Présentée Par

Vincent Vinel

Christopher Wool

Scenarii of light manipulation in dielectric nanostructures

Keywords: Nanophotonics, quasi-normal modes, nonlinear quantum optics, quantum entangled states, near-field couplings

Nonlinear, quantum and topological features in nanophotonics

This first chapter introduces the main tools and concepts that would help to navigate through the original work undertaken during my PhD research. The scattering of light by particles smaller than the wavelength is presented in the first section, in addition to the main tool used for modeling nanoresonators: quasi-normal modes. Then, I introduce the notions of nonlinear optics and quantum physics necessary to study biphoton states generated by parametric fluorescence, also known as spontaneous parametric down conversion. Finally, few elements of topological physics are given, to contextualize and highlight the stakes behind the study of nanoparticles and waveguides with a non-Hermitian approach.

Introduction

Over the nearly 40 millions books already digitized by Google in 2019, I selected the fraction of the total number of words that corresponds to the alternative spelling of the words 'nanoscience', 'metasurface', 'metamaterial' and 'nanophotonic'. All possible cases (singular/plural spelling) were included, and plotted as a function of the related book release year in Fig. 1. The data retrieved from Google only include the books referenced in Google Books English 2019 corpus, which exclude the scientific publications whose references are gathered in Google Scholar, yet highlighting two trends. These fields of research, although extremely present in the modern sciences, have only matured since the 2000s, which saw the rise of books (not publications) dealing with these subjects. If the term 'metamaterial' has entered the annals quite early, we observe only very recently an explosion of the use of the term 'metasurface' in the literature. Metasurfaces are a two-dimensional subclass of metamaterial, which more generally refer to materials that are engineered to have properties that are not found in naturally occurring materials. They are made from assemblies of multiple elements, arranged in repeating patterns, at scales that are smaller than the wavelengths of the phenomena they influence. Those materials can be encountered in various domains of physics: mechanics [START_REF] Lee | Micro-/nanostructured mechanical metamaterials[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF][START_REF] Surjadi | Mechanical metamaterials and their engineering applications[END_REF], acoustics [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF][START_REF] Zangeneh-Nejad | Active times for acoustic metamaterials[END_REF], or electromagnetism [START_REF] Itoh | Electromagnetic metamaterials: transmission line theory and microwave applications[END_REF][START_REF] Engheta | Metamaterials: physics and engineering explorations[END_REF][START_REF] Cui | Metamaterials[END_REF]. iii In the latter domain, a metamaterial is usually fabricated by modulating the permittivity of a medium. This implies patterning of subwavelength structures, as in one of the most famous scientific works published in Science in 2006, where a team from Dukes University successfully predicted and demonstrated an invisibility cloak in the microwave domain [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF]. This pioneering work is a direct application of transformation optics, where the change on the geometry of the copper split ring resonator affects the scattering of an RF wave (Fig. 2) by locally modifying the effective optical index of the material. The tour de force that represents the modulation of the optical index on each concentric cylinder of such metamaterial made it possible to redirect the light around a singularity of space, strongly decreasing the impact of the presence of a hidden copper cylinder on the propagation of a RF wave (Fig. 2 right panel). This work follows the first experimental demonstration of negative effective refraction index in a 2D array of repeated unit cells of copper strips and split ring resonators on interlocking strips of standard circuit board material [START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF].

Beyond the radio-frequency domain, the possibility of implementing metamaterials in optics has stemmed early [START_REF] Smith | Metamaterials and negative refractive index[END_REF], given that the modulation of the optical index could be achieved at nanoscale, below the wavelength of the visible or near infrared (NIR) light. The first class of materials envisaged was photonic crystals [START_REF] Govyadinov | Metamaterial photonic funnels for subdiffraction light compression and propagation[END_REF][START_REF] Povinelli | Toward photonic-crystal metamaterials: Creating magnetic emitters in photonic crystals[END_REF][START_REF] Lourtioz | Photonic crystals and metamaterials[END_REF][START_REF] Wehrspohn | Nanophotonic materials: photonic crystals, plasmonics, and metamaterials[END_REF]. A particularly promising sub-class of photonic [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF]. iv metamaterials are optical metasurfaces, which can be composed of arrays of nanoantennas with sub-wavelength size and separation. They enable light-matter interaction in compact optical components by confining light in sub-wavelength volumes [START_REF] Schuller | Plasmonics for extreme light concentration and manipulation[END_REF][START_REF] Baumberg | Extreme nanophotonics from ultrathin metallic gaps[END_REF] or control spontaneous emission [START_REF] Pelton | Modified spontaneous emission in nanophotonic structures[END_REF] from single molecules [START_REF] Anger | Enhancement and quenching of single-molecule fluorescence[END_REF] and quantum-dots [START_REF] Curto | Unidirectional emission of a quantum dot coupled to a nanoantenna[END_REF], while nanopatterned 2D or 3D arrays can manipulate light propagation [START_REF] Shaltout | Spatiotemporal light control with active metasurfaces[END_REF][START_REF] Kamali | A review of dielectric optical metasurfaces for wavefront control[END_REF][START_REF] Zhang | Multidimensional manipulation of wave fields based on artificial microstructures[END_REF] and group velocity [START_REF] Notomi | Manipulating light with strongly modulated photonic crystals[END_REF] thanks to periodic arrangements of metallic or dielectric nanoresonators. In recent years, the latter have drawn the attention of the scientific community, especially due to the phenomenological richness of highly multi-mode Mie-type resonators. During my PhD, we decided to opt for this paradigm: constructing of the metasystem from the properties of a single element, which is a optical nanoresonator. By modeling and engineering its modal properties and its interactions with the environment, we explored possibilities offered by dielectric metamaterials at the nanoscale to implement new functionalities and phenomena. Two fundamental challenges have oriented my work, at the interface between metamaterials and guided photonics: 1) the effect of interactions in finite-size systems, 2) the generation of quantum states of light via second-order nonlinearity in nanostructures.

The work reported in this manuscript was mostly carried out in the research group Dispositifs Optiques Non-linéaires (DON), within the laboratory Matériaux et Phénomènes Quantiques (MPQ) at Université Paris Cité, under the supervision of professor Giuseppe Leo and doctor Adrien Borne. During the last few years, the research work of the group has focused on the study of dielectric nanoantennas, developing state-of-the-art sample fabrication methods in the framework of international collaborations [START_REF] Gili | All-dielectric nonlinear nanophotonics[END_REF][START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF], but also deepening the modeling for nonlinear nanophotonic applications [START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF]. In the direct continuity of this research, we aimed to explore the two questions presented above, in addition to the fabrication of samples for international projects. Part of it has been the result of an international collaborative work involving several universities, namely Friedrich Schiller University of Jena, the King's College in London, the Polytechnic University of Milan, and the University Sapienza in Rome. I also had the chance to collaborate with the C2N laboratory in Palaiseau and the CEA of Grenoble, our close collaborators and suppliers of epitaxially grown AlGaAs wafers. My personal contributions range from the elaboration of a new theoretical model to describe quantum nonlinear phenomena at the nanoscale to the development of new fabrication processes for advance light manipulation with metasurfaces. I conceived original algorithms to automatize and scale-up our capability when systematically studying the resonances of nano-antennas. Finally, I implemented and worked on new experimental setup both for guided optics and measurements in the single-photon regime.

The manuscript, divided into five chapters and a general conclusion, is organized as follows:

Chapter I provides the fundamental tools needed to understand the manuscript. The first part focuses on the light-matter interaction at the nanoscale, and introduces an essential tool to describe open optical cavities: quasi-normal modes. In a second section, quantum physics notions allowing to describe two-photon states are introduced in the framework of nonlinear optics.

In particular, it will be explained what issues accompany the study of parametric fluorescence in nanoscopic systems, as well as the protocols and quantities to characterize the generated photonic states. Finally, topological physics notions motivating the study of coupling between elements of finite size systems are presented, in particular with respect to guided and nonlinear photonics.

Chapter II presents the essential techniques for the fabrication of the devices studied during my PhD. Our processes are based on a top-down approach involving electron-beam lithography and anisotropic etching to nanostructure epitaxial III-V semiconductors. The fabricated samples allow the study of linear and nonlinear properties of AlGaAs nanoresonators, reported in Chapters III and IV. The international projects I had the opportunity to take part in will serve as practical cases to illustrate the deployment of these state-of-the-art techniques to realize metasurfaces operating in reflection as well as in transmission, and exploiting symmetry breaking to explore new ways to confine or control light.

Chapter III presents the progress made in the modeling, design and measurement attempts of entangled photon sources. The studied states are generated by parametric fluorescence (or spontaneous parametric down conversion), a second-order nonlinear probabilistic process. The idea here is to take advantage of the strong nonlinearity of AlGaAs, coupled with the large possibilities offered by the nanostructuring of the epitaxial layers to control the spectrum, the emission pattern, as well as the polarization of the produced states. I will first present a new formalism, using quasi-normal modes and Green's tensors for the modeling of the nonlinear interaction and the generation of polarization states by parametric fluorescence. We identify these states by quantum tomography, and we quantify the degree of entanglement by their Schmidt number. In a second step, I will focus on our experimental efforts to provide a proof of the parametric generation of photon pairs in [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs metasurfaces. The last part of this chapter extends the theoretical study to [011]-AlGaAs nanoresonators, allowing frequency conversion processes where pump, signal, and idler photons copropagate at normal incidence to the substrate.

Chapter IV addresses the collective behaviors of the elements constituting a metasurface, or at least an organized set of nano-antennas. In order to study these structures, we have chosen a modal paradigm, relying on quasi-normal modes . We propose an approach in which the information extracted from the study of ensembles of a few nanoresonators allows to predict the modal behavior of larger systems of finite size, thanks to a non-Hermitian Hamiltonian formalism. To overcome the numerical limits relating to the study of resonator assemblies beyond Bloch's theorem, we have looked for an experimental validation of our model. Samples for AlGaAs-on-AlOx single nanoparticle spectroscopy were therefore fabricated, and measured on the bright/dark field spectroscopy benches and in the cathodoluminescence setup of our collaborators in Jena vi and London.

Chapter V extends the considerations of Chapter IV to guided optics, to design and engineer compact resonators based on whispering-gallery modes. Inspired by topological tight-binding Hamiltonian, we study ensembles of concentric-ring resonators, where the control on the gap between two rings modulates the couplings between each resonant mode. Working towards the design of topologically protected whispering-gallery modes, we introduced a new model based on input-output theory to understand the spectral behavior of those systems. The first experimental verification is performed on AlGaAs-on-AlOx rings fabricated on a mesa, who are probed thanks to the evanescent coupling to a bus waveguide in transmission measurements. This work constitutes the first step towards the implementation of compact topologically protected ring resonators, which promise reduction of losses and multiple-knob tuning of nonlinear frequency conversion in guided photonic systems. vii viii

Résumé long

Les métasurfaces sont une sous-classe bidimensionnelle des métamatériaux, qui désignent plus généralement des matériaux conçus pour avoir des propriétés qui n'existent pas dans les matériaux naturels. Ils sont constitués d'assemblages de plusieurs éléments, disposés selon des motifs répétitifs, à des échelles plus petites que les longueurs d'onde des phénomènes qu'ils influencent. Ces matériaux peuvent être rencontrés dans différents domaines de la physique : mécanique [START_REF] Lee | Micro-/nanostructured mechanical metamaterials[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF], acoustique [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF][START_REF] Zangeneh-Nejad | Active times for acoustic metamaterials[END_REF], ou électromagnétisme [START_REF] Itoh | Electromagnetic metamaterials: transmission line theory and microwave applications[END_REF][START_REF] Engheta | Metamaterials: physics and engineering explorations[END_REF][START_REF] Cui | Metamaterials[END_REF]. Dans ce dernier domaine, un métamatériau est généralement fabriqué en modulant la permittivité d'un milieu. Cela implique de modeler des structures sub-longueur d'onde, comme dans l'un des travaux scientifiques les plus célèbres publiés dans Science en 2006, où une équipe de l'université Dukes a prédit et démontré avec succès une cape d'invisibilité dans le domaine des micro-ondes [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF]. Ce travail de pionnier est une application directe de l'optique de transformation, où le changement de la géométrie du résonateur en anneau fendu de cuivre affecte la diffusion d'une onde RF en modifiant localement l'indice optique effectif du matériau. Le tour de force que représente la modulation de l'indice optique sur chaque cylindre concentrique d'un tel métamatériau a permis de rediriger la lumière autour d'une singularité de l'espace, diminuant fortement l'impact de la présence d'un cylindre de cuivre caché sur la propagation d'une onde RF. Ces travaux font suite à la première démonstration expérimentale d'un indice de réfraction effectif négatif dans un réseau 2D de cellules unitaires répétées de bandes de cuivre et de résonateurs en anneau fendu sur des bandes emboîtées de matériau de circuit imprimé standard : [START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF]. Au-delà du domaine des radiofréquences, la possibilité de mettre en oeuvre des métamatériaux en optique est apparue très tôt [START_REF] Smith | Metamaterials and negative refractive index[END_REF], étant donné que la modulation de l'indice optique pouvait être réalisée à l'échelle nanométrique, en dessous de la longueur d'onde de la lumière visible ou proche infrarouge (NIR). La première classe de matériaux envisagée était les cristaux photoniques [START_REF] Govyadinov | Metamaterial photonic funnels for subdiffraction light compression and propagation[END_REF][START_REF] Povinelli | Toward photonic-crystal metamaterials: Creating magnetic emitters in photonic crystals[END_REF][START_REF] Lourtioz | Photonic crystals and metamaterials[END_REF][START_REF] Wehrspohn | Nanophotonic materials: photonic crystals, plasmonics, and metamaterials[END_REF]. Une sous-classe particulièrement prometteuse de métamatériaux photoniques est constituée par les métasurfaces optiques, qui peuvent être composées de réseaux de nanoantennes de taille et de séparation inférieures à la longueur d'onde. Elles permettent l'interaction lumière-matière dans des composants optiques compacts en confinant la lumière dans des volumes sub-longueur d'onde [START_REF] Schuller | Plasmonics for extreme light concentration and manipulation[END_REF][START_REF] Baumberg | Extreme nanophotonics from ultrathin metallic gaps[END_REF] ou contrôlent l'émission spontanée [START_REF] Pelton | Modified spontaneous emission in nanophotonic structures[END_REF] de molécules uniques [START_REF] Anger | Enhancement and quenching of single-molecule fluorescence[END_REF] et de points quantiques [START_REF] Curto | Unidirectional emission of a quantum dot coupled to a nanoantenna[END_REF], tandis que les réseaux 2D ou 3D nanopatternés peuvent manipuler la propagation de la lumière [START_REF] Shaltout | Spatiotemporal light control with active metasurfaces[END_REF][START_REF] Kamali | A review of dielectric optical metasurfaces for wavefront control[END_REF] et la vitesse de groupe [START_REF] Notomi | Manipulating light with strongly modulated photonic crystals[END_REF] grâce à des arrangements périodiques de nanorésonateurs métalliques ou diélectriques. Ces dernières années, ces derniers ont attiré ix l'attention de la communauté scientifique, notamment en raison de la richesse phénoménologique des résonateurs de type Mie hautement multimodes. Au cours de ma thèse, nous avons décidé d'opter pour ce paradigme : construire le métasystème à partir des propriétés d'un seul élément, qui est un nanorésonateur optique. Par la modélisation et l'ingénierie de ses propriétés modales et de ses interactions avec l'environnement, nous avons exploré les possibilités offertes par les métamatériaux diélectriques à l'échelle nanométrique pour mettre en oeuvre de nouvelles fonctionnalités et de nouveaux phénomènes. Deux défis fondamentaux ont orienté mon travail, à l'interface entre les métamatériaux et la photonique guidée : 1) l'effet des interactions dans les systèmes de taille finie, 2) la génération d'états quantiques de la lumière via la non-linéarité de second ordre dans les nanostructures.

Les métasurfaces étudiées dans l'équipe DON du laboratoire MPQ forment un arrangement d'une multitude de nanoéléments, fabriqués à partir d'une approche "top-down" en structurant et gravant une couche épitaxiée de semi-conducteur III-V. L'étape de structuration, cruciale pour garantir la qualité de l'échantillon, repose sur l'utilisation d'un microscope électronique à balyage, ou MEB (SEM en anglais), dans laquelle un faisceau d'électrons balaie une surface pour en faire l'image. Les électrons du faisceau interagissent avec les atomes à différentes profondeurs dans l'échantillon, et perdent de l'énergie. L'énergie échangée est dissipée par le système par chauffage, ou sous forme d'électrons secondaires de faible énergie, d'électrons rétrodiffusés ou d'émission de lumière, qui produisent un signal traçable. Le signal récupéré porte des informations sur la topographie et la composition du matériau, et peut atteindre une résolution sub-nanométrique. Les SEM sont également utilisés comme alternative à la lithographie laser et UV : la lithographie par faisceau d'électrons (EBL) permet d'imprimer un masque de résine avec une précision nanométrique à l'aide d'un SEM. Au cours d'une EBL, le faisceau d'électrons focalisé balaie la résine, ce qui modifie localement ses propriétés chimiques. Une fois développé, le masque restant protège les futurs nano-éléments lors de la gravure de la couche épitaxiée semiconductrice. Cette approche versatile nous a permis à la fois de produire des échantillons pour des projets de recherche locaux, mais aussi dans le cadre de collaborations internationales avec des groupes de recherche britaniques, italiens, et allemands, développé dans le second chapitre du manuscrit.

Le premier chapitre du manuscrit présente les principaux outils et concepts permettant de s'orienter dans les travaux originaux entrepris au cours de ma recherche doctorale. La diffusion de la lumière par des particules plus petites que la longueur d'onde est présentée dans la première section, ainsi que le principal outil utilisé pour la modélisation des nanorésonateurs : les modes quasi-normaux. Les nanorésonateurs optiques composant les métasurfaces sont la plupart du temps des systèmes très fuyants, c'est-à-dire qu'ils confinent mal la lumière et se couplent facilement à leur environnement. En électrodynamique, les modes quasi-normaux sont les solutions harmoniques des équations de Maxwell sans source dans les systèmes ouverts pour les fréquences propres complexes, et constituent une base pour décrire les phénomènes x électromagnétiques linéaires et non linéaires dans les nanostructures. Les modes quasi-normaux sont utilisés de manière intensive dans les chapitres III et IV, où les mécanismes fondamentaux de la physique décrite nécessitent de modéliser l'interaction entre le champ confiné dans les nanorésonateurs. Ensuite, j'introduis les notions d'optique non linéaire et de physique quantique nécessaires à l'étude des états biphotons générés par la conversion paramétrique descendante spontanée (SPDC), qui favorise la désintégration d'un photon de pompe en deux photons dits signal et idler suivant la règle de sélection de conservation de l'énergie : ω p = ω i + ω s . Contrairement à d'autres processus non linéaires du second ordre qui sont la génération de fréquences de différence et de somme, le SPDC ne nécessite qu'un seul champ d'entrée. La génération de champs électromagnétiques à ω s et ω i provient de l'interaction de la pompe avec les fluctuations du vide à travers le tenseur non linéaire du matériau. La SPDC a la particularité d'être un processus spontané, donc non déterministe, et de générer des états quantiques à deux photons [START_REF] Klyshko | Photons and nonlinear optics[END_REF], qui sont un ingrédient clé pour la communication quantique [START_REF] Tanzilli | PPLN waveguide for quantum communication[END_REF]. Une autre condition affecte l'efficacité de conversion du SPDC : la conservation de la quantité de mouvement ou condition d'adaptation de phase [START_REF] Boyd | Chapter 1 -The nonlinear optical susceptibility[END_REF], c'est-à-dire k p = k s + k i , qui découle des équations d'ondes couplées des champs en interaction [START_REF] Boyd | Chapter 1 -The nonlinear optical susceptibility[END_REF]. Lorsque l'accord de phase est respecté, l'interférence constructive entre la pompe et les champs générés permet un transfert d'énergie efficace. Cependant, lorsque la condition d'adaptation de phase n'est pas satisfaite, les amplitudes des champs générés le long de la direction de propagation oscillent en raison de la présence d'un terme imaginaire, et ne s'accumulent pas. En général, la conservation de la quantité de mouvement n'est pas toujours satisfaite, ce qui réduit fortement l'efficacité de la conversion. L'idée principale derrière l'utilisation de nano-éléments est de réduire la longueur d'interaction en dessous de la longueur de cohérence L c = π(∆k) -1 , ce qui relaxe la condition de correspondance de phase. Enfin, quelques éléments de physique topologique sont donnés, pour contextualiser et mettre en évidence les enjeux de l'étude des nanoparticules et des guides d'ondes avec une approche non-hermitiennedans les chapitres IV et V, inspirés par une approche de liaison serrée.

Le chapitre III rapporte mes travaux la modélisation de la SPDC dans des nano-résonateurs. Du fait de la relaxation des conditions d'accord de phase à l'échelle sub-longueur d'onde, les nanostructures sont récemment apparues comme une plateforme très prometteuse pour l'obtention de sources de SPDC compactes aux propriétés spectrales, spatiales et de polarisation contrôlables via leurs nombreux degrés de liberté géométriques. Des travaux expérimentaux récents ont déjà démontré la génération large bande de paires de photons dans des métasurfaces et nanorésonateurs diélectriques non linéaires. Néanmoins, afin d'aller plus loin dans le contrôle des paires générées et de permettre l'ingénierie d'états quantiques, une description théorique de la SPDC dans de telles structures est nécessaire.Ainsi, nous présentons un modèle permettant l'étude du processus de fluorescence paramétrique dans des systèmes ouverts et de complexités arbitraires. Notre approche généralise un formalisme récent en développant les fonctions de Green du système sur la base des modes quasi-normaux (QNM), modes propres des systèmes ouverts. Cette approche nous permet d'exprimer le taux de génération de paires de photons comme : q i ,qs α,β,γ

d i,q i d s,qs n,m ξ n,m (ω i , ω s ) E m,q i (r i ) E n,qs (r s ) (1)
avec omega s et ω i les pulsations des photons signal et complémentaire générés, r s et r i leurs positions, q s et q i leurs directions de polarisation, Ẽm et Ẽn leurs QNMs associés, et ξ n,m un facteur de couplage correspondant à l'intégrale de recouvrement entre les champs en interaction dans le milieu non linéaire de susceptibilité χ (2) :

ξ n,m (ω i , ω s ) = 1 µ 2 0 1 (ω s -ω n ) ω n (ω i -ω m ) ω m ¢ V χ (2) αβγ (r)E p,γ (r) E n,β (r) E m,α (r)d 3 r (2)
Pour illustrer ce formalisme, nous l'appliquons à un nanocylindre d'AlGaAs de croissance (100) (esquissé en Fig. 3a). Le champ de pompe est une onde plane à 740 nm se propageant selon la direction de l'axe de révolution du nanocylindre et polarisée selon l'axe x. Les QNMs de cette géométrie sont calculés numériquement une seule fois, à l'aide d'un solveur de type éléments finis, et introduits dans le formalisme analytique pour déduire l'ensemble des propriétés des photons générés présentées sur la Fig. 3. La figure Fig. 3b représente le spectre du facteur de couplage des modes et révèle que seules quelques paires de QNMs se couplent efficacement au champ pompe lors du processus non linéaire. Proches de la dégénérescence, les photons signal et complémentaire sont majoritairement générés dans les résonances dipolaires magnétiques de la structure. La figure Fig. 3c présente la distribution angulaire du taux de génération correspondant à une détection conjointe de photons co-propagatifs. Ces paires sont essentiellement générées dans le plan (yz) et le long de deux de lobes privilégiés. Finalement nous présentons xii en Fig. 3d la matrice densité de l'état de polarisation des paires de photon à la dégénérescence reconstruite numériquement par une approche tomographique.

Les travaux rapportés au chapitre III soulignent que les résonances jouent un rôle central dans le contrôle des processus quantiques non linéaires en nanophotonique, lorsque la condition de correspondance de phase est relâchée [START_REF] Marino | Zeroorder second harmonic generation from AlGaAs-on-insulator metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF]. Couplages techniques dans les métamatériaux à l'échelle nanométrique [START_REF] Li | Plasmonic particle-on-film nanocavities: a versatile platform for plasmon-enhanced spectroscopy and photochemistry[END_REF][START_REF] Spinelli | Optical impedance matching using coupled plasmonic nanoparticle arrays[END_REF][START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Smirnova | Nonlinear topological photonics[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF][START_REF] Zhang | Magnetic and electric coupling effects of dielectric metamaterial[END_REF][START_REF] Vial | A coupling model for quasi-normal modes of photonic resonators[END_REF] offre un levier alternatif pour aller au-delà de l'optimisation de la géométrie d'un nano-élément : [START_REF] Rybin | High-Q supercavity modes in subwavelength dielectric resonators[END_REF][START_REF] Koshelev | Meta-optics and bound states in the continuum[END_REF]. L'ingénierie de leur symétrie et de leur facteur de qualité pourrait en principe faciliter la mise en oeuvre de processus non linéaires, par la médiation de l'ingénierie de couplage entre les nanoéléments. Dans le chapitre IV, nous présentons un modèle simple reposant sur le couplage à un bain effectif constitué d'un continuum de modes pour décrire des systèmes de résonateurs couplés, et le testons sur des chaînes de nanocylindres diélectriques accessibles aux expériences. Les constantes de couplage effectives, qui dépendent de manière non-triviale de la distance entre les résonateurs, sont extraites de simulations numériques dans le cas de seulement deux éléments couplés. Le modèle prédit avec succès la nature dispersive et réactive des modes pour les configurations à résonateurs multiples, comme validé par les solutions numériques. Il peut être appliqué à des systèmes plus grands, qui sont difficilement solubles avec des approches par éléments finis. Les défis expérimentaux et les tentatives de spectroscopie d'un tel système couplé seront également rapportés et analysés dans la perspective du modèle de bain non-hermitien.

Enfin, le chapitre V propose la mise en oeuvre de microrésonateurs topologiques reposant sur le couplage entre des guides d'ondes circulaires concentriques. Ce travail constitue la première étape vers un projet plus ambitieux : l'ingénierie de systèmes multimodaux riches pour des applications photoniques intégrées, comme les peignes de fréquence protégés [START_REF] Mittal | Topological frequency combs and nested temporal solitons[END_REF], la génération de paires de photons [START_REF] Yang | Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators[END_REF][START_REF] Steiner | Ultrabright entangled-photon-pair generation from an AlGaAs-on-insulator microring resonator[END_REF], ou portes logiques [START_REF] Espinosa-Ortega | Complete architecture of integrated photonic circuits based on and and not logic gates of exciton polaritons in semiconductor microcavities[END_REF][START_REF] Politi | Integrated quantum photonics[END_REF][START_REF] Fu | Silicon photonic crystal all-optical logic gates[END_REF][START_REF] Crespi | Integrated photonic quantum gates for polarization qubits[END_REF].

xiii xiv

Abstract English

Nanofabrication has played a key role in the development of metamaterials based on III-V semiconductors, which are increasingly studied to perform frequency conversion or wavefront control in phase or amplitude. Progress in this field has enabled the design of metamaterials at the nanoscopic scale, involving nano-antennas, sometimes with complex shapes and whose arrangement allows the generation of holograms, non-gaussian beams, or the implementation of topological phases. In this thesis, the constituting elements of the dielectric metasurfaces are studied in the framework of non-Hermitian physics, with the use of quasi-normal modes to model low-quality-factor cavities resonating in the near infrared. We are interested in the possibilities offered by structuring epitaxial thin films of AlGaAs, a non-centrosymmetric material with strong second-order nonlinearity. In particular, we explore their potential as compact sources of entangled photon pair generated by parametric fluorescence. This work demonstrates the possibilities brought by the modal study of dielectric nano-antennas applied to the engineering of two-photon states for communications or quantum computing. As a complement to this study, I have also proposed a new approach to model the interactions between the elements constituting finite size metamaterials: losses, radiation and interactions with the environment are described by a model based on the coupling to a continuum of modes. Our model has been used to demonstrate that long-range couplings exist in metamaterials, and that the control of interactions between nanoantennas is a lever to exacerbate the confinement of light in each nano-element. This modal approach has also been applied to coupled photonic waveguides, with a similar perspective: that of realizing compact topological phases by controlling the coupling between each element. For each of these axes, the modeling efforts are systematically coupled with measurement, as well as the fabrication of samples in order to validate our new theoretical approaches. The fabrication processes and skills acquired during this thesis have also been put to use in the context of international collaborations, aiming at designing and fabricating new nonlinear active metamaterials to precisely control the light emitted at both the fundamental and second harmonic wavelengths.

xv

Français

La nanofabrication a joué un rôle clé dans le développement de métamatériaux basés sur des semi-conducteurs III-V, de plus en plus étudiés pour réaliser des opérations de conversion de fréquences ou de contrôle de front d'onde en phase ou en amplitude. Les progrès en la matière ont permis de concevoir des métamatériaux à l'échelle nanoscopique, impliquant des nano-antennes aux formes parfois complexes et dont l'arrangement permet de générer des hologrammes, des faisceaux non-gaussiens, ou encore d'implémenter des phases topologiques. Dans cette thèse, les éléments constituant les métasurfaces diélectriques sont étudiés dans le cadre de la physique non-Hermitienne, avec l'utilisation des modes quasi-normaux pour modéliser des cavités à faibles facteurs de qualité résonantes dans le proche infrarouge. Nous nous sommes intéressés aux possibilités offertes par la structuration de couches minces épitaxiées d'AlGaAs, un matériau non-centrosymétrique à forte non-linéarité du second ordre. En particulier, nous explorons leur potentiel comme sources compactes de paires de photons intriqués générés par fluorescence paramétrique. Ce travail démontre les possibilités apportées par l'étude modale de nano-antennes diélectriques appliquée à l'ingénierie des états à deux photons pour les communications ou le calcul quantique. En complément de cette étude, j'ai aussi proposé une nouvelle approche pour modéliser les interactions entre les éléments constituant des métamatériaux de taille finie: les pertes, le rayonnement et les interactions avec l'environnement y sont décrites par un modèle reposant sur le couplage à un continuum de modes. Notre modèle a notamment permis de démontrer que des couplages longue distance existaient dans les métamatériaux, et que le contrôle des interactions entre nano-antennes constituaient un levier pour exacerber le confinement de la lumière dans chaque nano-élément. Cette approche modale a également été appliquée à des guides d'ondes photoniques couplés, avec une perspective commune: réaliser des phases topologiques compactes en contrôlant le couplage entre chaque élément. Pour chaque axe évoqué, les efforts de modélisation vont systématiquement de pair avec la fabrication d'échantillons et leur mesure dans le but de valider nos nouvelles approches théoriques. Les procédés de fabrication et les compétences acquises durant cette thèse ont aussi été mis à contribution dans le cadre de collaborations internationales, visant à concevoir et fabriquer de nouveaux métamatériaux non-linéaires actifs pour contrôler avec précision la lumière émise à la fois aux longueurs d'onde fondamantales et à la seconde harmonique.

Table of contents

Introduction iii

Résumé long ix

Abstract xv

I.1 Nanoresonators in photonics

This first section will address the problem of the interaction of light with nanoparticles. More specifically, I will focus on light scattering by sub-wavelength dielectric particles, and present the concepts and tools that enable us to perform efficient numerical modeling of such systems. To simplify the presentation, we will consider the case of a particle suspended in a uniform and loss-less medium.

I.1.1 Light-matter interaction at nanoscale

In classical electrodynamics, any phenomenon can be described by Maxwell's equations in matter, which read:

∇ × E(r, t) = - ∂B(r, t) ∂t (I.1) ∇ • B(r, t) = 0 (I.2) ∇ • D(r, t) = ρ(r, t) (I.3) ∇ × H(r, t) = ∂D(r, t) ∂t + J (I.4)
E and H are the electric and magnetic fields, D the displacement field, B the magnetic flux density, and J and ρ the current and charge densities. The two last terms act as sources for the electromagnetic field. By introducing the polarization of the field P and the magnetization M, two constitutive relations describe the fundamental properties of electromagnetic field:

B = µH = µ 0 (H + M) (I.5) D = E = 0 E + P (I.6)
where = r 0 the electric permittivity and µ = µ r µ 0 the magnetic permeability tensors of the medium in which the electromagnetic field propagates. Those tensors become scalar quantities in the case of an isotropic medium, defining = r 0 with the relative permittivity of the medium r and the vacuum permittivity 0 = 8.85 × 10 -12 F.m -1 , and the relative permeability of the medium and the vacuum permeability µ 0 = 4π × 10 -7 H.m -1 . Maxwell's equations can be written in the spectral domain both for space and time by applying Fourier transforms.

E inc (r, ω) = E 0 e i(k inc •r-ωt) (I.7)
with E 0 = E 0 e 0 . This impinging field interacts with the object, by exchanging energy with it: part of this energy is absorbed by the particle (then thermally dissipated), while some is scattered (linearly or non-linearly) in directions that are not collinear to the initial propagation direction. Thus, the resulting scattered E s field can be written as:

E s = S(n)E 0 e i(k inc •r-ωt) r (I.8)
where S is a second-order tensor called the scattering matrix, has the dimension of a distance, and is function of the scattering direction n, therefore of the spherical coordinates (φ, θ), and of the angular frequency of the field ω. We introduce the effective scattering cross-section σ s , defined such that the product σ s I inc (with

I inc = /µ/2|E 0 | 2
) is equal to the power scattered by the particle in all directions of space (see Fig. I.1a). This total scattered power being obtained by integration over a sphere of infinite radius, Eq. I.8 formally implies that (see Fig.

σ s = ¢ 4π |S(n)| 2 dΩ (I.9)
Likewise, we introduce the effective absorption cross-section σ a , by matching the product Chapter I I.1 Nanoresonators in photonics σ a I inc with the power absorbed by the particle. Note that in the absence of absorption, σ a = 0. Finally, let us introduce the effective extinction cross-section, so that the product σ ext I inc equals the energy transferred from the incident field to the particle. Considering scattering and absorption, it reads by definition:

σ ext = σ a + σ s (I.10)
This concept will be central to Chap. IV, where the linear scattering of white light is used to probe the spectral properties of dielectric nano-particles clusters. In those experiments, the absorption is neglected, yielding σ ext ≡ σ s .

I.1.1.2 Optical theorem

Poynting theorem establishes the energy conservation in the interaction between an electromagnetic field and a particle. At time t < t 0 , an external electromagnetic field excites the system in a steady-state and all the energy contributions are balanced. At t = t 0 , the external source is switched off, and the system enters a transient state at the end of which all the stored energy is lost in time by radiation or transformed into heat. Applying the divergence theorem on a closed surface S that encapsulates a volume V containing the particle, the stored energy W in V when an incident field E inc illuminates the particle is given by:

-W (V, t) = ¢ ∞ t 0 S (E × H) • ndsdt + ¢ ∞ t 0 ¢ V J • Edrdt (I.11)
In the time-harmonic domain, the complex Poynting theorem in presence of an external source gives the following relation:

- 1 2 ¢ V J * • Edr = 2iω ¢ V wdr + S Π • nds (I.12)
where the electromagnetic energy density w = E • D * /2 + B • H * /2 and the Poynting vector Π = E × H * have been introduced. Note that in the absence of absorption, the term in J vanishes. Therefore, the Poynting vector can be decomposed on the field components associated to the incident, scattered and extincted fields [START_REF] Bohren | Absorption and Scattering by an Arbitrary Particle[END_REF], expressed as:

Π = Π inc + Π s + Π ext = (E inc × H * inc ) + (E s × H * s ) + (E inc × H * s + E s × H * inc ) (I.13)
With < • • • > the time averaging operation, this allows to express the power associated to each field:

P = S < Π > •nds (I.14)
Chapter I or alternatively for the extinction power, with divergence theorem and Maxwell equations:

P ext = 1 2 S Re (E inc × H * s + E s × H * inc ) • nds = ω 2 ¢ V Im (P • E * inc ) d 3 r (I.15)
In the case a linearly polarized plane wave as defined in Eq. I.7, and by definition of the extinction cross section, it is possible to demonstrate that this extinction cross-section is related to scattering amplitude in forward direction (k inc ) [START_REF] Jackson | Classical electrodynamics[END_REF]. This relation is established by expressing the scattered field of Eq. I.8 as a spherical wave generated by point source (local polarisation). Using Eq. I.15, we obtain:

P ext = 2π µ 0 ω Im [E * 0 e 0 • S(u inc ) • E 0 e 0 ] (I.16)
With the definition P ext = σ ext I inc , this finally leads to:

σ ext = 4π k inc Im [e 0 • S(u inc ) • e 0 ] (I.17)
where u inc is the unit vector associated to k inc . This result constitutes the optical theorem: by measuring the complex amplitude of the forward-scattered field, one can access the extinction coefficient of the incident wave by the particle through scattering. The measure of a the forward amplitude of a wave therefore gives information on a energetic quantity, a concept on which the measurements and scattering simulations of Chap. IV heavily rely.

I.1.1.3 Influence of the particle size: towards the nanoscale

The concept of light scattering by a single particle introduced previously can be applied to the study of metamaterials in nanophotonics. In this context, the elements constituting the systems of study often have a characteristic dimension δ lower than the wavelength λ of the light with which they interact. Depending on the relative ratio between δ and λ, two regimes are distinguished (Fig. I.2): Rayleigh scattering when δ λ, and Mie scattering when δ λ.

Rayleigh scattering

In the limit where δ λ, it is possible to consider that the electromagnetic field inside the particle is uniform, and that the particle can be treated as a point source: it is the dipolar approximation. To simplify the situation, let us consider a spherical particle of radius r as sketched in Fig. I.2a. It is possible to demonstrate that the scattering cross-section of this particle is proportional to the modulus square of its polarizability α: With this formula, we retrieve Rayleigh scattering law (σ s ∝ ω 4 ) when α(ω) cst.

σ s = k 4 6π |α(ω)| 2 (I.18)
Mie scattering However, when the particle characteristic dimension is no longer small with respect to λ, then the dipolar approximation is no longer valid. There exists a rigorous theory for the scattering of spherical particles, Mie theory, which provides the scattered field in the form of a multipolar expansion [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. The size of typical elements that we study fits well into Mie scattering regime (Fig. I.2b), but they are not spheres. We rather consider cylinders for nanofabrication reasons, whose height and diameter is about a few hundreds nanometers. These particles present resonances from the near infrared (NIR) to the visible, which allow nonlinear frequency conversion given that those particles are made out of an optically nonlinear material [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF].

Although it is possible to model the behavior of these particles with Mie multipoles, other tools, like quasi-normal modes, offer more competitive solutions especially when considering particles of arbitrary shape, or ensembles of several particles.

I.1.2 Quasi-normal modes for nanophotonics

This section constitutes a brief introduction to the concept of quasi-normal modes (QNMs) applied to optical systems. The rigorous presentation of this formalism and its implementation is among others reported in the third chapter of Carlo Gigli's PhD manuscript [START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF] and in the reference [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF], granting the state-of-the-art for the utilization of the quasi-normal mode to describe the multi-modal nanoresonators in the Mie scattering regime in the context of nonlinear optics in metamaterials.

I.1.2.1 QNMs in optics and their normalization

Optical nanoresonators are most of the time very leaky systems, meaning that they confine poorly light and couple easily to their environment. In electrodynamics, QNMs are the timeharmonic solutions of the source-free Maxwell equations in open systems for the eigenfrequencies ωm [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF][START_REF] Doost | Resonant state expansion applied to two-dimensional open optical systems[END_REF][START_REF] Doost | Resonant-state expansion applied to three-dimensional open optical systems[END_REF]:

   0 i ∇× (r,ωm) -i ∇× µ(r,ωm) 0       Ẽm (r) Hm (r)    = ωm    Ẽm (r) Hm (r)    (I.19)
with and µ the permittivity and permeability tensors at position r and angular frequency ω, and the field eigenvectors Ẽm and Hm . The eigenfrequencies are complex-valued ωm = Ω m -iΓ m /2, with Ω m the resonance frequency and Γ m the spectral full width at half maximum of the resonance. With the notation ψm = ( Ẽm , Hm ) t , any solution ψ = (E, H) t of the scattering problem with a source term (i -1 J 0 , 0) t can be expanded on the QMN basis:

ψ(ω, r) = n α n (ω) ψm (r) (I.20)
which is exactly correct if the orthonormal basis is complete. Eq. I.20 highlights the efficiency of the method: once the modes of a given system are known, it suffices to retrieve the excitation coefficients α n of any driving field to solve the scattering problem.

Nonetheless, the situation is different from Sec. I.1.1.2, where the frequency of the field was real. Here, for complex eigenfrequencies, the exponential term exp [i(k • rωm t)] of the outgoing wave diverges in the limit t → ∞, even if the system is dissipating energy. For this reason, normalization procedures involving the conjugate complex of electromagnetic fields cannot be applied to QNMs. Instead, their normalization reads [START_REF] De Lasson | Roundtrip matrix method for calculating the leaky resonant modes of open nanophotonic structures[END_REF][START_REF] Yan | Rigorous modal analysis of plasmonic nanoresonators[END_REF]:

Ẽm Ẽn = ¢ Ẽm • ∂ω ∂ω Ẽn + Hm • ∂ωµ ∂ω Hn d 3 r = δ m,n (I.21)
where the integration is performed on all space, δ m,n is the Kronecker symbol, and all the derivatives are evaluated at ωm and ωn .

I.1.2.2 Numerical implementation of the QNMs for scattering of light by a nanoparticle

To illustrate the application of QNMs to problems related to light resonance and scattering by sub-wavelength particles, let us solve the case of the plane-wave scattering by a dielectric sphere suspended in air. The dispersion of the dielectric is modeled by the single-pole Lorentz- Chapter I

Drude model [START_REF] Gehrsitz | The refractive index of Al x Ga 1-x As below the band gap: accurate determination and empirical modeling[END_REF], which has proven to be a good approximation for several III-V semiconductors, including GaAs. The plane wave interacts with the QNMs and the scattered field is reconstructed following Eq. I.20. The excitation coefficient α n of each QNM is expressed thanks to the auxiliaryfield formalism [START_REF] Yan | Rigorous modal analysis of plasmonic nanoresonators[END_REF], which completes the vector ψ with the following auxiliary fields:

P = -0 ∞ ω 2 p ω 2 -ω 2 0 + iωγ E (I.22) J = -iωP (I.23)
in the time-harmonic domain. The coefficients ω p , ∞ and γ come from the specific dispersion model used. Then, the excitation coefficients are therefore given by:

α n (ω) = ω ωn -ω 0 [ (ω n ) -air ] + 0 [ (ω n ) -∞ ] ¢ V particle Ẽn (r) • E inc (r, ω)d 3 r (I.24)
Note that so far, the particle has been suspended in air. In practice, the probed systems will be supported by a substrate, which occupies the bottom half of the calculation space. Then, E inc will be replaced by E b , the background electric field, which corresponds to the field of the system in the absence of the nanoparticle. When suspended in air, E inc = E b . Eq. I.24 gives an analytical framework for expansion on the QNM basis: for a given incident wave, the calculation of the α n coefficient only depends on the QNMs of the structure. • The size of the particle entails many resonances, which cover a wide spectral range. To accelerate the convergence of the FEM simulation, the spectral span is often divided into several intervals, reducing the number of iterations to compute a satisfying number of QNMs. Here, we typically run three simulations (Fig. I.3b): 1) around the telecom wavelength, region of interest with a low spectral density for the particles of this dimension;

Numerical methods

2) at intermediary wavelengths, in the NIR; 3) in the visible, where the density of modes increases drastically. This method is particularly pertinent for nonlinear optics applications [START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF][START_REF] Gigli | Dielectric metasurfaces for second harmonic generation and wavefront shaping[END_REF], because fundamental and up/down-converted fields cover a wide spectral range.

• Some the computed modes are not the QNMs of the structure, but PML modes. They correspond to the numerous points for Im(ω n ) > 40 THz in Fig. I.3b, which account for the openness of the system. For them, the field is essentially concentrated at the edge between the inner sphere of air and the PML. Although those modes are not physical, they grant basis completeness.

• The large values of Γ m of the QNMs make them contribute to the scattered field over a large spectral range (Fig. I.3c), which needs to be taken into account when reconstructing the near field and the emission diagram of the nanoparticle.

One important point is that modeling the scattering with the QNMs or Mie multipoles basis should be in principal equivalent when using the full basis. In practice, only a restricted part of these bases is used [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF][START_REF] Yan | Rigorous modal analysis of plasmonic nanoresonators[END_REF]. For spherical particle, the Mie poles being the solution of the eigenmode problem, they are the QNMs of the system. However, when the shape of the particle changes (cylinders, cubes, bow-tie, or more exotic shapes) and its symmetry class becomes different from that of a sphere, QNMs becomes a more natural basis, more adapted to the new geometry. An accurate description is in this context more demanding when using the Mie multipole basis, which is why we often prefer QMNs for the modeling of the nanopatterned samples fabricated in our cleanroom.

I.2 Quantum nonlinear optics

I.2.1 Nonlinear processes in optics

Before presenting the work done during my PhD on SPDC in dielectric nano-elements, this first section aims at recalling the foundations of nonlinear optics, which is the study of phenomena that result from the modification of the optical properties of a material because of the presence of an intense light field [START_REF] Boyd | Nonlinear Optics, Third Edition[END_REF]. The invention of the laser by Maiman [START_REF] Th | Stimulated optical radiation in ruby[END_REF] in 1960 enabled the discovery and first demonstration of second-harmonic generation in a quartz crystal the next year [START_REF] Franken | Generation of optical harmonics[END_REF]. Since then, systematic studies of optical nonlinearities have opened the path for the discovery of new effects, including frequency comb generation, stimulated Raman scattering, and Kerr effect among others.

I.2.1.1 Nonlinear polarization and nonlinear wave equation

When an optical wave (characterized by its electric field E(t)) propagates in a material at low to moderate intensities (< 1 MW.cm 2 ), the response of this medium (the polarization P(t), dipole per unit of volume) is proportional to the E(t). The proportionality coefficient in this linear description is called linear optical susceptibility χ (1) . For some materials, a sufficiently intense light can affect their electrons dynamics, then their optical properties. A generalized form of the polarization expresses the nonlinear response of an isotropic homogeneous medium without losses and dispersion in the presence of an electric field E(t):

P(t) = 0 n χ (n) E n (t) (I.25)
where 0 is the vacuum permittivity, χ (n) is the n-order susceptibility of the material and

E n = E ⊗ E ⊗ • • • ⊗ E, n times.
The typical orders of magnitude for the linear (1), second-order (2) and third-order (3) susceptibilities are the unit, 10 -12 m.V -1 and 10 -24 m 2 .V -2 [START_REF] Armstrong | Interactions between light waves in a nonlinear dielectric[END_REF][START_REF] Boyd | Order-of-magnitude estimates of the nonlinear optical susceptibility[END_REF], and they decrease even further at higher orders. This implies that the nonlinear polarization of the material P N L (t) = 0 n>1 χ (n) E n (t) is much weaker than its linear counterpart in most cases. Nonetheless, this term can play a key role for an electromagnetic wave propagating in a medium with high nonlinear susceptibility, as it directly affects Maxwell's equation through the displacement field D = 0 E + P in Gauss's and Ampère's laws. This results in the nonlinear wave equation:

∇E - n 2 c 2 ∂ 2 E ∂t 2 = 1 0 c 2 ∂ 2 P N L ∂t 2 (I.26)
with n = √ r the optical index of the material and c the celerity of light in free space. The Chapter I I.2 Quantum nonlinear optics nonlinear polarization effectively acts as a source term, producing fields at frequencies that differ from that of the incident field E.

I.2.1.2 Second order nonlinear processes in non-centrosymmetric materials

In the rest of the manuscript, we will focus on second-order nonlinearites. For instance, let us consider the propagation of two electric fields of amplitudes E 1 (E 2 ) and angular frequencies ω 1 (ω 2 ). The resulting incident field will be expressed as:

E(t) = E 1 e -iω 1 t + E 2 e -iω 2 t + c.c. (I.27)
Once injected in Eq. I.25, the second order polarization P (2) reads:

P (2) (t) = 0 χ (2) E 2 1 e -2iω 1 t + E 2 2 e -2iω 2 t + 2E 1 ⊗ E 2 e -i(ω 1 +ω 2 )t +2E 1 ⊗ E * 2 e -i(ω 1 -ω 2 )t + c.c. + 2 0 χ (2) [E 1 ⊗ E * 1 + E 2 ⊗ E * 2 ]
(I.28)

The terms in Eq. I.28 can be labeled according to the physical process they describe. The two terms at 2ω 1,2 correspond to the generation of radiation at the second-harmonic frequency. It can be visualized by considering the interaction in terms of the exchange of photons between the various frequency components of the field: two photons of frequency ω merge into a photon of frequency 2ω in a single quantum-mechanical process: this is the second-harmonic generation (SHG). Similarly, the radiations at ω 1 +ω 2 and ω 1 -ω 2 are respectively sum-frequency generation (SFG) and difference-frequency generation (DFG). Finally, the two last terms of Eq. I.28 do not lead to the generation of electric waves, but to the generation of static fields across the nonlinear medium.

An important point is that centrosymmetric crystalline media (i.e. with an inversion center) will prohibit even-order optical susceptibilities. Indeed, a sign inversion for the electric field also results in a sign inversion for the polarization, i.e:

-P (2n) (t) = 0 χ (2n) [-E(t)] 2n = 0 χ (2n) [E(t)] 2n = P (2n) (t) (I.29)
which can only be true if χ (2n) = 0. Since spontaneous parametric down conversion, the quantum nonlinear process that has been studied during my PhD and will be introduced in the next section, is a second-order nonlinear process, the rest of the discussion will be restricted to noncentrosymmetric dielectric crystals, such as quartz, β-BaB 2 O 4 (BBO), KH 2 PO 4 (KDP), LiNbO 3 (LN), and GaAs. The latter is particularly interesting for potential applications because of the strong non-zero terms of its nonlinear susceptibility tensor: with i, j, k ∈ {x, y, z}. This value is more than hundred times superior to the one of BBO (d 22 = 2.2 pm.V -1 ) and ten times higher than to the largest element of LN (d 33 = -20 pm.V -1 ).

χ (2) GaAs =        0
An additional three-wave-mixing process exists: parametric fluorescence, or spontaneous parametric down conversion (SPDC), which foster the decay of a pump photon in two so-called signal and idler photons following ω p = ω i + ω s . Contrary to DFG and SFG, SPDC only requires one input field. The generation of electromagnetic fields at ω s and ω i arises from the interaction of the pump with vacuum fluctuations. SPDC has the particularity to be a spontaneous process, hence not deterministic, and to generate two-photon quantum states [START_REF] Klyshko | Photons and nonlinear optics[END_REF], which are key ingredient for quantum communication [START_REF] Tanzilli | PPLN waveguide for quantum communication[END_REF]. It was first observed in the late 60s [START_REF] Harris | Observation of tunable optical parametric fluorescence[END_REF], by pumping a LN crystal with an Ar laser at 488 nm. A 1 cm-long crystal was placed in an oven to control its refractive index, driving the phase matching condition to shift the wavelength of the generated photons.

I.2.2 Parametric fluorescence

I.2.2.1 Relaxation of the phase-matching condition

In the previous section, three classical second-order nonlinear processes have been introduced: SHG, SFG and DFG. They can all be qualified as three-wave-mixing process, because of the number of electric fields involved in the frequency conversion (see Fig I .4). All those processes obey energy conservation relation between the pump field (ω p ) and the idler and signal fields (ω i,s ). Note that another condition affects the conversion efficiency of the nonlinear process: the conservation of momentum or phase-matching condition [START_REF] Boyd | Chapter 1 -The nonlinear optical susceptibility[END_REF], i.e. k p = k s + k i , which arises from the coupled-wave equations of the interacting fields [START_REF] Boyd | Chapter 1 -The nonlinear optical susceptibility[END_REF]. When phase matching is fulfilled, constructive interference between the pump and generated fields allows an efficient energy transfer. However, when the phase-matching condition is not satisfied, the amplitudes of the generated fields along the propagation direction oscillate because of the presence of an imaginary term, and do not build up. In general, the conservation of momentum is not always satisfied, resulting in a wavevector mismatch ∆k = k p -k i -k s , which mitigates strongly the conversion efficiency, as illustrated by the SFG intensity I p at ω p from the input intensities I s at ω s and I i at ω i [START_REF] Boyd | Chapter 1 -The nonlinear optical susceptibility[END_REF]:

I p = 2 χ (2) 2 ω SF I s I i n s n i n p 0 c 2 L 2 sinc 2 ∆kL 2 (I.32)
The optical index n j at the angular frequency ω j (j ∈ {i, s, p}) is introduced, as well as the distance over which the nonlinear interaction occurs along the propagation direction. Maximum conversion is obtained ∆k = 0, and I p decreases fast as |∆k| increases because of the sinc function. From Eq. I.32, it appears that the shorter the nonlinear interaction distance L, the lesser the impact of ∆k = 0 on I p . From there, two strategies to enhance the nonlinear processes efficiency can be adopted: 1) operating at phase matching and extending the interaction length since I p ∝ L 2 when ∆k = 0; 2) reducing the interaction length below the coherence length L c = π(∆k) -1 . When considering nanosctructures, the latter is always satisfied.

I.2.2.2 From classical to quantum second-order nonlinear processes

Historically, the first systems that served as a platform to study SPDC were noncentrosymmetric crystals like BBO, LN or KDP [START_REF] Boitier | Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor[END_REF][START_REF] Neves | Characterizing entanglement in qubits created with spatially correlated twin photons[END_REF]. More recently, advances in nanofabrication techniques and tools allowing the fabrication of micro-or nanoscopic devices in III-V semiconductors have seen the emergence of nonlinear optical systems based on GaAs and its high second-order susceptibility [START_REF] Orieux | Semiconductor devices for entangled photon pair generation: a review[END_REF]. In particular, since the 2000s, guided devices have exploited the direction [1 10] of the GaAs zinc-blende crystal lattice to generate two-photon states that are co- [START_REF] Suhara | Generation of quantum-entangled twin photons by waveguide nonlinear-optic devices[END_REF] or counter-propagating [START_REF] Ding | Transversely pumped counterpropagating optical parametric oscillation and amplification[END_REF][START_REF] Rossi | Counterpropagating twin photons by parametric fluorescence[END_REF][START_REF] Lanco | Semiconductor waveguide source of counterpropagating twin photons[END_REF] with respect to pump photons in waveguides or cavities [START_REF] Kuklewicz | Time-bin-modulated biphotons from cavity-enhanced down-conversion[END_REF][START_REF] Zhang | Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion[END_REF]. This turnaround embraces the challenges of miniaturization of photonic systems to promote their integration into photonic circuits, first monolithic, then on hybrid platforms combining several materials, to take advantage of the strong nonlinearities of III-V materials and technologies of the silicon industry. During the last decade, the challenges of miniaturization and integration of nanoscale entangled photon sources have led to the consideration of metamaterials as nonlinear sources of entangled quantum light. Even if the measurement of SPDC-generated photons in nanoantennas and metasurfaces has already been accomplished in the few past years, full understanding and modeling of this phenomenon are still a challenge to which no solution is unanimously agreed. Several experimental and modeling difficulties arise. First, in order to observe two-photon state generation, the nonlinear conversion efficiency must be sufficiently high. Given the sub-wavelength size of the light-conferring elements in the systems shown in Fig. I.5, SPDC amplification over long distances is not possible. It is therefore desirable to optimize the resonances of the nano-elements to promote a strong nonlinear interaction between pump, signal and idler fields [START_REF] Parry | Enhanced generation of nondegenerate photon pairs in nonlinear metasurfaces[END_REF][START_REF] Santiago-Cruz | Resonant metasurfaces for generating complex quantum states[END_REF]. Secondly, because of the probabilistic nature of SPDC, a direct calculation of the generated signal and idler photons is not possible. It is possible to opt for two approaches to describe the phenomenon: 1) inferring the possible outcomes for SPDC signal and idler photons from the corresponding classical process, SFG (Fig. I.4), as done in [START_REF] Parry | Enhanced generation of nondegenerate photon pairs in nonlinear metasurfaces[END_REF][START_REF] Jin | Efficient single-photon pair generation by spontaneous parametric down-conversion in nonlinear plasmonic metasurfaces[END_REF], which is demanding in terms of numerical resources; 2) deriving the nonlinear interaction Hamiltonian. This approach includes the calculation of the Green's tensor describing the propagation of a the SPDC photons when the nanoresonator hosting the nonlinear frequency conversion acts as a source [START_REF] Santiago-Cruz | Resonant metasurfaces for generating complex quantum states[END_REF][START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF][START_REF] Marino | Spontaneous photon-pair generation from a dielectric nanoantenna[END_REF], and approach relies on the modal comprehension of the nano-element constituting the metamaterial. Although approaches based on Mie theory perfectly describe the resonances of spherically symmetric systems, that basis is not the best suited to represent the eigenmodes and properties of diffusion and radiation of nanoparticles of arbitrary shapes (see Sec. I. 1.1.3). For this reason, we then opted for a description relying on QNMs (see Sec. I.1.2), for which the near and far field are computed using finite-element method simulations. Those results will be presented in Chap. III of this manuscript.

I.2.3 Two-photon polarization state and entanglement

I.2.3.1 Quantum state and density matrix

A quantum state provides a probability distribution of the possible measurement on a quantum system. There exists two kinds of quantum state. On one hand, pure states correspond to vectors |ψ pure = (α 1 , ..., α n ) t an n-dimensional Hilbert space. For such states, the density operator, which allows to compute the probabilities of the outcomes of a quantum measurement, is simply the projector |φ pure φ| pure . On the other hand, mixed states consist of several possible outcome pure states |ψ j , each with probability p j . They are therefore a statistical mixture of pure states, characterized by the following density matrix:

ρ = n p n |ψ n ψ n | (I.33)
By definition, the density matrix is Hermitian. Since the diagonal elements of the density matrix form a probability distribution, its trace must be equal to 1 ( n p n = 1). Whatever the considered state |φ , the density matrix is positive semi-definite, i.e.:

φ| ρ |φ ≥ 0, ∀ |φ (I.34)
which implies that its eigenvalues are non-negative .

I.2.3.2 Quantum properties accessible through the density matrix

The density matrix fully characterizes any quantum state. It can also give access to the expectation value of an observable, the purity of a state, or the fidelity between two states.

State purity γ of a state |ψ can be determined by:

γ = T r(ρ 2 ) (I.35)
In the case of a pure state, the matrix being idempotent (ρ 2 = |φ φ|φ φ| = |φ φ| = ρ), the purity is always 1. In the case a mixed state, the statistical mixture implies that γ < 1.

Measure of an observable It is possible to evaluate the expectation value of an observable

A with respect to the density matrix via the trace operation:

A = ψ| A |ψ = T r(ρA) = T r(Aρ) (I.36)
where T r is the trace operator.

Fidelity

The fidelity is a way to quantify the degree of similarity (the distinguishability) between two quantum states. The fidelity between two states characterized with the density ma-Chapter I I.2 Quantum nonlinear optics trices ρ and σ is defined as:

F (ρ, σ) = T r ρ σ ρ 2 (I.37)
Note that the √ operator should be understood in the sense of the matrix product. By definition, the fidelity is a positive and symmetric function. If F (ρ, σ) = 0, the two states |ψ and |φ are orthogonal; if F (ρ, σ) = 1, they are identical. Fidelity is a central tool when it comes to comparing a given state to a basis of known states. Note that in the case of pure state |ψ and |φ , the fidelity is the projection of a state onto the other:

F (ρ = |ψ ψ| , σ = |φ φ|) = | ψ|φ | 2 (I.38)

I.2.3.3 Continuous and discrete variables for photon states

Two types of variables can be used to constitute a Hilbert space. First, continuous variables make use of observables whose numerical values span continuous intervals, such as frequency, intensity of a field, position or momentum of a particle or quasi-particle. In such case, the Hilbert space is infinite, and therefore it is more convenient to employ a phase space representation of the quantum state (i.e. quasi-probability Wigner functions). In the case of SPDC generated photons, it is possible to define continuous states based on the spectral or spatial degrees of freedom. This is especially true for photons generated in systems smaller than L c , where the relaxation of phase-matching condition allows for broader directional and spectral generation [START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF].

It is also possible to use the polarization degree of freedom, an otherwise discrete variable, to encode quantum information. The single photon polarization state is expressed in the following basis:

|H =    1 0    |V =    0 1    (I.39)
where |H (|V ) is the ket for a horizontally (vertically) polarized photon. Polarization state are particularly interesting for quantum communication and computing, because they define a two-level system, i.e. a qubit.

Similarly, the polarization states of a pair of photons are expressed in the basis:

|HH =           
This basis forms an ensemble of separable and maximally entangled states, the so-called Bell states [START_REF] Greenberger | Bell's theorem without inequalities[END_REF][START_REF] Pan | Greenberger-Horne-Zeilinger-state analyzer[END_REF][START_REF] Nielsen | Quantum computation and quantum information[END_REF]. The separable states are presented above, and the entangled state are linear combinations of them. In the rest of the manuscript, we will focus on these polarization states.

I.2.3.4 Reconstruction of the density matrix for a two-photon state

The density matrix describing the polarisation states of the photon pairs in our system takes the following form in this basis:

ρ = |HH |HV |V H |V V                   

Tomographic reconstruction of the density matrix

The tomographic approach allows to reconstruct the density matrix of a system from a set of finite number of measurements of outcomes {n j } j [START_REF] White | Nonmaximally entangled states: production, characterization, and utilization[END_REF][START_REF] James | Measurement of qubits[END_REF]. In this approach, the density matrix ρ is reconstructed as a simple linear combination of known matrices Mj (used as a basis), whose weights are directly given by each of the measured quantities n j . ρ = j n j Mj (I.42)

The full demonstration is given in the Appendix A, and its results is directly presented here, considering the set of tomographic states used to reconstruct the density matrix of the photons pairs, presented in Tab. I.1, where the states |D ,|L and |R are:

|D = 1 √ 2 (|H + |V ) (I.43) |L = 1 √ 2 (|H + i |V ) (I.44) |R = 1 √ 2 (|H -i |V ) (I.45)
and correspond respectively to diagonal, circular left and circular right polarizations of light.

The unknown constant N can be expressed as: which for our considered set of tomographic set reduces to:

N = ν n ν T
N =

I.2.3.5 Entanglement witness for biphoton states

The biphoton states generated by SPDC can be entangled states, meaning that they cannot be factored as a product of states of its local constituents: each photon of the pair cannot be described independently. Generating and controlling entangled states is a central challenge for future applications of quantum physics, since this mechanism serves as a base for quantum information, computing, and cryptography. Determining the entanglement of a state requires to compute an entanglement witness, such as the Schmidt number [START_REF] Sanpera | Schmidt-number witnesses and bound entanglement[END_REF], which is obtained by decomposing a given bipartite pure state |ψ AB in an orthogonal product basis of single particles modes |ψ A and |ψ B with minimal number of terms [START_REF] Peres | Quantum theory: concepts and methods[END_REF]. This decomposition reads:

|ψ AB = i λ i |ψ i A ⊗ |ψ i B (I.49)
Chapter I {|ψ i A } and {|ψ i B } are the Schmidt bases for A and B and √ λ i are the Schmidt coefficients satisfying λ i ≥ 0 and i λ i = 1. The reduced density matrices corresponding to the single particle wave functions are found as the two partial traces of ρ AB or can be deduced directly from the Schmidt decomposition:

ρA = T r B (ρ AB ) = i λ i |ψ i A ψ i | A (I.50) ρB = T r A (ρ AB ) = i λ i |ψ i B ψ i | B (I.51)
where T r B (T r A ) is an operator known as partial trace over system B (sytem A). Note that the eigenvalues of the reduced densities ρA and ρB are identical and equal to λ i . Considering the density matrix of the form Eq.(I.41), the reduced density matrices read:

ρA =    ρ 11 + ρ 22 ρ 13 + ρ 24 ρ 31 + ρ 42 ρ 33 + ρ 44    ρ =    ρ 11 + ρ 33 ρ 12 + ρ 34 ρ 21 + ρ 43 ρ 22 + ρ 44    (I.52)
The Schmidt decomposition is useful for the separability characterization of pure states:

• The state |ψ AB is separable (A and B not entangled) if and only if the densities matrices ρA and ρB possess only one non-zero eigenvalue equal to 1, i.e. if there is only one non-zero Schmidt coefficient {λ i = 1, λ j = 0 ∀j = i}.

• If more than one Schmidt coefficients are non-zero, then the state is entangled.

• If all the Schmidt coefficients are non-zero and equal, then the state is said to be maximally entangled.

• A Schmidt number is sometimes defined as the number of non-zero eigenvalues of ρA and ρB . An entangled state thus has a Schmidt number > 1 while a separable state has a Schmidt number equal to 1.

• The purity of the reduced states can be directly used as a quantifier of entanglement. Therefore it is possible to define a Schmidt entanglement parameter K, also called Schmidt number, by:

K = 1 T r(ρ 2 A ) = 1 T r(ρ 2 B ) = n λ 2 n -1 ≥ 1 (I.53)
K quantifies the amount of entanglement between systems A and B.

The above applies when considering bipartite systems which are in pure states. The method to quantify the degree of entanglement between parts that are in mixed states (that is, when not only ρ A and ρ B are mixed, but when ρ AB itself is already mixed) is still an open research topic. Under certain conditions, it is also possible to compute the Schmidt number associated to a density matrix [START_REF] Terhal | Schmidt number for density matrices[END_REF], but in the rest the manuscript, the study of entanglement will be limited to pure states (see Sec. III.2.1.3 and III.4.2).

I.3 About topology

This section recalls the basic concepts of topological phases in physics. Topological approaches has developed since the 80's, with an even more growing interest since the Nobel Price in Physics has been awarded to D. J. Thouless, F. Duncan. M. Haldane and J. M. Kosterlitz in 2016 for their 'theoretical discoveries of topological phase transitions and topological phases of matter'. This introduction to the field, oriented toward its applications in photonics, has benefited from the course of Jean Dalibard ('La matière topologique et son exploration avec les gaz quantiques', 2017-2018, Collège de France) and Nicolas Regnault ('Introduction to topological order', second year at ICFP master, 2017-2018).

I.3.1 Topological phases in modern physics

I.3.1.1 Basic concepts

Topological approach of physics is relatively new and has given a leverage towards unexplored phenomena localized at system edges. Topology corresponds to the study of space properties that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. It has been a purely mathematical field, yet the second half of the 20 th century has seen the emergence of a so-called topological physics. It has been named in opposition to classical physics, which relies on geometric order. Topological order establishes an equivalence between objects or phases distinguishable at first sight. Two objects are labeled as topologically equivalent if one can find a continuous transformation to go from one object to the other without crossing a singularity. For those geometrical shapes, their genus is a good topological invariant, since it only changes with their topology. For systems in physics, the geometrical phase, integrated from Berry curvature [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF], often gives a good topological invariant. Topological transitions are characterized by a gap closing to allow topological invariant modifications. Famous topological manifestations in physics are quantum Hall effect [START_REF] Laughlin | Quantized Hall conductivity in two dimensions[END_REF][START_REF] Arovas | Fractional statistics and the quantum Hall effect[END_REF][START_REF] Cage | The quantum Hall effect[END_REF], Kosterlitz-Thouless transition [START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF], and Majorana fermions [START_REF] Kitaev | Unpaired Majorana fermions in quantum wires[END_REF][START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF][START_REF] Leijnse | Introduction to topological superconductivity and Majorana fermions[END_REF][START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF]. They all have in commun that their physics happens not in the bulk but at edges of systems. Boundaries, defects, singularities are thus necessary to observe topological physics.

Berry phase and topological invariant

In physics, the study of topological properties essentially relies on the geometric phase, or Berry phase [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF], which is introduced when describing the adiabatic evolution of a system. Let us consider a system prepared at the initial time t = 0 in the eigenstate of a Hamiltonian Ĥλ depending on an external control parameter λ ∈ R N , N being an integer. λ is here time dependent, and its evolution describes an adiabatic closed loop in parameters space during the time T . Given an eigenstate |ψ(t) of Ĥλ at t = 0, the final state at t will be the same as the initial state except for its phase:

ψ(t ) = e iΦ |ψ(t) (I.54)
If λ was independent of time, the phase factor would only account for the evolution of energy with time: it is the dynamical phase. The eigenstate is qualified as a stationary state in this case. However, when λ is time-dependent, the state |ψ(t ) does not coincide with |ψ(t) because the eigenstates of Ĥλ , i.e. the basis of the system, is not the same at t than at t = 0. Defining the eigenstates ψ (n) λ of Ĥλ associated to the eigenenergies E n , the Schrödinger equation thus reads:

i d dt ψ λ (t ) = Ĥλ ψ λ (t ) (I.55) dΦ dt ψ (n) λ (t) + d dt ψ (n) λ (t) = E n ψ (n) λ (t) (I.56)
which, once projected on ψ (n) λ (t) and integrated for a given duration t leads to:

Φ = - 1 ¢ t 0 E n dt + i ¢ t 0 ψ (n) λ (t ) d dt ψ (n) λ (t ) dt (I.57)
The first term is the dynamical phase, and the second term defines the geometric phase associated to the Berry connection [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF][START_REF] Berry | The quantum phase, five years after[END_REF]:

A (n) λ = i ψ (n) λ d dt ψ (n) λ (I.58)
The geometric phase (or Berry phase) is defined by the integration over a loop in the λ parameter space of the Berry connection, and is invariant by gauge transformation (modulo 2π). In the case of periodic systems, the phase accumulated when integrating along the contour of the first Brillouin zone is called the Zak phase [START_REF] Zak | Berry's phase for energy bands in solids[END_REF], from which one can define the first Chern number [START_REF] Thouless | Quantized Hall conductance in a two-dimensional periodic potential[END_REF] as:

C = Φ Zak 2π (I.59)
Originally, Thouless introduced the Chern number as a bulk property that gives information on the number of conduction chiral channels that accounted for conductance plateau in quantum

Hall effect [START_REF] Thouless | Quantized Hall conductance in a two-dimensional periodic potential[END_REF]. More generally, the Chern number is a good topological invariant associated to a band in solid state physics systems. Band structure, topological transition and edge states Band structures results from the dispersion relation between the energy of a quasi-particle and its wavevector in periodic systems. Topological order and transitions can be apprehended through the band structure, as suggested in Fig. I.6. Coming back to the definition of a topological class, we have already stated that two objects share the same topological order if it is possible to continuously go from one to the other.

In the case of periodic systems, this implies that the transformation from an Hamiltonian H to H is a topological isomorphism (i.e. that its inverse transformation is continuous) and conserves the topological order if the gap stays open along the transformation path (Fig. I.6a). Conversely, if the gap closes during the transformation, the path followed in the parameter space will encompass a singularity, suppressing the isomorphic property of the transformation. Thereby, the topological invariant of the final Hamiltonian H may have changed along the transformation. More specifically in term of band structures, if states mix while the gap is closed, the number of states in the upper and lower bands will be affected, leading to non-zero Chern number difference between the two bands. The reminiscence of the gap closure are the so-called edge states (Fig. I.6b), that can be found at the interface between two phases of different topology.

Note that closing a gap in a band structure requires a strong modification of the system, generally related to the symmetry of the periodically repeated unit cell constituting the band structure. That can be due to a defect or a lattice mismatch [START_REF] Paulose | Topological modes bound to dislocations in mechanical metamaterials[END_REF][START_REF] Li | Higher-order topological states in photonic kagome crystals with long-range interactions[END_REF][START_REF] Liu | Bulk-disclination correspondence in topological crystalline insulators[END_REF], or the introduction of a symmetry breaking mechanism, like spin-orbit coupling in superconducting phases [START_REF] Bernevig | Topological insulators and topological superconductors[END_REF][START_REF] Sato | Topological superconductors: a review[END_REF]. While the next section focuses on some implementation of topological order in guided photonics and optical metamaterials, the reader is invited to refer to Appendix B for a derivation of the topologically protected edge states in the case of the simplest 1D topological dimerized tightbinding Hamiltonian: the Su-Schrieffer-Heeger (SSH) model [START_REF] Su | Solitons in polyacetylene[END_REF].

I.3.2 Promises of topological phases in photonics

In this last section, I will review recent articles that illustrate the stakes of photonic systems based on topological phases. In particular, I will focus on metamaterials and guided optics where topological laser and conversion frequency based on protected edge states have been achieved.

I.3.2.1 Guided optics

The control of the phase acquired by a photon when propagating in a matrix of coupled ring resonators has allowed to demonstrate an equivalent of quantum Hall effect modeled by the Harper-Hofstadter model [START_REF] Harper | Single band motion of conduction electrons in a uniform magnetic field[END_REF][START_REF] Hofstadter | Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields[END_REF]. The key challenge was to engineer an effective synthetic magnetic field to mimic the physics of electron, but this time with photon. In two articles by Hafezi et al. [START_REF] Hafezi | Imaging topological edge states in silicon photonics[END_REF][START_REF] Hafezi | Robust optical delay lines with topological protection[END_REF], the equivalent of the tunnel coupling between sites in the strong bond model is obtained by inserting between two rings a rectangle-shaped link with rounded edges. The experiment was done with a square array of 10 × 108 rings (site resonators) connected by links resonators (Fig. I.7). The vertical link resonators are identical. However, because of the vertical shift of the horizontal link resonators, a photon acquires a non-zero phase when it hops between resonators 1 and 2, and 3 and 4. Therefore, a photon progressing in the anticlockwise (clockwise) direction around the unit cell acquires a 2πα (-2πα) phase. This phase difference has the same effect for the photon as the application of a magnetic field for the electron in the quantum Hall effect. In the experiment, the light was injected in one of the rings located at the periphery (port 1/3 or 2/4 in Fig. I.7). The experiment confirmed the existence of edge channels that propagate without interruption even in the presence of defects. By changing the wavelength of light, one can excite a channel of positive or negative group velocity, i.e. propagating in the clockwise or counterclockwise direction, in good agreement with the predictions (bottom panel in Fig. I.7). Note that the simulation parameters, (κ e , κ i , J) = (31, 0.57, 26) GHz, are extracted from experimental measurement of simpler devices. These orders of magnitude will serve as reference for our study systems of concentric ring resonators in Chapter V.

The concept of photonic topological insulators, as previously described, has recently been used to implement lasing based on protected chiral edge channels in a square matrix of coupled ring resonators [START_REF] Bandres | Topological insulator laser: Experiments[END_REF][START_REF] Harari | Topological insulator laser: theory[END_REF]. The topological insulator laser system is an aperiodic array of 10 × 10 unit cells of coupled ring resonators on an InGaAsP quantum wells platform. In this design, the lasing steady state is the edge state of the ring resonator matrix (Fig. I.8 top panel). The cross section of each ring (500-nm width and 210-nm height) is designed to ensure single transverse-mode conditions at a wavelength of operation of 1550 nm. The nominal separation between the ring resonators and off-resonant links is 150 nm, thus leading to two frequency band gaps, each having a width of 80 GHz (0.64 nm), an order of magnitude that will guide us for the work reported in Chapter V. The output power plotted against the pump power shows that the topological laser operates with better efficiency than its non-topological counterpart (Fig. I.8 bottom panel). This is because in the non-topological array, the modes involved in lasing extend into the core of the material, resulting in significant losses. The topological edge mode occupies the outer edge channel in a uniform way and thus has a significant amplitude at the output coupler, contrarily to the trivial modes that tend to minimize the amplitude at the output coupler to maximize the gain. Moreover, the spectrum of the emitted light is narrower for the lasing arising from topological modes, which shows that it is single mode [START_REF] Bandres | Topological insulator laser: Experiments[END_REF][START_REF] Harari | Topological insulator laser: theory[END_REF]. Conversely, the spectrum of the non-topological laser indicates that it is multimode, because different regions of the sample lase at different frequencies due to the inevitable disorder that occurs during sample fabrication.

I.3.2.2 Metamaterials and nanophotonics

Tight-binding models can offer an intuitive comprehension of 1D or 2D metamaterials based on a resonant single element and constitute tools to help toward the implementation of exotic properties of solid-state physics in photonic systems. In this section, we will review two recent articles presenting implementations of SSH models based on cylindrical GaAs micropillars with quantum wells [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF] and silicon nanocylinders [START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF][START_REF] Smirnova | Third-harmonic generation in photonic topological metasurfaces[END_REF][START_REF] Slobozhanyuk | Near-field imaging of spin-locked edge states in all-dielectric topological metasurfaces[END_REF].

First, the implementation of a SSH chain by St-Jean et al. [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF] relies on the in-plane p-modes of the quantum well that have a dipolar near field profile (Fig. I.9 top panel). Depending on their polarization, those modes induce a symmetry breaking in the chain, that dimerizes the unit cell. In other words, the system formed by the coupling of s-modes can be mapped by the Hubbard model, whereas the one generated from the p-modes is of the same class as the SSH Hamiltonian. When performing photoluminescence experiments at 4 K, it is possible to probe the bands of the chain displayed in Fig. I.9, and determine the localization of the light emitted in the SSH chain (Fig. I.9 bottom panel). In compliance with the theoretical description of SSH chain (see Appendix B), the authors observed that both bulk and edge present the same bands, respecting bulk-edge correspondence. Interestingly, the presence of edge states in the p-band gap characterizes the topological behavior of the chain. The authors of the article have demonstrated topologically protected lasing in those modes, with advantages similar to those demonstrated in [START_REF] Bandres | Topological insulator laser: Experiments[END_REF][START_REF] Harari | Topological insulator laser: theory[END_REF].

Finally, topological phases can also be applied tononlinear metamaterials to engineer enhance frequency conversion in topologically protected edge states [START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF][START_REF] Smirnova | Third-harmonic generation in photonic topological metasurfaces[END_REF]. The authors have fabricated a SSH zigzag chain of silicon cylinders on a fused-silica substrate (500 µm thick), as shown in Fig. I.10a, and measured the third-harmonic generation (THG). Based on dipolar modes of the cylinders, the same symmetry breaking as in [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF] operates. The topological protection is strength- ). It appears that for maximal distortion amplitude lower than 18 • , topological protection holds [START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF]. Above this threshold, an increasing probability of generating distorted chains that breaks topological protection arises. The interest of topological phases is revealed when studying the nonlinear spectrum of the generated light (Fig.

I.10d):

The localization of THG can be controlled with the pump wavelength when impinging light arrives from the substrate, whereas it is invariably localized on the first unit cell when the chain is illuminated from air. This is the signature of a non-reciprocal process, which opens new ways of engineering controllable nonlinearity in metamaterials, taking advantage of the multimodal properties of nanoparticles and possibilities for topological phases. Note that THG has also been achieved in metasurfaces [START_REF] Smirnova | Third-harmonic generation in photonic topological metasurfaces[END_REF][START_REF] Slobozhanyuk | Near-field imaging of spin-locked edge states in all-dielectric topological metasurfaces[END_REF], this time relying on the inclusion of 2D topological patches in a non-topological honeycomb lattice of multimodal silicon nanoparticles.

CHAPTER II

Technological development and nanofabrication based on III-V materials

Nanofabrication played a key role in the development of devices based on III-V semiconductors, such as LEDs or more recently photonic circuits. This chapter aims at describing some top-down approaches for the fabrication of AlGaAs nanostructures that I have developed in my PhD. They rely on electron-beam lithography, reactive ion etching, material deposition or selective oxidation. This important part of my work served two scopes: 1) supporting the experimental realizations of Chapters III, IV and V, where the studies of dielectric nonlinear metasurfaces, coupled nanoresonators and waveguides are reported; 2) provide state-of-art samples to a few group abroad, within the framework of international research projects.

Contents

II.1 Fondamentals of cleanroom fabrication

In this section, I present the techniques involved in the fabrication of the nanostructured dielectric samples that I studied during my PhD. They include epitaxial growth of III-V semiconductors thin layers (performed by our collaborators in C2N and CEA Grenoble), electron-beam lithography and reactive ion etching.

II.1.1 Epitaxy of AlGaAs nanofilm for nanophotonic application

Epitaxy is a type of crystal growth in which new crystalline layers are formed with a controlled orientation from a crystalline seed layer. Epitaxial techniques can be based on deposition [START_REF] Köhler | Scanning tunneling microscopy study of lowtemperature epitaxial growth of silicon on Si (111)-(7× 7)[END_REF][START_REF] Ramesh | Epitaxial cuprate superconductor/ferroelectric heterostructures[END_REF][START_REF] Frey | Effect of atomic oxygen on the initial growth mode in thin epitaxial cuprate films[END_REF][START_REF] Krockenberger | Superconductivity phase diagrams for the electron-doped cuprates R 2-x Ce x CuO 4 (R= La, Pr, Nd, Sm, and Eu)[END_REF] or selective elimination/desorption of atoms [START_REF] Emtsev | Interaction, growth, and ordering of epitaxial graphene on SiC {0001} surfaces: A comparative photoelectron spectroscopy study[END_REF][START_REF] Hibino | Stacking domains of epitaxial few-layer graphene on SiC[END_REF][START_REF] Borovikov | Step-edge instability during epitaxial growth of graphene from SiC (0001)[END_REF]. It is a central tool for the fabrication of semiconductors components, and III-V dielectric-based photonics has strongly benefited from mature techniques of molecular beam epitaxy (MBE, [START_REF] Cho | Molecular beam epitaxy[END_REF][START_REF] Arthur | Molecular beam epitaxy[END_REF][START_REF] Bhattacharya | Comprehensive semiconductor science and technology[END_REF]) and metal-organic chemical vapor deposition (MOCVD, [START_REF] Jakomin | High quality tensile-strained n-doped germanium thin films grown on InGaAs buffer layers by metal-organic chemical vapor deposition[END_REF][START_REF] De Kersauson | Direct and indirect band gap room temperature electroluminescence of Ge diodes[END_REF]). In this section, I will briefly recall the bases of these two techniques, on which relies the growth of the crystalline thin films of [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs used during my PhD. Such epitaxial layers are the first step of top-down nanofabrication, which involves in a second time patterning or structuring methods.

II.1.1.1 Growth of AlGaAs from GaAs wafer

AlGaAs on GaAs wafers has a zinc-blende crystalline structure, and is grown along the direction [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF] (see Fig II.1). Two layers of gallium, aluminum and arsenic, with different aluminum concentration x are grown on the GaAs substrate. In order to tune the direct band-gap of semiconductor, the concentration of Al atoms is controlled during the growth process. The lattice constant a of these layers varies linearly with the aluminum concentration x according to the following equation:

a = 5.6533 + 0.0078x Å (II.1)
The first epitaxial layer is composed of Al 0.98 Ga 0.02 As (AlAs in the following), and the second of Al 0.18 Ga 0.82 As (AlGaAs in the following). The latter is meant to be nano-structured (see Sections II.1.2.1 to II.1.2.3), while the former layer will allow to improve the confinement of light after an oxidation step, detailed later (see Sec. II.1.2.4).

Molecular beam epitaxy MBE is based on the absorption and organization of atoms on a heated crystalline substrate. It is performed in a ultra high vacuum chamber to grow thin layers of semiconductor crystals with very low impurities concentration (see Fig II.2a). A source

Metal-organic chemical vapor deposition

The essential difference between MOCVD and MBE is that the mechanism involved in the MOCVD is not physical as in MBE, but rather chemical [START_REF] Stringfellow | Organometallic vapor-phase epitaxy: theory and practice[END_REF]. The growth takes place from a gas phase at moderate pressures (10 to 760 Torr), where precursor molecules for the elements of group III and V are injected. As these molecules approach the wafer surface, they undergo a pyrolysis reaction, whose products are absorbed by the surface. MOCVD enables an abrupt transition between layers of different Al concentration, which aim for better light confinement in the nanopatterned structures.

II.1.1.2 Wafer bonding for transparent samples

In addition to the previously described monolithic approaches, some of our samples are obtained by reporting a MBE-grown AlGaAs thin layer on a large-bandgap transparent sapphire substrate. This process, called wafer bonding, provides samples with high-index contrast that allow photons to be collected both in transmission and reflection in the visible and NIR wavelengths ranges. Wafer bonding between two flat surfaces is obtained by applying an external pressure.

In our case, the bonding of AlGaAs and sapphire requires the mediation of an adhesive polymer (here, epoxy resist). Our wafer-bonded samples are supplied by the group of Jean-Michel Gérard (CEA Grenoble), and have already been used to demonstrate the vertical emission and wave-front control in nonlinear metasurfaces [START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF]. Fabrication process for those samples is presented in Sec. II.2.2.

II.1.2 Overview of the major steps of the fabrication process

II.1.2.1 Sample cleaning and spin-coating

When we receive an epitaxial wafer, its surface is protected by a layer of resist. To obtain a workable surface, this protection layer is first dissolved in an acetone bath heated between 35°C and 50°C, to increase the solubilizing power of the solvent, for 5 min. This step is carried out in an ultrasonic bath to mechanically eliminate possible impurities more resistant to the sole solvent power of acetone. The impurities left by acetone are eliminated by a bath of isopropanol (IPA), also heated to a temperature ranging between 35°C and 50°C, for 2 min. As in the case of acetone, this step is performed in an ultrasonic bath. It is essential to dry the sample with a dry gas gun (in this case N 2 ) which expels the remaining solvent from the sample surface. This prevents the organic solvent from evaporating and depositing the impurities it contains on the sample. Finally, before depositing the resist required for the next fabrication steps, the sample is placed in a O 2 plasma for 2 min, which finalizes the cleaning and allows to get rid of the last organic components.

For the electronic lithography that defines the nano-or micro-structures, it is necessary to deposit a resist, that is used as a mask. This step is carried out by spin-coating, which allows the formation of thin and uniform layers of resist by using centrifugal force. To this end, the sample is placed on a spinner on which it is held by suction. A drop of resist is deposited on the sample and then is subjected to a high rotation speed to remove the excess resist (Poiseuille's law of flow for viscous fluids). To ensure the adhesion and the uniformity of the resist, an adhesion promoter can be deposited before the spin-coating. The promoter depends on the nature of the resist.

II.1.2.2 Scanning electron microscopy and mask implementation via electron beam lithography

In scanning electron microscopy (SEM), a focused electron beam is used to scan a surface and produce images of it. Electrons from the beam interact with atoms at various depths within the sample, and lose energy. Exchanged energy is dissipated by the system by heating, or as lowenergy secondary electrons, back-scattered electrons, or light emission, which produce a traceable signal. The retrieved signal carries information on both topography and composition of the material, and can reach sub-nanometric resolution. In the common SEM, electrons produced by an electron gun (generally a heated tungsten wire) are accelerated at a voltage ranging from 0.2 to 40 kV. SEMs are also used as an alternative to laser and UV lithography: electron beam lithography (EBL) allows to imprint a resist mask with a nanometric precision using an SEM. During an EBL, the focused beam of electrons (see Fig. The exposed zones, reinforced, remain once the mask developed by selective chemicals. We use negative resist for our during our lithography processes. The drawback of EBL is that it requires longer patterning time than photolithography. This difference in time is caused by the need to divide the sample in write fields, because the deflection of the electron beam is not possible over millimetric areas. Each write field thus corresponds to a position of the SEM stage, displaced by piezo-electric actuator. The position of the stage is controlled thanks to a laser interferometer.

A key parameter in EBL optimization is the insulation dose, which corresponds to the number of electrons that have to hit the sample within an area to alter the electro-sensitive resist mask. The dose D is given by the formula:

D = I beam T dwell s x s y (II.2)
where I beam is the intensity of the electron beam, T dwell is the dwell (or exposure) time, and s x (s y ) is the translation step along x (y) during the insulation process. In practice, the operator measures I beam , and chooses s x and s y according to the implemented pattern. Therefore, the dose is directly proportional to T dwell , which constitutes the knob to control the exposure of the resist. Before the fabrication of sample, it is often required to perform a dose test, which consists in the exposure of the targeted pattern while sweeping the dose D around a nominal dose D 0 .

The ratio D/D 0 is called the dose factor, and its optimal value during the dose test is chosen to fabricate the final sample. Examples of such dose tests are shown in Sec. II.2.

The EBL implementation of AlGaAs metasurfaces, ensembles of nanoresonators, or waveguides relies on the use of two types of negative resists: • maN-240X series: an industrial negative photoresist produced by Micro Resist Technology. The 'X' references the nominal thickness of the spin-coated maN. With suppliersuggested exposure dose of 120 µC.cm -2 , maN-240X series allows to pattern large surfaces with reasonably short process times. However, its resolution is limited by proximity effect due to the backscattered electrons in the sample. Ti-prime promoter is applied on the wafer (spin rate of 4000 rpm followed by 2 min soft bake at 120°C) to improve adhesion. We then spin the thinner resist of the series, the Man-2401, at 4000 rpm to obtain a thickness of 120 nm. The resist is finally soft-baked at 95°C for 1 min. This resist is particularly adapted for the fabrication of waveguides (see Chapter V).

• HSQ/Medusa 82: hydrogen silsesquioxane (HSQ) is commercialized by Dow Corning Corporation. It is a high-resolution silicon-based polymer, especially designed for dense patterning with low proximity effect, because its exposure dose is nominally much higher than maN-2401 (1 mC.cm -2 or above). Since this resist adhesion is low on GaAs surfaces, a 10 nm SiO 2 layer is deposited by plasma enhanced chemical vapor deposition (PECVD) at 280°C as adhesion promoter. Successively, HSQ is spun at 4000 rpm followed by a double-step baking (150°C for 2 min, then 200°C for other 2 min). This resist is particularly adapted for the metasurfaces and the isolated nano-antennas reported in Chapter III and IV. Alternatively, another silicon-based resist from the E-Beam Resist SX AR-N 8200 (Medusa 82) series of AllResist can be used. It presents the same characteristics as HSQ, and leads to the same sample quality once optimized. Note that future fabrication will involve more and more Medusa, because the production of HSQ has been stopped.

Once its developed, the mask is transferred to the sample by reactive ion etching.

II.1.2.3 Reactive ion etching

Reactive ion etching (RIE) is a directional etching process utilizing ion bombardment to remove material (see Fig II.5a). This etch process is commonly used in the manufacturing of printed circuit boards and other microfabrication procedures. RIE process relies on a chemically reactive plasma created in a high vacuum chamber by a strong radio-frequency (RF) electromagnetic field. This plasma aggressively etches in the vertical direction to remove, selectively or not, matter from a wafer. When the RF field is applied, it ionizes the gas molecules present in the chamber by stripping them of electrons. Through RF cycles, electrons are accelerated and driven up and down in the chamber, until they strike the sample surface. The most massive ions move relatively little. Deposited electrons create negative charges, while a high concentration of positive ions is formed in the gas, much denser than the free electrons one. The positive ions tend to drift toward the wafer due to the difference in electric potential created by the deflected electrons. As they collide with the sample surface, ions can remove matter from the material by transferring kinetic energy, or react chemically with it.

In the MPQ cleanroom, a dedicated RIE is used to etch III-V materials. This machine includes two high-vacuum chambers: one in which the plasma is created, and another one in which the sample is etched. Plasma ions are injected from the first chamber to the second, in order to separately control the RF fields creating the plasma and the ions bombarding the sample. This process is called inductively-coupled plasma RIE (ICP-RIE). The etching process can be broken down into four steps: 1) the ions first dislodge the atoms from the sample (physical etching); 2) the plasma gases react with the sample (chemical etching); 3) the ions then modify the surface of the sample by increasing its temperature or by changing the electronic configurations (this makes the surface more sensitive to chemical attack, which mainly occurs in the vertical direction); 4) a protective layer is formed on the exposed surfaces and the incident ions destroy only

II.1.2.4 AlGaAs oxidation for light confinement enhancement

When using MBE or MOCVD GaAs wafer, one last step is performed after etching the sample: the oxidation of the AlAs layer. This step allows the selective deep oxidation of the Al 0.98 Ga 0.02 As layer into non-stoichiometric aluminum oxide Al 2 O 3 (AlOx). The related reduction of its optical index from 2.9 to 1.6 improves the confinement of the light in the nanostructures. The oxidation of AlAs follows the simplified chemical reaction:

2AlAs + 3H 2 O → Al 2 O 3 + 2AsH 3 (II.3)
This fabrication step is carried out in an oxidation furnace under wet hydrogenated atmo-sphere, for which two gases are used: nitrogen (N 2 ) and hydrogenated nitrogen (N 2 + H 2 ). The pure nitrogen does not intervene directly in the oxidation. It is used to pressurize the water and to maintain a dry and oxygen-free atmosphere in the furnace chamber. The hydrogenated nitrogen and water form the wet oxidizing gas. Dihydrogen is added to the dinitrogen to remove the remaining arsenic products. In addition, the wet atmosphere is realized by injecting a N 2 + H 2 and water vapor, prepared in a controlled evaporation and mixing system. During oxidation, the sample is heated to 390°C for 40 min with a temperature rise of 20°C.min -1 . The sample is then allowed to cool to room temperature before being removed from the oven.

II.2 Design and fabrication of samples for international collaborations II.2.1 Optimization of fabrication process with high-resolution resist for dielectric metasurfaces implementation

The optimization of EBL-imprinted patterns is essentially done by adjusting the exposure dose. Since no electron propagation simulation tools are available, we perform a dose test, i.e. exposure of the same design while sweeping the dose. The result of such a dose test is presented in Fig. II.6, with SEM images of an AlGaAs-on-AlOx sample taken after development of the HSQ mask, ICP-RIE etching of the AlGaAs layer, and oxidation of the AlAs layer. Let us focus on the first line of metasurfaces, where the single element is a square of 600 × 600 nm 2 , centered in a unit cell of 750 × 750 nm 2 . Here, the thickness of the AlGaAs layer is 100 nm, on a 2 µm thick AlOx layer. From left to right, the dose factor D/D 0 increases, with D 0 = 1 mC.cm -2 . For dose factor lower than 0.8, the implemented single element is not resolved. The pattern is largely underdosed, and only the center of each square remains, because the cooperative action of ballistic and back scattered electron (proximity effect) results in a higher dose compared to the edges of the squares. The optimal dose is obtained for D/D 0 ∈ [0.9; 1.1]. Above these values, edges of the structures start to roughen, and the geometry is enlarged until squares merge when D/D 0 > 1.3. The resist mask is then overdosed.

Dose tests have to be regularly performed to compensate the aging of the resists, the fluctuation of the SEM electron beam over months of usage, or when new materials or new structures are fabricated. Optimization and updating of the EBL parameters are crucial over time, because they allow to shorten the time allocated to technological development and fabrication test, and provide a reliable starting point for the fabrication of more complex nanostructures.

II.2.2 C 1 symmetric meta-atoms for advanced resonance engineering

The most commonly used nanostructures for nonlinear metasurfaces are [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocylinders [START_REF] Marino | Zeroorder second harmonic generation from AlGaAs-on-insulator metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Smirnova | Third-harmonic generation in photonic topological metasurfaces[END_REF][START_REF] Slobozhanyuk | Near-field imaging of spin-locked edge states in all-dielectric topological metasurfaces[END_REF]. However, the symmetry of such structures generate symmetric second harmonic fields which cannot couple to the zero diffraction order under normal incidence of the pump beam. Fortunately, symmetry breaking can perturb the second harmonic mode profile, in such a way that the second harmonic field can also radiate in zero diffraction order. In collaboration with Franz Löchner and Frank Setzpfandt (Friedrich Schiller University, Jena), I fabricated asymmetric metasurfaces for vertical SHG. Such symmetry breaking can foster Fano resonances [START_REF] Campione | Broken symmetry dielectric resonators for high quality factor Fano metasurfaces[END_REF] or bound states in the continuum [START_REF] Koshelev | Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum[END_REF], whose high quality factors promise better conversion efficiency at the nanoscale. In the case of an asymmetric element, the optimal dose is more sensitive and requires finer tuning than in the case of a square element. Depending on the size of the protrusion, the optimal dose has to be adjusted at 50 µC.cm -2 (5% of D 0 ). Such sensitivity makes large samples hard to fabricate, because of the electron beam intensity fluctuations over EBL durations exceeding 12 hours. Measurements on a sample fabricated after this dose test are currently on the way at Friedrich Schiller University.

II.2.3 Two-step lithography process for metamaterials

The fabrication techniques for the dielectric devices described so far have only involved onestep lithography. This the limits design and geometric parameters tuning to the horizontal plane, leading to nano-elements invariant by translation in the epitaxial-growth direction. Breaking this invariance introduces a new way to engineer the resonances and scattering in metasurfaces. In particular, on-axis nonlinear flat lenses and holograms have been fabricated relying on so-called nanochairs, which are nanocylinders from which a part has been etched [START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF].

II.2.3.1 Nanochairs fabrication for vertical second harmonic generation

The fabrication process of such sample is depicted in Fig. II.9. A 400 nm thick layer of Al 0.18 Ga 0.82 As is grown by MBE and reported on a sapphire substrate by wafer bonding. The nanochair, which is a notched cylinder, requires to double the steps of lithography and etching. First, the top part of the chair is made by etching 200 nm of Al 0.18 Ga 0.82 As (step c to e in Fig. II.9). A second step allows to fabricate the bottom part of the nanochair (step f and g in Fig. II.9). During this second lithography, the mask must be superposed on the existing structures, which requires to align write fields one those of the previous lithography steps. For this purpose, alignment markers are added to the mask of step d. The alignment procedure relies on the detection of the marker within a single write field: the SEM software deduces the correction that must be brought to the mask coordinates to match the implemented structures with respect to the beam focus and write field alignment.

Note that maN-2403, a 220 nm thick version of the maN-2401, was used in this fabrication process, in order to reduce exposure time and facilitate the mask alignment of step f. Indeed, the SiO 2 adhesion layer deposited when using Si-based high-resolution resist absorbs too much charges from the electron beam, which prevents the detection of alignment mark and therefore hampers the alignment procedure. Using a similar process, I fabricated an anisotropic metasurface for ultrafast polarization control based on asymmetric notched gratings, called nanofins in the following sections. We carried out this work in collaboration with Giulia Crotti and Michele Celebrano from Politecnico di Milano.

II.2.3.2 Nanofins metasurfaces

This section details the fabrication of an all-optically reconfigurable AlGaAs anisotropic metasurface showing giant modulation of dichroism and birefringence. This design allows to exploit a sharp resonance of the structure in the desired spectral range, where the optically-induced From [START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF].

band-filling effect is the dominant process presiding over the ultrafast change of permittivity. I will not detail the physics behind this operation, but rather focus on the fabrication process of such a metasurface. of the first layer of the mask, because the depth etched back in step 2 does not offer a sufficient contrast to align the mask during the second lithography. Each fabrication step was controlled via both optical and scanning-electron microscopes. While width W and period P can vary on the same sample, all the structures of a given sample have the same step height S. For the first characterization, we chose the following parameters: W ∈ {150 + 5n} 0<n<6,n∈N nm, P ∈ {430, 450} nm, S ∈ {0, 35} nm. Due to the sample geometry and dense patterning, its morphological characterization is essentially performed with SEM observation. Acceleration voltage is set to 10 kV, and a 30 µm aperture is chosen to obtain a good contrast on the back-scattered electrons image. The step height is characterized by following the variation of contrast on the SEM image along the direction transverse to the principal axis of the array. The success of the fabrication process is assessed by two criteria: 1) the dimension of the implemented patterns compared to the mask design; 2) the relative alignment quality for the successive maN-2401 lithographic steps 2 and 3 (see Fig.

II.2.3.3 Ultrafast polarization control via all-optical modulation in an anisotropic metasurface

A promising device is shown in Fig. II.11a. In this case, the targeted (W, P, S) on the mask are [START_REF] Parrain | Origin of optical losses in gallium arsenide disk whispering gallery resonators[END_REF]430,[START_REF] Spinelli | Optical impedance matching using coupled plasmonic nanoparticle arrays[END_REF] nm. The inspection of contrast variation along the transverse direction to the waveplate stripes in the SEM image proves a reasonable alignment, and the measured dimensions correspond to the mask design. The edge quality can be improved by changing the resist, for instance using low-proximity resist like HSQ. The latter offers a better resolution than maN-2401, which stems both from a slightly reduced thickness and a lower sensitivity to e-beam exposure (typical used doses are around 2000 C.cm -2 ), reducing the impact of secondary electrons and in turn of proximity effects. This comes at the price of a significantly higher exposure time, up to 10 times longer than for the maN-2401. However, as fabrication expectation for this fabrication are slightly beyond the re-alignment capability guaranteed by Raith Nanofabrication on the available SEM, some samples will statistically suffer from misalignment between the EBL of step 2 and 3 of the fabrication protocol, as detailed for example in Fig. II.11b. To compensate this lack of accuracy, each device is repeated on the mask and thus fabricated several times.

In order to further assess the fabrication procedure, we carried out a static linear optical measurement of the reflection spectra of the samples. The reflection spectra are recorded with an Invenio R Fourier Transform spectrometer coupled to a Hyperion Microscope equipped with a 15X Cassegrain objective (NA = 0.4). The light source is a 150 W water-cooled halogen lamp, the beamsplitter is in quartz and the detector is a Si photodiode. Between the sample and the detector, an intermediate image is formed in the plane of an adjustable field stop that enables to select the area of interest. We firstly compared the promising (a) and misaligned (b) samples of Fig. 16, under unpolarized light source. The related spectra are reported in Fig. II.12. In order to highlight the polarization-dependent expected response, we acquired three additional spectra under linearly polarized illumination along three directions: perpendicular to the nanofin axis (x, blue curve), parallel to the nanofins (y, orange curve), and at 45°with respect to them (green curve). These spectra are reported in Fig. II.12b. Note that a giant optical dichroism is achieved, with a contrast as high as 3:1 between x-polarized and y-polarized reflectivity at around 760 nm. Also, note the very sharp and asymmetric dip observed at 770 nm, with a width of approximately 15 nm. In agreement with our numerical simulations, this sharp resonance is achieved under ypolarization, and can be ascribed to the resonant excitation of a bound state in continuum (BIC).

CHAPTER III

Spontaneous parametric down-conversion in nanopatterned resonators

Due to the relaxation of the phase-matching condition at the sub-wavelength scale [START_REF] Boyd | Chapter 3 -Quantum-mechanical theory of the nonlinear optical susceptibility[END_REF][START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF], nanostructures have recently emerged as a very promising platform for obtaining compact photonpair sources with controllable spectral, spatial and polarization properties via their many geometric degrees of freedom. Recent experimental works have already demonstrated the broadband generation of photon pairs in nonlinear dielectric metasurfaces and nanoresonators [START_REF] Jin | Efficient single-photon pair generation by spontaneous parametric down-conversion in nonlinear plasmonic metasurfaces[END_REF][START_REF] Marino | Spontaneous photon-pair generation from a dielectric nanoantenna[END_REF][START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF][START_REF] Santiago-Cruz | Entangled photons from subwavelength nonlinear films[END_REF][START_REF] Santiago-Cruz | Resonant semiconductor metasurfaces for generating complex quantum states[END_REF][START_REF] Santiago-Cruz | Photon pairs from resonant metasurfaces[END_REF]. Nevertheless, in order to go further in the control of the generated pairs, a theoretical description of the spontaneous parametric down-conversion in such structures is necessary. Thus, we present a model allowing the study of the parametric fluorescence process in open systems of arbitrary complexity. Our approach generalizes a recent formalism by developing the Green's functions of the system on the basis of quasi-normal modes (QNMs) [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF][START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF][START_REF] Muljarov | Brillouin-Wigner perturbation theory in open electromagnetic systems[END_REF], eigenmodes of open systems. Our formalism efficiently describes the set of spectral, spatial, and polarization properties of photon pairs generated by parametric fluorescence from nanostructures. It also highlights the modal interaction of the nonlinear process and allows the engineering of quantum states for these structures.

The elaboration of our formalism has been achieved in collaboration with Romain Dezert, post-doctoral fellow in our team, and Maximilian Weissflog, Sina Saravi, Frank Setzpfand and Thomas Pertsch, from the Friedrich Schiller University of Jena.

Contents

III.1 Spontaneous parametric down-conversion formalism based on quasi-normal modes

As seen in the first chapter of this manuscript, SPDC is a probabilistic nonlinear process. This implies that the properties of the generated photon pairs cannot be unequivocally computed, as would be those of its corresponding classical process: SFG. In macroscopic systems, SPDC requires phase matching ( [START_REF] Boyd | Chapter 3 -Quantum-mechanical theory of the nonlinear optical susceptibility[END_REF]. However, when the nonlinear interaction length becomes smaller than the coherence length in the material

- → k p = - → k s + - → k i )
(L c = π/|∆ - → k |, with ∆ - →
k the wave-vector mismatch), as it is the case at nanoscale, the longitudinal phase-matching condition is relaxed [START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF]. Although the relaxation of the phase-matching condition may lead to a broader spectral generation with larger emission angles, it will inevitably lead to very low conversion efficiency. To enhance the interaction, resonances of the confined photons in the system will induced by nanopatterning [31-33, 53, 62, 137].

Directly computing the quantum state of the generated photon pair is impossible. One could envisage to infer it through the corresponding SFG process, by considering the efficiency associated to the recombination of various signal and idler pairs. However, with the relaxation of phase matching supplanted by light resonances, this calculation would become very challenging because of the resonance richness of the nanoresonators commonly studied in nanophotonics [START_REF] Petrov | Second harmonic generation and spontaneous parametric down-conversion in Mie nanoresonators[END_REF]. The ensemble for the possible signal and idler photon becomes too large. Therefore, to describe SPDC in nanopatterned nonlinear resonators, we opted for a probabilistic formalism relying on the Green's tensor of the propagation of the photon pairs generated in the nonlinear nanoresonator [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF][START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF].

III.1.1 Derivation of the formalism

III.1.1.1 Two-photon transition amplitude

In order to set the ideas, let us consider a non-centrosymmetric dielectric material with χ (2) susceptibility. For the γ component of a pump field of amplitude E p,γ at the angular frequency ω p , the SPDC nonlinear interaction obeys the following Hamiltonian (in SI units) [START_REF] Drummond | The quantum theory of nonlinear optics[END_REF]:

ĤNL = -0 ¢ dω 1 dω 2 (2π) 2 ¢ d 3 rE † α (ω 1 , r)E † β (ω 2 , r)χ (2) αβγ (r, ω 1 , ω 2 , ω p )E p,γ e -iωpt + h.c. (III.1)
where † denotes the Hermitian conjugate (here, the creation of a photon associated to the field E). The three components α, β, γ of the signal and idler (E † α , E † β ) and pump E p,γ electric fields are coupled through the second order nonlinear susceptibility χ [START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF] αβγ (r, ω 1 , ω 2 , ω p ) at r. ĤNL can be seen as a source term, which spontaneously converts photons from the pump field to the signal and idler fields. The generated photons propagation is accounted for by a linear Hamiltonian term Ĥlin that involves their electromagnetic Green's tensor G(r, r', ω). It is defined as the impulsional response of the wave equation [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF]:

∇ ∧ ∇ ∧ -ω 2 µ 0 ε 0 ε r G(r, r , ω) = Iδ(r -r ) (III.2)
introducing the spatial Dirac function δ(r -r'), the vacuum permeability µ 0 , and the vacuum permittivity 0 . The detections of the signal and idler photons are modeled by polarized excitonic two-level systems. The following detection term must be added to the Hamiltonian of the system:

Ĥdet ≡ ω i,s â † i,s âi,s -di,s • E(ω i,s , r i,s ) (III.3) â † i,s (â i,s
) being the creation (annihilation) operator associated to the creation of an exciton when detecting signal or idler photons, and di,s the dipole momentum operator accounting for the detection of a photon in photon counting regime on our photodetectors, along a given polarization direction and at a given position in our experimental setup [START_REF] Del Valle | Theory of frequency-filtered and time-resolved n-photon correlations[END_REF]. di,s is defined as â † d i,s + âd * i,s .

To calculate the two-photon transition amplitude T is ∝ ψ| Û |0 , we must apply the evolution operator Û associated to the above full Hamiltonian system (nonlinear source, propagation and detection on the two-level system) [START_REF]Nonperturbative Calculation of Transition Amplitudes[END_REF]. Using the time domain definition of the Green's tensor [START_REF] Landau | Statistical Physics: Theory of the Condensed State[END_REF], the calculation leads to the following expression, which corresponds to Eq. 1 in [START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF]:

T (r i , ω i , d i ; r s , ω s , d s ) = q i ,qs α,β,γ d i,q i d s,qs ¢ V χ (2) αβγ (r)E p,γ (r, ω i + ω s )G q i ,α (r i , r, ω i )G qs,β (r s , r, ω s )d 3 r (III.4)
Eq. III.4 corresponds to the joint probability amplitude of exciting two atoms, one at position r i with angular frequency ω i and detected along d i , and the other one at position r s with angular frequency ω s and detected along d s , and gathers the information on the photon pair generation (2 nd order susceptibility, pump field, and Green's tensors), propagation (Green's tensors) and detection (dipoles d i,q i and d s,qs ). All the involved input variables fully describe the state of the SPDC photons, accounting for their propagation direction, energy, and polarization.

Remark: α, β and γ are summed over {x, y, z}; the sums over q i and q s runs over the different components of the vector (d i , d s ), which are defined by the user depending on the probed polarization state. This is why the dependence of T on q i and q s is not explicit in T (r i , ω i , d i ; r s , ω s , d s ).

III.1.1.2 QNM-based expression for the two-photon amplitude

Even if the expression of T of Eq. III.4 is valid in the general case, deriving an analytical form for the Green's tensor associated to the generation and propagation of a photon pair by SPDC is only possible in a few systems, such as dielectric spheres. For a given nanoresonator, Green's tensor can be decomposed on the QMNs basis without any assumption on the material permittivity. This method, reported in [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF][START_REF] Muljarov | Brillouin-Wigner perturbation theory in open electromagnetic systems[END_REF][START_REF] Muljarov | Exact mode volume and purcell factor of open optical systems[END_REF], relies on Mittag-Leffler's theorem, which stipulates that the Green's tensor can be expanded as a series of poles [START_REF] Muljarov | Brillouin-Wigner perturbation theory in open electromagnetic systems[END_REF][START_REF] More | Theory of decaying states[END_REF][START_REF] More | Properties of resonance wave functions[END_REF], leading to the formula of Eq. ( 7) in [START_REF] Muljarov | Exact mode volume and purcell factor of open optical systems[END_REF]:

G q,α (r, r , ω) = - 1 µ 0 ∞ m=1 E m,q (r) E m,α (r ) (ω -ω m ) ω m (III.5)
Therefore, the expression of the Green's tensor in the QNMs basis results in the following form for the two-photon transmission amplitude:

T (r i , ω i , d i ; r s , ω s , d s ) = - 2ω 2 i ω 2 s 0 c 4 T (r i , ω i , d i ; r s , ω s , d s ) (III.6)
where:

T (r i , ω i , d i ; r s , ω s , d s ) = q i ,qs α,β,γ d i,q i d s,qs n,m ξ n,m (ω i , ω s ) E m,q i (r i ) E n,qs (r s ) (III.7)
and:

ξ n,m (ω i , ω s ) = 1 µ 2 0 1 (ω s -ω n ) ω n (ω i -ω m ) ω m ¢ V χ (2) αβγ (r)E p,γ (r) E n,β (r) E m,α (r)d

Physical meaning of the two-photon transition terms and their implication

The term d i,q i d s,qs is reminiscent of the two-level detection model. Then, the term ξ n,m (ω i , ω s ) characterizes the spectral efficiency of the nonlinear process. It actually contains an overlap integral between the pump field and the fields of all possible modes for the signal and idler photons: frequency conversion is efficient only for certain mode combinations that foster strong electric field overlap (Fig. III.1a). It therefore determines which QNMs E m,q i (r i ) and E n,qs (r s ) will contribute to the pair generation. Since it is possible to compute the far-field of the QNMs of a given structure [START_REF] Yan | Rigorous modal analysis of plasmonic nanoresonators[END_REF], T is takes into account the emission of all the computed QNMs to reconstruct the radiation pattern of the nanoresonator due to the nonlinear interaction (Fig. III.1b).

III.1.1.3 From two-photon amplitude to generation rate and characterization of the two-photon state

d i,q i and d s,qs in Eq. III.6 correspond to components of the dipole moment modeling a twolevel sensor. They are intrinsically related to the systems that will be used to detect SPDC biphotons in our experimental setup (see Sec. III.3.1). To relate the measured photon counts to the theoretical prediction, we need to link d i,q i and d s,qs to the detection efficiency of our setup. The detection efficiency can be calculated by taking the inverse ratio between the number of photons incident on the detectors and the number of photons actually detected by the two-level system. Assuming that in the vicinity of the detector, the flux of photons arriving on the detector can be approximated as monochromatic plane waves, the amplitude of the Poynting vector S j of a plane wave polarized along q j reads:

|S j | = cn j ε 0 2 |E j | 2 (III.9)
where j ∈ {i, s}, and c/n j the velocity of the photon j of angular frequency ω j . The related photon flux through the surface element dS can be calculated as:

dN phot,j dtdS = |S j | ω j = c n j ε 0 2 ω j |E j | 2 (III.10)
On the other hand, the Fermi's golden rule gives the count rate detected by the two-level system. For an incident wave polarized along q j (j ∈ {i, s}), it reads as (cf. Eq. 59 in the Supplementary Material of [START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF]):

d 2 N abs,j dt = 2π δ( ω -ω j ) |d j • E j | 2 (III.11)
From this we can define the detection efficiency Q j of detector j, expressed in m 2 •s -1 , by:

dN abs,j dt = Q j d 3 N phot,j dtdSdω (III.12)
Integrating both sides of Eq. III.12 over dω leads to the following expressions of Q j , where we considered the particular case of dipole moment detector oriented along q j (the polarization direction of the plane wave j):

Q j,q j = 4πω j |d j | 2 cn j ε 0 (III.13)
In the following we will consider that all the detection directions are equivalent in the sense that Q j,q j = Q j ∀q j (this assumption is compatible with our detectors, see section III.3.1). Eq. III.13 is useful as it might be easier to link Q j , rather than d, to the effective experimental detector efficiency. From here, it is even possible to directly normalize the two-photon count rate by the efficiency of the two detectors Q j in order to get rid of the dependence on the detection configuration [START_REF] Poddubny | Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials[END_REF]:

d 5 N pair dtdω i dω s dS i dS s = 1 Q i Q s W is (III.14)
where

W is = 2π δ( ω i + ω s -ω p ) |T is | 2 (III.15)
and with T is is given by Eq. III.6. The integration of Eq. III.14 over dω i leads to a normalized count rate expressed independently of the detection apparatus:

d 4 N pair dtdω s dS i dS s (r i , ω i , d i ; r s , ω s , d s ) = n i n s ω 3 i ω 3 s 2πc 6 T is (r i , ω i , d i ; r s , ω s , d s ) 2 (III.16)

Summary of the formalism

To sum up, we write the main expressions that we will use to calculate the normalized photon-pair generation rate (per frequency and surface elements) in our numerical implementation. They do not depend on a specific detection setup:

d 4 N pair dtdω s dS i dS s (r i , ω i ; r s , ω s ) = n i n s ω 3 i ω 3 s 2πc 6 T is (r i , ω i , d i ; r s , ω s , d s ) 2 (III.17)
The generation rate can alternatively be expressed per frequency and solid angle and in this case reads:

d 4 N pair dtdω s dΩ i dΩ s (r i , ω i ; r s , ω s ) = n i n s ω 3 i ω 3 s 2πc 6 r 2 i r 2 s T is (r i , ω i , d i ; r s , ω s , d s ) 2 (III.18)
since dS = r 2 dΩ. The expression of T is with the QNM formalism reads:

T is (r i , ω i , d i ; r s , ω s , d s ) = q i ,qs α,β,γ d i,q i d s,qs n,m ξ n,m (ω i , ω s ) E m,q i (r i ) E n,qs (r s ) (III.19) with ξ n,m (ω i , ω s ) = 1 µ 2 0 1 (ω s -ω n ) ω n (ω i -ω m ) ω m ¢ V χ (2) αβγ (r)E p,γ (r) E n,β (r) E m,α (r)d 3 r (III.20)

III.1.2 From photon population to density matrix: tomography reconstruction of quantum polarization state

The previous section introduced a formalism to derive both spatial, spectral and polarization properties of a biphoton state generated by SPDC in a χ (2) nano-element. The generation rate computed with Eq. III.16 depends on the QMNs of the nanostructure, so the signal and idler photon emission diagram will be anisotropic. We perform quantum tomography of the generated biphoton, and calculate two quantities from the density matrix of the polarization state: the Schmidt number, which will be our entanglement witness, and the fidelity with respect to a target state. In particular, entangled Bell states encoded in polarization are central building blocks of quantum technologies [START_REF] Wang | Integrated photonic quantum technologies[END_REF][START_REF] Georgescu | How the Bell tests changed quantum physics[END_REF][START_REF] Luo | Quantum teleportation in high dimensions[END_REF][START_REF] Llewellyn | Chip-to-chip quantum teleportation and multi-photon entanglement in silicon[END_REF]. In order to reconstruct a physical density matrix (i.e. normalized, Hermitian, and semi-positively defined), we implemented the maximum likelihood estimation of density matrices introduced in [START_REF] James | Measurement of qubits[END_REF]. This method corrects the violation of density-matrix characteristics due to statistical count errors (dark counts, fluorescence), or angular uncertainty with respect to the polarization state (wave-plate misalignment and imperfect separation by polarization beam splitter), which could lead to negative eigenvalue for the density matrix. In practice, the maximum likelihood estimation of density matrices relies on a user-generated density matrix ρ gen , whose dimensions match the quantum-state basis on which the tomography is performed. Then, a likelihood function is computed. It statistically accounts for the similarities between ρ gen and the measured data. Once maximized, the likelihood function provides a set of components for ρ gen which are the best estimation for the measured data that respect Hermiticity and positive semi-definiteness.

The formalism and methods introduced above will be applied in the next section to AlGaAs nanocylinders. We will compute the spectrum of the generated photon pairs, their emission diagram, as well as the density matrix associated to the polarization of biphoton states.

III.2 Toward the implementation of metasurfaces for quantum state generation

In this section, we apply the above formalism to AlGaAs nanocylinders and nanochairs. Such meta-atoms have already been studied in our group for up-conversion applications [START_REF] Marino | Zeroorder second harmonic generation from AlGaAs-on-insulator metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF][START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF], and constitute the building blocks to fabricate AlGaAs-based metasurfaces. The exploration of SPDC in these structures follows three steps, related to the terms of the two-photon transition amplitude: 1) the spectral efficiency of the down-conversion process; 2) the tomography of the generated biphoton states following various detection schemes; 3) the tomography of the biphoton states generated with different pumping conditions.

Remark on notation and color codes. In the following section, directions or vectors associated to cartesian coordinates x, y, and z will be depicted in red, green and blue, respectively. Angles are given in the spherical coordinates θ (with respect to z-axis) and φ (rotation around z-axis).

III.2.1 SPDC on [100] AlGaAs nano-cylinder with normal incidence

III.2.1.1 Spectral characterization and efficiency of the SPDC process

To evaluate the properties of biphoton states generated by a [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocylinder, we first focus on the computation of the spectral efficiency of the nonlinear process, denoted ξ n,m (ω i , ω s ) in the expression of the two-photon transition amplitude (Eq. III.6). As previously detailed, this term is written as an overlap integral between the pump field at ω p and the QMNs E n and E m through the AlGaAs nonlinear tensor. We consider a dielectric cylinder of radius r = 220 nm and height h = 400 nm, to reproduce the conditions of Ref. [START_REF] Marino | Spontaneous photon-pair generation from a dielectric nanoantenna[END_REF]. The cylinder is pumped by a generated off resonance from the orthogonal magnetic dipole, because of the spectral width of the low Q-factor magnetic dipoles of a nanocylinder.

One notable fact is the presence of strong values for some ξ n,m far from the degeneracy. Because the structure is highly multimodal, it is possible to find higher-order modes with significant field overlap through the AlGaAs nonlinear tensor. For instance, the term ξ 122,2 fosters the nonlinear interaction between the pump field and the magnetic dipole along z and an octopole of the cylinder. The narrow resonance of the higher-order mode dominates ξ 122,2 , which reflects a strong conversion efficiency for the generation of a photon pair at (878 nm, 6 µm), extremely far from the degeneracy. The term ξ n,m in the case of a nanostructure therefore enables SPDC over a large spectral window. By computing the SPDC generation rate, it is possible to confirm this conjecture, as demonstrated in Fig. III.3b and c. The generation rate is integrated over all directions to quantify the maximal dN pair / (dtdω s ) for a range of pump wavelengths λ p . If the probability of generating biphotons is overall small, some values of λ p induce a stronger signal. If we now look at the scattering efficiency spectrum of the nanocylinder under the same excitation conditions (Fig. III.3a), it appears that a favorable case for high SPCD rate is obtained when λ p matches one of the resonance wavelengths. Moreover, the resulting local maxima of the emission rate match the presence of resonances for signal and idler wavelength (Fig. III. 3b andc). The next section will focus on the tomography of those state, introducing detection schemes and spatial filtering.

III.2.1.2 Emission diagram and quantum tomography of the measured two-photon states

The decomposition of Green's tensors on the QNMs basis allows to calculate the emission diagram of biphoton states, because all the far-field propagation properties are encompassed in the Green's function as long as a sufficient number of QNMs are computed for a given nano-element. optical axis with a pump beam impinging at normal incidence is expected to be experimentally challenging.

It is then possible to characterize the polarization properties of the generated biphoton states. For this, we use quantum tomography to compute the associated polarization density matrices, Schmidt number, and fidelity to targeted entangled polarization state. The results of this approach are reported in Fig. III.5, where the color bar of the emission diagram now represents the Schmidt number in grey scale (d-f). This representation combines the information on generation efficiency and entanglement. Quantum state tomography is performed on an ensemble of discrete emission directions. If the fidelity of the emitted biphoton states with respect to any Bell state exceeds 0.9, the direction(s) is(are) identified and labeled. Note that from now on, copropagating signal and idler photons will not be degenerate, in order to ensure distinguishability of the two photons and will be labeled as colored dots. Therefore, signal (idler) photons will be generated at 2λ p + 20 nm (-20 nm), or as close as possible, depending on the frequency sampling of the simulation. Since we have already demonstrated that SPDC in nanocylinders is a broadband process, this will not affect much the computed generation rate. andd), four states are identified with maximum fidelity. The |HH state is mainly emitted on the sides of the four principal lobes, where the Schmidt number is close to 1. It means that the photon pairs co-detected at this angle will not be entangled, contrary to those emitted at the apex of one of the six lobes of the emission diagram. On the four main lobes, the |HH + |V V states are emitted with a fidelity of 0.93. Note that for all the detection protocols that filter signal and idler photons in two unique directions, the trace of the squared density matrix is equal to 1, The color scale indicates the Schmidt number of the calculated states for each direction of the pairs. The colored dots identify few separable or maximally entangled states having a high fidelity (above 0.9) with the states calculated in the different directions. The red, green and blue arrows represent respectively the x, y and z directions. The top panels reveal the real and imaginary parts of the density matrices of highly entangled photon pairs generated in 3 sets of directions (indicated by the black dots on each diagrams). The fidelity of these 3 states with the nearest Bell state is reported above each panel. therefore they are pure states. This will not necessarily be true when integrating the generated pairs over a solid angle. On the secondary lobes along the x-axis, the biphoton state has a maximal fidelity with |HV + |V H . Finally the state |HH -|V V is generated at φ = 45 • in the xy plane. Although the coincident rates of the two latter are significantly lower that of |HH + |V V , the tomography of SPDC co-propagating photons generated with a pump at λ p = 735 nm does not give a homogeneous state, easy to manipulate. The situation is different with counter-propagating (Fig III .5b and e) and z-axisymmetric (Fig III .5c andf) detection protocols. In the first case, photons emitted with opposite wave-vectors are co-detected, the yz-plane photons are suppressed. The only possible signal and idler couples are exclusively generated by the magnetic dipoles orthogonal to the pump polarization direction. This fact is reflected also by the symmetry of the emission pattern, which exactly matches the far-field overlap between those two modes. In this situation, even though the generation rate is lower than on the main lobes of the co-propagating case, the entanglement of the photon pair is higher. The generated states include a mixture of |HV + |V H and |HH -|V V , with fidelities exceeding 0.99. As the two states are generated along two orthogonal directions, it would therefore be possible to generate and use two different biphoton states from a single nanocylinder with two counter-propagating detection directions: one along (-x, x) to access |HV +|V H ; the second along (-y, y) to access |HH -|V V . Lastly, in the case of z-axisymmetric detection (Fig III .5c andf), photon pairs are collected in two directions rotated by φ = π. For this detection protocol, only one biphoton state is measured: |HV -|V H , with maximal entanglement and fidelity.

The conclusion of Fig III .5 analysis is that the emission pattern and generation rate are the same for the z-axisymmetric and counter-propagating signal and idler detection, which means that adapting the detection protocol could in principle allow to switch from an entangled state to another. It therefore demonstrates the working principle for a new dynamically controllable source of entangled biphoton states. According to this calculation, the generation rate of such a source could exceed 10 kHz, if integrated over all admissible emission directions. However, dynamically controlling a detection protocol would in practice prove challenging. Therefore, we now see how the choice of the properties of the pump beam (wavelength, incidence angle, and polarization) affects the properties of the generated photon-pairs, as it would be much easier than changing collection scheme.

III.2.1.3 Control of the entanglement of photon pair with pump wavelength

In order to have on-demand control on the properties of SPDC biphotons generated by [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocylinders, we took advantage of the pump degrees of freedom. In the previous section, the pumping conditions were set and different detection protocols of the biphotons were implemented. This time, attention is brought to co-propagating biphotons, and tomography of polarization state is performed when the pump wavelength varies from 680 nm to 780 nm. The other properties of the pump remain the same: i.e. an x-polarized plane wave that propagates towards -z. In particular, we focus on the direction D = (θ = 135 In addition, we have computed the pair generation rate in the direction D (Fig. III.6a, purple curve). It exhibits two maxima in the spectral region, already reported when integrating over all possible directions of space in Fig. III.3. The first maximum of generation rate in the direction D is obtained for a pump wavelength of 738 nm, only red-shifted by 8 nm from the pump wavelength that generates a maximally entangled state (730 nm, B marker). The second maximum is found slightly above the scanned spectral window and is obtained when the |HH state is predominantly generated by the SPDC process. This means that it is possible to design an entanglement switch with two pump wavelengths with nearly optimal generation efficiencies for co-propagating biphoton states in the direction D, both exhibiting a generation rate per solid angle and frequency dN pair / (dtdω s dΩ s dΩ i ) = 2.3 × 10 -7 m -4 . The first spectral position of such a pump switch would be 734 nm, at the intersection of the black and purple curves in Fig. III.6a. This configuration results in the best compromise between entanglement, with a Schmidt number above 1.9 and a fidelity F (|φ biphoton , |HH + |V V ) > 0.9. The second would be 775 nm, where the generated state is mostly separable. Two continuous-wave lasers at 734 nm and 775 nm with fast amplitude modulation (order of MHz) would in theory enable to sequentially control the entanglement of the biphoton generated from the AlGaAs nanocylinder.

III.2.2 On-axis generation: SPDC on [100]-AlGaAs nanocylinder and nanochairs

A downside of the previously characterized [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocyliner is the complexity of detection schemes and spatial filtering of the SPDC generated biphotons: in all studied cases, the nonlinear tensor of the material entails an efficient generation far from normal incidence, which is challenging to implement. To overcome it, we predicted the generation of photon pairs from a single [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocyliner pumped at oblique incidence by a linearly polarized plane wave. In addition, we explored SPDC in advanced nanostructures with asymmetric design, to redirect emission pattern toward normal direction [START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF][START_REF] Campione | Broken symmetry dielectric resonators for high quality factor Fano metasurfaces[END_REF]. along the direction (0, 0, -z) is possible over a large angular range of pump incidence. More precisely, the generation rate is reported for s and p-polarized pump plane waves vs. angle of incidence. We describe by s(p)-polarized wave the plane wave that was x(y)-polarized, whose incidence angle rotates around the y-axis. The results of this calculation show that:

III.2.2.1 Control of biphotons directivity with pump incidence on nanocylinders

• p-polarized pump waves lead to less efficient SPDC in [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocyliner for this wavelength, whatever the incidence angle.

• the maximum of co-propagating generation rate is obtained for an angle of incidence of 45 • .

• a wide range of angles is admissible for generation rates above

d 2 N pair / (dtdω s ) = 0.7×10 -8 .
Consequently, our model predicts a favorable generation of co-propagating biphoton state at normal of the substrate for tilted pump impinging on a [100]-AlGaAs nanocylinder, with a maximal generation rate at θ = 45 • . Another option to foster on-axis generation would be to rely on symmetry-breaking of the nano-antenna to direct the emission from the nanostructure toward the z-direction.

III.2.2.2 Emission of the biphoton from a nanochair

Previous works both in up- [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF][START_REF] Campione | Broken symmetry dielectric resonators for high quality factor Fano metasurfaces[END_REF] and down-conversion [START_REF] Santiago-Cruz | Resonant semiconductor metasurfaces for generating complex quantum states[END_REF] based on GaAs nanostructures and metasurfaces demonstrated that C 1 -symmetrical objects can emit light at normal incidence, whereas objects with a symmetry axis in the z-direction cannot. We expect that the peculiar emission of optically nonlinear C 1 symmetry objects allows to compensate the off-axis emission of photon pairs in [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanostructures when pumped at normal incidence. The scattering spectrum of the nanochair imaged in ). However, the symmetry breaking performed to fabricate this particular nanochair from a nanocylinder is not enough to obtain a strong directive co-propagating photon pair generation in transmission. Further engineering and optimization on the geometry of the nano-element will be required to improve the directivity of SPDC generated photon pairs.

Take-home message

In this section, we have applied the formalism presented in Sec. III.1.1 to [100]-AlGaAs nanocylinders and nanochairs, demonstrateing that the generation of biphoton states by SPDC, enhanced by the resonances of the structure, is in principle a broadband process in all its degrees of freedom. The many QMNs of the nanostructures results in a rich emission diagram, in which many different states are jointly generated. It is possible to spatially filter the SPDC photons to obtain pure states, whose nature depends on the detection directions. The nature and emission of the SPDC biphoton states are controllable with the pump wavelength and its incidence angle, which confirms that χ (2) nanomaterials may be potential candidates as quantum sources of entangled photon pairs. The objective is now to setup an experiment to verify the prediction of our formalism.

III.3 Experimental realization

According to the literature on parametric fluorescence in nanostructured materials [START_REF] Santiago-Cruz | Resonant metasurfaces for generating complex quantum states[END_REF][START_REF] Jin | Efficient single-photon pair generation by spontaneous parametric down-conversion in nonlinear plasmonic metasurfaces[END_REF][START_REF] Marino | Spontaneous photon-pair generation from a dielectric nanoantenna[END_REF][START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF][START_REF] Santiago-Cruz | Entangled photons from subwavelength nonlinear films[END_REF][START_REF] Santiago-Cruz | Resonant semiconductor metasurfaces for generating complex quantum states[END_REF][START_REF] My | Broadband photon pair generation from a single lithium niobate microcube[END_REF], the first testbed towards the implementation of a two-photon-state source is to measure the second-order correlation function g (2) (τ ), τ being the temporal delay between the detected signal and idler photons, which should exhibit a correlation peak at τ = 0. In the following, after detailing the experimental setup for this measurement, we report on its tests with LN bulk crystal and film. The (unfruitful) attempt of SPDC generation in AlGaAs film and metasurfaces are finally reported in this section.

III.3.1 Experimental setup

As mentioned, the first signature of SPDC is the coincidental detection of the two generated photons on two different detectors. In our case, the latter are two ID Quantique InGaAs/InP single-photon avalanche photodiodes (APD, see Fig. III.9) which amplify the photocurrent created by an single impinging photon and convert its arrival in a digital signal readable by a time tagger for photon counting or correlation histograms (Fig. III.9b). The quantum efficiency of an APD is lower than superconducting nanowire-based photodetectors (SNSPD), with typically 0.10 detection efficiency at telecom wavelength (blue squared region in Fig. III.9a for used spectral window) against 0.9 for the superconducting technology. Nonetheless, the APD work temperature (T = -90°C for maximal detection efficiency) is reachable with simple Peltier cooling, and does not require any helium-based cryogenics. Note that multi-mode fibers guide light toward the APDs in order to have the best modal acceptance possible. In order to perform measurements in the quantum regime in our lab, we have integrated our APDs in the experimental setup described in Fig. III.10, where two pump and collection lines around the already existing sample stage have been implemented. Our idea was to take advantage of the flexibility of this stage, namely its microtranslation controlled with piezoelectric actuators, and the possibility to work both in transmission and reflection. Two laser sources can be used: an optical parametric amplifier (OPA) that delivers 200 fs pulses at a rate of 1 MHz, and a CW Ti:Sa laser. The beam polarization is controlled by a half-waveplate, and some neutral density filters can be added to attenuate the few tens of mW of the source. CCD cameras are used to adjust focus on the sample and select the nanostructure cluster or metasurface to pump. In order to optimize the collection of scattered light, either in transmission or reflection, it is essential to use a high NA objective or lens, optimized for IR signal, in order to collect as many SPDC photons as possible. Indeed, any losses on the detection line will affect the g (2) correlation measurement as the square of its value (for example, 0.7 transmission divides by two the coincidence rate). In the transmission configuration, light is injected either by an aspheric lens or a low-magnification microscope objective. Its transmission is not critical, since in our case the available pump power is not a limitation. In reflection configuration, photon-pairs are separated from the pump and directed towards the APD thanks to a short-pass (SP) dichroic mirror placed before the injection objective. Note that residual pump photons and SPDC photons are filtered before their splitting with long-pass filters (LP filters in Red lines correspond to the pump path, whereas blue lines represent the collection paths for SPDC photon pairs. For both free-space and fibered configurations, collection of the SPDC photons can be realized in reflection or in transmission, in the case of a transparent non-scattering and/or non-absorbing substrate.

The first setup was implemented in free-space (see Fig. III.10a). The light collected from the sample is split by a 50/50 beam splitter (BS), leading to 50% detection of photon pairs in correlation counting mode. In practice, the broadband beamsplitter used is not perfectly 50/50, with slight dependence on the wavelength and polarization of the impinging wave, which leads to a coincidence event probability p < 0.5 after the beam splitter. Therefore, the collection line of the setup was modified from the free-space configuration of Fig. III.10a in the semi-fibered configuration shown in Fig. III.10b. In this case, a single mode fibered 50/50 BS replaces the previous one after spectral filtering step, which increases the compactness and practical handling of the setup. Moreover, it reduces the number of selective optical components required to perform correlation between forward and backward emitted photons in the case of non-collinear photon-pair generation, therefore increasing the overall versatility and efficiency of the excitation and collection lines.

In both configurations, the total transmission T corr = |t corr | 2 of the collection line associated to the measured coincidence rate can be estimated in the same way:

t corr = P AP D (λ i )P AP D (λ s ) × T (λ i )T (λ s ) 1 2
(III. [START_REF] Curto | Unidirectional emission of a quantum dot coupled to a nanoantenna[END_REF] where P AP D is the quantum efficiency of the ADP, T accounts for the linear losses of the setup (reflection coefficients of the mirrors, the SP dichroic mirror, transmission of the collection objective, coupling coefficient to the input fiber, and probability that the splitting of photon-pairs by the BS yields a coincidence event), both calculated for the wavelength of the signal and idler photons. Because the system is spectrally limited by the upper detection limit of the APD, and because a strict LP filtering is performed, the spectral window in which the SPDC photons are collected spans from few tens to a few hundred nanometers. Over this sensing range, the quantity involved in Eq. III.21 can be considered to be independent of the photon wavelength, resulting of the following numerical application for T corr 10 -3 . Even in the best scenario, and without taking into account the limitation on SPDC photons emission diagram by the numerical aperture of the collection objective, slightly less than one over a thousand pairs will be detected. The principal limiting factor being the quantum efficiency of the APD, using SNSPD could lead to much shorter integration times.

III.3.2 Measurements of photon pairs from nonlinear material at the nanoscale

III.3.2.1 Validation of the experimental setup with a LiNbO 3 thin film

In order to validate our setup, a first correlation measurement has been performed, pumping a LiNbO 3 (LN) film supplied by our collaborators in Friedrich Schiller University, Jena. The interest of the 640 nm thick film is found in its nonlinear susceptibility, whose largest contributing element is the d 33 = 40 pm.V -1 (Fig. III.11a), fostering the frequency conversion between co-polarized photons. It means that signal and idler photons spontaneously generated through the χ

(2) xxx susceptibility will be co-propagating, easing up significantly the setup alignment and the characterization of the generated photon pairs. In the case of a film thinner than few micrometers, C. Okoth et al. [START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF] predict a relaxation of phase-matching condition, leading to a broader emission angle for SPDC photons. characterization on a LiNbO 3 film, sequentially dealing with our detection capacity, the influence of the pump, and the influence of filtering process. In compliance with the theory [START_REF] Burlakov | Polarization state of a biphoton: Quantum ternary logic[END_REF], the coincidence rate is proportional to the square of the cosine of the angle between the polarization of the pump and the LiNbO 3 axis.

Lastly, the impact of the filtering of collected light on detection efficiency is assessed by increasing the cut-off wavelength of the LP filters used before the BS. The results of this ad-justment, reported on Fig. III.13, prove that the narrower the spectral window of the collected IR photons, the better the quality of the measurement up to a certain cut-off limit. This is explained by the measurement process of the APD. Whenever a photon is detected, the APD is frozen for a given period, the dead-time, during which no photon is detected. Because non-SPDC photons are more filtered with LP filters of higher cut-off wavelengths, the non-correlated events contributing to the noise are less detected, leading to a more efficient detection window for correlation. This leads to an enhancement of the correlation detection. Such improvement results in a significant increase of the signal-to-noise ratio, as shown in Fig. III.13b. When changing from a cut-off wavelength of 1400 nm to 1500 nm, some of the generated parametric photons are filtered, and therefore the coincidence rate declines. This situation has similar implications when the degeneracy wavelength becomes too high for the ADP detection range and one photon of the pair is no longer detected. This test highlights the importance of optimizing the spectral width of the detection window in order to collect as many photon pairs as possible.

III.3.2.2 SPDC attempt on [100]-AlGaAs unstructured thin films and metasurfaces

Unfortunately, we were not able to measure any correlation peak from [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs unstructured thin films, meta-atoms (nanocylinders or nanochairs), or metasurfaces, regardless of the substrate (aluminum oxide or sapphire). In this section, I will report the elements gathered during the experimental attempts, and try to draw conclusions and indications for future experiments. Two types of samples were probed:

• metasurfaces of AlGaAs nanocylinders on AlOx, with a setup that enables to perform measurements in reflection.

• nanochairs metasurfaces of AlGaAs on sapphire [START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF], meant to favor on-axis nonlinear conversion in transmission.

Fabrication of AlGaAs on AlOx metasurfaces. The fabrication of metasurface of AlGaAs nanocylinders (as displayed in Fig. III.14a

) requires two careful steps: an electron-beam lithography (EBL), in order to implement design with dimension below 100 nm; and an advanced reactive ion etching (RIE) process for III-V materials. Beforehand, an epitaxially grown sample of AlGaAs(400 nm)/AlAs(1000 nm) on a GaAs substrate is cleaned with the standard procedure described in Chapter II, and covered with a high-resolution electro-sensitive resist, the HSQ. The lithography is performed with a combined EBL/SEM hybrid, with an acceleration voltage of 20 kV, and a beam aperture of 10 µm to ensure a satisfactory in-plane resolution, as well as a fast writing speed. AlGaAs is etched through ICP-RIE commonly adopted in III-V nanofabrication. A 35 W -RF source is used to create a high-density plasma through a metallic coil (the inductive component). The plasma is injected inside the sample chamber and accelerated with another 15 W -RF source. The gas mixture inside the chamber is composed by Argon (30 sccm) and SiCl 4 (30 sccm). Ar + ions provide highly directional and poorly selective mechanical etching. SiCl 4 and SiCl + 4 free radicals, produced in the plasma, chemically activate GaAs and AlGaAs surfaces, featuring high material selectivity. The combined action of free radicals and Ar + ions results in perfectly vertical and smooth sidewalls. The final step of monolithic fabrication protocol is the selective oxidation of the AlAs layer, performed with an AET Technlogies wet-oxidation oven. The sample is introduced in a chamber under vacuum and slowly warmed up to 390°C (20°C/min) under controlled gas conditions (N 2 at 500 mbar). For the three displayed periods, the spectral acceptance of the system is maximal for pump wavelengths centered around 1480 nm, which corresponds to the excitation of the in-plane magnetic dipoles at the fundamental frequency. However, we were not able to observe any signature of the corresponding down-conversion process on the same metasurfaces. To avoid saturating the two ADPs, the sample was pumped with less than 10 mW and the measurement duration extended overnight, both when illuminating the metasurfaces from the top and from the substrate. No coincidence peak was measured when illuminating the metasurfaces, despite the many single photons detected by the APDs in the telecom wavelength range. Therefore, to rule out the possibility that fluorescence from the GaAs, whose gap wavelength is 872 nm, pollute our measurement, we moved to new samples with nanochairs on sapphire. As demonstrated in section III.2.2.2, these structures boost onaxis emission compared to nanocylinders, and have already proven successful for on-axis SHG in transmission [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF].

Fluorescence from a [100]-AlGaAs/sapphire film. To draw a complete picture of the situation, we measured the dispersion of the generated photons in the most favorable case, ie the [100]-AlGaAs/sapphire sample. We performed the measurement on a film without nanostructure, to single out just the material contribution. Two protocols were implemented: 1) the summation of the counts on an APD when adding more LP filters on the collection line (see Fig. III.10a); 2) the use of long optical fibers with monotonous dispersion to establish an arrival time / wavelength scale. For the first protocol, we used the CW Ti:Sa laser. The luminescence from sapphire substrate was negligible compared to the IR signal from the nanochair metasurface. Nonetheless, the result remained the same: no correlation peak was observed. For the second protocol, we used the 200 fs pulses generated by our OPA at a repetition rate of 1 MHz. Since the pulse width allows the wave packet to spread when propagating in the 100 km fiber, the photons arrive on a 200 ps time interval (Fig. III.15a). This way, the spectral spread of the photons covers the maximal acquisition window of our APDs. In (c), we also observe a strong luminescence from the AlGaAs film on sapphire, on a spectral window over which SPDC should be measured when pumping at 775 nm. Both measurement protocols yielded equivalent spectra. However, no g (2) peak is observed in coincidence counting regime on this sample, which suggests that either no photon pair was generated, or that one of the photon was not detected. It is difficult for us to conclude, because the [100]-AlGaAs nonlinearity corroborates both hypotheses: the photons are expected to be generated in two different directions if no nanostructure shapes the emission diagram of SPDC photons. In addition, the presence of the substrate (AlOx or sapphire) might also have an effect on the emission diagram of SPDC photons in the nanostuctures. Therefore, replacing the QMNs set used in Sec. III.2 by a new one, taking into account the substrate, could bring new elements to understand these experimental results.

Our results are presented in

III.4 A general scheme for dielectric quantum metamaterials

So far, the only material that has been studied was epitaxial [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs, which presents a strong χ (2) nonlinearity of 200 pm/V at 1550 nm [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF]. The advantages of this strong non-linearity are mitigated by the challenging polarization coupling rules resulting from the χ (2) xyz term in the nonlinear interaction. Other materials have already been used to achieve SPDC at the nanoscale [START_REF] Okoth | Microscale generation of entangled photons without momentum conservation[END_REF][START_REF] Santiago-Cruz | Entangled photons from subwavelength nonlinear films[END_REF][START_REF] Santiago-Cruz | Resonant semiconductor metasurfaces for generating complex quantum states[END_REF], favoring nonlinearities that couple collinear polarization fields to obtain on-axis emission of photon pairs. In this section, we will apply our model to [011]-AlGaAs nanocylinder [START_REF] Xu | Forward and backward switching of nonlinear unidirectional emission from gaas nanoantennas[END_REF][START_REF] Camacho-Morales | Infrared upconversion imaging in nonlinear metasurfaces[END_REF], and demonstrate that they constitute a promising dynamically controllable biphoton source at the nanoscale.

III.4.1 Exploration of alternative growth direction: [011]-AlGaAs

For the epitaxial III-V materials with zinc-blende structure (Fig. III.16), it is possible to exploit the nonlinear tensor and generate on-axis photons through nonlinear processes even though their χ (2) couples three waves of orthogonal polarization. In particular, on-axis SHG at normal incidence has been demonstrated in nano-antennas when changing the growth orientation from [START_REF] Xu | Forward and backward switching of nonlinear unidirectional emission from GaAs nanoantennas[END_REF]) or to [START_REF] Sato | Topological superconductors: a review[END_REF]-AlGaAs [START_REF] Sautter | Tailoring second-harmonic emission from (111)-GaAs nanoantennas[END_REF]. For instance, [011]-AlGaAs behaves similarly to LN in terms of nonlinear properties, except with a much higher nonlinear interaction [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF]. The spectral characteristics of SPDC process in [011]-AlGaAs nanocylinders will differ from the one in [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF]-AlGaAs nanocylinders, because the crystal orientation impacts the projection of the pump, signal and idler fields and their interaction through the nonlinear tensor of the material. For SFG, the nonlinear polarization in the laboratory coordinate system is expressed as [START_REF] Valencia-Caicedo | Second harmonic generation from gaas-au subwavelength relief gratings[END_REF]:

       P 2ω x P 2ω y P 2ω z        = 0 d 14        -2E x E y E 2 x -E 2 z 2E y E z        (III.22)
Besides, the QNMs of the structure depend on the optical index of the material and its geometry, .17, and verify that the materials have comparable spectral behaviors, ie that they constitute broadband photon-pair sources. However, two differences must be noted. 1) The projection of the interacting fields involved in the down-conversion process is not the same, therefore the ξ n,m are different. It affects the spectral efficiency of the process, which results in an overall higher biphoton generation rate in the telecom band. This is especially true when λ p > 730 nm.

2) The SPDC dominant spectral contributions are not associated to the same QNMs for [START_REF] Beenakker | Search for Majorana fermions in superconductors[END_REF] and [011]-AlGaAs. This has two direct consequences: the maximum generation rate is not obtained for the same pump wavelengths, and the emission pattern will be significantly different. To illustrate this, this section will describe a practical application case, where we took advantage of the nonlinear interaction in

III.4.2 Dynamic control of two-photon quantum state entanglement with pump polarization

To demonstrate that dielectric nanostructures constitute a promising platform to engineer photon pairs with dynamically controllable tailored properties, we have empirically considered a [011]-AlGaAs nanocylinder of the same dimensions as in Sec. III.2. It is pumped with a lin- (c,f,i,e,h,k When rotating the polarization of the pump from α = 0 to 90 • , the generated biphoton state turns from a state with a given Schmidt number to a maximally entangled state |HV + |V H . This observation is valid for all the computed wavelength, and results from the coupling of several QNMs, in particular the in-plane magnetic dipole, with favorable polarization through the nonlinear tensor of [011]-AlGaAs. Indeed, the radiation pattern of the dipole (see Fig. III.2c) imposes on-axis biphoton generation to conserve polarization components during SPDC process. Depending on λ p , the evolution of the generated state with α towards |HV + |V H varies, because different resonances are solicited by the pump plane-wave. At λ p = 735 nm, the generated state shifts from an entangled state close to |HH + |V V to the cross-polarized final biphoton state. Note that, for this λ p , the generated photon pair is perfectly separable for α = 60 • , which thus entails that the nanocylinder can be switched on demand from a source of separable to entangled state with a 30 • -rotation of pump polarization. This discussion demonstrates that linear polarization of a plane wave impinging at normal incidence on a [011]-AlGaA nanocylinder constitutes a knob to switch from a maximally entangled |HV +|V H state to a separable biphoton state, or from an entangled state to a different one. We have also proven that the generation of the photon pairs is systematically achieved on axis, and that such a nonlinear nanostructure allows to work both in reflection and transmission configuration. The above phenomenon has been empirically observed by trying out our formalism on materials other than the one our team conventionally works with. Therefore, it would be (according to me) of the utmost importance to implement and verify the prediction of this last calculation, because of the potential applications of a controllable compact source of entangled photons in the perspective of future quantum technologies.

III.5 Conclusion and perspectives

In this chapter, we have introduced a new model to predict the spectrum, spatial pattern, emission rate, and polarization of biphoton states generated by SPDC in nanostructures. The model relies on the use of the QNM basis, i.e. the exact complex solutions of Maxwell equations in nanoresonators. It has enabled us to perform directional quantum tomography of the generated photon pair, and therefore to design on-demand sources and detection protocol for dynamically controllable sources of entangled photons at the nanoscale. The experimental demonstration of the possibilities offered by our formalism remains to achieve, with potential large scope of applications and materials to test. Our approach, being based on an Hamiltonian description of the SPDC in nonlinear nano-elements, is not limited to second-order optical susceptibility: one could envisage describing higher-order nonlinearity to create more complex frequency conversion operations in the quantum regime, at the price of a lower efficiency. This work has illustrated the well-known importance of resonances to control and enhance nonlinear processes in nanosctructures. Some groups have reported the implementation of quasi-BIC resonances [START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF][START_REF] Rybin | High-Q supercavity modes in subwavelength dielectric resonators[END_REF][START_REF] Koshelev | Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum[END_REF] which increases the mode quality factor by a factor 10 at least. Other groups have opted for topological protection [START_REF] Smirnova | Nonlinear topological photonics[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF], which localizes the nonlinear interaction in photonic gaps (spectrally) and at the interface between phases of different topology (spatially). The implementation of such phases can be based on coupling engineering between nano-elements composing those systems [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF], which opens a more general perspective to modify modal and spectral properties of a resonator by fostering interactions and collective behaviors. In the next chapter, the coupling between several nanoresonators will be studied, in the perspective of mode engineering and metasystems design.

IV.1 On the eigenmodes of a chain of nanoresonators

IV.1.1 Strategies to describe coupling

In this chapter, we will study a toy system consisting of dielectric nanocylinders, with characteristic radius and height of about few hundred nanometers. Such geometry is closely related to the implementation of most optical nanoresonators based on epitaxially grown crystalline dielectric thin film [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF] or evaporated metallic film [START_REF] Yu | A broadband, background-free quarter-wave plate based on plasmonic metasurfaces[END_REF][START_REF] Genevet | Ultra-thin plasmonic optical vortex plate based on phase discontinuities[END_REF][START_REF] Lin | Nanostructured holograms for broadband manipulation of vector beams[END_REF], which are intensively studied in the nanophotonic community. We have purposely selected the cylindrical geometry to better convey the main message of our study. This first section lays the foundation of my research on coupled nanoresonator systems, and details on my numerical analysis and algorithms for the systematic study of hybrid modes in coupled systems.

One possibility could be to adapt Coupled Mode Theory (CMT, see Fig. IV.1), a perturbational approach for analyzing the coupling of oscillating systems through a set of coupled equations, to the coupled nanoresonator problem. In our case, the set of coupled equations is based on Maxwell's equations. The coupling constant κ B→A traducing the action of the field leaking from an element B on an element A is analytically expressed as the overlap integral between the electric field leaked by dielectric waveguides or resonators [START_REF] Hardy | Coupled mode theory of parallel waveguides[END_REF]:

κ B→A ∝ ω A ¢ V A 0 [ r ( - → r ) -b ] E A • E B N dr (IV.1)
where ω A is the angular frequency of the element A of volume V A , r the relative permittivity of the material constituting A, b the background relative permittivity, -→ E A,B the electric field associated to A or B, and N the mode's normalization coefficient given by the condition:

E A • E B = ¢ E A ∂ω ∂ω E B + H A ∂ωµ ∂ω H B dr = δ A,B (IV.2)
where the notation X designates the amplitude of a complex field. Besides being a perturbation theory, CMT relies on a few hypotheses, namely linearity of the equations, time-reversal symmetry, energy conservation, and time invariance. Being mindful that the Q-factor of dielectric resonators under study is very low compared to the systems to which CMT is conventionally applied, we opted for another approach: the mapping of open coupled systems by a non-Hermitian tight-binding model, involving a dissipative bath in order to keep trace of the open-system behavior of low Q-factor Mie scatterers.

IV.1.2 Preliminary considerations: eigenmodes of a nanocylinder

Eigenmodes of cylindrical structures are close to those of spherical resonators, which provides a basis of modes close to Mie resonances. Their size determines the spatial and spectral properties of their eigenmodes [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF]. For equal diameter and height, in-plane magnetic dipoles (MD x and MD y , Fig. IV.2d) and normal dipole (MD z , Fig. IV.2e) are degenerate in energy. For a given height, varying the radius of the cylinder shifts the eigenfrequencies of dipolar modes, and lifts their degeneracy. Nonetheless, due to the cylindrical symmetry of the considered system, MD x and MD y are degenerate in energy for an unique isolated particle. The distinction between them is only relevant to complete the basis of p-orbital-like modes, or when it comes to nonlinear optics in the case of a nonlinear tensor that breaks the cylindrical symmetry. Furthermore, nanocylinders are relatively easy to implement with e-beam lithography and are experimentally accessible. Therefore, they give a large flexibility and represent a convenient toy system for more advanced characterizations, designs and future experiments. In practice, Al 0.18 Ga 0.82 As nanoantennas are supported by a low optical index semiconductor substrate like aluminum oxide (n AlOx = 1.6). However, since the presence of such substrate only lightly affects the nanocylinders eigenmodes, here we neglect it and suppose the nanostructures to be surrounded by air.

As introduced in Chapter I, FEM full-vectorial electromagnetic calculations are our main tool to simulate the behavior of a single nanoelement. A major advantage of this approach is that it allows to sweep extensively the parameter space for the considered particle. [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Tomaszewska | Detection limits of DLS and UV-Vis spectroscopy in characterization of polydisperse nanoparticles colloids[END_REF][START_REF] Tomaszewska | Detection limits of DLS and UV-Vis spectroscopy in characterization of polydisperse nanoparticles colloids[END_REF]. In-plane and vertical geometric parameters are controlled by the design of the implemented lithographic mask and by the thickness of the epitaxially grown dielectric layer. The latter can be thinned via etching, however inducing the creation of surface defects, and therefore enhancing the dissipative character of the nanostructures.

IV.1.3 Mode tracking algorithm

Before extensively investigating any model for coupled systems, we need a trial numerical dataset, gathering the spectra of hybrid targeted modes as a function of the distance d between the cylinders of a N -link chain. These FEM simulations will serve as reference, since numerically solving Maxwell's equation gives the exact complex values of the eigenmodes in those systems. To compute and identify the eigenmodes of finite size ensembles of nanoresonators, two algorithms were implemented. The first one relies on near-field spatial correlations that enable us to track any given mode when the gap d is varied. At each iteration, the electric field of each computed mode from FEM simulation is interpolated inside and in the close vicinity of each nanocylinder, and stored as a gridded array as plotted in Fig. IV.3. Spatial correlation coefficients are calculated by comparing the interpolated field from two successive iterations. To achieve this correlation, the function corr2 from Matlab Image Processing Toolbox is used, which computes the correlation coefficient r corr between two arrays A and B of the same size:

r corr = m n (A mn -A)(B mn -B) ( m n (A mn -A) 2 )( m n (B mn -B) 2 ) (IV.3)
where X is the mean value of the array X. Even if mode hybridization perturbs locally the field inside the cylinder, assuming the gap step is small enough, one can reconstruct the evolution of a given mode when the gap is varied, by tracking the maximum of r corr . This algorithm can be initialized with the single-element simulation: the modes of an N -element chain are then tracked by groups of N hybrid modes stemming from a single-element mode. This approach entails several advantages, such as allowing the adiabatic sweep to be initialized from the largest values of gap, which makes RAM management easier. Moreover, it offers an intuitive comprehension of the hybrid modes, where the main contribution stems from monomer modes featuring the same symmetry. However, this initialization labels the same the N hybrid modes stemming from a single-element mode. In order to ensure the separability of the modes in the tracking algorithm, the first step of the adiabatic gap sweep can be performed for a small gap, because the effect of the coupling is stronger on the near-field of the nanoparticles [START_REF] Krenn | Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles[END_REF].

The second algorithm relies on a direct identification of modes based on the evaluation of their complex field components in the whole integration space, whose principle is exposed in ), and two equivalent E y and E x components. PML modes are discarded due to their higher field density in the PML layer. Each calculation of this algorithm is thus independent of the others. This procedure is very versatile, assuming that the user can properly extract sorting conditions from physics of Maxwell's equa-

Is Q-factor > 5 ?
PML or surface mode no yes What is the power stored in the cylinder by each cartesian component of E?

(E x , E y , E z ) ≡ (1,1,0) (E x , E y , E z ) ≡ (0,1,1) (E x , E y , E z ) ≡ (1,0,1) MD z MD x MD y
Is there local belly or a node of field in the median plane between the nearest-neighbor cylinders? tions. It proves more efficient with longer chains, for which eigenmode coupling to PML becomes more prominent in a finite calculation volume.

Overall, the two algorithms are fully functional and perform similarly, provided that the FEM simulation and the different parameters and threshold involved in the algorithm are well adjusted. The first algorithm requires no prior knowledge of the system, thus its implementation is less demanding. However, any parameter or numerical error within the simulation will be amplified, or even worse, unintelligible from the produced dataset. In practice, a hybrid algorithm has finally been deployed to solve the eigenmode problem with FEM calculation.

IV.1.4 1D chain of N = 2 and N = 3 coupled dielectric nanocylinders

We have chosen to focus on hybrid MDs, because they constitute a central component of nanophotonic research [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF][START_REF] Marino | Spontaneous photon-pair generation from a dielectric nanoantenna[END_REF][START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF]. We adiabatically swept distance between the two resonators, first treating the simplest system possible: a chain of N = 2 dielectric cylinders. Fig. IV.5 reports the resolution of the eigenvalues problem with FEM in this case. The spectra of each pair of hybrid modes is plotted as a function of the edge-to-edge gap d. A pair of bonding and antibonding modes is formed, corresponding to two complex eigenfrequencies in phase opposition. In the following, we will neglect the coupling between two MDs oriented along different directions, since the field overlap between two non-collinear MDs is three orders of magnitude smaller than the one between two collinear MDs. We identify two coupling regimes: an exponential-like decay of the frequency splitting for small gaps; and a pseudo-periodic variation of the complex eigenfrequencies for larger gaps. Outside each leaky resonator, the electromagnetic field decreases much slower than for high-Q systems like micropillars [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF] or whispering-gallery mode resonators [START_REF] Armani | Ultra-high-Q toroid microcavity on a chip[END_REF][START_REF] Baker | Photoelastic coupling in gallium arsenide optomechanical disk resonators[END_REF][START_REF] Parrain | Origin of optical losses in gallium arsenide disk whispering gallery resonators[END_REF][START_REF] Roland | Frequency doubling and parametric fluorescence in a four-port aluminum gallium arsenide photonic chip[END_REF]. When more than one resonator is involved, the corresponding eigenfrequencies oscillate and result in non-zero energy splitting. A good figure of merit to assess the range of this interaction is the scattering cross-section of each resonator mode, which can be as large as ten times the cylinder cross-section in the case of MDs [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF]. The eigenfrequency spectra of a pair of MD eigenmodes exhibit degeneracy points which occur at different gap values for real and imaginary parts and depend on the specific mode. This feature has already been reported recently [START_REF] Pichugin | Interaction between coaxial dielectric disks enhances the q factor[END_REF] and offers interesting perspectives for meta-optics design via the dispersion of the optical modes. Additionally, the coupling between MDs modifies the quality factor of those modes. For a single AlGaAs nanocylinder of the same dimensions, the Q factor of the MD z is 7 and that the MD x and MD y is 5.5: our simulations thus confirm that even long-range coupling between nanoresonators can increase Q factor, although light remains poorly confined in those structures.

In the case N = 3 (Fig. IV.6), the study is restricted to the MD x hybrid modes, because in the perpective of future experimental work, in-plane dipoles are easily coupled to linearly polarized light, and their emission diagrams exhibit non-zero out-of-plane radiations [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF][START_REF] Rocco | Tuning the second-harmonic generation in AlGaAs nanodimers via non-radiative state optimization[END_REF].

Examining the spectrum of Fig. IV.6a, the same comments can be made. Three hybrid modes exhibit the features of each MD, with oscillating real and imaginary parts for the complex eigenfrequencies of the triplet of resonators. The pseudo-period of those oscillations is the same as in the case N = 2, and the amplitude of the modes 1 and 3 corresponds to the one of the N = 2 hybrid modes. The eigenmodes near-fields displayed in Fig. IV.6b highlight the analogy between the modes of the trimer and the atomic p orbital of conjugated π electron system of an allyl anion [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Woodward | The conservation of orbital symmetry[END_REF]: the bonding and anti-bonding modes (1 and 3 in Fig. IV.6) correspond to two situations in which the p orbital lobes of the central sp 2 carbon atom are inverted, thus leading to a bonding or and anti-bonding state. Furthermore, the molecular diagram of the acetylene is composed of a third non-bonding mode at intermediary energy, which is the analogous of the mode 2 in the case of a dielectric nanocylinder chain with three meta-atoms. Note that this analogy is relevant to interpret qualitatively the results of the FEM calculations as long as the hybridized modes manifest the same symmetry class as the one of atomic orbital with respect to group theory.

The enhancement of the quality factor for gaps smaller than 400 nm spurs the interest for such coupled systems: from Q = 7 in the case of a the above AlGaAs cylinder, the Q-factor of the hybrid MD x mode 1 can reach 36 when d tends to zero. Since longer lifetime of photon in nonlinear cavities foster nonlinear phenomena [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF], the N = 3 chain illustrates the scope of controlled coupling at nanoscale: break through current limitation of metamaterials when in comes to nonlinear nanophotonics. Capitalising on the atomic analogy, an integral formulation of the near-field interaction based on the QNMs could be an interesting approach to describe the coupling mechanism governing the physics of finite size ensemble of nanophotonic resonators. Now that reference datasets have been produced with our mode-tracking algorithm, the next step of this work consists in finding a model to be predictive beyond numerical limitations related to FEM full vectorial simulation.

IV.2 Non-Hermitian bath model for arrays of coupled nanoresonators

In this section, I report the analytical implementation of the non-Hermitian quantum model that I employed to describe one-dimensional (1D) chains of N dissipative dielectric nanoresonators. Our approach is to feed this model with complex coupling parameters obtained from a numerical simulation in the case N = 2, and then validate it in the case of longer chains by comparison with numerical calculations. The model thereby becomes an essential predictive tool for overcoming computational limitations associated to the numerical study and design of larger systems. This model includes both direct coupling between resonators and coupling mediated by the electromagnetic continuum acting as a reservoir. Such couplings are determined by fitting the numerical solution of Maxwell's equations in the case of two coupled resonators. The latter is obtained via finite-element-method (FEM) simulations on our trial system, which consists of chains of equidistant aluminum gallium arsenide (AlGaAs) nanocylinders along x. Their radius and height are set at 300 nm and 400 nm respectively, to lift the spectral degeneracy between out-of-plane (MD z ) and in-plane (MD x and MD y ) magnetic dipolar modes, the coupling strength being dictated by the gap d between the nanocylinders. The numerical integration domain is bounded by a perfectly matched layer (PML) that suppresses spurious electromagnetic-field reflection at the edges of the integration domain and emulates the openness of the system. We focus on the hybrid modes stemming from MDs of single nanocylinders.

IV.2.1 Derivation of the analytical framework

To describe the dynamics of an ensemble of leaky resonators, let us consider a Hamiltonian consisting of a tight-binding term for the nanoresonators coupled to an effective bath. Its non-Hermitian character is obtained by implementing non-conservative coupling to the bath degrees of freedom (Fig. IV.7). Our basic idea is to extract the effective bath parameters from exact FEM results for two coupled resonators, and then use the formalism to predict the modes for an arbitrary number and arrangement of nanoresonators. In the rotating-wave approximation, the Hamiltonian reads (with = 1):

Ĥ = Ĥsys + Ĥbath + Ĥint , (IV.4) Ĥsys = j ω j â † j âj + j,j J(d jj )(â † j âj + h.c.), (IV.5) Ĥbath = ¢ dη ω η α † η αη , (IV.6) Ĥint = j ¢ dη i(g jη â † j αη -g * jη α † η âj ) (IV.7)
Ĥsys is the bare resonators' Hamiltonian, where each nanoresonator has one mode with annihilation operator âj and frequency ω j , j and j denoting different resonators. In the following, we will consider equally spaced identical resonators (ω j = ω 0 , d jj = d). J(d) describes the coherent coupling between two resonators via evanescent field, and is essentially determined by the distance d between them for a given set of resonators. The second term Ĥbath describes the continuum of radiation modes represented by the annihilation operators αη indexed by η, and Ĥint is the interaction Hamiltonian between the system and the bath, where g jη is the coupling between the j th nanoresonator and the radiation mode αη . The operators âj and αη obey bosonic commutation relations, i.e. [â j , â † j ] = δ jj and [ αη , α † η ] = δ(η -η ). All commutators between âj or â † j and αη or α † η are taken to be zero.

From the Hamiltonian and the commutation relations, the equation of motion for the continuum bath modes in the Heisenberg picture writes:

d αη dt = -i[ αη , Ĥ] = -iω η αη - j g * jη âj (IV.8)
which can be formally solved as

αη (t) = e -iωη(t-t 0 ) αη (t 0 ) - j g * jη ¢ t t 0 dt e -iωη(t-t ) âj (t ) (IV.9)
This leads to the dynamical equation for âj :

dâ j dt = -i[â j , Ĥsys ] + ¢ dη g jη αη = -i[â j , Ĥsys ] + ¢ dη g jη e -iωη(t-t 0 ) αη (t 0 ) - ¢ dη k g jη g * kη ¢ t t 0 dt e -iωη(t-t ) âk (t ) (IV.10)
Defining αin η = αη (t 0 )e iωηt 0 , we can introduce the damping kernel :

Γ jk (τ ) = Θ(τ ) ¢ dη g jη g * kη e -iωητ (IV.11)
and the Langevin force [START_REF] Ciuti | Input-output theory of cavities in the ultrastrong coupling regime: The case of time-independent cavity parameters[END_REF]:

Fj (t) = ¢ dη g jη e -iωηt αin η (IV.12)
with Θ(τ ) the Heaviside step function. In the limit t 0 → -∞, Eq. IV.10 can be rewritten as

dâ j dt = -i[â j , Ĥsys ] - k ¢ ∞ -∞ dt Γ jk (t -t )â k (t ) + Fj (t). (IV.13)
Finally, the commutator can be calculated as

âj , Ĥsys = ω 0 âj + j J(d jj )â k (IV.14)
which, together with the nearest-neighbor-coupling assumption, completes the derivation of the quantum Langevin equation for the 1D chain in the Heisenberg picture for âj [START_REF] Ciuti | Input-output theory of cavities in the ultrastrong coupling regime: The case of time-independent cavity parameters[END_REF]:

dâ j dt = -i   ω 0 âj + J(d) j âj   - k ¢ ∞ -∞ dt Γ jk (t -t )â k (t ) + Fj (t) (IV.15)
If we apply Fourier transform with the convention Ã(ω) = ¡ ∞ -∞ dte iωt A(t) to Eq. (IV.15), we obtain the equations in the frequency domain:

ωã j (ω) = ω 0 ãj (ω) + J(d) j ãj (ω) -i k Γjk (ω)ã k (ω) + i Fj (ω) (IV.16)
which can be cast into the following matrix form

M(ω, d)        ã1 (ω) . . . ãN (ω)        + i        F1 (ω) . . . FN (ω)        = 0 (IV.17)
with eigenvalues λ (i) that can be obtained by diagonalization of M. We can then solve for the resonant frequencies. Their real part minimizes (arg min ω ) the norm of M(ω, d) eigenvalues:

Ω res = ω i ∈ R | ∃i, ω i = arg min ω |λ (i) (ω)| 2 (IV.18)
which, by definition, gives local maxima of the amplitude of the frequency response. The corre-sponding damping reads:

γ i = -Im[λ (i) (ω i )] (IV.19)
For a compact notation, we can assign the complex frequency ω res i = ω i -iγ i to the i th resonant mode. In the case N = 2, with identical resonators, the matrix can be explicitly written as :

M(ω, d) =    ω 0 -ω -i Γdiag (ω, d) J(d) -i Γoff (ω, d) J(d) -i Γoff (ω, d) ω 0 -ω -i Γdiag (ω, d)    (IV.20)
where we have further assumed Γ11 = Γ22 ≡ Γdiag and Γ12 = Γ21 ≡ Γoff by symmetry.

The coupling functions J, Γdiag and Γoff can be fitted from the simulation results presented in Fig. IV.5. To simplify the treatment, we expand them to first order in ω, which allows us to determine a set of possible reservoir functions from the simulation of the N = 2 system. For an N -resonator chain, the matrix in Eq. IV.17 can be written as:

M N (ω, d) = (ω -ω 0 )I(N ) + J(d)Sec(N ) -i Γjk (ω, d) {j,k}∈ 1;N 2 (IV.21)
with I(N ) being the N × N identity matrix, and Sec(N ) the N × N secondary diagonal matrix.

IV.2.2 Determination of coupling constants from numerical simulation

The eigenvalues of Eq. IV.20 are given by

λ ± (ω, d) = ω 0 -ω -i Γdiag (ω, d) ± J(d) -i Γoff (ω, d) , (IV.22)
and therefore

|λ ± (ω, d)| 2 = [ω 0 ± J(d) -ω] 2 + Γdiag (ω, d) ± Γoff (ω, d) 2 (IV.23)
We will determine the coupling functions in our analytical model with the simulation data of MD x for N = 2 shown in Fig. IV.5d. For a simplified treatment, with no a priori knowledge on the frequency dependence of the reservoir functions, let us expand them up to the first order in ω:

Γdiag (ω, d) = A 1 (d) + A 2 (d)(ω -ω 0 ) (IV.24) Γoff (ω, d) = B 1 (d) + B 2 (d)(ω -ω 0 ) (IV.25)
Since this assumption makes Eq.IV.23 quadratic in ω, the resonant frequencies ω res ± = ω ± -iγ ± can be expressed analytically in terms of our model parameters:

ω + = -A 1 A 2 -A 1 B 2 + A 2 2 ω 0 -A 2 B 1 + 2A 2 B 2 ω 0 -B 1 B 2 + B 2 2 ω 0 + J + ω 0 A 2 2 + 2A 2 B 2 + B 2 2 + 1 (IV.26) ω -= -A 1 A 2 + A 1 B 2 + A 2 2 ω 0 + A 2 B 1 -2A 2 B 2 ω 0 -B 1 B 2 + B 2 2 ω 0 -J + ω 0 A 2 2 -2A 2 B 2 + B 2 2 + 1 (IV.27) γ + = A 1 + B 1 + (A 2 + B 2 )(ω + -ω 0 ) (IV.28) γ -= A 1 -B 1 + (A 2 -B 2 )(ω --ω 0 ) (IV.
S + = {A 1 , A 2 , B 1 , B 2 , J} and S -= {A 1 , A 2 , -B 1 , -B 2 ,
-J} give the correct resonant frequencies ω res ± , one giving an odd wavefuntion for the eigenmode, and the other an even wavefunction. This perfectly matches the two FEM solutions, since they can be identified as either bonding (non-zero electric field at midpoint between nanocylinders) or antibonding (zero electric field at midpoint) for each gap value. The even parity (ω + ) binds the two resonators, as a bonding state. Conversely, the odd parity (ω -) places their field in phase opposition, similarly to the antibonding case. This allows us to determine the fitted functions with no sign ambiguity, as shown in Fig. IV.8.

IV.2.3 Super-modes in chains of coupled resonators

The same model is used to predict the modes behavior vs d in the case of three resonators (Fig. parts of the eigenfrequencies. All this confirms that a relatively simple analytical non-Hermitian formalism can predict the physics of non-trivial nanophotonic systems.

Strikingly, degeneracy points for real parts on one hand, and for imaginary parts on the other hand, arise for the same gaps in the cases N = 2, 3 and 4. The fact that they are independent of N indicates that the they only depend on field leaked by the single resonator. The numerical spectra of the eigenmodes in the cases N = 3 and 4 are fairly described by our formalism when nearest neighbor (NN) coupling is implemented (see the relative error in Fig. IV.11). Furthermore, following the correspondence between the probability density and the power stored inside each resonator, the eigenmodes solution of the non-unitary dynamics in the case N = 3 in does not vanish for distances that exceed the wavelength of the modes (approximately 2 µm). Therefore, in order to refine our model, we explore coupling to the next-nearest neighbor (NNN).

It is important to note that the coupling between two resonators j and j + 2 should be different whether a resonator j + 1 is present or not. This implies that next-nearest-neighbor coupling cannot be extracted from the N = 2 case, but from the N = 3 at least. Therefore, the knowledge of long-range coupling in a chain of a N nanoresonators depends on the knowledge of the corresponding N -1 chain. However, the resolution of such a model would prove tedious, with hard-to-extract coupling constants through iterative calculation processes, resulting in numerical challenges to predict modes of longer chains. From the information extracted in the case N = 2, we performed an analytical calculation of the N = 3 (resp. N=4) chain with simplified next-nearest-neighbor coupling (Figs. IV.9 and IV.10) to clarify whether it could improve our model. For this purpose, we introduce J(2r + 2d) + i Γoff (2r + 2d), the coupling between two nanoresonators separated by 2r + 2d , where r is the radius of the nanocylinder. Conservative and dissipative coupling constants, J(2r + 2d) and Γoff (2r + 2d), were extracted from Maxwell's equations solutions in the case N = 2. This simplified next-nearest-neighbor coupling improves the agreement of probability densities as seen in Fig. IV.9b and spectra of Figs. IV.9a and IV.10 with FEM simulations (see the relative error in the Fig. IV.11). Such a refinement confirms that those modes involve long-range interaction between coupled resonators. From Figs. IV.9 and IV.10, it also appears that the agreement between the analytical non-Hermitian model and the numerical Maxwell's equations resolution is more significant for larger gaps. This can be ascribed to the hypothesis of linear dependence of the coupling constants on ω, which enabled us to fit the complex eigenfrequencies. For a narrower gap between nanocylinders, the field overlap grows stronger, which implies a wider frequency splitting of the eigenmodes [START_REF] Zhang | Magnetic and electric coupling effects of dielectric metamaterial[END_REF][START_REF] Vial | A coupling model for quasi-normal modes of photonic resonators[END_REF], as well as a deformation of near-field both outside and inside the resonators. In this regime, the coupling can no longer be expressed as a perturbation, increasing the deviation of our analytical model from brute-force calculations. To provide a measure of the efficacy of the approach, we quantify in figure IV.11 the relative prediction error of the analytical model without and with the simplified next-nearest neighbor coupling for the cases N = 3 and N = 4, benchmarked against the FEM numerical solutions, which confirms the good predictability of the analytical model. From those plots, the better predictive capability of the NNN model is again emphasized, as well as the limitation of this approach when the gap shortens below 200 nm.

Finally, we computed several spectra for 1D equidistant chains with different number of sites N (Fig. IV.12). For a chain of N resonators, N hybrid modes are expected for a given MD. While analytical non-Hermitian calculations gives all eigenmodes of the N -site chains, numerical simulations are quickly limited by the size of the calculation space, which is defined by the PML size and the meshing of the system. Indeed, for N = 6, FEM simulations require up to 100 GBytes of RAM, which meets the upper limit of our local calculation resources. Additionally, hybridization of the eigenmodes with the PML modes prevents us to identify them properly. This is less prominent for MD y and MD z hybrid modes, whose symmetries differ more from spherical PML than MD

IV.2.4 Perspective: 2D clusters of nanocylinders

At variance with 1D chains of dielectric resonators, the case of 2D arrays of nanoparticles cannot be directly treated with the above methods. The reason of this limitation is that 2D arrangements add a rotative degree of freedom for all the eigenmodes of each meta-atom. Even in quasi-1D systems [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF], where zigzag chains have been implemented to design robust topologically protected edge states, the relaxation of strict 1D alignment allows in-plane MDs to rotate so as to minimize the energy of the chain under the constraint of reciprocal coupling. Hexagonal, triangular and square matrices, as well as periodically closed lines or disordered cluster are attractive finite-size systems to explore the formation of band gaps [START_REF] Monsarrat | Pseudo-gap and localization of light in correlated disordered media[END_REF] and the emergence of exotic phases if symmetry-breaking modulations perturb the N resonators. Similarly to our approach to generate input dataset for the 1D non-Hermitian bath model for coupled nanoresonators, we extended our FEM simulations to 2 × 2 square ensemble of AlGaAs nanocylinders ( ): one like the MD y -MD y interaction (J xx ) at a distance d in the case of dimer; one like the MD x -MD x interaction (J yy ) at a distance d; one as two dipoles, each rotated of 45°from each other (J diag ), at a distance √ 2d. Note that the combination of those three couplings rotate slightly the MD axis. It would not be the case in the lattice seeded by this unit cell. This effect is therefore attributed to the finite size of the system. Unfortunately, I did not have time to elaborate the treatment of such a cluster of resonators with a non-Herminitian tight-binding approach. However, solving the problem of the 2 × 2 square ensemble of dielectric nanoresonators might be a future step in the numerical exploration of more complex systems with this new approach.

IV.3 Experimental probing of couplings and coupled supermodes

This section focuses on the verification of the non-Hermitian bath model for coupled nanoresonators. Here, the challenges are to find an appropriate spectroscopy method for isolated or clustered dielectric nanoparticles, and design a sample that allows to reveal the coupling physics

IV.3.1 Sample design and fabrication

IV.3.1.1 Isolated AlGaAs nanoparticles on AlOx

The fabrication of chains or clusters of nanoresonators requires two delicate steps: EBL, in order to implement design with dimension below 100 nm; ICP-RIE for III-V materials. The technological developments of finite-size ensembles of coupled dielectric nanoparticles with varying geometrical parameters have directly benefited from the previous work on the fabrication of AlGaAs-on-AlOx metasurfaces [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Rocco | Tunable second harmonic generation by an all-dielectric diffractive metasurface embedded in liquid crystals[END_REF][START_REF] Carletti | Controlling secondharmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas[END_REF]. High-resolution, low-proximity-effect resist has been chosen to reduce proximity effect in compact structures, sidewall defects and, therefore, optical losses inside the resonators. All the lithographic processes are performed with a combined EBL/Scanning Electron Microscope hybrid (Pioneer Two by Raith Nanofabrication). The latter provides an acceleration voltage up to 20 kV, and a beam aperture of 10 µm is chosen to ensure a satisfying in-plane resolution, as well as a fast writing speed. The lithographic pattern is transferred to the AlGaAs layer through RIE. AlGaAs is etched through ICP-RIE commonly adopted in III-V nanofabrication. A 35 W -RF source is used to create a high-density plasma through a metallic coil (the inductive component). The plasma is injected inside the sample chamber and accelerated with another 15 W RF source. The gas mixture inside the chamber is composed by Argon (30 sccm) and SiCl 4 . Ar + ions provide highly directional and poorly selective mechanical etching. SiCl 4 and SiCl + 4 free radicals, produced in the plasma, chemically activate GaAs and AlGaAs surfaces, featuring high material selectivity. The combined action of free radicals and Ar + ions results in perfectly vertical and smooth sidewalls. The final step of monolithic fabrication protocol is the selective oxidation of the AlAs layer, which creates the high-index contrast necessary to tightly confine the electromagnetic field inside the resonator volume. This process is performed with an AET Technlogies wet-oxidation oven. The sample is introduced in a chamber under vacuum and slowly warmed up to 390°C (20°C/min) under controlled gas conditions (N 2 at 500 mbar). In the meantime, the oxidation gas (a mixture of N 2 and H 2 with water vapor) is prepared in a controlled mixer. When the steady state is achieved, the gas is injected in the oven and oxidation starts. When such vapor mixture interacts with AlAs it triggers a thermodynamically favorable (negative Gibbs free energies) reaction chain at high temperatures. With respect to metasurface technology, the spacing between each chain of nanocylinders imposes significantly different parameters during the lithography process. This happens because here we have to avoid long-range coupling (Fig. IV.5), as well as the excitation condition and signal collection from an unique nanoresonator cluster. More precisely, the exposure time required to properly resolve and implement masks for isolated nanostructures is much higher than in the case of metasurfaces, since the coercive proximity effect of back-scattered electrons from nearby structures is absent. Therefore, the optimal dose to fabricate such particles is much higher than for metasurfaces. As an example, when using HSQ resist for metasurfaces such as the one in Fig . IV.16c, the optimized dose essentially depends on the matrix arrangement, its period, and the size of the single elements. For a cylinder with r = 220 nm fabricated with HSQ, varying the period of a square matrix from 900 nm to 1500 nm shifts the dose from 1.1 mC.cm -2 to 1.5 mC.cm -2 . For isolated particles, which can be seen as the extreme case of an infinitely large period in the EBL framework, a strong increase of nominal dose is required (2.2 mC.cm -2 ), and even more as the resist ages.

IV.3.1.2 Mask design for reproducible parameters variations

Taking in consideration the long exposure time of high-resolution negative resists, and the wide range of the geometric parameters of interest (radius and gap), the mask presented on Fig. IV.17a serves as a matrix for a set of 1D chains of nanoresonators with a number N of cylinders. The chain axis is oriented along the horizontal direction, where the matrix separation is larger in order to avoid crosstalk between two chains with the same gap but different dose factors. For a given geometry, duplication of the structure accompanied with variation around the optimal exposure dose ensures that at least one device will be perfectly implemented, as in the SEM images of

IV.3.2 Linear spectroscopy of chains of dielectric meta-atoms

This section focuses on the measurement efforts with DF and BF spectroscopy. I fabricated the related samples in the MPQ cleanroom, and I performed the measurements on a spectroscopy setup of Abbe Center of Photonics in Jena.

IV.3.2.1 Scattering of arrays of coupled resonators

Before reporting the results of my measurements in Jena, let us recall the context of light scattering by (Mie) resonators. This will clarify which modal information can be accessed by BF/DF spectroscopy. When an incident plane wave impinges on a sub-wavelength scatterer, its potential to excite the resonances of the object can be assessed by the coupling between the near field of the scatterer eigenmode and the electromagnetic field of the exciting wave. In this sense, controlling the polarization of the incident plane wave plays a central role to probe such mode.

In order to further clarify the picture, let us first detail the excitation of MDs of a single nano-cylinder. Under normal incidence, the plane-wave electric and magnetic fields oscillate orthogonally to the cylinder's axis. Consequently, the magnetic field of the exciting light will be collinear to the in-plane magnetic dipole, resulting in the excitation of the latter. Conversely, the incident field will have a little overlap with the near-field of the vertical MD, which will not be efficiently excited. With this symmetry analysis, it appears that vertical MD modes cannot be probed with a plane wave at normal incidence, contrary to in-plane MDs. the case N = 2 for in-plane MDs hybrid modes. Here, the coupling breaks the C ∞ symmetry, and two situations must be considered :

• MD x hybrid modes, Fig. IV.18a and c: light polarized orthogonally to the dimer's axis couples to the magnetic dipole. For the interferences between the hybrid mode and the incident plane wave to be constructive, MDs on both resonators need to be collinear and oriented in the same direction, constituting a bonding hybrid mode. Conversely, the antibonding mode, featuring MDs oriented in opposite directions, is not excited by the planewave.

• MD y hybrid modes, Fig. IV.18b and d: light polarized along the dimer's axis couples to the magnetic dipole. For the interference between the hybrid mode and the incident plane wave to be constructive, MDs on both resonators need to be collinear and oriented in opposite directions, constituting an bonding hybrid mode. Conversely, the anti-bonding mode, featuring MDs oriented in the same direction, is not excited by the plane wave.

This analysis is supported by scattering simulations of AlGaAs dimers with varying gaps excited at normal incidence by a linearly polarized plane wave (Fig. IV.19). The scattering efficiency is represented as a function of the dimer gap and the pump wavelength λ. The results obtained by mode tracking are also reported, providing information on the eigenmodes of the dimer. For light polarized along the dimer axis (a), the scattering plot reveals a major contribution of the anti-bonding MD y hybrid mode. For light polarized orthogonally to the axis of the dimer (b), the scattering plot reveals a major contribution of the bonding MD x hybrid mode. Moreover, in regard of Fig. IV.5, we note that higher Q-factors for the eigenmodes coincide with a stronger scattering efficiency of the AlGaAs dimer. This sets the framework of plane wave probing at normal incidence.

IV.3.2.2 Experimental setup

The setup I used in Jena is depicted in Fig. IV.20. An unpolarized white light source emits a beam, which is collimated and then focused on the sample by an epiplan-neofluar ×100 high numerical aperture microscope objective. This objective is meant to work in the visible range, but its transmission does not vanish in the IR. Albeit non optimal, it has two practical advantages. First, it collects light emitted in a cone of 64°from normal incidence, which is adapted to the radiation pattern of electric and magnetic dipoles of a cylinder [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Rocco | Vertical second harmonic generation in asymmetric dielectric nanoantennas[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF]. In addition, such an objective allows to perform in the same configuration both bright-and dark-field spectroscopy, only by placing a beam stop on the beam path before the beam splitter. In this configuration, the impinging light couples differently to the modes of the nanocylinders: we model this situation by imposing a tilted incidence for the impinging light, will collecting light at the normal of the substrate.

The scattered light collected from the sample is sent toward the spectrometer slit. The latter is calibrated with a fibered Xe lamp. Reflected light is spatially filtered in the image plane of the microscope tube lens in order to reduce substrate contribution and to ensure only light emitted by the targeted ensemble of nanoparticles is measured. Whenever this spatial filter is modified, the signal from the substrate has to be acquired again in order to properly normalize the measurement. Moreover, a linear polarizer mounted on a motorized rotation stage can be placed between the adjustable diaphragm and the flipping mirror, in order to filter the signal by polarization. Note that the white lamp is always used at full power in order to run measurements with a spectrally reproducible probe, and only the acquisition time is adjusted to optimize the detected counts on the spectrometer. Depending on the diaphragm aperture (i.e. on the length of the N -mer under test) the acquisition time can range from a few hundred milliseconds to a few seconds. For every measurement, the plotted signal S plot is obtained with the formula:

S plot = S raw -S dark S sub -S dark (IV.30)
where S raw is the signal acquired from the N-mer, S sub the signal from a portion of substrate with the same diaphragm aperture, and S dark the dark counts for the same intregation time. The noise of the produced data is then reduced by applying low-pass forward-backward filter wih the Gustafsson method (see Fig.

IV.21a and [START_REF] Gustafsson | Determining the initial states in forward-backward filtering[END_REF]). For all the measured samples, the height of the cylinder is set to 400 nm by the thickness of the Al 0.18 Ga 0.82 As epitaxial layer. ing scattering simulations are presented in panels d and e. In the case of DF, measurements and simulations for the smallest radius r = 200 nm present a large peak around 1500 nm, which corresponds to the MD of the cylinder. This peak is asymmetric and broadened by the presence of the very low Q-factor electric dipole of the cylinder, centered around 1400 nm [START_REF] Carletti | Enhanced secondharmonic generation from magnetic resonance in AlGaAs nanoantennas[END_REF]. In the corresponding scattering simulation, the response of the in-plane MD induces a similar peak at 1500 nm, which is redshifted when the radius of the cylinder increases. If the peak measured at 1500 nm in Fig. IV.21b for r = 200 nm was a signature of the scattering by an in-plane MD, it would appear at high wavelength for the larger geometries. However, it does not exhibit such a behavior, implying that it does not correspond to the MD response. Overall, it is actually very challenging to connect the scattering simulations to the experimental results: it seems that the DF response of the monomer is hidden in weak signal variations, too tenuous to be fitted and connected to eigenmodes or scattering simulations.

IV.3.2.3 Scattering by a single AlGaAs nanocylinder

When we compare the scattering simulation results with different radii in Fig. IV.21d and e, the hallmark of dipolar modes is again a strong redshift with increasing radius, whereas the higher-order modes below 1200 nm are strongly affected but remain in the same spectral range. Two spectral regions must therefore be considered: 1) around telecom wavelength, where the dipole study is physically meaningful provided that one is able to extract the modal information from the reflection signal; 2) the near-IR region, with seemingly stronger measurable signal, but more complex and dense modal pattern. The latter is even more difficult to study with the developed mode tracking algorithm, since it requires finer interpolation and more distinction criteria to segregate hybrid modes stemming from a given monomer eigenmode.

Unfortunately, the BF spectra reported in Fig. IV.21c do not give better results: they present no similarity with the simulated scattering spectra of Fig. IV.21e. As in the DF measurement, it seems that meaningful variations of the signal, concentrated above 1500 nm are too weak to be properly interpreted in terms of modal content. Moreover, the polarization analysis of the signal reveals different behaviors between two orthogonal polarizations. This is unexpected considering the cylindrical symmetry of the probed resonators: the in-plane dipoles, to which a normally incident unpolarised plane wave is expected to couple, should not differ in the C ∞ configuration, without any symmetry breaking attributed to the environment. All these facts suggest that the response of an isolated particle in the reflection configuration of the setup of Fig. IV.20 is too weak to be analyzed. To overcome this difficulty, we might opt for a less constraining technological platform, for instance changing from AlGaAs-on-AlOx to AlGaAs-on-sapphire. This technological shift would allow to measure in transmission, and to reduce the contribution of the substrate at NIR and at telecom wavelengths [START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF]. Alternatively, we can also study longer chains, hoping for a significant signal increase when light scattered by more coupled cylinders is collected. Based on this intuition, responses of dimers, tetramers, longer chains and 2 × 2 square matrices from the same sample are discussed the three next sections. Note that a weak signal (green circle in Fig. IV.22), varying with the gap d, can be observed both on BF (around 1540 nm) and DF (around 1510 nm) spectra for d < 300 nm. In the two plots, the peak position increases with the gap, which could suggest a signature of the antibonding MD x mode has been observed. Nonetheless, this observation contradicts the previous analysis for two reasons: first, a plane wave should not be able to excite to this mode. Then, the signal wavelength depends on the measurement technique (BF or DF spectroscopy), which suggests that it is unlikely to arise from the excitation of the coupled nanoresonators. Therefore, our DF and BF spectral measurements are not conclusive.

IV.3.2.5 Response of longer chains and 2D clusters

In the case N = 4, DF and BF spectra in the near IR are fairly similar to the N = 2 case, but they are more dependent on the gap d. BF tetramer spectra of [START_REF] Pelton | Modified spontaneous emission in nanophotonic structures[END_REF], where is has been demonstrated that higher Q factor (boosted by hybridization) increases scattering efficiency. Therefore, this proves that a collective contribution boosting the Q factor of the resonances of four nanocylinders around 1550 nm is observed. However, it is still impossible to extract any relevant quantitative spectral information from the measurement: one can only conclude that the gap d indeed has a slight impact on the modes of the chain.

In order to improve the comprehension of the system, complementary measurements on the [760; 1230] nm spectral range have been performed on the same chain, and reported on Fig. IV.23c and d. In this case, and for both analyzer positions, similar spectral features are observed:

• a widening of the dark band around 960 nm for gaps shorter than 200 nm;

• an intensity evolution with gap around 1130 nm;

• symmetric features around 1050 nm: above this wavelength for gaps shorter than 400 nm on In addition to the series of N -mers with varying gap and geometry, some longer chains with identical elements and constant gap have been implemented atoms, whatever polarization is filtered by the analyzer. This indicates that the response of the chain scales linearly with the number of meta-atoms, when it is excited by a plane wave. No new feature appears when elements are added, which confirms the conclusions of Section IV.2.3 on longer chains.

Finally, the measurements run on 2 × 2 square matrix of nanocylinders are reported in Fig. IV.25. Beyond the typical aspect of the spectra, it seems that more resolved spectral features are affected by gap variation in this series of devices. The initial idea of probing isolated dielectric particle has proven to be too challenging for the reported spectroscopy protocol. The fact that larger systems presents a stronger signal-to-background ratio suggests that collecting signal from several identical N -mers could help to boost the spectral contribution of their resonances. The constraint to fabricate such a sample would be double: N -mers should be sufficiently separated so that the field scattered by the structures would not affect their neighbors and create long-range couplings or interference; the number of N -mers and their spread must be optimized in order to keep the sample size and the e-beam lithography time reasonably small.

IV.3.3 Beyond plane-wave spectroscopy: cathodoluminescence for chains of coupled nanoresonators

As recalled in Fig. IV.18, plane waves cannot excite all the modes of one or several of coupled nanostructures. To overcome this limit, and to collect light emitted at even broader angles, we resorted to cathodoluminescence imaging spectroscopy to probe and characterize the modal response of 1D chains of AlGaAs resonators. This measurement attempt has been realized in

IV.3.3.1 Experimental setup

The CL imaging setup is provided by DELMIC, a Delft-based microscopy specialized company, and schematized on Fig. IV.26. Electrons accelerated by a 30kV voltage (blue beam) are focused on the sample (red inset) placed in the SEM chamber to excite all the electronic transitions of a dielectric material. When the material electrons get back to the ground state, photons are emitted, which act as an internal broadband light source, and they couple to the resonator modes. The emitted light, carrying the spectral information of the resonator, is reflected on a parabolic mirror and directed toward the spectrometrer. A camera is used to align and optimise the signal by translating the sample stage and the parabolic mirror, and to autocollimate and center the lenses in the arm which connects the SEM chamber to DELMIC spectrometer. Raw data measured on the CL setup were treated applying the following formula:

S plot = (S raw -S sub ) × N cal (IV.31)
where the notations are the same as in Eq. IV.30, and N cal is a normalization factor accounting for detector calibration. The resulting data are filtered using the same method as for BF and DF spectroscopy. In the following section, each spectrum has been acquired from the same cylinder in each chain, after making sure that no spectral feature arising from possible fabrication defects appears locally in the CL maps. We focused on two geometries: r = 200 nm and r = 215 nm, because it appeared that larger cylinders entail a more intense signal.

IV.3.3.2 Results and discussion

Spectra extracted from the CL measurements for r = 200 nm are reported in Fig. IV.27, where we can identify the AlGaAs gap peak (750 nm) and another resonance near 770 nm. I computed the higher-order eigenmodes of the corresponding dielectric nanocylinder with FEM calculations, and plotted in Fig. IV.27d the near field of the mode whose wavelength was the closest to the CL resonance. This eigenmode might be a candidate for the resonance observed in the chains. However, the AlGaAs gap peak appears to fluctuate as well when d varies (Figs. IV.27a-c), which could be interpreted in two ways. The FEM simulations predict the existence of whispering-gallery modes, that broadens spectrally the CL signal around 750 nm. The effect would necessarily be gap-dependent, thus would make the intensity around the dielectric bandgap fluctuate when d varies. However since the Q-factor of such mode is greater than 1000, there are unlikely to contribute to a broadening of several nanometers in the CL signal. It could also a defect in Another important remark is that for higher order modes, with much more confined light, the coupling constant is expected to drop faster than for dipolar modes as the gap d increases. Consequently, the amplitude of the eigenfrequencies modulation is lesser when higher order modes hybridize. The mode splitting due to coupling is to be compared with another mechanism of frequency shift: the variation of geometry that can occur when implementing chains of nanocylinders with varying gaps. The mode studied in Fig. IV.27d for r = 200 nm has a resonance wavelength of 776 nm, whereas when the radius of the cylinder is increased by just 15 nm, such a resonance moves to 800 nm. We can deduce that variations of 8 to 20 nm in diameters, which can occur within the same array in the mask presented in

IV.4 Conclusion and perspectives

In this chapter, I have demonstrated numerically the capability of an innovating method for design of coupled dielectric metamaterials. Overall, the analytical non-Hermitian quantum formalism used here offers adequate means to compute modal information on 1D chains of N resonators, and it can be naturally extended to nonlinear quantum optics [START_REF] Li | Dissipation-induced antiferromagneticlike frustration in coupled photonic resonators[END_REF]. By increasing the length of the chain, one could study the transition between a discrete ensemble of nanoresonators and a photonic crystal. Considering the wide variety of geometries and properties accessible via nanostructured dielectric materials, we envisage that such nanophotonic systems could be a promising model for implementing more complex interactions in 1D, 2D or 3D metasystems. They could also constitute toy systems to study for example topological edge states [START_REF] Smirnova | Nonlinear topological photonics[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF]. A direct application for such a predictive model could be the improvement of Q factor optimization of leaky nanoresonators through non-Hermitian coupling.

Unfortunately, the experimental validation of this method and its application to the design of more advanced systems yet remain to be achieved. If no shift of fabrication paradigm is made, cathodoluminescence seems to be the measurement method to push: regrettably, no measurement run could be performed near telecom wavelength, which leaves plenty room for improvement of the characterization with AlGaAs-on-AlOx sample. For instance, having one 10 × 10 µm 2 square patch of the epitaxialy grown layer in order to deconvolute the signal of the material from the raw spectrum could augment to quality of data analysis, as well as facilitate the fine tuning of the CL setup. Another lead to explore would be to change the wafer in favor of AlGaAs reported on sapphire, in order to remove any spurious signal from eventual defect in the AlOx layer, and to reconstruction the full scattering cross-section by performing measurements both in transmission and reflection.

CHAPTER V

Coupled-system architecture with guided optics

Topological edge states hold great promises in term of light confinement and robustness to fabrication defects or disorders in the case of nanostructured implementations of photonic systems. From the previous chapter, it is clear that the issue of coupling calculation cannot be trivially solved in the case of nanoresonators. However, a wide range of coupled photonic systems has been analytically studied, both in the framework of coupled-mode theory and input-output theory, with experimental demonstrations. In particular, the last decade has seen a rise in the inquiry of topological systems to foster non-reciprocal [START_REF] Ma | All-Si valley-Hall photonic topological insulator[END_REF][START_REF] Poo | Experimental realization of self-guiding unidirectional electromagnetic edge states[END_REF][START_REF] Gao | Topologically protected refraction of robust kink states in valley photonic crystals[END_REF], or protected [START_REF] Hafezi | Imaging topological edge states in silicon photonics[END_REF][START_REF] Rituraj | Photonic chern insulators from two-dimensional atomic lattices interacting with a single surface plasmon polariton[END_REF][START_REF] Cheng | Robust reconfigurable electromagnetic pathways within a photonic topological insulator[END_REF][START_REF] Mittal | Topologically robust transport of photons in a synthetic gauge field[END_REF] light transport. In this chapter, we will explore the possibility of implementing topological microresonators relying on the coupling between concentric circular waveguides. This work constitutes the first step toward a more ambitious project: engineering rich multimodal systems for integrated photonics applications, such as protected frequency combs [START_REF] Mittal | Topological frequency combs and nested temporal solitons[END_REF], photon-pair generation [START_REF] Yang | Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators[END_REF][START_REF] Steiner | Ultrabright entangled-photon-pair generation from an AlGaAs-on-insulator microring resonator[END_REF], or logical gates [START_REF] Espinosa-Ortega | Complete architecture of integrated photonic circuits based on and and not logic gates of exciton polaritons in semiconductor microcavities[END_REF][START_REF] Politi | Integrated quantum photonics[END_REF][START_REF] Fu | Silicon photonic crystal all-optical logic gates[END_REF][START_REF] Crespi | Integrated photonic quantum gates for polarization qubits[END_REF].

This work has been carried out in collaboration with Pr. Claudio Conti and Dr. Laura Pilozzi from the University Sapienza and the National Research Council (ISC-CNR), discussing the theoretical background and the design of samples; Mounir Rezig, who helped during his internship with the implementation of our in-house input-output theory model; Andrea Gerini and Nathan Kieffer, doctoral student and undergraduate student from our team, for the common development of the AlGaAs-on-AlOx platform for guided photonics. Epitaxial AlGaAs wafers for the sample fabrication have been supplied by Dr. Isabelle Sagnes, from C2N.

V.1 Motivations: engineering and advantages of topological phases V.1.1 Circular cavities and whispering-gallery modes

Whispering-gallery modes (WGM) are a class of waves traveling around concave surfaces, most generally disk-or ring-shaped, which are resonant when an integer number of the propagating light wavelength λ 0 fits into the circumference of the ring. For a ring of radius r and optical index n, this resonant condition reads:

2πnr = mλ 0 (V.1)
where m ∈ N * is the azimuthal mode number. When Eq. V.1 is fulfilled, light can be injected in the WGM through evanescent coupling from a tapered fiber [START_REF] Armani | Ultra-high-Q toroid microcavity on a chip[END_REF][START_REF] Harun | Theoretical analysis and fabrication of tapered fiber[END_REF][START_REF] Ding | Ultralow loss single-mode silica tapers manufactured by a microheater[END_REF][START_REF] Lu | Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature[END_REF] or a waveguide [START_REF] Baker | Critical optical coupling between a GaAs disk and a nanowaveguide suspended on the chip[END_REF][START_REF] Hagness | FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whisperinggallery-mode disk resonators[END_REF][START_REF] Cai | Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system[END_REF] at a coupling rate κ e (in Hz). In those systems, absorption of the material, radiation and scattering phenomena can induce intrinsic losses, conventionally labeled in the literature with the rate κ i . Together, these two rates account for all the exchanges and losses of the mode with its environment, and thus define the total loss rate or bandwidth of the resonance. Depending on the ratio between κ e and κ i , three possible regimes can be distinguished.

If κ e > κ i , the cavity is then overcoupled, as external coupling to the light input exceeds the occurrence of internal losses in the cavity. The more κ e exceeds κ i , the more the spectral width of the mode resonance broadens. Conversely, when κ e < κ i , the system is undercoupled to the photon input. In the limit regime separating overcoupling and undercoupling, i.e. κ e = κ i , the destructive interference between the photons that did not enter the cavity and those which leave the cavity annihilate perfectly the transmission of light at the cavity output [START_REF] Yariv | Universal relations for coupling of optical power between microresonators and dielectric waveguides[END_REF]. This case is particularly interesting for two experimental reasons: 1) it enables a direct determination of the intrinsic losses of the cavity provided that κ e is known; it corresponds to the regime where the resonances have the largest contrast.

V.1.2 Concentric geometries and topological Hamiltonian mapping

Whispering-gallery modes are excellent systems for applications in sensing [START_REF] Foreman | Whispering gallery mode sensors[END_REF][START_REF] Vollmer | Whispering-gallery-mode biosensing: label-free detection down to single molecules[END_REF] and integrated nonlinear optics [START_REF] Roland | Frequency doubling and parametric fluorescence in a four-port aluminum gallium arsenide photonic chip[END_REF][START_REF] Savchenkov | Low threshold optical oscillations in a whispering gallery mode CaF2 resonator[END_REF]. They constitute a core element for frequency combs generation [START_REF] Savchenkov | Tunable optical frequency comb with a crystalline whispering gallery mode resonator[END_REF][START_REF] Chang | Ultra-efficient frequency comb generation in AlGaAs-on-insulator microresonators[END_REF][START_REF] Miller | Tunable frequency combs based on dual microring resonators[END_REF], optical parametric oscillators [START_REF] Fürst | Low-threshold optical parametric oscillations in a whispering gallery mode resonator[END_REF], or quantum state preparation [START_REF] Dutt | On-chip optical squeezing[END_REF][START_REF] Dutt | Tunable squeezing using coupled ring resonators on a silicon nitride chip[END_REF], and they have been recently integrated in hybrid systems that foster the coupling to one or several WGM [START_REF] Hafezi | Imaging topological edge states in silicon photonics[END_REF][START_REF] Dutt | Tunable squeezing using coupled ring resonators on a silicon nitride chip[END_REF]. Recently, some interest has emerged for the study of concentric coupled dielectric ring resonators [START_REF] Sun | Design and optimization of silicon concentric dual-microring resonators for refractive index sensing[END_REF][START_REF] Zhang | Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling[END_REF]. The authors report that supermodes can arise from the coupling of two whispering-gallery modes, and that the signature of this coupling, namely the splitting of the resonance dip, can be observed on the transmission spectrum of such a system. The question is therefore: can this coupling be characterized and engineered in order to create new resonators featuring several concentric coupled whispering-gallery modes? One of the motivations for this work would be the implementation of integrated photonic topological insulators [START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF], which holds great promises applications such as protected frequency combs [START_REF] Mittal | Topological frequency combs and nested temporal solitons[END_REF], or the implementation of topological laser. One of the advantages of topologically protected edge modes is their robustness to fabrication defects or impurities, which prevents scattering and thus losses. Translationally-invariant coupled waveguides can be approached as 1D systems, since the propagation direction is related to time through the definition of the propagation constant β. For those systems, the transverse direction can be framed as an effective 1D chain, whose physics is governed by the coupling constant and eigenenergies of the propagating modes. The objective of the work presented in this chapter is to explore the possible implementation of topologically protected guided modes, for frequency conversion in dielectric ring resonators. As in Chapter 4, the first task will be to conceive a protocol to quantify the coupling rate between concentric rings of arbitrary dimension, coupling rate on which an eventual implementation relies on. The targeted systems will be based on André-Aubry-Haper (AAH) modulation [START_REF] Harper | The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals[END_REF][START_REF] Aubry | Analyticity breaking and Anderson localization in incommensurate lattices[END_REF][START_REF] Hofstadter | Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields[END_REF][START_REF] Ganeshan | Topological zero-energy modes in gapless commensurate Aubry-André-Harper models[END_REF], whose symmetry breaking is especially attractive for band engineering in photonic systems.

The starting point of this work is an article by Pilozzi et al. [START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF], in which the optical index of the cladding of a photonic crystal fiber (PCF) is modulated to create a topological photonic insulator and reduce the propagation losses. Their concept and main results are presented in [START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF]). The AAH modulation η cos(2πγj + φ) is governed by three parameters: 1) its amplitude η; 2) its period γ = p/q, where p and q are two co-prime numbers; 3) the topological phase φ of the system, which can be interpreted as the presence of a synthetic magnetic field along the radial direction that acts on the optical index as a gauge transformation. Topologically protected modes are expected to be observed in the opened gaps. Interestingly, this article states that topological protection can be achieved in 2D optical resonators such as the one depicted in Fig. V.1b. AAH modulation can be apprehended through a 1D tight-binding Hamiltonian formulation. The previously presented AAH modulation [START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF][START_REF] Pilozzi | Anisotropic circular topological structures[END_REF] of the ring radii of the 2D structure is here modeled as a modulation of the coupling between two adjacent ring resontors. The Hamiltonian reads as follows:

ĤAAH = µ N + j -t [1 + η cos(2πγj + φ)] ĉ † j ĉj+1 + hc. (V.2)
where t is the coupling strength, and ĉ † j is the creation operator associated to site j. The chemical potential µ is set to 0 to simplify calculations. Similarly to other topological models, an analytical diagonalisation of ĤAAH exploiting Bloch theorem cannot predict topologically protected edge states, since it masks bulk-edge correspondence by making the system artificially infinite. Such edge states actually appear for numerical resolutions of the Hamiltonian problem, with finite size chains (Fig. V.2). The AAH modulation opens q -1 gaps in the band structure. Even though we arbitrarily display calculations for modulation of the form γ = 1/n, with n ∈ N * , other admissible combinations display similar behaviors. In addition, the two top plots display the same AAH modulation with different phase offsets. The fact that gap states appear in panel a and are absent in panel b is a clear sign that this phase offset is a knob that tunes the topology of the system. An interesting aspect of the 1D AAH model is that it can be exactly mapped to a 2D model or, in other words, it can be derived from a 2D ancestor lattice [START_REF] Hofstadter | Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields[END_REF][START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF][START_REF] Poshakinskiy | Radiative topological states in resonant photonic crystals[END_REF].

Eigenmodes profiles of an AAH chain are displayed in Fig. V.3 when the system is in a topological phase. For each gap, the AAH modulation can foster up to two gap states, depending on the value of φ. For γ ∝ 1/3, the energy of those gap states is localized on two of the three sites that constitute one AAH unit cell. This calculation also shows an asymmetry in the wavefunction localization within the pair of gap states, which is independent of φ or η and reminiscent from the asymmetry of the AAH unit cell. This result stresses out that control on both topological phase and modulation strength is important for sample design. For a given φ, topological states are expected to have significantly different energies and protection. Thus, an important care on localization of those states is required.

V.1.3 Guided optics to probe dielectric resonators

In this framework, we aim to fabricate a dielectric concentric-ring resonators, with modulation in the ring spacing such that a topologically protected WGM is located in the outer ring of the concentric system. The technological platform to enable this experiment is schematized on Fig. V.4a: the ring resonator will be fabricated on a mesa, and light wil be injected thanks to a bus waveguide [START_REF] Roland | Frequency doubling and parametric fluorescence in a four-port aluminum gallium arsenide photonic chip[END_REF][START_REF] Baker | Critical optical coupling between a GaAs disk and a nanowaveguide suspended on the chip[END_REF] through micro-lensed fibers similar to the ones displayed in Fig. V.4b. Before presenting the fabrication process, its improvement, and the first measurements related to such a sample, the modeling and analysis helping the design of the concentric-ring resonator is reported in the next section of this chapter.

V.2 Input-output theory for concentric ring resonators

Prior to the experimental implementation on AlGaAs-on-AlOx samples, I developed a model based on the input-output theory [START_REF] Zhang | Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling[END_REF][START_REF] Carmichael | Quantum trajectory theory for cascaded open systems[END_REF] in order to describe the system under study of ring resonators. This model enable us to evaluate key parameters, in particular the coupling rates, and to engineer them through the geometrical design of the system.

V.2.1 Single ring resonator coupled to a bus waveguide

Starting from the literature [START_REF] Gardiner | Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation[END_REF], we first choose to address the problem of a single ring resonator of mean radius r and width w coupled to a bus waveguide. This textbook system is essential to out study since the coupling between the such a ring and the bus waveguide will be our gateway to probe the outer ring resonator and the access spectral properties of a coupled system of concentric rings. In this case, light is injected from the waveguide into the ring with the evanescent coupling rate κ e (d, w, r), d being the distance separating the waveguide and the ring (see Fig. V.5). For the rest of the section, the field in the ring, which is resonant at the angular frequency ω = ω 0 , is denoted a(t). By introducing the lifetime of a photon inside the ring, τ int = Q int 2ω 0 = κ i (w, r) -1 , Q int being the intrinsic quality factor of the resonance, we can write the dynamic equation for a(t) from Hamiltonian formalism [START_REF] Carmichael | Quantum trajectory theory for cascaded open systems[END_REF]. At steady state, when the injection rate in the waveguide from the laser source κ s is much smaller than κ e,i , the derivative of the field a in the ring vanishes, and we retrieve the following relation between a and the input source field S 1+ :

da dt = 0 = [i (ω -ω 0 (w, r)) -κ tot ] a -2i κ e (d, w, r)κ s S 1+ (V.3)
where the first term of the right-hand side (RHS) corresponds to the variation of a(t) as a result of the resonance and the total losses of the ring (with κ tot = κ e + κ i ), while the second term of the RHS represents the coupling to the bus waveguide (see Fig. V.5). If we consider that only an incident wave propagating with a constant β from port 1 to port 2 is injected in the system, with no back-scattered wave from port 2, then the system obeys the following input-output relation:

S 2-= e iβL S 1+ -i κ e (d, w, r) κ s a (V.4)
Coupled equations V.3 and V.4 define the dynamics of the system.

In this calculation, the dimensions of the waveguide and the ring are set, thus κ e (d, w, r) ≡ κ e (d), κ i (w, r) ≡ κ i and ω 0 (w, r) ≡ ω 0 . Note that all these parameters are real by definition. It Coupling between the two elements is denoted as κ e , d being the distance between the guide and the ring. The ring is resonant at ω 0 , with its intrinsic losses denoted κ i . Photons that enter (+) or leave (-) the segment of the waveguide of length L coupled to the ring resonator are represented by the fields S i± (i = 1, 2). is possible to derive an analytical expression of the transmission of the system, T (ω) = | S 2- S 1+ | 2 , which is the most experimentally accessible quantity for such a device:

T (ω) = 1 - κ e (d) i(ω 0 -ω) + κ tot 2 (V.5)
As such, (κ i , κ e ) are undetermined, since two pairs of solutions can satisfy Eq. V.5 for a measured transmission. The relevant pair (κ e , κ i ) can be determined from the measurement of two transmission spectra of two different resonances of the same device. One of these measurements has to be realised at critical coupling condition (κ e = κ i ), which is defined by a vanishing transmission at resonance. By definition, this measurement gives a direct access to κ i , which depends only on the resonator characteristics. Then κ e can be inferred from a second measurement.

V.2.2 Pair of concentric ring resonator coupled to a bus waveguide

In this section, let us consider a pair of dielectric concentric ring resonators, of central radii r 1 and r 2 (Fig. V.6a). Light is injected into the rings from a bus waveguide at a distance d from the outer ring. The field in the first (second) ring is denoted as a 1 (a 2 ), and is resonant at ω 1 (ω 2 ). The intrinsic photon lifetime in each ring is labeled τ int,1 (τ int,2 ). Following the same model as in the case of a single ring resonator, we obtain a set of coupled equations describing the dynamics of a 1 (t) and a 2 (t) at steady state: da

1 dt = 0 = [i (ω -ω 1 ) -κ tot,1 ] a 1 -2i κ e,1 (d)κ s S 1+ -iu(r 1 , r 2 , w)a 2 (V.6) da 2 dt = 0 = [i (ω -ω 2 ) -κ i,2 ] a 2 -iu(r 1 , r 2 , w)a 1 (V.7)
where u(r 1 , r 2 , w) is the coupling rate between the rings. This parameter depends essentially on their geometries, namely their radii, which would be the key user-controllable parameter in implemented structures. Light being injected as in the single ring resonator case, the inputoutput Eq. V.4 still holds. Coupled equations V.6 and V.7 together with the input-output Eq.V.4 allow to derive the transmission of the system:

T (ω) = 1 - 2κ e,1 (d) κ tot,1 -i(ω -ω 1 ) + u 2 (r 1 ,r 2 ,w) κ i,2 -i(ω-ω 2 ) 2 (V.8)
The coupling strength u(r 1 , r 2 , w) can be accessed from transmission measurements, provided that the coupling of the outer ring to the waveguide κ e (d), the resonance frequencies ω 1 and ω 2 , and the intrinsic quality factor of the rings are known. Note that u ∈ R, because all dissipation sources, like scattering due to fabrication defects and interface with the substrate, will be accounted for through κ i .

We can further simplify the determination of the coupling constant from Eq. V.8. In practice, the radius of the rings being of the order of 50 µm, and the gap between them of the order of 300 nm, their intrinsic dissipation rate will be very close. As the intrinsic quality factor of ring devices depends on their fabrication process (lithography, oxidation, wafer epitaxy...), we can reasonably assume that κ i,1 = κ i,2 = κ i and u(r 1 , r 2 , w) ≡ u(g), with g = r 1 -r 2 -w . This results in:

T (ω) = 1 - 2κ e (d) i(ω 1 -ω) + κ tot + u(g) 2 κ i -i(ω-ω 2 ) 2 (V.9)
Consequently, the knowledge of the resonance frequency, the intrinsic quality factor, and the coupling of the outer ring to the waveguide, as well as the resonance frequency of the inner ring, are enough to determine the coupling u between the two rings. In the case of strong coupling, the transmission plot on Fig. V.6b shows an anti-crossing at zero detuning. The key information of the figure is that it is possible to capture the spectral features of the two supermodes of the coupled system only when the detuning ω 1 -ω 2 is not orders of magnitude above the coupling constant u. Otherwise, the presence of two resonances in the transmission spectra is not guaranteed. Having two modes with very close frequencies helps considerably the detection of the hybridization, and the degeneracy case (ω 1 = ω 2 ) allows to infer directly the coupling constant from transmission measurement.

V.2.3 Multiple concentric ring resonator coupled to a bus waveguide

V.2.3.1 Formalism

Hhaving modeled a pair of coupled rings, let us now address the problem of a system composed of N concentric ring resonators coupled together. We limit ourselves to the case where the n th ring is coupled only to the (n + 1) th and (n -1) th ring, i.e. coupling to the nearest neighbors. In many cases, the gap g between two WGM, or a WGM and a waveguide, is of the order of the characteristic transverse dimension of the mode. The N resonators are labeled by increasing integers from the outer to the inner rings, and the coupling constants {u n } n∈ 1;N correspond to the coupling between the n th and (n + 1) th rings. In this framework, the dynamic equation of the field inside the n th ring reads:

0 = [i (ω -ω n ) -κ i,n ] a n -iu n a n+1 -iu n-1 a n-1 (V.10)
The field in the most outermost and innermost rings obey different equations, with only one conservative coupling u and a source term for the first ring:

0 = [i [ω -ω 1 ) -κ tot,1 ] a 1 -iu 1 a 1 -2i κ e (d, w, r)κ s S 1+ 0 = [i (ω -ω N ) -κ i,N ] a N -iu N -1 a N -1 (V.11) 141
V.2 Input-output theory for concentric ring resonators Chapter V

The resulting set of equations can thus be cast into a matrix form:

M(N )A = S (V.12)
with A = (a n ) t n∈ 1;N and S = (2i κ e (d)κ s , 0, • • • , 0) t , and:

M(N ) =                       [i (ω -ω 1 ) -κ tot,1 ] iu 1 0 • • • • • • 0 iu 1 [i (ω -ω 2 ) -κ i,2 ] iu 2 0 . . . 0
• • • • • • 0 iu N -1 [i (ω N -ω) -κ i,N ]                      
The system described by the matrix M(N ) is analogous to a tight-binding 1D chain with open boundary conditions. To compute the transmission spectra of concentric ring resonators, we can adopt a numerical approach from now on, assuming that {u n } n∈ 1;N and {(ω n , κ i,n )} n∈ 1;N are provided by either from experiment or simulation.

V.2.3.2 Implementation with Python

In order to understand the effect of coupling on a system of concentric microrings, a toy-model has been implemented based on the above calculation and hypotheses. For the N coupled rings we impose that κ i,n∈ 1;N = κ i is a constant parameter, depending only on the material and fabrication processes. The conservative coupling strength between each ring is also constant, i.e. u n∈ 1;N = u. This hypothesis may seem drastic, but is reasonable for two rings of close geometries.

The main function, whose variables are described in Table V.1, calls the following functions, implemented as blocks for the calculation:

• Numerical description of the individual resonators for the chosen geometry. Their spectral resonances are considered within a target spectral range.

• Description of the uncoupled problem. It returns the diagonal terms of the matrix M, as well as a unique source term S coherently coupled to all modes presenting field on the outer ring. However, nothing ensures that each ring admits the same number of modes in a given spectral range for a set of chosen geometries. This means that some dimension mismatches can occur when assembling M. To prevent them, a test is implemented to ensure that the number of considered modes is the same in each microring. This test is a try-and-correct algorithm to enlarge incrementally the spectral admittance window until the matrix M becomes square.

• Creation of the off-diagonal terms of matrix M and resolution of MA = S. After normalization, it returns a square matrix that encodes the transmission of the system at a given wavelength and for a given coupling strength u.

The whole algorithm returns a 2D map that displays the transmission of the system as a function of the coupling strength and the wavelength of the injected light.

An important comment has to be made on the computation of the transmission. The microrings WGM are spectrally well separated, their contributions are incoherently summed. Here, we sequentially sum the square modulus of the field resulting from the interference between the photons leaving the outer ring and the injected field S 1+ . Yet, this process introduces N -1 replicas of the input field that must be subtracted in the end to ensure the proper normalization of the transmission.

To validate the matrix formalism and its capability to compute experimentally measurable quantities, the case of N = 2 was first considered, since it serves as a benchmark with the analytical formula in Eq. V.9. In this case, the system to be solved is: [START_REF] Govyadinov | Metamaterial photonic funnels for subdiffraction light compression and propagation[END_REF] where:

M(2)A -S = O 2,1 (V.
M(2) =    i ω -ω 1 -κ tot,1 iu 1 iu 1 i ω -ω 2 -κ i,2   
This system is solved with the Python tools presented above, and results in a perfect adequacy with Fig. V.6b transmission spectrum when the same input initializes the calculation. After this sanity check, we generalized the computation of the transmission to an ensemble of N concentric ring resonators, and explore a potential implementation for SSH-like configuration, by progressively increasing the complexity of the system.

V.2.3.3 Evenly spaced concentric coupled ring resonators

For different geometries, the transmission spectra of multi-ring systems were computed with our in-house Python algorithm, and reported in Figs. V.7 to V.11. Each depicted situation reveals the increasing difficulties when dealing with ideal systems of increasing complexity.

Let us first compare the case of two coupled ring resonators: one in the limit case where g = 0 (Fig. V.7a) and the other where g = 100 nm (Fig. V.7b). The first situation could correspond to two rings with identical in-plane (xy) geometries, but at different altitude (z). In Fig. V.7a, large coupling strength is associated with a splitting of the hybrid coupled modes. In addition, the transmission at resonance goes from 0 to 0.4 when the coupling is swept from 0 to 50 GHz, although the outer ring was critically coupled to the bus waveguide (κ i = κ e = 2.5 GHz). This stresses out that hybridization of the modes of each ring resonator modifies their individual field properties and the way they interact with their environment. In other words, the hybrid modes should present different field characteristics from the isolated ring modes. They share characteristics of both single ring modes, one being probed at critical coupling on the outer ring, and the other not on the inner ring. The same observation is made when two rings of different radii interact (Fig. u. It is due to the spectral mismatch between the modes, caused by the consideration of two rings with different radii in the case of two rings with different radii, and which competes with the inter-ring coupling u. In practice, the interaction between two whispering-gallery modes depends on a large number of Chapter V geometric and material parameters. It implies that the methodology developed in Chapter 4 to construct a coupling function from experimental data (numerical or acquired from measurements on available samples) is most out-of-reach with our numerical capabilities. With a 2.6 GHz processor, a 3D Finite-Difference Time-Domain (FDTD) simulation taking into account the whole geometry (bus waveguide and ring resonators) of such a system takes two days to complete.

The same calculation is performed for five equispaced concentric AlGaAs ring-resonators, with g = 10 nm (Fig. V.8a) and g = 100 nm (Fig. V.8b). The results underpin the crucial role of the spectral mismatch between the resonance frequencies when probing spectra associated of concentric multi ring structure. The situation where five hybrid modes are visible in the transmission spectra requires both low mismatch and large coupling rate. This observation orients future sample design towards low separation distance between two adjacent concentric rings, because of the exponential decay of the optical field out of dielectric waveguides [START_REF] Little | Estimating surface-roughness loss and output coupling in microdisk resonators[END_REF].

V.2.3.4 Modulated coupling/spacing and topological protection

In addition to systems of equispaced concentric microring resonators, our model allows to explore in a simple way more exotic coupling configurations and geometries inspired by the 1D tightbinding Hamiltonian presented in the first section of this chapter. Namely, a topological mode localized in the outer ring is expected to appear in the transmission spectrum, with a strong feature within optical gaps, induced by symmetry changes from the unmodulated case. The difficulty to obtain a model that accurately depicts the spectral features of a fabricated sample, where the spacing between each ring is not strictly constant, is double. First, the coupling function u(r i,j , g i,j , ω) is not a trivial function of the radius of the ring resonators r i,j , the interring distance g i,j , and the frequency. With infinite time and numerical resources allocated, we could consider an analytical form of u as a function of an overlap integral involving the electric field of each coupled mode. This integral would depend on the geometry (cross-section and interring distance), the material of the resonator, as well as the probing wavelength and polarization. This approach being too complex, a simplification of the parametrization was chosen:

• The modulation of the coupling of the system is dictated by a change of the inter-ring distance, following the evanescent decay of the electric field leaking out of a high-Q microring (Bessel and Hankel functions [START_REF] Little | Estimating surface-roughness loss and output coupling in microdisk resonators[END_REF]) and what has been demonstrated in Chapter 4.

• Taking into account both the exponential decay of the electric field outside of the waveguide and the proximity of each waveguide to its neighbors, the coupling function u(r i,j , g i,j , ω) is expanded to first order in g i,j as:

u(r i,j , g i,j , ω) ∝ g × u 0 (r i,j , ω) (V.14)
• The coefficient u 0 (r i,j , ω) ≡ u, because the geometric differences between two successive lower band transmission signal from (a) to (b) is characteristic of this protection: photons do not propagate easily toward the bulk modes if they enter the outer ring, but remain essentially located in the vicinity of the edge of the multi-ring resonator due to the presence of the localized mode.

Note that if each resonance can still be resolved in the transmission spectrum of five-unit-cell coupled-ring ensembles, the addition of rings will only densify the same spectral window, as evidenced in Fig. V.10. This is consistent with the predictions of the extended non-Hermitian formalism developed in Chapter 4, and may complicate the analysis of experimental data: if too many modes are coupled together, the comparison between Fig. V.9 and V.10 teaches us that a single broad dip can be interpreted as either a low quality factor mode, or many non-resolved hybrid modes. As in the two previous figures, even and odd dimerisation cases are plotted respectively on panel (a) and (b). This time, no topologically protected edge states appears in the spectrum, and none of these configurations exhibits the formation of a gap. This can be explained by the fact that the resonance frequency of each mode is no longer unique in a geometry consisting of rings with different radii. The Hamiltonian framework should therefore be corrected as follows:

Ĥcorrected = j ω a,j â † j âj + ω b,j b † j bj + j u (1 + δ)â † j bj + (1 -δ) b † j âj+1 + hc. (V.17)
where the resonance angular frequency of each mode on rings 'a' and 'b' is labeled as ω a/b,j . This situation differs from conventional SSH chain since now, because both the coupling function u and the set of resonances have to be parametrized and controlled if the goal is to engineer a topologically protected edge mode. An important optimization work would therefore be required in order to achieve this goal, following two possible algorithm strategies:

1. parametrize the geometry of the concentric rings (width and radius) in order to match the resonances of the inner rings to the one of the outer ring. To achieve this task, a genetic algorithm [START_REF] Mitchell | Genetic algorithms: An overview[END_REF] could help scanning a large parameter space if employed jointly to a 2.5D Finite Time Frequency Domain simulation (on Lumerical or a similar software). The drawback of this approach is that the resonators geometry dictates the coupling constants as well. Thus finding a proper configuration with this approach is likely to be very demanding, if even the calculation converges.

2. iteratively add a new inner ring to the system, taking into account the previously stated constrains. An analytical approach with Bessel and Hankel functions to model the electric field leaking out of the waveguide [START_REF] Little | Estimating surface-roughness loss and output coupling in microdisk resonators[END_REF] could be envisaged, even if it would be very demanding in terms of resources and conditioned to the convergence of each iteration.

Take-home message

All the calculations reported in this section prove one point: no matter how appealing the idea of designing a compact topological system withstanding potential ground-breaking applications for integrated photonics, concentric ring resonators are a challenging problem to parametrize, even under a simplified form. Although it was not possible to experimentally implement any topological systems during my PhD, the building blocks for future work, namely the possibility of coupling several concentric rings, and the optimization of the fabrication processes of AlGaAs-on-AlOx platform applied to guided optics can still be explored.

V.3 Technological development and characterisation of coupled concentric ring resonators

Remark In the following section, the word 'device' refers to the ensemble of a waveguide and a resonator, whereas the word 'sample' refers to the dielectric chip on which several devices are fabricated.

V.3.1 Design and fabrication of the samples

V.3.1.1 First design consideration: coupling through a bus waveguide

As explained previously, the targeted topological modes should appear at the edges of the system, i.e. in the outer and inner rings of the 2D concentric-ring system. Even though we do not know yet how to measure topological invariants of such a system, we can start looking for evidence of topological phases by comparing AAH-modulated and non-modulated resonators. In the modulated case, strong localization of the gap modes and their robustness to fabrication defects make them less susceptible to scattering. Consequently, we expect quality factors of those modes to be larger than their non-topological counterparts. As detailed above, photonss will couple to the concentric ring resonators via the evanescent field of a bus waveguide [START_REF] Steiner | Ultrabright entangled-photon-pair generation from an AlGaAs-on-insulator microring resonator[END_REF][START_REF] Roland | Frequency doubling and parametric fluorescence in a four-port aluminum gallium arsenide photonic chip[END_REF][START_REF] Baker | Critical optical coupling between a GaAs disk and a nanowaveguide suspended on the chip[END_REF], and propagate around them if they match the frequency of one of the resonators modes [START_REF] Gardiner | Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation[END_REF]. The wavevector is conserved from the waveguide to the resonator, and the coupling rate is determined by the geometry of the ring and the distance of the bus waveguide to the resonator. For now, we try to keep the sample as simple as possible, so we choose to use straight waveguides to approach the outer ring.

Chapter V

The bus waveguide must be tapered at its end facets, to avoid parasitic reflection turns it into a cavity. Fabry-Perot fringes would affect strongly transmission spectrum, although the experimental probing of the resonances would still be possible. We first designed a taper of the waveguide with the effective index method [START_REF] Hocker | Mode dispersion in diffused channel waveguides by the effective index method[END_REF] to inject light in the TE 0,0 mode of a (h, w) = (400, 320) nm waveguide. We estimated the tip of a 25 µm-long taper with n ef f = 2.15 to be 170 nm wide. We then optimized this geometry to maximize light injection into the guide with Lumerical. This led to a design of 220 nm-wide tips, resulting in the 30% fiber-to-waveguide transmission of a Gaussian beam of 2 µm diameter to the waveguide around 1.55 µm wavelength. In addition, the bus waveguides are typically longer than 500 µm, so direct fiber-to-fiber transmission becomes negligible compared to the transmission of the sample (see Fig. V.15 for more insight on this configuration).

V.3.1.2 Overview of the fabrication process

This project benefits from the technological developments related to the fabrication of AlGaAson-AlOx metasurfaces [START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Carletti | Controlling secondharmonic generation at the nanoscale with monolithic AlGaAs-on-AlOx antennas[END_REF][START_REF] Celebrano | Optical tuning of dielectric nanoantennas for thermo-optically reconfigurable nonlinear metasurfaces[END_REF][START_REF] Ghirardini | Polarization properties of second-harmonic generation in AlGaAs optical nanoantennas[END_REF]: the overall protocol is known, as well as the involved materials. The starting point of sample fabrication is a cleaved die of GaAs wafer, on which a 400 nm-thick Al 0.18 Ga 0.82 As layer has been grown by MOCVD on a 1000 nm-thick Al 0.98 Ga 0.02 As layer. The latter is oxidized during fabrication, along the guidelines presented in Chapter II and IV, so as to increase optical index contrast and favor light confinement in the Al 0.18 Ga 0.82 As waveguides. Once this die has been cleaned, the negative resist maN-2401 is spin-coated. A mask is then implemented on the sample using e-beam lithography. Once lithography is performed, the sample is developed with AZ726MIF and water. The resist edge is then softened and smoothed thanks to an UV exposure and a hardbake to improve side wall aspect during the ICP-RIE with SiCl 4 and Ar plasma. Etched samples are then oxidized following the process described in Chapter II. The last fabrication step allows us to inject light at both facets of the bus waveguides with micro-lensed fibers, by etching a mesa, i.e. an elevated area with flat top and steep edges, which contains the nano-structures. This mesa etching is performed with acid solutions: diluted HF(1.25%) and a (H 3 PO 4 /H 2 O 2 /H 2 O):(1/1/1) mixture, as presented in Chapter II. Before moving on to the measurements, a few comments of the fabrication process and its improvement arise, especially on the e-beam lithography step and on the critical step of mesa etching.

V.3.1.3 Mask implemention by e-beam lihtography

Compared to low proximity effect silicon-based resists (HSQ and Medusa) used to implement either metasurfaces [START_REF] Gili | Monolithic AlGaAs second-harmonic nanoantennas[END_REF][START_REF] Rocco | Tunable second harmonic generation by an all-dielectric diffractive metasurface embedded in liquid crystals[END_REF][START_REF] Celebrano | Optical tuning of dielectric nanoantennas for thermo-optically reconfigurable nonlinear metasurfaces[END_REF] and isolated dielectric particles (cf Chapter IV), the maN-2401 resist presents several advantages for this application. Since here the quality of the waveguides edges directly dictates the amount of scattered light (linear losses), HSQ or Medusa might seem preferable at first thought. However, since resist reflow processes can be applied to maN-2401 after the development of the resist with AZ726MIF, it is possible to obtain relatively smooth edges, such as in Fig. V.12, once the nominal dose and the dose factor have been optimized. The combination of these parameters is found to be much lower than for HSQ or Medusa, which are conventionally exposed with doses around 1 mC.cm -2 , with an optimal dose around 0.2 mC.cm -2 . The relatively low exposure time of the maN240X series is an advantage during the technological development phase, since it allows to produce more test samples, and therefore strengthens the compliance of maN-based technology to cleanroom fabrication hazard, especially when chemical etching is also involved.

A choice that was made regarding mask transfer on the sample during the e-beam lithography is the encoding of shapes in GDSII files. Indeed, either a polygon or a circle can be used to design a ring. In the first case, the ring will be interpolated by a polygon, and its shape will be exposed following the default beam displacement mode. This results in a given number of ridges, as seen in SEM images of Fig. V.12a. This discretization can even induce discontinuities in the rings because of the periodicity required to link all the nodes of a polygonal shape in this specific class of object (going from 0 to 2π for the outer circle, and from 2π to 0 for the inner circle). Such discontinuities happen at 2π, where the edges of the polygons are not welded, but happen to be placed side by side. Thus, a gap can emerge because of the discretization of the mask during the stitching step. To balance this effect, one can create a polygon overlap of the size of the exposure pixel, i.e. the xy-step parameter of the lithography. Instead of using the polygonal class, the circle class enables vectorial curve drawing, with direct circular patterning instead of the raster patterning of the polygon class. The circle class is thus more adapted to curvy design, as illustrated on Fig. V.12b, and this has been your choice for e-beam lithography of concentric ring-resonator masks.

V.3.1.4 Chemical etching and reproducibility of the fabrication process

The etching of the mesa of Fig. V.13a, which allows the light injection in the setup described in the previous section, causes a reproducibility issue during the last fabrication step. Indeed, after the chemical etching of GaAs bulk in the (H 3 PO 4 /H 2 O 2 /H 2 O):(1/1/1) mixture, floating AlOx decks remain on both sides of the sample. In order to approach the micro-lensed fiber close enough to the waveguide, they must be removed. The first fabrication protocol deployed in the laboratory cleanroom to get rid of those decks was HF chemical etching, similarly to the AlOx removal sketched in the first step of Fig. V.13a. Although this process is convenient and fast, with typical etching time of few tens of seconds, inevitable side-etch occurring during this last step statistically damages the devices. SEM flat and tilted observations of Figs. V.13b and c illustrate the effect of HF side-etch on nanopatterned samples. Because of volume shrinking of the AlAs layer induced during the oxidation step [START_REF] Choquette | Advances in selective wet oxidation of AlGaAs alloys[END_REF], the action of the acidic solution leads to a relaxation of constraint at the interface between AlOx and GaAs bulk, thereby favoring the delamination of the oxide layer. This detachment happens at large scale, up to 250 µm in the transverse direction of the mesa, and can be tracked thanks to the variation of the back-scattered electron signal intensity on the SEM observations. The delamination of the AlOx layer at the GaAs interface feed through to the AlGaAs interface, and echoes in similar detachments or even breaking of the nanopatterned waveguides. When this occurs, the device becomes unusable to any measurement. Out of ten samples fabricated with this protocol, the survival rate of a device lays below 20%, which is not satisfying for systematic measurements.

In order to improve fabrication outcome and minimize the changes in the fabrication process, the last HF etching step was replaced by a mechanical removal of the AlOx floating decks. This avoids degrading chemically the interface between the GaAs wafer and the oxide layer, taking advantage of the stiffness of the mesa edge to cleave the floating decks. To achieve this task, the fabrication process is followed up to the mesa etching with the (H 3 PO 4 /H 2 O 2 /H 2 O):(1/1/1) solution. The floating parts are then eliminated by applying pressure on the vertical direction with tungsten microtips (PTT-250/4-25) mounted on a 3-axis stage. The apex of those tips being 25 µm, they are thin enough to be placed at will on the at least 60 µm wide floating part. Although this method is far from being perfect, its yield is superior to the final HF etching step, with approximately 60% of measurable devices over two fabricated samples. The final aspect of one of those samples is presented on Fig. V.14. Optical microscopy reveals that most of the devices implemented on this particular sample seem measurable with the protocol sketched in the previous sections of this chapter, with close-to-edge or outcropping waveguides terminations. Possible issue occurring with this protocol are of two types:

• remaining of AlOx dangling pieces, as observed on the top side of the third device from the left of Fig. V.14a. In this case, it becomes impossible to approach the micro-lensed fiber close enough to the tapered waveguide.

• expelled AlOx flakes that land on the mesa. An example of such a defect is shown in Fig.

V.14a, on the fourth ring resonator from the right of the image. In this case, no signal can be measured in transmission.

Despite these issues in the mechanical elimination of the floating AlOx at the end of fabrication process: this protocol is more reproducible than the HF etching, but is still too user-dependent for being sustainable on long long-term scale. However, it constitutes an improvement of the fabrication process, and has allowed us to fabricate and systematically measure transmission spectra on coupled ring resonators with different coupling distances from the same sample. As indicated before, micro-lensed fibers are used to inject and collect light in the sample. Manufactured by OZ Optics, they are optimized to work at 1550 nm. These fibers (ref TSMJ-3A-1550-9/125-0.257-2-12-1-AR) have a numerical aperture NA = 0.13, a waist diameter of 2.0 ± 0.5 µm, and a working distance of 12 µm. Note that the design of the waveguide taper mentioned above was realized considering the specifications of these micro-lensed fibers.

V.3.2.2 Spectral characterisation of concentric ring resonators

Optimization of the bus-waveguide distance to the ring resonator A first series of devices has been fabricated to determine the gap for critical coupling between the waveguide and the outer ring resonator. This condition is the most favorable for spectral features analysis, since it offers the best contrast for the resonances of the resonator. The devices for this task were single ring resonators coupled to a waveguide and with varying gap. Fig. V.16 summarizes the main features of those measurements in two different devices. In both cases, a Fabry-Perot response can be observed, which result from spurious reflection. The period of its oscillation corresponds to an approximately 500 µm long cavity, which matches the width of the mesa. Fortunately, the dimension of this cavity is much larger than the perimeter of the probed microring resonator, thus its presence is not critical to extract useful information to characterize resonance of the resonators. In Fig. V.16a, the resonances are visible in the fast Fourier transform (FFT) histogram but not in the signal, because their amplitude is too small compared to the Fabry-Perot signal. This says that the ring resonator is not critically coupled to the bus waveguide, unlike in Fig. V.16b where this condition is almost achieved.

The above spectra allow to characterize resonators quality factor: Q = 26000 deduced from Fig. V.16b, which is representative of the possibilities offered by maN2401 e-beam lithography processes for AlGaAs-on-AlOx guided optics. The free spectral range (FSR) can also be computed from those data, by applying FFT. In all measured cases where resonances were present, a 1.88 nm FSR was measured, which corresponds to a 46 µm radius WGM and is consistent with the nominal radius of the outer ring. The resonance finesse can also be deduced:

F = ∆λ δλ = Q × ∆λ λ 30 (V.18)
where Q is the quality factor, λ the wavelength of the resonance, ∆λ the FSR, and δλ the full-width at half-maximum of the resonances. One final remark arises from the amplitude of the transmission of the system. For 0.4 mW out of the fibered laser, only a few hundred nW are measured on the setup of Fig. V.15. Considering that the inverted tapers are designed to match the fiber mode to the waveguide mode with a transmission of 0.3, it means that the implemented devices is extremely lossy. More precisely, over the propagation along the 500 µm

α = λ Q i × F SR × r (V.19)
where r is the radius of the ring, and device are comparable to that of Fig. V.16b, albeit with a lower transmission: quality factor of 36000 is measured, with the same FSR. The rest of the fabricated devices are pairs of concentric rings, with the outer ring unchanged, and the inner ring radius being shorter of w + g, with effective gaps g ∈ {145, 236, 269, 308, 328} nm. Except for device (b), for which poor transmission hinders resonance analysis, devices (c)-(f) present clear split paired minima, a signature of hybrid modes. We observe that not only the resonance splitting is heterogeneous across the scanned spectral range, but the closest pair of resonances are not centered around the resonance wavelength of the single ring.

Q i = Q/(
According to the analytical model of the previous section, the relative depth of the paired resonance suggests that the order of magnitude of the coupling is few hundred GHz. It exceeds the splitting of paired minima on the plots of Fig. V.17, which is of few tens of GHz, depending on the considered doublet. If the coupling rate exceeds the FSR, as suggested by our model, the mode sequence should be affected: the permutation of the resonances would lead to counterintuitive mode labeling. This circumstance prevented us from establishing any direct comparison to the resonances of the single ring resonator. A way to overcome the mixing of resonances would be to implement rings with larger FRS, i.e. shorter radii. In addition, when concentric rings are added, the effective index of the coupled mode should be affected since light is confined in a significantly different volume than a bare dielectric waveguide. For the hybrid resonant modes, the resonance condition of Eq. V.1 is changed, which should result in a resonance wavelength shift for a given azimuthal order [START_REF] Zhang | Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling[END_REF]. This discussion highlights the need of developing a strong numerical approach to clarify how each parameter affects the behavior of the whole system.

V.4 Perspectives for topologically protected whispering-gallery modes

My preliminary results have shown the difficulty of designing a concentric ring resonator. Coupling mechanisms, although their origin is well understood, are not trivial to compute or extract from experimental data. The problem has several levels of complexity: the design of the rings to match the waveguide modes, the computation of the coupling rates, the effect of the coupling on the effective index of the system, the experimental implementation on a non-optimal platform. On the experimental side, a first step to facilitate the interpretation of the transmission spectra would be to increase the FSR, therefore to reduce the radius of the ring resonator. This would effectively increase the spread of the resonances, facilitating the mode labeling and the study of coupling constants in concentric ring resonators.

My work opens a path toward the design of 2D coupled systems, and discloses some perpectives to design 3D arrangement of concentric ring resonators as the one presented on Fig. V.7, V.8, V.9 and V.10 (d = 0). This last perspective will require more complex epitaxial structures, alternating layers of different optical indexes. Overall, a deeper numerical analysis based on FDTD techniques would be of great benefit for anyone who wants to explore more in depth those designs. entanglement of biphoton states generated from SPDC in AlGaAs nanoresonators can be controlled with the probing conditions, the detection scheme, and the design of the resonator. With the current functional setup, experimental realization of SPDC on AlGaAs metasurface may soon be achieved, but would require more work from the material side: 1) prior to nanopatterning the epitaxially grown dielectric layer, we must understand the microscopic mechanisms that hinder SPDC while up-conversion is possible; 2) exploring new growth directions, either [011]-or [START_REF] Sato | Topological superconductors: a review[END_REF]-AlGaAs, which allow on-axis nonlinear operations. The broadband spectrum of the SPDC photons, essentially due to the relaxation of the PM condition, opens a lot of possibilities for quantum communication networks, for which entangled, spectrally non-degenerated photon pairs could provide a single communication channel between two nodes of networks [START_REF] Appas | Flexible entanglement-distribution network with an AlGaAs chip for secure communications[END_REF].

Modeling of coupling between open-cavities at the nanoscale

We have introduced a new method to study the coupling between nano-antennas, mapping them on a non-Hermitian tight-binding formalism. This approach offers a new modal perspective on finite-size ensemble of dielectric nanoresonators, bringing the evidence that local coupling edge protection can be used beyond the restrictive physics of Bloch theorem and periodic bondary conditions. Nonetheless, an experimental validation is still required to fulfill our demonstration. Hybridization of modes may offer interesting possibilities to tune nonlinear properties or promote topological effects, which are compatible with the leaky character of the studied nano-element. Central to this work, the use of exact numerical methods does not limit its scope to 1D, or even 2D metamaterials composed of one unique material. Indeed, the combination of fabrication techniques, such as epitaxial growth by MBE and metal evaporation, would create hybrid metallo-dielectric systems, in which interaction could allow to induce a fast-switching response, leveraging the development of active metamaterials.

The use of QNMs to predict the SPDC response from nano-antennas and the control of the interaction between nano-elements in metamaterial or waveguide in guided photonics might be a source of inspiration to push forward the activities of our group. For instance, SFG and DFG can be approached with our Green's tensor formalism, and studied in the same system as the one we used, the questions of resonator design and choice of epitaxial materials being shared between these subjects. Finally, it will be essential to study the interaction between different resonant elements in hybrid guided-wave photonics systems combining III-V semiconductors and silicon. The merging of technological platforms seems very promising, also in the perspective of exploiting of coupled strong χ (2) nonlinearities of the III-V and χ (3) nonlinearity of low losses of the silicon based devices.

u ± k = 1 √ 2    1 ∓e iφq    (B.5)
where φ q is the argument of the anti-diagonal term of H(k). The nature of the possible trajectories in the complex plane for h(k) = (1 -δ) + (1 + δ) e ikl when k sweeps the first Brillouin zone (k ∈ [-πl; πl]) depends on the sign of δ. If δ < 0, h(k) describes a circle full in the real positive semi-plane, and the associated Chern number is 0: the topology of the system is trivial. If δ > 0, h(k) encompasses the origin of the plane, and C = 1: the SSH chain will present edge states. As long as the limit case δ = 0 is not reached during any transformation that affect the parameters of the system, no gap closes, and the topology of the SSH chain remains the same.

B.2 Edge states in SSH chain

Edge states appear at the frontier between two phases of different topology, and are reminiscent of band mixing when the energy gap of a system closes. In the case of an SSH chain, this can happen in the presence of a defect in the chain, at the interface between two chains of different Chern numbers, or at the limit of a finite size chain, given that the vacuum is a topologically trivial phase.

B.3 Topological protection and robustness to disorder

As seen previously, edge states are protected by the bulk-edge correspondence, arising from changes in the symmetry of the system. To illustrate the robustness of topologically protected states, we computed the spectrum and eigenstates of the previous SSH chain, yet introducing random disorder r j on the coupling constants, which becomes 1 ± δ + r j , with r j randomly valued in [-r, r], and r the disorder strength. The effects of such a disorder on the spectrum and the edge states are exhibited in As long as no overlap between protected states or symmetry-breaking leading to the closing of the band gap occur, the state will remain protected. Note that the disorder has been introduced only on the coupling constant in this example. In practice, inhomogeneity on the site or local changes in the environment could affect both the coupling constants and the chemical potential of each site of the chain.

Figure 1 :

 1 Figure 1: Normalized occurrence of the terms 'metamaterial', nanoscience', 'metasurface' and 'nanophotonic' in the books of the 'English 2019' corpus referenced by Google, considering all case variations. The corpus is composed of a total of 200 billions words. Data queried from Books Ngram Viewer.

Figure 2 :

 2 Figure 2: Metamaterial electromagnetic cloak at microwave frequencies Left panel: D microwave cloaking structure (background image) with a plot of the material parameters that are implemented: µ r (red line) and µ θ (green line) are the component of the magnetic permeability in cylindrical coordinates; z (blue line) the dielectric constant. The split-ring resonator geometry of cylinder 1 (inner) and cylinder 10 (outer) are shown in expanded schematic form (transparent square insets). Right panel: snapshots of time-dependent, steady-state electric field patterns when microwaves propagates towards a cloaked Cu cylinder. Black stream lines indicating the direction of power flow. The cloak lies in the annular region between the black circles and surrounds a conducting Cu cylinder at the inner radius. The fields shown are (A) the simulation of the cloak with the exact material properties, (B) the simulation of the cloak with the reduced material properties, (C) the experimental measurement of the bare conducting cylinder, and (D) the experimental measurement of the cloaked conducting cylinder. Figures from [10].

Figure 3 :

 3 Figure 3: : a) Schéma d'un nanocylindre (r = 220 nm, h = 400 nm) générant une paire de photon signal et complémentaire par fluorescence paramétrique. b) Principales contributions modales au facteur de couplage en fonction de la fréquence du photon signal. En encart : diagrammes de rayonnement de la paire de modes présentant la contribution la plus importante. c) Distribution spatiale du taux de génération de photons signal et complémentaire co-propagatifs et dégénérés spectralement. d) Partie réelle de la matrice densité de l'état de polarisation des paires de photons générées.

Figure I. 1 :I. 1 . 1 . 1

 1111 Figure I.1: Scattering of light by a particle. (a) Schematic representation of the scattering and extinction processes, with I inc the power per surface unit of the incident field. (b) Calculation frameworkfor the computation of the differential effective scattering cross-section σ s , integrating the scattered field power through a surface element dS defined by solid angle dΩ in the direction n for an incident electric field E inc propagating in the z direction.

  I.1b):

Figure I. 2 :

 2 Figure I.2: Scattering of a particle at the nanoscale. (a) Schematic representation of the scattered field by a sphere of radius r of sub-wavelength dimension. Two regimes considering an incident plane wave of wavelength λ are distinguished: 1) Rayleigh scattering (2r λ/10); 2) Mie scattering (2r λ/10). (b) Titled scanning electron microscopy image of the typical nanoparticles (AlGaAs nanocylinder on Al 2 O 3 substrate) fabricated during my PhD. Their dimensions make them small Mie scatterers.

Figure I. 3 :

 3 Figure I.3: Scattering by an AlGaAs sphere. (a) FEM computation space and its mesh, with the PML (orange) and the dielectric sphere (green) of radius r = 210 nm. The impinging linearly polarized field and its wavevector are labeled E inc and k inc . (b) Complex plane dispersion of the 336 QNMs computed in three sets of modes: 15 around 1600 nm (red), 135 around 1000 nm (blue), 250 around 650 nm (green). The grey circles represents the relative contribution to the scattered field at 1550 nm. (c) Extinction efficiency σ ext /σ geo (with σ geo = πr 2 ) as a function of the incident wavelength, reconstructed with the whole set of QNMs (grey shaded), with the QNM main contributions plotted in dashed colored lines (corresponding eigenwavelengths in legend).

Figure I. 4 :

 4 Figure I.4: Schematic of second-order nonlinear optical processes. Figure adapted from [69].

2 )

 2 of the isotropic crystal of the class 43m. At telecom wavelength, the non-zero terms that couple the field along three orthogonal directions are all equal in this case: χ (ijk = 2d 14 = 2d 25 = 2d 36 = 240 pm.V -1 (I.31)

Figure I. 5 :

 5 Figure I.5: Overview of the recent experimental and theoretical works on SPDC at the nanoscale.

  Fig. I.5 reports the main pioneering works achieved combining nanophotonic systems and SPDC to design very compact controllable biphoton states sources.

  expression for the tomographic reconstruction of ρ:

Figure I. 6 :

 6 Figure I.6: Topology and band structures. (a) (i) Gap remaining open during Hamiltonian transformation means that the systems described by the two Hamiltonians are in the same topological phase, while (ii) a topological phase transition will result in the gap closing during the transformation. (b) For a finite system comprising a topologically non-trivial bulk surrounded by a trivial background, a topological phase transition occurs on the surface and the gap closes. Gapless topological edge states exist on the surface, crossing the gap. (c) Band inversion due to a mechanism such as spin-orbit coupling. Figure from [106].

Figure I. 7 :

 7 Figure I.7: Imaging topological edge states in silicon photonics. Top panel: principle and sample to realize a photonic equivalent to quantum Hall effect. (a) A single unit cell consisting of four link resonators (white rectangles) and four site resonators (grey resonators). (b) Scanning electron microscope (SEM) image of the device. Bottom panel: evidence of edge channels. Depending on the wavelength of the light, the chirality of the excited edge channel is positive (long edge, b and d) or negative (short edge, a and c). The images on the left correspond to experimental results and the right images to numerical simulations. Figure from [113].

Figure I. 8 :

 8 Figure I.8: Topological insulator laser. Top panel: schematic view of the topological array when pumped along the perimeter to promote lasing of the topological edge mode. The pump couples to the array through a grating coupler. Bottom panel: (a) lasing power as a function of input power for topological and trivial configurations; (b) measured spectrum for a pump power of 23.5 kW.cm -2 at 1064 nm with 10 ns pulses. Figure from [117].

Figure I. 9 :

 9 Figure I.9: Lasing in topological edge states of a one-dimensional lattice. Top panel: (a) SEM images of SSH Hamiltonian implemented with a zigzag chain of micropillars containing GaAs quantum well (QW) placed between two AlGaAs Bragg mirrors, with (b) the mode dispersion and near field in the QW. (c) Schematic representation of the field in the QW for the p x (trivial) and p y (topological) hybrid modes. Bottom panel: (a-b) momentum space image of emission from the bulk and the edge of a zigzag chain. (c-f) Spatial images of the photoluminescence intensity at the energies of the upper (c) and lower (e) p-bands, the s-band (f) and the orbital gap state (d), the position of the micropillars being indicated by blue circles. Figure adapted from [36].

Figure I. 10 :

 10 Figure I.10: Nonlinear light generation in topological nanostructures. (a) SEM images of SSH Hamiltonian implemented with a zigzag chain of silicon nanocylinders for third harmonic generation (THG). (b) Contrast of the THG intensity in one of the edge modes for various lengths of zigzag array. (c) Effect of a random distortion to the zigzag angles of maximal amplitude ∆φ on the protected edge state. For ∆φ > 18 • , configurations where the topological protection is suppressed (red) appear. (d) Measured distributions of the third-harmonic field along the zigzag array of 11 nanoparticles for various pump wavelengths, excited from the air (left) and the substrate (right). Insets: representative cases of third-harmonic distribution for wavelengths 1590 nm, 1620 nm and 1650 nm, respectively.Figure adapted from [38].

Figure II. 1 :

 1 Figure II.1: Zinc-Blende crystalline structure of GaAs, with a lattice parameter of a = 5.6533 Å: (a) 3D representation and (b) projection of the crystalline structure along the [100] direction. As atoms are represented with blue spheres, and Ga atoms with brown spheres.

Figure II. 2 :

 2 Figure II.2: Working principle of a MBE station. (a) Top view of a simple MBE chamber showing the essential growth sources, shutters, beam flux detector and the RHEED system for monitoring structure during growth. (b) Schematic diagram of the correlation of surface coverage of 2-D clusters with idealized RHEED oscillations. Figures from [129].

  II.3) scans the resist, which locally changes its chemical properties. Two families of resists are used for the implementation of these masks. 1) In positive resists (Fig. II.4d'-f'), like poly(methyl methacrylate) (PMMA), some chemical bonds are broken by the electron beam. The exposed zones are eliminated thereafter to obtain the final mask. 2) Negative resists like maN-240X and HSQ/Medusa crosslink more under the effect of the beam (Fig. II.4d"-f".

Figure II. 3 :

 3 Figure II.3: Working principle of a SEM. (a) Scheme of the SEM of MPQ cleanroom. Electrons from the filament are accelerated by a cathode and emitted toward the SEM column. Electron beam goes through an electromagnetic aperture and a condenser lens to obtain a collimated beam, which is focused on the sample by magnetic and electrostatic lenses. The beam booster protects the electron beam from ambient magnetic stray fields, and the scan coils allow to deflect the electron beam over several tens of micrometers. (b) Equivalent simplified scheme, where the electron beam is depicted in yellow. Figure supplied by Raith Nanofabrication.

Figure II. 4 :

 4 Figure II.4: EBL steps. Distinction made for positive (x') and negative resist (x"). A lift-off (evaporation of material on a nanopatterned mask made with solvatable resist, d'-f') and an etching procedure (d"-f", see Sec. II.1.2.3) are presented.

Figure II. 5 :

 5 Figure II.5: Working principle of ICP-RIE. (a) Schematics of an ICP-RIE machine. A high-density plasma is generated by sending a RF signal into a coil. The plasma is then injected into a separated chamber in which a second RF field accelerates electrons and ions in the plasma towards the sample. (b) Interferogram measured during the etching of a 400 nm thick AlGaAs-on-AlAs layer by a plasma of Ar and SiCl 4 , measured at 632 nm.

Figure II. 6 :

 6 Figure II.6: Optimization of the exposure dose for the fabrication of metasurfaces with square single elements. SEM images of a dose test sample, constituted of 600 × 600 nm 2 squares in 750 × 750 nm 2 unit cells. The nominal dose of 1 µC.cm -2 is modulated by a dose factor ranging from 0.3 to 1.3.

  The implemented design is presented in the inset of Fig. II.7. A protrusion of 120 × 80 nm 2 breaks the C 4 symmetry of the square. It results in the appearance of the Fano resonance when the structure is illuminated by an x-polarized plane wave impinging at normal incidence, which proves interesting for SHG. The dose test performed prior to the fabrication of a C 1 symmetry sample is presented in Fig. II.8. It was performed during the same lithography as in Fig. II.6.

Figure II. 7 :

 7 Figure II.7: Linear transmission spectrum of [100]-AlGaAs metasurfaces with modified square elements. Full-wave simulation with linearly polarized plane wave. Comparison between the geometry presented in Fig. II.6 (solid line) and the C 1 geometry of the inset (dashed lines). Courtesy of Franz Löchner.

Figure II. 8 :

 8 Figure II.8: Optimization of the exposure dose for the fabrication of metasurfaces with square single elements. SEM images of a dose test sample, constituted of the C 1 symmetric elements of Fig. II.7 in 750 × 750 nm 2 unit cells. The nominal dose of 1 µC.cm -2 is modulated by a dose factor ranging from 0.3 to 1.3. Insets: mask design at scale 1:1 with the fabricated structures.

Al 0 .

 0 Figure II.9: Nanochairs-on-sapphire fabrication. SEM images refer to steps e to g, respectively.From[START_REF] Gigli | Second harmonic generation and control in dielectric metasurfaces[END_REF].

Fig 1 .Figure II. 10 :

 110 Fig. II.10 presents the aspect (a) and the main fabrication step (b) of the nanofins metasurface. The optimized fabrication process for this sample consists of the 3 following steps: 1. Preparation and deposition of gold cross-makers for the alignment-dependent lithography. (Fig. II.4a-f') After cleaning the AlGaAs sample with warm (T > 40°C) acetone and IPA in a sonic bath, the sample surface is prepared in an oxygen plasma. Negative resist (PMMA, 200 nm thick) is deposited on the sample, and irradiated by the e-beam to implement the cross-marker pattern. After development of PMMA with a solution of methylisobutylketone (MIBK) and IPA, 5 nm of Ti (adhesion metallic layer) and 75 nm of Au are evaporated and lifted-off from the sample in acetone. Contrary to the manufacturing of chairs, the alignment marks must be implemented before the lithography

2 . 3 .

 23 Implementation of the W/2-wide section. (Fig. II.4a-f") Repeating the same cleaning process, maN-2401 is deposited on the sample. Aligning on the previously implemented Au cross-markers, a first W/2-wide pattern is imprinted on the sample by e-beam lithography (see Fig. II.10b). After developing the mask in tetramethyl-ammoniumhydroxide (TMAH) in water (AZ 726 MIF by Microchemicals), we improve the quality of resist edge with a reflow (oxygen plasma, then UV exposure) and annealing. 35 nm of the AlGaAs layer are then etched using RIE ICP. Implementation of the W-wide section. maN-2401 is once again used to imprint a mask of width W (see Fig. II.10b) with the realignment procedure described above. After development and reflow of the resist, the remaining of the AlGaAs layer is fully etched and the AlAs is oxidized to obtain AlOx.

  II.10b).

Figure II. 11 :

 11 Figure II.11: Morphological charcterization of the AlGaAs nanofin metasurface. (a) SEM images obtained for (a) promising and (b) misaligned samples. Inset show the mapping of cross-sections in contrast grey scale along the orange segment in the SEM images. Schematic cross-view of the nanofins (right) refers to the fabrication steps of Fig. II.10.

Figure II. 12 :

 12 Figure II.12: Static optical response in reflection. (a) Reflectivity spectra of the two samples of Fig. II.11 under unpolarized illumination. (b) Reflectivity spectra of the sample in Fig. II.11a, under three linearly polarized illuminations: parallel, perpendicular and at 45°with respect to the AlGaAs nanofins.

Figure III. 1 :

 1 Figure III.1: Physical meaning of the two-photon transition terms (Eq III.6) and their implication. (a) Example of spectral term ξ n,m (ω i , ω s ): the excitation of an electric octopole at the pump wavelength induces a decay of the resonant photon in two daughter photons, generated in an electric quadrupole (signal) and a magnetic dipole (idler). The norm of the electric field for the involved QMNs is plotted. (b) Schematic view of T (r i , ω i , d i ; r s , ω s , d s ), adapted from [152].

Figure III. 2 :Figure III. 3 :

 23 Figure III.2: SPDC efficiency of a [100]-AlGaAs nanocylinder with (r = 220 nm, h = 400 nm), pumped by an x polarized plane wave at normal incidence. (a) Schematic view of the nanocylinder and the pump. Cartesian axes (x, y, z) are collinear to the AlGaAs crystalline axis (x c , y c , z c ). (b) Spectral efficiency ξ n,m associated to given QNM pairs at a pump wavelength of 750 nm.(c) Near-field and far-field of the two resonances contributing to the main ξ n,m near degeneracy.

  By filtering the photons emitted by SPDC, we can compute the coincidence rates for specific detection protocols. Examples of such protocols are presented on Fig. III.4, with a, b and c corresponding respectively to the detection of: a) co-propagating signal and idler photons; b) signal photon on the -z direction (transmission) and a idler photon elsewhere; c) signal photon detected over a solid angle of 44 • in transmission while the idler can be detected anywhere. The amplitude of the radiation diagram corresponds to the detection rate of the two-photon states in a given direction. The nanocylinder is the same as in Fig. III.2 and III.3, and is pumped by a linearly x polarized wave at 735 nm. For the detection protocol depicted in Fig. III.4a, the detection rate presents two local maxima in reflection (upper hemisphere) at 45 • in the yz plane, and two significantly stronger directions of emission in transmission (lower hemisphere) in the same directions. This pattern results from the nonlinear tensor that couples three orthogonally polarized waves, as presented in Fig. III.2. Similarly, when the detection direction of signal photon is set to -z (Fig.III.4b), the idler photon is necessarily emitted by a mode which is polarized orthogonally, namely the z polarized magnetic dipole (mode 2 in Fig. III.2). In this configuration, the generation rate vanishes along the z direction, both in reflection and transmission, because of the emission pattern of mode 2. Moreover, the wider angular detection for the signal photon does not change much the radiation pattern. The relaxation of signal photon detection condition only seems to increase the coincidence rate by a factor 1.5 in this case, with maximum emission in the xy plane. Therefore, the coincidental detection of biphoton generated from a [100]-AlGaAs nanocylinder along the

Figure III. 4 :

 4 Figure III.4: Spatial filtering of the detection joint amplitude for biphoton states generated in an AlGaAs nanocylinder. Different spatial selectivity rules to detect signal and idler photons with normal-incidence linearly polarized plane wave: (a) collinear detection;(b) on-axis signal photon detection; (c) integrated detection of the signal photon on a solid angle to simulate the presence of a collection objective with finite numerical aperture (here NA = 0.7).

For

  the collinear detection protocol of photon pairs (Fig. III.4a, Fig. III.5a

Figure III. 5 :

 5 Figure III.5: Spatial mapping of the polarization state and the degree of entanglement of photon pairs generated for a pump at λ p = 735 nm. a,b,c Normalized generation diagram d 4 N pair / (dtdω s dΩ s dΩ i ) corresponding to (a) copropagative and nondegenerate photon pairs (λ s = 1450 nm, λ i = 1491 nm), (b) counterpropagative and spectrally degenerate photon pairs (λ s = λ i = 1470 nm) and (c) z-axysymmetric and spectrally degenerate photon pairs (λ s = λ i = 1470 nm).The color scale indicates the Schmidt number of the calculated states for each direction of the pairs. The colored dots identify few separable or maximally entangled states having a high fidelity (above 0.9) with the states calculated in the different directions. The red, green and blue arrows represent respectively the x, y and z directions. The top panels reveal the real and imaginary parts of the density matrices of highly entangled photon pairs generated in 3 sets of directions (indicated by the black dots on each diagrams). The fidelity of these 3 states with the nearest Bell state is reported above each panel.

Figure III. 6 :

 6 Figure III.6: From separable to highly entangled states by tuning the pump wavelength. (a) Evolution of the Schmidt number (black) and the generation rate per solid angle and frequency d 4 N pair / (dtdω s dΩ s dΩ i ) (purple) calculated for non-degenerate signal and idler photons co-propagating along the direction (θ = 135 • ,ϕ = 90 • ). The wavelength of the signal photon is chosen as λ s = λ deg -20 nm, while the idler photon wavelength varies according to the energy conservation. The considered direction (θ = 135 • ,ϕ = 90 • ) is the direction for which the generation rate is maximized at a pump wavelength of 735 nm. (b) Evolution of the fidelity between the state generated in the (θ = 135 • ,ϕ = 90 • ) direction and the |V V , |HH and |HH +|V V states. The colored rectangles identify the spectral bands for which the fidelity with these states is greater than 0.95 (threshold represented by the grey horizontal line). (c-d) Normalized generation diagram dN pair / (dtdω s dΩ s dΩ i ) corresponding to copropagative and nondegenerate photon pairs (such as λ s = λ deg -20 nm) obtained for (c) λ p = 680 nm, (d) λ p = 730 nm, (e) λ p = 775 nm. The direction (θ = 135 • ,ϕ = 90 • ) studied on panel (a) is represented by the A,B and C black points. The color scale indicates the Schmidt number of the states calculated in each direction.The colored dots identify a few common separable or entangled states having a high fidelity (above 0.9) with the actual states calculated in a given direction.

Fig. III. 7 Figure III. 7 :

 77 Fig. III.7 reports the effect of the pump angle of incidence on the directivity of photon pairs generation in a [100]-AlGaAs nanocylinder. The dimensions of the cylinder remain identical, and the pump incidence varies from normal (along the z-axis) to orthogonal to the cylinder axis (along x-axis). The cross section in the xy plane, where the maximum of emission has been observed in the calculation in Fig. III.5 and III.6, is plotted in Fig. III.7a. In this computation, d 3 N pair / (dtdω s dΩ s ) is displayed as the probability of generating the signal photon in a direction, conditioned to the detection of its idler photon in transmission. The emission pattern for an incidence of 0 • corresponds to the situation of Fig. III.4b, where the second photon of the pair is emitted by the z-polarized magnetic dipole of the nanocylinder. As the incident angle increases, we notice that the emission lobes for d 3 N pair / (dtdω s dΩ s ) rotate as well, in order to satisfy the coupling between the polarization components via the second-order susceptibility tensor of [100]-AlGaAs nanocylinder. When the pump impinges at θ = 90 • , the signal photon is exclusively emitted by the y-polarized magnetic dipole, whereas higher order modes enable the emission of the idler photon. All the other pump incidence angles involve contribution from various QNMs of the structures, which explains the distorted emission pattern for signal photons.

  Fig. III.8b is presented in Fig. III.8a, when excited by a linearly polarized plane wave propagating in the direction (0, 0, -z). Even if this spectrum resembles the nanocylinder spectrum, the near-field of the resonances is strongly affected by the changes in the object geometry. Understanding and manipulating the resonances of nanochairs is challenging, because the complexity of their geometry makes them different from the well-known multipolar Mie resonances (Fig. III.8e). Therefore, our formalism presents another advantage in this case: in addition to fully calculating the biphoton states with a probabilistic approach, resorting to the QMNs basis of an arbitrarily complex nano-element ensures the exactitude of this calculation. The generation diagram of the rate d 3 N pair / (dtdω s dΩ s ) is plotted in Fig. III.8c and d for two pump wavelengths λ p = 915 nm and λ p = 870 nm. As in Fig. III.4b, the idler photon is detected in transmission along the z direction, and we compute the pair generation rate for signal photons emitted in every direction. In this case, the generation of the signal photon is

Figure III. 8 :

 8 Figure III.8: Directivity of SPDC signal and idler photons in [100]-AlGaAs nanochair pumped by an x-polarized plane-wave at normal incidence (a) Scattering efficiency of the nanochair (grey shaded curve), reconstructed with 442 QNMs. The thirteen principal modal contributions are plotted, as well as their summed scattering efficiency (black solid line). (b) Tilted scanning electron microscopy image of the simulated chair with (r = 220 nm, h = 400 nm, h = h/2), adapted from [135]. The generation diagram dN pair / (dtdω s dΩ s ) calculated for spectrally non-degenerate (λ i,s = 2λ p ± 20 nm) photon pairs generated by the nanochair pumped by a x-polarized plane-wave at (c) λ p = 915 nm (d) λ p = 870 nm. The detection protocol is the same as in Fig. III.4b. The generation diagram are essentially resulting from combinations of QNMs at λ res ∈ {1281, 1518, 1670, 1872} nm (e).

Figure III. 9 :

 9 Figure III.9: Characteristic of the IDQuantique ADPs and time tagger. (a) Detection efficiency in the single-photon counting regime of the ADP as a function of the detected photon wavelength. Several efficiency regimes are available, but we chose to work with 10% detection efficiency (blue box) to reduce the amplification of noise. (b) Principle of time correlation measurements: the relative difference between arrival time of photons on two APDs for a given time window is computed and reported on a count histogram. ch 1 and ch 2 refer to the communication channels with the two APDs, as depicted on Fig. III.10.

  Fig. III.10) in order to reduce parasitic counts and consequently reduce noise threshold in the correlation histogram.

Figure III. 10 :

 10 Figure III.10: Experimental setup for the detection of two-photon coincidences. Picture and schematics of the two implemented setups: all free-space (a) and partly fibered (b).Red lines correspond to the pump path, whereas blue lines represent the collection paths for SPDC photon pairs. For both free-space and fibered configurations, collection of the SPDC photons can be realized in reflection or in transmission, in the case of a transparent non-scattering and/or non-absorbing substrate.

  Figs. III.11 to III.13 present the results of our setup

Figure III. 11 :

 11 Figure III.11: Validation of the SPDC setup with g 2 measurement on photon generated from a LN film. (a) Schematics of the SPDC process fostered by the χ (2) xxx nonlinearity of a thin LN film: pump photons (red beam) generate two collinear photons of the same polarization (blue beam). The optical axis of he LN is represented with the dotted double arrow. (b) Correlations peak from [140], pumping a 300 nm film at 685 nm with 9 mW for 10 minutes, and sensed with superconducting nanowire single photon detectors. (c) Optimized correlation detection on the setup described in Fig. III.10. Acquisition time of 15 minutes for an input power of 43 mW at 774 nm with 10ps bin. Black curve: experimental coincidence histogram. Blue curve: lorentzian fit of the correlation peak to extract the coincidence delay (100 ps) and its width (240 ps).

Fig

  Fig. III.11 presents the first optimized correlation detected when pumping such LiNbO 3 crystal at 774 nm (Fig. III.11c) with a CW Ti:Sapphire laser. When pumping the sample with 15 minutes for an input power of 34 mW, and two LP filters with cut-off wavelengths of

Figure III. 12 :

 12 Figure III.12: Influence of the pump on the SPDC process in LiNbO 3 films. (a) Pump power dependence on SPDC, for a 15 minute-long acquisition at 774 nm. (b) Normalized SPDC correlation peaks for five pump wavelengths, again integrated over 15 minutes. (c) Pump polarization dependence, pumping at 753 nm for 15 minutes with 37 mW. Note that the filtering of the collected light differs for each experiment: FELH 900 and 1500 for (a); FELH 900 and 1400 for (b); FELH 1100 and 1250 for (c).

Figure III. 13 :

 13 Figure III.13: Impact of LP filtering on collected photon pairs generated from a LiNbO 3 film. Coincidence peak (a) and correlation noise (b) when the LiNbO 3 film is pumped for 15 minutes with a 37 mW pump CW laser at 753 nm. For each curve, a different LP filter has been used, with cut-off wavelength of 900 nm, 1100 nm, 1250 nm, 1400 nm, and 1500 nm.

Figure III. 14 :

 14 Figure III.14: Second-harmonic generation from [100]-AlGaAs on AlOx metasurfaces. (a) SEM image of the typical fabricated [100]-AlGaAs/AlOx metasurface. (b) SGH signal as a function of the fundamental wavelength from metasurfaces of the same nanocylinder (r = 190 nm, h = 400 nm) with various periodicities.

  Fig. III.15 for a LiNbO 3 film (b) and a AlGaAs film on sapphire (c). In (b), the results are consistent with the wavelength dependence obtained in Fig. III.12b.

Figure III. 15 :

 15 Figure III.15: Fluorescence of the LiNbO 3 film and AlGaAs/sapphire film when pump by a linearly polarized plane-wave. (a) Calibration of the time/wavelength conversion scale for the two APDs when using 100 km long fiber with a monotonous dispersion. Normalized counts (mW -1 .min -1 ) measured with a single APD on (b) the previously characterized LiNbO 3 film and (c) a 400-nm-thick [100]-AlGaAs/sapphire film (c). The samples are pumped with linearly polarized plane wave at 753 (b) and 775 nm respectively. The black curve is obtained by sorting out the counts by adding LP filters (FELH) in the collection path, whereas the red data are measured when spreading the arrival times of the collected photons through a 100 km fiber.

  [100]-AlGaAs (Fig. III.16b) to [011]-AlGaAs (Fig. III.16c and

Figure III. 16 :Figure III. 17 :

 1617 Figure III.16: Zinc-blende crystalline structure of GaAs, with a lattice parameter of 5.6533 Å: 3D representation (a) and projection of the crystalline structure along the [100] (b) and [011] (c) directions. As atoms are represented with blue spheres, and Ga atoms with brown spheres.

  [011]-AlGaAs to model the generation of on-axis co-propagating biphoton states, whose entanglement is subjected to the pump polarization (Fig. III.18).

Figure III. 18 :

 18 Figure III.18: Switching from an initial state to the |HV + |V H Bell state in [011]-AlGaAs by rotating the pump beam polarization. (a) Schematic view of the crystalline axes for [011]-AlGaAs. The pump polarisation makes an angle α with the x axis corresponding to one of the crystal axes x c . (b) Evolution of the Schmidt numbers calculated for non-degenerate pairs co-propagating along the forward direction (θ = 180 • ) for three different pump wavelengths as function of the pump polarisation angle. The wavelength of the signal photon is chosen as λ s = λ deg -20 nm, while the idler photon wavelength varies according to the energy conservation.(c,f,i,e,h,k) Normalized generation diagram d 4 N pair / (dtdω s dΩ s dΩ i ) calculated for non-degenerate co-propagating pairs generated by the nanopillar pumped by a linearly polarized plane wave with α = 0 • (c,f,i) and α = 0 • (e,h,k), at λ p = 735 nm (c-e), λ p = 780 nm (f-h), and λ p = 830 nm (i-k). The color scale indicates the Schmidt number of the states calculated in each direction. The polarization state of the co-propagating pairs generated in the θ = 180 • direction is indicated below the diagram. (d,g,j) Evolution of the fidelity between the state generated in the θ = 180 • direction and the indicated separable or maximally entangled state when tuning the pump polarisation between 0 • and 90 • and pumping the structure at (d) λ p = 735 nm (g) λ p = 780 nm and (j) λ p = 830 nm.

)

  Figure III.18: Switching from an initial state to the |HV + |V H Bell state in [011]-AlGaAs by rotating the pump beam polarization. (a) Schematic view of the crystalline axes for [011]-AlGaAs. The pump polarisation makes an angle α with the x axis corresponding to one of the crystal axes x c . (b) Evolution of the Schmidt numbers calculated for non-degenerate pairs co-propagating along the forward direction (θ = 180 • ) for three different pump wavelengths as function of the pump polarisation angle. The wavelength of the signal photon is chosen as λ s = λ deg -20 nm, while the idler photon wavelength varies according to the energy conservation.(c,f,i,e,h,k) Normalized generation diagram d 4 N pair / (dtdω s dΩ s dΩ i ) calculated for non-degenerate co-propagating pairs generated by the nanopillar pumped by a linearly polarized plane wave with α = 0 • (c,f,i) and α = 0 • (e,h,k), at λ p = 735 nm (c-e), λ p = 780 nm (f-h), and λ p = 830 nm (i-k). The color scale indicates the Schmidt number of the states calculated in each direction. The polarization state of the co-propagating pairs generated in the θ = 180 • direction is indicated below the diagram. (d,g,j) Evolution of the fidelity between the state generated in the θ = 180 • direction and the indicated separable or maximally entangled state when tuning the pump polarisation between 0 • and 90 • and pumping the structure at (d) λ p = 735 nm (g) λ p = 780 nm and (j) λ p = 830 nm.

Figure IV. 1 :

 1 Figure IV.1: Overview of coupled mode theory. Left panel: CMT applied to planar waveguide, from [174]. (a) Refractive index (top plot) and differential functions used in the analytical formula for the coupling coefficient calculation. (b) Real and imaginary parts of the coupling coefficient as a function of the distance between the two waveguides. [4] refers to an article by Taylor and Yariv, 'Guided wave optics', published in 1974. Right panels: Application of the coupled QNM theory to symmetric gold nanorod dimer. Circles reports exact numerical simulations, whereas solid lines correspond to the hybrid spectra calculated with coupled QNM theory. Top right panel: hybrid mode spectrum when the gap between two co-axial nanorods varies: (a) schematics of the system; real (b) and imaginary (c) parts of the eigenfrequencies. Bottom right panel: extinction cross section of different dimers, illuminated by a plane-wave with right circular polarization at normal incidence. The gap is fixed at 60 nm (a-c) and 20 nm (d-f). All resonances display a strong spectral shift when the gap closes. Figures from the PhD manuscript of Kevin Cognée [175].

Figure IV. 2 :

 2 Figure IV.2: Magnetic dipoles of a single AlGaAs nanocylinder. Real part of the eigenwavelengths (µm) of the in-plane (a) and vertical (b) magnetic dipoles are reported when sweeping radius and height of the cylinder. (c) Cut of the previous spread for h = 400 nm, the conventional AlGaAs thickness used by us. (d) and (e): norm of the electric field inside the dielectric cylinder, respectively for in-plane and out-of-plane MDs.

  Fig. IV.2 reports the results of such analysis in the case of the magnetic dipoles in an Al 0.18 Ga 0.82 As nanocylinder. The MD wavelengths, which corresponds to the real part of their eigenwavelength, are plotted in Fig. IV.2a and b as a function of the radius and the height of the cylinder. This calculation emphasizes a general rule of thumb for the dimensioning of cylindrical nanoresonators: the larger the resonator, the lower the eigenmode frequencies

Figure IV. 3 :

 3 Figure IV.3: Interpolation of the electric field norm (Color bar, V.m -1 ) inside the dielectric cylinder provided by the mode-tracking algorithms. The four plots display the representative modes that can be encountered: the three magnetic dipoles, and a perfectly matched layer (PML) mode. Those modes have been computed with FEM simulations.

Fig. IV. 4 .

 4 For example, MD z has no E z component (Fig. IV.4a

Figure IV. 4 :

 4 Figure IV.4: Tracking of the eigenmodes based on the knowledge of their physical characteristics, applied to the MDs. (a) Decomposition of the electric field on the cartesian basis. The complex field phase reflects into the positive and negative values of the projected complex field component. (b) Simplified decision tree applied in the second mode-tracking algorithm.

Figure IV. 5 :

 5 Figure IV.5: FEM simulations for MDs in the case N = 2 with r = 300 nm and h = 400 nm. (a) Schematic view of the system: Al 0.18 Ga 0.82 As nanocylinders (dark blue); PML (light blue). (b), (c) and (d) Left: real parts of the eigenfrequencies ω vs. gap for MD y , MD z and MD x , respectively. The color bar provides the quality factor Q = Re(ω)/2 Im(ω) and the insets are 3D plots of the electric field norm in the nanocylinders. Right: complex-plane representation of the same data, with the color bar providing the gap d.

Figure IV. 6 :

 6 Figure IV.6: FEM simulations in the case N = 3, for r = 300 nm and h = 400 nm. (a) Real parts of the eigenfrequencies ω vs. gap for MD x . The color bar provides the quality factor Q = Re(ω)/2 Im(ω). (b) Norm of the electric field inside each resonator of the chain for each with d = 750 nm, black vertical line in (a). Normalized powers inside the resonators for each mode are 1: (0.27, 0.47, 0.26); 2: (0.50, 0, 0.50); 3: (0.22, 0.56, 0.22). (c) π orbitals of the allyl system, where the coefficients c i weight the linear combination of the individual p z orbital in the molecular orbital ψ = i c i φ i (from [183]): ψ 1 bonding state; ψ 2 nonbonding state; ψ * 3 antibonding state.

Figure IV. 7 :

 7 Figure IV.7:Schematic view of the non-Hermitian Hamiltonian on which the dynamics of the nanoresonator chain is mapped: dielectric cylinders separated by a gap d interact with each other through a coupling function J(x) between two cylinders separated by an edge-to-edge distance x. The system is coupled to a radiating bath accounting for the openess of the system at a rate g.

Figure IV. 8 :

 8 Figure IV.8:Fit parameters for MD x obtained from the FEM calculation when the dimer of hybrid MD x modes is mapped with the non-Hermitian model.

29 )

 29 Solving this for each value of the gap d, we obtain a possible set of fit parameters (plotted in Fig. IV.8) as a function of the gap, resulting in the same curves as in Fig. IV.5d. Two sets

  IV.9) and four resonators (Fig. IV.10), using only what we learned from the N = 2 case. The main features of the fully numerical simulations are captured by our model: initial exponentiallike decay, subsequent pseudo periodicity, and degeneracy points of both imaginary and real

Figure IV. 9 :

 9 Figure IV.9: Comparison between the analytical model and the FEM simulations in the case N = 3. (a) Real part (top panel) and imaginary part (bottom panel) of the eigenfrequency ω as a function of gap for MD x . (b) Probability density map in the case N = 3, depending if next-nearest neighbor coupling is implemented (right, NNN) or not (left, NN). Modes are labeled according to Fig. IV.6.

  Fig. IV.9b displays more symmetric results than the FEM simulations (Fig. IV.6). It seems that some ingredients are missing to capture the physics of the nanocylinder chain in the case N = 3 and N = 4. Moreover, as analyzed in Section IV.1.4, the coupling regime characterised with pseudo-oscillation of the eigenvalues of the resonators' chain suggest that coupling in the numerical simulation for N = 2 (Fig. IV.5), N = 3 (Figs. IV.6 and IV.9) and N = 4 (Fig. IV.10)

Figure IV. 10 :

 10 Figure IV.10: Case N = 4: real part (top panel) and imaginary part (bottom panel) of the eigenfrequency ω as a function of gap for MD x .

Figure IV. 11 :

 11 Figure IV.11: Relative error of the analytical model's prediction with only nearest-neighbor coupling as a function of the gap for the N = 3 (top panel) and N = 4 (bottom panel) systems, respectively for the pseudo-NNN (a) and the NN (b) models.

Figure IV. 12 :

 12 Figure IV.12: Comparison between numerical simulations and analytical model for d = 750 nm. (a) Real part (top panel) and imaginary part (bottom panel) of the eigenfrequencies ω vs. the number N of resonators in the chain. Blue dashes: spectra calculated with the Hamiltonian model; grey crosses: FEM simulations. (b) Complex-plane representation of the analytically calculated eigenfrequencies for chains of N = 9 (black triangles) and N = 200 (red crosses) with the NNN Hamiltonian.

  x modes. This observation underpins the necessity of finding alternative means to model and design coupled nanophotonic systems of larger size. However, analytical resolution of the tight-binding non-Hermitian problem is possible for large N , as illustrated in Figs. IV.12b and IV.13 for N = 200, where we assumed no frequency dependence in the coupling constants for this specific calculation. In the latter case, the eigenfrequencies of the 1D chain tend to form a continuum of modes, whose spectral localization depends on the coupling constants of the system. Comparing the spectral span of MD x in the complex plane obtained with FEM simulation plotted in Fig. IV.5d with the results of Fig. IV.13, the location of the extremal modes in the case N = 200 are the same as in case N = 2. This substantiates the validity of the bottom-up approach based on the extraction of information on coupling functions from the simplest case of a pair of coupled nanocylinders.

Figure IV. 13 :

 13 Figure IV.13: Complex-plane representation of the analytically calculated eigenfrequencies in the case N = 200 with the NNN Hamiltionian. Color stands for the gap d between value, with indigo coding for the smaller gap values, and yellow for the largest. The eigenvalues plotted in Fig. IV.12b are reported with red crosses and black triangles.

  Fig. IV.14a), and we applied logical tree mode tracking algorithm (Fig. IV.4) as the gap d between each cylinder slowly varies from 0 to 3 µm. Results of this calculation are summarized in Fig. IV.14b for MD z and Fig. IV.15 for in-plane MD arrangements.

Figure IV. 14 :

 14 Figure IV.14: FEM simulations of a 2 × 2 square ensemble of dielectric cylinders, for r = 200 nm and h = 400 nm. (a) Schematic view of the system: Al 0.18 Ga 0.82 As nanocylinders (blue); PML (green). (b) Real part of the eigenfrequencies ω vs. gap for MD z hybrid modes. The color bar provides the quality factor Q = Re(ω)/2 Im(ω) and the insets are cross section plots of the electric field norm in the horizontal plane at half-cylinder height, calculated for d = 180 nm (indigo vertical line).

Figure IV. 15 :

 15 Figure IV.15: FEM simulations of a 2 × 2 square ensemble of dielectric cylinders, for r = 200 nm and h = 400 nm, focusing on the in-plane MDs. The color bar provides the quality factor Q = Re(ω)/2 Im(ω) and the insets are cross sections of the electric field norm in the horizontal plane at half-cylinder height, calculated for d = 180 nm (indigo vertical line). (a): cross-polarized MDs; (b): co-polarised MDs. (c) Schematic view of the MD polarization in the cases of the cross-polarized and co-polarized configurations, showing the mode associated to the inset of (a). (d) Same in the co-polarized case, indicating the three couplings constants for this MD orientation: J xx , J yy and J diag .

Fig. IV. 16 Figure IV. 16 :

 1616 Fig. IV.16 presents the typical aspect of nanoresonators in the end of the above fabrication process. Both for a slightly under-exposed cylinder (a) and a closely spaced ensemble of cylinders (b), the top edge of the structures are beveled, as evidenced in the tilted SEM image of Figs. IV.16c and e. This can be explained by the inhomogeneous thickness of the developed resist: the mask is thinner on the edges of the cylinders as suggested in Fig. IV.16e, therefore it does not protect the AlGaAs layer along the whole ICP etching. Unfortunately, this limitation can be difficult to overcome: for our smallest gap values, 10 minutes of over-etching are applied to ensure that the cylinder is well resolved, unlike in Fig. IV.16e, where the ICP etching has been stopped at the very end of bulk AlGaAs layer. In addition to a typical 5%-height bevel,

  Fig. IV.17b-d. The 1D arrangements related to the calculations of Section IV.2, namely monomers (N = 1), dimers (N = 2), trimers (N = 3) and tetramers (N = 4), are implemented on the presented gridded mask, whereas longer chains of Fig. IV.17c and d are isolated, and elongated for a given (d, r) parameter set. In the same fashion as the non-Hermitian bath model, the gap d between two cylinders remains homogeneous in a given chain, except in the case of the first (unfruitful) attempt of obtaining a topologically protected edges states (Fig. IV.17d).

Figure IV. 17 :

 17 Figure IV.17: Typical sample structure. (a) Mask organization for systematic spectral measurements. (b-d) Images of some samples, respectively a tetramer, a 30-mer, and a 15-cells SSH chain, in addition to examples of Fig. IV.16.

  Fig. IV.18 summarizes

Figure IV. 18 :

 18 Figure IV.18: In-plane MD electromagnetic field configuration for a dimer. (a,c) Dipoles aligned along the dimer axis, with MD orientations resulting in constructive (a) or destructive (c) interference, i.e. bonding or anti-bonding modes respectively. On (b,d) Case of hybrid MD y , resulting in anti-bonding (b) and bonding (d) modes respectively.

Figure IV. 19 :

 19 Figure IV.19: Scattering efficiency (σ s /σ geo ) of a dielectric nanocylinder dimer (r = 300 nm, h = 400 nm) excited by a linearly polarized plane wave at normal incidence. The circles correspond to eigenvalues extracted from FEM simulations for the anti-bonding MD y hybrid modes (a) and for the bonding MD x hybrid mode (b).

Figure IV. 20 :

 20 Figure IV.20: Schematic view of the setup used for bright-and dark-field spectroscopy of dielectric meta-atoms at Abbe Center of Photonics. The source beam is depicted in blue, whereas the collected signal is represented by the purple beam (courtesy of Maximilian Weissflog).

Figure IV. 21 :

 21 Figure IV.21: Measured spectra (S plot from E. IV.30) of a single isolated nanocylinder. (a) Filtering methods applied on the DF signal from a r = 316 nm cylinder. (b) DF spectroscopy of a series of single nanocylinders. (c) BF spectroscopy of a the same nanocylinders, when the analyzer main axis is oriented along (solid line) and orthogonally (dashed line) to the existing N -mers fabricated on the same sample. (d) DF backscattering simulated within N A = 0.9 upon plane-wave excitation at 60°. (e) BF backscattering simulated within N A = 0.9, excitation at normal incidence.

IV. 3 . 2 . 4 Figure IV. 22 :

 32422 Figure IV.22: BF (a) and DF (b) spectra of a dimer with r = 200 nm as function of the gap d and the wavelength. Color bar provides the filtered signal amplitude calculated with Eq. IV.30. FEM simulated bonding and anti-bonding hybrid MD x and MD y are also plotted. Low contrast signal at small gaps is circled in green.

  Fig. IV.23a and b are representative of this tendency. For d < 600 nm, some variations seem to arise, in particular a more intense signal. It corroborates the results of Fig. IV.

  Fig. IV.23c; below this wavelength for d ∈ [300; 800] nm on Fig. IV.23d.In the spectral region around 1100 nm, some resonances are expected for an AlGaAs nanocylinder with (r, h) = (200, 400) (Fig. IV.21e), which correspond to the magnetic quadripoles of the structure. Beyond the scope of this thesis, an extensive study of those eigenmodes might be performed with mode-tracking algorithm and our non-Hermitian bath model. Their experimental detection should be easier because of the much better detection efficiency in the wavelength range of their resonance.

123

 123

Figure IV. 23 :

 23 Figure IV.23: BF spectroscopy of a tetramer (r = 200 nm) from telecom to visible wavelengths. (a) and (c): analyzer oriented parallelly to the chain axis; (b) and (d): analyzer oriented orthogonally to the chain axis.

  . The aim was to observe a signature of the mode densification predicted in Figs. IV.12 and IV.13. DF spectra of chains of cylinders with (r = 200 nm, g = 230 nm) are presented in Fig. IV.24, as a function of the number of nanocylinders composing the system. The related intensity triples when passing from 10 to 30

Figure IV. 24 :

 24 Figure IV.24: DF spectroscopy of a chain of dielectric nanocylinders (r = 200 nm, g = 230 nm) as a function of the number of nano-elements. The collection analyzer is parallel (a) or orthogonal (b) to the chain axis.

Figure IV. 25 :

 25 Figure IV.25: Spectroscopy of 2 × 2 cluster of nanocylinders (r = 200 nm, r = 400 nm), rotating the analyzer on collection line for 0 (a), 45 (b), and 90°(c). Angular origine is chosen parallel to the axis of the N -mers.

Figure IV. 26 :

 26 Figure IV.26: Schematic view of the CL setup. Left: SEM images and schematics view of a 1D-chain of AlGaAs cylinders excited by the electron beam. Right: schematic of the CL setup for spectral imaging, the SEM chamber (left box) being connected to a spectrometer (right, with the grating). Schematic from DELMIC website.

Figure IV. 27 :

 27 Figure IV.27: CL spectra of nanocylinders N -mers with (r = 200 nm, h = 400 nm), with different chain lengths: (a) N = 2, (b) N = 3, (c)N = 4. Color bar codes for the signal treated with Eq. IV.31 in arbitrary units. (d): cross-sections (from top to bottom: xy, xz, and zy planes) of the electric filed norm of the isolated cylinder eigenmodes corresponding to the resonance at 776 nm.

Figure IV. 28 :

 28 Figure IV.28: CL spectra of nanocylinders N -mers with (r = 215 nm, h = 400 nm), with different chain lengths: (a) N = 2, (b) N = 3, (c)N = 4. Color bar codes for the signal treated with Eq. IV.31 in arbitrary units. (d): near-field of the isolated cylinder eigenmodes in the vicinity of the resonances observed in the reported spectra.

  Fig. IV.17a, entail misleading shift and oscillation of the eigenfrequency on the CL signal. It is thus unlikely that the resonances observed are related to eigenmode hybridization.

Figure V. 1 :

 1 Figure V.1: (a) Topological PCF, (b) optical resonator, and (c) dielectric index profile for a cladding with γ = 1 3 , taking on the values n c for the core, and n a , n b for the cladding. Asymptotic bands for (d) unmodulated and (e) modulated cladding with φ = π 6 , η = 0.2 π , n a = 4.6 (tellurium), n b = 1.6 (polystyrene), s a = 0.33d 0 , and s a = 0.67d 0 ; n c = 1. Filled regions are TE (orange) and TM (blue) propagating modes. Figure from [221].

γ=1/ 3 , 1 Figure V. 2 :

 312 Figure V.2: Eigenenergies of an AAH 1D chain for various parameters, obtained after diagonalisation of a finite size chain of 300 sites. The x-axis stands for the mode wavevector amplitude in the first Brillouin zone, while the y-axis represents the relative energy of the associated mode, 0 corresponding to the Fermi level of the chain. For (a) and (b) top panels, γ = 1 3 . For (c) and (d) bottom panels, γ = 1 4 and 1 2 , respectively.

Fig. V. 1 ,

 1 Fig. V.1, where topological gaps are opened in the dispersion of the PCF guided modes upon the AAH modulation (see Fig. V.1d and e, as well as[START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF]). The AAH modulation η cos(2πγj + φ) is governed by three parameters: 1) its amplitude η; 2) its period γ = p/q, where p and q are two co-prime numbers; 3) the topological phase φ of the system, which can be interpreted as the presence of a synthetic magnetic field along the radial direction that acts on the optical index as a

Figure V. 3 :

 3 Figure V.3: Diagonalisation of the Hamiltonian of a AAH 1D chain (η = 0.9, Φ = 0.6π, γ = 1 3 , 33 sites). (a) 2D plot of the local density for all modes (x-axis) along the chain (y-axis). Density of states for topologically protected mode is displayed for the 10 th and 11 th modes (b) and 21 st and 22 nd modes (c).

Figure V. 4 :

 4 Figure V.4: (a) Schematics of an AlGaAs ring resonator coupled to a tapered bus waveguide. The device is placed on a mesa so that once on the sample stage, micro-lensed fibers (b) can be approached from its sides to inject/collect light in/from the waveguide.

Figure V. 5 :

 5 Figure V.5:Schematic view of a single AlGaAs ring of radius r coupled to a bus waveguide of the same width w. Coupling between the two elements is denoted as κ e , d being the distance between the guide and the ring. The ring is resonant at ω 0 , with its intrinsic losses denoted κ i . Photons that enter (+) or leave (-) the segment of the waveguide of length L coupled to the ring resonator are represented by the fields S i± (i = 1, 2).

Figure V. 6 :

 6 Figure V.6: (a) Schematic view of a paired-ring resonator coupled to a bus waveguide, with u the coupling between the two rings, and g the distance between the two rings of width w. (b) Transmission of the system calculated from Eq.(V.9). For this illustrative example, we chose κ e /(2π) = κ i /(2π) = 4 MHz, κ s /(2π) = 1 MHz and the coupling strength between the two concentric rings u/(2π) = 40 MHz.

  V.7b): the augmentation of u fosters the transfer of a photon from the outer to the inner ring, resulting in a modification of the spectral weight of each resonance in the transmission spectra. Moreover, according to what was already shown from Fig. V.6b, the coupling readout directly from transmission spectra possible only if ω 1 -ω 2

Figure V. 7 :

 7 Figure V.7: Transmission of a pair of concentric coupled ring resonators as a function of the coupling strength and the probing wavelength (left panel), and spectrum of the device along the red vertical line (right panel). (a) Limit case of two rings with same radius, and (b) separated by 100 nm. In both situations, κ i /(2π) = κ e /(2π) = 2.5 GHz and κ s /(2π) = 1 GHz is used, and the outer radius is fixed to 46.7 µm.

Figure V. 8 :Figure V. 9 :Figure V. 10 :

 8910 Figure V.8: Transmission of five concentric coupled ring resonators as a function of the coupling strength and the probing wavelength (left panel), and spectrum of the device along the red vertical line (right panel). We compare rings separated with a constant distance g = 10 nm (a), and g = 100 nm (b). In both cases, κ i /(2π) = κ e /(2π) = 2.5 GHz and κ s /(2π) = 1 GHz is used, and the outer radius is 46.7 µm.

  Finally, I used my model to compute the transmission spectrum of the SSH modulated concentric ring resonator system, with rings of different radii. The results are plotted in Fig. V.11.

Figure V. 11 :

 11 Figure V.11: Transmission of 30 concentric coupled ring resonators, dimerised to reproduced an SSHlike system for the expected trivial (a) and topological (b) configuration as a function of the coupling strength and the probing wavelength, and spectrum of the device along the blue vertical line (right panel). The radius of the outer ring is fixed to 50 µm, with a modulated spacing of ḡ = 50 nm. Both coupling rate and distance are modulated to engineer the intracell and intercell coupling rates u 1 = u 2 and u 2 = 3u 2 .

Figure V. 12 :

 12 Figure V.12: SEM characterisation of a sample from the technological development phase. (a) SEM images of a test geometry based on polygonal shape in the mask. (b) Same as (a), using the circle class in the lithography mask, with an optimal dose of 1.6 × 120 µC.cm -2 . (c) Same as (b), under tilted observation.

Figure V. 13 :

 13 Figure V.13: SEM characterisation of a sample after a full fabrication run. (a) Schematic cross-view of the sample during chemical etching. Color code: resist (S1818) in red, AlOx in green, GaAs in grey, side-etched volume in hatched blue. (b) Flat SEM observation of the top of the mesa after the steps schematized in (a), where the implemented devices has been partly damaged during the last HF underetching. Zoom: bent waveguide, detached from the AlOx layer because of mechanical strain. (c) Tilted SEM observation of the same kind of sample. The contrast variation on the AlOx surface indicates a different scattering process for the collected electrons. This is due to the partial delamination of the AlOx layer (left image) under the HF action.

Figure V. 14 :

 14 Figure V.14: Example of a sample fabricated by mechanical removing the floating Aluminum oxide after mesa chemical etching. (a) Optical microscopy images of the full mesa. Ring resonators and waveguides have been implemented on the central turquoise Almuminum oxide mesa. (b) SEM images of one of the devices implemented on the same sample.

Figure V. 16 :

 16 Figure V.16: Transmission spectra of ring resonators, fabricated with full chemical etching process. Top plots: raw transmission spectra of single ring resonators coupled to a bus waveguide for an input power of 0.4 mW. Bottom plots: FFT of the transmission signals. The edge-to-edge gaps between the waveguide and the ring is 220 nm (a), and 359 nm (b).

Figure V. 17 :

 17 Figure V.17: Transmission spectra from the sample of Fig. V.14: (a) single-ring resonator with h = 400 nm, w = 305 nm and r = 46.2 µm; (b)-(f) paired ring-resonators with the same outer radius and increasing gap between them. 1 mW is sent to the sample from the laser.

Figure B. 1 :

 1 Figure B.1: SSH chain with periodic boundary conditions. (a) Schematic representation of the SSH Hamiltonian (see Eq. B.1) for a dimerized 1D chain. (b) Band structure of the infinite SSH chain computed with periodic boundary conditions and different coupling asymmetry. Schematic view of the chains are included for the extreme case of δ = ±1 and δ = 0.

  Fig. B.2 presents the eigenmodes of semi-finite SSH chain of 40 unit cells (80 sites), obtained by diagonalizing the full Hamiltonian matrix of the system. Unlike for the infinite chain, states appears in the gap when δ > 0, i.e. when the topology of the system is not trivial (Fig. B.2a). Two states are degenerate in energy at E = 0 with strong localization on the first (A) and last (B) sites of the chain: one has an even wavefunction, the other an odd wavefunction. It is possible to recreate a protected state localized exclusively on the site A by summing those two eigenstates (Fig. B.2b). The weight of this wavefunction on each site is given by: a j ∝ -1 -δ 1 + δ j (B.6)

  Fig. B.3. Even if the upper and lower bands are strongly affected by the disorder, edge states remain localized near zero energy (Fig. B.3a). However, increasing the disorder strength weakens the localization of the edge state (Fig. B.3b).

Figure B. 3 :

 3 Figure B.3: Robustness of edge states of the finite-size SSH chain with µ = 0. (a) Spectrum of a 80 sites SSH chain when varying the disorder strength r. The splitting of the edge state (red area) is exponentially dumped. (b) Density probability of the wavefunction of the edge state localized on the site A of the first unit cell of the chain for δ various disorder strengths r.

  Overview of the major steps of the fabrication process . . . . . . . . . . . II.2 Samples for international collaborations . . . . . . . . . . . . . . . . . . . . . . .

				3
	I.1.1	Light-matter interaction at nanoscale . . . . . . . . . . . . . . . . . . .	3
		I.1.1.1	Linear scattering of light by a particle . . . . . . . . . . . . .	4
		I.1.1.2	Optical theorem . . . . . . . . . . . . . . . . . . . . . . . . .	5
		I.1.1.3	Influence of the particle size: towards the nanoscale . . . . . .	6
	I.1.2	Quasi-normal modes for nanophotonics . . . . . . . . . . . . . . . . .	7
		I.1.2.1	QNMs in optics and their normalization . . . . . . . . . . . .	8
	II.2.1	Optimization of fabrication process with high-resolution resist for dielectric
		metasurfaces implementation . . . . . . . . . . . . . . . . . . . . . . . . .
	II.2.2	C 1 symmetric meta-atoms for advanced resonance engineering . . . . . .
	II.2.3	Two-step lithography process for metamaterials . . . . . . . . . . . . . . .

Table of contents xvii I Nonlinear, quantum and topological features in nanophotonics I.1 Nanoresonators in photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1.1 Light-matter interaction at nanoscale . . . . . . . . . . . . . . . . . . . . I.1.2 Quasi-normal modes for nanophotonics . . . . . . . . . . . . . . . . . . . I.2 Quantum nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.1 Nonlinear processes in optics . . . . . . . . . . . . . . . . . . . . . . . . . I.2.2 Parametric fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.3 Two-photon polarization state and entanglement . . . . . . . . . . . . . . I.3 About topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.3.1 Topological phases in modern physics . . . . . . . . . . . . . . . . . . . . I.3.2 Promises of topological phases in photonics . . . . . . . . . . . . . . . . . II Technological development and nanofabrication based on III-V materials II.1 Fondamentals of cleanroom fabrication . . . . . . . . . . . . . . . . . . . . . . . II.1.1 Epitaxy of AlGaAs nanofilm for nanophotonic application . . . . . . . . . II.1.2 III Spontaneous parametric down-conversion in nanopatterned resonators III.1 SPDC with QNMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Contents I.1 Nanoresonators in photonics . . . . . . . . . . . . . . . . . . . . . . . . . I.1.2.2 Numerical implementation of the QNMs for scattering of light by a nanoparticle . . . . . . . . . . . . . . . . . . . . . . . . . 8 I.2 Quantum nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 I.2.1 Nonlinear processes in optics . . . . . . . . . . . . . . . . . . . . . . . 12 I.2.1.1 Nonlinear polarization and nonlinear wave equation . . . . . . 12 I.2.1.2 Second order nonlinear processes in non-centrosymmetric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I.2.2 Parametric fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . 15 I.2.2.1 Relaxation of the phase-matching condition . . . . . . . . . . 15 I.2.2.2 From classical to quantum second-order nonlinear processes . 15 I.2.3 Two-photon polarization state and entanglement . . . . . . . . . . . . 17 I.2.3.1 Quantum state and density matrix . . . . . . . . . . . . . . . 17 I.2.3.2 Quantum properties accessible through the density matrix . . 18 I.2.3.3 Continuous and discrete variables for photon states . . . . . . 19 I.2.3.4 Reconstruction of the density matrix for a two-photon state . 20 I.2.3.5 Entanglement witness for biphoton states . . . . . . . . . . . 21 I.3 About topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 I.3.1 Topological phases in modern physics . . . . . . . . . . . . . . . . . . . 23 I.3.1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 I.3.2 Promises of topological phases in photonics . . . . . . . . . . . . . . . 26 I.3.2.1 Guided optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 I.3.2.2 Metamaterials and nanophotonics . . . . . . . . . . . . . . . . 29

Table I .1: The

 I set of 16 tomographic measurements to compute/measure the density matrix of a polarization two-photon state.

	Chapter I					I.2 Quantum nonlinear optics
	measurement	1	2	3	4	5	6
	Mode 1	|H	|H	|V	|V	|R	|R
	Mode 2	|H	|V	|V	|H	|H	|V
	measurement	7	8	9	10	11	12
	Mode 1	|D	|D	|D	|D	|R	|H
	Mode 2	|V	|H	|R	|D	|D	|D
	measurement	13	14	15	16
	Mode 1	|V	|V	|H	|R
	Mode 2	|D	|L	|L	|L
				r( Mν )			(I.46)

  Growth of AlGaAs from GaAs wafer . . . . . . . . . . . . . . 35 II.1.1.2 Wafer bonding for transparent samples . . . . . . . . . . . . . 36 II.1.2 Overview of the major steps of the fabrication process . . . . . . . . . 37 II.1.2.1 Sample cleaning and spin-coating . . . . . . . . . . . . . . . 37

II.1 Fondamentals of cleanroom fabrication . . . . . . . . . . . . . . . . . . . 35 II.1.1 Epitaxy of AlGaAs nanofilm for nanophotonic application . . . . . . . 35 II.1.1.1 II.1.2.2 Scanning electron microscopy and mask implementation via electron beam lithography . . . . . . . . . . . . . . . . . . . . 38 II.1.2.3 Reactive ion etching . . . . . . . . . . . . . . . . . . . . . . . 41 II.1.2.4 AlGaAs oxidation for light confinement enhancement . . . . 42 II.2 Samples for international collaborations . . . . . . . . . . . . . . . . . . 43 II.2.1 Optimization of fabrication process with high-resolution resist for dielectric metasurfaces implementation . . . . . . . . . . . . . . . . . . . 43 II.2.2 C 1 symmetric meta-atoms for advanced resonance engineering . . . . 44 II.2.3 Two-step lithography process for metamaterials . . . . . . . . . . . . . 46 II.2.3.1 Nanochairs fabrication for vertical second harmonic generation 46 II.2.3.2 Nanofins metasurfaces . . . . . . . . . . . . . . . . . . . . . . 46 II.2.3.3 Ultrafast polarization control via all-optical modulation in an anisotropic metasurface . . . . . . . . . . . . . . . . . . . . . 49

III.1 SPDC with QNMs

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 III.1.1 Derivation of the formalism . . . . . . . . . . . . . . . . . . . . . . . . 55 III.1.1.1 Two-photon transition amplitude . . . . . . . . . . . . . . . . 55 III.1.1.2 QNM-based expression for the two-photon amplitude . . . . . 57 III.1.1.3 From two-photon amplitude to generation rate and characterization of the two-photon state . . . . . . . . . . . . . . . . . 58

	III.1.2	From photon population to density matrix: tomography reconstruction

of quantum polarization state . . . . . . . . . . . . . . . . . . . . . . 60 III.2 Metasurfaces for quantum states . . . . . . . . . . . . . . . . . . . . . . . 61 III.2.1 SPDC on [100] AlGaAs nano-cylinder with normal incidence . . . . . . 61 III.2.1.1 Spectral characterization and efficiency of the SPDC process . 61 III.2.1.2 Emission diagram and quantum tomography of the measured two-photon states . . . . . . . . . . . . . . . . . . . . . . . . . 64 III.2.1.3 Control of the entanglement of photon pair with pump wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 III.2.2 On-axis generation: SPDC on [100]-AlGaAs nanocylinder and nanochairs 69 III.2.2.1 Control of biphotons directivity with pump incidence on nanocylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 III.2.2.2 Emission of the biphoton from a nanochair . . . . . . . . . . . 71 III.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 III.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 III.3.2 Measurements of photon pairs from nonlinear material at the nanoscale 76 III.3.2.1 Validation of the experimental setup with a LiNbO 3 thin film 76 III.3.2.2 SPDC attempt on [100]-AlGaAs unstructured thin films and metasurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 III.4 A general scheme for dielectric quantum metamaterials . . . . . . . . 83 III.4.1 Exploration of alternative growth direction: [011]-AlGaAs . . . . . . . 84 III.4.2 Dynamic control of two-photon quantum state entanglement with pump polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 III.5 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Table V . 1 :

 V1 Input variables for the Python computation of the transmission of a waveguide coupled to an ensemble of concentric ring resonators.

Mot-clés : Nanophotonique, modes quasi normaux, optique quantique non linéaire, état quantique intriqué, couplage en champ proche.xvi

CHAPTER IV

Coupled dielectric non-Hermitian nanoresonators

Nanophotonic systems have recently been studied under the perspective of non-Hermitian physics [START_REF] Gigli | Quasinormal-mode non-Hermitian modeling and design in nonlinear nano-optics[END_REF][START_REF] Alaeian | Non-Hermitian nanophotonic and plasmonic waveguides[END_REF][START_REF] Lupu | Switching using PT symmetry in plasmonic systems: positive role of the losses[END_REF][START_REF] Cortes | Non-Hermitian approach for quantum plasmonics[END_REF]. Given their potential for wavefront control [START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Gigli | Tensorial phase control in nonlinear meta-optics[END_REF][START_REF] Decker | High-efficiency dielectric Huygens' surfaces[END_REF][START_REF] Liu | Resonantly enhanced second-harmonic generation using III-V semiconductor all-dielectric metasurfaces[END_REF][START_REF] Koshelev | Dielectric resonant metaphotonics[END_REF], nonlinear optics [START_REF] Marino | Zeroorder second harmonic generation from AlGaAs-on-insulator metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF] and quantum optics [START_REF] Jin | Efficient single-photon pair generation by spontaneous parametric down-conversion in nonlinear plasmonic metasurfaces[END_REF][START_REF] Santiago-Cruz | Photon pairs from resonant metasurfaces[END_REF], it is crucial to develop predictive tools to assist their design. As seen in the previous chapter, resonances play a central role in the control of nonlinear quantum processes in nanophotonics, when the phase-matching condition is relaxed [START_REF] Marino | Zeroorder second harmonic generation from AlGaAs-on-insulator metasurfaces[END_REF][START_REF] Gigli | All-dielectric nanoresonators for χ (2) nonlinear optics[END_REF][START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF]. Engineering couplings in metamaterials at nanoscale [START_REF] Li | Plasmonic particle-on-film nanocavities: a versatile platform for plasmon-enhanced spectroscopy and photochemistry[END_REF][START_REF] Spinelli | Optical impedance matching using coupled plasmonic nanoparticle arrays[END_REF][START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Smirnova | Nonlinear topological photonics[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF][START_REF] Zhang | Magnetic and electric coupling effects of dielectric metamaterial[END_REF][START_REF] Vial | A coupling model for quasi-normal modes of photonic resonators[END_REF] offers an alternative lever to go beyond the geometry optimization of a nano-element [START_REF] Rybin | High-Q supercavity modes in subwavelength dielectric resonators[END_REF][START_REF] Koshelev | Meta-optics and bound states in the continuum[END_REF]. In this chapter, we present a simple model relying on the coupling to an effective bath consisting of a continuum of modes to describe systems of coupled resonators, and test it on dielectric nanocylinder chains accessible to experiments. The effective coupling constants, which depend non-trivially on the distance between resonators, are extracted from numerical simulations in the case of just two coupled elements. The model predicts successfully the dispersive and reactive nature of modes for configurations with multiple resonators, as validated by numerical solutions. It can be applied to larger systems, which are hardly solvable with finite-element approaches. Experimental challenges and attempts of spectroscopy of such coupled system will be reported as well, an analyzed in the perspective of the non-Hermitian bath model. Because of the self-energy term in the master equation Eq. V.6 and the resonance condition of Eq. V.1, spectral mismatch between the modes of several rings with different geometries will not be constant over large wavelength spans. Therefore, we restricted the study to few nanometers in the telecom window. Then the radius of the ring being more than ten times the telecom wavelength, and the azimuthal order of the resonant mode will be greater than 200, considering the typical effective refractive index of a dielectric waveguide (slightly below 3.2 in the case of TE modes of Al 0.18 Ga 0.82 As waveguides). It would be extremely tedious to establish the dependence in ω and r i,j for the interaction strength u. It is important to note that the considerations discussed in the previous paragraph are crucial for the implementation of tolopogical systems, but can as well prove thorny to deeply understand the case of an unmodulated ensemble of coupled concentric ring resonators. Before addressing AAH-like modulated systems, as exposed in the articles of Pilozzi and Conti [START_REF] Pilozzi | Topological photonic crystal fibers and ring resonators[END_REF][START_REF] Pilozzi | Anisotropic circular topological structures[END_REF], let us focus on the implementation of a simpler model, the 1D SSH tight-biding Hamiltonian [START_REF] Su | Solitons in polyacetylene[END_REF], which I recall from Eq. B.1:

where u still stands for the hopping term, µ the chemical potential, δ ∈ ]-1; 1[ a dimerization parameter, and âj and bj the two annihilation operators of each site present in the unit cell of a dimerized 1D chain. This model is a particular case of the AAH depicted in Eq. V.2, with γ = 1/2 and no phase offset φ. Applying the previous working hypothesis, we implemented the following modulation of the coupling strength to reproduce an SSH chain:

with ḡ the average distance separating two rings. This new value affects the off-diagonal terms of the matrix M(N ), but the Python algorithm remains the same.

Similarly to Fig. V.7, the transmission spectra obtained in the limit case of conceptually superposed coupled ring resonator (g = 0) are plotted in Figs. V.9 and V.10. The two figures differ in the number of rings composing the system: ten for In each figure, panel (a) presents the computed data in the case of an even dimerisation, when intra-cell coupling is stronger than the intercell coupling, whereas panel (b) refers to the odd dimerisation case. For both scenarios, the dimerization of the chain results in two groups of resonances on the transmission spectra, separated by an energy gap. This is characteristic of the SSH model and the way dimerisation operates on the eigenstate of the modulated chain. The expected difference between the two panels is the presence of a strong interband signal in the odd case, characteristic of a topologically protected mode. The relative weakening of the upper and The measurement setup of Fig. V.15 makes use of tunable lasers Tunics Plus, with wavelength ranges from 1490 to 1620 nm that can be scanned in two ways. The first one is a fast scan, during which piezo actuators control the laser cavity length and sweep the desired frequency range. This scanning mode is useful to run fast tests, and acquire qualitative insights on samples. However, as the response of the piezo actuator of the laser cavity is slightly nonlinear over the window operation, sweep scans introduce uncertainty on wavelength. For this reason, we prefer to use the second mode, step scan, which allows to perform point-by-point data acquisition. Here, the nonlinear response of laser cavity has no impact since the acquisition of data is made after reaching a target wavelength. Time averaging is also possible with this operation mode, which makes it the best choice for precision measurements. Transmitted power from the sample is sent to a photodiode, calibrated to act as a powermeter.

CHAPTER VI

Conclusion and outlook

My PhD thesis has been focused on the fabrication and modeling of dielectric metamaterials. An important effort has been made to understand and find innovative ways to explore the possibilities offered by these nanoscale systems, at least in four respects:

• the derivation of new theoretical approaches to model complex phenomena in non-Hermitian linear and quantum nonlinear nanosystems;

• the related implementation of new numerical tools and algorithms;

• the related development and optimization of cleanroom nanofabrication processes;

• the use of a wide panel of measurement methods, sometimes non-conventional and exploratory, to characterize the properties of such complex systems.

My work was carried mostly in the MPQ laboratory, partly with the ANR-founded 'NOMOS' project, partly in the framework of international collaborations, with the EU FET-OPEN 'METAFAST' project. While this manuscript mainly focuses on my most ambitious goals concerning SPDC and the modeling of couping between optical nanoantennas (see below), I have also fabricated several samples for our collaborators in Politecnico di Milano, which have enabled that the demonstration of the SHG control mediated by liquid crystal or based on Fano resonances in metasurfaces. These contributions are part of forefront research in reconfigurable optical metasurfaces.

Quantum nonlinear optics in metamaterials

We implemented a QNM formalism to study SPDC in dielectric nanoresonators, a system that promises a great versatility in term of generated quantum states. We have demonstrated the control over the spectral properties and the

APPENDIX A

Quantum tomography

The density matrix ρ is reconstructed as a simple linear combination of known matrices Mj (used as a basis), whose weights are directly given by each of the measured quantities n j . ρ = j n j Mj (A.1)

In our case, the ρ matrix being a 4 × 4 matrix (16 elements), a total of 16 measurements corresponding to different polarization configurations are needed to reconstruct the density matrix. The starting point is to expand the density matrix ρ into a set of 16 linearly independent 4 × 4 matrices Γν :

These Γ matrices form a basis for the following scalar product:

and can be simply obtained from the 2 × 2 Pauli matrices as the ensemble of matrix { σi ⊗ σj } with (i, j = 0, 1, 2, 3) and:

The coefficient of the expansion are thus given by:

The next step consists in establishing the link between the coefficients r ν of the expansion (the unknown of the problem) and the set of measurement. The average number of coincidences counts that will be observed in a given experimental run ν is:

where N is a constant depending on the photon flux and detector efficiency and |ψ ν is the projection state corresponding to the specific polarisation configuration chosen for the detection. Injecting Eq.(A.2) in Eq.(A.6) the link between the coincidence counts n ν and the weights r µ is obtained as:

If the set of tomographic states {|ψ ν } has been properly chosen, the matrix Bνµ is non singular and Eq.(A.7) can be inverted:

Knowing the coefficients r ν , the density matrix can now be calculated directly through the expansion of Eq.(A. One of the simplest model that has topological properties is the Su-Schrieffer-Heeger (SSH) model [START_REF] Su | Solitons in polyacetylene[END_REF]. It is a tight-binding discretized 1D model, in which a particle can jump from a given site to its two adjacent sites, to the left or to the right. Historically, this model has been introduced to study the formation of solitonic waves in long polymer chains, such as polyacetylene [START_REF] Su | Solitons in polyacetylene[END_REF].

The SSH model constitutes a dimerized version of the Hubbard model (Fig. B.1a), which is a 1D chain with equal hopping term u between each sites. Here, the dimerization creates cells composed of two sites (A on the left, B the right). For a chain of N cells, the SSH Hamiltonian reads:

where u still stands for the hopping term, µ the chemical potential of the chain, δ ∈ [-1; 1] a dimerization parameter, and âj and bj the two annihilation operators of sites A and B. δ breaks the symmetry of the couplings A-B and B-A. When applying the Bloch theorem to the SSH chain with periodic boundary conditions, it is possible to derive the analytical form of the eigenenergies of the chain. Rewriting the Hamiltonian in Fourier space, and introducing the spinor

, the problem can be cast into the matrixial form, choosing µ = 0: where j is the position of the unit cell in the SSH chain. Therefore, its probability of presence P j = |a j | 2 decreases exponentially fast when we move away from the edge, with a characteristic distance l c given by:

l c diverges if δ → 0, which corresponds to the limit case of the Hubbard chain. This localization is the hallmark of the bulk-edge correspondence that protects edge states in topological systems. The localization length gives a limit criterion: when the chain length tends toward 2l c , the overlap between the edges states lifts the degeneracy of their energy.

List of abbreviations

•

List of Contributions

Publications:

• V. Vinel, Z.