current

I ∆n T (I) = ∆n T (I)| z∈S k = 2πn g,k λ 2 0 m r=1 ν r k I r (2.8)
Where m is the total number of laser sections. The term ν r k is the cross-heating coefficient that determines a wavelength shift of the laser due to an increase in current I r within each laser section S k .

Optical gain dispersion is considered in the TW model via operator D, which expresses the frequency dependence of the material gain and the coupled refractive index and allows for a selective wavelength mechanism. Through this operator, the material gain can be approximated using a Lorentzian function, which can reflect the semiconductor material's finite gain bandwidth. This function obeys the following linear differential equations in the time domain:

DE ± = ḡ 2 (E ± -P ± ), ∂ t P ± = γ 2 (E ± -P ± ) + iωP ± (2.9) DE ± = ḡ 2 (E ± -P ± ), ∂ t P ± = γ 2 (E ± -P ± ) + iωP ± (2.

10)

Where P ± (z, t) are the polarization functions. ḡ, ω and γ, are the Lorentzian gain amplitude, the peak frequency detuning and the full width at half maximum (FWHM) relative to the reference frequency ω 0 = 2πc 0 /λ 0 , respectively. Eqs (2.4) is usually coupled with an equation that describes the distribution of carrier density N in each active section (gain and SA), which reads the following rate equation (RE):

∂ t N = ∂ t N (z, t) = J(I, N ) -R(N ) -v g ℜe[E * (g(N ) -2D)E] (2.11) With R(N ) = AN + BN 2 + CN 3 ,

the cubic spontaneous recombination function where

A, B and C are the defects, the bi-molecular recombination, and the Auger recombination coefficient, respectively. To quantify the longitudinal spatial hole burning (SHB) effect caused by the non-uniform distribution of photons, the carrier density is treated as a spatially distributed variable i.e. N varies along the z-axis. It is important to consider the SHB effect since it affects the optical gain spectrum profile, which could impact the laser pulse dynamics. This results in a self-distribution of the injected current density J(I, N ), defined as

J(I, N )| z∈S k = I k eV k - U ′ F eV k r s (N (z, t) -Nk (t)), k = a1, a2, s (2.12)
With I k , e, V , U ′ F and r s , the sectional applied current, the electron charge, the active volume of the considered section, the derivative of the Fermi level separation with respect to N and the area-related total series resistivity, respectively, [31]. Nk (t) is the sectional averaged carrier density. Note that, for r s → ∞, the usual model of the average current density is recovered. Note that, the last term on the right in the equation (2.10) is the spatially distributed stimulated recombination function that consumes carriers.

For convenience, in the following, we will refer to the two gain sections S a1 and S a2 with the subscript a.

Model parameters

We choose to simulate a 1310 nm AlGaInAs three-section semiconductor laser, schematically described in Fig 2 .14. This laser is composed of two amplifiers (or gain) sections and an intracavity SA section that is modeled by setting I s = B = C = 0 and A -1 = τ s , with τ s the absorber carrier lifetime or recovery time. Different values of τ s (ranging from 5 to 50 ps) are used to reproduce the carrier sweep-out effect caused by a reverse voltage applied to the SA section (see Eqs (2.1)). Aside from that, we assume that the two symmetric gain sections are driven by a forward bias current I a (which can be tuned as well) rather than a voltage. Also, both gain sections are distinguished from the SA section by having a longer carrier lifetime τ a = 1 ns which implies that τ a > τ s , fulfilling a fundamental criterion for spiking regimes. In addition, the laser has a total length of L = 2076 µm, which includes L Act = l a1 + l s + l a2 = 600 A shallow-ridge architecture is assumed in order to stay consistent with fabrication constraints. Note that, the refractive indices of all simulated layers are the same as those presented in Table 2.2, including the averaged indices used for the active layers. Supported modes are computed for different wavelength values. From this simulation, we can also extract the optical confinement factor Γ M QW , which is another significant parameter. It represents the fraction of the guided mode energy contained in the MQW region. Typically, for the considered waveguide consisting of an averaged thickness layer d M QW , the basic expression for Γ M QW is defined as follows [26]:

Parameters

Γ M QW = d M QW /2 -d M QW /2 |E(z)| 2 dz ∞ -∞ |E(z)| 2 dz , ( 2

.13)

As we only want the light contained within the quantum wells, we calculate our laser's confinement factor using the following approximation:

Γ ef f ≃ Γ M QW × R W , (2.14)
Where R W = 0.4 is the ratio of the sum of the well thicknesses to the MQW total thickness. As a result, the confinement factor is estimated to be 12.35% for the considered optical TE mode and geometry, indicating a high optical modal gain. CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING

Gain medium parameters

The radiative recombination parameters of the MQW structure are estimated by performing a 4x4 k•p band structure calculation taking into account effects such as valence mixing, spectral broadening, and band gap renormalization. This is achieved using the MQW Gain Solver from Lumerical Software. For a detailed description of this model, please refer to the reference [35].

Through this microscopic simulation, we first extracted the MQW's material gain and refractive index spectra for different carrier densities and temperatures. The carrier-induced refractive index change was derived from a Kramers-Krogin transformation [36]. Using these two parameters (g ′ and ∆n ′ ), we can now calculate the linewidth enhancement (or Henry) factor α H , which represents the coupling strength between the gain and the refractive index through the relation (2.14). This parameter is essentially responsible for the frequency chirp (a temporal change in the instantaneous frequency) in semiconductor lasers and can have a dramatic impact on pulse characteristics i.e. amplitude and width [32].

α H = - 4π λ ∆n ′ g ′ (2.15)
In Fig 2 .16.e, we plot the α-factor as a function of injected carrier density for our MQW structure. As expected, the α-factor decreases as the injected carrier density increases. The calculated values are all lower than 1, which can be compared with those reported in strained AlGaInAs MQW materials [37]. For subsequent simulations, we choose α H = 0.1. Additionally, this factor may differ depending on whether the section is an absorber (S s ) or amplifier S a , resulting in α Ha ̸ = α Hs . Furthermore, some studies such as the one of ref [1] have suggested that α Ha > α Hs would yield a more stable ML regime. In our case, considering the fact that the material used for both the gain and SA sections is the same, we assume that α Ha = α Hs = α H , for simplicity.

Lorentzian fit parameters

As mentioned in section 2.3.1, in order to account for the dispersion of the material gain and refractive index, we use a Lorentzian filter to approximate and limit the bandwidth of our gain profile. Consequently, our model's peak gain and refractive index change are corrected as follows, [31]:

G(N, ω) = g(N ) - ḡ(ω -ω) 2 ( γ /2) 2 + (ω -ω) 2 , ∆N (N, ω) = ∆n(N ) + ḡ 4 γ(ω -ω) ( γ /2) 2 + (ω -ω) 2
(2.16) Here, G(N, ω) and ∆N (N, ω) are the dispersive peak material gain and refractive index change functions at the frequency ω, respectively. In our case, these parameters can be directly extracted by fitting our computed gain spectrum, as illustrated in Fig 2 .17. The Lorentzian fit displayed here is calculated for a fixed carrier density N. In addition, the x-axes indicate the relative frequencies ω -ω 0 .

Numerical results

As part of our analysis, we used LDSL-tool (Longitudinal Dynamics in multisection Semiconductor Lasers) [31] software to numerically solve the TW equations (2.4), (2.5), and (2.10). This software has been developed at the Weierstrass Institute and has allowed us to explore and identify the dynamic ranges of our laser system.

Considering the symmetry of our cavity, the fundamental mode spacing is expected to be around f = v g /L, namely ∼ 40 GHz, which corresponds to a colliding pulse mode-locked (CPM) regime. Consequently, to distinguish this ML regime and its harmonics from other pulsed regimes (especially Q-switching), the radio frequency (RF) spectra (the Fourier transform of the time trace of |E(z, t)| 2 ) and sampling diagram method have been implemented as described in [34] and [38]. Thus, we assessed the ML frequency and the corresponding signal-to-noise ratio (SNR) using the RF spectrum as shown in Fig 2 .18.a. We then assumed that a proper ML operation is found for an SNR of at least 25 dB. Furthermore, the pulse width, amplitude, and time jitter are extracted from the eye diagram sampled with the mean period of ML pulses (see Using the LDLS tool and this characterization protocol, we were able to analyze our laser's dynamic behaviour. The results are described below.

Spiking regimes analysis

We first investigate the overall dynamic trend of the FP laser by varying the parameters I a and τ s . The results of this two-parameter simulation are shown in that can be divided into several regions such as QS ML regime (region A), regular ML regime (region B), 2-nd harmonic ML (region C), and CW regime (region D). We note that some of these regimes are consistent with theoretical predictions described in section 2.1. An insight into the transition between these four regions and their stability can be seen in Fig 2 .19.c,d. These figures illustrate two bifurcation diagrams as a function of injection current, I a , for two different values of τ s . In order to obtain these graphs, we plotted the pulse peak powers observed after the laser reaches a steady state over a 5 ns time window for each value of I a . This allows us to illustrate the stability of the mode-locking regime. Below, we compare the two most important dynamical states we are most interested in, namely the QS regime and the ML regime.

Q-switching and instability

The underlying dynamics of the Q-switching instability (QSI) can be understood by looking at Let's take a close look at the time trace of the electric field (normalized by T rep ) corresponding to a current bias of I a = 60 mA in Fig 2 .20.a. We can see that the envelope of the QS pulses strongly modulates the so-called mode-locked pulses (having different amplitudes) with a fixed period which is about 10 times greater than the cavity round trip time. This behaviour is further confirmed by looking at the inset from this figure, which displays a QSI envelope (showing a bunch of pulses) on a shorter time scale. This image shows pulses with a repetition frequency of approximately 40 GHz, which corresponds to the free-running cavity's CPM operating regime. Fig 2 .20.b illustrates carrier density dynamics in the gain and SA sections. Based on this plot, several net gain windows may appear during a QS period (instead of a single net gain window as can be expected in pure QS) with gradually changing amplitudes, resulting in the generation of a modulated pulse train. This can be explained by poor current injection, which can lead to partial saturation of the gain (staircase-like saturation) and (highly absorbing) SA in a single round trip. In Fig 2 .20.c, you can see the corresponding RF spectrum of the QS instability, which shows a frequency comb with distinct peaks corresponding to QS harmonics and CPM frequencies. In particular, the highest RF peak is found at about 2 GHz, which is the signature of a QS frequency component. At this point, it is useful to recall that, normally, a pure QS regime should occur just before the establishment of the QS instability (as experimentally demonstrated in [39] or [40] ). This regime, however, was not captured by our simulations.

Finally, by increasing the current value, the QS envelope period gradually decreases until a pure ML regime is reached (see the bifurcation diagrams). This transition is illustrated with

Mode-Locking and instability

In order to make a qualitative analysis of this phenomenon, we will refer to 20.j-l shows a special case of CPM that takes place at higher I a and τ s values. Here, although the gain and SA carrier densities exhibit similar saturation dynamics, the temporal trace of the electrical field displays irregular satellite pulses alongside the ML pulses, giving rise to the modulated CPM-regime as shown in Fig 2 .20.j. Consequently, the laser has a complex chaotic behaviour which is represented by region C in our simulation results. The spectral trace of this regime reveals strong side modes accompanying the main CPM frequency, which is indicative of poor CPM performance. Thus, by looking at the frequency map, this regime might subsequently lead to ML harmonics, which are detected at frequencies such as ∼ 80 GHz and ∼ 120 GHz. Finally, for very high currents, a continuous wave CW state (region D) could be established.

(g) (h) (i) (j) (k) (l) (c) (d) (e) (f)

Influence of absorber's parameters

In our laser configuration, the absorber (SA) strength varies both with its recovery time (thereby the bias voltage applied) and its length, which varies from 2.5 % to 10 % of the overall active cavity volume. This section analyzes how these parameters affect the spiking regime of our laser model, especially the QS regime. In fact, our laser will only operate near its turn-on threshold, so it is more likely to operate within this regime. shows simulation results for two lasers with absorber lengths of 20 (∼ 3 %) µm and 60 µm (10 %), respectively. In both cases, the active section is 600 µm long. For these two figures, we varied I a while fixing all other parameters, including the SA's recovery time, CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING τ s . We then extracted the laser power-current characteristic curve for each device. Each plot exhibits the local variation of the laser output power, which allows to identification of stable and unstable zones. The region where the min and max values coincide corresponds to a stable ML operation (region B), whereas the other corresponds to a QS instability (region A).

First, we can see from these two figures that a longer SA leads to a larger QS instability region and a shift in the ML region. This is accompanied by an increase in the laser threshold, which means higher energy is needed to excite it. Indeed, longer SAs introduce additional optical losses into the laser cavity, resulting in higher threshold currents. In addition, although not evident in these graphs, this increase in intracavity losses may also result in a decrease in laser output power, which can affect the laser neuron's efficiency.

Spiking laser performance can also be affected by the SA recovery time. As a reminder, recovery time refers to the time required for the absorber to switch from its highly absorbing state to its low or transparent state. Longer recovery times lead to instabilities in the laser output, causing fluctuations in pulse amplitude and timing. This results in the apparition of unstable spiking regimes as seen in Fig 2 .19.c,d). From these same figures, we can also notice that the QSI region tends to be smaller, as opposed to the ML region, where stability decreases more significantly. This suggests that self-pulsing operation (despite not being a pure QS operation) can be ensured for a wide range of SA parameters. CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING

Chapter summary

This chapter has described the design and dynamics of our pulsed laser.

First, we covered the theoretical aspects related to the generation of short and ultrashort pulses in a semiconductor laser. Different techniques such as QS or ML were discussed, as well as the conditions required to achieve them. As we have seen, these techniques can be implemented with semiconductor laser structures that integrate a saturable absorber. Indeed, the saturable absorb exhibits nonlinear properties that enable the laser to generate short and ultrashort pulses cost-effectively.

Next, we reviewed the design of our spiking neuron based on a hybrid laser cavity with three sections operating at telecom wavelengths, namely 1310 nm. The optimization of the cavity, as well as its implementation in CEA Leti's III-V/Si process, have been studied. In particular, we focused on designing the saturable absorber, which is essential in realizing spiking dynamics in the semiconductor laser.

Finally, we conducted theoretical analyses of the overall spiking dynamic of our laser model. For this purpose, we have implemented the TW model well adapted to the specific case of semiconductor lasers with multiple sections. For defined parameter ranges, simulations have revealed unstable and stable pulse regions corresponding to a self-pulsing regime (or QS regime) and a ML regime, respectively. Additionally, we investigated the influence of SA parameters on laser dynamics. Our results provide a better understanding of the dynamics of the spiking laser system and can be used to optimize the laser performance.

In Chapter 3, we will show how this laser can be used to emulate the main features of a LIF neuron. A small system demonstration will be conducted to illustrate this laser system's potential for neuromorphic computing. Additionally, we will discuss the potential applications of this technology in the future.

Chapter 3

Adaptive photonic neural network for hardware implementation As we saw in the previous chapter, a hybrid laser configured to emit ultra-short pulses is used to build a photonic neuron on a silicon platform. In this chapter we will investigate its ability to function as a neuron and how this neuron model can be implemented in a largescale photonic neural network. In Section 3.1, we will review a photonic processing network node (PNN) to illustrate our networking approach. This section will also present a numerical analysis of the isomorphism between our photonic node and a traditional LIF neuron. Then, in Section 3.2 we will focus on the description of a supervised learning algorithm adapted to train our photonic network off-chip. In this part, the exact calculation of the gradient as well as the introduction of an approximate LIF neuron model of the laser are the main topics addressed. Finally, the chapter ends with a case study example and a general analysis of our model. As shown in the photonic configuration, the considered PNN features a weighted summation scheme based on a broadcast-and-weight (B&W) protocol, which is implemented with integrated lasers and a photodetector system. Using this architecture, an excitable laser (postneuron j) can be stimulated by incoherent signals from other lasers and produce a typically spiking response. Here, presynaptic signals are weighted optically before being converted into electrical energy by a photodetector system. In this case, a balanced photodetector is employed to provide both excitatory and inhibitory stimulation to the laser. However, It should be emphasized that, while no particular synaptic device architecture is studied in this thesis, a number of possibilities have been considered, including MMRs and MZI meshes. Indeed, IMPLEMENTATION both types of synaptic configurations are compatible with on-chip integration but most importantly with the B&W protocol. Besides, this section will focus on the photonic processing node (i.e. the excitable laser) and its ability to emulate a LIF-like behavior in response to a stimulus. This includes temporal integration, spiking threshold, spike generation, refractory period and cascadability which is essential for large-scale neural networks.

Method and model

To demonstrate the LIF-like dynamics of our photonic neuron, we consider here a single laser neuron j that processes outputs signal from other lasers corresponding to presynaptic neurons, just like the one presented Fig 3 .1 (bottom scheme). Similar to the voltage-driven behaviour of a LIF neuron (top scheme), the laser's spiking dynamics are governed by the strength of the total input signals, which modulate only the gain medium.

In this specific stage, the interaction between the gain and SA regions of the laser is now described using the traditional rate equations (RE) model (rather than the TW model) to minimize computing time. Basically, it means that the optical field and carrier densities are assumed to be uniform throughout the laser cavity. Although less accurate than the TW model, this model can still describe laser dynamics well while being less computationally demanding [2]. Thus, in the case of a laser j operating in single mode, the RE can be defined by the following expressions:

dN ph,j dt = [Γ a 2 m=1 g am (N am,j )V am + Γ s g s (N s,j )V s ] N ph,j V - N ph,j τ ph + 2 m=1
V am βB r N 2 am,j (3.1)

dN am,j dt = η a (I am,j + i em,j (t)) eV am - N am,j τ a -Γ a g am (N am,j ) N ph,j V (3.2)
dN s,j dt = η s I s,j eV s - N s,j τ s -Γ s g s (n s,j ) N ph,j V (3.3)
Logarithmic models are used to define the optical peak gain and losses, but the gain saturation factor is not taken into account:

g am (N am,j ) = g ′ a N tr ln( N am,j N tr ) g s (N s,j ) = g ′ s N tr ln( N s,j N tr )
The optical power of the laser neuron j is finally evaluated as such:

P out,j (t) ≃ η c hc τ ph λ j N ph,j (t) (3.4)
Here, N ph is the total photon number in the cavity. N am and N s are the carrier densities in the gain (m = 1 or 2) and SA sections, respectively. N tr is the carrier density evaluated at IMPLEMENTATION transparency. V am is the volume of the relevant gain section, V s the volume of the SA section and V = 2 m=1 V am + V s is the total volume of the laser active region. The photodiode system produces an electrical current modulation i em (t) which is added to the bias current I a applied to the gain section. This modulation represents the weighted sum of the inputs received by the laser j and is, therefore, a function of the spectral responsivity of the photodetector system R λ , i, the optical power P out,i of each input laser i and the strength of the optical signal considered, which is related to the synaptic weight ω ij .

i em,j (t) = f (R λ,i , P out,i , ω ij )
To keep things simple, we decided to treat the two (identical) gain sections as a single section of volume V a =

V am and a carrier density N a (t) = N am V am /V a in the following. In this case, the total injected bias current is I a = I am , and the electrical modulation is i e (t) = i em . Table 3.1 provides all the parameters and their corresponding values. In order to evaluate its excitability and inhibition properties, the laser is biased close to the self-pulsing threshold, which is near 50 mA (the gain section is biased at I a = 48 mA). We then send a pair of excitatory and inhibitory input pulses (i.e. the inhibitory pulse is first integrated) of increasing energy to the laser j and record its output response. Note that synaptic plasticity is implemented in our model by varying the optical signal amplitude, thereby adjusting the signal strength accordingly. Aside from that, both input pulses have a Gaussian shape with a FWHM of 10 ps and the SA is biased at I s = 0 mA. Here, for each inhibitory pulse (of five different energies), we measure the laser's response when the excitatory pulse is sent versus the inhibitory pulse first. It can be seen that all curves have a nonlinear shape with IMPLEMENTATION an apparent excitability threshold that rises from 5pJ when E inh = 0 to 8.1 pJ for E inh = 5 pJ. Accordingly, as the inhibitory pulse energy increases, it becomes harder for the laser system to stimulate an output spike. This can be explained by the fact that an excitatory pulse serves to build up the carrier density in the gain medium, allowing the absorber to reach saturation during the pulse time. As a result, the laser responds by firing a spike that leads to a fast depletion of carriers followed by the resetting mechanism of the gain (as well as the recovery of the absorber). On the contrary, in the presence of an inhibiting pulse, the carrier density decreases, which can significantly reduce or cancel the action of an excitatory pulse. Thus, the laser is able to selectively respond to positive or negative weight input spikes, providing a way to process information similar to that of a LIF neuron.

Parameters

We also recorded the laser's spike latency, by only considering the strength of an excitatory input pulse, as displayed in Fig 3 .2.b. The spike latency property can be defined as the time difference between the moment the input pulse is applied and the laser's first firing event. We can see from this graph that stronger inputs lead to shorter latency, an attribute also found in the LIF neuron. This feature confirms laser neurons' ability to perform temporal coding schemes, a fundamental aspect of information processing.

Leaky (temporal) integration

In this section, we investigate the laser's ability to integrate several consecutive spikes. Toward this, we perform a coincidence detection experiment, in which we observe the laser's spiking response to two identical excitatory pulses with different inter-spike intervals (ISI) times. Each pulse is a sub-threshold stimulus, and as such cannot independently trigger a spike. The ISI time varies from 0 to 100 ps and the laser polarization conditions remain unchanged, i.e. I a = 48 mA and I s = 0 mA. Fig 3.3.a shows the output pulse energy as a function of the correlation time between the two input pulses. As seen here, the laser produces a high-energy spike as long as the ISI time remains below 30 ps. Beyond this temporal separation, the two pulses are too far apart to trigger a strong spike from the laser. In fact, the temporal integration property is characterized by the accumulation of carriers in the gain medium that results from successive incoming spike perturbations. Therefore, if the spikes are temporally close enough (as in Fig .c (when the two pulses have a large ISI time), a peak is less likely to occur due to the carrier natural recombination causing the mechanism known as leakage in a LIF type neuron.

Refractory period

Our next focus is on the laser neuron's refractory period. As a reminder, the refractory period is the time it takes for the neuron to return to its resting state after it has been activated by an excitatory pulse. For this purpose, the laser is stimulated with two excitatory pulses of IMPLEMENTATION As illustrated in Fig 3 .4.a, the laser exhibits a single spike event at t out,1 (E out,1 > 0 pJ) for temporal separations of less than ∼ 3.5 ns despite the occurrence of a second triggering pulse (E out,2 = 0 pJ). This indicates absolute refractory time, where the laser cannot fire another spike immediately following its first firing. Then, after this period, the laser is once again excitable and enters a relative refractory period, where a second spike event t out,2 (E out,2 > 0 pJ) is induced by the second input pulse. Furthermore, the amplitude of the second output spike varies with the temporal separation between the two input pulses. As this temporal separation becomes larger, the output spikes converge to the same energy (E out,2 = E out,1 ).

The time traces corresponding to the absolute and relative refractory period are given in .c, respectively. The refractoriness of a laser is essentially related to the recovery dynamics of carriers in the gain medium subsequent to a previous excitation. As a result, the faster the carriers return to equilibrium, the lower the laser's refractoriness, and the faster it will respond to external signals.

Cascadability's demonstration

An imperative property to consider when designing any large-scale neural network is cascadability. This property is directly linked to the neuron's ability to regenerate and transmit an incoming signal without degradation. Thus, by preserving signal quality, cascadability en-IMPLEMENTATION t out,2 : firing time of the second spike sures reliable and efficient communication between layers of neurons. As reported in [4], a high level of cascadability can be achieved with photonic neurons with O/E/O links since coherence between signals is not necessary. Amplification, however, is essential to compensate for losses caused by signal degradation at the conversion (O/E/O) stages.

We evaluate our processing node's cascadability using an autapse model. Essentially, it consists of a single neuron receiving its own output signal as an input, forming a feedback loop. This allows us to measure the neuron's performance in terms of signal regeneration and cascadability. The simulation setup is shown in Fig 3 .5.a. To begin with, an external input signal is used to trigger the first output spike. Afterward, this output spike is fed back to the neuron as an input to stimulate the system again. As part of feedback, the signal is delayed by τ and amplified by a unitless control weight, W . Note that, the photocurrent induced by the recurrent signal, P out , is of the form: Hence, by injecting this output spike (which has been amplified) back into the system, a recursive process is initiated, resulting in a pulse train that repeats ad infinitum with a period close to τ . In this case, the system exhibits bistable dynamics and is capable of regenerating IMPLEMENTATION the properties of the input pulse endlessly. In addition, the stability of the induced recurrence is also evidence that our PNN can cascade and that information such as the amplitude or rate of spikes can be stored.

i a,rec = R λ 0 W P out (t -τ )

Towards the multilayer photonic SNN

The previous sections have allowed us to outline several similarities between spiking lasers and LIF neurons. These similarities include the ability to process weighted stimuli inputs as well as encode and store information in spiking events, in particular. Also, both systems exhibit recovery mechanisms, allowing them to reset their internal state variable (the gain of the laser vs. the LIF's membrane voltage) in preparation for the subsequent inputs. Otherwise, it has been emphasized that the photodetector front-end system should support multi-wavelength inputs fan-in and should facilitate cascadability of laser neurons. Thus, considering the scalability of the O/E/O concept along with the LIF analogy of spiking lasers, hardware implementation of a photonic neural network becomes achievable. with photodetector units. In addition, these nodes are fully interconnected by synaptic weight matrices, in a hierarchical manner (with input, hidden and output layers) allowing for a deeplearning approach. However, as mentioned in Chapter 1, learning multilayer SNNs directly on a chip is a complex task that would require an extensive hardware platform. To address this difficulty, we developed a hybrid supervised learning approach, which converts a trained LIF neural network into an equivalent laser-based network for inference. This is only possible by leveraging the commonalities found between the spiking laser and LIF neuron while minimizing their discrepancies, which are mainly related to:

1. The vast disparity between the dynamic time scales of neurons and lasers, the former operating in milliseconds, and the latter in a characteristic time of sub-nanoseconds.

2. The absence of an input bias in the conventional LIF model, a feature inherent to laser models that control their excitability thresholds.

3. There is a significant difference in their energy transfer function, as shown in Fig 3 .7. This plot indicates that laser output energy varies with the input energy, resulting in a non-ideal all-or-nothing transfer function. On the other hand, the LIF neuron exhibits a step-like transfer function as expected, with the output being the same regardless of the input.

To address these concerns, we propose an adapted LIF neuron (ADLIF) model which aims to represent and capture the most prominent parameters of the spiking laser model and would allow for better prediction of its behaviour.

Next, we will describe the full procedure of our learning approach, which is fully compatible with spike encoding and possibly hardware implementation. IMPLEMENTATION

A hybrid supervised learning approach for training photonic SNNs

Our learning approach is directly inspired by the ANN-to-SNN conversion-based scheme [5], [6]. This technique indirectly trains a spiking neural network using an equivalent pre-trained artificial neural network (ANN), thus avoiding the problems inherent in spiking-based learning. In our case, conversion occurs between a LIF neural network and a photonic neural network with realistic parameters and vice versa. In order to maximize the efficiency of the conversion process, we introduce an adapted LIF neuron (ADLIF) model that addresses output discrepancies between the basic LIF and the spiking laser neuron model. Finally, we employed the EventProp learning method [7], for accurate training of SNNs. The entire procedure is described in the next sub-sections.

The proposed LIF neuron model

We developed an adapted LIF neuron model to bridge the gap between the outputs of a spiking laser and a LIF neuron. Specifically, two major modifications were made to the conventional LIF model. Firstly, the overall dynamic of the LIF model is enhanced by introducing a bias current term I bias to adjust the output along with the synaptic input current I(t). Mathematically, this results in the following free dynamics equations

τ m ∂V j ∂t = -V j + R m I j (3.5) CHAPTER 3. ADAPTIVE PHOTONIC NEURAL NETWORK FOR HARDWARE IMPLEMENTATION τ s ∂I j ∂t = -I j + I bias,j (3.6)
With j the index of the corresponding post-synaptic LIF neuron. Equation 3.5 still describes the temporal dynamics of the membrane potential V (t), characterized by an integration time constant τ m = R m C m , with R m and C m representing the membrane resistor and capacitor, respectively. The intrinsic resting potential V 0 is set to zero in Equation 3.5. Aside from that, I(t) is the total injected current flowing into the neuron, which is subject to its own dynamics given by equation 3.6, where τ s is the corresponding synaptic time constant. The additional bias term in this equation acts similarly to the laser's current bias, bringing the neuron closer or farther away from its activation threshold, thereby modulating its activation function. It is also an adjustable term that can be trained to set up a desired firing pattern in neurons. Hence, the bias term provides an additional degree of freedom to tune the network performance. It should be mentioned that the conventional LIF model is obtained by setting

I bias = 0.
Otherwise, With this system of equation, in the absence of external inputs, the neuron's current and potential stabilizes to I j (t) = I bias,j and V j (t) = R m I bias,j . We also assume an internal resistor R m = 1Ω.

The other modification we made becomes apparent during a spike event. Essentially, the latter is no longer considered as a binary event but as a signal with a certain amount of energy. By doing so, the neuron's response can be tailored to different input spike amplitudes, thus exhibiting amplitude-dependent behaviour analogous to a laser. Thus, whenever the membrane potential of the neuron V j reaches a fixed threshold ϑ and the condition ∂V j ∂t > 0 is met, a post-synaptic spike of energy e j is discharged at the time t post j and sent to subsequent neurons. e j can be defined as a fitting function of the laser's energy transfer function, determined by the derivative of the LIF neuron's potential under spike conditions. Finally, after the spike generation, the synaptic current jumps and the potential is reset to its resting value, namely zero. These phenomena can be summarized as follows: if V j ≥ ϑ and ∂V j ∂t > 0 :

         V j (t post j ) + = V 0 = 0 I j (t post j ) + = I j (t post j ) -+ i ω ij e i , ω ij ∈ R e j (t post j ) = η( ∂V j ∂t ) α , η ∈ R, α ∈ [0, 1] (3.7)
otherwise : e j = 0

The signs -and + represent parameter values before and after a spike, respectively. The fitting parameters η and α are used to adjust the output of e j to a desired dynamic behavior. It can be seen that e j is being modulated by the synaptic weight ω ij linking the j -th postsynaptic neuron to the i -th pre-synaptic neuron. Also, ω ij can either be positive or negative, indicating inhibition or excitation, and can be adjusted accordingly. IMPLEMENTATION Thus, by taking into account both a bias term and a variable pulse energy function, the LIF neuron output can be matched to the laser output model. Detailed results of the parameters mapping we derived from fitting the two models are provided in the following subsection.

Parameters fitting

The ADLIF is characterized by a set of parameters (see Table 3.2) that can be tuned to achieve the desired dynamic behaviour specific to our spiking laser neuron. Thus, by comparing the two models, it is possible to find the best combination of parameters that minimizes the discrepancies between the LIF output and laser output. Then, for a given fitting parameter, its optimal value is determined by minimizing the following mean square error (MSE)

ε 2 = 1 N N n=1 (f n (x) -fn (x)) 2 (3.8)
where N is the number of samples, f n is the function of the laser model and fn (x) the function of the LIF model. A larger ε 2 indicates a poorer fit.

Our fitting procedure is conducted as follows: We first considered an adapted LIF model with a fixed output energy and let the four parameters τ m , τ s , ϑ, and δt vary. Note that, δt is the response delay of the laser when stimulated above its threshold excitability. A pulse of increasing energy was then applied to both the laser and LIF neurons and their responses were recorded under different bias conditions. Here, bias currents are kept below the threshold current in both neuron models. As a third step, all four parameters are adjusted simultaneously, till both models are comparable (the MSE is minimized). Subsequently, threshold parameters η and α are estimated using the optimized values of τ m , τ s , ϑ and δt. These two parameters were determined using the relationship of the variable pulse energy function defined in equation As can be seen from the fitting graphics, the adapted LIF neuron closely matches the excitability properties of our spiking laser model after fitting. This is especially true when we look at the relationship between the spiking energy threshold and the bias shown in c), probably due to shared parameters that may result in some data being underfitted. This suggests that further adjustments may be necessary to improve the ADLIF model's accuracy. These results, however, provide a reliable and realistic representation of the laser's underlying dynamic.

Adaptation of EventProp algorithm

As mentioned above, our training strategy involves implementing an off-chip learning algorithm to effectively train a photonic SNN. Therefore, ensuring accuracy during this step is crucial to maintaining and guaranteeing efficiency in the on-chip inference phase. Through our in-depth research, we have found that the EventProp algorithm [7], developed by Wunderlich et Pehle, is the most suitable method for accurately computing SNN gradients in a way that is analogous to the Backpropagation method. So, in line with our approach, we adapted this algorithm as follows:

To begin with, we consider a global loss function like the one introduced in the EventProp IMPLEMENTATION learning process. This function combines two differentiable functions, namely a discrete loss function denoted l p and a continuous loss function,l V :

L = l p (t post k ) + T 0 l V (V (t), t)dt (3.9)
where j(k) refers to the index of neuron emitting the k-th spike and T is the total trial duration. Here, if the loss function only requires spike times, we simply have l V = 0 -Otherwise l p = 0 and the non-zero components of the partial derivative of the loss function l V ( ∂l V ∂V ) must be considered during the training process.

According to Wunderlich et Pehle, the gradient of the loss function L(equation 3.8) can be calculated using the adjoint method. This method employs the Lagrange multipliers concept, which involves global operators that show similar dynamics to neurons' state variables (V , I). However, they are computed backward in time, i.e. from t = T to t = 0. Then, for a postsynaptic LIF neuron j ∈ [1, N ] with adapted parameters (I bias,j , e j ), the adjoint variables to this neuron exhibit the following free-dynamics:

τ m λ ′ V,j = -λ V,j - ∂l V,j ∂V (3.10) τ s λ ′ I,j = -λ I,j + λ V,j (3.11)
Where λ V and λ I are the Lagrange multipliers that are referred to in this system as the adjoint variables, respectively, for potential V and current I. At each spike times t post k , these variables are subject to a transition given by

λ - I,i = λ + I,i (3.12
)

λ - V,j(k) = λ + V,j(k) + 1 τ m V - j(k) (t) [ϑλ + V,j(k) e j(k) (t post k ) j̸ =i ω ij (λ + V -λ I ) j(k) - τ s ∂e j(k) (t post k ) ∂t post k j̸ =i ω ij λ + I j(k) + ∂l p ∂t post k ] (3.13)
In this equation, the adjoint variables backpropagate from the postsynaptic neuron j to a presynaptic neuron i thanks to ω ij (λ + V -λ I ) j(k) at spike time t post k , where j(k) is the neuron index causing the kth spike. One can notice that the variable pulse energy function particularly affects the transition expression of λ V,j , leading to logical modifications in gradient calculations. Thus, the gradient of the loss function L with respect to ω ij and I bias is derived by computing this system of adjoint equations with respect to each tunable parameter. This results in:

∂L ∂ω ij = -τ s i e i (t post i )λ I,j (3.14)
∂L ∂I bias,i = - +∞ -∞ λ I,i dt (3.15) IMPLEMENTATION
Interestingly, both gradients are determined by sampling the adjoint variable λ I solely at spike times. Thus, in contrast to traditional backpropagation techniques which calculate gradients for each neuron, at every time step, this approach can be significantly less energyintensive. Furthermore, its asynchronous nature allows for continuous learning while avoiding large memory and computing requirements, making it an ideal fit for our hardware application.

Finally, after applying the gradient descent, Weights and bias currents are adjusted in the network with a similar update rule, which takes the following form:

ω ij = ω ij -η r ∂L ∂ω ij (3.16) I bias,i = I bias,i -η r ∂L ∂I bias,i (3.17)
With η r being the learning rate. Appendix A provides detailed calculations of this adaptation of the EventProp algorithm.

Our learning framework

In this section, we present a simple framework for our training process that can be visualized by the diagram shown in Fig 3.9. First, the photonic network is converted into a spiking network (with the same topology) composed of adapted LIF neurons. In this way, the equivalent spiking network can be pre-trained to get close to the optimum values of the learnable parameters, namely weights and biases.

The spike latency strategy is used for pre-encoding and processing input data in the network. In a forward pass, neurons in the hidden layer process incoming information by numerically integrating equations 3.5, 3.6 and 3.7, and spike timing t post are stored as well as corresponding currents I -(t post ) whenever a threshold is crossed. In the final layer, the firing time of the first spike determines the prediction of the network. Thus, after a forward pass, the modified version of EventProp is used to backpropagate network errors in a backward pass via the adjoint variables. This process is then repeated until the weights and bias terms reach their optimal values, minimizing the network's global loss function.

Therefore, after the learning parameters have been pre-trained, the optimized values are reinserted into the photonic network for the inference task. Note that, no regularization or normalization strategy is required to convert between the two networks due to the fitting stage.

Case study: The Yin-Yang dataset

In order to evaluate the effectiveness of our training approach with the photonic network, we numerically implemented the experiment with the Yin-Yang classification task. Please note IMPLEMENTATION that this case study will only cover the training phase.

Dataset description and methodology

The Yin-Yang dataset is a commonly used dataset to evaluate the effectiveness of classification algorithms on non-linearly separable data. It contains a number of samples of 2-dimensional data points, which are labelled as either "Yin", "Yang" or "dot". Where T max is the time spread of the input spikes, which is set here to 0.1 ns. Then, we used a loss function based on time-to-first-spike, as introduced in [7]:

with l V (V (t), t) = 0 and L(t O ) = - 1 N batch N batch m=1 log[ exp(-t O m,l(m) /τ 0 ) N class j=1 exp(-t O m,j /τ 0 ) ] + α exp -t O m,l(m) τ 1 -1 (3.18)
Where N batch is the number of samples in a mini-batch, N class is the number of classes and m is the index of each sample. t O j is the time of the first spike from output neuron j and l(m) is the index of the neuron that should have fired first for the m -th sample. α, τ 0 and τ 1 are hyper-parameters that regulate learning. According to this definition, this function is minimal when the target neuron fires the earliest spike prior to all other neurons. Thus, we trained ADSNN on 32 batches of 100 samples from the Yin-Yang dataset. Samples are pre-encoded as spike times (see Fig 3.10.c) and sent to the network for classification, which assigns a label to each sample. Also, the network is simulated for a duration of T = 1 ns. Weights and biases are initialized randomly and updated via the back-propagation algorithm. Furthermore, we evaluated the network's performance using accuracy metrics which measure the overall correctness of the network's predictions. Besides, the optimization of the network was done using the Adam algorithm, [10]. You can find the simulation parameters used for training in Table 3.3. Note that, it displays the curve average over the 10 trials. Based on this graph, we can see that the ADLIF network correctly predicted about 80 % of the 3200 training data. This performance is slightly lower than that of [7], which reaches around 95% for a standard SNN network with the same number of hidden neurons. A sample with the highest classification accuracy (about 85 %) achieved with the network is presented in Fig 3 .10.e. It shows the spike times collected from the three labeled neurons ("yin", "yang", and "dot"), with red dots representing data that does not belong to the correct class while green dots represent correctly classified data. Here, as expected, incorrect data tend to cluster around the boundaries between classes, especially near the dot class. This is a sign that our model needs to be further optimized as we will explain in the next section.

Simulation results

Parameters

Discussion

In the previous sections, we intended to evaluate the effectiveness of our learning protocol for photonic SNNs. Overall, this study demonstrates that our algorithm with adapted neurons offers good performance in learning nonlinear tasks, such as classifying yin-yang datasets, which is challenging for most traditional neural networks. However, compared to similar models presented in the literature, our model did not achieve the same performance. This underperformance can be attributed to several factors. First of all, it might be directly related to our adaptation of the LIF model. Indeed, as we fitted the LIF model to our laser model, we changed its activation function, which is no longer an ideal step-like function. Consequently, the neuron's excitability and reliability might have been altered, producing errors in the learning algorithm. Secondly, our network accuracy could be further improved by adding more neurons to the hidden layer. Nevertheless, this solution has some limitations, since increasing IMPLEMENTATION the network size might be challenging to implement on hardware platforms. Furthermore, a larger network might also be more prone to overfitting, which could lead to a decrease in the network's generalization capabilities. Therefore, future research will need to be able to strike a balance between accuracy and complexity.

Otherwise, it is pertinent to note that these are preliminary results and further optimization of the network is necessary to achieve even better results. Furthermore, although the inference phase was not implemented in this case study, the isomorphism previously established between the LIF and the laser neurons can be used to assess the performance of the laser network.

Chapter summary

This chapter provides a comprehensive overview of our approach to co-designing photonic spiking neural networks, focusing on a hardware architecture and a learning algorithm.

The chapter begins by introducing the fundamental concept of our processing network node based on spiking lasers which is the building block of our large-scale photonic neural network. As pointed out in this section, the laser is used to implement a LIF neuron function as both exhibit similar dynamics i.e. through excitability property, the laser can be tuned to emit optical spikes that mimic the firing action potential of a LIF neuron. We also harnessed the O/E/O framework that can enable to interconnect several processing nodes using waveguides, modulators, and photodetectors for introducing weight-summation functionality. With this framework, a wide range of network topologies can be implemented with high performance and reliability without sacrificing scalability.

In addition to scalability, the network must also be capable of performing a variety of tasks via training, which is challenging. As discussed in part two of this chapter, our training approach relies on an adaptive LIF neural model that is tailored to fit spiking laser dynamics. As laser and LIF functions are highly analogous, the photonic network can be trained indirectly using a robust gradient-based algorithm that exploits the exact spike timing to provide highly accurate error functions. Finally, we have demonstrated the effectiveness of our learning approach in the classification of the Yin-Yang dataset, achieving an accuracy of up to 85%. Additionally, the chapter discusses the limitations and challenges associated with this approach, as well as potential areas for further research.

Chapter 4 Conclusion and future work 4.1 An overview of our contributions

During the past few years, there has been extensive work on developing neuromorphic computing that mimics the complex and parallel nature of the human brain. In this context, photonic technology has gained significant interest due to its ability to process data at the speed of light, consume minimal power, and be easily integrated with other technologies. This thesis poses the problem of developing high-density photonic-based neuromorphic systems to enable real-time applications that demand dense neuronal networks. Thus, my work over the last three years has mainly focused on the spiking neuron, the building block of these circuits, as well as solutions for training them.

Below is a summary of what has been accomplished so far:

1. First photonic neuron architecture designed on Leti's III-V/silicon platform. We have developed a novel photonic-based neuron using the inherent nonlinear dynamics of semiconductor lasers. The proposed architecture is highly scalable, and as it incorporates a saturable absorber directly within the gain section, the complexity and fabrication process has been simplified.

2.

A complete modelling and simulation environment for multi-section lasers. With the aid of both commercial and local simulation tools, we have conducted a detailed study of the free internal dynamics of a multisection laser. As a result of this study, valuable insights were gained into the laser's performance and its spiking dynamics characteristics. Furthermore, this is the most in-depth study conducted by the CEA's silicon photonics integration laboratory (LIPS) to date.

3.

A first solution for training photonic neurons off-chip with realistic parameters. We have proposed a functional learning method that exploits the isomorphism established between a LIF-type neuron and our laser model. Specifically, we have demonstrated up to 70% accuracy in classifying a Yin-Yang dataset with only a small number of spiking lasers. Our approach is also easily extendable to more complex tasks, such as image recognition.

Limitations

Overall, one of the major challenges we faced was the lack of functional experimental laser devices. Indeed, due to several factors, including the COVID-19 pandemic, the fabrication of spiking lasers took longer than expected and is still ongoing at the time of this writing. One major impact of this reality is that the design of a large-scale photonic network model, comprising lasers and photonic synapses, was put on hold. Also, without a functional laser prototype, we were unable to consolidate our simulation models, thus limiting our ability to develop more accurate models. Also, we did not manage to evaluate how much energy our spiking lasers consumed, which is another weakness of this thesis. As a result, to compensate for the lack of experimental devices, my research had to focus on exploring the potential of spiking lasers and developing theoretical and computational models that could provide a deeper understanding of their capabilities.

Future work

Despite a number of limitations, our achievements have allowed us to reach a first step toward the realization of a dense photonic neural network. This project, however, requires some improvements and further investigation to be successful. Some of these key improvements include:

1. Characterisation of the spiking lasers. A thorough assessment of the spiking laser behavior is necessary to understand and optimize the performance of the future photonic network. Additionally, this characterization will as well strengthen the learning algorithm by consolidating the laser numerical model.

2.

Developing/Finding a reliable numerical tool for laser gain spectrum extraction . This will allow for more accurate predictions of laser performance, allowing for better optimization of the laser parameters. While this might seem like a trivial thing, it has been a challenging point in this thesis, mostly due to the lack of information on laser materials and the immaturity of the numerical tool used for this purpose.

3. Development of silicon-based synaptic functions. Designing scalable architecture of synaptic functions is also necessary to introduce learning and memory in the photonic network. We have considered using MRR devices for their low energy consumption, scalability, and compatibility with our hybrid lasers. However, we did not have the opportunity to develop this aspect, therefore an in-depth investigation is still needed.

4.

Reconsidering the excitability mechanism of the photonic neuron. As we saw in Chapter 1, photonic neurons can be electrically (O/E/O) or optically activated. Initially, O/E/O neurons were the focus of our research due to their ease of integration. However, due to the quadratic relationship between optical power and electrical current, we believe that our algorithm's efficiency is limited. Consequently, we have explored the possibility of integrating an all-optical photonic neuron to preserve computation mostly in the optical domain. For a detailed description of the configuration and dynamics of this all-optical neuron, please refer to our paper [1]. There are clear benefits to an all-optical neuron, such as increased speed, power consumption, and potential for network learning, but they would also require much more laborious and controlled integration.

where f V and f I are vectors of size N (the total number of neurons). We introduce the Lagrange multipliers λ V and λ I such that the loss function (equation 3.9) can be written as:

L = l P t post + T 0 [λ V • f V + Rλ I • f I ] dt (A.4)
where t post is the output first pulse timing vector. As, by definition, f V and f I are null when

V and I comply with the neuron dynamics, λ V and λ I can be chosen arbitrarily. Now we separate the integral into a sum of integrals between spike times t post similarly, we have (V + ) s = 0. We thus obtain for V + :

∂t post k ∂I bias,n = - 1 ( V + ) s ∂(V + ) s ∂I bias,n
Thus, the pre and post-firing voltages obey the following relationship:

∂(V + ) s ∂I bias,n = ( V + ) s ( V -) s ∂(V -) s ∂I bias,n
Finally, as this neuron did emit a spike, we have τ mem. ( V + ) s = I s (t post k ) and τ mem. ( V -) s = -V + I s (t post k ). As a result, we have:

( V + ) s ( V -) s = ( V + ) s -( V -) s ( V -) s + 1 = V τ mem. ( V -) s + 1
As a result, can write the final equation for the transition of the membrane potential as:

∂(V + ) s ∂I bias,n = ∂(V -) s ∂I bias,n V τ mem. ( V -) s + 1
For a non spiking neuron m, we have (V + ) m = (V -) m , thus: For the pulsing neuron (W s,s = 0), we get:

∂(V + )
∂(I + ) s ∂I bias,n = ∂(I -) s ∂I bias,n
We now replace these expressions into equation A.12, after developing the dot products: The above expression allows us to set the transitions of the adjoint functions at the spike times for the non-spiking neurons:

ξ k = m̸ =s τ mem. λ - V -λ + V m ∂(V -)
(λ - V ) m = (λ + V ) m (λ - I ) m = (λ + I ) m
and for the spiking neuron: ∂I bias,n = 0 when s ̸ = n. Indeed, the pulse energy from neuron s does not depend on the bias of neuron n. Thus, the term ξ k can be written: where N n is the number of pulses emitted by neuron n.

(λ - V ) s = (λ + V ) s + 1 τ mem. ( V -) s V(λ + V ) s +
ξ k = -Rτ syn.
Coming back to the yinyang example with a purely spike time based loss function, we can compute analytically the bias gradient in equation A. 14. In between two successive spike events happenning at t post n,i and t post n,i+1 (t post n,i < t post n,i+1 ) , the integral of (λ I ) n is :

t post n,i+1

A.0.2 derivation of the weight gradient

We now compute the derivative of the loss function with regards to the synaptic weights W i,j between pre-synaptic neuron i and post-synaptic neuron j ([i, j] ∈ [1, N ] 2 ): The computation is now very similar to the one performed above and in the paper up to the application of the implicit funciton theorem where the energy depend function introduces additional terms. So, after some calculaiton we obtain (equation 33 and 34 in the paper): .18) with :

∂L ∂W i,
∂L ∂W i,j = Npost k=0 ξ k (A
ξ k = ∂l P ∂t post k ∂t post k ∂W i,j + τ mem. λ - V • ∂V - ∂W i,j -λ + V • ∂V + ∂W i,j (t=t post k ) + Rτ syn. λ - I • ∂I - ∂W i,j -λ + I • ∂I + ∂W i,j (t=t post k ) (A.19)
The adjoint functions are unchanged compared to the previous calculations. The application of the implicit function theorem for the membrane potential is similar to that of the I bias case and leads to, for a spiking neuron s:

∂(V + ) s ∂W i,j = ∂(V -) s ∂W i,j V τ mem. ( V -) s + 1
For a non spiking neuron m, we obtain:

∂(V + ) m ∂W i,j = ∂(V -) m ∂W i,j + RW s,m e s (t post k ) τ mem. ( V -) s ∂(V -) s ∂W i,j
For the current, the computation differs from before as we have (I + ) m = (I -) m +W s,m e s (t post k ). We derive, again using the implicit function theorem: For the pulsing neuron (W s,s = 0), we still get:

∂(I + ) m ∂W i,
∂(I + ) s ∂W i,j = ∂(I -) s ∂W i,j
We now replace these expressions into equation A.19, after developing the dot products:

ξ k = m̸ =s τ mem. λ - V -λ + V m ∂(V -) m ∂W i,
) ∂t post k 1 ( V -) s ∂(V -) s ∂W i,j (A.20)
The above expression shows that the transitions of the adjoint functions at the spike times are the same as computed before and finally we obtain: Note that, in the above expression, the k th spike at t post k is emitted by neuron s. The first term is only non-null when this spike is the pre-synaptic spike for W i,j (i = s) and the post-synaptic neuron is j (j = m). This term thus simplfies to -Rτ syn. e i (t post k ) λ + I j . The second term is only non-zero when W i,j is a weight that precedes the spiking neuron s (s = j). Indeed, the spiking behiour of neuron s cannot be influenced by a synaptic weight W i,j that is not directly preceding that neuron. This means that, in this term, the neurons m are thus post-synaptic to neuron j. So, when neuron j emits a spike, the adjoint current variables of the post-synaptic neurons back-propagate. Finaly, the expression for ξ k is: where N i is the number of pulses emitted by neuron i and N j the number of pulses emitted by neuron j. Abstract : Today, neuromorphic networks play a crucial role in information processing, particularly as tasks become increasingly complex: voice recognition, dynamic image correlation, rapid multidimensional decision making, data merging, behavioral optimization, etc... Neuromorphic networks come in several types; spiking networks are one of them. The latter's modus operandi is based on that of cortical neurons. As spiking networks are the most energy-efficient neuromorphic networks, they offer the greatest potential for scaling. Several demonstrations of artificial neurons have been conducted with electronic and more recently photonic circuits.

ξ k = -Rτ syn.
ξ k = -
The integration density of silicon photonics is an asset to create circuits that are complex enough to hopefully carry out a complete demonstration. Therefore, this thesis aims to exploit an architecture of a photonic spiking neural network based on Q-switched lasers integrated into silicon and an ultra-dense and reconfigurable interconnection circuit that can simulate synaptic weights. A complete modeling of the circuit is expected with a practical demonstration of an application in solving a mathematical problem to be defined.

Université Bourgogne Franche-Comté 32, avenue de l'Observatoire 25000 Besançon

CHAPTER 1 .

 1 INTRODUCTION AND STATE OF ART 1.1 Introduction to spiking neural networks 1.1.1 Biological background

  Fig 1.1.b.

Fig 1 . 2

 12 Fig 1.2.b depicts the neuron's spike response. A spike signal is a short voltage signal that

Fig 1 .

 1 Fig 1.3.a illustrates a chemical junction between an excitatory neuron (pre-synaptic) and a target neuron (post-synaptic). In fact, through the exocytosis process[6], a pre-synaptic action potential or spike signal is transmitted to a post-synaptic axon terminal using neurotransmitters, which are charged ions in synaptic vesicles. When ions are released, they diffuse into the post-synaptic neuron's extracellular fluid through a small gap (the synaptic cleft[4]), where they bind to specific receptors or gates. In response to these neurotransmitters, either

Figure 1 . 2 :

 12 Figure 1.2: (a) Distribution of ionic charges in a plasma membrane of a neuron. Species present in large quantities are marked by large square blocks, while species present in small quantities are marked by small square blocks. (b) Schematic reproduction of a neuron's spike with its different transitions. Spike has generally a maximum peak amplitude of around 40 mV,[5].

Figure 1 . 3 :

 13 Figure 1.3: (a) An illustration of a chemical synapse[8]. Synaptic vesicles contain positive or negative charges or neurotransmitters. A post-synaptic neuron's terminal contains specific receptors on which neurotransmitters bind after a spike has occurred. (b) Characterization of EPSP (depolarization) and IPSP (hyperpolarization), induced by the stimulation of receptorgated ion channels permeable to Na + (left) and Cl -+ (right), respectively [7].

Figure 1 . 4 :

 14 Figure 1.4: Schematic diagram of artificial neuron units. The three-generation differ in their activation function which can be either binary (a), analog (b), or spiking (c).

CHAPTER 1 .Figure 1 . 5 :

 115 Figure 1.5: Neuron models spectrum

Figure 1 . 7 :

 17 Figure 1.7: (a) An equivalent RC circuit modeling the passive membrane of LIF neuron. (b)A typical LIF neuron's membrane potential evolution over time[19]. The neuron integrates several spikes, resets its potential to zero after a spike is released, and enters a refractory period before returning to its resting potential[22].

Figure 1 . 8 :

 18 Figure 1.8: Visualisation of different coding techniques. (a) A single neuron's frequency coding and (b) Temporal coding with dashed lines showing stimulus input [2]

Fig 1 . 8

 18 .a shows an illustration of the count rate coding based on a mean firing rate, v, associated with a single neuron.

CHAPTER 1 .

 1 INTRODUCTION AND STATE OF ART 1.1.2.4 SNN most common architecture

CHAPTER 1 .

 1 Figure 1.11: Overview of backpropagation error training involving a network consisting of two matrices of synaptic connectivity, W 1 and W 2 .

Figure 1 .

 1 Figure1.13: (a) (left) Photonic synaptic implementation using PCM elements and tapered waveguide. An optical circulator is used to modulate light intensity in one direction (port 1), and allow the weight signal to be transmitted from the pre-neuron to the post-neuron through port 2 to port 3. (right) A scanning electron microscope (SEM) of the PCM synapse in the red box[START_REF] Cheng | On-chip photonic synapse[END_REF]. (b) A silicon nitride (Si 3 N 4 ) photonic integrated PCM cross-bar implementation for parallel matrix multiplication[START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF]

Figure 1 .

 1 Figure 1.14: (a) Tunable spectral filter based on microring resonator (MRRs) weight bank. Each ring targets a single WDM frequency, resulting in an array of signals with different amplitudes that are incoherently summed in a balanced photodetector. (b) Optical micrograph of a silicon four-channel MRR weight bank showing metal heaters for thermal tuning[START_REF] Tait | Multi-channel control for microring weight banks[END_REF]. (c) An example of a synaptic dot product implementation with N × M MRRs units[START_REF] Yang | On-chip optical matrix-vector multiplier for parallel computation[END_REF]. Note that, the multiplier coefficients are all positive, whereas they can be either positive or negative in designs (a) and (b). MD: modulator, PD: photodetector

Figure 1 . 15 :

 115 Figure 1.15: An example of an Optical Interference Unit (OIU) with multiple unitary matrices based on MZIs, which can perform non-unitary optical matrix multiplications [69]. MZIs arranged in (b) rectangular, (c) triangular, and (d) diamond mesh topologies[START_REF] Srouji | Photonic and optoelectronic neuromorphic computing[END_REF]

Figure 1 .Fig 1 . 16

 1116 Figure 1.16: (a) A micrograph of an O/E/O photonic neuron based on a two-section DFB laser (highlighted in black) which is driven by a pair of balanced photodetectors (highlighted in blue). (b) Response of the laser neuron to increasing width input pulses. (c) Coincidence detection: Laser neuron's response to two pulses with varying center-to-center separation. (d) The laser neuron's response to successive input pulses as a function of their temporal separation, showing its refractoriness. For fig (b)-(d), simulated and experimental results areshown in green and red, respectively. Taken from[START_REF] Peng | Temporal Information Processing With an Integrated Laser Neuron[END_REF]

  and in experiments (cf Figs 1.16.b-d). Fig 1.16.b illustrates the laser's nonlinear response with a constant excitability threshold. Input pulse energy beyond this threshold saturates the SA, which results in an output spike generation. The laser can also release a spike in the case where multiple weak stimuli arrive within a short period of time. This behavior highlights the leaky temporal integration property of the neuron laser, which is usually exploited to perform coincidence detection as in Fig 1.16.c. Another important spiking property is presented in Fig 1.16.d; the laser shows an absolute and relative refractory period,

  .

Figure 1 .

 1 Figure 1.17: (a) Cutaway structure of the proposed integrated GEL. (b) Response of both fiberbased (measured) and integrated GEL (simulated) to two excitatory pulses, for coincidence detection experiment. (c) Response of both fiber-based and integrated GEL to an input pulse with increasing energy. (d) Refractory periods of both excitable lasers: response to a second excitatory pulse following a spike generation induced by a first identical pulse. For fig (b)-(d), simulated integrated GEL and measured response of the fiber-based GEL are shown in pink and blue, respectively. Taken from[START_REF] Shastri | Spike processing with a graphene excitable laser[END_REF]

CHAPTER 1 .Figure 1 .

 11 Figure 1.18: (a) Schematic cross-section of a VCSEL-SA. (b) Excitable behaviour of a VCSEL-SA in response to optical input spikes (green dashed lines), showing the variation of the gain carrier density region (top) and its output power (bottom) [102].

  Figure 1.19: (a) Sketch and scanning electron microscope (SEM) image of the micropillar laser with SA. (b) Amplitude response to an optical pulse perturbation versus perturbation energy for bias pump P relative to the self-pulsing threshold PS.[108]

Figure 1 .

 1 Figure 1.20: (a) Scanning electron micrograph (SEM) of the PCM-based photonic neuron; The PCM element is implemented on top of a low-loss waveguide crossing section. (b)The transfer function of the photonic neuron at a fixed wavelength shows the highest contrast between the output states.[START_REF] Feldmann | Alloptical spiking neurosynaptic networks with self-learning capabilities[END_REF]

  Fig 1.21.a. Here, an array of N optical input signals (unicolor) with distinct wavelengths are multiplexed onto a single bus waveguide (multicolor). These signals are then distributed CHAPTER 1. INTRODUCTION AND STATE OF ART

Figure 1 .

 1 Figure 1.21: (a) Silicon architecture of a broadcast-and-weight network. An O/E/O link is employed, consisting of a multi-channel of tunable MRR for weight configuration, each connected to a pair of balanced photodetectors for incoherent summing, and an E/O converter that generates spikes at a specific wavelength [77]. (b) A typical B&W photonic implementation based on lasers [118].

CHAPTER 1 .Figure 1 .

 11 Figure 1.22: (a) The architecture of an all-optical neural network. (b) Single-layer photonic network. (c) Experimental design of a single-layer spiking neural network based on a bank of MRR sharing a single bus waveguide. (d) Vowel recognition task using four neurons. The figure shows the change in output spike intensity for the four learn pattern[74].

Fig 1 .

 1 Fig 1.22.b shows an experimental implementation of a single layer from the spiking neural photonic circuit. This prototype consists of four neurons, each connected to 15 synapses, an integrated distributor that feeds the neurons, and an off-chip collector that gathers input data. The entire network has been tested on a pattern recognition task including four vowels: A, B, C, and D. Specifically, each synapse represents a pixel of a 3-by-5 image and codes a wavelength associated with a ring multiplexer. According to the results (see Fig 1.22.c), the network can successfully distinguish the 4 patterns since each neuron is only activated by the pattern it

Figure 1 .

 1 Figure 1.23: (a) General architecture of an all-optical integrated neural network formulated with the coherent approach. (b) Optical micrograph of an on-chip optical interference unit. (c) Confusion matrix showing the results of 180 cases on vowel recognition test set [78].

CHAPTER 1 .

 1 INTRODUCTION AND STATE OF ART ture can be seen in Fig 1.23.a.

Figure 2 . 1 :

 21 Figure 2.1: Principles of saturable absorption in semiconductor material.(a) Saturable absorption due to Pauli Blocking effect [2], (b) Electric field-induced quantum confinement Stark effect [3].

Figure 2 . 2 :

 22 Figure 2.2: Illustration of the formation of a short light pulse under QS operation [10].

Figure 2 . 3 :

 23 Figure 2.3: Portrayal of mode-locking operation. (a) Mode-locked laser output signal showing phase-matching between multiple cavity modes [16]. (b) Dynamic representation of gain and losses inside the cavity during passive mode-locking [12].

CHAPTER 2 .

 2 PHOTONIC SPIKING LASER: DESIGN AND MODELING 2.1.3.2 Q-Switched Mode-Locked

Figure 2 . 4 :

 24 Figure 2.4: illustration of a hybrid III-V/Si laser integration. Longitudinal (a) and cross-section (b) schematic views of the CW laser [17]. (c) A top view of a silicon rib Taper designed to transfer only the even mode from the III-V region and the silicon region adiabatically [18].

Figure 2 . 5 :

 25 Figure 2.5: (a) Optical microscope image of one of a fabricated hybrid III-V/Si laser. Measured light-current-voltage (LIV) curve at room temperature of 25°C (b) and spectrum of the laser (c) emitting a CW near 1310 nm. The spectrum is realized at a driving current of 147 mA [18].

Fig 2 . 6 Figure 2 . 6 :

 2626 Figure 2.6: Longitudinal, top (a) and cross-section (b) views of the laser photonic neuron. The laser is a three-section cavity with a gain-SA-gain configuration. H+ refers to the hydrogenimplanted sections of the active waveguide. The underneath silicon waveguide has a 200-nmthick rib and a 300-nm-thick slab. (c) Cross-section view of the integrated mirrors used to define optical cavities.

Figure 2 . 7 :

 27 Figure 2.7: Diagram depicting two laser configurations: (a) colliding pulse mode (CPM), and (b) self-colliding pulse mode (SCPM). In both cases, the gain section(s) is (are) forward-biased while the absorber is reverse-biased.

Figure 2 . 8 :

 28 Figure 2.8: A longitudinal view of the laser, showing one isolation cavity of length, L isol and depth d isol . Hydrogen protons are implanted within the isolation section to ensure efficient isolation between the gain and absorber sections.

  Fig 2.9.a,b. A three-cases analysis of the isolation cavity length is displayed here: L isol = 1, 5 and 10 µm.

CHAPTER 2 .Figure 2 . 9 :

 229 Figure 2.9: Optical and electrical simulations of the isolation cavity. Illustrations of the transmission (a) and reflection (b) coefficients of the fundamental III-V optical mode when a cavity is etched along the active waveguides. Both cases involve varying lengths and depths of the cavity. The inset shows the fundamental mode field intensity profile of a 5-µm-wide III-V waveguide. (c) Evolution of the leakage current between two sections as a function of the length and depth of the isolation cavity. (d) Leakage current vs. isolation cavity length at a fixed depth of d isol = 0.4 µm. The red point represents the leakage of the isolation cavity with an H + implant.

Figure 2 .

 2 Figure 2.10: (a) Longitudinal view of the simulated region defined in the TRIM program. (b) Distribution of multi-energy H+ implant protons into the III-V layers resulting from the TRIM simulation. Grey and blue regions represent the top surface of the isolation cavity and the MQWs region, respectively.

  Results for protons implanted at 100, 170, and 230 keV are shown in Fig 2.10. The three cases are simulated with an implant dose of 2 × 10 14 atoms/cm 2 . Simulated III-V layers are similar to those in Fig 2.8, with an additional 200 nm silicon nitride (SiN) hard mask on the top surface of the isolation cavity, representing the simulation's starting point (see Fig 2.10.a).

CHAPTER 2 .Figure 2 .

 22 Figure 2.11: (a) Top-view of the spiking laser mask layout. (b) III-V epitaxy bonded on a 200 mm SOI wafer before III-V patterning.

Figure 2 . 12 :

 212 Figure 2.12: Overview of the process flow developed for the fabrication of the three-section laser. Bold arrows indicate the main fabrication sequences, while narrower arrows point to the sub-steps.

Figure 2 . 13 :

 213 Figure 2.13: SEM images of dry-etched III-V ridge waveguides in small (1.3 µm) (a) and large (5 µm) (b) III-V ridge waveguide configurations.

Figure 2 . 14 :

 214 Figure 2.14: Numerical scheme of the three-section laser model in the Traveling-wave model.Each section is discretized into segments of length ∆z. E ± (z, t) corresponds to the slowly varying complex amplitudes of the two counter-propagating optical fields travelling along the laser cavity. S a1/2 : gain section. S s : SA section. l a1/2 : length of the corresponding gain section.

Figure 2 .

 2 Figure 2.15: (a) Cross-section of the hybrid laser used for optical simulation. A 2D field profile of the fundamental mode computed using the FDE method.

  Fig 2.15.b shows a plot of the transverse electric (TE) fundamental optical mode of the structure. The respective effective index n ef f = 3.2586 and group index n g = 3.418 were determined at the main wavelength of 1310 nm.

  Fig 2.16.a,bshows the resulting spectra for a corresponding 6 nm thick AlGaInAs MQW operating around 1310 nm at 293K. As can be seen from these two figures, with the increase in carrier density injection, the peak wavelengths of the gain material (index) shift towards shorter (longer) wavelengths, indicating the band gap-filling (shrinking) effect.Besides, by looking at the dependence of material gain and index changes on carrier density at 1310 nm, one can determine the theoretical differential gain g ′ and differential index ∆n ′ , needed for our model. The results are shown in Fig 2.16.c,d. By using equations 2.6 and 2.7, we approximated the peak material gain and index functions at a fixed wavelength, as shown by the solid lines on the graphs. Accordingly, these models show good agreement with the microscopic simulations (open dots). The parameters a and b are then evaluated at the transparency density given by g(N ) = 0 on Fig 2.16.c, namely N tr = 1.47 × 10 24 m -3 . Thus, at N tr , we found approximately g ′ = 2.7 × 10 -19 m 2 and ∆n ′ = 2.5 × 10 -27 m 3 .

Figure 2 .

 2 Figure 2.16: (a) Stimulated emission (material gain) and (b) refractive index change spectra as a function of wavelength for various carrier densities at 293K. Carrier densities range from N = 2.083 × 10 23 m -3 to N = 6.875 × 10 24 m -3 in both cases. (c) Peak gain (blue) and differential gain (orange) versus carrier density at operating wavelength λ 0 = 1310 nm. The transparency carrier density N tr is depicted on the graph by a dashed line crossing the horizontal axis. (d) Refractive index change (blue) and differential refractive index (orange) versus carrier density at λ 0 . (e) α-factor as a function of carrier density at a fixed wavelength λ 0 . Solid line: analytic models, Open dots: k•p simulations.

Figure 2 . 17 :

 217 Figure 2.17: An example of a Lorentzian filter approximating a gain spectrum. The fit presented here (red line) is modeled with ω = 0, γ = 104 nm, ḡ = 2270 cm -1 and for N = 2.70 × 10 24 m -3 .

CHAPTER 2 .Figure 2 . 18 :

 2218 Figure 2.18: Automated detection of the ML regime using (a) RF spectrum and (b) the sampled eye diagram. (c) The number of pulses per cavity round-trip time observed in the CPM regime (at the bottom) and its harmonic regime (at the top).

Fig 2 .

 2 18.c). Additionally, harmonics are identified by monitoring the number of pulses emitted from the laser within a cavity round-trip time, T rep , where the CPM-regime is expressed by two pulses per T rep (see Fig 2.18.b).

Fig 2 . 19 .

 219 The first two panels, representing a frequency (a) and a maximal power (b) diagram respectively, have been generated by analyzing the RF spectrum and the time traces of the local electric field E + (L, t) at the right laser's facet. They both show a dynamic pattern of the operating state of the laser CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING

  Fig 2.19 and Fig 2.20.a-f. As expected, this regime appears to self-start as the laser is biased near its turn-on threshold, which is evaluated here at ∼ 50 mA. As we can see in Fig 2.19.c,d, the disparity in pulse amplitudes indicates a highly unstable oscillatory regime, which is characterized by low repetition frequencies ranging from 1 to 20 GHz on Fig 2.19.a).

Fig 2 .

 2 20.d-f where I a = 80 mA.

CHAPTER 2 .Figure 2 . 19 :

 2219 Figure 2.19: Three-section semiconductor laser dynamic analysis. Two-dimensional map depicting the mean frequency (a) and maximal output power (b) as functions of injection current I a and the SA's recovery time τ s : (A) Q-switching instability; (B) CW ML operation; (C) 2-nd harmonic ML; (D) CW single mode operation. The white area corresponds to the laser's offstate. Bifurcation diagrams for increased injection current at τ s = 5 ps (c) and τ s = 25 ps (d).

Fig 2 .

 2 19 and Fig 2.20.g-l. On bifurcation diagrams (cf Fig 2.19.c,d), the ML operation is represented by a stable T-branch (transition to a new oscillating regime), indicating regular pulse generation from the laser. These pulses have a repetition frequency of ∼ 40 GHz (see Fig 2.19.a), and according CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING (a) (b)

Figure 2 . 20 :

 220 Figure 2.20: An analysis of the different laser pulsed regimes. (a,d,g,j): Time traces of the laser's output power. (b,e,h,k) : Temporal evolution of carrier densities inside one gain section (dotted lines) and the SA section (solid lines). (c,f,i,l) : RF spectra in the 0-100 GHz range. τ s = 5 ps, I a = 60 mA (a,b,c), I a = 80 mA (d,e,f), I a = 100 mA (g,h,i). τ s = 25 ps, I a = 150 mA (j,k,l).

Fig 2 .

 2 Fig 2.20.j-l shows a special case of CPM that takes place at higher I a and τ s values. Here, although the gain and SA carrier densities exhibit similar saturation dynamics, the temporal trace of the electrical field displays irregular satellite pulses alongside the ML pulses, giving rise to the modulated CPM-regime as shown in Fig 2.20.j. Consequently, the laser has a complex chaotic behaviour which is represented by region C in our simulation results. The spectral trace of this regime reveals strong side modes accompanying the main CPM frequency, which is indicative of poor CPM performance. Thus, by looking at the frequency map, this regime might subsequently lead to ML harmonics, which are detected at frequencies such as ∼ 80 GHz and ∼ 120 GHz. Finally, for very high currents, a continuous wave CW state (region D) could be established.

Figure 2 . 21 :

 221 Figure 2.21: Simulated power-current characteristic curves for SA lengths of 20 µm (a) and 60 µm (b). Red lines: Maxima of the optical power. Dotted lines: Minima of the optical power. Grey area: QS instability.

Fig 2. 21

 21 Fig 2.21 shows simulation results for two lasers with absorber lengths of 20 (∼ 3 %) µm and 60 µm (10 %), respectively. In both cases, the active section is 600 µm long. For these two figures, we varied I a while fixing all other parameters, including the SA's recovery time,

Contents 3 . 1

 31 Processing network node (PNN) model . . . . . . . . . . . . . . . . . . 90 3.1.1 LIF-like dynamics of the photonic spiking neuron . . . . . . . . . . . 91 3.1.2 Towards the multilayer photonic SNN . . . . . . . . . . . . . . . . . 97 3.2 A hybrid supervised learning approach for training photonic SNNs . 99 3.2.1 The proposed LIF neuron model . . . . . . . . . . . . . . . . . . . . . 99 3.2.2 Adaptation of EventProp algorithm . . . . . . . . . . . . . . . . . . . 102 3.2.3 Our learning framework . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2.4 Case study: The Yin-Yang dataset . . . . . . . . . . . . . . . . . . . . 104 3.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

CHAPTER 3 .

 3 ADAPTIVE PHOTONIC NEURAL NETWORK FOR HARDWARE IMPLEMENTATION 3.1 Processing network node (PNN) model Our PNN model follows an incoherent O/E/O framework similar to the one proposed in ref [1]. As mentioned in Chapter 1, this approach offers the possibility to cascade multiple processing nodes without worrying about optical signal mismatches. A direct analogy between a LIF-type neural network and the proposed PNN is illustrated in Fig 3.1.

Figure 3 . 1 :

 31 Figure 3.1: Direct analogy between the proposed photonic processing network node (bottom) and a LIF neuron-based computation (top). Node: Photonic spiking neuron. PRE: pre-synaptic photonic node. POST: post-synaptic photonic node. ω ij is the reconfigurable synaptic weight between an i-th pre-synaptic node and a j-th post-synaptic node.

3. 1

 1 .1 LIF-like dynamics of the photonic spiking neuron 3.1.1.

Figure 3 . 2 :

 32 Figure 3.2: (a) Energy transfer functions of the laser neuron for different inhibitory pulse energies.(b) Neuron firing time as a function of the energy of the excitatory pulse. E inh : Inhibitory input pulse energy

3. 2

 2 .b), their integrated sum may exceed the gain threshold level, leading to a firing mechanism. However, in the case illustrated byFig 3.

2

 2

Figure 3

 3 Figure 3.3: (a) Evolution of the output spike energy as a function of the inter-spike interval (ISI) time of two excitatory input pulses. Carrier density in gain (green line) and intensity (orange dot) time traces for ISI = 20 ps (b) and ISI = 80 ps (c). t out,1 : firing time

Fig 3 . 4

 34 .b and Fig 3.

4

 4

Figure 3

 3 Figure 3.4: (a) Laser response to two excitatory pulses of the same energy as a function of their inter-spike interval (ISI) time. Time traces of the carrier density in gain (green line) and input pulse intensity (orange dot) for ISI = 3.3 ns (b) and ISI = 4.5 ns (c). E out,1 : energy of the first output pulse; E out,2 : energy of the second output pulse; t out,1 : firing time of the first spike;

Fig 3 . 5

 35 Fig 3.5.b,c shows simulation results with W = 35 and τ = 6 ns, under the same bias conditions. Simulations are conducted over a timescale of 100 ns. As can be seen in the plot (see Fig 3.5.c), the laser produces a first spike at t = 10 ns in response to the external input signal.Hence, by injecting this output spike (which has been amplified) back into the system, a recursive process is initiated, resulting in a pulse train that repeats ad infinitum with a period close to τ . In this case, the system exhibits bistable dynamics and is capable of regenerating

Figure 3 . 5 :

 35 Figure 3.5: (a) Setup of the autapse concept with a single laser neuron with O/E/O link. (b) Input current induced by an external signal pulse at t = 10 ns. (c) Self-sustained spikes caused by the recurrent connection.

Fig 3 . 6

 36 Fig 3.6 illustrates our conceptual model of a multilayer photonic neural network in a feedforward architecture. The network consists of cascaded laser nodes eventually implemented

Figure 3 . 7 :

 37 Figure 3.7: Comparison of energy transfer functions between the LIF neuron and the spiking laser.

3. 7 .

 7 The results of the most reliable fit are displayed in Fig 3.8 and the corresponding fitted values are reported in

Figure 3 . 8 :

 38 Figure 3.8: The spiking laser neuron fitting results. (a) Input energy threshold vs. applied bias current. (b) Spike latency as a function of input threshold energy under different current biases. (c) The neuron's energy transfer functions at different current biases. Solid lines indicate the fit, while dots indicate the laser data. The bias current varies between 40 and 48 mA with 1 mA steps.

Figure 3 . 9 :

 39 Figure 3.9: An overview of our hybrid supervised learning approach. The photonic spiking network (top) is composed of three layers (an input layer (I), a single hidden (H) layer and an output layer (O)), each of which has a specific number of processing units (PUs) or nodes, which typically consist of a pair of balanced photodetectors and a spiking laser. The communication between layers proceeds only one way (feed-forward topology), and the weight matrices, W IH and W HO have respectively, N I × N H and N H × N O elements. The network is transposed (T F ) in the LIF domain (bottom) and pre-trained with the EventProp algorithm using the same architecture. Inference is then performed in the photonic domain via reverse transposition (T F -1 ) of the weight and bias terms. Spike latency and first-to-spike strategies are used to encode input data and denote the final result, respectively. PSNN: Photonic spiking neural network; ADSNN: Adapted spiking neural network. N I : Number of elements in the input layer;N H : Number of elements in the hidden layer; N O : Number of elements in the output layer;

Fig 3 .

 3 Fig 3.10.a displays the input data locations on a disk of centre [0.5, 0.5] and a radius of 0.5 in the xy-plane. The data is divided into three regions or classes, each having a distinct color

Fig 3 .

 3 Fig 3.10.d,e shows the prediction performance of the ADLIF network when configured with the laser parameters. The network is trained over 300 epochs and 10 random trials are run to measure reproducibility. Fig 3.10.d illustrates the result of the prediction in terms of accuracy.Note that, it displays the curve average over the 10 trials. Based on this graph, we can see that the ADLIF network correctly predicted about 80 % of the 3200 training data. This performance is slightly lower than that of[7], which reaches around 95% for a standard SNN network with the same number of hidden neurons. A sample with the highest classification accuracy (about 85 %) achieved with the network is presented in Fig 3.10.e. It shows the spike times collected from the three labeled neurons ("yin", "yang", and "dot"), with red dots representing data that does not belong to the correct class while green dots represent correctly classified data. Here, as expected, incorrect data tend to cluster around the boundaries between classes, especially near the dot class. This is a sign that our model needs to be further optimized as we will explain in the next section.

k. 5 ) where t post 0 = 0

 500 with k ∈ [0, N post ]:L = l P t post + • f V + Rλ I • f I ] dt(Aand t post Npost+1 = T . Note that N post represents the total number of spikes emitted in the whole neural network.A.0.1 derivation of the bias current gradientWe now compute the derivative of the loss function with regards to the bias I bias,n of neuron n (n ∈ [1, N ]):

τk

  mem. λV = λ V τ syn. λI = λ I -λ V then one integral vanishes in equation A.10. We can rewrite the main equation as :∂L ∂I bias,,n= now consider two cases: the one of neuron s that is responsible for the spike at t post k and the one of neuron m that does not spike. We first compute the jumps of the adjoints functions (λ V ) s and (λ I ) s at the spike times t post k . At the spike time we have (V -) s = V. We consider the function f I bias,n , t post k = (V -) s -V and the implicit function ψ (I bias,n ) = t post k defined by the solutions of equation f = 0. As f is differentiable, the implicit function theorem leads to : ∂ψ ∂I bias,n = -∂I bias,n = -1 ( V -) s ∂(V -) s ∂I bias,n

  m̸ =n W n,m (λ + I ) m ∂e n (t post k ) ∂I bias,n with k only covering the pulses emitted by neuron n. Finally, the gradient function A.11 becomes : ∂L ∂I bias,n = -Rτ syn. Nn kn=1 m̸ =n W n,m (λ + I ) m ∂e n (t post kn ) ∂I bias,n -

  m̸ =s δ is δ jm e s (t post k ) λ + I m -Rτ syn. m̸ =s W s,m (λ + I ) m ∂e s (t post k ) ∂W i,j

Titre:

  Réseau photonique de neurones impulsionnels Mots clés : photonique sur silicium, réseaux de neurones impulsionnels, intelligence artificielle, laser à semi-conducteur, calcul neuromorphique Résumé : Les réseaux neuromorphiques pour le traitement d'informations ont pris une place importante aujourd'hui notamment du fait de la montée en complexité des tâches à effectuer : reconnaissance vocale, corrélation d'images dynamiques, prise de décision rapide multidimensionnelle, fusion de données, optimisation comportementale, etc... Il existe plusieurs types de tels réseaux et parmi ceux-ci les réseaux impulsionnels, c'est-à-dire, ceux dont le fonctionnement est calqué sur celui des neurones corticaux. Ce sont ceux qui devraient offrir le meilleur rendement énergétique donc le meilleur passage à l'échelle. Plusieurs démonstrations de neurones artificielles ont été menées avec des circuits électroniques et plus récemment photoniques. La densité d'intégration de la filière photonique sur silicium est un atout pour créer des circuits suffisamment complexes pour espérer réaliser des démonstrations complètes. Le but de la thèse est donc d'exploiter une architecture de réseau neuromorphique impulsionnel à base de lasers à commutation-Q (Q-switching) intégrés sur silicium et d'un circuit d'interconnexion ultradense et reconfigurable apte à imiter les poids synaptiques. Une modélisation complète du circuit est attendue avec, à la clé la démonstration pratique d'une application dans la résolution d'un problème mathématique à définir. Title : Photonic spiking neural network Keywords : silicon photonics, spiking neural networks, semiconductor laser, leaky integrate and fire (LIF) neuron, neuromorphic computing

  Introduction to spiking neural networks . . . . . . . . . . . . . . . . .

	Chapter 1
	Introduction and state of art
	Contents
	1.1

8 1.1.1 Biological background . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 ANN to SNN: Third generation of neural network . . . . . . . . . . . 10 1.1.3 SNN learning approaches: State-of-the-art . . . . . . . . . . . . . . . 17 1.2 SNNs implementation in photonics: state-of-the-art . . . . . . . . . . 22 1.2.1 Related works: neuromorphic photonics network . . . . . . . . . . . 22 1.2.2 Photonic synaptic functions . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.3 Photonic spiking neurons . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.4 Photonic neural network implementation . . . . . . . . . . . . . . . 35 1.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Table 1 .

 1 1.

	Method	Ref Neuron model Coding	Key features
	SpikeProp (2000)	[53] SRM	Time-based Surrogate gradient
	ReSuMe (2005)	[54] -	Rate-based Teacher signal
	Direct training (2018) [55] LIF	Rate-based Neuron normalization
	BP-STDP (2019)	[56] IF	Rate-based Backpropagation + STDP
	STDBP (2020)	[57] ReL-PSP	Time-based Rectified linear SRM
	S4NN (2020)	[58] IF	Time-based IF approximation + One spike per neuron
	TempCoding (2020) [59] SRM	Time-based Alpha Synaptic Function
	STiDi-BP (2021)	[60] Linear SRM	Time-based SRM approximation + One spike per neuron
	PLIF (2021)	[61] PLIF	Time-based Optimization of weights and time constant
	SSTDP (2021)	[62] IF	Time-based Backpropagation + STDP + TTFS
	EventProp (2021)	[63] LIF	Time-based Adjoint method + Partial deriv.jump

Table 1 .

 1

1: Comparison of different spike-based direct training algorithm

Table 1 .

 1

	Devices	Ref	Injection scheme	CMOS compatibility	Power consumption	Speed	Footprint
	VCSEL-SA	[102],[103],[113],[89]	Optical	Yes	mW	Sub-ns	Large
	DFB laser	[94],[95]	Electrical	Yes	mW	Sub-ns	Large
	Micropillar laser	[108],[110]	Optical	No	mW	Sub-ns	Large
	PCM-based cavity	[74],[111]	Optical	Yes	mW	ns	Small
	Graphene laser	[97],[114]	Electrical and Optical	Yes	mW	ns	Small
	PS-based VCSEL	[105],[115]	Optical	No	mW	Sub-ns	Large
	Microresonator	[112]	Optical	Yes	mW	Sub-µs	Medium
	Nanobeam	[112]	Optical	Yes	µW	ns	Medium
	2D Photonic crystal nanocavity	[116]	Optical	Yes	µW	µs	Medium

4: State of art of current photonic neurons, inspired from

[START_REF] Guo | Integrated Neuromorphic Photonics: Synapses, Neurons, and Neural Networks[END_REF]

,

[START_REF] Xiang | All-Optical Spiking Neuron Based on Passive Microresonator[END_REF]

.

  Short pulse generation in semiconductor laser . . . . . . . . . . . . . 50 2.1.1 Saturable absorption in semiconductor . . . . . . . . . . . . . . . . . 50 2.1.2 Passive Q-switching in semiconductor laser . . . . . . . . . . . . . . 51 2.1.3 Passive Mode-locking in semiconductor laser . . . . . . . . . . . . . 52 2.2 Design of the laser photonic neuron . . . . . . . . . . . . . . . . . . . . 54

	Chapter 2
	Photonic spiking laser: design and
	modeling
	Contents
	2.1 2.2.1 CEA Leti's technology platform for continuous-wave lasers integration 54
	2.2.2

The laser spiking Neuron: overall design . . . . . . . . . . . . . . . 56 2.2.3 Saturable Absorber's design . . . . . . . . . . . . . . . . . . . . . . . 59 2.2.4 Laser neuron fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3 Modelling of integrated semiconductor spiking lasers . . . . . . . . . 69 2.3.1 Travelling Wave model for three section laser . . . . . . . . . . . . . 69 2.3.2 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Table 2 .

 2 3. In addition, lasers are designed into 5 different ridge widths which are 1.3, 2, 4, 5, and 7 µm. The width of the gain region illustrated in Fig 2.6.b is 5 µm. Electrical p-and n-type contacts are placed on top of the ridge, and symmetrically on both sides of the n-InP layer.• III-V epitaxy of the active medium Each epitaxy used in this thesis was designed and manufactured by ALMAE Technologies. Detailed information about the III-V epitaxial layers in the case of 8-quantum well lasers is presented in table 2.1. Grey lines indicate InP substrates that will be removed during the molecular bonding process. The 120 nm MQWs region is based on AlGaInAs quaternary alloy and is designed to have a photoluminescence peak (PL) near 1310 nm.

	Layer	Material	Thickness (nm) Optical index Doping (cm -3 )
	Substrate	InP
	Transition	InP	50	3.197
	Stop etch	In (0.53) Ga (0.47) As 300
	Sacrificial layer	InP	300	3.197
	p-contact	In (0.53) Ga (0.47) As 200		3 × 10 19
	Transition	InGaAsP	50	3.3525	5 × 10 18
	p-cladding layer	InP	2000	3.197	2 × 10 18 to 1 × 10 18
	SCH over	InGaAsP	90	3.309	1 × 10 18
	MQW barriers (×9) AlGaInAs	8	3.4115	undoped
	MQW well (×8)	AlGaInAs	6	3.608	undoped
	SCH under	AlGaInAs	90	3.366	2 × 10 18
	n-contact layer	InP	110	3.197	2 × 10 18
	Super-lattice (×2)	InGaAsP	7.5	3.3525	2 × 10 18
	Super-lattice	InP	7.5	3.197	2 × 10 18
	Bonding interface	InP	17.5	3.197	2 × 10 18

Two 90-nm thick confinement heterostructures (SCHs) made of InGaAsP and AlGaInAs materials separate this MQW structure from the p-and n-doped layers. SCH layers are basically implemented here to increase the confinement of the optical mode in this region. A thick (2000 nm) InP cladding layer with a graded p-doping profile is used to CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING 1: Epitaxial composition of the III-V material for emission length around 1310 nm prevent mode absorption by the electrical p-contact. This layer is completed by 50 nm and 200 nm of heavily doped InGaAsP and InGaAs layers in order to ensure that the electrical contacts are ohmic.

Table 2 .

 2 2: Opto-electrical properties of the simulated III-V layers

Table 2 .

 2

3: Modelling parameters for the three-section semiconductor laser. CHAPTER 2. PHOTONIC SPIKING LASER: DESIGN AND MODELING 2.3.2.1 Optical mode parameters

Table 3 .

 3

	Symbol Unit	Value	Refs

Table 3 .

 3 2.

	Symbol Description	Unit	Optimum value
	τ m	Membrane time constant ps	99.6
	τ s	Synaptic time constant	ps	111
	ϑ	Membrane threshold	mA	53
	δt	Neuron latency	ps	55
	η	Energy fitting coefficient W.ps 2 /V 10.8
	α	Energy fitting coefficient -	1

Table 3 .

 3 2: Tunable parameters of the ADLIF model

Table 3 . 3

 33

		Symbol Unit	Value	Refs
	ADLIF neurons parameters
	Membrane time constant	τ m	ps	99.6
	Synaptic time constant	τ s	ps	111
	Threshold	ϑ	mA	53 × 10 -3
	Neuron latency	δt	ps	55
	First energy fitting coefficient	η	W.ps 2 /V	10.8
	Second energy fitting coefficient	α	-	1
	PSNN and LIF Network architecture
	Type			Deep-forward
	Input size	N I		4
	Hidden size	N H		200
	Output size	N O		3
	Training parameters
	Simulation time	T max	ns	10
	Regularization factor	α r		3 × 10 -3	[7]
	First loss time constant	τ 0	ps	2.77
	second loss time constant	τ 1	ps	55.5
	Optimizer			Adam	[10]
	Adam parameter	β 1		0.9	[7]
	Adam parameter	β 2		0.999	[7]
	Adam parameter	ϵ		1 × 10 -8	[7]
	Weight learning rate	η w		0.001
	Bias learning rate	η i		0.0001

: Simulation parameters used for Yin-Yang classification task

  where δ n is a N -element vector with a 1 in the n th element and 0 in all others.Replacing the above expressions in A.6, we obtain: V -τ mem. λV ∂V ∂I bias,n + R λ I -λ V -τ syn. λI ∂I ∂I bias,n dt -R

	we obtain:
	∂L ∂I bias,n	=	Npost k=0	∂l P ∂t post k	∂t post k ∂I bias,n	+ τ mem. λ V •	∂V ∂I bias,n	t post k+1 t post k	+ Rτ syn. λ I •	∂I ∂I bias,n	t post k+1 t post k	+
	t post k+1 t post k	k+1 λ t post k t post	λ I δ n dt
													(A.10)
	we now chose λ
													t post k+1 t post k	λ V •	∂f V ∂I bias,n	+ Rλ I •	∂f I ∂I bias,n	dt	(A.6)
	with:
				∂f V ∂I bias,n	= τ mem.	d dt	∂V ∂I bias,n	+	∂V ∂I bias,n	-R	∂I ∂I bias,n	(A.7)
				∂f I ∂I bias,n	= τ syn.	d dt	∂I ∂I bias,n	+	∂I ∂I bias,n	-δ n	(A.8)
	∂L ∂I bias,n	=	Npost k=0	∂l P ∂t post k	∂t post k ∂I bias,n	+ τ mem.	t post k+1 t post k	λ V •	d dt	∂V ∂I bias,n	dt+
				t post k+1 t post k	λ V •	∂V ∂I bias,n	dt + Rτ syn.	t post k+1 t post k	λ I •	d dt	∂I ∂I bias,n	dt+
												t post k+1 t post k	R (λ I -λ V ) •	∂I ∂I bias,n	dt -	t post k+1 k t post	Rλ I δ n dt (A.9)
	We now use partial integration:
		t post k+1 t post k	λ V •	d dt	∂V ∂I bias,n	dt = λ V •	∂V ∂I bias,n	t post k+1 t post k	-	t post k+1 t post k	λV •	∂V ∂I bias,n	dt
		t post k+1 t post k	λ I •	d dt	∂I ∂I bias,n	dt = λ I •	∂I ∂I bias,n	t post k+1 t post k	-	t post k+1 t post k	λI •	∂I ∂I bias,n	dt

k ∂I bias,n + V and λ I such that:

  we also have τ mem. ( V + ) m = τ mem. ( V -) m + RW s,m e s (t post k ), thus:∂(V + ) m ∂I bias,n = ∂(V -) m ∂I bias,n + RW s,m e s (t post k ) τ mem. ( V -) s ∂(V -) s ∂I bias,nFor the current, as we have (I + ) m = (I -) m + W s,m e s (t post k ), we derive, again using the implicit function theorem: as we also have τ syn. ( İ+ ) m = τ syn. ( İ-) m -W s,m e s (t post k ), we obtain:∂(I + ) m

					m ∂I bias,n	+	∂(V + ) m ∂t	∂t post k ∂I bias,n	=	∂(V -) m ∂I bias,n	+	∂(V -) m ∂t	∂t post k ∂I bias,n
	∂(I + ) m ∂I bias,n	+ ( İ+ ) m	∂t post k ∂I bias,n	=	∂(I -) m ∂I bias,n	+ ( İ-) m	∂t post k ∂I bias,n	+ W s,m	∂e s (t post k ) ∂I bias,n	+	∂e s (t post k ) ∂t post k	∂t post k ∂I bias,n
	∂I bias,n	=	∂(I -) m ∂I bias,n	+	W s,m e s (t post k ) τ syn.	∂t post k ∂I bias,n	+ W s,m	∂e s (t post k ) ∂I bias,n	+	∂e s (t post k ) ∂t post k	∂t post k ∂I bias,n
	which, finally, leads to :
	∂(I + ) m ∂I bias,n	=	∂(I -) m ∂I bias,n	-	W s,m e s (t post k ) τ syn. ( V -) s	∂(V -) s ∂I bias,n	+ W s,m	∂e s (t post k ) ∂I bias,n	-	∂e s (t post k ) ∂t post k	1 ( V -) s	∂(V -) s ∂I bias,n

  Re s (t post k ) Finally, we have ξ k = -Rτ syn. m̸ =s W s,m (λ + ∂I bias,n . Without loss of generality for the pulse energy function, we can assume that

						W s,m λ + V -λ + I m -
					m̸ =s
	Rτ syn.	∂e s (t post k ) ∂t post k	m̸ =s	W s,m (λ + I ) m +	∂l P ∂t post k
	(λ -I ) s = (λ + I ) s
			I ) m	∂es(t post k	)
		∂es(t post

k

)

  (λ I ) n dt = τ syn. (λ I ) n (t post n,i+1 ) 1 -e

			-	t	post n,i+1 τsyn. -t post n,i	+
	t post n,i
	τ mem.	(λ -V ) n (t post n,i+1 ) τmem. 1 -τsyn.	1 -e -	t post n,i+1 τmem. post -t n,i	-
					τ syn.	(λ -V ) n (t post n,i+1 ) τmem. 1 -τsyn.	1 -e	-	t post n,i+1 τsyn. -t	post n,i

  j =

	with:
	∂f V ∂W i,j	= τ mem.	d dt	∂V ∂W i,j		+	∂V ∂W i,j	-R	∂I ∂W i,j	(A.16)
	∂f I ∂W i,j	= τ syn.	d dt	∂I ∂W i,j	+	∂I ∂W i,j	(A.17)
	Npost k=0	∂l P ∂t post k	∂t post k ∂W i,j	+	t post k+1 t post k	λ V •	∂f V ∂W i,j	+ Rλ I •	∂f I ∂W i,j	dt	(A.15)

  + δ is δ jm e s (t post k ) as we also have τ syn. ( İ+ ) m = τ syn. ( İ-) m -W s,m e s (t post k ), we obtain:∂(I + ) m ∂W i,j = ∂(I -) m ∂W i,j -W s,m e s (t post k ) τ syn. ( V -) s ∂(V -) s ∂W i,j+ δ is δ jm e s (t post k )

									∂(V -) s ∂W i,j	+
							W s,m	∂e s (t post k ) ∂W i,j	-	∂e s (t post k ) ∂t post k	1 ( V -) s
	j	+ ( İ+ ) m	∂t post k ∂W i,j	=	∂(I -) m ∂W i,j	+ ( İ-) m	∂t post k ∂W i,j	+
									W s,m	∂e s (t post k ) ∂W i,j	+	∂e s (t post k ) ∂t post k	∂t post k ∂W i,j
	∂(I + ) m ∂W i,j	=	∂(I -) m ∂W i,j	+	W s,m e s (t post k ) τ syn.	∂t post k ∂W i,j	+
									W s,m	∂e s (t post k ) ∂W i,j	+	k ∂t post ∂e s (t post k )	∂t post

k ∂W i,j + δ is δ jm e s (t post k )

which, finally, leads to :

  j + Rτ syn. λ - I -λ + Rτ syn. δ is δ jm e s (t post k ) λ + I m +

										I m	∂(I -) m ∂W i,j	-
	τ mem. λ -V -λ + V s +	1 ( V -) s	Re s (t post k )	m̸ =s	W s,m λ + I -λ + V m -	∂l P ∂t post k	-V(λ + V ) s	∂(V -) s ∂W i,j	+
	Rτ syn. λ -I -λ + I s	∂(I -) s ∂W i,j	-Rτ syn.	m̸ =s	W s,m (λ + I ) m	∂e s (t post k ) ∂W i,j	-	∂e s (t post k

  Rτ syn. e i (t post k ) λ + I j -Rτ syn.Summing all the ξ k variables, we obtain:

			m̸ =j	W j,m (λ + I ) j	∂e j (t post k ) ∂W i,j
	∂L ∂W i,j	= -Rτ syn.	N i k i =1

e i (t post k i ) λ + I j -Rτ syn. N j k j =1 m̸ =j W j,m (λ + I ) j ∂e j (t post k j ) ∂W i,j (A

.21)

Remerciements

Remerciements Motivation and Objectives

IMPLEMENTATION

representing the corresponding label: "Yin" (blue), "Yang" (red) or "dot" (green). Therefore, the objective of the classification task is to assign each sample to one of these three classes using the PSNN. It was determined in [8] that to achieve the best classification accuracy, at least one hidden layer was required in conjunction with backpropagation techniques. We used the adapted Eventprop algorithm from the previous section to train a multilayer photonic SNN model on this dataset. Fig 3.10.b shows the network topology used for this particular task, with N I = 4, N H = 200, and N O = 3. Also, one can notice that unlike other studies [7], [9], there is no need here to add a fifth neuron to account for a fixed bias term since it is already included in the ADLIF-free dynamics. Thus, input samples are read from the disk area and defined by a set of coordinates (x, y, x', y'), where x' = 1-x and y' = 1-y. To encode these points into the spike time of the four input neurons (N = 4) the following scheme is implemented:

Adapted EventProp algorithm

Introduction of a bias term I bias and an energy dependent function e

We rewrite the neuron dynamics differential equations adding a current bias vector denoted I bias for the neurons:

At any pre-synaptic spike time, we have:

With this system of equations, in the steady state and without external stimuli, the current of neuron k stabilizes to I k = I bias,k and its voltage to V k = RI bias,k . We assume an internal resistor R = 1Ω.

In between two successive spike events happening at t pre i and t pre j (t pre i < t pre j ) and provided the variable V k remains under the threshold, the evolution of V k and I k can be computed analytically (∀t ∈ [t pre i , t pre j ]):

τsyn. We consider in the following, as in the previous section, an error function purely based on the times to the first spike (i.e. l V (V (t), t) = 0). In order to compute the backpropagation of the error function, we introduce the following functions to solve the system of differential equations:

Conclusion

In conclusion, the introduction of both a bias term and a variable pulse energy function lead to significant changes to the computation of the gradient functions. The gradient of one neuron is affected by the values of the weighted adjoint current functions of all its post-synaptic neurons. However, the computation of the gradient remains possible as the values of the adjoint current functions are anyway kept in memory for the computation of the integrals.