Andrea De Marchi

Andrea Barone

Anirudh, Maxime, Gabriel Bruno

Alexandre, Léa, Gladys, Franck, Théo, Elias, Malik, Ji, Sukania, Lena Melissa

Jeremie Amir

Toniq Permament Members Ariel

Kamel

Alejandro Sylvain

Khannan, Tirth, Sampo, Tasnimul, Moham- mad, Corentin, Mohamed, Hossein Byoung-Moo Simon

Sarah Benchabane

Mehdi Alouini

Simon Gröblacher

Dries

Keywords: cristaux optomécaniques, ondes acoustiques, cristaux phononiques, nanofabrication, oscillateurs. vii optomechanical crystals, acoustic waves, phononic crystals, nanofabrication, oscillators. ix

the amazing Arancine we cooked together with Francesco for one repas de Noël at C2N!).

First of all, I would like to thank my supervisor Rémy Braive for the amazing opportunity that he gave me when he accepted my application for this PhD. If I look back, I cannot trust all the achievements that we have been able to reach. We built the lab from zero after the move from Marcoussis to Palaiseau and we have been working in three different cleanrooms to fabricate our nanostructures (I can still remember our "meeting" in Paris at Gare du Nord really early in the morning to go to Lille). It has been an honor working with you Rémy, you are the best supervisor that a PhD student can have, thanks also for the time we enjoyed outside the lab and I hope we can work again together in the near future! Thanks a lot to Rui Zhu for the really nice time that we spent working together and also outside the lab. Rui was a PhD student when I joined the optomechanics group. She was so welcoming and she made me feeling home since the beginning. She is a great researcher and a really nice person. I wish you all the best for your future career Rui! I would also like to thank Robert Horvath that is a post-doc now in our optomechanics group. Robert helped me a lot, especially in the last period of my PhD when he was always there if I needed something. It was great working with you Robert, you are a talented guy and I wish you the best for the future! A special mention to Francesco Manegatti, PhD in the ToniQ group, with whom I shared this adventure since the beginning. With Francesco we studied the same master, we started the PhD at the same time and we shared a flat (actually three flats) in Paris for three years. I think that we have been the psychological support of each other in these last few years of our life and also during the confinement for the pandemic (I hope Covid will be over soon. . . ). Thanks Fra for all the memories, I wish you the best for your career and I'm sure we will still enjoy some parisian time together.

Thanks a lot also to Fabrice Raineri, permanent member in the ToniQ group and actually heading soon to new adventures in Nice. He was always there every time I needed help in cleanroom, thanks for the useful discussions we had and for the nice time we spent outside the lab as well (people in the ToniQ group still remember Dans un monde de plus en plus connecté, les oscillateurs ont un rôle primordial dans les systèmes de télécommunications et de navigation ainsi que pour les applications de détection et de métrologie. Dans l'ensemble de ces systèmes, la tâche d'un oscillateur est relativement simple: créer un signal périodique qui sont à la base des technologies de l'information et des télécommunications (ICTs) modernes. La sortie de l'oscillateur peut devenir par exemple une horloge permettant le rafraîchissement des informations dans un microprocesseur ou une référence temporelle pour la synchronisation des données sur Internet. L' "oscillateur idéal" peut être défini à partir de trois facteurs de mérite afin de répondre aux besoins des ICTs modernes :

(i) La fréquence du signal généré : un oscillateur idéal doit fournir une fréquence de travail directement dans le régime Ultra Haute Fréquence (UHF), c'est-à-dire entre 3 et 10 GHz.

(ii) La pureté du signal autrement dit le bruit de phase : ce dernier doit être le plus faible possible et il est mesuré par la fonction L(f ). Ce dernier est défini comme L(f ) = Sϕ(f ) 2 où S ϕ (f ) est la densité spectrale unilatérale de la fluctuation de phase aléatoire ϕ(t) et il est exprimé en [dBc/Hz]. Les oscillateurs de l'état de l'art ont une phase inférieure à -100 [dBc/Hz] à 10 kHz de la porteuse.

(iii) L'encombrement de l'ensemble : il ne doit pas dépasser quelques cm 3 pour faciliter leur intégration dans les technologies modernes. Il s'agit d'un enjeu crucial pour les oscillateurs, en particulier pour les technologies transportables.

Plusieurs types d'oscillateurs, basés sur différent concepts technologiques, sont utilisés pour la génération de référence de temps stable. De par leur maturité technologique, les oscillateurs à quartz, basé sur les vibrations mécaniques, sont depuis longtemps les plus utilisés. Cependant, leur fréquence de travail ne dépasse pas les quelques MHz représentant une énorme limitation. Pour contrer cette limitation et travailler dans le régime des UHF, il est indispensable d'utiliser plusieurs étages de multiplication de fréquence. Chacune de ces multiplications induit un bruit suppléxi Résumé substantiel mentaire réduisant ainsi les performances de bruit de phase de cette technologie. Par la suite, une autre technologie utilisant l'électronique est apparue. Ces oscillateurs électroniques offrent une bonne compacité et une fréquence de travail directement dans le GHz mais les objectifs en termes de bruit de phase ne sont pas remplis. Pour leur représentant emblématique, les Voltage Controlled Oscillators (VCOs), le bruit de phase typique est supérieur à -100 [dBc/Hz] à 10 kHz de la porteuse. De plus, le fait que la porteuse du signal de référence soit dans le domaine RF les rend sensibles aux perturbations électromagnétiques, en limitant du même coup une intégration aisée avec les composants optiques. Enfin, les oscillateurs qui ont dominé la scène au cours des dernières décennies sont les oscillateurs opto-électroniques (OEO). La porteuse du signal de référence est directement dans le domaine optique avec de bonnes performances de bruit de phase directement aux fréquences GHz mais leur grande taille, principalement due à la ligne à retard optique présente dans la boucle de rétroaction, mise en oeuvre avec des kilomètres de fibres, limite leur compacité.

Il existe donc un manque dans la technologie actuelle des oscillateurs afin d'atteindre l' "oscillateur idéal" en remplissant simultanément les trois besoins décrits précédemment. La motivation derrière cette thèse est donc de présenter un oscillateur basé sur le concept innovant de l'optomécanique et de l'acoustique pour démontrer un oscillateur optique micro-onde et compacte. Au coeur de cette thèse se trouve un développement technologique innovant pour renforcer et utiliser l'interaction entre dégrée de liberté optique et acoustique au sein d'une cavité nanostructurée. La colocalisation au sein d'une nanostructure à la fois d'un mode optique et d'un mode de vibration mécanique permet d'imprimer la fréquence de ce dernier sur la porteuse optique. L'exploitation cette forte interaction au sein d'une telle nanostructure permet d'envisager de nouvelles applications comme des oscillateurs optomécaniques GHz. Les oscillateurs optomécaniques (OMO) ont été récemment démontrés. Parmi eux, la géométrie la plus prometteuses est celle des cristaux optomécaniques unidimensionnels fabriqués à partir de matériaux semi-conducteurs III-V. Ces derniers suscitent un fort intérêt à cause de leur fréquence de résonance au GHz et leur compacité intrinsèque. Leurs performances en termes de bruit de phase restent néanmoins un enjeu important encore peu étudié. Cela requiert une investigation en profondeur des limitations physiques ainsi que la mise en place de rétroactions efficaces. Pour cela, il est indispensable d'adresser et de contrôler le mode mécanique de l'OMO. Ces rétroactions permettant la stabilisation des oscillations peuvent être obtenues par différent moyens exploitant par exemple des effets électrostatiques, piézoélectriques ou acoustiques.

La plate-forme développée durant cette thèse a pour objectif de démontrer l'intégration sur puce à la fois d'un générateur de fréquence optomécanique directement aux fréquences GHz avec l'actionnement ou excitation acoustique résonante et le délai nécessaire à la stabilisation des oscillations. Plus précisément, le cristal optomécanique unidimensionnel, siège de l'oscillation GHz, sera intégré sur des guides d'ondes silicium sur isolant (SOI) où le couplage évanescent permettra à la lumière en résonance avec la cavité d'interagir avec les vibrations GHz de la structure. Parallèlement, en profitant des propriétés piézo-électriques du semiconducteur III-V, l'excitation résonante d'ondes acoustiques de surface est réalisée grâce à des transducteurs interdigités (IDTs). Afin de contrôler la direction ainsi que la vitesse de propagation entre les IDTs et l'oscillateur optomécanique, un guide d'onde phononique peut en plus être ajouté sur la même membrane semi-conductrice où l'OMO a été fabriqué. En travaillant sur la dispersion d'un tel guide d'ondes, une faible vitesse de groupe du mode propageant peut-être conçu. Cela permet d'introduire précisément le retard nécessaire dans le système en jouant sur la longueur du guide d'onde. Ainsi, le délai nécessaire pour la stabilisation des oscillations peut être mis en oeuvre sur puce avec une taille micrométrique. Les performances de bruit de phase de l'OMO bénéficieront de ce schéma pour finalement essayer d'approcher les mêmes performances que pour les OEO mais dans volume plus compact. Le but non trivial de cette proposition est de faire correspondre les trois fréquences mécaniques qui entrent en jeu. Plus précisément, l'objectif est de concevoir un transducteur capable de générer une onde acoustique se propageant à Ω acoustic étant à la même fréquence que celle du mode mécanique du cristal optomécanique Ω m . L'onde ainsi générée doit également être guidée du transducteur vers le cristal à l'aide d'un guide d'onde phononique dont le mode propageant Ω prop doit être égale à Ω acoustic et Ω m . L'ensemble de ces éléments constitutifs de l'architecture vont être discutées séparément dans les chapitres de cette thèse afin de démontrer puis contrôler leurs fréquences et leurs performances respectives.

Dans le chapitre 2, les cristaux optomécaniques unidimensionnel sont présentés comme une solution possible pour les oscillateurs optomécaniques (OMOs) intégrés. La théorie et la simulation des structures photoniques sont d'abord discutées, montrant comment les modes optiques et mécaniques peuvent être finalement confinés dans un très petit volume au centre de la structure. Ceci peut notamment être réalisé grâce à une conception appropriée et contrôlée de la périodicité au sein du cristal. Le défaut de périodicité introduit permet un confinement doux de la lumière grâce à un ajustement progressif. Les outils de calcul de couplage entre le mode optique et mécanique sont ensuite introduits. A partir de ces simulations, nous montrons comment un couplage optomécanique élevé g 0 /2π de l'ordre de quelques centaines de kHz est finalement attendu entre le mode fondamental optique (à ≈ 1550 nm) et les page xiii Résumé substantiel modes localisés mécaniques (à Ω m ≈ 2-3 GHz selon le matériau). Dans cette configuration, les deux modes sont situés au centre du cristal optomécanique suspendu. Par la suite, la fabrication de ces cristaux optomécaniques intégrés est réalisée sur des circuits photoniques incluant des guides d'ondes silicium. Le couplage évanescent entre la cavité photonique et les guides d'ondes est ensuite démontré avec des résultats optiques expérimentaux montrant comment les cristaux optomécaniques fabriqués sont en effet capables de confiner des modes optiques aux longueurs d'onde des télécommunications (≈ 1550 nm) avec un facteur de qualité mesuré élevé (1x10 5 ). L'étude expérimentale du comportement mécanique de ces mêmes structures montre comment, pour les deux matériaux étudiés -GaAs et GaP-des modes mécaniques localisés directement dans le régime UHF existent et peuvent être mesurés expérimentalement. Dans le cas du GaAs, les modes mesurés ont une fréquence mécanique Ω m autour de 2 GHz mais finalement ils ne sont pas capables d'entrer dans le régime d'auto-oscillation caractérisé par une réduction de la largeur de bande du mode mécanique avec en même temps une augmentation de l'amplitude du pic. Pour le GaP, nous mesurons un mode à Ω m = 3.25 GHz qui est quant à lui capable d'atteindre le régime d'auto-oscillations. Avec ce système, des mesures de bruit de phase sont effectuées montrant des performances prometteuses avec -84 [dBc/Hz] à 100 kHz de décalage par rapport à la porteuse sans aucune mise en oeuvre de boucle de rétroaction. Ces résultats confirment que les cristaux optomécaniques sont effectivement de bons candidats pour la mise en oeuvre d'oscillateur intégré sur puce.

Dans le chapitre 3, nous nous concentrons exclusivement sur la génération des ondes acoustiques avec l'aide de transducteurs interdigités (IDTs). Ce travail se base sur des simulations, la conception et la fabrication des IDTs directement intégrés sur les mêmes matériaux que ceux utilisés dans le Chapitre 2 pour l'intégration des oscillateurs optomécaniques. L'optimisation des méthodes de fabrication des électrodes et la suspension des IDTs est ainsi réalisé. Cela permet de fonctionner à la fois dans les régimes d'ondes acoustiques de surface (SAWs) et d'ondes de Lamb. Pour ce dernier cas, il est montré comment la suspension totale du transducteur améliore drastiquement l'efficacité de transduction. Cela permet un confinement du mode de propagation acoustique uniquement dans la membrane de semiconducteur III-V. La comparaison des simulations avec les résultats expérimentaux des IDTs conçus montrent une preuve tangible de la création et la propagation d'ondes acoustiques à la fréquence Ω acoustic d'intérêt. La fréquence de fonctionnement peut être ajustée en changeant simplement les paramètres géométriques des transducteurs. Une correspondance de fréquence des ondes acoustiques créées à Ω acoustic avec les modes mécaniques des cristaux optomécaniques fonctionnant à Ω m présentés dans le chapitre précédent est montrée pour GaP (∼ 3 GHz) et GaAs (∼ 2 GHz). En parti-culier, les mesures de transmission démontrent expérimentalement la propagation de signaux acoustiques aux alentours de 3 GHz pour GaP et 2 GHz pour GaAs. Cela suggère qu'une excitation acoustique résonante des oscillateurs optomécaniques est donc possible avec les transducteurs intégrés présentés.

Dans le chapitre 4, nous démontrons le contrôle des ondes acoustiques dans le GHz avec des guides à cristaux phononiques. Ces derniers sont un moyen efficace de maîtriser le comportement acoustique d'une onde, à savoir sa direction et sa vitesse de propagation. Plus précisément, en introduisant un motif périodique spécifique inspirée du motif dit Kagome, une bande phononique interdite peut éventuellement être ouverte pour tous les vecteurs d'ondes possibles. En introduisant un défaut dans le guide d'onde phononique, il est alors possible de guider un unique mode acoustique à une fréquence spécifique Ω prop tel que Ω prop = Ω acoustic = Ω m . L'ingénierie de la dispersion des modes de propagation trouvés ici montre comment il est en effet possible d'adresser la vitesse de groupe. La fabrication des structures phononiques intégrées avec les IDTs, précédemment introduites, est présentée démontrant un processus de nano-fabrication optimisé. Avec cette technologie, nous sommes capables de démontrer expérimentalement la propagation et le filtrage d'un mode acoustique à 2.06 GHz. Au-delà de ces considérations spectrales, nous démontrons, à cette fréquence, une vitesse de groupe mesurée inférieure à 1000 m/s à travers un guide d'ondes phononique en GaAs suspendu. Nous avons ainsi démontré qu'une implémentation possible d'un retard sur puce entièrement intégré de l'ordre de quelques ns/µm est à portée de main. Des mesures sur différentes longueurs de guides d'ondes phononiques ont montré qu'il sera à terme possible de remplacer par exemple 25 m de fibres optiques par 75 µm de guides d'ondes phononiques.

Dans le dernier chapitre, nous montrons la fabrication d'un oscillateur optomécanique intégré au-dessus de guide d'onde silicium avec un transducteur IDT et un guide d'onde phononique. Des caractérisations préliminaires de ces composants sont ici discutées démontrant l'interaction entre les différents domaines impliqués (électrique, acoustique et optique). Plus précisément, les oscillateurs optomécaniques intégrés sont caractérisés optiquement et mécaniquement, montrant des modes mécaniques dans le régime UHF. La transduction de l'acoustique vers l'optique est démontrée. Une tonalité MHz généré par l'IDT est lue lorsque le laser est à résonance avec la cavité optomécanique. Malheureusement, l'interaction complète acoustique vers mécanique puis optique nécessaire à la mise en oeuvre du retard acoustique n'était pas visible dans cette première génération de dispositifs. Cela est due au régime basse fréquence des ondes acoustiques générées. Dans le même temps, une architecture plus simple a été développée pour mettre en oeuvre une

Chapter 1

General introduction 1.1 Oscillators in modern society

In an increasingly connected world, oscillators are the main characters behind the scenes. They are in fact basic ingredients for telecommunications and navigation systems [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF] as well as sensing and metrology applications [START_REF] Fan | Highprecision thermal-insensitive strain sensor based on optoelectronic oscillator[END_REF][START_REF] Li | Femtometer-resolution wavelength interrogation of a phase-shifted fiber bragg grating sensor using an optoelectronic oscillator. In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides[END_REF][START_REF] Xu | Dual-frequency optoelectronic oscillator for thermal-insensitive interrogation of a fbg strain sensor[END_REF][START_REF] Lee | Displacement measurement using an optoelectronic oscillator with an intra-loop michelson interferometer[END_REF]. The task of an oscillator is relatively simple: create a periodic signal. Periodic signals are the basis of modern Information and Telecommunications Technologies (ICTs). Here, the oscillator's output can become for instance a clock allowing the refresh of the information in a microprocessor or a time reference for synchronization of data over the internet.

Ideal Oscillator

The frequency selective component of an oscillator is a resonator. It is usually a system able to resonate at a specific frequency. The ability of the resonator to store energy will determine the quality of the signal produced [START_REF] Kh Drexhage | Influence of a dielectric interface on fluorescence decay time[END_REF]. A resonator is normally a lossy system. We need something to compensate losses, that's why the implementation of a feedback path is usually needed to ensure stationary oscillations at a given frequency for the output signal. We can define three main figures of merit for a modern oscillator: the output frequency, the phase noise (i.e. its stability) and the package size. Let's describe them to understand how an ideal oscillator should look like.

Ideal Oscillator's natural working frequency: The operating frequencies of oscillators can go from few kHz to several hundreds of GHz. Nowadays, ICTs need higher and higher operating frequency, that's why an oscillator directly working in the frequency of interest is theoretically the best possible option, avoiding multiplication stages that usually degrade the signal quality and the size of the complete Chapter 1. Introduction system. In particular, even for the best available multiplication stage, the phase noise increases by a factor 20log(f f /f i ) where f f is the frequency after multiplication and f i is the frequency before multiplication. The natural frequency of an ideal oscillator should be then directly in the GHz range. Ideal Oscillator's phase noise: If we think about an ideal oscillator, this delivers a perfectly stable periodic signal. But in a real system perturbations appear. They can be of various kind such as additive noise (thermal noise, shot noise..) or parametric noise (Flicker noise, temperature fluctuation, power supply noise..) [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF]. These can degrade the signal and both amplitude and phase noise can affect the oscillator output. If the former, i.e. amplitude noise, has not big impact in modern oscillators and it can be suppressed by gain compression, the latter, i.e. the phase noise, can become an issue especially for the long-term stability of the oscillator. Phase noise characteristic is actually the most important figure of merit of modern oscillators. The most used tool to measure it is the L(f ) function defined as

L(f ) = Sϕ(f ) 2
where S ϕ (f ) is the one-sided spectral density of the random phase fluctuation ϕ(t) [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF]. It is expressed in [dBc/Hz] and represents the noise power with respect to the carrier contained in a 1 Hz bandwidth and centered at a certain offset from the carrier. State of the art oscillators have phase below -100 [dBc/Hz] at 10 kHz from the carrier [START_REF] Metcalfe | Applications of cavity optomechanics[END_REF].

Ideal Oscillator's packaged size:

The size is also an important parameter for oscillators. The possibility to be easily integrated in modern technologies is in fact a crucial feature of modern oscillators, especially for transportable and wearable technologies where a compact occupied space is the main requirement to fulfill. An ideal oscillator should follow this size reduction philosophy in order to match with modern ICTs needs. An ideal oscillator's packaged size should be no more than few cm 3 .

State of the art

Crystals oscillators have been the most used oscillators for decades. Among them, quartz oscillator has been one of the most used since its invention in 1921 [START_REF] Guyton | The piezo-electric resonator[END_REF]. It shows good compactness and high spectral purity, especially with its upgraded versions where the temperature dependence of the quartz is balanced with specific techniques, namely TCXO (temperature controlled crystal oscillator) and OCXO (oven controlled crystal oscillator). The drawback of this kind oscillators is that the oscillating frequency is here in the order of MHz. This means that to get the high frequency standards that modern technologies need, the natural frequency of the resonator must be multiplied but for each multiplication stage the phase noise increases as well. A trade-off is needed for this category of oscillators between the page 2 1.1. Oscillators in modern society targeted frequency and the related phase noise level.

Another category are electronic oscillators. Voltage Controlled Oscillators (VCO) are an example of electronic oscillators whose oscillating frequency is controlled by a voltage input. They are rather small in size but their phase noise performances are not the best (from -80 to -90 [dBc/Hz] at 10 kHz from the carrier). Other examples are integrated microwaves resonators made with metallic walls [START_REF] Khalil | Mmic cavity oscillators[END_REF] or dielectric resonators [START_REF] James | Application of dielectric resonators in microwave components[END_REF]. In this case the frequency can go up to several hundreads of GHz. Due to their nature, they suffer from electromagnetic interference and the fact that the carrier is in the RF domain means that translation stage towards other physical domain (mainly optics) for distribution of the signals will bring losses as well.

Another interesting category are oscillators in the optical domain, namely optical oscillators. The main advantage of this class is that the carrier can directly be implemented in the optical domain, avoiding losses for the distribution of the signal towards for instance photonic integrated circuitry or optical fibers network. Among them the most interesting are the Opto-Electronic Oscillators (OEOs). They can in fact provide working frequency directly in the Ultra High Frequency regime (UHF) and high spectral purity (from -140 to -160 [dBc/Hz] at 10 kHz from the carrier) by taking advantage of ultra-low loss fiber as energy storage element [START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF]. However, kilometers of fiber are usually needed to introduce the proper time delay to reach high quality factor for good phase noise performances. This bulky element, around 10 3 cm 3 in size, expose the OEO to environmental perturbations such as thermal [START_REF] Kaba | Improving thermal stability of optoelectronic oscillators[END_REF][START_REF] Eliyahu | Improving short and long term frequency stability of the opto-electronic oscillator[END_REF] and mechanically induced fluctuations [START_REF] Hati | Cancellation of vibration-induced phase noise in optical fibers[END_REF]. Figure 1.1 from [START_REF] Metcalfe | Applications of cavity optomechanics[END_REF] shows a comparison of phase noise performances with respect to the package size of relative small volumes microwave oscillators. All the carriers have been scaled to 10 GHz for direct comparison. We can resume the comparison between the different kind of oscillators in table 3.1 in order to point out the missing point and to evidence the work towards an ideal oscillator.

Oscillators comparison

Crystals oscillators Electronic oscillators OEOs

Working frequency directly in the GHZ Phase noise below -100 [dBc/Hz] Compactness Table 1.1 -Comparison between most famous microwave oscillators categories and missing points with respect to an ideal integrated oscillator.

Delay implementation in OEO: a feature limiting the compactness

What we have learned so far is that one of the best solution in the microwave oscillators scenario are OEOs. Especially the fact that they work directly in the optical domain makes them easy to be combined with recent technologies, where a migration 1.1. Oscillators in modern society from electronic to optics is underway. As we already mentioned, optoelectronic oscillators take usually advantage from a delay line implemented in the system thanks to kilometers of optical fibers. A schematic of a general implementation of an OEO with optical delay line is represented in figure 1.2. The principle is that the OEO signal drives an RF signal which modulated an optical carrier with an electro-optical modulator (EOM). The modulated optical signal is sent through an optical delay line made by low loss optical fiber, acting as a high Q energy storage element of the OEO. The signal is then converted to the electrical domain with a photodetector (PD) and at this point the signal is amplified, filtered and phase shifted before closing the loop [START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF]. Figure from [START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF].

The initial motivation to insert time delay in first architecture of OEOs came from new theoretical insights in the area of non-linear dynamics. In early architecture, the time to go through the feeback path was negligible with respect to the other time constants of the system. To see the effect of the time delay an additional delay line was inserted on purpose [START_REF] Yanne K Chembo | Optoelectronic oscillators with time-delayed feedback[END_REF]. We can state then that OEOs are time-delayed dynamical systems because the round-trip time T in the loop is not negligible with respect to the other time constants of the systems. In modern OEOs, the purity is achieved thanks to the optical delay line inserted in the loop as stated and explained in two seminal papers of 1996 [START_REF] Yao | Optoelectronic microwave oscillator[END_REF][START_REF] Yao | Converting light into spectrally pure microwave oscillation[END_REF]. In fact, for OEOs the Q factor is directly related to the time delay τ present in the feedback loop, and so to the fiber length, by the following formula: Q ≈ 2πf τ = 2πf nL/c [START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF], being f the working frequency, τ the delay time, n the fiber refractive index, L the fiber length and c the speed of light. Even though the achieved performances are remarkable, the compactness is an issue for OEOs, where the needed bulky element for delay implementation limits their possible size reduction. Moreover, environmental perturbations such as thermal of mechanically induced fluctuation are another issue coming from the bulky Chapter 1. Introduction nature of the system.

Our system to be developed

The motivation behind this thesis comes in the presented framework of integrated microwave oscillators, in order to approach the ideal one. To do so, several requirements need to be fulfilled. We need an oscillator directly working in the frequency of interest (GHz), with the carrier signal possibly directly in the optical domain and with a compact size (few cm 3 when already packaged). In order to achieve this, we need several ingredients. Specifically, we need (1) a resonator acting as a frequency selective element for the oscillator, (2) a resonant excitation technique in order to stabilize the resonator oscillations and compensate its loss as well, (3) a way to introduce in the feedback path a time delay for increasing the spectral purity and, moreover, (4) we want to develop a platform where all the components are integrated on the same material in order to create a fully-integrable compact device (see figure 1.3 for a summary scheme). We will in the following describe possible solutions for the discussed needed components, to finally present our proposal for an integrate time-delayed oscillator.

Optomechanics for a new generation of integrated oscillators

In the recent past, technological and scientific progress in the field of nano-technologies allowed the creation of new paradigms in the field of integrated micro and nanosystems. Among them, micro and nano-optomechanical systems, in which optical and mechanical elements are resize to the nano-scale, gained a lot of importances 1.2. Our system to be developed in the last two decades. Here, the light-matter interaction gives rise to new phenomena that can eventually be exploited for a new generation of ICTs based on this photon-phonon interaction. In the following, we will describe the basic principles of optomechanics to lately introduce a new generation of integrated resonator and oscillators created out of them.

General notion of Optomechanics

Light-matter interaction is certainly a fascinating subject. The idea that light can exert a force on a physical object thanks to a momentum exchange is not trivial and finds its origin in the 17 th century when Kepler was the first observing how the comets tails were pushed away from the sun when they were transiting close to it. This force is the so called radiation pressure force and it can be therefore described as consequence of the momentum exchange between the particles of light, i.e. photons, and the interacting object, the dust of the tails in that specific case.

It is interesting to notice how this fascinating phenomenon does not bring any visible effect to our daily life, since its consequences are too small to be appreciated in a classical metric system, but it becomes really important at the two opposite limits: space and micro/nano-systems. For the former case, the radiation pressure is the evidence of how a tiny force can become important when accumulated over a long time period, and it is a crucial parameter that must be taken into account when one wants to accurately calculate for instance a spacecraft trajectory for space travelling. For the latter case, i.e. micro/nano-systems, the interaction of light with µm or nm size objects can play indeed an important role and recent advances in laser and optical technologies represent a concrete example of how light can be used to manipulate or probe tiny objects like molecules, cells or atoms. The increasing importance of this field (and its possible applications) is also underlined by two recent awarded Nobel Prizes. More precisely, in 2017 the Nobel Prize in Physics was awarded to Rainer Weiss, Barry C. Barish and Kip S. Thorne for gravitational waves detection, which were predicted by Einstein a hundred years ago, performed with the Laser Interferometer Gravitational-Wave Observatory (LIGO). One year after, in 2018, another important contribution related to lasers technologies was awarded with the Nobel Price in Physics, this time was the turn of Arthur Ashkin that was awarded for the development of the trapping of objects with light also called optical tweezers.

The photon-phonon interaction at the core of modern optomechanical system and its physics are explained by cavity optomechanics theory that has been exhaustively discussed in [START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. We will report here the basic idea of optomechanical interaction, giving lately in the manuscript other important details for understanding the page 7

Chapter 1. Introduction platform to be developed during this thesis.

The principles behind optomechanics can be explained with the help of a classical Fabry-Perot cavity system in which one mirror is free to move (see figure 1.4 for a schematic). The system can be described as the interaction between two subsystems, an optical resonator, the cavity, and a mechanical resonator, the movable mirror. Let's define with ω cav and κ the optical resonant frequency and the decay rate of the cavity and with Ω m and Γ m the mechanical resonant frequency and the dumping rate of the mechanical resonator. The associated optical and mechanical quality factors can be written as Q opt = ω cav τ , where τ = κ -1 is the photon lifetime, and Q m = Ωm Γm respectively.

Figure 1.4 -Schematic of a laser-driven optical cavity. Figure from [START_REF] Aspelmeyer | Cavity optomechanics[END_REF].

If one injects light into the cavity at the resonance frequency ω cav , the light will be confined inside the cavity and it will apply a radiation pressure force to the movable mirror. This latest will move and it will change the resonance frequency of the cavity. At this point the injected light is no more at resonance, the applied radiation pressure force decreases and the mirror is back to its initial position. This will let the process starting again, creating at the end an "optomechanical" loop. The cavity is used to store the photon such that the latter can probe many time the mechanical displacement due to its numerous round-trips. A first consequence of this is that a high Q-factor is needed in order to repeat the effect.

Optomechanical resonators

By taking advantage of the just explained optomechanical principles, in the last decade a new generation of optomechanical systems have been developed, where one page 8 1.2. Our system to be developed can imprint mechanical frequencies directly on a optical carrier. Several geometries can be used in order make this optomechanical phenomenon happening. The effective mass of the optomechanical resonator can range here from gram to femtogram (microscopic mirrors [START_REF] Favero | Optical cooling of a micromirror of wavelength size[END_REF][START_REF] Gröblacher | Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity[END_REF][START_REF] Gröblacher | Observation of strong coupling between a micromechanical resonator and an optical cavity field[END_REF], microtoroid [START_REF] Hossein | Observation of optical spring effect in a microtoroidal optomechanical resonator[END_REF][START_REF] Tj Kippenberg | Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity[END_REF][START_REF] Ruesink | Nonreciprocity and magnetic-free isolation based on optomechanical interactions[END_REF], microdisk [START_REF] Liu | Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators[END_REF][START_REF] Srinivasan | Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator[END_REF][START_REF] Sun | High-q silicon optomechanical microdisk resonators at gigahertz frequencies[END_REF], ..) or even attogram (suspended 1D or 2D photonic crystal (PhC) cavities) [START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. In our case, we will focus our attention to this last category which dimensions can be pushed towards the micro and nano scale.

An example of a silicon-based 2D optomechanical resonator is reported in [START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF] (figure 1.5.a). It shows how, by playing on the optomechanical coupling between confined photons and phonons in a slab of material (silicon here) is indeed possible to read mechanical modes in the MHz (151 MHz in this case). The experimentally measured optical quality factor was in this case quite high (Q = 1.2 • 10 6 ). Another example of 2D PhCs integrated this time in III-V materials (InP) has been reported in [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF] (figure 1.5.b). Here, it has been shown how it is is indeed possible to read mechanical modes both in MHz and in the GHz (∼ 1 GHz) regime and, moreover, integrability with standard photonic circuitry has been shown as well [START_REF] Viktor Tsvirkun | Integrated iii-v photonic crystal-si waveguide platform with tailored optomechanical coupling[END_REF] (figure 1.5.c). The low mechanical quality factor (∼ 180) prevented in that case to enter the self-sustained oscillations regime [START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. In [START_REF] Eichenfield | Optomechanical crystals[END_REF] a very early demonstration of 1D PhC has been proved. The suspended nanobeam was here in silicon pierced with rectangular holes (figure 1.5.d). The optical quality factor was here around 38000 and the sustained mechanical mode was in this case at 2 GHz. In [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF] an AlN-based nanobeam with high optical Q (up to 120000) sustaining a mechanical mode at 3.18 GHz has been presented. In this case the PhC design was implemented with elliptical etched holes (figure 1.5.e). ) and mechanical modes at 151 MHz [START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF]. b) 2D optomechanical resonator in InP sustaining both MHz and GHz modes [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF]. c) Same system integrated on SOI waveguides [START_REF] Viktor Tsvirkun | Integrated iii-v photonic crystal-si waveguide platform with tailored optomechanical coupling[END_REF]. d) Silicon-based 1D PhC with optical Q=38000 and sustained mechanical mode at 2 GHz [START_REF] Eichenfield | Optomechanical crystals[END_REF]. e) AlN-based 1D PhC with optical Q up to 120000 and sustained mechanical mode at 3.18 GHz [START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF]. page 10 1.2. Our system to be developed

Optomechanical oscillators

Under certain conditions, optomechanical crystals fuelled by optical injection power sustain oscillations, also called optomechanical oscillators (OMOs). The first demonstration of an OMO was realized in a silica micro-toroid optomechanical resonator [START_REF] Tj Kippenberg | Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity[END_REF] (figure 1.6.a) and its phase noise measured in [START_REF] Hossein-Zadeh | Characterization of a radiation-pressure-driven micromechanical oscillator[END_REF]. In this case the sustained mechanical mode was at 54.7 MHz with a mechanical quality factor of 2100 and a phase noise around -90 [dBc/Hz] at 10 kHz from the carrier at 54.7 MHz. Later, another example of low phase noise microwave architecture has been demonstrated base on a planar silica-on-silicon microdisk with a sustained mechanical mode at 21.7 GHz. The measured phase noise was -90 [dBc/Hz] at 10 kHz from the carrier at 21.7 GHz [START_REF] Li | Microwave synthesizer using an on-chip brillouin oscillator[END_REF][START_REF] Li | Low-noise brillouin laser on a chip at 1064 nm[END_REF][START_REF] Schwab Gigan | Self-cooling of a micromirror by radiation pressure[END_REF]. In [START_REF] Mercadé | Microwave oscillator and frequency comb in a silicon optomechanical cavity with a full phononic bandgap[END_REF] an example of 1D optomechanical crystal on silicon was presented with a mechanical mode at 3.895 GHz with a quality factor of 2400 (figure 1.6.b). Even though the frequency is directly in the GHz, the phase noise was not the best in this case, with -75 [dBc/Hz] at 10 kHz from the carrier at 3.895 GHz. Similar results have been reported in [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF], where an InGaP 1D optomechanical crystal was able to self-oscillate at 2.924 GHz. The mechanical quality factor was 2300 with a phase noise at -70 [dBc/Hz] at 10 kHz from the carrier at 2.924 GHz (figure 1.6.c). The working frequency directly in the GHz, the good compactness and the wide choice of materials where they can be implemented make optomechanical crystals Chapter 1. Introduction good candidates for our envisioned platform (see yellow star in figure 1.6.d for state of the art OMOs performances compared to the other types of oscillators previously discussed). Specifically, we will try to exploit the 1D version of optomechanical crystal for our integrated time-delayed oscillator. The choice of the material is also important. Most of the presented structures were integrated in silicon, but this implies a limited choice in the possible actuation technique that can be exploited for oscillations stabilization. Other presented examples took advantage of III-V semiconductor materials. In order to open the way towards several possible actuation solutions for oscillation stabilization (such as piezoelectric or electrostatic actuation) we will focus on III-V materials for the integration of our optomechanical oscillator. For the presented free running OMOs, the drawback is that the amount of power that can be injected is limited by thermo-optical instabilities and the lack of a feedback loop prevents frequency and phase noise to be furthermore optimized in the long-term stability. To push these limits beyond, another way than optics is required to control photon-phonon interactions at the heart of OMOs working principle. In the next section we will discuss possible resonant excitation techniques to be integrated together with 1D optomechanical crystals in III-V materials, opening the way towards a feeback path implementation in order to stabilize the OMO output.

Resonant excitation for optomechanical crystals

A tool to master the mechanical behaviour of the 1D optomechanical resonator is needed in order to stabilize and eventually optimize its output signal. As mentioned before, relying only on radiation pressure effects has in fact several limitations due to thermo-optical instabilities when high optical power is injected into the system. A resonant technique is required to reach an efficient actuation of the mechanical mode for enhancing its spectral purity and stability. By taking advantage from the properties of III-V materials, several techniques can be exploited in order to master the photon-phonon interaction at the core of the 1D optomechanical crystal, such as electrostatic, piezoelectric and acoustical actuations.

Electrostatic actuation is a commonly used technique in Micro-Electro-Mechanical Systems (MEMS) which takes advantage of integrated electrodes to apply an electrostatic force to a resonator. When an electrical signal is applied to the system, the resonator will be strained by the capacitive force of the created electric field and with a properly chosen RF input signal mechanical resonances can be eventually excited. In [START_REF] Quirin P Unterreithmeier | Universal transduction scheme for nanomechanical systems based on dielectric forces[END_REF] electrostatic actuation of a 1D SiN suspended resonator has been performed with a pair of integrated electrodes on its side (figure 1.7.a). The natural frequency of the system is between 5 and 9 MHz. In [START_REF] Chowdhury | Superharmonic resonances in a two-dimensional non-linear photonic-crystal nano-electro-mechanical oscillator[END_REF] Piezoelectric actuation takes advantage of the piezoelectricity to induce a mechanical strain that will deform the piezoelectric material when a voltage difference is applied to it. Also in this case integrated electrodes can be used in order to applied the needed electric field. In [START_REF] Bochmann | Nanomechanical coupling between microwave and optical photons[END_REF] a 1D AlN optomechanical crystal has been resonantly excited up to GHz frequencies thanks to a pair of integrated electrodes deposited directly on top and on bottom of one nanobeam's side (figure 1.8). The system has been integrated together with a photonic circuit and simultaneously electromechanical and optomechanical coupling effects have been shown. In [START_REF] Han | Triply resonant cavity electro-optomechanics at x-band[END_REF] an AlN microwheel cavity fabricated on silicon substrate is piezoelectrically actuated by a microstrip resonator aligned close to it. This system showed self-sustained microwave oscillations around 8 GHz. Acoustic actuation takes always advantage of piezoelectricity but the main difference with respect to piezoelectric actuation is that in this case one wants to exploit the propagation of the created mechanical waves, usually by using interdigitated transducers (IDTs) as acoustic generators, in order to enhance the generated displacement field effects on the mechanical resonator. IDTs have recently been used to create acoustic waves in the GHz to adress mechanical modes of optomechanical crystal for quantum applications [START_REF] Forsch | Microwave-to-optics conversion using a mechanical oscillator in its quantum ground state[END_REF] (figure 1.9.a) and strong coherent photon-phonon interaction [START_REF] Balram | Coherent coupling between radiofrequency, optical and acoustic waves in piezooptomechanical circuits[END_REF][START_REF] Li | Nanophotonic cavity optomechanics with propagating acoustic waves at frequencies up to 12 ghz[END_REF] (figure 1.9.b). Here IDTs were deposited on the same piezoelectric material where the 1D optomechanical resonators were fabricated (GaAs and AlN). Recently, injection locking of an optomechanical oscillator via acoustic waves has been also demonstrated, where the OMO has been locked to the frequency and phase of an external oscillator thanks to the acousto-optical interaction [START_REF] Huang | Injection locking of optomechanical oscillators via acoustic waves[END_REF]. Here the resonator was a microtoroidal silica OMO linked to an external piezoelectric transducer (PZT) with resonant frequency in the MHz. page 14 1.2. Our system to be developed Figure 1.9 -Examples of acoustic actuation of 1D optomechanical crystals with localized mechanical modes in the GHz. Here the acoustic generators are IDTs directly deposited on top of the piezoelectric membranes [START_REF] Forsch | Microwave-to-optics conversion using a mechanical oscillator in its quantum ground state[END_REF][START_REF] Balram | Coherent coupling between radiofrequency, optical and acoustic waves in piezooptomechanical circuits[END_REF][START_REF] Li | Nanophotonic cavity optomechanics with propagating acoustic waves at frequencies up to 12 ghz[END_REF].

All the presented techniques were able to address mechanical modes of integrated oscillators of various geometry and all of them could theoretically be able to be implemented in a feedback path together with a 1D optomechanical crystal for our envisioned device. By taking into account that the final aim is not only to create a feedback loop, but also to implement inside it a time delay for enhancing the final spectral purity, the more suitable technique in our case is the last one, i.e. acoustical actuation. This is in fact the only technique with enough degree of freedom allowing us to add a further element in the feedback loop for delay implementation, since in this case one can take advantage of the propagation of the acoustical waves by engineering their path towards the resonator. In the other two cases, in fact, the interaction between the resonant technique and the mechanical resonator is based on a direct actuation. In other words, no space for additional engineering is here available to design a further element for delay introduction. In the next section, a possible solution to introduce delay in micro-system by taking advantage of acoustical waves will be discussed.

Phononic crystals for guiding acoustic waves

Control of acoustic waves in micro and nano-systems can be obtained by implementation of phononic crystals in the material where the propagation takes place. Also in this case, technological improvements in the last decades allowed to implement guiding of high frequency mechanical waves (∼ GHz) in wavelength scale structures by introducing periodic defects. In [START_REF] Paul H Otsuka | Broadband evolution of phononic-crystal-waveguide eigenstates in real-and k-spaces[END_REF] evolution of phononic crystal waveguide eigenstates in real and k-space have been shown for frequencies in a 0-1 GHz range. Here the phononic pattern (a square lattice) is obtained by milling circular holes on (100) silicon substrate by dry reactive ion etching. By removing lines of holes, straight and L shaped waveguides are obtained and guiding of acoustic waves over page 15 Chapter 1. Introduction time was experimentally measured (figure 1.10.a). The transducers were not integrated on chip, an optical pump-and-probe technique was in fact used together with an interferometer to excite the structures and recover the waves field. In [START_REF] Benchabane | Guidance of surface waves in a micron-scale phononic crystal line-defect waveguide[END_REF] guiding of surface acoustic waves in a micron-scale phononic crystal line-defect waveguide has been shown. Air holes are milled in a square lattice on X-cut lithium niobate substrate. This time, the piezoelectric properties of the material allowed the implementation of chirped aluminum IDTs directly integrated on the lithium niobate substrate for an on-chip mechanical excitation (figure 1.10.b). The defect-free phononic crystal showed a band gap from 650 MHz to 950 MHz and transmission and guiding of acoustic waves have been experimentally demonstrated around this range of frequencies by imaging the produced elastic field with laser interferometry technique.

Figure 1.10 -a) Fabricated straight and L waveguides in (100) substrate by milling circular holes with etching techniques (right). Frequency filtered images at 322 MHz for straight and L waveguide (left) [START_REF] Paul H Otsuka | Broadband evolution of phononic-crystal-waveguide eigenstates in real-and k-spaces[END_REF]. b) Fabricated chirped aluminum IDTs together with straight waveguide obtained by milling circular holes in a square square lattice with one period line defect in a X-cut lithium niobate substrate (right). Measured elastic wave field amplitude for excitation at 820 MHz (left) [START_REF] Benchabane | Guidance of surface waves in a micron-scale phononic crystal line-defect waveguide[END_REF].

Interaction between Surface Acoustic Waves (SAWs), produced by integrated IDTs, and a phononic crystal resonator obtained by varying the size of grooves etched page 16 1.2. Our system to be developed on the 128 degree Y cut lithium niobate substrate was observed in [START_REF] Shao | Phononic band structure engineering for high-q gigahertz surface acoustic wave resonators on lithium niobate[END_REF] (figure 1.11.a). The working frequency of the created high-Q phononic crystal cavity were ranging from 0.5 to 5 GHz and the particular adopted design is a proof of the scalability and adaptability of phononic crystals for mastering the acoustic behaviour of a material with a relatively simple approach. Focusing and transmission of Lamb waves through wavelength-scale structures has been demonstrated in [START_REF] Siddiqui | Lamb wave focusing transducer for efficient coupling to wavelength-scale structures in thin piezoelectric films[END_REF] in suspended membranes of AlN at around 2 GHz (figure 1.11.b). Focused IDTs were used in order to prove transmission through waveguides with widths ranging from 2.5 to 50 µm. Recently, similar results have been presented in [START_REF] Yanni D Dahmani | Piezoelectric transduction of a wavelength-scale mechanical waveguide[END_REF] where transmission through a wavelength scale suspended phononic waveguide designed in a X-Cut lithium niobate on silicon has been shown. Integrated IDTs have been here used to show transmission through a 1-µm-wide waveguide at 1.7 GHz (figure 1.11.c). A time domain approach was used to recover the transmission spectra of the main signal and its echos as well.

Figure 1.11 -a) Fabricated phononic crystal cavity for SAWs with resonance frequency from 0.5 to 5 GHz [START_REF] Shao | Phononic band structure engineering for high-q gigahertz surface acoustic wave resonators on lithium niobate[END_REF]. b) Fabrication of AlN wavelength-scale structures for focusing and transmission of Lamb waves at 2 GHz [START_REF] Siddiqui | Lamb wave focusing transducer for efficient coupling to wavelength-scale structures in thin piezoelectric films[END_REF]. c) Wavelength scale suspended phononic waveguide designed in a X-Cut lithium niobate on silicon. A time domain approach was used here to recover the transmission spectra of the main signal and its echos as well for a working frequency equal to 1.7 GHz and a waveguide width equal to 1 µm [START_REF] Yanni D Dahmani | Piezoelectric transduction of a wavelength-scale mechanical waveguide[END_REF].

The previously discussed examples are a tangible proof of how phononic crystals can open the way towards the acoustic control of a material. Moreover, this page 17

Chapter 1. Introduction approach shows high degree of freedom in terms of design and tunability in order to match for instance the targeted frequency range. Also, by introducing defects phononic waveguides can be eventually created in order to guide phonons in all the real space directions. This is exactly what we need in order to master the propagation of acoustic waves coming from the IDT towards the 1D resonator in our envisioned device. In addition, by taking advantage of the well-known low velocity of acoustical phonons in material as well as their immunity to environmental disturbances, an on-chip acoustical feedback loop could be implemented. Here, an acoustic delay implementation in a compact and controllable manner can be achieved thanks to a dispersion engineering of the acoustical properties of the medium under investigation.

III-V materials as a solution for integrated OMO with on-chip acoustic delay

The final intent is to achieve a fully-integrated time-delayed oscillators where all the needed components have to be integrated on the same material (figure 1.3). In order to properly choose a suitable candidate, several aspects need to be taken into account. What we have learned so far is in fact that we need a material on which we can design an optomechanical crystal where we can master the behaviour of both photons and phonons as desired and, moreover, where an acoustic resonant excitation is eventually exploitable. In other words, a material with good piezolectric properties (enabling the acousto-actuation), high enough refractive index (for light confinement) and a large band gap (preventing any two photons absorption process and allowing a good phonon-photon interaction [START_REF] Zhou | Ultraviolet optomechanical crystal cavities with ultrasmall modal mass and high optomechanical coupling rate[END_REF]) is needed. In this context, III-V piezoelectric semiconductor materials can be considered as good candidates since they show at the same time interesting optical and mechanical properties together with an easy integration with standard photonic circuitry already proved [START_REF] Viktor Tsvirkun | Integrated iii-v photonic crystal-si waveguide platform with tailored optomechanical coupling[END_REF]. Table 3.2 shows a comparison of some of the most used semiconductor materials for optomechanical crystals and/or piezolectric applications. As we can see from Table 3.2, even though InP and InGaP have been succeffully used for integrated optomechanical crystals implementation ( [START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF], [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF]) they show weak piezoelectric properties. While GaN and AlN, even if they have the highest piezoelectricity among the presented materials, their refractive index is quite low and integration in standard photonic circuitry is still challenging to achieve as well ( [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF]).

Besides III-V materials, another possible solution would theoretically be LiN bO 3 . This new material shows really high piezoelectric coefficients, relatively high refractive index and it is now available also on insulator (LION). However, its processing is still a challenge in standard cleanroom facilities. page 18 Thus, in the following we will focus on GaAs and GaP in order to develop our integrated time-delayed oscillator. Both materials show in fact high refractive index, good piezoelectric properties and a well established process flow has been already used in our group for integration on such semiconductor membranes on SOI circuitry [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF].

Our proposal

The "scientific recipe" proposed in the scheme of figure 1.3 in order to approach the ideal microwave oscillator is now filled with the chosen "scientific ingredients" in figure 1.12. Specifically, we have chosen a suspended 1D optomechanical crystal as a resonator, IDTs as a resonant excitation method for oscillations stabilizations and phononic waveguides as a technique to introduce the needed time delay in the feedback path. Moreover, we want to integrate all the components on the same semiconductor slab of material (GaAs or GaP). The detailed version of the envisioned device is shown in figure 1. [START_REF] Eliyahu | Improving short and long term frequency stability of the opto-electronic oscillator[END_REF]. In particular, our proposal consists of a 1D semiconductor photonic crystal (PhC) nanobeam suspended on top of a Silicon-on-Insulator (SOI) waveguide with an IDT and a phononic waveguide integrated on the same semiconductor membrane. The nanobeam will act as resonator for our integrated time-delayed OMO. This optomechanical crystal will be able to sustain both localized optical and mechanical modes. For the former the resonant optical frequency λ opt will be in the telecommunications wavelengths (∼ 1550 nm) while for the latter the targeted mechanical frequency Ω m will be directly in the Ultra High Frequency (UHF) regime (∼ GHz). In this configuration, both photons and phonons will be confined in the PhC cavity and the lightmatter interplay will allow us to read the mechanical vibrations of the nanobeam as a modulation of the optical carrier. By injecting light to the SOI waveguide at PhC cavity resonant frequency, this will be evanescently coupled to the PhC cavity and by collecting the light from the waveguide output the modulations on the optical carrier will give us information about the nanobeam's motion. These oscillations can be furthermore stabilize thanks to a resonant feedback path integrated on the same semiconductor membrane where the 1D PhC lies. In particular, an IDT will be deposited onto the slab of the piezoelectric material for generation of acoustic waves at RF frequency Ω acoustic = Ω m . The propagation path of the generated acoustical waves will be acoustically engineered by designing a phononic waveguide sustaining a mechanical propagating mode at Ω prop = Ω acoustic = Ω m to properly control and delay the acoustic waves propagation. In particular, specific group velocity values of these latest can be addressed by acoustically engineering the dispersion relation of the phononic path. This will allow us to introduce the needed time delay for oscillations stabilization. Several strategies can be then studied in order to close the path for a feedback loop implementation in order to finally achieve an ultra-stable time-delayed OMO where the needed time delay for spectral purity enhancement is integrated on the same chip. The aimed performances of such a system could drastically improve the state of the art performances of OMOs without feedback loop (yellow star in the inset map) in order to approach the ideal oscillator area (yellow circle in the same map) in term of phase noise performances and package size. In this proposal, there are specifically two big challenges to face. The first one is the necessity to match the three frequencies Ω acoustic , Ω prop and Ω m of the three different integrated components. This can be achieved by designing and optimizing singularly each element before integrating them all together. The second one is a technological challenge related to the fabrication of the platforms under study in the cleanroom facilities. In this case, what we need is to create and optimize specific process flows for each case, allowing us to fabricate and test the desired heterogeneously integrated structures.

Organization of the manuscript

The manuscript will be divided into 4 chapters. In the first 3 chapters, each of the 3 elements composing the complete envisioned fully-integrated time delayed OMO (1D PhC nanobeam, IDT and phononic waveguide) will be singularly introduced and discussed with simulations, design, fabrication and measurements. In the last chapter, all the components will be integrated together in a first prototype of the page 21 Chapter 1. Introduction complete device for a first proof of interaction of the different involved physical domains.

In Chapter 2, design, fabrication and measurements of OMOs integrated on chip working at a mechanical frequency Ω m around 2-3 GHz (in GaAs and GaP) will be introduced and discussed. Specifically, the component acting as a resonator, the suspended 1D PhC nanobeam, will be firstly described. Here, photonic crystals cavity design will be presented as a recent example of materials engineering for manipulating and confining light at nano-scale. Moreover, it will be shown how, for suspended PhC cavities, also phonons can be actually properly confined and an enhanced intra-cavity photon-phonon interaction can be finally achieved. Tools to quantify the optomechanical coupling will be thus presented and discussed together with simulation of the system under study. After, fabrication of the integrated 1D PhCs nanobeams onto a standard photonic circuitry (SOI waveguides) will be presented. Finally, measurements to characterize optically (first) and mechanically (after) the structures under study will be shown with a first phase noise measurement of the obtained oscillations in the GHz.

In Chapter 3, we will demonstrate generation and transmission of acoustical waves at 2-3 GHz by taking advantage of properly designed IDTs integrated directly onto semiconductor membranes of GaAs and GaP. In more details, finite element simulations will help us to finally come up with properly designed IDTs in order to work both in the Surface Acoustic Waves and Lamb waves regimes at the frequency of interest. An optimized process flow will be then shown for fabrication of suspended face-to-face transducers. Finally, experimental demonstration of transmission of acoustical waves will be presented thanks to specifically applied time-gating strategies. Here, tunability of the acoustic frequency Ω acoustic of the transducer will be demonstrated as well with several acoustic modes able to propagate at the same frequency Ω m of the previously presented mechanical modes of the integrated OMOs.

In Chapter 4, delay control of acoustical waves at 2-3 GHz will be experimentally demonstrate. At the beginning of the chapter, phononic crystals will be discussed as a way to master the mechanical behaviour of a material (similar to photonic crystal for optics). Several design possibilities for integrated phononic crystal lattices will be presented and compared with the aim to finally introduce a phononic band-gap around the frequency of interest. After, it will be shown how, by breaking the introduced lattice periodicity, one can create phononic waveguides with accurately placed defects in order to guide phonons as desired. Tunability of the propagating mode frequency Ω prop in the GHz will be shown using several examples of possible phononic waveguides implementations. Then, acoustic engineering of the dispersion behaviour of phononic waveguides will be discussed. This allows us to address spepage 22 1.3. Organization of the manuscript cific group velocity values of the found propagating modes. Fabrication of phononic waveguides together with integrated IDTs will be then presented with related optimizations. Finally, transmission and filtering of acoustical waves in the GHz will be demonstrated with group delay extrapolation of the slow propagating modes. Here, with the matching between Ω prop of the phononic waveguide with Ω acoustic of the transducers plus the slow features of the propagating mode we will demonstrate that a delay implementation in a completely integrated device is at reach.

In Chapter 5, an OMO integrated together with an IDT and a phononic waveguide will be presented as a first prototype of a fully-integrated OMO with an on-chip acoustic delay. Fabrication of the complete device will be shown, underlying how the integration of the different components on standard photonic circuitry is indeed possible from a fabrication point of view. Characterization of the single elements and interaction between them will be presented with some preliminary results of the interaction between the different involved physical domains. An example of capacitive actuation will be presented as well in another fabricated OMO integrated on the same chip. After, optical packaging of 1D PhCs in InP will be shown as a first demonstration of compatibility of the structures under investigation with standard IC packaging technique. Finally, conclusion and perspectives towards new physics with optomechanical oscillators will be discussed.

Chapter 2

Optomechanical crystal oscillator integrated on-chip Since the beginning, humankind used to model what nature spontaneously offered them in order to increase the quality of the life. Engineered materials are a modern and tangible example of this attitude, showing how, by playing on their properties, it is indeed possible to optimize either a single material or a combination of them to achieve a specific target. Artificial materials with properties that do not exist in nature can be eventually created and properly controlled.

Photonic Crystals (PhCs) are a modern example of artificial material with the final goal of confining and guiding light. As for engineered electronic materials, where Chapter 2. Optomechanical crystal oscillator integrated on-chip a periodic arrangement of the atoms and molecules can bring a periodic change in the potential to guide electrons, also photonic crystals present periodic features to guide, this time, photons. For the photonic crystals case, the periodic modulation interests the dielectric constants of the combined materials. In fact, by playing with materials showing different dielectric constants, and low losses regarding light propagation, one can create a periodic media where a periodically change in the refractive index prevents light propagation in one, two or three dimensions for a certain frequency range, enabling light control [START_REF] John D Joannopoulos | Molding the flow of light[END_REF] (see figure 2.2 for a schematic of 1D, 2D and 3D photonic crystals principle). These forbidden frequency regions for light are the so called photonic band gaps. A photonic band gap is said to be complete if the forbidden frequency region is valid for light with any polarization and propagating direction. Once a band gap in the desired range of frequency is found, one can start to break the introduced periodicity to create localized regions where light can be locally confined. Besides electronics and photonics, one can address also mechanics in order to engineer the mechanical properties of a material. Mechanical resonators are examples of mechanical systems that have been modified in order to naturally vibrate at specific frequencies. The system will vibrate with greater amplitudes at these resonant frequencies that can be then adjusted and tuned as desired by playing mainly on the density and on the geometry of the resonator. Musical instruments are, for instance, examples that use acoustic resonators to produce sound waves of specific tones [START_REF]Resonator -Wikipedia, the free encyclopedia[END_REF]. The simplest example are the guitar strings (figure 2 Recent improvements in fabrication technologies allowed to reduce the size of mechanical suspended resonators to nano-scale. The small volume that can be now addressed allows the sustained mechanical modes to achieve really high natural frequencies (∼ GHz).

By merging improvements in both photonics and mechanics at nano-scale, new classes of artificial materials can be now well mastered. Optomechanical crystals are an example of engineered materials able to confine both optical and mechanical modes within a very small volume, enhancing light-matter interaction. In the last decade, among the possible exploitable geometries, 1D photonic crystal cavities have gained a lot of importance for their small volume and the really high confinement that can be achieved in such a structure. Several examples can be encountered in the field of nano-optomechanics [START_REF] Tg Tiecke | Nanophotonic quantum phase switch with a single atom[END_REF][START_REF] Kim | Photonic crystal cavities from hexagonal boron nitride[END_REF][START_REF] Eichenfield | Optomechanical crystals[END_REF][START_REF] Quan | Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide[END_REF][START_REF] Bayn | Triangular nanobeam photonic cavities in single-crystal diamond[END_REF][START_REF] Whp 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], non-linear dynamics [START_REF] Leijssen | Nonlinear cavity optomechanics with nanomechanical thermal fluctuations[END_REF][START_REF] Navarro-Urrios | Nonlinear dynamics and chaos in an optomechanical beam[END_REF] and integrated nano-photonics [START_REF] Crosnier | Hybrid indium phosphide-on-silicon nanolaser diode[END_REF]. In this chapter, we will discuss 1D suspended photonic crystals cavities able to sustain both localized optical modes in telecommunication wavelengths (≈ 1.55µm) and localized mechanical modes at Ω m in the ultra high frequency (UHF) regime (∼ GHz) (see 2.1 for a schematic of the 1D photonic crystal cavity platform under investigation).

More in detail, our 1D engineered semiconductor photonic crystal nanobeam will sustain localized and delocalized vibrational eigenmodes. The localized ones will lie, specifically, in the photonic cavity region, i.e. at the center of the nanobeam, enabling a possible interplay with the optical localized modes. This light-matter interaction between photons and phonons that coexist in the same small volume could bring interesting exploitable features. In our case, the idea is to inject light into the system that will be evanescently coupled to the photonic crystal cavity. If the optomechanical coupling between the injected localized optical modes and the localized mechanical modes is high enough, the light will allow to read these vibrations as modulations of the laser signal. The optical signal can be then extracted page 27

Chapter 2. Optomechanical crystal oscillator integrated on-chip and the imprinted mechanical signatures properly read. With accurate design, the interaction can be so strong to be able to optically drive vibrations and eventually entering the self-sustained oscillations regime [START_REF] Tobias | Cavity optomechanics: back-action at the mesoscale[END_REF]. This is the principle of a photonic crystal optomechanical oscillator (OMO).

In this chapter we will present the main results achieved so far with 1D PhC nanobeams. Design, fabrication and measurements of GaP and GaAs 1D PhCs integrated on chip, with working frequency around 2-3 GHz depending on the used material, will be discussed. In particular, 1D photonic crystal design and possible optimizations will be presented followed by an explanation of the optomechanical coupling principles. After that, fabrication of the integrated optomechanical structures will be presented, explaining the process flow developed in the clean room facilities. Measurements of the found mechanical modes will be finally shown with phase noise results confirming the potential of the system under study.

1D photonic crystal cavity

Light can be confined temporally and spatially in a material, if this latter has been properly engineered. This new class of materials is called photonic crystal cavity [START_REF] John D Joannopoulos | Molding the flow of light[END_REF][START_REF] Yablonovitch | Inhibited spontaneous emission in solid-state physics and electronics[END_REF][START_REF] John | Strong localization of photons in certain disordered dielectric superlattices[END_REF]. As we mentioned at the beginning of this chapter, a photonic crystal shows a periodic dielectric function. If we take as an example a dielectric slab in air, light can be guided thanks to total internal reflection. Only a discrete number of guided field profiles can in fact exist and they are called modes. An existing optical mode is described by its field profile distribution and by the in-plane wave number along the propagation direction [START_REF] Prorok | Nanophotonics and integrated optics: Photonic crystal cavities[END_REF]. Optical modes with in-plane wave number k satisfying k > ω/c, being ω the pulsation and c the speed of light in the medium, will lie below the so called light cone and they will be completely internally reflected and confined in the material. While, optical modes showing in-plane k values with k < ω/c will lie inside the light cone and they will be able to leak outside the material.

The same principle can be applied to 1D structure such a silicon waveguide where light can only propagate towards one direction. If properly designed, a waveguide can confine light at a certain wavelength comparable with the waveguide dimension. If now one wants to confine light in a specific region of the waveguide an efficient way is to introduce defects such as air holes. By playing on their periodicity as well as on local modification of the dispersion both transmission losses and out-of-plane scattering can be minimize for a proper confinement of the light [START_REF] Amir | Quantum optomechanics with silicon nanostructures[END_REF] For designing a photonic crystal cavity both temporal and spatial confinement must be taken into account for confining light at the desired resonance frequency in a proper way. On one hand, the temporal confinement is described by the optical quality factor Q o that is directly linked to the decay time of photons inside the cavity. A cavity with a high Q o shows long decay time. On the other hand, the spatial confinement is described by the effective mode volume V ef f that describes the volume occupied by the optical mode in the material. A small mode volume V ef f means strong optical confinement of the concerned mode. page 29 Chapter 2. Optomechanical crystal oscillator integrated on-chip Q o is related to the energy decay time e -ω 0 t Q 0 of the cavity. It can also be written as

Q o = ω 0
∆ω , where ∆ω is the the full-width at half max (FWHM) in the frequency spectrum of the cavity. Here, Q o can be regarded as the sum of two contributions losses: the in-plane losses with related Q , and the out-of-plane losses, with related Q ⊥ . The first is related to the strength of the mirror, or in other words to the number of the used holes to confine the mode, and to geometrical imperfections coming from the fabrication of the crystal under study. The second, they are losses mainly happening in the edges of the system due to light scattering. One can write then the total intrinsic Q o of the cavity as:

1 Q o = 1 Q + 1 Q ⊥ (2.1)
Concerning the mode volume, it gives information about the actual volume occupied by the electromagnetic field related to the cavity. It can be defined as:

V ef f = v ( r| E( r)|) 2 d 3 r max( | E( r)| 2 ) (2.2)
Being ( r) the dielectric constant and | E( r)| the electric field strength. The relation between these two quantities, optical quality factor and mode volume, is of crucial importance since we cannot increase both at the same time. On one side, a larger spatial distribution leads to smaller wave vector spectrum and thus less wave vector components inside the light cone, hence a better confinement. On the other side, a too large spatial distribution would limit the miniaturization of the system and, most importantly, it would decrease the optical field strength inside the cavity [START_REF] Prorok | Nanophotonics and integrated optics: Photonic crystal cavities[END_REF]. A trade-off between these two quantities must be found at the end for a proper PhC cavity design.

In the following we will discuss how this trade-off can be approached by engineering a 1D suspended PhC and working on the design of such system. It brings the question of how addressing this system. Thus we will introduce the evanescent coupling scheme and its founding principles.

Cavity design

For the semiconductor materials used for the envisioned platform presented in this thesis, the 1D PhC nanobeam design was well established in our group and it has been already carefully discussed in R. Zhu thesis [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF]. Specifically, the design takes inspiration from 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]. In this subsection, we will explain the general concepts in order to give the idea of how the design has been created and optimized. page 30

1D photonic crystal cavity

In our case, the 1D PhC is made by a semiconductor nanobeam that has been pierced with either circular or rectangular holes. The periodic dielectric function discussed at the beginning of this chapter, enabling light control, can be here reproduced by playing with the difference in the refractive index between the semiconductor media on one side and the surrounding area and the introduced holes on the other side (these latest made by air). By carefully choosing the holes dimension and the distance between them, i.e. the lattice constant a n , a photonic band gap, for TE-like modes, can be created and centered around the desired photonic frequency (around 1.55 µm in our case, figure 2.6 bottom left). This area will be used as mirror for the future cavity. Once the band gap is there, one can start to break the introduced periodicity of the lattice constant. By properly adjusting the lattice constant, a photonic crystal cavity can be created, where an optical mode at the previously forbidden frequency can now exist and be properly confined. In order to achieve a mode at 1.55 µm the edge of the band gap should reach this wavelength allowing a propagating mode (figure 2.6 bottom right). In a first step, this is possible just by gradually varying the lattice constant from two specifically chosen values a n and a 0 . Top part of figure 2.6 shows this design idea for a nanobeam with circular holes. The way of changing the lattice constant to form the cavity region is crucial. If one injects light into the nanobeam at the resonance frequency of the cavity, photons will be confined and the electric field related to the optical mode will spatially decay as e -qx where x is the propagation distance along the nanobeam and q the spatial decay rate. An optimization of this decay rate can be done in order to finally shape a cavity with a Gaussian profile to achieve a gentle confinement for photons. The idea is here to find a relation to link the decay rate q to the lattice constant a. A Gaussian apodisation for a can be finally reached. With this technique 1D photonic crystal cavity nanobeams with intrinsic optical quality factor around 10 7 for temporal confinement and a mode spatial extension of around 3 µm for spatial confinement can be properly designed.

Despite the apodisation of the holes position to shape the cavity profile, another interesting optimization can be done in order to reduce the optical losses. Specifically, this time the holes shape will be gradually varied along the direction of the beam to increase the final cavity quality factor. From the center to the edge, the holes width H x is unchanged and equals 250 nm; the holes length is tuned from H y0 = 400 nm to H yn = 700 nm following a Gaussian distribution. Figure from [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF].

By taking advantages of this latest optimization, the simulated optical quality factor can be pushed towards 10 8 . More details about both techniques can be found in [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF].

System efficiency: coupling with SOI waveguides

Our 1D PhC nanobeams will be fabricated and integrated onto a commercially available photonic circuitry made by SOI waveguides. Hence, the theoretical final efficiency of our system will be the sum of two main contributions. On one side, page 32 2.1. 1D photonic crystal cavity the already discussed quality factor Q o of the designed optical cavity itself. On the other side, the coupled quality factor Q c describing the coupling efficiency between the PhC and the waveguide. We can then write the total Q of a coupled cavity as:

1 Q = 1 Q 0 + 1 Q c (2.3)
Let's define better the second term of this sum. The coupled quality factor Q c is related to the coupling between the PhC and the waveguide. In our case, the 1D PhC nanobeam will be suspended on top of the SOI waveguide. The coupling between the cavity and the waveguide specifically relies on the evanescent coupling between the TE-like mode of the waveguide and the cavity mode, see figure 2.8 for a schematic. More precisely, the coupling between the waveguide mode and the cavity mode relies on the phase-matching conditions, imposing same transverse electric field mode profile and momentum for both modes.

We can write the coupling efficiency in term of the intrinsic loss rate γ 0 and the coupling loss rate γ c as:

η = γ c γ 0 + γ c (2.4) being γ 0 = ω 0 Q 0 and γ c = ω 0
Qc . The coupled quality factor can be tuned by playing on the distance between the optical cavity and the waveguide. In our case, this dimension can be carefully selected by using a proper sacrificial layer dimension for the suspension of the structure (see fabrication section for more details). This is a crucial parameter because, as it has been carefully investigated in [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF], a trade-off between the quality factor and the coupling efficiency must be found to avoid a degradation of the intrinsic Q c because of a coupling too strong. In our case, the distance between the cavity and the waveguide has been established to be 280 nm. This would theoretically allow us to have a coupling efficiency η close to 85%. page 33 Chapter 2. Optomechanical crystal oscillator integrated on-chip

Optomechanical interaction in 1D PhCs

In an optomechanical system, the hamiltonian describing the uncoupled optical and mechanical modes can be written as:

Ĥ0 = ~ωcav â † â + ~Ωm b † b (2.5)
Being â † and â the photons annihilation and creation operators and b † and b the phonons annihilation and creation operators respectively. We already discussed the optical part in the previous section. Here we will first describe the mechanical part and then how they can interact. We can write the displacement operator x as x = x ZP F ( b † + b) where x ZP F is the zero-point fluctuation amplitude of the mechanical oscillator, i.e. the spread in the coordinate in the ground state:

0| x2 |0 = x 2 ZP F
where |0 is the mechanical vacuum state. We can write x ZP F as:

x ZP F = 2m ef f Ω m (2.6)
Where m ef f is the effective mass. In fact, similarly to optics, we can determine the quality factor as well as the volume (or m ef f ). The effective mass relates to the potential energy of the system as

U (t) = m ef f Ω 2 m < x 2 (t) > /2.
This effective mass can be also written as:

m ef f = ρ( r) d r | u| max(| u|) 2 (2.7)
where u( r, t) is the time dependent vector field, ρ( r) the mass density of the material and r the position vector.

Mechanical modes of 1D PhC

For both GaAs and GaP 1D PhC nanobeams, two different categories of mechanical modes arise from eigenfrequency studies of the suspended mechanical structures. In a low frequency range, delocalized mechanical modes with a displacement profile affecting the full nanobeam's geometry are present (figure 2.9.a), usually from few kHz up to hundreds of MHz. While, in a higher frequency range (2-3 GHz), localized mechanical modes are here present with a mechanical displacement that interest mainly the cavity region (figure 2.9.b for a representation of the first order breathing mode). The same figure shows the fundamental optical mode profile on the same scale (figure 2.9.c). By intuition, the better overlap between this latest and the localized mechanical modes category should bring a better photon-phonon interaction due to a better overlap with the field profile of the cavity mode. A lower page 34 2.2. Optomechanical interaction in 1D PhCs interplay between photons and phonons should be expected for the second category, the delocalized mechanical modes, since the overlap with the cavity optical mode is here restricted to a small percentage of the mechanical displacement under consideration.

Coupling between mechanical and optical modes in 1D PhC

We will now introduce the tools in order to give a quantitative description of this coupling.

The optical resonance frequency is modulated by the mechanical motion amplitude by:

ω cav (x) ≈ ω cav + x∂ω cav /∂x + ... (2.8)
and the interaction part of the hamiltonian can be written as:

Ĥint = ~∂ω 0 ∂ x x ZP F â † â( b † + b) = -~g 0 â † â( b † + b) (2.9)
In this expression g 0 = -(∂ω 0 /∂ x)x ZP F is the optomechanical coupling strength. This is an important parameter quantifying the coupling between a single photonphonon pair. Specifically, it quantifies the shift of the optical resonance frequency when this latest is displaced by the zero point fluctuation amplitude of the mechanical oscillator. The interesting fact is that this g 0 can be applied to any optomechanical system with any geometry, making it a really versatile and useful parameter for quantifying the optomechanical strength of a system. If we now think about the specific case of a photonic crystal, two main parameters contribute to this g 0 . page 35 Chapter 2. Optomechanical crystal oscillator integrated on-chip They are the moving interfaces effect and the photoelastic effect. Specifically, we can write g 0 as:

g 0 = (g M I + g P E )x ZP F = - dω 0 dx M I + dω 0 dx P E x XP F (2.10)
Let's define better these two contributes. On one hand, the moving interfaces contribution g M I describes the change in the dielectric tensor ¯ ( r) at the boundaries of the optical resonator because of a deformation induced by a specific mechanical mode. It can be defined as:

g M I = - ω 0 2 ( u. n)(∆¯ |E || | 2 -∆¯ -1 |D ⊥ | 2 max(| u|) ¯ | E| 2 d r (2.11)
where u( r, t) is the time-dependent vector field, E( r, t) the electric field profile, n the unit normal vector on the surface of the unperturbed cavity, D = ¯ E, ∆¯ = ¯ mat -¯ air with ¯ mat and ¯ air being respectively the dielectric constant of the material forming the photonic crystal and the dielectric constant of the surrounding medium.

On the other hand, the photoelastic contribution g P E comes from the local changes in the refractive index due to the strain caused by the deformation of one of the sustained mechanical mode. We can write it as:

g P E = - ω 0 2 Eδ¯ Ed r max(| u|) ¯ | E| 2 d r (2.12)
where δ¯ is the dielectric pertubation given by:

δ¯ = - p s 0 ¯ (2.13)
where p is the photoelastic tensor and s the strain tensor represented by a unitless 3x3 matrix with component s ij = (δ i u j +δ j u i )/2. The photoelastic tensor is a fourthrank tensor with components p ijkl and it can be then written as a 6x6 matrix. Depending on the specific crystal, the relation between the coefficients can change. For the case of a crystal with hexagonal lattice, such as GaAs and GaP, one can write: From the figure, the two categories of modes, delocalized and localized, are underlined with a yellow and a green shaded area respectively. It is here obvious how, the highest optomechanical coupling rate g 0 /2π occur for the second category of modes, the localized ones, due to the better overlapping between the optical field profile and mechanical field profiles in the middle region of the nanobeam. More specifically, for the delocalized modes spanning from 0 to 1 GHz the optomechanical coupling rate g 0 /2π ranges from few kHz up to almost 100 kHz in both GaAs and GaP cases. For localized modes in a 2-3 GHz range, the optomechanical coupling rate g 0 /2π reaches values in a 200-300 kHz range in several cases. Displacement profiles of the fundamental breathing mode M1 and of the higher order breathing mode M2 showing the highest optomechanical coupling rate g 0 /2π are shown in the same figure. Table 3.1 shows, for both GaAs and GaP cases, the optomechanical coupling rate g 0 /2π (and related photoelastic and moving boundaries contributions g P E /2π and g M I /2π) for the two just discussed modes M1 and M2. For GaAs and Chapter 2. Optomechanical crystal oscillator integrated on-chip GaP 1D PhC nanobeams the M1 mode reachs an optomechanical coupling rate of 8.05 and 12.17 kHz respectively. While M2 shows a g 0 /2π equal to 215.05 kHz for GaAs and -279.1 kHz for GaP. Moreover, we can notice how, in the presented cases, the highest contribution always comes from the g P E /2π related to photoelastic effect. Furthermore, the calculated values are in the same order of magnitude with respect to recent studies with similar structures in literature [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF][START_REF] Mercadé | Microwave oscillator and frequency comb in a silicon optomechanical cavity with a full phononic bandgap[END_REF].

Eδ¯ E = -0 n 4 [2ReE * x E y p 44 xy + ReE * y E z p 44 xy + 2ReE * z E x p 44 zx + |E x | 2 (p 11 xx + p 12 ( yy + zz )) + |E y | 2 (p 11 yy + p 12 ( xx + zz )) + |E z | 2 (p 11 zz + p 12 ( xx + yy ))]
M1 GaAs M2 GaAs M1 GaP M2 GaP Ω m [GHz] 2.

Fabrication of 1D PhC cavities

In this section we describe the optimized process flow to fabricate 1D PhC cavities.

The shown III-V semiconductor material described is here GaP but the steps are exactly the same for the GaAs case. The final intent is to obtain a 1D photonic crystal nanobeam suspended on top of a SOI waveguide integrated in a silicon micro-chip. One of the interesting feature of our optomechanical structure is in fact that it can be heterogeneously integrated on top of a standard commercially available photonic circuitry micro-chip. Specifically, the used circuitry in our case is composed by Silicon-On-Insulator (SOI) optical waveguides presenting grating couplers at both ends, enabling an easy light injection through external optical fibres. A 3D schematic of the system is depicted in figure 2 The used process flow is represented in figure 2.12. In particular, the fabrication process starts with an adhesive mechanical bonding between the semiconductor slab and the silicon micro-chip with the already integrated photonic circuitry. Before the bonding, on the semicondutor membrane, a SiN layer of 280 nm has been previously deposited by sputtering technique in order to be used at the end as sacrificial layer for suspension.

For the adhesive bonding a polymerized divinylsiloxaane-benzocyclobutene (BCB) solution is used (thin green layer in the figure). The BCB is spin-coated on the SOI side and then baked at 80 • C for 15 minutes in order to evaporate the solvent and harden the BCB layer. At this point the III-V part is flipped on the substrate and the adhesive bonding of the obtained sandwich of materials is performed in a high precision wafer bonder. Here a vacuum environment will be reached (< 5.10 -3 mbar) with temperature reaching 340 • C while a constant force (400 N) is vertically applied to the sample. The entire bonding process takes around 3 hours. Chemical removal of the substrate where the GaP has been growth is performed at this point. The final obtained sandwich of material is shown on the right side of figure 2.12.a. After, a negative resist (HSQ) is spin coated on top of the III-V material and an electronic lithography step, with subsequently development, is performed in order to transfer the PhC pattern to the HSQ resist (figure 2.12.b). Afterwards, an Inductively Coupled Plasma (ICP) etching technique is used to transfer the pattern to the semiconductor membrane (first step in figure 2.12.c). Finally, the suspesion of the structure is achieved with an underetching step performed with dry plasma based on SF 6 (last step in figure 2.12.c). This latest step allows to get rid of the underneath sacrificial layer of SiN and of the residuals of the used resist. A suspended 1D PhC nanobeam aligned on a SOI waveguide is finally achieved. page 40

Optomechanical Measurements of 1D PhC nanobeams

In this section we will show the main results that have been achieved so far related to both GaAs and GaP 1D PhC nanobeams. The specific intent is here to look, firstly, for cavity resonances in the optical domain and, secondly, for mechanical modes in the UHF regime evidencing self-sustained oscillations.

The experimental setup that has been used in order to perform this kind of measurement is represented in figure 2.14. We are specifically using a fiber-to-fiber transmission configuration taking advantages of the integrated photonic circuitry discussed previously. Figure 2.14 -Fiber-to-Fiber experimental set-up used for 1D PhC optomechanical measurements. A tunable laser is here used to inject light into the device under test (DUT). The light is properly amplified and polarized before injection. The collected output signal is mostly sent before to a photo-diode and then to an electrical spectrum analyzer (ESA) for the detection of the mechanical spectrum. The remaining output optical signal is conveyed towards an optical spectrum analyzer (OSA) for detection of the optical spectrum.

More precisely, the idea is here to inject light into the system at the cavity resonant frequency with a tunable laser. The signal will be amplified if needed and properly polarized before being injected into the integrated SOI waveguides. Specifically, the light will be coupled in and out to the waveguide gratings from external optical fibers mounted on a high precision fiber alignment translation system (PI F-712) with an xyz-axis nanopositioner (PI P-616 Nanocube). Once the light will be evanescently coupled to the PhC cavity of the fabricated nanobeams, the interplay between the injected confined photons and the phonons belonging to mechanical modes will allow us to read the nanobeam motion. More specifically, the cavity mechanical displacement will be imprinted to the injected light as a modulation of the optical signal. At this point, the imprinted mechanical signatures can be extracted from the laser signal coming out from the output optical fiber. Specifically, part of the output optical signal is here used to recover the optical spectrum, with the help of an optical spectrum analizer (OSA) and the remaining part is translated into the electrical domain with a photo-diode (PD) and the mechanical spectrum is recovered with an electrical signal analyzer (ESA).

Optical measurements of 1D PhC cavities

A fiber-to-fiber transmission configuration is used to study the transmission through the SOI waveguides with the aim to eventually extract the information related to the 1D PhC nanobeams. About these latest, we use to integrate 4 different nanobeams with different geometrical parameters onto a same optical waveguide. The geometrical designs are chosen in order to avoid overlap of the cavity resonances, and, more page 42 2.4. OM measurements specifically, this technique is used to balance the imperfections that might come during the fabrication process that could induce an unwanted optical frequency shift. At the end, this allows us to always have a 1D PhC nanobeam with a fundamental mode working at the desired targeted wavelength (around 1.55 µm).

Regarding the optical injection, two light sources have been used for this set of optical measurements. A broadband light source (Exalos broadband SLED), with a 140 nm bandwidth centered at 1570 nm, and a more precise tunable laser (EXFO T100S-HP) with a 1500-1630 nm wavelength span. The idea is to use the former to have a general feedback about the working waveguides and nanobeams and the latter to scan in a more precise way a targeted cavity resonance in order to precisely extract information such as the optical quality factor of the cavity. Regarding the output signal, for this set of measurement this will be collected by an OSA (Anritsu MS9740A) which will give the final optical transmission spectrum.

Figure 2.15.a shows a retrieved transmission spectrum example from the OSA of a SOI waveguide where no 1D PhC nanobeam is present. In this case, the broadband light source is used to inject power into the system. This typical transmission shape comes from the used gratings. The center of the transmission can be adjusted by changing the coupling angle between the external optical fibers and the gratings. An angle of 11°is here used in order to center the transmission in the wanted wavelength range. Chapter 2. Optomechanical crystal oscillator integrated on-chip fundamental and higher order optical modes of the fabricated nanobeams. This a good feedback in order to validate the used design idea and to experimentally prove that the fabricated nanobeams are indeed working in the wanted wavelength range.

A second measurement has been carried out in order to experimentally estimate the total quality factor of the fabricated photonic crystal cavities. More specifically, the tunable laser has been used now to inject light into the system to scan the cavity at 1550 nm (red dashed circle in figure 2.15.b) with high resolution wavelength steps (0.002 nm step). The OSA peaks values have been recorded to reconstruct the high resolution cavity profile (see figure 2.15.c). After this, a Lorentzian fit is used to experimentally extract the quality factor of the targeted cavity. In this case, the measured quality factor was in the order of 1x10 5 , confirming us that high quality factor photonic crystal cavities can be indeed achieved with this GaP structures heterogeneously integrated on SOI.

Mechanical spectrum of 1D PhC cavities

The aim of this subsection is to investigate the presence of mechanical modes sustained by the 1D PhC nanobeams in the UHF regime. Specifically, there are two main objectives here: (1) verify the presence of these high frequency mechanical modes and (2) see if they can eventually entered the self-oscillation regime. This latest is characterized by a reduction of the mechanical mode linewidth with at the same time an increase of the peak amplitude [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF]. This is basically a regime of regenerative oscillations which can be addressed when the mechanical loss equates the optical antidamping and its threshold is given therefore by Γ m + Γ om = 0 where Γ m is the mechanical damping and Γ om the optomechanical antidamping coefficient. Its behaviour is similar to a single frequency laser. The limit for the short term linewidth follows in fact the Schawlow-Townes limit for lasers and it can be eventually calculated [START_REF] Hossein | An optomechanical oscillator on a silicon chip[END_REF][START_REF] Vahala | Back-action limit of linewidth in an optomechanical oscillator[END_REF][START_REF] Rokhsari | Brownian noise in radiation-pressure-driven micromechanical oscillators[END_REF]:

Γ ef f,L = Γ m nth 2n + 1 2n ≈ Γ m nth 2n (2.15)
where Γ ef f,L is the effective linewidth, nth is the number of phonons at thermal equilibrium and n is the number of phonons in the system. In our case, if light will be injected to the cavity at its resonant frequency the number of photons will increase. The interplay between photons and phonons due to the optomechanical coupling will let these latest to increase as well and the final effective linewidth will eventually decrease.

We will report here results related to GaAs 1D PhC cavities that has been exhaustively discussed in R. Zhu thesis [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF] and new results that we have been page 44 2.4. OM measurements recently able to achieve with GaP 1D PhC cavities.

For this set of experiments, the injected light will come from the tunable laser (EXFO T100S-HP). This will be locked to one specific resonant value of the cavity under study in order to fuel the cavity with photons at its resonant frequency. The interplay inside the cavity between the injected photons and phonons related to specific mechanical modes will allow to read the mechanical motion of the nanobeam as modulations of the optical signal due to this photon-phonon interaction. The optical signal will be then sent to a high-sensitive photodiode and the resulting electrical signal will be collected by the ESA (ROHDE&SCHWARZ Spectrum Analyzer 10 Hz -7.5 GHz) finally getting the mechanical spectrum of the 1D PhC cavity under investigation to look for possible mechanical modes.

Figure 2.16.a shows the retrieved power spectrum density related to the first case, GaAs 1D PhC nanobeams studied in R. Zhu thesis [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF], in a frequency range going from 0 to 3 GHz. Interesting features appeared in that case around 2.3 GHz. From the figure, we can clearly identify several mechanical modes that appear close to each other in a specific frequency range going from 2.2 to 2.4 GHz. The figure also shows the closest simulated mechanical eigenmodes. The measured mechanical quality factor was in that case around 800 and the highest measured optomechanical page 45

Chapter 2. Optomechanical crystal oscillator integrated on-chip coupling rate g 0 /2π was 145 kHz using the calibration technique described in [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF].

This generation of GaAs 1D PhC nanobeams gave us a tangible proof of the potential of our platform, being able to read the mechanical motions of the fabricated nanobeams by a commercially available SOI circuitry where they have been integrated. However, these mechanical modes were not able to enter the self-sustained oscillations in that case, due most probably to the high losses of the material and a competition between modes since they were really close to each other in frequency.

Thus, a second generation of 1D PhC nanobeams have been developed in GaP, material showing a larger band gap at room temperature and fully compatible with the fabrication process flow used for GaAs. Results related to the GaP 1D PhC cavities are displayed in figure 2.17.

Figure 2.17 -Mechanical spectrum of 1D GaP PhC cavities. a) Mechanical spectrum between 0 and 4 GHz when the tunable laser input it at resonance with the targeted cavity. We are able here to read delocalized modes in a low frequency range (0-1 GHz) and an interesting localized mode around 3 GHz. b) Zoom of the high frequency mode at low injected optical power. c) Zoom of the high frequency mode at higher injected optical power. The mode is here able to enter the self-sustained oscillations regime. page 46 2.4. OM measurements Specifically, figure 2.17.a reports the measured mechanical spectrum in a frequency range going from 0 to 4 GHz when the injected optical power from the tunable laser has been set to 4 mW. In this case, it is possible to notice that several modes are present. In a lower frequency range, from 0 to 1 GHz, we can distinguish several modes most belonging to the delocalized modes category discussed in the simulation section. In a higher frequency range, an interesting mode appears, at a mechanical frequency Ω m of 3.35 GHz, able to reach this time the self-sustained oscillation regime. This can be appreciated from the high amplitude and small linewidth shown.

Figure 2.17.b shows the same mode in low optical power regime (tunable laser set to 1 mW) displaying a mode that is not yet self-oscillating. In this case, the measured mechanical quality factor is around 1000 and the measured optomechanical coupling rate g 0 /2π is equal to 160 kHz. This is in the same order of magnitude with respect the one measured for the previous nanobeams in GaAs and comparable with literature values of similar nanobeams implemented with different materials [START_REF] Ghorbel | Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor[END_REF][START_REF] Gomis-Bresco | A one-dimensional optomechanical crystal with a complete phononic band gap[END_REF][START_REF] Davanço | Si3n4 nanobeam optomechanical crystals[END_REF][START_REF] Balram | Moving boundary and photoelastic coupling in gaas optomechanical resonators[END_REF][START_REF] Schneider | Optomechanics with one-dimensional gallium phosphide photonic crystal cavities[END_REF]. In figure 2.17.c, higher optical power is now injected from the laser, 3.5 mW, and the mode is entered now in a self-sustained oscillation regime. In this latest case, a detuning in wavelength is also needed to enhance the peak power to its maximum, with a related small shift in the mechanical frequency as well due to optical spring effect.

A really good feedback comes from this generation of GaP 1D PhCs nanobeams. The recovered mechanical mode signal in the UHF range shows interesting properties, being able to enter the self-oscillations regime. As it was mentioned before, this different behaviour with respect to GaAs 1D PhC nanobeams comes most probably from the fact that GaP has a larger bandgap than GaAs (2.26 eV instead of 1.42 eV) meaning a lower two-photon absorption for a more efficient final photon injection in the photonic crystal cavity. In the next subsection we will characterize deeper the proprieties of this oscillation by looking at its phase noise.

Phase noise measurement of GaP mechanical modes

We will now present results about phase noise measurements of the GaP 1D PhCs previously introduced to characterize the stability of these GHz oscillations. The most frequently used tool to describe oscillator phase noise is the L(f) function, mentioned in the introduction of the manuscript as one of the most important figures of merit of modern oscillators. Let's remind that it is expressed in [dBc/Hz] and represents the noise power with respect to the carrier contained in a 1 Hz bandwidth and centered at a certain offset from the carrier. In order to extract the L(f) curve, in our case we use an heterodyne downconversion method with a final phase noise analysis based on the fast fourier transform (FFT) of the measured time-domain traces. The used experimental setup is represented in figure 2.18. It is almost the same as the one used for mechanical spectrum measurements with the difference that here part of the signal coming out from the photodiode is filtered and amplified before being mixed with a low phase noise local oscillator signal at 50 MHz higher offset with respect to the OMO signal in order to downconvert the OMO signal to 50 MHz. This latest is sent to an oscilloscope and we record the time domain signal (for 4 ms) that is eventually analyzed in frequency domain to recover the phase noise. The downcoversion is here used because a direct sampling would be challenging as would require a sampling rate in the GHz (at least the double of the signal frequency) and larger amount of memory to record the same time-span as well. The best result is here achieved when the tunable laser is set to 1550.625 nm. For this case, the phase noise at 100 kHz offset from the carrier is equal to -84 dBc/Hz which corresponds to a 326 Hz linewidth. The linewidth was here extracted by averaging the related frequency noise in a range up to 500 kHz. Chapter 2. Optomechanical crystal oscillator integrated on-chip to have a clear vision of how the phase noise behaves as a function of the change in the two parameters, in order to identify their optimal values avoiding possible local minima problems. In the map, the dark-blue areas are the regions with lowest linewidths, the orange areas are region where the signal is not self-oscillating while the dark-red are out-of-resonance regions where no mechanical mode was found. The lowest measured linewidth is in this case 326 Hz with 3.5 mW laser input power and 1550.9 nm wavelength. The alternating narrow and wide regions in the map are mainly due to the reflections of the grating couplers bringing interferences along the waveguide. This effect, that can be modelled as a Fabry-Pérot cavity [START_REF] Ismail | Fabrypérot resonator: spectral line shapes, generic and related airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity[END_REF], is eventually changing the light coupling to the cavity. The phase noise measurements performed in this section allow us to approach the state of the art performances of OMOs discussed in the introduction of the manuscript (see dashed red circle in figure 2.20) but for the first time integrated on standard silicon circuitry and, moreover, with no feedback implementation yet (free running oscillator).

Conclusion of the chapter

In this chapter we have introduced 1D optomechanical crystals as a possible solution for integrated optomechanical oscillators (OMOs). Theory and simulation of 1D PhCs have been firstly discussed, showing how, for a suspended 1D PhC page 50 2.5. Conclusion of the chapter nanobeam, both optical and mechanical modes can be ultimately confined in a really small volume at the center of the nanobeam. This can specifically be achieved thanks to appropriate design where the introduced defects periodicity allows a gentle confinement of the light thanks to the adopted Gaussian-shaped cavity profile. Tools for optomechanical coupling calculation has been then introduced and with simulations we have shown how a high optomechanical coupling g 0 /2π in the order of few hundreds of kHz is finally expected between the optical fundamental mode (at ≈ 1550 nm) and the mechanical localized modes (at Ω m ≈ 2-3 GHz depending on the material) both lying at the center of the suspended nanobeam. After, fabrication of such 1D PhCs integrated onto commercially available photonic circuitry has been demonstrated. The evanescent coupling between the PhC cavity and the SOI waveguides has been then proved with experimental optical results showing how the fabricated nanobeams are indeed able to sustain optical modes in the telecommunication wavelengths (≈ 1550 nm) with high measured quality factor (1x10 5 ). Furthermore, experimental investigation of the mechanical behaviour showed how for both investigated materials, GaAs and GaP, localized mechanical modes directly in the UHF regimes actually exist. For the former case, GaAs, we measured promising modes with a mechanical frequency Ω m around 2 GHz but eventually they were not able to self-oscillate. For the latter case, GaP, we measured a mode at Ω m = 3.25 GHz that was finally able to sustain oscillations. Moreover, phase noise measurements have been carried out showing a promising phase noise of -84 [dBc/Hz] at 100 kHz offset from the carrier without any feedback loop implementation. These results confirmed us that the proposed 1D PhCs nanobeams are actually good candidates to be implemented as a core oscillating element in our envisioned time-delayed OMO integrated on chip. page 51

Chapter 3

GHz acoustic waves on III-V semiconductor on insulator The objective of this chapter is to achieve a robust and tunable way in order to create acoustic waves at nano-scale in a fully-integrable manner by keeping at the same time fully-compatibility with respect to our photonic crystal cavity process flow. The frequency Ω acoustic we are interested in is in the range of 2-3 GHz, in order to properly match the mechanical modes frequency Ω m of the suspended 1D PhC nanobeams either in GaP or GaAs presented in the previous chapter.

By taking advantage of the piezoelectricity of the chosen III-V materials (GaP and GaAs) integrated transducers can be created in order to generate acoustic waves Chapter 3. GHz acoustic waves on III-V semiconductor on insulator propagating within the material at a defined frequency. This can be then adjust by changing the geometrical properties of the transducers, allowing to match the frequency of the PhC mechanical modes.

If we look at the world of micro and nano-systems, there are mainly 3 types of acoustic waves that can be generated in a solid medium by integrated transducers via inverse piezoelectric effect [START_REF] Fu | Engineering inclined orientations of piezoelectric films for integrated acoustofluidics and lab-on-a-chip operated in liquid environments[END_REF]. Specifically, we can classify them with respect to their propagating features: Bulk acoustic waves (BAWs) propagating through the bulk [START_REF] Ba Auld | Acoustic fields and waves in solids[END_REF], Surface acoustic waves (SAWs) propagating through the surface [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF][START_REF] Delsing | The 2019 surface acoustic waves roadmap[END_REF] and Lamb waves propagating through thin plates [START_REF] Lamb | On waves in an elastic plate[END_REF][START_REF] Viktorov | Rayleigh and Lamb Waves: Physical Theory and Applications[END_REF] (see figure 3.2 for a schematic of these three configurations).

BAWs are waves that propagate through the bulk of a solid medium and they are usually created by applying an electrical signal to electrodes placed on the top and on the bottom of the material [START_REF] Royer | Elastic waves in solids I: Free and guided propagation[END_REF][START_REF] Aigner | Bulk-acousticwave filters: performance optimization and volume manufacturing[END_REF] (figure 3.2.a). The generated particle motion is then well confined inside the solid medium for a resulting acoustic wave propagating through the bulk.

SAWs and Lamb waves are acoustic waves created by interdigitated transducers (IDTs) placed on the surface of the chosen material [START_REF] White | Surface elastic waves[END_REF][START_REF] Farnell | 2 -elastic wave propagation in thin layers[END_REF][START_REF] Kojima | Two-port saw resonator using series connected idts[END_REF]. Specifically, IDTs are transducers made by a comb-shaped arrays of metallic electrodes deposited onto the surface of the material (see figure 3.1). By applying an electrical signal to these electrodes and by taking advantage of both inverse piezoelectric effect and periodicity of such a structure an acoustic wave propagating in both side perpendicularly to the electrodes length can be generated. In the former configuration, SAWs (figure 3.2.b), the particles motion is confined on the surface of the material in a region comparable with the acoustic wavelength and its amplitude disappears with distance from the surface. This is the so called Rayleigh wave, from Lord Rayleigh who was the first showing the existence of this kind of surface acoustic waves in 1885 [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF]. For the latter configuration, Lamb waves (figure 3.2.c), the wavelength of the generated acoustic wave is larger than the thickness of the chosen solid medium and the particles motion affects the full thickness of the chosen solid thin plate.

Our final intent is to generate acoustic waves for a proper mechanical excitation of our PhC cavities. In the following sections of this chapter design, fabrication and measurements of IDTs on different stack of materials will be presented, always with the aim of creating efficient acoustic waves in the same range of frequency of the 1D PhC nanobeams mechanical modes (around 3 GHz for GaP and around 2 GHz for GaAs). page 54

Simulation and design of integrated IDTs

In this section we will perform simulation and design of integrated IDTs by taking advantage of the piezoelectricity of the chosen semiconductor materials. The final intent is to study the transmission between two face-to-face Aluminum IDTs deposited directly onto the piezoelectric semiconductor membrane. When an RF signal is applied to the first IDT, the sender, an acoustic wave is expected to be created and lately collected by a second IDT, the receiver. An artistic view of this simple configuration is shown in figure 3.3. Here the design of the transducer plays an important role since it allows us to master the actuation frequency Ω acoustic of the created acoustic waves. In particular, this latest is fixed by the fingers pitch, i.e. the center-to-center distance between the electrodes. In this work, the pitch will be always equal to twice the used finger width e (see bottom left inset of figure 3.3).

When we are in the SAW regime, i.e. the acoustic wavelength is smaller than the thickness of the underneath material, an isotropic approximation of the first order acoustic resonance can be given by SAW theory from the following formula 3.1 [START_REF] David P Morgan | A history of surface acoustic wave devices[END_REF]:

f 0 = c R λ = c R 4e (3.1)
Being c R the Rayleigh velocity of the chosen material, λ the acoustic wavelength and e the width of electrodes with a fixed pitch of 2e. For linear elastic materials with a Poisson ratio ν ∈ [0, 0.5] a good approximation for the Rayleigh velocity is given by 3.2 [START_REF] Av Pichugin | Approximation of the rayleigh wave speed[END_REF] : Being c s the shear velocity defined as c s = µ ρ where µ is the shear modulus and ρ the density of the material.

c R = c s 0.87 + 1.12ν 1 + ν (3.
Table 3.1 presents the calculated Rayleigh velocity for GaAs and GaP (materials properties taken from [START_REF] Gaas | Available on line[END_REF] and [START_REF] Gap | Available on line[END_REF]). From this point of view, GaP seems more interesting with its lower Rayleigh velocity. So far, we have treated GaP and GaAs as isotropic materials to have a first simplified comparison between them. This is actually not true but it helped us to quickly understand the range for the finger width dimension in order to get acoustic resonances in the GHz. The calculated Rayleigh velocity is in the order of few page 56 3.1. Simulation and design of integrated IDTs thousands meters per second for both cases, meaning that transducers with finger width in the order of few hundreds of nanometers would be needed in order to get acoustic resonances in the GHz.

GaAs GaP

From the next paragraph we will perform FEM simulations for modelling our devices. In this case, the used materials must be defined in a more precise way. GaAs and GaP are in fact anisotropic materials and it is well known that the precise Rayleigh/shear velocities depend on crystallographic orientation and therefore the actuation frequencies for a given geometry. For a first generation of transducers, we will deal with IDTs showing electrodes following the cleaving angle of the thin semiconductor membrane. For this, the chosen simulated crystallographic direction will be 0 degrees in the following.

Limit between Lamb waves and Surface Acoustic waves

In this subsection we will try to simulate the behaviour of an IDT deposited onto a semiconductor membrane to have a better understanding of the principles behind the piezoelectric transduction in different situations. This will help us to come up with realistic design ideas. We will take as example of semiconductor material GaP with a 0 degree crystallographic orientation but all the conclusions from this set of studies will be still valid and can easily be translated for GaAs as well.

At this point, it would be interesting studying the transition between SAWs and Lamb waves regime of an IDT in order to see how the changing in the thickness of the underneath material affects the frequency shift of the created acoustic modes. From this, we will be then able to think about possible design configurations of our platform. Finite element simulation (FEM) can be performed in order to investigate this situation. In our case, COMSOL Multiphysics has been used to have a better understanding of how, for a fixed dimension of electrodes, the changing of the underneath material thickness affects the piezoelectric transduction of an unchanged IDT.

Let's recall that for us the thickness of GaP membrane is 300nm and it is fixed because of optical purpose. From a fabrication point of view we are able to either finally release it for a total suspension (as for PhC cavities presented in the previous chapter) or we can choose to leave it unreleased. For the latter, we can try to engineer the stack of material underneath if needed.

Eigenfrequency studies have been performed over a span of thickness λ ratio ranging from 0.1 (i.e. strong Lamb wave regime) to 1.5 (i.e. a SAW regime) when λ is kept equal to 2 µm (i.e. finger width e = 500 nm, chosen as a reasonable achievable value from a fabrication point of view). Two different cases have been studied, one where page 57 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator the bottom is fixed and one where the piezoelectric membrane is suspended. The COMSOL 2D models used for these two kinds of situations are represented in figure 3.4. The two electrodes have been defined as ground and terminal and periodic boundary conditions have been applied on both left and right side of the unit cell in order to simulate an infinite number of fingers pairs. Figure 3.4 -2D Comsol models used for studying the transition between lamb waves to SAWs regime. For both cases the finger width is kept equal to 500nm and the unit cell dimension λ is kept equal to 2µm. While, the GaP thickness will span over a specific range in order to have the thickness λ ratio ranging from 0.1 (i.e. strong lamb wave regime) to 1.5 (i.e. a SAW regime). For the first case, on the left, the bottom is fixed, while for the second case, on the right, the membrane is suspended.

Eigenfrequency results of the first case, bottom fixed, are shown in figure 3. [START_REF] Lee | Displacement measurement using an optoelectronic oscillator with an intra-loop michelson interferometer[END_REF]. In this situation it was possible to find for each case the first mode, symmetric and antisymmetric, and its shift in frequency according to the change in the thickness λ is presented in the graph on the left side of the figure. Total displacement profiles for the symmetric first order mode for ratio equal to 0.1, 0.5 and 1.5 are displayed as well. We can notice how as long as we stay in a SAW regime ( thickness λ > 1) the frequency stays stable around 1.65 GHz. As soon as the ratio becomes lower than 1 the frequency starts to increases to finally reach 6.76 GHz for thickness λ = 0.1. This study helps us understanding that as long as we stay in a ratio close to (or higher than) 1 the SAW regime should be fulfilled and the frequency of the first mode should remain stable for a bottom fixed situation. On the right side of figure 3.5 is depicted the geometrical change of the 2D model when we jump to one limit case ( thickness λ = 0.1) to the other ( thickness λ = 1.5). In this latter case we can notice how the dimension of the bulk starts to be consistent with respect to the acoustic wavelength λ and this helps the confinement of the Rayleigh mode solely in the surface of the material.

In this set of simulation, the case with thickness λ = 0.1 might be unrealistic from a fabrication point of view because the material underneath has also acoustic properpage 58 3.1. Simulation and design of integrated IDTs ties which will allow displacement, even small one. To avoid it, suspended structure needs to be investigated. Eigenfrequency results related to a suspended membrane, are shown in figure 3.6. Two kinds of modes appear here, the extensional symmetric mode S0 and the flexural antisymmetric mode A0. The symmetry classification for these two classes of modes has to be taken here with respect to the horizontal axis. Each class of mode can then show symmetry with respect to the vertical axis as well. We can appreciate again how, as long as we stay in a region where thickness λ > 1, their frequency remains stable around the same value as before, 1.65 GHz.

As soon as the ratio becomes lower than 1, the two modes split in two branches. Specifically, A0 goes towards a lower frequency range (reaching around 500 MHz for thickness λ = 0.1) and S0 goes towards a higher frequency range (reaching around 2.5 GHz for the symmetric and 2.75 GHz for the antisymmetric when thickness λ = 0.1). Total displacement profiles for antisymmetric S0 at 0.1, 0.2, 0.3, 0. changes. For a ratio > 1 the frequency for both modes is stable around 1.65 GHz. As soon as the ratio becomes < 1 they split into two branches, S0 goes towards higher frequencies and A0 goes towards lower frequencies. Total displacement profiles for several cases are displayed in the same graph while on the right side of the figure the geometrical change of the simulated model when the ratio increases for a fixed thickness of the membrane is shown. in the mode shape, always affecting the full membrane. Total displacement profiles for antisymmetric A0 are shown for ratio equals to 0.1, 0.6 and 1 as well. The geometrical change of the 2D model when we jump from one limit case to the the other is again shown here on the on the right side of figure.

The main goal is being able to predict the frequency shift of the acoustic modes when we jump from a bottom fixed SAW regime to a suspended Lamb waves regime, since this is exactly what we do in a real device when we decide to release the semiconductor membrane. In fact, if we think about the real case, before suspension the semiconductor membrane is in a "bottom fixed" situation with thickness λ > 1. Once it is released, we jump to a suspended situation with thickness λ much lower than 1 (0.15 for GaP and 0.13 for GaAs for finger width of 500 nm).

If we combine results of these two sets of simulations we can appreciate how the frequency working range of the simulated transducers is basically the same when the the constrictions limits are less important and/or stringent (i.e. for thickness λ > 1). While their behaviour differs when we push them to specific limits. An interesting comparison merging these results is shown in figure 3.7. Here is evident how for a page 60 From this comparison is evident how as soon as the SAWs regime is achieved ( thickness λ > 1) all the cases tend to converge to the same first order mode frequency, close to the theoretical prediction. While, if we jump from a bottom fixed SAWs regime to a suspended lamb waves regime, we should expect a split into two branches of the original rayleigh mode, one going towards higher frequencies and one going towards lower frequencies.

thickness λ

> 1 all the results converge to the same frequency. This frequency, around 1.65 GHz, is close to the theoretical estimation of the first SAW mode in a bulky GaP when e = 500 nm is chosen as finger width. Specifically, equation 3.1 tells us that in this case the first order mode should be around 1.46 GHz. Moreover, these results tell us that for an unchanged IDT, if we switch to a bottom fixed SAW regime to a suspended Lamb waves regime we should expect a split of the Rayleigh mode into two branches: an A0 mode should appear in a lower frequency range with respect to the Rayleigh mode and a S0 mode should appear in an higher frequency range.

Design of acoustic transducers in engineered stack of materials

So far we have investigated how integrated IDTs can be implemented in microsystems with the main intent to define the limit between Lamb waves regime and SAWs regime for the case of a semiconductor membrane that can ideally change its thickness as desired. The main limitation of this approach was the definition of the bulk, that has been always encountered as the semiconductor membrane itself.

page 61

Chapter 3. GHz acoustic waves on III-V semiconductor on insulator This helped us to shape the model in the easiest way to understand the difference between Lamb waves and surface acoustic waves creation. Now that we have a better idea of how these two regimes coexist, let's try to replicate and test this behaviour in micro devices that can be actually fabricated and tested. In particular, to be consistent with the PhC crystals on SOI process flow, we have designed three different kinds of structures in order to study acoustic features of IDTs on GaP in different situations: (a) GaP on SiN/Si, (b) GaP on SiN/SiO 2 /Si and (c) GaP suspended on SiO 2 /Si. A cross section of each of them is displayed in figure 3.8. Here, it should be noticed that the GaP thickness is 300 nm, i.e. this is exactly the same semiconductor membrane that we used to fabricate the 1D PhCs nanobeams. The visible thin green layer in the structures is a polymerized divinylsiloxaane-benzocyclobutene (BCB) solution used for the adhesive bonding (thickness ≈ 50nm) (more details of bonding technique in the following fabrication section).

The first two types of structures, (a) and (b), are bulky systems where the intent will be to work in a SAWs regime. While with the third one, structure (c), the intent is to study a Lamb waves regime where the IDTs will lie exactly on the same semiconductor membrane as the 1D PhC nanobeams. For the first two cases, the interest in studying different layered structures lies on the fact that besides the piezoelectric properties mentioned before, other features might influence the piezoelectric transduction. The creation of acoustic waves depends in fact also on the mechanical properties of the underneath materials and/or how well the produced electric field will be confined when an RF signal is applied to the electrodes. dimension over a defined range to see how this affects the frequency of the created acoustic waves by keeping the chosen materials thickness the same. As previously, the two electrodes have been defined as ground and terminal and periodic conditions have been applied on both left and right side to simulate an infinite number of fingers pairs. Here the horizontal dimensions change for each finger width in order to keep the dimension of the repeated unit cell always equal to λ = 4e (see figure 3.9.b). Eigenfrequency studies have been performed for finger width ranging from 475nm to 600nm with a 25nm step. Figure 3.9 -a)COMSOL 2D models used to perform an eigenfrequency study of the three different stack of materials related to the design ideas proposed in figure 1.8 . Here the materials thickness will be kept the same (300nm). What is changing is the electrode thickness creating the IDTs fingers pair. Periodic boundary conditions have been applied in order to simulate an infinite number of fingers pairs like in the previous set of simulation. b) The unit cell horizontal dimension λ is here changing to be always equal to 4e in order to be consistent with the different simulated finger width dimensions.

Results related to structure (a), GaP on SiN/Si, are shown in figure 3.10. Here, figure 3.10.a shows a lower frequency range (1-2 GHz) and it is possible to identify the 1st SAW mode (symmetric and antisymmetric). In particular, by taking as reference the two extreme values for finger width, 475nm and 600nm, the first mode page 63 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator is ranging from 1.685 to 1.258 GHz. Figure 3.10.b shows results related to higher frequency range, from 2.5 to 4 GHz, where the second order mode can be identified (symmetric and antisymmetric) and it is ranging from 3.916 to 2.925 GHz. In the same range a shear mode appears as well ranging from 3.350 to 2.742 GHz. For structure (b), GaP on SiN/SiO 2 , simulations give a really similar situation as the previous case. Results are plotted in figure 3.11. In particular, in figure 3.11.a we can see that the first order mode is now ranging from from 1.616 to 1.174 GHz. In figure 3.11.b the second order mode is ranging from 3.925 to 3.050 GHz. Also in this case a shear mode appears exactly in the same range as before, from 3.354 to 2.756 GHz. Here we evidence the weak impact of the substrate on the acoustical frequency.

However, if we look at the total displacement profile shown on the bottom side of each of the two figures just described, we can see how for the GaP on SiN/Si and GaP on SiN/SiO 2 the modes tends to leak towards the bulk, where a well defined particle motion can be noticed for several cases especially for the latter. For this, the bulk is SiO 2 and this may help the confinement of the produced electric field in a real case for a better transduction of the modes, even if, since there is no real mechanical isolation between the piezoelectric membrane and the bulk, a leakage of the displacement field should always be expected. For the last case, structure (c) with a GaP membrane suspended on SiO 2 , the membrane is now released and results are shown in figure 3.12. Here, it is possible to appreciate how A0 (in figure 3.12.a) and S0 (in figure 3.12.b) are respectively at lower and higher frequency with respect to the first Rayleigh mode of the previous not suspended case, ranging from 0.762 to 0.463 GHz and from 2.781 to 2.089 GHz respectively. While a flexural ansisymmetric higher order mode, A1, is ranging from from 2.454 to 1.624 GHz.

For this last case, the GaP membrane is now completely released and suspended on top of the SiO 2 bulk. The mechanical confinement of the produced acoustic modes is here clear and well defined uniquely in the piezoelectric membrane (see modes profiles on the bottom part of the figure).

This kind of study confirmed us the prediction of the frequency shift made by the previous set of simulations, i.e. two modes, A0 and S0, should appear after the release of the membrane and their frequency should be respectevely lower and higher with respect to the Rayleigh first order mode of the not suspended counterpart. Moreover, a further useful feedback coming from these results is that working with a suspended GaP membrane should help us to properly focus the acoustic mode exclusively in the semiconductor membrane where PhC nanobeams will lie. Since our intent is to lately excite them with the created acoustic wave, minimizing the losses in the electro-acoustic transduction can be an important point to take into page 65 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator Figure 3.12 -Eigenfrequency results related to the GaP suspended on SiO 2 design. Here A0 and SO are respectively at lower and higher frequency with respect to the previous first Rayleigh mode. a) In the lower frequency range A0 is ranging from 0.762 to 0.463 GHz for finger width changing from 475 nm to 600nm. b) In the higher frequency range, A12 and S0 can be identified ranging from 2.454 to 1.624 and from 2.781 to 2.089 GHz respectively. On the bottom side of the figure total displacement profiles are shown for each case.

account. This study also confirmed us that acoustic modes close to 3 GHz should exist if the fingers of the IDTs are in the range of 475 to 600 nm. page 66 3.2. Fabrication of suspended IDTs on GaP on Silicon

Fabrication of suspended IDTs on GaP on Silicon

In this section we will describe the process flow in order to fabricate the three structures (a), (b) and (c) discussed earlier and represented in figure 3.8. Let's remember that here the shown semiconductor material is GaP, but, as for simulations, the process flow will be still valid in the case of a GaAs membrane. Chapter 3. GHz acoustic waves on III-V semiconductor on insulator Specifically, figure 3.13 shows that the fabrication process starts with an adhesive bonding between the chosen substrate (a thick SiO 2 layer on silicon wafer in this case) and the semiconductor membrane. In this latter a SiN layer of 280 nm has been previously deposited by sputtering technique, similarly to what it has been done for PhC fabrication. For the adhesive bonding a polymerized divinylsiloxaanebenzocyclobutene (BCB) solution is used (thin green layer in the figure). The BCB is spin-coated on the SiO 2 substrate side and then baked at 80°C for 15 minutes in order to evaporate the solvent and harden the BCB layer. At this point the III-V part is flipped on the substrate and the adhesive bonding of the obtained sandwich of materials is performed in a high precision wafer bonder. Here a vacuum environment will be reached (< 5 • 10 -3 mbar) with temperature reaching 340°C while a constant force (400N) is vertically applied to the sample. The entire bonding process takes around 3 hours. Chemical removal of the substrate where the GaP has been growth is performed at this point. The final obtained sandwich of material is shown on the right side of figure 3.13.a. After that, a positive resist (PMMA) is spin-coated on top of the semiconductor membrane for electronic lithography (figure 3 Two large contact pads are here visible for each IDT. These will be the contact area for the ground-source (GS) high frequency electrical probes that will be used for electrical injection. In order to optimized the adhesion with the probes and to ensure a proper reusable electrical contact, an additional metal deposition of 250 nm of Aluminum has been performed in this area. The zoom inset of the same figure shows a SEM image of one face-to-face IDTs configuration that will be used to study acoustic transmission. The space between the IDTs is here kept fixed at 20 µm for all the IDTs pairs (i.e. 10 times the acoustic wavelength λ acoustic when e = 500 nm).

Fabrication of interdigitated fingers

Calibration of the process

As discussed previously, a positive resist (PMMA) is used to create the IDTs fingers pattern. In this case, an important electron dose calibration must be done in order to be sure that the wanted dimensions will be finally obtained. In fact, on one hand, an excessively high dose would enlarge the fingers dimensions with a possible shortcut between them. On the other hand, an excessively low dose wouldn't be enough to properly modify the resist, preventing an optimal metal deposition and lift-off page 68 Here the pitch between them is fixed at 1.1 µm, i.e. twice the targeted finger width, while the actual width of the fabricated fingers can change according to the electron beam dose. From the graph we can see how for an electron dose ranging from 800 to 1500 µC/cm 2 the actual measured finger width changes from 540 nm (a) to 600 nm (b) respectively. From the SEM images above the graph it is possible to observe the change in the shape of the fingers just discussed. The pitch between them is always respected (the alignment is not impacted by the change in the dose) but their dimensions and definition change according to the dose value. When a 1500 µC/cm 2 is used, point b in the graph, as electron dose the fingers look really well defined even if their width is now close to 600 nm instead of the designed 550 nm.

We investigate at this point the change in the finger width, by keeping the pitch the same, to see if this will impact, or not, the actuation frequency of the created acoustic mode. Figure 3.16 shows simulations results of eigenfrequency study of 2D IDTs model. The model is the same as the one used for simulations in the previous section but in that case we were varying both pitch and finger width dimensions and we have appreciated a shift in the created acoustic mode. Now for this simulation, in order to properly simulate the electron dose calibration discussed above, the pitch is kept fixed while only the finger width dimensions will vary. From the graph, the results tell us that, if the pitch is kept fixed for the periodic structure, the change in the finger width does not affect almost at all the frequency of the produced acoustic mode. The frequency shift is in fact in this case of just 1.7 MHz, ≈ 0.3% from 543 MHz (a) to 544.7 MHz (b), when the finger width changes from 530 nm to 600 nm From this last results, we can conclude that using a higher dose is more suitable for a better definition of the electrodes constituting the IDTs fingers since the final actuation frequency will not be impacted by the enlargement of the finger width as long as the pitch between them will be kept constant. page 70 3.2. Fabrication of suspended IDTs on GaP on Silicon

Underetching of the semiconductor membrane

From the presented fabricated structure is then possible to achieve the suspension of the semiconductor membrane, in order to get the third envisioned system of figure 3.8. The SiN sacrificial layer can be in fact removed through small holes carefully positioned and opened between the IDTs fingers. The optimized process flow to perform the suspension is shown in figure 3.17. Specifically, the process starts with a Plasma-Enhanced Chemical Vapor Deposition (PECVD) of a SiN layer (first step in figure 3.17.a) that will be used as a hard mask for the patterning of the holes as it will be explained in the following. A positive resist (PMMA) is at this point spin-coated on top of the SiN layer and after page 71 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator lithography exposure and development (second and third steps in figure 3.17.a) the pattern of the holes is transferred from the PMMA layer to the SiN layer by Reactive Ion Etching (RIE) technique (first step in figure 3.17.b). Inductively Coupled Plasma (ICP) is used to transfer pattern to the semiconductor membrane that is etched with a gas mixture based on HBr by using the SiN as hard mask (second step in figure 3.17.b). At the end, dry plasma based on SF 6 gas is used to underetch SiN sacrificial layer underneath through the small holes just opened and to remove the SiN hard mask as well. A fully suspended IDTs platform is finally achieved (figure 3.17.c). Here, in the top left inset the holes between the fingers used for the underetching are clearly visible (darker spots) as well as the suspended area thanks to the difference in the color contrast. The top right inset of the same figure shows a SEM cross section after that the SiN sacrificial layer has been removed for suspension. These presented fabrication results show how, after a proper calibration, it is possible to master the underetching expansion in order to match the required dimensions.

Measurements of acoustic waves in the GHz

In this section we will test the fabricated samples in order to look for acoustic resonances in the GHz. Reflection and transmission measurements will be performed with the aim to extract the S-parameters to characterize the integrated IDTs. A TOSM (through-open-short-match) calibration has been used to calibrate a two ports Vector Network Analyzer (VNA, ZND Rodhe&Schwarz 0-8.5 GHz) when two RF Ground-Source (GS) electrical micro-probes are properly connected to it. The used setup is shown in figure 3 When an electrical signal is sent to an IDT an acoustical wave is expected to be created thanks to the inverse piezoelectric effect in the material. During this electro-acoustic transduction part of the energy of the original electrical signal is then transduced to an acoustic mechanical wave. This effect will induce a dip in the reflection at a specific frequency meaning that part of the energy signal is not coming back because transduced to an acoustic wave that has been created. This dip page 73 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator is a signature of an acoustic resonance and, according to simulations of the previous section, we should expect a shift in frequency of all the existing acoustic modes if different finger widths will be used. In particular, as the finger width is reduced the frequency should shift towards an higher range and this should be true for both SAWs and Lamb waves regime. Measurements for all the three GaP fabricated samples (see figure 3.8) will be here shown.

Reflection in the SAWs regime of GHz acoustical waves

The two samples designed to work in a SAW regime, i.e. GaP on SiN/Si and GaP on SiN/SiO 2 /Si, have been firstly tested.

Measured reflection for the former configuration is shown in figure 3.20 for all the fabricated finger widths (from 475 nm to 600 nm with a 25 nm step). Here we see that unfortunately there are no evidence of acoustic resonances in the GHz shifting with finger width dimensions. The only interesting behaviour is in the range of few hundreds of MHz where several dips are present (see inset figure 3.20). These are not changing with finger widths dimensions, suggesting that they might not be surface acoustic waves. Moreover, in this range of frequency FEM simulation suggests that a bulky mode should be present (as shown in the figure) and its working frequency is not affected by the changing in the transducers design, i.e. it is not possible to tune it. Its questionable origins and the lack of tunability make it not useful for our purpose, since we are looking for acoustic resonances that can be master and tune in order to properly match the PhC mechanical modes frequency as desired. The three involved mechanical frequencies Ω acoustic , Ω prop and Ω m have to be in fact the same for a correct feedback implementation. Being able to tune them as desired is then a crucial degree of freedom to achieve this purpose.

Results related to IDTs fabricated in the second stack of materials, GaP on SiN/SiO 2 /Si, are shown in figure 3.21 and they are splitted in two different frequency range, from 0 to 2 GHz, left panel, and from 2 to 4 GHz, right panel. In the reflection related to the lower frequency range, figure 3.21.a, several dips can be identified for each finger width. Specifically, the found acoustic resonant frequency Ω acoustic goes from 1.067 to 1.405 GHz when finger width dimension ranges from 600 to 475 nm respectively. This confirmed us the expected trend of having an higher actuation frequency when a smaller finger width is used. If we now look at the second frequency range, right panel in figure 3.21, we can appreciate how also in this case several acoustic resonances are present and their frequency Ω acoustic is changing again with the dimension of the finger width. In particular, this measured second order mode ranges from 2.720 to 3.275 GHz for the same span of finger width as before. A comparison with the simulated second order mode (blue and violet triangles for symmetric and symmetric) and the shear mode (green diamond and dark blue triangle) is displayed in 3.21.d. Here, the measured second order mode is slightly shifted towards lower frequency with respected to the simulated one being finally closer to the shear mode.

The direct comparison of the 2 stacks of material with or without SiO 2 layer evidences an obvious difference. This suggests that the SiO 2 substrate can play indeed a role in the confinement of the produced electric field with respect to the Si substrate. The injected electrical signal can leak and dissipate more easily. By introducing a thick isolating layer acoustic resonances are present around 3 GHz and their actuation frequency can be actually tuned and mastered by changing the transducer design in agreement with the previous simulations. However, the amplitude of these dips does not evidence a large depth, meaning a low transduction efficiency. This could be due to various phenomena but first of all the excessive damping induced by the underneath layer.

Reflection in the Lamb waves regime of GHz acoustical waves

Results related to IDTs on GaP suspended on SiO 2 /Si are shown in figure 3. [START_REF] Hossein | Observation of optical spring effect in a microtoroidal optomechanical resonator[END_REF]. In this configuration the membrane is released and we should now work in a suspended Lamb waves regime. From the results is possible to appreciate how also in this case it is possible to measure acoustic resonances shifting with finger width dimensions. In the frequency range from 0 to 2 GHz (figure 3.22.a) we see acoustic resonances ranging from 0.795 to 0.575 GHz when finger width changes from 475 to 575 nm. No resonance was found for the 600 nm in this case. These results are really close to the simulated A0 acoustic mode. It was in fact ranging from 0.762 to 0.538 GHz (symmetric) and from 0.710 to 0.501 GHz (antisymmetric) for 475-575 nm range of finger width dimensions (see figure 3.22.c for a comparison). Figure 3.22.b displays results page 76 3.3. Measurements of acoustic waves in the GHz related to a 2-4 GHz frequency range. Also in this case acoustic resonances have been found, this time for all the finger widths. In particular, the measured mode here is ranging from 3.150 to 2.544 GHz being close to the simulated S0 mode, ranging from 2.781 to 2.216 GHz (antisymmetric) and from 2.631 to 2.089 GHz (symmetric). Figure 3.22.d shows the comparison between them where measurements are always at higher frequencies than simulations (most probably for difference in the actual thickness, acoustic properties used for simulations etc.) but they always respect the trend. Moreover, depths are larger and widths smaller compared to figure 3.21 evidencing reduction of damping due to the SiN sacrificial layer that has been now removed.

From SAWs to Lamb waves regime: experimental conclusion

The two previously presented sets of measurements gave us information about the acoustic actuation frequency Ω acoustic related to IDTs in two different situations: SAWs regime, i.e. when the semiconductor membrane is not released yet, and Lamb waves regime, i.e. when the semiconductor membrane where the transducers lie has been suspended. It is interesting now to compare such results between each other in order to experimentally understand the frequency jump between the two just cited situations. Figure 3.23 shows a comparison between the measured results and the simulated expectations for SAWs and Lamb waves behaviour for 500 nm finger width transducer. Here, the predicted split into the two branches for the SAW mode after the suspension of the membrane is now evident also from measurements (stars in the figure). Specifically, the measured SAW mode is in this case at 1.302 GHz (light blue star in the figure). If we jump to Lamb waves regime, the measured A0 and S0 modes are at 726 MHz and 3.014 GHz respectively (yellow and orange stars). Let's specify that here in the graph the measured SAW mode is plotted at thickness λ = 1.5 for convenience since this is the maximum simulated value. The real fabricated device has in fact a much bigger ratio if we take into account the whole bulk material, but as along as we enter the SAWs regime the frequency should remain always stable. While for the other two cases, the 300 nm thick GaP membrane and the 500 nm finger width give us an actual ratio of 0.15. Overall, from the figure we can observe that measurements and simulations are in good agreement for all the cases, with the SAW mode that splits into a lower frequency A0 mode and a higher frequency S0 mode when we pass from SAWs to Lamb waves regime. Further evidence confirming our observations come from literature where similar results have been found as well in similar integrated IDTs [START_REF] Zhu | Switchable lamb wave delay lines using algan/gan heterostructure[END_REF]. This agreement between the expected simulated behaviour and the actual fabricated device gives us a good feedback for the mastering of the IDTs actuation frequency Ω acoustic when transducers are integrated on different stack of materials.

Keeping in mind that the final goal is to have acoustic resonances around 3 GHz for a proper acoustic excitation of the PhC mechanical modes in GaP, two candidates are available so far: (1) second order Rayleigh mode when the membrane is not released and (2) the S0 mode once the membrane has been released. Figure 3.24 gives a closer look at these two kinds of mode when 525 nm finger width is chosen as a finger width. In this comparison we can see that the bandwidth of the S0 mode is three times smaller with respect to the second Rayleigh order mode but, at the same time, the acoustic signature appears deeper as well, for a better transduction efficiency. The occurred frequency shift between these modes is here around 170MHz. page 78

Measurements of acoustic waves in the GHz

Impact of crystallographic orientation

As mentioned before, the IDTs tested so far were always designed with electrodes following the cleaving angle of the thin GaP membrane. To experimentally test the anisotropy of the used material, results related to IDTs oriented at 30, 45, 60 degrees with respect to the cleaving angle will be here shown. This will give us a complete view about all the exploitable possibilities of IDTs integrated on a thin piezoelectric III-V semiconductor membrane.

A new set of IDTs at 0, 30, 45, 60 and 90 degrees with respect the cleaving angle has been fabricated and tested with 475, 500, 525 and 550 nm finger widths dimensions. The structures have been directly suspended on top of 2 µm of SiO 2 and they have been tested in the frequency range of more interest (around 2-3 GHz). For the sake of simplicity here we discuss frequency evolution only for one finger width (= 475 nm). The behaviour will be the same for the other widths. The conclusion that we can draw from this type of study is that the crystallographic orientation plays indeed a role for the piezoelectric transduction of the IDTs. Both the frequency and the efficiency seem in fact to be affected when the pair of IDTs is oriented at different angles. Yet, by taking into account the studied finger widths dimensions, the 0 • (and 90 • ) looks still the best option in order to achieve acoustic actuation with Ω acoustic = Ω m around 3 GHZ. With the smaller finger width (475 nm) at 0 • and 90 • we are in fact already over 3 GHz. This is not the case for 475 nm at 45 • (even though with a better transduction) and at 30 • and 60 • (similar efficiency but lower in frequency). Some effort on the fabrication needs page 80 to be done to be able to achieve at the same time higher efficiency and 3 GHz as a working frequency. From figure 3.22.b ∆Ω acoustic ∼ 400 MHz when ∆f inger = 50 nm. This implies that to achieve Ω 45 • acoustic = 3 GHz we should fabricate IDTs with finger width e ∼ 375 nm, i.e. critical for the performed lift-off technique and a precise optimization of the process would be expected to be developed in this case. page 81 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator

Calculation of the acoustic waves power efficiency

Calculation of the acoustic waves power efficiency can be performed in order to estimate the efficiency of the different tested IDTs. In particular, the power of the acoustic waves that is created from the piezoelectric transduction occurring in the transducer can be calculated from the measured reflection parameter S 11 and the used input RF power (always 0 dBm in our case). At any frequency Ω m the power dissipated at the IDT is [START_REF] Li | Nanophotonic cavity optomechanics with propagating acoustic waves at frequencies up to 12 ghz[END_REF]:

P dis = P in (1 -|S 11 (Ω m )| 2 ) = 2P wave (Ω m ) + P R (Ω m ) = P in [2η wave (Ω m ) + η R (Ω m ) (3.3)
we can then finally write:

1 -|S 11 (Ω m )| 2 = 2η wave (Ω m ) + η R (Ω m ) (3.4)
Where P in is the total input RF power, P wave (η wave ) is the power (electromechanical coupling efficiency) of the traveling acoustic wave and P R (η R ) is the power loss (power loss ratio) at the IDTs. The factor of 2 takes into account that the IDT is launching an acoustic waves in both direction. We can then calculate the electromechanical coupling efficiency η wave from the reflection S 11 . Figure 3.26 shows the measured spectrum of |S 11 | 2 for all the previous presented cases at 475 nm plus the case at 0 degree not suspended.

Table 3.2 shows the extracted electromechanical coupling efficiency η wave for all the discussed cases. Two observations can be done here: (1) the suspension of the IDTs boosts the efficiency of the transducer, in fact η wave goes from 0.15 % for a 0 • IDT that is not suspended to 0.81 % for a 0 • IDT that is suspended. (2) The 45 • IDT shows the better efficiency (2.36 %), at least three times higher with respect to all the other angles. In the future, this information may modify the architecture of the targeted device by implementing a specific angle between the 1D PhC nanobeam and the acoustic transducer.

Electromechanical coupling efficiency

0 • not susp. 0 • 30 • n1 30 • n2 45 • 60 • n1 60 • n2

Transmission measurements of GHz acoustical waves

In this subsection we will study the transmitted S-parameters in order to prove acoustic transmission through the face-to-face IDTs configuration in GaP. All the investigated IDTs pairs are spaced by 20 µm (10λ acoustic for e = 500 nm). If an IDT shows acoustic signatures in reflection, as the acoustic resonances presented in the previous section, what would be interesting to study is the propagation of such created acoustic wave. In fact, this will help us to fully characterize our IDTs in order to have a clear view of all the features that we could exploit for our final resonant excitation platform.

In a transmission process an RF signal is sent to a first IDT, an acoustic wave is created at a resonant frequency Ω acoustic and it will propagate through the semiconductor material. If the wave now faces a second IDT working at the same resonant frequency, it will be transduced back to a RF signal by the second IDT thanks to the direct piezoelectric effect. By collecting and studying this signal, i.e. the transmitted S 21 , information related to the propagating wave can be extracted.

Let's take at this point the 475 nm IDTs oriented at 45 • , the one showing so far the deeper S 11 reflection, and let's try to study its transmission. As it can be seen from the figure, at 2.215 GHz a transmission signal seems to be there but it looks somehow masked at this stage. In fact, for properly reading the transmitted peak, a time gating technique is needed in order to get rid of the page 84 3.3. Measurements of acoustic waves in the GHz electromagnetic radiation (EMR) between the electrical probes present in our experimental setup. This is in fact a well known technique in literature used to reveal transmission signals in similar integrated structures [START_REF] Petroni | Gan-based surface acoustic wave filters for wireless communications[END_REF][START_REF] Yu | Acoustic delay lines to measure piezoelectricity in 4h silicon carbide[END_REF][START_REF] Udo | Aln/sio Bibliography 2/si 3 n 4/si (100)-based cmos compatible surface acoustic wave filter with-12.8-db minimum insertion loss[END_REF][START_REF] Rỳger | An algan/gan based ghz-range surface acoustic wave oscillator for sensor applications[END_REF][START_REF] Takagaki | Guided propagation of surface acoustic waves in aln and gan films grown on 4 h-sic (0001) substrates[END_REF]. Specifically, the EMR manifests in frequency domain as a flat wide-band signal severely reducing the out-of-band insertion loss. In time domain, the EMR shows itself as an intense narrow peak after a very short time delay (electromagnetic signal through the substrate or air propagates much faster than acoustic wave [START_REF] Clement | Saw characteristics of aln films sputtered on silicon substrates[END_REF]) (figure 3.28). In order to investigate this, let's have a look about the time domain features of the S 21 just presented in figure 3.27. transforming them back to frequency domain to reveal the transmission peak. The selection of the time window is crucial since it will allow us to filter out the EMR and the reflection coming from the probes, i.e. the components masking the signal. As it is possible to see from the figure, in this case the time gating has been defined in a 20-500 ns time window (orange rectangle in figure 3.28).

Figure 3.29 shows the gated S 21 spectrum transformed back to frequency domain (red trace). Here we can appreciate how the peak is now revealed and unmasked from the wide flat-band related to the EMR, that now is properly filtered out. Moreover, the unmasked signal shows great out-of-band rejection of almost 50 dB and, furthermore, the level of the gated signal is comparable to that of the non-gated signal (dash-dotted black trace), both being around -50 dB.

Let's now focus on the 0 • oriented IDTs. This orientation evidences in fact the best trade-off between frequency matching, Ω acoustic = Ω m at around 3 GHz, and trasduction efficiency in the reflection parameters. The bottom part of figure 3

IDTs in GaAs

All the previous discussions about GaP can be easily translated to GaAs. On one hand, the change in the piezolectric material will induce in this case a shift of the acoustic resonances if we use the same finger width dimensions. On the other hand, the discussed origins and features of the modes are still consistent with GaP simulations and their behaviours follow the same rules and are also reproducible for GaAs.

Reflection and Transmission measurements

Let's recall that the mechanical modes frequency Ω m of PhC in GaAs is around 2 GHz. This is then the target frequency Ω acoustic we should look for with IDTs on GaAs. In order to test acoustic reflection and transmission with GaAs a new series of IDTs have been fabricated. The 300nm thick GaP membrane has been replaced with a 260nm thick GaAs membrane to be consistent also in this case with the PhC design shown in the previous chapter. Face-to-face IDTs have beed fabricated following the same process flow as before. The structures have been finally suspended on top of the bulky SiO 2 used as a substrate. Figure 3.31 shows reflection and gated transmission when 600 nm is used as finger width. Here the IDTs have been spaced by 24 µm (i.e. 10λ acoustic when e = 600 nm). As it is possible to appreciate from the figure, a couple of dips are present in the reflection spectrum while several peaks appear in the gated transmission. The first two the A0 ans S0 observed in GaP as well but shifted towards lower frequency with respect to this latter because of the change in the semiconductor piezoelectric membrane. Another interesting acoustic resonance appears here at around 6.5 GHz. Its transmission peak is present as well. This is a nice feature that can be for instance exploited in the future to stabilize an OMO working at higher frequency.

A nice feedback comes from the fact that 600 nm is apparently a good candidate in order to excite PhCs with Ω m equal to 2 GHz, showing the S0 reflection and transmission frequency Ω acoustic at 2.046GHz. Some of the peaks don't have apparently a reflection counterpart. This is most probably because the reflection dips are too small and a specific VNA frequency calibration around them would be needed in order to properly reveal them.

Losses of propagating acoustical waves

Face-to-face 600 nm finger width IDTs spaced by different distances have tested in order to measure the acoustic losses in the material where the waves are travelling through. Specifically, IDTs spaced by 20λ acoustic (= 48 µm), 40λ acoustic (= 96 µm) and 80λ acoustic (= 192 µm) have been fabricated and tested. Figure 3.32 shows the gated transmission related to the discussed situations. From the figure, we can identify the three transmission peaks at 2.060 GHz (black trace), 2.063 GHz (red trace) and 2.069 GHz (blue trace) for 48 µm, 96 µm and 192 µm spaced IDTs page 88 3.4. Conclusion of the chapter respectively. We notice as a first remark the robustness of the design, able to work at almost the same reproducible actuation frequency Ω acoustic if the same finger width is used. Secondly, we can see how the peak amplitude is different for each case, signature indeed of losses during the acoustic waves propagation. For a noise floor around -100 dB for all the cases in fact, the peak value is at -53.5, 62.8 and -76.2 dB for 48 µm, 96 µm and 192 µm respectively. Here, all the transmission traces S 21 have been time-gated with the same time interval for a fair comparison between them (specifically it goes from 5 to 450 nm, large enough to contain all the acoustic components for each case). The final calculated loss rate is then of 0.158 ± 0.009 dB/µm. From this set of results we can see how, even though the acoustic losses in the semicondcuctor membrane are indeed consistent, we are able to read transmission of acoustical waves that have traversed a path up to 192 µm. This is a good feedback to take into account in order to finally calculate the maximum possible acoustic delay that can be introduced in the envisioned device.

Conclusion of the chapter

In this chapter we have shown simulation, design, fabrication and measurements related to IDTs directly integrated on the same semiconductor membranes used in Chapter 2 for 1D PhCs integration. An optimized process for both fingers realization and suspension of IDTs has been achieved, with proper calibration of the used fabrication steps, in order to work in both SAWs and Lamb waves regimes. Both page 89 Chapter 3. GHz acoustic waves on III-V semiconductor on insulator simulated and experimental results of the designed IDTs gave us a tangible proof that creation and propagation of acoustical waves at the frequency Ω acoustic of interest is indeed possible and its frequency actuation point can be tuned by simply changing the geometrical parameters of the transducers. A frequency matching of the created acoustic waves at Ω acoustic with the PhCs mechanical modes working at Ω m presented in previous chapter has been shown for both GaP (∼ 3 GHz) and GaAs (∼ 2 GHz) suggesting that a resonant mechanical excitation of the 1D PhCs nanobeams is therefore theoretically possible with the presented integrated transducers. More finger widths dimensions for IDTs in GaAs will be investigated in the next chapter, together with a phononic engineering of the material for integrated phononic waveguides design. In the previous two chapters we have demonstrated integrated OMO with well defined mechanical modes at Ω m in the GHz and integrated tunable IDTs for creation of acoustical waves Ω acoustic also in the GHz. A third ingredient is now missing in order to complete the recipe and to make the acoustical waves created by the IDTs properly interacting with PhC mechanical modes. One candidate to make this interaction possible to achieve, and tunable at the same time, are Phononic Crystals (PnCs). With them it will be possible in fact to achieve creation of phononic waveguides sustaining propagating mode at Ω prop = Ω acoustic = Ω m .

Chapter 4 Control of GHz acoustical waves

Phononic crystals have gained a lot of importance in the last decade in integrated nano-systems thanks to continuous improvements in nano-fabrication technologies allowing an easier integration in different situations. Similarly to photonic crystals Chapter 4. Control of GHz acoustical waves discussed earlier, phononic crystals are engineered materials where periodic features have been introduced in order to periodically change the elastic properties as well as the density of the chosen solid medium. In fact, the controlled periodic modulation of these characteristics will open the way to control the acoustic dispersion at the chosen wavelength scale [START_REF] Gorishnyy | Sound ideas[END_REF][START_REF] Srivastava | The Physics of Phonons[END_REF]. One of the most interesting features of this technique is the creation of acoustic band-gaps in precise frequency range that can be then mastered by controlling the pattern of chosen phononic crystals. Specifically, by changing their geometrical proprieties, the acoustic behaviour of phononic crystals can be easily tuned in frequency according to the desired target [START_REF] Mohammadi | Complete phononic bandgaps and bandgap maps in two-dimensional silicon phononic crystal plates[END_REF].

In this chapter, we will firstly describe how phononic crystals lattices can be used in order to introduce acoustic band-gaps in a material with a tunable frequency range. We will then try to furthermore engineer the introduced lattice for phononic waveguide creation in order to guide phonons at nano-scale and engineer their velocity. Finally, we will end by integrating the presented phononic platform with the integrated IDTs presented in the previous chapter in order to experimentally prove phononic engineering for acoustic delay control of phonons at nano-scale (see figure 4.1 for an artistic scheme). In this chapter we will deal with GaAs as reference material but all the presented studies can be easily translated to GaP.

Phononic crystal lattices

As mentioned before, phononic crystals are able to create acoustic band-gaps in a material by creating well defined frequency range where phonons are not allowed to propagate. The dispersion relation describing the phonons behaviour with respect to frequency can be then tuned by changing the geometrical properties of the introduced phononic crystal pattern.

For GaAs the frequency range of interest, i.e. where PhC mechanical modes appeared, is around 2.2 GHz. It would be interesting at this point to look for a phononic crystal lattice allowing the creation of a band-gap around this frequency. Also, we look moreover for a phononic crystal lattice able to shield mechanical vibrations in all the real space directions, i.e. with a complete phononic band-gap. This will allow us to potentially create a phononic waveguide in any direction of space, which is an important degree of freedom for the future complete device.

Last but not least, we should look for a phononic crystal lattice with a fullycompatible process flow with respect to the other integrated structures presented so far. page 92 4.1. Phononic crystal lattices

PhN crystals lattices design

By taking into account the reasons discussed in the previous paragraph, three main candidates have been studied in order to look for interesting phononic features around 2.2 GHz: triangular lattice (conventionally used for complete photonic band-gap [START_REF] Imada | Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure[END_REF][START_REF] Claudio | Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method[END_REF][START_REF] Qiu | Design of a channel drop filter in a twodimensional triangular photonic crystal[END_REF]), hexagonal lattice (usually discussed for phononic bandgap [START_REF] Mohammadi | Complete phononic bandgaps and bandgap maps in two-dimensional silicon phononic crystal plates[END_REF][START_REF] Wu | Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy[END_REF]) and a new case related to Kagome lattice. Related unit cells are highlighted in figure 4.2. In fact, by taking inspiration from these three different types of patterns, phononic crystal lattices can be integrated by introducing pattern of circular empty holes directly in the semiconductor membrane. This will allow an integration together with PhCs and IDTs presented in the previous chapter easy to achieve. FEM simulations have been carried out with COMSOL Multiphysics with the intent to extrapolate the dispersion relation for all the k vectors propagating in all directions in the frequency range of interest in order to look for possible phononic band-gap creation.

The principle is here similar to the photonic case. Periodic modulation of the elastic properties of the material can bring phononic interferences that can eventually create forbidden frequency regions for mechanical waves. By looking at the phonons behaviour between the highest symmetry points in the first irreducible Brillouin zone of the specific lattice, one can look for complete phononic band gaps creation. In our simulations, periodic conditions will be applied to the model in order to span all the possible eigenvalues between the highest symmetry points M , Γ and X describing the phonons propagation in all directions of the specific unit cell. 3D COMSOL models have been created for each case. The simulated semiconductor material is GaAs with a thickness of 265nm fixed because of optical purpose (needed thickness page 93 Chapter 4. Control of GHz acoustical waves to have a single optical mode in the membrane). Figure 4.3 shows the used unit cells where periodic conditions have been applied in x and y direction (top-right part of figures 4.3.a, 4.3.b, 4.3.c) and the obtained phononic band diagram for triangular, hexagonal and Kagome lattice when the lattice constant has been defined equal to 400 nm for the first two cases and 370 nm for Kagome lattice. The r a ratio has been chosen equal to 0.43 for all the three cases studied in a frequency range from 1.5 to 3 GHz. From the figure it is possible to see how for the first case, i.e. triangular lattice, there is no band-gap opening in the frequency range of interest since k vectors are able to propagated for all the frequency values in all directions. For the second case, i.e. hexagonal lattice, a band gap is here opened from 2.020 to 2.337 GHz in the related band diagram. Here the absence of propagating k vectors for a 317 MHz frequency range around 2.2 GHz makes the Hexagonal lattice a first possible candidate suitable for phononic bandgap implementation in our case. The last chosen design, i.e. Kagome lattice, shows an even better total band gap of 424 MHz from 2.023 to 2.447 MHz for all the direction of k vectors, making it the best candidate among the three for phononic bandgap implementation around 2.2 GHz. Triangular lattice does not show any bandgap in the frequency range of interest while the other two candidates, exploiting hexagonal and Kagome lattices, show interesting features with phononic band gaps opening with bandwidth of several hundreds of MHz around 2.2 GHz. Specifically, the Kagome implementation would allow a 424 MHz band gap creation, 107 MHz larger than the hexagonal one. Even though the presented results are focused on specific geometrical parameters, this comparison is always true as long as we remain in the same wavelength scale. Finally, even if for some k vectors the hexagonal lattice shows a larger pseudo bandgap (specifically from Γ to X direction) with respect to the Kagome one, it is preferable for us to work with a phononic crystal lattice showing a complete band gap covering page 94 4.1. Phononic crystal lattices all the possible k vector directions. This will allow to have a larger mechanical shielding in any situation, giving us for instance more degree of freedom in the case of a change in the waveguide direction would be needed. For this reason we will from now on focus on Kagome crystal lattice to phononically engineer our semiconductor membranes.

Mastering the tunability

At this point, it is interesting to show how is possible to tune in frequency this band gap just by playing with the geometrical parameters of the introduced pattern of holes. In our case the thickness of the GaAs membrane is fixed to 265 nm. The remaining geometrical degrees of freedom that we can change are the lattice constant a and the radius r (figure 4.4.a).

FEM simulations have been carried out in order to evidence that changing the lattice constant a indeed shifts the band gap center frequency, while the ratio r a is kept constant (= 0.43 in this case). For this set of simulation we spanned the a value from 300 to 450 nm. In figure 4.4.a unit cells for a = 300 nm and a = 450 nm are shown at the same scale, showing how consistent is the change in the unit cell dimension when we pass from one limit to the other of the considered span. Figure 4.4.b shows the shift in frequency of the phononinc band gap edges when the lattice constant a changes. In particular, it is possible to appreciate how the center of the phononic band gap is shifted of more than 1.2 GHz towards lower frequency for a changing from 300 to 450 nm, with its central frequency that passes from 2.942 GHz to 1.707 GHz respectively. Overall, the shift in frequency is about 8 MHz/nm even if it does not follow an exact linear trend. If we take into account that the maximum control of the holes position during electron beam lithography is around 1 nm, this should not bring a significant impact for the central frequency of the fabricated device. The band gap bandwidth changes as well presenting a width of 574 MHz for the first case and 322 MHz for the second case. This is a relative small change with respect to the shift of the center band gap frequency. We will explain it better with the next simulated example. Furthermore, the best candidate to work around the PhCs mechanical modes at Ω m (∼ 2.2 GHz, see red bracket in the y-axis of the figure) turned out to be a = 370 nm. For this lattice constant value, the bandgap is in fact well centered around 2.235 GHz with a 425 MHz bandwidth. Another set of simulations has been carried out to show how by changing the r a ratio, for a fixed at 370 nm, the created band gap can mainly changes this time its bandwidth. Figure 4.5.a shows the geometrical change in the unit cell holes dimensions when, for a fixed a = 370 nm, we pass from a r a ratio equal to 0.30 (i.e. radius = 111 nm) to a r a ratio equal to 0.45 (i.e. radius = 166 nm). This value, 0.45, has been chosen as a limit for simulation because it is an actual limit from a fabrication point of view. In fact, for a fixed a = 370 nm, when the holes radius is set to be 166 nm the minimum space between holes edges is ∼ 35 nm, i.e. the fabricated suspended structure is going to lack already in robustness with the risk of collapsing. Figure 4.5.b shows how the change of the bandwidth is evident when we pass to one limit to the other of the considered span. Specifically, from 0.30 to 0.37 only the bottom edge of the band gap is actually affected while the top one does not show a frequency shift. In this range, the band gap width changes from 212 MHz when r a = 0.30 to 405 MHz when r a = 0.37. After this, the band gap width remains 405 MHz till r a = 0.45. In the full considered span, for a band gap that almost doubled its width, this time the band gap center frequency shift was equal to just 263 MHz, passing from 2.458 to 2.195 GHz for r a = 0.30 and r a = 0.45 respectively. More specifically, the radius changed from 111 nm to 165 nm in the considered span, giving a frequency shift of ∼ 4.7 MHz/nm. A good calibration of the used electron beam dose (more details in the fabrication section) will allow to master the holes page 96 4.2. Guiding phonons dimension in order to respect the targeted frequency. To sum up, the Kagome design showed good tunability in frequency by changing its geometrical parameters. This means that in the future if one wants to address PhC mechanical modes present in other frequency range this will be possible just by changing and adjusting the lattice constant and the radius of the chosen Kagome crystal lattice. From now on, we will focus on a Kagome lattice with a lattice constant a = 370 nm and with a r a ratio equal to 0.43. This will in fact give us a complete phononic band gap of 424 MHz and a 600 MHz band gap along the Γ -X direction centered around 2.2 GHz (see figure 4.3.c) and thus exploitable for the GaAs PhCs mechanical modes excitation. For the discussed case, the gap-tomidgap ratio is 26%, wider with respect to other possible design implementations with 2D geometry such as triangular or hexagonal lattices [START_REF] Mohammadi | Complete phononic bandgaps and bandgap maps in two-dimensional silicon phononic crystal plates[END_REF].

Guiding phonons

Once a phononic crystal lattice with a specific frequency band gap has been found, i.e. shielding of phonons in a specific frequency range has been proved, one can start to engineer the lattice in order to guide phonons as desired. A commonly used solution is to remove holes to introduce a defect or waveguide allowing phonons to propagate for certain k-vectors. A phononic waveguide guiding phonons at a specific frequency Ω prop is exactly what we need to guide the acoustic waves created by an IDT at Ω acoustic towards a 1D PhC nanobeam sustaining a mechanical mode page 97 Chapter 4. Control of GHz acoustical waves at Ω m in order to properly mechanically excite and stabilize its oscillations (with

Ω prop = Ω acoustic = Ω m ).
In this section we will present two techniques for creating a phononic waveguide. With the first one, we will directly remove lines of holes from a perfect phononic crystal lattice in order to create phononic crystal waveguides. In fact, by breaking the symmetry of the pattern, new eigenvalues will appear in the dispersion relation, and they will be now able to propagate in the previously forbidden frequency range. With the second technique, we will take inspiration from the phononic crystal pattern to create corrugated phononic waveguides. In this case, the inspired corrugated design will shape the sides of the waveguide and it will be shown how, also in this case, it is possible to engineer propagating modes for specific frequency values.

Phononic crystals waveguide

In our case, in order to create phononic crystals waveguides, we will start from a perfect Kagome crystal lattice. From this, we will remove some of the introduced lines of holes in order to break the pattern periodicity and open a phononic waveguide along the perpendicular direction of the lattice.

Waveguides design

Two versions of phononic crystal waveguides will be here discussed, one when 3 lines of holes have been removed (3ML design) and one when 5 lines of holes have been removed (5ML design). The final intent is always to finally open a waveguide in the Γ -X k-direction. An odd number of removed lines has been chosen here to keep the waveguide symmetric. The schematics of two designs are represented in figure 4.6.

Guiding phonons

The intent is now to simulate new unit cells related to the two waveguides design to study the new eigenvalues born from the break in the periodicity in one direction, in order to look for possible propagating modes. FEM simulation with COMSOL multiphysics have been carried out to investigate this situation. Two unit cells have been designed to simulate the dispersion relation of the 3ML and 5ML waveguides when the lattice constant a and the r a ratio have been fixed according to the chosen design in section 1.1, i.e a = 370 nm and r a = 0.43. Figure 4.7 shows the results related to the 3ML design. Specifically, in figure 4.7.a several dispersion relations are plotted. With white hexagons we have the band diagram for a perfect Kagome phononic crystal lattice along the Γ -X direction showing the already discussed band gap due to the periodicity of the structure. With shaded orange pentagons we have every computed mode sustained by the unit cell of the waveguide under investigation. Among them, we can identify two propagating modes: mode 1 in orange pentagons and mode 2 in yellow pentagons. They can be identified till a certain value of k, after they mix with many others modes. At k = X the mode 1 shows a frequency Ω prop equal to 2.318 GHz while the mode 2 equal to 2.2 GHz. The difference between these two propagating modes with respect all the others modes is underlined in figure 4.7.b. Here, the displacement profiles at k = X for mode 1, mode 2 and the closest 6 other modes is displayed, showing that these latest are indeed non propagating through the waveguide. While the profiles of the two propagating modes show a displacement well confined inside the waveguide. They show also similar slope (in opposite direction) meaning that similar group velocity values should be expected for both. We will focus more about this aspect in the next section. 4.8 shows the results related to the 5ML design. Specifically, in figure 4.8.a it is possible to see again how two main propagating modes are now present and able to propagate through the waveguide for k=X: propagating mode 1 (orange pentagons) at Ω prop = 2.128 GHz and propagating mode 2 (yellow pentagons) at Ω prop = 2.207 GHz. Figure 4.8.b shows total displacements profile of these two propagating modes and the 6 closest ones showing how, also in this case, these are indeed non propagating through the waveguide. If on one side 5ML mode 1 and 2 show a similar displacement profile with respect to 3ML modes 1 and 2 respectively, on the other side this is not particularly true for the slope. In fact, even though the 3ML and 5ML modes 1 present somehow a similar slope just shifted in frequency, 3ML and 5ML modes 2 they present different slopes, with the latest showing an almost constant behaviour, signature of a really low group velocity. The different dispersion relations with related different slopes open the way to group velocity engineering of the propagating mechanical vibrations. This subject will be carefully discussed in the next section. For discussion, similar behaviour it has already been studied in photonic waveguides and it has been pushed even further by changing for instance the size of holes in the direction perpendicular to the waveguide [START_REF] Baba | Slow light in photonic crystals[END_REF]. In summary, it has been shown how, by starting from a perfect Kagome crystal lattice, it is possible to remove some of the introduced holes in order to create phononic crystals waveguides. Two different designs have been presented here, where 3 and 5 lines of holes have been removed respectively. In both cases it was possible to find two well confined propagating modes in the frequency range of interest (around 2.2 GHz) theoretically able to propagate through the opened waveguides.

Time domain simulations for dispersion engineering validation

A time domain FEM simulation can be used in order to validate the design discussed in the previous paragraph. In fact, with a time domain approach is possible to validate both the expected frequency band gap range and the propagating modes by observing the effects of an acoustic actuation at a defined frequency value on an accurately designed phononic crystal waveguide.

More precisely, the idea is to create a small phononic crystal waveguide model in COMSOL and looking at its behaviour over time when a prescribe displacement, simulating the acoustic actuation of an IDT, is applied to its side. By looking at the total displacement profile when different acoustic frequencies are used as input it will be possible to discriminate if both the expected forbidden frequencies and the propagating modes are consistent with the previous shown simulations, i.e. if they are actually able to guide and shield phonons in the real space as expected. Figure 4.9 describes the performed time domain simulations carried out to validate the Kagome phononic crystal lattice implementation in a 265 nm thick GaAs semiconductor membrane. Figure 4.9.a shows the specific 3D COMSOL model of the Kagome 5ML phononic crystal waveguide. To be consistent with the previously presented design, the lattice constant is here equal to 370 nm and the r a ratio equal to 0.43. Hence, we should expect a 600 MHz band gap along the Γ -X k direction centered at around 2.2 GHz for a perfect Kagome crystal lattice and two main propagating modes around the same frequency if a 5ML waveguide is openened in the lattice. The inset shows details of the waveguide. This has been chosen to have 5 unit cells per side, to properly confine the propagating modes, and it is 9 unit cells long. Moreover, in the same inset it is possible to see how the prescribe displacement area has been defined, simulating the last finger of an ideally perpendicularly positioned IDT. Two different input frequencies have been chosen, 2.7 and 2.2 GHz, in order to test two different situations and see the results when a prescribe displacement is applied along the direction of the waveguide over a 10 ns time period. This latest, it has been chosen as a reasonable time value in order to study the effect of the page 102 4.2. Guiding phonons mechanical waves propagation through the entire waveguide.

With the first frequency value, 2.7 GHz, the intent is to excite the phononic crystal lattice out of the frequency band gap. This means that in the dispersion relation several k values would theoretically allow the propagation of phonons through the entire lattice, i.e. the designed phononic crystal lattice shouldn't provide any mechanical shielding for this acoustical frequency. While with the second case, 2.2 GHz, the provided frequency value should provide a proper acoustical pumping for the 5ML propagating modes and it should be perfectly shielded by the perfect lattice all around. Figure 4.9.b. shows the total displacement profile after 10 ns that a prescribed displacement at 2.7 GHz has been applied along the perpendicular direction of the waveguide and from the figure the deformation is evident along the full area of the phononic crystal lattice. In other words, the 5ML phononic crystal waveguide is not working as expected since the lattice around is not providing the needed mechanical shielding and vibrations are not guided as desired. This is a good feedback because the system is here driven out of its designed frequency range where any acoustical guiding can be theoretically provided at the frequency of interest. Different is the case depicted in figure 4.9.c, showing the total displacement profile after a 10 ns time period when the input frequency is now at 2.2 GHz. Here vibrations are now well confined within the waveguide and they are pumping propagating modes that are now able to get the output of the waveguide in a clean way in the 10 ns time period of the simulation. At the same time we can appreciate a good phononic shielding of the perfect Kagome crystal lattice, which unit cells are just slightly deformed only at the entrance of the waveguide, i.e. the closest area to the prescribed displacement actuation input.

With the presented model we have been able in fact to demonstrate that the dispersion engineering design for phononic crystal waveguide is furthermore validated from a time-domain simulation point of view.

Corrugated phononic bridge

A second technique will be now presented to create phononic waveguide. In particular, by taking inspiration from phononic crystals described earlier, corrugated phononic waveguides can be created. Here, we will take inspiration from the 5ML design in order to create a phononic bridge directly integrated in the semiconductor membrane (see figure 4.10 for the related schematic and unit cell). Specifically, the width of the corrugated phononic bridge is given by 5 missing lines of holes as for the 5ML case while a Kagome inspired corrugated design will shape the two sides of the bridge. The related unit cell of the obtained phononic waveguide is represented on the right side of the same figure. FEM simulations have been carried out also in this case in order to investigate the presence of eigenfrequencies able to propagate through the corrugated phononic waveguide. Figure 4.11 presents the eigenfrequency results of such a structure. In particular, figure 4.11.a shows the comparison between the dispersion relation of a perfect Kagome crystal lattice (white hexagons) and every computed mode of the phononically engineered corrugated waveguide (shaded orange pentagons). In this case, only one mode is able to propagate through the waveguide and it is highlighted with filled orange pentagons in figure 4.11.a. Its total displacement profile for k = X is displayed in figure 4.11.b (orange frame) with Ω prop equal to 2.061 GHz. Specifically, it takes advantage from the full width of the waveguide, being the total displacement profile well focused in the central region of the bridge. Figure 4.11.b shows also the profiles all the other existing modes which are actually non propagating modes since their total displacements is mainly limited only in the corrugated sides. Compared to the 5ML waveguide, from which the bridge took inspiration, only mode is here propagating instead of the two that we found in the 5ML waveguide. A quick look at the total displacement profile of this propagating mode suggests us that this mode might be the "corrugated" counterpart of the 5ML mode 1. page 104 With this second technique, we demonstrated a good alternative to classic phononic crystals waveguides design presented in the previous section where holes were removed from a perfect crystal lattice. Here, with a corrugated design, we show that it is in fact possible to engineer a dispersion relation of a phononic bridge in order to guide phonons at a specific frequency. We have furthermore shown that, in this case, only one mode is able to propagate in the frequency range of interest. This feature can be useful in our case, since in the envisioned platform the implementation of an integrated acoustic delay will rely, at least as a first approach, in the optimization of a specific single propagating mode. Avoiding modes mixture can be then a really good property to take into account for a precise delay implementation.

Corrugation vs Non corrugation

The proper isolation of a unique propagating mode in the frequency range or interest it was achievable thanks to the corrugated design inspired from a Kagome crystal lattice. In order to prove this and to furthermore stress the importance of the corrugations, a new set of simulations have been carried out looking for total displacement profiles of eigenfrequencies at k = X for a non corrugated waveguide in a 265nm thick GaAs membrane having the same width of the corrugated one. From figure 4.12.b it is evident how, if no corrugation is applied, several propagating modes appear around 2.2 GHz with total displacement occurring in the full area of the waveguide, i.e new modes will be able to propagate in the same range of frequency. While figure 4.12.a recalls us that in the case of a corrugated design, only one mode is actually able to propagate for k=X. Its role is in fact to allow the isolation of a unique propagating mode in a specific frequency range. In fact, the non-existence of other propagating modes will avoid leakage and will improve the efficiency when a mechanical pumping will be applied with IDTs in order to excite at a precise frequency value the PhC mechanical modes.

Group velocity engineering

In the previous sections it has been shown firstly how a Kagome phononic crystal lattice can open a phononic band gap at 2 GHz and secondly how it is possible to modify the introduced holes pattern in order to guide phonons in the previously forbidden frequency range. This was achievable thanks to creation of phononic crystals page 106 4.3. Gv engineering waveguide or corrugated phononic waveguides. In particular, several propagating modes have been found for 3ML, 5ML and the corrugated bridge able to propagate by taking advantage of the full cross section area of the created waveguides. These modes are then good candidates to be mechanically pumped by proper designed IDTs discussed in Chapter 3 for acoustic excitation of PhC mechanical modes discussed in Chapter 2.

Let's recall that a crucial parameter that must be mastered for a proper mechanical excitation of the PhC mechanical modes is the acoustic delay. In fact, a proper acoustic delay value will allow an appropriate stabilization of the mechanical modes with a decrease of their phase noise characteristics. In order to master the group delay implementation, the group velocity must be studied for a better understanding of the presented propagating modes and their possible utilization for the envisioned platform. By calculating the derivative of the reduced wave-vector k values for the propagating mode, the group velocity can be extracted as following:

V g = ∂ω ∂k (4.1)
Being ω the angular frequency. In the following, group velocities of the designed propagating modes will be studied in order to have an idea of the modes propagating behaviour with respect to frequency. In particular, a fitting of the point-by-point mode profile describing the dispersion relation in a specific frequency range will be performed, with subsequent study of the slope's features.

Phononic crystals waveguides

For the two different designs of phononic crystals waveguide have been presented so far (3ML and 5ML design), two propagating modes exhibit interesting propagating features and we will now extract for each case the group velocity from (4.1).

Figure 4.13 display the extracted group velocities for the 3ML phononic crystals waveguide propagating modes. Particularly, figure 4.13.a shows the 3ML mode 1 group velocity values versus frequency. At k = X, i.e. at 2.318 GHz, the group velocity for this specific propagating mode is equal to 140 m/s. Figure 4.13.c shows the group velocities values for 3ML mode 2. In this case for k = X, i.e. at 2.2 GHz, the group velocity is 144 m/s at it is actually constant for all the studied frequencies values, features coming from the constant slope in the dispersion diagram. From the figure it is possbile to notice how at k = X, i.e. for frequency equal to 2.061 GHz, the group velocity value is 606 m/s for the mode under investigation. This value is close to 5ML mode 1, confirming us the hypothesis that this might be its corrugated counterpart.

Also in this case, the group velocity shows slow propagating features exploitable for possible acoustic delay implementation in a fully integrable manner for the final platform. We are in fact around five order of magnitude lower with respect to velocity of photons in optical fibers. Moreover, compared to the fact that only one mode is present in the studied frequency range might allow a more precise acoustic delay implementation based on its uniqueness.

IDTs + Phononic platform

In the previous three sections of this chapter design and simulations of phononic crystals and waveguides in the GHz have been presented and discussed. Specifically, it has been shown how it is possible to guide phonons in a specific frequency range around 2 GHz thanks to a Kagome lattice implementation. Group velocity studies of the found propagating modes have been finally presented to emphasize their slow mode features useful for a possible acoustic delay implementation.

In this section we will demonstrate fabrication of such structures together with the IDTs presented in Chapter 3 in order to experimentally prove acoustic transmission and filtering only for specific propagating modes at precise frequency values. Moreover, we will experimentally measure the group velocity for group delay extrapolation.

Fabrication of suspended phononic waveguide between two face-to-face IDTs

In this subsection the process flow to achieve phononic waveguides fabrication between two already fabricated face-to-face IDTs (which process flow has already been shown in Chapter 3) will be presented.

The steps involved for phononic waveguides fabrication are based on the same main steps as for IDTs suspension presented in Chapter 3. Let's recall in fact that the suspension of the IDTs is achieved thanks to the opening of small holes between the electrodes. This will open a path for the dry plasma underetching for the complete suspension of the transducers. If the intent is now to fabricate a phononic waveguide between the two IDTs, we can take advantage of the just discussed technique to introduce with one single fabrication step both the holes for IDTs suspension and the phononic pattern. This latter can be either a phononic crystals waveguides or a corrugated design. Both are in fact mastered at the same way.

Figure 4.16 shows the fabrication steps involved for suspension of two already fabricated IDTs and creation of a phononic waveguide between them. In figure 4. [START_REF] Yao | Optoelectronic microwave oscillator[END_REF].a the process starts with a Plasma-Enhanced Chemical Vapor Deposition (PECVD) of a SiN layer (dark pink layer) that will be used as a hard mask for the patterning of both the holes between the electrodes and the phononic features. A positive resist (PMMA) is at this point spin-coated on top of the SiN layer and after lithography exposure and development holes and phononic patterns are transferred to the PMMA layer. After that, in figure 4.16.b Reactive Ion Etching (RIE) and Inductively Coupled Plasma (ICP) techniques are used for transferring the patterns to the SiN layer before and to the semiconductor membrane after. Specifically, this last ICP step uses a gas mixture based on HBr to etch the semiconductor membrane. At the end, dry plasma based on SF 6 gas is used to underetch the SiN sacrificial layer underneath through the opened holes and phononic patterns and to get rid of the SiN hard mask as well. A fully suspended phononic waveguide between two fully suspended face-to-face IDTs is finally achieved (figure 4.16.c). 4.17 shows SEM images of the fabricated phononic devices between a pair of suspended IDTs. In particular, figure 4.17.a shows the 3ML (on the left) and the 5ML (on the right) phononic crystal waveguides based on a Kagome lattice with lattice constant a = 370 nm. While figure 4.17.b shows a fabricated corrugated phononic bridge with a design inspired also in this case by a Kagome lattice with a = 370 nm. From the SEM images it is also possible to appreciate the small holes opened between the electrodes (highlighted in yellow) used for the IDTs suspension.

Calibration for phononic crystals fabrication

The calibration of the process flow is one of the most important and non trivial step in this fabrication process. As for the IDTs fingers in Chapter 3, also for phononic crystals a proper calibration is needed in order to match the dimensions of the fabricated samples with the designed features.

Here, the dimension of phononic crystals must be perfectly mastered. In fact, a non-calibrated process flow could completely destroy the sample (a wrong holes dimensions might in fact bring the fabricated device to collapse during the last step of suspension if the holes are oversized) or making it not working at the targeted frequency. shows SEM images of fabricated phononic crystals after the full process. Three different electron beam doses have been here used for the three depicted cases. Specifically, 900, 1200 and 1500 µC/cm 2 dose values have been used for frame 1, 2 and 3 respectively. From the figure we can appreciate how the diameter of the holes increases accordingly. In this situation, on one hand, an excessively low dose will bring smaller holes than expected (frame 1) preventing to reach the right r/a ratio, important parameter for precisely opening and centering the phononic band gap as desired (as discussed previously in the simulation part of this chapter). On the other hand, an excessively high dose would enlarge too much the holes for finally breaking the connections between them (frame 3). Frame 2 shows how a right choice for the dose can eventually replicate exactly the desired design. If, on one side, the low dose case would bring a change in the expected frequency band gap but it would not destroy the sample, this is not the case for the overdosed case. In fact, this last situation would prevent a proper suspension of the phononic crystal pattern with a collapse of the semiconductor membrane during the underetching step. A tilted SEM image showing this last situation is reported in figure figure4. [START_REF] Aspelmeyer | Cavity optomechanics[END_REF].b where GaAs "nano-stars" lie on the bottom of the SiO 2 substrate as a consequence of an page 113 Chapter 4. Control of GHz acoustical waves excessively high dose value.

Transmission with and without filtering

The intent is here to prove acoustic transmission and filtering in two different situations. In a first step, we will test transmission through a 265nm thick GaAs membrane that has not been phononically engineered. We expect to recover acoustic transmission for all the three different finger widths as we did when we studied the S 21 parameter of IDTs on a 300nm thick GaP membrane in Chapter 3. In a second step we will introduce phononic features in the semiconductor membrane and we will again measure acoustic transmission with the intent to look for the designed propagating modes only in a specific frequency range.

A set of IDTs on GaAs has been fabricated as described in the previous chapter. Three different finger widths (550, 600 and 650 nm) have been chosen in order to prove transmission and filtering of acoustical waves around 2 GHz. Here the traces have been properly time-gated in order to reveal the transmission peaks that are indeed present for all the tested finger widths as expected. Specifically, for 650, 600 and 550 nm finger widths transmission peaks are present at Ω acoustic equal to 1.9, 2.06 and 2.24 GHz respectively showing that, as for GaP, also in this case creation and transmission of acoustical waves is possible thanks to the designed IDTs and that the frequency of the actuated transmission waves is shifting according to the finger width dimensions. The distance between the two transducers is here 24 µm for all the three IDTs pairs. This value has been chosen since it is 10 times the acoustic wavelength for the 600 nm finger width case (let's recall that the acoustic wavelength λ is in fact page 115 Chapter 4. Control of GHz acoustical waves λ = 4e, where e is the finger width dimension). While the phononically engineered part, i.e. the waveguide length, is 20 µm.

The time-gated transmission of the corrugated phononic waveguide at different mechanical pumping frequency is shown in figure 4.20.c. Here we can observe a transmission peak only when the provided acoustical pumping frequency Ω acoustic (2.06 GHz) is matching the frequency Ω prop of the designed propagating mode (2.061 GHz) of the waveguide. There is no evidence of transmission for the other two provided acoustical pumping frequencies at 1.91 and 2.245 GHz. This is a really interesting result showing that only one propagating mode exists in the range of frequency of the fabricated IDTs (confirming the FEM simulations) and that it can be indeed properly pumped and transmitted through the corrugated phononic waveguide. In other words this means that the phononic waveguide is acting as an acoustic filter, filtering out the acoustic transmissions of the IDTs when Ω acoustic = Ω prop and allowing the transmission only when Ω acoustic = Ω prop . This set of measurements experimentally validates the developed design for acoustic transmission and filtering of acoustical propagating waves in a specific and tunable frequency range.

Group Delay extrapolation

It is now interesting to experimentally measure the group delay of the transmitted mechanical mode at 2.06 GHz, in order to extrapolate the group velocity and compare it with simulated results.

Figure 4.21.a shows the time domain trace of the transmitted propagating mode. From the trace it is possible to recognize the EMR at t ≈ 0 ns as well as two main acoustic signals. The first one is the main acoustic signal arriving to the output IDT from the input IDT and the second one is the triple transit echo (TTE) which has traversed three times the path between input and output IDTs. This latest is the echo that contributes the most to the transmitted S 21 [START_REF] Warren P Mason | Physical Acoustics V14: Principles and Methods[END_REF] and it is in fact the only visible in the time domain trace.

Time gating technique is here used to distinguish the two signals with two time windows properly defined around them (shaded orange and violet frames in 4.21.a). By Fourier-transforming back to frequency domain the related time components the EMR is now filtered out and the two transmission peaks can be revealed (see inset 4.21.a). As it is possible to appreciate from the inset, the two signals are exactly at the same frequency, evidence that we are actually looking at the same signal at different instants of time. page 116 At this point, let's study the group delay with the intent to ultimately calculate the group velocity of the propagating mode. The time delay can be defined as:

τ g = - ∂φ rad ∂ω (4.2)
where φ rad is the phase of the S 21 transmission response. By fixing the starting point of the time-gating windows at 20 ns and 65 ns for the two acoustical signals and gradually increasing by 5 ns the width of these windows, we extract the group delay for both signals as a function of the gated time (4.21.b). For each window, the extracted time delay increases as a function of time. The reason of this is because as the window width increase, we include new time components containing delay information with respect to the window starting point. We calculate at this point the average over the extracted values. The mean τ g value for the first acoustic signal and the triple transit echo is 32.54 ns and 78.46 ns respectively. Being 24 µm the distance between the two IDTs (20 µm of waveguide plus 2 µm distance between page 117 Chapter 4. Control of GHz acoustical waves waveguide and IDT on each side) the calculated group velocity is then 737 ± 123 m/s for the first acoustic signal and 917 ± 153 m/s for the TTE that has traversed a 72µm path. Here the uncertainty value belongs to the velocity uncertainty in the 2 µm per side ( 17% of the total path) that have not been phononically engineered. These experimental results are in the same order of magnitude of the simulated value of 606 m/s for the group velocity discussed before. It follows that this combination of IDTs and phononic waveguide can be used to introduce a delay of 1.36 ± 0.23 ns/µm in a fully-integrable manner [START_REF] Modica | Slow propagation of 2 ghz acoustical waves in a suspended gaas phononic waveguide on insulator[END_REF].

Acoustic delay introduction with phononic bridge of different lengths

In the previous paragraph, transmission and filtering for a 2.06 GHz propagating acoustical mode has been demonstrated when the IDTs are spaced by 10λ acoustic = 24 µm. The experimentally measured group velocity allowed us to estimate a time delay implementation of 1.36 ± 0.23 ns/µm, i.e. 32.64 ± 5.52 ns for the considered phononic path. Face-to-face 600nm finger width IDTs with phononic waveguide of different lengths has been fabricated to investigate the losses of the propagating acoustic mode in order to estimate the maximum achievable delay introduction with the platform under investigation. Specifically, IDTs spaced by 36 µm (= 15λ acoustic ) and 48 µm (= 20λ acoustic ) have been studied and compared to the 24 µm case. If we now calculate the introduced delay for the 36 and 48 µm path, this is equal to 48.96 ± 8.28 ns for the former and 65.28 ± 11.04 ns for the latter. We can also notice that for the longest path, 48 µm, we still have a peak amplitude value of ∼ 12 dB from the noise floor (at ∼ -100 dB). If we take into account the calculated loss rate of ∼ 0.408 dB/µm, an achievable delay introduction of 100 ns would be theoretically possible with the presented system. In this case in fact a ∼ 75 µm page 119 Chapter 4. Control of GHz acoustical waves phononically engineerd path would be needed, for a final theoretically estimated 2 dB transmission signal amplitude from the noise floor.

If, for instance, we want to compare it with usual used delay values in OEOs that implement fibers delay lines for oscillations stabilization, the same amount of time delay introduced with 25 meters of optical fibers (100 ns), it can be replaced by a 75 µm long corrugated phononic waveguide, gaining at the same time in size and stability to environmental perturbations. Different possibilities can be implemented in the future in order to furthermore improve both the occupied size and the efficiency of the presented devices. About the former, the occupied size, this can be reduce by implementing 3ML and 5ML phononic crystals waveguides, whose design and fabrication have been already shown in this chapter. This can be specifically achievable thanks to the simulated lower group velocity of the sustained propagating modes with respect to the corrugated bridge. In particular, 3ML mode 2 showed a simulated V g ≈ 90m/s at 2.207 GHz, i.e., if measurement will follow the same trend, the same delay introduced by the corrugated bridge (V g ≈ 606 m/s) can be implemented in a ∼ 6 times smaller space. About the latter, the efficiency optimization of the presented system, this can be increased by playing on both the shape of the transducers and the clamping part of the corrugated bridge on both sides. More specifically, focused IDTs can be used to create high focused mechanical waves. These can be then guided to the corrugated bridge through a mechanically engineered clamping part for the input/output of the waveguide to reduce mechanical scattering. A proposed schematic for this optimized next generation of the device is depicted in figure 4.23.

Conclusion of the chapter

To sum up, in this chapter we have shown how is possible to acoustically engineer the used semiconductor membrane in order to master the mechanical waves behaviour. Properly phononic crystal lattices can in fact be used to introduce a phononic bandgap with specific center frequency and aperture. From this, one can create phononic waveguides by breaking the introduced defects periodicity. By playing on the design, propagating mechanical modes can be indeed found and, with an accurate dispersion engineering, their velocity can be tuned in frequency as desired. The experimental results presented at the end confirmed us that the designed platform made by IDTs and phononic waveguides can actually achieve interesting results really close to simulations. This can open the way to a new method for delay introduction in micro and nano-systems.

In the next conclusive chapter, we will bring all the integrated systems presented so far together, showing some preliminary results and new ideas in order to get closer to the aimed fully-integrated optomechanical oscillator discussed at the beginning of this thesis.

Chapter 5

On-chip time-delayed OMO with integrated acoustic feedback control and perspectives where the needed time delay for oscillations stabilization is integrated on the same platform. By taking advantage of both IDT and phononic waveguide the needed delay can in fact be implemented in a µm size area exploiting the acoustic features of the same semiconductor membrane where the OMO made by a 1D PhC nanobeam lies (see figure 5.1 for a schematic). The non trivial goal in this proposal is to match the 3 mechanical frequencies that come into play: Ω acoustic , Ω m and Ω prop . Specifically, the intent is to design a transducer able to generate mechanical propagating wave at Ω acoustic at the same frequency Ω m of the localized GHz mode of the 1D PhC cavity. Thus, this wave has to be guided from the transducer towards the suspended nanobeam with the help of a properly designed phononic waveguide sustaining a mechanical propagating mode at Ω prop = Ω acoustic = Ω m . Moreover, by engineering the dispersion of such a waveguide, specific group velocity of the propagating mode can be addressed. This will allow to precisely introduce the needed time delay value into the system by playing for instance on the waveguide length. The phase noise performances of the OMO will benefit from this scheme to finally trying to approach the same OEO performance but in a smaller integrated area (see yellow circle in inset of figure 5.1). In the previous three chapters we have carefully introduced the 3 building blocks constituting the envisioned device: the 1D PhC nanobeam, the integrated IDT and the phononic waveguide.

In Chapter 2, we have discussed the OMO part made by a 1D PhC nanobeam directly integrated onto a standard photonic circuitry. The proposed system was able to sustain mechanical modes directly in the UHF regime for both materials that we have presented, GaAs and GaP. For the former, we measured promising mechanical modes around 2 GHz but eventually they were not able to self-oscillate. For the latter case, i.e. GaP, it was finally able to generate a coherent oscillating signal at a mechanical frequency Ω m of 3.25 GHz. This specifically happens when the optical cavity, working at telecom wavelength (≈ 1550nm) with a measured quality factor around 10 5 , is fuelled with photons at the resonance optical frequency. This OMO showed already a promising phase noise of -84 dBc/Hz at 100 kHz offset from the carrier without feedback loop implementation.

In Chapter 3, IDTs directly integrated onto the semiconductor membrane have been presented. There were two main achievements presented here. A first one is related to the optimization of the used process flow in order to fully suspend the transducers for a better transduction efficiency and for a confinement of the mechanical propagating mode solely in the III-V membrane. The second one is the experimentally tested acoustic propagating modes at Ω acoustic in the same range of frequency Ω m of the 1D PhCs mechanical modes, opening the way for a possible mechanical pumping of these latest. Specifically, it has been shown how, by carefully changing the transducer design, it is indeed possible to tune the actuation frequency page 124 of the created propagating modes for precisely addressing a desired target frequency. Transmission measurements showed how, by properly time-gating the S 21 traces, propagating acoustical signals at around 3 GHz for GaP and 2 GHz for GaAs have been eventually experimentally demonstrated.

In Chapter 4, we have demonstrated acoustical control of mechanical waves in the GHz. Phononic crystals have been introduced how a tangible way to master the acoustic behaviour of the used semiconductor membrane. More precisely, by introducing specific periodic pattern, a phononic band-gap can be eventually opened in the material for all the possible k-directions. By introducing defects in the perfect introduced patterns, phononic waveguides can be lately created with the final intent to guide mechanical propagating modes at specific frequencies values Ω prop = Ω acoustic = Ω m . Dispersion engineering of the found propagating modes showed how it is indeed possible to address specific group velocity values as well. Fabrication of phononic structures integrated together with the previously introduced IDTs has been presented to show that also in this case a well optimized process flow was found for the fabrication in the cleanroom facilities. Finally, we have experimentally demonstrated propagation and filtering of a designed slow acoustical mode in the UHF regime (2.06 GHz) with a measured group velocity below 1000 m/s through a suspended GaAs phononic waveguide. With this, we have thus demonstrated that a possible fully integrated on-chip delay implementation in the order of few ns/µm is at reach. Specifically, experiments on different phononic waveguide's lengths have shown that it will be eventually possible to replace for instance 25 m of optical fibers with 75 µm of corrugated phononic waveguide.

In the following, we will bring all the previously presented building blocks together in a first prototype of a complete system. Fabrication of a 1D PhC nanobeam integrated together with an IDT and a phononic waveguide on a commercially available photonic circuitry will be presented. Measurements to singularly characterize each component will be discussed with preliminary results on the interaction between the different involved domains (electrical, acoustical and optical). In the meantime, a simpler architecture has been developed to implement electro-optomechanical interaction but this time without integrated acoustic delay. The design and the preliminary results of such architecture will be introduced evidencing electro-static actuation of 1D PhC optomechanical oscillators. One important challenge of on-chip coherent RF signals is their efficient and easy distribution. Thus, optical packaging of devices with waveguides arrays to fiber transposer is a recently developed solution.

In collaboration with STMicroelectronics, on-chip optomechanical crystals have been connected to fibers arrays demonstrating a packaged unit. This implementation of a standard telecom system opens the way towards interconnected multi-systems networks. We will finally conclude with future perspective towards new physics and page 125 Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives applications with the platforms introduced in this thesis.

Fabrication of an on-chip OMO with integrated phononic waveguide and IDT

The process flow to fabricate the envisioned complete device gathers all the fabrication techniques discussed so far together. The gained experience during the development of each single sub-system allows us to develop a complete fabrication process for finally achieve a fully suspended structure. We are not going through the entire process flow to avoid repetitions of the same steps already discussed in the previous chapters (adhesive bonding, electron beam lithography, metal deposition with subsequent lift off and etching techniques). By taking advantage of these fabrication steps, a complete suspended structure integrating at the same time one 1D PhC nanobeam, an IDT and a phononic waveguide has been fabricated (5.2.b for optical microscope image of a single integrated device).

Independent characterization of building blocks

In this section we will characterize singularly each component of the system to lately show a first visible interaction between the different physical involved domains in the next section. For this first generation we have integrated 6 different structures on a same SOI waveguide. The IDT and phononic parts are kept the same for each waveguide, while the 1D PhCs nanobeams slightly change their geometry in order to ensure the correct wavelength range at least with one of them, balancing possible imperfections coming for instance from fabrication (similarly technique was used in Chapter 2).

Optical measurements of 1D PhC nanobeam

Optical measurements have been carried out in order to optically characterize the 1D PhCs nanobeams. Similarly to what we did in Chapter 2, also in this case a fiber-to-fiber transmission configuration allows us to extract the information related to the different photonic crystal cavities (the setup scheme is shown in the top-left inset of figure 5.4.c). The main intent is here to show that the nanobeams integrated in the complete structure are indeed still working from an optical point of view as the single counterparts presented in Chapter 2. The same Exalos broadband SLED used in Chapter 2 is here adopted to inject light into the system. The light has been properly polarized before being coupled to the SOI waveguide and the output signal is finally sent to the OSA from which we can retrieve the optical transmitted spectrum. integrated (only one visible here). Figure 5.4.b shows a view from an infrared camera when the broad band light is injected into the system. All the 6 structures are here visible and they can be easily recognized from the blinking associated nanobeams (white spots in the figure). In figure 5.4.c two different optical spectra are presented. The red trace is a spectrum recorded before the structures have been suspended, i.e. before removing the SiN sacrificial layer with a dry plasma etching technique. Several optical cavities are here visible and the transmission amplituded is similar to the results shown in Chapter 2. While the blue trace is a transmission spectrum recorded after suspension of the structures. Several optical resonances are here recognizable although the total spectrum shape is less clean with respect to before. This comes page 128 5.2. Characterization most probably from fabrication issues related to the underetching technique. This is the first prototype where the phononic structures have been fabricated on top of photonic circuitry. This means that for the first time we have integrated the structures that need more time to be suspended, i.e. the phononic waveguides, with the structures that are the most sensible to underetching techniques and so the easiest to be degraded, i.e. the SOI waveguides. A better trade-off between the wanted suspended area and the minimum possible degradation must be found for the next generation. Even though the degradation affected the optical Q factor of the PhCs cavities, this is still high enough to let us observing mechanical modes. In the following section we will show the mechanical behaviour especially related to a cavity located at 1558.5 nm (shown in the inset of figure 5.4.c) which measured quality factor was 9x10 3 .

Mechanical spectrum in the GHz of 1D PhC nanobeam

In the previous section we have optically characterized the 1D PhC nanobeams integrated in the complete structure. Let's try at this point to characterize them mechanically to check if we are able to recover the same mechanical spectrum as for single integrated nanobeams presented in Chapter 2. We will now use a tunable laser (EXFO T100S-HP) to inject light into the system at one cavity resonant wavelength and the signal will be sent first to a photodiode and later to an ESA from which we recover the mechanical spectrum. The used setup is shown in the inset of figure 5.5.b.

The cavity that we will use in the following has a mode located at around 1558.5 nm. In order to identify to which 1D PhC nanobeam it belongs, we looked at the 6 structures integrated onto the same SOI waveguide with and Infrared Camera. When we inject light at the cavity resonant wavelength we should observe more light coming from the cavity. Figure 5.5.a and 5.5.b show the field of view from the Infrared Camera in two different situations. In the former, the tunable laser is set at 1558 nm and in this case all the 1D PhC nanobeams look the same from the Infrared Camera. For the latter, the tunable laser is now set at 1558.74 nm, i.e. at cavity resonance. In this case, more light is seen from the nanobeam number 3 than from the others (see red frame figure 5.4.b) suggesting that we are most probably confining light in its photonic crystal cavity.

The goal is now to see if the photon-phonon interplay inside the cavity will allow us to read the mechanical motion of the nanobeam when the tunable laser is set at cavity resonance. Figure 5.5.c shows the recovered mechanical spectrum in this situation. From the figure it is clear how several mechanical modes are here visible. In a low frequency range, around 0-1 GHz, several delocalized modes are and perspectives present. In a higher frequency range, two main peaks are visible at around 3.5 and 4.5 GHz respectively signatures that mechanical modes do exist in the UHF regime. The former, giving its quite large bandwidth and irregular shape, might contains actually several modes with low Q m . In this case, we were not able to enter the selfsustained oscillating regime as the single integrated counterpart in GaP presented in Chapter 2. This comes most probably to the slightly lower quality factor, due to some waveguide degradation during the last fabrication step of suspension. Higher optical power could have been used in order to reach self sustained oscillation at the risk of degrading the whole structure.

Acoustic resonances of IDTs

In order to perform a mechanical pumping of the GHz mode around 3 GHz belonging to the 1D PhC nanobeams, IDTs with different finger widths have been implemented on this first prototype. Specifically, IDTs with finger width of 500, 513, 525, 538 page 130 5.3. First implementation of OMO with integrated acoustic delay and 550 nm have been properly designed.

For this first try, we didn't suspend the IDTs to simplify the process flow, avoiding the opening of small holes between the electrodes fingers for underetching. In fact, if we would have done this, it would have degraded the SOI waveguides even more and no optical transmission would have been seen. The setup used for IDTs characterization was the same as in Chapter 3, with an electrical probe injection from the VNA in order to recover the S-parameters. Unfortunately, despite the indication in Chapter 3 of acoustic resonances in the GHz even for a non suspended IDT, this was not the case for this first prototype. Specifically, no resonance was found in the GHz. The only acoustic signatures appearing in this case were in the MHz. Figure 5.6 shows the reflection for all the tested finger widths in a frequency range going from 100 to 400 MHz. In the figure, we can see how the resonances are not shifting with the finger width dimensions. This might be again a signature of a bulky mode, like discussed in Chapter 3. Since this is the only acoustic signatures that we were able to extract from this first generation, in the following we will try to use them to show a first interaction between the transducers and the 1D PhCs nanobeams.

First implementation of OMO with integrated acoustic delay

Two set of experiments have been carried out in order to prove a first interaction between the different systems integrated together in this first prototype, and so page 131 Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives between the different involved physical domains.

Acousto-optical transduction with a 2-ports VNA

In a first experiment, we use a tunable laser to inject light into the system at a precise wavelength while at the same time electrical injection is performed to the IDT from VNA's port 1. The signal coming out from the SOI grating is collected by an optical fiber, sent to a photodiode and finally to VNA's port 2 to recover the transmission spectrum. The schematic of the setup is depicted in figure 5 Here, the acoustic signatures that we discussed in the previous paragraph appear again together with two transmission peaks at the same frequency in the tramission spectrum. This might be a first evidence of acousto/optical modulation from the IDT to the PhC cavity read as modulation of the transmitted optical signal. The fact that the transmission peaks are not present in the transmission spectrum of figure 5.8.a in fact is a proof that those peaks are not natural eigenfrequencies of the nanobeam, and so no acousto/mechanical interaction between the mechanical waves created by the IDTs and mechanical modes of the 1D PhC nanobeam is here achieved. This, it is most probably due to the fact that we are here in a low frequency regime, and we need most probably to be able to excite mechanical modes in the GHz to see a complete acoustical/mechanical/optical interaction between the mechanical waves coming from the IDT and the localized mode of a 1D PhC nanobeam readable as modulation of the optical signal. Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives the laser is now out of cavity resonance. Here, the transmitted peaks disappeared, confirming us that they were actually signature of the acoustic resonances imprinted to the optical signal. Once photons are no more localized in the cavity in fact, the peaks disappeared and no more modulations are visible in the optical collected signal.

Acousto-optical transduction with a RF signal generator

In a second experiment, we substitute the VNA with a RF signal generator (for electrical injection) and an ESA (for collecting the transmitted mechanical spectrum from the photodiode). This has been done in order to see Brownian motion and see the impact of acoustic on mechanics. A schematic of the used setup is shown in figure 5.9.a. The main difference is that now we fix not only the optical signal but also the RF page 134 5.3. First implementation of OMO with integrated acoustic delay injection to one specific input value. Figure 5.9.b shows the collected transmitted spectrum in three different situations. Specifically, the RF probe is always touching the IDT in all the three presented situations and its RF frequency is always kept equal to 195 MHz (frequency at which we noticed a mechanical mode of the nanobeam). While the tunable laser is changing its wavelength for each case. In the top part (black trace), the tunable laser is out of cavity resonance and in this case we do not see any signal in the transmission spectrum. In the middle part (brown trace), the tunable laser is now close to resonance and a mechanical mode of the nanobeam is here visible at 195 MHz together with the acoustic tone coming from the IDT. In the bottom part (orange trace), the laser is at cavity resonance and the mechanical mode is now well visible together with the acoustic tone. Although there is no visible interaction between the mechanical mode of the nanobeam with the acoustic tone coming from the IDT, the fact that this latest is visible as a modulation of the optical signal only when we are at cavity resonance is another proof that an acousto/optical modulation is indeed experimentally tested in our system. Unfortunately, also in this case no direct access to the mechanics of the sustained mechanical modes was achieved.

Conclusion of first preliminary results of OMO with integrated acoustic delay

To sum up, with this set of experiments, we have shown that an acousto → optical interaction is visible in this first generation of complete device, with a RF modulation on the optical carrier only occurring when the laser signal is set at the cavity resonant wavelength. Unfortunately, the complete acoustical → mechanical → optical interaction needed for acoustic delay implementation was not visible in this first generation of devices, most probably because of the low frequency regime of the acoustic signatures. At this frequency we may imagine that we are modulating the distance waveguide-cavity and thus the optical coupling. In a next generation, with a fully suspension of the IDT, the acoustic resonance in the GHz should come back opening for a possible interaction between the mechanical waves properly guided by the phononic waveguide and the localized modes of the 1D PhC nanobeam. This will allow to study several feedback loop schemes in order to correctly implement the acoustic delay for improving the phase noise performance of such integrated OMO. In this next generation, the fully suspension of the IDTs can be achieved for instance by implementing a multi-steps underetching process flow. With this, we can in fact protect by lithography techniques the most sensible structures (i.e. the SOI waveguides) during the first steps of suspension.

In the next section, we will show how addressing the mechanical modes of a page 135

Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives 1D PhC nanobeam integrated on a SOI waveguide it is eventually possible with integrated electrodes by playing with electrostatic actuation. This simpler configuration, where no acoustic delay implementation is possible, allows us to complete the demonstration of interaction in the involved physical domains in our platform, underline once again how a complete OMO with integrated acoustic delay is indeed possible to achieve once a new generation of prototypes with suspended IDTs will be ready to test.

Electrostatic actuation of OMO without delay

Electrostatic actuation is another possibility to perform a resonant excitation of the 1D PhC nanobeam mechanical modes. For this technique, a pair of Gold electrodes are deposited on the side of a suspended photonic crystal cavity (see figure 5.10.a for a 3D schematic of the system). These electrodes will act as a capacitor when an electric potential is applied to them. The created inhomogeneous electric field distribution due to the AC potential will result in an electrostatic force acting to the nanobeam (see 5.10.b for field distribution depicted with dashed black lines). The system thus behaves as a capacitor embedding a semiconductor suspended nanobeam.

If an AC voltage signal is applied to the system, the nanobeam made by a polarizable material (GaP in this case) will experience an electrostatic force and the cavity region will be then deformed (see 5.10.c). By playing with the involved parameters (mainly the electrodes area A and the electrodes distance d) the coupling between the field distribution and the nanobeam can be changed and so the resulting electrostatic force with related nanobeam deformation ( [START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF], [START_REF] Chowdhury | Phase stochastic resonance in a forced nanoelectromechanical membrane[END_REF], [START_REF] Quirin P Unterreithmeier | Universal transduction scheme for nanomechanical systems based on dielectric forces[END_REF]). Specifically, A is given by the optical extension of the mechanical mode. While d should be the smallest but the presence of metal too close to the cavity degrades its optical quality factor. A trade-off must be found and in the presented case d is equal to 1.5 µm.

We have fabricated several 1D PhC nanobeams with integrated electrodes for capacitive actuation on the same SOI circuitry that we have used to integrate the complete OMO structures with acoustic delay. Figure 5.11.a shows the optical microscope image already shown at the beginning of this chapter, where actually now we can also evidence the presence of the capacitive structures. The same setup shown in figure 5.9.a has been used to test the mechanical behaviour of a 1D PhC nanobeam when an electrical input signal is sent to the pair of electrode when the laser signal is at cavity resonance. This is a demonstration of a electric/capacitive → mechanical → optical interaction, with the integrated electrodes able to directly address a mechanical mode sustained by the suspended 1D PhC nanobeam for a frequency locking region of 2 MHz. Another set of measurement has been carried spanning this time the power of the RF driving signal. Results of this are shown in figure 5.12.b. Here, we can appreciate how the locking region increases from around 0.5 MHz to around 4 MHz when the injected power changes from 5 dBm to 17.5 dBm respectively. This shows how the locking range is actually tunable with respect to the injected power but a trade-off between the locking width and the injected power should be taken into account to avoid to damage the structure under investigation.

To sum up, with this configuration no acoustic delay can be implemented, but it is indeed an important step in the framework of this thesis in order to prove that we can actually access the mechanics of an OMO with integrated actuation techniques. It could open new architecture for delayed feedback of OMO. page 138 5.5. Packaging

Towards interconnected multi-systems network

With the words "Internet of Things" (IoT) nowadays we describe all the physical objects (the "things") that contain sensors, software and other technologies with the final intent to exchange data over the internet thanks to specific protocols [START_REF] Keyur | Internet of things-iot: definition, characteristics, architecture, enabling technologies, application & future challenges[END_REF].

In [START_REF] Wortmann | Internet of things[END_REF] it has been reported a recent definition for IoT given by the International Telecommunication Union (ITU): IoT is "a global infrastructure for the Information Society, enabling advanced services by interconnecting (physical and virtual) things based on, existing and evolving, interoperable information and communication technologies". What is even more incredible is that nowadays this world wide interconnection of "things" addresses all the the aspects of our life, where everything is basically connected to a global network.

What made this possible was also a huge improvement in the development of Information and Communications Technologies (ICTs) in the last few decades, where integrated micro and nano systems are crucial components. In modern ICTs in fact, mechanical, electrical and optical elements are integrated on the same chip allowing unthinkable performances with respect to the recent past. What makes the communication possible between the integrated systems and the outside world (where this latest can be another integrated system on the same chip or somewhere else in the connected network) is the packaging. This latest in fact is performed in order to protect and possibly power the integrated elements as well as provide the needed mechanical/electrical/optical interconnections between the different components.

In the framework of this thesis, an interesting collaboration with STMicroelectronics has been carried out in order to prove that the presented structures can be indeed packaged with standard Integrated Circuit (IC) packaging techniques. This will allow us to get a step closer towards the demonstration of fully integrability of our developed systems in commercially available ICTs.

In the following, it will be shown how an optical packaging of an SOI chip with several integrated 1D PhC nanobeams has been successfully implemented. In particular, optical characterization of the integrated systems will be discussed, with some preliminary demonstration of interconnected OMOs with chaotic-like behaviour.

Optical packaging and measurements

An optical packaging has been performed in a micro-chip integrating several 1D PhC nanobeams fabricated on top of SOI waveguides. The used material is here InP. Its fabrication is in fact much easier to master and here the final intent is not to achieve high quality factor cavities and/or perform a piezoelectric actuation but to achieve indeed a first demonstration of properly working 1D PhCs nanobeams page 139 Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives in a packaged unit. Figure 5.13.a shows a picture of the chip where the structures have been already fabricated (shaded white area). Figure 5.13.b shows a schematic representing how the structures have been integrated (4 nanobeams per waveguide on 29 waveguides). Here, the dashed yellow frames in figure 5. [START_REF] Eliyahu | Improving short and long term frequency stability of the opto-electronic oscillator[END_REF].a indicate the area where the grating couplers lie in the chip, and so the area where two fibre blocks (one for each side of waveguide, i.e. for input and output optical injection) will be installed. Figure 5.13.c shows the same chip after that the optical packaging has been performed with the installation of two fibre blocks by STMicrolectronics engineers. In order to optically characterize the integrated 1D PhC nanobeams and see how both the fabrication steps and the optical packaging affect the photonic crystal cavities, we performed transmission measurements after removal of the SiN sacrificial layer (i.e. after suspension) and after the optical packaging performed by STMicroelectronics. shows the measurements results after that the optical packaging has been performed. The used setup to recover in this case the transmitted optical spectrum , from the same waveguide measured before, is shown on the left side of the picture. This time, the coupling between the "outside world" and the packaged structures is performed thanks to optical fibers directly linked to the APC connectors of the fibre blocks. The right side of the figure shows the measured optical spectrum (orange trance) showing that the resonances are still there, they didn't shift in frequency and we actually gained in term of transmitted power after that the packaging was performed. The highlighted shaded yellow part in the figure is a region where two cavities are present at really close resonant frequencies. Interesting features that can come from this will be discussed in the next subsection.

Two useful feedbacks come from this demonstration. Firstly, we can see that the packaging didn't damage the integrated suspended 1D PhC nanobeams. Optical page 141 Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives resonances are in fact still present and they are exactly at the same wavelengths as before. Secondly, the fact that we can now access the desired waveguide through external connectors already perfectly aligned on the gratings couplers, avoiding a semi-manual alignment for each waveguide as before, it helped us to gain in the transmitted power level (and in the no more needed alignment time as well).

Mechanical measurements and chaotic behaviour

In order to mechanically characterize the 1D PhCs nanobeams of this packaged sample, measurements have been performed where the collected optical output have been sent to a photodiode and then to the ESA in order to recover the mechanical spectrum. In this sample, unfortunately no mechanical modes were found in the GHz range. This is most probably due to the high loss in the used material (InP). While in a lower frequency range, we were able to easily enter the self-sustained oscillations regime. Figure 5.15 shows the recovered mechanical spectrum where the laser input wavelength is set at one cavity resonance (λ = 1545.5 nm in this case) with an input power of 10.5 dBm. Here, in a frequency range going from 0 to 30 MHz, we can identify a self-oscillating mode at ∼ 4 MHz and several equally spaced replica of it. From the figure we can understand also that the frequency is not stable, a wide linewidth characterizes in fact the present mechanical modes. Here, the mechanical modes show chaotic signatures. Their behaviour in fact bring to bifurcations in this specific wavelength range. This is most probably a proof of interaction between the two different involved cavities belonging to two different nanobeams. For a better understanding, figure 5.17 shows a schematic explaining the undergoing phenomenon.

Specifically, in the figure is reported the case of two 1D PhCs nanobeams, integrated on the same waveguide, with cavity resonances existing at two really close wavelengths λ 1 ≈ λ 2 (3D schematic on top right of the figure). More in detail, the figure shows the behaviour when they are optically pumped with an increasing injected power P . At a low power value P a the cavities behave normally and they don't show signs of interaction. If now we slightly increase the power, P b (> P a ) situation in the figure, more light gets coupled to the cavities bringing a temperature increase of the system that will induce a change in the refractive index for a final resonance redshift of the two cavities. If we keep going increasing the injected power this phenomenon is more and more evident and at some point the resonances region may overlap between each other (see situations at power levels P c and P d in the figure). Here, we can see how the overlapping regions is larger for a higher injected power (∆λ d > ∆λ c for P d > P c ). In other words, the two different cavities, with optical resonances λ 1 ≈ λ 2 , belonging to two different nanobeams fabricated on the same packaged waveguide are now optically coupled. In fact, in the wavelength range where this happens, the injected light is confined at the same time in both cavities bringing the chaotic behaviour that we observed in the measurements.

In the future, further studies will be carried out in order to have a better understanding of the undergoing phenomenon and to characterize the chaos of the system. For instance, numerical calculation of the Lyapunov exponent will permit to furthermore characterize and validate the chaotic signature that we experimentally observed in the bifurcation-like behaviour of the measured spectrogram. and perspectives

Future perspectives and applications

In this section we will give future perspectives and possible applications where the platform developed during this thesis could be exploited.

Synchronization of several OMOs

We ended the previous section with the demonstration of two optically coupled 1D PhCs nanobeams fabricated on the same packaged waveguide. In particular, this is possible when two optical resonances, belonging one each nanobeam, exist at two close wavelengths λ 1 ≈ λ 2 . In this case, the coupling between the two cavities can be tuned just by playing on the input laser power and wavelength. Another degree of freedom can be also added in order to tune the coupling in a more complete way. This is the phase control of the light coupled inside the cavities and propagating through the waveguide. In the previsously presented example this was not possible because of the fixed physical distance of the nanobeams integrated on the same waveguide. But, as mentioned before, in our device 29 waveguides have been packaged with four fabricated 1D PhCs nanobeams each. Thus, we can think to couple two different nanobeams belonging to two different waveguides, as long as the condition λ 1 ≈ λ 2 (belonging one each nanobeam) is true. A 3D schematic of two optically coupled suspended 1D PhCs nanobeams integrated on two different waveguides is represented in figure 5.18.a. Here, the laser input and output come from the two different waveguides. In this configuration, a phase control of the optical path linking the two waveguides can be achieved thanks to specifically tunable optical fibers linking the two waveguides under investigation. This can finally enhance the optical coupling between the two different optomechanical oscillators and a synchronization between them when they are in their respective chaotic regime can be eventually expected. Here, the easy access to all the packaged waveguides thanks to the fibre block connectors will allow to achieve synchronization of optomechanical crystals integrated on the same chip in a really compact and efficient way. Furthermore, the number of synchronized optomechanical crystals is not limited to two with this architecture. In fact, n different optomechanical crystals could be synchronized as long as the condition λ 1 ≈ ...λ i ≈ ...λ n will remain always valid.

Neuromorphic computing and coherent Ising model

The world of information and telecommunication technologies shows a constant ongoing improvement. More and more new technologies are developed in order to push the efficiency of the way of treating information to its limit. Recently, several approaches inspired by the brain's ability to process information [START_REF] Rabinovich | Transient dynamics for neural processing[END_REF] have been introduced such as 'Reservoir Computing' [115,[START_REF] Jaeger | The "echo state" approach to analysing and training recurrent neural networks-with an erratum note[END_REF][START_REF] Verstraeten | An experimental unification of reservoir computing methods[END_REF][START_REF] Dean | State-dependent computations: spatiotemporal processing in cortical networks[END_REF][START_REF] Maass | Computational aspects of feedback in neural circuits[END_REF] or 'Coherent Ising Machine' [START_REF] Yamamoto | Coherent ising machines-quantum optics and neural network perspectives[END_REF]. The main idea behind this philosophy is to try to mimic neural network [START_REF] Verstraeten | An experimental unification of reservoir computing methods[END_REF] in order to process information in a more efficient way by using complex network of dynamical systems where a non linear transformation of the input signal is normally performed (see figure 5.19.a for a input-output transformation example of coherent Ising machine implementation with optical parametric oscillators (OPOs) [START_REF] Yamamoto | Coherent ising machines-optical neural networks operating at the quantum limit[END_REF]). Several results have been already obtained for processing empirical data (chaotic time series prediction [START_REF] Miguel | Optoelectronic reservoir computing: tackling noise-induced performance degradation[END_REF], speech recognition [START_REF] Larger | High-speed photonic reservoir computing using a time-delay-based architecture: Million words per second classification[END_REF] . . . ).

Innovative nanoscale systems such as our integrated nano-optomechanical oscillators might be also a possible solution in order to map primitive function for neuromorphic computing. This can be done for instance by achieving an on-chip network of interconnected and coupled optomechanical oscillators in order to demonstrate the computing of elementary functions based on collective-states of the system where a non linear input-output relation is required (see 5.19.b for a schematic).

Ultrasensitive detection

All types of sensors have a threshold, defined by the smallest signal amplitude or phase that they can detect. Choosing an optimal noise level that maximizes mutual information between sensor input and output, however, requires calibration. A highly sensitive detection scheme based on phase transitions could be demonstrated with our optomechanical crystals integrated on the same chip. Specifically, by playing on the coupling between them, two distinct states can be achieved: [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF] unsynchronized and (2) synchronized regime and/or oscillating/chaotic regime. The injected signal can be used to perform a proper calibration of the system and, once page 147 Chapter 5. On-chip time-delayed OMO with integrated acoustic feedback control and perspectives Figure 5.19 -a) Principle and sequence of operations of coherent Ising machines with a non linear realtion between input and output signal. Figure from [START_REF] Yamamoto | Coherent ising machines-optical neural networks operating at the quantum limit[END_REF]. b) Scheme for a possible complex network implementation with a non linear input-output with our integrated optomechanical oscillators. the sensitivity is calibrated, the device might be used to detect an arbitrary external trigger. High sensitivity can be expected from this type of detection scheme (see figure 5.20 for a schematic).

4Contents 4 . 5

 45 Control of GHz acoustical waves 4.1 Phononic crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 PhN crystals lattices design . . . . . . . . . . . . . . . . . . . 4.1.2 Mastering the tunability . . . . . . . . . . . . . . . . . . . . . 4.2 Guiding phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Phononic crystals waveguide . . . . . . . . . . . . . . . . . . . 4.2.2 Corrugated phononic bridge . . . . . . . . . . . . . . . . . . . 4.3 Gv engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Phononic crystals waveguides . . . . . . . . . . . . . . . . . . 4.3.2 Corrugated phononic bridge . . . . . . . . . . . . . . . . . . . 4.4 IDTs + Phononic platform . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Fabrication of suspended phononic waveguide between two face-to-face IDTs . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Calibration for phononic crystals fabrication . . . . . . . . . . 4.4.3 Transmission with and without filtering . . . . . . . . . . . . . 4.4.4 Group Delay extrapolation . . . . . . . . . . . . . . . . . . . . 4.4.5 Acoustic delay introduction with phononic bridge of different lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page xviii Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . 5 On-chip time-delayed OMO with integrated acoustic feedback control and perspectives 5.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Optical measurements of 1D PhC nanobeam . . . . . . . . . . 5.2.2 Mechanical spectrum in the GHz of 1D PhC nanobeam . . . . 5.2.3 Acoustic resonances of IDTs . . . . . . . . . . . . . . . . . . . 5.3 First implementation of OMO with integrated acoustic delay . . . . . 5.3.1 Acousto-optical transduction with a 2-ports VNA . . . . . . . 5.3.2 Acousto-optical transduction with a RF signal generator . . . 5.3.3 Conclusion of first preliminary results of OMO with integrated acoustic delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Electrostatic actuation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Optical packaging and measurements . . . . . . . . . . . . . . 5.5.2 Mechanical measurements and chaotic behaviour . . . . . . . . 5.6 Future perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Synchronization of several OMOs . . . . . . . . . . . . . . . . 5.6.2 Neuromorphic computing and coherent Ising model . . . . . . 5.6.3 Ultrasensitive detection . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 -

 11 Figure 1.1 -Comparison of phase noise performances with respect to the package size for state of the art microwave oscillators. All the carriers have been scaled to 10 GHz for direct comparison. Figure from [7].

  Figure 1.2.b shows a spectrum example of OEO.

Figure 1 . 2 -

 12 Figure 1.2 -Fundamentals of an OEO. a) General schematic. b) Spectrum example.Figure from[START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF].

Figure 1 . 3 -

 13 Figure 1.3 -Summary scheme of the system to be developed.

Figure 1 . 5 -

 15 Figure 1.5 -Examples of PhCs resonators. a) Silicon-based 2D PhC showing high optical quality factor (Q = 1.2 • 10 6) and mechanical modes at 151 MHz[START_REF] Amir | Optomechanics in an ultrahigh-q two-dimensional photonic crystal cavity[END_REF]. b) 2D optomechanical resonator in InP sustaining both MHz and GHz modes[START_REF] Gavartin | Optomechanical coupling in a two-dimensional photonic crystal defect cavity[END_REF]. c) Same system integrated on SOI waveguides[START_REF] Viktor Tsvirkun | Integrated iii-v photonic crystal-si waveguide platform with tailored optomechanical coupling[END_REF]. d) Silicon-based 1D PhC with optical Q=38000 and sustained mechanical mode at 2 GHz[START_REF] Eichenfield | Optomechanical crystals[END_REF]. e) AlN-based 1D PhC with optical Q up to 120000 and sustained mechanical mode at 3.18 GHz[START_REF] Fan | Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors[END_REF].

Figure 1 . 6 -

 16 Figure 1.6 -Examples of Optomechanical oscillators. a) First demonstration of an OMO in a silica micro-toroid optomechanical resonator with working frequency equal to 54.7MHz and phase noise around -90 [dBc/Hz]. [23],[33]. b) 1D optomechanical silicon resonator with a mechanical mode at 3.895 GHz and with -75 [dBc/Hz] at 10 kHz from the carrier [37]. c) 1D optomechanical crystal OMO in InGaP able to self-oscillate at 2.924 GHz and a phase noise at -70 [dBc/Hz] at 10 kHz from the carrier [38]. d) State of the art OMOs performances.

12 1. 2 .

 122 superharmonic resonances were shown, always in the MHz, this time in a two-dimensional photonic crystal InP page Our system to be developed membrane with interdigitated electrodes integrated underneath (figure 1.7.b).

Figure 1 . 7 -

 17 Figure 1.7 -Examples of electrostatic actuation. a) 1D SiN suspended resonator with resonaces frequencies between 5 and 9 MHz excited by a pair of integrated electrodes [39]. b) Two-dimensional photonic crystal InP membrane with interdigitated electrodes integrated underneath with excited superharmonic resonances in the MHz [40].

Figure 1 . 8 -

 18 Figure 1.8 -Example of piezo-actuation. a) 1D AlN nanobeam integrated together with photonic circuitry [41]. b) Microwheel cavity piezoelectrically actuated by a microstrip resonator [42].

Figure 1 . 12 -

 112 Figure 1.12 -Schematic of the needed components for the envisioned device.

Figure 1 . 13 -

 113 Figure 1.13 -Envisioned device for ultra-stable time-delayed integrated OMO with on-chip delay.

Figure 2 . 1 -

 21 Figure 2.1 -Schematic of a suspended 1D photonic crystal cavity sustaining both optical modes and high frequency mechanical modes at the center of the nanobeam.

Figure 2 . 2 -

 22 Figure 2.2 -Schematic of 1D, 2D and 3D photonic crystals. Different colours represent materials with different dielectric constants. Figure from [54].

  .

  3.a), they can be seen in fact as 1D suspended resonators whose resonant frequencies depend on the string tension, length and mass. Specifically, the fundamental frequency can be calculated as f 1 = T m/L 2L where T is the string tension, m the string mass and L the string length. The other harmonics will be integer multiples of the fundamental one [56] (figure 2.3.b).

Figure 2 .

 2 Figure 2.3 -a) Guitar strings as examples of 1D suspended resonator. b) Fundamental frequency and harmonics of a string. Figure from [56].

(see figure 2 Figure 2 . 4 -

 224 Figure 2.4 -Example of construction of a 1D photonic crystal cavity. Figure from [70].

Figure 2 . 5 -

 25 Figure 2.5 -Example of a 1D photonic crystal nanobeam. a) PhC band gap. b) Tranmission spectra calculated with FDTD for different grating numbers N. Figure from [71].

Figure 2 . 6 - 31 Chapter 2 .

 26312 Figure 2.6 -(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 = 634 nm and a 0 = 485 nm. Figure from [53].

  Figure 2.7 shows an example of concept for a 1D PhC nanobeam with rectangular holes where both optimization techniques, gentle shaping of a gaussian cavity profile and apodisation, have been implemented in order to increase the final quality factor.

Figure 2 . 7 -

 27 Figure 2.7 -Example of design optimization by varying the holes size along the direction of the beam.From the center to the edge, the holes width H x is unchanged and equals 250 nm; the holes length is tuned from H y0 = 400 nm to H yn = 700 nm following a Gaussian distribution. Figure from[START_REF] Zhu | Integrated nano-optomechanics in photonic crystal[END_REF].

Figure 2 . 8 -

 28 Figure 2.8 -Coupling scheme between a suspended 1D PhC nanobeam and the SOI waveguide where it has been integrated.

Figure 2 .

 2 Figure 2.9 -a) Example of delocalized mechanical modes. b) Example of localized mechanical modes. c) Fundamental opticla mode profile.

( 2 . 14 ) 36 2. 2 .

 214362 Figure 2.10 -a) Optomechanical coupling for GaAs 1D PhC nanobeam b) Optomechanical coupling for GaP 1D PhC nanobeam.

Figure 2 . 11 -

 211 Figure 2.11 -Schematic of the final desired 1D PhC nanobeam.

Figure 2 . 12 -

 212 Figure 2.12 -Process flow for 1D PhC nanobeam fabrication. a) Adhesive BCB mechanical bonding. The green thin layer is BCB here. b) Electronic lithography for PhC patter with negative resist (HSQ). c) ICP ethching of the III-V material and dry plasma underetching of the SiN sacrificial layer. A 1D PhC nanobeam suspended on top of a SOI waveguide is finally obtained.

  Figure 2.13 -a) Top view of a SEM image showing several fabricated GaP 1D PhC nanobeams suspended and aligned on top of SOI waveguides (light grey lines). b) Tilted SEM image of a single suspended 1D PhC nanobeam with rectangular holes. Inset: zoomed image of rectangular holes.

Figure 2 .

 2 Figure 2.13 shows a SEM image of several real fabricated 1D PhCs nanobeams suspended and aligned on top of SOI waveguides (2.13.a). The fabricated nanobeams can present either elliptical or rectangular holes.Figure 2.13.b depicts a tilted SEM image showing a single suspended nanobeam with rectangular holes (see inset of the same figure for a zoomed view).

Figure 2 .

 2 Figure 2.13 shows a SEM image of several real fabricated 1D PhCs nanobeams suspended and aligned on top of SOI waveguides (2.13.a). The fabricated nanobeams can present either elliptical or rectangular holes.Figure 2.13.b depicts a tilted SEM image showing a single suspended nanobeam with rectangular holes (see inset of the same figure for a zoomed view).

Figure 2 . 15 -

 215 Figure 2.15 -Optical measurements of 1D PhC cavities. a) Retrieved transmission spectrum from a SOI waveguide where no PhCs have been integrated. b) Retrieved transmission spectrum from a SOI waveguide where 4 1D PhCs nanobeams have been integrated.

Figure 2 .

 2 Figure 2.15.b shows a transmission spectrum for a case of a waveguide where four 1D PhC nanobeams have been placed. As it is possible to notice from the figure, several optical resonances are present this time, clearly signatures of the

Figure 2 .

 2 [START_REF] Yao | Optoelectronic microwave oscillator[END_REF].b shows a zoomed view of this range of frequency.

Figure 2 . 16 -

 216 Figure 2.16 -Mechanical spectrum of GaAs 1D PhC cavities. a) 0-3 GHz range. Several mechanical modes around 2.3 GHz are present when the tunable laser is at cavity resonance (blue trace) with respect to the out of resonance case (red trace). b) Zoom in the spectrum from 2.15 to 2.45 GHz. Figure from [53].

Figure 2 .

 2 Figure 2.18 -Experimental setup for phase noise measurements. The setup is basically the same as the one presented in figure 2.14 with the difference that here, part of the signal coming from the photodiode is used for phase noise measurement. It particularly mixed with an high stable local oscillator after a proper filtering and amplification stage. The mixed signal is then read by an oscilloscope and the final phase noise curve is recovered from the FFT.

Figure 2 .

 2 Figure 2.19 -Phase noise measurements for 1D PhCs nanobeam in GaP. a) Mechanical spectrum for the high frequency mode in self-oscillating regime as a function of wavelength detuning. b) Phase noise measurements recovered with heterodyne downconversion method. c) 2D phase noise mapping as a function of laser power and wavelength detuning.

Figure 2 .

 2 Figure2.[START_REF] Favero | Optical cooling of a micromirror of wavelength size[END_REF] shows phase noise measurement related to the fabricated GaP 1D PhC nanobeam presented previously. Figure2.19.a shows the recovered mechanical spectrum for the high frequency mechanical mode in self-oscillating regime when the input injected light (at 3.5 mW of power) have been set to different wavelengths in the resonance frequency range of the optical cavity. In figure2.19.b the recovered phase noise L(f) is shown for the four spectra shown in figure2.19.a. The curves are following a 1/f 2 (white FM noise) slope until 20 MHz offset where it reaches the noise floor of the instrument. The best result is here achieved when the tunable laser is set to 1550.625 nm. For this case, the phase noise at 100 kHz offset from the carrier is equal to -84 dBc/Hz which corresponds to a 326 Hz linewidth. The linewidth was here extracted by averaging the related frequency noise in a range up to 500 kHz.

  Figure2.[START_REF] Favero | Optical cooling of a micromirror of wavelength size[END_REF] shows phase noise measurement related to the fabricated GaP 1D PhC nanobeam presented previously. Figure2.19.a shows the recovered mechanical spectrum for the high frequency mechanical mode in self-oscillating regime when the input injected light (at 3.5 mW of power) have been set to different wavelengths in the resonance frequency range of the optical cavity. In figure2.19.b the recovered phase noise L(f) is shown for the four spectra shown in figure2.19.a. The curves are following a 1/f 2 (white FM noise) slope until 20 MHz offset where it reaches the noise floor of the instrument. The best result is here achieved when the tunable laser is set to 1550.625 nm. For this case, the phase noise at 100 kHz offset from the carrier is equal to -84 dBc/Hz which corresponds to a 326 Hz linewidth. The linewidth was here extracted by averaging the related frequency noise in a range up to 500 kHz.

Figure 2 .

 2 Figure 2.19.c presents a 2D mapping of measured phase noise values as a function of different input laser powers and wavelengths detuning. The intent of the map is

Figure 2 . 20 -

 220 Figure 2.20 -Comparison of our developed 1D PhC OMO with respect to the state of the art microwave oscillators.

Figure 3 . 1 -

 31 Figure 3.1 -artistic view of integrated IDTs on a piezoelectric membrane.

3. 1 .

 1 Figure 3.2 -a) Schematic of BAWs configuration. A top and a bottom electrodes are used to generate acoustic waves well confined inside the bulk of the material. b) Schematic of SAWs configuration. An IDT is used to create surface acoustic waves propagating through the surface of the material in a region comparable with the acoustic wavelength. c) Schematic of Lamb waves configuration. Here the acoustic wavelength of the produced propagating waves is bigger than the thickness of the material. The particles motion affects the full thin membrane.

2 ) page 55 Chapter 3 .Figure 3 . 3 -

 255333 Figure 3.3 -Schematic of two face-to-face IDTs deposited onto a piezoelectric seminconductor membrane. By applying an RF signal to the first IDT, the sender, an acoustic wave is generated and lately collected by a second IDT, the receiver. Bottom left Inset: cross section of an IDT finger pair showing the two important dimensions that allow to master the actuation frequency, the finger width and the pitch between them.

Figure 3 . 5 -

 35 Figure 3.5 -Eigenfrequency simulation results of "Bottom fixed" configuration. The graph on the left shows the change in frequency for the first order mode as the thickness λ changes. For a really low ratio the frequency of the mode is quite high, 6.75 GHz for ratio equals to 0.1. As the ratio increases the frequency stabilizes around 1.65 GHz. Total displacement profiles of the symmetric first order mode are shown in the graph for ratio equals to 0.1, 0.5 and 1.5. The right side of the figure shows the geometrical change of the simulated model when the ratio increases for a fixed thickness of the membrane.

59 Chapter 3 .

 593 Figure 3.6 -Eigenfrequency simulation results of "Suspended" configuration. The graph on the left shows the change in frequency for S0 and A0 modes as the ratio thickness λ

3. 1 .

 1 Figure 3.7 -Comparison between "bottom fixed" and "suspended" configurations. From this comparison is evident how as soon as the SAWs regime is achieved ( thickness

Figure 3 . 8 - 62 3. 1 .

 38621 Figure 3.8 -Cross section of the three design ideas chosen to test IDTs on GaP in different situations. a) GaP on SiN/Si (SAWs regime). b) GaP on SiN/SiO 2 /Si (SAWs regime). c) GaP suspended on SiO 2 /Si (Lamb waves regime). The small green layers here represents the BCB used for the bonding, thickness ≈ 50nm.

Figure 3 . 10 -

 310 Figure 3.10 -Eigenfrequency results related to the GaP on SiN/Si design. a) Lower frequency range, from 1 to 2 GHz. Here, it is possible to see how the first order mode is ranging from 1.685 to 1.258 GHz for finger width changing from 475 nm to 600nm. b) Higher frequency range, from 2.5 to 4 GHz. Here, a second order mode and a shear mode can be identified ranging from 3.916 to 2.925 and from 3.916 to 2.925 GHz respectively. On the bottom side of the figure total displacement profiles are shown for each case.

page 64 3. 1 .

 641 Figure 3.11 -Eigenfrequency results related to the GaP on SiN/SiO 2 design. a) Lower frequency range, from 1 to 2 GHz. Here, it is possible to see how the first order mode is ranging from 1.616 to 1.174 GHz for finger width changing from 475 nm to 600nm. b) Higher frequency range, from 2.5 to 4 GHz. Here, a second order mode and a shear mode can be identified ranging from 3.925 to 3.050 and from 3.354 to 2.756 GHz respectively. On the bottom side of the figure total displacement profiles are shown for each case.

Figure 3 .

 3 Figure 3.13 shows the process flow to fabricate IDTs on GaP on SiN/SiO 2 /Si substrate. For IDTs on GaP on SiN/Si substrate the steps that are involved are exactly the same, the only change is in the used substrate (pure Si instead of SiO 2 /Si).

Figure 3 . 13 -

 313 Figure 3.13 -Process flow for fabrication of suspended IDTs on GaP. Three main steps are schematized here: a) Adhesive wafer bonding involving BCB (thin green layer in the figure). b) Electron beam lithography for IDTs pattern. c) Metal deposition and lift off of 50 nm of Aluminum.

  .13.b). With lithography exposure and development the IDTs pattern (fingers and contact pads) is transferred to the PMMA layer. Metal deposition of 50 nm of Aluminum and subsequently lift-off are carried out (figure 3.13.c) and the IDTs are now completely defined on the III-V membrane.

Figure 3 .

 3 Figure 3.14 shows a set of fabricated IDTs with different finger width dimensions.Two large contact pads are here visible for each IDT. These will be the contact area for the ground-source (GS) high frequency electrical probes that will be used for electrical injection. In order to optimized the adhesion with the probes and to ensure a proper reusable electrical contact, an additional metal deposition of 250 nm of Aluminum has been performed in this area. The zoom inset of the same figure shows a SEM image of one face-to-face IDTs configuration that will be used to study acoustic transmission. The space between the IDTs is here kept fixed at 20 µm for all the IDTs pairs (i.e. 10 times the acoustic wavelength λ acoustic when e = 500 nm).

3. 2 .Figure 3 . 14 -

 2314 Figure 3.14 -Optical microscope image of the fabricated set of IDTs pairs with with finger width ranging from 600 to 475nm. Inset zoom: SEM image of a IDTs pair.

Figure 3 .

 3 Figure 3.15 shows an electron dose calibration for a targeted 550 nm fingers pair.Here the pitch between them is fixed at 1.1 µm, i.e. twice the targeted finger width, while the actual width of the fabricated fingers can change according to the electron beam dose. From the graph we can see how for an electron dose ranging from 800 to 1500 µC/cm 2 the actual measured finger width changes from 540 nm (a) to 600 nm (b) respectively. From the SEM images above the graph it is possible to observe the change in the shape of the fingers just discussed. The pitch between them is always respected (the alignment is not impacted by the change in the dose) but their dimensions and definition change according to the dose value. When a 1500 µC/cm 2 is used, point b in the graph, as electron dose the fingers look really well defined even if their width is now close to 600 nm instead of the designed 550 nm.

Figure 3 . 15 -

 315 Figure 3.15 -Electron dose calibration for 550 nm fingers width with a 1.1 µm fixed pitch. Graph: Measured finger width for dose values changing from 800 µC/cm 2 (a) to 1500 µC/cm 2 (b). Top part: SEM images of finger width for 800 µC/cm 2 (a) and 1500 µC/cm 2 (b) e-beam dose.

Figure 3 . 16 -

 316 Figure 3.16 -Frequency evolution for first order mode mechanical mode of COMSOL simulation for dimensions spanning from 530 nm (a) to 600 nm (b). Top part: total displacement profile for 530 nm (a) and 600 nm (b).

Figure 3 . 17 -

 317 Figure 3.17 -Process flow steps for membrane suspension. a) SiN layer deposition (first image) and electron beam lithography with positive resist (PMMA) where the holes pattern is transferred. b) RIE and ICP etching process to transfer the holes pattern to the SiN layer before and to the GaP membrane after. c) Underetching with dry plasma technique for suspension of the fabricated transducer.

Figure 3 . 18 -

 318 Figure 3.18 -Optical image of the suspended structures. a) Pair of suspended IDTs. b) Zoom image showing the suspended area around the electrodes.

Figure 3 . 72 3. 3 .

 3723 Figure 3.18 shows optical microscope images of a fabricated pair of suspended IDTs. In figure 3.18.a the underetched area is already visible as darker lines around the deposited electrodes. A zoom of one of the transducer is presented in figure

  .19.

Figure 3 . 19 -

 319 Figure 3.19 -Experimental setup used for IDTs characterization.

Figure 3 . 74 3. 3 .

 3743 Figure 3.20 -Measured reflection for different finger widths dimensions of IDTs on GaP/SiN/Si. Zoom in the 200-300 MHz region (see inset). Displacement profile of a simulated bulky mode at 354 MHz is shown in the figure as well.

page 75 Chapter 3 .Figure 3 . 21 -

 753321 Figure 3.21 -Measured reflections coefficients of IDTs on GaP on SiN/SiO 2 /Si structure. a) and b) Reflections for 0-2 GHz and 2-4 GHz frequency ranges. c) and d) Comparison of measured reflection signals and simulated acoustic modes in the same range of frequency.

Figure 3 . 22 - 77 Chapter 3 .

 322773 Figure 3.22 -Measured reflections coefficients of IDTs on GaP suspended on SiO 2 /Si. a) and b) Reflections for 0-2 GHz and 2-4 GHz frequency ranges. c) and d) Comparison of measured reflection signals and simulated acoustic modes in the same range of frequency.

Figure 3 . 23 -

 323 Figure 3.23 -Comparison between simulated and measured (stars) results for 500nm finger width for SAWs regime and Lamb waves regime.

Figure 3 . 24 - 79 Chapter 3 .

 324793 Figure 3.24 -Direct comparison of 525nm finger width unreleased IDT and the suspended counterpart in a frequency range around 3GHz showing the second order Rayleigh mode (for unreleased) and S0 mode (for released).

Figure 3 .

 3 Figure 3.25 shows reflection measurements of the tested pairs of IDTs at different crystallographic orientations in the frequency range of interest (1.9-3.3 GHz). From the figure we can appreciate how the 0 • oriented IDTs (orange trace) follow the result presented previously, with an acoustic resonance at 3.154 GHz, showing good reproducibility of the fabricated transducers. The 90 • oriented IDTs (brown trace) show really similar results as expected, with an acoustic resonance located at 3.141 GHz. The blue trace in the middle of the figure displays the result of 45 • oriented IDTs. Here the found acoustic resonance is at 2.216 GHz. The acoustic signatures in this case present deeper dips, meaning that a better transduction might happen for 45 • orientation (please note that in the figure the scale of the y-axis is the same for all the subplots for a fair comparison between all the discussed acoustic resonances). Finally, on the bottom part of the figure, results related to 30 • (olive green trace) and 60 • (light green trace) oriented IDTs are presented. Here two resonances appears in the presented frequency range. The first acoustic signature is located at 2.035 GHz and 2.079 GHz for 30 • and 60 • respectively while the second one is at 2.951 GHz for 30 • and at 3 GHz for 60 • oriented IDTs.

3. 3 .

 3 Figure 3.25 -Reflection measurements for IDTs oriented at 0, 30, 45, 60 and 90 degree with respect to the cleaving angle.

Figure 3 . 26 - 83 Chapter 3 .

 326833 Figure 3.26 -Measured spectrum of |S 11 | 2 .

  Figure 3.27 show the S 21 transmission of the IDTs pair just discussed above.

Figure 3 .

 3 Figure 3.27 -S21 transmission coefficient for 45deg oriented IDTs with 475nm finger width. A transmitted signal is here present at the resonance working frequency of the IDTs but it is partially masked by the EMR between the electrical probes used in the setup.

Figure 3 .

 3 Figure 3.28 -S 21 transmission trace in frequency domain for the 45 • oriented IDTs with 475 nm finger width. Black trace: S 21 trace when electrical tips are in the air. Orange trace: S 21 trace when electrical tips touch the face-to-face IDTs.

Figure 3 . 85 Chapter 3 .

 3853 Figure3.28 shows the time domain S 21 transmission for the 45 • oriented IDTs with 475 nm finger width. Two traces are here shown: the orange one, when the electrical probes are touching the two face-to-face IDTs, and the black one, when the probes are at the same distance but in the air. As it is possibile to see from the figure, the EMR is here evident at t ≈ 0s in both cases. A tips reflection, due to the used cables, is here evident also for both traces at t ≈ 20ns. Most importantly, it is also evident how when the electrical tips are touching the IDTs the signal amplitude of the S 21 increases and this is a clear signature that an acoustic signal has been detected by the IDT, as the difference with respect to the black trace suggests.The time gating technique consists now in properly selecting a time domain window of the time trace containing all the acoustic components of interest and Fourier-

86 3. 3 .

 863 Figure 3.30 -S 21 time-gated transmission spectra for 0 • unrealead and released IDTs configurations.

Figure 3 .

 3 Figure 3.31 -IDTs on GaAs suspended on SiO 2 /Si substrate. Reflection and transmission are here shown when 600 nm is used as finger width dimension. The gated transmission shows several transmission peaks. The first two can be identified most probably as the A0 and S0 mode already encountered in GaP.

Figure 3 . 32 -

 332 Figure 3.32 -Gated transmission for face-to-face IDTs pairs on GaAs at different separation distances.

Figure 4 . 1 -

 41 Figure 4.1 -Artistic view of a phononic waveguide integrated between two IDTs.

Figure 4 . 2 -

 42 Figure 4.2 -Crystal lattices design for phononic crystals implementation. a) Triangular lattice b) Hexagonal lattice c) Kagome lattice with their respective unit cells.

Figure 4 . 3 -

 43 Figure 4.3 -Phononic crystals band gap simulations. a) Triangular lattice. b) Hexagonal lattice. c) Kagome lattice.

Figure 4 . 4 -

 44 Figure 4.4 -Kagome phononic crystal lattice lattice constant a sweep for a fixed r/a ratio (=0.43) and a fixed thickness d = 265 nm. a) Geometrical description of Kagome phononic crystal lattice geometry and related changes according to a value. b) Band gap variation according to change in the a value.

Figure 4 . 5 -

 45 Figure 4.5 -Kagome phononic crystal lattice r/a sweep for fixed a = 370 nm. a) Geometrical description of Kagome phononic crystal lattice geometry and related changes according to r/a value. b) Band gap variation according to change in the r/a ratio.

Figure 4 . 6 -

 46 Figure 4.6 -Schematic representation of phononic crystals waveguides creation from perfect Kagome crystal lattice. The one on the left has been created by removing 3 lines of holes while the one on the right by removing 5 lines of holes.

Figure 4 . 7 -

 47 Figure 4.7 -3ML phononic crystals waveguide dispersion relation. a) Comparison between the perfect kagome crystal lattice (white hexagons) and the new k vectors (shaded orange, orange and yellow pentagons) related to the 3ML waveguide. b) Total displacement profiles of the propagating (orange and yellow frame) and non propagating modes for k=X. The colour scale is the same for all the modes.

Figure

  Figure 4.8 shows the results related to the 5ML design. Specifically, in figure4.8.a it is possible to see again how two main propagating modes are now present and able to propagate through the waveguide for k=X: propagating mode 1 (orange pentagons) at Ω prop = 2.128 GHz and propagating mode 2 (yellow pentagons) at Ω prop = 2.207 GHz. Figure4.8.b shows total displacements profile of these two propagating modes and the 6 closest ones showing how, also in this case, these are indeed non propagating through the waveguide. If on one side 5ML mode 1 and 2 show a similar displacement profile with respect to 3ML modes 1 and 2 respectively, on the other side this is not particularly true for the slope. In fact, even though the 3ML and 5ML modes 1 present somehow a similar slope just shifted in frequency, 3ML and 5ML modes 2 they present different slopes, with the latest showing an almost constant behaviour, signature of a really low group velocity. The different dispersion relations with related different slopes open the way to group velocity engineering of the propagating mechanical vibrations. This subject will be carefully discussed in the next section. For discussion, similar behaviour it has already been studied in photonic waveguides and it has been pushed even further by changing for instance the size of holes in the direction perpendicular to the waveguide[START_REF] Baba | Slow light in photonic crystals[END_REF].

Figure 4 . 8 -

 48 Figure 4.8 -5ML phononic crystals waveguide dispersion relation. a) Comparison between the perfect kagome crystal lattice (white hexagons) and the new k vectors (shaded orange, orange and yellow pentagons) related to the 5ML waveguide. b) Total displacement profiles of the propagating (orange and yellow frame) and non propagating modes for k=X. The colour scale is the same for all the modes.

Figure 4 . 9 -

 49 Figure 4.9 -5ML phononic crystals waveguide dispersion relation. a) Comparison between the perfect kagome crystal lattice (Time domain simulations for dispersions engineering validation. a) Created 5ML waveguide 3D COMSOL model. Inset: waveguide details. b) Prescribed displacement profile at 10 ns when a 2.7 GHz prescribed displacement input is applied. c) Prescribed displacement profile at 10 ns when a 2.2 GHz prescribed displacement input is applied.

Figure 4 . 10 -

 410 Figure 4.10 -Schematic representation of corrugated phononic waveguides inspired from Kagome lattice on the left. Corrugated phononic waveguide unit cell on the right.

4. 2 .

 2 Figure 4.11 -Corrugated phononic waveguide dispersion relation. a) Comparison between the perfect kagome crystal lattice (white hexagons) and the k vectors (shaded and filled orangepentagons)related to the corrugated phononic waveguide. b) Total displacement profiles of the propagating (orange frame) and non propagating modes (black dashed frames) for k=X. The colour scale is the same for all of them.

Figure 4 .

 4 Figure 4.12 shows the comparison between total displacement profiles for k = X of eigenfrequencies in the frequency range of interest for corrugated (figure 4.12.a) and non corrugated (figure 4.12.b) phononic waveguides.

Figure 4 . 12 -

 412 Figure 4.12 -Comparison between corrugated and non corrugated total displacement profiles for k=X in the frequency range of interest. a) Total displacements profiles of existing modes for corrugated design. b) Total displacements profiles of existing modes for non corrugated design.

Figure 4 . 13 -

 413 Figure 4.13 -Group velocity VS Frequency for 3ML propagating modes. a) Group velocity values for 3ML propagating mode 1. b) Group velocity values for 3ML propagating mode 2.

Figure 4 . 14 -

 414 Figure 4.14 -Group velocity VS Frequency for 5ML propagating modes. a) Group velocity values for 5ML propagating mode 1. b) Group velocity values for 5ML propagating mode 2.

Figure 4 . 108 4. 4 .

 41084 [START_REF] Hati | Cancellation of vibration-induced phase noise in optical fibers[END_REF].a show that 5ML mode 1 at k = X, i.e. at 2.128 GHz, exhibit a group velocity value of ≈ 509m/s while in 4.14.a it is possible to see that 5ML modes 2 manifests a group velocity of ≈ 90m/s for k = X, i.e. at 2.207 GHz. All the modes related to both phononic crystals waveguide designs show interesting slow mode features. These results represent a good feedback for possible acoustic delay implementation in an integrated platform where only few hundreds of µm are available for a fully-integrated feedback loop implementation.page IDTs + Phononic platform4.3.2 Corrugated phononic bridgeRegarding the corrugated phononic bridge, only one mode revealed propagating features in the frequency range of interest after eigenfrequency studies. Group velocity results related to this propagating mode with respect to frequency are shown in figure4.15.

Figure 4 . 15 -

 415 Figure 4.15 -Group velocity VS Frequency for corrugated phononic waveguide propagating mode.

Figure 4 . 16 -

 416 Figure 4.16 -Fabrication of suspended phononic waveguide between two face-toface IDTs. Three main steps are schematized here. a) Electron-beam lithography of holes and phononic features. b) RIE and ICT etching of SiN hard mask and semiconductor membrane. c) Suspension of phononic waveguide and IDTs.

Figure

  Figure 4.17 shows SEM images of the fabricated phononic devices between a pair of suspended IDTs. In particular, figure 4.17.a shows the 3ML (on the left) and the 5ML (on the right) phononic crystal waveguides based on a Kagome lattice with lattice constant a = 370 nm. While figure 4.17.b shows a fabricated corrugated phononic bridge with a design inspired also in this case by a Kagome lattice with a = 370 nm. From the SEM images it is also possible to appreciate the small holes opened between the electrodes (highlighted in yellow) used for the IDTs suspension.

Figure 4 .

 4 Figure 4.17 -SEM images of fabricated phononic waveguides. a) 3ML phononic crystal waveguide (on the left) and 5ML phononic crystal waveguide fabricated between a pair of IDTs (false color) in yellow. b) Corrugated phononic bridge fabricated between a pair of IDTs (false color) in yellow.

Figure 4 .

 4 Figure 4.18 -Calibration for phononic crystals fabrication. a) SEM images representing phononic crystals with electron beam dose value equal to 900 µC/cm 2(frame 1), 1200 µC/cm 2 (frame 2) and 1500 µC/cm 2 (frame 3). b) Tilted SEM image of GaAs phononic crystals degenerated to "nano-stars" lying down on the SiO 2 substrate as a results of an excessively high electron beam dose.

Figure 4 .

 4 Figure 4.18 will help us to clarify better this situation. In particular, figure4.[START_REF] Aspelmeyer | Cavity optomechanics[END_REF].a shows SEM images of fabricated phononic crystals after the full process. Three different electron beam doses have been here used for the three depicted cases. Specifically, 900, 1200 and 1500 µC/cm 2 dose values have been used for frame 1, 2 and 3 respectively. From the figure we can appreciate how the diameter of the holes increases accordingly. In this situation, on one hand, an excessively low dose will bring smaller holes than expected (frame 1) preventing to reach the right r/a ratio, important parameter for precisely opening and centering the phononic band gap as desired (as discussed previously in the simulation part of this chapter). On the other hand, an excessively high dose would enlarge too much the holes for finally breaking the connections between them (frame 3). Frame 2 shows how a right choice for the dose can eventually replicate exactly the desired design. If, on one side, the low dose case would bring a change in the expected frequency band gap but it would not destroy the sample, this is not the case for the overdosed case. In fact, this last situation would prevent a proper suspension of the phononic crystal pattern with a collapse of the semiconductor membrane during the underetching step. A tilted SEM image showing this last situation is reported in figure figure4.18.b where GaAs "nano-stars" lie on the bottom of the SiO 2 substrate as a consequence of an

Firstly, transmission

  of IDTs in a classic face-to-face configuration has been tested through a 265nm thick GaAs membrane.

Figure 4 .

 4 19.a shows the schematic with the membrane that has not been phononically modified yet. Transmission results are diplayed in figure 4.19.b.

Figure 4 . 19 - 114 4. 4 .

 4191144 Figure 4.19 -Acoustic transmission of face-to-face IDTs on a 265nm thick GaAs membrane. a) Schematic of the measured structure. b) Time-gated transmission peaks for 550, 600 and 650 nm finger widths.

Figure 4 . 20 -

 420 Figure 4.20 -Acoustic transmission of face-to-face IDTs on a 265nm thick GaAs corrugated phononic waveguide. a) Schematic of the measured structure. Inset zoom: details of the corrugated suspended phononic waveguide. b) Acoustic reflections of IDTs with 550, 600 and 650 nm finger width. c) Time-gated transmission traces through a corrugated phononic waveguide for 550, 600 and 650 nm finger widths. Shaded green, red, and blue lines in the three graph are guides to the eye.

4. 4 .

 4 IDTs + Phononic platform

Figure 4 .

 4 Figure 4.21 -a) S 21 in time-domain. The shaded red part contains the EMR peak in proximity of t = 0 ns. The shaded orange time window contains the first acoustic signal, the shaded violet time window contains the TTE. Inset: Orange and violet time-windows transformed back to frequency domain revealing the related transmission peaks. b) Group delay at peaks value. Time-gate window starts at 20 ns for the first acoustic signal and at 65 ns for the TTE. Values are sampled every 5 ns.

Figure 4 . 22 -

 422 Figure 4.22 -Top part: two examples of fabricated devices at 24 and 48 µm distance respectively. Graph: gated transmission signals for face-to-face IDTs spaced by different distances.

Figure 4 .

 4 Figure 4.4 shows a comparison between them. More in detail, the top part of the figure shows two examples of fabricated devices at 24 and 48 µm distance while the graph on the bottom depicts the time-gated transmission for all the three cases (same time-gating window has been used for all of them for a fair comparison). Here, on one hand, it is possible to notice how the three set of transducers are working basically at the same frequency (2.061, 2.062 and 2.066 GHz) demonstrating a good reproducibility of the designed device. On the other hand, the peak amplitude is indeed decreasing as the distance increases, signature of the losses in the material. Peak values are in fact at -78.6, -82.6 and -84.4 dB for 24, 36 and 48 µm spaced IDTs respectively. The calculated loss rate due to mechanical losses in the material is then ∼ 0.408 dB/µm.

Figure 4 . 23 - 120 4. 5 .

 4231205 Figure 4.23 -Design for an high efficient next generation phononic corrugated waveguide between two focused IDTs.

Figure 5 . 1 -

 51 Figure 5.1 -Schematic of the proposed OMO with integrated acoustic delay. Inset: Phase noise vs Package size state of the art map for most famous Oscillators.

  Figure 5.2.a shows an optical microscope image with a broad view of the SOI chip where the structures have been implemented together with other integrated systems.

Figure 5 . 2 -

 52 Figure 5.2 -Optical microscope images of the fabricated devices. a) Broad view of the chip where several complete devices have been integrated together with other integrated systems. b) A single integrated complete device.

Figure 5 .Figure 5 . 3 -

 553 Figure 5.3 displays SEM images of the final fabricated device. In particular, figure 5.3.a shows a tilted SEM image of the complete integrated device while figure 5.3.b is a top view SEM image where is possible to appreciate the suspended area around the phononic crystals, the phononic waveguide and even the SOI waveguide is here visible through the holes of the phononic pattern.In the next section first experimental results coming from this first generation of OMO implementing an acoustic delay line will be discussed.

Figure 5 .

 5 Figure 5.4 -a) Top view of the setup from visible camera. b) Top view of the setup from infrared camera. c) Red trace: transmission spectrum before suspensions of the integrated structures. Blue trace: transmission spectrum after suspension of the integrated structures.

Figure 5 .

 5 Figure 5.5 -a) Infrared Camera view when the tunable laser is set at 1558 nm. b) Infrared Camera view when the tunable laser is set at 1558.74 nm. Red frame: excited 1D PhC nanobeam. Inset: photonic crystal cavity profile.

Figure 5 . 6 -

 56 Figure 5.6 -Acoustic resonances of IDTs integrated on the SOI chip.

  .8.a while figure 5.8.b and 5.8.c show a picture of the setup implementing together an optical and electrical injection and an optical microscope image with the DUT respectively.

Figure 5 .

 5 Figure 5.7 -a) Schematic of the setup for electro-optical transduction with VNA. b) Picture of the setup. c) Optical image showing the device under test.

  Figure 5.8.b shows the page 132 5.3. First implementation of OMO with integrated acoustic delay reflection and transmission spectra when the RF probe is now touching the IDT, i.e. providing electrical injection, while the laser is always at cavity resonance.

Figure 5 . 8 -

 58 Figure 5.8 -VNA measurements of the complete structure. a) Reflection and transmission spectra with laser at cavity resonance without electrical injection. b) Reflection and transmission spectra with laser at cavity resonance and electrical injection. c) Reflection and transmission spectra with laser out of cavity resonance and electrical injection.

Figure 5 .

 5 Figure 5.8.c depicts the situation when the RF probe is always touching while

Figure 5 .

 5 Figure 5.9 -a) Schematic of the setup for electro-optical transduction with a RF signal generator and an ESA. b) Transmission spectra collected by the ESA in different situations: out of optical resonance (black trace), close to optical resonance (brown trace) and at optical resonance (orange trace). For all the three cases the RF input signal was always kept equal to 195 MHz.

Figure 5 .

 5 [START_REF] Liu | Stable and low-spurs optoelectronic oscillators: A review[END_REF].b shows a SEM image of 1D PhC nanobeam integrated together with a pair of electrodes on both side for capacitive actuation.

Figure 5 .

 5 Figure 5.10 -a) 3D schematic for electrostatic actuation of 1D PhC nanobeam. b) Cross section. c) Top view.

Figure 5 .

 5 Figure 5.11 -a) Optical microscope images of the fabricated devices. b) Tilted SEM image of 1 system with capacitive actuation

Figure 5 .

 5 [START_REF] Kaba | Improving thermal stability of optoelectronic oscillators[END_REF].c shows a cut-line profile of the spectrogram at 3558.2 MHz (related to white dashed line number 1 in figure5.12.a) where all the discussed components are identified. We can see moreover that the behaviour of the mechanical mode changes for a 2 MHz region area (see underlined locking region in the figure5.12.a) where the mechanical mode frequency is locked to the electrical input frequency. A cut-line profile of this situation is shown in figure5.12.d (related to white dashed line number 2 in figure and perspectives 5.12.a).

Figure 5 .

 5 Figure 5.12 -a) Measured spectrogram where an electrostatic actuation and locking of the 1D PhC mechanical mode to the RF input signal is visible for a 2 MHz region. b) Changing in the locking region for different injected power values. c) Cut-line profile of white dashed line 1 in a). d) Cut-line profile of white dashed line 2 in a).

Figure 5 . 13 -

 513 Figure 5.13 -Fabricated and packaged device. a) Chip after fabrication in our cleanroom facilities. b) Schematic of fabricated structures. c) Chip after optical packaging by STMicroelectronics.

  Figure 5.14.a shows the results related to the first case. On the right side of the figure is depicted the used setup. the light injection is here performed page 140 5.5. Packaging with the SLED and the coupling to the SOI waveguides is performed as we always did so far, i.e. with optical fibers coupled to the integrated grating couplers. The signal is recovered by an OSA and the results are shown on the left side of the figure (dark yellow trace) where we can see that several optical resonances appear, signature that the nanobeams are able to properly work and confine light in the correct range of wavelength.

Figure 5 . 14 -

 514 Figure 5.14 -Optical tranmission measurements before and after optical packaging. a) Right side: transmitted optical spectrum left side: used setup for recovering the transmitted spectrum, here the coupling between the injected light and the DUT is performed with optical fibers and grating couplers as usual. b) Right side: transmitted optical spectrum left side: used setup for recovering the transmitted spectrum, here the coupling between the injected light and the DUT is performed with via the fibre blocks connectors.

Figure 5 .

 5 Figure 5.14.b shows the measurements results after that the optical packaging has been performed. The used setup to recover in this case the transmitted optical spectrum , from the same waveguide measured before, is shown on the left side of the picture. This time, the coupling between the "outside world" and the packaged structures is performed thanks to optical fibers directly linked to the APC connectors of the fibre blocks. The right side of the figure shows the measured optical spectrum (orange trance) showing that the resonances are still there, they didn't shift in frequency and we actually gained in term of transmitted power after that the packaging was performed. The highlighted shaded yellow part in the figure is a region where two cavities are present at really close resonant frequencies. Interesting features that can come from this will be discussed in the next subsection.

Figure 5 . 15 -

 515 Figure 5.15 -Mechanical spectrum from packaged devices at cavity resonance input signal with 10.5 dBm of input power.

Figure 5 .Figure 5 . 17 -

 5517 Figure 5.16 -a) Spectrogram of mechanical behaviour of one packaged waveguide. b) Zoom of the region with chaotic signatures.

Figure 5 .Figure 5 .

 55 Figure 5.18 -a) 3D schematic of two optically coupled suspended 1D PhCs nanobeams integrated on two different waveguides. b) Possible coupling implementation scheme with our packaged device.

Figure 5 . 20 -

 520 Figure 5.20 -Possible high sensitive detection scheme based on calibrated phase transitions. a) Calibrating step. b) Arrays of integrated optomechanical oscillators used for ultrasensitive detection.

  Our system to be developed . . . . . . . . . . . . . . . . . . . . . . . .2.5 Our proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the manuscript . . . . . . . . . . . . . . . . . . . . .2 Optomechanical crystal oscillator integrated on-chip 2.1 1D photonic crystal cavity . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Cavity design . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 System efficiency: coupling with SOI waveguides . . . . . . . . 2.2 Optomechanical interaction in 1D PhCs . . . . . . . . . . . . . . . . Fabrication of 1D PhC cavities . . . . . . . . . . . . . . . . . . . . . OM measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Optical measurements of 1D PhC cavities . . . . . . . . . . . 2.4.2 Mechanical spectrum of 1D PhC cavities . . . . . . . . . . . . 2.4.3 Phase noise measurement of GaP mechanical modes . . . . . . 2.5 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . 3 GHz acoustic waves on III-V semiconductor on insulator 3.1 Simulation and design of integrated IDTs . . . . . . . . . . . . . . . . 3.1.1 Limit between Lamb waves and Surface Acoustic waves . . . . 3.1.2 Design of acoustic transducers in engineered stack of materials 3.2 Fabrication of suspended IDTs on GaP on Silicon . . . . . . . . . . . 3.2.1 Fabrication of interdigitated fingers . . . . . . . . . . . . . . . 3.2.2 Calibration of the process . . . . . . . . . . . . . . . . . . . . 3.2.3 Underetching of the semiconductor membrane . . . . . . . . . 3.3 Measurements of acoustic waves in the GHz . . . . . . . . . . . . . . 3.3.1 Reflection in the SAWs regime of GHz acoustical waves . . . . Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . .

	Contents
	2.4
	xvii

1 Introduction 1.1 Oscillators in modern society . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Ideal Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Delay implementation in OEO: a feature limiting the compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 1.2.1 Optomechanics for a new generation of integrated oscillators . 1.2.2 Resonant excitation for optomechanical crystals . . . . . . . . 1.2.3 Phononic crystals for guiding acoustic waves . . . . . . . . . . 1.2.4 III-V materials as a solution for integrated OMO with on-chip acoustic delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Mechanical modes of 1D PhC . . . . . . . . . . . . . . . . . . 2.2.2 Coupling between mechanical and optical modes in 1D PhC . 2.3 3.3.2 Reflection in the Lamb waves regime of GHz acoustical waves 3.3.3 Transmission measurements of GHz acoustical waves . . . . . 3.3.4 IDTs in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4

Table 2 .

 2

		34	2.39	2.85	2.92
	g P E /2π [kHz]	9.8	243.81	13.12	-289
	g M I /2π [kHz]	-1.75	-28.76	-0.95	9.9
	g 0 /2π [kHz]	8.05	215.05	12.17	-279.1

1 -Table showing simulated optomechanical coupling rate g 0 /2π and related photoelastic and mouving boundaries contributions g P E /2π and g M I /2π for optical fundamental modes M1 and for the mode with the highest coupling M2 for both GaAs and GaP 1D PhC nanobeams.

Table 3 .

 3 1 -Calculated Rayleigh velocity for GaAs and GaP

	Poisson ratio	0.31	0.31
	Shear modulus µ [N/m 2 ]	3.92 • 10 10	3.285 • 10 10
	Density ρ [kg/m 3 ]	4138	5317
	Rayleigh velocity c R [m/s]	2860	2311

Acknowledgment