Keywords: Integrated nonlinear optics with hybrid integration of 2D materials Nonlinear optics, photonics, Silicon Integrated photonics offers solu-

The author took complete responsibility for conducting a thorough literature review, which formed the basis of

Titre: Optique intégrée nonlineaire avec l'intégration hybride de matériaux bidimensionnels Mots clés: Photonique, Optique non linéaire, Silicium Résumé: La photonique intégrée offre des réponses aux limitations de la microélectronique mais certains effets tel que les effets d'optique nonlinéaires sont difficile à générer. En intégrant des matériaux 2D aux propriétés non linéaires exceptionnelles, ces limitations peuvent être surmontées. Les outils numériques sont utilisés pour la conception et l'optimisation des guides d'ondes, en tenant compte de l'impact des matériaux 2D sur les paramètres sensibles de la structure photonique. De premiers essais s'appuyant sur l'exfoliation mécanique permet de fabriquer et transférer des matériaux 2D sur les guides d'ondes. Au regard des résultats, et malgré une première demonstration des développements supplémentaires sont nécessaires pour surmonter les limitations liées à la taille des feuillets de matériau, aux dépôts involontaires et à la faible répétabilité des transferts.

Différentes strategies sont ainsi mises en place pour l'optimisation des structure hybrid. Cela s'effectue en trois axes distincts. Tout d'abord, l'optimisation des paramètres optogéometriques des tructures guidantes permet l'augmentation de l'interaction entre les matériaux bidimensionnels et le mode transverse améliorent les performances non linéaires des guides. Ensuite, les méthodes de transfert peuvent être améliorié afin d'augmenter la fiabilité des transfert ainsi que la surface couverte par les matériaux 2D. Le dévelopment de méthodes permettant le transfert ciblé de matériaux fabriqués par CVD offrent des surfaces de meilleure qualité et une plus grande homogénéité, facilitant le transfert de larges surfaces de matériaux. Des résultats prometteur de dépôt en large surface (2.8mm x 3.2 mm) de matériaux 2D ont été obtenus. Enfin, l'utilisation de cavités en anneau permet une amélioration notable de l'interaction entre les matériaux 2D et l'onde en exploitant les effets de résonance. Des études expérimentales de résonateurs en anneau hybrides ont été réalisées, démontrant le potentiel de cette approche.

Ces avancées ouvrent de nouvelles perspectives dans la photonique intégrée, améliorant les performances non linéaires des guides hybrides et ouvrant la voie à de nouvelles applications dans les communications optiques, le traitement optique de l'information et d'autres technologies photoniques avancées.

tions to the limitations of microelectronics, but certain effects, such as nonlinear optical effects, are difficult to generate. By integrating 2D materials with exceptional nonlinear properties, these limitations can be overcome. Numerical tools are used for the design and optimization of waveguides, taking into account the impact of 2D materials on the sensitive parameters of the photonic structure. Initial experiments based on mechanical exfoliation allow for the fabrication and transfer of 2D materials onto waveguides. However, despite some initial demonstrations, further developments are necessary to overcome limitations related to the size of material flakes, unintentional depositions, and low repeatability of transfers.

Different strategies are being implemented for the optimization of hybrid structures, focusing on three distinct axes. Firstly, optimization of the optogeometric parameters of the waveguide structures enables increased interac-tion between the 2D materials and the transverse mode, improving the nonlinear performance of the waveguides. Secondly, transfer methods can be improved to enhance transfer reliability and increase the surface coverage of 2D materials. The development of methods allowing targeted transfer of materials fabricated by chemical vapor deposition (CVD) offers better surface quality and greater homogeneity, facilitating the transfer of large material surfaces. Promising results have been obtained for the deposition of 2D materials in large surface areas (2.8mm x 3.2 mm). Finally, the use of ring cavities enables a significant improvement in the interaction between the 2D materials and the wave by exploiting resonance effects. Experimental studies of hybrid ring resonators have been conducted, demonstrating the potential of this approach.

These advancements open up new perspectives in integrated photonics, enhancing the nonlinear performance of hybrid guides and paving the way for new applications in optical communications, optical information processing, and other advanced photonics technologies.

As I am writing this manuscript, I am brought to reflect on the past four years and the adventure this PhD has been. Through all the hardship and challenges, failure, and success, I was able to count on the support of plenty of wonderful people, making the completion of this journey possible. I am profoundly grateful to my dedicated supervisors, Eric, and Zhipei. Your continuous guidance, insightful advice, and supervision were the bedrock of this thesis. Your unwavering support has been invaluable throughout this academic journey. Thank you for giving me the incredible opportunity to work in this joint project and for meeting the challenge of supervising even at a distance. Both of you have provided me food for thought, thus allowing me to develop further my knowledge and skills. You offered the necessary support and feedback all along the PhD, helping me to grow as the project went on, and for that I am very grateful. Special gratitude is extended to: Delphine and Laurent, your tireless efforts in the lab and readiness to offer guidance and advice greatly enriched the research process. You were always ready to have a discussion on various aspects of the thesis and generous at offering your help. This has been especially valuable at the start of my project where your support and advice helped get things started. Carlos, your help in multiple aspects of my work was instrumental in the success of this thesis and the teachings I am bringing out with me. You introduced me to the design of photonic chips and gave constructive feedback to my multiple drafts of designs, thus helping me in my learning. Thank you as well for the time you put in the fabrication of samples, allowing me to push my project forward. Xavier, whose untimely passing left a void in our team. His technical expertise and ever-present willingness to assist with various aspects of this research remain a cherished memory. His legacy endures. Christian and Jianhao, your support and guidance, especially during the early stages of my project was invaluable. Your assistance with algorithms and simulations as well as the training you provided on experimental benches was pivotal in building a strong foundation for my project. Yuchen: Your role in kick-starting the project in Finland cannot be overstated. You played a crucial role in setting my journey on the right path, and for that, I am profoundly grateful. Mikko: Beyond being a steadfast comrade, you have been a beacon of support and a source of invaluable laughter. Our shared moments during coffee breaks, hikes (hype), and even the occasional home crafted beer have created cherished memories that have brightened all the PhD experience. It has been a true highlight of this journey. Suvi, for not only contributing to a positive work atmosphere at the lab in Aalto but also for being a steadfast and supportive friend. Complaining to you about PhD life offered me a good outlet for my frustration and helped seeing this thing through. Thank you for listening and thank you for helping out whenever possible. Henri: Your camaraderie and support have added a special dimension to our time in the lab. Our shared coffee breaks, hikes, and moments of relaxation (Tappen ääyyy!) have been invaluable. Your good mood and bad jokes (sorry but I have to say the truth) are always welcome. Andreas, for sharing your extensive knowledge, experience, and for your valuable contributions to the success of this thesis. Additionally, your skill as a belayer and climbing partner was greatly appreciated. I also extend my thanks to Olli, thank you for being a good friend, always happy to see me and always ready to play. You are a good boy. Xueyin, your expertise in the transfer of 2D materials and invaluable assistance played a pivotal role in the success of this thesis. Thank you for your providing materials for my work as well as for the teachings you provided along the way. Henry, for your expertise in perfecting transfer methods, your teaching, and your invaluable advice. My work would not have been the same without your help and advice. I would also like to acknowledge the following colleagues, whose camaraderie, support, and contributions were instrumental in creating a productive and positive work environment: Lucas, Miguel, Dinh, Alicia, Zhengrui, Jonathan, Mohsen, Fedor, Gius, Xiaoqi, Juan, Hoon, and Faisal. Your collective efforts have left an indelible mark on this research, and I am truly fortunate to have had the privilege of working alongside such a dedicated and supportive team.

Conclusion and perspectives

Résumé en francais

Ce résumé de quelques pages décrit de manière synthétique le travail mené au cours de cette thèse effectuée dans le cadre d'une coopération entre deux équipes recherche de l'Université Paris-Saclay (C2N, UMR 9001) et de l'Université Aalto (Centre of Excellence in Quantum Technology), et qui a donné lieu à une cotutelle internationale de thèse. Ce travail s'inscrit dans le contexte de la recherche de solutions innovantes en termes d'intégration photonique pour la réalisation future de fonctions tout-optiques de traitement de l'information ou la métrologie exploitant l'intégration de matériaux bidimensionnels nonlinéaires avec/dans des circuits optiques réalisés par des technologies microélectroniques de couches minces, comme décrit ultérieurement.

Certains domaines des micro-nano-technologies appliqués aux communications ou aux calculateurs ont atteint des limites, un exemple emblématique de ces limitations étant celui des interconnexions électriques dans les circuits microélectroniques. La levée de ces verrous et la réalisation d'un large panel d'autres fonctions (pour la métrologie sur puce, l'optique quantique sur puce, etc) nécessitent des solutions innovantes pour lesquelles la photonique intégrée, basée sur des plateformes compatibles avec celles de la microélectronique, présente des opportunités intéressantes du fait de la maturité technologique des solutions développées précédemment et de la possibilité d'un réemploi de plateformes déjà matures en vue d'un redéploiement vers ces nouvelles applications. La photonique intégrée, exploitant les propriétés de la lumière et son panel d'effets physiques associés (interférences, propriété classiques ou quantiques de la lumière, effets optiques nonlinéaires, etc) associé à la maîtrise de la fabrication de circuits optiques permettant de guider, séparer, combiner, multiplexer, etc, le champ optique, offre ainsi de nombreuses réponses potentielles aux problématiques ou domaines précédents. En particulier, l'exploitation des effets optiques nonlinéaires s'avère extrêmement intéressante pour la réalisation de nombreuses fonctions, effets d'ordre 2 ou d'ordre 3, pour la génération de nouvelles fréquences, la génération de peignes de fréquences, etc. Dans le contexte de la filière d'intégration photonique dominante, celle de la photonique silicium -ceci incluant principalement les guides et circuits silicium sur isolant (SOI : Silicon On Insulator) ou à base de nitrure de silicium (SiN sur SiO 2 ), la déclinaison de ces effets prend une forme un peu particulière : les effets optiques nonlinéaires à l'ordre 2 y sont à peu près inexistants, et ceux à l'ordre 3 sont forts dans le silicium (guides SOI) mais relativement faibles dans la famille SiN/SiO 2 , l'utilisation de guides SOI soulevant par ailleurs d'autres problèmes dans le proche infra-rouge, comme détaillé ultérieurement (génération de porteurs libres, au détriment des effets optiques nonlinéaires). Ce travail de thèse s'inscrit dans ce contexte où la filière photonique dominante (photonique silicium), présente une limitation intrinsèque relativement à l'exploitation des effets optiques d'ordre 3 aux longueurs d'onde des télécoms. Pour le contournement de cette difficulté, l'approche explorée s'est inspirée de solutions émergeantes basées sur l'intégration de matériaux bidimensionnels aux propriétés non linéaires exceptionnelles dans les structures de la photonique (guides d'ondes, résonateurs diélectriques) et conservant par ailleurs de larges bandes interdites lorsque préparés sous forme monocouche (1,6 eV typiquement). La partie évanescente des modes optiques se propageant dans les guides d'onde optiques se retrouvant en interaction avec les matériaux bidimensionnels déposés en surface des structures guidantes, ces modes bénéficient des propriétés exceptionnelles de ces matériaux, ceci conduisant finalement à une nonlinéarité effective élevée allant -particulièrement dans le cas des guides SiN/SiO 2 , bien au-delà de la nonlinéarité de guides SiN/SiO 2 témoins (non recouverts par les matériaux 2D). Plus précisément, la gamme exploitable de matériaux 2D inclut en particulier le graphène, les métaux de transition dichalcogénures (TMD) tels que M oS 2 et W S 2 , ou le phosphore noir. L'un des principaux avantages des matériaux 2D est leur épaisseur extrêmement fine, ce qui les rend particulièrement adaptés à l'intégration dans des composants photoniques. Le rapport surface/volume élevé, qui les rend très sensibles à la lumière, se révèle extrêmement adapté à leur intégration, bien que soulevant simultanément une difficulté liée au volume d'interaction potentiellement très faible avec le champ optique des modes guidés. L'objectif des travaux menés au cours de cette thèse de doctorat, qui s'est focalisée sur l'intégration de monocouches ou de multicouches MoS2 et W S 2 , a été d'apporter une démonstration expérimentale du bénéfice de l'intégration de tels matériaux actifs nanométriques et d'étudier les mécanismes en oeuvre associés au travers de simulations et d'expériences menées dans le but d'optimiser les structures hybrides et d'améliorer ces interactions. Ainsi dans un premiers temps une étude numérique pour l'anticipation de l'impact des matériaux 2D sur les propriétés des guides d'onde optiques a été conduite conjointement à des premières démonstrations expérimentales. Dans un second temps, plusieurs approches pour l'optimisation des structures photoniques hybrides ont été explorées. Afin d'étudier précisément l'impact des matériaux 2D déposés, nous avons tout d'abord développé des outils numériques adaptés nous permettant de simuler des guides d'onde hybrides. Les outils numériques traditionnels tel que la FDE (finite difference eigenmode) peuvent être utilisés pour la conception et l'optimisation de guides d'onde classiques. Cependant, lorsqu'il s'agit d'intégrer des matériaux bidimensionnels, il est essentiel de faire preuve d'une attention particulière lors de leur prise en compte dans la description des structures en raison de leur impact potentiellement significatif sur les propriétés des guides hybrides et des artefacts potentiels ainsi rencontrés si leur la description n'est pas effectuée proprement (difficile par essence compte tenu de leurs épaisseurs sub-nanométriques). Par conséquent, il s'est avéré crucial d'inclure ces matériaux dans les simulations en calculant leur conductivité de surface, qui joue le rôle équivalent à la susceptibilité dans les simulations tridimensionnelles utilisées par les solveurs de modes optiques. Cette conductivité peut être obtenu de manière analytique pour des matériaux tel que le graphene où la structure de bandes en cône de Dirac permet une formulation analytique des calculs. Pour les autres matériaux 2D -M oS 2 et W S 2 notamment -, la conductivité de surface des matériaux bidimensionnels est généralement obtenue par mesure expérimentale. Il nous a ainsi été nécessaire d'implémenter une approximation analytique (fit des données) des propriétés de ces matériaux afin de les inclure aux simulations numériques modales faites, en pratique, avec la suite logicielle LUMERICAL. L'étude des résultats obtenus grâce à ces simulations a révélé l'impact des matériaux 2D sur les guides d'onde. Un impact limité des cristaux monocouche a été constaté sur les propriétés du guide telles que le profil de dispersion et la distribution transverse des modes optiques. En revanche, même pour des épaisseurs de matériaux ne dépassant pas quelques monocouches (exemple : ∼ 0, 65nm pour une monocouche de M oS 2 ), l'impact de leur ajout est en général significatif concernant la nonlinéarité effective des guides, et peut par ailleurs s'avérer rapidement perceptible dès lors que l'empilement dépasse quelques monocouches. Naturellement, nous avons apporté un soin particulier à la prise en compte du recouvrement entre le mode optique guidé et les matériaux bidimensionnels couvrant les guides lors du calcul de la distribution transverse du mode (contraintes de maillage des structures transverses). Ce recouvrement, estimé proprement à partir d'une comparaison à des résultats antérieurs (« benchmarking »), nous a notamment permis d'effectuer le calcul du coefficient non linéaire des guides étudiés en tenant compte des matériaux ajoutés, ce qui s'est avéré essentiel pour la suite des études menées.

Dans un second temps, nous avons consacré un certain nombre d'efforts à expérimenter le transfert de matériaux 2D sur des guides d'onde SiN/SiO 2 en vue de répondre à l'enjeu de la thèse d'intégrer ces matériaux sur des zones ciblées des circuits optiques. La fabrication des matériaux bidimensionnels a représenté l'un des enjeux à surmonter pour la démonstration de l'efficacité des structures optiques hybrides. Dans une première approche directe de fabrication de guides hybrides recouverts de matériaux bidimensionnels, nous avons exploité la méthode classique d'exfoliation mécanique consistant à retirer successivement des couches ultra-minces d'un matériau tridimensionnel par le biais de ruban adhésif afin de le réduire en un cristal bidimensionnel. La couche mince de cristal obtenue a ensuite pu être transférée en utilisant un tampon de polymère. Ce processus a ainsi permis la fabrication et le transfert de matériaux bidimensionnels sur des guides d'ondes SiN/SiO 2 , mais s'est néanmoins avérée difficile à mettre en oeuvre du fait d'un faible taux de succès. En effet, les polymères utilisés pour les tampons de transfert sont sensibles à de nombreux paramètre tel que les contraintes mécaniques ou la température. De ce fait, le contrôle de l'adhésion entre les différentes couches de polymère impliquées dans le transfert et le matériaux 2D peuvent être difficiles à contrôler. Cela introduit des incertitudes dans le transfert qui réduit drastiquement le taux de succès. L'utilisation de cette méthode a permis la fabrication de guides hybrides en vue de démontrer le renforcement du phénomène d'auto-modulation de phase dans des guides SiN/SiO2 grâce à l'ajout d'un feuillet de cristal bidimensionnel. Un feuillet de WS2 de longueur 14, 8µm a été déposé sur un guide SiN droit. Le matériau a été caractérisé à l'aide des méthodes de spectroscopie Raman, de photoluminescence et d'un microscope à force atomique utilisé pour mesurer les dimensions du feuillet (longueur d'interaction, épaisseur, etc.). Des régions différentes ont été identifiées, tel qu'une zone de ∼ 3.3nm d'épaisseur avec une longueur d'interaction avec le guide de ∼ 6.5µm, une zone de ∼ 7.9nm d'épaisseur avec une longueur d'interaction avec le guide de ∼ 6.3µm, et une zone de ∼ 5.6nm d'épaisseur avec une longueur d'interaction avec le guide de ∼ 2µm . À partir de ces données, une moyenne de ∼ 5.57nm d'épaisseur a été calculée. Grace a cela, le feuillet complexe a été approximé en un feuillet uniforme de ∼ 14.8µm de longueur d'interaction pour des besoins de simplification pour le modèle numérique.

• Un laser en régime impulsionnel, centrée à 800 nm, délivrant un train d'impulsion avec une duration de 100 fs pour une fréquence de 84. • Dans un premier temps, en se basant sur l'approche numérique établie précédemment pour la simulation des guides optiques hybrides, nous avons cherché à optimiser la structure transverse des guides optiques en augmentant l'interaction entre le matériau bidimensionnel et le mode optique. Par rapport à la configuration initiale des guides d'ondes hybrides, nous avons d'une part proposée et considéré l'intérêt du dépôt final d'un matériau présentant un indice de réfraction élevé par rapport aux autres matériaux (par exemple Al 2 O 3 , Hf O 2 , ...) de la structure transverse (Fig. 2 a). Cette approche s'est avérée capable de déplacer le centre de gravité du mode optique plus près du matériau bidimensionnel, augmentant ainsi l'interaction entre le mode propagé et les matériaux fortement linéaires (Fig. 2 d). La deuxième proposition a consisté à augmenter le nombre de couches de matériau bidimensionnel intégrées à la structure en vue également d'en augmenter l'interaction avec le mode transverse (Fig. 2 c). Afin de conserver un matériau actif à large bande interdite (afin de limiter le phénomène d'absorption à deux photons dans la bande télécoms), nous avons considéré l'intégration d'une série de monocouches de MoS2 séparées par des espaceurs diélectriques (Al 2 O 3 ). Sur ces deux bases, les simulations numériques conduites ont permis d'estimer une augmentation du facteur nonlinéaire effectif des guides d'onde hybrides peuvent atteindre plus de 120W -1 m -1 (voir Fig. 2c)), c'est-à-dire conduisant à des valeurs comparables à celles obtenues pour des guides SOI mais ne présentant pas d'absorption à deux photons (vers 1, 55µm et autour). Nous avons par ailleurs évalué les limitations de cette approche multi-monocouches, d'ailleurs observables en Figs. 2c) et 2d), par une saturation de l'augmentation du coefficient nonlinéaire des guides hybrides au-dessus d'un certain seuil (dans l'exemple Fig. 2c cela intervient au-dessus de 40 couches) du fait de l'interaction limitées du mode optique avec le matériau actif pour des épaisseurs trop élevées, ceci ne remettant pas en cause l'intérêt de la nonlinéarité effective très significative déjà atteinte par cette approche (> 100W -1 m -1 ).

• Dans une deuxième direction plus expérimentale, nous avons conduit un ensemble de tests menés en vue de l'améliorations des méthodes de transfert des matériaux bidimensionnels sur les guides et résonateurs optiques, les techniques traditionnelles de transfert par exfoliation mécanique présentant certains problèmes déjà mentionnés précédemment. Il s'est donc avéré intéressant d'explorer la possibilité de réaliser des transferts localisés de matériaux fabriqués par dépôt chimique en phase vapeur (CVD). Ces matériaux offrent en effet des surfaces aux dimensions bien supérieures à celles des matériaux 2D obtenus par exfoliation et une plus grande homogénéité en termes d'épaisseur. Ainsi, un processus a été développé en combinant les méthodes de transfert en solution habituellement utilisées dans les méthodes de transfert quasi-aléatoires de matériaux 2D obtenus par CVD avec les approches utilisées pour les matériaux exfoliés. Cette combinaison a permis de transférer de larges surfaces de matériaux relativement homogènes avec une surface recouverte à entre 80% et 90% d'un feuillet monocouche avec des taux de succès du transfert élevés et de garantir une grande précision dans les transferts (à quelques microns prêt). Ces avancées sont prometteuses pour améliorer la fiabilité et la reproductibilité des transferts de matériaux bidimensionnels, ouvrant ainsi de nouvelles possibilités pour la conception et la fabrication de dispositifs photoniques intégrés. Dans une dernière approche, nous avons exploré l'utilisation de cavités en anneau pour le renforcement de l'interaction champ électromagnétique/matériau 2D par 'recyclage des photons'. Ces structures photoniques offrent en effet une amélioration notable de l'interaction entre les matériaux bidimensionnels et l'onde se propageant en exploitant les effets de résonance. Le potentiel de ces structures a été étudié dans un premier temps par simulation, révélant des perspectives prometteuses pour de nombreuses applications.

Une étude expérimentale d'un résonateur en anneau hybride a par ailleurs été réalisée. Un feuillet de MoS2 (pour une longueur d'intéraction de 42, 5µm) est transféré sur un résonateur en anneau (guide de SiN au dimensions h = 600 nm, w = 1700 nm pour un résonateur en anneau de référence avec un facteur de qualité de 192 000). Le phénomène de mélange à quatre ondes a ensuite été caractérisé en injectant une pompe (200 mW en continue centrée à 1559 nm) et un signal (centrée à 1561 nm). La fréquence générée en sortie du résonateur en anneau a été mesurée et son intensité a été comparée à celles des autres fréquences présentes (Fig. 3). Cette analyse a permis de déterminer le coefficient non linéaire effectif du guide d'onde, à la fois pour le guide de référence et pour le guide hybride. Un coefficient nonlinéaire de 0, 74W -1 m -1 a été mesuré pour le résonateur de référence. Pour le résonateur hybride, le coefficient non linéaire de la section hybride a été estimé à 9, 19W -1 m -1 . Une amélioration significative a ainsi pu être observée, confirmant le potentiel de telles méthodes d'hybridation.

Pour conclure, ce résumé de quelques pages synthétise un travail mené pendant 4 ans concernant l'intégration de matériaux 2D sur des guides SiN/SiO 2 en vue de la réalisation d'une plateforme photonique à forte nonlinéarité effective à l'ordre 3 mais préservée de la génération de porteurs libres aux longueurs d'onde des télécoms. La démarche suivie a associé des aspects de modélisation numérique et différentes approches et étapes expérimentales (fabrication et report des matériaux 2D, caractérisations optiques linéaires et nonlinéaires, caractérisation des matériaux 2D). Les avancées produites par ce travail ouvrent de nouvelles perspectives dans le domaine de la photonique intégrée, offrant des possibilités d'amélioration des performances non linéaires des guides hybrides et ouvrant la voie à de nouvelles applications dans les domaines des communications optiques, du traitement optique de l'information, ou de la métrologie sur puce.

Acronyms

Introduction

The past decades have seen tremendous changes in the field of micro and nanotechnologies. From the development of powerful computers to the rise of the internet and all subsequent evolutions, technological achievements have progressed rapidly and have transformed multiple industries (e.g., Data science, AI, medicine, telecommunication, . . . ). However, physical boundaries still exist and are increasingly becoming a limiting factor for further technological progress. The drive for more efficient and compact devices has also been increasing, pushing the requirements beyond the limits set by current platforms. This is particularly evident in the field of microelectronics and telecommunications. Many limiting properties of electronic circuits, such as the management of the temperature of the bandpass and electrical consumption in optical interconnects, are starting to have a detrimental effect on performance. As these limits set by quantum mechanics are reached, Moore's empirical Law [START_REF] Gordon E Moore | Cramming more components onto integrated circuits[END_REF], which states that the number of transistors in an integrated circuit is bound to double every year has strongly started to be challenged. Solutions could be found in the field of telecommunication which is facing similar issue. The shift from traditional copper telecommunication cables towards optical fibres provided useful guidelines and perspectives on how these issues could be solved.

For that purpose, photonics, the study of the generation, control, and detection of light has risen as a promising field. Unlike electronics, which is based on the control of a flow of electrons, photonics studies the flow of photons. Thus, it offers many advantages over traditional electronics, such as lower power consumption, no energy dissipation through the Joule effect, faster data transfer speeds, and the ability to process multiple signals simultaneously. Given these advantages, developing further photonics as a solution might be key to trigger a new paradigm and answer growing concerns such as energy consumption and environmental impact while addressing the physical boundaries of the current electronic platforms. In that regard, integrated photonics specifically seems to be very promising. The integration of photonic components, such as lasers, detectors, modulators, and more optical functions, onto a single chip can be the corner stone for the next generation of on-chip devices. One platform particularly wellsuited for the integration of photonic components is the complementary metal-oxidesemiconductor (CMOS) one. CMOS is a widely-used technology in the production of electronic devices, including microprocessors, memory chips, and image sensors. Its popularity is due to its low cost, high reliability, and scalability, making it the go-to platform to produce integrated circuits. Although these advantages make the platform appealing, many obstacles exist to the development of fully integrated photonic chips. Because of these limitations and the properties of the main compatible materials (e.g., Si, SiN , SiO 2 ) several integrated optical functions, particularly in the field of nonlinear optics, remain a challenge.

The design of nonlinear optical functions in view of all-optical signal processing is indeed a main issue. Due to the presence of two-photon absorption (TPA) in silicon in the near infra-red around 1.5 µm wavelength despite a huge third-order Kerr nonlinearity, the performance of the silicon device is negatively impacted in most nonlinear optical processes. Additionally, weak nonlinear responses for most other common materials compatible with its clean room processes, such as SiO 2 and SiN are observed. Therefore, leveraging these limitations to unlock the possible use of ultrafast nonlinear effects in the Si platform needs to overcome all the limitations set by the platform. To that end, a promising solution to the development of new integrated optical functions has gained significant attention in recent years due to its potential applications in electronics and optoelectronics and photonics. It is the use of 2D materials (e.g., graphene, transition metal dichalcogenides (TMD) such as M oS 2 and W Se 2 , or black phosphorus). One of the main advantages of 2D materials is their extremely thin thickness, making them highly suitable for integration with photonic components. The high surface-to-volume ratio, making them highly sensitive to light, has also shown to be extremely suitable for their use in photodetectors and other optoelectronic device schemes. 2D materials also have unique electronic and optical properties, which can be often tuned by applying an external electric field. This allows for the design of highly efficient and customizable photonic devices. For all these reasons, they are suitable candidates to enhance the CMOS platform for the realization of low-power nonlinear integrated optical functions.

In this context, this thesis contributes to exploring the hybrid integration of highly nonlinear 2D materials in the Si and SiN photonic platforms. Introducing a reliable methodology for the integration and characterization of these nanomaterials on integrated photonic devices is the major focus of the work presented here. The first chapter focuses on the fundamentals of integrated optics and nonlinear optical waveguides and their state of the arts. Accordingly, this chapter will provide an introduction to CMOS-compatible photonics and 2D materials. It will serve as the base line upon which the following chapters will expand. In the second chapter, an account a direct approach to the hybrid integration will be presented with the introduction of Chapter 1

Introduction and motivation

1.1 The silicon photonics platform

A brief introduction to Silicon photonics

Silicon is one of the main building blocks of the semiconductor industry. It was first established in 1963 by Frank Wanlass with the introduction of the first CMOS circuit [START_REF] Michael J Riezenman | Wanlass's cmos circuit[END_REF][START_REF] Singh | Evolution of cmos technology[END_REF]. The overall low power consumption compared to other technologies (e.g., the off state of the transistor draws very little current compared to other technologies such as bipolar junction transistors), low leakage current and the wide array of functions it allows on a single chip, rapidly made it a corner stone of micro-technologies. It ultimately sparked a revolution with the rise of computers and other portable electronic devices. Nowadays, CMOS is everywhere, from the digital sensor in cameras to the micro-processor in hand-held cellular devices. Nothing is avoiding the hegemony of this platform. For 50 years, it has been improved upon research. Therefore, it is a reliable, time-tested technology. A natural transposition of the Si technology to photonics was thus envisaged as early as about for fourty years [START_REF] Soref | All-silicon active and passive guided-wave components for λ= 1.3 and 1.6 µm[END_REF], with yet a dramatic acceleration of the Si photonics progress in the last 15 years. Taking advantages of the accumulated knowledge on the CMOS platform, and the already established foundries, a rapid and efficient development has been observed. High levels of integration and low fabrication costs at high volumes as well as compatibility with the microelectronics (Fig. 1.1 b.) for optical-electrical-optical devices have been capitalized as critical assets. As such, integration of photonics components in Si is increasingly taking a central role in the development of the next generation of optical telecommunication receivers and transceivers (as illustrated with Fig 1 .1 e. the comparison of data rates between the different platforms) In addition with telecommunications, plenty of emerging applications (internet of things, sensors everywhere, take benefit of Silicon technologies, etc). Their development needs low-cost, high-performance, and low-power circuits for which Si photonics is also involved. [START_REF] Thomson | Roadmap on silicon photonics[END_REF].

Figure 1.1: a. Schematic of a fully integrated silicon photonic chip [START_REF] Thomson | Roadmap on silicon photonics[END_REF] b. Interface between the photonic platform and an electronic circuit (source Intel)c. Large-scale silicon quantum chip for two-qubit processing [START_REF] Qiang | Large-scale silicon quantum photonics implementing arbitrary twoqubit processing[END_REF] d. SEM image of a typical silicon photonic chip including a grating coupler for light injection and collection in and out of the chip [START_REF] Gunn | Silicon photonics: Poised to invade local area networks-by integrating optical functions into a microprocessor fabrication process, engineers can achieve the performance of optical links within the[END_REF] e. Electrical data rate compared to Silicon photonics and optical data rate [START_REF] Zuffada | The industrialization of the silicon photonics: Technology road map and applications[END_REF].

Many optical functions and integrated devices have already been developed for the Silicon platform (Fig. 1.1 a.). The potential of the platform is huge, specifically with recent development and the rise of interest in quantum computing applications. The Silicon platform has seen an impressive development in that field with, for example, the development of a large-scale arbitrary two-qubit processor (Fig. 1.1 c. [START_REF] Qiang | Large-scale silicon quantum photonics implementing arbitrary twoqubit processing[END_REF]). Furthermore, despite shortcomings (discussed in Section 1.1.3), convincing nonlinear optical functions have already been successfully implemented, such as supercontinuum sources [START_REF] Adrea R Johnson | Octave-spanning coherent supercontinuum generation in a silicon nitride waveguide[END_REF][START_REF] Halir | Ultrabroadband supercontinuum generation in a cmos-compatible platform[END_REF][START_REF] Marco | Two-octave spanning supercontinuum generation in stoichiometric silicon nitride waveguides pumped at telecom wavelengths[END_REF], frequency combs [START_REF] Martin Hp Pfeiffer | Octave-spanning dissipative kerr soliton frequency combs in si 3 n 4 microresonators[END_REF][START_REF] Tobias J Kippenberg | Dissipative kerr solitons in optical microresonators[END_REF][START_REF] Li | Stably accessing octave-spanning microresonator frequency combs in the soliton regime[END_REF][START_REF] Brasch | Photonic chip-based optical frequency comb using soliton cherenkov radiation[END_REF], and photon pair sources through spontaneous four-wave mixing (FWM). This has motivated the development of fully integrated, nonlinear optical devices, aiming at various applications (e.g., all on-chip spectroscopy [START_REF] Picqué | Frequency comb spectroscopy[END_REF], on-chip quantum computations and communications [START_REF] Oser | High-quality photonic entanglement out of a stand-alone silicon chip[END_REF][START_REF] Qiang | Large-scale silicon quantum photonics implementing arbitrary twoqubit processing[END_REF] (Fig. 1.1 c.), efficient multiplexing for data communications [START_REF] Marin-Palomo | Microresonator-based solitons for massively parallel coherent optical communications[END_REF], on-chip metrology [START_REF] Tobias J Kippenberg | Microresonatorbased optical frequency combs[END_REF], bio-sensing [START_REF] Karpov | Photonic chip-based soliton frequency combs covering the biological imaging window[END_REF][START_REF] Hc | Graphene oxide integrated silicon photonics for detection of vapour phase volatile organic compounds[END_REF], or LIDARs [START_REF] Trocha | Ultrafast optical ranging using microresonator soliton frequency combs[END_REF]). The rapid progress, promising results and several advantages of the Si platform have brought attention to the silicon platform with high investment levels and made it a really active research field.

Advantages of the Silicon platform

There are many advantages in choosing Silicon as the main component of integrated photonics devices. The intrinsic properties of the material offer specific conditions that can be exploited in a favourable way to expand the range of possibilities. The first of these advantages can be linked to its high refractive index in the C-band, between 1530 nm and 1560 nm. Indeed, in this range, silicon has a refractive index of 3.48 [START_REF] Daniel | Electronic structure of amorphous si from photoemission and optical studies[END_REF][START_REF] Edward | Handbook of optical constants of solids[END_REF], which offers high contrast with respect to the oxide layer (SiO 2 substrate with a 1.44 refractive index [START_REF] Ian H Malitson | Interspecimen comparison of the refractive index of fused silica[END_REF][START_REF] Cz Tan | Determination of refractive index of silica glass for infrared wavelengths by ir spectroscopy[END_REF]). Hence, building photonic circuits in Silicon on insulator (SOI) enables high confinement of the light, therefore reducing the footprint of the circuits on a chip. The possible use of reduced bending radii without excessive additional losses [START_REF] Aalto | Open-access 3-µm soi waveguide platform for dense photonic integrated circuits[END_REF][START_REF] Van Thourhout | Recent progress in soi nanophotonic waveguides[END_REF] is a major upside originating from this high contrast between the Si core and the air/SiO 2 cladding. On top of that, other optical properties of Silicon can provide useful outcomes such as an ultra-wide transparency window above 1100 nm in optical wavelength, up to ∼ 3.5µm (limit set by the transparency of SiO 2 ). This feature is particularly interesting when considering broadband applications such as multiplexing for telecommunication or spectroscopy. This is even more attractive as the transparency window spans over the C band and higher wavelength making it a versatile building block of an integrated photonic circuit.

Other CMOS-compatible materials are available to develop various integrated optical functions, among which SiN is the immediate follower of Si as the core material or SiO 2 cladded waveguides. The success of SiN can be partly attributed to its low propagation losses (lower index contrast than Si/SiO 2 and low propagation loss ( 3 dB/m [START_REF] Bauters | Ultra-low-loss high-aspect-ratio si 3 n 4 waveguides[END_REF][START_REF] Hubert | Photonic damascene process for low-loss, high-confinement silicon nitride waveguides[END_REF])). Additionally, the transparency window expanding from 500 nm to 2400 nm offers a good alternative to SOI waveguides. All these features are exploited for developing most of the building blocks necessary for the development of on-chip devices. Operations such as modulation [START_REF] Cutolo | Silicon electrooptic modulator based on a three terminal device integrated in a low-loss singlemode soi waveguide[END_REF][START_REF] Angulo Barrios | Electrooptic modulation of silicon-on-insulator submicrometer-size waveguide devices[END_REF], switching [START_REF] Fischer | Integrated optical waveguide switches in soi[END_REF][START_REF] Harjanne | Sub-µs switching time in silicon-on-insulator mach-zehnder thermooptic switch[END_REF], filtering [START_REF] Oser | Coherency-broken bragg filters: overcoming on-chip rejection limitations[END_REF][START_REF] Morand | Ultra-compact microdisk resonator filters on soi substrate[END_REF] or amplification [START_REF] Mark | Broad-band optical parametric gain on a silicon photonic chip[END_REF][START_REF] Gajda | Highly efficient cw parametric conversion at 1550 nm in soi waveguides by reverse biased pin junction[END_REF] are being developed and optimized following different strategies. Convincing demonstration and development have been achieved in the building of several essential integrated functions such as optical interconnects [START_REF] Jonathan | The evolution of silicon photonics as an enabling technology for optical interconnection[END_REF], optical amplifiers [START_REF] Van Gasse | 27 db gain iii-v-onsilicon semiconductor optical amplifier with> 17 dbm output power[END_REF] or filters [START_REF] Pérez-Galacho | Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures[END_REF]. This shows the tremendous progress that has been achieved so far.

Limitations of the Silicon platform

Despite the many advantages of Si photonics, several impediments are still holding the platform from implementing even more disruptive approaches, relying especially on all-optical signal processing. The intrinsic properties of the material, mostly related to the nature of its band structure (indirect bandgap at ∼ 1.12 eV), indeed bring several limitations. The case of light sources is thus emblematic in that regard as most solutions still reside in the integration of III-V components, thus generating a whole other degree of problems and complexity [START_REF] Keyvaninia | Demonstration of a heterogeneously integrated iiiv/soi single wavelength tunable laser[END_REF][START_REF] Roelkens | Iii-v/silicon photonics for on-chip and intra-chip optical interconnects[END_REF]. III-V compound crystal structures, lattice constants and thermal expansion coefficients are indeed not directly compatible with Si. The epitaxial growth of these materials is difficult on Si and results in the limited quality of the grown materials. On the other hand, good results relying on wafer-bonding technologies have reported good results for the integration of III-V on SOI. However, the complexity involved with that process makes it challenging to scale to industrial requirements. Hence, the quest for an integrated CMOS-compatible efficient light source is still crucial to the development of Si photonics. Another optical function that could benefit from new developments is optical modulation. Although optical modulators have now been demonstrated at >50 GHz with a large extinction ratio, improvements beyond current performances would require all-optical ultra-fast processes. As these processes (e.g., Kerr-like/third-order nonlinearity-based mechanisms) are hampered by the presence of free carriers in Si, there is now a challenge to alleviate these problems. This highlights a fundamental problem with nonlinear optics in SOI and adjacent platforms (e.g., SiN/SiO 2 ) that expands further than just the limitations it sets to the design of all-optical modulators. It is a recurring issue when operating in the C-band (essential to telecommunication or quantum applications). The presence of two-photon absorption in this range is damaging to performance in most cases. Alternatives such as SiN or doped SiO 2 do not provide satisfying results as they do not present second-order nonlinearity due to their centro-symmetry and simultaneously present weak third-order nonlinear response. Therefore, the ultrafast Pockels effect (originating from secondorder nonlinearity) would be preferable, but is not available in the natural state of the platform. The introduction of such an effect relying on the hybrid integration of III/V semiconductors on Si was proven efficient for that purpose [START_REF] Alloatti | 100 ghz silicon-organic hybrid modulator[END_REF], but once again, can present difficulties the same difficulties as in the implementation of light sources on SOI. It appears that developing these nonlinear optical functions could be key to solving many problems locking access to fully integrated Si-photonics chips. Hence, the field is in need of new developments to enable these interesting perspectives.

Integrated nonlinear optical functions

A key component of the integrated photonics platform, is the optical waveguide. To grasp fully the principle behind the development of integrated optical functions, understanding the relationship between the physical properties of materials (e.g., free carrier (FC), nonlinear effects, photolumisnescence (PL)), the geometry in which these materials are set and the effect they have on the propagating electromagnetic waves is necessary. The following is an introduction to the fundamentals of waveguides and integrated optical functions bringing a focus on the light matter interaction controlled by mode confinement and the overlap between nonlinear materials and propagating modes.

Propagation in basic optical waveguides.

The main building block of a photonic circuit is the waveguide. A waveguide is an optogeometrical structure in which an electromagnetic wave is confined and can propagate. The higher refractive index medium guides the light. Waveguides are more widely known in the form of optical fibers. Yet, smaller-scale waveguides with different properties offer the best opportunities in the design of the next generation of photonic devices. In a dielectric medium (which is the relevant setting in the description of integrated waveguides in the Silicon platform), Maxwell's equations can be written as follows:

∇.E = 0 (1.1)
∇.H = 0 (1.2)

∇ × E = -µ 0 ∂ ∂t H (1.3) ∇ × H = ∂ ∂t D (1.4)
, where D is the flux density such as:

D = ϵ 0 E + P (1.5)
with P the induced electric polarization of the medium, E is the electric field, H is the magnetic field, µ 0 is the vacuum permeability, ∇ is the vectorial differential operator (∇.U for divergence of the U field and ∇ × U is the curl of the U vector field). Combining the curl equations Eq. 1.3 and Eq. 1.4 one can write the propagation equation of an electromagnetic wave in a dielectric medium:

∇ 2 E -ϵ 0 µ 0 ∂ 2 E ∂t 2 = µ o ∂ 2 P ∂t 2 (1.6)
While in the case of a linear medium, P can be expressed as P L = ε 0 χ (1) E(t), the general case of a nonlinear medium necessitate P to be expressed as a power series over the field strength E(t) as follows:

P = ε 0 χ (1) E(t) + χ (2) E 2 (t) + χ (3) E 3 (t) + ... (1.7)
, with χ (n) the n th order susceptibility. We can break down the polarization into 2 main components such as P = P L + P NL . The susceptibility is related to the refracted index of the propagation medium such as:

n = 1 + χ (1) (1.8)
, hence, Eq. 1.6 can be written as:

∇ 2 E - n 2 c 2 ∂ 2 E ∂t 2 = 1 ϵ 0 c ∂ 2 P NL ∂t 2 (1.9)
Integration of the propagation equation can be performed with help of numerical mode solvers, calculating the distribution of the optical transverse eigenmodes for any kind of cross section. Numerical methods such as finite difference eigenmodes (FDE) or finite element method (FEM) can resolve high degrees of complexity in the optical structure, thus enabling the calculation for small variations in the waveguide's cross section [START_REF] Philipp | Computational Physics[END_REF]. The optical and geometrical properties of a waveguiding structure are essential in shaping the transverse optical mode propagating through the waveguide. Relating this with our quantum analogy, it is identical to finding the wavefunctions for a particle stuck in a potential well. These numerical methods are also able to derive the dispersion relations for the eigenmodes of a waveguide. Parameters such as the effective index n ef f , the group index n g or other derivatives can be obtained from the numerical simulations. Accordingly, a full information on the waveguides's dispersion is directly obtained, which allows to extract a full expansion of any mode propagation constants around a given pump frequency ω P :

β(ω) = n(ω) ω c = ∞ k=0 1 k! β k (ω -ω P ) k (1.10) β k = dβ dω k ω=ω P (k = 1, 2, 3, ...) (1.11)
It is important to note that different orders of this Taylor expansion correspond to important quantities for the guided modes. The first order parameter is related to group velocity v g :

β 1 = dβ dω = 1 v g (1.
12) v g is the velocity at which the envelope of an optical pulse is propagating. The second order is the derivative of the first :

β 2 = 1 c 2 dn dω + ω d 2 n dω 2 (1.13)
β 2 expresses the group velocity dispersion (GVD) and is responsible for pulse broadening along propagation. When trying to assess the chromatic dispersion, this is often the parameter being first considered. It is often characterized by its value of around 0. The regime is called anomalous for β 2 < 0 and normal for β 2 > 0. The value of the GVD, the regime it finds itself in and the shape of its profile versus the wavelength are critical to describe the effect of the chromatic dispersion on nonlinear mechanisms, specifically in the case of broad-band applications such as spectral broadening, supercontinuum and frequency combs. A good approximation of these parameters is essential to produce the expected outcome and thus is required in the design of nonlinear optical functions [START_REF] John M Dudley | Supercontinuum generation in photonic crystal fiber[END_REF][START_REF] Zhang | Stretching the spectra of kerr frequency combs with self-adaptive boundary silicon waveguides[END_REF]. Coming back to the analogy with quantum physics, the dispersion does, in fact, correspond to the different bands of the band structure.

Description and simulation of nonlinear optical functions

Historically introduced in the experiment by Franken et al. in 1961 with the demonstration of a second harmonic generation (SHG) phenomenon [START_REF] Eg Pa Franken | Generation of optical harmonics[END_REF], nonlinear optics is the study of the intensity dependant response of light while propagating through a specific medium. When the light is powerful enough (typically only enabled with Laser sources), the medium induces a change in the electromagnetic field. An electrical field composed of multiple frequencies mixed together can be written in the complex form:

E(t) = n E n e -iωnt + c.c. (1.14)
, with c.c. the complex conjugate. Typical nonlinear responses are FWM, self-phase modulation (SPM), SHG , and third-harmonic generation (THG) (Fig. 1.3 a.). These processes have been described thanks to the consequences of the binomial theorem when treating nonlinear systems confronted with these multiple frequencies mixed together. Injecting Eq. 1.14 in Eq. 1.5 allows to describe the various processes occurring in the context of a nonlinear medium [START_REF] Boyd | Nonlinear optics[END_REF]. Consequently, k th order of the polarization can be written as:

P (k) (t) = n P (ω n )e -iωnt (1.15)
This summation covers positive and negative frequencies ω n . The order k of the polarization sets the number of frequencies mixed in a single process. For the 2 nd order, two frequencies, ω 1 and ω 2 are involved. Thus, the complex amplitude of the nonlinear polarizations for the different possible nonlinear processes are:

P (2ω 1 ) = ϵ 0 χ (2) E 2 1 (SHG) (1.16) P (2ω 2 ) = ϵ 0 χ (2) E 2 2 (SHG) (1.17) P (ω 1 + ω 2 ) = 2ϵ 0 χ (2) E 1 E 2 (SF G) (1.18) P (ω 1 -ω 2 ) = 2ϵ 0 χ (2) E 1 E * 2 (DF G) (1.19) P (0) = 2ϵ 0 χ (2) (E 1 E * 1 + E 2 E * 2 ) (OR) (1.20)
, with SHG, the second harmonic generation, the sum frequency generation (SFG ), difference fequency generation (DFG) and the optical rectification (OR). Third-order nonlinear processes involve 3 frequencies ω 1 , ω 2 and ω 3 and can also be described following the same approach:

P (ω 1 ) = ϵ 0 χ (3) (3E 1 E * 1 + 6E 2 E * 2 + 6E 3 E * 3 )E 1 P (ω 2 ) = ϵ 0 χ (3) (6E 1 E * 1 + 3E 2 E * 2 + 6E 3 E * 3 )E 2 P (ω 3 ) = ϵ 0 χ (3) (6E 1 E * 1 + 6E 2 E * 2 + 3E 3 E * 3 )E 3
(1.21)

P (3ω 1 ) = ϵ 0 χ (3) (3)E 3 1 , P (3ω 2 ) = ϵ 0 χ (3) (3)E 3 2 , P (3ω 3 ) = ϵ 0 χ (3) (3)E 3 3 , (1.22)
P (ω 1 + ω 2 + ω 3 ) = 6ϵ 0 χ (3) (3)E 1 E 2 E 3 , P (ω 1 + ω 2 -ω 3 ) = 6ϵ 0 χ (3) (3)E 1 E 2 E * 3 , P (ω 1 + ω 3 -ω 2 ) = 6ϵ 0 χ (3) (3)E 1 E 3 E * 2 , P (ω 2 + ω 3 -ω 1 ) = 6ϵ 0 χ (3) (3)E 2 E 3 E * 1 (1.23) P (2ω 1 + ω 2 ) = 3ϵ 0 χ (3) (3)E 2 1 E 2 , P (2ω 1 + ω 3 ) = 3ϵ 0 χ (3) (3)E 2 1 E 3 P (2ω 2 + ω 1 ) = 3ϵ 0 χ (3) (3)E 2 2 E 1 , P (2ω 2 + ω 3 ) = 3ϵ 0 χ (3) (3)E 2 2 E 3 P (2ω 3 + ω 1 ) = 3ϵ 0 χ (3) (3)E 2 3 E 1 , P (2ω 3 + ω 2 ) = 3ϵ 0 χ (3) (3)E 2 3 E 2 P (2ω 1 -ω 2 ) = 3ϵ 0 χ (3) (3)E 2 1 E * 2 , P (2ω 1 -ω 3 ) = 3ϵ 0 χ (3) (3)E 2 1 E * 3 P (2ω 2 -ω 1 ) = 3ϵ 0 χ (3) (3)E 2 2 E * 1 , P (2ω 2 -ω 3 ) = 3ϵ 0 χ (3) (3)E 2 2 E * 3 P (2ω 3 -ω 1 ) = 3ϵ 0 χ (3) (3)E 2 3 E * 1 , P (2ω 3 -ω 2 ) = 3ϵ 0 χ (3) (3)E 2 3 E * 2 (1.24)
Expressing the different configurations for the polarization in critical in establishing the occurring nonlinear processes depending on the incident frequencies and the nature of the medium. Although all these effects have first been established and identified in bulk materials, they can naturally occur during propagation in a waveguide, provided the pump is powerful, and the composite materials interacting with the propagating wave induces nonlinear responses. Indeed, different materials are required to build an optical waveguide, and some are bound to have a nonlinear response to the propagating mode. This is highlighted by the example of Fig. 1.3 with a pulse broadening occurring due to SPM through propagation in an integrated waveguide. In the following, the focus is put on nonlinear interaction of light with a waveguide in the context of guide propagation. Nonlinear optical phenomena are included in Maxwell's equations and the propagation equation as described in the previous section with Eq. 1.6. Thus, calculating these nonlinear effects in a waveguide is enabled by the derivation of the Maxwell's equations. However, in this case, a special care must be taken with the phase-matching conditions. Parametric nonlinear phenomena require conservation of energy and momentum (this will be discussed in detail when discussing FWM in a subsequent chapter) and as such, the different frequencies involved in a process need to be in phase. The additional phase shifting originating from the variation of the refractive index caused by the Kerr effect (third-order nonlinear effect) is intensity dependent and needs to be carefully considered. It can be written as:

n(ω) = n 0 (ω) + n 2 I (1.25)
,with n 2 the Kerr refractive index expressing the variation in the medium susceptibility originating from third-order nonlinearities. This index variation is responsible for the self-focusing phenomenon and factors significantly in SPM. To describe accurately the phase matching conditions necessary to achieve a nonlinear process, the chromatic dispersion discussed in a previous section also needs to be accounted for. Thus, it is usual to consider the phase matching conditions as [START_REF] Govind | Nonlinear fiber optics[END_REF]:

κ = ∆β L + ∆β N L = 0 (1.26) ∆β L = ∞ k=0 1 k! ∂β ∂ω k (ω -ω P ) k (1.27) ∆β N L = 2γP (1.28)
, with P the peak pump power, β the optical mode propagation constant, ∆β L the contribution of chromatic dispersion to the phase matching, as previously discussed in Eq.1.6, and ∆β N L the nonlinear component of the phase matching, which is induced by the nonlinear response of the material. This nonlinear response leads to an additional phase shift known as self-phase modulation (SPM). The nonlinear component ∆β N L is responsible for the change in the phase of the propagating waves due to the intensitydependent refractive index of the material (see Eq. 1.26). Together, these contributions, ∆β L and ∆β N L , play a crucial role in determining the overall phase matching behaviour in the waveguide system. The coefficient γ represents the effective nonlinear coefficient, which characterizes the strength of the nonlinear response that occurs during the propagation of light in the waveguide. The nonlinear behaviour of the waveguide is primarily determined by the real part of this coefficient, and therefore, it is desirable for it to have a large value. The effective nonlinear coefficient, γ, is derived from the effective nonlinear susceptibility tensor, Γ. This tensor takes into account the contributions of each material present in the cross-section of the waveguide, considering the fraction of the mode that overlaps with each material. Hence, Γ can be calculated as:

Γ = A 0 A N L e * (r; ω)χ (3)
ef f e(r; ω)e * (r; ω)e(r ⊥ ; ω)dA

( A∞ n 2 (r)|e(r; ω)| 2 dA) 2 (1.29)
, where A N L is the area of the waveguide's cross section where the materials exhibit a nonlinear response, A ∞ the whole cross-section area, A 0 the area of the waveguide's core, e(r; ω) the transverse electromagnetic mode distribution, and e * (r; ω) its complex conjugate. One can distinguish the real and imaginary part of Γ [START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF]:

Γ = Γ ′ + jΓ ′′ (1.30)
Γ ′ accounts for the refractive index variations, whereas Γ ′′ accounts for the nonlinear TPA, if any. The Kerr nonlinear refractive index n 2 and the TPA β T P A coefficients of the waveguiding structure can be calculated from this effective susceptibility tensor. The two parameters describing the Kerr effect and the TPA are related as follows:

ω c n 2 + i 2 β T P A = 3ω 4ϵ 0 c 2 n 2 Γ (1.31) n 2 = 1 cn 2 0 ϵ 0 3 4 Re(Γ) β T P A = -ω c 2 n 2 0 ϵ 0 3 2 Im(Γ) (1.32)
Although these parameters are essential for the description of nonlinear absorption and nonlinear refraction, allowing the modelling of SPM or saturable absorption, they do not provide a useful gauge to assess the nonlinear performance of different waveguide structures. A factor that describe more accurately the nonlinear behaviour in a waveguide is the effective nonlinear coefficient γ, mentioned in Eq.1.28 which considers the mode group velocity v g :

.4: a. Some of the most emblematic nonlinear processes. In yellow, a second-order nonlinear medium, in blue, a third-order nonlinear medium. Virtual energy levels are represented in dashed lines. i. Second-harmonic generation, a pump frequency ω is doubled to the second harmonic 2ω through a nonlinear process described in Eqs. 1.20 and 1.21 ii. THG. A frequency ω is tripled to the third-harmonic 3ω through a nonlinear process described in Eq.1.26 iii. FWM process. This process will be described more thoroughly in a subsequent chapter. FWM can be found in its non-degenerate state where ω P 1 ̸ = ω P 2 or in it's degenerate state ω

= 3ωΓ 4ϵ 0 A 0 v 2 g (1.33)
This factor encapsulates the information related to the nonlinear Kerr coefficient, which is represented by the real part of the effective nonlinear coefficient, γ, as well as the two-photon absorption (TPA), which is captured by the imaginary part of γ. When performing numerical integration over the entire cross-section of the waveguide, as described in Eq. 1.29, it is necessary to consider the properties of the different nonlinear materials present in the waveguide structure. This is critical to consider the overlap between the transverse mode and the huge nonlinearities of different materials that may be included in the structure.

It is thus possible to account in a straightforward added materials with interesting properties boosting standard dielectric waveguides (chalcogens, 2D materials,..). This method, known as waveguide hybridization, aims to enhance the overall effective properties of waveguides. To study different hybrid waveguide geometries, it is crucial to employ a methodology that provides intuitive and complementary measures of the contributions from various components of the waveguide to its nonlinear effective properties. One important measure in evaluating the performance of the active material is the confinement factor, which represents the total fraction of power passing through the active material [START_REF] Vincent Pelgrin | Boosting the sin nonlinear photonic platform with transition metal dichalcogenide monolayers[END_REF]. To calculate the confinement factor, we analyze both the Poynting vector through the effective thickness of the layered material and the Poynting vector through the entire structure. To determine the fraction of power that is confined within the active material:

S = E × H (1.34)
S is the Poynting vector, E is the electric field, and H is the magnetic field. To describe the propagating energy responsible for the nonlinear effects occurring in the waveguide, we derive the average power flow through the desired area from the vector S :

< P >=< S.u z ds > (1.35) CF = < P > N L < P > tot (1.36)
with <> is a time average and CF the confinement factor. P N L is the power flow through the component of the waveguide the impact on the overall effective nonlinearity is being assessed and P tot the power flow through the all cross-section of the waveguide. This approach offers a versatile framework that can be applied in two directions. Firstly, experimental characterization of the nonlinear coefficient for the added materials is possible. This can be achieved by fitting experimental spectra obtained from measurements conducted in the hybrid waveguide, such as self-phase modulation (SPM) or four-wave mixing (FWM) measurements [START_REF] Zhang | Enhanced fourwave mixing with mos2 on a silicon waveguide[END_REF]. The specific impact of the added materials on the nonlinear behaviour of the waveguide can then be determined knowing its contribution to the overall structure. [START_REF] Wang | Enhancing si3n4 waveguide nonlinearity with heterogeneous integration of fewlayer ws2[END_REF]. Secondly, it can be utilized as a tool for predicting and optimizing the effective performance of waveguiding structures through simulations and optimization processes [START_REF] Vincent Pelgrin | Boosting the sin nonlinear photonic platform with transition metal dichalcogenide monolayers[END_REF]. By incorporating the characteristics of the 2D materials into the simulations, the anticipated behavior of the hybrid waveguide can be assessed and optimized thus leading to optimal design of hybrid waveguides. This will be discussed in the following chapters with the addition of 2D materials to standard waveguides.

Generalized nonlinear Schrödinger equation (GNLSE)

As mentioned in the previous part, assessing the nonlinearity of a material added to a waveguiding structure is possible if the effective properties of the waveguide are known. Thus, fitting spectral measurements in the characterization of a sample is necessary. This purpose can be fulfilled by integration of the GNLSE [START_REF] Govind | Nonlinear fiber optics[END_REF][START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF][START_REF] John M Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. Other usages can be found for this equation like a predictive use to anticipate the outcome of a particular waveguide. It can simulate the propagation of a pulse through a waveguide, taking into consideration the various physical effects tempering with the spectral output. Several parameters discussed previously are essential for the accurate simulation of a nonlinear waveguiding structure. The equation considers these effects discussed in a previous section, such as the effective nonlinear optical properties (n 2 and β T P A ), FCs, Raman effect, chromatic dispersion (embodied by the Taylor expansion of Eq. 1.10 ) or linear passive losses. In a lot of cases, these effects need to be included in the model to match experimental results as closely as possible. For instance, in the case of Si and as depicted by Fig. 1.4, TPA in Si generates a lot of FC that induces refractive and absorbing effects on the propagative waves. The equation originates from the propagation equation discussed before. The electrical field can be written as:

E(x, y, z, t) = A(z)E(x, y)e j(ωt-β(ω)z) (1.37)
As such, an equation in A(z) where a few approximation have been made such as the slowly varying envelop approximation [START_REF] Govind | Nonlinear fiber optics[END_REF] (similar to the quasi static regime approximation in electronics) can be written as:

∂A ∂z - α 2 A - k≥2 β n i n-1 n! ∂ n ∂T n A - σ a 2 + iσ n N C A = iγ(1 + 1 ω 0 ∂ ∂T ) × (1 -f R )A|A| 2 + f R A ∞ 0 h R (τ )|A(z, T -τ )| 2 dτ (1.38)
with A the complex envelope, α the attenuation constant, β n the higher-order coefficients from the Taylor expansion of the propagation constant calculated around the central frequency ω 0 . The Raman response is considered with f R , the delayed Raman response to polarization [START_REF] Govind | Nonlinear fiber optics[END_REF][START_REF] Hult | A fourth-order runge-kutta in the interaction picture method for simulating supercontinuum generation in optical fibers[END_REF], and h R (t) the Raman response function. TPA can be expressed in this equation as the complex part of the nonlinear coefficient γ [START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF]. FCs are described using the following parameters: σ a and σ n are the FC absorption and refraction coefficients, and N C is the FC density and can be calculated as follows: [START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF][START_REF] Paul | The elements of nonlinear optics[END_REF][START_REF] Soref | Electrooptical effects in silicon[END_REF].

σ a = 2e 3 ϵ 0 n 0 ω 2 c E 2 g E 2 g -ℏ 2 ω 2 1 µem * 2 e + 1 µ h m * 2 h (1.39) σ n = -e 2 2ϵ 0 n 0 ω 2 E 2 g E 2 g -ℏ 2 ω 2 1 m * e + 1 m * h (1.40)
, with e the electric charge, ϵ 0 the vacuum permittivity E g the gap energy of the material in which the calculation is performed (usually silicon) ℏ the Plank constant, µ e the electron mobility, µ h the hole mobility, m * e the effective mass of the electron and m * h the effective mass of the hole. From Eq. 1.39 and Eq. 1.40, the carrier absorption and the carrier refraction can be expressed by introducing the carrier density N C :

α F C = σ a N C (1.41) δn F C = σ n N C (1.42)
, where N C is a time-dependent parameter that needs to be computed at each step of the propagation as it depends on the local mode envelop. Therefore at each step, the following differential equation needs to be solved:

∂N C ∂t = - -N c t c + P 2 0 Im(γ) ℏA 0 ω 0 |A| 4 (1.43)
This calculation of the carrier density can be simplified by introducing a numerical solution:

N C = t/2 -t/2 n g n 0 2 β T P A 2ℏω 0 |A(z, τ )| 4 dτ e -t/τ F C (1.44)
with τ F C the carriers life time. In the case of Silicon the values are : E g = 1.12eV , µ e = 0, 14m 2 /(V.s) µ h = 0, 045m 2 /(V.s) m * e = 0, 26 m 0 m * h = 0, 39 m 0 with m 0 = 9, 109.10 -31 kg the electron mass and τ F C = 10 -9 s. Fig. 1.5 shows the increase of the carrier density in the time domain. The time domain envelope of the propagating light pulse is also represented showing how a time dependant description of the phenomena is required at each step because of the relationship between the time envelop of the pulse and the carrier density.

Depending on the case some effects can be neglected or not. For instance, in most examples relating to Si-photonics in the C-band, the Raman response can be neglected for simulations involving Si or SiN waveguides [START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF][START_REF] Lafforgue | Broadband supercontinuum generation in nitrogen-rich silicon nitride waveguides using a 300 mm industrial platform[END_REF]. FCs and TPA can be further ignored in several cases. Many materials do not generate FCs in the frame of wavelengths used during the experiments (e.g., doped silica waveguides or SiN waveguides around 1.5µm). However, while ruling some effects out, it is important to remain cautious. Fig. 1.4 b. presents an example of integration of the GNLSE while accounting for various parameters. The simulation is calculating the outcome of a 100 fs pulsed 6.5 W of peak power centred around 1500 nm propagating in a Si wire (h × w = 220 × 360nm). The input of the simulation is the pulsed parameters, the waveguide dispersion (here are considered only the first 2 orders of the Taylor expansion β 2 = -4ps 2 /m and β 2 = -0.0915ps 3 /m), the nonlinear parameters (γ = 446.5W -1 m -1 ) and the propagation length. As seen in the comparison Fig 1 .4 b., it is essential to accurately assess which physical effects are pertinent to the model as the results can drastically vary depending on that. This is a concern when attempting to fit experimental results. As mentioned prior, a major appeal to the calculation of this equation is to retrieve the waveguide's nonlinear performance from experimental measurements. Using the nonlinear coefficient γ as the fitting parameter, multiple simulations can be performed until an output spectrum resembling the measured spectrum can be retrieved [START_REF] Wang | Enhancing si3n4 waveguide nonlinearity with heterogeneous integration of fewlayer ws2[END_REF][START_REF] Ishizawa | Optical nonlinearity enhancement with graphene-decorated silicon waveguides[END_REF][START_REF] Feng | Enhanced optical kerr nonlinearity of graphene/si hybrid waveguide[END_REF][START_REF] Lafforgue | Broadband supercontinuum generation in nitrogen-rich silicon nitride waveguides using a 300 mm industrial platform[END_REF] (this will also be demonstrated later on in the second chapter). Numerical methods for integrating this equation are standard such as the split-step Fourier method using a fourth-order Runge-Kutta scheme for the integration [START_REF] Govind | Nonlinear fiber optics[END_REF][START_REF] Hult | A fourth-order runge-kutta in the interaction picture method for simulating supercontinuum generation in optical fibers[END_REF][START_REF] Lafforgue | Broadband supercontinuum generation in nitrogen-rich silicon nitride waveguides using a 300 mm industrial platform[END_REF]. For pulse-broadening experiments in hybrid waveguides (covered with additional materials to boost the nonlinear effective properties), it allows to retrieve the effective nonlinear coefficient of the added material following methods described earlier. Eq. 1.29, and proves itself useful in that context as it provides the contribution of each material to the effective nonlinear properties of the whole waveguide. Additionally, a specific care must be taken while treating with the longitudinal variation of the nonlinear coefficient for more accurate results. Indeed, as will be discussed in the second chapter, some cases can present two different types of structures with different nonlinear properties consecutively between light injection and collection in and out from the chip. For that reason, a longitudinal profile of the nonlinear coefficient along light propagation must be considered for realistic numerical modelling covering all propagation. It can be particularly interesting in that case to monitor the temporal evolution of the envelope. Fortunately, as depicted in Fig. 1.4 c., the numerical integration method is iterative and thus allows to keep a record of the temporal evolution of the envelop. The GNLSE used in the context of waveguides covered with 2D material is a powerful tool to assist in the rapid development of this field.

2D materials and their properties

Generalities about 2D materials

A great variety of nanomaterials can arguably be placed under the denomination "2D materials". For our purpose, it designates a thin crystal layer with a thickness of just a few atoms called the effective thickness h ef f (usually a few nm). Usually, 2D materials are considered a single layer (monolayer) of crystal, but stacks up to a few atomical layers can also be qualified as being part of that family. Van der Waals crystals which are stacks of multiple different 2D materials [START_REF] Andre | Van der waals heterostructures[END_REF][START_REF] Castellanos-Gomez | Van der waals heterostructures[END_REF] can also be included under that denomination. 2D materials do not correspond to any physical reality, it is an umbrella term regrouping various crystals and compounds. No material is completely 2D and the effective thickness is an important parameter, especially while considering the interaction between a transverse mode propagating through a hybrid structure and a 2D material (as discussed in previous sections for the calculation of the effective nonlinear susceptibility).

2D materials can be recognized by the fact they exhibit many exceptional properties compared with traditional 3D materials. Due to the quantum confinement in the direction perpendicular to its plane, the electrical properties of the material find themselves to be greatly impacted when compared to those for similar bulk materials. For instance, the number of states an electron can occupy in a crystal is tremendously reduced due to Pauli blocking [START_REF] Fedor | Electronic states and optical transitions in semiconductor heterostructures[END_REF]. Thus, new properties such as saturable absorption can arise. The most emblematic nanomaterial belonging to the 2D material family is graphene, a honeycomb-structured layer of carbon atoms that was first demonstrated through mechanical exfoliation [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF]. Graphene's unique electronic and optical properties [START_REF] Bonaccorso | Graphene photonics and optoelectronics[END_REF][START_REF] Cheng | Third order optical nonlinearity of graphene[END_REF][START_REF] Ah Castro Neto | The electronic properties of graphene[END_REF] have led to the discovery of numerous potential applications. This breakthrough has sparked enormous interest in 2D materials and motivated the exploration of other families of thin-layer crystals with distinct physical properties [START_REF] Xia | Two-dimensional material nanophotonics[END_REF]. These 2D materials, such as transition metal dichalcogenides (TMDs), black phosphorus (BP), or hexagonal boron-nitride (h-BN), are also under thorough investigation. Although most 2D materials exhibit a "honeycomb-like" structure, they differ in their band gaps (as illustrated in Fig. 1.6 d.) and group symmetries (e.g., D6h for graphene, D3h for typical TMDs [START_REF] Ribeiro-Soares | Group theory analysis of phonons in two-dimensional transition metal dichalcogenides[END_REF]). The variety of 2D materials embodied by the variations in their band structure is introducing a range of electrical properties that can be exploited to engineer a wide variety of optical and electrical devices (field effect transistors [START_REF] Babu Mitta | Electrical characterization of 2d materials-based field-effect transistors[END_REF], photo-detector [START_REF] Jiang | Recent advances in 2d materials for photodetectors[END_REF],...). In that regard, hybrid integrated devices also benefit from this range of possibilities (e.g., switch operation, broadband light sources [START_REF] Jw You | Nonlinear optical properties and applications of 2d materials: theoretical and experimental aspects[END_REF]).

Hybrid integration

Integrating 2D materials with dielectric optical waveguides is a recent idea, and consists in depositing a thin layer crystal on a waveguiding structure. The propagating transverse mode overlap with the added 2D material, thus improving the general effective properties of the waveguide (Fig. a.). As previously stated, SOI (and adjacent platforms i.e., SiN/SiO 2 ) might be the most suitable for building integrated photonic chips but are lacking some features, such as a way to generate TPA-free powerful nonlinear response in the near infrared. Hence, the growth or deposition of 2D materials on

Integrated optics

Free space Material

n 2 /n 2Si n 0 TPA χ (3) χ (2) Band gap (eV) Si 1 [73] 3.5 0.5 × 10 -11
(n 2Si = 7.2 × 10 -18 m 2 W -1 ), TPA [m.W -1 ], χ (3) [m 2 V -2 ] at 1550 nm, χ (2) (m 1 V -1 )
second-and third-order nonlinear optical parameters of the commonly used 2D materials and primary materials of the CMOS-compatible platform employed in Si and Si-hybrid integration schemes at the telecom wavelengths. We differentiated between the results obtained in integrated settings and those obtained in free space, revealing certain discrepancies. In the context of our study, 'BP' refers to black phosphorous, 'GO' denotes graphene oxide, and n 2Si represents the Kerr refractive index value for Si at 1550 nm. The band gap can vary between 0.3 and 2 eV making it a versatile semiconductor iv. h-BN and its large bandgap make it an insulator.

waveguides is being investigated as a promising solution to resolve this issue. 2D materials have a lot of potential as they offer many advantages over other alternatives such as III-V (e.g., the deposition of 2D materials introduces arguably very few defects as they stick to the substrate by Van Der Waals forces, thus, no issue with lattice mismatch). Moreover, 2D layered materials have demonstrated very promising performance in their nonlinear responses [START_REF] Autere | Optical harmonic generation in monolayer group-vi transition metal dichalcogenides[END_REF][START_REF] Autere | Nonlinear optics with 2d layered materials[END_REF][START_REF] Jw You | Nonlinear optical properties and applications of 2d materials: theoretical and experimental aspects[END_REF]. Table 1 is an account of reports showing the investigation work that has been conducted on a selection of the most common 2D materials. It includes the characterized nonlinear response of various hybrid waveguides as well as free space characterization. However, assessing the impact of adding 2D materials to photonic devices can be challenging. The issue resides in the complexity related to the quantification of the nonlinear effects in 2D materials. Indeed, due to the influence of the layer, fabrication methods and the quality of the thin layered crystal and their properties [START_REF] Autere | Optical harmonic generation in monolayer group-vi transition metal dichalcogenides[END_REF][START_REF] Autere | Nonlinear optics with 2d layered materials[END_REF], a lot of discrepancies are found in the literature. For instance, the fabrication and deposition methods leave residues (e.g., polymer, water), defects in the monolayer crystal, or wrinkles of the flake [START_REF] Nan | Strong photoluminescence enhancement of mos2 through defect engineering and oxygen bonding[END_REF], thus affecting randomly performance in different samples. Other variations are found between the 2D material's fabrication method, a chemical-vapor-deposition (CVD)-grown sample (e.g., CVD grown graphene prompting χ (3) = 5 -10 × 10 -19 m 2 V -2 from THG measurements with a pump at 1566 nm [START_REF] Jiang | Gate-tunable third-order nonlinear optical response of massless dirac fermions in graphene[END_REF]) can have different performance than a mechanically-exfoliated flake (e.g., χ (3) = 4.2 × 10 -15 m 2 V -2 for exfoliated graphene measured from THG measurements with a pump at 1550 nm wavelength [START_REF] Saynatjoki | Rapid large-area multiphoton microscopy for characterization of graphene[END_REF]). The crystal doping can also impact the measured nonlinear response in various ways [START_REF] Lee | Doping-induced changes in the saturable absorption of monolayer graphene[END_REF][START_REF] Zhang | Doping-induced secondharmonic generation in centrosymmetric graphene from quadrupole response[END_REF][START_REF] Shi | Tuning the nonlinear optical absorption of reduced graphene oxide by chemical reduction[END_REF]. 2D materials are very susceptible to the environment they are being characterized in (e.g., the substrate [START_REF] Cory R Dean | Boron nitride substrates for high-quality graphene electronics[END_REF]), and induced strain [START_REF] Sh Rhim | Strain-induced giant secondharmonic generation in monolayered 2 h-mox2 (x= s, se, te)[END_REF][START_REF] Liang | Monitoring local strain vector in atomic-layered mose2 by second-harmonic generation[END_REF][START_REF] Mennel | Optical imaging of strain in two-dimensional crystals[END_REF][START_REF] Liang | Universal imaging of full strain tensor in 2d crystals with third-harmonic generation[END_REF]which can affect performance significantly.

Although these variations can be used as an advantage in some cases, it usually remains uncontrolled and can vary randomly. It is thus most often, a factor of uncertainty.

Besides variations of performance occurring as a result of the fabrication process, the characterization method itself can yield different results. It has been reported in several cases that an in-plane electromagnetic wave excitation can generate a different output compared to an out-of-plane one [START_REF] Autere | Optical harmonic generation in monolayer group-vi transition metal dichalcogenides[END_REF][START_REF] Dai | Broadband plasmon-enhanced four-wave mixing in monolayer mos2[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF][START_REF] Bao | Broadband graphene polarizer[END_REF].

The optical excitation mode of 2D sheets in the context of a hybrid waveguide has also a preferred polarization. For graphene and GO a quasi-TE polarization in the waveguide yielded to better recorded nonlinear response than the quasi-TM polarization [START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF][START_REF] Bao | Broadband graphene polarizer[END_REF]. It has been suggested in some preliminary studies that the same advantage for the TE mode might be confirmed in TMDs, such as M oS 2 [START_REF] Tan | Polarization-dependent optical absorption of mos2 for refractive index sensing[END_REF]. The study and deep understanding of the nonlinear properties of 2D materials remain challenging. So far, the study of most 2D materials have relied heavily on experimental measurements, without any strong link with the theoretical origin of the observed properties. Graphene [START_REF] Ji | Enhanced parametric frequency conversion in a compact silicon-graphene microring resonator[END_REF] characterized in a FWM experiement. The FWM spectrum from the bare Si waveguide is in black and from the Si-graphene waveguide is in red. c. Tunable FWM characterized in a gated graphene-SiN waveguide, [START_REF] Alexander | Electrically tunable optical nonlinearities in graphenecovered sin waveguides characterized by four-wave mixing[END_REF] The conversion efficiency (FWM) varies depending on the gate voltage (VGS) and the wavelength detuning between the pump and the signal. d. Graphene-covered silicon waveguide characterized in an optical pulse broadening experiment for several lengths (LGr) of graphene sheet covering the waveguides compared to a pure silicon waveguide (SWG. The experiment is performed using a pulsed pump at 1560 nm wavelength with a 250 MHz repetition rate, 80 fs pulse duration, and an input pulse energy of 120 pJ [START_REF] Ishizawa | Optical nonlinearity enhancement with graphene-decorated silicon waveguides[END_REF]. e. Saturable absorption in a graphene-SOI strip waveguide [START_REF] Wang | Cmos-compatible all-optical modulator based on the saturable absorption of graphene[END_REF]. characterized by transmission measurements at diffeent pump power. The experiment is done for different length of graphene. f. SHG enhancement generated in a M oSe 2 -Si waveguide [START_REF] Chen | Enhanced second-harmonic generation from twodimensional mose2 on a silicon waveguide[END_REF]. Measured spectrum with a significant increase in the SHG emission for the hybrid structure. Peak R1 corresponds to SHG, and peak R2 to PL. g. Raman response on graphene on a Si-L3 cavity [START_REF] Gomulya | Enhanced raman scattering of graphene using double resonance in silicon photonic crystal nanocavities[END_REF]. Raman spectrum of graphene excited with a laser source tuned at 532 nm for the different locations on the device: red for the cavity and blue for the SOI plane substrate. In the inset: zoomed-in on the normalized 2D peak for spectral comparison.

is in that regard an exception. Indeed graphene has specific properties, due to its Diraccone shape band structure. This enable analytical investigation of its third-order conductivity. The analytical solution for σ (3) , equivalent to the susceptibility tensor χ (3) , can be explicitly expressed for this material [START_REF] Cheng | Third order optical nonlinearity of graphene[END_REF]. In the case of 2D materials, the surface conductivity is equivalent to the susceptibility in a bulk material. Other families of 2D material have different type of band structures (Fig. 1.6). As such, building a theoretical approach is more difficult. It often requires iterative comparisons between experimental results and numerical modelling and analysis. This is raising at the same time a difficulty but also opening a very interesting field of investigation to evaluate their full potential.

The theoretical understanding of the optical properties of 2D materials is bound to expand as the interest surrounding the field is drastically increasing. As can be seen in Table 1.1, multiple materials have been investigated beyond graphene and are showing interesting results. In fact, there exits already some literature documenting the use of various 2D materials in the context of hybrid waveguides. Third order nonlinear processes have been successfully boosted in several works such as FWM (Fig. 1.6 b. and c.), or SPM (Fig. 1.6 d.). Beyond the benefit of adding highly nonlinear materials to waveguides to the third-order nonlinear response, other nonlinear effects present in 2D materials can also be exploited or studied. Thus, effects like saturable absorption [START_REF] Demongodin | Ultrafast saturable absorption dynamics in hybrid graphene/si3n4 waveguides[END_REF][START_REF] Wang | Cmos-compatible all-optical modulator based on the saturable absorption of graphene[END_REF] (Fig. 1.6 e.), SHG [START_REF] Mennel | Second harmonic generation in strained transition metal dichalcogenide monolayers: Mos2, mose2, ws2, and wse2[END_REF][START_REF] Ribeiro-Soares | Second harmonic generation in wse2[END_REF][START_REF] Chen | Enhanced second-harmonic generation from twodimensional mose2 on a silicon waveguide[END_REF] (Fig. 1.6 f.) or Raman emission [START_REF] Gan | Strong enhancement of graphene-light interaction in a photonic crystal nanocavity[END_REF][START_REF] Gomulya | Enhanced raman scattering of graphene using double resonance in silicon photonic crystal nanocavities[END_REF] (Fig. 1.6 g.) can be reinforced or introduced thanks to an improved interaction between the incident wave and the 2D material in the context of an hybrid structure. Notable results using gated 2D materials should also be mentioned. The electrical properties of 2D materials can be exploited, with for instance the possible tuning of the Fermi level of gated-graphene by shifting the biased current at the surface. This yielded several very promising results such as tunable FWM [START_REF] Alexander | Electrically tunable optical nonlinearities in graphenecovered sin waveguides characterized by four-wave mixing[END_REF] (Fig. 1.6 f.) or tunable frequency combs [START_REF] Yao | Gate-tunable frequency combs in graphene-nitride microresonators[END_REF]. Other materials such as W S 2 have shown potential in that regard as well [START_REF] Datta | Low-loss composite photonic platform based on 2d semiconductor monolayers[END_REF].

Objectives of the thesis

In spite of the progress made, research on the use of 2D materials on optical waveguides is still in its early stages, with significant gaps in both theoretical and experimental knowledge. The purpose of the work presented in the subsequent chapters is to contribute to this field. The primary objective is to demonstrate the enhancement of nonlinear effects through the integration of 2D materials with CMOS compatible waveguides, exploring materials other than graphene, which has been the main focus in existing literature. By exploring different materials, we aim to push the boundaries of what has been achieved so far in the field of hybrid integrated optics. In Chapter 2, we will describe a direct approach using accessible fabrication and characterization methods, aiming to establish a proof of concept and investigate various aspects of the field, such as simulation, fabrication, and characterization. Pursuing these goals will provide us with valuable experience and knowledge to drive further innovation while being mindful of the challenges involved.

The second goal of this research is to introduce innovations in the field. Drawing from the experience gained during the exploration in the second chapter, several ideas may emerge to propose new avenues for investigation. Building upon the work established in chapter 2, we can leverage the developed tools to introduce various methods and ideas aimed at enhancing the potential of hybrid integrated optics for nonlinear applications. The solutions derived can be categorized into three main areas identified in the first chapter: improving the overall design of the hybrid waveguide through simulation tools, exploring new techniques for transferring 2D materials in the fabrication process, and introducing novel methods of characterization using resonators. These improvements and ideas will be presented in the third chapter.

Chapter 2 Integration of 2D materials on photonic waveguides

This chapter describes an introduction to the field of hybrid integrated optics. We will here discuss a first approach to integrating 2D material on photonics waveguides by covering first the matter of the design of a hybrid waveguide. Then a first approach to fabrication will be discussed, and at last, the characterization of a hybrid straight waveguide will also be detailed.

Hybrib integrated waveguide design

The design of a waveguide is a crucial process that involves selecting the optimal dimensions, architecture, materials, and layout for a structure that guides and manipulates light. Each of these parameters plays a critical role in shaping the integrated optical functions of the device. To optimize performance, minimize the on-chip footprint, and tailor the optical function's output, careful consideration must be given to the waveguide's purposeful design and the overall chip layout. Numerical tools discussed in Chapter 1 can be leveraged for this purpose, but when dealing with 2D materials, additional care must be taken in integrating the thin layer of crystal into the structure, as it can significantly impact sensitive parameters. Therefore, in the first section, we will explore how 2D materials can be included in simulations of hybrid integrated waveguides. In the second section, we will examine the compromises that may be necessary for optimizing the hybrid waveguide.

Numerical estimation of the integration influence on waveguide mode properties

In the simulation of hybrid waveguides, accounting for 2D materials is crucial as their unique properties may have a significant impact on the waveguide's properties. Therefore, introducing 2D materials in numerical studies is necessary to anticipate their impact on key waveguide properties, such as dispersion, transverse eigenmode distribution, and effective nonlinear coefficient [START_REF] Okamoto | Fundamentals of optical waveguides[END_REF]. As described in the previous chapter, the optogeometrical parameters of a waveguide structure are critical in shaping various optical functions, especially in the case of nonlinear integrated functions. Hence, correctly modelling a 2D material's quasi-surface is necessary, yet it can present new challenges. Despite the extreme thinness of the added layer, the optical effect of a 2D material can be significant due to its huge surface conductivity.

There are two different ways to introduce 2D materials into the numerical model:

• The first approach considers the 2D material as a real 2D layer with surface properties characterized by a complex surface conductivity σ s .

• The second approach describes the 2D material as an extremely thin bulk layer of material, with the parameter employed being the "bulk surface conductivity." This is a bulk equivalent to the real surface conductivity σ s , which depends on the effective thickness h ef f (h ef f ∼ 0, 5nm) of the 2D material.

σ b = σ s h ef f (2.1)
Contrary to what the name "2D materials" implies, these materials have indeed a third dimension that is equivalent to the atomic thickness of the crystal lattice. The bulk surface conductivity approach takes this fact into account, whereas the surface conductivity does not. It is important to state that none of the approaches is physically wrong as they described both the electromagnetic behaviour using surface conductivity and not the bulk equivalent (i.e., bulk material), the permittivity. The reason for that is the electromagnetic properties of 2D materials are indeed related to the thin nature of the crystal restricting the electronic states to a 2D "quantum well" kind of problem which is also the reason these materials are considered "2D". Regardless, similar results can be found using either of these methods. They both present advantages and inconveniences.To reduce the computation time, the first approach may be preferable. In the case of the bulk approach, the Finite Difference Eigenmode (FDE) calculation requires a very dense mesh to accurately account for the boundary conditions imposed by the hybrid structure. Since 2D materials typically have a thickness of around 1 nm and the overall structure spans dimensions on the order of a few micrometers, achieving a tight mesh for the FDE calculation can be computationally intensive. On the other hand, using an equivalent bulk material, it becomes possible to construct a highly resolved transverse mode distribution that facilitates the calculation of the overlap between the transverse mode and the 2D material. This highly resolved region around the nanomaterial is essential for accurately evaluating the overlap integrals described in Eq. 1.29, which are necessary for determining the effective nonlinear parameters.

Wether the aim is to establish quickly a dispersion profile (in which case, one should prefer the real surface conductivity approach) or to accurately assess the interaction between the 2D material and the transverse mode (in which case the bulk approach is better suited), estimations for the surface conductivity of the 2D material needs to be established. This estimation can be made using different methods depending on the material of interest [START_REF] Weismann | Theoretical and computational analysis of second-and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers[END_REF]. For graphene, analytical solutions can be found. The Diraccone shape of its band structure indeed allows for analytical investigation [START_REF] Hwang | Dielectric function, screening, and plasmons in two-dimensional graphene[END_REF][START_REF] Wunsch | Dynamical polarization of graphene at finite doping[END_REF]:

σ s (ω) = e 2 E F πℏ 2 τ 1 -iωτ + θ(ℏω -2E F ) + i π ln ℏω -2E F ℏω + E F (2.2)
With ω the frequency, e the elementary charge, ℏ the reduced Planck constant, E F the Fermi energy level, τ the relaxation time, and θ the Heaviside step-like function [START_REF] Weismann | Theoretical and computational analysis of second-and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers[END_REF]. Surface conductivity for graphene can be calculated (E F = 0.6eV and τ = 0.25/2π) accordingly and is displayed in Fig. 2.1. Due to their more complicated band structures, the surface conductivity of other 2D materials cannot be described with an analytical approach. Instead, the use of experimental measurements is required. Materials such as TMDs have been thoroughly studied and their surface conductivity can be retrieved from the literature (e.g., M oS 2 , W S 2 , M oSe 2 , W Se 2 [START_REF] Zhang | Enhanced mid-to-near-infrared second harmonic generation in silicon plasmonic microring resonators with low pump power[END_REF]). Fitting the data by a sum of Lorentzian, one can then calculate the relative permittivity ϵ r :

ϵ r = ϵ(ω) ϵ 0 = 1 + N k=1 f k ω 2 k -ω 2 -iωξ k (2.3)
, where f k stands for the oscillator strength, ω k the resonance frequency, and ξ k is the spectral width of the k th oscillator. The material permittivity enables access to the surface conductivity, through the following expression:

ϵ(ω) = ϵ 0 1 + iσ b ϵ 0 ω = ϵ 0 1 + iσ s ϵ 0 ωh ef f (2.4)
, where h ef f is the effective atomic thickness of the thin layer 2D material, ϵ(ω) is the permittivity, ϵ 0 is the vacuum permittivity, σ b is the bulk conductivity and σ s is the surface conductivity. Fig. 2.1 displays the surface conductivity calculation for two of the most common TMDs (i.e., M oS 2 , W S 2 ) using the data from Ref. [START_REF] Li | Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: Mos 2, mo s e 2, ws 2, and ws e 2[END_REF].

To better grasp the impact of a 2D material on a waveguiding structure, a comparison is established in Fig. 2.2 using a buried strip SiN waveguide with dimensions such as h×w = 600×1000nm. The two parameters that are controlled are the transverse mode distribution for the TE mode and the GVD between 1µm and 2µm. The comparison is established between a uncladded "raw" waveguide (no 2D material on top) and a hybrid waveguide. To better visualize the shift in the TE mode distribution at 1550 nm, the plotted field distribution is calculated as follows:

E plot = E 2D -E ref (2.5)
The mode distribution values of E 2D and E ref are normalized. As such, the maximum intensity of the TE mode in this data is 1. Calculating the difference between these two modes yields a maximum of 2 orders of magnitude below (∼ 0, 01). This shows a minor shift in the distribution of the transverse mode. The comparison between the GVD profile of the hybrid waveguide and the GVD profile of the reference waveguide also shows very few changes. As a first approximation, one could thus assume that in an ideal case (strictly monolayer and perfect transfer of the 2D material), the addition of the 2D materials has no impact on the passive properties of the waveguide. However, this highly depends on fabrication, transfer, and post-treatment methods (e.g., annealing). Table 2.1: Comparison between key parameters for an increasing number of Layers. Is compared the maximum of the mode distribution difference E plot also represented in Fig. 2.2. The comparison between the dispersion profiles is also represented by the shift in wavelength of the zero dispersion wavelength (ZDW). For each number of layers, the total thickness is also indicated.

It remains difficult to account for these externalities in numerical studies. Additionally, it can be found from similar calculations performed with higher thicknesses that the passive properties can shift rapidly with an increasing amount of layers. This can especially be a concern for mechanically exfoliated materials that tend to have an uncontrolled amount of crystal layers. Table 2.1 presents the case of the shift of the transverse TE mode and the dispersion profile between 1 and 20 layers. The shift of the TE mode is represented by the increase in the maximum of E plot . The shift of the dispersion profile, is represented by a red shift of the zero dispersion wavelength (ZDW), the wavelength for which the GVD is zero. While in many cases the shift in the waveguide's properties can be neglected [START_REF] He | Low-loss integrated nanophotonic circuits with layered semiconductor materials[END_REF][START_REF] Vincent Pelgrin | Boosting the sin nonlinear photonic platform with transition metal dichalcogenide monolayers[END_REF], it is still necessary to consider the additional phase shift introduced by the 2D material in certain applications [START_REF] He | Low-loss integrated nanophotonic circuits with layered semiconductor materials[END_REF][START_REF] Datta | Low-loss composite photonic platform based on 2d semiconductor monolayers[END_REF]. Therefore, the ability to perform accurate calculations accounting for these phase shifts remains crucial. Furthermore, in nonlinear applications, a precise understanding of the interaction between the 2D material and the transverse propagating mode is essential, particularly due to the significant nonlinear properties of the added material. This knowledge is crucial for optimizing and designing devices that incorporate 2D layers to achieve desired nonlinear performances.

GVD profile engineering with the example of a SOI waveguide

In the continuity of the previous section, we describe the optimization of the GVD profile and how it articulates with an optimized interaction with the 2D material in the case of a SOI waveguide. The choice can be made between different typical types of waveguides, as shown in Fig. 2.4, and although most of these structures have been reported to be used in hybrid waveguide experiments involving 2D materials, it should be noted that planarized buried strip offer the better option for this kind of application. Indeed, deposition or growth of material on regular surfaces limits the number of defects and facilitates the adhesion through Van Der Waals bonds.

As illustrated in the previous section, an important parameter to monitor is the group velocity dispersion coefficient (GVD) β 2 , which impacts phenomena such as spontaneous FWM and modulation instability. As before, this can be studied using FDE mode solver calculation. So, in addition to dispersion engineering, analyzing the overlap between transverse modes and 2D materials is crucial. It's important to note that a design that appears optimal based on dispersion profiles might not necessarily perform well in nonlinear applications, and there can be a trade-off between dispersion properties and nonlinear performance depending on the specific application. This is illustrated in Fig. 2.4 c. and d. with the variation of the width for a 600 nm thick planarized burried SOI strip waveguide for a better illustration of the point.

It can be seen from Fig. 2.4 that the width has significant impact on the GVD profile. Depending on the targeted application and wavelength range, a suitable width can be selected, offering a desired dispersion profile. However, in the case of hybrid waveguides, the impact of the flake needs also to be considered. In Fig. 2.4 c. and d. the overlap between the flake and the 2D material and the effective nonlinear coefficient of the waveguide are plotted versus the waveguide's width. Although the trend for both parameters is quite similar, a shift between the two parameters optimal values can be noticed. The reason for the decrease in effective nonlinear properties with increased waveguide width can be attributed to the fact that the core material, which in this case is Si, is also a strong nonlinear material. Although increasing the waveguide width first seems to improve the overlap and the nonlinear coefficient with the 2D material, the mode confinement start decreasing, thus negatively impacting the effective nonlinear properties of the waveguide. Additionally, the optimal design for nonlinear performance is not necessary the optimal design for the GVD profile. Finally, while the general trend followed by the overlap versus the waveguide's width seems to be similar to the trend followed by the nonlinear coefficient, the optimal between the two is shifted which shows that both parameters are slightly decorrelated and thus, shows the importance of conducting these design studies.

Transfer and characterization of the mechanically exfoliated 2D materials

The fabrication of 2D materials is also a crucial step in the integration process. Mechanical exfoliation (ME) [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF][START_REF] Ottaviano | Mechanical exfoliation and layer number identification of mos2 revisited[END_REF], also known as the "Scotch tape" method, is one of the earliest and most widely used techniques for producing high-quality 2D materials. It involves peeling off a thin layer of material from a bulk crystal using an adhesive tape. However, the process is highly time-consuming, and it is challenging to obtain a large quantity of transferable material with consistent quality. This point is discussed later in this chapter with concrete examples of ME's limitations. Alternatives have been developed over the years allowing to circumvent some of the problems faced with ME. Another method, liquid phase exfoliation (LPE) [START_REF] Neill | Preparation of high concentration dispersions of exfoliated mos2 with increased flake size[END_REF], is a technique that involves dispersing bulk crystals in a solvent and applying sonication to produce 2D flakes. This approach can produce a large volume of material, but the quality is often compromised due to contamination from the solvent. Finally, chemical vapour deposition (CVD) [START_REF] Chen | Oxygen-assisted chemical vapor deposition growth of large single-crystal and high-quality monolayer mos2[END_REF] is a promising technique for growing high-quality 2D materials on substrates. The process involves heating a precursor gas to produce 2D flakes on a substrate, and the resulting material is of high quality and can be produced in large quantities. Each method presents upsides and downsides that need to be accounted for depending on the targeted 2D materials application. It is, however, important to note that the different methods have been consistently improved upon, increasing the flake size and consistency and the one hand and improving the material quality on the other hand.

Regardless of the fabrication method used, transferring the 2D materials onto the photonic chip is a critical step that requires careful handling to avoid damage or contamination. Direct transfer methods, such as wet transfer [START_REF] Chen | Progress and challenges in transfer of large-area graphene films[END_REF][START_REF] Gupta | A facile process for soakand-peel delamination of cvd graphene from substrates using water[END_REF], dry transfer [START_REF] Castellanos-Gomez | Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping[END_REF] , and semi-dry transfer [START_REF] Jung | Reduction of watermolecule-induced current-voltage hysteresis in graphene field effect transistor with semi-dry transfer using flexible supporter[END_REF], are commonly used in hybrid waveguide fabrication. The wet transfer involves fishing the 2D material in suspension in a solution and depositing it onto the substrate. The dry transfer involves depositing the material using a polymer stamp and will be the topic of the following section, while semi-dry transfer is a combination of the wet and dry transfer methods. Recently, attempts have been made to grow 2D materials directly on the chip using CVD [START_REF] Kuppadakkath | Direct growth of monolayer mos2 on nanostructured silicon waveguides[END_REF][START_REF] Liu | Silicon nitride waveguides with directly grown ws 2 for efficient secondharmonic generation[END_REF], eliminating the need for transfer methods. This approach could revolutionize the fabrication process by reducing the risk of contamination and improving the quality of the material. However, this method is still in its early stages and is not widely available. Therefore, the use of direct transfer methods remains the most commonly used approach for integrating 2D materials on photonic waveguides.

Mechanical exfoliation

ME is the historical [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF], the most widely available way to produce and transfer 2D materials. It consists in peeling off over and over layers of the crystal until reaching the ultimate thinness of the single layer. Because of the intrinsic properties of the bulk source materials (e.g., graphite, bulk M oS 2 ,...), the crystal breaks apart layer by layer, thus allowing the operator to thin the crystal down to a 2D material. The result of this process is a piece of tape covered with thin crystal layers. There are different methods requiring various types of polymers to perform the next step consists in transferring the material from the tape to the sample in a non-destructive way. This is indeed a delicate operation, as just a minor strain on the thin crystal layer can generate cracks and destroy the overall integrity of the sample. The method described down below is the more straight-forward, historical way to produce 2D materials. It was employed in the fabrication of several samples described in this chapter.

PDMS stamp method

The mechanical exfoliation method commonly employed throughout this chapter, is called "PDMS stamp" method. It is a standard process depicted in Fig. 2.5. This process, with both exfoliation and transfer steps, is ideally performed in a clean-room environment to avoid contamination. The full process can be broken down as follows: 1. A bulk crystal (e.g., graphite, bulk TMDs,...) is connected with a piece of tape.

The removal of the bulk piece of crystal from the tape results in some thick layers of material being peeled off from the main bulk piece of material due to the adhesive nature of the tape.

2. The piece of tape carrying thick layers of material is stuck against a second, piece of tape multiple times consecutively. This repeated pealing off both pieces of tape from each other effectively results in the exfoliation of the crystal layers, thus thinning them down. After several iterations, the remaining material present on the tapes is likely to include layers approaching a 2D crystal. This is done at the operator's discretion, and too few iterations can result in too thick material, while too many iterations can result in broken down small flakes.

3. A piece of Polydimethylsiloxane (PDMS) polymer is cut and stuck on a glass slide that serves as a support in the following steps. One of the two pieces of tape is put in contact with the PDMS and peeled back off, thus resulting in some of the material from the tape being exfoliated one last time onto the PDMS.

4. The PDMS stamp baring a multitude of flakes of various shapes, sizes and thicknesses, is put under an optical microscope to find a suitable flake for transfer. Under a bright field, the brightness of the flake, as well as its color can give an indication of the thickness. Moreover, cracks and splits can be identified, thus allowing for ruling some candidates out for the transfer.

5. Once a suitable candidate is identified, the PDMS can be cut around the areas where the suitable flake is located. This allows for limited unwanted deposits after transfer and facilitates an optimal contact between the target sample and the PDMS stamp by reducing the overall surface. However, we see will in a later section that unwanted deposits are inevitable using this kind of approach. as well as a heating plate allowing for control of the temperature, an essential parameter to transfer the 2D material successfully. Both stages are positioned under a microscope allowing for the alignment between the flake and the targeted deposition area. Control over the focus plane and the depth of field (thanks to a controllable aperture) allows for monitoring of the height of the flake and the sample to precisely perform the transfer. Once aligned, the PDMS stamp and the flake are put in contact, and the stamp is pressed down.

7. Additional heating is required for the PDMS to loose its adhesion to the 2D material and for the 2D material to bond with the targeted sample. After a few minutes, the stamp can be lifted very slowly while the transfer stage is being cooled down. At this point, the 2D material should stick to the targeted sample. Two additional steps are still necessary to guarantee the success and quality of the transfer. The sample is to be deposited on a hot plate for a few minutes to enhance bonds between the 2D material and the sample. The residual polymer may be washed away using an acetone bath. The sample can then be annealed in a furnace to remove all the OH liaisons that might be detrimental to optical performance.

Although relatively simple in theory, with a small number of steps to follow, this method presents several disadvantages that will be discussed in the conclusion of this chapter. However, it could be mentioned that one of the major issues with this method is the uncertainty around the success of the transfer. It is not always guaranteed that upon heating, the PDMS can release the 2D material, and thus, step 7 is bound to fail often. This additional degree of uncertainty makes this seemingly simple method very timeconsuming with a low success rate. There exist more complex methods to perform mechanical exfoliation with higher success rates that will be discussed in Chapter 3.

Characterization of deposited 2D materials.

Essential to the characterization of the hybrid waveguides is the identification of the 2D materials once the deposition process is over. Indeed, once deposited, optical microscopy might be limited to detect and identify the areas covered by 2D materials. The deposition surface being in the case of a waveguide uneven, the subtle variations introduced by the 2D material layers on the surface might not be visible in bright field microscopy. Moreover, identifying precisely the size and quality of the deposit is necessary for later steps in the characterization process (as will be discussed in the next section). To render a complete analysis of the sample, micro-spectroscopy over the deposition area offers a wide range of information. The principle of the microspectrometer, which was employed to that end, is represented in Fig. 2.7. In this tool, three different units are typically needed: (a) the microscope unit, (b) the laser unit and (c) the CCD unit.

The typical characterization of a sample goes as follows. A sample is deposited on the piezo stage, allowing for focus a location control for scanning and accurate measurements. A lamp allows bright field microscopy to align the microscope on the targeted area and perform a rough focus on the surface. Then, the probing wavelength is selected at the laser unit. Different wavelengths can trigger different responses depending on the material. For the purpose of interest, the source commonly used is a laser at 532 nm, as it allows for probing most of the 2D materials discussed in this manuscript (graphene, M oS 2 or W S 2 ). The source is then injected into the microscope and focused on the sample. There are two different responses from the material that can be leveraged to gather more information on the 2D material deposited: the Raman response and the photoluminescence.

Raman spectroscopy

The Raman spectroscopy originates from the eigen-vibrational states of the crystal. It is an inelastic scattering of the incident light interacting with the atomic or molecular vibrations of the probed material. The incident photons are raising the material's energy level to a virtual level and then scattered back after experiencing a shift in energy (up or down) corresponding to the energy of the vibrational mode's phonon. The resulting scattered frequency is thus characteristic from the eigen-vibrational modes of the probed material and thus can allow its identification as depicted in Table 2.2). The Raman energy shift translates into a frequency shift of the scattered light that can be expressed as:

ω S = ω -ω V (2.6)
,where ω S is the scattered frequency, ω in the probe frequency and ω V is the vibrational frequency of the probed material. The case of M oS 2 can serve as an example as it is emblematic of how this kind of measurement can bring a lot of information about the deposits. Figure 2.8 shows the case of TMD's vibrational states that can be leverage to identify the 2D material.

With a laser excitation at 532 nm wavelength, the observed characteristic peaks of the response spectrum are typically, the E 1 2g and the A 1g around 400cm -1 . Peaks for the other existing vibrational modes (E 1g and E 2 2g ) that are represented in Fig 2 .8 are typically not observed because of the selection rules for the scattered geometry [START_REF] Lee | Anomalous lattice vibrations of single-and few-layer mos2[END_REF] or due to the limited rejection of the Rayleigh scattered radiation. Additional information for the exact frequency of the scattered light can be analysed. The two main vibrational modes of interest, E 1 2g and A 1g can be identified and characterized. In TMDs, the intensity of the Raman response increases linearly up to four layers and then decreases for thicker samples, which can provide additional insight while analysing the results of a Raman mapping measurement. This can be seen in Fig. 2.9 a. with the Raman mapping of a mechanically exfoliated flake. The mapping represented here is the result of the integration of both characteristic peaks. It can be seen from this image that a stronger intensity is localized in the middle of the flake, thus suggesting thicker material. This can be confirmed or nuanced by additional measurements such as the characterization of the peaks positions. Indeed, Fig. 2.9 c. in the inset and Fig. 2.9 d. show the E 1 2g peak experiencing a red shift while the A 1g experiences a blue shift as the number of layer increases. The peak position ends up ultimately, converging towards the peak values exhibited by a bulk M oS 2 sample (see Fig. 2.9 d.). This can provide additional insight regarding the topography of the flake. Indeed, besides informing on the sample's thickness, it is known that the position of these peaks is also affected by other factors such as strain or doping. This can be seen Fig. 2.9 c. in the inset where a pattern appears where standard Raman measurements based on the intensity and the bright field image were rendering a uniform area. This pattern identified from variations in the peak position probably originates from wrinkles in the flake. Relying on Raman spectroscopy only does not provide enough information, and other photo-emission mechanisms are needed.

Photoluminescence

The common complementary alternative measurement to Raman spectroscopy is the characterization of photoluminescence (PL) response of 2D materials [START_REF] Tonndorf | Photoluminescence emission and raman response of monolayer mos 2, mose 2, and wse 2[END_REF]. Similarily to Raman spectroscopy, it involves illuminating the material with light at a specific wavelength, and then measuring the resulting photo-emission from the material. The PL process is distinct from Raman emission as it is based on the actual energy levels form the material's band structure. Electrons in the valence band of the material are excited to the conduction band by the probing source, generating holes in the valence band. The electrons and holes bound together by Coulomb interactions form excitons. When an exciton recombines, the electron falls back into the hole, releasing energy in the form of light. The energy of the emitted light is similar to the energy gap between the conduction band and the valence band, which is characteristic of the material (as depicted in Table 2.2). As such, the PL emission spectrum provides information about the bandgap energy, excitonic transitions, and defect states in the material by measuring the intensity, position, and shape of the PL peaks.

To better grasp how PL can provide useful information, we can once again take TMDs as an example with the case of M oS 2 (Fig. 2.9 b. and e.). As can be observed in Fig. 2.9 e., in bulk M oS 2 , no PL can be measured. However, in the case of ultrathin lay- ers, the photolumiscence shows an opposite dependence on the number of layers. For a monolayer flake of material, the PL is strongest, contrary to what could be observed with Raman up to a few layers. It can be seen by comparing Fig. 2.9 a. and Fig. 2.9 b. in the red dot region. For the Raman measurement, the thin layer is barely visible, while for the PL (outside the wrinkle area we will discuss here after), the single layer exhibits the strongest photo-emission. This is directly related to the physical principal behind PL, where a higher intensity is characteristic of a higher quantum efficiency in the emission process. This is characteristic of TMDs which have an indirect bandgap in their bulk form. Has the number of layers is reduced, the band structure finds itself affected until a point where, for single layer, a direct bandgap appears [START_REF] Splendiani | Emerging photoluminescence in monolayer mos2[END_REF]. Yet, monolayer M oS 2 does not exhibit the strongest recorded PL in the example Fig. 2.9 e. It can be seen from Fig. 2.9 b. and e. around the green dot that the region we identified as wrinkles using Raman's peaks position exhibits the strongest PL .

Similarly to the peak position of the Raman spectrum, the resulting spectrum of a PL is indeed greatly affected by the surrounding, strain or doping of the flake. It is thus an important tool to unveil details and features that cannot be seen under a bright field microscope. Of course, the photo-excitation of the flake does not provide a complete picture, and the gathered information might need to be supplemented with other types of measurements (i.e., atomic force microscope (AFM),...). In the context of hybrid waveguides in terms of nonlinear properties of the 2D materials it is unknown what features can provide the best performances. Hence, identification of the exact location of the flakes, characterization of their thicknesses and overall control of the flake quality (no cracks, wrinkle,...), is important. Yet, the use of these characterization methods does not provide additional information on the performances, and actual measurements of the sample are needed. This aspect is developed in the next section.

Characterization of hybrid samples

With most of the basis now established, we describe here the experimental work that was conducted using the described approaches. This section is divided into two distinct subsections. In the first one, we report on the successful hybrid integration of a W S 2 flake on a SiN waveguide using all the aforementioned methodologies. We will then discuss other failed attempts and critically analyse the lessons drawn from these experiments.

2D Material

Hybrid sample and characterization

Sample description

The first more straight forward approach when attempting to demonstrate the benefit of hybrid integration is to use accessible methods to possibly reach the nonlinear optical effect expected enhancement. This was the aim of the work reported in section 2.2. The process of hybridization was performed on a SiN waveguide using standard mechanical exfoliation and direct transfer, as explicitly described in a previous section. Although no preliminary study on the optimal waveguide design was conducted due to the collaborative nature of this work (the waveguides were obtained through a partnership with an other university), the design of the chip was suitable for hybrid integration and optical pulse broadening experiments. Indeed, the 330 × 760nm SiN ridged-waveguide provided a dispersion profile, suitable for this type of experiments (see next section).

After transferring a few-layer W S 2 flake deposited with the common "PDMS stamp method", the sample was characterized with a scanning micro-Raman spectrometer using the set-up presented in the previous section. This was meant to gauge the quality and layout of the deposit and confirm if the sample was suitable for further examination and characterization. Hence, a laser at 532 nm was used to trigger a Raman response in the deposited 2D material. The mapping of the deposition region was performed with a spatial resolution of 500 nm. This mapping was the result of a summation of the characteristic Raman spectral peaks. The first peak was taken from ∼ 348 to 358cm -1 (the integration area is identified in blue in Fig. 2.11 b.) and the second peak from ∼ 417 to 424cm -1 [START_REF] Barbosa | Straightforward identification of monolayer ws2 structures by raman spectroscopy[END_REF](the integration area is identified in yellow in Fig. 2.11 b.). These correspond to the vibrational states discussed in the previous section, which are respectively the E 1 2g and the A 1g peaks (similar to the case of M oS 2 ). This spectral mapping can be seen in Fig. 2.11 c. and several characteristics can be identified here. For instance, as the case for M oS 2 up to a few layers, it is visible that the thicker the flake, the more intense the Raman signal.

In addition to the Raman data, atomic force microscope (AFM) was used for thickness measurements (Fig. 2.11 d.). The obtained data together with that of the Raman mapping were put together to identify a thickness variation profile on the surface of the sample. Different regions were identified (Fig. 2.11 d.), such as a ∼ 3.3nm thick with a ∼ 6.5µm overlap, a ∼ 7.9nm thick with a ∼ 6.3µm overlap, and a ∼ 5.6nm thick with a ∼ 2µm overlap where measured. From this data, an average thickness of ∼ 5.57nm was calculated. Thanks to this calculation, the approximation of a uniform flake over the interaction length ∼ 14.8µm could be used for simplification in the model, as will be discussed in the part relating to simulation.

Passive measurements

The setup used to characterize the waveguide is shown in Fig. 2.12. A mode-locked Ti:sapphire laser with a repetition rate of around 84.5 MHz and a pulse duration of approximately 100 fs was used as the source, with a maximum average power of 300 mW. The pump power was controlled by a variable neutral density filter. To inject the pump, it was directed through a dichroic mirror (a long-pass filter with a cut-off wavelength of 750 nm) and a couple of gold mirrors before being focused into the waveguides using an aspheric lens with a focal length of 1.45 mm. A continuous-wave He-Ne laser emitting at 633 nm was also used to aid the waveguide coupling procedure and to measure passive waveguide losses.

The linear propagation losses of the waveguide were characterized using several spiral waveguides on the same chip, each having a different total length. It was estimated to be around 1.12 dB/mm. Coupling losses were also calculated and estimated to be in the order of 6 dB per facet. Butt-coupling the source through a cleaved facet of the waveguide resulted in important losses due to mode mismatch and scattering. An estimation of the additional losses introduced by the W S 2 flake was also retrieved, with ∼ 0.91 and 0.02dB measured at 633 and 800 nm, respectively. Taking into account the overlap length of the flake, this leads to an additional propagation loss of ∼ 1.35dB/mm at 800 nm. Additionally, higher losses at 633 nm can be expected due to a high absorption coefficient at that wavelength, originating from the nature of the material's band gap. This is visible in Fig. 2.1 b (blue curve) with the real component of the surface conductivity, which relates to absorption. The observed loss measurements are consistent with expected outcomes and compare favorably to former reports of additional propagation losses due to the integration of 2D materials in waveguide structures. For instance, hybrid graphene waveguides have shown results such as 2.05dB/mm in a graphene oxide/silicon hybrid waveguide [START_REF] Zhang | Enhanced fourwave mixing with mos2 on a silicon waveguide[END_REF] and 52dB/mm in a graphene/silicon hybrid waveguide [START_REF] Ishizawa | Optical nonlinearity enhancement with graphene-decorated silicon waveguides[END_REF]. This is to be expected, as the gapless nature of graphene is known for introducing high absorption. Working with alternative materials, such as TMDs, thus provides advantages in reducing additional propagation losses.

Nonlinear characterization

A pulse broadening experiment was chosen as the best candidate to characterize the waveguide effective nonlinear properties. It consisted in injecting a short yet powerful pulse of light. Due to the intrinsic nonlinear properties of the propagating medium (here a SiN waveguide), the injected pulse experiences spectral broadening attributed to the Kerr effect (see chapter 1). Interestingly enough, this broadening can be witnessed when looking at the top microscope pictures (Fig. 2.12). Indeed, at the input, the light scattered out of the waveguide can be seen as purple and corresponds to the 800 nm wavelength as recorded by the camera. As we look further along the waveguide, light shifts rapidly to orange/red color. This is directly related to the spectral broadening as part of the spectrum is now covering visible light towards the red. The measurement principle was to record the output spectrum at various pump power levels. The resulting spectrum is matched to simulated results from the integration of the GNLSE where the fitting parameter was the waveguide effective nonlinear coefficient γ. Nonlinear characterization of the sample was thus performed in two steps. First the waveguide was characterized without the integration of W S 2 . By controlling the orientation of a half-wave plate, the TE mode of the waveguide was excited. As a preliminary precaution, characterization of the TPA was also performed. This also later comes as a key parameter in the fitting of the results. As mentioned in Chapter 1, the different carrier effects have important consequences on the shape of the output spectra. Therefore, an accurate estimation of this parameter is required. To do so, the peak power of of the injected pumped was gradually ramped up.

For the waveguide in this experiment with a width of 700 nm, the overall input-tooutput transmission was measured from 1 mW to 30 mW of average pump power. The inverse transmission (1/T) was plotted against the injected peak power. The values of T were estimated as the measured total output power divided by the injected pump power. From the experimental results, we could retrieve the TPA coefficient β T P A according to the method reported in Ref. [START_REF] Hk Tsang | Two-photon absorption and self-phase modulation in ingaasp/inp multi-quantum-well waveguides[END_REF] using the following expression: where P in and P out are input and output powers, α is the linear absorption, A ef f is the effective area of waveguide mode and L ef f is the effective length:

L ef f = (1 -e -αL ) α (2.8)
The results of the measurements carried out at 800 nm on the SiN waveguide revealed a TPA coefficient of 2.9 × 10 -8 cm/GW for SiN. As expected, this value is quite small since the photon energy at 800 nm is only 1.55 eV, which is well below half of the energy bandgap of amorphous SiN that ranges between 4.6 to 5.2 eV. To investigate the Kerr effect, a pulse broadening experiment was then conducted with the characterization of the full width at half maximum (FWHM). Initially, the pump was measured with a FWHM of about 13 nm (see Fig. 2.14). After the pump propagated through the 4 mm long waveguide, the FWHM was measured again and a significant increase with a FWHM at 70,9 nm was observed at an injected peak pump power of around 1,35 kW (equivalent to about 123,6 nm at -20 dB level). Upon increasing the pump power further, a bigger increase in the pulse broadening was observed, which supported the hypothesis of the nonlinear origin of the broadening. As such, increasing the injected peak pump power to about 2,7 kW, affected the FWHM of the output pulses with a FWHM of about 97,9 nm (equivalent to about 163,5 nm at -20 dB level).

The same waveguide was then selected for the hybrid integration of a 2D material. After the deposition of the flake onto the waveguide, the hybrid W S 2 -SiN waveguide produced a much broader spectrum under the same excitation conditions. A comparison between the spectra before and after the W S 2 integration was made at two different pump power levels, which is depicted in Fig. 2.15 a. and b. At a pump peak power of around 1,35 kW, the FWHM of the output pulses was measured to be around 105.5 nm (equivalent to about 177,3 nm at -20 dB level). On the other hand, at around 2,7 kW of peak power, the spectrum was measured with a FWHM bandwidth of about 133,5 nm, spanning from around 730,0 to 863,5 nm. This covered a bandwidth of about 228,1 nm at -20 dB level, ranging from around 672,7 to 900,8 nm.

The addition of the W S 2 flake to the SiN waveguide results in a significant increase in the spectral bandwidth of the output pulses. At a pump power of ∼ 1, 35kW , the FWHM of the output pulses increased by ∼ 48, 8%, while at a pump power of ∼ 2, 7kW , the increase was ∼ 36, 4%. This translates to a maximum pulse broadening factor of ∼ 17, 5, where the 13 nm wide input pulses broaden to ∼ 228nm in the hybrid W S 2 -SiN waveguide of 4 mm total length. This is a noteworthy improvement compared to previously demonstrated on-chip pulse broadening compression results [START_REF] Dawn Th Tan | Nonlinear photonic waveguides for on-chip optical pulse compression[END_REF].

Result fitting and analysis

To quantify this improvement and attempt to estimate the nonlinear properties of the 2D materials, numerical tools described in Chapter 1 are now necessary. The first step is to calculate a numerical approximation of the waveguide's passive properties. It is essential to retrieve a dispersion profile and a transverse mode distribution to integrate these parameters in later calculations. As such, FDE calculation for a SiN raised-strip waveguide with a cross-section of h × w = 330 × 760nm was performed. The dispersion profile (Fig. 2.10 b.) can be directly re-injected as a parameter in the integration of the GNLSE equation, which aims at fitting as accurately as possible the obtained experimental output spectrum. To achieve that, every parameter has to be considered in the most accurate way possible. Intrinsic features of the experiment (length, pulse duration, peak power, pulse shape, wavelength, linear losses but also the calculated dispersion profile) are thus set to the appropriate values. The input pump pulse is measured at the output of the laser and fitted with a hyperbolic secant to get similar conditions in simulation (Fig. 2

.14):

A 0 = P 0 sech t T 0 (2.9)
, where P 0 is the pump peak power, T 0 is the pulse duration and sech(x) = 2 e x +e -x . The model also takes into account the variations of the waveguide's properties along the propagation. A section of the simulated waveguide, for an equivalent of the measured dimension of the W S 2 flake (with a length of ∼ 14.8µm and a thickness of ∼ 5.57nm) was set at its location along the waveguide (roughly 0.5 mm from the input facet). The change of properties at this location can thus be implemented in the model. The output pulse spectra can be estimated with respect to the measured results at two different peak powers to guarantee more accurate results. To calibrate the model and verify all the input parameters deliver accurate results, this fitting process is first completed for results recorded from the uncladded waveguide. The fitting parameters, n 2 and β T P A are modified until reaching a match between experimental and numerical spectra. This preliminary fitting can be seen in Fig. 2.15 a. and b. in red for the two distinct pump powers. The Kerr refractive index and the TPA coefficient of the SiN were retrieved by this approach and the calculated values were found to be consistent with the literature, 3.3.10 -20 m 2 .W -1 and 6.6.10 -15 m.W -1 for n 2 and β T P A respectively, thus supporting the validity of the fitting process. This first step is key, as it allows retrieving the experimental effective nonlinear coefficients of the base waveguide.

The second step is to reiterate the fitting process for the hybrid waveguide. Results of this second fitting process can also be seen in Fig. 2.15 in blue, where once again, a convincing fit was achieved. Figure 2.15 c-g depicts the false-color maps that showcase the evolution of the spectrum of the pulses travelling along the nonlinear waveguides. Notably, the impact of the W S 2 flake in enhancing the effect of the nonlinear waveguides is clearly evident. Despite the small flake length of only 14.8µm, which is positioned at z = 0.5 mm from the input, a significant amplification is observed. A zoomed-in map is also included in the figure, highlighting the rapid enlargement caused by the presence of the W S 2 flake. The hybrid section in the waveguide exhibits a high nonlinearity that largely anticipates the saturation of the nonlinear broadening effect. This outcome is not observed in the pristine SiN waveguide with a length of 4 mm. Therefore, the addition of a small W S 2 flake induces large and rapid nonlinear effects, which ultimately reduces the propagation length necessary to achieve a particular outcome relying on a nonlinear process. This demonstration highlights the potential of 2D materials in significantly enhancing nonlinear processes in waveguides. With the addition of small flakes, large and rapid nonlinear effects can be induced, leading to a reduction in the length of propagation required to achieve a desired outcome.

The analysis of the hybrid waveguide's nonlinear coefficient produced two different outcomes depending on the peak power of the pump. At a pump peak power of 1.35 kW, the real part of the effective nonlinear coefficient, γ hybrid , was approximately 650 W -1 m -1 . However, at a pump peak power of 2.7 kW, the coefficient was slightly lower at around 600 W -1 m -1 . This difference in the results is most likely due to the limitation of the broadening effect caused by the strong linear absorption in W S 2 at wavelengths close to 600 nm. This illustrates the limitations of the pulse-broadening experiment and highlights the importance of taking several measurements at different pump powers to obtain accurate results. Moreover, the analysis also successfully retrieved the two-photon absorption (TPA) coefficient, which is the imaginary part of the effective nonlinear coefficient, for both power levels. The uncladded SiN waveguide had a TPA coefficient of 6.6 × 10 -15 m.W -1 , while the hybrid waveguide had a TPA coefficient of 6.9 × 10 -13 m.W -1 . The fitting process revealed a remarkable improvement in the nonlinear performance of the hybrid waveguide compared to the uncladded waveguide, which had a nonlinear coefficient of approximately 1.75W -1 m -1 . The hybrid waveguide exhibited enhancement factors, of about 371.4 and 342.9 for input pump powers of 1.35 and 2.7 kW, respectively.

The Kerr coefficient of the few-layer W S 2 flake was retrieved using the waveguide mode overlap model and the waveguide effective susceptibility model, as described by Eq. 1.29 in Chapter 1. The retrieved Kerr coefficient, under the condition of 1.35 kW pump power, was estimated to be ∼ 2.18 × 10 -15 m 2 W -1 at 800 nm, while at 2.7 kW pump power, the estimated value was slightly lower at ∼ 2.01 × 10 -15 m 2 W -1 . These values are consistent with the experimental results of ∼ 1.15 × 10 -15 m 2 W -1 measured at 532 nm using a vectorial two-wave mixing method. [START_REF] Torres-Torres | Third order nonlinear optical response exhibited by mono-and few-layers of ws2[END_REF] Additionally, the TPA coefficient for W S 2 providing the best fitting was estimated to be ∼ 158cm/GW , which is in agreement with the experimentally measured values of ∼ 525 ± 205cm/GW reported in ref [START_REF] Dawn Th Tan | Nonlinear photonic waveguides for on-chip optical pulse compression[END_REF].

The effective third-order nonlinear coefficient of the hybrid waveguide was estimated to be approximately 1.7 at 1.35 kW pump power, by retrieving the real and imaginary parts of the coefficient. The definition used for the nonlinear figure-of-merit is given in a previous reference. Table 1 provides a comparison of the results from this experiment with previously reported results of 2D material hybrid waveguide structures. Graphene-Si hybrid structures have currently demonstrated the largest nonlinear coefficients in channel waveguides, exceeding 1000m 2 W -1 , as reported in Refs. [START_REF] Yonezu | Evaluation of graphene optical nonlinearity with photon-pair generation in graphene-on-silicon waveguides[END_REF] and [START_REF] Alexander | Electrically tunable optical nonlinearities in graphenecovered sin waveguides characterized by four-wave mixing[END_REF]. However, graphene also introduces high propagation losses, whereas graphene oxide (GO), as a dielectric material, does not absorb linearly and exhibits a large Kerr coefficient, a few orders of magnitude higher than Si, as reported in Ref. [START_REF] Wu | Graphene oxide for integrated photonics and flat optics[END_REF]. Not only transition metal dichalcogenides, such as M oS 2 and W S 2 , offer Kerr coefficients higher than GO, but, they provide better versatility, such as access to second-order nonlinearity with their monolayers and complex optoelectronic devices due to their semiconducting nature. The optimization of the waveguide dispersion profile and material properties in the W S 2 -SiN hybrid waveguide in this work has resulted in much more significant pulse broadening effects compared to those in the M oS 2 -Si hybrid waveguide previously reported (spectral bandwidth of approximately 2.2 nm at -20 dB level and a difference of 10% between the hybrid waveguide and the pristine waveguide), as stated in Ref. [START_REF] Liu | Enhanced optical kerr nonlinearity of mos 2 on silicon waveguides[END_REF]. The effective nonlinear coefficients of the W S 2 -SiN structure in this work, which are approximately 600m -1 W -1 , also reach the level achieved with GO-Si SiN structures (γ ef f = 167m -1 W -1 ), as reported in Ref. [START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF]. In summary, the study has successfully demonstrated the significant enhancement of nonlinear pulse broadening in a SiN waveguide through the integration of few-layer W S 2 . The enhancement in the waveguide Kerr nonlinearity was characterized using a mode-overlap model. We observed a remarkable increase in the effective third-order nonlinear coefficient, exceeding a factor of 300 compared to the pristine SiN waveguide. This enhancement opens up immediate applications in on-chip supercontinuum generation (SCG) and pulse compression. By employing a mode overlap and waveguide effective susceptibility model, the Kerr coefficient of few-layer W S 2 at 800 nm can be determined to be approximately 2.18 × 10 -15 m 2 W -1 . The hybrid integration method utilized in this experiment, along with the detailed numerical model for the waveguide, and the potential for a fully fiber-based alignment-free experimental setup, offer new opportunities for characterizing the linear and nonlinear optical responses of low-dimensional materials and their integration into on-chip nonlinear optical circuits.

λ (nm) γ (W -1 .m -1 ) n 2 (m 2 .W -1 ) β T P A (m.W -
One notable advantage of our heterogeneous integration method is that it eliminates the need for moving parts or free-space alignment, making it more stable and efficient compared to conventional z-scan methods. Additionally, by exciting the waveguide's transverse magnetic mode, we can access the out-of-plane components of the optical properties, which is not feasible in the z-scan method at normal incidence. Overall, the experiment demonstrated the great potential of hybrid waveguides for achieving enhanced nonlinear optical functionalities and pave the way for future advancements in on-chip nonlinear photonics utilizing low-dimensional materials.

Other first generation prepared by mechanical exfoliation samples and their performances

The experimentation involving mechanical exfoliation (ME) flakes on SiN waveguides underwent multiple iterations. The most convincing samples are depicted in Fig. 2.16, and some of the results obtained from these samples will be discussed in this section and in Chapter 3, section 3. The outcomes of these experiments provide useful lessons and guidelines for improvements. However, there were some major issues encountered with ME flakes in the context of hybrid integration on a waveguide, which are discussed below:

1. The size of the flake was a significant issue. Regardless of the exfoliation method used, it was challenging to obtain flakes larger than 20µm. Although designing the waveguide layout, such as in the spiral example, could allow for higher interaction length, the flake's size still limits the effectiveness of the approach.

2. While hybrid waveguides aiming to utilize third-order nonlinear effects do not require a strictly uniform monolayer of 2D materials, having even and controllable layers of 2D materials is advantageous. It makes the fitting process and assessment of the overlap between the transverse mode and the 2D material easier. Additionally, having an even surface is beneficial should further steps in the fabrication process be necessary. As discussed in the next chapter, adding additional cladding on top of the 2D material can improve performance, which is more feasible if the additional material deposit is performed on an even layer of materials. However, this is not possible with ME flakes, as they have inconsistent thickness.

3. One major issue with ME flakes is their uneven nature, which poses a challenge in managing the usable flake surface in relation to the waveguide's path. Since bulk areas should be avoided to prevent significant alterations to the waveguide's properties, the possible angles and layouts for placing the 2D material become restricted. For instance, in the flake shown in Fig. 2.9, the bright triangular section at the bottom is characteristic of a thick piece of the material, which is probably thick enough to be considered bulk (as confirmed by Raman and PL measurements). Therefore, aligning the waveguide with the flake's longest side is not possible or at least more challenging, as the bulk section blocks the way. This is a recurrent issue when attempting ME and direct transfer for fabricating hybrid waveguides.

4. From a broader perspective, the randomness and lack of control over the density of flakes in the surroundings of the targeted piece of crystal in ME pose a major problem for hybrid integration on waveguides. In the spiral example discussed here after, the deposition of the target flake was successful, but it also came with additional bulk flakes deposited on the surface at unwanted locations. These unwanted pieces of materials were left on the PDMS stamp during phase 3 of the exfoliation process (see Fig. 2.5) and subsequently deposited together with the flake of interest. Although it is not possible to fully fix this issue, the density of flakes on the stamp depends on the exfoliation cycle (steps 1 to 3 in Fig. 2.5) and can be optimized with proper training (as it is heavily operator-dependent). However, it should be noted that the lower the density, the lower the probability of finding a suitable candidate. Sometimes, interesting flakes appear in clusters, making it challenging to isolate a single good sample in an isolated location on the polymer.

5. As mentioned earlier in section 2.2.1, the release of 2D materials from the PDMS stamp during the fabrication process (stage 7 of Fig. 2.5) is a random and unreliable step. This leads to a low success rate, thereby reducing the reproducibility and efficiency of the process. However, some novel methods incorporating new polymers may provide a more promising solution to this problem, and these methods are discussed in Chapter 3.

6. Related to the question of unreliable release of the 2D materials, another significant disadvantage of ME materials is the lack of repeatability in sample fabrication. In addition to that latter point, the intrinsic randomness of the exfoliation process and the heavy dependence on operator skill makes it almost impossible to produce identical samples in subsequent experiments. While attempts can be made to reproduce the samples as closely as possible, the constraints imposed by the above issues lead to a very low yield of good samples. Consequently, it is extremely time-consuming to obtain a usable sample, and even then, unexpected issues may arise, such as the need to clean the sample with plasma, ultimately destroying some of its potential (as mentioned here after). Yet, a multitude of samples were fabricated using the ME approach as it is a rather simple step into the field of 2D materials. Indeed, despite all the disadvantages, it remains a really accessible process to use as a starting point in the study of hybrid waveguides. Some of these samples are presented and discussed in the following.

Sample description

The decision to repeat the W S 2 -SiN experiment with a different TMD presented an opportunity to not only compare different materials but also refine the approach. In this experiment, waveguides with slightly different parameters were chosen to potentially increase the success rate. The first chosen feature was the "spiral" layout (see Fig. 2.16 d. and 2.17), which offered a promising deposition area for the 2D material. The deposited flake covered the spiral on several round trips, increasing the interaction length. The second feature was the structure of the waveguide, which in this case was a planarized buried strip waveguide that allowed for a smooth deposition on the waveguide. This was intended to increase the transferability of the flake, as it was less likely to break due to the even surface.

For sample preparation, mechanical exfoliation of the 2D material was once again performed, followed by "direct" transfer. However, this time around, the necessity to target a more specific region of the waveguide introduced additional complexity. As shown in Fig. 2.17 a., the transfer introduced an unwanted bulk flake on the surface of the sample. While this was without consequence if the bulk flake landed outside of the waveguide's path, it became an issue when a bulk flake was visible on the waveguide. This not only rendered the process of isolating the properties of the hybrid waveguide difficult during characterization, but also affected the overall quality of the sample. TMDs have high refractive indices, and a thick layer of that material can have detrimental effects on losses and dispersion. For a high number of layers, a significant shift in the transverse mode can cause coupling losses along the propagation. Local perturbations to the dispersion profile are also expected to be introduced, making the fitting process difficult.

To address this issue, a process was implemented to characterize the sample. The desired flake was masked with Hydrogen silsesquioxane (HSQ) to protect it, while the rest of the sample was cleaned using O 2 plasma. O 2 plasma was reported as an effective etchant for TMDs or graphene [START_REF] He | Etching techniques in 2d materials[END_REF], making it a suitable choice to etch the undesired bulk flake off the surface of the sample. Comparison between the sample before and after the cleaning process showed the effectiveness of the method 2.17. This provided additional insight into the development of future processes and unlocked a new tool to facilitate experimentation with 2D materials. After cleaning, the sample could be characterized, and the results are discussed in the following section. Overall, the decision to repeat the experiment with a different TMD and waveguide structure, as well as the implementation of a new cleaning process, demonstrated the ongoing effort to refine and improve the approach for the integration of 2D materials with optical waveguides.

Results

The experiment revealed a high amount of losses in the hybrid sample, which could be attributed to multiple factors. While the introduction of the 2D material is known to cause losses, it may not be the sole cause. The addition of HSQ also contributed to significant losses, as observed through top-view microscope images that showed scattered light around the polymer square. It was however confirmed with additional loss measurement that extensive O 2 plasma etching is not responsible for any significant additional propagation losses along propagation in the SiN waveguide. Nevertheless, other properties of the 2D material could still be characterized under these conditions. For instance, it was observed that W S 2 exhibits a small amount of TPA around 800 nm. Hence, similar transmission measurements were performed using a setup presented in Fig. 2.18 a. and b. A pulsed source centred at 1550 nm with a pulse duration of 220 fs and a repetition rate of 1 MHz was injected into the sample, and the output was collected onto a power meter and a spectrometer. A comparison of the transmission measurement of the reference waveguide and the hybrid waveguide was presented in Fig. 2.18. No inflection can be seen in the curve, and the output power behaves linearly versus the input power, indicating that no measurable amount of TPA is being introduced with the addition of the thin layer of M oS 2 . Therefore, it can be inferred that characterization identical to the one performed in section 2.3.1 is not warranted. This is a critical result that confirms the absence of TPA in TMDs around 1.5µm, providing a strong argument in favor of using these materials for CMOS-compatible platforms.

Of all the examples presented in Fig. 2.16, only 2 yielded exploitable results. The characterization of the flakes revealed their small size, which limits their potential for use in practical applications. Furthermore, the fabrication and characterization of the flakes were complicated and time-consuming due to issues such as unwanted flakes. Although ME flakes represent a promising starting point for exploring hybrid integrated optics, their limitations make it difficult to justify further investment in their development. Nevertheless, the insights gained from this research may inform future efforts to improve the performance of 2D materials in optical devices.In light of these challenges, it is reasonable to consider alternative approaches for integrating 2D materials with optical waveguides. Chapter 3 discusses various strategies for improving the interaction between light and 2D materials, some of which can be applied to ME flakes, while others may require the development of new fabrication processes.

Chapter 3

Optimizing the interaction between 2D materials and the propagating light modes

Stemming from the results of chapter 2, we describe in this chapter different approches for the optimization of hybrid optical waveguides embedding 2D materials. The exceptional versatility of the deposition or growth techniques, as well as the recent progress in the transfer techniques, and the possibility to take benefit from optical resonance structures indeed open up a wide range of opportunities that we will report hereafter.

Optimizing the transverse spatial interaction

In Chapter 2, we introduce several different approaches for the hybrid integration of 2D materials with waveguides, each of which can result in varying levels of quality and consistency in the transfer process. However, in the case of the transfer of large-area CVD 2D materials, we can expect a consistent deposit of monolayer 2D materials that can serve as a foundation for more advanced manufacturing processes. Among these processes, layer deposition of materials using advanced technologies such as atomic layer deposition (ALD) has become a hot topic, particularly in the field of 2D materials [START_REF] Franssila | Introduction to microfabrication[END_REF][START_REF] Li | Atomic layer deposition of al2o3 directly on 2d materials for high-performance electronics[END_REF][START_REF] Gu | Atomic layer deposition on 2d materials[END_REF]. One application of ALD in the field of 2D materials is the deposition of an oxide layer such as Al 2 O 3 or Hf O 2 on top of the 2D material, which can provide several benefits. Firstly, it can act as a protective layer to prevent the thin layer crystal from degrading over time when exposed to a normal atmosphere. This is particularly important for some 2D materials prone to oxidizing and losing their performance over time. By encapsulating the thin layer crystal with a protective layer, such as a polymer or an other type of material, we can prevent this degradation from occurring.Moreover, the oxide layer can also enhance the properties related to the 2D material. Research has shown interesting results for the direct growth of oxides on 2D materials using ALD [START_REF] Li | Atomic layer deposition of al2o3 directly on 2d materials for high-performance electronics[END_REF][START_REF] Gu | Atomic layer deposition on 2d materials[END_REF]. In addition to the protective function, oxides such as Al 2 O 3 or Hf O 2 have suitable properties for waveguiding structures. Their linear refractive indices of 1.7 and 2.07, respectively around 1550 nm wavelength, making them ideal candidates for integration with the SiN platform. Therefore, the integration of ALD with 2D materials has significant potential for the development of hybrid waveguides. In this section, we will explore the benefits of these technologies in the field of hybrid waveguides, as well as investigate the possibilities that would arise from a more mature and efficient growth method of oxides on 2D materials.

There is another benefit related to the relatively high refractive indices of the top cladding materials. Indeed, adding a top cladding with a refractive index superior to 1 can modify the waveguide's passive properties significantly. The example presented in Fig. 3.1 shows a h × w = 600 × 1000nm planarized burried strip SiN waveguide with a monolayer sheet of M oS 2 deposited on top. The comparison between the structure without a top cladding and the structure with a 200 nm thick layer of Hf O 2 added on top shows a significant modification to the waveguide's properties. Calculating the GVD profile for both waveguides between 1.0µm and 2.0µm results in a significant shift of the ZDW from ∼ 1.45µm to ∼ 1.59µm thus expanding the range where nonlinear optical functions requiring an anomalous regime can be generated. Another important property of the waveguide being altered by structural modifications is the transverse distribution of the propagating TE mode. The relatively high refractive index of the top cladding introduces an upward pull on the TE mode and shifts the mode distribution center of mass closer to the 2D material. This can be seen in Fig. 3.1 in the insets. It is difficult to gauge how these changes can affect the propagation losses, however it certainly does benefit to the light interaction with the 2D material. Indeed, the spatial migration of the TE mode center of mass is bound to increase the overlap between the propagating mode and the 2D material. We already discussed in chapter 1 and chapter 2 how the interaction between the 2D material and the propagating mode could be characterized (see Eq. 1.36) and how this parameter and the effective nonlinear coefficient are related to each others (see Eq. 1.29). As such, in the case of the Fig. 3.1 example, the spatial overlap factor of the guided quasi-TE mode increases from 0.03% in the absence of Hf O 2 cladding, to 0.10% with a 200 nm top cladding of Hf O 2 . This is thus causing an improvement of the nonlinear coefficient from 2, 5m -1 .W -1 to 5, 0m -1 .W -1 which represents a 100% improvement.

To go further in the analysis of the benefit of this updated design and to assess the appropriate optogeometrical parameters, the improvement of γ has been more thoroughly studied with various cross sections. In this study, e.g. as a function of the design dimensions (width and height of the waveguide). The results are depicted in Fig. 3.2 with both the height and the width effects on the waveguide's nonlinear performance represented in a false color map. In Fig. 3.2 a., the base layout of the structure is represented. Then Fig. 3.2 b. provides the calculated nonlinear coefficient. It is clear here that similarily to what was previously observed, tighter cross-sections result in a more efficient nonlinear response of the waveguide as the density of the TE mode surrounding the nonlinear material is higher (due to higher confinement of the mode). The parameters calculated and represented in Fig. 3.2 c. and d. are the improvement of the structure of Fig. 3.2 a. over a bare SiN planarized waveguide (no cladding, no 2D material), and the overlap between the 2D material and the propagating TE mode. Significant improvement is to be noted with up to 220% improvement compared to the bare structure. This is more than 3,5 times higher improvement than for the 2D material only. In the same settings an identical structure with no Hf O 2 top cladding and a top M oS 2 monolayer exhibit a maximum of 60% improvement for the standard hybrid waveguide over the bare SiN waveguide.

Building upon the realization that increased interaction between highly nonlinear materials and the propagating mode can lead to a massive increase in the effective nonlinear response, additional steps can be taken to design highly nonlinear devices. In this regard, the main objective of the following waveguide design is to integrate a variable number of M oS 2 monolayers while ensuring the conservation of the 2D material's unique features. To ensure a realistic technological approach, the same planarized SiN waveguides used in previous studies are considered. An important aspect of this study is also to maintain the M oS 2 monolayer interesting feature of a large material bandgap and thus the quasi absence of TPA [START_REF] Lan | First-principles study of the excitonic effect on two-photon absorption of semiconductors: Theory and application to mo s 2 and w s 2 monolayers[END_REF][START_REF] Hee | Mos2 nanosheet phototransistors with thickness-modulated optical energy gap[END_REF].

Adding 2D materials on top of each others is obviously the more straightforward way to increase the quantity of 2D nonlinear layers. It is possible and generates Van der Waals crystals (or heterostructures) with interesting properties [START_REF] Gius Uddin | Engineering the dipole orientation and symmetry breaking with mixeddimensional heterostructures[END_REF][START_REF] Hao Chen | Transition-metal dichalcogenides heterostructure saturable absorbers for ultrafast photonics[END_REF]. However, the properties of these structures tend to diverge from the ones favoured in monolayer 2D materials, making them unsuitable for the targeted nonlinear applications. To overcome this issue, we propose the alternation of TMD monolayers separated by Al 2 O 3 b. The studied parameter is the effective nonlinear coefficient γ. c. The structure is compared with a reference uncladded burried strip SiN waveguide. As such the improvement from the reference to the structure presented here (planarized buried strip SiN waveguide topped with 2 monolayers of M oS 2 ) is calculated. d. The overlap between the 2D material TE mode is also calculated. Every parameter is calculated for a wide range of cross-section dimensions and represented in a false color map.

spacers of a few nanometers. A 5 nm spacer is considered as a compromise between a safe isolation of single M oS 2 mono-sheets and a reasonably thin spacer to minimize the possible thickness of the entire material stack.

The proposed approach is compatible with the growth of all materials (TMD, Al 2 O 3 spacers, Hf O 2 ) through ALD. It is important to note that the presented structure serves as a general framework that can be further optimized and tailored to specific requirements. For instance, materials such as BN can be used as alternative insulation layers instead of Al 2 O 3 and Hf O 2 , depending on the specific needs of the application. To illustrate the feasibility of the approach, the study presents results for a M oS 2 bilayer stack on top of a SiN planarized waveguide, which was previously considered in Figures 3.1 and 3.2. In this case, a 200 nm Hf O 2 top cladding is applied to guide the optical mode upwards and enhance the spatial overlap with the M oS 2 monolayers (see Fig. 3.3). In Figure 3.3, we observe the nonlinear coefficient γ calculated with equation 1.33 for varying cross-section dimensions, showing a five-fold increase as a function of the cross-section dimensions. The comparison of these two figures suggests that the increase in γ is strongly related to the mode/TMD layer overlap. Based on this assessment, further investigations can be conducted. Recent reports of direct growth of 2D materials on waveguiding structures have emerged, indicating that more advanced manufacturing methods can be expected to mature in the next few years, expanding the range of possible designs. This is also supported by the convincing feasibility elements related to the possibility of stacking a series of 2D material sheets [START_REF] Jia | Large third-order optical kerr nonlinearity in nanometer-thick pdse2 2d dichalcogenide films: implications for nonlinear photonic devices[END_REF]. It can be inferred from the previous design that the number of layers can be increased, further enhancing the interaction with the 2D materials and boosting the nonlinear performances of the waveguiding structure. To explore this possibility, the nonlinear coefficient was calculated in two scenarios with increasing numbers of layers (up to 55). These results can inform the design of future nonlinear photonic devices that exploit the nonlinear properties of 2D materials.

In Fig. 3.4, we present the results obtained for two different scenarios where the number of M oS 2 layers in the SiN/M oS 2 stack is increased up to 55. In the first case, similar to the design in Fig. 3.3, we introduce additional layers and cap the SiN/M oS 2 stack with a 200 nm layer of Hf O 2 to enhance the density of the mode around the 2D material. This technique effectively boosts the effective nonlinear coefficient γ as we observe an increase of about two orders of magnitude for the maximum value compared to a standard SiN planarized buried strip. However, we also observe that the improvement in γ saturates above a certain number of layers (around 40). This saturation occurs because the successive addition of Al 2 O 3 inter-layer buffers stretches the TE mode further, reducing its density around the 2D material layers. As the number of layers increases, the density of the mode around the 2D materials decreases, diminishing the benefit of each added layer. In the second case, we remove the top cladding layer, and observe that the increase in γ is initially much slower than in the first case because the TE mode does not migrate to the top of the 2D materials. However, as the number of added layers increases, the improvement in γ accelerates until it catches up with the results of the first case. The addition of a small amount of Al 2 O 3 inter-layer buffers for each layer generates a mode shift, which is also responsible for the improvement acceleration above 20 layers. However, because of the absence of cladding, the TE mode remains relatively dense, allowing this second design to outperform the first design above 50 layers.

In conclusion, it is important to balance all parameters when designing hybrid TMD-SiN waveguides for optimal performance. High mode density is crucial to achieve significant overlap and enhance the nonlinear coefficient γ. However, adjustments can be made to prevent excessive spreading of the transverse mode, such as using lower index materials like SiO 2 instead of Al 2 O 3 for the buffer layer. This can impact the coverage of the 2D material as the mode may not be shifted efficiently over the nonlinear region. Similarly, the thickness or material of the top cladding can also be modified to optimize the design. Ultimately, the trade-off must be found as mode spreading can be both an asset and a limitation in these designs. Finding the right equilibrium to guarantee the optimal interaction between the 2D material and the transverse mode is key. Despite these limitations, hybrid TMD-SiN waveguides have a significant potential for optimization, with nonlinear coefficients reaching values of over 100W -1 m -1 in both presented cases, approaching the nonlinear coefficients of silicon waveguides [START_REF] Salem | Signal regeneration using low-power four-wave mixing on silicon chip[END_REF], while offering avenues to minimize TPA. The development of such approaches shows great promise, but it heavily relies on the advancement of manufacturing methods. To realistically implement these approaches, significant improvements are still required. ALD growth on top of 2D materials is a relatively new topic and can experience significant development in the upcoming years. Additionally, the fabrication and transfer of 2D materials is also a key parameter in the realization of these types of designs. Being able to grow and transfer consistently large sheets of 2D material is required to make this possible. The development of new methods such as direct growth on the sample [START_REF] Kuppadakkath | Direct growth of monolayer mos2 on nanostructured silicon waveguides[END_REF][START_REF] Liu | Silicon nitride waveguides with directly grown ws 2 for efficient secondharmonic generation[END_REF] or the deterministic transfer of CVD-grown large-area 2D materials can provide solutions to this problem. In the next section, we propose to discuss our advances in the latter with our work on transfer methods.

Optimizing the longitudinal interaction, introduction of new transfer methods

The previous chapter highlighted the transfer method as a major factor for improvement in the integration of 2D materials on a chip, particularly for the fabrication of hybrid waveguides. However, the conventional method of mechanical exfoliation (ME) poses several challenges, such as flake size, flake consistency, and transfer reliability. These challenges can be addressed through the use of a new semi-dry transfer method that integrates CVD-grown 2D materials. The use of CVD materials aims to solve issues of size and consistency, while the transfer method will address process reliability. In this section, we will discuss the wet transfer method for CVD-grown materials, specifically for M oS 2 , which is an essential component of the semi-dry transfer method. We will also explore the introduction of new polymers such as Polypropylene carbonate (PPC), in the transfer and deposition of 2D materials, which solves the issue of reliability during the release of the 2D material during the transfer. Finally, we will combine these principles into a semi-dry transfer method that aims to provide reliable and deterministic transfers of large-area CVD-grown 2D materials. All these efforts aim at the optimization of the longitudinal interaction between light propagating modes and the 2D material sheets.

Transfer of CVD grown material: wet transfer

The transfer of CVD-grown 2D materials is typically accomplished using wet transfer methods, which involves the detachment of the material from its growth substrate and subsequent transfer onto a target surface. In this case, we will focus on the wet transfer of CVD-grown M oS 2 layers. The material is grown on a substrate made of either Si or Sapphire, and the primary challenge is to detach the material without causing any damage to its structural integrity, either from strain stress or chemical contamination. The wet transfer method has proven effective in achieving this goal, but it still requires careful attention to the specific transfer conditions to ensure reliable and repeatable results. Fig. 3.5 depicts the different steps of the wet transfer method as described down below: 1. a. Before transferring the 2D material, it is important to properly prepare the growth substrate to ensure a successful transfer. This includes cleaving the substrate into smaller pieces (typically between 5 mm square and one cm square), which not only facilitates the operation but also helps in sizing the amount of material necessary for the transfer. The substrate is then subjected to a quick rinse in acetone and Isopropyl alcohol (IPA) to remove any dust and impurities that may have accumulated during storage and the cleaving process. This step is crucial as any impurities left on the substrate can potentially contaminate the transferred material and affect its properties.

b. The next step is to apply a transfer layer on top of it. This is typically done using a polymer such as poly(methyl methacrylate) (PMMA). The PMMA is spincoated at 4000 rounds per minute for 1 min on top of the 2D material and then baked at 170 • C for 2 min. This is meant to prepare the transfer layer with an adequate thickness to be handled during the following steps. After the application of this new layer, the sides of the piece of the substrate are scraped with a knife to facilitate the next step.

2. The resulting PMMA/M oS 2 /Si stack is immersed in a potassium hydroxide (KOH) bath with a 45% concentration. The use of KOH is crucial as it selectively etches away the Si or sapphire substrate, allowing the M oS 2 to be detached and floated on the surface of the solution. The PMMA layer plays a vital role in this process as it provides mechanical support to the M oS 2 during the etching process. The PMMA layer also has enough buoyancy in KOH to keep the whole stack at an equilibrium on the surface.

3. The KOH solution slowly dissolves the Si substrate. Because the edges of the PMMA have been scraped away, the KOH can also dissolve the Si on the surface, allowing it to penetrate underneath the PMMA and 2D material. This is a crucial step in the transfer process, as it ultimately separates the 2D material from its original substrate. The KOH bath can also be kept at a temperature of around 60-80 • C, which helps to accelerate the Si dissolution and thus shorten the transfer time.

4. Because of the PMMA buoyancy, the 2D material and the polymer are thus lifted to the surface while the Si substrate sinks. This effectively facilitates the 2D material as it performs a delicate mechanical peeling off the substrate. The KOH eats away the smallest amount of Si and the 2D layers are then pulled away from each other very gently due to their buoyancy difference.

5. Once completely detached, the 2D material, attached to the PMMA, floats on the surface of the KOH solution while the substrate sinks to the bottom of the beaker.

6. The PMMA/M oS 2 stack can be picked up or "fished" using a bare substrate or a glass slide. Due to a small amount of the KOH solution remaining trapped between the 2D material and the glass slide, there is no immediate adhesion between the 2D material and the glass slide. This is important to avoid any potential damage to the transferred material during the lifting process.

7. The small amount of trapped KOH between the 2D material and the glass slide can be considered as contamination that needs to be removed before further use of the transferred material. To ensure the highest quality of the transferred material, several rinsing cycles are necessary to remove any unwanted chemicals. The rinsing process involves dropping the PMMA/M oS 2 stack into water baths for multiple cycles. This process is repeated anywhere between 2 to 5 times, depending on the desired cleanliness of the final deposited material. It is crucial to ensure that all traces of KOH and other contaminants are removed to avoid any adverse effects on the material's properties or performance.

8. The PMMA/M oS 2 stack is left floating in the bath of water to be picked up one final time. Similarily to the previous steps, this is usually done by "fishing" the PMMA/M oS 2 stack but the time the operator uses the targeted sample as the scoop. Compared to the "fishing" performed in the previous steps, a great care must be taken in draining most of the water off the sample to guarantee adhesion between the 2D material and the sample surface. Additionally, emphasis on the transfer region on the sample must be considered. Although extremely difficult due to the water surface tension to properly determine the exact transfer location, it can still be nudged in a general area of the targeted chip. Then the removal of all excess water can be ensured using a short annealing by putting the PMMA/M oS 2 /sample stack on a hot plate for a few minutes. 9. a. Accordingly to usual procedures, the samples can then be dropped in an acetone bath to get rid of most of the polymers. b.Additional annealing of the sample in a furnace is further required to get rid of the OH liaisons.

The "wet-transfer" method is commonly employed for transferring CVD-grown material and has proven to be effective in many cases. However, in some cases, such as in the production of hybrid waveguides, an accurate targeting of the deposition area is required. This can be challenging as 2D materials grown in CVD often form triangles that occupy only a portion of the substrate (as will be shown further in this section). Therefore, selecting the proper patch of 2D material and deterministically transferring it onto the waveguide is necessary but impossible with the basic "wet transfer" method. Thus, despite the large transferable areas allowed by this method (similar to the semidry transfer discussed in in its dedicated upcoming section), this method is not suitable for efficient targeted fabrication of hybrid waveguides. To refine the process and allow for the precise targeting of the deposition area, other methods such as the "PPC release" method need to be discussed.

Improvement of the dry transfer method: PPC release

The "PPC release" method is a process developed to circumvent the issues related to the unreliability of the direct PDMS stamp transfer method. It was stated in Chapter 2 that standard ME using a PDMS stamp is unreliable due to the difficulty for the PDMS to release the 2D material and for the 2D material to find proper adhesion with the targeted surface. The "PPC release" method aims at correcting that issue by introducing a new polymer and additional steps in the process. The whole process is depicted in Fig. 3.6 and can be detailed as follows:

1. A glass slide is first spin-coated with PPC. The parameters, (e.g., 2000 rounds per minute for 1 min) allow for the ideal uniform layer with a suitable thickness. The PPC is then annealed at 150 • C for 2 min. The side of the PPC layer is scraped off to allow for easier peel-off for a later step.

2. For the transfer of a mechanically exfoliated (ME) flake, the tape resulting from step 1 and 2 of the ME process presented in the previous Chapter (section 2.2.1) is carefully placed onto the PPC layer. The 2D material is then exfoliated once again onto the PPC/glass substrate in a manner similar to the PDMS stamp method. The resulting 2D/PPC/glass substrate is then inspected under an optical microscope to identify suitable areas for transfer.

3. The area of interest identified, tape can be placed all around it and stuck on the surface of the PPC. In the mean time, a PDMS stamp is prepared by placing a piece of the polymer on another glass slide.

4. The tape surrounding the targeted patch of 2D material can be gently peeled off, along with the PPC layer. Due to the lower adhesion between the PPC and glass, compared to that between the tape and PPC, the tape can be easily removed, resulting in the extraction of the PPC layer with the 2D material on top.

5. The extraction of the PPC with the 2D material allows for the transfer of the 2D material/PPC stack onto the PDMS/glass stack.

6. Using a scalpel, the excess polymer and tape around the flake can be carefully cut off, resulting in a smaller patch that limits the area of deposition and reduces the risk of contamination. This smaller patch also increases the effectiveness of the transfer process as it enables optimal contact between the stamp and the sample.

7. The remaining 2D material/PPC/PDMS/glass stack is set in the transfer stage described in the previous Chapter (fig. 2.6). The proper flake or patch of material is aligned with the targeted device and then the stamp is pressed down against the sample. The different steps of the "PPC release" transfer method. This method aims at performing the transfer of ME flakes with higher success rates than the basic ME direct transfer method.

8. Heating the sample to 60 • C facilitates the release of PDMS and PPC from each other, allowing the glass and PDMS to be peeled away and leaving the PPC and 2D material on the sample. This step is more reliable than in the case of the PDMS stamp method and much more effective due to the lower adhesion of PPC with PDMS at higher temperatures.

9. Additional heating of the sample is required to guarantee a better adhesion between the sample and the 2D materials. The PPC can then be dissolved in an acetone bath. Finally, annealing in a furnace is required for optimal performance. In this method the fact that the PPC can be released easily from the PDMS improves greatly the rate of success. However, it adds a few additional steps. There are also remaining issue that will be discussed in the conclusion of this chapter upon reviewing experimental results.

The PPC release method is a significant improvement over the initial ME transfer method. Although it involves additional steps and longer iterations, each iteration has a much higher success rate due to the almost guaranteed release of the PPC/2D material from the PDMS. This improved reliability leads to high yield for the process and reduces significantly the average time necessary to make a sample, ultimately resulting in better performance. This is an intermediate method that could be used for fabrication of hybrid waveguides. However several disadvantages of ME flakes are still present with this method and thus it was not extensively employed in the work presented in this manuscript.

Deterministic semi-dry transfer method

In this section, we will describe a semi-dry transfer method that combines the previously discussed transfer methods of "wet transfer" and "PPC release". The goal is to achieve effective and deterministic transfer of CVD-grown material. This method was primarily used for attempts to bring nonlinear response to the ion-exchange glass waveguide platform [START_REF] Broquin | Ion-exchanged integrated devices[END_REF][START_REF] Broquin | Glass integrated optics: state of the art and position toward other technologies[END_REF]. As these waveguides do not trigger any measurable nonlinear response, adding 2D material is an option to introduce nonlinearity to the platform. However, with no base nonlinearity (contrary to SiN or Si which have nonlinearities of their own), large length of interaction are required to generate nonlinear behaviour, hence the necessitate to use large area CVD grown materials. This process was also attempted as preliminary test on a SiN spiral.

Principle

The first specificity of the deterministic semi-dry transfer method has to do with modifications in the last steps of the wet transfer method. A specific "fishing" method is required as depicted in Fig. 3.7 and goes as follows:

1. After steps 1 to 7 of the "wet transfer" method, the PMMA/2D material finds itself in suspension in water.

2. The flake is "fished" out of the water using a glass slide or a piece of bare Si wafer. Some water remains trapped in between the PMMA and the glass preventing adhesion of the 2D material with the glass slide.

3. A PDMS/glass stack is turned upside down and brought delicately, towards the PDMS. Upon contact adhesion between the two polymers, PPC and PDMS is created.

4. The all stack, 2D material/PMMA/PPC/glass is lifted off the scoop. Adhesion between the 2 polymers is stronger than the water surface tension making this transfer possible.

5. The water can be dried off using a very gentle blow of nitrogen gas. The whole stack is then left to dry for 24 hours. This is necessary to get rid of most of the water without tempering with the adhesion of the different polymers by annealing.

The resulting 2D material/PMMA/PPC/glass stack is then introduced to a similar process as described with the "PPC release" method. This process is depicted in Fig. 3.8. The semi-dry transfer method is a hybrid approach of the "wet transfer" and "PPC release" methods, with the initial steps similar to the former and the latter steps similar to the latter. In summary, the semi-dry transfer method involves using a wet-transferred CVD-grown material instead of a tape with multiple 2D material flakes for the "PPC release" method. Although the most complex and time-consuming of the transfer methods discussed, it is also the most promising. The size of a patch of material in CVD growth can be up to a few cm squares, which can be divided into multiple transferable patches, allowing for higher volumes to be produced simultaneously. An operator can have several stamps bearing material ready for transfer at the same time, which, combined with high transfer reliability, makes the method highly efficient and scalable for the industry. Another significant advantage is the surface area that can be covered in a single transfer with no additional contamination. These advantages can also be seen by looking at some examples of this method being used for the transfer of M oS 2 .

Results

The semi-dry transfer method has been tested with various types of CVD-grown materials, including M oS 2 . In the case of M oS 2 , the material often forms triangular shapes of varying sizes on the growth substrate, ranging from 20µm to several hundred µm. This is a significant improvement over ME flakes, as the size of the 2D material is much larger. Moreover, the constellation of 2D material flakes, which was an issue with ME stamps requiring careful placement, is no longer an issue with CVD stamps. The CVD stamp is covered with multiple similar 2D material triangles with similar thicknesses, although the number of layers can still vary from one triangle to another (see Fig. 3.9 where the second flake from the top exhibits thinner edges and thicker center and where the first from flake is uniform and monolayer as suggested by a strong, uniform PL emission). The thickness is much more consistent with almost uniform Raman and PL response across each triangle. This can be assessed thanks to characterization methods established in the chapter 2 where the magnitude of the Raman and PL spectra can be analysed. No flake on the polymer patch is too thick to be used for hybrid integration. As a result, there is no longer a need for plasma cleaning or other measures to reduce sample contamination. However, a careful deterministic transfer is still required to optimize the surface covered by the 2D flakes. In the example shown in Fig. 3.9, four flakes have been strategically transferred to maximize the interaction length. Indeed the triangles are in a fixed position on the polymer stamp, as such, it is import to align it with the spiral waveguide to maximize the interaction length. Some growth processes for M oS 2 allow for the growth of large area monolayer material (up to several cm 2 ) on a substrate. In such cases, the sheet of 2D material is mostly monolayer, covering up to 90 percent of the surface with some holes and small patches of thicker material. However, to maintain the integrity of the 2D material sheet during transfer, the size of the stamp should not be too large. This facilitates handling and reduces the risk of wrinkles and cracks during transfer. As shown in Fig. 3.10, we realize the successful transfer of large M oS 2 area on ion exchange glass waveguides [START_REF] Broquin | Ion-exchanged integrated devices[END_REF][START_REF] Broquin | Glass integrated optics: state of the art and position toward other technologies[END_REF]. The size of the transferred sheet of 2D material corresponds to the size of the stamp used during the transfer (2.8mm × 3.2mm). The sheet of M oS 2 is delineated by the red dotted line in Fig. 3.10 a. Fig. 3.10 b. shows a zoomed-in bright field image of the sheet of 2D material, depicting the deposit with holes (darker areas) and thicker patches (brighter areas), while the material in between is strictly monolayer. The transfer was successful, with no cracks or wrinkles observed, demonstrating the efficiency of the method and its suitability for the large-scale transfer of 2D materials.

It should be emphasized here that while the presented methods have focused on the transfer of M oS 2 , similar approaches can be used for the transfer of other 2D materials. For example, graphene has also been successfully transferred using the same deterministic semi-dry approach. However, there may be some specificities depending on the material and the growth process. For instance, while TMDs are often grown on Si or sapphire substrates, graphene is grown on copper foil, which changes the initial steps of the transfer process. Regardless of the material or the type of growth, new transfer methods are continuously being developed, pushing the boundaries of previous ME methods and increasing the spacial interaction between the propagating mode and the 2D material. However, it is important to note that increasing the surface interaction is not the only way to achieve superior performance. An increase in the temporal interaction can also be accomplished with the help of cavities, which will be discussed in the next section.

Optimizing the time interaction with the help of cavities

In the context of lasers and many other optical/optoelectronic devices, the use of cavities is among the necessary and common tools to increase light-matter interactions and benefit from photon recycling effects. Similarly, the toolbox of available resonators/cavities [START_REF] Liapis | On-chip spectroscopy with thermally tuned high-q photonic crystal cavities[END_REF], or distributed feedback slow light waveguides [START_REF] Liam O'faolain | Loss engineered slow light waveguides[END_REF] can be considered to further enhance the nonlinear properties of hybrid optical waveguides made of a proper integration of 2D materials on standard optical waveguides. Among the large set of possible resonators, simple ring structures present several advantages, among which simple design, great flexibility in term of footprint / sizes / quality factors [START_REF] Bogaerts | Silicon microring resonators[END_REF]. Ring resonators are waveguide loops characterized by their length, dispersion, internal propagation losses, and quality factor (Q factor). Light is usually coupled into a ring resonator using a bus waveguide, which approaches the ring in a tangential manner. The feedback effect from the cavity generates several interesting properties that will be discussed in this chapter, particularly in the case of four wave mixing (FWM). FWM is a third-order nonlinear phenomenon discussed in Chapter 1, and it will be the focus of the study presented in this chapter. We will first discuss how the ring resonator can enhance the strength of this nonlinear phenomenon. Then, we will discuss an experiment that was conducted with a hybrid resonator and detail how the properties of the ring allowed for the monitoring the performances of the device and retrieving effective physical properties of the embedded 2D material layers.

Enhancement of the conversion efficiency using a ring resonator cavity

FWM is an important process to build all-optical parametric amplifiers [START_REF] Azzini | From classical four-wave mixing to parametric fluorescence in silicon microring resonators[END_REF] but also can be considered to generate whole new frequencies or for photon pair generation using spontaneous FWM [START_REF] Azzini | From classical four-wave mixing to parametric fluorescence in silicon microring resonators[END_REF] or for the generation of broad band Kerr frequency combs [START_REF] Wang | Advances in soliton microcomb generation[END_REF]. In the work presented here FWM is discussed in its degenerate form where:

2ω p = ω s + ω c (3.1)
, according to the conservation of energy and momentum between the 3 frequencies, ω p , ω s and ω c for the pump frequency, the signal frequency and the converted frequency (idler) respectively. This phenomenon is the most effective when the frequencies are phase-matched, as described in Chapter 1 (conditions where κ = 0). In a straight waveguide, the efficiency of the FWM process can be characterized in terms of conversion efficiency (in the case of TPA-free optical waveguides):

η straight = P C (L) P S (0) = |γP 0 L ef f | 2 e (-αL) (3.2)
, with P C (L) the power of the converted frequency (idler) after propagation in a waveguide of L length, P S (0) the input signal frequency power, P 0 the input pump power, α the propagation losses, and:

L ef f = 1 -e -(α+jκ)L α + jκ (3.3)
, where κ is the phase mismatch described by Eq. 1.26 in Chapter 1. Introducing FWM to a ring resonator allows for an increase of the conversion efficiency. The cavity exhibits a feedback effect and thus introduces boundary conditions for the propagating electric field [START_REF] Pp Absil | Wavelength conversion in gaas micro-ring resonators[END_REF]:

E 4 p,s,c = -jkE 1 p,s,c + rE 3 p,s,c (3.4)
E 2 p,s,c = rE 1 p,s,c -jkE 3 p,s,c (3.5)
, where E is the electric field, k is the cross (coupling) coefficient and r is the pass (transmission) coefficient of the coupling area between the ring and the bus waveguide.

Those two coefficients verify:

|r| 2 + |k| 2 = 1 (3.6)
In the case of the small-signal analysis

(|E p | ≫ |E s | ≫ |E c |)
the variations of the different propagating fields can be expressed as follows:

∂E p ∂z = - α P 2 E P (3.7) ∂E s ∂z = - α s 2 E s (3.8) ∂E c ∂z = - α c 2 E c + jγE 2 p E * s e jκz (3.9)
One can make the hypothesis that the propagation losses are similar between the different frequencies involved in the FWM process which translates into α p ≃ α s ≃ α c ≃ α.

In the case of our experiment, it is a reasonable assumption as the different frequencies are located at neighbouring resonances. Additionally, as demonstrated in Chapter 2, the TPA around 1.5µm can be neglected in a M oS 2 -SiN waveguide. The shift of the resonance due to the Kerr effect also needs to be small compared to the resonance linewidth according to the following relation:

Lγ(F E) 2 P p 2πm ≪ ∆λ λ m (3.10)
If both of these conditions are met, we can derive from Equations 3.4, 3.5 and 3.9 the following expression for the conversion efficiency in a ring resonator.

η resonator = P C (L) P S (0) = |γP 0 L ef f | 2 F E 4 p F E 2 c F E 2 s e (-αL) (3.11)
Where F E i is the field enhancement factor and can be expressed as follows in an all pass ring:

F E i = k 1 -re -αL 2 +j ω i L c
(3.12)

3.3.

The field enhancement factor is the factor responsible for the improvement of the FWM performances in a ring compared to a straight waveguide. Thus, the improvement can be expressed using the following expression:

η resonnator η straight = L resonnator ef f L straight ef f 2 (F E) 8 (3.13)
Frequency conversion through FWM can be employed for various purposes, but in the case of hybrid structures, it is particularly useful as a means of characterization. By measuring the gain at the converted frequency, one can extract the effective nonlinearities of the waveguide. This method has been widely used in different material systems due to its reliability and accuracy in determining third-order nonlinear properties of waveguides [START_REF] Wu | Enhanced four-wave-mixing with 2d layered graphene oxide films integrated with cmos compatible micro-ring resonators[END_REF]. Furthermore, the enhanced conversion efficiency enables the detection of the generated frequency (idler) that may not be detectable in a straight waveguide with identical properties. Figure 3.11 illustrates the power of the generated idler frequency as a function of the input pump power for a SiN waveguide with a typical nonlinear coefficient of γ = 1m -1 W -1 . The black line represents the case of a straight waveguide, where the idler power remains relatively low even at high pump powers in continuous wave operation. As a result, the idler signal would be buried in the background noise and may not be detectable by standard equipment such as an optical spectral analyzer (OSA). However, using a ring resonator (with r = 0.99 and t = 0.1) significantly boosts the idler power, making it readily measurable by OSA even for a fraction of the maximum pump power. It is important to note that this is an ideal case of a high-quality factor (Q factor) ring resonator, but as will be shown later, it remains valid for more realistic ring resonators.

To better grasp the possibilities enabled by the use of a ring resonator for operations such as parametric amplification or frequency generation, this conversion efficiency was compared in between straight waveguides and rings in different configurations. The micro-ring has a 100 µm radius, the considered unfolded propagation lengths being set to the same values to enable the comparison between the designs. The results of this study are presented in Fig. 3.12 with the case of degenerate FWM around 1.55 µm wavelength. In this study, the pump power level was set at a reasonably low 5 mW, suitable for continuous wave pumping of the optical source. Calculations were performed at neighbouring resonances, determined by the ring's transfer function (see Fig. 3.12 a.). The comparison was made using different parameters as variables, corresponding to the different types of waveguide designs discussed in Section 3.1 of this chapter, which translate into different effective nonlinear γ factors. The γ = 1W -1 m -1 case corresponds to a standard reference SiN buried strip planarized waveguide without The experimental results demonstrated a distinct advantage in favor of a strong transmission factor, as anticipated. This outcome aligns with the expectation that a high-quality factor corresponds to a longer photon lifetime within the cavity. Furthermore, when considering a ring with a high coupling transmission, the conversion efficiency maintains its superiority over a straight waveguide, even in scenarios involving significant losses. However, it is noteworthy that as the propagation losses escalate, the overall quality factor diminishes, leading to a faster reduction in conversion efficiency compared to that observed in a straight waveguide. Overall, a proper design with a high coupling transmission allows a conversion efficiency as high as -6.3 dB (see Fig. 3.12 c. and d. ), which compares well to similar studies conducted in the SOI platform [START_REF] Li | Modified model for four-wave mixing-based wavelength conversion in silicon micro-ring resonators[END_REF].

These results presented an ideal case to illustrate the potential of the methodology. However, to provide a more realistic framework and demonstrate how the ring resonator approach can be utilized for the characterization of hybrid waveguides and as a first step towards functional devices, we will now discuss experimental results obtained from the measurement of a hybrid ring resonator.

The SiN planarized buried strip ring resonator

A h × w = 600 × 1700nm SiN planarized buried strip ring resonator with a 200 µm radius (from STMichroelectronics) was selected for the FWM experiments. Several iterations of the same ring were measured on the same chip with various coupling gaps by performing a scan in wavelength in the C -band. The ring with a 500 nm gap was then selected as a good compromise between high-quality factor and high intracavity power. This ring resonator was subjected to the direct transfer of ME flake using methods discussed in Chapter 2. In fig. 3.13, on the left handside is a bright field image of the ring with a red rectangle highlighting the zone of the transfer. It can be seen that the flake of M oS 2 was deposited as far as possible from the coupling area to not impact the coupling conditions. On the right-hand side of Fig. 3.13 is a zoomed-in version of the bright field image with the Raman scan on top, recorded following the methods discussed in Chapter 2.2.2. This allowed to measure the length of interaction between the 2D material and the waveguide, which was estimated at 42.5µm.

The ring waveguide properties were then calculated using FDE. The results plotted in Fig. 3.14 a. revealed an overlap between the 2D material and the transverse TE mode at 1550 nm of 0, 023% . On the other hand the GVD profile showed a wide anomalous window spanning over the C-band 3.14 b., thus making the design suitable for the experiments conducted with the sample.

Quality Factor

Retrieving the various parameters of the ring resonator is essential for the accurate calculation of the nonlinear performances. Parameters such as propagation losses, Q factor, extinction ratio (ER), etc. are all important and need to be retrieved. Characterization of the Q factor plays a major role in this regard. The Q factor is calculated from the collected data from the cavity characterization. A tunable laser is injected into the cavity, and a sweep in wavelength is performed. The output transmission is recorded and correlated to the input sweep. As such, the transfer function of the cavity is experimentally measured, and the resonances appear as successive nearly equally spaced dips in transmission, as shown in Fig. 3.15. The interval between each resonance is the socalled free spectral range. A zoomed-in section of the recorded transfer function shows the three main components of the resonance being recorded: the depth of the resonance corresponding to the ER, the resonance wavelength λ 0 , and ∆λ which is the FWHM (Full Width at Half Maximum). Once these parameters are identified, they allow for the calculation of the quality factor:

Q = λ 0 ∆λ (3.14)

Propagation losses (intrinsic Q factor)

To characterise further the reference and the hybrid ring resonators, additional measurements can be performed. Reiterating the scanning measurement on identical rings with different coupling gaps on the reference chip, allows to retrieve different values of the Q factor versus the coupling gap. Using the properties of the Q factor together with this extended set of data, it is possible to extract the waveguide's propagation losses. In a ring resonator, we name τ the field life time inside the ring resonator and its decay rate 1/τ . The external decay rate 1 τext corresponds to the decay rate in a perfect ring where photons only escape through out coupling into the bus waveguide and the intrinsic decay rate 1 τ 0 corresponds to the decay rate in a ring without bus waveguide where photons only escape through the scattering from propagation losses. The actual ring resonator decay rate 1/τ is a sum of both decay rates such as:

1 τ = 1 τ ext + 1 τ 0 (3.15)
The quality factor is itself proportional to the field life time : Q = ω 0 τ /2. The total quality factor (the one we measure) can be expressed as the sum of the contribution of the external and intrinsic quality factors.

1 Q = 1 Q 0 + 1 Q ext (3.16)
, where Q is the total quality factor, Q ext is the external quality factor relating to the coupling efficiency, and Q 0 is the intrinsic quality factor relating to the ring's internal losses. The coupling losses can be reduced to a minimum by changing the nature of the coupling area (e.g., the increase of the distance between the bus waveguide and the ring). Reducing the coupling losses results in an increase of Q ext to a point where 1/Q ext can be neglected in Eq. 3.16, thus revealing the value of Q 0 . To retrieve this value, a fit of 1/Q is performed using the following function:

f (x) = a 1 e -a 2 x + a 3 (3.17)

, where x is the gap and a 1 , a 2 , a 3 are the fitting coefficients. The asymptote value of 1/Q toward +∞ can thus be recovered, as it corresponds to this virtual ring with no coupling losses. Indeed for x → +∞, 1/Q ext → 0 (hence 1/Q ≃ 1/Q 0 ). From the retrieved intrinsic quality factor, the propagation losses inside the ring resonator can be retrieved with the following expression:

1 Q 0 = αλ 0 2πn g (3.18)
, where α are the waveguide's linear propagation losses. This method was used to characterize the reference chip and the results are presented in Tab. 3.1 as well as in Fig. 3 In Fig. 3.15, each data point is represented with its corresponding Q factor value. A fit was found and allows to derive an asymptote represented in a dashed green line on the graph. Ultimately the calculated propagation loss for the reference waveguide is α = 33.3m -1 which is relatively low for an integrated waveguide. This highlight one of the advantages to work with the SiN platform as propagation losses in the C-band can be advantageous especially in the case of nonlinear applications. Compared to standard propagation losses characterization, this method has multiple merits. Measuring the transmission in various waveguide's with different length introduces high levels of uncertainty unless the sample size is very large. Indeed, it relies on optimizing the injection to the best of the operator's ability for each waveguide to estimate the best case transmission performance. In the case of the Q factor method, the results rely on the intrinsic properties of the ring and as such it does not require the level of care and a smaller sample size. Another advantage is the possibility to rapidly get an estimation of the propagation losses. 3.17 to retrieve the asymptote toward +∞. The asymptote is represented in a green dashed line and allows to retrieve the intrinsic Q factor Q 0,ref .

The intrinsic losses for the hybrid ring are bound to differ from the reference ring due to the added material. Moreover, in that particular example, no annealing of the sample after the transfer of 2D material was performed, leaving residues on the sample post transfer and thus increasing significantly propagation losses. Regardless, retrieving the hybrid ring internal losses is necessary yet challenging as contrary to the reference ring, we do not have identical hybrid rings with various gaps between the bus waveguide and the ring to perform the same method described earlier. Fortunately, it is still possible to retrieve the internal propagation losses in the hybrid ring by measuring its Q factor, Q hyb . From Eq. 3.16 the external Q factor Q ext in the case of the reference ring with an identical gap can be expressed as follows:

1 Q ext = 1 Q ref - 1 Q 0,ref (3.19)
Establishing Eq.3.16 for the hybrid ring resonator we get:

1 Q hyb = 1 Q ext + 1 Q 0,hyb (3.20)
Here, it is reasonable to assume the external losses are identical between the reference ring and the hybrid ring as the coupling area finds itself unaltered by the transfer of the flake. Indeed, as can be seen in Fig. 3.13, the flake has been deposited far away from the coupling area. In the frame of that assumption, 1/Q ext can be substituted in Eq. 3.20 by its expression established in Eq.3.19 from the reference ring resonator. This allows for the calculation of the intrinsic Q factor for the hybrid resonator:

1 Q 0,hyb = 1 Q hyb - 1 Q ref - 1 Q 0,ref (3.21)
Using Eq. 3.18 in the case of the hybrid ring resonator, the propagation losses of the hybrid waveguide are estimated at α = 91.6m -1 . The increase in propagation losses is, of course, related to the addition of the 2D material but also to the fact that for this particular sample, no post-transfer annealing was performed, thus leaving potential residues after the transfer. Nevertheless, as it will be shown in the next section, significant improvement in the conversion efficiency was recorded anyway, thus showing the strong potential of the approach.

Nonlinear characterization

Experimental setup

The FWM experiment was performed in both the reference ring resonator and the hybrid ring resonator with the experimental setup presented in Fig. 3.17. Two different tunable lasers were employed for the nonlinear experiment. One was dedicated to the pump, while the other was assigned to the signal. A polarization controller was set at the output of both sources to guarantee individual and independent control over the polarization of each source. Both sources were then amplified using erbium-doped fibre amplifiers (EDFA). To limit the background on the final measurement, the pump can also filtered. This can be necessary for the pump but not for the signal, as the pump is amplified to a higher degree than the signal resulting in an important amount of amplified spontaneous emission (ASE). Additionally, the combination of both sources was achieved using a 90:10 fibred coupler , the pump was plugged into the 90 % port, thus making the ASE from the amplified Pump laser much more impact-full than the ASE on the amplified signal laser. After the coupler, a third polarization controller was installed to adjust the global polarization before injection in the sample. The TE mode was selected and guaranteed thanks to a polariser set right before the injection. Injection and collection are performed in Butt-coupling through the cleaved facet of the bus waveguide using piezo-controlled lensed fibre. A top-view microscope and a power meter connected to the output of the collection fibre were used to optimise the injection and optimize the lensed fibres alignments. Depending on the measurement device, an attenuator was optionally be installed at the collection fibre output. The output spectrum was then recorded using an OSA.

FWM measurments

The reference ring resonator and the hybrid ring were both characterized using the set-up presented in Fig. 3.17. The output of EDFA 1 (pump) was set to 1 W, and the output of EDFA 2 (signal) was set to 0.316 W. Considering the losses introduced by the different components in the setup, plus the coupling losses at the interface between the lensed fibre and the waveguide's facet, the internal power inside the bus waveguide was estimated to be 0.2 W for the pump and 0.063 W for the signal. The coupling losses were estimated at 6 dB per facet, which also explains the difficulty in measuring the generated idler frequency and also justifies the use of powerful sources, these sources finding themselves severely depleted before encountering the ring.

The results from both ring resonators are displayed in Fig. 3.18. The spectrum from the hybrid ring resonator appears in red, while the spectrum from the reference sample appears in black. To better visualize the difference in frequency generation from the two rings, data are plotted in normalized transmission. What appears clearly is that although both spectra exhibit the presence of a weak idler frequency generated through FWM, the hybrid resonators exhibit a more prominent peak. In the inset, a zoomed-in spectrum of the idler peak reveals a difference of 1.82 dB between the two peaks. Accordingly the The spectrum from the hybrid ring resonator is plotted in red, and the reference spectrum from the reference ring resonator is plotted in black. In the top left corner, a zoomed-in spectrum on the idler frequency.

conversion efficiency has been estimated for both waveguides, and the values are displayed in Table 3.2. From the conversion efficiency, the γ coefficient for the reference and the hybrid rings were also retrieved and are presented in Fig. 3.18. It should be mentioned that the calculated γ factor for the hybrid ring is a factor encompassing the totality of the ring, whereas the flake, with its 42.5µm interaction length, only occupies 3% of the ring. Accounting for that, the γ factor of the hybrid section of the ring length is estimated to be around γ hyb,wg = 9.19W -1 .m -1 . Following the same method as in Section 2.3.1 of Chapter 2, the Kerr refractive index was also retrieved by accounting for the overlap between the 2D material and the propagating mode calculated with the TE mode from Fig. 3.14. With an overlap of 0, 023%, the nonlinear refractive index of the M oS 2 flake can be estimated to n 2,M oS2 = 4, 26 × 10 -15 m 2 .W -1 which is slightly higher than the n 2,M oS2 = 1.1 × 10 -16 m 2 .W -1 found in the literature [START_REF] Liu | Enhanced optical kerr nonlinearity of mos 2 on silicon waveguides[END_REF]. However this can be explained by the use of the TE mode polarization in our work compared to the TM mode polarization used in the reference work. It has been suggested that in plane interaction of the mode with the 2D material trigger a more significant nonlinear response than the TM polarization [START_REF] Dai | Broadband plasmon-enhanced four-wave mixing in monolayer mos2[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF][START_REF] Bao | Broadband graphene polarizer[END_REF]. These results point toward the same conclusion. Moreover, a similar order of magnitude is found with results obtained in the pulse broadening experiment of chapter 2 which is also expected as TMDs have similar nonlinear susceptibility [START_REF] Autere | Optical harmonic generation in monolayer group-vi transition metal dichalcogenides[END_REF]. From these results, it appears that the combination of highly nonlinear materials with cavities such as ring resonators could be a way to achieve highly efficient nonlinear devices. The nonlinear coefficient measured inthe hybrid section of the waveguide is 12.43 times higher than in the reference waveguide. This opens opportunities for more advanced applications such as Kerr modulator [START_REF] Deckoff-Jones | Enhancing sin waveguide optical nonlinearity via hybrid gas integration[END_REF] or even device for the generation of pair photons [START_REF] Wang | Integrated photon-pair sources with nonlinear optics[END_REF].

It also appears that lots of improvements could be made upon the results presented here. For one, the losses introduced by the transfer of 2D material could be reduced by completing the process with annealing of the sample, which was not performed at the time of the fabrication and characterization. Moreover, the newly introduced transfer methods also discussed in this chapter could also provide cleaner and larger deposits of 2D material layers and thus boost further the ring's performances. This experiment is representative of the work attempted all throughout this manuscript as it constitutes a convincing first step toward more advanced results. To better grasp the way paved by all the work established here, we will now provide a conclusion of the work presented. Then, we will discuss the outlooks for future investigations that may be conducted based on this work in the future.

Conclusion and perspectives

The field of hybrid integrated optics using 2D materials is still in its early stages of development. Most of the existing literature focuses on experimental investigations that explore the potential benefits of integrating highly nonlinear 2D materials into waveguide structures. Additionally, for the most part, reported results were obtained using graphene and very little results exist for other 2D materials. Theoretical investigations in this area are relatively scarce, and there is a wide variation in the methodologies employed for the fabrication of waveguide structures and in their characterization in experiments. As a result, there are significant discrepancies in the reported results. This can be attributed to the fact that the field is still evolving, and many guidelines and best practices have yet to be established based on accumulated knowledge and experience. Furthermore, the combination of two challenging fields, namely 2D materials and nonlinear integrated optics, presents additional obstacles that need to be overcome in order to advance the field and realize functional hybrid integrated devices.

In this manuscript, the primary focus has been on integrating the methods from both the fields of 2D materials and nonlinear integrated optics to demonstrate the potential of hybrid optical integrated waveguides embedding M oS 2 or W S 2 2D monolayers. Chapter 2 of the manuscript specifically details the pragmatic approach taken to achieve hybrid integration of 2D materials on photonic waveguides. This involved becoming familiar with various tools and techniques, as well as conducting numerical investigations of the hybrid waveguides giving insight in the light-mode/2Dmaterial interaction. Several well-established methods were employed in this part of the work, including mechanical exfoliation, Raman spectroscopy, finite difference eigenmode calculations, and optical pulse broadening experiments (both measurements and simulations). Through these efforts, significant progress was providing a solid foundation for further improvements. The measurements revealed an enhancement factors, of about 371.4 and 342.9 for input pump powers of 1.35 and 2.7 kW, respectively from the reference waveguide to the hybrid waveguide.

In Chapter 3, three main approaches were investigated to progress one step further and demonstrate enhanced light/TMD-layers interaction. The first axis involved conducting numerical investigations to optimize the cross-sectional design of the waveg-uide. The numerical tools to conduct this optimization while introducing specifically new approaches to enlarging the design parameter space (top cladding material, multimono layer stacks) were described in chapter 2. The second axis focused on improving the transfer of 2D materials onto the waveguide. Issues related to the use of mechanically exfoliated flakes were addressed, and alternative methods were introduced. This included utilizing chemical vapor deposition (CVD) grown materials and implementing semi-dry transfer methods. These advancements led to increased material coverage on the waveguide and improved the success rate of the transfer process. Consequently, the interaction length between the 2D material and the propagating light along the waveguide was also enhanced. The third axis explored the use of ring resonator cavities to further improve the results obtained from hybrid waveguides. Initially, this investigation was carried out by relying on simulations to explore the potential benefits of incorporating ring resonators into the hybrid integrated structures. Subsequently, experimental results were obtained, demonstrating the potential of ring resonators to enhance the performance of the hybrid waveguides in practical settings. The nonlinear coefficient measured in the hybrid ring is 12.43 times higher than in the reference ring, opening opportunities for subsequent enhancement in several ring resonator based nonlinear optical functions (e.g., Kerr modulators, pair photon source, frequency combs,...). Overall, Chapter 3 introduced new guidelines and improvements based on the lessons learned from Chapter 2. The optimization of waveguide cross sections, advancements in material transfer techniques, and the incorporation of ring resonator cavities collectively contributed to the progress of hybrid integrated optics using 2D materials.

Perspectives

The work presented in this manuscript has not only established a proof of concept of the potential of 2D materials capped optical waveguides for on-chip nonlinear applications, but also laid down guidelines to facilitate future groundbreaking research in the field.

Building upon these foundations, there are several immediate avenues to explore that hold significant potential. One direction is the utilization of large-area 2D materials on SiN spiral waveguides. Further investigation can be conducted to explore the possibilities of atomic layer deposition (ALD) on hybrid waveguides, with the aim of improving the cross-sectional interaction between the waveguide and the 2D material. Attention can be brought to less studied 2D materials in the context of hybrid waveguides (e.g., BP, h-BN,...) offering alternatives and opening more opportunities. Additionally, exploring the coverage of ring resonators with larger patches of 2D materials can be pursued to enhance their performances. Moreover, combining 2D materials with other types of integrated photonic structures could also expand the range of possibilities. Such is the case of photonic crystals, which have a huge potential regarding the possibilities when combined with 2D materials (e.g., slow light waveguides, cavities with high Q and low volumes, L3 cavities,...). Finally, another immediate possible use of 2D materials would be to exploit them to introduce nonlinear effects in a platforms that does not usually exhibit nonlinear behaviour. Such is for example the case of ion exchange glass waveguides that could become nonlinearly active using appropriate waveguide design and large area transfer of 2D materials (over cm square footprints).

There is no doubt that experimentation on hybrid integrated waveguides will continue to advance, bringing us closer to the realization of fully functional and efficient integrated devices. In this pursuit of all-optical integrated devices, 2D materials are expected to play a crucial role as key enablers. The potential applications in various fields are extensive, including photodetectors, pulsed laser sources, broadband light sources, parametric amplifiers, modulators, pair photon light sources, biosensing, and more. To fully harness the benefits of 2D materials, further progress is needed, particularly in expanding our theoretical understanding and developing more accurate models for improved designs. Attention should also be given to the fabrication methods, with a focus on enhancing the quality and surface area of materials through advancements in chemical vapor deposition (CVD) growth processes. Additionally, transfer methods need to be perfected to minimize contamination during the transfer process, making them suitable for industrial-scale production. Encouragingly, there have been recent reports of direct growth of transition metal dichalcogenides (TMDs) on waveguides, indicating ongoing efforts to overcome these challenges. Given the compelling experimental results and the growing interest in this field, we can anticipate rapid progress in the coming years.

CONTENTS 2 . 3 . 2

 232 Other first generation prepared by mechanical exfoliation samples and their performances . . . . . . . . . . . . . . . . . . . 3 Optimizing the interaction between 2D materials and the propagating light modes 3.1 Optimizing the transverse spatial interaction . . . . . . . . . .

Figure 1 :

 1 Figure 1: Résultats de la propagation à travers le guide d'une longueur de 4mm pour des dimensions de h = 330 nm et de w = 760 nm. En point pointillé les résultats numériques issues de l'intégration de la GNLSE. Le tracé continue est le spectre mesuré expérimentalement. Les résultats apparents en rouge correspondent aux mesures de références dans le guide avant le recouvrement par le feuillet de matériaux 2D. Les résultats tracés en bleu correspondent aux mesures faites dans les mêmes conditions après recouvrement par le feuillet 2D. La mesure est effectuée à deux puissances crètes distincts, a) à 1,35 kW et b) à 2,70 kW

Figure 2 :

 2 Figure 2: Résultats des simulations menées pour l'optimisation de l'interaction entre le matériau 2D et le mode se propageant dans un guide d'onde SiN/SiO 2 de dimensions h = 600 nm et w = 1000 nm à la longueur d'onde de 1, 55µm. a) Schéma de la structure établie montrant la couche HfO2 supérieure ainsi que plusieurs couches de matériaux 2D séparées par une fine couche de Al 2 O 3 . b) Coefficient nonlinéaire du guide représenté en a) pour différentes dimensions du guide. c) Étude de l'ajout successif de couche de matériaux 2D sur le modèle de la figure a). d) Coupe du mode quasi-TE se propageant dans le guide pour différent nombres de couches ajoutées sur le guide par intégration en surface.

Figure 3 :

 3 Figure 3: Spectre normalisée d'un mélange à quatre ondes mesurées dans deux résonateurs. En noir, le résonateur de référence et en rouge, le même résonateur couvert par un feuillet de M oS 2 .

Figure 1 . 2 :

 12 Figure 1.2: Propagation through a waveguide. a. Schematic of a strip waveguide . The light beam is represented in a geometrical configuration on the side of the waveguide. On the top, some propagation effects, such as pulse broadening, is depicted. Propagation occurs along the z-axis . b. Dispersion profile for a Si strip waveguide (h × w = 220 × 360 nm). The green background indicates the anomalous dispersion regime (β 2 < 0), and the red background, the normal dispersion regime (β 2 > 0) c Transverse TE0 mode at 1550. d. Slice of the transverse TE0.

Figure 1 . 3 :

 13 Figure 1.3: Effect of the nonlinear propagation in a waveguide. The phenomenon depicted here is SPM, where the input pulse (red) has been broadened during propagation. It results in a wider spectral output (blue). The nonlinear effects in the waveguide originate from the properties of the core material, which the propagating wave overlaps with significantly (see inset, top left )

P 1 =

 1 Figure1.4: a. Some of the most emblematic nonlinear processes. In yellow, a second-order nonlinear medium, in blue, a third-order nonlinear medium. Virtual energy levels are represented in dashed lines. i. Second-harmonic generation, a pump frequency ω is doubled to the second harmonic 2ω through a nonlinear process described in Eqs. 1.20 and 1.21 ii. THG. A frequency ω is tripled to the third-harmonic 3ω through a nonlinear process described in Eq.1.26 iii. FWM process. This process will be described more thoroughly in a subsequent chapter. FWM can be found in its non-degenerate state where ω P 1 ̸ = ω P 2 or in it's degenerate state ω P 1 = ω P 2 . b. Outcome of the integration of the GNLSE by including different effects in the simulation. The simulated waveguide is a h×w = 220×360nm strip Si waveguide. The injected pulse (black line) is a 100 fs pulse with a 6.5 W peak power centred at 1500 nm in optical wavelength. i. Resulting spectrum after propagation where TPA and FC have been ignored. ii. Resulting spectrum after propagation where FC have been added to the model. iii. Resulting spectrum after propagation where both FC and TPA have been added to the model. c. Evolution of the spectral envelop in a logarithmic scale along light propagation, obtained from simulation of pulse propagation in the Si strip waveguide. Corresponds to the evolution for the outcome presented in b.i. d. Dispersion curves for the TE and TM modes. The radiation cone is highlighted in yellow. The transverse T E 0 is plotted in the inset.

  γ

Figure 1 . 5 :

 15 Figure 1.5: Results of the calculated density of free carrier generated by the light pulse for the input pulse of the simulation presented in Fig. 1.4.

Figure 1 . 6 :

 16 Figure 1.6: a. Representation of the most common 2D materials (graphene, BP, TMD and h-BN). b. Brillouin zone for a hexagonal lattice. c. 3D representation of the crystal layout for a TMD. M stands for metal (tungsten W, molybdenum Mo), and X stands for the chalcogene (selenide Se, sulfide S,...). d. Band structure for the different monolayer materials presented in this figure.The blue background is the conduction band, and the yellow background is the valence band. i. graphene, the gap is closed, the "gapless" nature of graphene is here visible ii. In this example, M oS 2 , different TMDs have different gaps between 1.0 and 2.5 eV, making this family of versatile semiconductors. iii. BP, here the band-gap features different directions in the reciprocal space (X, S, Y) due to the "wavy" layout of the phosphorous atoms (see a). The band gap can vary between 0.3 and 2 eV making it a versatile semiconductor iv. h-BN and its large bandgap make it an insulator.

Figure 1 . 7 :

 17 Figure 1.7: 2D material on dielectric waveguide hybrid structure. a. Principle of a hybrid waveguide. A 2D material is deposited on a waveguide. The transverse mode of the propagated wave overlaps with this material, thus taking advantage of the 2D material's properties. b. Graphenecoated Si ring-resonator[START_REF] Ji | Enhanced parametric frequency conversion in a compact silicon-graphene microring resonator[END_REF] characterized in a FWM experiement. The FWM spectrum from the bare Si waveguide is in black and from the Si-graphene waveguide is in red. c. Tunable FWM characterized in a gated graphene-SiN waveguide,[START_REF] Alexander | Electrically tunable optical nonlinearities in graphenecovered sin waveguides characterized by four-wave mixing[END_REF] The conversion efficiency (FWM) varies depending on the gate voltage (VGS) and the wavelength detuning between the pump and the signal. d. Graphene-covered silicon waveguide characterized in an optical pulse broadening experiment for several lengths (LGr) of graphene sheet covering the waveguides compared to a pure silicon waveguide (SWG. The experiment is performed using a pulsed pump at 1560 nm wavelength with a 250 MHz repetition rate, 80 fs pulse duration, and an input pulse energy of 120 pJ[START_REF] Ishizawa | Optical nonlinearity enhancement with graphene-decorated silicon waveguides[END_REF]. e. Saturable absorption in a graphene-SOI strip waveguide[START_REF] Wang | Cmos-compatible all-optical modulator based on the saturable absorption of graphene[END_REF]. characterized by transmission measurements at diffeent pump power. The experiment is done for different length of graphene. f. SHG enhancement generated in a M oSe 2 -Si waveguide[START_REF] Chen | Enhanced second-harmonic generation from twodimensional mose2 on a silicon waveguide[END_REF]. Measured spectrum with a significant increase in the SHG emission for the hybrid structure. Peak R1 corresponds to SHG, and peak R2 to PL. g. Raman response on graphene on a Si-L3 cavity[START_REF] Gomulya | Enhanced raman scattering of graphene using double resonance in silicon photonic crystal nanocavities[END_REF]. Raman spectrum of graphene excited with a laser source tuned at 532 nm for the different locations on the device: red for the cavity and blue for the SOI plane substrate. In the inset: zoomed-in on the normalized 2D peak for spectral comparison.

Figure 2 . 1 :

 21 Figure 2.1: Calculated surface conductivity for various 2D materials.Here σ 0 = e 2 /4ℏ = 6, 0853 × 10 -5 A.V -1 a. Surface conductivity for graphene calculated with Eq. 2.2. Both the real and imaginary parts are represented. b. Surface conductivity for two of the most common TMDs. The real and imaginary parts are retrieved from a fit of the experimental data from Ref.[START_REF] Li | Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: Mos 2, mo s e 2, ws 2, and ws e 2[END_REF]. The fit is performed by the sum of Lorentzian of Eq. 2.3.

Figure 2 . 2 :

 22 Figure 2.2: FDE simulation of a hybrid waveguide. a. Schematics of the planarized SiN strip waveguide layout. b. Comparison between the transverse mode of a SiN planarized buried strip waveguide at 1550 nm (h×w = 600nm×1000nm) and the same waveguide, cladded on the top by a 2D material (monolayer M oS 2 ). Is plotted the difference between the TE mode distribution such as E plot = E 2D -E ref . c. Comparison between the GVD profile of the waveguide in different configurations, uncladded, with one layer of M oS 2 on top and twenty layers of M oS 2 on top. The zero dispersion wavelength for the uncladded waveguide is indicated here by the red dot.

Figure 2 . 3 :

 23 Figure 2.3: ZDW shift for a planarized burried strip SiN waveguide with dimensions h × w = 600nm × 1000nm for various number of crystal layers on top

Figure 2 . 4 :

 24 Figure 2.4: a.Typically met types of integrated waveguide cross sections (independently of their material). In this depiction, the darker the color, the higher the refractive index. b. Typical geometrical parameters can be leveraged to optimize a waveguide's properties (independently of its material). c. Mapping the dispersion profile for an SOI waveguide with 220 nm thickness between 1.36µm and 3.5µm. The anomalous regime is characterized by the blue regions, while the normal regime is characterized by the red regions. d. Calculation of the overlap between a single layer of M oS 2 added on top of the waveguide and the transverse TE mode versus the widths of a 220 nm thick SOI waveguide. The same graphic shows the calculated waveguide effective nonlinear coefficient.

Figure 2 . 5 :

 25 Figure 2.5: Mechanical exfoliation process and transfer using a classical "scotch tape method" for the exfoliation and a "PDMS stamp" for the direct transfer.

Figure 2 . 6 :

 26 Figure 2.6: Schematics of the transfer stage. The objective is connected to a camera and a monitor to control the transfer. Both stages are controlled with micro-metric screws.

Figure 2 . 7 :

 27 Figure 2.7: Micro-spectrometer used for Raman and PL measurements of samples. a. Microscope unit, including several input for the bright field source, and various laser wavelengths. The sample is placed, and an XYZ piezo-controlled stage. The microscope it self also has a z-dependent control stage to optimize focus on the surface of the sample. b. Laser unit that includes several sources at different wavelengths with tunable optical power. c. CCD unit for detection of the output signal. The signal injected into the CCD unit has been cleared from the source using polarization filters.

Figure 2 . 8 :

 28 Figure 2.8: Raman process. On the right, the most important vibrational modes for TMDs at the origin of the Raman effects in TMDs.

Figure 2 . 9 :

 29 Figure 2.9: Micro-spectroscopy analysis of a M oS 2 ME flake. a. Raman mapping of the flake depicting the intensity of the two characteristic peaks E 1 2g and A 1g . b. PL mapping of the flake showing the intensity of the emitted light for a photon energy between 1.8 and 2.1 eV. c. Bright field microscope image of the flake. The pink rectangle shows the scanning area for a. and b. and the orange rectangle the scanning area for the insets. In the inset, left: frequency shift for the A 1g peak. The shift is characterized with respect to the median peak position for the scanned area. The red shift areas (signifying thinner layer) are in red and the blue shift area (signifying thicker layer) in blue. Right: Frequency shift for the E 1 2g peak. The red shift areas (signifying thicker layer) in red and the blue shift area (signifying thinner layer) are in blue. d. Raman spectra of the characteristic peaks of M oS 2 recorded at the pin-pointed locations in c. and d. e. Comparison between different PL and Raman spectra recorded at the pin-pointed locations in b.

Figure 2 . 10 :

 210 Figure 2.10: Calculated input parameters for the simulation of the pulse propagation experiment. a. Calculated TE mode at 800 nm of the SiN raised strip with a cross-section of h × w = 330 × 760nm . The layout of the M oS 2 flake in the cross-section is represented in Red. The purpose of this simulation is to obtain the overlap for the later characterization of the 2D materials nonlinear properties. b. Calculated GVD profile of the SiN raised strip with a crosssection of h×w = 330×760nm. The profile curve has an inflection around the pump wavelength, resulting in a relatively flat dispersion around the pump wavelength.

Figure 2 . 11 :

 211 Figure 2.11: Characterization of the deposited W S 2 flake. a. Bright field microscope image. The waveguide can be identified in the middle. On top of the waveguide, a slightly brighter area (inside the red rectangle) can be identified as the deposited W S 2 flake. b. Raman spectrum of the W S 2 flake. The two characteristic peaks E 2 2g and A 1g with their respective integration areas in blue and yellow. c. Raman mapping obtained through the integration of the characteristic peaks of the Raman spectrum represented in b. The waveguide cause a more intense Raman response on the edge due to variation of refractive index or to a different crystal orientation. d. AFM measurements of the flake at different points allow to identify the overall thickness of the flake.

Figure 2 . 12 :

 212 Figure 2.12: Experimental setup used for the characterization of the hybrid waveguide sample. The beam path is represented as a red line. The injection is monitored with a microscope giving a top view of the sample. The captured image of the sample can be seen in the inset on the top right side.

Figure 2 . 13 :

 213 Figure 2.13: Inverse of the transmission versus the injected peak power. Recorded data is fitted to calculate the TPA coefficient with Eq. 2.7

Figure 2 . 14 :

 214 Figure 2.14: Calculated input parameters for the simulation of the pulse propagation experiment.Fit of the input pulse, the pump spectrum is measured before injection (red curve) and fitted with a hyperbolic secant (blue curve) to render it in the simulation.

Figure 2 . 15 :

 215 Figure 2.15: Characterization of the W S 2 -SiN waveguide. a. Spectral output for a 1.35 kW peak power pump. b. Spectral output for a 2.7 kW peak power pump. The output of the hybrid waveguide is in blue, and the output of the uncladded waveguide is in red. The spectra have been fitted by integration of the GNLSE. The results of the simulation appear in dashed lines. c-f. Spectral evolution along the propagation with the intensity represented in false color. c. Uncladded waveguide at 1.35 kW peak power. d. Uncladded waveguide at 2.70 kW peak power. e. Hybrid waveguide at 1.35 kW peak power. f. Hybrid waveguide at 2.70 kW peak power. g. Enlarged section of f. showing the impact of the flake on the spectral evolution.

Figure 2 . 16 :

 216 Figure 2.16: Several examples of ME flakes transferred on SiN waveguides.

Figure 2 .

 2 Figure 2.17: Hybrid SiN spiral waveguide fabricated with ME . a. Sample after deposition, the flake of interest has been masked with HSQ to protect it from the next step. b. The same sample after O 2 plasma cleaning. Other flakes are removed, and the protected flake remains. In the inset, Raman mapping of the deposited flake is obtained through integration of the E 1 2g and A 1g peaks.

Figure 2 . 18 :

 218 Figure 2.18: Results from the characterization of the spiral waveguide. The experimental setup was used for the characterization of the samples. a. Transmission measurements for the hybrid waveguide. b. Transmission measurements for the reference waveguide.

Figure 3 . 1 :

 31 Figure 3.1: Improvement in the passive properties of a h × w = 600 × 1000nm SiN planarized burried strip waveguide. The comparison is made between a hybrid waveguide without any top cladding and a waveguide with a 200 nm layer of Hf O 2 of top cladding. The left figure depicts both GVD profiles, while the right-hand side shows the cross-section structure and the TE mode distribution.

Figure 3 . 2 :

 32 Figure 3.2: a. Monolayer cases with a 200 nm top cladding of Hf O 2 b. Evolution of the effective nonlinear parameter γ of a planarized buried strip SiN waveguide. c. The structure is compared with a reference uncladded burried strip SiN waveguide. As such, the improvement from the reference to the the structure presented here (planarized buried strip SiN waveguide topped with a monolayer M oS 2 ) is calculated. d. The overlap between the 2D material and the propagating TE mode is also calculated. Every parameter is calculated for a wide range of cross-section dimensions and represented in a false color map.

Figure 3 . 3 :

 33 Figure 3.3: a. Bi-layer case: optogeometrical study for a planarized buried strip SiN waveguide.b. The studied parameter is the effective nonlinear coefficient γ. c. The structure is compared with a reference uncladded burried strip SiN waveguide. As such the improvement from the reference to the structure presented here (planarized buried strip SiN waveguide topped with 2 monolayers of M oS 2 ) is calculated. d. The overlap between the 2D material TE mode is also calculated. Every parameter is calculated for a wide range of cross-section dimensions and represented in a false color map.

Figure 3 . 4 :

 34 Figure 3.4: Numerical study of the benefit of a multilayer stack of 2D material sheet for the effective nonlinear performance of the SiN planarized single-mode waveguide at λ = 1550nm being considered here. a. Comparison between 2 different cases. The first case, in Red: corresponds to the multilayer layer stack topped with a 200 nm cladding of Hf O 2 similar to the structure presented in Fig. 3.3 with additional layers. The second case, in blue, is the same stacking method without a top cladding. b. TE mode distribution for various amounts of stacked monolayers of M oS 2 for the Hf O 2 cladded structure. c. TE mode distribution for various amounts of stacked monolayers of M oS 2 for the uncladded structure.

Figure 3 . 5 :

 35 Figure 3.5: The different steps of the wet transfer process. This process doesn't allow for the deterministic transfer of material but allows the deposition of large sheets large area CVD grown 2D materials (e.g., graphene, M oS 2 , ...).

Figure 3 . 6 :

 36 Figure 3.6:The different steps of the "PPC release" transfer method. This method aims at performing the transfer of ME flakes with higher success rates than the basic ME direct transfer method.

Figure 3 . 7 :

 37 Figure 3.7: Fishing of the PMMA and 2D material for the semi-dry transfer method. In dark blue is water, and light blue is a glass slide. PMMA is in orange, PPC is in green, and the 2D material is in red.

Figure 3 . 8 :

 38 Figure 3.8: The different steps of the semi-dry transfer process. Dark blue is the tape, light blue is the glass slide. PMMA is in orange, PPC is in green, SiN is in yellow, PDMS is in purple, and the 2D material in red.

Figure 3 . 9 :

 39 Figure 3.9: SiN burried strip spiral waveguide chosen as the test bed for the deterministic transfer of CVD grown M oS 2 flakes in this case, the CVD grown material has a standard triangular shape and is highlighted in Red. Each flake was characterized with the Raman ad PL methods described in Chapter 2, and the mapping is shown down below. Raman mapping of the flake depicting the intensity of the two characteristic peaks E 1 2g and A 1g . PL mapping of the flake showing the intensity of the emitted light for a photon energy between 1.8 and 2.1 eV.

Figure 3 . 10 :

 310 Figure 3.10: Result of a semi-dry transfer of a large area M oS 2 sheet on ion exchange glass waveguides from [146, 147]. a. Bright field image of the transferred sheet of material. The image is the result of stitching multiple captures since a powerful objective is required to distinguish the materials from the bare surface. The 2D material field is delimited by the red dotted line. b. Zoomed in bright field image of the deposited sheet. The darker areas are characteristic from the surface uncovered by 2D materials as the transfer sheet does not cover strictly 100 % of the surface, and small holes in the 2D material can be observed. The red square represents the area chosen for Raman and PL measurements. c. Raman measurement showing the various thicknesses of M oS 2 and the holes in the 2D material sheet. d. PL measurement confirming the very thin nature of the 2D material (cf chapter 2) and the monolayer character of the majority of the sheet surface.

Figure 3 . 11 :

 311 Figure 3.11: Power of the converted frequency (idler) after propagation over L. Are compared the results out of a straight waveguide with results out of a ring resonator. This is contextualized with the typical detection level of an OSA represented in the black dashed line (Corresponding to the background noise level).

Figure 3 .Figure 3 . 13 :

 3313 Figure 3.12: a. Transfer function of a 100 µm radius ring resonator around 1.55 µm. b. Depiction of a FWM experiment with the key parameters of the ring represented. k and r are the cross and pass coefficients respectively. c.Conversion efficiency η in dB for straight waveguides (dashed lines) and rings (full lines)versus the propagation losses (for r =0.99) d. Conversion efficiency η in dB for straight waveguides (dashed lines) and rings (full lines) versus the coupling transmission coefficient r (for α = 0.7cm -1 )

3. 3 . 2

 32 Linear characterization of the ring resonator 3.3.3 FWM experiment in a hybrid ring resonator.

Figure 3 . 14 :

 314 Figure 3.14: Properties of the ring resonator waveguide calculated with FDE. a. TE mode distribution at 1550 nm b. GVD profile of the waveguide.

Figure 3 . 15 :

 315 Figure 3.15: Sweep measurement for the hybrid ring cavity measured by injecting TE mode polarization. a. Transmission measurement of the cavity, different dips in transmission at nearly regular intervals are characteristic of the ring's resonances. b. Zoomed-in Transmission spectrum for the ring cavity corresponding to the boxed area of the transmission spectrum depicted in a. The difference between the two levels depicted in blue allows for retrieval of the extinction ration ER. The FWHM can also a measured for the purpose of calculation of the Q factor.

Figure 3 .

 3 Figure 3.16: 1/Q versus the coupling gap. Different points have been calculated from Eq. 3.14 for identical rings with different coupling gaps. The different data points are fitted using Eq.3.17 to retrieve the asymptote toward +∞. The asymptote is represented in a green dashed line and allows to retrieve the intrinsic Q factor Q 0,ref .

Figure 3 . 17 :

 317 Figure 3.17: Schematic of the experimental setup employed for the characterization of a ring resonator cavity. PC stands for polarization controller, EDFA stands for erbium-doped fibre amplifier, DUT stands for the device under testing, and OSA stands for the optical spectral analyser.

Figure 3 . 18 :

 318 Figure 3.18: Results of the FWM characterization for the ring resonators. Both spectra are plotted in terms of the normalized transmission for easier comparison between the two results.The spectrum from the hybrid ring resonator is plotted in red, and the reference spectrum from the reference ring resonator is plotted in black. In the top left corner, a zoomed-in spectrum on the idler frequency.

  . . . . . 3.2 Optimizing the longitudinal interaction, introduction of new transfer methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Transfer of CVD grown material: wet transfer . . . . . . . . . . 3.2.2 Improvement of the dry transfer method: PPC release . . . . . . 3.2.3 Deterministic semi-dry transfer method . . . . . . . . . . . . . 3.3 Optimizing the time interaction with the help of cavities . . . . . . . . . 3.3.1 Enhancement of the conversion efficiency using a ring resonator cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Linear characterization of the ring resonator . . . . . . . . . . . 3.3.3 FWM experiment in a hybrid ring resonator. . . . . . . . . . . . 3.3.4 Nonlinear characterization . . . . . . . . . . . . . . . . . . . .

Table 1 .

 1

	1.6 × 10 -21 -	1.12

1:

Table 2 .

 2 2: Characteristics of main 2D materials and some common bulk materials in terms of their Raman response and PL emission.

		Typical exci-	Raman Response	PL	PL
		tation wave-		(eV)	(nm)
		length (nm)
	Graphene	514-532	G band at 1580 cm -1 -	-
			2D band at 2700 cm -1
	MoS 2	532-633	E 1 2g at 383 cm -1	1.8	690
			A 1g at 408 cm -1
	WS 2	405-514	E 1 2g at 356 cm -1	1.9	650
			A 1g at 418 cm -1
	MoSe 2	532	A 1g at 286 cm -1	1.5	820
			A 1g at 286 cm -1
	WSe 2	532	E 1 2g at 249 cm -1	1.7	730
			A 1g at 286 cm -1
	MoTe 2	633	E 1 2g at 165 cm -1	1.1	1125
			A 1g at 214 cm -1

Table 2 .

 2 3: Comparison between results from this work

Table 2 . 4

 24

: Comparison between results from this work

  .[START_REF] Picqué | Frequency comb spectroscopy[END_REF]

	Sample	gap (nm)	Q factor
	R1	300	103000
	R2 (reference)	500	192000
	R3	700	215000
	R4	900	270000
	hyb	500	94000

Table 3 .

 3

1: Comparison between results from this work

Table 3 .

 3 2: Comparison between results recorded in the reference ring resonator and the hybrid ring resonator, including the quality factor, the propagation losses and the nonlinear coefficient.

Acknowledgements

Author contribution

The author of this manuscript has been extensively involved in all aspects of the research presented. Here is a breakdown of the author's contributions in different areas: