Résumé

Dans la société moderne, l'importance des oscillateurs à l'intérieur des dispositifs technologiques de pointe retient de plus en plus l'attention. La production d'un signal périodique continue et stable, définition d'un oscillateur, constitue l'élément central pour que toute machine ou périphérique continue de fonctionner correctement. En plusieurs décennies, le développement des technologies autour des oscillateurs sont de plus en plus tributaires des applications et poussent au marketing. De manière générale, trois paramètres principaux permettent d'évaluer les performances élevées d'un oscillateur : sa fréquence de travail élevée ; sa compacité et sa stabilité. Cependant, les oscillateurs actuels ont tous certaines limites pour remplir simultanément ces trois critères. Ils sont soit trop volumineuse (telle que TCXO, oscillateur à cristal compensé en température), soit fonctionne à trop basse fréquence (telle que FBAR, résonateur acoustique en vrac autoportant), ou avec une valeur trop élevée de bruit de phase (tel que VCO, oscillateur contrôlé en tension). Parmi toutes ces technologies, un oscillateur fonctionnant sur les principes de l'optomécanique (OMO) pourrait être la solution pour surmonter les limitations actuelles et atteindre à la fois des fréquences de travail élevées, une forte compacité et un faible bruit de phase. L'oscillateur optomécanique (OMO) fonctionne sur la base de l'interaction et le couplage entre un résonateur optique et un résonateur mécanique. Du fait du couplage entre ces deux degrés de liberté, le déplacement mécanique va induire une modification de la résonance optique. Cette dernière va pouvoir alors être détecté. C'est aussi la motivation de ce travail, qui utilise l'optomécanique pour construire une OMO à base de cristaux photoniques intégrée à faible bruit de phase. Le principe de base de cette technologie est similaire à une autre technologie appelée oscillateur optoélectronique (OEO). L'OMO et l'OEO convertissent tous deux le signal optique en signal RF, une fréquence de travail élevée jusqu'à GHz et une très bonne performance en bruit de phase (inférieure à -110 dBc/Hz). Cependant, pour OEO, pour obtenir un bruit de phase faible, il est indispensable d'introduire une longue ligne à retard optique. Un tel retard optique nécessite plusieurs kilomètres de fibre optique, ce qui rend le dispositif OEO complet très volumineux et limité en applications. Au contraire, l'OMO peut utiliser différents types de ligne à retard basé sur une physique différente. Ainsi le retard peut être contrôlé acoustiquement. Cela permet de profiter de la relative faible vitesse des ondes acoustiques par rapport celles optiques. La vitesse de propagation du signal acoustique (3000 m/s) est de l'ordre de cinq ordres de grandeur inférieure à la vitesse de la lumière (3x108m/s). Grace à cette différence, la ligne à retard peut être réduite du même ordre de grandeur pouvant passer de plusieurs kilomètres à quelques centimètres. De la sorte, l'architecture envisagée de l'OMO peut permettre d'atteindre une compacité bien supérieure et intégrée sur puce. Néanmoins, le retard acoustique requière, pour des questions d'efficacité, l'utilisation de matériaux avec iii d'importantes propriétés piézoélectriques. Les matériaux choisis dans le cadre de ce travail sont deux matériaux semiconducteurs III-V : le GaAs et le GaN.

Dans ce manuscrit, nous aborderons tous les composants nécessaires à l'obtention d'un oscillateur optomécanique intégré sur puce avec notamment: la conception de cavité à cristal photonique, l'intégration de matériaux semiconducteur III-V sur silicium ainsi que la réalisation de ces structures, la caractérisation optique et optomécanique et enfin les méthodes d'actionnement par résonance.

Dans le cadre de ce travail, le résonateur optomécanique est une cavité à cristal photonique siège à la fois de mode optique aux longueurs d'ondes télécom et de modes mécaniques avec des fréquences allant jusqu'au GHz. Ce choix a été dicté, du point de vue optique, par le fait que ces structures permettent un confinement optique élevé permettant ainsi une forte sensibilité aux déplacement mécanique. Du point de vue mécanique, la très faible masse (femtogramme) et les hautes fréquences de vibration mécanique de ce type de structures les rendent également intéressantes. La première partie de ce travail a été d'augmenter le facteur de qualité optique des cristaux photoniques en jouant sur la géométrie de ces derniers. Deux géométries ont été étudié : une cavité induite par l'introduction d'un défaut dans un cristal photonique bidimensionnel parfait ou bien un cristal photonique unidimensionnel. Dans les deux configurations étudiées et après des améliorations pour le contrôle de la position et la forme des trous constituant le cristal, des facteurs de qualité supérieur à 104 pour les structures bidimensionnelles et à 108 pour les cristaux bidimensionnels ont été atteint. En plus du confinement optique, ces structures sont également le siège de modes mécanique avec un fort confinement pour les modes GHz localisées au coeur de la cavité optique. Cette colocalisation permet d'atteindre un fort recouvrement entre les deux modes et donc un fort couplage optomécanique nécessaire pour améliorer la sensibilité de la mesure.

Au-delà du fort confinement optique des cristaux photoniques, la possibilité d'intégration de ces structures est également un enjeu important. Cette technologie de nano-fabrication en salle blanche a été développé afin de sonder d'abord les propriétés optiques puis ensuite mécaniques des structures précédemment développées. Grace à des techniques de pointe en nano-fabrication comme l'intégration hétérogène des matériaux III-V sur silicium, la gravure plasma et le sous-gravure chimique, il a été possible de fabriquer des cristaux photoniques suspendues en GaAs ou GaN au-dessus de guides d'ondes silicium.

L'intégration de cette structure réalisée par intégration hétérogène de matériaux semiconducteur III-V sur silicium permet simultanément de sonder les modes optique et mécanique ainsi que de contrôler ces derniers. Il a ainsi été possible de d'observer des modes optiques pour l'ensemble des structures fabriquées. Néanmoins les facteurs de qualité optique pour les structures bidimensionnelles étant trop faible (inférieur à 500) n'ont pas permis d'observer des modes mécaniques. En ce qui concerne les structures unidimensionnelles, les mesures optiques ont permis de mettre en évidence des facteurs de qualité entre 104 et 105. Une étude et la compréhension fine du couplage optique entre la cavité photonique et le guide d'onde en silicium ont permis de mettre en évidence les différentes contributions au couplage optique. Il a ainsi été possible de connaître les limitations et améliorations possibles. En utilisant les structures unidimensionnelles, il a été possible d'observer les modes mécaniques aux fréquences GHz de ces structures avec des couplage optomécanique . Une étude fine de la dynamique de ces modes avec la puissance et la longueur d'onde d'excitation a v été réalisé. Malheureusement, avec ces différents paramètres, il n'a pas été possible d'observer l'auto-oscillation d'un des modes mécaniques. Plusieurs explications sont avancées (Mode optique fondamental inaccessible d'où un couplage optomécanique plus faible, compétition entre les modes mécanique trop proche les uns des autres. . . ).

Malgré la difficulté d'atteindre le domaine des auto-oscillations de manière purement optique, l'utilisation de l'excitation résonante des modes mécanique peut permettre de surmonter cette difficulté. De ce point de vue, l'utilisation de matériaux piézoélectrique est indispensable. Plusieurs géométries sont étudiées : l'excitation capacitive, piézoélectrique et acoustique. Pour ces trois propositions les principes physiques de base sont expliqués. Des simulations numériques démontrent la possibilité d'excitation résonante des modes GHz et l'efficacité de chacune des techniques envisagées. Des premières réalisations technologiques démontrent la faisabilité de telles structures.

Honestly, I am always a pessimistic person, towards almost everything in this world. I can't objectively or optimistically think about things. The decision I made to shift from chemistry to physics indeed was not a easy one. Despite having countless incompatibility and anxiety, I am still standing here with this manuscript. I believe that there are so many people I want to thank in the following that prompted me to do this. Their trust and support for me has accompanied me for a full four-year doctoral career.

I would like to start with my thesis director Isabelle Robert-Philip. Her profound knowledge in physics, fluent and graceful speaking ability and her concise and straightforward writing skill always inspire and encourage me. And I will always remember her kindness in saving a hopeless girl from so much fussy administrative work, and enabled her to go back to her family during the Chinese New Year. Even though she had to move the lab from Paris to Montpellier in the middle of my thesis, her cares and support always reach in me.

I would like to thank Rémy Braive, who is the co-director of my thesis. Rémy is not only a divide support for every issues or questions during my work, but he is also a kind friend who always listens to my complaints and doubts. During my PhD, we have faced so many challenges, and every times Rémy is the one who has faith in finding solutions. His organization and methodical good habits also deeply affects me. Working with Rémy may be the best thing that every PhD student could dream of. For me, Rémy is a real physicists, rigorous, modest, full of ideas and always ready to go beyond himself. Thank you so much Rémy! I also would like to thank Fabrice Raineri, who gave me so much help and guide in building nanobeam photonic crystal cavities. His advanced research in hybrid III-V photonic crystal nanolaser inspired my work in 1D structures. Whenever I came to him for questions, he is always patient and supportive to me. Besides being a brilliant researcher, he is also a wonderful and humorous friend. He always organized interesting activities to unite everyone in Toniq Group together. He presents a good example that adviser can really be a good friend of PhD students. I would like to show my great appreciation to Giuseppe Modica, whom may be the best colleague to work with in the world. When I was stuck in deep depression and self-doubting, Giuseppe always showed his caring and consideration to me. He is a hardworking researcher, a clever student, a generous friend and a good listener. I also want to acknowledge my appreciation to Dorian Sanchez, a former post-doc of the group who currently working as researcher engineer in company. As a researcher, he taught me so much skills and knowledge in building and scripting simulation models. As a friend, he gave me so many important advises on my research and career. vii viii I also want to acknowledge Viktor Tsvirkun and Avishek Chowdhury, two former PhDs who made the foundation on optomechanical crystal including design, fabrication and experimental set-up. I benefits so much from their legacy of experiences.

As a optomechanics team, I would like to thank Inès Ghorbel for her rich experience in experiments that help us have a better understanding in optomechanics crystal. I would like to acknowledge her for the following work on measurement and analysis. I also would like to thank the other researchers from Thales: Alfredo De Rossi, Sylvain Combrié and Martin Aude. The numerous scientific discussions we have during last two years help me better understand this project. With their support, I could have access to Thales to operate characterization on GaN nanobeam samples. Without these measurements, my knowledge on GaN material won't be accomplished.

There are so many colleagues and friends in my group that I spend a really nice time with. With Léa Constans (one of the most beautiful vocal in the group), Gabriel Marty (always very thoughtful and gentleman) and Maxime Delmulle (super sportive and energetic) and Jérémie Schuhmann (I am amazed by your courage and braveness in traveling), we shared the same office and also spent lots of time together. The various chats among us make up the precious daily routine of my doctoral career. David Barrel, you are really funny person and also possess a pure goodness. Francesco Manegatti, I wish one day our cats could meet again and play football together. Antu Gortari, you are a very unique guy and you can always follow your heart. I wish you good luck for finishing your thesis in few months. Quentin Chateiller, a real geek who truly puts thoughts into action. The others in Toniq Group I would like to thank include Ariel Levenson, Sylvain Barbay, Alejandro Giacomotti, Nadia Belabas (the other most beautiful vocal), Kamel Bencheikh, Dimitris Fitsios, Franck Correria, Anirudh Pammi, Bruno Garbin, Lorenzo Procopio. I am sorry that I couldn't be more precisely for each of you here, but I am always with great appreciation to be one of Toniq people! Outside Toniq group, there are thousands of people I would like to acknowledge for. Firstly, I want to show my special thank to Martin Esmann and his wife Lichen (and baby Clara) for their kindness and hospitality. When I was very homesick, it is their comfort as well as delicious Chinese food made me fully charged. I would like to thank Daniel Kimura for his trust on me for organizing student seminar (even I failed to continue just after 2 months), and our 'optomechanical team' altogether with Julien Chaste, Martin Esmann, Omar Ortiz and Fabrice Lamberti for several scientific meetings and fruitful discussion.

I could like to acknowledge Isabelle Sagnesm and Grégoire Beaudoin for their vital work on III-V (GaAs, InGaP) material expitaxy during my PhD. Their pretty and strong membranes are the foundation before we discuss about this optomechanical oscillator. As another important III-V material epitaxy group, I also would like to acknowledge Fabrice Semond (also invited as my Jury member) and Stéphanie Rennesson for their high quality expitaxy on GaN/AlN membranes. I would like to thank Sophie Bouchoule for her unyielding support for me in cleanroom fabrication with GaN/AlN material. Without her, we would be never succeed in realizing N-based optomechanics crystal. I would like to acknowledge all the cleanroom researchers and engineers for their fabrication support during my PhD, especially Laurent Couraud, Stéphane Guilet, Christian Ulysse and Etienne Herth. I would like to thank all the directions of the lab, system information departments, admin department... for their support on my PhD.

The last but not the least for acknowledgement list, I would like to thank my father Hengwei Zhu and my mother Juhui Xu, without their love, support and understanding, I couldn't have courage to come to France for study and research. It is my father who inspired me to continue PhD after master degree, and it is also him who showed me in real example that even at his complicated period, a will to go forward could never be overthrown.

Thank you for all these people I mentioned above! Without you and your support, a pig will never imagine to fly one day.

Reference oscillators for time-frequency measurements and generation

An oscillator is a device that produces a periodic signal [START_REF] Metcalfe | Applications of cavity optomechanics[END_REF] . Oscillators can be of very different natures and can be found in all areas in science. Their fields of applications is also very broad, among which accurate time and frequency measurements or generation. Such reference timing oscillators, that generate highly stable reference frequencies and can measure time, are used in a wide range of applications including wireless communication, high-speed digital electronics, radars, timing systems, modern navigation. The market growth in this field [START_REF]Global crystal oscillator market[END_REF] , combined to the evolution of current applications, calls for smaller oscillators, decreased costs, and integration with other electronics. For decades, quartz crystal oscillators have been used for the generation of high spectral purity carrier signals. If such oscillators (and related versions such as TCXOtemperature controlled crystal oscillator-or OCXO -oven controlled crystal oscillator-) provide today an unmatched combination of compactness and spectral purity, they oscillate at frequencies below a few hundred MHz. In systems operating at higher frequencies, the oscillator frequency must be multiplied, but at the penalty of an increased oscillator noise. Other types of electronic oscillators (such as dielectric resonator oscillators and sapphire oscillators) produce directly signals at high frequencies in the GHz range. These oscillators essentially rely upon resonant devices that feature large storage times or equivalently high quality factors Q. These high Q elements determine both the frequency of operation and the achievable spectral purity. However, the best dielectric resonators exhibit a limited Q-factor at high frequency, while sapphire oscillators are bulky and complex. OptoElectronic microwave Oscillators (OEO) are a promising alternative. Indeed, these oscillators, developed for instance in OEWaves (USA), Selex (Italy) or Thales (France), provide a direct generation of high-frequency microwave signals and have already demonstrated performances in terms of spectral purity compatible with stringent applications such as radars and high-end communications systems.

Purity of microwave signal in OEOs is achieved thanks to an optical delay line (usually an optical fiber) inserted into the loop. The Q-factor of the oscillator is then given by L/Λ, with L denoting the fiber length and Λ the RF signal wavelength. Q factors of the order of 10 5 are required to obtain high spectral purity (typically around -145dBc/Hz at 10 kHz carrier frequency offset) for an oscillation at 10 GHz (Λ =3 cm), which imposes the use of few km-long fibers. Such km-long fiber delay lines are however sensitive to the environment (vibrations, temperature...), limiting the oscillator close-to-the-carrier noise performance. In addition, OEO architecture, based on individual components (laser, lithium niobate modulator, photodiode) and including 5-10 cm diameter and 1 cm-thick fiber coils, results in bulky devices (typically few liters), preventing their smooth and rapid integration into systems. A challenge here lies in the implementation of miniaturized, compact, transportable and integrated oscillators, especially for on-board applications and on-chip signal processing. This high level of miniaturization is not only important from a compactness point of view, but also for a better insensitivity to environment. The phase noise is expected to be reduced down to -110dBc/Hz while operating at a high output frequency in the GHz range. Most importantly, the package size of the OMO can be as low as 1cm [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF] which is the most compact microwave oscillators.

Figure (1.1) gathers the phase noise performance of various small-volume microwave oscillators presently developed. Voltage Controlled Oscillators (VCO) are compact, but suffer from limited phase noise performance, while quartz oscillators (OCXO, TCXO) are less compact and also suffer from frequency multiplication. Another alternative pursued in this research field consists in replacing the fiber in the OEO's architectures by an optical resonator. These optical cavity-based versions allow for volume reduction, at the cost however of a high manufacturing complexity; the difficulties are here threefold: (i) realizing the optical resonator itself (10 9 optical Q-factor is required) (ii) tuning precisely the optical coupling to the cavity and (iii) maintaining the laser's wavelength locked on the cavity's resonance. To date, these points are addressed by complex manufacturing steps (high-end handmade polishing of the resonator, ultra-precise adjustment of the laser source and coupling optics relative to the resonator in the oscillator package). This leads to a high oscillator cost, that is neither compatible with integration in systems requiring a large number of oscillators (e.g. many-elements radar antennas) or in communication and positioning portable devices.

The PhD project comes within this framework. It aims at taking up the challenges created by these two scaling laws of performance, that is increasing frequency and decreasing size.

Some Figures of merit of reference oscillators

Among all figures of merit and requirements, three are of crucial importance: phase noise which has to be low, operating frequency which has to lie above GHz, and compactness with a small footprint enabling integration.

Low phase noise

One crucial figure of merit is the oscillator's stability either in phase or in amplitude. Practical oscillators are all subjected to noise which can take various forms. When this noise is superimposed on the output periodic signal of the oscillator, it induces a random variation in the output signal's amplitude and frequency, limiting the precision and accuracy of time-frequency measurements by the oscillator [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF] . Noises in oscillators include amplitude noise and phase noise. In general, the amplitude noise can be suppressed by gain compression. Phase noise can be divided into two main classes : additive and multiplicative noise. The additive noise includes thermal noise and shot noise to name a few. Main contribution to multiplicative noise arises from flicker noise and environment noise such as temperature fluctuation or power supply. Multiplicative noise causes a convolution of the signal with the noise in the frequency domain and is therefore harder to remove than additive noise. Generation of high purity signal thus requires a deep investigation of all noise sources, in order to implement appropriate strategies to eliminate amplitude noise and reduce as possible phase noise.

High output frequency

Generally, current oscillators output frequency signal is in kHz and MHz range. In order to shift the output signal in GHz range, frequency multipliers are commonly used. The frequency multiplier is a device that can multiply the driving source to higher-order harmonic frequency [START_REF] Saeedkia | 1 -Optoelectronic techniques for the generation and detection of terahertz waves[END_REF] . However, a frequency doubler will increase the phase noise by a factor of roughly 6dB and a tripler will increase it by about 9.54dB [START_REF] Rouphael | Chapter 4 -System Nonlinearity[END_REF] . Thus, reaching GHz frequency from a MHz oscillator will dramatically reduce its performances in terms of stability and purity.

High compactness

Size reduction is another challenge in the oscillators design. The oscillators market is governed by P words: price, package, performance and power [START_REF] Rosenberg | Static and dynamic wavelength routing via the gradient optical force[END_REF] . The scalability and integrability of the oscillators may largely reduce their production cost and may also allows for cheaper packaging solutions. Even though, the integration also brings complexity in terms of design and fabrication.

Photonic crystal based optomechanical oscillators (OMO)

Recent advancements in micro-optical and micro-mechanical resonator technologies has enabled to exploit the coupling between the optical field and mechanical motion in an optical cavity to control mechanical motion. The investigation of this mutual coupling of optical and mechanical degrees of freedom of optical resonant structures that harbor both optical and mechanical resonant modes, is the core in the field of cavity optomechanics. The cavity-enabled enhancement of the phonon-photon coupling in such structures, combined to their miniaturization down to the nanoscale, has opened the way, within the timeframe of the last decade, to numerous breakthrough experiments. In the context of our work, a notable contribution is the manifestation of parametric instability, resulting in mechanical amplification and thereby oscillation of the mechanical mode driven purely optically. This ability to achieve self-sustained oscillation with no need for feedback electronics makes optomechanical resonators (OMRs) compelling for on-chip applications such as microwave clocks, in which directed light energy from a laser is available to fuel the oscillation. In this section, I will first introduce the optomechanical resonators commonly used and then focus on photonic crystal based OMRs whose mechanical vibrations can lie directly in the GHz frequency range. Finally I will introduce the state-of-art of reference optomechanical oscillators (OMOs).

Cavity optomechanical resonators

During the last decade, a vast number of optomechanical systems has been developed. Basically five different types can be identified:

• Fabry-Pérot type: This configuration corresponds to the canonial configuration in which one of the mirror forming the cavity is free to move or vibrate. These optomechanical systems are usually large, with mirrors diameters ranging from micrometric scale up to decimeter scales.

• Superconducting microwave resonators: such OMRs combine a mechanical oscillator and a microwave resonator.

• Membrane in the middle: Here, the mechanical oscillator consist of a microscale dielectric, possibly partly reflective membrane embedded into a Fabry-Pérot cavity.

• Whispering gallery mode cavities: these OMRs make use of microscale optical resonators in a toroidal or spherical shape. These sustain optical whispering gallery modes.

• Photonic crystal cavities: In such suspended 2D or 1D photonic crystal cavities, the optical modes can couple to mechanical modes of the entire membrane, or to localized modes within the cavity core. Such latter mechanical modes usually features very high optomechanical coupling rates and frequencies in the low GHz range.

Figure 1.2 -State-of-art of optomechanical resonators of different geometries and ranked according to their mass [START_REF] Aspelmeyer | Cavity optomechanics[END_REF] . The red dash rectangle indicates photonic crystalbasec devices (2D photonic crystals cite15 and 1D photonic crystals [START_REF] Eichenfield | Optomechanical crystals[END_REF] ). The effective mass of photonic crystal devices vibration modes are usually in the range of few femtogram (10 -15 ) down to the attogram (10 -18 g).

Figure (1.2) displays current OMRs, ranked accordingly the mass of their vibrational modes, ranging from g -mg (such as macroscopic mirrors or micro pillars), down to femtograms and below such as microtoroid, microdisk, photonic crystal 2D, 1D and even smaller mass as nanorod and cold atoms). Reduced effective masses usually translate into higher optomechanical coupling strength and higher mechanical frequencies. Several OMRs reach the GHz frequency range.

Photonic crystal-based optomechanical resonators

Among the OMRs with vibration modes in the GHz, photonic crystal-based optomechanical resonators have attracted lots of interest in recent decade, due to several advantages they provide compared to other OMR's geometries. First, photonic crystals provide a extremely versatile toolbox for guiding and confining light at the nanoscale, and even confining phonons. These can also be combined with phononic crystals geometries acting as a phonon shield, up to forming phoxonic crystal devices. This opens to the way to advanced optomechanical architectures on a single chip providing enhanced functionality. Second, these photonic crystal-based OMRs can confine both photons and phonons in diffraction-limited volumes, with a high optomechanical spatial overlap and consequently enhanced optomechanical coupling strength. Third, due to the small mass, the coupling strength is further enhanced and the vibration modes frequency can reach frequencies up to GHz range, enabling to implement microwave oscillators without frequency multiplication stages.

In the context of cavity optomechanics, the photonic crystal cavity geometries that are used, are usually two-dimensional (2D) or on-dimensional (1D) photonic crystal cavities etched on a suspended membrane. These cavities provide a high optical confinement and the mass of some of their vibration modes, is small with resonance frequencies in the GHz. These also feature strong optomechanical coupling not only because of the high frequency and small mass of the vibrational modes, but also because of the co-localization of the photons and photons inside the cavity.

1D-photonic crystal platforms, usually named nanobeam cavity, enable strong confinement in diffraction-limited volumes, thanks to a gradual tuning of the photonic crystal geometry (holes size, shape or position) along the line of holes. The very early demonstration of an optomechanical nanobeam cavity, has been provided on a silicon platform [START_REF] Eichenfield | Optomechanical crystals[END_REF] in 2009; the photonic crystal consisted of rectangular holes; it harbored an optical mode at 1550 nm with a quality factor around 38000 and a mechanical mode at 2GHz with mechanical quality factor over 1000 (see Figure (1.3a)). This nanobeam also exhibited high optomechanical coupling. These first demonstration was then extended to photonic crystals formed by circular holes [START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF] implementing an optical cavity with a quality factor of 80000 and an on-resonance transmission through the nanobeam of 70% (see Figure (1.3b)). Various strategies have then been developed to reduce the optical losses. These include by tuning the holes arrangement and geometries so as to implement a Gaussian spatial confinement of the light along the beam [START_REF] Leijssen | Nonlinear cavity optomechanics with nanomechanical thermal fluctuations[END_REF] (see Figure (1.3c)). This improvement enabled to reach a thermo-mechanically induced nonlinear regime. Another strategy proposed a more advanced design combining holes and rod-like shapes [START_REF] Navarro-Urrios | Nonlinear dynamics and chaos in an optomechanical beam[END_REF], [START_REF] Gomis-Bresco | A one-dimensional optomechanical crystal with a complete phononic band gap[END_REF] , the holes periodicity insuring photon confinement while the rod-like periodicity insured phonon confinement (see Figure (1.3d)).

All previous designs have been processed in silicon membranes. Such implementations have also been extended to other host materials like diamond or III-V materials. [START_REF] Eichenfield | Optomechanical crystals[END_REF] . (b). silicon-based nanobeam cavity with circular holes from Q. Quan et al [START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF] . (c). siliconbased zipper cavity formed by two identical nanobeams within the near field of each other from R. Leijssen et al [START_REF] Leijssen | Nonlinear cavity optomechanics with nanomechanical thermal fluctuations[END_REF] . (d). Phoxonic crystal cavity enclosed by two periodic structures exhibiting band gaps for the propagation of photons and phonons from D. Navarro-Urrios et al [START_REF] Navarro-Urrios | Nonlinear dynamics and chaos in an optomechanical beam[END_REF], [START_REF] Gomis-Bresco | A one-dimensional optomechanical crystal with a complete phononic band gap[END_REF] . (e). diamond-based s nanobeam cavity with rectangular holes etched in a beam of triangular cross-section from Bayn et al [START_REF] Bayn | Triangular nanobeam photonic cavities in single-crystal diamond[END_REF] . (f). AlN-based cavity made of circular holes from W. H. P. Permice et al [START_REF] Pernice | High-q aluminum nitride photonic crystal nanobeam cavities[END_REF] . (g). AlN-based cavity etched in a slightly wider nanobeam at the cavity location from L. Fan e et al [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] . I. Bayn et al [START_REF] Bayn | Triangular nanobeam photonic cavities in single-crystal diamond[END_REF] reported on a 1D photonic crystal-based cavity etched in a triangular cross-section nanobeam in single-crystal diamond (see Figure (1.3e)). Such nanobeam cavity can potentially feature optical Q's as high as 22000. W. H. P. Pernice et al [START_REF] Pernice | High-q aluminum nitride photonic crystal nanobeam cavities[END_REF] developed AlN-based nanobeam cavities with circular holes featuring an optical quality factor of 146000 (see Figure (1.3f)). The nanobeam fully integrated the photonic waveguide and waveguide circuitry to funnel light to or extract light from the cavity. Last, L. Fan et al [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] reported on an AlN-based nanobeam system with both high optical Q up to 120000 and a mechanical mode at 3.18GHz with Q m of about 10000 (see Figure (1.3f)). This cavity is formed by tapered elliptical holes etched in a nanobeam whose width is slightly increased in the center.

Besides 1D-photonic crystal, 2D-photonic crystals can also sustain simultaneously highly coupled optical and mechanical modes. In 2011, in the framework of a collaboration between our group and the EPFL, Gavartin et al [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] developed such a 2D suspended photonic crystal defect cavity (a L3 cavity) made by Indium Phosphide and featuring an optical quality factor of 10 4 (see Figure (1.4a)). This platform also sustains mechanical modes, in particular mechanical modes localized within the optical cavity core and with resonance frequencies reaching 1GHz. This architecture has been further integrated over SOI photonic circuitry l [START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF] for implementing a stable and controlled optical access. Such integrated system allows for unprecedented control on the optomechanical coupling and for the development of multi-element optomechanical circuits in view of signal processing applications (see Figure (1.4b)). In parallel, some other works on 2D systems proposed novel designs to insure higher optical confinement or higher mechanical frequency. For instance, Safavi-Naeini et al [START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF] built a 2D photonic crystal cavity in silicon with an ultra-high optical Q=1.2 × 10 6 and a fundamental mechanical frequency at 146.1MHz (see Figure (1.4c)). Meanwhile, in the same group, they proposed a new design in view of implementing a 2D phoxonic resonator [START_REF] Amir | Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab[END_REF], [START_REF] Amir | Two-dimensional phononic-photonic band gap optomechanical crystal cavity[END_REF] (see Figure (1.4d)). Instead of relying on conventional circular or rectangular holes, they proposed to rely on snowflake patterns. The mechanical frequency in this device reached 9.3GHz with an optomechanical coupling strength g 0 /2π = 220kHz.

Reference optomechanical oscillators

The first reported functional on-chip optomechanical oscillator is based on the radiation-pressure-driven self-sustained oscillations in a microtoroid resonator in 2005 [START_REF] Rokhsari | Radiationpressure-driven micro-mechanical oscillator[END_REF], [START_REF] Rokhsari | Brownian noise in radiation-pressure-driven micromechanical oscillators[END_REF] . The phase noise above threshold power in such OMO was around -90dBc/Hz at 10kHz offset from the 54MHz carrier a . In this experiment, the operation frequency was in MHz yet the carrier frequency can be increase to GHz by simply modifying the cavity geometry. Several years later in 2013, another low phase noise microwave oscillator architecture was proposed, based on cascaded stimulated Brillioun scattering [START_REF] Li | Microwave synthesizer using an on-chip brillouin oscillator[END_REF], [START_REF] Li | Low-noise brillouin laser on a chip at 1064nm[END_REF] . The optomechanical resonator is here a ultra-high Q (8.75 × 10 9 ) planar silica-on-silicon microdisk [START_REF] Gigan | Self-cooling of a micromirror by radiation pressure[END_REF] , with a phase noise of -90dBc/Hz at 10kHz offset on the 21.7GHz carrier. This device's performances are comparable to mid-range commercial devices which exhibit both low phase noise and high operation frequency performance. Therefore, OMOs enable to tackle three main challenges of other conventional oscillators: low phase noise, high operating frequency directly in GHz range a Note the unit dBc/Hz stands for dB below the carrier in a 1 Hz bandwidth and the offset frequency at 10kHz is the typical characterstic flicker noise for oscillators. (a) InP-based L3 photonic crystal defect cavity with an optical Q up to 10 4 and mechanical modes in the GHz from Gavartin et al [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] ; (b) Integration of the previous cavity on SOI-based photonic circuitry from Tsvirkun et al [START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF] . (c) silicon-based 2D photonic crystal cavity with high Q 1.2×10 6 from Safavi-Naeini et al [START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF] ; (d) snwoflake phoxonic crystal cavity from Safavi-Naeini et al [START_REF] Amir | Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab[END_REF], [START_REF] Amir | Two-dimensional phononic-photonic band gap optomechanical crystal cavity[END_REF] and compactness. Moreover, due its planar geometry, photonic crystal-based OMO may pave the way towards more complex circuits providing various functionality such as electrical or acoustic actuation, amplification, signal processing and so on. This could enable the integration of a single-chip scale setting of the OMO as well as feedback loops insuring the oscillator stability. Indeed, in a free running oscillator, phase shift or drift is the main issue which influence the stability of the output signal. This can be reduced by using a resonator with a high mechanical quality factor. In any case, phase noise suppression techniques can be used to reduce even further phase noise [START_REF] Ahmed | On phase noise suppression in full-duplex systems[END_REF] . The most common one is a feedback phase-locked loop (PPL) which tends to stabilize the output signal by correcting the phase noise [START_REF] Ivanov | Ultra-low-noise microwave oscillator with advanced phase noise suppression system[END_REF], [START_REF] Kinget | Integrated GHz Voltage Controlled Oscillators[END_REF] . Such OMOs platform, integrating actuation tools for implementing such feedback loop, is the ultimate objective of the PhD work. The envisioned platform rely on multiple elements (see Figure (1.5)). The electrical or acoustical actuation will allow us to resonantly drive the mechanical mode within the photonic crystal cavity. At the same time, the mechanical displacement will be probed optically. The optical output could then be used to apply the necessary feedback on the mechanical drive in view of reducing the phase noise.

The material used for such fully integrated OMOs is of prime importance. Most of the works in optomechanics, make use of silicon. In order to enlarge the capabilities of the OMO, piezo-electric and large bandgap materials would be more favorable.

III-V materials-based optomechanical oscillators with integrated actuation tools

Beside the most of current works on OMRs usually use silicon material, there are also many researches in this field extended to other host materials, among which III-V semiconductors [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF], [START_REF] Ghosh | Piezoelectric actuation of aluminum nitride contour mode optomechanical resonators[END_REF], [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF], [START_REF] Schneider | Optomechanics with one-dimensional gallium phosphide photonic crystal cavities[END_REF], [START_REF] Xiong | Cavity piezooptomechanics: Piezoelectrically excited, optically transduced optomechanical resonators[END_REF], [START_REF] Guha | High frequency optomechanical disk resonators in iii–v ternary semiconductors[END_REF] . The use of such materials may indeed enable novel functionalities, such as resonant excitation of the mechanical modes via actuation tools exploiting the material piezoelectricity. In addition, the integration of such III-V-based platforms on silicon-on-insulator (SOI) photonic circuitry [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF], [START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF] opens up new architectures of OMOs, associating optical, electrical or acoustic probing and actuation tools.

Mechanical modes can be driven optically; yet the efficiency of optical actuation schemes usually remains rather low. As a consequence, the frequency stability of the optically-driven OMOs is poor, limited by the fluctuations of the thermal bath as well as the thermodynamical noise and drifts of the mechanical modes. The implementation of reference OMOs thus calls for novel actuation strategies for driving resonantly the mechanical mode. Several schemes can here be envisioned. These include electrostatic, piezoelectric and acoustic actuation. In order to resonantly excite mechanical vibrations in GHz range, the piezoelectric and acoustic actuation are more preferable. In this case, the material's choice is governed by two main requirements: piezoelectricity and large band gap for preventing any two photons absorption processes.

Piezoelectric materials already found many applications, in particular for implementing transducers applied to sensing or driving of vibrations with low damping and high detection accuracy. In the context of cavity optomechanics, piezoelectric transducers can be integrated on OMRs, in order to drive or probe the mechanical modes in an integrated manner. In this context, piezoelectric wide bandgap semiconductors are preferable. Indeed the phonon-photon interaction is higher in wide bandgap semiconductors [START_REF] Zhou | Ultraviolet optomechanical crystal cavities with ultrasmall modal mass and high optomechanical coupling rate[END_REF] thank to their broadband optical transparency and limited two-photon absorption.

Different semiconductor materials can be considered (see Figure (1.6)). Yet some materials such as InAs, InSb and InP feature low piezoelectrical property (piezoelectric constant lower than -0.05C/m 2 ). Conversely, III-V materials like GaAs, InGaP, GaP, GaN and AlN feature a piezoelectric response (see table (1.1)). [START_REF] Fujita | Wide-bandgap semiconductor materials: For their full bloom[END_REF] . In this work, we will mainly focus on these materials combining high piezoelectric property and large bandgap materials for implementing external actuation in the resonator. These are GaN and AlN in the first place; other III-V materials such as GaAs , GaP(InGaP) will also be considered due to their high refractive index and good piezoelectric property.

Organization of the manuscript

This manuscript is divided into 4 scientific chapters and a conclusion chapter.

Chapter 2 deals with the design of optomechanical photonic crystal cavity system from two dimensional to one dimensional photonic crystal. The design includes the optical and mechanical simulation of the OMR's response as well as the optomechanical coupling strength calculation. This chapter will start with a brief introduction on the basics of photonic crystals, including the optimized design of optical cavities in 2D photonic crystal membrane and of 1D nanobeam cavities. To evaluate the optical response, besides the identification of the modes harbored by the structure, its ability to confine light in time will be evaluated. The coupling efficiency and related optical losses when coupling the cavity to an integrated optical waveguide will be estimated. After identifying the mechanical modes sustained by the platform, the different processes coupling photons and phonons will be discussed and their relative contribution to the optomechanical coupling strength will be simulated. This will enable us to propose optimized designs of OMRs, taking into account the specificities of the host materials (GaAs, AlN and GaN).

Chapter 3 concerns the processing of the OMRs. The process will differ depending on the coupling waveguide architecture, either integrated in the suspended III-V semiconductor membrane or integrated underneath in a SOI substrate. The SOI integration with different III-V materials is always challenging, requiring an optimization of the processing steps.

Chapter 4 is focused on the optical and optomechanical characterization of the fabricated OMRs. The optical measurement aimed at identifying the resonance frequencies and optical losses, via transmission or reflection measurements. The coupling strength is then probed optically. In nanobeam cavity systems, we also observe some nonlinear dynamics properties, including thermo-bistability, and study the impact of laser detuning as well as laser power on the optomechanical response.

Chapter 5 develops the three different resonant actuation strategies we followed. External electrodes will be introduced on the optomechanical photonic crystal system. These three strategies include : electrostatic actuation, piezoelectric actuation and acoustic actuation.

Chapter 6 gathers a conclusion and some perspectives of this work. Appendix A and B describe the development of III-V materials tension, strain, photoelasticity and piezoelectricity for calculating the optomechanical coupling strength as well as piezoelectric actuation mechanism. Appendix C concerns the piezoelectric actuator and sensor calculation, in order to calculate the strain (electricity) induced by the electric (strain) field.

Chapter 2

Photonic crystal cavities and optomechanical coupling

A photonic crystal is a periodically-structured dielectric media. The by-construction periodicity of the refractive index in photonic crystals forms an energy band structure for the photons, defining frequency ranges for which light propagation is allowed or forbidden. The ranges of frequency in which light cannot propagate through the structure, are named photonic bandgaps. The periodicity can be one-dimensional (1D), two-dimensional (2D) or even three-dimensional (3D). By tailoring light propagation and enabling light confinement in very small volumes, photonic crystals have become a very efficient and versatile toolbox for controlling light emission, shaping light propagation and enhancing light-matter interactions, including optomechanical interactions.

This Chapter will first introduce the band structure induced in photonic crystals, based on a description of the Bloch modes they sustain. The strategies for confining light in very small volumes and forming optical cavities, will then be described, with a particular focus on defect cavities in 2D and 1D suspended photonic crystals. Numerical simulation of their optical response will enable to quantify the most important figures of merit in terms of light confinement in space and time. These simulations will be extended to their mechanical response, with the specific objective to enhance optomechanical coupling. This will allow to identify optimized designs adapted to III-V materials based photonic crystals for applications in optomechanics.

Photonic crystals: general notions

Photonic crystal band structure

In 1987, E. Yablonovitch and S. John [START_REF] Yablonovitch | Inhibited spontaneous emission in solid-state physics and electronics[END_REF], [START_REF] Johnson | Photonic Crystals: The Road from Theory to Practice[END_REF] independently introduced for the first time the concept of three dimensional photonic crystal based on an analogy between electromagnetism and solid-state physics. Yet, the study of wave propagation in threedimensional periodic media without scattering was first proposed by Felix Bloch in 1928 [START_REF] Bloch | Über die quantenmechanik der elektronen in kristallgittern[END_REF] . The periodic medium is here regarded as a non-magnetic medium with a periodic dielectric constant ε( r) = ε( r + R i ), where R i is the primitive lattice vector and i equals to the dimension for photonic crystal (i = 1, 2 or 3 for 1D-, 2D-or 3D-photonic crystal). The electromagnetic modes sustained in this medium can be solved as eigensolutions of the following wave equations:

× ε -1 × H = ( ω c ) 2 H, (2.1) × × E = ( ω c ) 2 ε( r) E, (2.2)
with E and H denoting respectively the electric and magnetic fields. ω refers to the pulsation and c to the speed of light in the medium. These are eigenvalue equations, with eigenoperators × and × ε -1 . These operators are Hermitian under the inner product H * • H (or E * • E ) between the two fields H and H (or E and E ).

In the case of a periodic dielectric constant, the Floquet-Bloch theorem stipulates that the electromagnetic modes inside the periodic dielectric medium are pseudoperiodic electromagnetic eigenmodes named Bloch modes in the form:

H k ( r) = e i k• r H n, k ( r), (2.
3)

E k ( r) = e i k• r E n, k ( r), (2.4)
H n, k ( r) and E n, k ( r) represent the periodic envelope functions that satisfy the following equations:

( + i k) × ε -1 ( + i k) × H n, k = ( ω n ( k) c ) 2 H n, k , (2.5)
( + i k) × ( + i k) × E n, k = ( ω n ( k) c ) 2 ε E n, k , (2.6)
with k referring to the Bloch wave vector. For each Bloch vector, such equations are different eigenproblems over the primitive cell of the lattice yielding eigenvalues ω n ( k). These eigenvalues, which are a continuous function of k, define a dispersion relation linking the mode momentum to the mode frequency. Moreover, the eigensolutions are periodic functions of k: the solution at k is the same as the one at k + K j , where K j is a primitive reciprocal lattice vector defined by R i K j = 2πδ ij . Thanks to this periodicity, one needs only to compute the eigensolutions within the primitive cell of this reciprocal lattice around k = 0. This restricted space of reciprocal lattice vectors is referred to as the Brillouin zone. By plotting the relation dispersion within this first Brillouin zone, one obtains the band diagram of the periodic medium. Two different zones appear:

• allowed bands: these correspond to propagative modes within the medium.

• bandgaps: in these specific energy frequencies range, no mode can be sustained in the medium.

Photonic crystal slab band structure

Due to its easier process, one very successful geometry in photonic crystals realizations, consists in 2D-or 1D-photonic crystals etched within a dielectric slab. This approach was first proposed in1994 [START_REF] Meade | Novel applications of photonic band gap materials: Low-loss bends and high q cavities[END_REF] , implemented in 1996 [START_REF] Thomas | Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths[END_REF] , and further extended to photonic crystal waveguides [START_REF] Kuchinsky | 3d localization in a channel waveguide in a photonic crystal with 2d periodicity[END_REF], [START_REF] Lončar | Design and fabrication of silicon photonic crystal optical waveguides[END_REF] . Slab structures consist of a photonic crystal etched in a thin dielectric layer sandwiched between two layers of lower effective refractive index, providing an index guiding by total internal reflection in the direction normal to the plane of the crystal. One example, which is at the core of our work, is a 2D or 1D photonic crystal made of arrays of holes etched in a thin high-index membrane surrounded by air. In this case, the index contrast between slab and the air is very high and light is strongly confined within the slab. Beyond providing a threedimensional confinement of light, one important difference between infinite 2D-or 1D-photonic crystals and photonic crystal slabs, lies in the role of the light line. If the slab is suspended in air, then the eigensolutions of the bulk air are ω = c | k | 2 + k 2 ⊥ , with k referring to the Boch wavevector in the plane of the slab and k ⊥ to the wavevector perpendicular to the slab. This relation defines the light cone as ω ≥ c| k |. Modes lying beneath this light cone, cannot couple to any vertically radiating mode since they are totally internally reflected at the interfaces between the slab and the air. They are thus pure Bloch modes confined in the slab, they are lossless as in the case of an ideal structure where no scattering occurs. Conversely, modes lying above the light line of the cladding material, couple to the continuum of leaky modes of the slab. They are resonant or 'quasi-guided', and have intrinsic radiation losses related to outof-plane diffraction. It should also be noted that all guided bands can be divided into two groups depending on the electric field symmetry with respect to the plane in the middle of the slab: even and odd (or equivalently transverse electric [TE]-like or transverse magnetic [TM]-like). In case of a 2D-photonic crystal made of a triangular lattice of air holes, a band-gap exists only for TE-like modes. Figure (2.2) displays the TE-like band structure diagram for a photonic crystal modes of holes arranged into a square lattice and a triangular lattice. This band-gap of TE-like modes is also usually wider in triangular lattices compared to rectangular lattices. [START_REF] Englund | General recipe for designing photonic crystal cavities[END_REF] .

Photonic crystal cavities

Photonic crystal slabs provide a very elegant method for implementing both waveguides as well as mirrors. Two important features should be noted here: the presence of a photonic band-gap, and the presence of modes with very low group velocities near the edge of the Brillouin zone below the light cone. By working with these two features, one can build mirrors (if the mode frequency lie within the band-gap), or waveguides with very low group velocity (for mode frequencies just above or below the band-gap), or photonic crystal cavities. Two strategies are mainly used to design such optical resonators.

A first strategy relies on an engineering of the dispersion relation in photonic crystal waveguides to form heterostructure cavities. To make matters concrete, let's consider a perfect 2D-photonic crystal consisting of an array of holes 'pierced' in a suspended membrane. By removing one line of holes, one can form a waveguide which features a certain dispersion relation. The frequency of the waveguide mode depends notably on the photonic crystal geometry, in particular its lattice constant or hole radius. The construction of a cavity then consists in forming a waveguide along which the geometry of the surrounding photonic crystal is very locally modified to lower the frequency of the waveguide mode. Such mode at lower frequency can exist in the region featuring a modified geometry but can not propagate since its frequency lies below the dispersion curve of the nearby waveguides. The mode is thus confined within this local region, forming a cavity. Such cavities currently feature record values in the solid state in terms of light confinement in space and in time [START_REF] Combriè | Gaas photonic crystal cavity with ultrahigh q: microwatt nonlinearity at 1.55 µm[END_REF], [START_REF] Han | High quality factor in a two-dimensional photonic crystal cavity on silicon-oninsulator[END_REF], [START_REF] Kuramochi | Ultrahigh-q photonic crystal nanocavities realized by the local width modulation of a line defect[END_REF] .

A second strategy consists in removing one or few adjacent holes in a perfect 2Dor 1D-photonic crystals to form defect cavities [START_REF] Yoshie | High quality two-dimensional photonic crystal slab cavities[END_REF], [START_REF] Painter | Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab[END_REF], [START_REF] Riedrich-Möller | One-and twodimensional photonic crystal microcavities in single crystal diamond[END_REF], [START_REF] Noda | Spontaneous-emission control by photonic crystals and nanocavities[END_REF] . For instance, by removing n holes in a line in a perfect array of holes, one creates a short waveguide close at its both ends by the photonic crystal. If the frequency of the waveguide mode lies within the photonic bandgap , the photonic crystal then acts as a mirror, which reflects back and forth light within the 'spacer' region. Such cavities, usually named Ln cavities, will be the one used in this work. One issue here relates to the mirror reflectivity which impacts the mode confinement. Such reflectivity can be enhanced by engineering the region around the 'spacer' such that the mode gets 'gently' confined within the spacer. This engineering makes use of small perturbations of the photonic crystal geometry at the edge of the cavity, by slightly and adequately modifying the holes radii or period. Such perturbations, which will be more detailed in the next section, translate into a higher light confinement, due to a slow-wave effect and an increase of the impedance wave matching at the cavity edges [START_REF] Sauvan | Slow-wave effect and mode-profile matching in photonic crystal microcavities[END_REF] . [START_REF] Combriè | Gaas photonic crystal cavity with ultrahigh q: microwatt nonlinearity at 1.55 µm[END_REF] . The photonic crystal cavity is made of GaAs and the measured quality factor reaches 7 × 10 5 . (c) and (d): SEM images of a L3 optical resonator done by E. Gavartin [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] . The photonic crystal cavity is made of InP and the measured quality factor is up to 10 4 .

The optical response of such cavities can be described via three main parameters: the resonance frequency ω 0 or wavelength, the ability of the cavity to confine light in time, and its ability to confine light in space. These two last features are related to different figures of merit, which include the cavity quality factor Q o and the cavity mode volume V ef f . The quality factor of the cavity is defined in terms of the energy decay of the cavity, which goes as e (-ω 0 t/Qo) . Using a Fourier transform of the decaying electric field at frequency ω, the quality factor can also be defined as Q o = ω 0 /∆ω, where ∆ω is the full-width at half-max (FWHM) of the cavity spectrum in the frequency domain. Main limitations here are the material absorption losses and light radiation losses. These losses can also be classified as in-plane losses and out-of-plane losses. The in-plane losses are mainly affected by the number of holes around the cavity and imperfections induced during the crystal processing; their contributions to the quality factor writes as Q . Out-of-plane losses occurs mainly at the cavity edges due to light scattering by the mirror around cavity; their contribution writes as Q ⊥ . The total optical Q o then writes:

1 Q o = 1 Q + 1 Q ⊥ (2.7)
In total, the Q o factor is a measure of the temporal confinement of the photons. On the other hand, the mode volume of the cavity is a measure of the volume that the photonic field occupies when confined to the cavity. It is defined as [START_REF] Claudio Andreani | Strongcoupling regime for quantum boxes in pillar microcavities: Theory[END_REF] :

V ef f = V ε( r| E( r)|) 2 d 3 r max{ε| E( r)| 2 } (2.8)
where ε( r) is the dielectric constant, | E( r)| is the electric field strength. In general, increasing Q o and decreasing V ef f increase the light-matter interaction inside the photonic cavity.

Due to their complex geometry, the computation of the optical response of photonic crystals usually relies on numerical simulations. These fall mainly within two categories: in the frequency domain (FD) or in the time domain (TD). Time-domain methods compute the time-evolution of the fields in the periodic medium with arbitrary starting conditions, while frequency-domain methods directly extract the Bloch modes and frequencies by diagnosing the eigenoperator. Our work relies on a time-domain method called Finite Difference Time Domain (FDTD) provided by the Lumerical company. This program solves Maxwell's equations in 2D or 3D nanophotonic structures. This tool can compute the photonic band structures, can simulate the modes profile inside the periodic medium, and can extract the quality factor of photonic crystal cavities. This will be done in the next section, with a focus on two cavity designs that have been used in this work: L3 defect cavities in a 2D triangular lattice of holes pierced in a suspended slab and 1D photonic crystal cavities etched in a suspended beam.

2D photonic crystal defect cavities: design and optimization

In this section, we will focus on 2D photonic crystal defect L3 cavities based on a triangular lattice of holes pierced within a suspended slab surrounded by air (see Figure (2.4)), eventually coupled to an integrated ridge waveguide placed underneath.

We will precise the geometry to achieve a resonance around 1.55µm, optimize the design to achieve strong light confinement and then analyze the coupling efficiency to the waveguide.

Resonance frequency in a L3 cavity

An important feature of a photonic crystal slab is the existence of higher-order guided modes. If the slab is thick enough, these higher-order modes will lie in the band gap, which makes the cavity a multimode resonator. The maximum thickness h for favoring a single mode operation, is given by h = λ/2ε membrane . For III-V material such as GaAs (ε = 3.37) or GaN (ε = 2.31), the maximum height is 245nm and 335nm respectively. For the next simulations, in order to optimize the light confinement, the height of photonic crystal membrane will be chosen equal to h = 265nm for GaAs, and h = 360nm for GaN a .

a The chosen thickness of III-N membrane is slightly larger than the maximum simulated value for expitaxy reason. To grow GaN, additional AlN layer (ε = 2.02) is mandatory. This will be discussed in Chapter 3. First simulations aimed at revealing the spectral extent and location of the photonic band-gap. Finite-difference time-domain (FDTD) simulation methods were then employed to find the exact resonant wavelength of the modes sustained by the cavity, but also their quality factors (by computing the inverse Fourier transform of the cavity field), and their spatial profiles. Such profiles are shown on Figure 2.5 for the fundamental mode as well as the second-order mode. One can already note here that the amplitude of the fields associated to the second-order mode is much lower, with a light confinement that spreads more in the holes than in the material. This suggests already a lower quality factor and high volume associated to this higher-order mode. Two parameters can be varied to adjust the resonance frequency of the fundamental mode sustained by the cavity. These are the photonic crystal lattice constant a and the hole radius r. The dependence of the resonance frequency of the fundamental mode as a function of these two parameters is shown on Figure 2.6. Increasing the lattice constant a, tends to shift the optical resonance towards higher wavelength. Conversely, increasing the holes radius tends to lower the resonance wavelength. For instance, in the case of the GaAs membrane, the optical resonance blueshift is about 4.3 nm every 1nm increment in the lattice constant a, while its redshift is about 4 nm per 1nm increment in the holes radius r. Optimized values for a and r are gathered in the following table.

Material of the slab

a r h Q o GaAs 420nm 110nm 265nm 4900 GaN 630nm 200nm 360nm 500
From these values, one can note that the optical losses in such cavities are high. Yet, as mentioned earlier, an engineering of the photonic crystal geometry can enable an increase of the quality factor. The strategy employed to reduce to optical losses will be described in the next subsection.

Optical confinement in a L3 cavity

Several approaches have been proposed and implemented in the literature to decrease the radiation losses and increase the Q o of a Ln cavity. In 2001, Vuckovic et al. [START_REF] Suman | Grid sensitivity and role of error in computing a lid-driven cavity problem[END_REF] showed that by varying the radius and shape of the holes at the edges of a L1 cavity, the Q o could be increased from few hundreds up to 2800 with a Q o /V ef f figure of merit around 10 5 /λ 3 [START_REF] Yoshie | High quality two-dimensional photonic crystal slab cavities[END_REF] . The principle was extended to L3 cavities by Akahane [START_REF] Akahane | High-q photonic nanocavity in a two-dimensional photonic crystal[END_REF] in 2003. By perturbing the shape and/or position of the holes at the edges of the cavity along the direction of the missing holes, a drastic enhancement of the quality factor was also observed with a Q o up to 45000 and a Q o /V ef f figure of merit around 2.10 6 /λ 3 [START_REF] Srinivasan | Fabrication-tolerant high quality factor photonic crystal microcavities[END_REF], [START_REF] Srinivasan | Momentum space design of high-q photonic crystal optical cavities[END_REF], [START_REF] Srinivasan | Fourier space design of high-q cavities in standard and compressed hexagonal lattice photonic crystals[END_REF] . This increase results from two effects [START_REF] Sauvan | Slow-wave effect and modeprofile matching in photonic crystal microcavities[END_REF] . The first effect is an increase of the mirror reflectivity due to a better mode-matching: The perturbation at the cavity edges ensures a gentle matching of the Bloch mode profiles in the cavity and in the mirror. Second, the group velocity of the trapped-defect mode is modified by the perturbation, that leads to slower waves confined inside the cavity. The perturbations can be of different nature and these can even be combined. They include shifting the neighbouring holes, modifying their shape, modifying their radius. We solely focused on changes in the position and radius of the first neighbouring holes at the cavity edges (see Figure (2.7)). Figures (2.8) and (2.9) display the dependance of the cavity quality factors as a function of the outward shift of the holes and reduction of their radius for GaAs-based and GaN-based cavities respectively, starting from the geometric parameters given in the previous table. One can observe a bell-shape dependence of the quality factor as a function of the hole displacement as theoretically expected [START_REF] Sauvan | Slow-wave effect and modeprofile matching in photonic crystal microcavities[END_REF] . This bell shape results from two concomitant effects. First the hole tuning induces a progressive increase in the reflectivity at the cavity edges, followed by a rapid drop. This effect is related to the mode-profile mismatch between the propagative Bloch mode of the photonic crystal cavity and the evanescent Bloch mode of the photonic crystal mirror. Second, the highly dispersive nature of the Bloch mode and the slightly longer penetration depth into the mirrors due to the broken periodicity translate into an increase and then a slow decrease of the group index for increasing holes tuning. These effects give rise to an optimum in the changes of the holes displacement. Same occurs for the hole radius modification. In our work, best parameters for the modified L3 cavities are:

Optical coupling with an optical waveguide

One advantage of using photonic crystal resonators etched in a slab is that they can be easily integrated with waveguide structures to access the optical cavity. The waveguide can either be an integrated photonic crystal waveguide etched in the slab, or a ridge waveguide placed underneath the slab in a 3D-integration scheme. We relied on this second approach, using a 500µm-long Silicon-On-Insulator (SOI) waveguide placed at a distance d below the slab. Efficient light coupling to the cavity then requires an efficient light coupling between the waveguide and the free space, as well as an efficient light coupling between the cavity and the waveguide. The efficiency of the coupling between the waveguide and the air is achieved by relying on a taper a both end of the waveguide. In our work based on silicon waveguides with a thickness of 200nm and width of about few hundreds nm, this taper consists of a smooth increase of the waveguide width up to 10µm, enhancing the coupling efficiency up to about 30.9% for TE-polarization.

The efficiency of light transfer between the cavity and the waveguide strongly depends on the overlap between the evanescent cavity field and the waveguide mode, which can be optimized by adjusting the position and orientation of the waveguides with respect to the cavity. A schematic of the geometry is given on Figure (2.10a). The L3 cavity and the SOI waveguide are parallel with each other in xy-plane. The evanescent coupling involves a pair of TE-like modes (cavity mode in yellow and waveguide mode in blue) that shall feature similar transverse modes profile in electric field polarization and same momentum component (so called phase matching condition [START_REF] Barclay | Design of photonic crystal waveguides for evanescent coupling to optical fiber tapers and integration with high-q cavities[END_REF] ) to ensure a good overlap. Two geometrical parameters play here a crucial role: the distance d between the waveguide and the slab and the waveguide dimension w W G . The efficiency of the energy transfer between the cavity and the waveguide can then be quantified as follows: η = Rate of the loss from cavity to waveguide Total cavity loss = κ c κ 0 + κ c (2.9) where κ 0 is the intrinsic loss rate of the uncoupled cavity, and κ c is the loss rate induced by the coupling between the waveguide and the cavity. The total decay rate then writes (κ 0 + κ c ). Knowing that Q 0 = ω 0 /κ 0 and denoting Q c = ω 0 /κ c , the efficiency η then writes:

η = Q 0 Q 0 + Q c = Q Q c (2.10)
We notice here that an efficient energy transfer between the cavity and the waveguide degrades the quality factor of the coupled cavity. The total Q of a coupled cavity indeed relates to the uncoupled cavity loss rate Q -1 0 and to the loss rate into the waveguide Q -1 0 according to:

1 Q = 1 Q c + 1 Q 0 (2.11)
A trade-off between a high coupling efficiency and a high Q, has then to be found. The simulation was done by V. Tsvirkun [START_REF] Viktor Tsvirkun | Optomechanics in hybrid fully-integrated two-dimensional photonic crystal resonators[END_REF] The study of the optical cavity Q and coupling efficiency η as function of different cavity-waveguide distances d and waveguide widths w W G was performed by FDTD simulations. The overlap of electromagnetic fields enhances for decreasing values of d: as d reduces from 1µm to 100nm, the coupling efficiency η increases from 20% to nearly 100%. Yet Q drops rapidly from 10 5 to 10 2 . A trade-off here consists in choosing d smaller than 200nm to insure a good coupling efficiency η (around 98%) while keeping moderate optical losses with Q factors around 10 2 to 10 3 . The waveguide width also plays an important role, mainly to insure the phase matching condition, with an optimum achieved for w W G = 350nm. The optimum parameters are: w W G 350nm at 200nm air gap. At this condition, we expect Qs as high as 10 2 to 10 3 with coupling efficiencies η higher than 98%.

Our numerical simulations provided the optimized geometrical parameters of the coupled waveguide-defect cavities. For GaAs-based cavities, the quality factor can still reach high values. Yet, due to the lower refractive index of GaN, the achieved optical losses in GaN-based cavities still remain high, preventing any efficient optomechanical coupling. One strategy here is to rely on another cavity design based on a 1D-photonic crystal cavity etched in a suspended nanobeam. The investigation of such geometry is the focus of the next section.

1D photonic crystal: design and optimization

In recent years, 1D-photonic crystal cavities etched in a suspended nanobeam have gained much attention due the ultra-high confinement they provide in space and time, with a competitive benefit for various fields of research including: nanophotonics [START_REF] Crosnier | Hybrid indium phosphide-on-silicon nanolaser diode[END_REF] , cavity quantum electrodynamics [START_REF] Tiecke | Nanophotonic quantum phase switch with a single atom[END_REF], [START_REF] Kim | Photonic crystal cavities from hexagonal boron nitride[END_REF] and optomechanics [START_REF] Eichenfield | Optomechanical crystals[END_REF], [START_REF] Leijssen | Nonlinear cavity optomechanics with nanomechanical thermal fluctuations[END_REF], [START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF], [START_REF] Navarro-Urrios | Nonlinear dynamics and chaos in an optomechanical beam[END_REF], [START_REF] Bayn | Triangular nanobeam photonic cavities in single-crystal diamond[END_REF], [START_REF] Pernice | High-q aluminum nitride photonic crystal nanobeam cavities[END_REF], [START_REF] Gomis-Bresco | A one-dimensional optomechanical crystal with a complete phononic band gap[END_REF], [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] . These in general consist of a line of holes etched along a suspended waveguide. Again, by introducing a gradual perturbation in the photonic crystal geometry, one can implement nanoscale cavities with very small mode volume and ultra-low losses in such a platform. These cavities are usually referred to as "nanobeam cavities". In order to engineer such cavities, the same tools as the one used in 2D-photonic crystals are used: changing locally the shape of the holes [START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF] , modifying locally the unit lattice cell sizes, increasing locally the nanobeam width [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] , or removing one or few holes to form a defect [START_REF] Pernice | High-q aluminum nitride photonic crystal nanobeam cavities[END_REF] . One example of implementation is a silicon-based nanobeam cavity based on a line of rectangular holes that featured an optical Q 0 around 3.8×10 4 at 1500 nm [START_REF] Eichenfield | Optomechanical crystals[END_REF] (see Figure (2.12a)). Such cavities have also been processed in III-V material, with AlN-based cavities featuring an optical Q 0 up to 1.2 × 10 5 [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] . This nanobeam cavity consisted of a line of tapered elliptical holes with a slight increase of the nanobeam width in its center (see Figure (2.12b)).

Figure 2.12 -Examples of nanobeam cavities (a) etched in silicon [START_REF] Eichenfield | Optomechanical crystals[END_REF] and (b) etched in AlN [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] . In this work, the design of the cavities builds upon the work done by O. Painter et al [START_REF] Eichenfield | Optomechanical crystals[END_REF] and F. Raineri et al [START_REF] Crosnier | Hybrid indium phosphide-on-silicon nanolaser diode[END_REF] . The nanobeam cavity is designed by tuning gently the lattice constant a in the line of holes with rectangular or elliptical cross-sections. We will describe here in more details the nanobeam design and the optimization of the optical confinement to reach high Q 0 in the next section.

Light confinement in nanobeam waveguide

The nanobeam cavities are etched in a suspended waveguide insuring light confinement and propagation along the beam. As such, a prerequisite is to optimize the electromagnetic confinement inside the dielectric material, without the presence of any holes. This confinement is directly related to the waveguide geometry but also to the relative permittivity of the dielectric material. One parameter enabling to quantify the confinement is the effective refractive index n ef f of the mode sustained in the waveguide that equals the ratio of the propagation constant in the waveguide to the free space propagation constant. Such effective refractive index for the fundamental TE mode is plotted in Figure (2.13) as a function of the waveguide width w W G . The thickness t has been chosen to insure a single mode operation. In order to achieve efficient guiding of the modes, such refractive index has to be higher than the one of the surrounding (air in our case) and to approach its maximum value. We observe that the effective refractive index grows with the waveguide width. Yet w W G shall not be taken to high in order to prevent a multimode operation. Best geometrical parameters are consequently:

Cavity optical losses reduction by tuning the lattice constant

The cavity is then built in the nanobeam by piercing a 1D-photonic crystal made of holes into which the lattice constant is subtly varied from a 0 to a n (see Figure (2.14)). The choice of these two extreme values is governed by the bandgap frequency of the 1D-photonic crystal: the TE-like mode sustained in the beam at 1.55µm, shall lie in the bandgap in the mirror region, and should correspond to a valence band edge mode at the center of the beam where the cavity is created. Such valence band edge mode corresponds to propagating modes in the photonic crystal for which the field is mainly confined in the material between the holes. These two situations (within the bandgap or at the edge of the valence band) are displayed on the band diagrams obtained by FDTD simulations and shown on Figure (2.14). This leads to the following lattice constant values:

Material of the slab a 0 a n GaAs 315nm 400nm GaN/AlN 500nm 634nm

The lattice constant is then gradually varied to build the cavity at the center of the beam. Before describing the procedure to retrieve the relevant change in the lattice constant, let's first consider a nanobeam pierced by a line of holes with constant period a. For frequencies within the bandgap of the photonic crystal, no propagating modes exist: If light with a frequency within the bandgap is injected into the beam, the associated electric field will decay exponentially as e -qx with x referring to the propagation distance along the beam. This decay rate q strongly varies with the lattice constant a. In order to build a cavity with a Gaussian profile, the underlying idea is to tune the value of q by tuning the value of the period a along the beam, so that to obtain a decay of the mode envelope in the form of e -Bx 2 . This imposes to induce a linear variation of q with the distance x from the cavity center, i.e. to have q = B|x| for a cavity centered in x = 0. In order to precise the associated photonic crystal geometry, one needs first to retrieve the dependence of q on the value of a and then to tune properly q(x) along the 1D-photonic crystal to obtain a linear dependence via an adequate change in the lattice constant a(x). Figure (2.15a) displays the dependence of the decay rate q as a function of the frequency f , for different lattice constants. The decay rate features a bell shape whose maximum redshifts as expected for increasing values of a. For a given value of the period a, the dispersion relation q(f ) inside the gap can be approximated near the dielectric band-edge mode frequency by a parabolic function:

f = f cut a + A a q 2 (x) (2.12)
Figure 2.15 -(a) decay factor q(f ) as a function of the frequency for different lattice constants a. In the figure, we called the minimum frequency when q = 0 as f cut , the frequency when q at its maximum as f max ; (b) Variation of the cut-off frequency f cut and of the resonance frequency f max of q(f ) as a function of the inverse of the lattice constant a -1 . The fitting equation used is equation(2.14), with a K value of 6.935 × 10 4 (nm) for f cut ; (c) Proportionality coefficient A -a of the fitting equation(2.13) as a function of lattice constant a. The fitting parameter α is 0.001607.

where A a is a fitting parameter linearly dependent on the period a:

A a = A 0 + α(a(x) -a 0 ) (2.13)
and where the cut-off frequency f cut of the dielectric band-edge mode can be expressed, in the period range of interest, as:

f cut a = K 1 a - 1 a 0 + f cut a 0 (2.14)
K is a fitting parameter and f cut a 0 is a constant corresponding to the cut-off frequency for a lattice parameter equal to a 0 . This cut-off frequency is set at our target frequency for the cavity mode (193THz, that is a wavelength of 1.55µm). Using previous equations, for f cut a = f cut a 0 the decay rate q can then be rewritten as:

q(x) = Bx = K 1 a(x) -1 a 0 A a (2.15)
This corresponds to a quadratic equation of a(x) whose solution writes:

a(x) = -(A 0 -αa 0 )B 2 x 2 -K a 0 - (A 0 -αa 0 )B 2 x 2 -K a 0 2 -4αB 2 x 2 K 2αB 2 x 2 (2.16)
Knowing the values of α, K and B, this equation gives the a(x) function specifying the change in the lattice period to implement. α and K values can be extracted from previous fits. In order to determine the lattice constant for all the holes, we put a 0 for the first positioning holes at x = 0. Then the holes positions a i are fixed by discretizing the apodisation function a(x). Figure (2.16) shows an example for a quadratic function a(x) with the discretized approximated holes position.

In order to fix the B value, one has to remind that the B factor relates the optical confinement of the mode in space. It is linked to the spatial full-width-half-maximum F W HM quantifying the mode spatial extension as follows:

F W HM = 2
In(2) B (2.17)

This B factor not only impacts the effective mode volume but also the quality factor. This dependence can be seen on Figure (2.17). These curves reveal an optimum value for F W HM in order to maximize the quality factor. Same effects as in the case of the 2D-photonic crystal cavity come here into play: change in the amplitude of the reflectivity of the mirror and change in the group velocity of the mode, when spreading the mode along the 1D-photonic crystal. The associated quality factor of the fundamental mode can reach values up to about 2 × 10 7 and 3.8 × 10 4 for respectively GaAs-based and GaN-based cavities. Such geometries thus provide quality factors at least two orders of magnitude higher compared to modified L3 cavities.

Cavity optical losses reduction by tuning the holes shape

Though the achieved quality factors are high, there is still room for improvement, by slightly change the holes geometry. We focused in particular on the holes cross-section along two lines: rectangular cross-sections and elliptical cross-sections. Let's first consider holes with rectangular cross-section whose dimensions along the beam (x axis) will be kept constant and denoted H x and whose dimensions along the axis perpendicular to the beam will be varied and denoted H yn (see Figure (2.18)). We will consider either a Gaussian or a linear variation of the H yn values. H yn is set close to its upward limit fixed by the beam width (that is 430nm for GaAs-based cavities) and H y0 is chosen so that to achieve a cavity resonance around 1.55µm (that is around 200nm for GaAs-based cavities). While the cavity quality factor tends to keep rather constant in the case of a linear change of H yn whatever the value of H y0 , it drops rapidly for increasing H y0 in the case of a Gaussian tuning of the H yn values. The maximum achieved quality factor is however higher for a Gaussian tuning of H yn . In the case of a linear tuning, the maximum Q 0 factor around 1.41 × 10 7 , is even lower compared to holes with constant shape along the beam. As a result, in the case of a GaAs-based cavity, the geometry that optimizes the quality factor is the following:

Material Holes variations H x H y0 H yn λ 0 Q 0 F W

3.0µm

The Q 0 values of elliptical nanobeams are slightly lower compared to rectangular nanobeams for both GaAs and GaN material, though in the same order of magnitude reaching values up to 1.8 × 10 8 and 2.0 × 10 4 . As such, we will consider both elliptical and rectangular nanobeams as possible candidates for investigating the optomechanical coupling, since they shall features different mechanical properties due to the holes pattern difference.

In the following, regarding the outputs of our simulations, the geometries that will be implemented will involve 1D-photonic crystal with a tuning of the holes lattice inducing a Gaussian profile of the fundamental edge-mode. The holes cross-section will be either circular, rectangular or elliptical. In the case of the two latters, a Gaussian tuning of the holes dimensions will be applied. In order to route the optical field in and out of the cavity, again one can rely on integrated waveguides coupled to the cavity. Various schemes can be implemented (see Figure (2.19)). A first one involves 3D-integration with the waveguide placed underneath the nanobeam at a distance d in a similar manner to the schemes studied for the L3 cavity. A second one, here investigated, relies on neighboring nanobeams separated in the plane by a distance d, one hosting the cavity and the other acting as a waveguide funneling light (see Figure (2.19b)). This waveguide is usually slightly bent to approach the cavity so that it lies in its evanescent field. Again, the coupling to the waveguide impacts the optical properties of the cavity, in particular its quality factor. In the same manner as for the previously investigated coupled L3 cavities, a trade-off has to be found between the efficiency of light transfer and the optical losses. The variations of the coupling efficiency and of the quality factor as a function of the waveguide width w W G and as a function of the distance d, are shown on Figure (2.20). As expected, the quality factor evolves inversely compared to the coupling efficiency. It increases with d and with w W G . In order to preserve the quality factor while insuring a good coupling efficiency, the chosen parameters for the GaAs-based cavity will be set as follows: d = 200nm and w W G = 700, corresponding to a quality factor of 7 × 10 5 and a coupling efficiency of 83%. Previous simulations considered that the nanobeam cavity and the waveguide lie in the same horizontal plane. In order to estimate the impact of any vertical misalignment, we extracted a 2D map of the coupling efficiency and quality factors for different waveguide-cavity in-plane distance d h and different vertical misalignment d v (see Figure (2.22)). We observe that the variations of the quality factor and of the coupling efficiency is much more impacted by d h than d v .

Optical coupling with an optical waveguide

The analysis and optimization of the optical properties of the photonic crystal slab based cavities as described in this section, was a prerequisite to evaluate the performances of such platform for implementing optomechanically-driven effects. The investigation of the interaction between light and mechanical motion will be the subject of the next section. Main goal will be to evaluate the amplitude of the optomechanical coupling strength and then optimize the geometry of the platform to achieve enhanced optomechanical interactions.

Cavity optomechanics and optomechanical coupling

In this section, we will first recall some general notions related to cavity optomechanics, in particular the definition of the vacuum optomechanical coupling strength g 0 that quantifies the amplitude of the light-mechanical interaction. In a second step, we will identify the mechanical modes sustained in a L3 and a nanobeam cavity. We will then estimate the g 0 amplitude achieved in these platforms.

Cavity optomechanics: general notions

Vacuum optomechanical coupling strength

The field of optomechanics studies the interaction between electromagnetic and mechanical degrees of freedom. The idea that light can exert a force on a moving material object, was first postulated by Kepler in 1619 who believed that comets tails were caused by the outward pressure of sunlight. Yet, the foundations of optomechanics were laid in 1967 by Vladimir Braginsky and Anatolii Manukin, who demonstrated that the momentum imparted by the reflection of photons on a movable mirror could alter that mirror's mechanical properties. This interaction between light and a moving mirror can be enhanced when both electromagnetic and mechanical degrees of freedom are normal modes of resonators. A canonical cavity-enhanced optomechanical system is a Fabry-Pérot cavity in which one mirror is mounted on a spring, thereby constituting a mechanical element (see Figure (2.23)). Vladimir Braginsky and Anatolii Manukin showed in particular that in that context, the motion of the moving mirror could be damped or amplified depending on the frequency of the light inserted in the cavity.

Let's describe in more details the response of such linear cavity optomechanical system. The cavity is characterized by its optical resonance (at a frequency ω 0 and with a decay rate κ) and by its mechanical resonance (at a frequency Ω m and with a decay rate Γ m ). The mirror is constrained to move along the cavity axis denoted x. Let's denote x(t) the time-varying displacement of the mirror from its equilibrium po-Figure 2.23 -Schematics of an optomechanical system based on a Fabry-Pérot cavity. One end mirror of the cavity is fixed and the other end mirror is connected to a spring. Light is injecting from the right side of the cavity. Due to the radiation pressure force, the mechanical oscillator will be put in motion consequently modifying the cavity property.

sition when the intracavity field is not excited. The uncoupled optical and mechanical modes can be represented by two harmonic oscillators:

Ĥ0 = ω 0 â † â + Ω m b † b (2.18)
Operators â † and â are the creation and annihilation operators of the cavity photon field respectively, while b † and b denote the motional creation and annihilation operators. The displacement operator x then writes x = x ZP F ( b † + b) with:

x ZP F = 0|x 2 |0 = 2m ef f Ω m (2.19)
where x ZP F denotes the zero-point fluctuation amplitude of the mechanical oscillator, i.e. the spread of the coordinate in the ground-state and |0 > denotes the mechanical vacuum state, while the effective mass m ef f of the displacement amplitude relates to the potential energy of the mass-spring system U (t) = m ef f Ω 2 m < x 2 (t) > /2. If we denote u( r, t) the time-dependent displacement vector field, this effective mass is then defined as:

m ef f = ρ( r) d r   | u| max(| u|)   2 (2.20)
with ρ( r) the mass density of the material and r the position vector. We can now introduce the coupling between the optical cavity and the mechanical oscillator by relating the optical resonance condition to the mechanical displacement. In the canonical case of a Fabry-Pérot cavity with a movable end mirror, the cavity resonance frequency is modulated by the mechanical amplitude as follows:

ω 0 (x) = ω 0 + ∂ω 0 ∂ x x + ... (2.21)
The interaction part of the Hamiltonian can then be written as:

Ĥint = ∂ω 0 ∂ x x ZP F â † â( b † + b) = -g 0 â † â( b † + b) (2.22)
where g 0 = -(∂ω 0 /∂ x)x ZP F is the vacuum optomechanical coupling strength, expressed as a frequency. It quantifies the interaction between a single phonon and a single photon: it corresponds to the optical frequency shift induced by the zeropoint fluctuation amplitude of the mechanical oscillator. This figure of merit is very useful for characterizing the optomechanical coupling strength since it applies to any optomechanical platform independently of its geometry and it does not depend on the amplitude of the possibly complex displacement of the mechanical oscillator. In the case of optomechanical photonic crystals, mainly two mechanical effects can contribute to the optical frequency shift and thus to g 0 . These effects are the moving interface effect and the photoelastic effect. The vacuum optomechanical coupling strength thus develops as:

g 0 = (g M I + g P E )x ZP F = - dω 0 dx M I + dω 0 dx P E x ZP F (2.23)
where g M I is the frequency shift per displacement induced by the moving interface effect and g P E is the frequency shift per displacement induced by the photoelastic effect. These two contributions can be calculated in a perturbation theory framework [START_REF] Eichenfield | Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals[END_REF] , whose results will be summarized in the following. When the optical resonator is deformed due to mechanical motion, it affects the dielectric tensor at the interfaces between the different materials. In the case of a photonic crystal surrounded by air and pierced by holes, this translates into an abrupt change of the dielectric tensor ¯ ( r) whose profile gets shifted in space by the deformation of the material. If we denote u( r, t) the time-dependent displacement vector field and E( r, t) the electric field profile, the shift in frequency induced by the deformation then writes:

Moving interface effect

g M I = - ω 0 2 ( u. n)(∆¯ |E | 2 -∆¯ -1 |D ⊥ | 2 ) max(| u|) ¯ | E| 2 d r (2.24)
where n is the unit normal vector on the surface of the unperturbed cavity and D = ¯ E. ∆¯ = ¯ mat -¯ air with ¯ mat and ¯ air being respectively the dielectric constant of material forming the photonic crystal and the dielectric constant of the surrounding medium. To calculate such frequency shift g M I , one strategy that we applied consists in simulating the optical fields when applying a know deformation which is large enough to be detectable numerically but kept small enough to prevent the contribution of any higher order dispersion to the frequency shift.

Photoelastic effect

The photoelastic contribution to the optomechanical coupling arises from local changes in the refractive index due to the strain induced in the structure by the mechanical motion. This contribution strongly depends on the photoelastic properties of the material. The induced frequency shift here writes:

g P E = - ω 0 2 E.δ¯ Ed r max(| u|) ¯ | E| 2 d r (2.25)
where δ¯ is the dielectric perturbation given by:

δ¯ = -¯ p. s 0 ¯ (2.26)
where p is the photoeleastic tensor and s the strain tensor represented by a unitless 3 × 3 matrix with components

s ij = (∂ i u j + ∂ j u i )/2.
The photoelastic tensor is a fourth-rank tensor with components p ijkl and can thus be written as a 6 × 6 matrix.

The relations between the different coefficients involved in the photoelastic tensor depend strongly on the symmetry of the crystal. Considering these relations and the symmetries in the atomic lattices (see details in Appendix A), on can end up with the following equations :

• for a cubic lattice (such as GaN or AlN)

E.δ¯ E = -0 n 4 2Re {E * x E y } p 44 ε xy + 2Re E * y E z p 44 ε yz + 2Re {E * z E x } p 44 ε zx + |E x | 2 (p 11 E xx + p 12 (ε yy + ε zz )) + |E y | 2 (p 11 ε yy + p 12 (ε xx + ε zz )) + |E z | 2 (p 11 ε zz + p 12 (ε xx + ε yy ))]
(2.27)

• for an hexagonal lattice (such as GaAs, GaP or InP)

E.δ¯ E = -0 n 4 Re {E * x E y } (p 11 -p 12 ) ε xy + 2Re E * y E z p 44 ε yz + 2Re {E * z E x } p 44 ε zx + |E x | 2 (p 11 ε xx + p 12 ε yy + p 13 ε zz ) + |E y | 2 (p 12 ε xx + p 11 ε yy + p 13 ε zz ) + |E z | 2 (p 33 ε zz + p 13 (ε xx + ε yy ))] (2.28)
with n being the refractive index of the material. The table below gathers the values of the photoelastic coefficients for the materials here investigated.

In order to compute g 0 , g M I and g P E , one needs to retrieve the mechanical displacement field inside a L3 or nanobeam photonic crystal cavities. The computation of the mechanical displacement field is done by numerical simulation with Comsol Multiphysics software, while the optical electromagnetic field profile is retrieved by FDTD simulations as done in previous sections. The mechanical displacement field is recovered by first extracting the eigenfrequency solutions and then computing the mechanical modes. The investigation of the mechanical response of the photonic crystal cavities is described in the following.

Mechanical modes and optomechanical coupling strength in L3 cavities

Photonic crsytal slab L3 cavities can sustain two families of mechanical modes [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] . A first family, whose mechanical frequency lies in the MHz range, corresponds to drum modes of the whole suspended slab. A second family, with mechanical frequencies in the GHz range, corresponds to localized vibrations within the photonic crystal cavity.

In this study, we will only focus on the second family, due to its high frequency of oscillation and its higher optomechanical coupling strength. The enhanced interaction here between the optical and mechanical degrees of freedom, results from the strong confinement of both the mechanical and optical modes, combined to their high overlap.

We considered two cavity geometries: the standard L3 cavity and the modified L3 cavity with tuned neighboring holes at the cavity edge. Figure (2.25) displays the first mechanical mode profiles of the localized modes. The M1 mode is the fundamental cavity mode. Its mechanical frequency is about 900 MHz. Higher-order modes also exist, at higher frequencies from 1 up to 4 GHz. Importantly, these localized mechanical modes coincide spatially fully with the optical defect cavity mode. Therefore, the photonic crystal not only offers strong optical and mechanical confinement, but simultaneously high phonon-photon modes overlap. This localization occurs even in the absence of a phononic band gap. This co-localization of the optical and mechanical mode within the defect cavity, and high overlap of the mechanical displacement with the distribution of electromagnetic energy, promise high optomechanical coupling. This enhancement of the optomechanical coupling will strongly depend on the symmetries of the mechanical mode profile with respect to the optical mode profile. This dependence is confirmed on Figure (2.26) which displays the g 0M I /2π = g M I × x ZP F /2π, g 0P E /2π = g P E × x ZP F /2π and g 0 /2π values for the different modes. One observes that up to the fifth-order modes, the coupling strength is rather modest with equivalent contributions of the moving interface effect and the photorelastic effect. Three modes however feature a strongly dominant contribution of the photoelastic effect, resulting in very high optomechanical coupling strength up to 580 kHz. This strong contribution comes from the symmetry of this mode, or more precisely the profile of the associated strain profile with respect to the optical mode. All these results are gathered in the following table and complemented with the associated effective mass and zero-point fluctuations:

Let's note that the effective mass keeps the same order of magnitude for all the modes, in the pg range. Same occurs for the zero-point fluctuations that lie in the few fm range.

Yet, as shown in the previous section, standard L3 cavities provide a modest confinement of light, which can be enhanced by implementing modified L3 cavities with tuned neighboring holes. Tuning and shrinking the holes at the edge of the optical cavity will slightly modify the mechanical response of the platform. These changes are shown on Figure (2.27) for the M6 mode that features the highest optomechanical coupling strength in standard cavities. We notice that the resonance frequency, the effective mass and zero-point fluctuations keep rather constant with the change in geometry. The mechanical resonance wavelength and the mechanical effective mass slightly increase for higher displacement and smaller radius of the holes due to the increased volume of the cavity. Conversely, more important changes are observed for the coupling strengths. In particular, we observe a strong decrease of g 0P E and g 0M I with increasing displacement of the holes and smaller reduction of the holes radius. It is also observed for the higher M8 and M9 modes (see Figure (2.28). This decrease is likely to arise from the change in the optical mode profile that translates into a degraded overlap with the mechanical mode when modifying the cavity. The simulation have been carried for GaAs-based and GaN-based cavities. Results are gathered in the Table (2.3). Again, g 0 values are higher for the M6 mode for both materials, yet with a decrease by one order of magnitude for GaN-based cavities, in particular due to the low optical confinement they provide.

Mechanical modes and optomechanical coupling strength in nanobeam cavities

Compared to the L3 cavities, nanobeam cavities provide generally a higher optical confinement even for low index materials such as GaN. The high Q 0 of the optical modes promises higher optomechanical coupling strengths. Similarly to the L3 cavity, the mechanical modes of a nanobeam cavity can also be distinguished into two families: modes associated to the vibration of the entire nanobeam (occurring at low frequencies from few kHz till hundreds of MHz and that correspondsto Group A modes on Figure (2.29)) and modes associated to the vibration within the nanobeam cavity (with resonant frequencies higher than the GHz). These latter can again be differentiated in two categories: modes related to deformations on the nanobeam edges usually referred to as breathing modes [START_REF] Eichenfield | Optomechanical crystals[END_REF] (group B) and modes associated to a deformation between the holes usually referred to as pitch modes (group C). We will mainly focus on Group B modes in this work due to their high optomechanical coupling strength as will be shown in the following. Figure (2.29) displays the optomechanical coupling g 0 /2π of the different modes as a function of the frequency. The three groups can be here easily distinguished: Group A modes (yellow part) feature an alternation of the g 0 sign and an exponential decay with the frequency. Even if the coupling strength is rather high for these modes (with g 0 /2π as high as few kHz), their frequency is low (in the kHz range). In the GHz frequency domain, the pitch modes feature the highest resonance frequencies, with rather high g 0 /2π values up to 650kHz/nm. The group B modes lie at a slightly lower frequency around 2GHz. These are the ones that provide the highest optomechanical coupling strength. The displacement associated to these breathing modes can be either symmetric or anti-symmetric with respect to the in-plane y axis perpendicular to the nanobeam. The displacement profile of the symmetric modes (denoted Sn) and anti-symmetric modes (denoted ASn) are shown on Figure (2.30).

The mechanical characteristics of these symmetric and anti-symmetric breathing modes are gathered in the following Table (2.4). It appears that the vacuum optomechanical coupling rate is higher for the symmetric modes, probably due to their stronger confinement at the beam center, resulting in a higher overlap with respect to the optical cavity field.

It could be beneficial to extend the investigation of the geometry modification, in view of enhancing not the Q 0 factor but the optomechanical coupling strength g 0 . We considered here a photonic crystal made of squared holes with constant dimension along the nanobeam H x and constant period a. We then considered a parabolic distribution of the hole dimensions along y H ym , increasing from H y0 at the beam center to H yn on the sides in the mirror regions.

In the first series of simulation on a GaAs-based nanobeam cavity, the value of H x is varied. A parabolic dependence of H ym is applied, with H y0 = 200nm and H y0 = 430nm; a is fixed to 315 nm and the optical mode FWHM is about 3µm. The resonance frequency of the mechanical mode naturally decreases with increasing holes width. The g 0 value is also impacted by the change in holes size yet except for H x = 170nm. The changes in g 0 /2π keeps rather modest, with all values in the few hundreds kHz range. Interestingly, the lowest value of g 0 is achieved for H x = 170nm, which is the optimal dimension for enhancing Q 0 . This illustrates the fact that there is no bijection between a high Q 0 and a high g 0 since the relation between the two parameters is not straightforward. In order to simultaneously obtain low optical losses and high optomechanical interactions, a trade-off consists in choosing H x = 180nm. The shape of the holes can also be changed from a rectangular to an elliptical crosssection. In Figure (2.33), the vacuum optomechanical coupling rate of the different mechanical modes (associated to groups A, B and C) are plotted as a function of the modes frequencies, in the case of a photonic crystal with rectangular and elliptical cross-sections. The dimensions of the holes are the same, H x and H ym denoting either the side lengths of the rectangle or the axis lengths of the ellipse. Since Group A modes are related to the entire nanobeam vibration, the Group A modes do not display much difference depending on the holes geometry. This is much different concerning group B and C modes. First their frequency range differs: Group B modes for rectangular and elliptical holes resonate around 1.8 and 2.1GHz respectively. The optomechanical coupling strength is also higher in cavities formed by elliptical holes, with g 0 /2π values up to 1357kHz and 1378kHz for the S1 and S3 modes, while it 'only' reaches -729kHz and -807kHz in cavities with rectangular holes. In cavities with rectangular holes, the moving interface contribution g M I to the optomechanical interactions is much higher than in cavities with elliptical holes (see Table (2.5)), due to larger boundaries area. However the photoelastic effect contribution g P E is much higher in the case of cavities with elliptical holes, due to the smooth edges of the elliptical shape resulting in a more homogeneous elastic strain inside the material forming the cavity. This study, carried on GaAs-based cavity, can be extended to other materials, in particular GaN-based cavities. Again, 3 groups of mode are identified. The modes feature the same displacement profiles. Again, same trends are observed in terms of optomechanical coupling strengths, with higher g 0 values for Group B breathing modes whose frequencies lie around 2.7 GHz (see Figure (2.33)). Especially, the photoelastic coefficients in GaN materials are much higher than in GaAs, enhancing the relative contribution of the photoelastic effect g P E in the optomechanical interaction. Similarly to L3 cavities, the optomechanics coupling strength is still limited by the low optical confinement achieved in GaN-based cavities. Yet, the nanobeam cavity geometry provides a much higher optical confinement with Q 0 factors up to 10 5 and much higher vacuum optomechanical coupling rates reaching values up to 160.56kHz, more than three times higher than the values achieved in modified L3 cavities.

Summary and conclusion on the design of the optomechanical photonic crystal based platform

This chapter was dedicated to the design of photonic crystal based platform for optomechanics. Optomechanical interaction gets enhanced for increasing mechanical and optical modes confinement, concomitantly with an increased overlap between the The optimization of the platform required to compute both the optical and mechanical response of the cavities. The optical properties were simulated by FDTD methods, providing the optical resonance of the modes, their optical losses and spatial profile. In both geometries, the optical losses were tailored thanks to an engineering of the holes surrounding or forming the cavities, in terms of holes positioning, dimensions and shapes. Comparison between both geometries revealed the benefits of the nanobeam cavity, featuring higher optical quality factors in particular in GaN-based platforms. When coupling the cavities to integrated waveguides enabling to efficiently funnel the light in and out of the cavity, a trade-off has yet to be found between high coupling efficiencies and high optical quality factors. All these simulations provided optimum values for the cavities geometries given in the related section. The mechanical modes were also simulated numerically by use of Comsol Multiphysics software. Both cavities sustain different families of modes. We focused on the ones lying in the GHz range and featuring enhanced optomechanical interaction. These are the localized modes in the L3 cavity and the breathing modes in the nanobeam cavity. Both their frequency spatial profiles and mechanical quality factors were quantified. Again, the impact of a perturbation in the cavity design on the mechanical response of these modes was analyzed.

The optomechanical interactions in such platform involve two mechanisms: the moving interface effect and the photoeleastic effect. Both were quantified and their relative contribution to the vacuum coupling strength was determined. These simulations allowed to identify the modes featuring an enhanced optomechanical coupling. In L3 cavities, the highest vacuum optomechanical coupling strength g 0 /2π is achieved for mode M6 with a value of -579kHz at a mechanical frequency of 2.2GHz in GaAs based cavities, and a value of -50kHz for a mechanical frequency of 2.7GHz in GaN based cavities. These values are enhanced in nanobeam cavities, especially for the S1 and S3 breathing modes featuring larger optical and mechanical field overlap. The values of g 0 /2π reach respectively -729kHz and -807kHz for GaAs based cavities for instance. The table below gathers the optimized values for the different investigated cavity designs and different hosting materials.

Next step will be focused on the processing of such platforms. The development of the fabrication process and its different steps will be described in the next chapter.

Chapter 3

Fabrication of III-V photonic crystal cavities system

In the previous chapter, we introduced two different III-V photonic crystal resonators of interest for our work : two-dimensional L3 resonators and one-dimensional nanobeam resonator. We in particular discussed their optical properties such as the mode volume, the optical quality factor Q o as well as their coupling to integrated waveguides. We also discussed about their mechanical properties and the strength of the coupling between optics and mechanics in such resonators. We identified optimal designs in view of enhancing the optomechanical coupling strength.

In this chapter, we will mainly focus on the processing of these different resonators based on III-V materials, in particular GaN/AlN, GaAs and InGaP. The fabrication process will differ depending on the host material. The materials that will form the suspended membrane, also differ. Each of the involved heterostructures are displayed on For GaAs-and InGaP-based resonators, the material forming the membrane will be the GaAs and InGaP top layers, grown on a sacrificial layer made of AlGaAs and GaAs respectively. This sacrificial layer will be further used to suspend the membrane. These heterostructures were grown by Metalorganic Chemical Vapor Deposition (MOCVD) at C2N by Isabelle Sagnes and Gregoire Beaudoin. The thickness of the GaAs and InGaP membrane is 265nm and 250nm respectively. Both heterostructures are grown on a GaAs substrate.

The GaN-based heterostructures were grown on a Silicon (111) wafer by molecularbeam epitaxy (MBE) at CRHEA (Nice) by Fabrice Semond and Stéphanie Rennesson. Generally, direct growth of GaN on silicon suffers from the strong chemical reactivity of gallium and silicon, whose products introduce some defects and dislocations. These defects and dislocations can accumulate gradually layer by layer, and may introduce high surface roughness or even cracks. The solution proposed by CRHEA [START_REF] Mante | Proposition of a model elucidating the aln-on-si (111) microstructure[END_REF], [START_REF] Gommé | A combined growth process for state-of-the-art gan on silicon[END_REF], [START_REF] Lucci | Universal description of iii-v/si epitaxial growth processes[END_REF], [START_REF] Le Louarn | Aln buffer layer growth for gan epitaxy on (111) si: Al or n first?[END_REF] consists of growing a buffer AlN layer firstly on the silicon substrate, since Aluminum react more hardly with silicon. The final GaN-base heterostructure is thus composed of 200nm of AlN and 160nm of GaN. Such heterostructure-based growth provides a high purity and a good mechanical quality of the GaN membrane. This may lead to some potential fabrication issues that will be discussed later in this chapter. This chapter will be divided into two parts, corresponding to the two photoniccrystal resonators designs, starting with the one dimensional resonators. We will first describe the fabrication steps of one dimensional GaAs/InGaP and GaN nanobeam. In a second part, integration of the photonic crystal resonators on a SOI photonic circuitry will be described. Both cavity designs will be here considered : the nanobeam cavity and the two-dimensional L3 cavity on GaAs/InGaP and GaN substrate. In each part, I will discuss about all the fabrication process flows as well as the fabrication specificities related to each material.

Fabrication of one-dimensional nanobeam photonic crystal resonators

The processing of one-dimensional nanobeam cavity, involves only few fabrication steps including: electron beam lithography, dry etching and underetching of the membrane as shown in Figure (3.4). All these steps will be described in the next section.

Additional developments had to be implemented for GaN/AlN-based resonators and these will be described in a distinct section.

Processing steps

When considering platforms based on one-dimensional nanobeam resonators, the cavity and the corresponding access optical waveguide can be integrated on the same suspended membrane, involving simply a 2D-integration process. The III-V access waveguide is then parallel to the nanobeam cavity and the distance between the cavity center and the waveguide determines the optical coupling efficiency. Figure (3.3) shows the first generation of nanobeam cavity. This system includes two major elements: the nanobeam resonator described earlier and the optical access waveguide. The latter element is composed by 4 sub-parts. The injection waveguide (yellow line) from left to right includes the injection tip, a two dimensional photonic crystal waveguide, a curved waveguide as well as a photonic crystal mirror. In this first generation design, the optical signal was supposed to be injected from the tip by use of a lens fiber or an objective. The whole process flow is shown in Figure (3.4). The nanobeam and waveguide are patterned by electron beam lithography. These patterns are then transferred by a dry etching technique suited to the III-V materials used. The process ends with the under-etching chemical step that will release the platform and make it suspended. In the following parts, I will discuss these steps with more details.

Electron beam lithography

Photonic crystal resonators are artificial periodic structures whose mechanical and optical response are highly sensitive to any geometry deviation. In order to achieve high optical quality, factors the photonic crystal patterns need to be realized with high precision and resolution. In our work, in order to pattern the photonic crystal, we choose the electron beam lithography that can provide very precise resolution with very small spot size, enabling to reproduce the targeted photonic crystal pattern. The electron beam lithography system in our laboratory is provided by a Leica EBPG5000+ beamwritter system. The working field in which the system can write is as large as 520 × 520µm 2 , with a writing resolution as small as 1nm. With such electron beam lithography system, we can achieve very high quality patterning. Figure 3.5 -Left: Schematic of the electron beam propagation when it hits the substrate covered with a resist layer on top [START_REF] Boden | Helium ion beam lithography for sub-10nm pattern definition[END_REF] . Right: Lithography minimum feature size as a function of the beam energy and resist thickness. At low beam energy, the feature size is limited by forward scattering; at high beam energy, the limitation is imposed by the beam spot size.

The principles of electron beam lithography is the interaction between the e-beam and the resist. During electron beam writing, different kind of electrons trajectories are involved. Forward electrons coming directly from the e-beam column interact with the resist and the layers underneath. At each interface, part of direct electrons are reflected and these so-called back reflected electrons also contribute to the interaction between the e-beam and the resist, thus increasing the effective dose (i.e.number of electrons per area) exposing the resist. These reflected electrons spread over a large distance compared to forward electrons. Such effect dramatically reduces the resolution which can be achieved. By increasing the energy of the incoming electrons, back reflected electrons can penetrate deeper in the substrate, which reduces the amount of reflected electrons.

The achieved resolution also depends on the resist layer. Two families of resist can be used: positive and negative resists. When relying on positive resists, the written patterns in the resist will be removed after development in solvent. Conversely, when relying on negative resist, the areas that were not exposed to the e-beam will be removed. In the electron beam lithography process involved in this section, negative resist is used for several reasons. First, the resist used, namely HSQ (standing for Hydrogen Silsesquioxane), is hard enough to be used as a mask for semiconductor dry etching. As a consequence, it simplifies the fabrication flow since no extra mask is required. Second, the surface which has to be exposed, is much smaller, thus reducing the exposition time. In our experiments, HSQ resist is spin-coated on the substrate and baked at 90 • C for 40 minutes to harden the resist. Then the sample is ready for electron beam lithography. The unexposed are then removed by the developer solution (AZ400K),.

Importantly, the main parameter to be optimized during this step is the electron dose used during the exposition. The dose is defined as the amount of direct electrons per exposed area (unit µC/cm 2 ). This value is set in the lithography machine but it doesn't take into account the back reflected electrons. Electron dose calibration is thus necessary in order to achieve :

• homogeneity of holes in 2D photonic crystal

• hole diameters equal to the targeted ones • distance between nanobeam cavity and waveguide equal to the targeted one.

The homogeneity of the patterns is mainly influenced by back-scattering electrons. This requires to rely on proximity effect correction to improve the homogeneity. The proximity effect results from current crowding induced by back scattered electrons. This effect will be more prominent, depending on the exposed area of the pattern : the inner part of one pattern will be exposed by more back scattered electron than the edge. Locally, the effective electrons dose will be higher. As a consequence, in this case of 2D photonic crystal with negative resist, inner hole will be smaller than the one at edge of structure (see Figure (3.6)). In order to eliminate this proximity effect, correction has to be applied to the lithography step. Basically, it reduces the dose in the inner area proportionally to gradually achieve the nominal dose of the outer area. In order to avoid patterns distortion, it is also important to choose a proper electron beam dose. Figure(3.7)-left schematically illustrates the impact of the dose on the holes diameter. When increasing the dose, the radius of the holes is decreased.

Holes deformation can shift the photonic crystal band structures. For increasing doses, the nanobeam cavity features smaller holes and the resonance wavelength increases by 0.8nm per 1nm narrowing in the holes diameter. For two-dimensional photonic crystal waveguides, the propagating wavelength shifts by 1.3nm accordingly. The lattice constant a is conversely not much affected by the dose value since the position of the each hole is determined by the lithography writing field. However, for holes radius, to achieve the targeted values of 110nm, the proper dose should be even larger than 4500µC/cm 2 . If the dose is not optimized, the photonic crystal will largely deviate from the designed geometry. Figure (3.8) shows SEM images of the different part of the nanobeam for three different doses. For very high doses, the nanobeam holes are too small. On the other hand, for a too low dose, the holes become too large, up to merge. The SEM images in the middle correspond to structures drawn with the proper beam dose around 5000µC/cm 2 . The holes forming both the mirror and the cavity feature diameters equal to the targeted values.

ICP-RIE dry etching of the semiconductor

After the e-beam lithography step and resist development, the patterns have to be transferred to the semiconductor made of III-V material. In order to etch III-V material, we use a dry etching technique namely Inductively Coupled Plasma Reactive Ion Etching (ICP-RIE). The dry etching, also called plasma etching, produces an anisotropic reaction with the material. It requires the use of a mask to cover the part we don't want to remove. It is driven by two kinds of processes: either physical or chemical. The physical process, which leads to anisotropic etching, is controlled by the density of involved ions (related to the ICP power) and the acceleration of the ions (related to the RF power). The chemical process which leads to isotropic etching, is mainly controlled by the composition of plasma and pressure. Depending on the material, the gas composition needs to be modified as shown in Table (3.1). The GaAs/InGaP/GaP alloys are etched in plasma composed of oxygen and hydrogen bromide, while the etching of nitrite-based materials makes use of a mixture of oxygen, argon and dichlorine. In our etching process, the etch rate is of the order of few hundreds of nm per minute.

As HSQ resist is a hard resist, that can be used as a mask for transferring the patterns during the III-V etching ICP-RIE process. The ICP-RIE process will slightly etch the resist, yet with a much slower etching rate compared to the III-V materials. It is important here to pay attention to the induced roughness of the side walls in the semiconductor, since it would bring extra light scattering. In order to achieve smooth side walls in the III-V membrane, some of the parameters during ICP-RIE etching, such as the temperature, pressure and etch rate, need to be optimized accordingly.

Underetching of the membrane

The nanobeam and waveguide being transferred to the III-V membrane, the next and last step consists in releasing the membrane. This is achieved by removing the sacrificial layer under the membrane. The suspension height of the membrane is then usually around 1µm. Depending on the material of the sacrificial layer, the underetching can be either achieved by wet etching or by dry etching.

Wet etching is here more favorable since it is an isotropic reaction conversely to normal dry etching technique. However, after wet etching, on needs to take care during the transfer of the devices from the solution to the air, unless the membrane gets destroyed due to capillary adhesion. One strategy here is to use the critical point drying technique, which releases the membrane from liquid phase directly to gas phase by changing temperature and pressure. Depending on the material under the membrane, another technique can be used for isotropic etching. Instead of using a chemical solution, one can use a plasma without any bias. By use of such technique, it is possible to etch the sacrificial Si 3 N 4 layer very smoothly and isotropically. The entire etching process is done in gas atmosphere which can maintain the suspended membrane without damaging it. It combines the functionality of both wet etching and critical point drying in even simpler and more efficient way. Another advantages of dry etching technique is the fast etching rate around 400 to 500 nm/min: the release of the membrane by wet etching technique usually takes more than 2 hours in total, while the plasma-based dry etching technique only takes less than 3 min.

In our group, we use both wet etching and dry etching techniques depending on the material to be etched. For GaAs/InGaP-based membranes, the sacrificial layer is made of AlGaAs which can be removed easily and selectively by chemical etching using the critical point drying technique. For GaN/AlN-based membranes, the underetching is carried out by the dry etching technique.

During the whole process flow involving e-beam lithography, dry etching and underetching, some issues appeared and differed depending on the material forming the membrane. In the next sections, I will point these issues and will detail the strategies we implemented to overcome them.

Processing of GaAs/InGaP nanobeams: underetching of the membrane

The GaAs and InGaP-based devices processing follows exactly the same steps, except the solution used for the underetching since the composition of the sacrificial layers differ. Figure (3.10) shows SEM images of InGaP-based nanobeam formed by either rectangular or elliptical holes. The top images are taken on the surface of the HSQ resist after the electron beam lithography and the bottom images are taken on the semiconductor surface after the ICP-RIE process. The photonic crystal pattern has been well transferred to the InGaP membrane with smooth ICP-RIE etching edges. The next step is to release the nanobeam by under-etching process. During this last step, hundreds of nm of sacrificial layer is chemically removed in order to isolate the nanobeam resonator from the substrate and to avoid optical loss. For InGaP-based membranes, the sacrificial layer is made of GaAs. The chemical solution enabling to selectively etch GaAs is usually a mixture of ammonia (N H 3 • H 2 O) and hydrogen peroxide (H 2 O 2 ) in ratio 1:19 for nearly one hour etching duration. This solution is highly selective and doesn't etch the InGaP membrane and substrate. As shown in Figure (3.11), after removing the sacrificial layer, the InGaP nanobeam system is well suspended with a suspension height around 1µm. The membrane is held by large pads which are also underetched at their edges due to the isotropic etching process.

For GaAs-based membrane, the underetching process is similar, but involves a different chemical solution based on ammonia fluoride (AF) , since the sacrificial layer is made of AlGaAs. However, we faced some issues here during under-etching. The sacrificial layer thickness is much thinner than 1µm. Removing solely the sacrificial layer to release the nanobeam is not sufficient to preserve the optical properties of the resonator. Mechanically, the membrane can also collapse. Therefore, during the ICP-RIE etching step, we etched slightly deeper beyond the sacrificial layer. However, since the GaAs substrate is slightly overetched (over a depth of few hundreds nm) and because of the selectivity of the chemical solution, the imprinted patterns during the ICP-RIE process, still remain on GaAs substrate after underetching. As shown in Figure (3.12), the distance between GaAs nanobeam and GaAs substrate is still too small to avoid any optical leakage in the substrate. Figure 3.12 -SEM images of suspended GaAs-based nanobeams. The entire heterostructures involve two GaAs layers, one on top (the membrane to be suspended), one on the bottom the substrate). In between, the sacrificial is removed but the thickness is not sufficient to suspend adequately the membrane.

In order to improve the GaAs-based nanobeam suspension, a first strategy lies in increasing the sacrificial AlGaAs layer thickness up to 1µm during growth. A second strategy is to bond the GaAs membrane on silicon wafer via an sacrificial interlayer (usually Si 3 N 4 or SiO 2 ). The suspension height is then fixed by the sacrificial layer thickness. Both solutions could be helpful for releasing GaAs-based nanobeam. However, the bonding solution may be more interesting for our experiments. When bonding the III-V membrane on a SOI substrate embedding an integrated optical waveguide, the optical access to the resonator can be located directly below the cavity by such 3D-integration technique. I will detail such technique in the following section.

Processing of GaN-based nanobeams: ICP-RIE and underetching

Two main challenges appeared during the processing of GaN/AlN-based nanobeams. The first one is the smooth and sharp etching of the holes in the semiconductor during the ICP-RIE in order to preserve high optical and mechanical quality factors. The second one springs from the large lattice mismatch between silicon and GaN/AlN. This supposes to control the induced stress which is released during under-etching process. To efficiently etch the GaN/AlN membrane, the gas composition in the ICP-RIE process is Cl 2 /Ar/O 2 mixture. The Cl 2 gas chemically reacts with the gallium and aluminum compounds, while the Ar gas is used to physically etch the Nitrite compound. One important parameter here is the control of the ions acceleration via the applied RF power. When increasing the RF power, the physical etching contribution is favored compared to the chemical etching process. In this case, we expect more vertical etching profiles with sharp edges and a smooth surface. The main drawback is that the mask will also be etched faster. However, this is not an issue if we rely on a thick and hard HSQ mask.

ICP-RIE optimization

Figure (3.13) shows the etch profiles in the GaN/AlN membrane for different RF powers. At a 35W RF power, the photonic crystal holes are tilted with an angle of about 30 • . This is strongly detrimental for our experiments. For instance, If the photonic crystal holes feature tilted sidewalls and a trapezoidal shape, the optical confinement will be decreased due to the coupling between the TE and TM modes in the membrane, leading to extra optical losses. Moreover, the etched sidewalls present some roughness. When we increased the RF power up to 48W, the etching is improved and provides smooth, vertical and sharp etching sidewalls.

Under-etching: different injection geometries

In order to preserve the optical and mechanical properties of the resonators, it is also crucial to suspend the membrane at a distance of about 1µm from the silicon substrate underneath the GaN/AlN membrane. This process is performed by a dry etching technique with a purely chemical etching. The gas composition is a mixture of CF 4 and N 2 , operated at 130 • C with 500W ICP power, an etching rate of 400 to 500nm/min. However, due to high stress induced in the GaN/AlN membrane during epitaxy, the nanobeam experiences a strong deformation once the under-etching is performed. Figure (3.14) shows some SEM images of the processed samples. We observe that the nanobeams embedding the resonator tend to be pushed upward, while the access waveguides bend downward. This induces two major consequences. First, the light injected in the waveguide can couple more efficiently to the substrate. Furthermore, the vertical misalignment of about 400nm between cavity and waveguide will reduce coupling efficiency η to a value lower than 20% (see simulation described in previous chapter). As a consequence, a new geometry has to be used in order to obtain an access waveguide and a cavity in the same plane.

One strategy consists of using the nanobeam also as the access waveguide. A priori, the injected light should not reach the cavity when funneled into the nanobeam since the cavity is enclosed by two highly-reflective photonic crystal mirrors. Yet, by removing few holes forming the mirror on one side of the cavity, the mirror transmission increases, enabling the coupling of the injected light to the cavity through this mirror without inducing a significant degradation of the cavity optical losses. The main disadvantages of this configuration are the less efficient optical injection and the optical quality factor reduction. Yet, it allows to overcome the issue of misalignment between the separated nanobeam and optical waveguide as in previous configuration. An example of such GaN/AlN-based nanobeam cavity and waveguide system is shown on Figure (3.15). In this section, we described the different steps involved in the process of the nanobeam platforms on GaAs, InGaP and GaN/AlN materials. The nanobeam is directly fabricated on the III-V substrate and excited optically either via the nanobeam or via an additional curved waveguide located on the cavity side. The process involves e-beam lithography, ICP-RIE etching and chemical underetching. For all three material systems, some of these steps had to be optimized, in particular the underetching step and the ICP-RIE step. In particular, the design of the GaN nanobeam platform had to be revised due to the high stress induced during growth in the membrane. In the next section, we will consider another configuration to optically access the cavity. This configuration relies on a 3D-integration scheme, enabling to bond the III-V material membrane on a silicon waveguide. This new configuration is applied to both one-dimensional GaAs/InGaP-based nanobeam resonators and two-dimensional GaN/AlN-and GaAs/InGaP -based L3 resonators.

Fabrication of one-and two-dimensional photonic crystal resonators on SOI waveguides

Figure (3.16) shows the schematic of a nanobeam resonator integrated on a SOI waveguide. Light is coupled to the SOI waveguide via gratings etched at the edges, is guided to the cavity location and coupled to the cavity by evanescent coupling. In the following parts, some general basics of 3D SOI integration technology will be introduced; its specificities for GaAs/InGaP-based nanobeams devices will then be described. This section will end with the details of the process used for the integration of 2D photonic crystal resonators made of GaAs/InGaP and GaN.

SOI wafer preparation and bonding

SOI waveguides have been widely used in integrated silicon-based nanophotonics as well as for hybrid optical devices based on the merging of III-V material and silicon photonics. The SOI structure consists of a high index silicon waveguide on top of a 2µm silicon dioxide (low index). This structure allows optical signal propagation through the silicon waveguide with high optical confinement. In our case, our goal is to bond the III-V material membrane on the SOI substrate and to build the photonic crystal resonators above the waveguide. The coupling strength between the cavity and the waveguide will be determined by cavity-waveguide distance which can be controlled during the bonding process. The SOI substrate is designed so as to integrate dozens of 8mm long waveguides with optical losses around 2.4dB/cm. The central waveguide width varies from 250 to 500nm with steps of 50nm. At the two ends of each waveguide, couplers gratings are etched enabling optical access to the waveguide from outside. The coupling efficiency through these gratings is around -5.1dB within a 40nm wide spectral range for TE polarization.

Figure 3.17 -Optical image of the SOI waveguide circuit with metallic marks, enabling to reliably spot the waveguides. Outside the waveguide circuit, two lines enable to position the metallic marks area. These marks are made of Ti/Au with a thickness of 22 nm, enabling to achieve a high contrast during the e-beam lithography.

Before bonding the SOI waveguide platform and the III-V substrate, metallic marks are deposited on the SOI substrate, in order to insure a good tracking of the SOI waveguides and to insure a good alignment between the waveguides and cavities during the electron beam lithography. These marks consists of 10 × 10µm 2 squares made of titanium and gold and provides a high contrast between semiconductor and metal. They are located at the outer part of the SOI waveguide circuit and cover each corner of the wafer. After bonding, all the SOI waveguides are invisible and only metallic marks on the SOI wafer can be found. In Figure (3.17), we can clearly see both SOI waveguides and marks. These marks are deposited using a usual lift-off process (resist spin-coating, patterning with electron beam lithography, resist developing, metallisation and lift-off). Marks already present on the SOI substrate are used as a reference for locating properly the new metallic marks. The marks are written on a positive resist (namely polymethyl methacrylate, i.e. PMMA) by electron beam lithography. These marks consist of a bilayer of Ti and Au with respective thicknesses of 2nm and 20nm; these are deposited by electron beam physical vapor deposition (EBPVD). After a lift-off process used to remove the unexposed areas, metallic marks are available on the SOI waveguide wafer. After depositing the marks on the SOI waveguide wafer, we can start to bond the III-V material sample. The process is based on adhesive bonding by use of polymerized divinylsiloxaane-benzocyclobutene (BCB) oligomers. The BCB layer acts as a planarized adhesion layer between the SOI waveguide and the III-V material substrate. The thin BCB layer is spin-coated on the SOI waveguide wafer up to reach a thickness of about 200nm in thickness and then baked at 80 • C for 15 minutes in order to evaporate the solvent and harden the BCB layer.

The surface of the III-V substrate should be slightly larger than 8×8mm 2 in order to cover entirely the SOI waveguide circuits. A thin Si 3 N 4 layer (with a thickness of 200nm) is deposited beforehand on the III-V surface, and this layer will latter forms the sacrificial layer during the under-etching process. The thickness of Si 3 N 4 layer will as such determine the distance between the waveguide and cavity. Moreover, the Si 3 N 4 layer helps the adhesion of the III-V substrate on the BCB during the bonding. The bonding is then performed in a high precision wafer bonder (SU SS Micro tech). The III-V membrane is placed face to face with the SOI waveguide wafer as shown in Figure (3.19). The bonder machine will increase the temperature up to 320 • C in a vacuum environment (≤ 5 × 10 -3 mbar). Meanwhile, a constant force (300N ) is applied vertically on the sample. The applied force will last 30 min and the entire process including temperature increase and vacuum pumping will last 2 hours.

After III-V substrate bonding on SOI waveguide, the backside III-V substrate needs to be removed, leaving solely the III-V membrane on the SOI waveguide wafer. The removal process is different depending on III-V substrate. The GaN/AlN samples were grown on a Si (111) wafer, and the GaAs and InGaP heterostructures were grown on a GaAs substrate. In the next section, I will first discuss the fabrication of GaAs samples and then of the GaN/AlN samples. The latter being on silicon wafer similar to SOI, and due to high stress in the GaN/AlN membrane, a new fabrication process had to be developed.

Integration of GaAs or InGaP nanobeam resonators on the SOI platform

In order to remove the GaAs substrate , one usually relies on wet etching techniques [START_REF] Desalvo | Wet chemical digital etching of gaas at room temperature[END_REF] . The chemical solution used here is a mixture of ammonia (70% N H 3 • H 2 0) and hydrogen peroxide (30% H 2 O 2 ) in volume ratio of 1:19. The chemical reaction between the GaAs substrate and the solution writes :

GaAs + H 2 O 2 -→ Ga(OH) 3 As(OH) 3 -→ GaO 2 + As 2 O 3 + H 2 O
in which N H 3 • H 2 0 acts as a partial oxidizing agent removing the oxidized As 2 O 3 and GaO 2 layers. The total GaAs substrate thickness is around 300 to 350 µm and the etch rate is around 5 to 6 µm/min. The etching will thus last around 60 min in total. The reaction will stop at the etch-stop layer interface which is set on purpose between substrate and membrane. Generally speaking, this wet etching method provides quite gentle and selective etching with homogeneous and smooth etched surface.

Figure 3.20 -Schematic of the steps involved during the GaAs substrate removal after SOI bonding. First the GaAs thick substrate is removed, then the thin sacrificial layer, both by wet etching.

After removing GaAs substrate, the etch-stop layer is chemically etched. The composition of this layer differs depending on the membrane material (GaAs and InGaP membrane). For GaAs membrane, the etch-stop is made of AlGaAs layer of thickness around few hundreds of nm thick. To remove it, ammonia fluoride (AF) solution is used. For InGaP membrane, the etch-stop layer is a bilayer made of InGaP and GaAs. To remove first etch-stop layer, HCl and H 3 P O 4 solution is used and provides a fast etch rate (0.6µm/min). The second etch-stop layer made of GaAs can be removed by use of the same solution as the one used to remove the GaAs substrate.

The photonic crystal patterning steps are shown in Figure (3.21). They are very similar to the one used in previous section. For electron beam lithography step, we make use of a negative resist (HSQ). The dose is chosen as 5000µC/cm 2 to have holes patterns close to the targeted patterns. After the e-beam lithography, the III-V membrane is etched by ICP-RIE. The nanobeam resonator is then released by dry underetching. Indeed the sacrificial layer in this SOI integrated platform is now made Figure (3.22) shows SEM images of final GaAs or InGaP nanobeam resonators integrated over a SOI waveguide. If such 3D structures can be processed on GaAs or InGaP samples, it is challenging to transfer this process to GaN samples. Due to the high stress inside the GaN/AlN membrane, the GaN/AlN membrane will bend and cause membrane stripping and peeling off. Even worse, as the nanobeam is a small volume and low mass system, as soon as the GaN/AlN membrane is released, the entire nanobeam system will be stripped and split from the holding substrate.

Integration of GaAs or InGaP L3 resonators on the SOI platform

The process here is quite similar to previous one.. The sole differences are the photonic crystal design and the larger surface of the suspended structures. This involves slight modifications of two steps : the electron-beam lithography step and the membrane suspension via underetching . Unlike the nanobeam cavity, the patterning of two-dimensional photonic crystal requires much larger writing field, with concomitant possible issues related to holes shape deviations with respect to the targeted geometry. The photoresist used here is a positive resist (PMMA) on which the e-beam exposure imprints the photonic crystal patterns. There are several parameters during electron beam lithography that can influence the photonic crystal holes shape. The first one is the electron beam dose. As mentioned in previous section with negative resist, increasing doses translates into smaller holes diameter. Conversely, for positive resists, higher doses will enlarge the holes diameters as shown on the left in Figure (3.24).

Photonic crystal patterning

Figure (3.24) right display the holes radius dependence as a function of the applied dose. Considering a targeted radius of 120nm, the optimal dose is about 1000 to 1100 µC/cm 2 . As we also discussed in the chapter 2 on the optical simulation part, as the holes radius vary, the optical resonance of photonic crystal will be largely modified. With the electron beam dose optimization, we can ensure the L3 cavity will feature resonances around 1.55µm.

Besides the holes diameter, an important parameter to achieve high quality resonators is the holes shape. The holes shape can be improved by edge contour techniques. The principle of the edges contouring is to redefine the electron beam writing path at every hole edge. Normally the electron beam writes patterns lines by lines, from left to right or from top to bottom in each writing field (520 × 520µm 2 ). The circular holes are thus divided into discretized lines, leading to some roughness around the edges (see Figure ( contour technique can then be applied when writing the edges of the holes, the usual writing along lines being solely applied to the holes central area. With this holes contour technique, we can improve the smoothness and sharpness of the photonic crystal holes. On Figure (3.25) left, we can see clearly that the edges of the holes processed without holes contouring are very blur and rough and the holes shape also slightly deviates from a perfect circular shape. When applying holes contouring (see Figure (3.25) right), the holes shape gets close to the perfect circular shape with clear and sharp edges.

A last parameter of importance when patterning large areas is the proximity effect we already discussed. As mentioned previously, this effect can be corrected by redistributing the beam dose as a function off the spatial position. In our case, the dimensions of the 2D photonic crystals, though larger than the nanobeam designs, are small enough not to be impacted by proximity effects. All these correction techniques enabled us to improve the photonic crystal patterning as much as possible, with holes featuring a nearly perfect circular shape, correct holes radius as well as homogeneous holes shape distribution. During the etching process, the PMMA mask will however be damaged. The best solution is to add an a harder sacrificial layer between PMMA and III-V material, that will acts as a mask during the semiconductor etching. This sacrificial layer is a Si 3 N 4 thin layer with a 300nm thickness. Consequently, we first write the photonic crystal patterns in the PMMA resist and develop the patterns. Then we transfer the patterns in the Si 3 N 4 layer by Capacitively Coupled Plasma Reactive Ions Etching (CCP-RIE). Last, the photonic crystal patterns are transferred from the Si 3 N 4 layer to the III-V layer by ICP-RIE technique using the Si 3 N 4 layer as a mask. Figure (3.26) shows the final GaAs photonic crystal platform. The distribution of holes radius is homogeneous. The average fitted holes radius is 112.8nm, close to the targeted value of 110nm.

Mesa structure patterning and under-etching

The suspension of the L3 photonic crystal is again achieved by underetching the layer below the membrane. The photonic crystal is in that case anchored to mesa structures on the sides of the photonic crystal. Besides providing enough support for the suspended photonic crystal, these mesa structures enable to optically isolate the SOI waveguide underneath and to preserve the optical confinement in the waveguide. Indeed, if the III-V material layer is kept all over the SOI substrate, light funneled into the waveguide could optically leak in the semiconductor, reducing the optical confinement in the waveguide and increasing the propagation losses. The mesa structures are thus designed as large rectangular pads in dozens of µm in size that surrounding all the photonic crystal platform.

To fabricate the mesa structures, the process is similar to the photonic crystal patterning step but with less constrains. Due to the size of mesa, the resolution is not an issue. In this case, negative resists are more suitable. As shown in Figure (3.27), the steps here include negative resist spin-coating, electron beam lithography and ICP-RIE. After etching the mesa structure down to the III-V material, we can start to under-etch the semiconductor membrane. The sacrificial layer is here made of Si 3 N 4 removed as previously by dry etching. The etching process is operated at 130 • , with a 250W RF power by use of a mixture of oxygen, SF 6 and argon. It is here worth noticing that the etching duration needs to be optimized to have both well suspended photonic crystal and non-damaged SOI waveguides. Indeed the dry etching can also etch the SiO 2 and BCB material though with a slower etch rate (around 1 or 2 nm/min). Consequently, once the Si 3 N 4 layer is completely etched, the reactive ions can etch the BCB as well as SiO 2 layers. If the etching duration is too long, the BCB layer will be further etched and most importantly, the SOI waveguide could be exposed and especially the gratings may be damaged. The most detrimental consequence with etched SOI waveguide and gratings is the induced optical losses. In our case, the etching duration has been fixed to 10 minutes to avoid this issue.

Final realization of GaAs L3 cavities

Integration of GaN L3 resonators on the SOI platform

As seen previously for the nanobeam configuration, high stress in GaN/AlN is a major issue for the fabrication of suspended photonic crystals. In this section, we will discuss the realization of suspended 2D PhC made of GaN/AlN. One of the main challenge is to bond such layer on the SOI substrate without inducing cracks or striping. Different strategies have been envisioned and tested for removing the growth substrate : dry etching and mesh patterns protection.

Substrate removal by dry etching

The bonding of the GaN sample on the SOI substrate is similar to the one used for bonding GaAs samples. However, for GaN/AlN substrate removal, the process is more complicated since the GaN/AlN membrane is grown on a silicon wafer, which is the same as the SOI wafer. Removing the silicon substrate by wet etching will also etch part of the SOI substrate (at least on the its sides). Another selective and directional etching is here required. In this context, the GaN sample substrate removal can be performed by dry etching : XeF 2 -based etching or deep reactive-ion etching (DRIE).

For XeF 2 -based etching, the reaction is highly selective to silicon. The induced reaction writes:

XeF 2 + Si -→ Xe(↑) + SiF 4 (↑)
Both reaction products are gas-like and harmful for human beings. Especially, the XeF 2 etching rate doesn't depend on the silicon crystalline plane. The selectivity is very high and efficient. Moreover, the XeF 2 etching requires dehydrated substrate surface to avoid silicon fluoride polymer layer forming on the surface that would stop the etching reaction completely. However, XeF 2 etching is isotropic.

In our work, DRIE etching is more preferable. The DRIE is an etching method used to etch silicon structures with high aspect ratio.This technique is anisotropic and widely used in three-dimensional silicon structure fabrication. The etching process is based on the repetition of cycles involving two different plasma exposures. In each cycle, the SF 6 plasma is first used to etch silicon at the surface, then the plasma changes to the C 4 F 8 that enables to passivize the etched surfaces. The passivized layer made of CF-polymers covers all the exposed surfaces. Then the cycle is repeated : the SF 6 plasma remove the CF-polymer deposited on the bottom, and etch the exposed silicon surface. The C 4 F 8 plasma is switched back again for passivizing the etched surface. By repeating several times this cycle, the silicon substrate can be etched vertically with scalloping shape on the sidewall. By controlling the number of etching cycles, we can stop the silicon etching as soon as the silicon substrate is etched away and the AlN membrane is exposed in air. However, stress still remains a fabrication issue here. As soon as the silicon substrate is removed, adhesion between the BCB and the GaN layer is too weak to tackle the high tension in GaN/AlN layer. As a consequence, the GaN/AlN layer no longer adheres on the SOI wafer and starts to release the force by rolling and folding as shown in Figure (3.32). By measuring the thickness of the membrane, we can confirm that such stripped membrane detached at AlN/Si 3 N 4 interface. It proves that as soon as the silicon substrate is removed, the GaN/AlN alone cannot hold a planar shape due to the released stress. The observed cracks also follow the crystalline direction. This effect starts firstly at the corner and then propagate to the entire membrane. Extra fabrication strategy is required to to overcome such issue. The mesh pattern is obtained by spin-coating a resist on the silicon substrate and by UV lithography. The transfer of mesh pattern to the substrate is done by CCP-RIE. During the etching, the resist works as a mask. We etch around 10µm deep in silicon substrate and then remove the resist mask. We then continue etching the silicon wafer by DRIE technique followed by thermo-plasma dry etching. We chose not rely here on XeF 2 -based etching since it is isotropic and may damage the mesh pattern on top of the silicon substrate. Using the DRIE technique, we can remove most of the silicon substrate within 150 cycles. As soon as some the AlN membrane is exposed, the etching process is stopped. A very thin layer of silicon substrate around few µm still remains and the mesh pattern is more than 10µm in height.

Substrate removal via mesh patterns protection

With these extra steps developed for GaN/AlN samples, we finally succeeded in substrate removal leaving an unstrapped membrane with a smooth surface. Figure (3.35) shows the final AlN surface after substrate removal. Some dots on the surface are still visible and cannot be removed chemically. Being at the interface between the silicon wafer and the AlN epitaxial layer, this might be due to complex compound that plasma chemistry cannot remove. In general, the surface quality is Figure 3.34 -Schematic of the steps for bonding the GaN membrane on the SOI waveguide substrate and for removing the growth silicon substrate with meshing protections. The GaN/AlN is grown on a silicon substrate. To remove it without damaging the SOI wafer, the dry etching technique is favorable. The mesh acts as a framework protecting the GaN/AlN membrane and is patterned by UV lithography. The mesh pattern is then transferred by CCP-RIE with a depth around 10µm. The silicon substrate is removed first mainly by DRIE or XeF 2 and then with thermo-plasma dry etching to achieve smooth GaN surface.

Final realization with membrane issues

It is now possible to pierce the photonic crystal inside the GaN/AlN membrane. Using the same process as for the GaAs membrane with dry etching technique, we achieved to suspend the 2D PhC membrane. However, the pierced holes are slightly distorted and not vertical, with variations in the holes radius from the top to the bottom. This is caused during the ICP-RIE process. Compared to GaN/AlN nanobeam geometry, the L3 cavity features a higher density of holes. Moreover, even with ICP-RIE optimization, due to the high thickness of the GaN/AlN membrane (360nm), it becomes difficult to obtain vertical sidewalls, in particular at the bottom of the holes. Increasing the acceleration of the ions may be risky since it could damage the GaN/AlN surface. A consequence of the trapezoidal shape of holes, is the probable blueshift of the optical resonance wavelength and optical losses increase.

Another issue is still related to the high tension in GaN/AlN membrane, even if the mesh patterns can help to hold the tension. Indeed, the mesh patterns are removed by dry etching as soon as the mesa structures are etched. Since the mesh patterns no longer act as holding pads, the GaN/AlN membrane gets unstable.

The worse issue happens during the photonic crystal suspension, when part of the Si 3 N 4 sacrificial layer at the corner of the mesa structure is removed. Some exfoliation of the GaN/AlN membrane happens at these corners as shown in Figure (3.38a). However, sometimes, the adhesion of the GaN/AlN membrane on the wafer works better like in (d), with most of the membrane being flat.

The high tension on GaN/AlN membrane still brings cracks on the photonic crystal. The cracks appear randomly and may destroy the cavities. As shown in Figure (3.38c), the photonic crystal holes broke and the cracks propagated along the cavity. All of these issues may come from the fragility and high tension in the GaN/AlN membrane. To solve these issues, changing the growth substrate to achieve GaN/AlN layers with less strain or changing the thickness and ratio of the GaN/AlN bilayer may be a solution. It could also be possible to improve the bonding by relying on thicker adhesion and sacrificial layers, to create also mesh pattern on the mesa structure to hold the membrane.

Summary and conclusion

In this chapter we discussed the processing of the photonic crystal resonators, either coupled to waveguides in the same plane as the photonic crystal platform or located underneath on a SOI substrate.

In order to obtain smooth and vertical etching sidewalls as well as holes geometries close to the targeted ones, both the e-beam lithography and the etching of the semiconductor by ICP-RIE had to be optimized. For the e-beam lithograph step, this has been carried out by investigating several strategies including electron beam dose correction, proximity correction, holes contouring and so on.

When considering 3D integration of the photonic crystal resonators on a SOI optical waveguide, one important step is the growth substrate removal. The substrate removal process is easy for most of III-V materials by using wet etching. By choosing the proper acids or bases with oxidizing agent, the etching process can be well controlled with moderate etching rate at etch-stop layer. For the III-V material grown on in silicon wafer similar to the host wafer, dry etching seems to be the only solution. DRIE etching is especially advantageous. If the substrate thickness is small (few µm), the CCP-RIE (with large bias) and plasma-therm dry etching can also be considered.

GaAs/InGaP are strain-less membranes and the realization on both nanobeam and L3 cavity are simple. For GaN membrane, the fabrication process is much more complex. The high tension in GaN/AlN membrane caused several issues during all the steps. In this case, the GaN/AlN nanobeam integrated over a SOI waveguide was not possible to achieve.

The following table summaries the advantages and drawbacks of each platforms.

Chapter 4

Optical and optomechanics measurement of the photonic crystal cavities

This chapter is dedicated to both the optical and optomechanical characterization of our III-V material photonic crystal cavity systems whose design and fabrication process have been described in previous chapters.

After briefly describing the basic experimental methods used, including the different experimental configurations for funneling light to the optical resonator (side coupling and gratings coupling), we will fully characterize the optical response of the two-dimensional L3 cavities and the one-dimensional nanobeam cavities. For L3 cavities, transmission measurements can be carried out thanks to the integration of the SOI waveguide, enabling to fully analyze the optical response of the resonator. For nanobeam cavities, the transmission measurement can be carried out either by relying on the gratings coupling set-up or by use of the side coupling set-up. Since the injection tip as well as the optical waveguide are fabricated on the same III-V substrate, the optical property of the system can also be tackled by optical reflection methods.

For L3 cavity, as the Q values is comparably low as 10 2 to 10 3 , the interaction of photons and phonons is quite low. It is very difficult to find optomechanical modes at room temperature and atmosphere environment. With our current experimental set-up, optomechanical measurement can solely be carried out on the nanobeam cavities. The optomechanical interaction can not only be probed but also be controlled optically. This will allow us to further study the dynamical property of the cavities. In particular, by modifying the injected laser power or tuning the laser wavelength, non-linear mechanical behavior will be observed.

Experimental set-ups for optical and optomechanical characterization

Two independent experimental set-ups will be used to analyze the response of the platforms, depending on the presence or not of the SOI waveguide underneath the photonic crystal slab. For platforms that do not integrate any SOI waveguide, such as GaN nanobeam cavities, we will rely on side coupling configurations for injecting light inside the cavity: The light is directly injected into the integrated III-V waveguide via an injection tip and the optical properties of the cavity are then probed on the reflected optical signal. For platforms integrated above a SOI waveguide, such as GaAs L3 and nanobeam cavities, the optical properties are retrieved on the transmitted optical signal via the SOI waveguide, referred to as "gratings coupling configuration":

The light is coupled to the SOI waveguide via a grating coupler, propagates along the waveguide, is extracted to free space via the opposite grating coupler and then measured. The details of both experimental configurations will be discussed in the following.

Side coupling configuration

The schematic of the side coupling configuration is shown in Figure (4.1). The setup includes the tunable light source, a circulator, an objective and a power meter.

The objective and the sample, fixed on a translation stage, are placed in a box filled with nitrogen gas, in order to maintain the measurement environment with stabilized moisture level and air composition. Light is focused on the integrated tip via an objective and funneled to the cavity via the integrated III-V waveguide; the light source used in this measurement is a tunable laser source with a wavelength ranging from 1490 to 1630nm. Part of the laser intensity is reflected back to the objective and sent to a power-meter thanks to the optical circulator. By sweeping the laser wavelength and meanwhile plotting the reflected intensity, the cavity optical resonance can be probed: At resonance, a dip should appear in the spectrum of the reflected power.

Grating coupling configuration

The experiment set-up for the grating coupling configuration can be simplified as shown in Figure (4.2). Two light sources can be used: a broadband light source (Exalos broadband SLED -super luminescent diode) and a narrow bandwidth tunable laser (Photonetics Tunic 1550). The SLED features a wide spectrum lamp centered at 1570nm with a 140nm bandwidth. The wide spectrum of the SLED can be coupled to one waveguide, allowing for a fast transmission measurement over a large spectral window. A tunable laser source with narrow bandwidth is yet necessary for a better wavelength resolution and finer optical measurement. The tunable laser wavelength can be tuned from 1490 to 1590nm, with steps of 0.01nm and a maximum power of 2mW. The laser signal can then be amplified by an erbium doped fiber optical amplifier -EDFA (Keopsys Continuous Erbium fiber amplifier CEFA-C-PB-HP). The EDFA can efficiently amplify the laser signal with 20dB power. In order to remove the amplified spontaneous emission (ASE) from the EDFA, a tunable optical filter (Exfo XTM-50) is added. The polarization of optical signal is controlled via a polarizer placed just after the filter.

The optical signal, propagating into a fiber system, is then coupled to the SOI waveguide via the grating couplers. The grating coupling is performed by use of a 3D translation stage system enabling high precision fiber alignment system (PI F-712) with xyz-axis nanopositioner (PI P-616 Nanocube). An infrared camera is placed above the sample station stage in order to observe the optical resonance inside the cavities. The output optical signal is then analyzed on an optical signal analyzer (Anritsu MS9740A, resolution 0.03nm).

One of the most delicate part in such experiment is the coupling of light from the fiber to the waveguide and vice-versa. The efficiency of this coupling via the grating couplers, depends on the fiber angle θ with respect to the sample surface. In order to achieve efficient light coupling, θ needs to be adjusted accordingly to the grating design (such as gratings period and its index). In our set-up, this incident angle should be 10 • . It can be adjusted by use of a fiber bracket with fixed angles: 9 • , 10 • or 11 • . Figure (4.3) displays transmission spectra obtained for these three angles, when injecting broadband light from the SLED. The light is filtered by the gratings over a FWHM of about 40nm. The spectrum shifts to lower wavelengths for increasing angles θ. With slightly larger angles around 11 • , it is shifted around 1550nm which is the targeted wavelength for our photonic crystal cavities platforms. Consequently, the incident angle in all following transmission-based measurements will equal 11 • . The coupling efficiency between the single-mode fiber and the SOI waveguide gratings is then around -5.1dB for TE polarization.

L3 cavity optical characterization

In this section, optical transmission spectra measurement are carried out on L3 cavities coupled to SOI waveguides. From these measurements, we can extract the cavity resonance wavelength as well as its quality factor. The transmission spectrum of the waveguide when fed by broadband source, is shown in Figure (4.4c). This spectrum features several dips which correspond to the cavities resonances. The five photonic crystal cavities possess identical lattice constant (a =420nm) but different hole radii equally distributed between 100 and 120nm with 5nm steps; their resonance wavelength should as such lie between 1470 and 1550nm. However, during the fabrication process, the achieved photonic crystal holes diameters appeared to be slightly smaller than the targeted values. As such, the actual resonance frequencies are all shifted to longer wavelength.

Optical transmission spectrum of GaAs-based L3 cavities

In order to extract more acutely the quality factors of the cavities, we repeated the experiment though by replacing the broadband source by the tunable laser, whose wavelength λ L can vary from 1520 to 1580nm. Figure (4.5) shows examples of the obtained transmission spectra. This more acute measurement reveals two resonances in two different wavelength region around 1525nm and 1565nm (indicated by grey areas). If we assume that these two resonances correspond to two different cavities with two different corresponding holes radii r 1 and r 2 , the spectral separation between these two modes ∆λ = λ 2 -λ 1 ≈ 43nm, shall correspond to ∆r = r 2 -r 1 ≈ 10 -12nm for identical lattice constants of a = 420nm. By fitting these resonances with a Lorentzian curve, one can infer the resonance linewidth κ/2π = 3.1nm which corresponds to Q 0 values around 500. Solely two cavities are observed. Two possible reasons may account for not observing the three other resonances: the quality factor of the other cavities is too low or, more probably, their resonance is shifted out of the laser tunability wavelength range.

Optical transmission spectrum of GaN-based L3 cavities

The optical transmission spectrum measured on the waveguide coupled to the GaNbased L3 cavities and fed with the broadband light source, is shown in Figure (4.6). As a reference, the transmission spectrum of the SOI waveguide without any cavity is also displayed in blue line. The IR image with no injected laser reveals the position

Nanobeam cavity optical characterization 4.3.1 GaN nanobeam cavity optical characterization in the reflection mode configuration

The investigation of the optical response of the GaN-based nanobeam cavities is carried in the reflection mode configuration due to the platform architecture. The GaN nanbeam platform indeed combines an optical injecting tip, the nanobeam cavity, holding clamps as well as holding pads (see Figure ). It plots the amplitude of the reflected optical power as a function of the incident laser wavelength. Here the measured sample is with rectangles holes on GaN nanobeam. The shape of this spectrum is a flat line with a dip at the resonance, here around 1565nm. By fitting this dip with Lorenztian function, one can extract the coupled quality factor Q c , here around 3000. The measured resonant wavelength are all slightly lower that the simulated ones. If we assume that the accuracy of the holes positioning is given by the 1 nm resolution of the e-beam lithography, this deviation can be explained by variations in the holes diameters, with experimental reduced holes radii. The smaller dimensions of the holes (of about 10 to 15 nm reduction) can be due to the etching process as mentioned in the previous chapter.

As shown in Figure(4.7b) for cavities with fixed holes radii but tuned lattice constant a, the dependence of the experimental and simulated cavity resonance as a function of the lattice constant is very similar. A change of the lattice constant a by 10nm, results in a shift of the resonance wavelength by 23nm. Meanwhile, the coupled quality factor Q c is barely modified and kept constant about 1300. If the lattice constant a is fixed but the holes width H x is increased as shown in Figure(4.7c), the resonance wavelength tends to decrease by 0.9nm for every holes width variation of 1 nm while the coupled Q c tends to increase.

The achieved quality factors are moderate. Indeed, the optical confinement inside the GaN nanobeam cavity is limited by its low refractive index and also due to realization deviation in the geometry. With the optimum GaN cavity design, the maximum coupled Q c is lower than 3500. This low optical confinement prevents further optomechanical measurement.

GaAs nanobeam cavity characterization in the transmission configuration

As shown previously, the control over the GaAs-based cavities processing makes it possible to fabricate nanobeam cavities integrated over a SOI waveguide. Such architecture enables to fully characterize the optical cavity response via transmission measurements. The sample here combines four different cavities per waveguide. Due to their different geometries (in the hole shape), their optical resonances can be distinguished and these cavities are not coupled. Moreover, they are physically separated preventing any mechanical coupling as well. As a consequence, each cavity can be considered as independent. Each cavity has been designed to reach a resonance wave-length of their fundamental optical mode around 1550nm; the higher order modes are equally spaced by about 20nm. Figure (4.8) shows a typical transmission spectrum recorded when the waveguide is fed with the broadband light source. In this experiment, all the optical modes of the four cavities are excited simultaneously. This simultaneous excitation of the modes can also be observed on the spectrum that features several resonance dips corresponding to the fundamental and higher order modes of the different cavities.

A careful investigation has yet to be carried out to attribute each resonance dip to a single cavity and to discuss the evolution of these resonances as a function of geometrical parameters such as holes shape.

Identification of the modes

The identification of the modes sustained by the nanobeam cavities has been carried out by launching in the SOI waveguide tunable laser light. It assumes that all the cavities, due to their different geometries, sustain optical resonances at different wavelengths. As such, when the input laser wavelength matches the resonance of one cavity, the laser will be coupled to this cavity solely. When observing simultaneously the sample with the IR camera, solely this cavity should diffract light and not the three other ones. As a consequence, when sweeping the wavelength of the input laser, we should expect to see bright spots when the laser wavelength matches one optical resonance, yet at different positions corresponding to the different cavities sites. This is illustrated on Figure(4.9): when the laser pulsation is at resonance with cavity (a) (ω L = ω a ) or cavity (b) (ω L = ω b ), cavity a or b will diffract light, with an observed bright spot on the IR camera at the location of the cavities. By subsequently comparing the spectrum as shown in Figure (4.8) and the bright spot locations, the attribution of the modes to each cavity is made possible. The identification of the cavities optical modes has been carried out for nanobeam cavities featuring either rectangular holes or elliptical holes. Since the wavelength range of our tunable laser is limited from 1500 to 1590nm, modes outside this range cannot be observed. Previously we identified to which cavity the optical mode belongs to. We can now play with the geometrical parameters used in the design of these cavities and investigate their impact on the evolution of the optical resonances. Meanwhile, we will compare our experimental results with our theoretical expectations.

Evolution of the optical resonances as a function of the cavity geometry

As introduced in chapter 2, any variation in the lattice constant a 0 will translate into a shift of the optical resonance λ 0 . Here we chose four different lattice constant a 0 values and holes width H x values. In this case, the fundamental modes resonance should vary from 1520 to 1575nm. However, due to the deviations introduced during the processing, the holes dimensions are slightly larger and as a consequence, the resonances wavelengths are blue-shifted. By modifying the nanobeam geometry (such as the cavity holes width H x ), the optical resonances get modified. Figure(4.10a), displays the evolution of the resonance wavelengths for four different cavities with holes widths H x varying from 200 to 275nm by steps of 25nm. We observe that the resonance of the processed cavities are all blue-shifted compared to expected values, due to the larger widths of the holes in the processed cavities. The holes widths and optical resonances are related as follows ∆λ = 1.58∆H x . As such, the observed shift of ∆λ which is around 35 to 40nm, corresponds to a geometrical deviation of about 55 to 63nm in ∆H x . The holes width in the processed cavities is thus ranging from 260nm to 330 nm. When taking into the account of such geometrical deviation and correct the theoretical value as in Figure(4.10b), we find a good agreement between theory and experiment. Since the optical resonance are clearly identified, we can then focus on the other crucial parameters of the optical resonator, namely the optical quality factor of the cavity and the coupling efficiency between the waveguide and the cavity. We will first explain with one example how we extract these parameters and finally discuss their evolution with the geometrical parameters.

Procedure to extract the quality factor and coupling efficiency η

By fitting the resonance dips in the transmission spectra by a Lorenztian curve, we can extract the resonance wavelength λ and linewidth ∆λ of each mode. The loaded (total) quality factor Q T can then be obtained as follows Q T = λ/∆λ. As an example we focused on a nanobeam cavity with elliptical holes whose resonance wavelength λ lies around 1523nm as shown in Figure (4.11a). This optical mode has been identified as the second order optical mode. By fitting the resonance dip with Lorentzian curve, we can estimate that the coupled Q c reaches 11657 with a linewidth 9GHz. With these information, we can further estimated the intracavity power P c and intrinsic Q 0 as well as the coupled Q c .

The total losses combine coupling losses but also intrinsic losses due to light scattering, absorption and radiation losses. In previous chapter, we have introduced the relationship between the loaded Q T (which is also called total Q), the intrinsic quality factor Q 0 and the coupled quality factor Q c as:

1 Q T = 1 Q 0 + 1 Q c
Depending on the dominating loss process,three different coupling regimes can be distinguished: under-coupling regime (Q 0 < Q c ), critical coupling regime (Q 0 = Q c ) and over-coupling regime (Q 0 > Q c ). For a cavity-waveguide distance of 500nm inplane and 200nm out-of-place, we can assume that the involved coupling regime here is the under-coupling regime (Q 0 < Q c ) [START_REF] Viktor Tsvirkun | Optomechanics in hybrid fully-integrated two-dimensional photonic crystal resonators[END_REF] . In this context, the intrinsic Q 0 can be calculated from the following relation: [START_REF] Borselli | Beyond the rayleigh scattering limit in high-q silicon microdisks: theory and experiment[END_REF], [START_REF] Vanier | Intrinsic quality factor determination in whispering gallery mode microcavities using a single stokes parameters measurement[END_REF] . Here, T min is the fraction of the transmitted power at the resonance wavelength λ 0 . T min can be estimated by normalizing the transmission spectrum. The intensity outside the resonance is set as the maximum and the depth of resonance equals T min . Knowing the values of the loaded Q T and of the intrinsic Q 0 , the coupled Q c as well as coupling efficiency η can be calculated as

Q 0 = 2Q T /(1 + √ T min )
Q c = Q T × Q 0 /(Q 0 -Q T ) and η = Q 0 /(Q 0 + Q c ).
Figure (4.11b) displays the normalized transmission spectrum. The value at the resonance dip then equals T min . We can therefore infer intrinsic Q 0 as 25265, the coupled Q c as 11657 and the coupling efficiency η = Q 0 /(Q 0 + Q c ) as 68.43%. In this cavity, a value of η of 68.43% means that the coupled Q c is 2.22 times larger than the intrinsic Q 0 , confirming that the cavity-waveguide architecture is in the undercoupling regime. If η ≤ 50%, the system would be in the critical or over-coupling regimes, for which the main optical loss arises from the optical coupling.

This procedure can be applied to all observed modes which all correspond to higher order optical modes from the second order to the sixth order. The fundamental mode can not be probed for this cavity since its resonance lies out of the laser tunability range. As can be seen on Figure(4.12)a to d), for higher order modes, the normalized transmission T min and the quality factors decrease gradually, due to a higher coupling efficiency η.

Evolution of the quality factor and coupling efficiency as a function of the holes width

The quality factors as well as the coupling efficiency depend on different geometrical parameters such as the cavity center lattice constant value a 0 , holes width H x and air holes shape (rectangular or elliptical). A comparison is shown on Figure Generally speaking, cavities with higher lattice constant a 0 and holes widths H x feature generally higher optical quality Q 0 and Q c . As the order of optical modes increases, the optical quality also tends to decrease while the coupling efficiency η increases. The higher order modes are more coupled than the fundamental mode, and the coupling loss has a larger contribution for the intrinsic losses.

Evolution of the resonance wavelength, quality factor and coupling efficiency as a function of the holes shape

Up to now, we focused on one kind of hole shape namely elliptical holes. As explained in Chapter 2, the holes can also be rectangular. Here we compare and discuss the impact of the holes shape on the optical properties of the cavities featuring same geometrical parameters (a 0 and H x ).

Whatever the holes shape, the evolution of optical resonance wavelengths behave similarly with a linear increase of the resonance wavelength with the modes orders, with yet a slightly smaller resonance wavelength in cavities with elliptical holes compared to rectangular holes cavities. Theoretically, the resonance wavelength should also shift with the holes width, with a shift ∆λ for ∆H x = 25nm of 4 and 8nm for rectangular holes and elliptical holes respectively. In the experiments, the measured shift is much smaller, of about 2nm and 6nm respectively. These experiments yet confirm that elliptical holes shape makes the cavity more sensitive to the holes dimensions. Rectangular holes cavities also feature a slightly lower resonance wavelength compared to the elliptical holes cavities with the same a 0 and holes width H x , in accordance with the calculations developed in the Chapter 2. On the other hand, if we compare the intrinsic Q 0 between elliptical holes and rectangular holes cavities with the same geometrical parameters (a 0 315nm and H x 250nm) as shown in Figure (4.15), we can observe that elliptical holes cavities in general feature slightly higher intrinsic Q 0 . The coupling efficiency η is also slightly higher in elliptical holes cavities compared to rectangular holes cavity.

Up to now, we focused on the optical properties of the nanobeam cavities with different geometrical parameters and holes shapes. External geometrical parameter can also be at stake such as the waveguide width which influences the coupling between the waveguide and cavity. Figure 4.15 -Intrinsic Q 0 and coupling efficiency η for elliptical holes and rectangular holes cavities with the same geometrical parameter with a 0 315nm and H x 250nm, as a function of the modes orders. These values could not be measured for the fifth-order modes in rectangular holes cavity due to too low optical coupling.

Impact of the SOI waveguide width w W G on the optical coupling

In our nanobeam platform integrated on a SOI waveguide, the cavity-waveguide distance d is fixed to 200nm in order to achieve an optimum optical coupling as mentioned in chapter 2. However, there is another parameter which can largely influence the optical coupling between the SOI waveguide and the cavity, that is the width of waveguide w W G . To characterize such influence, we measured the optical properties of a rectangular holes cavity (with a 0 =315nm, H x =250nm) coupled to SOI waveguides of widths w W G ranging from 300 to 450nm by the steps of 50nm.

Theoretically the waveguide width w W G has little influence on the cavity optical resonances. If the cavity has the same lattice constants a 0 and holes width H x , its resonance wavelength λ 0 should remain unchanged regardless of the waveguide width w W G . Figure (4.16) displays the resonance wavelengths of the first five-order optical resonances as a function of w W G . As the width w W G increases, the optical resonances slightly shift to shorter wavelength. Yet, the shift is rather small (few nm), within the observed dispersion of the cavity resonance wavelengths of the processed cavities.

Conversely, the coupling efficiency varies with the waveguide width. Let's remind here that η > 50% corresponds to the over-coupling regime,while η < 50% corresponds to the under-coupling regime and η = 50% to the critical coupling regime. As η keeps higher than 67% for all modes regardless the waveguide widths under study, we can conclude that the cavity is over-coupled for all waveguide width. If we then compare the coupling efficiency η for all those optical modes, the highest η value is generally achieved for the higher orders optical modes as mentioned in the previous section. However, when increasing the waveguide width, the coupling efficiency decreases for most of the modes. As w W G increases from 350 to 450nm, the η of the fundamental optical mode decreases from 68.13% to 67.12%.

However, in order to reach a compromise between better optical coupling efficiency and higher quality factor, an optimal w W G value needs to be found. We chose to investigate here the forth-order mode whose resonance is around 1550nm. More precisely, we analyzed the evolution of its coupling efficiency and loaded quality factor As the w W G increases from 300 to 450nm, the coupling efficiency slightly decreases from 71.19% to 67.46%. Meanwhile, the loaded Q T increases from 6.5 × 10 3 to 2.8 × 10 4 . In order to have higher loaded quality factor as well as good optical coupling, the waveguide width should be around 400nm.

Optomechanical measurement on GaAs nanobeam

The optomechanical characterization of photonic crystal cavities systems can be implemented by optical means. When the tunable laser launched in the SOI waveguide is set close to the optical resonance of a nanobeam cavity, the mechanical displacement of the membrane hosting the cavity induces a modulation of the transmitted light at the frequency of the mechanical mode. The Fourier transform of the modulated light can then be transposed to a measurable electrical signal function by use of a fast photodiode and then analyzed by use of an electrical signal analyzer. This experimental set-up is shown on Figure (4.20).

Relying on optical transmission signal for probing the mechanical modes has several advantages in our system: firstly, each waveguide is coupled to four nanobeam cavities all featuring different optical resonance wavelengths. By shifting the input laser wavelength, we can access to any of these cavities without moving the sample position or optical alignment. Secondly, each cavity sustains several orders of optical modes lying between 1500 and 1590nm, enabling to excite not only the fundamental optical mode, but also higher order modes that can also be used to probe the mechanical response of the cavity.

In the following, I will focus first on the optical measurement of the mechanical spectrum of the GaAs nanobeam cavity; then we will extract the optomechanical coupling strength by introducing a phase modulation; next, I will discuss the influence of the probing laser detuning and power on the optical and mechanical response .

Mechanical spectrum of the nanobeam cavity

When the laser wavelength is close to the cavity resonance, it is possible to record optically the mechanical spectrum of the cavity. In our experiment, the nanobeam cavity under study is made of elliptical holes and its second-order optical mode is used to read the mechanical motion. Figure (4.19a) displays the power spectrum density of the optical signal transmitted via the waveguide when the input laser is tuned at resonance with the second-order mode. This spectrum features several peaks located around 2.3GHz. When comparing the frequency of these peaks and the mechanical frequencies expected on such cavities as predicted by our simulations, each peak can be attributed to different mechanical modes whose displacement field profile is also shown (see Figure (4.19b)). The first optomechanical modes (from) M 1 to M 3 ) are prevailing. We can also notice that the fundamental M 1 mode is split into two similar mechanical modes M 1a and M 1b with a tiny difference in their symmetrical distribution. These two modes feature very close resonance frequencies with only few hundreds of kHz difference. In our measurement, mode M 1 b is predominant over M 1 a and its linewidth Γ m /2π is around 5M Hz, a value much larger than the resonance difference ∆Ω m between M 1a and M 1b . We chose to discard mode M 1a and to only consider the contribution of mode M 1b named M 1 in the following discussion.

Optomechanical coupling strength g 0 in the nanobeam cavity

The optomechanical coupling strength g 0 which was introduced previously, can be measured experimentally again via transmission measurements. The experimental setup for g 0 measurement is shown in Figure (4.20); it is similar to the one used previously. Yet, it includes in addition an electro-optical phase-modulator and another polarizer.

In this experiment, a monochromatic electrical signal is sent from the electrical signal generator to the phase modulator at a frequency close to the mechanical frequency.

The second polarizer is here used to control the polarization of the phase modulator for an optimum coupling to the cavity. The laser light launched in the waveguide is now phase-modulated at a given frequency Ω mod and coupled to the cavity. The frequency Ω mod should be very close but slightly higher than the mechanical resonance frequency of the cavity Ω m to avoid to drive the cavity system at its resonance. The output optical signal, modulated both at Ω mod and Ω m , is then measured by the fast photodiode and analysed on the electrical signal analyzer. Knowing the frequency and amplitude of the phase modulation, we can calculate the strength of optomechanical coupling g 0 for a given mechanical mode. The optomechanical coupling strength is then given by [START_REF] Gorodetksy | Determination of the vacuum optomechanical coupling rate using frequency noise calibration[END_REF] :

g 0 = 1 2n β2 Ω 2 mod 2 Γ m 4RBW S xx (Ω m ) S xx (Ω mod ) (4.1)
where n is the average phonon occupancy in the mechanical oscillator at the given frequency Ω m , β is the phase shift factor induced by phase modulation, Γ m is the damping rate of the mechanical mode, RBW is the resolution bandwidth of the electrical signal analyzer, S xx (Ω m ) and S xx (Ω mod ) are the noise spectrum at the mechanical and modulated frequencies respectively. The ratio of S xx (Ω m )/S xx (Ω mod ) can be obtained by fitting the electrical signal spectrum of the mechanical modes by a Gaussian curve and the modulation peak with a Lorentzian curve.

In this measurement, the phase shift factor β is determined by the phase modulation. This factor describes the voltage needed to operate a π phase shift. It can be written as :

β = πV V π = π 2P rf Z V π (4.2)
where V is the voltage applied to the phase modulator, V π is the required voltage for inducing a π phase shift, P rf is the input power from the electrical signal generator, Z is the impedance of the RF cable between the electrical signal generator and the phase modulator. In our experimental set-up, the V π voltage of the used phase modulator, equals 3.5V below 10GHz and the impedance Z is 50Ω. However, the real power that is sent to the phase modulator P rf needs to be characterised preliminarily due to the reflectance as well as transmittance in the RF cable and the phase modulator.

In order to determine the real power P rf , we can use S-parameter measurements to investigate the power loss in our system. We firstly connected the RF cable on one Here the calibration is operated with the RF cable that is used in the g 0 measurement. The RF power P rf is set to 0dBm in this calibration measurement. port of the vector network analyzer and calibrated the reflectance of the RF cable to zero. We then connect the phase modulator at the end of RF cable and set the delivered RF power P rf at the value used for g 0 measurement. We can the record the reflection signal spectrum S 11 displaying the reflectance of the phase modulator with the RF cable. The reflectance of the phase modulator is frequency-dependent as shown in Figure (4.21). According to the datas sheets, the phase modulator should have a S 11 signal around -12dB, while in our experiments, the S 11 value within 3GHz is around -13 to -20dB. This discrepancy arises from the extra reflectance induced by the RF cable. Thus, knowing the reflectance R mod , we can deduce the applied RF power P rf as follows:

P rf = P rf × (1 -10 R mod /10 ) (4.3)
Hence, if the frequency Ω mod is set at 2.43GHz and P rf at -32dBm with R mod = -13.45dB , the applied P rf should be 6.02 × 10 -1 mW . The phase shift factor β thus equals:

β = π sP rf Z V π ≈ 6.97 × 10 -3 (4.4)
For the measurement of g 0 , the optical power is set to the lowest value as possible. In order to have more accurate S xx (Ω m )/S xx (Ω mod ) ratio measurement for all the modes from M 1 to M 6 , we divided the mechanical spectrum into two parts as shown in Figure (4.22a) and (b). For modes M 1 to M 3 , the injected laser is modulated at a frequency of 2.34GHz with -30dBm power. For the higher order modes, since their amplitude is lower, the injected power is also lower (-32dBm) and the modulation frequency is set at 2.47GHz.

Besides the coupling strength g 0 , another figure of merit can be used to quantify the strength of the optomechanical coupling, that is called the cooperativity C. The cooperativity quantifies the optomechanical systems efficiency in exchanging photons and phonons [START_REF] Aspelmeyer | Cavity optomechanics[END_REF] . It combines the optomechanical coupling strength g 0 , the cavity decay rate κ and the mechanical decay rate Γ m into a dimensionless constant as below: and Ω mod =2.47GHz. The cavity is made of elliptical holes and its Q 0 is around 16000.

C 0 = 4g 2 0 κΓ m (4.5)
where C 0 represents the single photon-phonon cooperativity. C 0 is also proportional to the multiphoton cooperativity C = ñcav × C 0 where ñcav denote the number of photons in the cavity. The calculation of ñcav will be introduced in a next section. Table (4.2) gathers the measured values of g 0 and C 0 for the first six modes.. We observe that modes M4 and M5 feature the highest mechanical quality factor Q m up to 806. However, their coupling strength g 0 is relatively low compared to the M6 mode with lower Q m . The measured coupling strengths g 0 are all of the same order of magnitude (hundreds of kHz) than the one observed in reference works dealing with optomechanical crystal cavities sustaining modes in the GHz [START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF], [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF] . Yet our measured g 0 are 3 to 4 times smaller, probably due to the fact that the optical resonance used here is not the fundamental mode but the second-order mode.

Nonlinear dynamics of the nanobeam optomechanical cavity

Up to now, the optomechanical properties of the nanobeam cavity were probed for a fixed laser detuning at low optical power, enabling us to infer the linear response of our optomechanical resonator. Yet, the dynamical behavior of the resonator strongly depends on the probing laser power and detuning. Yet, changing these parameters may also introduce novel processes such as thermo-optical effects. After describing these effects, we will analyse the impact of laser power and detuning on the optomechanical resposne, with the objective to reach optomechanical self-oscillation.

Thermo-optic effects

When increasing the laser power, more light gets coupled to the optical cavity, possibly leading to the emergence of non-linear phenomena such as thermal bistability: the absorption of the laser light in the cavity induces a temperature increase by ∆T and hence a change of the refractive index n translating into a cavity resonance redshift. There are different absorption mechanisms which can lead to a modification of n: the intrinsic absorption, free-carrier absorption, two photon absorption and Kerr effect absorption [START_REF] Weidner | Nonlinear and bistable behavior of an ultrahigh-q gaas photonic crystal nanocavity[END_REF] . In general, the thermal shift of the cavity resonance wavelength can be written as :

∆ω = - 1 n dn dT ∆T ω 0 (4.6)
where ω 0 is the resonance frequency, dn/dT is the thermo-optic coefficient of the material hosting the cavity.

With injecting high laser power into the cavity, the optical resonance linewidth is also modified, changing from its original Lorentzian profile to a broader and asymmetric shape when sweeping the laser wavelength from shorter wavelength to longer wavelength. In our system, we observed such thermo-optical effect when injecting high probing laser powers P in and sweeping the laser wavelength around the cavity resonance. The cavity under study features a resonance at λ 0 = 1530.5nm and an intrinsic Q 0 = 2.25 × 10 4 at zero detuning. When increasing the laser power, the optical resonance gets red shifted, highlighting the nonlinear behavior of the cavity. Figure(4.23a) displays the transmission spectrum recorded at four different probing laser powers P in . The red circles point the resonance edges for each measurement that shifts to higher wavelength for higher injected power. In order to calculate this wavelength shift as a function of the probing laser power, we gradually increased the injected laser power P in from 26 to 73µW step by step.

Figure (4.23a) shows a 2D color map plotting the normalized transmission value as a function of different laser powers and detuning. We can see a clear edge in the color map that occurs when the laser sweeps across the resonance edge. The cavity linewidth becomes wider and the resonance edge shifts to longer wavelength. The resonance of the cavity shifts from the so-called cold-cavity resonance to longer resonance wavelength.

The amplitude of the observed thermal shift enables us to extract the absorbed power P abs inside the cavity. Let's first introduce here the intracavity energy term U c . U c can be infered from the power transmitted through the waveguide at the resonance wavelength. The power dropped into the cavity should be equal to the power output from the cavity-waveguide system. With such relationship, the intracavity energy can be written as [START_REF] Vanier | Intrinsic quality factor determination in whispering gallery mode microcavities using a single stokes parameters measurement[END_REF] :

U c = Q 0 ω (1 -T min )P in (4.7)
where ω is the resonance frequency at this power and T min denotes the normalized transmission at the cavity resonance. In this equation, Q 0 is the intrinsic optical quality factor of the cavity; and (1-T min )P in is the output power. If we plot the intracavity energy U c with respect to the injected laser power P in as shown in Figure (4.24), the internal cavity energy is linearly related to the injected laser power and from the fit of this dependence, we can extract P in /U c = 15.37µW/pJ (or U c /P in = 0.065pJ/µW ). However, besides intrinsic absorption, the resonance shift induced by the thermal absorption could be related to several other mechanisms including free-carrier, two photon absorption, or Kerr effect. The intracavity energy U c and absorbed power P abs can then be estimated as follows [START_REF] Borselli | Beyond the rayleigh scattering limit in high-q silicon microdisks: theory and experiment[END_REF] :

P abs ∼ a F CA U 3 c + b T P A U 2 c + γ in U c (4.8)
where a F CA is the free carrier absorption coefficient, b T P A is the two-photon absorption coefficient and γ in is the linear injected laser power absorption coefficient. In order to estimate the contribution of the different absorption mechanisms in our system, we first have to find the relation between the power absorbed P abs and the resonance frequency shift ∆ω. Thanks to the time-independent thermal behavior of the cavity, the temperature change can be calculated as the product of the absorbed power and its thermal resistance. Therefore, in order to estimate the absorbed power P abs , we can approximate the resonance shift from the equation(4.6) as [START_REF] Borselli | Beyond the rayleigh scattering limit in high-q silicon microdisks: theory and experiment[END_REF] :

∆ω = - 1 n dn dT ∆T ω 0 = - 1 n dn dT R th P abs ω 0 (4.9)
where ∆T = R th P abs , and R th is the thermal resistance. Here we assume that our system is operating under a thermal bath environment with constant temperature (T=293K). With such assumption, we directly connect the cavity resonance shift ∆ω with the absorbed power P abs . In our system, the thermal resistance in GaAs material R th is 1.37 × 10 5 K/W and the thermo-optic coefficient dn/dT is 2.48 × 10 -4 /K [START_REF] Bass | Handbook of Optics, Third Edition Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments(Set)[END_REF] . The refractive index n in the linear regime is 3.37. With these parameters, we can calculate the resonance shift with the absorbed power as ∆ω/ω 0 = 10.08P abs . Now we can establish the dependance of the resonance shift ∆λ on the intracavity energy U c . The data points shown in Figure (4.25a), are all extracted from the 2D map shown in Figure (4.23b). These points reveal the dependance of U c on ∆λ and then the dependance of P abs on U c . As shown in Figure (4.25b), there is abrupt jump that occurs for an intracavity energy U c = 3pJ. This jump may be related to experimental issues. At high powers (grey area), the system is not stable, preventing any accurate fitting of the experimental data. Those data points will not be used in the following discussion. At much lower laser powers, P abs scales linearly with U c. Yet, for increasing laser powers, the onset of two-photon absorption (quadratic term) and free-carrier absorption (cubic term) significantly change the dependance of P abs on U c [START_REF] Borselli | Beyond the rayleigh scattering limit in high-q silicon microdisks: theory and experiment[END_REF], [START_REF] Weidner | Nonlinear and bistable behavior of an ultrahigh-q gaas photonic crystal nanocavity[END_REF] , which then follows a cubic polynomial law (P abs ∼ a Our nanobeam platform thus enters in a non-linear optical regime for increasing injected laser power and detuning, due to thermo-optical effects. Such bistablity can also be observed on the mechanical motion.

F CA U 3 c +b T P A U 2 c +γ in U c ).

Optomechanical response as a function of the laser detuning

In optomechanical systems, the optical force F induces changes in the cavity length and as such modifies the intracavity power. This induced modification strongly depends on the sign of the laser detuning. In the case of radiation pressure force, for negative detuning(∆λ < 0), the radiation force extracts mechanical energy enabling to cool the mechanical motion; for positive detuning (∆λ > 0), the mechanical motion is amplified with extra mechanical energy scattered to the system by the radiation force, corresponding to a 'heating' process of the mechanical mode.

To understand the dynamics of the mechanical motion, we can solve the optomechanical coupling as a linearized problem [START_REF] Aspelmeyer | Cavity optomechanics[END_REF] . The effective mechanical susceptibility χ -1 m,ef f can be expressed as the sum of the original susceptibility χ -1 m and some small modifications Σ(ω):

χ -1 m,ef f (ω) = χ -1 m (ω) + Σ(ω) = m ef f Ω 2 m + 2ωδΩ m (ω) -ω 2 -iω [Γ m + Γ opt (ω)] (4.10
) where δΩ m is the mechanical frequency shift and Γ opt is the optical damping rate. By taking real and imaginary parts from this equation, one can extract the mechanical resonance shift δΩ m (ω) and the extra optical damping rate Γ om (ω) as:

δΩ m (ω) = ReΣ(ω)/2ωm ef f = g 2 Ω m ω ∆ + ω (∆ + ω) 2 + κ 2 /4 + ∆ -ω (∆ -ω) 2 + κ 2 /4 (4.11) Γ om (ω) = -ImΣ(ω)/2ωm ef f = g 2 Ω m ω κ (∆ + ω) 2 + κ 2 /4 - κ (∆ -ω) 2 + κ 2 /4 (4.
12) where g is the light-enhanced optomechanical coupling in the linear regime as g = g 0 √ ñcav . ∆ denotes the optomechanical-coupling modified detuning. At the mechanical resonance ω = Ω m , δΩ m and Γ opt can be thus written as:

δΩ m = g 2 ∆ + Ω m (∆ + Ω m ) 2 + κ 2 /4 + ∆ -Ω m (∆ -Ω m ) 2 + κ 2 /4 (4.13) Γ om = g 2 κ (∆ + Ω m ) 2 + κ 2 /4 - κ (∆ -Ω m ) 2 + κ 2 /4 (4.14)
This yields a full effective mechanical damping rate Γ ef f = Γ m +Γ om . Since Γ om can be either positive or negative depending on the laser detuning, Γ ef f can be smaller than Γ m and reach a level where Γ ef f = 0, which allows to reach a self-sustained oscillations regime.

For positive detuning and large injected laser powers, the system enters its bistable regime. The cavity potential then features two equilibrium positions instead of one. The two equilibrium positions correspond to a low and high light intra-cavity intensity respectively. Figure (4.26) plots the mechanical spectrum observed at such high optical powers, for two different sweeping directions from lower to higher laser wavelength and vice-versa. We can observe the first three mechanical modes M 1 , M 2 and M 3 located around 2.24GHz, 2.28GHz and 2.32GHz. When the laser wavelength is swept at the edge of cavity resonance, the modes disappear. Yet, the laser detuning at which the modes disappear, depends on the sweeping direction due to the cavity optical bistability. The two different resonances edge occurs at (λ -λ 0 ) =5.1nm and 2.5nm respectively. Figure (4.27) plots the dependance of the mechanical frequency Ω m as a function of the normalized detuning ∆/κ a , for modes M 2 (blue) and M 3 (orange). As the laser detuning ∆/κ increases, the mechanical frequency Ω m /2π for both mechanical modes decreases linearly. The linear relationship can be written as Ω m ∝ ∆/κ. By fitting the experimental data, we can extract the mechanical frequency shift per frequency detuning as δΩ m = Ω m -Ω m0 . Here Ω m0 is the mechanical frequency at zero detuning that equals 2.288GHz and 2.328GHz for mode M 2 and M 3 respectively. Table (4.3) a The normalized detuning is the laser frequency detuning over the optical decay rate (ω L -ω 0 )/κ.

shows the fitting parameter of the frequency shift per detuning for both modes, with a larger shift for mode M 2 .

Table 4.3 -Laser detuning ∆/κ and mechanical frequency shift δΩ

m Ω m0 δΩ m /(∆/κ) M 2 2.288GHz -9.82 × 10 -2 GHz M 3 2.328GHz -8.60 × 10 -2 GHz
On the other hand, the dependance of the effective mechanical linewidth as a function of the laser detuning, as shown in Figure (4. 27c andd), is different for mode M 2 and mode M 3 Γ ef f /2π for M 2 decreases with the laser detuning while the dependence of Γ ef f /2π for M 3 features a bell shape. For mode M 2 , we observe an inflection point close to ∆/κ 50. Around this point, the linewidth starts to decrease more rapidly. However, above ∆/κ = 80, lit keeps constant. Conversely, the linewidth of mode M 3 (Figure(4.27)d) present a maximum as a function of the laser detuning yet varies very slightly. For both modes, the regime of self-oscillation is not reached, maybe due to a mode competition between M 2 and M 3 . As M 2 has slightly stronger coupling strength g 0 and the laser power is more efficient to couple with compared to M 3 . However, we still didn't know why M 3 behaves quite differently with M 2 mode.

Optomechanical response as a function of the laser power

The optomechanical interaction also strongly depends on the injected laser power. In the following experiments, we fixed the laser detuning at ∆/κ = 50, and varied the laser power P in between 42µW and 120µW . The spectra of the first three mechanical modes as a fonction of the laser power are shown in Figure (4.28a). All mechanical frequencies Ω m /2π keep unchanged regardless of the injected optical power; these are located at 2.24GHz, 2.28GHz and 2.32GHz respectively. When gradually increasing or decreasing the laser power, no non-linear behavior is observed with a symmetric response independent of the power sweeping direction. This indicates that the injected laser power has no delayed influence conversely to the laser wavelength detuning. As P in increases, both the mechanical modes amplitude and the noise level increase. Figure (4.29a) displays the evolution of the mode amplitude (subtracted from the noise, that is S xx (Ω m ) -S xx (noise)) as a function of P in . This evolution is similar for modes M 2 and M 3 . When the laser power P in is lower than 67µW , both modes amplitudes increase rapidly. When P in reaches 67µW , the modes amplitudes reach their maximum values and saturate, indicating that no additional heating is induced. . For laser powers increasing from 42µW to 67µW , we observe a Γ ef f /2π decrease for mode M 2 from 3.8MHz to 3.7MHz. Then above 67µW , Γ ef f /2π remains around 3.7MHz. On the contrary, for mode M 3 , the linewidth keeps almost unchanged around 3.67MHz with the injected laser power. This discrepancy between these two behaviors may due to the respective optomechanical coupling strength of the two modes. As we have shown in table(4.2), M 2 features a higher optomechanical coupling strength g 0 /2π = 143.9kHz compared to M 3 with g 0 /2π = 114.6kHz. However, the reason for which mode M 2 amplitude and linewidth saturate at 67µW is still unclear.

Again, whatever the laser power, there is no evidence that our system enters the self-oscillation regime. To successfully achieve self-oscillations in our optomechanical system, several possibilities are proposed in the next section.

Towards self-oscillations

Self-oscillation is the generation and maintenance of a periodic motion by a power source such as radiation power, the oscillator itself controlling the phase with which the external power acts on it [START_REF] Jenkins | Self-oscillation[END_REF] . It is also known as self-sustained or spontaneous vibration. Conversely to forced and parametric oscillations, the power that sustains the motion, is not modulated externally and even lacks any corresponding periodicity. The driving force is here controlled by the oscillation itself so that it acts in phase with the velocity, causing above a certain power threshold P th a negative damping that feeds energy into the vibration. These self-oscillations appear as a strong linewidth narrowing and amplitude increase with the source's power. In cavity optomechanical systems, they occur for a blue-detuned optical input laser and at pump power levels exceeding a certain threshold pump power P th [START_REF] Rokhsari | Radiationpressure-driven micro-mechanical oscillator[END_REF], [START_REF] Kippenberg | Cavity opto-mechanics[END_REF], [START_REF] Lin | Mechanical oscillation and cooling actuated by the optical gradient force[END_REF], [START_REF] Rosenberg | Static and dynamic wavelength routing via the gradient optical force[END_REF], [START_REF] Lin | Coherent mixing of mechanical excitations in nano-optomechanical structures[END_REF] . At this threshold power, the full effective optomechanical damping rate Γ ef f = Γ om + Γ m is null.

As shown in the previous sections, we were not able to observe any self-oscillation even at high laser powers. In order to understand the reasons, we introduced another parameter: n cav that denotes the average number of photons circulating inside the cavity. The number of photons inside the cavity is strongly dependent on the injected laser frequency ω L and power P in . It can be written as [START_REF] Aspelmeyer | Cavity optomechanics[END_REF] :

n cav = κ c ∆ 2 + κ 2 /4 P in ω L (4.15)
where κ c is the optical decay rate related to the coupling losses and here equals 1.32 × 10 10 Hz; κ is the total decay rate including both coupling losses and intrinsic losses, that equals 2.43 × 10 10 Hz; ∆ is the laser detuning. At fixed laser detuning ∆, the photons number is linearly related to the injected laser power. And at a given P in laser power, the number of photons reach a maximum value for zero detuning when the laser is strictly at resonance with the cavity mode. If we fixed the laser detuning far away from the resonance (∆/κ ≥ 50 for instance as in the previous section), the number of intracavity photons n cav is lower than 100. For much smaller laser detuning (∆/κ < 0.4, n cav ), n cav will be four orders of magnitude higher. The photon numbers circulating inside a cavity is then sufficiently high to achieve self-oscillation. Indeed, the self-oscillation threshold power can be written as [START_REF] Wei | High-frequency silicon optomechanical oscillator with an ultralow threshold[END_REF] :

P th = m ef f ω 0 2g 2 0 κ 0 Γ m κ c ∆ [(∆ -Ω m ) 2 + (κ c /2) 2 ][(∆ + Ω m ) 2 + (κ c /2) 2 ] (4.16)
From this equation, we can find the relationship between P th , the normalized laser detuning ∆/κ and the mechanical mode linewidth Γ m . For the cavity we investigated in the previous section, the κ 0 and κ c are 13.2GHz and 24.3GHz respectively, and the mode linewidth Γ m /2π is 3.8MHz at low laser injected power. The calculated coupling strength g 0 /2π is 143.9kHz. The evolution of P th (corresponding to Γ ef f = 0) as a function of the g 0 , laser detuning and photon numbers is plotted in Figure (4.31a). On these plots, the black dash lines indicate the threshold level (Γ ef f = 0) and n cav is set at 10 3 (a value which is achievable in our experiments). As we can see, in order to reach the self-oscillation regime, the minimum coupling strength g 0 /2π is 4100kHz, which is nearly 30 times higher than the actual value. The actual g 0 /2π = 143.9kHz in our case requires a minimum n cav of 8.2 × 10 5 . This value is still nearly three times more than the maximum n cav we can achieve at zero detuning. In both case, either by changing the laser detuning or increasing the laser power, our cavity system can hardly reach self-oscillation. Several possible limitations in our system could be improved in order to reach the self-oscillation regime. These are discussed in the following.

• Optical limitations

First of all, the optical quality factor of the cavity is not sufficient. In our case, the linewidth of the resonance κ/2π is 9GHz. The evolution of the threshold power P th as a function of optical linewidth κ/2π and injected laser power P in is shown in Figure (4.33). For a linewidth of 9GHz, reaching the threshold level requires laser powers higher than 420µW . However, as we mentioned in previous section, the mechanical mode amplitude and damping saturate at powers above 67µW .

The limitation of the optical quality factor in our system is partly due to the fact that we rely on higher-order optical modes. In general, the fundamental mode features the highest optical quality factor compared to other modes. The optical confinement of fundamental mode is moreover more spatially confined in the cavity center, resulting in a much larger overlap area with the mechanical displacement field. To overcome such optical limitations, relying on the fundamental optical modes could be one promising strategy for future study.

• Mechanical/optomechanical limitations Figure 4.32 -Evolution of the threshold power P th to reach self-oscillations as a function of the optical linewidth κ/2π and injected laser power P in . For optical modes with a linewidth of 9GHz, the required laser power is 420µW .

Another limitation in our cavity system arises from the low coupling strength. Compared to similar state-of-art III-V nanobeam cavity systems, InGaP and GaP nanobeams all reached g 0 /2π up to 400kHz [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF], [START_REF] Schneider | Optomechanics with one-dimensional gallium phosphide photonic crystal cavities[END_REF] . Figure (4.33) plots the effective optomechanical damping rate Γ ef f as a function of the mechanical linewidth Γ m /2π and coupling strength g 0 /2π. In order to reach self-oscillations in our experiments (with Γ ef f = 3.8M Hz), the g 0 /2π value needs to exceed 280kHz which is two times higher than our actual value. Once again, relying on the fundamental optical mode (and not the second-order mode) should allow to enhance the observed coupling strength. Another improvement relates to the fabrication process. As discussed in Chapter 2, the geometrical optimization can largely influence the coupling strength. However, during the realization of the nanobeam cavity, the holes may distorted, and some surface defects may be introduced. These deformations and surface defects lead to a degradation of the mechanical quality factor.

• Mechanical mode competition

This low optomechanical coupling strength could also arise from mechanical modes competition. At least 5 mechanical modes feature g 0 /2π values of same order of magnitude (around hundred kHz), all located around 2.3GHz. When injecting some laser power into the cavity, several vibrating modes are excited, leading to some competition between these modes in the optomechanical interaction. By adjusting the nanobeam cavity geometry, it could be possible to maintain the fundamental modes (such as M 1 modes), meanwhile suppressing some higher order modes (such as M 2 , M 3 modes).

Summary

In this Chapter, we first described the optical characterization of the photonic crystal cavity systems under study, including GaAs-and GaN-based L3 or nano beam cavities. We relied on two experimental configurations (transmission/reflection configurations) to probe the cavity resonance. L3 cavities feature optical quality factors Q around few hundreds to few thousands in GaN and GaAs system respectively, while nanobeam cavities feature Q value of 10 3 to 10 4 in GaAs and GaN system respectively. The mechanical modes sustained by GaAs-based nanobeam cavities were then probed optically. The frequency of these mode lies around 2.3GHz.The associated coupling strengths g 0 /2π are around hundreds of kHz.

When increasing the laser power or detuning, optical bistability has been observed, with a red-shift of the resonance for increasing injected powers. By studying the relationship between the intracavity energy and the absorbed power, we could extract the thermal absorption coefficients related to various absorption mechanisms including free-carrier absorption and two-photon absorption. The dependence of the mechanical modes frequency and effective damping as a function of the laser detuning and power, was probed experimentally. The self-oscillation regime was nevertheless not achieved, mainly due to several limitations that have been identified : (i). The optical linewidth κ/2π is too high; (ii). the coupling strength g 0 /2π is too low with respect to the intracavity photons number n cav . For reaching this self-sustained oscillation regime, improvement of both the optical and mechanical confinement is required. Moreover an optimization of the nanobeam design would enable to prevent any mechanical vibration modes competition.

Chapter 5

Resonant excitation of mechanical modes for 2D/1D cavities

The optomechanical coupling between an optical mode and a mechanical mode can enable to control mechanically the optical mode and reciprocally, to read and control optically the mechanical motion. Such optomechanical coupling can get enhanced when relying on nanoscale resonators confining light down to the diffraction limit and featuring a high overlap between the mechanical and optical modes. The optical resonator can then reach a sensitivity to the mechanical displacement down to the quantum limit [START_REF] Jasper Chan | Laser cooling of a nanomechanical oscillator into its quantum ground state[END_REF] . In such experiments, the mechanical actuation usually makes use of the radiation pressure effect which is quite weak; this imposes to deal with relatively high optical powers inducing contingently a detrimental rise in temperature of the entire system. As a consequence, other actuation means have to be implemented in order to reach an efficient actuation of the mechanical mode. Moreover, as shown before, optomechanical photonic crystal cavity can feature mechanical modes highly coupled to light directly in the GHz range. In order to develop an optomechanical oscillator with low phase noise based on photonic crystal cavity in such frequency range, a key element is still missing, namely the resonant excitation of the mechanical mode. Due to the piezoelectric properties of the III-V materials investigated here, several strategies can be envisioned to implement such resonant excitation, such as electrostatic, piezoelectric and acoustical actuation.

The electrostatic capacitive actuation has been widely used in radio frequency MEMS resonators and it enables direct processing of RF signals by integrated electrodes. In electrostatic capacitive actuation, an electric potential is applied to a capacitor (electrodes) surrounding the resonator. The attractive capacitive force between the electrodes strains the resonator and shifts the position of its optical resonances.

In conventional piezoelectric actuation, the metallic electrodes are integrated on the resonator made of piezoelectric material. By applying a voltage difference on the electrodes, the strain inside the piezoelectric substrate is induced by piezoelectric coupling and a complex, frequency-dependent displacement field is induced. As a consequence, different mechanical modes can be resonantly excited. The piezoelectric actuation generally provides higher operation frequency and lower motional impedances [START_REF] Ghosh | Piezoelectric actuation of aluminum nitride contour mode optomechanical resonators[END_REF] . One example of such approach has been reported in J. Bochmannet al [START_REF] Bochmann | Nanomechanical coupling between microwave and optical photons[END_REF] in which the mechanical platform consists of a one-dimensional AlN nanobeam cavity put in motion via piezoelectric actuation. The system was integrated with a photonic circuitry and microwave nano-electromechanical tools. The strong coher-Figure 5.1 -Artificially-coloured scanning electron microscope image of a cavity optomechanical system subjected to a resonant excitation via electrostatic actuation. It shows in blue the two dimensional L3 cavity InP membrane featuring mechanical motion in MHz range from A. Chowdhuryet al [START_REF] Chowdhury | Superharmonic resonances in a two-dimensional non-linear photonic-crystal nano-electro-mechanical oscillator[END_REF] . By applying a periodic electrostatic force at the mechanical resonant frequency on the electrodes shown in yellow, the system exhibited surperharmonic resonances. ent interactions between the electrical, mechanical and optical modes, that can be achieved on such platform, enable both electromechanical and optomechanical coupling up to the GHz frequency range. Figure 5.2 -State-of-the-art example of a cavity optomechanical system subjected to a resonant excitation via a piezoelectric actuation. It consists of a one-dimensional AlN nanobeam cavity investigated by J. Bochmannet al [START_REF] Bochmann | Nanomechanical coupling between microwave and optical photons[END_REF] . The high coherent interactions between the electrical, mechanical and optical modes enabled to implement simultaneously electromechanical and optomechanical coupling effects in the GHz range.

More recently, purely acoustic actuation has been investigated by several groups. The acoustic actuation makes also use of electrodes integrated on the piezoelectric material. The main difference between piezoelectric actuation and acoustic actuation is the generated displacement field. In acoustic actuation, surface acoustic waves (SAW) are generated by the inter-digital transducers (IDT) and then propagate to the resonator. The strong surface displacement field then excites the mechanical modes of the resonator. The IDT works as an acoustic generator. In this frame, H. Liet al [START_REF] Li | Nanophotonic cavity optomechanics with propagating acoustic waves at frequencies up to 12ghz[END_REF] and K. C. Balramet al [START_REF] Balram | Coherent coupling between radiofrequency, optical and acoustic waves in piezooptomechanical circuits[END_REF] have implemented a cavity optomechanical system consisting of high-Q one dimensional photonic crystal nanocavities integrated with SAW transducers respectively with AlN and GaAs. Their demonstration of an acoustic wave interference effect built up a new architecture for signal transduction between the optical, electrical and mechanical domains. As such, it could be used for quantum or microwave photonics. Figure 5.3 -State-of-the-art example of a cavity optomechanical system under resonant excitation via acoustic actuation. It consists of a one-dimensional GaAs nanobeam cavity enclosed by a two dimensional phononic shield and IDT electrodes done by K.C. Balramet al [START_REF] Balram | Coherent coupling between radiofrequency, optical and acoustic waves in piezooptomechanical circuits[END_REF] . The acoustics waves excited by the IDT were funneled to the mechanical cavity through integrated phononic waveguides.

In this chapter, we will focus on these different complementary electrical actuation tools, that provide a supplementary control channel besides optical control. We will first focus on electrostatic actuation on an optomechanical platform consisting of a one-dimensional photonic crystal. Then we will introduce actuation tools making use of the piezoelectricity of the hosted material along two schemes : piezoelectric and acoustical excitation schemes. The comparison between these three different actuation strategies will then enable us to find out the one providing the most efficient actuation of the mechanical modes as well as the highest electromechanical and optomechanical coupling.

Electrostatic actuation on a photonic crystal cavity

Electrostatic actuation relies on an electrostatic force generated by a capacitor formed by two electrodes on each side of the cavity. Such actuation strategy has already been reported by A. Chowduryet al [START_REF] Chowdhury | Phase stochastic resonance in a forced nanoelectromechanical membrane[END_REF] with an actuation of mechanical modes in MHz range sustained by a 2D-photonic crystal membrane. However, addressing the modes in the GHz range along this scheme on such platform is rather difficult. This difficulty can be overcome if relying on a one-dimensional photonic crystal. This is the strategy that we have pursued.

Figure 5.4 -Schematics of the electrostatic actuation scheme on a one-dimensional nanobeam cavity system. Two electrodes act as a capacitor and are placed on the two side of the cavity. In the rest state the distance between the two electrodes is denoted d and the nanobeam width is denoted x; when an AC potential is applied on the electrodes, the cavity deforms the shape of the nanobeam due to the electrostatic force; its width is then denoted (x + dx).

Some basics on electrostatic force and displacement motion

In the electrostatic actuation scheme, two electrodes act as a capacitor located in the surrounding of the resonator. An electric potential is applied to the capacitor and the electric field across the capacitor leads to a capacitive force on the resonator. The electric potential combines a continuous and alternative potential respectively denoted V ac and V dc . The capacitance forms a resonance circuit with the resonance frequency Ω C = 1/ √ LC, where L is the inductance. In the general case with a capacitor formed by two parallel plates, the capacitance depends on the distance between the two plates. It is given by the relationship C = ε 0 A/d, where ε 0 is the permittivity of the spacer, A is the electrodes area and d is the distance between the two electrodes. However, when a membrane is placed between the two electrodes, the capacitance C is modified. This model is the simplest dielectric between the capacitor case, the potential in the capacitor embedding a membrane can be written as:

V = - Q Aε 0 d -x(1 - 1 ε r ) (5.1) Q = - Aε 0 V d -x 1 -1 εr (5.2)
where Q is the charge, V is the applied voltage, ε r is the permittivity of the nanobeam material, x is the nanobeam width and d is the total distance between the electrodes. In the rest state, with no voltage applied on the electrodes, the static capacitance C of the system can be expressed as:

C = Q V = - Aε 0 d -x 1 -1 εr (5.
3)

The force acting on the nanobeam system will also change its position. If we denote the equilibrium position as x = x + dx, the capacitance then writes :

C(x) = Q V = - Aε 0 d -(x + dx) 1 -1 εr (5.4)
The derivative of the capacitance is thus:

dC(x) dx = Q V = - Aε 0 1 -1 εr d -(x + dx) 1 -1 εr 2 = C 1 -1 εr d -(x + dx) 1 -1 εr (5.5)
These equations enable us to introduce various parameters related to the electromechanical coupling strength. These are : the electromagnetic coupling parameter, the electrostatic force induced on the membrane and the mechanical displacement response .

Electromagnetic coupling parameter

The energy stored in the capacitor writes:

E C = Q 2 2 C - 1 2 Q 2 C2 dC dx | x δx (5.6)
with Q = Q + δQ,. This expression can be developed as follows :

E C = ( Q + δQ) 2 2 C - 1 2 ( Q + δQ) 2 C2 dC dx | x δx (5.7)
By neglecting the second order term δQ 2 , we obtain:

E C = Q2 2C - Q2 C dC dx δx + Q C δQ -Q C2 dC dx δQδx (5.8)
In this equation, the energy term related to the electromechanical coupling involving both variables δQ and δx, writes :

E EM ≈ -Q C2 dC dx δQδx (5.9)
As introduced by E. Zeuthen [START_REF] Zeuthen | Electro optomechanical equivalent circuits for quantum transduction[END_REF] , this enables us to introduce the electromechanical (EM) coupling parameter G as:

G ≡ -Q C2 dC dx (5.10)
G is given in V /nm 2 highlighting the strength of the applied voltage on the displacement of the resonator. With the charge term Q and the derivative of capacitance term from equation(5.5), G can be written as:

G = -Q C2 dC dx = V 1 - 1 ε r 1 d -(x + dx) 1 -1 εr 2 (5.11)
This equation highlights that the EM coupling strength depends on the induced potential as well as the electrode distance. As expected, the smaller the distance and the higher the voltage, the higher the electromechanical coupling.

Electrostatic force induced by the capacitor

The electrostatic force F induced by the capacitor on the system, can be written as the derivative of the energy stored in the capacitor as:

- dE C dx = - 1 2 dC(x) dx V 2 (5.12)
This force is proportional to the the square of the applied voltage V = V DC + δV . It may vary due to a change in the the applied voltage V and/or in the position of the system. Thus the change in the force can be written as follow:

δF = δF δV δV + δF δx δx = - dC dx V DC δV - 1 2
d 2 C dx 2 V 2 DC δx (5.13)
The first term in the derivative of the force relates to changes in the applied voltage, while the second term depends only on the DC voltage acting on the system.

Mechanical displacement spectrum

The equation of motion of the membrane, actuated by the electrostatic force, can then be written as :

δF m ef f = d 2 x(t) dt 2 + Γ m dx(t) dt + Ω 2 m x(t) (5.14)
where the displacement x is a time-dependent term as x(t). m ef f denotes here the effective mass of the mechanical mode, Ω m its natural frequency and Γ m its damping rate. By denoting x(t) = X(Ω)e -iΩt and δF = δF (Ω)e -iΩt with Ω the driving frequency applied to the capacitor, the equation(5.15) rewrites:

d 2 x(t) dt 2 +Γ m dx(t) dt +Ω 2 m x(t) = -Ω 2 X(Ω)e -iΩt -Γ m jΩX(Ω)e -iΩt +Ω 2 m X(Ω)e -iΩt = δF (Ω)e -iΩt m ef f -Ω 2 -Γ m iΩ + Ω 2 m X(Ω) = δF (Ω) m ef f X(Ω) = δF (Ω)/m ef f (Ω 2 m -Ω 2 ) 2 + Γ 2 m Ω 2
(5.15)

The frequency Ω can be adjusted by the input AC voltage frequency:

V ac = V 0 cos(Ωt). If Ω = Ω m , it corresponds to the maximum displacement with X = δF (Ω)/Γ 2 m Ω 2 m .
In the one-dimensional nanobeam cavity as shown in Figure (5.4), if we sweep the frequency Ω around Ω m , we will obtain the nanobeam cavity mechanical displacement spectrum.

Electrostatic actuation simulation in a nanobeam cavity system

The nanobeam studied here is with rectangular holes with GaAs material and other parameters as: a 0 = 315nm, H x = 170nm, F W HM = 3µm and H y = 200nm. Figure (5.5) shows the evolution of the displacement as a function of the frequency Ω.

The nanobeam sustains three symmetric mechanical modes located at Ω m frequencies of 1.842, 1.858 and 1.870GHz respectively. In this simulations , we considered : an input voltage V set at 1V, an electrodes distance of d = 0.7µm, electrodes surface area A = 0.2×2.5µm = 0.5µm 2 , ε of 8.85 for the air and 12.9 for the nanobeam cavity material, a nanobeam width x of 0.7µm; The m ef f of the modes, deduced from the simulation for the three symmetric resonance modes is about 1.2e -17 kg for all three modes. For a fixed input voltage V , changes in d naturally translated into changes in the motion amplitude X(Ω m ). As expected, the maximum displacement amplitude decreases with the electrodes distance d as shown in Figure (5.6) for three symmetric modes, following a trend in the form of 1/d2. We then fitted the data with equation X = a/(b + c × d) [START_REF]Global crystal oscillator market[END_REF] . From the fitting curves, the first and second symmetric modes Ω m = 1.842 and 1.858GHz have good agreement between displacement field X and electrodes distance d 2 . Same trend is observed for the electromechanical coupling factor G (see Figure Besides the electrodes distance, care also has to be paid to their length.The spatial expansion of the mechanical mode along the nanobeam indeed changes as a function of the mode order. For a given capacitor length, the overlap area between the electrodes and the mechanical modes thus varies. For a smaller electrodes lengths than the mode expansion along the nanobeam, the induced electrostatic force only acts on part of the mode which is contained inside the capacitor. The capacitance term C(x) has to be modified accordingly taking into account the finite plate area. In order to excite the mechanical modes with large mode spatial distribution along the nanobeam, the electrodes should be designed as long as possible.

Electrodes processing

Let's now consider the feasibility of such electrostatic actuation tools, formed by electrodes deposited close to a one-dimensional nanobeam cavity system. Two gold electrodes pads enable the electrostatic actuation and are located on the two sides of the nanobeam cavity. The fabrication process can be simplified in several steps including resist spin-coating and e-beam lithography, resist development, metallisation and lift-off, photonic crystal patterning and suspension as shown in Figure (5.8).

Figure 5.8 -Schematic of the fabrication steps for realizing the capacitive electrodes pads with nanobeam cavity system. The electrodes have to fabricated before nanobeam system, to avoid collapsing of suspended nanobeam system. This protocol relies on an electrodes deposition prior to the etching of the photonic crystal. Indeed, the deposition of the electrodes requires two steps : metal deposition and then a lift-off. However, the removal of all metal (here gold due to its high electrical conductivity) residues requires the use of an ultrasonic bath during the liftoff step. This may damage or even destroy the fragile etched suspended nanobeam if processed prior to the electrodes deposition. Figure 5.9 -SEM images of the electrostatic actuation pads enclosing the onedimensional GaAs nanobeam system. The blue parts in the SEM images indicate the nanobeam cavity, while the yellow parts are the electrodes. The entire nanobeam cavity system is suspended at a distance of few hundreds of nm above the substrate.

In the final realization, the electrode pads are 5µm in width, dozens of µm in length and 200nm in thickness. The electrode-to-electrode distance is set at 3µm. Since the electrodes are located on the two sides of the nanobeam cavity, the nanobeam cavity is designed with an in-line optical injection as mentioned in the Chapter 3. Further experimental characterization need to be finished.

Piezoelectric actuation on a photonic crystal cavity

In the piezoelectric actuation scheme, strain is generated according to the direction of the applied electric field with respect to the piezoelectric polarization in crystalline structure. For identical piezoelectric actuators geometry but different material structure, strain can be produced either in the same direction for Zinc-Blende structure, or in the perpendicular direction to the electric field for Wurtzite structure. Moreover, since the electrodes cannot be as large as possible, the induced electrical displacement is also limited by the electrodes area. For this reason, the nanobeam cavity photonic crystal geometry seems better suited than the L3 cavity photonic crystal geometry.

The strain is then generated from the two ends of nanobeam. The strain acts as an electrical displacement : it propagates through the nanobeam and then gets localized by the cavity in the center.

Some basics on piezoelectric properties

The converse piezoelectric effect refers to the pressure generating an electric field; conversely, the piezoelectric effect consists in the generation of an electric charge in a material subjected to an applied stress. Both spring from the electromechanical interaction between mechanical states and electrical states in crystalline materials with no inversion symmetry. They are closely related to the occurrence of electric dipole moments in the solids. Piezoelectric crystals are indeed usually composed of microscopic electric dipoles which have random orientation rendering the overall polarization of the material zero. These dipoles may correspond to ions on crystal lattice sites with asymmetric charge surroundings or may directly be carried by molecular groups. When applying a mechanical strain to the crystal, each element changes its dipole moment : the applied strain orients the domains with respect to the mechanical field direction, leading to a remanent polarization of the material. Conversely, when applying a bias voltage on the piezoelectric material, the external electric field induces elastic displacement of charges in the crystal, producing a deformation in the material. As such, the applied electrical energy is converted into a mechanical energy to the system [START_REF] Paufler | Fundamentals of piezoelectricity[END_REF] . Generally, the interaction between the electric field and displacement field is linear, in particular at low electrical field and stress. This interaction depends on the orientation of polarization within the crystal, the applied mechanical stress but also the crystal symmetry. For cubic symmetry in Zinc-Blende structures, the shear strain is generated in the same direction as the applied electric field. For hexagonal symmetry in Wurtzite structures, the strain can be generated in one or two different directions perpendicular to the applied electric field. This is quantified through the piezoelectric strain constants d ij defined as the ratio of the generated strain to the applied electric field. The subscript i refers to the electric field applied in i direction, and the displacement is generated in j direction. For i or j equals to 1 or 2 or 3, these directions refer to the x, y and z axes; The 4, 5 and 6 subscripts refer to the shear force generated along the x, y and z axes (see Figure (5.10)). Considering the materials properties (see Table (5.1), it is then possible to calculate the electric displacement D, strain ε or stress σ of a given material (see Appendix). Moreover, taking into account the piezoelectric constants matrix, one can predict the direction and the amount of generated strain or electric charge when applying an external electric field or force field in certain direction in the piezoelectric material. In this work, we rely on Zinc-Blende structure materials such as GaAs, InGaP, even if their piezoelectric coefficients are lower compared to Wurtzite structure materials such as GaN and AlN. However, the generated strain is in the same direction as the applied electric field, which could provide an easier implementation in the nanobeam cavity optomechanical geometry. For GaN and AlN-based platforms, the generated strain is perpendicular to the applied direction, requiring more sophisticated designs. However, thanks to their higher piezoelectric coefficients, one can also expect significant piezoelectric actuation. The piezoelectric actuator, which relies on the converse piezoelectric effect, is one of the most common application of piezoelectric materials. As shown in in Figure (5.10), the piezoelectric actuator is bonded on the two side of a piezoelectric substrate with thickness t, length l and width w. If we apply a voltage V to a piezoelectric actuator along the z axis, the electric filed in this direction writes as:

The piezoelectric actuator and sensor

E 3 = V t (5.16)
The electric field deforms the piezoelectric substrate with a generated deformation along the x axis that writes: ∆l = d 31 V l t (5.17)

ε 1 = ∆l l = d 31 V t (5.18)
where ∆l is the change in length along x axis, and d 31 is the piezoelectric constant for this material. Conversely, piezoelectric sensors are used to detect the displacement field by converting it to a measurable electric charge (see Figure (5.11)). The applied force F along the z and the induced charge q are related as follows :

q = d 33 F (5.19)
Here the constant d 33 refers to the ratio between the generated electric charge per unit area flowing perpendicularly to j direction with respect to the applied force in i direction. In our case, the generated electric charge will flow out along the same direction as the applied force in z axis and the current circuit can be formed as shown in Figure (5.11). The total voltage V created by the force can then be written as:

V = q C = d 33 σ 3 wl C (5.20)
where C is the capacitance of the piezoelectric substrate.

The electromechanical coupling factor k

The efficiency of the piezoelectric actuator and sensor devices, is directly related to the energy conversion between the electrical and mechanical states. It is quantified by the electromechanical coupling factors k ij , which are an indicator of the effectiveness with which a piezoelectric material converts electrical energy into mechanical energy, or converts mechanical energy into electrical energy. The first subscript i to k denotes the direction along which the electric field is applied; the second j denotes the direction along which the mechanical energy is applied, or developed. To determine a coupling factor k ij , one of the most interesting methods is based on a closed-loop quasi-static energy cycle. The effective energy conversion from mechanical to electrical and viceversa can then be computed [START_REF]Ieee standard on piezoelectricity[END_REF], [START_REF] Trindade | Effective electromechanical coupling coefficients of piezoelectric adaptive structures: Critical evaluation and optimization[END_REF] . The underlying principle of the method is firstly to apply a force to a piezoelectric body in an open-circuit condition to store both mechanical and electrical energies U oc inside the body as U oc (see Figure (5.12) line a to b), and then to constrain the deformation of the piezoelectric body (see Figure (5.12) line b to c). During Figure 5.12 -The closed-loop quasi-static energy cycle for a piezoelectric body with the electromechanical coupling factor evolution method. The figure is done by MA. Trindade and A. Benjeddou [START_REF] Trindade | Effective electromechanical coupling coefficients of piezoelectric adaptive structures: Critical evaluation and optimization[END_REF] . this second step, part of the energy stored in the body is converted into work U conv in the ideal electrical load connected to the electrodes. The electric load is here a resistance so that the electrical energy is converted into heat and dissipated in the surrounding environment. When all electrical energy is dissipated the difference of potential between the piezoelectric body electrodes should be zero. Finally, the piezoelectric body is short-circuited and released from its holders so that it deforms back to its original configuration, performing a purely mechanical work U sc. From this energy cycle, the electromechanical coupling factor can be defined as the stored electrical or mechanical energy with respect to the applied mechanical or electrical power.

k 2 ij = Stored electrical (mechanical) energy Input mechanical (electrical) energy = U conv U oc = U oc -U sc U oc (5.21)
Let's consider here that the applied force is oriented along the x direction. In such linear and uni-dimensional (direction 3) regime, the internal energy stored in the piezoelectric body can be written as :

U = 1 2 Ω (σ 3 ε 3 + E 3 d 3 )dΩ (5.22)
with the mechanical stress σ 3 and the electrical displacement D 3 expressed as:

σ 3 = C 33 ε 3 -d 33 E 3 (5.23) D 3 = d 33 ε 3 + e ε 33 E 3 (5.24)
In the first phase of energy cycle (from a to b), there is no displacement field D 3 = 0 so that :

E 3 = - d 33 ε 3 e ε 33 σ 3 = (C E 33 + d 33 e ε 33 )ε 3
As such the energy stored in this first phase U oc writes as:

123

U oc = 1 2 Ω C D 33 ε 2 3 dΩ (5.25)
In the third phase of energy cycle (from c to a), there is no electrical applied field E 3 = 0, so that σ 3 = C E 33 ε 3 and D 3 = d 33 ε 3 . And the energy stored U sc can thus be written as:

U sc = 1 2 Ω C E 33 ε 2 3 dΩ (5.26)
Considering homogeneous deformation throughout the piezoelectric body, the square of the electromechanical coupling coefficient can be expressed as:

k 2 33 = U oc -U sc U oc = C D 33 -C E 33 C D 33 = d 2 33
C E 33 e σ 33 (5.27) This situation corresponds to that of Wurtzite structures , for which the only non-null coupling factor is k 2 33 . This analysis can be extended to for Zinc-Blende structures, in which k 14 has to be considered and can be written as k [START_REF]Global crystal oscillator market[END_REF] 14 = d 2 14 /C E 44 e σ 11 . In Table (5.2) shows some bulk electromechanical coupling coefficient k 2 including III-V materials such as GaAs, GaP, GaN and AlN. The data is taken from the reference in [START_REF] Delsing | The 2019 surface acoustic waves roadmap[END_REF] . Among these materials, GaN and AlN have the highest electromechanical coupling due to their better piezoelectric property.

Piezoelectric actuation electrodes design and realization

In order to induce strain by piezoelectric actuation to the cavity optomechanical system, the one dimensional nanobeam cavity has been here considered due to easier electrodes implementation. As the generated strain is a direction-dependent term with the applied voltage, we first considered Zinc-Blende structure materials in order to simplify the realization. The electrodes are placed on both ends of the nanobeam cavity, with an applied electric field and generated strain propagating along the x axis (shown in Figure (5.13)). For the GaAs substrate, the considered piezoelectric constant is d 14 for which the generated shear strain follows the direction as the one of the same applied electric field. By applying a voltage (10V) to the piezoelectric actuator, different mechanical modes around 1.8GHz are actuated (see Figure (5.13)). Scanning electron images of processed structures are also shown on Figure (5.14).

From the simulation, the mechanical modes at GHz range are successfully excited. Moreover, the realization of the piezoelectric actuator pads is similar to the capacitive actuation pads shown in Figure (5.8). The final realization is done with GaAs nanobeam, as shows in Figure (5.14). The pads (yellow) locate on the two sides of the nanobeam. Due to the electromagnetic influence induced by the pads, the optical access (red) to the cavity has to be modified as using side-coupling mechanism. Eventually, the entire system is suspended in hundreds of nm. With piezoelectric actuation, we expect the modes in GaAs nanobeam to be actuated at 1.8GHz. What's more, the electromechanical enhancement on the modes should be exhibited as the amplitude increase in PSD spectrum compared with no actuation case. Further experimental characterization need to be finished. Figure 5.14 -Optical and SEM images of the GaAs nanobeam cavity system with piezoelectric actuation electrodes on the two ends of beam. The yellow part is the gold electrodes, the blue part is the nanobeam cavity and the red part is the optical waveguide. The entire cavity system is suspended at a distance of hundreds of nm from the substrate.

Acoustic actuation on a photonic crystal cavity

One of the broadest application of piezoelectric actuators is surface acoustic waves (SAW) transducers based on the piezoelectric effect and the electromechanical coupling via the electrodes. Different kinds of piezoelectric material are currently used for SAW devices such as LiN bO 3 , quartz and III-V materials. All of these materials feature significant piezoelectricity. However, as far as the photonic crystal system is concerned, III-V material-based platforms provide higher optical confinement (Q o up to 10 5 ) as well as higher optomechanical coupling (with g 0 around hundreds of kHz). In our work, we mainly focused on such materials, in view of implementing a resonant excitation of the mechanical modes via acoustic actuation.

The SAWs are then usually generated by inter-digital transducers (IDT). The fundamental characteristic parameters for material choice in SAW devices are: surface waves propagating velocity, the propagating loss as well as the electromechanical coupling factors k ij . Contrary to the two previous actuation methods, acoustic actuation mediated by surface acoustic waves may provide wider functionality. For instance, one can envision to engineer the speed of the waves in the propagating medium, namely the membrane in our case. This would enable to implement acoustic delays for instance in optomechanical circuits. Yet, in order to efficiently excite the mechanical modes embedded in the optomechanical crystal, several parameters have to be taken into account and will be discussed in the following. These parameters include the respective position of the IDT with respect to the cavity as well as propagation losses, due to the stack of the different layers. Before discussing these issues, a basic description of these SAW transducers will firstly be given. Surface acoustic waves (SAWs), also called elastic surface waves, can be distinguished along two types : Love waves and Rayleigh waves. We will here only focus on Rayleigh waves. Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They include both coupled longitudinal and transverse motions that decrease exponentially in amplitude as distance from the surface increases. The longitudinal component of the displacement u and the transverse component w can be described as follows :

Rayleigh waves velocities c R

u = A(re -qz -2sqe -sz )cos[k(x -c R t)]
(5.28)

w = Aq(re -qz -2se -sz )sin[k(x -c R t)] (5.29) q = 1 - c R 2c L 2 , s = 1 - c R c T 2 , r = 1 - c R c L 2 (5.30)
where A is the amplitude; k denotes the wave number of the surface wave; x is the displacement value on the surface along the wave propagation direction. The wave velocity in transverse and longitudinal direction are presented as c T and c L . In the case of Rayleigh waves, the velocity c R satisfies the following equation : Let's stress that the Rayleigh surface waves velocity c R is independent on the frequency. Moreover the amplitude of the surface waves in propagation direction decays as 1/x with x being the propagation distance in transverse direction. Thirdly, the relationship among the acoustical waves velocity in transverse, longitudinal direction always follows the inequalities : c L > c T > c R . Last, if the propagating medium is not homogeneous but contains more than one materials layers with different boundaries, the study of the surface waves becomes much complicated. Different waves forms will appear due to the layers interface: bending wave, shear wave, quasi longitudinal wave and Rayleigh wave. We will solely focus here on this last family.

c R = c T ζ ≈ 0.

Surfaces acoustic waves transducers design

The generation of these Rayleigh waves on a piezoelectric material, makes use of inter-digital transducers (IDT). IDTs consist of two interlocking comb-shaped arrays of metallic electrodes, made typically by dozens pairs of parallel fingers. The two electrodes array are with each other ( but not overlap) and will be applied by different voltage of potential between the two set of fingers. The operating principle of the IDT relies again on the converse piezoelectric effect. When applying a voltage difference between the two arrays, the IDTs function is to convert electric signals to surface acoustic waves by generating periodically distributed mechanical forces via piezoelectric effect (an input transducer). The generated waves then propagate in the perpendicular direction with respect to the fingers. The generated surface waves frequency and the conversion efficiency are then set, by the fingers width e and the pitch between two neighboring fingers. In our case, we design the IDTs with a finger width that equals to the finger pitch. The surface waves frequency then writes :

f n = c R • (2n + 1) λ = c R • (2n + 1) 4e
, n = 0, 1, 2... (5.33)

f 0 = c R λ = c R 4e (5.34)
where c R is the velocity of the Rayleigh waves. The frequency f 0 is the lowest order mode, that is also called the characteristic frequency. This frequency depends on the medium in which the SAW is generated. The f 0 frequencies for various II-V material are listed in Table (5.4). One can see that for a given IDT geometry, the generated frequencies differ. Another important parameter for designing IDTs is the surface wave coupling factor k s . k s quantifies the change in the surface acoustic velocities when the waves cross the interface between the IDT electrodes and the piezoelectric substrate. The electromechanical coupling factor k s can be expressed as [START_REF] Ingebrigtsen | Surface waves in piezoelectrics[END_REF] :

k s = 2(c R -c IDT R ) c R = 2 - 2c IDT R c R (5.35)
where c IDT R is the surface waves velocity at the surface of the metallic electrode. High coupling factor k s thus implies to rely on low c IDT R values with respect to c R . By choosing a material featuring a high electromechanical coupling and a high surface acoustic waves velocity c R with a smooth surface, the surface acoustic wave coupling factor k s can be improved with less internal loss in the waves propagation. At the same time, in order to reduce the relative value of c IDT R , the choice for electrodes material as well as the electrodes thickness should be taken into account.

Different metals can be used to process the IDT electrodes (see Table (5.5). Compared to platinum and gold, aluminum and titanium have higher surface acoustic waves velocity c IDT R . Furthermore, gold, copper and platinum are much heavier metal materials compared to aluminum and titanium, which induces a stronger mass loading effect on the IDT with much lower coupling factor k s [START_REF] Zou | Electrode design of aln lamb wave resonators[END_REF] . This effect may reduce the surface acoustic waves velocity [START_REF] Namdeo | Fem study on the effect of metallic interdigital transducers on surface acoustic wave (saw) velocity in saw devices[END_REF] . As a trade-off, IDT electrodes material is usually made by lighter material such as aluminum, and not titanium for its low conductivity. [START_REF] Nazarchuk | Propagation of Elastic Waves in Solids[END_REF] 2896 [START_REF] Grünwald | Simulation of acoustic wave propagation in aluminium coatings for material characterization[END_REF] 3.50E7 Titanium 4508 3182 2958 2.38E6 Gold 19281 1200 [START_REF] Nazarchuk | Propagation of Elastic Waves in Solids[END_REF] 1075 [START_REF] Bassoli | Surface brillouin scattering in polycrystalline gold[END_REF] 4 Another IDT parameter that has to be optimized is the electrodes thickness. To find a optimum IDT electrode thickness, simulations can be performed by finite element method (FEM) in Comsol multiphysics, which provides the deformation displacement of the substrate enabling to induce the electromechanical coupling between the IDT electrodes and the piezoelectric substrate. Since the IDT fingers are periodic, simulation can be performed solely on single pair of the IDT fingers with a periodic boundary condition applied on the two sides. With such simplified model, we can investigate the eigenfrequency, substrate displacement field and so on. To find the optimum IDT thickness, we can simulate the displacement field of the piezoelectric substrate which is deformed by the IDT. Figure (5.17) shows the piezoelectric substrate displacement field amplitude for the modes at f 0 as a function of IDT thickness varying from 5 to 200nm. The larger displacement field, the stronger strain and thus the higher electromechanical coupling is achieved. In general, the deformation induced by IDT is dominant along the x axis displacement. The higher displacement field is reached for an IDT thickness of 50nm. In the following, the IDT thickness will be fixed at this optimized value.

With the optimized IDT electrodes in terms of thickness and period, the next step is to find the IDT characteristic frequency which can be simulated by using the same FEM model. Figure (5.18) shows the induced deformation and the eigenfrequency simulations, for a finger width e of 600nm on top of GaAs substrate. The characteristic frequency f 0 locates at 1.20GHz which is not too far from the theoretical calculation of 1.30GHz. The difference may be due to the fixed constraint condition limited by the singe IDT pair model.

The analytic expression of these frequency also indicate a dependence on the finger width e : for increasing increasese values, the surface waves frequency will decrease linearly. Numerical simulations have again been carried out for e=500, 600 and 750nm IDT with a thickness of 50nm, and compared to theoretical values. The results are shown on Figure (5.19), highlighting a relatively good agreement.The characteristic mode frequency f 0 also depend on the Rayleigh velocity of the material, with higher frequencies achieved in materials featuring higher c R values such as GaN (see following table).

Two-port IDTs introduction

In practice, not only one IDT is used but two IDTs facing each other. These IDTs act as actuators or one as actuator and the other as receiver. In this configuration, the surface acoustic waves can be generated by the two IDTs. In the two actuators configuration, the efficiency of the actuation will then also depend on the distance between IDTs since the two generated surface acoustic waves may interfere destructively or constructively. In order to achieve an efficient excitation, the constructive interaction between the two surface acoustic waves should be favored. In the actuator and receiver configuration, the surface acoustic waves generated by one IDT will propagate in the transverse direction to the second IDT, where it will be detected electrically by again the piezoelectric effect. The distance between two IDTs will result into a delay. The delay time τ depends on the surface acoustic wave velocity c R as: τ = distance/c R . This delay will as such depend on the materials in which the SAW propagates. Time domain simulations have thus been carried out in the single actuator and double actuators configurations to optimize the distance between the IDTs. In this model, the IDTs consist of 50 pairs of fingers with finger width e set at 300nm. A minimum distance between the IDTs will be here considered, in view of locating in between the photonic crystal membrane of width 10µm and length 50µm. To simplify the calculation, we did not consider the presence of this optomechanical platform. This simplified model can briefly give us some trends regarding the surface acoustic waves propagation. More advanced calculations, that will not be carried out here, would be required to retrieve the precise surface acoustic wave decay loss inside the actual photonic crystal area. However, by comparing the displacement field at certain time between different IDTs, we can retrieve qualitatively the expected different propagation behavior. Some results are shown on Figure (5.21). In the single IDT configuration, the surface acoustic waves propagate along the x axis and the waves amplitude decays inside the membrane. This can be explained from the surface acoustic waves basics: the amplitude of surface waves decays as 1/x along the propagating length x. In the two IDTs configuration, the surface acoustic waves interaction between two waves propagating in opposite direction has to be taken into account. Figure (5.21) shows the time domain three dimensional simulation of the surface displacement field evolution from 0 to 8ns. In this simulation, two IDTs are identical. We can first notice the build-up of the interference during the first ns, in relation with the time delay introduced by the SAW propagation to the center between the two IDTs (in red on the Figure). Then, when the surface acoustic waves reach the center area, the two surface acoustic waves interference, resulting in a standing wave. The phase difference ∆ϕ between the two waves is determined by the IDT-to-IDT distance d. This phase difference will lead into constructive interference (with ∆ϕ = kλ = 4ke, k = 0, 1...) or destructive interference (with ∆ϕ = 2/λ + kλ = 2e + 4ke, k = 0, 1...). In our case, the constructive wave interference is preferable to the single wave propagation provided in the single IDT configuration, due to the higher amplitude of the displacement field. In order to directionally guide the generated surface acoustic waves towards the photonic crystal cavity, a extra waveguide funneling the surface acoustic waves, could be placed between the photonic crystal and the IDT transducers. With such phononic waveguide, the generated surface acoustic waves can be routed more efficiently to the cavity with optimized surface waves group velocity. I will discuss about the phononic waveguide introduction in the end of this chapter as the future prospective. The use of high resolution electron beam lithography was imposed by the small finger widths (few hundreds of nm) of the optimized IDTs arrays. High resolution is then achieved by taking care of the electron beam dose that is used. As can be seen on the Figure (5.25a), as the beam dose gets higher, the exposed resist area increases. As consequence, the fingers width gets larger, the gap separating two fingers gets smaller and the frequency of the generated SAW decreases. For doses of 900µC/cm 2 , the finger width is 659 nm and slightly wider than the gap width.

IDTs processing

Figure 5.26 -Evolution of the IDT fingers gap and pitch widths for increasing e-beam dose value. The pitch width is the sum of the finger width and gap, that is the period. To achieve 600nm in finger width with 1200nm pitch width, 900 or 1000µC/cm 2 dose should be used with target finger width 550nm instead of 600nm.

To quantify the correct beam dose value, we also investigate the finger widths and period for different e-beam dose values. For doses increasing from 900 to 1400µC/cm 2 , the finger width increases by 10 to 12nm per 100µC/cm 2 . As the gap width also decreases with the dose increment, the total pitch width remains to the constant value around 1250nm. In order to reach the targeted width of 550nm, we should reduce slightly the dose. Yet, he minimum e-beam dose is 800µC/cm 2 for exposing the resist. We thus chose an optimal dose of 900µC/cm 2 .

Another issue occurs during the IDTs metallisation, that can lead to electrodes stripping due to the bad adhesion contact between the metallic layer and the III-V substrate. There is several reasons that may explain such effect : bad metal adhesion, residues of resist on the substrate before metallisation or improper lift-off solution. For some metals like gold, the adhesion between the metal and the substrate is quite bad. Generally speaking, the common solution is to deposit another thin adhesion metal layer (like Titanium) before depositing the gold layer. In our case, the IDTs are made of Aluminum which usually does not require extra adhesion metal layer. The IDT stripping as shown in Figure (5.27b) and (c) are due to some residues of resist which remained on the III-V substrate. To avoid the IDTs stripping, the best solution is to run a quick and gentle etching process (such as CCP-RIE or thermplasma etching) to remove all the oxidized particles on the III-V substrate before the IDTs patterning. With all these optimization of the fabrication process, we can eventually fabricate the IDTs with the target widths and thickness. After depositing the IDTs and the contact electrodes, the photonic crystal as well as the mesa structure can be processed along the same steps as discussed in Chapter 3.

In order to characterize the efficiency of surface acoustic waves generation with the IDTs, the device has been first simplified with no SOI waveguide integration. The platform is simply bonded on the silicon substrate for purely electrical measurement. In the final devices, three series of IDTs arrays with different finger widths ranging from 500nm to 700 nm have been processed. In each series, some devices do not include any L3 photonic crystal cavity between the two IDTs and other do, in order to evaluate the impact of the photonic crystal on the surface acoustic waves transmission (see Figure (5.28)).

Electrical characterization of the IDTs

The electrical characterization of the IDTs was carried by use of a vector network analyzer (VNA). It enables to retrieve the network scattering parameters, or S-parameters S ij . The parameters S 11 and S 22 are electrical reflection coefficients. The parameters S 12 and S 21 represent transmission from source to load, and vice versa. The magnitude of the coefficients S ij can exhibit a resonance peak or drop at the IDTs resonance frequency. The diagram in Figure (5.29) shows the essential parts of a typical 2-port vector network analyzer (VNA). The two ports which correspond the two IDTs on the two sides of the device-under-test (DUT), are denoted port 1 (P1) and port 2 (P2). The RF frequency is generated by a variable frequency CW source and its power level is fixed at 0dB (1mW). Initially the test signal is incident on the DUT at P1 which measures the reflection and transmission signal S 11 and S 21 . When the test signal is applied to P2, we can measure the reflection and transmission signal S 22 and S 12 . Prior to the measurements of the S ij parameters, a calibration procedure is however required. The dissipation of the power is usually frequency dependent. In order to insure accurate measurement, we considered different frequency domains: 500MHz to 3GHz, 3GHz to 5GHz, 5GHz to 6GHz and 6 GHz to 8GHz. The noise level induced by the power loss for each domain is different.

The different finger width e

The S-parameter measurements features some resonances that the fundamental resonance frequency f 0 but also at higher order resonance frequencies f n for IDTs with width e equal to 500nm, 600nm and 750nm. Figure (5.30) shows the SAW modes resonance frequency f 0 and f 1 for different finger widths, observed on the reflection signal S 11 . The measured characteristic frequency f 0 for 500nm IDTs is 1.23GHz. By comparing with the simulation results, the measured resonance modes are around 200 to 400MHz lower in frequency. This difference is mainly due to the geometry deviation during fabrication. By checking the geometry from SEM images, the finger width is around 20% to 50% larger than the targeted value, thus, the measured characteristic frequency gets smaller than the target value. The possible reason for such lower measured f 0 value may be due to the defects inside the GaAs substrate, that would decrease the surface acoustic waves velocity c R . Table 5.7 -Comparison between simulation SAW modes frequency f si 0 and measured f ex 0 IDT e (nm) 500 600 750 f si 0 (GHz) 1.61 1.20 0.82 f ex 0 (GHz) 1.23 0.87 0.62 When plotting the full reflection spectrum S 11 as a function of frequency from 500 MHz up to 8 GHz, the different resonant frequencies of the fundamental and higher modes can be identified. For IDTs finger widths of 500 and 600nm, the first order mode at f 1 is located at 2.46 and 2.10GHz respectively. Let's here remind that in our GaAs nanobeam system, the optomechanical mode featuring the higher coupling strength g 0 /2π lies at Ω m = 2.277GHz which is close to our SAW frequency. By slightly adjusting the IDTs finger width, we could modify the f 1 frequency to put it in resonance with Ω m . We also notice that the dependence of the frequency of the modes as a function of the finger widths, feature a linear relationship f = A * (2n + 1) with n being the mode order and A = c R /4e (see Figure (5.32)). This dependence is consistent with equation (5.33). By fitting of the experimental data, the A values equal 0.493, 0.602 and 0.505 for IDTs widths of 500, 600 and 750nm respectively. The value of c R are then 0.99, 1.44 and 1.515km/s. These values are slightly lower than the theoretical one (3.11km/s), probably due to the slightly larger finger widths in the processed sample compared to the targeted widths.

When comparing the reflection dips depth of the different SAW modes, i.e. the SAW power in these modes, we can see that for the lower order of modes, the resonance Figure 5.31 -Reflection signal S 11 for different IDTs widths equal to 500 nm (blue line), 600 nm (red line) and 750 nm (yellow line) in the frequency range between 500MHz and 8GHz. The resonances are indicated by black dash rectangles. In higher frequency range, the resonance mode has a larger amplitude due to higher SAW power compared to the fundamental mode. digs are relatively shallow compared to the higher order modes. Higher orders modes have stronger SAW power. However, the high frequency also means faster SAW wave velocities, which may be less favored compared to lower order modes in our case. One can measure not only the S 11 parameter, but also the S 22 , S 12 and S 21 parameters. These are shown on Figure (5.33) in the frequency range corresponding to the third-order mode f 3 . In this figure, the same IDT fingers width is 500nm on both IDTs arrays. All spectra show a resonance at the SAW mode resonance located at 4.12 GHz. For both reflection signal S 11 and S 22 parameters, the resonance appears as a dip while for the transmission signals S 12 and S 21 parameters, the resonance spectrum features a more complicated shape. This may be due to the account parasitic effects which the electromagnetic signal through the electrical tips may has interference [START_REF] Clement | Saw characteristics of aln films sputtered on silicon substrates[END_REF] . As a perspective for the future measurement, since the electromagnetic signal propagates via air is much faster than the acoustic wave via the substrate, by using extra time gating method can eliminate the most of electromagnetic signal. With time gating, the transmission signal can exhibit its real shape.

SAW power conversion efficiency η SAW and electromechanical cou-

pling factor k 2 s SAW power η SAW

In Figure (5.31), one can observe that the noise level in the different frequency domain is different. In the lower frequency range from 500MHz to 5GHz, the noise level increases with the frequency. For higher frequencies ranging from 5 to 8GHz, the noise level varies slightly with the frequency and can be regarded as constant. The noise level is related to the power loss in the environment compared to the input power. Generally, in order to quantify the electric power that is converted into the acoustic power P a , one relies on the following relation : (5.36) where P rf is the RF power applied to the IDTs. From this relation, we obtain the following equation :

P a = |S ii | 2 P rf
P rf -P a = P rf (1 -|S ii | 2 )
(5.37)

The RF power can be split into three terms : the acoustic power P a , the power loss due to the radiation and which is the dissipated power into the environment P R and the power converted into the SAW P SAW . Consequently :

P rf -P a = 2P SAW + P R = P rf (2η SAW + η R ) (5.38) 1 -|S ii | 2 = 2η SAW + η R (5.39)
with η R = P R /P rf and η SAW = P SAW /P rf . Since the SAW modes propagate back and forward in reflection signal, there is factor of 2 in equation. The amplitude of η SAW relates to the overlap between the stress field of SAW mode and the stress field induced by the input power and the η R ratio is usually frequency dependent term. However, in some frequency range as shown in Figure (5.31), the η R varies slowly as the frequency increases and can be regarded as constant from 5 to 6GHz. For IDTs widths equal to 500, 600 and 750nm, the inferred SAW power conversion efficiency η SAW is 0.160, 0.134 and 0.114 respectively, with a power ratio η R dissipated in the environment around 0.06. The SAW power ratio η SAW can be used to characterize the efficiency of the conversion of the applied power into the surface acoustic waves modes. In the Table5.9, the η SAW values for different modes at finger widths e = 500nm. However, it is interesting to noticed that besides mode f 5 , the η SAW values of all the other modes are in the same magnitude. For mode f 5 , with still unknown reason, it has at least 10 times stronger in SAW power. This is validated for all the devices of different IDTs widths. Therefore, working with mode f 5 can be more efficient with larger power for the future resonant excitation study.

Coupling efficiency k 2 s

The IDTs performance is also characterized by the electromechanical coupling factor k 2 s given in equation (5.35). In the S-parameters measurement, the coupling factor k s is given by the maximum bandwidth of the acoustic signal and the amount of electrical loss between input and output. In practice, the measurement coupling factor can be written as:

k 2 s = G a 8f 0 C t N (5.40)
where G a is the radiation conductance of the SAW modes, C t is the capacitance of a single IDT pair, N is the number of IDTs pairs that equals 50 in our case. In order to infer these parameters, one has to remind that the reflection signal is related to the complex impedance Z = R + jX, where R is the IDT resistance (real part) and X is the reactance (imaginary part).

k 2 s = G a 8f 0 C t N = πG a 4N B a (5.41)
Figure (5.36) shows the admittance in the frequency range of the fifth-order mode at 5.73GHz for IDTs width of 500nm. The conductance value G a is 1.155 ohms and the susceptance value B a is 1.1827 ohms. The corresponding electromechanical coupling k 2 s is then 1.53%. For 600nm and 750nm IDTs width, this value for the fifth-order mode equals 1.46% and 1.33% at frequencies of 5.466 and 5.338GHz respectively. We can see that narrower finger width provides larger coupling factor k 2 s . If we focus on the IDTs width of 500nm and extract the coupling factor k 2 s for all modes (as shown in Table(5.10)), we observe that the highest k 2 s is achieved for fundamental mode. For higher order modes, the k 2 s tends to decrease.

Influence of the suspended membrane

In order to estimate the impact of the photonic crystal membrane on the SAW modes, we processed two kinds of devices. A first family consists of solely the the two IDTs arrays facing each other. A second family embed in addition the photonic crystal suspended membrane between the two IDTs (see Figure (5.37)). However, during the processing of this second family, some of the platforms have been 'accidentally' overetched, inducing an underetching below the IDTs. The polymer underneath the piezoelectric layer can also influence the IDTs property. If the membrane is not isolated from the SiO 2 and BCB, the generated surface waves will be partially absorbed in the BCB layer. In order to estimate the impact of the underetching and absorption, we measured the S-parameters on three different platforms (see Figure (5.38)) and extracted the the SAW power ratio η SAW (see Table5.11). Figure (5.38) displays the reflection S 11 spectrum for IDTs devices with the same width of 600nm. Two of them embed a photonic crystal membrane. One of them features released IDTs. First, one can observe a shift in the resonances. For platforms embedding a photonic crystal membrane, the first modes at f 0 and f 1 feature a resonance frequency at 0.87 and 2.10 GHz respectively, while for the device with no photonic crystal, these resonances slightly shift to 1.05 and 2.13GHz. The difference in resonance frequency may relate to the fabrication. For the IDTs devices with the same width and on the same substrate, the resonance frequency should be the same. If there is a shift in frequency, the most possible reason is due to the tiny difference in finger width. One can also observe on the S 11 spectra that the more membrane is released, the deeper is the reflection dip and the higher SAW power ratio. The depth of the reflection dip is an important parameter since it related to the power funneled in surface acoustic wave mode. For the devices integrating both released IDTs and the photonic crystal membrane, η SAW equals 0.019 and 0.033 for respectively the fundamental and first-order modes. These values are higher compared to the η SAW values achieved on the two other devices. When only the photonic crystal is suspended, the SAW power ratio is slightly lower and for the unreleased device with no photonic crystal, it is several orders of magnitude lower (see Table5.11). This is especially true for the characteristic mode at f 0 . The influence of the membrane suspension level on the surface acoustic waves power ratio gives us some indications on the propagation loss. For non-suspended membrane, the generated surface acoustics waves will propagate on the surface but a part of it will penetrate into the substrate. This penetration depth is a frequency dependent term: the modes with higher frequency will have shallower penetration depth. Thus, suspension has a fewer impact on η saw for the higher order modes. However, during the processing of the IDTs device, the piezoelectric substrate (GaAs or GaN) is bonded on silicon wafer by a BCB bonding technique (see Chapter 3). The BCB is a benzene-based polymer with no piezoelectric property. The BCB layer is less than 100nm thick under the piezoelectric layer which thickness is 265nm. In order to see the polymer influence on the surface acoustic waves, we changed the substrate heterostructures underneath the piezoelectric material in the simulation model. As shown in Figure (5.39), in order to see the influence of the BCB layer, we considered three different substrates : (a) a GaAs membrane on a silicon substrate, (b) a GaAs membrane bonded on a silicon substrate via a thin BCB layer and (c) a suspended GaAs membrane. We then plotted the field distribution at the frequency of the characteristic mode. For the structure (a), the SAW mode is not confined at the surface but expands smoothly in the silicon substrate. The presence of the BCB layer significantly reduces this penetration depth. The Rayleigh velocities c R inside the silicon substrate is indeed 5.02km/s [START_REF] Udo | Fem simulation of rayleigh waves for cmos compatible saw devices based on aln/sio2/si(100)[END_REF] , while c R is supposed to be very low inside BCB material. Despite this change in the penetration depth, the displacement intensity remains similar between the two structures. As a comparison, for the suspended membrane structure, the mode is dominantly confined in the piezoelectric layer (GaAs). In our processed devices, the devices featuring released IDTs and photonic crystals, are similar to the (c) structure while the platforms featuring unreleased IDTs behave like the (b) structure. As a consequence, If the IDTs arrays are not suspended, the BCB layer may cause huge power loss on SAW modes. Importantly, at least of the two IDTs arrays should be released to enhance the η SAW ratio. Figure (5.40) shows the S 11 and S 22 spectra of platforms exhibiting either one released IDTs or no released IDTs. We first observe that for each device, he S 11 and S 22 spectra are very similar as expected. We also observe that only the devices featuring released IDTs exhibit strong resonance dips, with η SAW equal to 0.134 and 0.114 for the modes at 5.47 and 5.73GHz respectively. These η SAW values are in the same range as the one observed on platforms with two released IDTs. Conversely, for the devices with non-released IDT, the reflection spectrum is almost flat with no resonance.

Processing of suspended IDTs arrays enclosing a suspended photonic crystal

In order to improve the SAW power, one important parameter is to release the IDTs membrane. One first strategy could be to increase the photonic crystal underetching time. This may be very risky since it could damage all the IDTs and photonic crystal devices. An alternative solution is to design small holes in between the IDT fingers. With proper design of such holes, we can release the IDTs structures during the underetching of photonic crystal. Figure (5.41) shows SEM images of the IDTs holes and of the photonic crystal. One can observe the holes related to the photonic crystal structures as well as a zoom on the holes between the IDTs fingers. In order to insure the IDTs release, an optimum on the underetching duration and holes radius has been found. The optimal radius, of the holes between the fingers for the underetching duration used for suspending the photonic crystal, is about 100 nm.

Figure 5.41 -SEM images of fully suspended IDTs devices. The small holes with a diameter of 100nm between IDTs fingers are used to ease the substrate underetching.

As a conclusion for this section, we processed different IDTs structures with different finger width, ether suspended or not. Large IDTs fingers widths (above 600nm) result in a shift of the resonance to lower frequencies and smaller electromechanical coupling factor k 2 s . Optimal fingers widths are about 500 nm -600 nm. Due to the fabrication deviation, the measured resonance frequencies on the first devices are slightly lower than the targeted values, which will be taken into account for the next series of devices. Our results also highlighted that the release of the IDTs enhance significantly the SAW power ratio. This release enables to overcome the penetration and absorption of the SAWs in the underlying layers (substrates or BCB).

Summary and conclusion

This chapter introduced three possible different actuation means including electrostatic, piezoelectric and acoustic actuation, in view of resonantly excite the mechanical modes of photonic crystal-based optomechanical resonators in the form of twodimensional L3 cavity or one dimensional nanobeam cavity, enabling to resonantly excite mechanical modes up to the GHz range.

The electrostatic actuation is based on a capacitor located around the nanobeam cavity, enabling to excite mechanical modes up to the GHz frequency range. Since the capacitive force depends on the electrode-to-electrode distance, this distance should be controlled within 1µm in order to insure efficient actuation. This actuation scheme doesn't need direct contact with the photonic crystal cavity, and as such does not influence on its mechanical properties such as its effective mass. Another advantage comes from its ease of fabrication : the electrodes and the photonic crystal can be easily integrated. However, due to fixed electrode-to-electrode distance, there is no possible way to adjust and control any mechanical delay.

The two other actuation means are all based on the piezoelectricity of the hosted material, such as GaAs, GaP, or InGaP. In this context, the induced external electrical field can generate strain inside the material either uni-directional or bi-directional. Two schemes can then be implemented : direct piezoelectric actuation and acoustical actuation, for which the generated strain takes the form of a periodic surface deformation (surface acoustic waves). Though piezoelectric actuation is more efficient, the direct contact of the electrodes with the nanobeam resonator can degrade the optical and mechanical properties of the resonators, in particular in terms of losses; moreover, the acoustic delay is here almost null due to the short distance separating the resonators and the electrodes. Compared to piezoelectric actuation, acoustic actuation provides advantages. A first advantage is that IDTs are not in contact with the photonic crystal cavity system, preserving their optical and mechanical properties. However, in order to achieve efficient acoustic actuation, the resonance frequency of the SAW Ω SAW has to to be tuned to the cavity mechanical frequency Ω m .. With an IDTs finger width e of 500nm, the SAW first-order mode frequency is 2.46GHz which is close to the measured frequency Ω m of the mechanical mode sustained by the GaAs nanobeam resonator (see Chapter 4). In addition, releasing the IDTs arrays largely enhances the SAW power, enabling more efficient acoustic actuation. In order to release the IDTs, small holes (100nm in diameter) have been pierced between the IDTs fingers to suspend the IDTs during the underetching of the photonic crystal membrane. Most importantly, the acoustic actuation allows to integrate a phononic circuitry on the platform in view of implementing an acoustic delay. It could take the form of a phononic waveguide modifying the surface acoustic waves velocities according to the different propagation modes. With such acoustical delay, the feedback loop could be integrated to the system for acousto-optomechanical oscillation and further signal stabilization.

All three actuation means could be interesting to achieve high electrooptomechanical or acousto-optomechanical coupling. The resonant actuation of the mechanical modes by those methods can allow to reach high mechanical motions as well as to enhance the photon-phonon interaction up to achieve the strong coupling regime directly in GHz range. This would enable to study the nonlinear dynamics of optomechanical resonator [START_REF] Xiong | Cavity piezooptomechanics: Piezoelectrically excited, optically transduced optomechanical resonators[END_REF] and potential applications for optomechanical cooling and amplification.

Chapter 6 Conclusion and perspective

Conclusion

This work aimed at exploiting the photon-phonon interaction in optomechanical resonators (OMR) in view of proposing a novel architecture of reference oscillators in the GHz. The proposed platform consists of photonic crystal-based cavities etched in suspended III-V piezoelectric semiconductor membranes, coupled to integrated optical waveguide for optically probing and controlling the mechanical modes. These cavities are either L3 cavities in a 2D-photonic crystal or 1D-photonic crystal nanobeam cavities. Both cavities not only provide high light optical confinement but also harbor mechanical modes directly in the GHz frequency range featuring a large spatial overlap with the optical mode. The stability of the oscillators would be insured via the resonant driving of the mechanical mode implemented by electrostatic, piezoelectric or acoustic transducers. Acoustic actuation would in addition allow for engineering an integrated acoustical delay, enabling to integrate on a single chip setting the OMR and the feedback loop circuit.

Chapter 2 described the design of the OMR architecture. We discussed in particular the various strategies to be implemented in order to enhance the optical confinement in time in L3 cavities and nanobeam cavities, hosted either in GaAs or GaN/AlN. Such strategies make use of slight modifications in holes radii, positions or even shapes up to elliptical or rectangular cross-sections. This enabled us to increase the optical quality factors Q 0 from 4900 up to 24000 in GaAs-based L3 cavities and from 500 to 1240 for GaN-based resoators. The optical quality factors of nanobeam cavities was also improved: our optimized nanobeam cavities formed by rectangular and elliptical holes, shall feature Q 0 up to 2.2 × 10 8 and 1.8 × 10 8 in GaAs respectively and 8.9 × 10 5 and 2.0 × 10 4 rin GaN. These cavities also feature mechanical modes; these modes have been identified; some of them are resonant in the GHz range and feature a high spatial overlap with the optical modes, translating into a high optomechanical coupling strength g 0 /2π. This coupling is governed by two main processes: moving interfaces and photoelastic effects. In our case, g 0 /2π reaches values up to 579.45kHz in GaAs L3 cavities and 807.11kHz and 1378.50kHz in GaAs nanobeam cavities formed by rectangular and elliptical holes respectively. Due to the lower optical confinement in GaN material, the optomechanical coupling strength is relatively low: the highest coupling g 0 /2π achieved reaches 50.79kHz and 160.56kHz for L3 cavities and nanobeam cavities respectively. The processing of these architectures is however challenging. The various hurdles we faced, were described in Chapter 3. Main difficulty arose from the full integration of the III-V semiconductor membranes upon SOI substrates. The crucial step here is the III-V substrate removal after adhesive bonding, especially for the high tension material such as GaN and AlN. By investigating different dry etching techniques, the process for GaN membranes substrate removal could be optimized and controlled. A second difficulty arose from the etching of the photonic crystal, in particular in view of achieving a photonic crystal pattern as close as possible to the targeted one and as such preserving the high optical confinement in the cavities. This required several optimization steps during the lithography and etching steps. Finally, the plasma-therm dry etching technique was used to suspend the photonic crystal system. Eventually, SOI integration of the L3 cavity on GaAs and GaN substrate, as well as SOI integration of the nanobeam cavity on GaAs substrate have all been successfully realized. However, SOI integration of the nanobeam cavity on GaN substrate was very difficult to achieve due to the GaN substrate exfoliation. In order to overcome the high tension issues in GaN substrates, the optical waveguide design was modified. We decided to rely on a curved GaN waveguide instead of SOI waveguide, which we succeeded to process. Yet such design limits the optical investigation to reflection measurements and do not enable any transmission measurements.

In Chapter 4, we investigated both the optical and optomechanical response of our OMRs. The optical analysis relied on transmission and reflection measurements. For GaN-based membrane integrating both the nanobeam cavities and optical access waveguide, optical reflection measurements enabled us to observe the fundamental optical resonances which revealed to be slightly higher in wavelength than the targeted resonances. The obtained measured Q 0 s are around 1000 to 3000, two orders of magnitude lower than the simulated Q 0 ∼ 10 4 to 10 5 . Such Q 0 values are however reasonable by taking into account the fabrication deviation. However, even with Q 0 ∼ 10 4 , the theoretical optomechanical coupling strength is quite weak. With Q 0 around 3000, it is not possible to observe any optomechanical coupling in GHz range. For cavities integrated upon the SOI substrates, technical issues during the realization process damaged the optical gratings enabling efficient light injection in the waveguides. The resulting limited optical transmission did not allow us to further study the optomechanical coupling on most of these platforms. We could however observe various optical resonances sustained by GaAs nanobeam cavities corresponding to the fundamental modes up to the sixth order modes in the range of 1500 to 1600nm. The intrinsic Q 0 of the fundamental mode can reach values as high as 3 × 10 4 with a coupling efficiency η around 67% to 75%. The intrinsic Q 0 of the fundamental mode can reach 3 × 10 4 with the coupling efficiency η around 67% to 75%(under-coupled regime). For higher-order modes, the coupling efficiency increases, at the cost of a reduction of the intrinsic Q 0 . By varying the nanobeam geometrical parameters such as lattice constant a and holes radius r, the evolution of the optical resonances revealed to be in the good agreement with the simulated values by taking into account the fabrication deviation. Thanks to the high achieved Q 0 , the optomechanical response of such cavities could further experimentally studied. The cavity harbors mechanical modes around 2.24 to 2.43GHz, with a corresponding coupling strength g 0 /2π around 69.2 to 143.9kHz. By tuning the injected laser wavelength λ as well as the power P in , we could observed a non-linear behavior. As the laser power increased, the cavity resonance shifted due to the temperature change due to the emergence of thermo-optic effects. Several absorption mechanisms take place here, including free carrier, two-photon, Kerr-effect and so on. In our platform, by studying the relationship between internal energy U c and absorbed power P abs , we assumed that the free carrier absorption and two-photon absorption are dominant. Moreover, as the laser was detuned from the cavity resonance, changes of the mechanical modes frequency Ω m and linewidth Γ m could be observed. With a fixed laser detuning and increase of the laser power, an amplitude enhancement and linewidth narrowing on one mechanical mode have been measured.

However, the regime of mechanical self-oscillation could not be reached. By studying the threshold power P in to reach self-oscillations, we identified several limitations in our experiment. First of all, the laser detuning is too high, which limits the number of photons inside the cavity. With less than 10 2 photons, a coupling strength g 0 /2π up to 4200kHz is required. Second, the optical mode linewidth is also too large, limiting again the number of photons circulating inside the cavity. Last, mechanical modes competition in the optomechanical coupling interaction also prevent to reach the regime of self-oscillation.

In Chapter 5, we focused on three different resonant excitation methods : electrostatic capacitive actuation, piezoelectric actuation and acoustic actuation. The two latter ones are more preferable in the GHz frequency range. Both make use of the piezoelectricity f our host materials. For electrostatic actuation, the modification of the electrode-to-electrode distance allowed to optimize the optomechanical coupling G for the modes in GHz range. For piezoelectric actuation, our simulation showed that the displacement induced by piezoelectric effect can effectively excite the nanobeam cavity mechanical resonances in GHz range. For acoustic actuation, electrical measurement on IDTs devices revealed different high-order SAW modes featuring comparable SAW power in the GHz frequency range. When integrated on the complete suspended architecture, the IDT devices exhibited a strong SAW power enhancement especially for the higher order modes. To be noticed that the first order SAW modes locate around 2.1 and 2.5GHz, which is quite close to our measured mechanical frequencies Ω m in GaAs nanobeam cavity-based OMRs.

To conclude, in order to build a low phase noise optomechanically-driven reference oscillator (OMO), we firstly designed a SOI integrated OMR based on III-V piezoelectric photonic crystal cavities with both high optical confinement and strong optomechanical interaction in GHz range. To further stabilize the frequency from frequency drifts and thermodynamics noise, three external actuation methods have been introduced : electrostatic, piezoelectric to acoustic actuation. In particular, acoustic actuation can help for further integration, enabling to build on chip acoustical delay and feedback loops for signal stabilization in OMOs.

Perspectives

As mentioned previously, in order to further reduce the phase noise, self-excited OMRs can be controlled with a delay line feedback loop circuit. With this purpose, several perspectives to our work could be pursued.

Architecture of the fulll optomechanical oscillator

The first generation of integrated acousticallly-driven L3 photonic crystal resonators (OMRs) has been realized (see Figure (6.1). The system is built on the GaAs substrate with five independent L3 cavities (in green) located on one single platform. A SOI waveguide (in red) is integrated underneath the suspended L3 cavities membrane (in blue). On the two sides of the photonic crystal, two SAW transducers (in yellow) are processed for acoustical excitation. The L3 cavity based OMR was designed to operate at 1.55µm optical wavelength with mechanical displacement field in the GHz range. The vibrational modes can be excited acoustically via the SAW transducers and read either optically through the SOI waveguide or acoustically through another SAW transducer acting as a signal receiver.

However, this first generation of OMRs, did not integrate any phononic waveguide. Some preliminary designs of phononic waveguide are proposed in the next section. With such devices, we could perform optomechanical measurement by use of an external electrical excitation on SAW transducer for generating surface acoustic waves. The SAW transducer would be isolated from the photonic crystal cavity in order to maintain high optical properties of the cavity. The efficient actuation requires both high electromechanical coupling from the electrodes as well as high optomechanical coupling to the photonic crystal cavity. The electromechanical coupling of the SAW transducer is measured around 3% which is comparable to state-of-art values. The optomechanical coupling rate for InP-based L3 cavity was measured as 90kHz [START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF] . The resonance frequency between the optomechanical cavity and SAW modes should also be matched.

This architecture presently introduced for 2D-photonic crystal platforms, can also be extended to nanobeam cavity-based architectures (see Figure (6.2)). Similarly to the L3 cavities, the nanobeam cavity OMO is also composed of a SOI waveguide (in red ), a nanobeam cavity (in red dash circles), SAW transducers (in yellow) as well as a phononic waveguide. The nanobeam is designed for harboring colocalized optical modes resonant at 1.55µm and GHz mechanical modes. To be noticed that here a single SAW transducer is proposed. To resonantly actuate the nanobeam cavity, a single SAW transducer is sufficient. The phononic waveguide is designed as a bent V-shape waveguide [START_REF] Li | Bending and branching of acoustic waves in twodimensional phononic crystals with linear defects[END_REF] .

Delay feedback loop integration

The required frequency stability and low phase noise level of the oscillator could be achieved by making use of a feedback loop circuit. This feedback loop circuit shall include a delay line in the feedbackloop determines the oscillator frequency and needs to be equivalent to the resonant frequency of OMR. The architecture of the feedback loop circuitry could drow from the architecture of delay line delay line based Light wave from a wavelength tunable laser is sent to a Mach-Zehnder modulator, which is modulated by the self-oscillating signal in response to the signal from the feedback loop. This modulated light is then coupled to the OMR and passes through an optical our acoustic delay line. The delay line here provides the needed quality factor for low noise oscillation. The laser signal is then detected by a fast photoodiode that converts the optical signal into an electrical signal to probe optically the mechanical vibration and to interlink the optical and the electrical part. This latter consists of a filter, that selects the desired frequency of interest, and a phase shifter. The electric signal is then fed back to the electric port of the modulator. This configuration supports self-sustained oscillations, at a frequency determined inter alia by the delay length. Such OMO's configuration shall enable to reach low phase noise levels down to -80dBc/Hz at 100Hz offset and -120dBc/Hz at 10kHz offset. The design of the delay line is however crucial in view of achieving such low phase noise and high frequency stability. Two kinds of delay loops can be implemented: optical or acoustic delay lines. Main advantage provided by acoustic delay lines is their compactness, due to the much slower velocity of SAWs compared to light.

Acoustical delay line design

Phononic crystals, similarly to the photonic crystals, are engineered composite materials featuring a periodic spatial modulation of elasticity, mass density and longitudinal and transverse speed of sound, at the scale of the acoustic wavelength. Typical structures consist of scattering centers with acoustic properties different to the homogeneous matrix surrounding the scatters. As in photonic crystals that can implement photonic bandgaps, interference in phononic crystals can give rise phononic band gaps, a range of frequencies in which elastic waves cannot propagate inside the crystal. This band structure can be engineered by tuning the geometrical properties such as the crystal pattern or the scatterers shape, dimensions or positions, holes and so on. Again similarly to photonic crystals, photonic crystals can be used as toolbox to tailor on chip the confinement, propagation, filtering or demultiplexing of acoustic or elastic waves [START_REF] Khelif | Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency[END_REF], [START_REF] Zhi | Acoustic wave propagation in one-dimensional phononic crystals containing helmholtz resonators[END_REF], [START_REF] Khelif | Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal[END_REF], [START_REF] Pennec | Tunable filtering and demultiplexing in phononic crystals with hollow cylinders[END_REF], [START_REF] Li | Bending and branching of acoustic waves in twodimensional phononic crystals with linear defects[END_REF], [START_REF] Qiu | Mode-selecting acoustic filter by using resonant tunneling of two-dimensional double phononic crystals[END_REF], [START_REF] Zhang | Broadband wave filtering of bioinspired hierarchical phononic crystal[END_REF] . In our proposed architecture of OMOs, a phononic crystal waveguide is introduced to guide the acoustic waves to the photonic crystal cavity. It could consist of a honeycomb lattice of air holes as shown in Figure (6.4) . The band structure of such lattice can be modified according to the geometrical ratio r/a between the holes radius and lattice constants, or filling factor β which refers to the ratio between the periodic cylinders area (air holes in our case) to the overall unit cell area. The filling factor β in our honeycomb phononic crystal can be calculated as: We in particularly focused on photonic crystals etched on GaAs suspended membranes. As shown in Figure(6.5a-b), the phononic crystal structure, for our chosen geometrical parameters, give rise to two different bandgaps located around 3.5 GHz and separated by two propagation modes. The width of these bandgaps varies with the the filling factor values. In particular the higher frequency bandgap disappears for increasing βvalues. The bandgap structure also depends on the ratio r/a. We also extended these calculations to GaN membranes (see Figure (6.5c-d)). Here the bandgap, for our chosen geometrical parameters, locates at 1GHz. By increasing the filling factor, the bandgap gets larger.

With the knowledge of the phononic crystal band structures and the bandgap location,phononic crystal waveguides can be designed. The phononic crystal waveguide is formed by removing one or several lines of holes as shown in Figure (6.6). From the left to the right, the SEM images show the phononic crystal forming a phononic shield (no missing line) and the phononic waveguide within the phononic crystal. More importantly, the traveling waves group velocities in such waveguides is frequency dependent and can be tuned by engineering the dispersion relation, enabling to implement slow wave phononic waveguides and a such acoustic delay lines. Compared to light speed of c ≈ 3×10 8 m/s, the surface acoustic waves c R is around 3000 to 5000 m/s in most of piezoelectric III-V materials, and the phononic crystal waveguide could reduce this velocity down to 1000m/s or even lower [START_REF] Kurosu | On-chip temporal focusing of elastic waves in a phononic crystal waveguide[END_REF] . This would afford for increased compactness of the delay line..

Reaching ultrastable microwave oscillation

We here described a first strategy to achieve high stability of the self-oscillation, that is delay line-based oscillator making use of a feedback loop. One could also envision other strategies. These include injection locking via electrostatic/piezoelectric actuation, or even synchronization of several OMR's. Synchronization of coupled oscillatory systems could pave the way towards the investigation of nonlinearities [START_REF] Bagheri | Photonic cavity synchronization of nanomechanical oscillators[END_REF], [START_REF] Qiao | Quantum synchronization of two mechanical oscillators in coupled optomechanical systems with kerr nonlinearity open[END_REF] , phase noise suppression [START_REF] Zhang | Conference on Lasers and Electro-Optics (CLEO)[END_REF], [START_REF] Zhang | Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light[END_REF] , the phase [START_REF] Shlomi | Synchronization in an optomechanical cavity[END_REF] or frequency locking [START_REF] Walter | Quantum synchronization of a driven self-sustained oscillator[END_REF] and collective behaviors [START_REF] Zhang | Conference on Lasers and Electro-Optics (CLEO)[END_REF], [START_REF] Heinrich | Collective dynamics in optomechanical arrays[END_REF], [START_REF] Ying | Quantum manifestation of a synchronization transition in optomechanical systems[END_REF] . Our fully integrated platform would be here an ideal testbed, enabling the coupling of several OMRs either optically (via the integrated optical waveguide), optomechanically or even acoustically (via an acoustic link) as shown in in Figure (6.7). where ε is the strain tensor and C is the photoelastic tensor. For the second-rank strain tensor ε, there is six coefficients in the form of an array with applying Voigt notation:

ε =    ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz    =           ε xx ε yy ε zz ε yz + ε yz ε zx + ε xz ε xy + ε yx           =           ε xx ε yy ε zz 2ε yz 2ε zx 2ε xy          
Strain in crystals is created by deformation and the elements in the strain tensor is defined as relative lattice displacement:

ε ij = ε ji = 1 2
dq i dx j + dq j dx i (6.3)
where q i is the displacement of the lattice point under the study along x i . In the photoelastic tensor, the elastic coefficients C ijkl have relationship with the strain and stress tensors as:

σ ij = - 3 k=1 3 l=1
C ijkl ε kl (6.4) where σ ij is the stress coefficients inside the second-rank stress tensor, and the equation(6.4) is also called stress-strain relation. Both stress and strain are secondrank tensors, to relate them, C ijkl has to be a fourth-rank tensor and can be wrote in The crystalline structures of these materials should be taken into account in photoelastic tensor calculation. Due to their different symmetries, the photoelastic tensor patterns can be reduced in matrix notation. For example, the Zinc blend structures is cubic crystal symmetry and Wurtzite structure is hexagonal symmetry. According to the J. F. Nye( [START_REF] Dobson | Physical properties of crystals and their representation by tensors and matrices[END_REF] ) study in crystal properties, the triclinic symmetry requires 21 coefficients while cubic symmetry requires only 3 coefficients. The reproduced image done by Okaya( [START_REF] Okaya | CALCRUST (1985) seismic reflection survey, whipple mountains detachment terrane, california: An overview[END_REF] ) is shown below. Figure 6.8 -The patters of photoelastic tensors for crystal with material symmetries(adapted from Nye 1985 [START_REF] Dobson | Physical properties of crystals and their representation by tensors and matrices[END_REF] ). The most of III-V materials are in cubic (Zinc) structure such as GaAs, InGaP and GaP, except GaN and AlN are in hexagonal (Wurtzite) structures. In the photoelastic tensors calculation, the crystalline structures should be taken into account. When we put photoelastic tensor and strain tensor together in the equation(6.2): When we put photoelastic tensor and strain tensor together in the equation( dV e|E 2 | (6.8) So far, we have calculated both equations for moving interfaces and photoelastic contributions. In photoelastic contribution, the elastic coefficients for different crystalline symmetries are important in calculating g P E values. The table below shows important properties for some materials. To be noticed that, in which we are interested in, GaN have two different permittivity constants with respect to the applied electrical field direction differs from longitudinal or transverse.

de ij dα (cubic) = -
With the knowledge of optomechanical coupling strength and the main contributions, we can study different mechanical motions by calculating their g 0 , g M I and g P E values. With the same 2D or 1D design, by comparing the optomechanical coupling strength for different III-V materials, we can have better understanding in III-V photonic crystal optomechanical systems. where ε and σ are the strain and stress vector, E is the electrical field, e is the permitivity with upper superscript σ to emphasize that is caused by stress, d is the piezoelectric strain constants in third rank matrix matrix, C is the elastic compliance coefficients in fourth rank matrix, D is the electric displacement. i, j are the indexes and m, k refer to the coordinate direction of the system.

The equations above can also be written as:

ε i = C D ij σ j + g mi E m (6.
11)

E i = g mi σ i + β σ ik D k (6.12)
where g is the piezoelectric constants that replaced the d piezoelectric strain constants, and β is the impermitivity in replace of the e permitivity. The equation(6.9) and (6.11) describe the converse piezoelectric effect that can deal with the piezoelectric actuators, while equations (6.10) and (6.12) describe the direct piezoelectric effect for the case in piezoelectric sensor. The piezoelectric constants d refers to the ratio between the strain in j direction to the applied electric field in i direction. Direct strain and stress are denoted by 3 axis from 1 to 3 and shear strain and stress denoted from 4 to 6. The electric field is denoted by only 3 axis from 1 to 3. The piezoelectric constants d ij is a 3 × 6 matrix. The elastic compliance constant C is the ratio between the strain in i direction to the stress in j direction. The strain and shear strain are denoted from 1 to 6. C ij is then a 6 × 6 matrix as also mentioned in the chanpter 2.

The strain tensor of the piezoelectric material can be wrote as:

          ε 1 ε 2 ε 3 ε 4 ε 5 ε 6           =         
             E 1 E 2 E 3   
The electric displacement can be developed as: where permitivity constants e ε ij determines the charge per unit area in a i direction due to the electric field applied in j direction. If we apply the notation to the shear stress to the stress vector, we can rewrite as: τ 23 = σ 4 , τ 31 = σ 5 and τ 12 = σ 6 . The same notation can be done for the shear strain to strain vector: γ 23 = ε 4 , γ 31 = ε 5 and γ 12 = ε 6 .

For a Zinc crystalline piezoelectric material such as GaAs, InGaP and GaP, the piezoelectric coefficients are in cubic symmetry. The components of strain tensor can be reduced with C 12 = C 13 , C 22 = C 33 , C 21 = C 13 and C 44 = C 55 = C 66 . The none-zero piezoelectric strain constant is d 14 and for permitivity e ε 11 = e ε 22 = e ε 33 . The piezoelectric strain tensor can be wrote as:

          ε 1 ε 2 ε 3 ε 4 ε 5 ε 6           =           C 11 C 12 C 12 0 0 0 C 12 C 11 C 12 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44                     σ 1 σ 2 σ 3 τ 23 τ 31 τ 12           +           0 0 0 0 0 0 0 0 0 d 14 0 0 0 d 14 0 0 0 d 14              E 1 E 2 E 3   
and the electric displacement: with e ε 33 . The piezoelectric strain tensor can be wrote as:

          ε 1 ε 2 ε 3 ε 4 ε 5 ε 6           =          
             E 1 E 2 E 3   
The electric displacement matrix can be wrote as:

Appendix C

We assume the strain distribution is in linear relationship for the piezoelectric substrate in z axis as ε(z) = αz. Here, the α is the relationship equation between strain and the stress. However, as the strain can be generated in two different directions with applied electric field in z axis for Wurtzite structures piezoelectric material, the strain distribution along the z axis can be further divided in xz-plane and yz-plane.

The distribution is then wrote as:

ε x (z) = α x z (6.13)
ε y (z) = α y z (6.14)

From equation (5.18), by applying voltage V , the stress distribution can be found in x and y directions as: where E p is the Young's modulus and ν is the Poisson's ratio of the substrate. To be noticed that the top electrode and the bottom electrode will experience the opposite stresses as shown in the figure (6.9). The stress inside top electrode can be presented as σ t

x and σ t y , while in the bottom electrode is presented as σ b x and σ b y . The stress distribution inside the top and bottom electrodes can be written in terms of the total strain as:

σ t x = E p 1 -ν 2 (ε x + νε y -(1 + ν p )ε 0 ), σ t y =
E p 1 -ν 2 (ε y + νε x -(1 + ν p )ε 0 ) (6.17)

σ b x = E p 1 -ν 2 (ε x + νε y + (1 + ν p )ε 0 ), σ b y =
E p 1 -ν 2 (ε y + νε x + (1 + ν p )ε 0 ) (6.18) where ε 0 represents the stress inside the electrodes. Since the strain distributes in the substrate is homogeneous. If we regard the electrode has the same length and width w = l, the stress in x and y direction can be the same as ε x = ε y = σ x z = σ y z. Due to the symmetrical stress field inside the piezoelectric substrate, the integration of the stress starts from z = 0 point to the top electrode z = (t/2 + t e ) is the same as to the bottom electrode z = -(t/2 + t e ). The moment of the piezoelectric substrate is defined by integrating the stress distribution inside the substrate. Since in x and y axis the moment is in equilibrium, the integration can be developed as: With the known amount of the applied voltage, the displacement field generated by this piezoelectric actuator can be calculated accordingly.

If we apply the reverse piezoelectric effect to a general case in piezoelectric sensor, with no extra electric field applying to the system, the electrical displacement field can be wrote as:

         σ 1 σ 2 σ 3 σ 4 σ 5 σ 6          
The electric charge can be determined as:

q = D 1 D 2 D 3    dA 1 dA 2 dA 3   
where the dA i is the electrodes area perpendicular to the i axis. If the stress is only applied to the x axis, the electric charge can be simplified as: tecture. These cavities harbor highly-confined optical modes around 1.55 µm and mechanical modes in the GHz and most importantly, feature a high phonon-photon spatial overlap, all resulting in an enhanced optomechanical coupling. This enhanced optomechanical coupling strength is here probed optically on photonic crystal structures with optimized design. These cavities are hosted in III-V semiconductor materials whose piezoelectricity enable us to integrate additional tools for probing and controlling mechanical vibrations via capacitive, piezoelectric or acoustic driving. This full control over the mechanical modes and optomechanical interaction, paves the way towards the implementation of integrated injection locking circuits of feedback loops for reducing the phase noise of the oscillator.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Figure 1 . 1 -

 11 Figure 1.1 -Comparison of the noise performance between various small-volume microwave oscillators, all carriers being scaled to 10 GHz for direct comparison. The red-shaded region indicates the expected performances of optomechanical devices.The phase noise is expected to be reduced down to -110dBc/Hz while operating at a high output frequency in the GHz range. Most importantly, the package size of the OMO can be as low as 1cm 3 which is the most compact microwave oscillators.

Figure 1 . 3 -

 13 Figure 1.3 -State-of-the-art of optomechanical nanobeam cavity designs. (a). siliconbased nanobeam cavity with rectangular holes from M. Eichenfield et al[START_REF] Eichenfield | Optomechanical crystals[END_REF] . (b). silicon-based nanobeam cavity with circular holes from Q. Quan et al[START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF] . (c). siliconbased zipper cavity formed by two identical nanobeams within the near field of each other from R. Leijssen et al[START_REF] Leijssen | Nonlinear cavity optomechanics with nanomechanical thermal fluctuations[END_REF] . (d). Phoxonic crystal cavity enclosed by two periodic structures exhibiting band gaps for the propagation of photons and phonons from D. Navarro-Urrios et al[START_REF] Navarro-Urrios | Nonlinear dynamics and chaos in an optomechanical beam[END_REF],[START_REF] Gomis-Bresco | A one-dimensional optomechanical crystal with a complete phononic band gap[END_REF] . (e). diamond-based s nanobeam cavity with rectangular holes etched in a beam of triangular cross-section from Bayn et al[START_REF] Bayn | Triangular nanobeam photonic cavities in single-crystal diamond[END_REF] . (f). AlN-based cavity made of circular holes from W. H. P. Permice et al[START_REF] Pernice | High-q aluminum nitride photonic crystal nanobeam cavities[END_REF] . (g). AlN-based cavity etched in a slightly wider nanobeam at the cavity location from L. Fan e et al[START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] .

Figure 1 . 4 -

 14 Figure 1.4 -State-of-art of optomechanical 2D photonic crystal-based cavity designs.(a) InP-based L3 photonic crystal defect cavity with an optical Q up to 10 4 and mechanical modes in the GHz from Gavartin et al[START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] ; (b) Integration of the previous cavity on SOI-based photonic circuitry from Tsvirkun et al[START_REF] Viktor Tsvirkun | Integrated III-v photonic crystal -si waveguide platform with tailored optomechanical coupling[END_REF] . (c) silicon-based 2D photonic crystal cavity with high Q 1.2×10 6 from Safavi-Naeini et al[START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF] ; (d) snwoflake phoxonic crystal cavity from Safavi-Naeini et al[START_REF] Amir | Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab[END_REF],[START_REF] Amir | Two-dimensional phononic-photonic band gap optomechanical crystal cavity[END_REF]

Figure 1 . 5 -

 15 Figure 1.5 -Schematic of an optomechanical oscillator (OMO) consisting of a high Q optomechanical resonator (OMR) with a feedback loop and external actuation circuits. Under the external actuation, the OMR can maintain self-sustained oscillation. With the feedback loop circuit, the phase noise can be suppressed to the minimum.

Figure 1 . 6 -

 16 Figure 1.6 -Bandgap energy of some semiconductor materials as a function of their lattice constant. Piezoelectric materials are highlighted a red rectangle. Black dots correspond to direct bandgap materials and the white dots to indirect bandgap materials. Figure plotted by Simone Montanari according to[START_REF] Fujita | Wide-bandgap semiconductor materials: For their full bloom[END_REF] .

Figure 2 . 1 -

 21 Figure 2.1 -Example of the TE-like photonic band diagram in a 2D hexagonal-lattice photonic crystal. The red lines indicate the propagative modes within the medium. The Brillouin zone in the form of an hexagon follows the sixfold symmetry of the crystal; the high symmetry points are given canonical names, where Γ denotes the origin k = 0, K is the nearest-neighbor direction, and M is the next-nearest-neighbor direction. The air holes (r = 0.317 × a) pierced in a dielectric ( = 2.3), where a is the center-center periodicity.

Figure 2 . 2 -

 22 Figure 2.2 -TE -like modes band diagram below the light cone for a square lattice of holes (left) and for a triangular lattice of holes (right) etched in a slab. The reciprocal lattice (red line) and the first Brillouin zone (blue line) are shown in each figure in the insets. The grey area in the band structure indicates the bandgap. Figure originated from[START_REF] Englund | General recipe for designing photonic crystal cavities[END_REF] .

Figure 2 . 3 -

 23 Figure 2.3 -(a) and (b): Schematics and SEM image of an heterostructure optical resonator done by S. Combrié[START_REF] Combriè | Gaas photonic crystal cavity with ultrahigh q: microwatt nonlinearity at 1.55 µm[END_REF] . The photonic crystal cavity is made of GaAs and the measured quality factor reaches 7 × 10 5 . (c) and (d): SEM images of a L3 optical resonator done by E. Gavartin[START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] . The photonic crystal cavity is made of InP and the measured quality factor is up to 10 4 .

Figure 2 . 4 -

 24 Figure 2.4 -Schematics of a L3 cavity.

Figure 2 . 5 -

 25 Figure 2.5 -FDTD simulations showing the electric and magnetic fields amplitude of the fundamental and second-order modes of a GaAs L3 cavity at 1.55µm, with a = 430nm, r = 120nm, and a slab thickness of 265nm. The color scale are not the same for each configuration. Note that the field intensity of the second-order mode is much lower with respect to the fundamental mode.

Figure 2 . 6 -

 26 Figure 2.6 -Simulated fundamental mode resonance wavelength in a L3 cavity as a function of the lattice constant a and holes radius r. Left: in GaAs with a slab thickness of h = 265nm. The star point at a = 420nm with r = 110nm features an optical Q o around 4900; Right: in GaN with a slab thickness of h = 360nm. The star points at a = 630nm with r = 200nm features an optical Q o around 500.

Figure 2 . 7 -

 27 Figure 2.7 -Schematics of a modified L3 cavity. Two holes next to the cavity are modified with the reduction of the hole radius normalized by r and relative outward displacement of the holes normalized by a.

Figure 2 . 8 -

 28 Figure 2.8 -Simulated quality factor of the fundamental mode in a GaAs L3 cavity as a function of the first neighboring defect holes displacement and radius reduction, normalized respectively to a and r. The white circle shows the best Q 0 = 24000 achieved for 0.19a and 0.08r.

Figure 2 . 9 -

 29 Figure 2.9 -Simulated quality factor of the fundamental mode in a GaN L3 cavity as a function of the first neighboring defect holes displacement and radius reduction, normalized respectively to a and r. The white circle shows the best Q 0 = 1243 achieved for 0.21a and 0.13r.

Figure 2 .

 2 Figure 2.10 -(a) Schematic of the evanescent coupling between the SOI waveguide (WG) and the L3 photonic crystal cavity; (b) and (c) FDTD simulations of the TElike cavity mode profile displaying the amplitude of (b) the magnetic field along z H z in xz-plane (y = 0) and of (c) the electric field along y E y in x -y plane (z = 0). The cavity-waveguide distance is set to d = 200nm and the waveguide width to w W G = 350nm. The mode is resonant at 1.55µm.

Figure 2 .

 2 Figure 2.11 -FDTD simulation of the optical quality factor Q total in (a) and the coupling efficiency η in (b) as a function of the SOI waveguide width w W G for different cavity-waveguide distances d.The simulation was done by V. Tsvirkun[START_REF] Viktor Tsvirkun | Optomechanics in hybrid fully-integrated two-dimensional photonic crystal resonators[END_REF]

Figure 2 . 13 -

 213 Figure 2.13 -FDTD simulation of the effective refractive index for TE modes in a bilayer GaN/AlN-based ridge (left) and GaAs-based ridge (right) as a function of the nanobeam width. The red lines are nanobeam width we chose in the final design. The n ef f for GaN/AlN and GaAs material are 1.6 and 2.7 respectively.

Figure 2 .

 2 Figure 2.14 -(Top) Schematic of the nanobeam cavity with tuned photonic crystal lattice constants a. (Center) Example of the variation of the lattice constant as a function of the hole position along the beam. (Bottom) Band diagram of a perfect 1D-photonic crystal for a n = 634nm and a 0 = 485nm.

Figure 2 . 16 -

 216 Figure 2.16 -Gaussian apodisation a(x) function and discretized approximated positions for the nanobeam holes.

Figure 2 . 17 -

 217 Figure 2.17 -Simulated optical resonance wavelength of the fundamental mode (left axis) and its quality factor (right axis) as a function of the F W HM for a GaAs-based (Left) and GaN-based (Right) nanobeam cavity.

Figure 2 . 18 -

 218 Figure 2.18 -Schematic of GaN nanobeam cavity formed by rectangular holes whose dimensions are tuned along the beam. From the center to the edge, the holes width H x is unchanged and equals 250nm; the holes length is tuned from H y0 = 400nm to H yn = 700 following a Gaussian distribution.

Figure 2 . 19 -

 219 Figure 2.19 -Coupling schemes between the nanobeam cavity and an access optical waveguide.

Figure 2 .

 2 Figure 2.20 -FDTD simulation along scheme (2.19b) of the coupling efficiency and the quality factor of a coupled GaAs-based nanobeam cavity as a function of the waveguide-nanobeam cavity distance d (Left) and as a function of the waveguide width w W G .

Figure 2 .

 2 Figure 2.21 -FDTD simulation along scheme (2.19)-b of the coupling efficiency and the quality factor of a coupled GaN-based nanobeam cavity as a function of the waveguide-nanobeam cavity distance d.

Figure 2 .

 2 Figure 2.22 -FDTD simulation along scheme (2.19b) of the coupling efficiency and the quality factor of a coupled GaAs-based nanobeam cavity as a function of cavity -waveguide distance in vertical plane d v and in the horizontal plane d h . The black part in the figure corresponds to missing data.

Figure 2 . 24 -

 224 Figure 2.24 -Schematic of perturbation theory with a moving interface and unit normal vector u and v.

Figure 2 . 25 -

 225 Figure 2.25 -Mechanical eigenmode displacement plots of the first mechanical modes sustained by a standard GaAs-based L3 cavity with a slab thickness of 265nm, a lattice constant a of 420nm and holes radius r of 130nm.The M6 and M8 modes are the higher order modes of respectively M2 and M3. The color represents total displacement amplitude, as does the deformation of the structure.

Figure 2 .

 2 Figure 2.26 -Values of the optomechanical coupling strength component for the different mechanical modes coupled to the optical fundamental mode in a standard GaAs-based L3 cavity with a slab thickness of 265nm, a lattice constant a of 420nm and holes radius r of 130nm. The contribution of g M I × x ZP F /2π (blue) and g P E × x ZP F /2π (red) to g 0 /2π (yellow) are plotted as a function of the mechanical frequency.

Figure 2 .

 2 Figure 2.27 -(Top) Optical field of the investigated mode and deformation profile of the M6 mechanical mode and related field components in a modified GaAs-based L3 cavity with a slab thickness of 265nm, a lattice constant a of 420nm and holes radius r of 130nm. The radius of the first neighboring holes at the cavity edge has been modified to 0.08r and their position has been outward-shifted by a distance 0.18a. (Bottom) Changes in the mechanical response (frequency, effective mass, zero-point fluctuations, g 0 /2π, g P E × x ZP F /2π and g M I × x ZP F /2π) as a function of the radius normalized to r and the holes displacement normalized to a of the first neighboring holes. The black stars indicate the optimum values of the holes displacement and radius reduction, regarding the optical response of the cavity.

Figure 2 .

 2 Figure 2.28 -(Top) Optical field of the investigated mode and deformation profile of the M8 and M9 mechanical modes in a modified GaAs-based L3 cavity with a slab thickness of 265nm, a lattice constant a of 420nm and holes radius r of 130nm. The radius of the first neighboring holes at the cavity edge has been modified to 0.08r and their position has been outward-shifted by a distance 0.18a. (Bottom) Changes in the value of g 0 /2π as a function of the radius normalized to r and the holes displacement normalized to a of the first neighboring holes. The black stars indicate the optimum values of the holes displacement and radius reduction, regarding the optical response of the cavity.

Figure 2 . 29 -

 229 Figure 2.29 -Mechanical and optomechanical characteristics of a GaAs-based nanobeam with a 0 = 315nm, a n = 485nm, H x =190nm, H y0 = 200nm, H yn = 430nm and an entire nanobeam length of 27.7µm (F W HM = 3µm). The Gaussian apodisation is applied to both a and H y tuning. (a): Simulation of the vacuum optomechanical coupling rate as a function of the frequency of the mechanical modes; their associated groups are indicated from Group A to C. On top left, are shown the mechanical band diagrams at the cavity center (b) and in the cavity mirrors (c). (d): Optical mode profile used for optomechanical coupling calculation. The mechanical displacement profiles of the different mechanical modes sustained by the nanobeam and belonging to the three different groups, are shown below for Group A in (e), Group B in (f) and Group C in (g).

Figure 2 . 30 -

 230 Figure 2.30 -Mechanical resonance profile of the first three symmetric and antisymmetric mechanical modes in a GaAs nanobeam cavity with a 0 = 315nm, a n = 485nm, H x =190nm, H y0 = 200nm, H yn = 430nm and an entire nanobeam length of 27.7µm (F W HM = 3µm). The Gaussian apodisation is applied to both a and H y tuning.

Figure 2 .

 2 Figure 2.31 -Left: Averaged mechanical frequency of the first three symmetric and anti-symmetric mechanical modes as a function of holes width H x . Right: Corresponding g 0 /2π values of these modes. The simulation was carried on GaAs nanobeam cavity with a=315nm, a parabolic distribution of H ym ranging from H y0 =200nm to H yn =430nm, and a F W HM of 3µm.

Figure 2 . 32 -

 232 Figure 2.32 -Optomechanical coupling strength g 0 /2π of a GaAs-based nanobeam cavity with a 0 =315nm, H x =150nm, H y0 =200nm, H yn =450nm and F W HM 3µm. The shape of the holes is either rectangular (blue points) or elliptical (yellow points).

Figure 2 . 33 -

 233 Figure 2.33 -Optomechanical coupling strength g 0 /2π of a GaN-based nanobeam cavity with a 0 =480nm, H x =450nm, H y0 =400nm, H yn =700nm and F W HM = 3µ. The shape of the holes is either rectangular (blue points) or elliptical (yellow points).

  Figure(3.1).

Figure 3 . 1 -

 31 Figure 3.1 -Schematic of the III-V materials epitaxy heterostructures involved in this work. From left to right: for GaN/AlN-, GaAs-and InGaP-based resonators.

  Figure(3.2) displays 10 × 10µm 2 and 2 × 2µm 2 AFM images of the GaN surface. The mean roughness for each area is 0.782 to 0.853nm respectively. The hill-like structures that distribute homogeneously on the GaN surface is the atomic composition. From Figure(3.2b), we can see clearly some hexagonal patterns which indicate the hexagonal crystalline structure. If relying on heterostructure-based growth provides high quality GaN materials, such GaN/AlN heterostructure inevitably features a lattice and thermal coefficient mismatch, which introduces tensile strain.

Figure 3 . 2 -

 32 Figure 3.2 -AFM image of the surface roughness of GaN/AlN membranes for (a)10 × 10µm 2 area and (b)2 × 2µm 2 area. The hill-like structure is the atomic composition with a mean roughness around 0.782 to 0.853nm on the GaN surface.

Figure 3 . 3 -

 33 Figure 3.3 -Schematic of the first generation of III-V waveguide and nanobeam cavity platform. The yellow line indicates the optical waveguide,consisting of a twodimensional photonic crystal waveguide and one nanobeam. The nanobeam waveguide features a curved shape, with a reduced distance to the nanobeam cavity in the resonator's vicinity. At both ends of the nanobeam waveguide, a two-dimensional photonic crystal shield is used to reflect the optical signal back to the waveguide.

Figure 3 . 4 -

 34 Figure 3.4 -Schematic of the different steps for processing the III-V waveguide and nanobeam cavity platform. Here GaN/AlN-based platforms are considered. The processing of the GaAs and InGaP-based platforms, involves the same basic steps.

Figure 3 . 6 -

 36 Figure 3.6 -SEM image of a two-dimensional photonic crystal waveguide pattern on HSQ resist. The 2D waveguide is not homogeneous from the inner area to the outer area. The darker color in the black circle indicates thicker HSQ resist compared to the two sides part. This is due to the proximity effect induced by back scattering electrons.

Figure 3 .

 3 Figure 3.7 -(Left) Schematics illustrating the variation of the holes diameter for increasing e-beam doses. (Right) Dependence of the holes radius as a function of the electron beam dose expressed in µC/cm 2 . The two SEM images show a zoom of the patterns drawn in the HSQ resist in the vicinity of the two dimensional photonic crystal waveguide. For high doses, the holes size tend to increase while the lattice constant changes slightly.

Figure 3 . 8 -

 38 Figure 3.8 -Top and middle: SEM images of the processed GaN/AlN cavity with a too high dose and a proper dose expressed in µC/cm 2 . Bottom: imprinted patterns on the HSQ resist with a too low dose expressed in µC/cm 2 .

Figure 3 . 9 -

 39 Figure 3.9 -SEM image of the ICP-RIE etched GaAs nanobeam. The nanobeam is etched down to a depth of about 1.2µm, reaching the sacrificial AlGaAs layer and thus enabling the further chemical underetching of the GaAs layer.

Figure 3 .

 3 Figure 3.10 -SEM images of a InGaP-based nanobeam with rectangular and elliptical holes on the HSQ resist (top) and on the InGaP membrane(bottom).

Figure 3 .

 3 Figure 3.11 -SEM images of suspended InGaP-based nanobeams. The nanobeam and waveguide are suspended at a distance of about 1µm above the substrate.

Figure 3 .

 3 Figure 3.13 -SEM images of different edge profiles in the GaN/AlN membrane obtained after ICP-RIE for different RF powers. At low RF power around 35W, the side walls are tilted with an angle of about 30 • . For RF powers of 48W, the etched side walls become vertical and sharp.

Figure 3 .

 3 Figure 3.14 -(a) Schematic of the GaN/AlN-based platform. The yellow line is the optical waveguide. (b-c-d-e) SEM images of processed GaN/AlN-based platforms. Due to the distortion of the beams, we observe on (e) a misalignment in the vertical direction of about 400nm between the cavity and the waveguide.

Figure 3 .

 3 Figure 3.15 -(a) Schematic of the final GaN/AlN-based nanobeam platform. The side coupling waveguide is replaced by injection tips and light is funneled to the cavity via the nanobeam. More than 10 holes from left side of nanobeam cavity are removed for enabling light propagation. (b) and (c) SEM images of the GaN/AlN-based nanobeam platform.

Figure 3 . 16 -

 316 Figure 3.16 -Schematic of a nanobeam resonator integrated on a SOI waveguide. The red lines indicate the SOI waveguide, the green part is the nanobeam resonator. Light is injected through gratings at the edges of the SOI waveguide and is then coupled to the cavity via evanescent coupling.

Figure 3 .

 3 Figure 3.18 -Schematic of the SOI waveguide process before bonding. The steps include resist spin-coating, electron beam lithography, metallisation, lift-off and BCB spin-coating as well baking.

Figure 3 . 19 -

 319 Figure 3.19 -Schematic of the steps for the SOI waveguide wafer and III-V substrate bonding by BCB adhesion. The III-V substrate is placed face to face to SOI wafer and bonded by use of a strong pushing force (300N ) at high temperature (320 • C) and in a high vacuum system (≤ 5 × 10 -3 mbar).

Figure 3 . 21 -

 321 Figure 3.21 -Schematic of the different steps involved during the nanobeam photonic crystal patterning including resist spin-coating, electron beam lithography, ICP-RIE etching, residual resist removal and under-etching by dry etching technique. The resist (HSQ) is a negative resist and the removed sacrificial layer is made of Si 3 N 4 .

Figure 3 .

 3 Figure 3.22 -SEM images of GaAs/InGaP nanobeam resonators integrated over a SOI waveguide. Top: GaAs nanobeam SOI integrated system; Bottom: InGaP nanobeam SOI integrated system. The blue part is the suspended nanobeam cavity, red line indicates the SOI waveguide and yellow part is to hold the suspended structure.

Figure 3 . 23 -

 323 Figure 3.23 -Schematic of the steps involved in the two dimensional photonic crystal L3 cavity patterning on GaAs/InGaP membranes bonded on SOI waveguide substrate. The steps include Si 3 N 4 deposition, resist spin-coating, electron beam lithography, CCP-RIE, ICP-RIE and the residual Si 3 N 4 removal. The resist (PMMA) here is a positive resist and the Si 3 N 4 layer acts as a hard mask during the patterns transfer in the semiconductor by ICP-RIE.

  3.25) left). By changing the writing path along several circular paths around the holes (see Figure(3.25) right), this roughness can be overcome. This

Figure 3 .

 3 Figure 3.24 -Left: schematic of the impact of the dose exposing a positive resist on the imprinted holes diameter. Right: Holes radius variation as a function of the electron beam dose. The targeted holes radius is 120nm. The three SEM images correspond two 1000, 1200 and 1300 µC/cm 2 doses from left to right. The optimal holes radius is obtained for doses around 1000 -1100µC/cm 2 .

Figure 3 .

 3 Figure 3.25 -Top: schematic of the edge contour technique. The colored part corresponds to the e-beam writing field. Without the holes contour technique, the e-beam writes the holes lines by lines with rough edges. The holes contour technique can help to achieve holes in perfect circular shape. Bottom: SEM images of GaAs-based photonic crystal cavities without and with holes contour technique. Both photonic crystal cavities patterns were written with an e-beam lithography resolution of 1.25nm.

Figure 3 .

 3 Figure 3.26 -Left: SEM images of a GaAs photonic crystal cavity. Right: Statistical distribution of the holes radius extracted from the SEM image analysis. The distribution of holes radius presents a Gaussian shape. The averaged holes radius is 112.8nm which is slightly larger than targeted value of 110nm.

Figure 3 . 27 -

 327 Figure 3.27 -Schematic of the mesa structures patterning steps and under-etching procedure to release the membrane. The steps include Si 3 N 4 deposition, resist spincoating, electron beam lithography, CCP-RIE, ICP-RIE and dry etching. The Si 3 N 4 layers act as the hard mask during ICP-RIE pattern transfer and as a sacrificial layer for suspending the membrane.

Figure( 3

 3 Figure(3.28) shows three L3 GaAs cavities processed with different under-etching process duration: 10 min, 20min and 30min. Since the etch rate is around 100nm/min, the underetching should at least last 2 to 3 minutes to completely underetch the 200nm thick Si 3 N 4 layer. It is here worth noticing that the etching duration needs to be optimized to have both well suspended photonic crystal and non-damaged SOI waveguides. Indeed the dry etching can also etch the SiO 2 and BCB material though with a slower etch rate (around 1 or 2 nm/min). Consequently, once the Si 3 N 4 layer is completely etched, the reactive ions can etch the BCB as well as SiO 2 layers. If the etching duration is too long, the BCB layer will be further etched and most importantly, the SOI waveguide could be exposed and especially the gratings may be damaged. The most detrimental consequence with etched SOI waveguide and gratings is the induced optical losses. In our case, the etching duration has been fixed to 10 minutes to avoid this issue.

Figure 3 . 28 -

 328 Figure 3.28 -Optical images of GaAs L3 photonic crystal cavities suspended following a 10 minutes, 20 minutes and 30 minutes dry etching. The etch rate is around 100nm/min with isotropic etching. The under-etching depth can be inferred by measuring the etched distance in transverse direction.

Figure( 3 .

 3 Figure(3.29) shows SEM images of final L3 photonic crystal platforms integrated over the optical SOI waveguide. Five cavities with different holes radius r 1 to r 5 are addressed by each waveguide. These five different cavities feature each different

Figure 3 .

 3 Figure 3.29 -Left: SEM images of a GaAs photonic crystal cavity with five different holes radius r 1 to r 5 . Middle: zoom in one photonic crystal cavity with holes radius r 3 . The dash line indicates the SOI waveguide located underneath the GaAs substrate. Right: SEM image on the edge of the photonic crystal cavity showing the mesa pad structure and the SOI waveguide.

Figure 3 .

 3 Figure 3.30 -SEM images of two silicon substrate after etching along two dry etching methods : XeF 2 -based etching and DRIE etching. The surface is homogeneous with little defects for both methods. To be noticed, the scalloping shape side wall is quite identical in DRIE etching.

Figure 3 . 31 -

 331 Figure 3.31 -Schematic of DRIE technique used to remove silicon substrate. The process includes two different plasma cycles. The C 4 F 8 plasma is used to form CFpolymers on the surface (b) that can be removed in a SF 6 plasma in the next step (c). These two cycles (b) and (c) will continue alternating until all silicon substrate is removed.

Figure 3 .

 3 Figure 3.32 -SEM images showing the rolling of the GaN/AlN membrane after removal of the silicon substrate. The cracks introduced by the strong tension from GaN/AlN bilayer damage the adhesion on the SOI wafer via the BCB layer.

Figure 3 . 33 -

 333 Figure 3.33 -Schematic of the mesh pattern used to avoid any detachment of the GaN membrane on the SOI waveguide substrate. The mesh is represented by the black lines. This mesh is 10 × 10mm 2 in size and 1µm in width. The SOI waveguide circuit is located inside the mesh.

Figure 3 .

 3 Figure 3.35 -SEM images of GaN/AlN final surface after the plasma-therm dry etching. The dots on the surface may come from the epitaxy and the big object may be some piece of dust .

Figure 3 . 36 -

 336 Figure 3.36 -Optical images of the final GaN/AlN membrane bonded on the SOI waveguide substrate after growth substrate removal. The height of the mesh patterns is around 5to8µm outside the waveguide circuits. The violet color indicates the GaN/AlN membrane since these materials are almost transparent in the visible light.

Figure 3 .

 3 Figure 3.37 -Left: SEM images of a GaN photonic crystal cavity. Right: Distribution of the holes radius extracted from the SEM image analysis. The holes distribution is rather homogeneous. The average holes radius is 186.5nm, slightly smaller than the targeted 195nm.

Figure 3 .

 3 Figure 3.38 -SEM images revealing some GaN/AlN photonic crystal fabrication issues. Upper pictures from (a) to (c) illustrate the several issues faced during the fabrication process, including membrane peeling off, damage as well as cavity break. Bottom pictures from (d) to (f) are photonic crystal platforms that we succeeded to process.

Figure 4 . 1 -

 41 Figure 4.1 -Optical reflection measurement set-up. The light is injected via an objective lens and the reflected optical signal is guided to a power-meter via a circulator.

Figure 4 . 2 -

 42 Figure 4.2 -Optical transmission experimental set-up with a 3D translation stage system. The laser source is provided by (a) a broadband light source (SLED) and (b) a tunable laser. The erbium doped fiber optical amplifier and the tunable filter enable us to amplify the injected optical power and to remove amplified spontaneous emission. The 3D translation stage system is controlled by a PI high precision fiber alignment system with nanopositioners. The optical transmission signal is analyzed by an optical signal analyzer.

Figure 4 . 3 -

 43 Figure 4.3 -Left: schematic of the coupling of the light from the fiber to the waveguide via the SOI grating couplers with incident angle θ. The coupling efficiency between the fiber and the SOI waveguide depends on the gratings period and refractive index. Right: normalized optical transmission spectra measured at the output of the SOI waveguide for three different incident angles θ = 9 • , 10 • and 11 • when injecting light from a broadband source (the SLED). When the angle increases, the highest transmission wavelength shifts to lower wavelength.

Figure( 4 . 4 )

 44 Figure(4.4) displays the optical transmission spectrum of the SOI waveguide coupled to the L3 cavities made of GaAs. Let's remind that each SOI waveguide is coupled to five L3 cavities with different holes radius r and resonance frequencies ω, as can be seen on the optical image of Figure(4.4a) and (b). From the infrared image we can see a bright spot corresponding to the diffracted light from the cavities out of the membrane plane.The transmission spectrum of the waveguide when fed by broadband source, is shown in Figure(4.4c). This spectrum features several dips which correspond to the cavities resonances. The five photonic crystal cavities possess identical lattice constant (a =420nm) but different hole radii equally distributed between 100 and 120nm with 5nm steps; their resonance wavelength should as such lie between 1470 and 1550nm. However, during the fabrication process, the achieved photonic crystal holes diameters appeared to be slightly smaller than the targeted values. As such, the actual resonance frequencies are all shifted to longer wavelength.In order to extract more acutely the quality factors of the cavities, we repeated the experiment though by replacing the broadband source by the tunable laser, whose wavelength λ L can vary from 1520 to 1580nm. Figure(4.5) shows examples of the obtained transmission spectra. This more acute measurement reveals two resonances in two different wavelength region around 1525nm and 1565nm (indicated by grey areas). If we assume that these two resonances correspond to two different cavities with two different corresponding holes radii r 1 and r 2 , the spectral separation between these two modes ∆λ = λ 2 -λ 1 ≈ 43nm, shall correspond to ∆r = r 2 -r 1 ≈ 10 -12nm for identical lattice constants of a = 420nm. By fitting these resonances with a Lorentzian curve, one can infer the resonance linewidth κ/2π = 3.1nm which corresponds to Q 0 values around 500. Solely two cavities are observed. Two possible reasons may account for not observing the three other resonances: the quality factor of the other cavities is too low or, more probably, their resonance is shifted out of the laser tunability wavelength range.

Figure 4 .

 4 Figure 4.4 -(a) Optical image of the full platform revealing the five GaAs-based L3 cavities (featuring different radius and thus different optical resonances) anchored to the mesa structures on top of the SOI waveguide. (b) Infra-red image taken above the sample featuring a bright spot corresponding to diffracted light out of the five cavities; (c) Transmission spectrum of the waveguide coupled to the L3 cavities and fed by the broadband light source without any cavity (in blue) and with the cavities (in orange).

Figure 4 . 5 -

 45 Figure 4.5 -Optical transmission spectrum of the waveguide coupled to the GaAs L3 cavities, when the waveguide is fed by the tunable laser at two different powers (Blue scatters: 1 mW; red scatters: 2.5mW). The laser wavelength λ L varies from 1520nm to 1580nm.

Figure 4 . 6 -

 46 Figure 4.6 -Top: Transmission spectrum of the waveguide coupled to the GaN-based L3 cavities and fed by the broadband light source without any cavity (in blue) and with the cavities (in orange). Bottom: infra-red images of the sample taken from above, without any light feeding the waveguide (left), and with the laser tuned at four different wavelengths corresponding to different dips (at resonances for a and d) and peaks (out of resonance for b and c) as indicated on the spectrum.

  (

  4.1)).A typical measured spectrum is shown on Figure(4.7a

Figure 4 . 7 -

 47 Figure 4.7 -(a) Example of measured power spectrum in the reflection mode configuration on a GaN nanobeam cavity whose resonance is around 1565nm. The fitting Lorenztian curve in red enables us to infer a linewidth of 0.5nm at resonance and thus a quality factor Q c around 3158. (b-c) Comparison between the simulated and experimental cavity resonance wavelength and quality factor, as a function of lattice constant a (b) and holes width H x (c). The nanobeam is with rectangular holes on GaN material.

Figure 4 . 8 -

 48 Figure 4.8 -Optical transmission of the SOI waveguide without any cavity (blue line) and coupled to four GaAs nanobeam cavities (red line) when fed by a broadband light source. The IR image inside the spectrum is the image taken by the IR camera when injecting the broadband light source in the SOI waveguide. The four spots in the image confirm that all the optical modes sustained by the four cavities are excited.

Figure 4 . 9 -

 49 Figure 4.9 -(a): Identification of the cavities modes by launching a narrow band tunable laser source in the SOI waveguide. The IR images below shows typical recorded images when the laser wavelength matches a resonance of cavity a, b, c or d. (b, c): Optical spectrum in a transmission measurement for nanobeam cavities featuring either (b) rectangular or (c) elliptical holes. The four different colors (green for cavity a, yellow for cavity b, red for cavity c and blue for cavity d) corresponds to the four different cavities (a, b, c, d). All the optical modes of each cavity can be distinguished within the tunable laser wavelength range from 1500 to 1590nm. The grey color correspond to an unidentified modes.

Figure 4 . 10 -

 410 Figure 4.10 -Optical resonance evolution of the fundamental and high-order modes sustained by GaAs-based nanobeam cavities for increasing targeted holes width H x ranging from 200nm to 275nm and comparison with expected theoretical values in (a); same figure yet taking into account the geometrical deviations induced during the processing for the experimental values, with achieved holes widths ranging from 260nm to 330nm (b). The lattice constant a 0 is fixed at 315nm for all the cavities.

Figure 4 .

 4 Figure 4.11 -(a).Transmission spectrum of a nanobeam cavity formed by elliptical holes and featuring a resonance wavelength of λ at 1523.0nm, a loaded or total Q T of 21643 and a linewidth κ/2π of 9GHz; (b) Normalized transmission spectrum of the same cavity with a minimum transmitted power at resonance T min of 0.51. The intrinsic Q 0 is then calculated as 25265 with a coupled Q c of 11657 The coupling efficiency η is then 68.43%.

  (4.13) for cavities formed by elliptical holes, same lattice constant yet different holes widths.

Figure 4 .

 4 Figure 4.12 -(a) to (d) Normalized transmission spectrum of the higher modes sustained by the GaAs-based nanobeam cavity ( second to the sixth order optical modes). (e) Evolution of the optical quality factors and coupling efficiency η as a function of optical mode orders in the same cavity (formed by elliptical holes with a 0 315nm and H x 250nm).

Figure 4 . 13 -

 413 Figure 4.13 -Evolution of the intrinsic Q 0 (a), coupled Q c (b) and coupling efficiency η (c) of the first six order optical modes sustained by nanobeam cavities featuring different holes width H x . The cavities are formed by elliptical holes with fixed lattice constant a 0 of 315nm.

Figure 4 . 14 -

 414 Figure 4.14 -Optical resonance wavelength of the nanobeam cavity as a function of the optical mode order for holes width H x 275 (in blue), 250 (in red) and 225nm (in yellow) for cavities featuring elliptical holes (star scatters) and rectangular holes (circle scatters).

Figure 4 .

 4 Figure 4.16 -(a)Optical resonance wavelengths and (b) coupling efficiency η of the first six-order modes sustained by a rectangular holes nanobeam cavity as a function of the waveguide width w W G . The GaAs-based cavity geometrical parameters are a 0 =315nm and H x =250nm.

Figure 4 . 17 -

 417 Figure 4.17 -Coupling efficiency η and loaded quality factor Q T as a function of w W G varying from 300 to 450nm, for a nanobeam cavity featuring rectangular holes with a 0 =315nm and H x =250nm,. The mode under study here is the forth optical mode.

Figure 4 . 18 -

 418 Figure 4.18 -Experimental set-up used to probe the optomechanical coupling via optical transmission measurements. The input laser wavelength is set at the cavity resonance wavelength λ 0 . The optical signal is coupled to the sample by use of fibers and grating couplers. The alignment of the fibers at the input and output of the SOI waveguide is implemented by use of an IR camera enabling to observe the intensity of the localized optical mode inside the nanobeam cavity. The transmitted optical signal is sent to a fast photodiode and analyzed by electrical signal analyzer.

Figure 4 .

 4 Figure 4.19 -(a) Power spectrum density of the transmitted optical signal via the SOI waveguide when launching a laser source tuned at the cavity resonance (blue line) and out of resonance (red line). The Group B mechanical modes can be found in the blue area. (b) Zoom in the spectrum around 2.3GHz. Each peak corresponds to different mechanical modes whose mechanical displacement field profiles deduced from simulations are shown. The measured nanobeam cavity is here made of elliptical holes and the second-order optical mode is used to probe the mechanical motion.

Figure 4 . 20 -

 420 Figure 4.20 -Optomechanical coupling strength g 0 measurement set-up combining an optical source (tunable laser Photonetic Tunics 1550), an optical amplifier (CEFA-C-PB-HP), an optical filter (Exalos XTM-50), a first polarizer for laser polarization, an electro-optical phase modulator (Thorlabs LN65SFC), a second polarizer for modulation polarization, a fast photodiode and an electrical signal generator/analyzer (Anritsu MS2830A).

Figure 4 .

 4 Figure 4.21 -S-parameter measurement on the S 11 reflection signal for the phase modulator.Here the calibration is operated with the RF cable that is used in the g 0 measurement. The RF power P rf is set to 0dBm in this calibration measurement.

Figure 4 . 22 -

 422 Figure 4.22 -Power spectrum density of the transmitted optical signal via the SOI waveguide when launching a laser source tuned at the cavity resonance. The spectrum features peaks at the mechanical mondes frequencies and at the laser modulation frequency. (a) for modes M 1 to M 3 with Ω mod =2.34GHz; (b) for modes M 4 to M 6and Ω mod =2.47GHz. The cavity is made of elliptical holes and its Q 0 is around 16000.

Figure 4 .

 4 Figure 4.23 -Thermo-optical bistability of the nanobeam cavity whose resonance wavelength equals λ 0 =1530.5nm and intrinsic Q 0 = 2.25 × 10 4 . (a) Normalized transmission spectrum as a function the laser detuning ∆λ = λ -λ 0 (in nm) and the probing laser power. (b) Normalized transmission spectrum as a function the laser detuning (in nm) recorded at four different probing laser powers of 26, 38, 52 and 73µW . The red circles show the resonance edges.

Figure 4 . 24 -

 424 Figure 4.24 -Injected probing laser power P in as a function of the internal cavity energy U c . The internal cavity energy features a linear relationship with the injected laser power with coefficients P in /U c = 15.37µW/pJ.

  The corresponding fitting parameters give us the values of a F CA , b T P A and γ in as 1.04 × 10 -4 µW/pJ 3 , 8.0 × 10 -5 µW/pJ 2 and 2.05 × 10 -3 µW/pJ respectively.

Figure 4 .

 4 Figure 4.25 -(a) Dependance of the intra-cavity energy U c on probing laser detuning ∆λ in the nanobeam cavity. (b) Dependance of the absorbed laser power P abs on U c in the nanobeam cavity.

Figure 4 . 26 -

 426 Figure 4.26 -Mechanical spectra of the M 1 , M 2 and M 3 mechanical modes for positive laser detuning (a) when sweeping the laser towards higher wavelengths; (b) when sweeping the laser towards lower wavelengths. Depending on the sweeping direction, the mechanical modes disappear at different laser detuning which indicates the cavity resonance edge shifts accordingly. In this study, the cavity features an intrinsic Q 0 =17562 at a resonance wavelength of λ 0 =1531.45nm.

Figure 4 . 27 -

 427 Figure 4.27 -Mechanical frequency Ω m and effective linewidth Γ m of mode M 2 in (a)(c) and M 3 in (b)(d) as a function of the normalized laser detuning ∆/κ. The black dash lines are linear fitting curves. The involved optical mode is here the second-order optical modes with intrinsic Q 0 17562 at a resonance λ 0 =1531.45nm. The injected laser power is fixed at 45µW .

Figure 4 .

 4 Figure 4.28 -(a) Mechanical spectrum of modes M 1 , M 2 and M 3 as a function of the injected laser powers for a laser detuning (∆/κ = 50). The laser power is swept from 42µW to 120µW and then from 120µW to 42µW . T(b) to (e) Power spectrum density recorded at different injected laser powers ranging from P in = 42µW to 70µW . The red line is the Lorentzian fitting curve.

Figure 4 .

 4 Figure 4.29 -(a) Mechanical modes amplitude as a function of the laser power for modes M 2 and M 3 . (b) Mechanical linewidth Γ ef f /2π as a function of the laser power for modes M 2 and M 3 . In this study, the involved optical mode is the second-order optical mode with intrinsic Q 0 =17562 and resonance wavelength λ 0 =1531.45nm.

  Figure(4.30a) shows the n cav evolution as a function of the normalized laser detuning ∆/κ. The resonance frequency ω 0 equals here 195.76THz and the injected power is 67µW . At zero detuning, the number of photons reach a maximum value of n cav = 3 × 10 5 .

Figure 4 .

 4 Figure 4.30 -(a) Evolution of n cav at a fixed laser power as a function of the laser detuning; (b) Evolution of n cav at the fixed laser detuning ∆/κ = 50 and 0.4 for increasing injected laser power. The calculation is done for a cavity featuring a resonance at ω 0 = 195.76T Hz(1531.45nm), and loss rates of κ c = 1.32 × 10 10 and κ = 2.43 × 10 10 .

Figure 4 .

 4 Figure 4.31 -(a) Effective damping rate Γ ef f as a function of the laser detuning and optomechanical coupling strength. The black line indicates the threshold for reaching the self-oscillation regime and corresponds to Γ ef f = 0. The n cav value here is set to 1000; (b) Effective damping rate Γ ef f as a function of the intracavity photon number and optomechanical coupling strength.

Figure 4 . 33 -

 433 Figure 4.33 -Evolution of the effective optomechanical damping rate Γ ef f as a function of the mechanical linewidth Γ m /2π and coupling strength g 0 /2π. The black dash line indicates the threshold condition to reach self-oscillations (Γ ef f = 0).

Figure 5 . 5 -

 55 Figure 5.5 -Frequency domain simulation of the electrostatic actuation of a nanobeam cavity system. The electrodes are fixed on the two side of the nanobeam with a distance d = 2000nm. The displacement field of the cavity center X(Ω) is a frequencydependent term. When the sweep AC voltage frequency Ω equals to the resonance frequency Ω m , the cavity has the largest displacement field.

Figure 5 . 6 -

 56 Figure 5.6 -Simulation of the displacement field along the x axis as a function of the electrode-to-electrode distance d for the first three symmetric mechanical modes Ω m = 1.842GHz in (a), 1.858GHz in (b) and 1.870GHz in (c) . The red lines are the corresponding fitting curve with equation X = a/(b + c × d) 2 as the maximum displacement field is proportional to 1/d 2 .

  (

  5.7) left). As the distance d decreases to less than 1000nm, the G starts to increase rapidly and proportionally to 1/d 3 . However, the reduction of the electrodes distance also induces optical losses in the nanobeam cavity (see Figure(5.7) right). For d lower than 1200nm, the coupled quality factor Q c of the fundamental optical mode decreases by more than two orders of magnitude from 1.4 × 10 7 down to 5 × 10 4 . As a trade-off, the electrode-to-electrode distance d should be chosen around 750 to 1000nm in order to have a high capacitance force actuation while preserving good optical property.

Figure 5 . 7 -

 57 Figure 5.7 -Evolution of the electromechanical coupling factor G (left y axis) and optical quality factor Q c (right y axis) as a function of the electrodes distance d. As d gets larger than 750nm, G begins to decrease rapidly and the Q c rises from 10 4 to 10 7 . The optimum distance d should be chosen between 750 and 1000nm to achieve a compromise between the electromechanical coupling strength and the optical losses.

Figure 5 . 10 -

 510 Figure 5.10 -Schematic of a piezoelectric actuator with two electrodes bonded on a poled crystalline piezoelectric material.

Figure 5 . 11 -

 511 Figure 5.11 -The schematic of the piezoelectric sensor with two electrodes bonded on a poled crystalline piezoelectric material.

Figure 5 . 13 -

 513 Figure 5.13 -Simulation of the piezoelectric actuation with electrode actuators on both ends of the GaAs nanobeam cavity system. When a voltage (10V) is applied to the electrodes, shear strain is generated along the same direction as the electric field. The spatial distribution of the first three mechanical modes around 1.8GHz are shown in the bottom.

Figure 5 . 15 -

 515 Figure 5.15 -The schematic of the Rayleigh surface waves on the substrate.

ζ 3 -

 3 8ζ 2 + 8ζ(3 -2η) -16(1 -η) = 0 (5.31) where ζ = (c R /c T ) 2 and η = c 2 T . The ratio of velocities of longitudinal and transverse waves depends on the Poisson's ratio ν of the medium. For example, in GaAs substrate, ν = c L /c T = ζ = 0.31. When solving equation(5.31), the only real solution, denoted c R , writes :

Figure 5 .

 5 Figure 5.16 -Left: schematic of the surface waves generated by IDT. The substrate is piezoelectric material with single layer in z direction. Right: schematic of IDT with finger and pitch width e and aperture length w.

Figure 5 .

 5 Figure 5.17 -Top: schematics of two different IDTs with a thickness 50 (left) and 200nm (right). The red squares are the IDT finger and the piezoelectric substrate in this study is GaN. Bottom : simulation of the substrate boundary displacement field as a function of IDT electrodes thickness, varying from 5nm to 200nm. The solid line indicates the displacement field in x axis, and dash line is the y axis. A potential of 1V is applied to one IDT electrode while the other IDT is connected to the ground.

Figure 5 . 18 -

 518 Figure 5.18 -Simulated eigenfrequencies and displacement field induced by IDTs with finger width e=600nm on a GaAs substrate. Three modes are shown: the characteristic mode at f 0 and higher order modes at f 1 and f 2 .

Figure 5 .

 5 Figure 5.19 -(a)The comparison between the simulation and theoretical values of IDT with different finger width e=500, 600 and 750nm. The IDT was built on GaAs substrate in the simulation. The blue line and red line show the characteristic f 0 mode between simulation and theoretical values as a function of finger width respectively. (b)the simulation models for three IDT finger width and their characteristic frequency f 0 .

Figure 5 .

 5 Figure 5.20 -Top: time domain simulation of platforms including either a single IDT (Left) or two IDTs face to face (Right) on GaAs substrate by Comsol multiphysics software. Bottom: the displacement field in z axis at time t = 8ns is shown in both configurations. The blue area indicates the center area between the IDTs, where the photonic crystal membrane will be located.

Figure 5 .

 5 Figure 5.21 -Left: The displacement field on time domain simulation at different time; Right: all the displacement distribution along x axis from t = 0 to t = 8ns. The simulation was done by two IDTs excitation on the GaAs substrate.

Figure 5 . 22 -

 522 Figure 5.22 -Time domain simulation of the displacement amplitude for two different IDT-to-IDT distance d. Blue line corresponds to d values leading to constructive interference at the center (d = 4ke) with a maximum displacement at x = 0; Red line corresponds to d values leading to destructive interference at the center (d = 2e+4ke), with minimum displacement at x = 0.

Figure 5 . 23 -

 523 Figure 5.23 -Schematic of the two dimensional photonic crystal cavity platform with two IDTs integrated on the same membrane. The two IDTs are located on the side side of the L3 cavity.

Figure 5 . 24 -

 524 Figure 5.24 -Schematic of IDT process including resist spin-coating and e-beam lithography, metallisation and the lift-off steps.

Figure 5 .

 5 Figure 5.25 -SEM images of the IDTs fabricated by lower beam dose 900µC/cm 2 in (a) and higher beam dose 1400µC/cm 2 in (b). The target finger width as well as the gap are 600nm in all the cases.

Figure 5 .

 5 Figure 5.27 -SEM images of various processed IDTs. (a) the IDTs geometries are close to the targeted geometry with no broken or stripping parts; (b) the IDTs slightly stripped from the two edges; (c) all IDTs are stripped and there is no adhesion between IDTs and the substrate.

Figure 5 .

 5 Figure 5.28 -SEM images of the IDTs devices embedding a L3 photonic crystal and without any photonic crystal. The IDT-to-IDT distance d in both cases are fixed to 20µm.

Figure 5 .

 5 Figure 5.29 -S-parameter components of a network analyzer on the device-undertest(DUT). The two sides SAW pads are connected with port 1 (P1) and port 2 (P2) of a network analyzer. Depending on the transmission or reflection signal, S ij is denoted with P1 or P2.

Figure 5 . 30 -

 530 Figure 5.30 -Reflection signal S 11 for IDTs widths of 500, 600 and 750nm in the frequency range between 500MHz and 3GHz.

Figure 5 . 32 -

 532 Figure 5.32 -Comparison of the resonance frequencies of the SAW modes for IDTs featuring different widths. The dash line is a linear fitting curve.

Figure 5 .

 5 Figure 5.33 -S-parameter spectra with IDTs width of e 500nm.

Figure 5 . 34 -

 534 Figure 5.34 -Measurement of η R = P R /P rf and η SAW = P SAW /P rf from the S 11 parameter spectrum for the 5-th mode. The dips occur at the resonance frequency.

  Figure(5.35) displays the impedance of IDTs in the frequency range of the fifth-order mode f 3 with resonance frequency of 5.73GHz. The amplitude of the mode is shown as |S 11 | in dB and the phase is also shown as arg(S 11 ) in degree.

Figure 5 .

 5 Figure 5.35 -Amplitude (in blue line) and phase (in red line) of the S 11 parameter in the frequency range of the fifth-order mode located at 5.73GHz for a device with IDTs width of 500nm.

Figure 5 . 36 -

 536 Figure 5.36 -Spectrum of the IDTs admittance (left) and the susceptance (right) in the frequency range of the fifth-order mode located at 5.73GHz for a device with IDTs width of 500nm.

Figure 5 .

 5 Figure 5.37 -Top: Optical images of different platforms integrated two IDTs arrays embedding a photonic crystal ()PhC) or membrane or without any PhC between the IDTs. Among the devices embedding the PhC, some feature released IDTs. Bottom: schematic of the different structures.

Figure 5 . 39 -

 539 Figure 5.39 -Simulation of the displacement field of the characteristic mode for three different substrates. (a) a GaAs membrane bonded on silicon substrate. (b) a GaAs membrane bonded on a silicon substrate via a thin BCB layer . (c) a suspended GaAs membrane. The displacement intensity are of the same order of magnitude in all structures. The SAW power is launched by a pair of IDTs on top of the GaAs membrane.

Figure 5 . 40 -

 540 Figure 5.40 -The reflection spectrum S 11 and S 22 in the frequency range of the fifthorder mode for IDTs widths of (a). 600nm with either (P1) or (P2) released IDTs and no released IDTs; (b). 500nm with either (P1) or (P2) released IDTs.

Figure 6 . 1 -

 61 Figure 6.1 -(a) Schematic of the OMR based on L3 cavity architecture. The piezoelectric substrate (in blue) is integrated on SOI wafer (in incarnadine) with the optical waveguide (in red) underneath the photonic crystal cavity. The IDT (in yellow) can generate surface acoustic waves guided via a phononic crystal waveguide to the optical cavity. (b) Optical image of the first generation of the final device based on on L3 GaAs photonic crystal cavity. The IDTs are show in blue, and the contact pads in white; the L3 photonic crystal is shown in green and the SOI waveguide in yellow. The phononic waveguide is missing in the realization.

Figure 6 . 2 -

 62 Figure 6.2 -Schematic of the OMR based on a nanobeam cavity system. Similarly to the L3 cavity-based OMR, the optomechanical photonic crystal resonator can be excited externally by SAW transducers and the mechanical modes can be probed optically via the SOI integrated waveguide. The V-shape phononic waveguide can guide the acoustical signal to one side of the nanobeam cavity in order to actuate the mechanical modes.

Figure 6 . 3 -

 63 Figure 6.3 -Schematic of OMO based on L3 cavity system with two different delay line feedback circuit from acoustical delay loop to optical delay loop. With acoustical delay, the advantage is the compactness and lower phase noise because of the fullyintegration.

Figure 6 .

 6 Figure 6.4 -a)Schematic of a phononic crystal composed of a matrix of cylinders in honeycomb pattern. The cylinders are made of air inside a solid medium. The unit cell is indicated in red dash line with lattice constant a and holes radius r. The corresponding first irreducible Brillouin zone with three symmetry points (Γ, M, K) is shown on the left.

Figure 6 . 5 -

 65 Figure 6.5 -(a)Band structure diagrams of a GaAs honeycomb phononic crystal with suspended membrane height of 200nm with r/a =0.21 and β = 0.30; (b) Evolution of the bandgap with the filling factor β from 0.3 to 0.44, and a values are fixed here; (c) Band structure diagrams of a GaN honeycomb phononic crystal of 400nm with r/a = 0.23 and β = 0.37; (d) Evolution of the bandgap with the filling factor β from 0.32 to 0.44, and a values are fixed here. Due to the nature of honeycomb lattice, the maximum β value can not be higher than 0.45.

Figure 6 . 6 -

 66 Figure 6.6 -SEM images of the GaAs-based phononic crystal (left) forming a phononic shield (middle) or a waveguide when removing a line of holes (right). These preliminary structures have been processed by G. Modica.

Figure 6 . 7 -

 67 Figure 6.7 -Schematic of optically and optomechanically coupled nanobeam cavities.

  Since we are working with different III-V materials, such as GaAs, In GaP, GaP, GaN and AlN. If the material is in cubic symmetry such as GaAs, InGaP and GaP, the photoelastic coefficients has such relationship C ijkl = C ijlk = C jikl . In cubic crystal with three axes are equivalent, we can get C 12 = C 13 , C 22 = C 33 , C 21 = C 13 , C 44 = C 55 = C 66 with only three independent components and the photoelastic tensor can be reduced to 6 × 6 matrix as :C ij (cubic) =

  For a Wurtzite crystalline material such as GaN, AlN and piezoelectric ceramics, the strain tensor are in hexagonal symmetry and we can have the relationship asC 11 = C 22 , C 13 = C 31 = C 23 = C 32 , C 12 = C 21 , C 44 = C55 and C 66 = (C 11 -C 12 )/2. There are only fives constants in the tensor matrix. For piezoelectric strain constants, the none-zero components are d 31 = d 32 and d 15 = d 24 and for permitivity e ε 11 = e ε 22

C 11 C 12

 1112

σ x (z) = E p 1 -Figure 6 . 9 -

 169 Figure 6.9 -The schematic of strain distribution in the piezoelectric actuator with two electrodes bonded on a poled crystalline piezoelectric material. The thickness for the substrate and electrode are t and t e in respective.

d 11 d

 11 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36

  q = d 31 E p C l w ε 1 dxdy = d 31 E p ε 1 wl C (6.22) Titre : Nano-optom écanique int égr ée dans les cristaux photoniques Mots cl és : Oscillateurs, optom écanique, nanophononique, nanostructures 2D/1D, circuits SOI, pi ézo électricit é, mat ériaux III-V, excitation r ésonante R ésum é : Les oscillateurs de r éf érence de haute puret é sont actuellement utilis és dans un grand nombre d'applications allant du contr ôle de fr équence aux horloges pour les radars, les GPS et l'espace ... Les tendances actuelles dans ce domaine requi èrent des architectures miniaturis ées avec la g én ération de signaux directement dans la gamme de fr équences d'int ér êt, autour de quelques GHz. R écemment, de nouvelles architectures bas ées sur les principes de l'optom écanique ont vu le jour dans ce but. De tels oscillateurs optom écanique g én èrent non seulement des signaux hyperfr équences directement dans la gamme de fr équences GHz avec éventuellement un faible bruit de phase, mais permettent également un degr é élev é d'int égration sur puce. Ce travail de th èse s'inscrit dans cette d émarche. L'oscillateur optom écanique étudi é se compose de cavit és à cristaux photoniques suspendues coupl ées à des guides d'ondes silicium sur isolant int égr és dans une architecture tridimensionnelle. Ces cavit és abritent des modes optiques fortement confin és autour de 1550nm et des modes m écaniques dans le GHz. De plus, ces structures pr ésentent un recouvrement spatial entre phonon et photon élev é. Il en r ésulte un couplage optom écanique am élior é. Cette force de couplage optom écanique am élior ée est ici sond ée optiquement sur des structures à cristaux photoniques de conception optimis ée. Ces cavit és sont r éalis ées dans des mat ériaux semi-conducteurs III-V dont la pi ézo électricit é nous permet d'int égrer des outils suppl émentaires pour sonder et contr ôler les vibrations m écaniques via un pilotage capacitif, pi ézo électrique ou acoustique. Ce contr ôle total des modes m écaniques et de l'interaction optom écanique ouvre la voie à la mise en oeuvre de circuits int égr és pour le verrouillage par injection et des boucles de r étroaction permettant de r éduire le bruit de phase de l'oscillateur. Title : Integrated nano-optomechanics in photonic crystal Keywords : Oscillators, optomechanics, nanophononics, 2D/1D nanostructures, SOI circuitry, piezoelectricity, III-V materials, resonant excitation Abstract : High purity reference oscillators are currently used in a wide variety of frequency control and timing applications including radar, GPS, space... Current trends in such fields call for miniaturized architectures with direct signal generation in the frequency range of interest, around few GHz. Recently, novel optomechanically-enhanced architectures have emerged with this purpose. Such optomechanically-driven oscillators not only generate microwave signals directly in the GHz frequency range with possibly low phase noise but also are amenable to a high degree of integration on single chip settings. This PhD work falls within this scope. The optomechanicallydriven oscillator under study consists of suspended photonic crystal cavities coupled to integrated siliconon-insulator waveguides in a three-dimensional archi-

Table 1 .

 1

	material	InP	GaAs	InGaP	GaP	GaN	AlN
	Index ε	3.16	3.37	3.17	3.07	2.31	2.1
	Gap energy	1.34 eV 1.42 eV 1.70 eV 2.26 eV	3.28 eV	6.03 eV
	Crystalline structure	Zinc	Zinc	Zinc	Zinc	Wurtzite	Wurtzite
	Piezoelectricity	-0.03 (e 14 )	-0.16 (e 14 )	-0.07 (e 14 )	-0.10 -0.30(e 15 ) -0.48(e 15 ) (e 14 ) -0.33(e 31 ) -0.58(e 31 )
	(Cm -2 )					0.65(e 33 )	1.55(e 33 )

1 -Some basic parameters of some III-V semiconductors

Table 2 .

 2 1 -Photoelastic coefficients and dielectric permittivity for some III-V materials at 300K

			GaAs InGaP GaP	GaN	AlN
		p 11 119	121	140.5 390 ± 15 410 ± 10
		p 12 53.4	59	62	145 ± 20 149 ± 10
	Photoelastic coefficients	p 13	/	/	/	106 ± 20 99 ± 4
	unit: GPa	p 33	/	/	/	398 ± 20 389 ± 10
		p 44 59.6	58	70.3 105 ± 10 125 ± 5
	Permittivity	¯	12.9	11.8	11.1	10.4(E ) 8.5(E ) 9.5(E

⊥ ) 11(E ⊥ )

Table 2 .

 2 2 -Characteristics of the mechanical modes in a standard GaAs-based L3 cavity with a slab thickness of 265nm, a lattice constant a of 420nm and holes radius r of 130nm Ω

	(kHz)	(kHz)	(kHz)

m m ef f x ZP F g 0M I /2π g 0P E /2π g 0 /2π (GHz) (pg) (fm)

Table 2 .

 2 3 -Optomechanical coupling strengths for the first mechanical modes sustained in a modified L3 cavity etched in GaAs and GaN slabs.

		GaAs L3 cavity GaN L3 cavity
		Ω m	g 0 /2π	Ω m	g 0 /2π
		(GHz) (kHz) (GHz) (kHz)
	M1	0.90	-1.20	1.06	0.09
	M2	1.12	-0.09	1.31	0.02
	M3	1.48	-0.63	1.74	-0.41
	M4	1.88	-0.66	2.24	-0.12
	M5	1.91	-0.28	2.26	0.07
	M6	2.19 -579.45	2.67	-50.79
	M7	2.52	-0.50	3.06	0.13
	M8	3.05	-84.63	3.84	-36.54
	M9	4.03 -583.99	4.49	-21.73
	M10 4.17 -521.42	5.02	-24.23

Table 2 .

 2

	4 -Group B mechanical modes with their mechanical and optomechanical
	characteristics
		S1	S2	S3	AS1	AS2	AS3
	Frequency (GHz) 1.8567 1.8672 1.8748 1.8563 1.8673 1.8776
	m ef f (pg)	1.30	2.21	2.26	1.47	1.54	1.59
	x ZP F (fm)	4.67	3.58	3.53	4.40	4.29	4.21
	g

0M I /2π (kHz) -234.10 -429.01 -412.63 -86.80 71.22 26.51 g 0P E /2π (kHz) -495.04 -239.82 -394.34 -42.44 -17.30 26.48 g 0 /2π (kHz) -729.22 -668.85 -807.11 -129.25 53.94 53.00

Table 2 . 5

 25

		unit: (kHz)	S1	S2	S3	AS1	AS2	AS3
	Rectangular holes	g 0M I /2π g 0P E /2π	-234.10 -429.01 -412.63 -86.80 71.22 26.51 -495.04 -239.82 -394.34 -42.44 -17.30 26.48
	@1.8GHz	g 0 /2π	-729.22 -668.85 -807.11 -129.25 53.94 53.00
	Elliptical holes	g 0M I /2π g 0P E /2π	-39.28 -172.83 160.23 1396.61 -304.56 1218.27 -541.04 481.66 303.42 62.04 160.99 24.69
	@2.1GHz	g 0 /2π	1357.33 -477.40 1378.50 -479.00 642.65 328.11

-Optomechanical coupling strengths for the first breathing modes of a GaAs-based nanobeam cavity with a 0 =315nm, H x =150nm, H y0 =200nm, H yn =450nm and F W HM = 3µm. The shape of the holes is either rectangular or elliptical.

Table 2 . 6

 26

		unit:kHz	S1	S2	S3	AS1	AS2 AS3
	Rectangular holes	g 0M I /2π g 0P E /2π -73.88 32.29 103.19 23.60 -4.26 -7.37 18.14 28.46 57.37 -18.80 -4.69 1.78
	@2.7GHz	g 0 /2π	-55.74 60.75 160.56	4.80 -8.95 -5.59
	Elliptical holes	g 0M I /2π g 0P E /2π -109.45 -32.86 -100.55 3.75 -1.77 1.44 4.22 -15.31 -15.61 2.20 0.17 0.86
	@2.9GHz	g 0 /2π	-105.23 -48.17 -116.16 5.95 -1.60 2.30

-Optomechanical coupling strengths for the first breathing modes of a GaNbased nanobeam cavity with a 0 =480nm, H x =450nm, H y0 =400nm, H yn =700nm and F W HM = 3µm. The shape of the holes is either rectangular or elliptical

Table 2 .

 2 7 -Optical Q 0 , mechanical frequencies and optomechanical coupling g 0 /2π in the different optimized cavity systems

			L3 cavity	Nanobeam cavity
			Standard Optimum Rectangular Elliptical
		Q 0	4900	24000	10 8	10 8
	GaAs	Ω m (GHz) 2.19 (M6) 2.13 (M6)	1.86	2.10
		g 0 /2π (kHz)	-579.45	-344.66	-807.11	1378.50
		Q 0	-	-	10 7	10 7
	InGaP	Ω m (GHz)	-	-	2.31	-
		g 0 /2π (kHz)	-	-	-1199.27	-
		Q 0	500	1243	10 5	10 5
	GaN	Ω m (GHz) 2.67 (M6) 2.59 (M6)	2.69	2.90
		g 0 /2π (kHz)	-50.79	-25.36	160.56	-116.16

two modes. Two geometries were here considered based on GaAs or GaN materials: 2D photonic crystal slab defect cavities (in particular L3 cavities) and nanobeam cavities.

Table 3 .

 3

	1 -ICP-RIE recipes used for etching some III-V materials
	height	GaAs	InGaP/GaP GaN/AlN
	HBr/He/O 2 (sccm) 10/15/0.4	8/90/0.9	-
	BCl 3 /Cl 2 /N 2 (sccm)	-	-	10/15/10
	RF power (W)	60	60	48
	ICP power (W)	600	600	800
	Pressure (mTorr)	1.5	5	9

  Table (4.1) gathers the measured and expected mechanical frequencies Ω m of the modes sustained by the cavity. These values are in good agreement. By fitting the

Table 4 .

 4

	1 -Optomechanical modes characterization
		M 1a	M 1b	M 2	M 3
	Simulated resonance frequency (GHz) 2.2331 2.2333 2.2520 2.2717
	Measured resonance frequency (GHz) 2.2439 2.2444 2.2831 2.3232
	Linewidth Γ m /2π(MHz)	|	4.476 3.810 3.664
	Q m	|	500	600	643

Table 4 .

 4 2 -Coupling strength g 0 /2π and single photon-phonon cooperativity C 0 of the first six mechanical modes sustained by the nanobeam cavity made of elliptical

	holes
		M 1	M 2	M 3	M 4	M 5	M 6
	Ω m /2π(GHz) 2.2386 2.2771 2.3183 2.4195 2.4255 2.4340
	g 0 /2π (kHz) 126.5 143.9 114.6	69.2	107.4 134.2
	C 0 (×10 -5 )	4.6	7.0	4.6	2.0	5.0	6.8

Table 5 .

 5 1 -Some basic material parameters for some III-V materials

			GaAs InGaP GaP	GaN	AlN
		p 11	119	121	140.5 390 ± 15 410 ± 10
		p 12 53.4	59	62	145 ± 20 149 ± 10
	Elastic coefficients	p 13	-	-	-	106 ± 20 99 ± 4
	GPa (300K)	p 33	-	-	-	398 ± 20 389 ± 10
		p 44 59.6	58	70.3 105 ± 10 125 ± 5
		d 14 -0.16	-0.07 -0.10	-	-
	Piezoelectric coefficients d 15	-	-	-	-0.30	-0.48
	Cm -2	d 31	-	-	-	-0.33	-0.58
		d 33	-	-	-	0.65	1.55
	Permittivity (300K)	e 11 12.9 e 33 -	11.8 -	11.1 -	10.4 9.5	8.5 11

Table 5 .

 5 2 -Electro-mechanical coupling k 2 material Si Quartz GaAs GaP GaN AlN LiN bO 3

	k 2 (%)	0	1.0	0.38	0.2	1.1	7.2	17

Table 5 .

 5 929c T(5.32) with c T being equal to 3.35km/s in GaAs substrate. The Rayleigh waves c R is then 3.11km/s. The following table lists he different Poisson's ratio ν and the wave velocities in different materials. 3 -Different materials and their associated waves velocities

			Si	GaAs InGaP GaP GaN AlN
	Poisson's ratio ν Calculated √ ζ	0.28 0.924 0.929 0.934 0.929 0.936 0.928 0.31 0.34 0.31 0.35 0.29
		Transverse c T	5.84	3.35	3.60	4.12 5.02 6.22
	Acoustic waves	Longitudinal c L 8.43	4.73	5.21	5.81 6.90 11.27
	(unit: km/s)	Rayleigh c R	5.40	3.11	3.36	3.83 4.70 5.77

Table 5 .

 5

	4 -Characteristic frequency f 0 (expressed in GHz) for IDT fingers width e
	equal to 500, 600 and 750nm
	Frequency	GaAs InGaP GaP GaN AlN
	Rayleigh wave velocity c R (km/s) 3.113 3.363 3.828 4.698 5.770
	IDT 500nm	1.557 1.682 1.914 2.349 2.885
	IDT 600nm	1.297 1.401 1.595 1.958 2.404
	IDT 750nm	1.038 1.121 1.276 1.566 1.923

Table 5 .

 5

		5 -Properties of some metals for IDT's processing
		Density Shear waves Surface waves Electrical conductivity
		ρ, kg/m 3	c T , m/s	c R , m/s	σ, S/m
	Aluminum	2710	3080

Table 5 .

 5

	6 -Simulated characteristic frequency f 0 for different IDT fingers width and
	piezoelectric substrate (unit:GHz)
	Substrate GaAs GaN
	IDT 500nm 1.61 1.91
	IDT 600nm 1.20 1.41
	IDT 750nm 0.82 0.96

Table 5 .

 5 8 -SAW modes resonance frequencies lying from 500MHz to 8GHz (unit:

	GHz)
	width (nm) f 0	f 1	f 2	f 3	f 4	f 5	f 6
	500	1.23 2.46 3.36 4.12	-	5.73 6.61
	600	0.87 2.10 3.30	-	-	5.46 6.57
	750	0.62 1.72 2.83 3.27 4.87 5.34 6.50

Table 5 .

 5 9 -SAW power η SAW for the device formed by IDTs (e=500nm)

		f 0	f 1	f 2	f 3	f 5	f 6
	Frequency (GHz) 1.23 2.46 3.36 4.12 5.73 6.61
	η SAW	0.017 0.035	-	0.019 0.160 0.017

Table 5 .

 5

	10 -Coupling factor k 2 s for the device formed by IDTs (e=500nm)
		f 0	f 1	f 2	f 3	f 5	f 6
	Frequency (GHz) 1.23	2.46 3.36 4.12	5.73	6.61
	k 2 s	1.71% 1.64%	-	1.48% 1.53% 1.47%

Table 5 .

 5 11 -The SAW power ratio η SAW for IDTs width 600nm in the three different devices (susp stands here for suspended)

	mode	no susp only PhC susp PhC+SAW susp
	f 0 (0.87GHz)	0.003	0.010	0.019
	f 1 (2.10GHz)	0.010	0.025	0.033
	f 4 (5.47GHz)	≈ 0	≈ 0	0.134

Figure 5.38 -Reflection spectrum S 11 for IDTs with width 600nm. Top: the IDTs are not released and a suspended PhC is etched in between; middle: the IDTs are released and a suspended PhC is etched in between; bottom: the IDTs are not released and a suspended PhC is not etched in between.

  C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66

			Appendix
			
	C ijkl =	        	        
	6 × 6 matrix as:
		157

  C 11 ε xx + C 12 ε yy + C 12 ε zz C 12 ε xx + C 11 ε yy + C 12 ε zz C 12 ε xx + C 12 ε yy + C 11 ε zz 2C 44 ε yz 2C 44 ε zx 2C 44 ε xyWhen we reverse the Voigt notation with equation(6.2.4), it becomes:C 11 ε xx + C 12 ε yy + C 12 ε zz C 44 ε xy C 44 ε zx C 44 ε xy C 12 ε xx + C 11 ε yy + C 12 ε zz C 44 ε yz C 44 ε zx C 44 ε yz C 12 ε xx + C 12 ε yy + C 11 ε zz x E y } C 44 ε xy + 2Re E * y E z C 44 ε yz + 2Re {E * z E x } C 44 ε zx + |E x | 2 (C 11 E xx + C 12 (ε yy + ε zz )) + |E y | 2 (C 11 ε yy + C 12 (ε xx + ε zz )) + |E z | 2 (C 11 ε zz + C 12 (ε xx + ε yy ))] dV e|E 2 | (6.7) If the material is in Hexagonal symmetry such as GaN and AlN, we have 5 independent coefficients in elastic tensor with C 66 = (C 11 -C 12 )2.

								
						e 2 e 0	C ijkl ε kl = -	e 2 e 0	        	        	(6.5)
	de ij dα	(cubic) = -	e 2 e 0	  		  
							(6.6)
	Let's put equation(6.6) inside the first-order correction function to calculate the
	photoelastic effects contribution:
	g P E (cubic) =	dω dα P E	x ZP F
		=	ω 0 e 2 2e 0		dV 2Re {E

*

  6.2):C 11 ε xx + C 12 ε yy + C 13 ε zz C 12 ε xx + C 11 ε yy + C 13 ε zz C 13 ε xx + C 13 ε yy + C 33 ε zz 2C 44 ε yz 2C 44 ε zx (C 11 -C 12 ) ε xy C 11 ε xx + C 12 ε yy + C 13 ε zz 0.5 (C 11 -C 12 ) ε xy C 44 ε zx 0.5 (C 11 -C 12 ) ε xy C 12 ε xx + C 11 ε yy + C 13 ε zz C 44 ε yz C 44 ε zx C 44 ε yz C 13 ε xx + C 13 ε yy + C 33 ε zz x E y } (C 11 -C 12 ) ε xy + 2Re E * y E z C 44 ε yz + 2Re {E * z E x } C 44 ε zx + |E x | 2 (C 11 ε xx + C 12 ε yy + C 13 ε zz ) + |E y | 2 (C 12 ε xx + C 11 ε yy + C 13 ε zz ) + |E z | 2 (C 33 ε zz + C 13 (ε xx + ε yy ))]

									
				de ij dα	(hexagonal) = -	e 2 e 0	C ijkl ε kl = -	e 2 e 0	        	        
	= -	e 2 e 0	  					  
	With the photoelatic contribution in optomechanical coupling strength as:
	g P E (hexagonal) =	dω dα P E	x ZP F
			=	ω 0 e 2 2e 0	dV Re {E

*

Table 6 .

 6 1 -Basics mechanical parameter for some III-V materialsD m = d mi σ i + e σ ik E k (6.10)

	GaAs InGaP GaP	GaN	AlN

⊥ ) 11(E ⊥ )



  C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 11 d 21 d 31 d 12 d 22 d 32 d 13 d 23 d 33 d 14 d 24 d 34 d 15 d 25 d 35 d 16 d 26 d 36

			σ 1			
	        	         τ 23 σ 2 σ 3 τ 31	        	+	        
		τ 12

d

  11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36

					 σ 1	
	   D 1 D 2 D 3	   =	  	  	         σ 2 σ 3 σ 4 σ 5	        	+	 e σ 11 e σ 12 e σ 13   e σ 21 e σ 22 e σ 23 e σ 31 e σ 32 e σ 33	  	 E 1   E 2 E 3	  
					σ 6

d

  In the end, the stress distribution in this piezoelectric actuator α is as:

		0	t 2	σ x zdz +	2 t	t 2 +te	σ t x zdz = 0	(6.19)
		0	t 2	σ y zdz +	2 t	t 2 +te	σ t y zdz = 0	(6.20)
	α =	3E

p ( t 2 + t e ) 2 -( t 2 ) 2 (1 -ν)ε 0 2E p ( t 2 + t e ) 3 -( t 2 ) 3 (1 -ν) + 2E p ( t 2 ) 3 (1 -ν) (6.21)

Acknowledgment

Appendix Appendix A

The photoelastic effect is due to the variation in strain induced by the mechanical motion that change the refractive index of the material. This contribution is material-dependent and different dielectric materials have their photoelastic properties. Usually, the materials with photoelastic properties are birefringent, which means depending on the strain in the material, they can refract light for different orientations [START_REF] Dally | Experimental stress analysis[END_REF] . In this case, the photoelastic contribution in frequency shift for a material concerning with its refractive index tensor e is given as:

Appendix B

In this section, I will briefly introduce the linear piezoelectricity for most of the III-V materials with either Zinc or Wurtzite crystalline structure. The piezoelectric actuators is formed as the two electrodes bonded on the two sides of a piezoelectric substrate with dipole alignment in the crystal. The two electrodes are connected to the voltage supply. Here, we assume that the total strain in this piezoelectric actuator system is the total mechanical strain induced by the applied electric field. In linear piezoelectric regime, the electromechanical relation can be wrote as:

where E p is the Young's modulus of the piezoelectric sensor. The generated voltage V is then as:

and the generated strain ε 1 can be found as : (6.24)