1 Indice de réfraction et coefficient d'absorption extraits de VO 2 en fonction de la température. Cette extraction est basée sur un fit de données ellipsométriques, où l'on arrive à un accord entre les données optiques et expérimentales . . . . . . . . . . . . . . .

2.2

Relation between tetragonal structure (high-temperature, black lines) and monoclinic (low-temperature, dashed blue lines) unit cells. This image has been adapted from [START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF]. Below table shows the dimensions of the unit cell of VO 2 at both its state [START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF] Afin de combler l'écart avec la demande croissante de données, l'industrie se tourne de plus en plus vers la photonique silicium pour relever ce défi. C'est pourquoi la photonique silicium est en passe de devenir une base essentielle pour la prochaine génération de systèmes de communication et les interconnexions basées sur le transport d'information par la lumière.

L'utilisation de la lumière comme vecteur d'information est une stratégie connue depuis longtemps. Elle a été largement utilisée pour les communications à grande distance grâce au développement de la technologie des fibres en silice, pour lesquelles la transmission optique présente de très faibles pertes. Cela a, en soi, révolutionné la dynamique de la demande actuelle de données, qui ne cesse d'augmenter. En fait, la fibre optique est devenue de plus en plus populaire pour les réseaux de communication à courte distance, en raison de sa peut être obtenue par l'application d'un champ électrique (E = 10 5 Vcm -1 ) [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF], par injection de porteurs ( n e ≈ 10 18 cm -3 [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF]), par voie optique [START_REF] Cavalleri | Femtosecond Structural Dynamics in VO 2 during an Ultrafast Solid-Solid Phase Transition[END_REF], impulsions térahertz [START_REF] Liu | Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial[END_REF] et chauffage thermique (au-delà de Tc ≈ 340 K à la pression standard) [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF]. C'est-à-dire en utilisant trois oscillateurs Tauc-Lorentz pour des valeurs d'énergie faibles à élevées, partageant une bande interdite optique Tauc commune [START_REF] Jellison | Parameterization of the optical functions of amorphous materials in the interband region[END_REF]. Pour décrire les propriétés optiques de VO 2 à l'état métallique, un modèle Drude [START_REF] Jellison | Parameterization of the optical functions of amorphous materials in the interband region[END_REF] est ajouté au modèle de dispersion Tauc-Lorentz afin de prendre en compte la contribution des électrons libres.

Pour plus de détails, nous renvoyons le lecteur au chapitre 2.

Comme le montre la figure 2.17 Il est intéressant de noter qu'en traçant les valeurs ∆n ef f et ∆k ef f calculées pour quelques tailles sélectionnées de VO Information has become the most valuable commodity in the world. This drive to the new information age has been propelled by the ability to transmit information faster, at the speed of light. This erupted the need for finer researches on controlling the information carriers more efficiently. Starting from Erbium-doped optical fibers for long haul communication, the photonic research has grown exponentially to the point where light has become a major information carrier for increasingly short distances communication. The exponential growth of telecommunication space and easy access to smart devices fueled the technological industry to be highly competitive, innovative and perceptive to exponential advancements. This increased data demand has led to the implementation of optical fibers and transceivers inside data centres interconnecting the servers, while putting intense researches into integrating optical components directly onto micro-scale chipsets.

The current generation chipsets are based on copper interconnects between the chips and within the chips components. These metallic interconnects are limited by the latency and bandwidth due to parasitic resistance and capacitance of the material. This has paved the way for silicon photonics as an emerging field using light for optical interconnection.

Silicon has been the backbone for the microelectronic industry, making silicon photonics CMOS compatible, which is an industrial standard. Even though the current technology allows electrical interconnects at 10 nm architecture [START_REF] Ghavam | Chip Power Scaling in Recent CMOS Technology Nodes[END_REF], the demand has created a bottleneck in regard with further miniaturisation of their electronic components, making it difficult in predicting and controlling the logic operations in nanoscale components. On the other hand, shifting towards photonics (relying on speed of light) can appear as an endless resource, 1 although it comes with a fair share of drawbacks. But if utilized properly, it may satisfy the increasing industrial data demand for decades. This has motivated the researches to invest in the field, making photonics technologies a foundation for the universal global data communications [START_REF] Baliga | Energy Consumption in Optical IP Networks[END_REF]. The ever increasing data bandwidth, falling power consumption requirements, and low cost margins of on-chip optical interconnects [START_REF] Miller | Device requirements for optical interconnects to silicon chips[END_REF] have paved the way for photonic components to be the new integrated platform for the information age [START_REF] Kirchain | A roadmap for nanophotonics[END_REF].

However, a complete shift to photonic devices integration in microelectronic chips is difficult, as the current optical components (lasers, modulators) are substantially larger compared to their electronic counterpart (transistor gates). Although researchers have demonstrated integrated on-chip photonic functional devices, the size remains comparatively large by industrial chip-architectural standards. With increased size comes increased costs and power consumption. Therefore, without further innovation, this subsequent mismatch in size proportions prevents the further integration of photonics components with electronic chips. This has lead to exponential increase in researches and technology for alternative technologies combining, for example electro-optic materials and plasmonics for realising new and functional devices with increased modulation index (modulators), frequency of operation as well as reduced footprints, energy consumption and insertion losses.

There are two main methods that are actively researched to create efficient photonic devices. First, by using the appropriate materials for each specific functionalities in an optical system (e.g. III-V materials for lasers, LiNbO 3 for modulators, etc.) [START_REF] Wang | Nanophotonic lithium niobate electro-optic modulators[END_REF] and second, by creating nanostructured materials, to increase light-matter interaction and control propagation [START_REF] Tae | CMOS-compatible hybrid plasmonic modulator based on vanadium dioxide insulator-metal phase transition[END_REF][START_REF] Kelvin | Ultracompact vanadium dioxide dual-mode plasmonic waveguide electroabsorption modulator[END_REF][START_REF] Ryan M Briggs | Compact silicon photonic waveguide modulator based on the vanadium dioxide metal-insulator phase transition[END_REF]. For the purpose of having an efficient control over the flow of light, there is a need for utilizing modern techniques to enhance light-matter interactions.

With that intention, the potential of nanophotonics for tailoring light-matter interaction at the nanoscale has gained considerable attention in recent years [START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Femius Koenderink | Nanophotonics: shrinking lightbased technology[END_REF][START_REF] Kivshar | Meta-optics with Mie resonances[END_REF][START_REF] Liu | Recent progresses on physics and applications of vanadium dioxide[END_REF]. It was made possible by a wealth of research in characterization and nano-fabrication techniques, understanding novel materials with unique properties and utilizing them in efficient and sometimes unnatural ways [START_REF] Arseniy | Optically resonant dielectric nanostructures[END_REF][START_REF] Femius Koenderink | Nanophotonics: shrinking lightbased technology[END_REF].

Engineering materials to accurately manipulate light has been at the forefront of most of the recent technological advancement in optics [START_REF] Kivshar | Meta-optics with Mie resonances[END_REF][START_REF] Femius Koenderink | Nanophotonics: shrinking lightbased technology[END_REF]. And in particular, there is a current strong interest in the structural engineering of materials through nanopatterning in order to modify or impart abrupt changes to the phase and amplitude of incident light over deep-subwavelength length scales. Such exotic optical scattering properties were found to occur in natural microstructures such as the wings of morpho-butterfly, which became an inspiration for controlling light-matter interaction through engineering the structure and arrangement of matter. In that spirit, the field of metamaterials was then created to go beyond what is naturally existing ('meta' means 'beyond' in greek). In order to achieve the desired interactions with light, the constituents of these metamaterials, called meta-atoms (usually smaller than wavelength of light), are user defined and are designed and arranged such that an averaged effect is encountered in terms of scattering and interference with light waves.

The unique optical properties of metamaterials are derived from engineering their multipolar resonances in subwavelength structures. Therefore, knowledge of the underlying behavior of these multipolar resonances is essential for a variety of metamaterial applications [START_REF] Coenen | Directional emission from a single plasmonic scatterer[END_REF][START_REF] Alù | Cloaking and transparency for collections of particles with metamaterial and plasmonic covers[END_REF].

Approaching the standard of the current technological industry, Si is still the backbone of all advancement primarily due to the existing infrastructure and knowledge base for Si processing residing in the microelectronics industry. The crystallographic structure of Silicon prevents it to have intrinsic electro-optic modulation properties, hence realizing Si photonic tunable devices has either required the use of resonant structures (e.g., ring resonators and photonic crystal nanocavities) [START_REF] Tanabe | All-optical switches on a silicon chip realized using photonic crystal nanocavities[END_REF][START_REF] Xu | Micrometre-scale silicon electro-optic modulator[END_REF] or large footprints in non-resonant Mach-Zehnder interferometer configurations [START_REF] William | Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator[END_REF]. Although current technology enables us to have a good light-matter interaction, the devices are still passive in terms of modulation parameters.

Nothing goes faster than the speed of light, however our ability to send information depends highly on how fast can we modulate this light.

Keeping the trend, this thesis aims to examine the introduction of additional tunability, leveraging the unique properties of VO 2 as a driving mechanism for integrated photonic and metamaterial applications. We propose new types of actively tunable devices with improved or extended modulation capabilities, tackling the drawbacks of material absorption and keeping the standard size and energy consumption to the minimum. The following section will be the introduction to the unique insulator-metal transition (IMT) of VO 2 , explaining the types of transition and their associated physical theory.

Theoretical aspects on insulator-metal transitions

Vanadium is a transition metal with electronic configuration [Ar]3d 3 4s 2 . It therefore has various oxidation states and have more that ten different stable states including single valent oxide, VO 2 and mixed valent oxide like V 8 O 15 Fig. 2.1. Most of these oxides exhibit reversible semiconductor to metal transitions, which is contrary to the state predicted by band theory [START_REF] Seitz | The modern theory of solids[END_REF]. Generally this transition for the oxides are triggered by an external stimulus and they are usually manifested as a drastic change of structure as well as in their optical and electrical properties. The trigger can be of various nature like temperature, pressure, doping and electromagnetic fields [START_REF] Muraoka | Large modification of the metal-insulator transition temperature in strained VO 2 films grown on TiO 2 substrates[END_REF][START_REF] Berglund | Hydrostatic-Pressure Dependence of the Electronic Properties of VO 2 Near the Semiconductor-Metal Transition Temperature[END_REF][START_REF] Morin | Oxides Which Show a Metal-to-Insulator Transition at the Neel Temperature[END_REF]. This class of material has been of great interest for researchers and engineers, not only for applications perspectives but also for understanding the complex physics at play in these intriguing materials.

In solid-state physics, the electronic band theory provides a general framework for materials, classifying them generally into metal or insulator, based on whether their electronic band is partially or completely filled. Metals can be defined as materials with partially filled electronic band and finite density of state at the Fermi level, thus it is able to conduct electricity even with slight input, whereas insulators have completely filled valence band, a zero density of states at the Fermi level and an empty conduction band. This powerful approach was found to be invalid for certain kind of materials like VO 2 or NiO [START_REF] Mott | The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals[END_REF][START_REF] Goodenough | The two components of the crystallographic transition in VO2[END_REF]. Indeed, VO 2 and some other complex materials involve complex physics for which it is necessary to introduce other concepts in addition to band theory. In the following sections, we will try to explain the reason why, starting with band theory, followed by its limitations and then introduce additional concepts, explaining the behaviour in correlated materials.

Band Theory

Band theory is the basis for behaviour and classification for practically everyday materials.

The equations that define the theory are based on a model described by the conventional non-relativistic Schrodinger equation, which is given by

ih.( δ δt )ψ = ψH
where :

-H is the Hamiltonian, -ψ is the wavefunction, H, the Hamiltonian in terms of quantum mechanics, can be described as the total energy operator of a system (in terms of components of momentum and coordinates of space and time). Solving the Schrodinger's equation can give us insight on the energy band distribution and thus help us to understand and classify the material. But it is difficult to solve Schrodinger's equation for a system of large values of particles. Therefore, for simplification, a number of assumptions are usually made. One of which is the Born-Oppenheimer approximation [START_REF] Born | Zur quantentheorie der molekeln[END_REF][START_REF] George A Hagedorn | A time dependent Born-Oppenheimer approximation[END_REF], which separates the total wavefunction of a solid (ψ) into the wavefunction of electron (ψ e ) and nuclei (ψ n ).

According to the Hartree-Fock approximation [START_REF] Arthur | Generalized hartree-fock approximation for the calculation of collective states of a finite many-particle system[END_REF][START_REF] Paul | Note on exchange phenomena in the Thomas atom[END_REF], the wavefunction of a collective system having N number of electrons can be represented by one determinant having the spin and position of all electrons in defined direction, representing them as a summation in terms of each single-electron wave functions. Therefore, keeping the symmetry of the system, the Hartree-Fock method determines the set of spin orbitals which minimize the energy and give us the best single determinant function. The assumption therefore implies that electrons are treated independently in classical theory, which means that the electrostatic potential experienced by each electron comes from the nucleus and the averaged distribution of other electrons, leading to consistency in their behaviour. This mean-field theory forms the basis of the classical band theory. However, the classical band theory completely disregards the interaction between the electrons, including Pauli's exclusion principle (spin-electron pairing). And instead implies the correlation between electrons is substituted by repulsion of each electron with an average electron cloud.

Mott insulators and Mott Transition Mechanism

Mott, in his study [START_REF] Mott | The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals[END_REF] found that Nickel Oxide (NiO), that was supposed to be metallic by the classical band theory, was interestingly insulating. He postulated that, for those materials which were expected to be metallic but were insulating in reality, were experiencing an additional electron-electron interactions within their lattice structure. Based on his theory, it is considered that, if an electron jumps from its initial site to a new site that has already been filled, it will undergo a Coulombic repulsion. If the energy of repulsion is higher than the energy of the incoming jumping electron, then this latter will be bound to their initial site, constricting their freedom of movement. This electron interaction phenomenon leads to splitting of the original single half-filled band into two subsequent bands. As identified, this is contrary to the fact that isolated atom arrays are not conducting because it neglects the Coulombic interaction between electrons. This neglected electron-electron interaction is known as electron correlation.

This inter-particle correlation and splitting of energy bands is the heart of the quantum

Peierls Transition Mechanism

Rudolf Peierls [START_REF] Ernst | Surprises in theoretical physics[END_REF] studied a 1-D lattice of periodically arranged chain of atoms and postulated that a structural deformation can create periodic perturbations in the lattice chain.

Keeping in regard with Bloch theorem, this perturbation can manipulate the band structure by coupling both electrons and phonons, thus introducing an IMT in 1-D systems. The discovered phenomenon was coined after his name: the Peierls Transition.

In order to picture the Peierls transition, we imagine a periodic string of 1D atoms with atomic space "a". The periodicity of the crystal produces an energy bandgap at the edges of the brillion zone k = π a . Now if we introduce a small distortion, say ∆, that propagates along the lattice, periodically perturbing the atoms and thus doubling the periodicity, it therefore opens up new band gaps at k = π 2a , leading to available electronics state. In order to attain equilibrium and to counter this instability, the ions tend to loose energy by rearranging or distorting the lattice. This lattice distortion leads to introduction of smaller band gaps than that of the materials. So if, the energy saved from the band gap is larger than the elastic energy thus generated by lattice distortion, the rearrangement will be initiated. This periodically modulating position of atoms and the lattice potential leads to band gap openings at the Fermi surface, increasing the overall stability of the whole system.

Therefore, overlooking atomic vibrations, Peierls [START_REF] Grüner | The dynamics of charge-density waves[END_REF] showed that a small perturbation ∆, can dominate the total energy of the system. This can be seen as a mechanism of IMT, where the perturbed state has additional bandgaps, so it is the insulating state, while the unperturbed has the fermi level in a partially filled band, so it is the metallic state. In the following sections we will analyze the interplay between the Peierls distortion and other 10 degrees of freedom and later we discuss how this theory can explain some of the physical properties of VO 2 .

1.3 Vanadium oxides and its transition theory

1.3.1 Vanadium oxide
VO 2 comes under the class of correlated metal oxides that has an insulator-metal phase transition (IMT) [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF]. This transition in VO 2 can be achieved using applied electric fields (E = 10 5 Vcm -1 ) [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF], injected carrier densities ( n e ≈ 10 18 cm -3 [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF]), optical [START_REF] Cavalleri | Femtosecond Structural Dynamics in VO 2 during an Ultrafast Solid-Solid Phase Transition[END_REF], terahertz pulses [START_REF] Liu | Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial[END_REF] and thermal heating (beyond Tc ≈ 340 K at standard pressure) [START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF]. This IMT is concomitant with a structural transformation, subsequently leading to an adverse change in material's electrical and optical properties. The IMT of VO 2 and its various oxides have been a known fact for a few decades. Thermally-induced IMT was first studied by

Morin [START_REF] Fj Morin | Oxides which show a metal-to-insulator transition at the Neel temperature[END_REF] [START_REF] Lee | Epitaxial stabilization and phase instability of VO 2 polymorphs[END_REF]. Therefore, it is not easy to explain the complex theory behind the transition of VO 2 , as the simple band theory of solids fails to predict its state. As explained before, different physical mechanisms could be responsible for an IMT, and more specifically the Mott transition and the Peierls transition mechanisms. The debate over which one is the prominent transition mechanism has been going on for a long time [START_REF] John B Goodenough | The two components of the crystallographic transition in VO2[END_REF][START_REF] Grandi | Unraveling the Mott-Peierls intrigue in Vanadium dioxide[END_REF]. On par with Mott, it could be that the electron correlation is the key factor for the transition behaviour. But the experimentally verified structural transition can be credited to a Peierls transition [START_REF] John B Goodenough | The two components of the crystallographic transition in VO2[END_REF] due to lattice displacement.

There have been some recent study suggesting an IMT of VO 2 without Peierls distortion [START_REF] Lin | Metal-insulator transition of monoclinic VO2 thin film without Peierls distortion[END_REF].

In a single crystal form, at room temperature Vanadium dioxide is semiconducting, with a monoclinic structure, having lattice constants a = 5.75Å b = 4.53Å, c = 5.38Å and a band gap of 0.6 eV [START_REF] John M Longo | A refinement of the structure of VO2[END_REF]. Upon heating above the transition temperature, the resistivity decreases about 4 to 5 orders of magnitude, indicating a metallic conductivity. This phenomenon is the starting point of puzzling physics: the conductivity of this material is strongly enhanced when we heat it. As we describe in the following, this electronic property changes is also accompanied by a structural change [START_REF] Marezio | Structural Aspects of the Metal-Insulator Transitions in Cr-Doped VO 2[END_REF].

In this new metallic phase, vanadium dioxide crystallizes into a rutile structure (referred to , because of the dimer formation. So the overall contribution of V in a crystal is 1 2 as the two original vanadium atoms in the dimer become non-equivalent.

Hence, the new unit cell in the insulating phase of VO 2 must be twice as large as the original rutile primitive unit cell to include both vanadium atoms of the dimer. The tilt of the dimers leads to vanadium atoms being displaced away from the rutile c axis and also out from rutile plane. The tilting and dimerization depreciates the crystal symmetry and the crystal system transitions from tetragonal to monoclinic. These phases related cell system i.e. insulating phase cell to metallic phase cell are related by the edge vector of the lattice relations: As people might see and extrapolate, the change in crystal structure can lead to change in the electronic band structure and thus act as a Peierls mechanism for the metal-insulator transition. Imada studied that the drastic change in conductivity across the metal to insulator transition could either be due to the vanishing free carrier density or the divergence of the effective mass of electrons [START_REF] Imada | Metal-insulator transitions[END_REF], which corresponds to a change in the carrier density or carrier mobility, respectively [START_REF] Imada | Metal-insulator transitions[END_REF].

a M 1 = 2 c R , b M 1 = b R , c M 1 = a R -c R .
In vanadium dioxide, it was observed that the conductivity change is specifically due to the change in the carrier density but the carrier mobility is almost unchanged. Hall measurements as measured on both single crystal and thin film VO 2 , show that the majority carriers are electrons in both phases and the carrier density increases from ∼ 10 18 cm -3

to ∼ 10 23 cm -3 from the Insulating to the metallic phase [START_REF] Li | Ultrafast All-Optical Graphene Modulator[END_REF][START_REF] Ruzmetov | Hall carrier density and magnetoresistance measurements in thinfilm vanadium dioxide across the metal-insulator transition[END_REF]. Whereas, the carrier mobility stays at a roughly constant value of ∼ 0.1cm 2 /V.s across the transition. Coming down to asking the central and main question in VO 2 , we want to understand whether the transition is due to electron-lattice interaction (Peierls transition), electron-electron correlation (Mott transition), or a collaborated Peierls-Mott transition [START_REF] Goodenough | The two components of the crystallographic transition in VO2[END_REF][START_REF] Zylbersztejn | Metal-insulator transition in vanadium dioxide[END_REF][START_REF] Rice | Comment on "VO 2 : Peierls or Mott-Hubbard\? A view from band theory[END_REF]. This question is under debate even to this day. In the next section we discuss the proposed transition mechanisms and corresponding electronic band structures.

Electronic Band structures of Vanadium oxide

We can understand the electronic structure and conductivity in rutile phase, simply by discussing the metal t 2g orbital of VO 2 [START_REF] Volker | The metal-insulator transitions of VO2: A band theoretical approach[END_REF], which fall into the one-dimensional d || band, and the isotropically dispersing e π g bands, since vanadium has a d 1 electron configuration in VO 2 . Therefore in order to understand why t 2g splitting happens, we can take a look at the distance between V -V atoms (vanadium -vanadium) along the a, b, and c axis (axis pointers). In the monoclinic phase, we observe that the V -V distance is less along the c axis as compared to a or b direction, because of the lattice distortion. This smaller distance can initiate electron-electron correlation, leading to an imbalanced energy system. The electronic energy of the orbitals which are pointing along the c axis are lower as compared to others' because of this cation -cation interaction. Therefore this orbital is almost one-dimensional and directed along the c axis (rutile) and, it is called the d // orbital. On the contrary, the other two t 2g orbitals are isotropic and known as the π * band. It has also been suggested that this V-V dimer leads to the transition from high temperature localised rutile state to a localised monoclinic state.

In rutile metallic phase, the d // , and the π * orbitals in figure 1 With all the researches in the previous decades that focused on understanding the properties of the IMT in VO 2 [START_REF] Goodenough | The two components of the crystallographic transition in VO2[END_REF][START_REF] Zylbersztejn | Metal-insulator transition in vanadium dioxide[END_REF][START_REF] Rice | Comment on "VO 2 : Peierls or Mott-Hubbard\? A view from band theory[END_REF], it came to a stale mate conclusion that the transition is a combined effect of lattice distortion and Coulomb correlation [START_REF] Biermann | Dynamical Singlets and Correlation-Assisted Peierls Transition in VO 2[END_REF][START_REF] Haverkort | Orbital-Assisted Metal-Insulator Transition in VO 2[END_REF][START_REF] Koethe | Transfer of Spectral Weight and Symmetry across the Metal-Insulator Transition in VO2[END_REF][START_REF] Belozerov | Monoclinic M1 phase of VO2: Mott-Hubbard versus band insulator[END_REF][START_REF] Kim | Correlation-assisted phonon softening and the orbital-selective Peierls transition in VO 2[END_REF]. However, the observation of different other phases of VO 2 under doping and stress have further complicated the interpretation [START_REF] Pouget | Dimerization of a linear Heisenberg chain in the insulating phases of V 1-x Cr x O 2[END_REF][START_REF] Pouget | Electron Localization Induced by Uniaxial Stress in Pure VO 2[END_REF] since neither Mott nor Peierls mechanism could alone explain the wide range of phenomenon in VO 2 .

The discussion concerning the physics behind these unique properties have always been a dilemma and is still been debated [START_REF] Kotliar | Strongly correlated materials: Insights from dynamical mean-field theory[END_REF][START_REF] Tomczak | Effective band structure of correlated materials: the case of VO2[END_REF][START_REF] Mw Haverkort | Orbital-assisted metal-insulator transition in VO 2[END_REF][START_REF] Qazilbash | Mott Transition in VO2 Revealed by Infrared Spectroscopy and Nano-Imaging[END_REF][START_REF] Grandi | Unraveling the Mott-Peierls intrigue in Vanadium dioxide[END_REF]. Reasoning to the presence of both structural and electronic transitions provides complications in theoretical modelling. In fact, the question still remains whether the driving force behind the transition is due to the changes in the structural symmetry favouring electron localization or whether the structural transition is driven by strong electron-electron correlations or a combination of both the mechanisms. Even though the exact physics is not completely understood, we will show in this thesis that we can exploit the IMT in many different ways, to induce a much needed optical tunability in nanophotonic devices.

Organization of the manuscript

My aim in this thesis is to study the unique physical properties of VO 2 (Vanadium dioxide)

and devise a strategy leveraging the IMT in order to induce a tunability factor in functional photonics devices ). One of the challenge is to exploit this unique property of VO 2 even though it comes with its fair share of disadvantages like high absorption which prevents the cascading of VO 2 based functional devices in an integrated manner. Therefore the main objectives of my thesis are to:

1. Study thoroughly the optical properties of VO 2 in all its states ranging from insulator to metal as well as all intermediate states;

2. Design devices that leverage the IMT while still circumventing the optical absorption;

3. Nanostructure VO 2 in a way that mitigates the optical losses;

4. Design devices that exploit both the refractive index modulation and absorption modulation (color printing and tunable absorber).

Overall, my thesis aims to demonstrate that we can design and fabricate an efficient hybrid VO 2 -based platform for different fields of photonics (guided optics, metamaterials and metasurfaces) overcoming its limitations and find an efficient way to have an additional control over the light.

The Organisation of chapters is as follows:

• Characterization of VO 2 : Chapter 2 presents the structural analysis of the VO 2 thin films fabricated in this thesis. We characterize the thin film's structural properties at both its states. Followed by optical characterization of the thin film investigating its electrical and optical properties. The samples under investigation here will be the basis for the the devices that will be theorised and demonstrated in the following chapters.

• VO 2 based waveguide modulator: Chapter 3 illustrates the design strategy and fabrication methodologies used in realising hybrid VO 2 based waveguide modulators.

Which is a device that is used to modulate an optical signal by varying several characteristics of the input light such as the amplitude, the phase or the polarization.

The active switching dynamics is presented in form of compact CMOS compatible integrated VO 2 photonic device that can manipulate light efficiently.

• VO 2 nanocrystals: We demonstrate in chapter 4 the dynamic control of multipolar resonances accommodated by VO 2 nanocrystals and successfully retrieve the complex refractive index of the effective medium using Mie extended Maxwell-Garnett theory.

In addition we demonstrate a further scope for designer tunable materials showing distinct optical tunability compared to unpatterned VO 2 . Thus opening up a new degree of freedom to design low-loss phase-change metamaterials with designer optical tunability and actively controlled light scattering properties.

• VO 2 metasurfaces: In Chapter 5, we investigate various methods of dynamic control of light interaction with metasurfaces. We try to introduce tunability factor leveraging the insulator to metal transition of VO 2 in order to dynamically tune the resonances of sub-wavelength scattering structures. We further design and demonstrate possible applications in the field of structural color generation and tunable perfect absorbers.

CHAPTER 2 STRUCTURAL AND OPTICAL CHARACTERIZATION OF VO 2 THIN-FILMS

Introduction

The fabrication of VO 2 thin-films is a difficult task. Owing to various stable and complex phases of vanadium oxide as shown in figure 2.1, it is difficult to fabricate a perfectly stoichiometric VO 2 layer. It is observed that even a small variation in the oxygen-tovanadium ratio and in the bonding structure, affects the thin-film characteristics and is comparable to introducing defects and doping, hence influencing the phase transition properties of VO 2 [START_REF] Pouget | Dimerization of a linear Heisenberg chain in the insulating phases of V 1x Cr x O 2[END_REF][START_REF] Ghedira | Structural aspects of the metal-insulator transitions in V0. 985Al0. 015O2[END_REF][START_REF] Tselev | Symmetry relationship and strain-induced transitions between insulating M1 and M2 and metallic R phases of vanadium dioxide[END_REF]. Additionally, any unidentified impurity, either in V-O system or stoichiometric phases usually lead to modifications in the optical dispersion and switching characteristics. Since the fabrication is complex, each samples thus fabricated may end up having different optical dispersions. This in turn makes the structural characterization even more challenging. Therefore, it is extremely important to perform accurate characterization to extract the optical properties that are needed to design devices.

VO 2 growth technique and deposition

Synthesis of VO 2 can be achieved via both chemical and physical methods such as, sol-gel method [START_REF] Chae | Highly Oriented VO[sub 2] Thin Films Prepared by Sol-Gel Deposition[END_REF], molecular beam epitaxy [START_REF] Tashman | Epitaxial growth of VO2 by periodic annealing[END_REF], chemical vapour deposition [START_REF] Maruyama | Vanadium dioxide thin films prepared by chemical vapour deposition from vanadium(III) acetylacetonate[END_REF] but the most common is the sputtering technique [START_REF] Jin | Formation and Thermochromism ofVO2Films Deposited by RF Magnetron Sputtering at Low Substrate Temperature[END_REF]. The advantage of sputtering process are film uniformity and size scalability of the sample substrate. Thin film growth of VO 2 using sputtering was first demonstrated at Bell laboratory in 1967 by Fuls, Hensler and Ross [START_REF] Fuls | REACTIVELY SPUTTERED VANADIUM DIOXIDE THIN FILMS[END_REF].

For our case, we utilized magnetron assisted reactive sputtering deposition technique, for the growth of VO 2 thin films. The sputtering system is an AC450 by alliance concept. The deposition of VO 2 thin-films in our laboratory at INL was led by B. Vilquin. The deposition This image has been adapted from [START_REF] Katzke | Theory of morphotropic transformations in vanadium oxides[END_REF][START_REF] Yang | Oxide electronics utilizing ultrafast metal-insulator transitions[END_REF] recipe for VO 2 has been standardized on many experimental trials and characterizations.

In the initial study, we manipulated the temperature of deposition (Room temperature and at 550 • C) and the concentration of O 2 in the chamber. Since the target used was V 2 O 5 , the standard procedure based on a previous study [START_REF] Shigesato | Thermochromic VO2 films deposited by RF magnetron sputtering using V2O3 or V2O5 targets[END_REF] suggests using only Ar gas or Ar + H 2 gas to maintain the partial pressure, but here we modified the concentration of O 2 to 2%. Later some samples were subjected to post deposition annealing at 550

Structural Characterization

XRD characterization and Results

X-Ray diffraction (XRD) is a nondestructive analysis for thin film materials in order to identify the film's crystallinity and the strain between the film and the substrate. In XRD a collimated monochromatic x-ray beam is directed on a sample at a known angle and the intensity of the reflected beam is measured by a detector. The intensity and position of the reflected beam is an indication of the crystallographic arrangement of atoms. For example, at particular angles, we see a strong constructive interference if the planes of atoms are parallel in the crystal lattice and offer a constructive phase difference. The working principle of XRD is governed by Bragg's law which is given by nλ = 2dsinθ, where λ is the wavelength, d is the atomic lattice spacing and θ is the diffraction angle. The sample is generally rotated at θ angles and reflection at each angle is recorded. The magnitude of the reflection provides information about the form factor of atoms (higher intensity for heavier element) and their arrangement in the lattice. Therefore, the angle and magnitude of reflection enable measuring the material's crystallographic structure, strain between the layers, the degree of crystallinity and the orientation of its phases [START_REF] Ertas Uslu | Selective IR response of highly textured phase change VO 2 nanostructures obtained via oxidation of electron beam deposited metallic V films[END_REF].

The figure below (Fig. 2.2) shows the crystallographic structure of VO 2 's unit cell at both 22 its states. The knowledge of the crystallographic orientation is important, so to understand the growth direction and the crystallographic plane orientation of the thin-film. Furthermore, the growth direction and crystallographic orientation of materials partially dictate the overall property of the deposited medium. Relation between tetragonal structure (high-temperature, black lines) and monoclinic (low-temperature, dashed blue lines) unit cells. This image has been adapted from [START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF]. Below table shows the dimensions of the unit cell of VO 2 at both its state [START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF].

We performed a so-called θ -2θ scan, that enables both the incident beam and the From the position and the amplitude of the diffraction peaks, we can identify the surface crystallographic orientation, in this case we measure the (011) plane of VO 2 from the first order reflection. This indicates that (011) face could be parallel to the c axis (C m -C p plane).

Identifying (022) face reflections, can suggest that the b-axis is perpendicular to the a-c plane, both suggesting monoclinic phase [START_REF] Wei | Growth mode and texture study in vanadium dioxide thin films deposited by magnetron sputtering[END_REF][START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF]. Thus we can identify the crystallinity in the growth direction. The peak intensity provides a qualitative estimation of the crystalline quality of the film: a high peak intensity indicates a good crystalline structure. The contrary is shown in figure 2.5, which shows multiple peaks indicating multiple crystallinity in different orientations.

Transmission electron microscopy (TEM) characterization

Transmission electron microscopy (TEM) is a powerful tool for microstructure characterization. It enables imaging a material with a sub-nanometer precision providing direct insights into a material system down to the atomic scale. Additionally, as compared to commonly used surface analysis technique like SEM (scanning electron microscope) and AFM (atomic force microscope), TEM enables analyzing internal and small size microstructures along with cross-section and plan-view.

TEM analysis was performed on two samples for VO In addition, we performed SAED (selected area electron diffraction) in order to closely observe the orientation of the atoms in their lattice structure. Analysing SAED of VO 2 on sapphire, we observe the diffraction patterns that are related to crystal lattice being observed.

The bright spots on the figure 2.12 corresponds to the crystal plane of VO 2 . The spacing between the spots is inversely related to the distance between atomic planes and here we observed the monoclinic lattice arrangement of VO 2 . that they present the correct stoichiometry and structures. We note the presence of an undesired oxide layer at the interface between the VO 2 and the Silicon substrate. This undesired layer seems to have negligible effects on the crystallographic growth, as we measure the structural change as expected from a VO 2 thin film with correct stoichiometry.

This structural analysis will serve as a basis for optical characterization and will help creating an optical model to mimic the optical properties of VO 2 . We now want to deeply characterize and analyze the optical properties of these VO 2 thin-films.

Spectroscopic ellipsometry: Measurements and analysis

The ellipsometric characterization was performed using a Jobin Yvon-UVISEL spectroscopic ellipsometer combined with a heating cell. The device is composed of a Xenon source, a polarizer, an analyser, and a monochromator that disperses and redirects the reflected light to different photo-detectors. The ellipsometer UVISEL is capable to continuously cover the spectral range from 250 to 2100 nm.

Working Principle

Spectroscopic ellipsometry is an optical technique used to measure the spectral amplitude tan(ψ) and phase (∆) of the reflected light, which are related to the optical properties and microstructure of the sample under study. The enabling principle of ellipsometry is that pand s-polarized light reflect differently.

R p = R s
With ellipsometry, we measure the complex reflectivity ratio of p-and s-polarized light and reports the result generally in ellipsometric units ψ and ∆,

R p R s = tan(ψ)e i∆
where tan(ψ) is the magnitude of the ratio of Fresnel reflection coefficients and ∆ is the phase difference. Spectroscopic ellipsometry therefore measures the change of phase and polarization of light, which enables the determination of the sample's complex dielectric function ( ) [START_REF] Humlicek | 1 -Polarized Light and Ellipsometry[END_REF].

Is and Ic parameters

In the case of phase-modulated ellipsometer, like the one we use, we do not measure the ellipsometric angles ψ and ∆ directly, but instead we measure functions of ψ and ∆. Defined Knowing the unique nature of VO 2 , it exhibits complex structural changes upon the transition, making it quite difficult to establish a unique dielectric function for all phases.

The IMT behaviour of VO 2 makes it harder to theorise the optical model specially at the intermediates stages of transition. An analytical representation for their different optical responses is therefore required to establish a correlation between nano film structure and its optical properties. In order to model the complex permittivity of VO 2 , we have used a

Tauc-Lorentz model with three oscillators and an additional Drude model for the metallic state, as explained in details in the next sections. Fig. 2. [START_REF] Ernst | Surprises in theoretical physics[END_REF] shows the ellipsometric multilayer model composed of three layers: the top layer (Layer 3) is the roughness layer (VO 2 model + Void), expressed by an effective medium approximation (EMA) with Bruggeman formula [START_REF] Fujiwara | Assessment of effective-medium theories in the analysis of nucleation and microscopic surface roughness evolution for semiconductor thin films[END_REF]. Layer 2 is the VO 2 thin-film and Layer 1 is the substrate. The model is then subject to free fit using least-squares curve fitting thanks to the Levenberg -Marquardt (LM) algorithm.

Oscillators model and dispersion function for VO 2

Depending on the properties of materials, the frequently utilized dispersion functions for transparent or weakly absorbing films include Cauchy and Lorentz oscillator as well as Amorphous, New Amorphous, and Tauc-Lorentz models for semi-transparent materials (dielectrics, polymers and semiconductors absorbing in the VIS/FUV), and Drude model for metals. A sum of three Tauc-Lorentz oscillators was used to model the optical properties of VO 2 . Each of which corresponding to inter-band transitions in the material [START_REF] Gavini | Optical Properties of Semiconducting VO 2 Films[END_REF], sharing a common Tauc optical bandgap [START_REF] Jellison | Parameterization of the optical functions of amorphous materials in the interband region[END_REF]. To describe the optical properties of VO 2 at the metallic state, a Drude model [START_REF] Jellison | Parameterization of the optical functions of amorphous materials in the interband region[END_REF] is added to the Tauc-Lorentz dispersion model in order to take the contribution of free electrons into account.

The Tauc-Lorentz theoretical model [START_REF] Chen | Perspectives in the characteristics and applications of Tauc-Lorentz dielectric function model[END_REF] was developed by Jellison and Modine in 1996.

They came up with the complex dielectric function using a Tauc joint density of states and the Lorentz oscillator, given by:

˜ T L = r,T L + i. i,T L = r,T L + i.( i,T × i,L )
In this equation, the imaginary part of the dielectric function i,T L , is given by the product of imaginary part of Tauc's dielectric function i,T with the Lorentz one i,L . While taking the approximation of parabolic bands the Tauc's dielectric function describes the interband transition above the band edge. The obtained equation is:

i,T (E > E g ) = A T .( E -E g E ) 2
where :

• A T is the Tauc coefficient,
• E is the photon energy,

• E g is the optical band gap,

The imaginary part of Tauc's dielectric function gives the response of the material caused by inter-band mechanisms only. Therefore i,T (E ≤ E g ) = 0. The imaginary part of the Lorentzian oscillator model is given by :

i,L (E) = A L .E 0 .C.E (E 2 -E 2 0 ) + C 2 .E 2
where:

• A L is the strength of the 2,T L (E) peak,

• C is the broadening term of the peak,

• E 0 is the peak central energy, Therefore by evaluating these two functions, Jellison set up a new equation to express the imaginary part as an amalgam of both [START_REF] Jellison | Parameterization of the optical functions of amorphous materials in the interband region[END_REF], i.e.

i,T L (E) = i,L × i,T =      ( 1 E ) A.E 0 .C.(E-Eg) 2 (E 2 -E 2 0 ) 2 +C 2 .E 2 , f orE > E g 0, f orE ≤ E g where: A = A T × A L
Now the real part of the r,T L is derived from the equation of i using the Kramers-Kronig integration. Which is given by:

r (E) = r (∞) + 2 π .P. ∞ Eg ξ. i (ξ) ξ 2 -E 2 dξ
where: ξ is the integration energy parameter (Cauchy's residue theorem analytics) and 'P' denotes Cauchy's principal values.

To take into account the effects of free carriers on a material's optical properties, we use the Drude model, that is given by:

= (ω p ) 2 -(ω) 2 + iΓ D ω
Where, ω p is the plasma frequency, ω is the frequency of the incident electromagnetic field and Γ D (in eV) is the collision frequency (damping factor), which increases the broadening of absorption tail. The damping effect is due to the absorption and collision processes of free carriers.

Extension to Multiple oscillators

In the case where multi-transitions occur in the measured wavelength range, we generally use the multiple Tauc Lorentz (TL) model, which is the sum of several single-TL terms [START_REF] Jai | Spectroscopic ellipsometry characterization of high-k dielectric HfO2 thin films and the high-temperature annealing effects on their optical properties[END_REF]. Increasing the number of oscillators thus increase the number of parameters that can be altered to model the permittivity of a material. To model the transition and the intermediate phases of VO 2 , we use a combination of 3 Tauc Lorentz oscillators.

So the equation for the real and imaginary parts of the permittivity r and r are given by:

i,T L (E) = i,L × i,T =      N i=1 ( 1 E ) A i .E i .C i .(E-Eg) 2 (E 2 -E 2 i ) 2 +C 2 i .E 2 , f orE > E g 0, f orE ≤ E g r (E) = r (∞) + N i=1 2 π .P. ∞ Eg ξ. i (ξ) ξ 2 -E 2 dξ
Where N denotes the number of oscillator used (3 in our case).

• r parameter is linked to the real part of the dielectric function.

• r (∞) = ∞ is the high frequency dielectric constant. This parameter prevents the ∞ from converging to zero for energies below the band gap ∞ > 1 .

• A i (in eV) express the strength of the i th absorption, and i refers to the number of oscillators.

• C i (in eV) is the term related to the broadening ; it is a damping coefficient linked to the FWHM of the i th peak of absorption.

• E g (in eV) is the optical band gap energy.

• E i (in eV) denotes the energy of maximum transition probability or the energy position of the peak of absorption. Always E g < E i

As compared to other models, the TL model can reflect more correct optical responses of matter, especially the sub-gap (mainly below E g , sometimes below E 0,T L ) refractive index, providing an insight into the electronic structure and the transitions within the material.

Optical modelling and characterization of extracted data

As we have learned from previous section, we can summarise that we first collect Is and Ic values using ellipsometry, following which we create (simulate) the optical model for the sample in question (VO 2 on SiO 2 ) using dispersion formula for all materials constituting each layers, individually. For, VO 2 we use 3TL to model the correct dispersion in semiconductors state and 3TL + Drude model for the metallic state. As displayed in figure 2.17 we obtain a very good agreement between the experimentally measured ellipsometric parameters Is/Ic and the simulated optical model described above, with a good fit factor χ 2 = 0.59. These fits confirm the evolution of the material towards a metallic state as we vary the temperature.

In our case, we experienced the emergence of metallic behaviour at 58 Another unique feature of VO 2 is the thermal hysteresis (See Figure 2.23). As it shifts its crystal structure from monoclinic to rutile during the transition, it acquires a semi-stable state and therefore it shows a delay in transiting back to its original state once the trigger is removed, thus showing the presence of a memory effect in the form of hysteresis. This acquisition of optical properties from VO 2 thin-films is fundamental for understand-ing the influence of the substrate and the thickness on the optical properties. We will see in the next chapters that this step is crucial to provide inputs to simulate, design and realise VO 2 based functional devices, which is the basis for this thesis.

Conclusion

In this chapter we thoroughly studied the structural and optical properties of VO 2 thin-films deposited in different laboratories. As mentioned before, there are multiple variations for the oxides of Vanadium, hence it is critical to control the environment and the condition for the growth of VO 2 during deposition. With further characterisation using XRD we observe the crystallinity of the deposited VO 2 and structural changes associated with it. This was followed by the optical characterization of the VO 2 thin-films, in order to measure their optical properties. Using ellipsometry, we extracted the optical dispersion of the materials and mapped its evolution as a function of temperature. Further, the thermal hysteresis of VO 2 coupled with the thermal evolution of its optical properties confirms the good quality of VO 2 thin-films, consistently with previous researches [START_REF] Li | Anomalous optical switching and thermal hysteresis behaviors of VO2 films on glass substrate[END_REF]. The optical dispersion of VO 2 thus extracted will be utilised as the primary input for the design, development and application of tunable nano-devices in the forth coming chapters.

Introduction on Silicon photonics

In this data driven world using light as the information carrier, Silicon-based photonics has become a promising technology for on-chip photonics. The success of Silicon as a standard platform is due to its compatibility to monolithically integrated electronics devices. Driven by the industrial demand, high purity silicon is now readily available in large quantities, including the ability to pattern nanoscale (< 10nm) features on silicon devices at a large scale [START_REF] Wagner | EUV lithography: Lithography gets extreme[END_REF]. Thus making Si, an industrial standard and a natural candidate for integrated photonics.

Adding to its success, Silicon On Insulator (SOI) architecture has become a technological advancement that has propelled Si-based integrated photonics. This structure of Silicon layer on SiO 2 buried oxide has enabled the fabrication of low-loss Silicon waveguides, taking advantage of the naturally available large refractive index contrast between Si and SiO 2 , achieving high optical confinement in sub-micron dimensions [START_REF] Yurii | Losses in single-mode silicon-on-insulator strip waveguides and bends[END_REF]. Contrary to most expensive systems like III-V based semiconductors, the economics incentive of SOI platform has been the main driver behind exploring silicon platform for on-chip integrated photonics.

Wavelength division multiplexing (WDM) has enabled the transmittance of many signals through an optical waveguide, using monolithically integrated scalable silicon on-chip component. Thus WDM allows for much broader bandwidth density compared to electronic interconnects, with a speed demonstrated at 100Gb/s fiber coupled transceiver [START_REF] Doerr | Single-chip silicon photonics 100-Gb/s coherent transceiver[END_REF]. In addition to the clear advantages of Silicon for efficient waveguiding and multiplexing 52 functionalities, there is a need for more than just passive waveguide components.

A complete and scalable on-chip monolithically integrated photonic system requires various additional components like light sources, modulators, detectors, routers etc. In fact, each of these components represent a field of intense research pushing the limits of device performance and miniaturization while still keeping the size and cost of the devices low [START_REF] Preston | Waveguide-integrated telecom-wavelength photodiode in deposited silicon[END_REF][START_REF] Liu | A high-speed silicon optical modulator based on a metal-oxidesemiconductor capacitor[END_REF][START_REF] Xu | 12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators[END_REF]. In terms of the limitations in optical properties and tunability factor for silicon there have been an active push to integrate new materials to create hybrid devices based on SOI platform [START_REF] Aya O Zaki | Hybrid plasmonic electro-optical modulator[END_REF][START_REF] Pandey | Terahertz plasmonic waveguides created via 3D printing[END_REF][START_REF] Sun | Experiment and modelling study of the modulation performance in the graphene-Si heterojunction waveguide[END_REF][START_REF] Baehr-Jones | Optical modulation and detection in slotted silicon waveguides[END_REF][START_REF] David | Experimental demonstration of a tunable transverse electric pass polarizer based on hybrid VO 2/silicon technology[END_REF].

Electro-optic modulators

The most important component for optical communication in a photonic link is an electrooptic modulator. Even though nothing travels faster than light, the real speed at which information can be carried by light is the speed at which we can modulate or control it.

The speed of data-communication on-chip is therefore governed by the response time of our modulators. By definition, an optical modulator is a device that is used to modulate an optical signal by varying several characteristics of the input light such as the amplitude, the phase or the polarization. There have been a large amount of research dedicated to the demonstration of high efficiency optical modulators, with many different phenomena being used, including carrier depletion in silicon waveguide modulators [START_REF] Reed | Recent breakthroughs in carrier depletion based silicon optical modulators[END_REF], optical pumping induced Pauli blocking in graphene [START_REF] Rosevear | Hall Effect in VO2 near the Semiconductor-to-Metal Transition[END_REF], and thermally, electrically and optically induced phase changes and metal-insulator transitions [START_REF] Joushaghani | Sub-volt broadband hybrid plasmonic-vanadium dioxide switches[END_REF][START_REF] Joushaghani | Wavelength-size hybrid Si-VO2 waveguide electroabsorption optical switches and photodetectors[END_REF]. Within the standard of the current technological industry, Si is still the backbone of all advancement, primarily due to the existing infrastructure and knowledge base for Si processing residing in the microelectronics industry.

The crystallographic structure of Silicon prevents it to have intrinsic electro-optic modulation properties. Hence, realizing Si photonic modulators has either required the use of resonant structures (e.g., ring resonators and photonic crystal nanocavities) [START_REF] Tanabe | All-optical switches on a silicon chip realized using photonic crystal nanocavities[END_REF][START_REF] Xu | Micrometre-scale silicon electro-optic modulator[END_REF] or large footprints in non-resonant Mach-Zehnder interferometer configurations [START_REF] William | Ultra-compact, low RF power, 10 Gb/s silicon Mach-Zehnder modulator[END_REF]. Therefore this chapter aims to examine integrated photonic applications utilizing the transition of VO 2 as a driving mechanism for a new type of active optical modulator. But before that, we briefly review the state-of-the-art in terms of functional materials integration on Silicon for optical modulation.

Hybrid integration of functional materials on Silicon for efficient modulators

Contrary to the large optical changes observed in VO 2 , the silicon photonics technologies primarily take advantage of small changes in the refractive index of the guiding medium (silicon) in response to the change in charge density, typically induced by a voltage. Even though, this small variation in the refractive index can result in a modulation of >5dB for a single wavelength in excess of modulation frequency greater than >10 GHz [START_REF] Xiao | High speed silicon photonic modulators[END_REF].

However these classical cavity based or interferometric modulator structure usually have linear dimensions exceeding few tens of micron with limited spectral range. In the effort towards the advancement in optical hybridization of Si modulators, researchers have looked into Si-based waveguide modulators utilizing III-V semiconductors and LiNbO 3 (Lithium niobate) [START_REF] Wang | Nanophotonic lithium niobate electro-optic modulators[END_REF] to extend its modulation capacity. Similarly, LiNbO 3 based modulators that employs MZI scheme (Pockel's effect), offers several advantages, like purely linear electrooptic response, no free carrier absorption loss, a simpler fabrication process, and higher bandwidths [START_REF] Macario | Full spectrum millimeter-wave modulation[END_REF], nevertheless they are larger in dimensions. On the other hand, compound modulators (InGaAIAs/lnAIAs/AlGaN), have performance capability matching LiNbO 3 modulators but they require longer coupling distance (few hundred microns) because of their limited absorption modulation and modal overlap [START_REF] Hao | 40 Gb/s electro-absorption-modulator-integrated DFB laser with optimized design[END_REF].

In theory, we can assume the switching energy for a standard modulator with a footprint of 1µm 2 will be a few orders of magnitude greater than the energy required by a VO 2 device of similar size [START_REF] Ko | Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry[END_REF] with a switching speed observed at 10 ns [START_REF] Jerry | Dynamics of electrically driven sub-nanosecond switching in vanadium dioxide[END_REF]. With meticulous device design the power requirement could be further reduced. Given all the attractive properties of VO 2 reviewed in the previous chapters, it is a very interesting material for achieving active tunability in photonic devices.

VO 2 based free space modulators have already been demonstrated both in infrared [START_REF] Chen | A novel structural VO 2 micro-optical switch[END_REF] and visible [START_REF] Jiang | Design, fabrication and testing of a micromachined thermooptical light modulator based on a vanadium dioxide array[END_REF] range wavelengths, both utilizing thermal induced transitions. Given the material's property and the devices' designs proposed so far, VO 2 based devices are often plagued by large insertion and absorption losses, making it difficult to cascade them in an integrated system. This absorption loss make them not viable for future applications. Here, we strategized a design that can fully exploit the refractive index tuning without suffering from the high absorption. We plan to create an integrated VO 2 based waveguide modulator that circumvent the absorption losses of VO 2 by using it as an advantage, and for future CMOS compatibility it is designed on a SOI platform.

In this chapter we demonstrate a design strategy that relies on a patch of VO 2 meticulously designed for modulation, leveraging the high absorption in the metallic phase of VO 2 . Other groups have previously proposed VO 2 -based plasmonic modulators but with a different architecture [START_REF] Tae | CMOS-compatible hybrid plasmonic modulator based on vanadium dioxide insulator-metal phase transition[END_REF][START_REF] Kelvin | Ultracompact vanadium dioxide dual-mode plasmonic waveguide electroabsorption modulator[END_REF][START_REF] Ryan M Briggs | Compact silicon photonic waveguide modulator based on the vanadium dioxide metal-insulator phase transition[END_REF][START_REF] Olivares | Optical switching in hybrid VO 2/Si waveguides thermally triggered by lateral microheaters[END_REF]. Our simple structure will be compact in terms of architectural standpoint and could be easily be integrated to the current Si based photonic infrastructure and could leverage the fast transition of VO 2 . Aided by the electromagnetic simulations we determine the device geometries required to achieve the deeper modulation depth and also show the waveguide based device structure can be utilised to leverage the intermediate states of VO 2 to act as power filter and further probe the fundamental transition properties.

In the following section, we provide a comprehensive review on the physics involved with optical waveguiding and mode coupling mechanisms. We then explain in detail our design strategy and the simulation study for the VO 2 -based waveguide modulator device.

Optical Waveguides: Theory and coupling mechanisms

When we talk about integrated photonics or integrated optics, we usually refer to optical circuits that guide light, realised on a chip. It is very similar to electronic circuits integrated in modern chipsets. In the next section we present the theory of waveguiding and understand the coupling mechanisms which are relevant to understand the different concepts used in this chapter.

Optical waveguide theory

The basic component of an integrated optical circuit is a waveguide that confines light and provides a path for its propagation. A typical integrated dielectric waveguide comprises a rectangular core with a refractive index (n c ) higher than the one of the surrounding cladding or surface (n s ). Therefore, when the condition for total internal reflection (n c > n s ) is met, we observe an optical guiding along the length of the waveguide, in the direction defined as z-axis such that x-and y-axis are the width and height of the waveguide core, respectively. The majority of the characteristics of waveguiding by dielectric waveguides can be understood through ray optics [START_REF] Bahaa | Fundamentals of photonics[END_REF]. Ray optics has its own limitation as it cannot predict the propagation characteristics when the light confinement is of the order or comparable to the wavelength of light. Therefore to get a complete picture of light guiding we have to take into account the wave nature of light [START_REF] Marcuse | Theory of dielectric optical waveguides[END_REF], hence the wave theory. This can be achieved by solving Maxwell's equations [START_REF] Marcuse | Theory of dielectric optical waveguides[END_REF][START_REF] Clerk | VIII. A dynamical theory of the electromagnetic field[END_REF] given the specific geometry of the guiding structures and understanding the characteristics of the modes.

For solving the general case of Maxwell's equation in 1D, we consider a homogeneous, linear, isotropic, charge free, dielectric medium. Inside the linear material we define the electromagnetic wave characteristics using Maxwell's equation, where electric field is defined as E (volts per metre), the magnetic field as H (ampere per metre), the electric flux density as B (ampere per square meter) and the current density J. In that case, Maxwell's equations are as follows;

∇ × E = -∂B ∂t (3.1) ∇ × H = -∂D ∂t + J (3.2) ∇.B = 0 (3.3) ∇.D = 0 (3.4)
Where 3.4 is Gauss's law in differential form and the constitutive relations are given below,

D = .E (3.5) B = µ.H (3.6)
Solving for the curl of equation 3.1 with certain assumption and variable separation method, we end up with the complete solution for the wave characteristics of light in 1D,

The electric field (E) evolution of the wave in time and in the direction of propagation is shown as,

E(z, t) = E • e i(ωt-kz) (3.7)
Provided the direction of propagation is fixed as +z direction, we observe that the light wave is a plane wave at a given time (t) and a constant z value. The ω represents the oscillating nature of electromagnetic (EM) wave propagating in Z direction with a phase 57 velocity given by ω k = v p , where k is the angular wave number. Similarly solving for the three dimensional case (3D), we can write the EM field equations as,

E(x, y, z, t) = E • e i(ωt-kxx-kyy-kzz) = E • e i(ωt-k. r) (3.8) H(x, y, z, t) = H • e i(ωt-k. r) (3.9)
where, k 2 = k 2 x + k 2 y + k 2 z and r = xx + y ŷ + z ẑ such that k x , k y , k z can assume any value provided there is no restriction on the direction of propagation.

Further solving for understanding the transverse property of the propagating EM field in z direction, we take the divergence of Maxwell's equations 3.4 and 3.3, we find that the solution of these equations showed no longitudinal component. Hence the wave is purely transverse: the E and H components of the wave lie in the x -y plane and no component exists in the z direction. In order to find the relationship between the E and H component, we solve 3.1 with a condition such that the field is expressed as E x,y,z = E •x,y e j(ωt-βz) .

Solving for the E component (E •x ) in the x direction, we get the following solution for H component;

H •x = 0 (3.10) H •y = k ωµ E •x (3.11) B • = k ω E • = E • v p (3.12)
where v p is the phase velocity, which is the speed of light (c) in free space. Therefore, in a linearly polarised wave the amplitude of the electric component E • is much higher compared to the magnetic component B • . 1

Poynting Vector

In order to calculate the energy associated with the electromagnetic wave we take the divergence of E × H, because if we compare the S.I units of E is volts(V ) M etre(m) and H is ampere s(A)

M etre(m) . So divergence will give P ower(V.A) Area(m 2 ) , which is technically an intensity. Therefore, if we take the entire cross sectional area then from divergence theorem [START_REF] Henry | XV. On the transfer of energy in the electromagnetic field[END_REF] we get the power using the equation; Substituting from Maxwell's equation,

∇.( E × H) = -µ H. ∂ H ∂t -J. E -E. ∂ E ∂t (3.15)
Rearranging the terms we get,

∇.( E × H) = - ∂ ∂t [ 1 2 µ H 2 + 1 2 E 2 ] -J. E (3.16)
Where, 1 2 µ H 2 represents the density of the energy stored in the magnetic field, 1 2 E 2 is the density of the energy stored in the electric field and J. E represents joule loss per unit volume. Now if we represent E × H as S and u as 1 2 µ H 2 + 1 2 E 2 , we can rewrite the 1 That is why cones and rods in our eyes are sensitive to the electric field of the incoming light.

59

equation while integrating over all volume as:

v ∇. Sdv = - ∂ ∂t v udv - v J. Edv (3.17)
Rearranging the terms,

s S.ds = - ∂ ∂t v udv - v J. Edv (3.18)
Where s S.ds represents the flux of S through the closed surface S enclosing the volume v. Which are equal to [-∂ ∂t v udv], the rate of decrease of energy stored in EM field and [ v J. Edv], the rate of Joule loss, respectively. Therefore, conclusively we can say that S is the energy per unit time per unit surface area, or the power per unit area. The direction of which is perpendicular to E and H field i.e. the z-direction. So we can say that power propagates on the direction of propagation of light and it is denoted by S Poynting vector.

The intricate theory for the boundary conditions and coupled mode theory are given in the Appendix A.1.

VO 2 -based waveguide modulator: Design and Working principles

The principle design strategy relies on the optical power sharing, in accordance with the directional coupler theory. According to this theory, when two waveguides having similar propagation constant are in close proximity, they tend to share the power of input light between them considering a modal overlap of each waveguide. By carefully designing the distance between the waveguides and the length of the coupling region, it is possible to couple any desired fraction of the light from one waveguide to the other. The spacing between the waveguides is designed such that the evanescent tail of the mode in either of the waveguide penetrates the other waveguide, and thus the wave from one guide gets coupled to the adjacent guide.

If the waveguides are identical in terms of phase propagation constants (β m = k 2 0 n 2 0 -

( mπ d ) 2
where m denotes the mode number, n 0 denotes the index of the medium of propagation), a complete transfer of wave happens over some length of the guide and then it transfers back to the original guide again. If the propagation constants of the waveguides are not identical then the transfer of the wave is not complete. In classical modulators, the electrodes are placed in specific arrangements near each guide such that, when the electric field is applied, the refractive index of the waveguides are changed leading to significant shift in propagation constants [START_REF] Ryan M Briggs | Compact silicon photonic waveguide modulator based on the vanadium dioxide metal-insulator phase transition[END_REF][START_REF] Graham | Silicon optical modulators[END_REF]. Thus influencing the power transfer from one guide to another.

The directional coupler strategy to our proposed VO 2 -based waveguide modulator consists of two silicon waveguides at close proximity of each other. One of the waveguide is hybrid, as it designed with a patch of VO 2 deposited on the top of the waveguide. We aim at designing its dimensions such that, when VO 2 is metallic, the hybrid waveguide is well index-matched with the Si bus waveguide, fulfilling the coupling conditions. Hence, the input light from the bus waveguide is coupled to the VO 2 hybrid waveguide. Owing to short coupling length and the high absorption of VO 2 , the re-coupling of light back to the bus waveguide is prevented. The figure 3.1 shows the architectural concept of the proposed VO 2 based waveguide modulator. In order to find the right coupling conditions between the two proposed waveguides, we 62 plot evolution of the effective indices in the Si and Si/VO 2 waveguides versus the width of the Si/VO 2 waveguide (see fig. 3.2). The width of the Si bus waveguide is set at 0.52 µm. Thicknesses of the Si and VO 2 layer are respectively 0.22 and 0.12µm. Calculation is performed at λ = 1.55µm and with a spacing between the two waveguides of 0.4µm. The advantage of our proposed design is the simple architecture employed and the scalability involved that can be considered as an add-on component for any linear waveguide system. And since, the VO 2 is deposited on top of the device rather than being embedded in the structure, the nanofabrication process is simpler. To design such a switchable directional coupler we need to match the propagation constant of bus waveguide β 1 to the propagation constant of waveguide with VO 2 patch on top β 2 at the condition when VO 2 is metallic.

Refractive indices of Si, SiO

Where we see the supermode condition when

β 1 ≈ β 2 .
The effective index of the mode in the strip Si waveguide is constant (n ef f,Si = 2.5555) for λ = 1.5 µm as the width stay constant for all the modelling. For small width (0.47 µm) of the strip waveguide with VO 2 , the effective index of the mode is lower (n ef f,2 = n ef f,Si/V O 2 = 2.5171). Mode profiles of the two modes are given in the inset of figure 3.2. As the width of the waveguide with VO 2 increases, a phase matching occurs for W = 0.494 µm. In this case, the two modes are degenerated into odd and even modes (see the two insets in the middle of the figure 3.2) with effective indices respectively equal to n ef f,1 = 2.5565 and n ef f,2 = 2.5544. For larger widths, the effective index of the waveguide with VO 2 becomes higher than this of the Si waveguide (n ef f,1 = n ef f,Si/V O 2 = 2.5913 for Width = 0.52 µm).

Close to the phase match condition, a power exchange between the two waveguides can be explained as follows.

Codirectional Coupling

Figure 3.4 represents co-directional coupling where at z=0, we assume that the entire power is in core 1 and as the wave propagates reaching a certain distance, the power is completely transferred to the adjacent waveguide. Since both the modes in co-directional coupling are travelling in the same direction and usually with a same propagation constant (β 1 > 0 and β 2 >0), the general solution for the coupled mode equation can be written as; Where the amplitude of mode propagating is given by the term [a 1 e jψz + a 2 e -jψz ], these terms are taken as a function of complex exponential because the power transfer between the guides follows a cosine pattern. Therefore with boundary conditions and acknowledging that in most case the power is injected to only one wageguide, then A(0) = A • and B(0) = 0.

A(Z) = [a 1 e jψz + a 2 e -jψz ]e -jψz (3.19)
Hence, our equations are reduced to

A(Z) = A • cos(ψZ) + j ∆β 2 ψ sin(ψz) e -j ∆β 2 z (3.21) B(Z) = -A • jκ ψ sin(ψz)e j ∆β 2 z (3.22)
Now we represent the above equations 3.21 and 3.22 in terms of the power in each waveguide [START_REF] Yariv | Coupled-mode theory for guided-wave optics[END_REF] and are given as;

P 1 (z) = P 1 (0) 1 - κ 2 ψ 2 sin 2 ψκ (3.23)
And the power variation in Core 2 can be represented as;

P 2 (z) = P 1 (0) κ 2 ψ 2 sin 2 ψκ (3.24)
where ψ 2 = κ 2 + ∆β 2 4 and ∆β = β 1 -β 2 the phase mismatch. κ is the strength of interaction which depends on the waveguide parameters, core separation and wavelength.

We observe that now the power follows a sin 2 profile, which is close to cosine. We can express the maximum power-coupling efficiency as F = ( κ ψ ) 2 . The coupling length is the maximum length over which the maximum power is transferred to the second waveguide. It is denoted by L c = π 2κ , when β 1 = β 2 or ∆β = 0.

Symmetric and Anti-symmetric field pattern

If we take two waveguides, such that they are well separated, the modes propagating in the waveguide will not interact. But if we bring closer both the waveguides, the modes in each waveguide start interacting, creating a new field or mode pattern for such composite system. This new field pattern corresponds to supermodes and they can be symmetric mode or asymmetric mode interaction. The schematics of which is shown below in figure 3.5b and 3.5.

67

(a) Well separated waveguide with isolated modes.

(b) Symmetric and anti-symmetric supermodes in the composite system when separation is less Now if we take the case for co-directional couplers, where we inject power in one waveguide, which is then later coupled to adjacent waveguide, the excited mode pattern for the input mode can be expressed in terms of symmetric and anti-symmetric field patterns (even and odd field patterns). Since they are mode patterns which remains the same throughout the propagation, therefore we can understand the propagation of mode in the composite system by understanding the propagation of symmetric and anti-symmetric mode.

68

Interestingly, as they propagate, since each mode has a different velocity and therefore a different effective index, they accumulate a phase shift between them. When the phase shift is π, the mode pattern are inverted such that the whole mode is transferred to the adjacent waveguide, this phenomenon explains the mode coupling as given by the equations below; At Z = 0, E(x) = E sym (x) + E antisym (x) (3.25)

At Z = L, E(x) = E sym (x) + E antisym (x)e [j(βsym-β antisym )L] (3.26) W hen (β sym -β antisym )L = π, E(x) = E sym (x) -E antisym (3.27)
where E(x) is the field of the mode propagating in the composite system, because of the phase shift, there is constructive and destructive interferences between the symmetric and 69 anti-symmetric modes which dictates the light transfer from one waveguide to the other.

The length of the coupling can be written in terms of n ef f , as shown in equation 3.28. The variation of the coupling length with the inter-waveguide gap is shown in figure 3.7.

L c = λ 2.(n ef f (sym) -n ef f (antisym) )
(3.28) We can notice in figure 3.7 that for gap between 0.2 to 0.7 µm, variation of Lc in the logarithmic scale versus the gap is linear. This means that only with two points, it is possible to have an analytic expression of the coupling length versus the gap as:

L c = 10 (40.4771)×gap+7.454 (3.29) This linear behavior can be explained by the fact that the exponential evanescent part of the fields of the two modes interact to create the even and the odd modes. Saturation effect for larger gap is due to the discretization of the modelling. The phase match condition is not strictly obtained. Following, in order to calculate the energy transfer of the input waveguide to the even and odd modes of the two coupled waveguides, we use the mode matching method (MMM). The intricate theory of MMM is explained in the appendix B.1. The theory involving the numerical analysis is explained in the next section and the figure 3.2 has been calculated employing the following numerical theory.

Simulation study and Numerical Analysis

2D Mode Solver

In a strip waveguide, the refractive index is a function of both transverse coordinates. In such a case, the analysis of the waveguide modes becomes a more complex task compared to slab guides as the dimensions for confinement increases. A starting point for the formulation of rigorous full-vectorial waveguide analysis are the Maxwell's equation, for the complex amplitudes and a time dependence in an anisotropic media.

∇ × H = jω • [ r ] E (3.30) ∇ × E = -jωµ • [µ r ] H (3.31)
This equation can easily be converted into matrix format that makes the tensor aspect of µ and .

      0 -∂ ∂z ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             H x H y H z       = jω •       xx xy xz yx yy yz xz yz zz             E x E y E z       (3.32)       0 -∂ ∂z ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             E x E y E z       = -jωµ •       µ xx µ xy µ xz µ yx µ yy µ yz µ xz µ yz µ zz             H x H y H z       (3.33)
We take only the diagonal values of the anisotropy because we can chose the axis that corresponds to the crystal axis of the anisotropic material. Which then reduces to only diagonal elements that arises when the tensor is rotated relative to the coordinate systems.

These are the first few steps towards eigen value matrix equations that can be solved thanks to finite difference schemes. We also normalise the magnetic field H, such that H = -jωµ • H and also we normalise the matrix grid like x = jω • x, y; = jω • y and z = jω • z. Hence the modified matrix are:

      0 -∂ ∂z ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             Hx Hy Hz       =       xx 0 0 0 yy 0 0 0 zz             E x E y E z       (3.34)       0 -∂ ∂z ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             E x E y E z       =       µ xx 0 0 0 µ yy 0 0 0 µ zz             Hx Hy Hz       (3.35)
Following, ignoring the loss, a mode in a waveguide has the following general mathematical form which is consistent with the Bloch theorem.

E(x, y, z) = A(x, y)e γz (3.36) unlike in previous section, here A is the complex amplitude (mode shape) and γ represents the phase accumulation in -z direction (γ = α + jβ) which is complex propagation constant with oscillation and decay factor. For a perfect waveguide, the refractive index and the permeability are invariant versus the z axis, therefore a 3D waveguide problem can be reduced to a 2D problem. And therefore, we normalise the modal equation to fit with the matrix grid and we have the following equation form;

E(x , y , z ) = A(x , y )e γz /k• (3.37)

H(x , y , z ) = B(x , y )e γz /k• (3.38)
Substituting the normalised to solution form of Maxwell's equations, like,

∂E z ∂y - ∂E y ∂z = µ xx Hx (3.39)
with some arithmetical calculation we get the following solution which are independent of E and H factors but with A and B which are the amplitude for the normalised mode equation and the z-component becomes γ which is accumulation of phase.

∂A z ∂y - γ k • A y = µ xx B x (3.40)
Following the similar analogy we can rewrite the matrix equations in the following form,

      0 γ ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             B x B y B z       =       xx 0 0 0 yy 0 0 0 zz             A x A y A z       (3.41)       0 γ ∂ ∂y ∂ ∂z 0 -∂ ∂x -∂ ∂y ∂ ∂z 0             A x A y A z       =       µ xx 0 0 0 µ yy 0 0 0 µ zz             B x B y B z       (3.42)
we normalise the propagation constant such as, γ = γ k• , γ = j ñ ef f , where ñ ef f = n + jk.

Following we convert the matrix into linear form so that we can discretize them using Dirichlet condition [START_REF] Lu | Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations[END_REF] and transform them into eigen value problems. So we can represent the matrix equations in linear matrix form as,

zz D h y + µ xx D e y -1 zz D h x       b x b y    = γ    a x a y    (3.53)    -D h x µ -1 zz D e y D h x µ -1 zz D e x + yy -(D h y µ -1 zz D e y + xx ) D h y µ -1 zz D e y       a x a y    = γ    b x b y    (3.54)
Where in we substitute,

P =    -D e x -1 zz D h y D e x -1 zz D h x + µ yy -(D e y -1 zz D h y + µ xx D e y -1 zz D h x    and Q =    -D h x µ -1 zz D e y D h x µ -1 zz D e x + yy -(D h y µ -1 zz D e y + xx ) D h y µ -1 zz D e y   
Therefore, we now can rewrite them as,

P    b x b y    = γ    a x a y    (3.55) Q    a x a y    = γ    b x b y    (3.56)
For the final step to convert the equation into an eigen value problem we substitute b from equation 3.55 into the equation 3.56 and we get the following single matrix equation as,

P Q    a x a y    = γ2    a x a y    (3.57)
Where PQ can be replaced as some factor Ω 2 , therefore giving us the the standard eigen value problem (λ = γ2 ), that can be easily be solved using a simple function (eig() / eigs())

in matlab, in a manner such as [V, D] = eig(A, B). Where V would provide with the eigenvectors that describes the mode of the amplitude profile and D describes the eigen-values representing phase accumulation. Eigs() function on the other hand helps to calculate all the modes with eigen-values close to the estimated effective refractive index.

Finite difference method

There are various numerical ways to analyse waveguides modes numerically. The most common technique used are Finite Element Method (FEM) and Finite Difference Method (FDM). Both these methods are equally powerful but the only difference is in their method of approach. Their main task is to solve complex differential equations. The FDM analyses the problem as a whole, solving the equation directly and reduces the complexity into a matrix form. Where as FEM lies more on a variation principle where it tries to minimize functions of the wave equation. In the scope of my thesis we will be focusing on FDM base mode matching method.

FDM was introduced by Stern in 1988 [START_REF] Ms Stern | Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles[END_REF] with an idea of discretizing the wave equation (eigen value equations) into eigen value matrix equations. Where in the second order derivatives are approximated using three point difference scheme, where the discontinuity of the medium is matched by averaging the refractive index across the interface. As seen in the schematics of basic FDM strategy 3.8, we start with an equation having a continuous time derivative, where in store the function values at discrete points and we use finite differences to calculate derivatives, where we only know the functions at discrete points. The derivation is found most commonly at the center of the two given points providing accuracy. The derivative of any order of a function at any given point can be approximated into a linear sum of known points of that function. The finite difference [START_REF] Jiang | Finite-difference-based mode-matching method for 3-D waveguide structures under semivectorial approximation[END_REF] scheme can be applied to find the almost exact solution of mainly second order differentials and can be summarised graphically based on the various finite difference schemes. We apply this technique in addition to Yee's grid [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[END_REF], which discretize a given structure into finite cube blocks and adequately apply electric and magnetic components along the vertices. These EM components are arranged in a manner such that it satisfies the Maxwell's divergence condition, follow the continuity and discontinuity of the EM components along the boundary conditions and provides an elegant arrangement to approximate Maxwell's curl equations. Together with these techniques we can accurately measure the EM field characteristics at each discrete points of the given photonic element structure. Following we show how using FDM and discretization using Yee's grid we can calculate modes of the waveguide numerically, thus getting a step closer toward realisation of the device.

In our case the grid mapping in terms of finite difference scheme have been performed

such that we discretize the areas near the boundary of two interfaces and calculate more accurately the E and H fields, in contrast to commercially available mode solver which averages at the boundary. The schematics of which is given in the figure 3.10. The whole system is discretized into nine layers vertically along x axis and into six sections in horizontal y axis. This gives the accurate solution for effective refractive index and propagation constants on the order of 10 -15 as opposed to an accuracy of 10 -7 achieved by the commercial ones. The only down side is that the calculation is process intensive, hence it takes more time when imaginary part of the refractive index is added. The eigen value problem in Matlab would return the following eigen values in the form λ i = γ2 i , which we solve for un-normalised γ i , giving us γ i = k • γ2 . Where in γ is represented as

γ i = α i + jβ i such that, α (α i = Re[γ i ])
is the attenuation coefficient and β

(β i = Im[γ i ]) is propagation constant (β = k • n ef f
). Therefore we can finally calculate the effective refractive index as,

n ef f = β i k • = Im[ γ2 i ] (3.58)
Thereafter, discretizing space and time so the electric and magnetic fields are separated in both space and time. We then solve the difference equations to obtain updated equations that express the unknown future fields in terms of known past fields. Using the updated equations enumerate future magnetic fields and future electric fields. We then keep repeating the steps until the result is acceptable. In our simulation, for defining the interfaces, we use the simplest method by overlapping orthogonal grids with different space steps [START_REF] Ardavan | Meep: A flexible free-software package for electromagnetic simulations by the FDTD method[END_REF].

For example, the space step around the nano-scale structure is small enough to describe its shape, whereas the space step in homogeneous space is bigger to obtain results in real time. The next section will focus on the simulations and the results thus obtained for the realization of the device followed by fabrication.

Simulation Results

As stated before, when the propagation constants of the two waveguides are nearly the same, the Si waveguide and the VO 2 patched waveguide (VO 2 metallic state) satisfies the coupling condition and therefore an efficient optical power transfer between them takes place. Our task therefore consists in adjusting the dimensions of the two waveguides so their propagation constants match when VO 2 is metallic. Therefore, we calculate the effective refractive indices of the Si waveguide and VO 2 (at metallic state) patched waveguide at a wavelength of 1.55 µm by home made mode solver. The width and height of the bus Si waveguide were 530 nm and 220 nm, respectively. We set the height of VO 2 patch on top of Si waveguide to 50 nm. In effectively making the height of VO 2 -Si active hybrid waveguide to be (220 nm + 50nm) 270nm. The trick used here to quickly design the system and reduce computing time is that for the simulation, the k value of VO 2 is kept at 0. This was done such that the computational time for simulation is reduced.

Following, in the figure 3.11, we see the synchronizing power transfer between two waveguides and the even and odd symmetric supermodes. In order to achieve the good coupling, the width of the VO 2 patched waveguide has to be determined such that the effective refractive indices of the two waveguides are nearly the same. The main complications involved with the correct width calculations for hybrid waveguide comes with high computational time when imaginary part of VO 2 's refractive index is included. altered to match the perfect index condition. The high absorption value of VO 2 slowed the computational time, additionally the dimensions for best index match came at resolution of 1nm accuracy. This was beyond the scope of the fabrication facility available at the lab.

After scores of the trials with cluster computations and reducing the computational step, we decided to create feasible device by keeping the imaginary part at 0. Such that, for experimental stand point we could dynamically change the input light properties and control the transition using the experimental setup to find the right working condition. Therefore the calculated width of the VO 2 patched waveguide is 510 nm at k = 0.

Efficiency of the modulation

In order to have an efficient modulation, as seen in figure 3.12 we need the ∆n for optical switching between off/on states. In the following simulation figures 3.15, we see the schematic cross-sectional view of the coupler section and we observe the coupling of the wave in the VO 2 patch upon its insulator-metal transitions. We calculate the power in each waveguide thus simulated by mode matching and calculate the power transfer as a function of change of phase in VO 2 , table 3.1. The power calculation is an integration of all the field at an area close to the core of the waveguide. VO In the simulation, we observe a modulation depth of 9dB -10dB, similar to previous architecture [START_REF] Tae | CMOS-compatible hybrid plasmonic modulator based on vanadium dioxide insulator-metal phase transition[END_REF]. With all the parameters thus obtained the only remaining value to be calculated is the coupling length which itself is proportional to the gap or the distance between the coupling waveguide. To isolate the optimized coupling length (Lc) which provides maximum optical power transfer, the whole structure of the coupler section must be analyzed. Lc is a function of the difference between the propagation constants of the even and odd supermodes,

Lc = π (β even -β odd ) = λ 2(n even -n odd )
where λ is the wavelength, and n even and n odd are the effective refractive indices of the even and odd supermodes, respectively. Similarly Lc can also be highly dependent on the distance between the co-directional waveguides which are responsible for the coupling. As we see in figure:3.16, the couplers with a wider separation gap require a longer coupling length due to a slight propagation constant difference between the even and odd supermodes. For a much larger gap (>380 nm), the mode coupling between the two waveguides becomes weak, as we see below.

Parameters

Value Simulated

Height of Waveguide 1 220nm

Height of Waveguide 2 220nm

Height of VO 2 patch 50nm

Width of Waveguide 1 530nm

Width of Waveguide 2 510nm

Gap between the waveguides (Varying) 350µm, 400µm, 450µm, 500µm In figure 3.16 above, we see the changes in the coupling length parameters Lc as a function of distance or gap between then. Based on our simulations we plan to fabricate four different modulators with similar design parameters where the only varying parameter is the gap between the two waveguides. In the following, we will talk about the fabrication steps thus undertaken to achieve the device based on the simulated results.

Fabrication

As outlined before fig3.1, the initial sample consists of a VO 2 thin-film of 50nm deposited on a standard 220nm thick SOI substrate.

Following the steps of fabrication, the first lithography step defined both the waveguide structures. In order to define the structure, ZEP 520A resist was spun and pre-baked and then the pattern was exposed using e-beam lithography system and then later developed.

The mask was made in house and the whole lithography procedure was led by P. Rojo-Romeo. After defining the patterns, we performed SEM imaging Fig3.18, in order to verify a successful pattern transfer and we observed a perfect pattern transfer and accurate waveguide gap distinction. The next step was to perform etching using RIE-ICP (Inductively Coupled Plasma Etching (ICP)) using a mixture of Ar and Cl 2 at room temperature. The formulation for etching recipe was done in house and the etching procedure was performed by C. Chevalier. After the etching procedure, we performed a SEM imaging in order to visualise the etched pattern. The SEM imaging reveled that contrary to the fabrication sketches, there were multiple imperfections following the fabrications.

Discussion

The primary results of the fabrication can be considered as clearly not optimized. Contrary to what was expected, the metallic mask and the resist either got completely etched thus leading to residual effect on the surface. In order to recuperate the device, it is necessary to analyse the sample thoroughly and understand the chemical composition on the surface topography.

This would pinpoint the material composition which would help us understand the etching failure stage. Secondly, performing a topographical study by AFM could give information about the roughness. Following the characterisation, we could proceed in caution with further etching and cleaning process in the hope of recovering the device. While on the other hand, we could rely on in house deposition technique to grow VO 2 and re-fabricate with bottom-up approach, where in VO 2 patch is directly grown on already designed Si waveguide.

Use of this modulator as a partial power transfer

The emergence of machine learning and artificial intelligence has opened up a new avenue for optical modulators. In that field, VO 2 could play a major role, leveraging its multi-level stages of phase transition. Following additional simulations as shown below fig: 3.20, we could see its potential applications as a partial power filter using intermediate stages of VO 2 state transition as shown in figure 3.21. Power in core 1 denotes the original wave power being transferred in the bus waveguide. As we slowly switch the VO 2 on the hybrid waveguide, the refractive index of the material at dielectric state gradually switches towards metallic value, thus affecting the coupled power transfer efficiency [START_REF] Tae | CMOS-compatible hybrid plasmonic modulator based on vanadium dioxide insulator-metal phase transition[END_REF]. The transition at later stages will be electrically assisted thus having finer control of the transitions.

We foresee that this dynamically controlled power transfer can act as a weight bank, which is one of the main functions performed by an algorithm in a neural network. And neural network is the core for the development of machine learning technologies and artificial intelligence. The optical modulation supported by our modulator design could aid in realising an all-optical neural network, propelling the computation capacity bottle-necked by our current transistors. The optical neural network system currently uses classic MZI passive modulators [START_REF] Woods | Optical computing: Photonic neural networks[END_REF][START_REF] David | Silicon photonics: Meshing optics with applications[END_REF]

PHASE-CHANGE MATERIALS

Introduction

Engineering materials to accurately manipulate light has been the forefront of most of the recent technological advancement in optics. Light interaction and propagation within finite-size particles has been a topic of research for decades and is still the basis for a wealth of interesting new physical phenomena. Indeed, the most unusual fact is that such a classical topic is still the foundation for many fundamentally new and unexpected scientific and technological advances like loss-free negative-index metamaterials [START_REF] Yoo | Loss-free negative-index metamaterials using forward light scattering in dielectric meta-atoms[END_REF][START_REF] Gholipour | Reconfigurable ultraviolet and high-energy visible dielectric metamaterials[END_REF], zero-index materials [START_REF] Silveirinha | Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials[END_REF][START_REF] Alu | Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern[END_REF][START_REF] Liberal | Near-zero refractive index photonics[END_REF], invisibility cloaking [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF][START_REF] Cai | Optical cloaking with metamaterials[END_REF]. With the advancement in nano-science and nano fabrication techniques, new nano-objects are now available for the enhancement and control of light-matter interaction at the nanoscale [START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Femius Koenderink | Nanophotonics: shrinking lightbased technology[END_REF][START_REF] Kivshar | Meta-optics with Mie resonances[END_REF].

The unique optical properties of metamaterials are derived from engineering their multipolar resonances in subwavelength structures. Therefore, knowledge of the underlying behavior of these multipolar resonances is essential for a variety of metamaterial applications [START_REF] Coenen | Directional emission from a single plasmonic scatterer[END_REF][START_REF] Alù | Cloaking and transparency for collections of particles with metamaterial and plasmonic covers[END_REF]. In dielectric metamaterials [START_REF] Zhao | Mie resonance-based dielectric metamaterials[END_REF], supported resonances are a combination of electric and magnetic dipoles. Mie [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF] in 1908 formulated the solutions for radiating and internal fields supported by the dielectric medium. These solutions were useful to understand and calculate the exact contribution of each modes of the nanostructure and to calculate the field distribution inside it.

These man-made Mie-resonance based metamaterials are static devices, with properties fixed after the fabrication process. It is therefore currently a great challenge to find efficient means to dynamically modulate such photonic devices at the nanoscale [START_REF] Ferrera | Dynamic nanophotonics[END_REF][START_REF] Nikolay | Obtaining optical properties on demand[END_REF]. One 95 such approach to induce tunability is to use phase transition materials like VO 2 [START_REF] Wuttig | Phase-change materials for non-volatile photonic applications[END_REF][START_REF] Ke | Vanadium dioxide: The multistimuli responsive material and its applications[END_REF][START_REF] Shi | Recent advances in fabrication strategies, phase transition modulation, and advanced applications of vanadium dioxide[END_REF][START_REF] Cueff | Dynamic control of light emission faster than the lifetime limit using VO 2 phase-change[END_REF]. Recently similar strategies have been used to demonstrate phase modulation [START_REF] Kim | Phase Modulation with Electrically Tunable Vanadium Dioxide Phase-Change Metasurfaces[END_REF],

tunable optical absorption [START_REF] Zhu | Dynamically reconfigurable metadevice employing nanostructured phasechange materials[END_REF][START_REF] Rensberg | Active optical metasurfaces based on defect-engineered phase-transition materials[END_REF] or switchable dielectric-plasmonic regimes [START_REF] Butakov | Switchable plasmonic-dielectric resonators with metal-insulator transitions[END_REF] in nanophotonic devices.

The starting idea of this chapter is to reduce the dimensions of VO 2 structures down to sizes comparable to the wavelength of operation, in order to tailor the electric dipole (ED)

and magnetic dipole (MD) Mie resonances supported by these crystals. The IMT of VO 2 could then be used as an additional degree of freedom to dynamically tune the ED and MD resonances of this correlated system.

Here, we experimentally demonstrate the multipolar resonances supported by VO 2 nanocrystals (NCs) can be dynamically tuned and switched thanks to the IMT of VO 2 . Using both Mie theory and Maxwell-Garnett effective medium theory, we retrieve the complex refractive index of the effective medium composed of a slab of VO 2 nanospheres embedded in SiO 2 and show that such a resulting metamaterial presents distinct optical tunability compared to unpatterned VO 2 . But more importantly, we show in the following that this metamaterial approach enables designing tunable composite metamaterials presenting a refractive index modulation without induced extinction, or in other words a zero-induced extinction phase-change metamaterial. The following section explains the theory and concepts and is followed by a section explaining the fabrication and characterization techniques involved with the VO 2 NCs.

Mie theory for wavelength-scale particles

When an object is illuminated by an electromagnetic wave, it reradiates parts of the energy while being lost from the original wave. The nature, shape and the size of an object highly influences the quantity and the direction of light that is scattered [START_REF] Noguez | Surface plasmons on metal nanoparticles: The influence of shape and physical environment[END_REF]. The complexity of scattering analysis increases substantially when the size of the object approaches the wavelength of the incident wave. In such a situation, we can no longer assume the field to be constant within the object and therefore the so-called "long-wavelength approximation" ceases to apply, and retardation effects must be considered. Accordingly, in order to compute the scattered field for such an arbitrary object, one would require a numerical approach.

Such a complex problem can however be simplified if the object is a sphere. In 1908, Mie found an exact solution to the problem, by calculating the scattered field as a series solution [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF]. The idea was to expand the electric and magnetic fields in terms of vector spherical harmonics. Then the field functions can be represented in a linear combination of terms that are products of separable functions of the three spherical coordinates.

Mie scattering theory indicates that light scattered by small (relative to the incident wavelength of light) spherical particles can be considered to consist of partial waves radiated by constituent electric charges of the spherical particle. The amplitude of these partial waves are central to Mie's solution and the coefficients terms are given by a n and b n . Where a n denotes the amplitude of the n th electric partial wave and b n is the amplitude of the n th magnetic partial wave. They are expressed numerically in the form of [START_REF] Roy | Calculation of Mie derivatives[END_REF][START_REF] Zhao | Mie resonance-based dielectric metamaterials[END_REF][START_REF] Christoffel | Light scattering by small particles[END_REF]:

a n = ψ n (x)ψ n (mx) -mψ n (x)ψ n (mx) ξ (1) n ψ n (mx) -mξ (1)
n (x)ψ n (mx)

(4.1) b n = ψ n (x)ψ n (mx) -mψ n (x)ψ n (mx) ξ (1) n ψ n (mx) -mξ (1) n (x)ψ n (mx) (4.2)
Where m is the complex refractive index, x is the size parameter (x= 2πmR λ ) and the prime symbol (') represents the derivatives in terms of the arguments of the function. And ψ n (z) and ξ

(1) n (z) are the Riccati-Bessel functions defined in terms of the spherical Bessel functions (z) [START_REF] Christoffel | Light scattering by small particles[END_REF][START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF].

The electromagnetic interaction of light with spheres can then be modeled following the Lorenz-Mie formalism for scattering and absorption of light by small particles [START_REF] Craig | Absorption and scattering of light by small particles[END_REF]. Within this formalism, the extinction, scattering and absorption efficiencies are given by:

Q sca = 2 x 2 ∞ n=1 (2n + 1)(|a n | 2 + |b n | 2 ) (4.3) Q ext = 2 x 2 ∞ n=1 (2n + 1)Re(a n + b n ) (4.4) Q abs = Q ext -Q sca (4.5)
where a n and b n are the so-called scattering coefficients [START_REF] Craig | Absorption and scattering of light by small particles[END_REF]. These depend on both the particle optical properties (relative to its surrounding medium) and size. Where as the size parameter x is defined as The colored lines represent the contributions of dipolar electric a 1 , dipolar magnetic b 1 , quadrupolar electric a 2 and quadrupolar magnetic b 2 to Q abs (absorption cross section).

x = 2πm med R

99

We note that, regardless of the state of VO 2 , below 700 nm the electromagnetic response is dominated by the dipolar electric (ED) and magnetic response (MD), with a small contribution of the quadrupolar electric term (QE). These resonances correspond to Whispering Gallery Modes (WGM) which resonate at the suitable wavelengths, the magnetic resonance redshifted with respect to the electric one [START_REF] Geffrin | Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere[END_REF]. On the other hand, when VO 2 is in its metallic phase, a DE resonance appears at a longer wavelength (1015 nm). The intensity of this resonance directly depends on the temperature, i.e. the higher the temperature, the more the metallic character and consequently the higher is the intensity of the resonance. This resonance is plasmonic and its physical origin is the negative value of the dielectric constant above 1000 nm in the metallic phase.

An ensemble of VO 2 nanospheres would therefore enable a dynamic tunability of multipolar resonances collectively [START_REF] Michael | Small Dielectric Spheres with High Refractive Index as New Multifunctional Elements for Optical Devices[END_REF] and to actively control the presence and the intensity of a plasmonic mode in the NIR. Reports on the synthesis and characterization of VO 2 NCs were first published in 2002 [START_REF] Lopez | Synthesis and characterization of size-controlled vanadium dioxide nanocrystals in a fused silica matrix[END_REF][START_REF] Lopez | Size effects in the structural phase transition of VO 2 nanoparticles[END_REF], followed by studies on their nonlinear optical properties [START_REF] Lopez | Optical nonlinearities in VO 2 nanoparticles and thin films[END_REF] in 2004 and their potential for ultra-fast modulation of optical transmission [START_REF] Rini | Photoinduced phase transition in VO 2 nanocrystals: ultrafast control of surface-plasmon resonance[END_REF] in 2005.

Here, the synthesis of a dense layer of isolated VO 2 nanocrystals embedded at about 100 nm depth below the surface began with a high fluence ion implantation of V + and O + with energies of 100 keV and 36 keV, respectively. These energies were chosen to produce overlapping concentration depth profiles of the elements. The formation of the VO 2 chemical compound was controlled by the fluence ratio of V to O (i.e. 8×10 16 cm -2 to 1.6×10 17 cm -2 and 4×10 16 cm -2 to 8×10 16 cm -2 for a ratio of 1:2 ). After ion implantation, nanocrystal growth was initiated by a rapid thermal annealing (RTA) step at 1000 • C for 10 min in an inert gas at atmospheric pressure [START_REF] Jostmeier | Optically imprinted reconfigurable photonic elements in a VO 2 nanocomposite[END_REF][START_REF] Jostmeier | Optically imprinted reconfigurable photonic elements in a VO 2 nanocomposite[END_REF]

Characterization

We analysed the optical properties and the IMT of VO 2 -NCs using a spectroscopic ellipsometer (UVISEL plus Horiba), ranging from 250 nm to 2100 nm. The system is coupled

Effective parameter retrieval of VO 2 Nanocrystals

In order to model the optical response of the slab composed of VO 2 -NCs embedded in quartz,

we have to calculate its effective refractive index m ef f . Because the conventional effective medium approximation is only valid for small inclusions compared to the wavelength, we therefore have to use a different framework to homogenize the medium as an effective medium. Here, to do so we use a Mie theory-based extension of Maxwell-Garnett effective medium approximation (EMA), as proposed by Doyle [START_REF] Ruppin | Evaluation of extended Maxwell-Garnett theories[END_REF][START_REF] William | Optical properties of a suspension of metal spheres[END_REF]. Using the Mie extension of EMA, through the calculation of the Mie dipolar electric a 1 and magnetic b 1 dipolar coefficients we can take into account the size effects which are ignored by the electrostatic approximation in Maxwell-Garnett EMA.

Derivation of the expressions for effective permittivity and permeability

In order to numerically derive the effective dielectric permittivity ( ef f ) and magnetic permeability (µ ef f ), we start with the Clausius -Mossotti equation, which describes the relation between the effective dielectric constant of suspended particle (¯ ) in a host medium ( h ) as a function of the size (radius a), polarizability (α) and the volume fraction (f ) of the embedded particles. The equation is given by:

¯ -h ¯ + 2 h = f a 3 α (4.6)
Under the electrostatic approximation, the particle dipolar polarizability can be written as:

α = i -h ¯ + 2 h a 3 (4.7)
Where i is the dielectric constant of the spherical material. Substituting and replacing the polarizability factor from the above two equations we get the classic Maxwell-Garnett 106 formula, given by:

¯ -h ¯ + 2 h = f i -h ¯ + 2 h (4.8)
Looking closely at Maxwell-Garnett formula, we notice that this equation is independent of the sphere radius a and is therefore only valid for sphere sizes much smaller than the illuminating wavelength λ. In our case, the light-matter interaction at play in the dielectric nanoparticles departs from the sole point-dipole approximation, due to the excitation of both electric and magnetic Mie resonances as well as the higher-order multipoles they can support. Therefore the electric dipole polarizability can be re-written in terms of the Mie coefficient a 1 by [START_REF] Mulholland | Light Scattering by Agglomerates: Coupled Electric and Magnetic Dipole Method[END_REF]:

α E = i3a 3 2x 2 a 1 (4.9)
Where the size parameter x is defined as:

x = 2πa √ h λ (4.10)
Now if we introduce the equations for α E and x in the Clausius -Mossotti equation, we can derive the equation for ef f defined as:

ef f = x 3 + 3if a 1 x 3 -3 2 if a 1 (4.11)
Similarly for µ ef f , we define the magnetic dipole polarizability α H in terms of dipolar magnetic Mie coefficient b 1 , shown as:

α H = i3a 3 2x 2 b 1 (4.12)
Now substituting α H and x in Clausius -Mossotti equation we can derive the equation for 107 µ ef f defined as:

µ ef f = x 3 + 3if b 1 x 3 -3 2 if b 1 (4.13)
Relating to the sample values, x is the size parameter of the VO 2 spheres and f the volume filling fraction of the VO 2 spheres in the slab. And from the values of ef f and µ ef f the effective complex refractive index of the slab can be calculated as:

m ef f = n ef f + ik ef f = √ ef f .µ ef f (4.14)
The dipolar coefficients are enough to model the response of the VO 2 particles since the quadrupolar terms are negligible, as seen in Figure 4.2.

Following the data fit procedure of experimental ellipsometry measurements, as mentioned in the last section, the respective thicknesses of SiO 2 top layer, VO 2 -NC layer and V inclusion layer were set as free fit parameters, in multi-layer model shown in figure 4.7. None of the material's dispersion were fit, rather we modify the size and density of VO 2 -NCs in our Mie-Maxwell Garnett model to adjust the fits to the measurements. Because there could exist correlations between thicknesses and permittivity in absorbing films, we simultaneously fit the two sets of measurements for VO 2 insulating and VO 2 metallic and we bind the different thicknesses. Meaning, the final fit will yield the same thicknesses for the different layers whatever is the phase of VO 2 -NCs. The only difference between the two models lies in the dispersion of VO 2 -NCs.

In Figure 4.8, we show the optimized fits together with the experimental data. We obtain an overall good correspondence between model and measurements for a VO 2 -NCs size of R = 35nm and a filling-factor f = 0.18. Especially, the fits are nearly perfect in the UV/Vis range but a slight discrepancy appears in the NIR.

The observed discrepancy in Fig. 4.8 between the model and the experimental results is due to different effects which cannot be included without losing simplicity in the proposed model. Two of the most probable important interactions could be the electromagnetic interaction between the particles and a potentially non-negligible degree of polydispersity in the size distribution of the VO 2 -NCs. The electromagnetic interaction between nanoparticles has a long standing history and a nice work on this effect can be found in Rechberger et al. [START_REF] Rechberger | Optical properties of two interacting gold nanoparticles[END_REF]. We have applied the results of this research to our case and the details can be found in the following section.

Modelling the effect of interactions between nanoparticles

The Mie theory-based extension of Maxwell-Garnett effective medium approximation doesn't take into account the possible interaction between nanocrystals (NCs). In principle,

given the mean separation of the NCs [190], interaction effects are expected to be small. However, to have a better estimation of how interaction effects may affect the plasmonic response of the NCs, we have calculated the absorption cross-section C abs of dimers made of insulating and metallic VO 2 spherical NCs for different inter-particle distances (gaps) embedded in quartz. The spheres that were used for the calculation of the effective optical properties have a size of 70 nm (R = 35nm) and the considered gaps range between 10 and 150 nm. These calculations have been performed for three polarizations, as shown in All the following calculations involving interparticle interactions were performed by Yael Gutierrez and Fernando Moreno from Department of Applied Physics, Universidad de Cantabria, Santander, Spain. These calculations have been performed using commercial software FDTD solutions (Version 8.16) from Lumerical Inc. Total field/scattered field light source conditions were used in all simulations and an illuminating linearly polarized plane-wave was set to propagate perpendicular to the dimer axis. The wavelength spectral range analyzed was from 260 to 1500 nm to mimic experimental conditions. A non-uniform mesh was used in the simulation region and a finer mesh was defined in the vicinity of the NPs. In this region, the mesh step was fixed at d x = d y = d z = 2nm. The absorption cross-section was calculated within the total-field/scattered-field formalism. Therefore, when we consider our sample under study, we know that the NCs are randomly distributed inside the fused silica substrate, we can on average consider, the effect of the 111 interaction has an intermediate effect between the two extreme situations: x-polarization and y-polarization. Similarly from Figure 4.10(c) we can observe how the absorption crosssection (C abs ) of the dimer is affected for different gaps when it is illuminated with 45 • linearly polarized light. As the gap decreases, the LSPR peak is redshifted of approximately about 40 nm and it is produced for the smallest gap.

VO 2 dimers in its insulating phase

Similarly like above, the Figure below (4.11) shows how the different interparticle distances (gaps) affects the overall Localized Surface Plasmon Resonance (LSPR) peaks that appear in the absorption cross-section (C abs ) of a dimer made of insulating VO 2 nanospheres. And again dashed black lines represent the spectral peak positions for a system of two noninteracting spheres (as in the previous case, they correspond to a coherent mixture of dipole resonances, of both electric and magnetic character). But in this case we look at resonance at 381nm rather that at 1176nm, whose origin is the superposition of a dipolar electric and a dipolar magnetic resonance. We study this resonance in the insulating phase in order to remove any parasitic effect coming from the dipolar electric resonance appearing at longer wavelengths in the metallic phase. For smaller gaps the resonance peak comes back to the starting point. In this case, the resonance has two dipole contributions, one electric and one magnetic. The dimer electromagnetic interaction shifts the electric to the red part of the spectrum whilst the magnetic one is blue shifted [START_REF] Albella | Low-Loss Electric and Magnetic Field-Enhanced Spectroscopy with Subwavelength Silicon Dimers[END_REF]. In this case, the blue shift dominates for intermediate gap distances (60-100nm) whilst for strong interaction (gaps smaller than 50nm) the dipole electric resonance redshift compensates that of the magnetic one to the blue.

In the next section we will tackle the second important discrepancy, which is the possible polydispersity in the size distribution of the VO 2 -NCs.

Modelling the polydispersity in size distribution of nanoparticles

In this approach we calculate the effect of the potential polydispersity of the NCs in the sample. Fig. 4.12 shows the difference in the absorption cross-section spectra (C abs ) between the reference system (R = 35 nm) and two other systems with well-defined polydispersities.

The first has a symmetric size distribution centered on the reference size. The other system has an asymmetric size distribution, having a majority of NCs with the reference size, but others with larger sizes. For this study we will focus on the resonance (LSPR) appearing at ≈ 1176 nm in the VO 2 -NCs in their metallic phase. For example Fig. 4.12(a) shows the C abs spectra for the reference system and the one with the symmetric NCs size distribution Fig. 4.12(b). In this case, there is no significant difference between both systems. On the contrary, in the case of the asymmetric NCs size distribution (Fig. 4.12(c)) there are significant differences. The LSPR appearing in the C abs spectrum of the polydisperse sample, as shown in Fig. 4.12(c), is redshifted (20 nm) and also broader than the reference system.

Discussion on the results and parameter retrieval of the system

Apart from the two important discrepancies that we have addressed, there could be other effects that could present minor contributions to the discrepancies observed in Figure 4.8.

For exmaple, we assume that VO 2 -NCs is composed of purely stoichiometric VO 2 and we neglected the presence of suboxides such as V 2 O 5 at the NCs surface. Furthermore, we directly used the dispersion as extracted from VO 2 thin films measurements to compute the different Mie coeffients. By doing so, we made the hypothesis that the complex permittivity of VO 2 inside the NCs has the exact same dispersion as VO 2 thin films, what could be subject to discussions [START_REF] Whittaker | Microscopic and nanoscale perspective of the metal-insulator phase transitions of VO 2 : some new twists to an old tale[END_REF].

Even though we have made all these approximations, we find a remarkably good qualitative agreement between measurements and calculations. We emphasize that this simple Mie-Maxwell-Garnett effective medium model is able to reproduce well the prominent features of the optical response for both states of VO 2 by simply changing the dispersion of VO 2 from insulating to metallic. Indeed, the calculations nicely reproduce the first peak in the visible, which is a combination of electric and magnetic dipole resonances and the second peak in the NIR which is a plasmonic resonance. This latter is only present when the VO 2 is metallic, e.g. when the real part of the permittivity is negative and is therefore a dynamically tunable resonance.

These calculations enable us to retrieve the effective permittivity of the active VO 2 -NCs layer and homogenize it as an effective metamaterial medium. states.

We see the clear differences in the optical dispersion as compared to un-patterned VO 2 layer. In particular, there are ranges of wavelength for which the IMT of VO 2 produces a large modulation of n ef f with a simultaneously low k ef f that is much lower than in bulk VO 2 . This is especially true in the NIR range, for which k ef f remains lower than 0.1. In the previous section, we calculated the main spectral characteristics of VO 2 -NCs based metamaterials using a combination of Mie theory and Maxwell-Garnett effective medium.

This has enabled us to predict the expected optical properties of VO 2 -NCs of arbitrary sizes.

Following, we show how this framework can be used to design tunable metamaterials with tailored properties in desired wavelength ranges. In order to have a general view about different regimes of tunability that we can obtain with this system, Figure 4.14 e-f displays the amplitude of refractive index modulation produced by the IMT of VO 2 as a function of NCs size. To do so, we plot ∆n ef f and ∆k ef f , which are defined as ∆n ef f = n ef f,insulating -n ef f,metallic and ∆k ef f = k ef f,insulating k ef f,metallic and which represent the amplitude of modulation of the effective refractive index and extinction coefficient, respectively. With this figure, we can pinpoint specific regions of the size versus wavelength map in which the switching of VO 2 either translates into a tunable refractive index or a tunable extinction coefficient.

Interestingly, by plotting the ∆n ef f and ∆k ef f values calculated for a few selected VO 2 -NCs sizes (as displayed in Figure 4.15), we immediately see that around the maximum ∆n ef f reached for each NCs size, we simultaneously have ∆k ef f =0. Furthermore, the wavelength at which this effect is obtained is directly governed by the size of VO 2 -NCs. In other words, this phase-change metamaterial platform enables tailoring a dynamical tuning of refractive index without modifying the optical absorption. As displayed in Figure 4.15, this 'zero-induced-extinction' refractive index tuning occur in a designer wavelength range via selecting an appropriate size of NCs.

Said otherwise, by plotting the figure of merit (FOM) ∆n ef f /∆k ef f for different sizes of NCs and comparing the obtained FOM to the corresponding one for VO 2 thin films, we immediately see that not only the FOM is enhanced by more than two orders of magnitude but we can design the system to select the wavelength at which the FOM is boosted via the size of VO 2 -NCs. This designer high FOM is much larger than in any other existing 'unpatterned' PCMs [START_REF] Abdollahramezani | Tunable nanophotonics enabled by chalcogenide phasechange materials[END_REF], and highlights the usefulness of our metamaterial approach. We also note that the opposite effect can be exploited, namely, close to the maximum ∆k ef f , we can simultaneously have negligible ∆n ef f . So this platform can be used to actively control the absorption without modifying the refractive index of the medium (e.g. for R=95nm at λ=1500nm we get ∆k ef f =-0.2 and ∆n ef f =0).

Perspective towards designer metamaterial

Another exciting opportunity that can be envisioned for this metamaterial is to be used as an active platform to control scattering properties and directional beaming of light. Indeed, when multipolar resonances are spectrally overlapping in nanoparticles, the scattered fields can interfere and produce directional scattering. In particular, spectrally overlapping ED and MD resonances yield the so-called Kerker conditions that either suppress the forward or the backward scattering of light [START_REF] Kerker | Electromagnetic scattering by magnetic spheres[END_REF][START_REF] Geffrin | Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere[END_REF]. Simply put, at specific wavelengths we switch from a dielectric MD resonance to a plasmonic ED resonance. This implies that this hybrid metamaterial enables an active and dynamical control of the Kerker conditions [START_REF] Kerker | Electromagnetic scattering by magnetic spheres[END_REF]. From that perspective, we foresee a dynamic modulation of the directionality of light scattering as well as actively tunable perfect reflection or absorption. We also note the presence of tunable quadrupolar modes (a 2 and b 2 coefficients, figure 4.16) that may be used for complex tunable interference phenomena between different multipoles, hence opening a wealth of different possibilities of spatially distributing the scattered light for directionality control purposes (see [START_REF] Michael | Small Dielectric Spheres with High Refractive Index as New Multifunctional Elements for Optical Devices[END_REF][START_REF] Liu | Generalized Kerker effects in nanophotonics and meta-optics[END_REF] for more details). This tunable multipoles could also find an interest in the recently proposed framework of spontaneous emission engineering through intereferences between higher-order multipoles [START_REF] Rusak | Tailoring the Enhancement of and Interference among Higher Order Multipole Transitions in Molecules with a Plasmonic Nanoantenna[END_REF].

Conclusions

In conclusion, we have demonstrated an active platform using VO 2 -NCs that can be integrated in SiO 2 as spherical nano-inclusions that enable exploiting the VO Contrary to bulk and thin-films VO 2 , such a metamaterial approach provides means to tailor the complex refractive index and tunability of phase-change materials. Indeed, we have shown that, by adjusting the VO 2 -NCs size, this system can be exploited as a designer phase-change metamaterial with adjustable refractive index modulation and absorption modulation. Especially, we can design the system to have a large refractive index tunability without inducing modulation of the extinction coefficient at specific wavelengths, or in other words a zero-induced-loss refractive index modulation. Furthermore, such VO 2 -NCs can be optically switched at femto-seconds timescale (∼150 fs) [START_REF] Rini | Photoinduced phase transition in VO 2 nanocrystals: ultrafast control of surface-plasmon resonance[END_REF] and present very large hysteresis (see [START_REF] Jostmeier | Optically imprinted reconfigurable photonic elements in a VO 2 nanocomposite[END_REF] and the supplementary information). We can therefore envision this platform to be used for ultra-fast all-optical integrated systems with memory effects.

We emphasize that the resulting metamaterial presents several benefits from a practical point of view, and most notably the nanostructures are embedded in a dielectric host that protects them from the environment and consequent processes that can affect its optical response (protection from oxidization [START_REF] Gutierrez | How an oxide shell affects the ultraviolet plasmonic behavior of Ga, Mg, and Al nanostructures[END_REF], chemical reaction [START_REF] Oates | The effect of atmospheric tarnishing on the optical and structural properties of silver nanoparticles[END_REF], mechanical scratches, etc.). This makes such a platform extremely easy to handle and to integrate in various environments, contrary to thin-films of phase-change materials that are very sensitive. The samples presented here were fabricated by ion implantation, a technique that is widely used in the microelectronics industry and could therefore be exploited for large-scale fabrication of samples. Furthermore, the size, density and depth of NCs in the host could be controlled through adjustment of implantation parameters and annealing process. This implantation process can be carried out in many different CMOS-compatible platforms such as thermal SiO 2 on silicon substrate and results in a flat surface that does not require any complex postprocess step such as CMP to planarize the device. A next logical step could be to integrate nanophotonic devices on top of this metamaterial for a wealth of tunable functionalities such as an active control of spontaneous light emission [START_REF] Bidault | Dielectric nanoantennas to manipulate solid-state light emission[END_REF][START_REF] Rusak | Tailoring the Enhancement of and Interference among Higher Order Multipole Transitions in Molecules with a Plasmonic Nanoantenna[END_REF][START_REF] Cueff | Tailoring the Local Density of Optical States and Directionality of Light Emission by Symmetry Breaking[END_REF] or dynamically tunable Huygen's metasurfaces [START_REF] Decker | High-efficiency dielectric Huygens' surfaces[END_REF] .

From a broader scientific perspective, the results presented in this chapter help define a new system with which to study tunable interactions between nanoparticles and tunable interferences between multipolar resonances. We envision it to serve as a testbed for dynamical modulation of light scattering, tunable Kerker effects and active control of light directionality. antennas. The scaling of these antennas down to lengths comparable or even smaller than the wavelength of light as well as the periodic arrangements of these antennas has led to great advantages in terms of scalability and efficiency compared to isolated antennas.

The light-matter interactions in natural materials are governed by the complex refractive index and not the structure. The interaction and control of light is produced through the modification of its polarisation, phase and amplitude. There are only a few exceptions for which the micro-structure in natural materials comes into play, the most well-known being the blue wings of the morpho-butterfly [START_REF] Gralak | Morpho butterflies wings color modeled with lamellar grating theory[END_REF]. These natural microstructures became an inspiration for controlling light-matter interactions through engineering the structure and arrangement of matter. Such artificially engineered materials are also known as Metamaterials, which were created to go beyond what is naturally existing (meta means 'beyond' in greek). In order to achieve the desired interactions with light, the constituents of these metamaterials, called meta-atoms (usually smaller than the wavelength of light), are user defined and are designed and arranged such that an averaged effect is encountered in terms of scattering 123 and interference with light waves. The overall property or response of the metamaterials depend on the individual unit structures as well as the arrangement and orientation of the meta-atoms. Therefore, by adjusting those properties, one take advantage to develop new materials, possessing unique properties that are not naturally available in nature. Such works have led to the development of negative index materials [START_REF] Brian | Negative refraction makes a perfect lens[END_REF][START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF][START_REF] Zhang | Experimental demonstration of near-infrared negative-index metamaterials[END_REF], zero-index materials [START_REF] Silveirinha | Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials[END_REF][START_REF] Alu | Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern[END_REF][START_REF] Liberal | Near-zero refractive index photonics[END_REF], invisibility cloaking [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF][START_REF] Cai | Optical cloaking with metamaterials[END_REF] and sub-diffraction imaging [START_REF] Fang | Sub-diffraction-limited optical imaging with a silver superlens[END_REF][START_REF] Liu | Far-field optical hyperlens magnifying sub-diffraction-limited objects[END_REF].

Recently, there is a growing interest for 2D planar metamaterials, namely metasurfaces.

These metasurfaces provide similar exotic properties as metamaterials, but their thickness are only a fraction of the wavelength and are therefore easier to fabricate and to integrate. While standard bulk components such as lenses rely on phase accumulation during propagation and therefore require to be curved, metasurfaces are flat components that can be integrated easily into most of the on-chip devices. This dramatically reduce the dimensions of the optical components. The physics behind metasurfaces is to locally control the electromagnetic fields through the abrupt modification of the phase and/or amplitude using nano-resonators. By spatially arranging these elements, one can mold the wavefront of light by design [START_REF] Yu | Light propagation with phase discontinuities: generalized laws of reflection and refraction[END_REF][START_REF] Yu | Flat optics with designer metasurfaces[END_REF].

And it is made possible using arrays of nano antennas with designed spacing and varying geometric parameters like shape, size, material type, orientation and their arrangement.

Unlike in conventional optics, where light travels through a medium of a particular refractive index to alter its path, metamaterials enable manipulating the effective complex refractive index of the medium achieving desired wavefront control. So far most of the experimentally demonstrated metasurfaces are static post-fabrication, which means that their optical functionalities are encoded into their design permanently after the fabrication. The potential application of these metasurfaces (all dielectric or plasmonic) can be increased exponentially if their optical response can be dynamically and reversibly modified. [START_REF] Montelongo | Plasmonic nanoparticle scattering for color holograms[END_REF][START_REF] Huang | Aluminum plasmonic multicolor meta-hologram[END_REF].

The type of metasurfaces that is based on metallic nano-structures or nano-resonators leverage surface plasmons (SP) [START_REF] Alexander V Kildishev | Planar photonics with metasurfaces[END_REF] to drive the optical response of the devices; hence known as plasmonic metasurfaces [START_REF] Genevet | Recent advances in planar optics: from plasmonic to dielectric metasurfaces[END_REF][START_REF] Chen | A review of metasurfaces: physics and applications[END_REF]. Researchers have demonstrated that plasmonic metasurfaces can enable controlling polarization, amplitude and phase of electromagnetic radiation [START_REF] Zhao | Manipulating light polarization with ultrathin plasmonic metasurfaces[END_REF][START_REF] Ni | Metasurface holograms for visible light[END_REF] for applications as flat lenses [START_REF] Pors | Broadband focusing flat mirrors based on plasmonic gradient metasurfaces[END_REF], holograms [START_REF] Zheng | Metasurface holograms reaching 80% efficiency[END_REF] or directional couplers [START_REF] Alexander | Visible-frequency hyperbolic metasurface[END_REF]. On the other hand, dielectric metamaterials and metasurfaces can support multipolar resonances such as electric and magnetic dipoles as well as higher-order modes. Mie theory as discussed in Chapter 4 is therefore useful to understand and calculate the exact contribution of each modes of the nanostructure and to calculate fields distribution inside it.

Despite the growth in static Mie resonance based metasurfaces, making them tunable remains a major issue. In recent years, considerable efforts have been made towards tunable nanophotonic devices made of metallic nanostructures coupled to various phasechange materials such as GeSbTe and VO 2 [START_REF] Wuttig | Phase-change materials for non-volatile photonic applications[END_REF][START_REF] Cueff | Dynamic control of light emission faster than the lifetime limit using VO 2 phase-change[END_REF][START_REF] Kim | Phase Modulation with Electrically Tunable Vanadium Dioxide Phase-Change Metasurfaces[END_REF]. However, all-dielectric tunable nanostructures remain largely unexplored. In this chapter we show that the IMT of VO 5.1 VO 2 metasurface for tunable structural color.

Introduction

Sci-fi movies and comics have shaped the way we see the future and are sometimes precursors to innovations, e.g. the giant touch-screen transparent displays. The past few decades have seen exponential advancement in color display technologies which are crucial for devices ranging from televisions to smartphones. As the current trend is moving from high contrast 2D displays to 3D or heads-up displays, it is crucial for the pixel technology to keep up the pace. But the majority of these technologies uses either LCD's (Liquid Crystal Displays) or LED's (Light emitting diodes) which are larger in dimension compared to wavelength of operation, therefore it limits the boundaries for scaling down the device size.

Additionally, the majority of the time when we want to produce color we use pigments for visual display both static (posters, paints, printers etc,) and some in dynamic displays like LCD's. Synthetic colorants are expensive to produce, harder to recycle and environmentally damaging. Furthermore they decay and are not stable, they lack resolution what further limits their range of applications. Therefore, in search for suitable replacement, researchers have given considerable attention to structural coloration. As previously mentionned, a famous example of structural color can be seen in nature: the metallic blue color of Morpho butterfly wings. Morpho butterfly helped us realize that there is an alternative way of producing colors without any colorants or pigments, by simply arranging transparent materials in a periodic structure [START_REF] Gralak | Morpho butterflies wings color modeled with lamellar grating theory[END_REF] shaping the interference between the incident and reflected light.

However, the high quality of structural colors exhibited by nature requires complicated and bulky structures [START_REF] Okada | Rendering Morpho butterflies based on high accuracy nano-optical simulation[END_REF][START_REF] Xu | Structural colors: from plasmonic to carbon nanostructures[END_REF] which makes the size of the structure much larger than the wavelength [START_REF] Gu | Color generation via subwavelength plasmonic nanostructures[END_REF].

In this regard, utilizing the artificial structures of 2D metasurface appears favourable for the generation of high resolution structural color beyond the diffraction limit. Structural color generation can be achieved in various number of ways. The phenomenon revolves around the interaction of nanostructures with the incident light to the reflected or transmitted light [START_REF] Inoue | Polarization independent visible color filter comprising an aluminum film with surface-plasmon enhanced transmission through a subwavelength array of holes[END_REF][START_REF] Miyata | Full-color subwavelength printing with gap-plasmonic optical antennas[END_REF][START_REF] Duan | Dynamic plasmonic colour display[END_REF].

In the field of plasmonic structural color reproduction, the plasmonic resonance has shown tremendous ability to selectively filter visible colors in transmission and reflection modes by coupling SP with the incident light. Additionally, the ease of fabrication and high flexibility at nano scale have further propelled its applications in creating high resolution structural color filters beyond the diffraction limit [START_REF] Chen | High transmission and low color cross-talk plasmonic color filters using triangular-lattice hole arrays in aluminum films[END_REF][START_REF] Alexander S Roberts | Subwavelength plasmonic color printing protected for ambient use[END_REF][START_REF] Li | All-metal structural color printing based on aluminum plasmonic metasurfaces[END_REF][START_REF] Duempelmann | Four-fold color filter based on plasmonic phase retarder[END_REF]. Alternatively, using high-index dielectric materials as meta-structures generates electric and magnetic dipole resonances [START_REF] Genevet | Recent advances in planar optics: from plasmonic to dielectric metasurfaces[END_REF] that can induce novel optical phenomenon [START_REF] Krasnok | Towards all-dielectric metamaterials and nanophotonics[END_REF][START_REF] Arseniy | Optically resonant dielectric nanostructures[END_REF]. The application and further explanation regarding the structural color production using dielectric metamaterial will be discussed in a later section.

Color perception

First, it is important to understand that the color of an object, as we see it, depends both on the light source and on the human perception of color. Indeed, the human eye has a limited spectral bandwidth and only detects the values of wavelength that lie in the range of 0.38-0.78 µm. There are two receptors in human eyes, rod cells and cone cells and it is the cone cells that are responsible for color vision. Cone cells are further divided into 3 types, on the basis of which wavelength of color they are more sensitive to: namely, the short (S, short wavelength (blue)), medium (M, greenish yellow), and long (L, yellowish red) cone types [START_REF] Schanda | Colorimetry: understanding the CIE system[END_REF]. Additionally, human eye is able to distinguish up to a few hundred of hue, i.e.

it can detect a change of 1nm in the blue-green and yellow wavelength ranges up to 10nm and more in longer red wavelengths.

RGB color space

Most displays generate colors using a combination of red, green and blue pixels, which are primary colors defining the RGB color space. This is the most traditional way for defining colors of pixels in LDR (Light Dependent Photo Resistor) images. However, the definition for red, green and blue can vary from devices to devices, and are thus called devicedependent. In reality, the RGB need not be defined by their spectral power distribution but The spectral power distribution shown in figure 5.3 is similar to a core of the light source -it provides the fundamental data about a light from which color temperature, chromaticity and other photo-metrics are calculated from. Therefore, in color representation, the D65 white illuminant for each structure is crucial, as the RGB color produced will be a function of the white color the structure can represent.

Structural coloration has proven to generate bright colors with transparent nanostructures.

One could therefore imagine this technology to replace conventional color paint that uses toxic or non-recyclable chemicals. So far, it is however difficult to envision this structural coloration to be the basis for new color displays because this would require to dynamically tune each color pixels. Research into tunability of structural coloring is increasing thanks to its practical applications such as dynamic displays, cryptography and camouflage. In the following, we investigate how VO 2 could be used to dynamically modulate structural colored pixels. and electric dipolar modes. The enhancement of magnetic response caused when incident light interacts with particles of size similar to the wavelength can provide an opportunity to design color filters that uses Mie resonance with higher order multipoles [START_REF] Kivshar | Meta-optics with Mie resonances[END_REF].

In addition we can further adjust the optical properties of metasurfaces using collective effects through coupling between resonators with a more general resonance category called

Fano resonance [START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF] that is based on the constructive/destructive interferences between the radiated and reflected fields [START_REF] Fan | Analysis of guided resonances in photonic crystal slabs[END_REF]. This kind of resonance can be used as another degree of freedom to engineer the structural color generation by interference between the directly reflected light and the guided resonances mode on a surface structure. This type of resonance gives us the flexibility in material choice, scalability for large scale fabrication and angular dependency.

Hence, in recent years, all dielectric nano-materials have been extensively used to realise a broad range of effects and functionalities [START_REF] Decker | High-efficiency dielectric Huygens' surfaces[END_REF][START_REF] Wu | Spectrally selective chiral silicon metasurfaces based on infrared Fano resonances[END_REF]. However, in the field of color filters, the application of dielectric materials was quite challenging because of the high loss in the visible range for standard high refractive index materials (ex: a-Si) which limited the higher saturation attainable by the dielectric metasurface. To overcome this limitation, TiO 2 appears as a promising alternative dielectric material with negligible visible range losses [START_REF] Khorasaninejad | Achromatic metalens over 60 nm bandwidth in the visible and metalens with reverse chromatic dispersion[END_REF][START_REF] Sun | All-dielectric full-color printing with TiO2 metasurfaces[END_REF].

Therefore, we studied square lattice of nano-rods of TiO 2 for structural coloration, as shown in figure 5.4a [START_REF] Jh Lee | Investigation and evaluation of structural color of TiO2 coating on stainless steel[END_REF][START_REF] Sun | All-dielectric full-color printing with TiO2 metasurfaces[END_REF]. TiO We chose to simulate the structure such that, keeping the thickness h constant at 100nm, we vary the period a and the filling factor F F (F F = d a , where d is the diameter) and map out the color thus obtained on a CIE diagram, as shown in the figure 5.2. The reflectance spectra for structures with different period a and different filling factor F F are computed with the rigorous coupled wave analysis method. On the other hand, to avoid complications related to representing colors in a 2D plot, we represent them in terms of CIE-LAB color space values [START_REF] Zhang | A Spatial Extension of CIELAB for Digital Color Image Reproduction[END_REF][START_REF] Connolly | A study of efficiency and accuracy in the transformation from RGB to CIELAB color space[END_REF] In the next section we discuss design strategy for dynamic tunability of the nano pillars, 135 where we replace the TiO 2 nano-pillars with VO 2 nano-pillars in a similar pixel array arrangement.

Tunable color generation using VO 2

Traditional transmissive and reflective displays typically have three sub-pixel regions with static red, green and blue color filters. These sub-pixels control the intensity of each basic color transmitted or absorbed to create arbitrary colors through a process called color mixing.

On the contrary, a display built from a dynamic color changing surface can eliminate the need for individual sub-pixels, increasing resolution without reducing pixel dimensions. While all displays are based on light emission, here the display would be a reflective or transmissive one. And would therefore only require daylight or ambient lighting, reflected off the device [START_REF] Sun | Large-scale nanophotonic phased array[END_REF]. Another possibility is to use it as a transmissive device for augmented reality.

The need for tunable structural color arises from the limitations faced by the current industry. Apart from larger pixel size, most of the tunable pixels in displays industry uses liquid crystal's (LCs), which are big (hundreds micron-scale), have a slow response time to dynamic color switch and are difficult to recycle. The tunablility of structural color has been previously studied and demonstrated but they involved liquid crystal's (LCs) [START_REF] Franklin | Polarization-independent actively tunable colour generation on imprinted plasmonic surfaces[END_REF] or chemical transformation [START_REF] Franklin | Actively addressed single pixel full-colour plasmonic display[END_REF] or mechanical changes [START_REF] Gutruf | Mechanically tunable dielectric resonator metasurfaces at visible frequencies[END_REF]. In our case we demonstrate through simulation the tuning of structural color using VO 2 as the tuning element and we further discuss two different ways for achieving the tunability using VO 2 in different arrangement.

VO 2 nanostructures

Similar to previous nano-pillars structure of TiO 2 , we simulate the reflectivity of VO 2 nanopillars on silica for structural color generation and leverage the IMT of VO 2 to dynamically tune these structural colors. The idea behind is that, in addition to the transition property, compared to TiO 2 , the complex refractive index of VO 2 is larger in the visible range with lower absorption value. As a result we can achieve good index contrast compared to external environment. Secondly, at the transitioned metallic state, the refractive index of VO 2 is changed along with the absorption value, affecting Mie resonance by shifting the electric and magnetic dipole hosted by dielectric VO 2 . Additionally, the radiated field is highly influenced by the increasing absorption during IMT for metallic VO 2 , thus altering Fano resonance value. And therefore we expect to see some variation in the color. Comparing the color production in insulating (5.7) and metallic (5.8), we observe a minor shift in the chromaticity of the reproduced colors. It could be clearly observed in figure 5.9, where the white dots represents the colors reproduced when VO 2 is metallic and black when it is insulating. We see that, the points covered when VO 2 is insulating is much far spread as when it is metallic. This change in chromaticity can be owed to the change in optical properties during the transition, affecting both Mie and Fano resonances.

In figure 2.18 in Chapter 2, we see the evolution of n and k of VO 2 thin-film. We observe greater index contrast at higher wavelengths than at visible range. Therefore, this shows a promising direction for the advancements in dynamic color tunability exploiting VO 2 material's properties as foundation rather than involving additional gases or chemicals for 137 We observe zero reflection from the nanostructures at normal incidence from the figure The ultra-thin size of the color filters has the potential to be integrated on-chip. We believe this research will be important for the realization of optical display, imaging, data storage.

And have potential applications like in cryptography, security, imaging, optical data storage and further optical devices that rely on efficient and simple color tuning mechanisms.

Although, mass production remains still a challenge and, there is still a large space to explore in relation to structural colors before commercialization. We believe colors generated from metasurfaces can be extensively used to take the place of chemical pigments, and dramatically improve quality and sustainability of the environment in the future.

VO 2 metasurface for tunable perfect absorption

Metamaterial based perfect absorbers have gained quite a traction in the recent years [START_REF] Jy Rhee | Metamaterial-based perfect absorbers[END_REF][START_REF] Baqir | Hyperbolic metamaterial-based UV absorber[END_REF]. An electromagnetic (EM) perfect absorber is a device in which all the incident radiation is absorbed at the operating frequency. The initial applications for absorbers were in stealth operation in range of RADAR or microwave/ radio wave operations, reducing radar cross section [START_REF] Ben | Frequency selective surfaces: theory and design[END_REF][START_REF] Ronald | Reflection properties of the Salisbury screen[END_REF]. More recently EM absorbers are used to reduce EM interference by absorbing spurious EM radiations. The control of light absorbance plays a major role in our current photonic industries and can have strong impact in the solar energy harvesting [START_REF] Harry | Plasmonics for improved photovoltaic devices[END_REF][START_REF] Stéphane | Nanostructure arrays in free-space: optical properties and applications[END_REF], bolometers, chemical and biological sensors. Metamaterials are designed to modify the permeability (µ) and the permittivity ( ) such that their impedance match the free space impedance, creating reflection-free artificial medium. This can be highly useful in creating perfect filters with limited losses. In this section we design and experimentally demonstrate design strategy for a VO 2 based tunable metasurface perfect absorber, composed of Si resonators, supporting both electric and magnetic Mie mode.

Concept

As explained in the previous chapter, in some specific cases, dielectric nano-particles can support both electric and magnetic resonances [START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF][START_REF] Andrey B Evlyukhin | Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region[END_REF]. In a dielectric nano-particle, a

Mie magnetic resonance originates from an excitation of a particular electromagnetic mode in the particle. This excitation occurs when the effective wavelength of light inside the particle becomes comparable to the particle's size, exhibiting a circular displacement current of electric field [START_REF] Zhao | Mie resonance-based dielectric metamaterials[END_REF][START_REF] Ahmadi | Physical configuration and performance modeling of all-dielectric metamaterials[END_REF]. The capability of exciting both electric and magnetic responses is of great interest, but simultaneously exciting them gives an additional advantage for accurate light manipulation. Moreover, the spectral positions of electric and magnetic dipole resonances can be tuned by varying the geometry of a dielectric resonator so that desired scattering properties are achieved. This could be key as it gives another degree of freedom to engineer a metamaterial's response by design [START_REF] Miroshnichenko | Optically induced interaction of magnetic moments in hybrid metamaterials[END_REF][START_REF] Zhao | Mie resonance-based dielectric metamaterials[END_REF]. More importantly, constructive or destructive interferences between the fields of electric and magnetic dipole resonances, can affect the particle's scattering properties [START_REF] García-Cámara | Light scattering by an array of electric and magnetic nanoparticles[END_REF]. This phenomenon was studied by Milton Kerker in 1983, his study revealed that one can achieve an asymmetric field radiation with zero backward scattering or zero forward scattering under proper combinations of electric and magnetic dipoles in subwavelength magneto-dielectric particles. According to Kerker's first condition, when µ = , the incident wave experiences a zero backward scattering or in other words, no reflection. The second Kerker condition produces a zero forward scattering (i.e. no transmission) when, µ and satisfy the condition = (4-µ) 2µ=1 . Kerker's conditions could occur in small high-index dielectric nano structures by enabling the spectral overlap between the electric dipole (ED) and magnetic dipole (MD)

sustained by Mie resonances. It has been shown that silicon nano-particles can support electric and magnetic dipole [START_REF] García-Etxarri | Strong magnetic response of submicron silicon particles in the infrared[END_REF][START_REF] Andrey B Evlyukhin | Optical response features of Si-nanoparticle arrays[END_REF] and it has also been demonstrated that using cylinders rather than spheres allows for tuning of the spectral spacing between the electric and magnetic lowest order resonances [START_REF] Staude | Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks[END_REF][START_REF] Andrey B Evlyukhin | Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation[END_REF]. A full overlap between the ED-lattice resonance and MD resonances of nanoparticles in certain spectral range realize the resonant lattice Kerker effect (resonant suppression of the scattering or reflection) [START_REF] Viktoriia | Resonant lattice Kerker effect in metasurfaces with electric and magnetic optical responses[END_REF][START_REF] Geffrin | Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere[END_REF]. Where a strong suppression of reflected light is observed due to the destructive interferences between EM waves scattered by the ED and MD of every nanostructures [START_REF] Van De Groep | Designing dielectric resonators on substrates: Combining magnetic and electric resonances[END_REF][START_REF] Staude | Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks[END_REF].

In our work, we assemble specifically designed Si nano-resonators into two dimensional periodic arrays fabricated on VO 2 thin-film. The principle is as follows: when VO 2 is insulating, the ED and MD resonances overlap and we thus fulfill the Kerker conditions and obtain a zero backward scattering. As VO 2 is lossy even in its insulating state, most of the light is absorbed and we therefore obtain a near-perfect absorber. On the other hand, when VO 2 becomes metallic, the MD resonance disappears and the Kerker condition is no longer fulfilled. However, as the absorption of VO 2 largely increases, we still maintain a nearperfect absorption regime, but spectrally shifted due to the refractive index modulation of VO 2 . The phase state of the VO 2 layer therefore defines the interaction between the incident light and two resonant modes and control the spectral position of the perfect absorption.

We further demonstrate experimentally that, leveraging the reversible tuning of VO 2 we measure a regime of tunable coherent perfect absorption reducing metasurface reflection by more than two orders of magnitude at 1555nm wavelength. The proposed design concept is shown in figure: 5.13. Supporting numerical calculation related to metasurfaces were performed in CST Microwave Studio. This work was done in collaboration with Yuri Kivshar's team from Australian national university and Shriram Ramanathan's team from Purdue University, US.

Metasurface Design and Simulation

Design

The silicon nano-disks were designed on the basis of EM simulation study on absorption, transmission and reflection as a function of Si nano-disk diameter. The simulation was rendered with Si nano-disks fabricated on VO 2 thin-film with SiO 2 as substrate. As shown in the figure 5.14, we see a perfect absorption region in the telecom range.

Metasurface Fabrication

The fabrication of the Si resonator metasurface starts with a 25 nm thin VO 2 layer deposited on quartz by magnetron sputtering from a V 2 O 5 target under Ar/O 2 partial pressure, similar to the conditions and techniques discussed in Chapter 2. The deposition was performed at an elevated temperature of 750 • C, in order to maintain the crystalline growth of VO 2 . After a thin layer of SiO 2 is deposited to protect the VO 2 layer. Following, a 240 nm thick layer of amorphous silicon (a-Si) was deposited by PECVD (plasma-enhanced chemical vapor deposition). The precursor gas used is SiH 4 at a temperature 300 • C. figure: 5.17. The next step involves e-beam lithography for patterning the metasurface; Hydrogen Silsesquioxane (HSQ) resist is spun on the sample and patterned using E-beam lithography.

Subsequently, TMAH developer is used to develop the HSQ resist which is then used as a hard mask to transfer the patterns into a-Si via ICP-RIE (plasma) dry etching with Cl 2 /Ar. Finally, a 400-nm-thick layer of PMMA is spun on the sample, in order to provide an appropriate index contrast and protective layer. Finally, a small part of the sample is covered by a 200-nm-thick gold layer serving as a reference layer for the reflection measurements.

Metasurface Characterization

The first step for characterization was done on 25nm VO 2 on SiO 2 , prior to the fabrication of metasurface. The results of which are mentioned in Chapter 2. Spectroscopic measurement were performed at various stage of deposition as to analyse the effect of the fabrication process on the VO 2 thin-film's transition properties. It was observed that direct deposition of a-Si on VO 2 thin-film, inhibits the IMT, therefore we chose to modify the fabrication process to add a protective layer of SiO 2 circumventing the effect of a-Si deposition on the transition. Following the deposition of metasurface and the subsequent lithography, temperature controlled spectroscopic ellipsometry was performed at 25 • C and 85 • C to check the VO 2 switch quality after each technological steps.

Conclusions and outlook

We have demonstrated, for the first time to our knowledge, tunable Mie-resonant silicon metasurfaces leveraging the transition of VO 2 to control the spectral overlap between the ED and MD resonances. This dynamic tuning was shown to produce a two-order-of-magnitude modulation of the metasurface reflection. We have also demonstrated the metasurface optical response further features a hysteresis behavior, hinting towards the possibility of memory state devices. Although we faced some fabrication issues with the IMT of VO 2 , which we believed were caused by the deposition procedure of the amorphous-silicon, we were able to overcome this issue, by carefully fabricating a protective layer of SiO 2 on VO 2 and evaluated the switching of VO 2 at each steps of fabrication. Therefore the successful fabrication of this device, paves the way to nanoscale optical switches, modulators, and sensors based on all-dielectric resonant metasurfaces.

CHAPTER 6 CONCLUSION AND FUTURE PERSPECTIVES

Nanophotonics have an incredible potential that should play an important role in the coming technological revolution. With the growth in the artificial intelligence and neural computing fields, the demand for on-chip computation and autonomy has propelled the nano-designing to improve the computing efficiency per unit area for the current architecture. In lieu for such revolution, the microelectronics industry is currently facing a strong bottleneck, due to the intrinsic limitations of electronic interconnects. Nanophotonics is one of the possible solution to improve the overall performance of microelectronic systems e.g. by using photonic links as an on-chip information carrier. If we consider the hardware standards, Silicon plays a major role in both the microelectronics and the photonics sector, but the electro-optical modulation properties are inherently weak in Silicon. Therefore, the majority of demonstrated devices in Silicon photonics are passive and numerous researches are now focusing on hybrid technologies. The aim of my thesis was to tackle this problem by introducing active materials in hybrid platforms to design and demonstrate dynamically reconfigurable nanophotonic devices. VO 2 has been at the forefront of active research for understanding the physics behind its phase change properties and its adaptability to modulation devices for various application.

Even though VO 2 has intriguing physics and extraordinary properties it also presents drawbacks for photonics. The high inherent absorption of VO 2 in the telecommunication range represents a major disadvantage for its application in integrated or cascaded nano/micro devices. My task in this thesis aimed at investigating the potential of VO 2 for nanophotonics, with a specific emphasis on how to circumvent these drawbacks and to design and demonstrate efficient integrated devices. Chapter 1 and 2 are dedicated to understand the phase transition in VO 2 and its associated physical mechanisms. Following this study, we were able to create an optical model which successfully describes the permittivity dispersion of VO 2 in all of its states, ranging from insulating to fully metallic as well as all the mixed metal-insulator partial states in between.

Even though VO 2 has been widely studied, its full potential has not really been exploited due to its high inherent optical absorption losses. Through the design strategy in Chapter Showing, that such a resulting metamaterial presents distinct optical tunability compared to unpatterned VO 2 and can indeed be exploited as a designer phase-change metamaterial with adjustable refractive index modulation and absorption modulation. This provides us the ability to design a system having large refractive index tunability without inducing modulation of the extinction coefficient at specific wavelengths. Hence paving the way for low-loss tunable metamaterials.

Finally, in the last chapter, we explore the use of VO 2 as a means to create tunable metasurfaces in two important domains of application: tunable structural coloration and tunable perfect absorption. In the field of color generation technologies, we are expected to move past our current generation of stack displays and pigment dependencies to more simple surface displays aiming for more power efficient, ecological and vibrant display technology.

One solution is to use structural color generation. Initial structural colors were created using designed TiO 2 dielectric nanostructures invoking Mie resonances and Fano resonances, but most nano-based devices studied so far are passive [START_REF] Yang | Structural colors in metasurfaces: principle, design and applications[END_REF]. My preliminary simulations results

show us the potential of VO 2 to induce dynamic tunability for the structural color generation.

We try to achieve the tunability in mainly two ways; First, imprinting TiO 2 nanostructures on VO 2 thin films, manipulating the Fano resonance via the IMT transition of VO 2 , thus promoting a tunable color. The second way is to construct directly VO 2 nano-structures designed specifically to support Mie resonances such that by controlling the state of VO 2 , we could achieve an original color switching paradigm. These results indicate that the IMT of VO 2 may have a future in tunable color displays.

The second section of this chapter is focused on Mie-resonant silicon metasurfaces tunable by VO 2 . Using arrays of a-Si nano-pillars, specifically designed to sustain overlapping ED and MD resonances, we have experimentally demonstrated that the reversible tuning of the VO 2 state enables switching the Kerker condition and introduces a two-order-ofmagnitude modulation of the reflection at telecom wavelength ranges. The metasurface optical response further features a hysteresis behavior due to the IMT of VO 2 , thus adding an interesting possibility for a non-volatile memory of its state. We believe that our work uncovers a great potential for tunable dielectric metasurfaces, paving the way for nanoscale optical switches, modulators, and sensors based on all-dielectric resonant metasurfaces.

The majority of the work in this thesis was focused on incorporating an additional tunable degree of freedom leveraging VO 2 for different passive nanostructure devices, ranging from integrated guided optics to free-space metasurfaces. Many different adaptations, tests, and experiments have been left for the future due to lack of time (i.e. the experimental demonstration of the modulator device and the tunable structural color). Future works concern better implementation of VO 2 for its tunability, because in order to beat the virtue of electronics, i.e. integrability and re-programmability, it is necessary for photonics to evolve into next generation miniaturized systems, that can incorporate reconfigurable platforms with adjustable properties and are able to completely manipulate the key features of the information carrier (photons).

Additionally, future work also concerns with adapting the designer VO 2 NCs: to harness the advantage of optics in realization of on-demand adaptive functionalities, design of dynamically reconfigurable meta-surface based tunable devices having unprecedented performance is indispensable. Keeping in mind the energy, the robustness, the memory and the endurance of multi-cycle performance of the photonic device in order to outperform its electronic counterparts, we may also have to look into new design paradigm and use new material approaches [START_REF] Salinga | Monatomic phase change memory[END_REF][START_REF] Zhang | Single-element glass to record data[END_REF]. In order to achieve such a grand feat we might also take into account the emerging field of Deep Learning as powerful platform for designing, intelligent analysis and optimisation of meta-surfaces leveraging VO 2 tunable properties or any adaptable PCMs by reducing computational loads and solving multi dimensional EM simulation problems and thus helping us to find optimized solutions for each applications. • Normal components of the magnetic flux densities are continuous, such that

B 1n = B 2n (A.6)
So in a summary, these boundary conditions, where the tangential components of E and H must be equal and continuous and normal components of D and B must be equal or continuous, must be followed whenever we encounter an interface between two dielectric media provided there are no free charges at the interface.

A.2 Total internal Reflection

We are well versed from ray theory that, when light is incident on an interface between two medium, at a given angle the light will be totally reflected internally. The conditions being that medium of propagation should have refractive index n 1 greater than n 2 , the outside medium. The schematic of which is shown in figure A.1. In terms of wave theory, the question thus arises, the existences of a transmitted wave and its nature. Therefore in order to find the wave characteristics of transmitted wave, we assume an existence of transmitted wave at an angle i 2 with respect to normal n 2 having a propagation constant k 2 , then the electric field associated with the wave is given by;

E 2 = E 20 exp[i(k 2x x + k 2z z -ωt)] = E 20 exp[i(k 2 xcosi 2 + k 2 zsini 2 -ωt)] (A.7)
Now if we apply Snell's law of refraction, we know see that,

sini 2 = n 1 n 2 sini 1 (A.8) cosi 2 = n 1 n 2 n 2 2 n 1 2 -sin 2 i 2 (A.9)
Since, i 1 is greater than φ c , which is the critical angle, then equation A.11 become imaginary. Based on the schematic, we can define the wave equation in the region 0<x<d as, This shows the relationship between mode number, β and n e f f . Although using planar mirror waveguide is useful for understanding, it is more practical to understand and evaluate the TE modes for step index dielectric waveguide as it is closer to objective of the thesis. Therefore, we evaluate TE modes for the schematics as shown in the figure A.4. Until now, we have concerned our self in understanding the properties of EM waves in an confined medium and its properties while propagating in the medium. Based on this understanding we have developed the concept of modes, its associated power and its propagation properties. One of the most important property of an optical waveguide is its ability to share power with 100 % when certain condition are fulfilled. This sharing of power is ubiquitous and are known as mode coupling. This is one of a common physical principle that underpins the behaviour nearly all perturbed waveguide structures; like waveguide tapers, bends, grating and beam couplers [START_REF] Yariv | Coupled-mode theory for guided-wave optics[END_REF]. The principle of coupled mode theory can easily be understood with one of the most profound and versatile device in integrated optics, i.e. the directional coupler. The directional coupler works by coupling two modes traveling in the same direction such there is a power redistribution between the two cores of the waveguide and after a certain distance, there is complete transfer of power.

d 2 E y dx 2 + [k • 2 n 2 -β 2 ]E y = 0 (A.
Based on our previous section, we know that, there is a evanescent field extending outside any dielectric waveguide, whose field is exponentially decaying. If we bring two similar and symmetric parallel waveguides sufficiently close together, the evanescent part overlap spatially. Usually, the inter-waveguide gap required for this overlap to be significant is of the order of the width of the waveguide, the schematics of which is shown in figure A.5. To understand the coupling power transfer mechanism, rather than using similar waveguide, we take 2 waveguides such that the propagation constant (β) is different for each waveguide, as shown in the schematic figure 3.4. This case is similar to the scope this thesis.

We start with the following solutions; Here, unlike previous section κ represents mode coupling coefficient, c represents butt coupling coefficient and χ propagation constant. These generalised coupled-mode solutions are further solved to describe the coupling between two waveguides, irrespective of their shape. These equations are important as we can further evaluate them to calculate coupling coefficient like, butt coupling coefficient, mode coupling coefficient and change in average propagation constant [START_REF] Miller | Coupled wave theory and waveguide applications[END_REF]. The coupling mechanism usually entails the transfer of power from one waveguide to another. This transfer also influences the mode in the adjacent waveguide modifying its shape. This interaction can happen in one of the two ways, either evanescent waves of symmetric modes interacts or asymmetric modes interact. This interaction create a new mode pattern and are called supermodes. The power transfer in super modes can be calculated using the power equation for the waveguide (waveguide 1) which is given by, With assumption of complex conjugate symmetry of c 21 = c 12 and χ 1 = χ * 2 , where δ = β 2 -β 1 2 . Following we now try to analysis the power transfer that takes place during mode coupling.

E = A(z) E 1 + B(z) E 2 (A.
P = 1 2 ∞ -∞ ∞ -∞ ( E ×
In order to do that we assume that there is no loss of power along the propagation direction ( dP dz = 0) but only a transfer of power from one guide to another happens. So we can rewrite the equation as; In MMM, the optical structure is discretized in to various sections along the propagation direction and replacing the structure to be modeled by an equivalent structure made up of a succession of elementary straight guides of variable sections. The advantage of this equivalent structure is that we know how to calculate the guided and radiated modes of each straight portion. These modes will be called local modes of the portion considered. In our case for 3D structure, the effective indices and the profiles of the various components of the electromagnetic field of the local modes are calculated using matlab modeling tools, whose the simulation numerical theory was discussed earlier in this chapter was developed in the For our case, we consider a plane interface between two portions of straight guides denoted 1 and 2 in z = 0, as shown in figure B.2. We seek to calculate the field reflected and transmitted to this interface when it is lit by a guided mode of index p in guide 1, as shown in figure 2. Based on the boundary conditions we obtain the following relations: The negative sign (-) for the tangential components of magnetic induction is imposed by the conservation of the Lorentz reciprocity principle for a counter-propagating wave.

a 1,+ p E
Therefore in order to calculate the reflection coefficients R i,p and transmission T j,p of the interface, we construct a vector products between the fields E and H so as to obtain the energy transferred to each mode. For equation B.1, we represent it with respect to the magnetic field with modes k propagating in guide 1, whereas for equation B.1, we represent the electric field with modes l propagating in the guide 2. We thus obtain the relations: where k varies from 1 to n, and l varies from 1 to m. And thus we obtain a square matrix of dimension n + m. The resolution of this system makes it possible to obtain the coefficients a 1,- i and a 2,+ j corresponding to the excitement of p mode. The PEH and PHE conventions are used when the variable indices i or j are applied respectively to the field E or H. It is to note that we can represent n i=1 P EH 1,1 i,k = δ i,k .P EH 1,1 k,k and m j=1 P HE 2,2 j,l = δ j,l .P HE 2,2 l,l are the diagonal matrix. And P EH 1,1 p,k = δ p,k .P EH 1,1 p,p vector containing a single non-zero element at index p.

a 1,+ p E 1 p , H 1 k + n i=1 a 1,- i E 1 i , H 1 k = m j=1 a 2,+ j E 2 j , H 1 k (B.
The reflection coefficient r 1,2 i,p and t 2,1 j,p are obtained by the relations;

r 1,2 i = a 1,- i (B.7) t 2,1 j = a 2,+ j (B.8) With    a 1,- i a 2,+ j    =    -n i=1 E 1 i , H 1 k m j=1 E 2 j , H 1 k n i=1 E 2 l , H 1 i m j=1 E 2 l , H j 2    -1    E 1 p , H 1 k E 2 l , H 1 p   
By varying the index p of the incident field in the medium 1, we obtain the set of reflection and transmission coefficients of the structure when the incident field is one of the modes of the guide for region 1. We can notice the set of reflection coefficients constitutes a square matrix of dimension n×n, while the transmission coefficients form a rectangular matrix of dimension n×m. Similarly, it should be noted that the same reasoning can be used when the incident field comes from region 2. Therefore we obtain the coefficients a 1,- The reflection coefficient r 2,1 j and t 1,2 j are obtained by the relations; r 2,1 j = a 2,+ j (B.11)

t 2,1 i = a 1,- j (B.12) With    a 2,+ i a 1,- j    =    -m j=1 E 2 j , H 2 k n i=1 E 1 i , H 2 k m j=1 E 1 l , H 2 j n n=1 E 1 l , H i 1    -1    E 1 p , H 1 k E 2 l , H 1 p   
By varying the index p of the incident field in the medium 2, we obtain all the reflection and transmission coefficients of the structure when the incident field is one of the modes of the guide for region 2. We can notice the of reflection coefficients constitutes a square matrix of dimension m×m, while the transmission coefficients form a rectangular matrix of dimension m×n.

B.1.2 Interface matrix

To obtain the interface matrix of refractive index i, it is necessary to obtain a matrix relation which connects the propagating and counter-propagating waves propagating in the guide of refractive index i-1 with those propagating in the guide of refractive index i. When we try to express the relationships between the reflection and transmission of light at an interface, we consider the case shown in figure B.3 for which, we must determine the link between propagating waves and counter-propagating in both media. For this, we determine the link between the two incoming waves a 1,-and a 2,+ , which were not taken into account in the above figures. With the calculation for the reflection and transmission coefficient for the mode matching method, we have laid the foundation for the theory for all the numerical calculation that would enable us to simulate and fabricate the novel modulator.

Therefore, if the optical modulator is properly designed, when VO 2 is semiconducting, the propagation constant doesn't match with the bus waveguide, consequently, light from the input Si waveguide propagates through without coupling to the VO 2 patched waveguide.

However, when the phase of VO 2 is changed from dielectric to metal, the wave changes direction and couples into the VO 2 patched waveguide. The propagation loss of the hybrid VO 2 patched waveguide abruptly increases due to the VO 2 lossy metallic characteristics. As a result, the light amplitude at the output Si waveguide decreases after being absorbed along the hybrid VO 2 waveguide. By repeating this operation back and forth, we can modulate the optical signals at the output of the Si bus waveguide. This simple architecture has shown to be a promising solution for on-off operations. When the device is "ON" there is not a single photon that is lost in the device, as no light is coupled in the VO 2 -based WG. On the contrary, in the "OFF" state, we could potentially absorb all of the incoming light.

In order to implement this design strategy, we have to compute the light propagation and its interaction with the strip waveguide structure and its confinement. So to understand that wave, we have to perform numerical analysis for the E and H components in terms of material properties, design and boundary conditions. Theory involving the numerical analysis is explained in the main text section and the figure 3.2 has been calculated employing the following the numerical theory.

  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . .

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Résume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 0.1 Part 1: Guide d'onde modulateur hybride VO 2 /Si . . . . . . . . . . . . . . xxi 0.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 0.1.2 La photonique sur silicium: Avantages et inconvénients . . . . . . . xxii 0.1.3 Oxyde de Vanadium (VO 2 ): Introduction et science de la modulationxxiii 0.1.4 Caractérisation du VO 2 . . . . . . . . . . . . . . . . . . . . . . . . xxvi 0.1.5 Modulateur guide d'onde hybride VO 2 /Silicium: Conception et principe de fonctionnement . . . . . . . . . . . . . . . . . . . . . . xxx 0.1.6 Simulation, fabrication et résultat . . . . . . . . . . . . . .

3. 5 . 1 Chapter 4 :

 514 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Use of this modulator as a partial power transfer . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multipolar Resonances with Designer Tunability Using VO 2 Phase-Change Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mie theory for wavelength-scale particles . . . . . . . . . . . . . . . . . . 4.3 VO 2 Nanocrystals: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 VO 2 Nanocrystals: Fabrication and Characterization . . . . . . . . . . . . . 4.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Effective parameter retrieval of VO 2 Nanocrystals . . . . . . . . . . . . . . 4.5.1 Derivation of the expressions for effective permittivity and permeability106 4.5.2 Modelling the effect of interactions between nanoparticles . . . . . viii LIST OF TABLES 2.1 Parameters for Reactive Sputtering Technique for the growth of VO 2 thinfilms on quartz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Power transfer simulation calculated around the core area of the waveguide. 80% of the input power is calculated at the central area of the core, rest is dissipated at the edges of the waveguide. . . . . . . . . . . . . . . . . . . . 3.2 Geometric parameters for VO 2 waveguide modulator. . . . . . . . . . . . . xi 1.3 (a) Crystal structure of VO 2 in its metallic phase. It has a rutile structure with small atoms being vanadium and large atoms being oxygen. (b) The cationcation interaction further splits the t g and e g orbitals. Crystal structure of (c) metallic rutile phase and (d) insulating monoclinic phase of VO 2 viewing form the b axis. (e) Proposed band structure of the metallic and insulating phase based on the Goodenough and Mott pictures, respectively. . . . . . . 2.1 Experimental phase diagram of VOx. This image has been adapted from [116, 117] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 14 4 . 2 120 5. 1 128 5. 2 129 5. 3 130 xvii 5 . 4 176 B. 3

 1442120112821293130541763 Evolution of the output power (P out) versus the variation of the wavelength with losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Schematic cross-sectional view of the coupler section in the optical device as simulated with light modulation as function of transition in VO 2 . . . . . . 3.16 Intensity of the guided mode as a function of position along the device. . . . 3.17 Fabrication steps involved for the realization of VO 2 modulator . . . . . . . 3.18 SEM image of patterned hard mask before the first ICP etching. . . . . . . . 3.19 SEM image of patterned hard mask after ICP etching. . . . . . . . . . . . . 3.20 Simulation for VO 2 waveguide modulator as partial power filter using the intermediate state of VO 2 transitions. . . . . . . . . . . . . . . . . . . . . . 3.21 Gradual power variation in the bus waveguide (Core 1) as function of changing refractive index. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Real and imaginary part of the dielectric permittivity of a VO 2 thin-film in the Vis/NIR range as a function of temperature. . . . . . . . . . . . . . . . xv Calculated absorption efficiency of VO 2 spheres with radius 50 nm in their a) insulating and b) metallic states embedded in quartz. With colored lines are represented the dipolar electric a 1 (red), dipolar magnetic b 1 (yellow), quadrupolar electric a 2 (light blue) and quadrupolar magnetic b 2 (dark blue) contributions to Q abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 VO 2 NCs TEM image (top) and sketch (bottom) . . . . . . . . . . . . . . . 4.4 Evolution of ellipsometric parameters Is and Ic measured as a function of temperature. The accusation was performed on VO 2 NC at an incident angle of 65• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Thermal Hysteresis loop of VO 2 NCs. . . . . . . . . . . . . . . . . . . . . 4.6 Dynamic optical measurements of the VO 2 NCs IMT. The rate of temperature ramp is 1 • per minute. The initial stating temperature was 23 • C . . . . 4.7 3D representation of optical stacks for VO 2 NC embedded in fused silica. .4.8 Fitting (solid line) and experimental measurements(circles) of the ellipsometric parameters Is (blue) and Ic (red) for both states of VO 2 -NCs: a) metallic and b) insulating for a VO 2 -NCs for size of R = 35nm and a filling-factor f = 0.18. (incident angle: 55 • ). . . . . . . . . . . . . . . . . 4.9 Scheme of the analyzed system: dimer formed by spheres (R = 25nm) separated by a gap and illuminated with x-(left) and y-polarized beams (right). With arrows are represented the polarization of the illuminating beam (E, blue arrow) and the wave vector (k, red arrow). . . . . . . . . . . 4.10 Absorption cross-section C abs of VO 2 dimers in its metallic phase (R = 35 nm) when illuminated with (a) x-(b) y-and (c) 45 • polarizations (4.9) for different interparticle distances (gaps). As an inset in (a) and (b) is indicated the maximum spectral shift between the interacting and non-interacting (dashed black lines) dimer. In (c) the spectral shift ∆λ is represented as a function of the gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Absorption cross-section C abs of VO 2 dimers in its insulating phase (R = 35 nm) when illuminated with (a) x-(b) y-and (c) 45 • polarizations (4.9) for different interparticle distances (gaps). As an inset in (a) and (b) is indicated the maximum spectral shift between the interacting and non-interacting (dashed black lines) dimer. In (c) the spectral shift ∆λ is represented as a function of the gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 4.12 Absorption cross-section C abs spectra for the reference monodisperse system (R = 35 nm, black line) and others with (a) symmetric and (c) asymmetric distribution of NC sizes. Size distribution of the (b) symmetric and (d) asymmetric systems. With the light dotted line is represented the size of the monodisperse distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.13 a) Effective dielectric function ( ef f ), b) magnetic permeability (µ ef f ), and c) real and d) imaginary parts of the effective refractive index (m ef f = n ef f + ik ef f ) of a slab of quartz with VO 2 spheres with radius R = 35 nm and a filling fraction f = 0.18 in their insulating (low temperature) and metallic (high temperature) states. . . . . . . . . . . . . . . . . . . . . . . 116 4.14 Variation of effective optical parameters for different sizes of nanocrystals in both states of VO 2 a) and b) n ef f and k ef f for insulating VO 2 with a filling fraction f = 0.18. c) and d) n ef f and k ef f for metallic VO 2 . e) and f) represent the amplitude of modulation of the effective refractive index of the metamaterial as a function of NCs size. e) ∆n ef f and f) ∆k ef f . . . . . . 117 4.15 Spectral variation of the amplitude of modulation of the effective refractive index (∆n ef f and ∆k ef f ) of the metamaterial for selected NCs size. Bottom graph: Figure of Merit (FOM=∆n ef f /∆k ef f ) of the VO 2 -NCs platform for different sizes of nanocrystals and compared to the FOM for a VO 2 thin-film.119 4.16 Calculated absorption cross-section efficiency Q abs of NCs of different sizes in their insulator and metallic states embedded in quartz. With colored lines are represented the dipolar electric a 1 (red), dipolar magnetic b 1 (yellow), quadrupolar electric a 2 (light blue) and quadrupolar magnetic b 2 (dark blue) contributions to Q abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized response spectra of human cones, to monochromatic spectral stimuli, with wavelength given in nanometers [213]. . . . . . . . . . . . . . The chromaticity diagram of the CIE 1931 color space. The curved outer limit is the spectral locus, with wavelengths shown in nanometers. The colors your monitor displays in this image are specified using sRGB, so colors outside the sRGB range are not displayed correctly. This diagram shows the maximum saturated vivid colors that can be perceived by human eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This the D65 sunlight spectrum simulated by D series daylight simulator by CIE standard. It cover the whole visible spectrum and the roughness shows the various points of absorption and diffusion by earth's atmosphere. This figure is taken from [215] . . . . . . . . . . . . . . . . . . . . . . . . . . . TiO 2 nanostructures and the constituent color reproduction with varying dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Real and simulated image based on structural color of TiO 2 . . . . . . . . . . 5.6 TiO 2 D65 white reference and the variation of color and L value as the function FF and a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The structural color reproduction by VO 2 in semiconductor state. . . . . . . 5.8 The structural color reproduction by VO 2 in metallic state. . . . . . . . . . 5.9 VO 2 nano structure color gamut comparison . . . . . . . . . . . . . . . . . 5.10 TiO 2 metasurafce fabricated on VO 2 thin film deposited on SiO 2 . . . . . . 5.11 TiO2 nanostrutures on VO 2 , the color switch observed at both the states of VO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Photographs of TiO 2 nano pillars viewed at (a) normal incident and (b) at an angle. For the images (c) and (d) we observe the different structural colors at different angles, as they are highly sensitive to angle. . . . . . . . . . . . 5.13 a) Design concept: a silicon meta-atom is embedded into glass/PMMA surrounding with a buried thin VO 2 layer. VO 2 thickness: 25 nm, Si disk height: 240 nm, disk radius: 270 nm, square lattice period: 800 nm, PMMA layer thickness: 400 nm and b) Electron microscope image of the fabricated Si-VO 2 metasurface prior to the PMMA coating. c) Functionality concept: at low temperatures, the metasurface reflects light at a designed wavelength, and d)at high temperature the light is absorbed by the metasurface. . . . . . 5.14 EM simulation study on absorption, transmission and reflection as a function of Si nano disk diameter, with a period of 830nm and VO 2 is insulating. The red dotted circle shows perfect absorption (transmission is near 0) region due ED and MD overlap and black dotted circle shows the split in the dipole overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 EM simulation study on absorption, transmission and reflection as function of Si nano disk diameter, with a period of 830nm and VO 2 is Metallic. The red dotted circle shows disappearance of the MD when VO 2 becomes metallic but we keep a strong absorption. . . . . . . . . . . . . . . . . . . . xviii 5.16 EM simulation study on absorption, transmission and reflection of Si nano disk diameter 540nm scanned as function of temperature, with a period of 830nm. The dotted line shows the wavelength of interest 1.55 µm, for our tunable near-perfect absorption device. . . . . . . . . . . . . . . . . . . . . 5.17 Device sketch ofthe a-Si metasurface on VO 2 thin films. The layers are defined as: a. PMMA, b. a-Si nano disks, c. SiO 2 protective layer, d. VO 2 thin film and e. SiO 2 substrate layer . . . . . . . . . . . . . . . . . . . . . 5.18 Is and Ic values as a function of temperature at the final stage of the fabrication of the metasurfaces. Analysing the figure, we see the effect of VO 2 transition, suggesting successful completion of all the stages of fabrication. 5.19 Reflection measurements versus temperature of the metasurface. We observe a perfect absorption at 1550 nm when the phase of VO 2 is metallic. And we also observe the tunability and reversibility as due to temperature induced phase modulation of VO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Experimental results on the tunable reflection of the Si-VO 2 metasurface. (a) Reflection spectra for two temperature points: 40 • C (a dielectric phase of VO 2 ) and 80 • C (a metallic phase of VO 2 ). (b) Reflection contrast between the two temperature points. (c) Reversible temperature tuning of the metasurface reflection at 1550 nm wavelength exhibiting a hysteresis-like behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Transmitted wave characteristics in total internal reflection (TIR), the direction of propagation in z direction. . . . . . . . . . . . . . . . . . . . . . . . A.2 Schematics of 1-D planar waveguide and 2-D rectangular planar waveguide . A.3 Planar mirror waveguide with reflective metal surface on the boundary and the index variation map as function of x-axis value. . . . . . . . . . . . . . A.4 Symmetric step-index planar waveguide structure and the index variation map as function of x-axis value. . . . . . . . . . . . . . . . . . . . . . . . A.5 Overlap of transverse evanescent field between two adjacent parallel symmetric slab waveguide structure and the index variation map of guide as function of x-axis value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 a) Structure to model. b) Decomposition of the structure to be modeled into elementary straight wave guides. . . . . . . . . . . . . . . . . . . . . . . . xix B.2 Schematic diagram of the calculation of reflection and transmission coefficients at an interface in the case of an incident wave equal to (a). a n 1,+ and (b) a n 2,-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between the amplitudes of electromagnetic waves at an interface.180 xx RÉSUME 0.1 Part 1: Guide d'onde modulateur hybride VO 2 /Si 0.1.1 Introduction Les communications optiques, internet, la détection, l'informatique, les applications de type "machine learning" ou "cloud computing" font l'objet d'une demande toujours croissante en termes de bande pasante et de débit de données. Pour les communications sur puce, les solutions électroniques existantes basées sur des semi-conducteurs et des technologies CMOS sont actuellement mises à rude épreuve et sollicitées à leur limite maximale. La vitesse maximale de la nouvelle connexion mobile 5G est de 10 Gbits/s. Lorsque la vitesse dépasse les normes 5G de 10 Gbits/s, les technologies actuelles d'interconnexions métalliques transportant le courant entre puces déforment et atténuent fortement le signal de données en raison de la résistance parasite, de l'inductance et de la capacité de la piste métallique.

Figure 1 :

 1 Figure 1: Indice de réfraction et coefficient d'absorption extraits de VO 2 en fonction de la température. Cette extraction est basée sur un fit de données ellipsométriques, où l'on arrive à un accord entre les données optiques et expérimentales

Figure 3 :

 3 Figure 3: Variation spectrale de l'amplitude de modulation de l'indice de réfraction effectif (∆n ef f et ∆k ef f ) du métamatériau pour la taille des NCs sélectionnés. Graphique du bas : Figure de mérite (FOM=∆n ef f /∆k ef f ) de la plateforme VO 2 -NCs pour différentes tailles de nanocristaux et comparé au FOM pour une couche mince VO 2 .

  devient métallique, la résonance MD disparaît et la condition Kerker n'est plus remplie. Cependant, comme l'absorption du VO 2 augmente largement, nous maintenons toujours un régime d'absorption quasi-parfait, mais décalé spectralement en raison de la modulation de l'indice de réfraction du VO 2 . L'état de phase de la couche VO 2 définit donc l'interaction entre la lumière incidente et deux modes de résonance et contrôle la position spectrale de l'absorption parfaite. Il est possible d'approcher un régime d'absorption parfaite cohérente réduisant la réflexion des métasurfaces de plus de deux ordres de grandeur à une longueur d'onde de 1555 nm. Le concept proposé est illustré dans la figure : 5.13. La fabrication d'un tel échantillon est une tâche fastidieuse car il y a de multiples empilements de couches soigneusement ajustées, déposées et gravées. Nous avons mesuré et caractérisé l'échantillon plusieurs fois au cours de sa fabrication pour nous assurer que l'échantillon reste intact et que son fonctionnement peut être répété sur de nombreux cycles. Toutes les étapes importantes sont mentionnées en détail dans le chapitre 5. li Finalement, nous avons caractérisé le dispositif ainsi fabriqué et démontré, pour la première fois à notre connaissance, des métasurfaces hybrides silicium/VO 2 présentant des résonances de Mie accordables dynamiquement. En jouant sur le recouvrement spectral des ED et MD et en tirant parti de la transition de phase du VO 2 , nous avons démontré expérimentalement une modulation de la réflexion de deux ordres de grandeur à λ = 1550nm. Par la suite, nous avons également démontré que la réponse optique de la métasurface présente un comportement d'hystérésis et pourrait donc être utilisée comme mémoire non volatile. Conclusion Cette partie correspond au chapitre 5 de la thèse. Nous y avons démontré deux types de métasurfaces activement accordables en espace libre, exploitant la propriété de transition du VO 2 . Le concept qui sous-tend les deux démonstrations est similaire, mais le spectre d'application peut être varié. Nous avons utilisé des matériaux diélectriques structurés en métasurfaces et présentant des résonances ED et MD accordables permettant de contrôler le chemin de diffusion du rayonnement et son absorption. Dans un cas, ce principe permet de moduler les résonances de manière à générer des couleurs structurelles; dans l'autre cas, il permet de créer un absorbant parfait avec un avantage d'accordabilité grâce au VO 2 . Ceci montre l'impact potentiel du VO 2 et ses différentes domaines d'applications possibles.

  mechanics analytics. Similar to the Kronig Penny interacting potential-well problem, where we discretize the original band into multiple bands because of the increased interaction by nearby energy systems. Like in the case for two interacting well problem, analytical solution gives us two solutions where the original energy band splits into one higher energy band and one lower energy bands. Which is similar when electrons face large enough Coulombic repulsion U , the electron correlation effectively splits the single half filled bands in two: one containing electrons occupying the site, called a full lower band (lower Hubbard band, LHB) and an empty higher band, known as upper higher band (upper Hubbard band, UHB). The magnitude of the coulomb repulsion U is the energy experienced between the jumping electron and the electron that are already present at the lattice position, which dictates the overall proportion of band splitting. This model is known as the Hubbard model[START_REF] Hubbard | Electron correlations in narrow energy bands. II. The degenerate band case[END_REF] and the materials that experience such strong electron correlations are referred to as "Mott insulators". The competition between the carrier localisation and delocalisation can drive a transition between the metallic and insulating states, a phenomenon also known as "Insulator-to-Metal Transition" (IMT) . Most transition metal oxides (partially filled 4d, 4f, 5f orbitals) should behave as metals according to band theory. But because of electron correlation and the band splitting, they act as Mott insulators. There are different ways to induce phase transition, one such way is to manipulate the delocalisation energy by introducing stress. Compressing lattice constants by an external stress can modify the bandwidth (W) leading to bandwidth-induced IMT. Another way is by manipulating the Coulombic repulsion energy U by doping carriers charges, hence reducing the repulsion energy. In some cases like temperature driven transitions, both the bandwidth and carrier-induced mechanisms come into play. The density of states with varying relative magnitude of Coulomb repulsion (U) to bandwidth (W) is shown in Figure 4b(1.1).Additionally, in strongly correlated oxides, oxygen orbitals also play an important role in dictating the transition property of the material. Following the assumption that there are no hybridization between the transition metal 3d band and the oxygen 2p band (O2p), it is possible to categorize transition metal oxides based on the relative level of O2p band[START_REF] Taguchi | Bulk screening in core-level photoemission from Mott-Hubbard and charge-transfer systems[END_REF][START_REF] Zaanen | The electronic structure and band gaps in transition metal compounds[END_REF]. The first category is a Mott-Hubbard type, where the O2p level is under the lower Hubbard band. And the second category is a charge transfer type, where the O2p lies in between two Hubbard bands Figures(1.

Figure 1 . 1 :

 11 Figure 1.1: (a) Splitting of a normal band into upper and lower Hubbard band due to electron correlations. (b) Bandwidth controlled metal-insulator transition. Changing the bandwidth W could induce a metal-insulator transition. (c) and (d) Oxide Mott insulators could be categorized into two types based on the relative position of the oxygen band and Hubbard bands. (c)Mott-Hubbard insulator where the oxygen p-band lies under the lower Hubbard band; and (d) charge-transfer insulator where oxygen p-band is in between the lower Hubbard band and upper Hubbard band.This figure is reproduced from Ref.[START_REF] Taguchi | Bulk screening in core-level photoemission from Mott-Hubbard and charge-transfer systems[END_REF][START_REF] Zaanen | The electronic structure and band gaps in transition metal compounds[END_REF]

Figure 1 . 2 :

 12 Figure 1.2: Schematic illustration of a Peierls transition. Energy versus density of states a) for a 1D lattice of equally spaced atoms and b) for a dimerized 1D lattice. The horizontal dotted line represents the Fermi energy c) The changes in the lattice across the Peierls transition. The dotted lines represent the original atomic position and the solid sphere represents the perturbed state.

  as R phase from body-centered tetragonal) with lattice constant a = b = 4.53Å, c = 2.87Å as shown in figure: 1.3a. The Vanadium atoms occupy the vertices and the center of the tetragonal unit cell. Each of which is surrounded by six oxygen atoms sitting at the centre of an octahedron made of neighbouring oxygen atoms. Along the c axis, these octahedron share their edges with their neighboring octahedron within the (001) plane 1.3b. While crossing the transition temperature back into the insulating state, the dimerization of vanadium atoms occurs along the original rutile c axis and the tilt of the dimer is with respect to the rutile c axis, while the oxygen atoms roughly stay at the same positions, represented in the figure 1.3 c and d. At the body center of the unit cell is the vanadium atom with the c coordinate equal to 1 2

  (a) VO 2 rutile crystal structure (b) VO 2 monoclinic crystal structure (c) b R axis view of lattice arrangement in Metallic rutile phase (d) b M axis view of lattice arrangement in Insulating monoclinic phase (e) Band structure of the metallic.(f) Band structure of the insulating.

Figure 1 . 3 :

 13 Figure 1.3: (a) Crystal structure of VO 2 in its metallic phase. It has a rutile structure with small atoms being vanadium and large atoms being oxygen. (b) The cation-cation interaction further splits the t g and e g orbitals. Crystal structure of (c) metallic rutile phase and (d) insulating monoclinic phase of VO 2 viewing form the b axis. (e) Proposed band structure of the metallic and insulating phase based on the Goodenough and Mott pictures, respectively.

  .3e, broaden and overlap to form the electronic band. The band structure as shown in figure1.3 is generally accepted for the representation of the metallic phase, but exact band structures which includes the Fermi level, the bandwidth, and the offset between d // and π * band are not well characterized. It has been estimated that the total bandwidth of d // and π * bands is ∼ 2eV and the Fermi level is located at 0.6 eV above the minimum of conduction band[START_REF] Ruzmetov | Hall carrier density and magnetoresistance measurements in thinfilm vanadium dioxide across the metal-insulator transition[END_REF][START_REF] Biermann | Dynamical Singlets and Correlation-Assisted Peierls Transition in VO 2[END_REF]. An interesting point to note here is that because of the quasi one dimensional character of the d // band, conductive anisotropy can be observed in VO 2 . The measured conductivity, parallel along the rutile c axis is larger than along the perpendicular a axis at room temperature. This conductivity difference is not too large, but, it is probably due to contribution from the π * band. 1.3.4 Discussion on Vanadium oxide transition The tilting of the vanadium pairs causes a shifting of the π * band. Collectively this causes a band gap opening, following Peierls transition. To be noted, these above discussions are solely based on how the structural changes can modify the band structure and for simplicity we have disregarded the electron-electron correlation. On the other hand, Mott et al., explained the Metal-to-Insulator transition in terms of electron correlation [106]. It was discussed that in the Insulating phase, the d // , band splits into two Hubbard bands because of electron correlation. When transformed into the metallic phase, the π * band is energetically lowered and becomes partially filled. Thus the electrons in the π * band could overwhelm and overpower the electron correlation in the d // band and fuse the two Hubbard bands into a single band.

Figure 2 . 1 :

 21 Figure 2.1: Experimental phase diagram of VOx. This image has been adapted from[START_REF] Katzke | Theory of morphotropic transformations in vanadium oxides[END_REF][START_REF] Yang | Oxide electronics utilizing ultrafast metal-insulator transitions[END_REF]

Figure 2 . 2 :

 22 Figure 2.2:Relation between tetragonal structure (high-temperature, black lines) and monoclinic (low-temperature, dashed blue lines) unit cells. This image has been adapted from[START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF]. Below table shows the dimensions of the unit cell of VO 2 at both its state[START_REF] Pj Fillingham | Domain structure and twinning in crystals of vanadium dioxide[END_REF].

23 detector

 23 to rotate simultaneously, to collect specular reflection and form a 2θ based intensity results. Since the Bragg's condition is only satisfied by certain combinations of θ and d-spacing, the intensity versus 2θ provides a quantitative measurement of the crystallinity of the material. The x-ray diffraction pattern for a VO 2 thin-film grown at INL is displayed in the figure 2.3. Analysing the pattern we observe intensity peaks measured along 2θ.

Figure 2 . 3 :

 23 Figure 2.3: XRD of VO 2 on Si, home-made deposition (Reactive ion sputtering).

Figure 2 . 4 :

 24 Figure 2.4: XRD of VO 2 on Al 2 O 3 , home-made deposition (Reactive ion sputtering).

Figure 2 . 5 :

 25 Figure 2.5: VO 2 on SiO 2 home-made (failed deposition).

  (a) Transverse XRD on VO 2 /Al 2 O 3 sample. We can observe the peak shift in relation with the temperature change. (b) Peak shift indicates change in the crystalline structure thus indicating structural change.

Figure 2 . 6 :

 26 Figure 2.6: Transverse XRD and peak shift observed during the transition of VO 2 indicating the structural transition.

Figure 2 . 7 :

 27 Figure 2.7: XRD intensity peak shift and width increase showing structural changes in the crystalline structure of VO 2 during the temperature induced phase transition.

  (a) 30 • C (b) 50 • C (c) 70 • C. (d) 90 • C.

Figure 2 . 8 :

 28 Figure 2.8: RSM on VO 2 on Al 2 O 3 as a function of temperature. We observe the gradual shift in in-plane parameter.

  (a) Lattice Parameters shift in in-plane direction. (b) Lattice Parameters shift in out-of-plane direction.

Figure 2 . 9 :

 29 Figure 2.9: In-plane and out-of-plane Lattice parameter shift as a function of temperature

3 Figure 2 .

 32 Figure 2.10: TEM cross section reveals the presence of additional native oxide layer in between the VO 2 layer and Si substrate.

Figure 2 .

 2 Figure 2.11: EELS and TEM cross of the intermediate layer for Sample 1 (VO 2 100nm) reveals that at the interface of SiOx we observe 2-3 nm of V(3 + ) or V(2 + ) reduction V atoms.

Figure 2 .

 2 Figure 2.12: SAED pattern VO 2 deposited on sapphire indexed as a monoclinic structure on the growth direction.

Figure 2 .

 2 Figure 2.13: (a) SAED surface plane analysis and (b) TEM cross section for Sample (VO 2 25nm) on Al 2 O 3 revealing the interface of substrate and thin film. (c) EELS cross section of intermediate layer

  as: Is = sin(2ψ).sin(∆) Ic = sin(2ψ).cos(∆) These trigonometric functions Is and Ic are directly related to ψ and ∆. They depend on the measurement conditions, in our case these above equations are only valid for the situation when the analyzer is at 45 • and the modulator at 0 • . Throughout this work, we directly fit the theoretical models to the Is and Ic values.

Figure 2 .

 2 Figure 2.14: HORIBA-Jobin Yvon UVISEL Ellipsometry setup

Figure 2 . 15 :

 215 Figure 2.15: Linkam THMSEL350V Heating stage temperature controller

Figure 2 . 16 :

 216 Figure 2.16: Optical model: Layer by layer modelling of VO 2 on SiO 2 .

  (a) Experimental and simulated Is/Ic parameters for semiconductor state of VO 2 (b) Experimental and simulated Is/Ic parameters for metallic state.

Figure 2 . 17 :Figure 2 . 18 :

 217218 Figure 2.17: Fitting the optical model and experimental data.

Figure 2 . 19 :

 219 Figure 2.19: Fit parameter values obtained for both the states of VO 2 .

Figure 2 . 20 :

 220 Figure 2.20: Varying parameters of VO 2 ellipsometric models calculated as a function of temperature.

Figure 2 . 21 :

 221 Figure 2.21: Band structure of the a. Insulating and b. Metallic VO 2 .

Figure 2 . 22 :

 222 Figure 2.22: Evolution of refractive index as a function of temperature on VO 2 /Al 2 O 3 .

Figure 2 . 23 :

 223 Figure 2.23: Thermal hysteresis loop of VO 2 at λ = 1200nm

CHAPTER 3 VO 2

 32 BASED ACTIVE OPTICAL MODULATOR: DESIGN, FABRICATION AND EXPERIMENTAL RESULTS

v

  div( E × H)dv (3.13) So we use Maxwell's equations 3.2 and 3.1 to calculate the divergence of E × H as shown, ∇.( E × H) = H.( E × H) -E.( E × H) (3.14)

Figure 3 . 1 :

 31 Figure 3.1: Schematic view of the active PCM modulator based on vanadium dioxide insulator-metal phase transition: Top Figure : VO 2 is insulating, wave propagation is unaltered. Figure below shows when VO 2 is metallic, the wave diverts its path of propagation into the adjacent waveguide leveraging the absorption of VO 2 .

  2 and VO 2 in metallic and insulator states are respectively of 3.4799, 1.4448, 2.0245 and 3.2430. The schematics for which are shown in figure 3.3. As a result we can observe the difference in the field and the effective refractive index, giving us an advantage to manipulate physical conditions for adequate coupling.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Evolution of the effective indices in the guide to guide coupler versus the width of the waveguide with the VO 2 layer.

Figure 3 . 4 :

 34 Figure 3.4: Schematics for mode coupling between two dissimilar waveguides.

B

  (Z) = [b 1 e jψz + b 2 e -jψz ]e jψz (3.20)

Figure 3 . 5 :

 35 Figure 3.5: Schematics for symmetric and anti-symmetric super modes in directional couplers

Figure 3 . 6 :

 36 Figure 3.6: Schematics for π order phase between symmetric and anti-symmetric mode leading to co directional mode coupling.

Figure 3 . 7 :

 37 Figure 3.7: Variation of Lc versus the gap between the two coupled waveguides.

D e y a z -γa y = µ xx b x ( 3 . 43 )

 343 γa x -D e x a z = µ yy b y (3.44) D e x a y -D e y a x = µ zz b z (3.45)D h y b z -γb y = xx a x (3.46) γb x -D h x b z = yy a y (3.47) D h x b y -D h y b x = zz a z (3.48) We now solve equations 3.45 and 3.48 for the longitudinal components such that we can replace the E z and H z . After substituting and replacing, we now remain with equations that just contain the transverse field components E x , E y , H x and H y . These equations are then rearranged into common terms that are multiplying the field components, and are shown below, x µ -1 zz D e y a x + (D h x µ -1 zz D e x + yy )a y = γb x (3.51) -(D h y µ -1 zz D e y + xx )a x + D h y µ -1 zz D e y a y = γb y (3.52) we now convert the linear matrix equation into block matrix form as shown,

Figure 3 . 8 :

 38 Figure 3.8: Basic Finite Difference Method

Figure 3 . 9 :

 39 Figure 3.9: Finite Difference Method

Figure 3 . 10 :

 310 Figure 3.10: Finite difference scheme discretization of VO 2 waveguide modulator.

( a )

 a Simulation of wave propagation in bus waveguide when VO 2 is insulating. (b) Co-directional coupling of wave in VO 2 patch waveguide when VO 2 is metallic without loss. (c) Co-directional coupling of wave in VO 2 patch waveguide when VO 2 is metallic with loss.

Figure 3 . 11 :

 311 Figure 3.11: Top simulation view of wave propagation and modulation in the proposed device when VO 2 is insulating and when it is metallic (with and without loss).

Figure 3 . 12 :

 312 Figure 3.12: Efficient coupling of VO 2 waveguide modulator.

Figure 3 . 13 :

 313 Figure 3.13: Evolution of the output power (P out) versus the variation of the refractive index of the VO 2 layer.

Figure 3 . 14 :

 314 Figure 3.14: Evolution of the output power (P out) versus the variation of the wavelength with losses.

( a )

 a Device design when VO 2 patch is dielectric. (b) Device design when VO 2 patch is metallic. (c) Simulated incident wave power in bus waveguide. (d) Simulated incident wave power in hybrid waveguide.

Figure 3 . 15 :

 315 Figure 3.15: Schematic cross-sectional view of the coupler section in the optical device as simulated with light modulation as function of transition in VO 2 .

  (a) 'ON' state of VO 2 with gap of 350µm. (b) 'ON' state of VO 2 with gap of 400µm. (c) 'ON' state of VO 2 with gap of 450µm. (d) 'ON' state of VO 2 with gap of 500µ m.

Figure 3 . 16 :

 316 Figure 3.16: Intensity of the guided mode as a function of position along the device.

Figure 3 . 17 :Figure 3 . 18 :

 317318 Figure 3.17: Fabrication steps involved for the realization of VO 2 modulator

Figure 3 . 19 :

 319 Figure 3.19: SEM image of patterned hard mask after ICP etching.

Figure 3 . 20 :

 320 Figure 3.20: Simulation for VO 2 waveguide modulator as partial power filter using the intermediate state of VO 2 transitions.

Figure 3 . 21 :

 321 Figure 3.21: Gradual power variation in the bus waveguide (Core 1) as function of changing refractive index.

3. 6 Conclusion

 6 Our presented VO 2 based waveguide modulator, if fabricated successfully can be a great component for future integrated systems. The simulations for the modulators show that, compared to classical modulators, a simple trigger (optical, electrical or thermal) can have a profound impact on the transmission values in the bus waveguide. In opposition to the high absorption based VO 2 devices, this device architecture is compatible with CMOS technology and can be cascaded with low absorption loss. In future works, we plan to realise the device, overcoming the fabrication hurdles and obtain experimentally the modulation performance of the device. Furthermore, in regard to the controlled power transfer filter, it find its usefulness in the creation of artificial all-optic neural networks (AONN), specifically as a weight bank. It is an emerging field with groundbreaking applications in improving Machine learning circuit and in sense for Artificial Intelligence (A.I) advancements[START_REF] Shen | Deep learning with coherent nanophotonic circuits[END_REF]. In the previous decade the the Neural network technology were dependent on electrical circuit / transistors for logic 93 operation. Followed by increased demand, the neural training networks are systematically switched to optical circuits, component by component. One of the most important of them are the weight banks. The electrical circuits used resistance as a weight to change the current in a electrical circuit. Optical weight banks depend on the absorption of an input light and amount of non-absorbed light that is detected is considered as weight.

  λ wherein m med is the refractive index of the surrounding medium, λ the wavelength of the incident light in vacuum and R the sphere's radius. Materially, a n and b n are the weighting factors of the different excited electric and magnetic multipolar contributions. For example, a 1 and b 1 represent the electric and magnetic dipolar modes, and a 2 and b 2 correspond to the electric and magnetic quadrupolar modes, respectively.4.3 VO 2 Nanocrystals: Theory

Figure 4 .

 4 Figure 4.1 shows the real and imaginary part of the dielectric permittivity of a VO 2 thin-film in the Vis/NIR range as a function of temperature, as obtained by ellipsometry characterization and modelling of VO 2 thin film.

Figure 4 . 1 :

 41 Figure 4.1: Real and imaginary part of the dielectric permittivity of a VO 2 thin-film in the Vis/NIR range as a function of temperature.

Figure 4 .

 4 Figure 4.2 shows the absorption cross section Q abs of a VO 2 sphere of radius 50 nm in its insulating and metallic states embedded in fused silica ( m med = 1.45 for λ = 1 µm).

Figure 4 . 2 :

 42 Figure 4.2: Calculated absorption efficiency of VO 2 spheres with radius 50 nm in their a) insulating and b) metallic states embedded in quartz. With colored lines are represented the dipolar electric a 1 (red), dipolar magnetic b 1 (yellow), quadrupolar electric a 2 (light blue) and quadrupolar magnetic b 2 (dark blue) contributions to Q abs .

100

 100

4. 4 VO 2

 42 Nanocrystals: Fabrication and Characterization 4.4.1 Fabrication

  figure 4.3b.

  with a heating cell to thermally induce the IMT. The incident broadband light source is a Xenon lamp polarized at 45 • . After reflection off the sample under study, the light is directed to a spectrometer, dispersed by appropriate gratings and measured with UV/Vis and NIR detectors. The acquired ellipsometric parameters Is and Ic of the samples have been collected for varying angles 55 • -75 • over a spectral range of 260 -2100 nm. The

  (a) TEM image of the VO 2 -NCs in fused silica. (b) 3D sketch representing a cross-sectional top view of the VO 2 -NCs and deposition procedure.

Figure 4 . 3 :

 43 Figure 4.3: VO 2 NCs TEM image (top) and sketch (bottom)

Figure 4 .

 4 Figure 4.4 shows the evolution of the I s and I c parameters of ellipsometry measurements

Figure 4 . 4 :

 44 Figure 4.4: Evolution of ellipsometric parameters Is and Ic measured as a function of temperature. The accusation was performed on VO 2 NC at an incident angle of 65 •

Figure 4 . 5 :

 45 Figure 4.5: Thermal Hysteresis loop of VO 2 NCs.

Figure 4 . 6 :

 46 Figure 4.6: Dynamic optical measurements of the VO 2 NCs IMT. The rate of temperature ramp is 1 • per minute. The initial stating temperature was 23 • C

Figure 4 . 7 :

 47 Figure 4.7: 3D representation of optical stacks for VO 2 NC embedded in fused silica.

Figure 4 . 8 :

 48 Figure 4.8: Fitting (solid line) and experimental measurements(circles) of the ellipsometric parameters Is (blue) and Ic (red) for both states of VO 2 -NCs: a) metallic and b) insulating for a VO 2 -NCs for size of R = 35nm and a filling-factor f = 0.18. (incident angle: 55 • ).

figure 4 .

 4 figure 4.9: x-polarization (i.e. along the dimer axis), y-polarization (i.e. perpendicular to the dimer axis) and at 45 • (This angle is chosen because the incident polarization used in the

Figure 4 . 9 :

 49 Figure 4.9: Scheme of the analyzed system: dimer formed by spheres (R = 25nm) separated by a gap and illuminated with x-(left) and y-polarized beams (right). With arrows are represented the polarization of the illuminating beam (E, blue arrow) and the wave vector (k, red arrow).

VO 2

 2 dimers in its metallic phase The Figure below(4.10) shows how the different interparticle distances (gaps) effect the overall Localized Surface Plasmon Resonance (LSPR) peaks that appears in the absorption cross-section (C abs ) of a dimer made of metallic VO 2 nanospheres. Focusing on the figure, the dashed black lines represent the peak positions for a system of two non-interacting spheres (the two most left lines correspond to the electric and magnetic dipole resonances, respectively). Particularly, we pay attention to the LSPR peak appearing at 1176 nm which 110 has a pure dipolar electric character.

Figure 4 .

 4 10(a) shows how, for the dimer illuminated with x-polarized light, the LSPR peak is red-shifted as the gap becomes smaller, reaching a maximum shift of 110 nm for a gap of 5 nm. On the contrary, Figure4.10(b) shows that the LSPR peak is blue-shifted when the dimer is illuminated with an y-polarized beam. The maximum blue-shift is 30 nm for a gap of 10 nm. These results are consistent with those reported by Rechberger et al[START_REF] Rechberger | Optical properties of two interacting gold nanoparticles[END_REF].

Figure 4 . 10 :

 410 Figure 4.10: Absorption cross-section C abs of VO 2 dimers in its metallic phase (R = 35 nm) when illuminated with (a) x-(b) y-and (c) 45 • polarizations (4.9) for different interparticle distances (gaps). As an inset in (a) and (b) is indicated the maximum spectral shift between the interacting and non-interacting (dashed black lines) dimer. In (c) the spectral shift ∆λ is represented as a function of the gap.

Figure ( 4 .

 4 Figure (4.11) below shows how the resonance peak is shifted with decreasing the interparticle gap for (a) x-(b) y-and (c) 45 • polarizations, respectively. For the three studied polarizations the overall effect is the same. For the largest gap (150 nm), the resonance is slightly blueshifted with respect to the non-interacting system. With decreasing interparticle distances the resonance peak further blueshifts reaching a maximum spectral shift for an intermediate gap of 60 nm approximately.

Figure 4 . 11 :

 411 Figure 4.11: Absorption cross-section C abs of VO 2 dimers in its insulating phase (R = 35 nm) when illuminated with (a) x-(b) y-and (c) 45 • polarizations (4.9) for different interparticle distances (gaps). As an inset in (a) and (b) is indicated the maximum spectral shift between the interacting and non-interacting (dashed black lines) dimer. In (c) the spectral shift ∆λ is represented as a function of the gap.

Figure 4 . 12 :

 412 Figure 4.12: Absorption cross-section C abs spectra for the reference monodisperse system (R = 35 nm, black line) and others with (a) symmetric and (c) asymmetric distribution of NC sizes. Size distribution of the (b) symmetric and (d) asymmetric systems. With the light dotted line is represented the size of the monodisperse distribution.

Figure 4 .

 4 13 a-d shows the calculated ef f , µ ef f and m ef f for a slab with VO 2 spheres with radius R = 35 nm and a filling fraction f = 0.18 in their insulating (low temperature) and metallic (high temperature)

115

 115

  Theoretical effective parameters for insulating and metallic VO2, R = 35 nm FF = 0.18

Figure 4 .

 4 Figure 4.13: a) Effective dielectric function ( ef f ), b) magnetic permeability (µ ef f ), and c) real and d) imaginary parts of the effective refractive index (m ef f = n ef f + ik ef f ) of a slab of quartz with VO 2 spheres with radius R = 35 nm and a filling fraction f = 0.18 in their insulating (low temperature) and metallic (high temperature) states.

4. 6

 6 Tunable multipolar resonances based on VO 2 -NCs system: Future Perspective 4.6.1 Discussion

Figure 4 .Figure 4 . 14 :

 4414 Figure 4.14: Variation of effective optical parameters for different sizes of nanocrystals in both states of VO 2 a) and b) n ef f and k ef f for insulating VO 2 with a filling fraction f = 0.18. c) and d) n ef f and k ef f for metallic VO 2 . e) and f) represent the amplitude of modulation of the effective refractive index of the metamaterial as a function of NCs size. e) ∆n ef f and f) ∆k ef f .

Figure 4 . 15 :

 415 Figure 4.15: Spectral variation of the amplitude of modulation of the effective refractive index (∆n ef f and ∆k ef f ) of the metamaterial for selected NCs size. Bottom graph: Figure of Merit (FOM=∆n ef f /∆k ef f ) of the VO 2 -NCs platform for different sizes of nanocrystals and compared to the FOM for a VO 2 thin-film.

Figure 4 . 16 :

 416 Figure 4.16: Calculated absorption cross-section efficiency Q abs of NCs of different sizes in their insulator and metallic states embedded in quartz. With colored lines are represented the dipolar electric a 1 (red), dipolar magnetic b 1 (yellow), quadrupolar electric a 2 (light blue) and quadrupolar magnetic b 2 (dark blue) contributions to Q abs .

Figure 4 .

 4 [START_REF] Ernst | Surprises in theoretical physics[END_REF] displays the multipolar decomposition of light fields within VO 2 -NCs as a function of size and VO 2 phase, either insulating or metallic. We see that each size of NCs produces overlapping ED and MD resonances which would coherently interfere at specific wavelengths. Then, when the VO 2 is switched from 120 its insulating phase to the metallic one, we can suppress the DM contribution associated to the b 1 coefficient and strongly enhance the DE contribution associated to the a 1 coefficient.

2

 2 IMT to tune and switch multipolar modes in the visible and near-infrared ranges. Particularly, the presence of a plasmonic mode in the NIR whose gradual appearance and intensity are directly controlled by the VO 2 state. The presence of two simultaneous dipole resonances in two different spectral regions, Vis and NIR, could open the possibility of new multiplexing based techniques for SERS applications.

CHAPTER 5 VO 2

 52 BASED METASURFACES FOR TUNABLE FREE-SPACE NANOPHOTONICS Many modern day conveniences, which we take for granted, has been possible through the understanding of light as an electromagnetic wave. Devices like radio, lasers and more recent things such as wireless communication, flat-screen color displays and LEDs were achievable because we understood and controlled electromagnetic waves. The breakthrough in wireless technology has been one of the turning point in the human civilization. This made the world a smaller place as it enabled instantaneous communications affecting the global trade and economics. The credit for the development goes to the decades of research in the field of transmission and detection of electromagnetic waves through the study of

2

 2 can be exploited to dynamically tune the resonances of sub-wavelength scattering structures and the overall optical properties of dielectric metasurfaces. This concept is used for two different targeted applications: first we propose a platform for tunable structural coloration based on TiO 2 nanostructures on VO 2 thin-film. Second, we experimentally demonstrate a VO 2 -assisted modulation of electric and magnetic Mie resonances supported by amorphous silicon nano-pillars, achieving tunable perfect absorption or transmission modulation depth of about three orders of magnitude at λ = 1555nm.

  Second it is important to understand how we scientifically classify different colors. For that, we usually use CIE diagrams. CIE stand for Commission internationale de l'éclairage, which is a 100 years old organization that creates international standards related to light and color. CIE 1931 is a Color Matching System. Color matching does not try to explain how colors look to humans, color matching teaches us how to determine a color calculated numerically, and then accurately replicate the color measured (e.g. in print or digital displays). To reiterate the concept, Color Matching Systems are not based on explaining color with attributes such as hue or intensity, they simply tell us what light configurations seem to be the same color to most people (they "match"). Color matching helps to provide us with a simple color reproduction system. In order to give them some mathematically useful properties, the 1931 XYZ Color Matching Functions (CMFs) are a linear transformation of the 1931 RGB Color Matching Functions. You can plot the primaries of other color spaces onto the XYZ chromaticity diagram to see the triangles they form. The area inside a primary triangle is known as a color gamut. The figure 5.1, displays the sensitivity of different types of cone and the figure 5.2 shows a CIE graph of all the colors that human eye can perceive and also the sRGB chromaticity triangle showing the colors of our monitor displays. Therefore, for future structural color-based displays employing metasurfaces it is desirable to create or filter distinct colors as human eyes are quite sensitive to the accuracy of the colors.

Figure 5 . 1 :

 51 Figure 5.1: Normalized response spectra of human cones, to monochromatic spectral stimuli, with wavelength given in nanometers [213].

Figure 5 . 2 :

 52 Figure 5.2: The chromaticity diagram of the CIE 1931 color space. The curved outer limit is the spectral locus, with wavelengths shown in nanometers. The colors your monitor displays in this image are specified using sRGB, so colors outside the sRGB range are not displayed correctly. This diagram shows the maximum saturated vivid colors that can be perceived by human eyes.

  four chromatic points in the CIE-xy 5.2 diagram are sufficient. Three of which corresponds to RGB colors but the fourth point is white and it is obtained when we mix equal intensities of red green and blue (i.e. when R=G=B). The most common white point in CIE standard is illuminant D65[START_REF] Tm Goodman | International standards for colour[END_REF] and it is used extensively in industries for the simulation of natural daylight for visually demanding applications like digital art, film, photography, colorimetry 129 etc. The idea behind the D65 is for industrial standardisation of lighting. D65 is a particular embodiment of the D series of illuminants defined by the CIE and attempt to simulate the spectral power distribution of natural daylight represented in figure 5.3.

Figure 5 . 3 :

 53 Figure 5.3: This the D65 sunlight spectrum simulated by D series daylight simulator by CIE standard.It cover the whole visible spectrum and the roughness shows the various points of absorption and diffusion by earth's atmosphere. This figure is taken from[START_REF] Ohta | Colorimetry: fundamentals and applications[END_REF]

130 5. 1 . 4

 13014 All dielectric tunable structural color generationStructural color generation by metallic nanostructures can operate beyond the diffraction limit but it has drawbacks[START_REF] Arbabi | Fundamental limits of ultrathin metasurfaces[END_REF] like restrained compactness of the functional elements and high resolution is backed by high loss, leading to peak broadening, implying limited number of distinctive colors. Whereas dielectric structures overcome these optical loss by exploiting Mie resonances. As explained in chapter 4, under some specific conditions, the resonances hosted by high refractive index dielectric nanoparticles is a combination of both magnetic

2

 2 is a material of interest because it is transparent in the visible range, but compared to Silicon it has a lower refractive index and the resonant response of each nano-pillar is therefore weaker. But this situation can be altered by fabricating arrays of multiple resonators together. Furthermore, using a periodic arrangement of TiO 2 unit cells we can enhance Mie resonances by overlapping through phase matching the radiated Fano and Mie scattering resonances. Therefore with the following arrangements, it is possible to generate a collective enhanced structural color, that can be tuned by design, through adjusting the geometry of arrangement. The simulation relating to the following work was aided by my fellow doctorate colleague Raphaël Mermet-Lyaudoz.

Figure 5 . 4 :

 54 Figure 5.4: TiO 2 nanostructures and the constituent color reproduction with varying dimensional parameters .

1 .

 1 figure 5.4b that for a given height (h) of 100nm the TiO 2 nanopillars simulates colors at FF > 0.2 and at a > 0.4µm, meaning the overall diameter of the pillars are large. In next section, keeping the same dimensional condition, we observe the color reproduction of VO 2 nano-pillars. Following, we now reproduce a colored picture using the previously defined TiO 2 structures arranged in a pixel array. As shown in figure 5.5, we can observe that the colors of the picture represented are overpowered by white hue because the structural colors reproduced by TiO 2 nano pillars are close to white reference region. As we see from the figure 5.4c the color gamut L* value ranges between 40 -100, which indicates the contrast of the color compared to white illuminant. From the above observations we can conclude that,

  (a) The original image depicting the flag of Auvergne-Rhône-Alpes region (b) The replicated image using the TiO 2 nano pillars.

Figure 5 . 5 :Figure 5 . 6 :

 5556 Figure 5.5: Real and simulated image based on structural color of TiO 2 .

  The variation in the structural reproduction by each state of VO 2 nano-pillars can be observed in Figure: 5.7b and Figure: 5.8b, comparing the simulated color palettes. As displayed in Figure 5.7, the variation in FF and period of the nano-pillars, produces a significant range of colors. Even at the natural insulating state we observe sharp contrast in the colors produced by the VO 2 nano-pillars when compared to TiO 2 . The dimensions of the structures were kept the same and we observe that VO 2 nano-pillars simulate colors at lower values of FF and a, meaning smaller diameter for the nano-pillars. Therefore in addition to bright colors and the transition, we have the size advantage. Because VO 2 has a higher refractive index compared to TiO 2 , the color thus produced by VO 2 nanostructures are more dispersed, covering larger area in CIE diagram compared to the TiO 2 structures.

  (a) VO 2 colors on CIE map (b) Color reproduction as function FF and a (c) CIE-LAB color space of VO 2 nanopillars. (d) The replicated image.

Figure 5 . 7 :

 57 Figure 5.7: The structural color reproduction by VO 2 in semiconductor state.

Figure 5 . 8 :

 58 Figure 5.8: The structural color reproduction by VO 2 in metallic state.

Figure 5 . 9 : 139 TiO 2

 591392 Figure 5.9: VO 2 nano structure color gamut comparison 139

Figure 5 . 10 :Figure 5 . 11 :

 510511 Figure 5.10: TiO 2 metasurafce fabricated on VO 2 thin film deposited on SiO 2

Figure 5 . 12 :

 512 Figure 5.12: Photographs of TiO 2 nano pillars viewed at (a) normal incident and (b) at an angle. For the images (c) and (d) we observe the different structural colors at different angles, as they are highly sensitive to angle.

Figure 5 .

 5 Figure 5.13: a) Design concept: a silicon meta-atom is embedded into glass/PMMA surrounding with a buried thin VO 2 layer. VO 2 thickness: 25 nm, Si disk height: 240 nm, disk radius: 270 nm, square lattice period: 800 nm, PMMA layer thickness: 400 nm and b) Electron microscope image of the fabricated Si-VO 2 metasurface prior to the PMMA coating. c) Functionality concept: at low temperatures, the metasurface reflects light at a designed wavelength, and d)at high temperature the light is absorbed by the metasurface.

Figure 5 .

 5 Figure 5.14: EM simulation study on absorption, transmission and reflection as a function of Si nano disk diameter, with a period of 830nm and VO 2 is insulating. The red dotted circle shows perfect absorption (transmission is near 0) region due ED and MD overlap and black dotted circle shows the split in the dipole overlap.

Figure 5 .

 5 Figure 5.15: EM simulation study on absorption, transmission and reflection as function of Si nano disk diameter, with a period of 830nm and VO 2 is Metallic. The red dotted circle shows disappearance of the MD when VO 2 becomes metallic but we keep a strong absorption.

Figure 5 .

 5 Figure 5.16: EM simulation study on absorption, transmission and reflection of Si nano disk diameter 540nm scanned as function of temperature, with a period of 830nm. The dotted line shows the wavelength of interest 1.55 µm, for our tunable near-perfect absorption device.

Figure 5 . 17 :

 517 Figure 5.17: Device sketch ofthe a-Si metasurface on VO 2 thin films. The layers are defined as: a. PMMA, b. a-Si nano disks, c. SiO 2 protective layer, d. VO 2 thin film and e. SiO 2 substrate layer

Figure 5 .Figure 5 . 18 :

 5518 Figure 5.18: Is and Ic values as a function of temperature at the final stage of the fabrication of the metasurfaces. Analysing the figure, we see the effect of VO 2 transition, suggesting successful completion of all the stages of fabrication.

Figure 5 . 19 :

 519 Figure 5.19: Reflection measurements versus temperature of the metasurface. We observe a perfect absorption at 1550 nm when the phase of VO 2 is metallic. And we also observe the tunability and reversibility as due to temperature induced phase modulation of VO 2 .

Figure 5 . 20 :

 520 Figure 5.20: Experimental results on the tunable reflection of the Si-VO 2 metasurface. (a) Reflection spectra for two temperature points: 40 • C (a dielectric phase of VO 2 ) and 80 • C (a metallic phase of VO 2 ). (b) Reflection contrast between the two temperature points. (c) Reversible temperature tuning of the metasurface reflection at 1550 nm wavelength exhibiting a hysteresis-like behavior.

3 ,

 3 we try to leverage the inherent absorption of VO 2 as an advantage to create an efficient optical modulator. The theoretical studies performed in Chapter 1 and Chapter 2 have been fundamental in designing such a device. The electromagnetic simulation has helped us designing optimal dimensions and geometries for the proposed modulator. Despite the un-optimized fabrication process, we have seen that it is possible to realise an active modulator leveraging the transition properties of VO 2 . While most VO 2 -based modulators proposed so far rely on the end states of VO 2 , which means either ON or OFF state, we propose an optional advantage of such a tunable modulator device in the field of neural network computing, wherein the intermediate states of VO 2 plays a major role in controlling the power coupling, creating a power filter. Seen in terms of Neural computing, this filter can act as a weight bank, which is an essential component for realising a machine learning algorithm in an all-optical system.In chapter 4, we manipulate the material dimensions influencing the ED and MD resonances sustained by the material. Contrary to the previous chapter, where tunability depends on the states of VO 2 , here, additional external dimensional control over ED and MD along with the inherent IMT of VO 2 , give us a better control over the manipulation of light.And thus demonstrating that, by creating a composite metamaterial of VO 2 nanocrystals (NCs), one can tailor the multipolar resonances hosted by VO 2 NCs and dynamically tune them by leveraging the insulator-to-metal transition of VO 2 . In addition to that, we retrieve the complex refractive index of the effective medium composed of a slab of VO 2 -NCs embedded in SiO 2 , using Mie theory and Maxwell Garnett effective medium theory.

•

  The normal components of the electric flux densities are continuous if there are no charge on the surface, D 1n = D 2n (A.4) But if there are chage on the surface then the flux densities are discontinuous and related to charge density ρ s D 1n -D 2n = ρ s (A.5)

Figure A. 1 :

 1 Figure A.1: Transmitted wave characteristics in total internal reflection (TIR), the direction of propagation in z direction.

11 )

 11 figure A.3.

Figure A. 3 :

 3 Figure A.3: Planar mirror waveguide with reflective metal surface on the boundary and the index variation map as function of x-axis value.

27 ) where we define [k • 2 n 2 -β 2 ] = κ 2 ., β m 2 = k • 2 n 2 - mπ d 2 ; m = 1 , 2 , 3 (A. 30 )

 272222212330 For the region where light is confined 0<x<d, the β < k • n, then κ 2 > 0. Therefore the wave equation can be represented as,E y = Asinκx + Bcosκx (A.28)Applying boundary conditions, i.e. E y = 0 at x=0 and at x=d. Therefore, B=0 and κd = mπ, we get the TE (E y ) modes;E y = Asin mπx d ; m = 1, 2, 3 (A.29)The propagation constant β can be then calculated from[k • 2 n 2 -β 2 ] = κ 2 ,therefore, we getWe can observe from the equation A.30 that it can sustain only discrete modes or it can entertain certain sets of values. Further evaluating the TE mode propagation we can discretize it into two constituent plane waves propagating in +xz and -xz direction make an angle θ m with z axis, similar to TIR plane wave. From which we can evaluate the effective index of the material as observed by various TE modes can be represented as; n ef f,m = β m k • = ncosθ m (A.31)

Figure A. 4 : 1 ; 2 n 2 ; 2 (A. 35 )• 2 n 2 n 1 ; 2 n 2 ;

 41222352122 Figure A.4: Symmetric step-index planar waveguide structure and the index variation map as function of x-axis value.

Figure A. 5 :

 5 Figure A.5: Overlap of transverse evanescent field between two adjacent parallel symmetric slab waveguide structure and the index variation map of guide as function of x-axis value.

41 )H

 41 = A(z) H 1 + B(z) H 2 (A.42)Here A and B are the amplitude of the wave propagating in the waveguide 1 and 2 respectively, where as E and H are the electric and magnetic field of the light wave. Inco-directional couplingWe then compute the equations A.41 and A.42 by substituting them in to Maxwell's curl equations (3.2 and 3.1), which is followed by substituting the enforced Maxwell's equation into general integral equations describing orthogonality in lossless isotropic waveguides to obtain generalised coupled-mode equations, as shown; dA dz + c 12 dB dz e -j(β 2 -β 1 )z + jχ 1 A + jκ 12 Be -j(β 2 -β 1 )z) = 0 (A.43) dB dz + c 21 dA dz e -j(β 2 -β 1 )z + jχ 2 A + jκ 21 Ae -j(β 2 -β 1 )z) = 0 (A.[START_REF] Zhang | Experimental demonstration of near-infrared negative-index metamaterials[END_REF]

2 +

 2 |B| 2 + A * Bc 21 e -j2δz + AB * c * 21 e -j2δz (A.46)

  jA * B(κ * 21 -κ 12 -2δc 21 )e -j2δz -jAB * (κ * 21 -κ 12 -2δc * 21 )e j2δz = 0 (A.47)In order to make the power independent of Z-direction, we must have κ * 21 = κ 12 + 2δc * 21 .Keeping it in mind, we can simply write the reviewed coupled mode equations, assuming that c 12 = χ = 0;dA dz = -jκ 12 Be -j(β 2 -β 1 )z) (A.48) dB dz = +jκ 21 Ae j(β 2 -β 1 )z) (A.49) APPENDIX B ENERGY TRANSFER AND MODE MATCHING METHOD B.1 Mode Matching Method Mode matching method (MMM) is computational electromagnetic technique [249, 158] formulated in a Cartesian coordinate to simulate the evolution of the light wave which propagates in a guiding structure, by expanding the EM field into eigen modes in local area cross section, which varies according to the axis of propagation figure B.1. This method consists in replacing the structure to be modeled by an equivalent structure made up of a succession of elementary straight guides of variable sections.

Figure B. 1

 1 Figure B.1: a) Structure to model. b) Decomposition of the structure to be modeled into elementary straight wave guides.

1

 1 Mode matching at an Interface The propagation of the light wave in the equivalent structure shown in figure B.1b is calculated from interface matrices which is expressed by boundary conditions. Considering the continuity of the tangential components of the electric field E and of the magnetic induction H on either side of the interface between two straight guides and propagation matrices P which represents the phase shift introduced by the propagation of the modes in the equivalent straight guide portion. It should be noted that the method gives a rigorous solution of the problem when the thicknesses of the different portions of straight guide converge towards 0.

Figure B. 2 :

 2 Figure B.2: Schematic diagram of the calculation of reflection and transmission coefficients at an interface in the case of an incident wave equal to (a). a n 1,+ and (b) a n 2,-.

2 (B. 4 )= 1 . 5 )

 2415 With P a,b i,j = E a i , H b j = S (E a i,x .H b j,y -E a i,y .H b j,x )dxdy, where i and j represent the indices of the modes considered, a and b the numbers of the layers and n and m the total numbers of modes considered in the 2 guides. The coefficients a 1,- i and a 2,+ j are obtained by solving the linear system by equating a 1,+ p We can thus represent the equations in matrix form and can be represented as;Which can be written in term of Power P is shown as;

Figure B. 3 :

 3 Figure B.3: Relationship between the amplitudes of electromagnetic waves at an interface.

a 1 , 1 = a 1 ,-(z 1 ) a 1 ,+ (z 1 ) = I r 1 (

 1111111 -= r 1,2 a 1,+ + t 2,1 a 2,- (B.13) a 2,+ = t 1,2 a 1,+ + r 2,1 a 2,- (B.14) Equations B.13 and B.14 make it possible to obtain the interface transfer matrix: The layer matrix simply describes the propagation of the light wave in the homogeneous guide of layer with refractive index i. It is simply expressed as a function of the phase shift of the light wave: Generalization for N interfaces We are trying to calculate the reflection and transmission coefficients in Figure B.1 which consists of a large number of interfaces. The calculation begins by describing the boundary conditions of the last interface. By applying the Sommerfield radiation conditions [250]which says that in the presence of an excitation coming from the first medium, the wave a n,-= 0. By referring on the conditions in equation B.[START_REF] Grandi | Unraveling the Mott-Peierls intrigue in Vanadium dioxide[END_REF] to the interface, we obtain the following relations:a n-1,-(z n-1 ) = r n 1 ,n a n-1,+ (z n-1 ) (B.17) a n-1,+ (z n-1 ) = t n 1 ,n a n-1,+ (z n-1 ) (B.18)By adding the propagation constant to the interface z n-1 , we can rewrite the equation B.17 as;a n-1,-(z n-2 ) = I r n-1 a n-1,+ (z n-2 ) n = [e -jβ n-1 i e n-i ].[r n-1,n ].[e -jβ n-1 i e n-i ] (B.20) a n,+ (z n-1 ) = I t n-1 a n-1,+ (z n-2 ) (B.21)With,I t n-1,n = [e -jβ n-1 i e n-i ].[t n-1,n ] (B.22)We perform this iterative process up to interface 1, which allows us to obtain the relationship:a 1,-(z 1 ) = I r 1 a 1,+ (z 1 ) (B.23)From the equations B.13 and B.14, we can calculate the amplitudes for i + 1 in different layers using the equations:a i+1,+ (z i+1 ) = T i a 1,+ (z i ) (B.24)where,T i = [[I d ] -[r i+1,i ].[I r i+1 ]] -1 .[t i,i+1 ].[e -jβ i+1 e i+1 ] (B.25) a i+1,-(z i ) = I r i+1 a 1,+ (z i ) (B.26)The reflection and transmission coefficients at the different interfaces are obtained by solving a linear system comprising a number of equations equal to the number of unknowns, which ensures the uniqueness of the solution. In the end, we can obtain the reflection and transmission coefficients using the relations B.23 and B.24: r

TABLE OF CONTENTS

 OF

  Théorie de Mie et extraction des paramètres effectifs des nanocristaux de VO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii 0.2.3 Discussion et perspectives futures . . . . . . . . . . . . . . . . . . xl 0.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii 0.3 Part 3: Métasurfaces à base de VO 2 pour la nanophotonique accordable en espace libre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv 0.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv 0.3.2 Métasurface de VO 2 pour la génération de couleur structurelle accordable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlvi 0.3.3 Métasurface de VO 2 pour une absorption totale accordable. . . . . . Electronic Band structures of Vanadium oxide . . . . . . . . . . . . 1.3.4 Discussion on Vanadium oxide transition . . . . . . . . . . . . . . 1.4 Organization of the manuscript . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 VO 2 growth technique and deposition . . . . . . . . . . . . . . . . . . . . 2.3 Structural Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 XRD characterization and Results . . . . . . . . . . . . . . . . . . 2.3.2 Reciprocal space mapping (RSM) . . . . . . . . . . . . . . . . . . 2.3.3 Transmission electron microscopy (TEM) characterization . . . . . 2.4 Spectroscopic ellipsometry: Measurements and analysis . . . . . . . . . . Hybrid integration of functional materials on Silicon for efficient modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 VO 2 -based waveguide modulator: Design and Working principles . . . . . 3.3.1 Codirectional Coupling . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Symmetric and Anti-symmetric field pattern . . . . . . . . . . . . . 3.4 Simulation study and Numerical Analysis . . . . . . . . . . . . . . . . . . 3.4.1 2D Mode Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . . .

	. . . . . xxxiii
	0.2 Part 2: Résonances multipolaires accordable par design utilisant des nanocristaux
	de VO

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii 0.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii v 0.2.2 xlix Chapter 1: Introduction and Background . . . . . . . . . . . . . . . . . . . . . .

1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical aspects on insulator-metal transitions . . . . . . . . . . . . . . 1.2.1 Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Mott insulators and Mott Transition Mechanism . . . . . . . . . . . 1.2.3 Peierls Transition Mechanism . . . . . . . . . . . . . . . . . . . . 1.3 Vanadium oxides and its transition theory . . . . . . . . . . . . . . . . . . 1.3.1 Vanadium oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Crystallographic structure of Vanadium oxide . . . . . . . . . . . . 1.3.3 Chapter 2: Structural and Optical characterization of VO 2 thin-films . . . . . . vi 2.1 2.4.1 Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Is and Ic parameters . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Optical permitivity modeling of VO 2 . . . . . . . . . . . . . . . . . . . . . 2.5.1 Oscillators model and dispersion function for VO 2 . . . . . . . . . 2.5.2 Extension to Multiple oscillators . . . . . . . . . . . . . . . . . . 2.6 Optical modelling and characterization of extracted data . . . . . . . . . . . 2.6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3: VO 2 based active optical modulator: design, fabrication and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction on Silicon photonics . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Electro-optic modulators . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 3.2 Optical Waveguides: Theory and coupling mechanisms . . . . . . . . . . . 3.2.1 Optical waveguide theory . . . . . . . . . . . . . . . . . . . . . . . vii 3.2.2 3.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  In-plane and out-of-plane Lattice parameter shift as a function of temperature 2.10 TEM cross section reveals the presence of additional native oxide layer in between the VO 2 layer and Si substrate. . . . . . . . . . . . . . . . . . . . 2.11 EELS and TEM cross of the intermediate layer for Sample 1 (VO 2 100nm) reveals that at the interface of SiOx we observe 2-3 nm of V(3 + ) or V(2 + ) reduction V atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SAED surface plane analysis and (b) TEM cross section for Sample (VO 2 25nm) on Al 2 O 3 revealing the interface of substrate and thin film. (c) EELS cross section of intermediate layer . . . . . . . . . . . . . . . . . . . 2.14 HORIBA-Jobin Yvon UVISEL Ellipsometry setup . . . . . . . . . . . . . 2.15 Linkam THMSEL350V Heating stage temperature controller . . . . . . . . 2.16 Optical model: Layer by layer modelling of VO 2 on SiO 2 . . . . . . . . . . . 2.17 Fitting the optical model and experimental data. . . . . . . . . . . . . . . . 2.18 Thermal evolution of refractive index of VO 2 extracted from the optical models after the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Fit parameter values obtained for both the states of VO 2 . . . . . . . . . . . Evolution of the effective indices in the guide to guide coupler versus the width of the waveguide with the VO 2 layer. . . . . . . . . . . . . . . . . . 3.3 Refractive index distribution of the coupled waveguide and electric field of subsequent waveguides. The width of the VO 2 /Si waveguide is adjusted to manipulate the effective index in order to match the bus waveguide effective index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematics for π order phase between symmetric and anti-symmetric mode leading to co directional mode coupling. . . . . . . . . . . . . . . . . . . . 3.7 Variation of Lc versus the gap between the two coupled waveguides. . . . . 3.8 Basic Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Finite difference scheme discretization of VO 2 waveguide modulator. . . . . 3.11 Top simulation view of wave propagation and modulation in the proposed device when VO 2 is insulating and when it is metallic (with and without loss).

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 XRD of VO 2 on Si, home-made deposition (Reactive ion sputtering). . . . . 2.4 XRD of VO 2 on Al 2 O 3 , home-made deposition (Reactive ion sputtering). . 2.5 VO 2 on SiO 2 home-made (failed deposition). . . . . . . . . . . . . . . . . 2.6 Transverse XRD and peak shift observed during the transition of VO 2 indicating the structural transition. . . . . . . . . . . . . . . . . . . . . . . 2.7 XRD intensity peak shift and width increase showing structural changes in the crystalline structure of VO 2 during the temperature induced phase transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 RSM on VO 2 on Al 2 O 3 as a function of temperature. We observe the gradual shift in in-plane parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 2.12 SAED pattern VO 2 deposited on sapphire indexed as a monoclinic structure on the growth direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 2.13 (a) 2.20 Varying parameters of VO 2 ellipsometric models calculated as a function of temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.21 Band structure of the a. Insulating and b. Metallic VO 2 . . . . . . . . . . . . 2.22 Evolution of refractive index as a function of temperature on VO 2 /Al 2 O 3 . . 2.23 Thermal hysteresis loop of VO 2 at λ = 1200nm . . . . . . . . . . . . . . . 3.1 Schematic view of the active PCM modulator based on vanadium dioxide insulator-metal phase transition: Top Figure : VO 2 is insulating, wave propagation is unaltered. Figure below shows when VO 2 is metallic, the wave diverts its path of propagation into the adjacent waveguide leveraging the absorption of VO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 3.4 Schematics for mode coupling between two dissimilar waveguides. . . . . . 3.5 Schematics for symmetric and anti-symmetric super modes in directional couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 3.6 3.12 Efficient coupling of VO 2 waveguide modulator. . . . . . . . . . . . . . . . 3.13 Evolution of the output power (P out) versus the variation of the refractive index of the VO 2 layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Rien ne va plus vite que la lumière. L'avantage clair des commmunications basées sur la lumière est donc la vitesse et la bande passante. Grâce à la richesse des recherches dans ce domaine, il est de plus en plus possible d'intégrer des circuits photoniques et électroniques sur la même puce. Les technologies photoniques basées sur le silicium sont en passe de devenir la norme pour répondre à la demande actuelle de données. Cependant, la vitesse de transmission des données dépend de la rapidité avec laquelle un dispositif peut contrôler ou moduler la lumière.

large bande passante et de son débit élevé. Ces différents avantages des commmunications optiques paraissent tout à fait prometteurs pour surmonter le goulot d'étranglement lié aux xxi limites des interconnexions métalliques auquel est confrontée l'industrie technologique actuelle. 0.1.2 La photonique sur silicium: Avantages et inconvénients Les dispositifs fonctionnels classiques de la photonique silicium sur puce comme les modulateurs, les filtres, les guides d'ondes et les commutateurs, sont largement utilisés pour les technologies intégrées depuis un certain temps. Avec tous leurs avantages, nos dispositifs actuels basés sur la lumière sont cependant considérés comme encombrants (quelques centaines de micromètres) par rapport aux interconnexions métalliques et autres composants microélectroniques de la génération actuelle. Même si la technologie actuelle permet des interconnexions électriques à une architecture de 10 nm [1], la demande de données a créé un goulot d'étranglement en ce qui concerne la poursuite de la miniaturisation de leurs composants électroniques pour les raisons déjà mentionnées ci-dessus. Cette incompatibilité de taille entre les composants photoniques et l'architecture électronique a été au centre de nombreuses recherches récentes. La majorité d'entre elles comprennent l'utilisation de matériaux alternatifs en conjonction avec des dispositifs à base de silicium pour amplifier les capacités de dispositifs hybrides. Cette hybridation de matériaux avec le silicium est nécessaire à cause des limites du Silicium pour la photonique. Comme le dispositif photonique est conçu pour être suffisamment petit pour être intégré sur une puce, on suppose que sa fonctionnalité devrait être comparable à celle des circuits xxii Introduction Le dioxyde de vanadium (VO 2 ) appartient à la classe des oxydes métalliques corrélés qui présentent une transition de phase métal-isolant (IMT). Cette transition de phase du VO 2

  ThéorieLe vanadium est un métal de transition de configuration électronique [Ar]3d 3 4s 2 . Il présente donc différents états d'oxydation et compte plus de dix états stables différents, dont l'oxyde monovalent VO et l'oxyde mixte comme V 8 O 15 , figure: 2.1. La plupart de ces oxydes présentent des transitions réversibles entre semi-conducteurs et métaux, ce qui est contraire Afin de réaliser un dispositif actif, nous devons analyser les caractéristiques structurelles et optiques du VO 2 en fonction des différentes techniques de dépôt et du substrat. En raison des différentes stoechiométries stables et complexes des phases de VO 2 , comme le montre la figure 2.1, le dépôt de VO 2 en couche minces n'est pas une tâche simple.

D'autre part, la nature sousjacente de cette IMT est qu'elle est suivie d'une transformation structurelle, entraînant par la suite un changement drastique des propriétés électriques et optiques du matériau. L'IMT du VO 2 et ses différents oxydes sont connues depuis quelques décennies. Les IMT induites thermiquement ont été étudiées pour la première fois par Morin

[START_REF] Fj Morin | Oxides which show a metal-to-insulator transition at the Neel temperature[END_REF]

en 1959. Plusieurs oxydes de vanadium intermédiaires et un certain nombre d'autres oxydes de métaux de transition possèdent cette propriété, mais le VO 2 est le plus intéressant d'un point de vu pratique, car sa température de transition est de loin la plus proche des températures ambiantes, ce qui xxiii signifie qu'il est plus facilement accessible pour une application à température proche de la température ambiante. à l'état prédit par la théorie des bandes. En général, cette transition pour les oxydes est déclenchée par un stimulus externe et elle se manifeste généralement par un changement radical de structure, suivi d'une variation drastique des propriétés optiques et électriques. Le déclenchement peut être de nature diverse comme la température, la pression et les champs électromagnétiques (EM)

[START_REF] Muraoka | Large modification of the metal-insulator transition temperature in strained VO 2 films grown on TiO 2 substrates[END_REF][START_REF] Berglund | Hydrostatic-Pressure Dependence of the Electronic Properties of VO 2 Near the Semiconductor-Metal Transition Temperature[END_REF][START_REF] Morin | Oxides Which Show a Metal-to-Insulator Transition at the Neel Temperature[END_REF]

.

La raison pour laquelle la théorie des bandes ne parvient pas à prédire la nature du VO 2 , est que la théorie générale des bandes

[START_REF] Seitz | The modern theory of solids[END_REF]

est basée sur différentes hypothèses quant aux matériaux étudiés. Ces hypothèses ne prenent pas en compte la corrélation électron-électron et la corrélation électron-réseau. En outre, le VO 2 est connu pour être un matériau polymorphe, c'est-à-dire qu'il existe dans différentes structures crystallographiques: VO 2 (A), VO 2 (B), VO 2 (M1) et VO 2 (R)

[START_REF] Lee | Epitaxial stabilization and phase instability of VO 2 polymorphs[END_REF]

. Il n'est donc pas facile d'expliquer la théorie complexe qui sous-tend la transition de VO 2 . De nombreux débats existent sur les différents mécanismes de transition majoritaires au sein du VO 2

[START_REF] John B Goodenough | The two components of the crystallographic transition in VO2[END_REF][START_REF] Grandi | Unraveling the Mott-Peierls intrigue in Vanadium dioxide[END_REF]

. Mott, dans son étude sur l'oxyde de nickel (NiO), a découvert que, selon la théorie, le NiO était censé être métallique mais qu'expérimentalement il se révèlait être isolant. A partir de cette observation, Mott a théorisé que certains matériaux, censés être métalliques selon la théorie des bandes, étaient en fait isolants à cause d'interactions électron-électron supplémentaires au sein de leur structure. En suivant cette théorie, il a suggéré que, si un électron se déplace de son site d'origine vers un nouveau site déjà rempli, il subira une xxiv répulsion coulombienne. Et si l'énergie de répulsion est supérieure à l'énergie de l'électron entrant, cet électron restera alors lié à son site initial, limitant ainsi sa liberté de mouvement.

Ce phénomène d'interaction des électrons conduit à la division de la bande initiale à moitié remplie en deux bandes séparées. Cette interaction électron-électron, négligée dans la théorie des bandes, est connue sous le nom de corrélation électronique.

Au contraire, Rudolf Peierls

[START_REF] Ernst | Surprises in theoretical physics[END_REF]

dans ses recherches, a étudié un réseau 1-D d'atomes en chaîne disposés périodiquement et a postulé qu'une déformation structurelle peut créer des perturbations périodiques dans la chaîne du réseau. Selon le théorème de Bloch, ces perturbations peuvent modifier la structure de bande en couplant à la fois l'électron et le phonon, introduisant ainsi des transitions isolant-métal (IMT) dans les systèmes 1-D. Le phénomène découvert a été nommé Distorsion de Peierls. La transition structurelle peut ainsi être attribuée à une distorsion de Peierls due au déplacement du réseau

[START_REF] John B Goodenough | The two components of the crystallographic transition in VO2[END_REF]

. Des études récentes suggèrent cependant qu'une IMT du VO 2 pourrait également se faire sans distorsion de Peierls

[START_REF] Lin | Metal-insulator transition of monoclinic VO2 thin film without Peierls distortion[END_REF]

.

De nombreux débats existent quant aux différents mécanismes physiques qui gouvernent les propriétés uniques de ces matériaux fortement corrélés et notamment sur les mécanismes majoritairement responsables de la transition isolant-métal.

[START_REF] Kotliar | Strongly correlated materials: Insights from dynamical mean-field theory[END_REF][START_REF] Tomczak | Effective band structure of correlated materials: the case of VO2[END_REF][START_REF] Mw Haverkort | Orbital-assisted metal-insulator transition in VO 2[END_REF][START_REF] Qazilbash | Mott Transition in VO2 Revealed by Infrared Spectroscopy and Nano-Imaging[END_REF][START_REF] Grandi | Unraveling the Mott-Peierls intrigue in Vanadium dioxide[END_REF]

. En effet, dans le cas du VO 2 , la présence d'une transition à la fois structurelle et électronique complique la modélisation théorique. Une des questions fréquemment posée dans les recherches est de savoir si la force motrice de la transition est due aux changements de symétrie structurelle favorisant la localisation des électrons ou si la transition structurelle est motivée par de fortes corrélations électrons-électrons. Ou s'il s'agit plutôt d'une combinaison des deux mécanismes. En d'autres termes: l'IMT du VO 2 est-elle une transition de Mott, une transition de Peierls, ou un mélange de ces deux phénomènes? Même si la réponse exacte à cette question ne fait pas encore l'objet d'un consensus scientifique, nous montrerons dans les sections suivantes que nous pouvons exploiter l'IMT de nombreuses manières différentes, en induisant l'accordabilité des matériaux dans les dispositifs nanophotoniques. La théorie xxv détaillée est expliquée dans le chapitre 1. 0.1.4 Caractérisation du VO 2 Ce qui rend la caractérisation structurelle encore plus difficile. En outre, on observe que même une petite variation du rapport oxygène/vanadium et de sa structure de liaison, affecte les caractéristiques de la couche mince, influençant ainsi les propriétés optiques et la transition de phase du VO 2 [22, 23, 24]. Et toute impureté non identifiée, que ce soit dans le système V-O ou dans les phases stoechiométriques, entraîne généralement une modification de la dispersion optique et des caractéristiques de commutation. Comme la fabrication est complexe, chaque échantillon ainsi fabriqué peut présenter des dispersions optiques différentes. Il est donc extrêmement important de procéder à une caractérisation précise de chaque couche mince pour en extraire les propriétés optiques nécessaires à la conception des dispositifs. Dans notre laboratoire, nous avons utilisé la technique de dépôt par pulvérisation réactive assistée par magnétron pour la croissance de couches minces VO 2 . Le système de pulvérisation est un AC450 de chez Alliance Concept. Les détails du système et de la technique de dépôt sont donnés dans le chapitre 2. De plus, nous avons reçu des échantillons de VO 2 déposés sur les substrats Si, SiO 2 , Al 2 O 3 et SOI, de S. Ramanathan de l'université de Purdue (USA). Le VO 2 déposé a été analysé structurellement à l'aide de techniques telles que XRD, RSM et TEM, puis analysé optiquement à l'aide de l'ellipsométrie spectroscopique. Les résultats sont détaillés au chapitre 2. L'importance de cette caractérisation est que ces échantillons seront utilisés comme base pour la réalisation de dispositifs VO 2 hybrides mentionnés ci-dessus. Par conséquent, sur la base de cette caractérisation optique, nous xxvi extrayons les valeurs des propriétés du matériau, comme les changements de l'indice de réfraction en fonction de la température, qui sera un paramètre d'entrée important pour la conception des dispositifs. Afin d'extraire les propriétés optiques, nous avons caractérisé les couches minces de VO 2 par ellipsométrie spectroscopique. Il s'agit d'une technique optique utilisée pour mesurer l'amplitude spectrale tan(ψ) et la phase (∆) de la lumière réfléchie, qui sont liées aux propriétés optiques et à la microstructure de l'échantillon étudié. Dans le cas d'un ellipsomètre à modulation de phase, comme celui que nous utilisons, nous ne mesurons pas directement les angles ellipsométriques ψ et ∆ mais nous mesurons plutôt des fonctions de ψ et ∆. Définies comme : Is = sin(2ψ).sin(∆) Ic = sin(2ψ).cos(∆) Ces fonctions trigonométriques Is et Ic sont directement liées à ψ et ∆. L'indice de réfraction analysé de VO 2 à l'état d'isolant (en dessous de T c ) est de 3,243 + 0,25367i, et il passe à 2,02 + 2,9393i pour l'état métallique à une longueur d'onde de 1,55 µm (au-dessus de T c ) [25]. L'approche ellipsométrique extensive est expliquée en détail dans la section sur l'ellipsométrie du chapitre 2. Nous pouvons résumer que les étapes impliquées dans le calcul des propriétés optiques de VO 2 en utilisant l'ellipsométrie comme première étape sont la collecte des valeurs Is et Ic par ellipsométrie, après quoi nous créons (simulons) le modèle optique pour l'échantillon en question (VO 2 sur SiO 2 ) en utilisant une formule de dispersion pour tous les matériaux constituant chaque couche individuellement. Pour VO 2 , nous utilisons un modèle d'oscillateur dit "3TL" pour modéliser la dispersion du VO 2 , où 3TL représente trois oscillateurs de Tauc-Lorentz. Le modèle théorique Tauc-Lorentz [26] a été développé par Jellison et Modine en 1996. Ils ont modélisé la fonction diélectrique complexe en utilisant la densité d'états joints de Tauc et l'oscillateur de Lorentz.

xxvii La somme de trois oscillateurs Tauc-Lorentz, chacun correspondant à des transitions interbandes dans le matériau

[START_REF] Gavini | Optical Properties of Semiconducting VO 2 Films[END_REF]

, a été utilisée pour modéliser les propriétés optiques du VO 2 .

  de longueurs d'onde infrarouge[START_REF] Chen | A novel structural VO 2 micro-optical switch[END_REF] et visible[START_REF] Jiang | Design, fabrication and testing of a micromachined thermooptical light modulator based on a vanadium dioxide array[END_REF], via des transitions induites par la chaleur. Compte tenu des propriétés du matériau, les dispositifs basés sur le VO 2 sont souvent confrontés à de grandes pertes d'insertion et d'absorption, ce qui rend difficile leur mise en cascade dans un système intégré. Ces pertes d'absorption les rendent inutilisables pour des applications futures. Dans le cadre de cette thèse, nous avons mis au point une stratégie qui permet d'exploiter pleinement l'accordabilité de l'indice de réfraction sans que la forte absorption affecte les performances du dispositifs. Nous prévoyons de créer un modulateur intégré à guide d'ondes basé sur le VO 2 compatible CMOS qui utilise les pertes d'absorption du VO 2 comme un avantage. proposé permet un nouveau type de modulateur compact qui peut être facilement intégré dans l'infrastructure photonique actuelle basée sur le Si et pourrait tirer parti de la transition rapide du VO 2 . Les simulations électromagnétiques indiquent des géométries de dispositif de 600µm de longueur avec un patch de modulation VO 2 de 300µm avec un espace entre les guides d'ondes de 500 nm pour obtenir une plus grande profondeur de modulation. En outre, elle montre que la structure du dispositif basée sur le guide d'ondes peut être utilisée pour exploiter les états intermédiaires de VO 2 afin d'agir comme un filtre de puissance et de sonder davantage les propriétés fondamentales de la transition. les conditions de couplage pour les deux guides d'ondes, de sorte que les conditions de propagation dans les deux guides soient identiques. Dans tous les calculs, nous nous sommes basés sur un film mince de VO 2 sur SOI. Nous avons calculé la largeur correcte du guide d'ondes hybride de sorte que l'indice effectif des deux guides d'ondes soit identique lorsque le VO 2 est dans sa phase métallique.En outre, dans notre cas, la cartographie de la grille en termes de schéma de différences finies a été réalisée de telle sorte que nous discrétisons plus finement les zones proches de la limite de deux interfaces et sans utiliser de moyennage de l'indice de réfraction à la frontière Ar et de Cl 2 à température ambiante. La formulation de la recette de gravure a été faite en interne et la procédure de gravure a été réalisée par C. Chevalier. d'ondes, comme filtre de puissance, en exploitant les différentes étapes de transition de VO 2 . Cette application est particulièrement utile dans le domaine des réseaux neuronaux, dans le but de développer un réseau neuronal tout optique (AONN). Cet AONN est un réseau complexe de dispositifs multifonctionnels fonctionnant en synergie pour fournir un calcul similaire à ceux ayant lieu au sein des neurones. Notre modulateur VO 2 pourrait être utilisé comme une "weight bank" optique qui est un composant de l'AONN.Le modulateur à guide d'ondes hybride VO 2 /Si, s'il est fabriqué avec succès, peut être un composant prometteur pour les futurs systèmes optiques intégrés. Les simulations 'évolution calculée des valeurs n ef f et k ef f des VO 2 -NCs dans leurs états isolants et métalliques pour différentes dimensions de NCs. Nous avons ensuite tracé ∆n ef f et ∆k ef f , qui sont définis comme ∆n ef f = n ef f,isolant -n ef f,mtallique et ∆k ef f = k ef f,isolant -

	0.2 Part 2: Résonances multipolaires accordable par design utilisant des nanocristaux pour calculer le champ diffusé par un objet de forme arbitraire, il faudrait une approche TEM des sections transversales de l'échantillon. Nous utilisons un fichier de dispersion de Nous obtenons une bonne correspondance globale entre le modèle et les mesures pour une avec des propriétés adaptées pour des gammes de longueurs d'onde souhaitées. La figure
	le motif gravé. L'observation MEB a révélé que contrairement à ce qui était attendu, il y avait de multiples imperfections suite à l'étape de gravure. En comparaison directe avec les images MEB avant (fig:3.18) et après (fig:3.19), nous observons un mauvais état général de la surface de l'échantillon. Comme nous pouvons presque distinguer les différents guides d'ondes, on peut supposer que le problème est soit de VO 2 0.2.1 Introduction Dans cette partie, nous étudions les propriétés de nanocristaux de VO 2 implantés dans de la silice fondue. Nous y démontrons expérimentalement que les résonances multipolaires des nanocristaux (NC) de VO 2 peuvent être accordées dynamiquement grâce à la transition isolant-métal du VO 2 . En utilisant à la fois la théorie de Mie et la théorie du milieu effectif composé d'une couche de VO 2 -NCs immergés dans du SiO 2 et nous montrons qu'un tel métamatériau présente une accordabilité optique distincte par rapport à du VO 2 non structuré. numérique. Un problème aussi complexe peut cependant être simplifié si l'objet est une sphère. En 1908, Mie a trouvé une solution exacte à ce problème, en calculant le champ diffusé comme une solution en série [39]. L'idée était d'étendre les champs électriques et magnétiques en termes d'harmoniques sphériques vectorielles. Ensuite, les fonctions de champs peuvent être représentées dans une combinaison linéaire de termes qui sont les produits de fonctions séparables des trois coordonnées sphériques. Nous avons calculé numériquement le coefficient de diffusion en nous basant sur le particules [40]. En traçant ainsi les coefficients de diffusion calculés, nous avons observé que, quel que soit l'état de VO 2 , en dessous de 700 nm, la réponse électromagnétique est référence pour le SiO 2 (Palik : [41]). Afin d'extraire l'indice de réfraction effectif numérique, nous utilisons l'approximation du milieu effectif (EMA) de Maxwell-Garnet assistée par la théorie de Mie. Ce mélange des théories de Mie et de Maxwell-Garnett (Mie-MG-EMA), nous permet de calculer une permittivité diélectrique effective ( ef f ) et une perméabilité magnétique (µ ef f ) qui sont directement déterminées par les coefficients dipolaires de Mie électrique a 1 et magnétique b 1 respectivement. Ainsi, l'ajout de la théorie de Mie dans les calculs de milieux effectifs électrostatique dans l'EMA classique de Maxwell-Garnett. Nous avons ainsi pu obtenir des formules analytiques permettant le calcul des les valeurs de ef f , de µ ef f et ainsi de l'indice taille de VO 2 -NCs de R = 35nm et un facteur de remplissage f = 0, 18. En particulier, les fits sont presque parfaits dans la gamme UV/Vis mais une légère divergence apparaît dans le proche infrarouge. Cet écart entre le modèle et les données expérimentales peut être expliqué par les quelques interactions interparticulaires possibles qui n'ont pas été prises en compte par l'approximation de Mie-MG-EMA. Dans le chapitre 4, nous essayons d'expliquer et d'étudier en détail les deux phénomènes physiques potentiellement responsables de cet écart degré de polydispersité potentiellement non négligeable dans la distribution de taille des VO 2 -NCs. 4.14 montre lk ef f,mtallique et qui représentent l'amplitude de modulation de l'indice de réfraction effectif et du coefficient d'extinction effectif, respectivement. Avec cette figure, nous pouvons identifier des régions spécifiques de la taille en fonction de la longueur d'onde pour lesquelles la commutation de VO 2 se traduit soit par un indice de réfraction accordable, soit par un Stratégie Après la procédure de gravure, nous avons effectué une observation MEB afin de visualiser de Maxwell-Garnett, nous retrouvons l'indice de réfraction complexe du milieu effectif formalisme de Lorenz-Mie pour la diffusion et l'absorption de la lumière par des petites nous permet de tenir compte des effets de taille qui ont été ignorés par l'approximation entre calcul et expérience, à savoir l'interaction électromagnétique entre les particules et un coefficient d'extinction accordable.
	Mais surtout, nous montrons dans ce qui suit que cette approche métamatériau permet de dominée par la réponse dipolaire électrique (DE) et la réponse dipolaire magnétique (DM), de réfraction complexe effectif en fonction de la taille des NCs et du facteur de remplissage.
	concevoir des métamatériaux composites accordables présentant une modulation de l'indice avec une petite contribution du terme quadrupolaire électrique (QE). comme le montre la Ces formules analytiques ont ensuite été utilisées pour fitter les données expérimentales 0.2.3 Discussion et perspectives futures
	de réfraction sans extinction induite, ou en d'autres termes un métamatériau à changement figure 4.2. mesurées par ellipsométrie. Dans ce chapitre, nous avons pu constater un accord qualitatif remarquable entre les mesures
	de phase à extinction induite nulle. La section suivante explique la théorie, suivie des La procédure de fit des mesures expérimentales de l'ellipsométrie a été la suivante: et les calculs, malgré les approximations mentionnées ci-dessus. Nous avions également
	techniques de fabrication et de caractérisation des VO 2 -NC. Extraction des paramètres effectifs des nanocristaux de VO 2 (détails dans le chapitre 4), les épaisseurs respectives de la couche supérieure SiO 2 , de souligné le fait que ce modèle simple est capable de reproduire correctement les principales
	Avant de se pencher sur les méthodes d'extraction de l'indice de réfraction effectif du la couche d'inclusion VO 2 -NCs et de la couche d'inclusion V ont été définies comme caractéristiques de la réponse optique pour les deux états de VO 2 en changeant simplement
	0.2.2 Théorie de Mie et extraction des paramètres effectifs des nanocristaux de VO 2 milieu composite, nous avons analysé les propriétés structurales et de transition des VO 2 des paramètres d'ajustement libre, dans le modèle multicouche présenté sur la figure 4.7. la dispersion de VO 2 entre isolant et métallique. Ce que l'on peut effectivement observer
	Théorie de Mie NCs intégrés dans la silice. Nous avons appliqué les mêmes tactiques d'ellipsométrie Aucune des dispersions de la matière n'a été fittée, nous avons plutôt modifié la taille et sur la figure 4.2, est que les calculs reproduisent bien le premier pic dans le visible, qui
	Nous commençons par analyser les propriétés de particules sphériques diélectriques lorsqu'elles que celles mentionnées au chapitre 3 pour extraire les propriétés physiques inhérentes à la densité des VO 2 -NCs dans notre modèle Mie-Maxwell Garnett pour ajuster les fits aux est une combinaison de résonances dipolaires électriques et magnétiques, et le second pic
	sont illuminées par une onde électromagnétique et en particulier, nous étudions la diffusion l'échantillon. Selon les résultats, (qui sont expliqués en détail au chapitre 4), nous observons mesures. Comme il pourrait exister une corrélation entre les épaisseurs et la permittivité dans le proche infrarouge qui est une résonance plasmonique. Cette dernière n'est présente
	(scattering) correspondante de manière à évaluer les composantes magnétiques et électriques. les changements de propriétés des matériaux comme on peut s'y attendre d'un matériau à des films absorbants, nous ajustons simultanément les deux séries de mesures pour VO 2 que lorsque le VO 2 est métallique, c'est à dire lorsque la partie réelle de la permittivité est
	La complexité de l'analyse de la diffusion augmente considérablement lorsque la taille changement de phase. Pour les mesures expérimentales d'ellipsométrie, un modèle optique isolant et VO 2 métallique et nous lions les différentes épaisseurs. Cela signifie que le fit négative. Il s'agit donc d'une résonance accordable dynamiquement.
	de l'objet s'approche de la longueur d'onde de l'onde incidente. Dans notre cas le rayon multicouche a été créé (similaire au chapitre 2) qui comprend quatre couches, comme le final donnera les mêmes épaisseurs pour les différentes couches, quelle que soit la phase Le modèle Mie-MG-EMA nous permet ainsi de reproduire par calcul les principales
	sphérique était de 50 nm. Dans une telle situation, nous ne pouvons plus supposer que le montre la figure 4.7. (i) une couche mince de SiO 2 , (ii) la couche effective VO 2 -NCs + SiO 2 , de VO 2 -NCs. La seule différence entre les deux modèles réside dans la dispersion des caractéristiques spectrales des métamatériaux basés sur VO 2 -NC. Nous pouvons ainsi prédire
	champ est constant à l'intérieur de l'objet: l'"approximation aux grandes longueurs d'onde" (iii) une couche intermédiaire contenant des inclusions et des impuretés de vanadium et (iv) VO 2 -NCs. les propriétés optiques attendues des VO 2 -NC de tailles arbitraires. Nous montrons ci-
	cesse de s'appliquer et les effets de retard doivent être pris en compte. Par conséquent, un substrat semi-infini de SiO 2 . Ce modèle à quatre couches est justifié par les observations Dans la figure 4.8, nous montrons les fits optimisés avec les données expérimentales. dessous comment ce modèle peut être utilisé pour concevoir des métamatériaux accordables
	xxxvi xxxvii xxxviii xxxix xl xli

[START_REF] Xu | Micrometre-scale silicon electro-optic modulator[END_REF]

, mais ils nécessitent une plus grande distance de couplage (quelques centaines de microns) en raison de leur modulation d'absorption limitée et du chevauchement modal

[START_REF] Hao | 40 Gb/s electro-absorption-modulator-integrated DFB laser with optimized design[END_REF]

.

Idéalement, en termes d'énergie requise, un modulateur typique d'une empreinte de 1µm 2 aura une consommation d'énergie de plusieurs ordres de grandeur supérieure à celle d'un dispositif de taille similaire basé sur VO 2

[START_REF] Ko | Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry[END_REF]

avec une vitesse de commutation observée en régime sub-nanoseconde

[START_REF] Jerry | Dynamics of electrically driven sub-nanosecond switching in vanadium dioxide[END_REF]

. Compte tenu de toutes les propriétés intéressantes du VO 2 présentées dans le chapitre 2, il s'agit à ce jour d'un matériau très intéressant pour obtenir une accordabilité efficace dans les dispositifs photoniques. Les modulateurs en espace libre basés sur le VO 2 ont déjà fait l'objet de démonstrations xxx dans les gammes de conception Nous passons maintenant à la conception du dispositif optique, dont la théorie a été formulée en détail dans le chapitre 3. Le design du guide d'onde modulateur hybride VO 2 /Si proposé est basé sur les composants de type coupleur directionnel, conformément à la théorie des modes couplés. La stratégie de conception repose sur un patch de VO 2 intégré sur un guide Si, dont les dimensions sont conçues afin d'optimiser la modulation. Le schéma de la figure: 3.1, montre le patch de VO 2 conçu sur un guide d'onde hybride en Si, qui est placé stratégiquement à proximité du guide d'onde bus en Si. Le principe est le suivant: lorsque le VO 2 est isolant, le signal optique se propage dans le guide d'onde bus sans se coupler au guide d'onde hybride VO 2 /Si. Puis lorsque le VO 2 est porté dans sa phase métallique, l'indice effectif du mode guidé dans le guide d'onde hybride se modifie et remplit les conditions pour le couplage directionel. S'ajoute à cela les pertes optiques induites par le VO 2 , permettant ainsi l'absorption totale du mode optique, et ainsi la modulation du signal dans le guide d'onde bus. Même si d'autres groupes ont déjà proposé des modulateurs plasmoniques basés sur le VO 2 avec une architecture différente, le dispositif que nous proposons va au-delà des anciens dispositifs proposés en terme de fonctionnalité et d'intégration. [36, 37, 38]. xxxi (a) Le VO 2 est isolant, la propagation des ondes est inaltérée et reste dans Si-WG1. (b) Le VO 2 est métallique, l'onde dévie son chemin de propagation dans le guide d'onde adjacent et est absorbée par le VO 2 .

Figure 2: Vue schématique du modulateur hybride actif basé sur la transition de phase isolant-métal du VO 2 xxxii Le dispositif hybride 0.1.6 Simulation, fabrication et résultat Simulation La simulation du dispositif nécessite des moyens numériques pour calculer et analyser les modes des guides d'ondes dans la structure. Dans notre cas, nous utilisons la méthode des différences finies pour discrétiser le modulateur de manière à pouvoir calculer l'évolution des composantes électriques et magnétiques du champ éléctromagnétique en chaque point afin de pouvoir prédire avec précision le comportement de la lumière à l'intérieur et aux limites de la structure de guidage. Les informations importantes requises pour analyser le modulateur par calcul sont les propriétés du matériau, qui ont été extraites précédemment à l'aide des techniques de caractérisation optique. L'explication détaillée de la théorie de simulation est donnée dans le chapitre 3, où les méthodes de discrétisation et le solveur de mode conçu en interne sont également expliqués. Grâce à la simulation numérique, nous avons pu calculer les dimensions souhaitées du guide d'onde bus Si et du guide d'onde hybride ainsi que leur espacement. Les dimensions obtenues, comme mentionné dans le paragraphe précédent, sont de la plus haute importance car le couplage de la lumière du guide d'ondes bus au guide d'ondes hybride est très sensible à l'espace entre les guides d'ondes. Il est également important de faire correspondre les xxxiii modes effectifs et contrairement aux outils du commerce, de manière à calculer plus précisément les champs E et H. Les schémas pour lesquels nous avons effectué cette discrétisation sont donnés dans la figure 3.10. L'ensemble du système est discrétisé en neuf couches verticalement sur l'axe des x et en six sections sur l'axe horizontal des y. Cela permet d'obtenir des valeurs plus précises en termes de l'indice effectif et des conditions d'orthogonalité entre mode de l'ordre de 10 -15 par opposition à 10 -7 pour les outils commerciaux. Le seul inconvénient est que le calcul est un processus intensif qui prend plus de temps lorsqu'une partie imaginaire de l'indice de réfraction est ajoutée. Fabrication Nous cherchons par la suite à fabriquer la structure simulée précédemment. La première étape vers la réalisation du dispositif hybride VO 2 /Si est l'étape de lithographie définissant les deux structures de guide d'ondes. Afin de définir la structure, le motif a été exposé sur une résine ZEP 520A à l'aide d'un système de lithographie par faisceau électronique et ensuite développé. Le masque a été fabriqué en interne et toute la procédure de lithographie a été réalisée par P. Rojo-Romeo. Après avoir défini le motif, nous avons effectué une imagerie MEB Fig3.18, afin de vérifier le transfert du motif et nous avons observé un transfert parfait du motif et une distinction précise des intervalles entre les guides d'ondes. L'étape suivante a consisté à effectuer une gravure par RIE-ICP (Inductively Coupled Plasma Etching (ICP)) xxxiv en utilisant un mélange d'lié à des restes de résine ou soit des résidus de masque dur sur la surface supérieure des guides d'ondes. Ce qui indique que le processus doit avoir été incomplet, entraînant des résidus de résine, ou que la gravure a pu endommager le masque métallique (atteignant le VO 2 ), contrairement à ce qui était prévu, en raison de la faible vitesse de gravure de la région couverte par le masque dur. Résultats et discussion Contrairement à ce que l'on attendait, le masque métallique et la résine ont été complètement gravés, ce qui a entraîné un effet résiduel sur la surface. Afin de récupérer le dispositif, il est nécessaire d'analyser l'échantillon en profondeur et de comprendre la composition chimique de la topographie de la surface. Cela permettrait d'identifier la composition du matériau en surface, qui nous éclairerait sur les problèmes liés à la gravure. Deuxièmement, une étude topographique par AFM pourrait nous donner des informations sur la rugosité. Après la caractérisation, nous pourrions procéder à une nouvelle gravure et à un nouveau processus de nettoyage afin d'éventuellement réussir à finaliser le dispositif. D'autre part, nous pourrions utiliser une technique de dépôt et de fabrication alternatives, par le biais d'une approche dite de lift-off pour permettre le dépôt de VO 2 uniquement sur les guides d'ondes déjà patternés, évitant ainsi l'étape de gravure qui est problématique. Plus loin dans le chapitre, j'ai également montré un nouveau concept pour l'application xxxv du modulateur guide pour les modulateurs montrent que, par rapport aux modulateurs classiques, un simple déclencheur (optique, électrique ou thermique) peut avoir un impact profond sur les valeurs de transmission dans le guide d'ondes du bus. Contrairement aux dispositifs VO 2 basés sur une absorption élevée, cette architecture de dispositif peut être mise en cascade avec de faibles pertes d'absorption.

  Dans notre travail, nous assemblons des nano résonateurs Si spécialement conçus en réseaux périodiques bidimensionnels fabriqués sur un film mince VO 2 . Le principe est le suivant : lorsque VO 2 est isolant, les résonances ED et MD se chevauchent et nous remplissons ainsi les conditions de Kerker et obtenons une diffusion arrière nulle. Comme le VO 2 est partiellement absorbant même à l'état isolant, la majeure partie de la lumière est absorbée et nous obtenons donc un absorbant quasi parfait. D'autre part, lorsque VO 2

	observé qu'autour du maximum ∆n ef f atteint pour chaque taille de NCs, nous avons simul-lumière, les effets Kerker accordables et le contrôle actif de la directivité de la lumière. 0.3 Part 3: Métasurfaces à base de VO 2 pour la nanophotonique accordable en es-Dans cette section, nous nous concentrerons davantage sur les métasurfaces 2D, qui être une technologie alternative aux technologies d'affichage actuelles car la majorité de Génération de couleurs structurelles accordables par des structures diélectriques mais aussi par la géométrie dans laquelle elles existent. Par conséquent, pour créer une absorbant les rayonnements EM parasites. Le contrôle de l'absorbance de la lumière joue du résonateur de manière à obtenir les propriétés de diffusion souhaitées. Milton Kerker
	tanément ∆k ef f =0. Ce qui signifie que cette plateforme de métamatériaux à changement de Le type de matériau utilisé classifie la métasurface et son interaction. Par exemple, les pace libre présentent un phénomène similaire à celui attendu des métamatériaux, mais dont les épais-ces technologies utilisent soit des LCD (écrans à cristaux liquides) soit des LED (diodes reproduction précise des couleurs, il est essentiel de déterminer les mécanismes de couplage un rôle majeur dans nos industries photoniques actuelles et peut avoir un fort impact dans a étudié ce phénomène en 1983 et a révélé qu'il est possible d'obtenir un rayonnement de
	phase permet d'obtenir par design une accordabilité dynamique de l'indice de réfraction seurs sont de l'ordre d'une fraction de la longueur d'onde et sont donc relativement plus électroluminescentes) dont les dimensions sont plus grandes que la longueur d'onde de de la surface avec la lumière extérieure et de pouvoir les concevoir de manière à obtenir les la collecte de l'énergie solaire [71, 72], les bolomètres ainsi que les capteurs chimiques et champ asymétrique avec une diffusion arrière nulle (zero back scattering) ou une diffusion
	sans modifier l'absorption optique, figure 3. nanostructures métalliques utilisent les plasmons de surface (une interaction collective entre 0.3.1 Introduction simple à fabriquer. En outre, les métasurfaces permettent de fabriquer des composants plats fonctionnement, ce qui limite la taille des pixels et des dispositifs. En outre, la plupart propriétés souhaitées. Ce type de résonance nous donne la flexibilité nécessaire pour le choix biologiques. Les métamatériaux sont conçus pour modifier les valeurs de µ et de telle sorte avant nulle (zero forward scattering) en combinant correctement les dipôles électriques et
	les électrons libres et le photon) (SP) [58] pour piloter la réponse optique des méta-structures
	De plus, cet ajustement de l'indice de réfraction à "perte induite nulle" peut être conçu Le domaine des métamatériaux et des métasurfaces connaît une croissance considérable. qui peuvent être intégrés presque partout. Cela permet de réduire considérablement les du temps, lorsque nous voulons produire de la couleur, nous utilisons des pigments pour du matériau, l'extensibilité pour la fabrication à grande échelle et une faible dépendance que leur impédance corresponde à celle de l'espace libre, créant ainsi un milieu transparent magnétiques des particules magnéto-diélectriques. Selon la première condition de Kerker,
	métalliques ; d'où le nom de métasurface plasmonique [59, 60]. Des recherches récentes
	pour se produire dans toute la gamme du proche infrarouge en sélectionnant spécifiquement Cela est dû aux limites des matériaux et structures naturels où l'interaction et l'accordabilité dimensions des composants optiques (les composants en optique classique reposent sur l'affichage visuel, qu'ils soient statiques (affiches, peintures murales, imprimantes, etc.) angulaire. et sans réflexion. Cela peut s'avérer très avantageux pour créer des filtres parfaits et limiter lorsque µ = , l'onde incidente subit une diffusion vers l'arrière nulle (ZBS). Alors que
	une taille appropriée de NCs. Cet effet ne peut pas être obtenu avec des films minces de sont limitées par la diffraction et sont principalement régis par l'indice de réfraction et non l'accumulation de la phase optique pendant la propagation, ce qui nécessite, par exemple, ou dynamique, comme les écrans à cristaux liquides. Les colorants synthétiques sont ont démontré expérimentalement des métasurfaces plasmoniques permettant de contôler la polarisation, l'amplitude et la phase du rayonnement électromagnétique [61, 62]. Celles-ci Dans notre cas, nous avons démontré à l'aide de simulations, deux stratégies uniques les pertes. Malgré la croissance des métasurfaces statiques basées sur la résonance de pour une diffusion avant nulle (ZFS), µ et doivent satisfaire la condition = (4-µ) 2µ=1 , soit la
	matériaux à changement de phase et souligne l'utilité de les modifier par une approche type par la structure. Sauf dans certains cas, comme celui du papillon Morpho, où les nano-d'avoir des surfaces courbes pour les lentilles). Contrairement à l'optique classique, où coûteux à produire, plus difficiles à recycler et nuisibles à l'environnement. En outre, ils pour mettre en oeuvre l'accordabilité en créant une couleur structurelle accordable. Notre Mie, les rendre accordables reste un problème majeur. Ces dernières années, des efforts deuxième condition de Kerker.
	ont joué un rôle essentiel dans divers domaines d'application tels que les lentilles plates
	métamatériaux. Nous notons également que l'effet inverse peut être exploité, à savoir, proche structures complexes de ses ailes reproduisent une couleur bleue vibrante. Ce phénomène a la lumière traverse un milieu d'un indice de réfraction particulier pour modifier son trajet, se décomposent et ne sont pas stables, ils manquent de résolution, ce qui limite encore première stratégie consiste à utiliser des nano-structures cylindriques de TiO 2 sur des films considérables ont été déployés pour mettre au point des dispositifs nanophotoniques accord-
	[63], les hologrammes [64], les coupleurs directionnels [65] etc.
	du maximum ∆k ef f , nous pouvons avoir simultanément une valeur ∆n ef f négligeable. Cette motivé les scientifiques à comprendre et à reproduire une telle couleur vibrante en laboratoire les métamatériaux permettent de manipuler l'indice de réfraction du milieu par le biais de leur champ d'application. C'est pourquoi, dans la recherche d'un remplacement approprié, minces de VO 2 , de telle sorte que pendant la transition, la résonance fano de la couleur ables constitués de nanostructures métalliques couplées à divers matériaux à changement de
	Dans les métamatériaux diélectriques, les résonances supportées sont une combinaison
	plate-forme peut donc être utilisée pour contrôler activement l'absorption sans modifier en utilisant des matériaux naturels. nanostructures pour obtenir le contrôle du front d'onde souhaité. Jusqu'à présent, la plupart les chercheurs ont accordé une attention considérable à la coloration structurelle. Un structurelle est modulée, ce qui entraîne un léger décalage dans l'interférence de l'onde phase tels que GeSbTe et VO 2 [73, 74, 75], mais les nanostructures accordables entièrement
	de dipôles électriques et magnétiques, autrement appelées résonances de Mie. La théorie de
	l'indice de réfraction du milieu (par exemple, pour R=95nm à λ=1500nm, on obtient Cette limitation des matériaux naturels est surmontée en créant essentiellement des des métasurfaces démontrées expérimentalement sont statiques après fabrication et, par exemple célèbre de couleur structurelle peut être observé dans la nature : la couleur bleue incidente et de l'onde rayonnée. Ce qui conduit à un changement de couleur de manière diélectriques restent largement inexplorées. Dans cette section, nous démontrons la stratégie
	Mie est une solution analytique des équations de Maxwell pour la diffusion du rayonnement
	∆k ef f =-0,2 et ∆n ef f =0), comme le montre la figure 4.16. matériaux artificiels qui peuvent se comporter de manière similaire à un milieu naturel ou conséquent, leurs applications sont pour le moment limitées. [52, 53]. métallique des ailes du papillon Morpho. Le papillon Morpho nous a permis de réaliser collective. Alors que la seconde stratégie implique la mise en oeuvre directe de nanocylindres de conception d'un absorbant parfait à métasurface accordable basé sur le VO 2 , composé de
	électromagnétique par des particules de toute taille (également appelée diffusion de Mie).
	même présenter des propriétés supérieures à ces matériaux. Cette catégorie de matériaux ou Notre travail vise donc à conférer de l'accordabilité aux métasurfaces en exploitant la qu'il existe une autre façon de produire des couleurs sans aucun colorant ni pigment, de VO 2 , où la transition IMT du VO 2 affecte à la fois les résonances de Mie et les résonances résonateurs en Si, supportant à la fois des modes de Mie électriques et magnétiques.
	Cela a permis de comprendre que l'utilisation d'un matériau diélectrique à indice de réfrac-0.2.4 Conclusion de dispositifs fabriqués par l'homme est connue sous le nom de métamatériaux (où méta transition isolant-métal du VO 2 . Pour cela, nous cherchons à moduler dynamiquement les en disposant simplement des matériaux transparents dans une structure périodique [54] de Fano. Les résultats de la simulation dans les deux cas sont présentés en détail au chapitre 5.
	En conclusion, nous avons démontré une plateforme active basée sur des VO 2 -NCs intégrés signifie "au-delà" en grec). Afin d'obtenir les interactions souhaitées avec la lumière, les résonances de structures de dimensions sous-longueurs d'onde. En utilisant cette stratégie de façonnant l'interférence entre la lumière incidente et la lumière réfléchie. Cependant, la tion élevé nous permet de contrôler à la fois le dipôle électrique et magnétique contrairement Le résultat sur l'implémentation des nanostructures de VO 2 montre des résultats prometteurs Principe et théorie de conception des absorbants parfait
	au seul dipôle électrique dans un système de matériau plasmonique. L'amélioration de la
	dans du SiO 2 sous forme de nano-inclusions sphériques qui permettent d'exploiter l'IMT constituants de ces métamatériaux, appelés méta-atomes (généralement plus petits que la contrôle de la résonance, nous proposons d'abord une modulation de couleurs structurelles au haute qualité des couleurs structurelles que présente la nature nécessite des structures par rapport à la structure en couche mince et pourrait être un avantage pour les futures Contrairement à la section précédente, nous nous concentrons ici sur l'accordabilité du
	réponse magnétique provoquée par l'interaction de la lumière incidente avec des particules
	du VO 2 pour accorder et commuter des modes multipolaires dans le visible et le proche longueur d'onde de la lumière), sont définis par l'utilisateur et sont conçus et disposés de sein de nanostructures de TiO 2 sur un film mince de VO 2 , créant ainsi un champ d'application compliquées et volumineuses [55, 56] qui rendent la taille de la structure beaucoup plus technologies d'affichage colorés basées sur des structures accordables. VO 2 dans le spectre des télécommunications (proche infrarouge) [76, 77]. Dans une nano-
	de taille similaire à la longueur d'onde peut permettre de concevoir des filtres de couleur qui
	infrarouge. Contrairement aux couches minces de VO 2 non patternées, cette approche méta-telle sorte qu'un effet moyenné se produise en termes de diffusion et d'interférence avec futur pour les affichages de couleurs structurelles accordables. Nous démontrons également grande que la longueur d'onde [57]. particule diélectrique, une résonance magnétique de Mie provient d'une excitation d'un
	matériau permet d'adapter l'indice de réfraction complexe et l'accordabilité des matériaux à les ondes lumineuses. La propriété ou la réponse globale des métamatériaux dépend des la manipulation de résonances de Mie électriques et magnétiques induites par l'IMT du La technologie des écrans est très utile à l'humanité. Il est donc très important de utilisent différentes résonances de Mie avec des multipôles d'ordre supérieur. [66] 0.3.3 Métasurface de VO 2 pour une absorption totale accordable. mode électromagnétique particulier dans la particule. Cette excitation se produit lorsque la
	De même, nous pouvons ajuster davantage les propriétés optiques des métasurfaces en
	changement de phase. En effet, nous avons montré qu'en ajustant la taille des VO 2 -NCs, structures unitaires individuelles, mais surtout des propriétés physiques, de la disposition et VO 2 , supportée par des nano-piliers de silicium, qui permettent d'obtenir une modulation de comprendre comment l'homme perçoit les couleurs. Dans le chapitre 5, nous avons expliqué Introduction longueur d'onde effective de la lumière à l'intérieur de la particule devient comparable à la
	utilisant des effets collectifs grâce au couplage entre les résonateurs avec une catégorie de
	ce système peut être exploité comme un métamatériau à changement de phase dont les de l'orientation de la ou des méta-structures/méta-atomes. Par conséquent, en ajustant les transmission de l'ordre de trois ordres de grandeur à λ = 1555nm. en détail la perception des couleurs et la carte des couleurs RGB (Red Green Blue), qui résonance plus générale appelée résonance de Fano [67]. Elle utilise l'interférence entre Les absorbants parfaits utilisant des structures de type métamatériaux ont gagné en popularité taille de la particule, présentant un courant de déplacement circulaire du champ électrique
	propriétés de modulation d'indice de réfraction et d'absorption sont ajustables par design. Dans une perspective scientifique plus large, les résultats présentés dans ce chapitre aident dimensions et arrangements de ces méta-atomes, il est possible de développer de nouveaux matériaux, possédant des propriétés uniques qui ne sont pas naturellement disponibles sont très importantes pour comprendre et représenter les différentes couleurs sous forme la lumière rayonnée et réfléchie de manière constructive ou destructive. Ce phénomène a [78, 79]. La capacité d'exciter simultanément les réponses électriques et magnétiques au sein ces dernières années. Un absorbant parfait est un dispositif dans lequel tout le rayonnement 0.3.2 Métasurface de VO 2 pour la génération de couleur structurelle accordable numérique. été étudié par Fano et al [68]. Cette théorie de la résonance peut être utilisée pour générer incident est absorbé à la fréquence de fonctionnement. Les premières applications des d'une nanoparticule est d'un grand intérêt pour diverses applications et donne un autre degré
	à définir un nouveau système permettant d'étudier les interactions accordables entre les dans la nature. De tels travaux ont conduit par exemple au développement de matériaux à Introduction une couleur structurelle par interférence entre la lumière directement réfléchie et le mode absorbants éléctromagnétiques (EM) ont été l'exploitation furtive dans la gamme des opéra-de liberté pour ajuster la réponse d'un métamatériau par conception [80, 78]. Il a également
	nanoparticules et les interférences accordables entre les résonances multipolaires. Nous indice de réfraction négatif [42, 43, 44], de matériaux à indice zéro [45, 46, 47], de "capes La couleur structurelle est un champ de recherche dont l'origine a été inspirée par la de résonance guidée sur une structure de surface. Puisque les réponses électromagnétiques tions RADAR ou micro-ondes/ondes radio, réduisant la section transversale du radar [69, été démontré que les emplacements spectraux des résonances dipolaires électriques et
	l'envisageons comme un banc d'essai pour la modulation dynamique de la diffusion de la d'invisibilité" (optical cloaking) [48, 49] et imagerie sub-diffraction [50, 51]. couleur bleue vibrante des ailes du papillon Morpho. La couleur structurelle pourrait de ces structures ne sont pas seulement régies par les permitivités complexes du matériau, 70]. Plus récemment, les absorbants EM sont utilisés pour réduire les interférences en magnétiques d'un résonateur diélectrique peuvent être ajustés en faisant varier la géométrie
	xlii xliii xliv xlv xlvi xlvii xlviii xlix l

2 -NCs (comme le montre la figure 4.15), nous avons

Table 2 .

 2 Following the deposition trials, to understand the crystallinity of deposited materials, we performed XRD to analyze the thin-film structural properties and phase of VO 2 . The following successful recipe is shown in the table 2.1. In addition to this homemade VO 2 , the majority of the VO 2 thin-films forming the basis of my research have been provided by the Ramanathan group from Purdue university (USA). 1: Parameters for Reactive Sputtering Technique for the growth of VO 2 thin-films on quartz.

	Temperature	550 • C
	Flow rate Ar gas	50 sccm
	Flow rate O2 gas	2%
	RF power	200 W
	Rate of Deposition	2 nm/min
	Rapid Annealing	10min
	Annealing Temp	550 • C

• C and Ar concentration was fixed at 50 sccm (Standard Cubic Centimeters per Minute) for the procedure.

Table 3 .

 3 1: Power transfer simulation calculated around the core area of the waveguide. 80% of the input power is calculated at the central area of the core, rest is dissipated at the edges of the waveguide.

Table 3 .

 3 2: Geometric parameters for VO 2 waveguide modulator.

  which are larger in size and limited tunability. Thus VO 2 seems to a hopeful replacement and efficient candidate, and also since the VO 2 tunability can be triggered in multiple ways, therefore future devices can use electrical or optical stimulus to trigger the transition states of VO 2 performing at much shorter timescales. Our modulator is quite broadband compared to the ring resonators in these linear modulators, which requires extremely precise wavelengths

	CHAPTER 4
	MULTIPOLAR RESONANCES WITH DESIGNER TUNABILITY USING VO 2
	to function
	94

  5.12a and when observed from certain angles we see vibrant structural colors, represented in figure[START_REF] Stefanovich | Electrical switching and Mott transition in VO2[END_REF].12b. The structural colors in the figures 5.12c and figure 5.12d can be observed with perfect distinction from each pixel, even though the experiment was done to measure the temperature dependence of the color switching but we observed stronger angle dependence 142 compared to temperature dependence. Therefore it might seem the color switching is due to IMT but it is mainly due to the angle of observation. We need to further evaluate the temperature dependency of the sample with help of a steadier camera observation setup. 5.1.6 Discussion and future perspectives In summary, Mie resonances supported by dielectric nanostructures can be used to create highly saturated structural colors. Additionally, we have presented two different approaches for achieving tunable structural color generation with VO 2 as a tunable element. By tuning the VO 2 transition state, the recorded color switch were presented in a CIE diagram. Based on the properties of Fano resonances as well as Mie resonances, the reflected colors have also been achieved from VO 2 metasurface and thus can generate bright, high-contrast, and high resolution structural colors and even colorful images. Even though these preliminary results are promising, there is still challenge of achieving perfect tunability in color reproduction.

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2020LYSEI044/these.pdf © [J. John], [2020], INSA Lyon, tous droits réservés

CIE-LAB stands for CIE L*a*b*, where L* for the lightness from black (0) to white (100), a* from green (-) to red (+), and b* from blue (-) to yellow (+). International Commission on Illumination (CIE) in 1976 defined the color space so that the same amount of numerical change in these values corresponds to roughly the same amount of visually perceived change.

After we have defined the characteristics of the EM wave properties in the main text. We now try to understand the EM field behaviour at the boundary edges of the propagating medium.

It is important because the modern devices are not composed of only one medium but several, stacked in a manner necessary for its application. Hence it is required to understand the behaviour of EM wave at the interfaces of the different medium and the environment. To do so we hypothesize two mediums with 1 and 2 and then solve the Maxwell's equation to equate a relation between the wave characteristics between the two mediums. Therefore, by solving Maxwell's equation based on the wave characteristics we observe the boundary conditions, as follows;

• The tangential components of the electric field should be continuous across the boundary.

• If there are no current flowing on the surface of the boundary then, tangential magnetic components are continuous, that is

But if the current flows on the surface, then the magnetic fields are discontinuous and are related by current density on the surface J s ,

following equation,

where α = k 2 n 1 n 2 γ. Which means with TIR, there is an associated transmitted wave in the region with lower index medium, whose amplitude decreases exponentially w.r.t to x and this wave is known as evanescent wave. Later we'll show that this evanescent wave will play a major role in realization of photonic devices.

A.3 Optical Waveguide and transmission analysis

We have understood the wave characteristics inside a medium and also at interface of an medium. These conditions were an ideal since there was no restriction in direction for wave propagation and the medium was homogeneous. For practical applications, this not the case, hence we try to derive the equation for an in-homogeneous medium. Therefore we solve the Maxwell's equation like before, but this time the constitutive relation swill be, as follows;

Solving for wave equation, we take curl of equation 3.1 but contrary to previous calculation, we don't assume ∇. E = 0 since depends on the x, y, z direction. Therefore we derive the wave equation in an inhomogeneous medium as;

Considering a general case similar to classical waveguide where there is an index discontinuity in both x and y direction, i.e. n 2 (x, y), eg: optical fiber or rectangular slab waveguide. Then the z-part (longitudinal component) of the wave equation and t-part of the wave equation can be separated out, and equations can be separated written as;

The aim is to understand the field values as function of x and y and their respective propagation constants. In order to understand the concepts, we again, try to simply the conditions by taking a planar waveguide rather than classical waveguide, as the index discontinuity is only in one direction for the slab. In order to understand the characteristics of field functions based on index discontinuity, we try to simplify, by considering the case where the refractive index only varies in one transverse direction i.e. n 2 (x) as shown in figure A.2. Now since the discontinuity of the index is only in x direction therefore the y-part of the wave equation can also be separated out, such that if the direction of propagation is z-direction, then propagation constant in ydirection is zero..

where j = x, y, z. Solving for equations 3.1 and 3.2, we get in total 6 resultant equations corresponding to x, y and z. These equations relate the E and H components corresponding

x,y,z directions and are function of n(x), these equations are as shown;

•

A.4 TE mode

If we assume the form of a mode where the electric field is not in the longitudinal direction, i.e. E y = 0 and E x = 0. Then we invoke equations A.20, A.22 and A. [START_REF] Tselev | Symmetry relationship and strain-induced transitions between insulating M1 and M2 and metallic R phases of vanadium dioxide[END_REF], where E y , H x