List of Tables

Despite the importance of the fabrication development, a theoretical method was needed to completely study this new field and certainly to solve the Maxwell's equations. Many numerical methods have been developed to present a full study of the electromagnetic field's propagation and properties. The Finite Differential Time Domain method (FDTD), most used method nowadays, was invented in 1966 when Yee proposed an algorithm [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media[END_REF] to solve the Maxwell curl equations directly in time domain on a space discretized window. After that, Taflov and Broduin [3] reported the numerical stability criterion for Yee algorithm. This method has continued to develop and many researchers proposed

Introduction

different tools to ameliorate it in order to have accurate results, of the electromagnetic problems, similar to the realistic ones.

However, in 1960 the first laser light was invented [START_REF] Theodore H Maiman | Stimulated optical radiation in ruby[END_REF] which was the reason behind the appearance of new optics field which was completely unknown: the nonlinear (NL) optics. In fact the important of the laser in this field comes from its high density of focused injected beam, which was not possible using other injection methods. Due to this highly dense and focused beam, many NL effects appeared, these effects are related to the susceptibility tensor χ2 , χ3 or to a higher order tensor. The first NL phenomena observed was the second harmonic generation, after that many other phenomena were discovered such as the third harmonic generation, four wave mixing, Kerr effect, Pockels effect (electro-optic (EO) effect) and many others. In fact, the appearance of the NL effect is related to the material used to fabricate the optical structure or it could be observed due to an application of other external physical parameters; for example the application of an electric field into a non-centrosymmetric substrate induce the appearance of the EO effect.

Light confinement inside some particular nano structure and the enhancement of light transmitted from the structure have lead to the appearance of both linear and non-linear effects, each depending on the overall condition of the study. Confinement of light in a subwavelength area, high optical transmittance factor and the resonance of the structure are the main interesting parameters of a nano-structure and define its importance to achieve different applications. Those nano-structure properties have attracted many researches to study the optical response of various structures and many geometrical shapes have been proposed in a way that cover a wide range of applications.

In our work, we are interested in both linear and NL effects, that appear due to the light confinement, for different structures. In the first part of our work, we will

show a new application using a well known nano-structure, the Bowtie Nano-Aperture antenna (BNA). This structure is widely discussed and shows interesting characteristics (resonance with high electric field confinement) making it useful in various domains and applications. However, in the second part of this thesis, we will discuss the electro optical effect, an effect related to the second order susceptibility tensor and appears in noncentrosymmetric material. The EO effect or Pockels effect appears due to the application of an external electric field into the NL medium which induces a linear variation of the medium refractive index. In this part, a different type of photonic crystal (PhC) will be proposed and different FDTD algorithms will be used to simulate the optical response of these structures that exhibit large electro-magnetic confinement.

Bowtie Nano-Aperture (BNA) antenna

BNA antenna has become a very well known structure in the field of nano-photonics due to their ability to confine light in a nano-metric region. This light confinement was the reason behind the usability of the BNA in a very wide range of fields and for various applications.

State of the art of the BNA

The appearance of nano-antennas dates back to the end of the 20th century. The idea of optical nano-antennas comes from their counterparts: the antennas in radio-frequency.

Similar to the radio frequency antenna which transforms the electromagnetic waves into an electrical signal and vice versa, the optical nano-antenna is responsible for transforming a propagating beam of light into a focused light in a nano-metric zone and inversely. This

Introduction

double functionality of the nano-antennas make them, similarly to the radio-frequency antenna, able to operate in two functional modes: emission and detection.

From the first appearance of the nano-antenna, many geometrical shapes were proposed in order to enhance the property of the antenna in a way that corresponds to a specific desired application. The beginning was with the study of the optical properties of a single nano-particle and nano-wires metallic structure [START_REF] Weeber | Plasmon polaritons of metallic nanowires for controlling submicron propagation of light[END_REF][START_REF] Ol Muskens | Optical scattering resonances of single and coupled dimer plasmonic nanoantennas[END_REF], by studying the role of metal cluster for surface-enhanced processes like Raman scattering [START_REF] Kreibig | Optical properties of metal clusters[END_REF][START_REF] Palpant | Optical properties of gold clusters in the size range 2-4 nm[END_REF][START_REF] Kelly | The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment[END_REF]. Searching for a better enhancement of optical field and confinement of light have conducted researchers into developing more complex geometry, nano-antennas have not been fabricated only from one nano-structure as in the case of nano-wire, researchers have proposed the combination of two structures which have lead to the appearance of the dimers [START_REF] Aizpurua | Optical properties of coupled metallic nanorods for field-enhanced spectroscopy[END_REF][START_REF] Ol Muskens | Optical scattering resonances of single and coupled dimer plasmonic nanoantennas[END_REF] , Bowtie [START_REF] David P Fromm | Gap-dependent optical coupling of single bowtie nanoantennas resonant in the visible[END_REF][START_REF] Fischer | Engineering the optical response of plasmonic nanoantennas[END_REF], diabolo [START_REF] Zhou | Complementary bowtie aperture for localizing and enhancing optical magnetic field[END_REF][START_REF] Grosjean | Diabolo nanoantenna for enhancing and confining the magnetic optical field[END_REF] and BNA [START_REF] Sendur | Near-field radiation of bow-tie antennas and apertures at optical frequencies[END_REF][START_REF] Wang | High transmission nanoscale bowtie-shaped aperture probe for near-field optical imaging[END_REF][START_REF] Flauraud | Large-scale arrays of bowtie nanoaperture antennas for nanoscale dynamics in living cell membranes[END_REF][START_REF] Lu | Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures[END_REF][START_REF] Fadi | Double-way spectral tunability for the control of optical nanocavity resonance[END_REF]. Combined structures have the advantage of confining light in the vicinity of the nano-antenna center (gap between the two structures) [START_REF] Novotny | Antennas for light[END_REF][START_REF] Lu | Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures[END_REF] and these nano-antennas present a very interesting sensitivity to the light polarization [START_REF] Park | Effect of incident beam width on light transmission enhancement by bow-tie-shaped nano-aperture[END_REF][START_REF] Mathieu Mivelle | Light funneling from a photonic crystal laser cavity to a nano-antenna: overcoming the diffraction limit in optical energy transfer down to the nanoscale[END_REF][START_REF] Elie M Atie | Remote optical sensing on the nanometer scale with a bowtie aperture nano-antenna on a fiber tip of scanning near-field optical microscopy[END_REF]. Various geometrical shapes were proposed in order to fabricate a nano-antenna showing optical properties that fit with multiple applications see figure 1.1.

The difference between a single nano-structures and paired ones was studied by Cubukcu et al [START_REF] Cubukcu | Plasmonic laser antennas and related devices[END_REF]. The authors discuss the cases of a nano-rod and a triangular nano-structure and compare them, respectively, to the dimers (composed of two nano-rods) and Bowtie (composed of two triangular shapes) nano-antennas. The study demonstrates that the field enhancement calculated in the cases of paired nano-structures is more pronounced than the field enhancement calculated using the single nano-structure as shown in figure

1.2.
The difference in terms of the light confinement in the vicinity of a 2 nm gap of a

Introduction

Bowtie antenna and a dimers one was discussed in the study conducted by Hao et al [START_REF] Hao | Electromagnetic fields around silver nanoparticles and dimers[END_REF]. The authors show that the enhancement in the case of a Bowtie antenna is higher than the one recorded by using a dimers antenna. Comparison between Bowtie shaped nano-antennas and other structures has been discussed in several other studies [START_REF] Guo | Optical resonances of bowtie slot antennas and their geometry and material dependence[END_REF][START_REF] Bedir | Modeling of optical nanoantennas[END_REF].

However, the first appearance of the Bowtie antenna as an optical nano-antenna returns to the study presented by R. Grober et al [START_REF] Robert D Grober | Optical antenna: Towards a unity efficiency near-field optical probe[END_REF] in 1997. In their study they have relied on the well-known characteristic of Bowtie antenna in the radio frequency domain, whereby a Bowtie is well-known to be an efficient planar antenna [START_REF] Richardc Compton | Bow-tie antennas on a dielectric halfspace: theory and experiment[END_REF]. Their study demonstrated that the Bowtie antenna represents a new paradigm for near-field optical probes that combines spatial resolution well below the diffraction limit with transmission efficiency. Based on the study presented by R. Grober et al. [START_REF] Robert D Grober | Optical antenna: Towards a unity efficiency near-field optical probe[END_REF], E. Oesterschulze et al. [START_REF] Oesterschulze | Transmission line probe based on a bow-tie antenna[END_REF] conducted a study three years later in which they discussed a high transmission probe for scanning near-field optical microscopy that is based on a Bowtie antenna. The study investigates theoretically the transmission behavior of a Bowtie antenna probe based on a batch-fabricated silicon dioxide tip. In parallel, the study discussed the transmission efficiency and the field confinement, as functions of the antenna geometrical parameters and the gap dimension. This study has opened the way towards using the Bowtie aperture for various nano-optical applications due to its strong electric field energy localized in the gap and to its ability to be implemented at the apex of a fiber tip.

Thanks to the very interesting properties presented by E. Oesterschulze et al. [START_REF] Oesterschulze | Transmission line probe based on a bow-tie antenna[END_REF],

BNA antenna has attracted many researchers and many studies were published discussing its ability in various applications and domains. In 2002, K. Sendur et al [START_REF] Sendur | Near-field radiation of bow-tie antennas and apertures at optical frequencies[END_REF] conducted a study in which they investigated the ability of the Bowtie aperture antenna to generate intense optical spots below the diffraction limit. It also discussed the effect of various parameters on the Bowtie aperture optical response, and compared the enhancement of the field using square and circular apertures. An intensity enhancement of factor 5 was recorded numerically by using the FDTD simulation method. In addition, the study presented the Full-width at half-maximum (FWHM) and the maximum of the field intensity |E| 2 as a function of the aperture width and film thickness. They showed the variation of the FWHM and |E| 2 for a rectangular aperture, circular aperture and for a Bowtie aperture fig. 1.3(a). The optimum square aperture response was 1.45 V 2 .m -2 for a 35 nm FWHM spot size while the optimum Bowtie antenna configuration gave a response of 55.7 V 2 .m -2 for the same spot size. This corresponds to a peak field intensity that is 38.4 times larger for the Bowtie slot antenna than the square aperture on a gold, film. Furthermore, the study discussed the cases of two film metals: silver and gold as shown in figures 1.3(c) and (d), as well as the relation between the incidence angle and the field intensity |E| 2 as in figure 1.

3(b).

As mentioned before, the geometrical shape of the structure has a great effects on the optical response and the confinement intensity. However, some geometrical parameters of the nano-structure also have a major effect on the optical response. The relation between the geometrical parameters of a BNA and: the resonance and the electric field confinement intensity, was discussed in the work conducted by I. Ibrahim [1] in 2010.

In addition, the authors have demonstrated that the optical resonances of the BNA and the light confinement are due to the combination of a guided mode inside the aperture and Fabry-Perot modes along the metal thickness. Further more, the authors deduce an analytical formula from which the cut-off resonance wavelength (FP0) could be calculated:

λ c (G, D) = αD + βln(G) + δDln(G) + γ (1.1)
where α, β, δ and γ are given in Table 1 of [1]. The study also presents the effect of the gap dimension and the lateral length of the BNA on its resonance. In figure 1.4(c) we see On the other hand, other researchers were interested in studying the ability of using the BNA in many fields and various applications. A BNA used in nano-lithography was discussed by L.Wang et al [START_REF] Wang | Nanolithography using high transmission nanoscale bowtie apertures[END_REF]. The study demonstrates that holes of sub-50 nm dimension could be produced in photo-resist, using a BNA with a 30 nm gap size fabricated in aluminum thin films coated on quartz substrates and illuminated by a 355 nm laser beam polarized in the direction across the gap. In 2014 X. Wen et al [START_REF] Wen | Optical nanolithography with λ/15 resolution using bowtie aperture array[END_REF] published their own research about using a BNA in nanolithography. The study is based on combining the BNA nano-lithography system with an interferometric-spatial-phase-imaging (ISIP) system. The ISIP system gives a high precision control over the distance between the aperture and the photo-resist. The authors conclude by demonstrating that using this combination a resolution of 22 nm is achieved, as shown in figure 1.5.

The optical properties of the BNA has lead as well to study the ability of using the nano-antennas for optical trapping of small particles [START_REF] Righini | Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range[END_REF][START_REF] Chen | Enhanced optical trapping and arrangement of nano-objects in a plasmonic nanocavity[END_REF][START_REF] Pang | Optical trapping of a single protein[END_REF][START_REF] Roxworthy | Application of plasmonic bowtie nanoantenna arrays for optical trapping, stacking, and sorting[END_REF][START_REF] Kimani | Femtosecond-pulsed plasmonic nanotweezers[END_REF][START_REF] Huang | Optical trapping of a colloidal quantum dot[END_REF]. A theoretical study, based on the calculation of Maxwell's stress tensor and completed through the FDTD simulation, was achieved by N. Hameed et al. [START_REF] Nyha | Standalone three-dimensional optical tweezers based on fibred bowtie nanoaperture[END_REF]. The authors demonstrate that the optical force exerted by a BNA on a latex bead due to the light confinement in the BNA vicinity is responsible for trapping the bead. The study also discusses the strength of the optical force as a function of the bead radius and from the distance between the BNA and the bead. On the other hand, an experimental study conducted by A. Eter et al. [START_REF] Eter | Fiberintegrated optical nano-tweezer based on a bowtie-aperture nano-antenna at the apex of a snom tip[END_REF], presents the validation of the theoretical study presented in [START_REF] Nyha | Standalone three-dimensional optical tweezers based on fibred bowtie nanoaperture[END_REF]. Due to the enhancement of the electric field in the gap center, the BNA, which has a lateral width L = 165 nm and a gap of G = 35 nm engraved at the apex of aluminum fiber tip, was able to trap a particle of radius R = 250 nm see figure 1.6.

BNA were discussed in many other studies and covered a wide range of applications and scientific fields. Studies have either discussed the advantage of using a BNA in a specific application or studied the optical response of the nano-aperture in a specific situation.

Introduction

High field confinement of the electric field at the gap of the BNA of 47 mW/µm 2 is promising to achieve ultrahigh near-field optical recording, and the small spot size corresponds to storage densities up to 150 Gbites/in 2 as reported in [START_REF] Rao | High-intensity bowtieshaped nano-aperture vertical-cavity surface-emitting laser for near-field optics[END_REF]. However, high field confinement, resulting from the coupling between the BNA and individual fluorescence molecules, leads to enhanced molecules signal [START_REF] Mathieu Mivelle | Ultrabright bowtie nanoaperture antenna probes studied by single molecule fluorescence[END_REF]. While coupling between the nano-aperture and a PhC [START_REF] Mathieu Mivelle | Light funneling from a photonic crystal laser cavity to a nano-antenna: overcoming the diffraction limit in optical energy transfer down to the nanoscale[END_REF] provides a path to overcome the diffraction limit in optical energy transfer to the nano-scale.

Thesis novelty

In all the applications in which the BNA is used, the later has to be placed in front of a sample or in a media with a relative refractive index. Therefore, the dependency between the optical response of the BNA and the sample position was not well discussed and in the applications in which the distance might be accurately measured, different types of mechanical tools such as AFM or shear force system have been used. In the third chapter of this thesis we will present the optical response of the BNA as a function of the distance from a sample placed in front of it. The results collected show a high sensitivity of the BNA to the distance from a sample. The study paved the way towards using the BNA as an optical tool promising an alternative to the mechanical nano-positioning techniques.

Electro-Optic effect

The EO effect is the change in refractive index of a material induced by the presence of a static (or low-frequency) electric field. In some materials, the change in refractive index depends linearly on the strength of the applied electric field. This change is known as the linear EO effect or Pockels effect [START_REF] Boyd | Nonlinear optics[END_REF]. The EO effect involves on the second order NL susceptibility tensor, thus, it can only occur for non-centrosymmetric material such as Lithium Niobate (LN).

State of the art of the Pockels effect

NL effects were widely studied and they are the reason behind several applications used nowadays. NL effects consist on the interaction of a light beam when passing through a NL medium which leads to the modification of the beam frequency. The modulation of light occurs either due to an intrinsic property of the medium itself, as in the case of second harmonic generation SHG [START_REF] Bouhelier | Near-field second-harmonic generation induced by local field enhancement[END_REF][START_REF] Barakat | Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate[END_REF] , Raman effects [START_REF] Loudon | The raman effect in crystals[END_REF][START_REF] Karpov | Raman self-frequency shift of dissipative kerr solitons in an optical microresonator[END_REF], four-wave mixing [START_REF] Aso | Four-wave mixing in optical fibers and its applications[END_REF][START_REF] Razzari | Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals[END_REF] or due to the application of an external motif as in the case of Pockels effect [START_REF] Wu | Free-space electro-optic sampling of terahertz beams[END_REF][START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF][START_REF] Chmielak | Pockels effect based fully integrated, strained silicon electrooptic modulator[END_REF] and Kerr effect [START_REF] Razzari | Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals[END_REF][START_REF] Rr Subkhangulov | Terahertz modulation of the faraday rotation by laser pulses via the optical kerr effect[END_REF]. SHG an Pockels effects are both depending on the second order NL susceptibility tensor thus they occur only for non-centrosymetric materials. However the Kerr effect and four-wave-mixing are related to the third order susceptibility tensor, therefore, they are present for all materials.

NL devices have been developed for applications in various domains such as communications, analog and digital signal processing, information processing, optical computing, and sensing. Among the NL devices, we mention: phase and amplitude modulators [START_REF] Mohammed F Saleh | Plasma-induced asymmetric self-phase modulation and modulational instability in gas-filled hollow-core photonic crystal fibers[END_REF][START_REF] Venkataraman | Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing[END_REF], multiplexers [START_REF] Olshansky | Subcarrier multiplexed lightwave systems for broad-band distribution[END_REF], switch arrays [START_REF] Hoon | Selector-free resistive switching memory cell based on bifeo3 nano-island showing high resistance ratio and nonlinearity factor[END_REF][START_REF] Huang | Reconfigurable nonvolatile logic operations in resistance switching crossbar array for large-scale circuits[END_REF], couplers [START_REF] Dutta | Design of efficient photonic coupler structures for lumped raman amplification in silicon waveguides[END_REF][START_REF] Covey | All-optical switching with 1-ps response time in a ddmebt enabled silicon grating coupler/resonator hybrid device[END_REF], polarization controllers [START_REF] Zheng | Switchable multi-wavelength erbium-doped photonic crystal fiber laser based on nonlinear polarization rotation[END_REF], deflectors [START_REF] Sasaki | Resolution enhancement of kta1-xnbxo3 electro-optic deflector by optical beam shaping[END_REF][START_REF] Mhaouech | Low drive voltage electro-optic bragg deflector using a periodically poled lithium niobate planar waveguide[END_REF] and sensors for detecting temperature [START_REF] Zu | Temperatureinsensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber[END_REF][START_REF] Lu | Integrated temperature sensor based on an enhanced pyroelectric photonic crystal[END_REF] etc... Due to the NL effects, those devices allow much higher modulation frequencies than other methods, such as mechanical shutters [START_REF] Tan | A femtosecond laser-induced periodical surface structure on crystalline silicon[END_REF][START_REF] Robert | Wavelength selective cross-connect switch using a mems shutter array, mars 18[END_REF], moving mirrors [START_REF] Yu-Xin Qin | Design and experimental research on a visible-near infrared spatial modulating fourier transform spectrometer based on micro multi-step mirrors[END_REF], or acousto-optic devices [START_REF] David K Biegelsen | Acousto-optic modulation device[END_REF][START_REF] Semere A Tadesse | Acousto-optic modulation of a photonic crystal nanocavity with lamb waves in microwave k band[END_REF], due to the faster time response.

The NL effects have been widely studied and many researchers were interested in the for different applications, based on the NL effect, as most of the photonic sensors (the electric field sensor [START_REF] Jung | Electro-optic electric field sensor utilizing ti: Linbo 3 symmetric mach-zehnder interferometers[END_REF], temperature sensor [START_REF] Geng | Compact and ultrasensitive temperature sensor with a fully liquid-filled photonic crystal fiber mach-zehnder interferometer[END_REF],..). Despite the efficiency of the Mach Zehnder structure, they suffer from a relatively big dimension (long interaction length due to the small index change). Therefore, the integration of the NL optical devices in a photonic integrated circuits has become a real question that lead many researchers to use the PhC as a promising nano-structure able to enhance different NL effects due to its optical properties (slow light phenomena and the confinement of light in the structure [START_REF] Baba | Slow light in photonic crystals[END_REF]).

Many materials were proposed to be used in the fabrication of the optical modulators based on the PhC, such as silicon [START_REF] Gu | High speed silicon photonic crystal waveguide modulator for low voltage operation[END_REF], organic polymer [START_REF] Bosshard | Polymers for Photonics Applications I[END_REF][START_REF] Enami | Hybrid polymer/sol-gel waveguide modulators with exceptionally large electro-optic coefficients[END_REF][START_REF] Koos | All-optical high-speed signal processing with siliconorganic hybrid slot waveguides[END_REF] Using silicon or polymer material to fabricate the optical modulator [START_REF] Soref | Electrooptical effects in silicon[END_REF][START_REF] Xu | Micrometrescale silicon electro-optic modulator[END_REF] represented a good step towards achieving a real photonic integrated optical modulator. These materials show very interesting characteristics in terms of interaction length, area and low-power operation. Despite the progress achieved using these materials, the proposed optical modulators based on PhCs are not considered for commercial use because these materials still present temperature limitation, and rely on the relatively weak free carrier dispersion effect and they are typically long and require high driving powers for obtaining a significant modulation depth and thus exhibit an intrinsic NL phase modulation characteristic. Therefore, the LN is considered as a promising material to be used in the fabrication of an optical modulator and overcome the disadvantages presented by other materials. The LN presents some very interesting physical properties [START_REF] Volk | Lithium niobate: defects, photorefraction and ferroelectric switching[END_REF] such as EO, acousto-optic, piezo-electric and pyro-electric effects. However, one of the major limitations of the LN modulator technology is the strong etching resistance [START_REF] Volk | Lithium niobate: defects, photorefraction and ferroelectric switching[END_REF] and thus the missing ability to monolithically integrate modulators with other photonic components.

Studying the NL effects has been widely developed through many experimental studies.

However, the theoretical studies, and certainly for the EO effects face some constraints.

Many studies can be found in the literature presenting theoretical studies based on various simulation techniques for the SHG, the Kerr effect, or even for the four-wave-mixing.

However, it is rare to find a theoretical study, discussing the EO effect or the modulation of nano-structure in order to enhance this effect, aside from the studies achieved by our research group. In fact, most of the research discussing the Pockels effect make their theoretical studies without taking into account the confinement of light inside the structure, as they are working in a bulk NL media. In our group, studying the electro optical effect started in 2005. Modulation performance of the photonic crystal. The input is a sinusoidal signal at 13.5 V and 10 kHz. The output transmitted signal presents an extinction ratio of 5.2 dB. The measurement has been performed at a wavelength of 1550 nm for which the waveguide presents monomodal behavior.

The first study was achieved by Roussey et al. [START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF] In 2012, a second study by H. Lu et al [START_REF] Lu | 6-micron interaction length electro-optic modulation based on lithium niobate photonic crystal cavity[END_REF][START_REF] Lu | Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut thin film[END_REF] theoretically and experimentally demonstrated the feasibility of a compact EO modulator on a LN PhC cavity structure with an active length of only 6 µm made on an Annealed Proton Exchange (APE) LN waveguide with vertically deposited electrodes. The study showed that by using this configuration one can reach a tunability of 0.6 nm/V and a relatively low driving electrical power (10 dBm) (see figure 1.8). Based on their results, the authors state that more compact and low energy-consuming modulators may be realized by optimizing both the optical confinement of high quality factor PhC cavity and the electrical performance (approaching the electrodes and matching the impedance).

Moreover H. Lu et al. in 2013 [65] demonstrated that by using a PhC and by enhancing the pyro-electric effect, a temperature sensor could be realized. The enhancement of the pyro-electric effect generates an electric field in the PhC that depends on the temperature variation, which in its turn modifies the refractive index of the medium (due to the EO effect). The study presents a PhC cavity, of 5.2 µm interaction-length on a LN air suspended membrane, used as a photonic temperature sensor with a temperature sensitivity of 0.359 nm/ 0 C (see figure 1.9).

Introduction

Furthermore, a theoretical study discussing the case of a thin film LN rectangular lattice PhC has been discussed as well by Qui et al [START_REF] Qiu | Guided resonances on lithium niobate for extremely small electric field detection investigated by accurate sensitivity analysis[END_REF] (see figure 1.11). The authors discuss the case of variable PhC geometrical parameters in order to ensure the best resonance quality factor and to enable the high light confinement inside the structure. The structure approved to be a good candidate for electric field sensing. The study paves the way towards the improvement of the LN bulk devices such as sensors and modulators.

Theoretical studies of all works done by our research group since 2005 were based on the FDTD simulation method. The method used in [START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF][START_REF] Lu | Integrated temperature sensor based on an enhanced pyroelectric photonic crystal[END_REF] was based on the calculation of the optical field factor in all the structure and a uniform modification of the refractive index of the media is then estimated and integrated into a second study that leads to a new optical response as a function of the applied electric field or heating temperature.

However, in [START_REF] Qiu | Guided resonances on lithium niobate for extremely small electric field detection investigated by accurate sensitivity analysis[END_REF] a new method has been introduced in which a calculation of the variation of the refractive index is calculated at each grid cell of the simulation window which leads to a more accurate method for calculating the sensitivity of the structure to the applied electric field.

Thesis novelty

In our work, presented later in chapter 4, we will present different estimation methods to calculate the refractive index variation of the refractive index of the NL medium. In addition, we will showcase a new algorithm of an FDTD code in which we define a new self-consistent algorithm that is able, and during one simulation, to estimate and take into account the variation of the refractive index as a function of the applied electric field.

The code is designed to modify the second order susceptibility tensor as a function of the applied electric field, and the obtained results show the optical response of the structure by taking into account the Pockels effect. Both NL and linear FDTD code algorithms will

Thesis partition

In addition to this present chapter, this thesis contains three other chapters. In the second chapter we will present the basic theory of Maxwell's equations which is the basis of the FDTD simulation. In addition we will present the algorithm of the FDTD code and the problematic and assumptions we took to develop a new FDTD code able to simulate, in a self-consistent way, the variation of the refractive index due to the EO property of the medium.

In chapter three, theoretical and experimental studies discussing the optical response sensitivity of a BNA to the distance separating it from a given substrate will be presented.

Firstly, a theoretical study, using a homemade code 3D-FDTD will be shown. In the theoretical study we will discuss both operating modes of the BNA (collection and emission).

The study will takes into account two types of substrate, with low and high refractive index, and the influence of the substrate optical index will be discussed. As well, both results, collected in near field and in far field, will be shown. However, the experimental study will only consider the case of a BNA operating in collection mode and facing a high optical index substrate. Further more, in the experimental study we show the results for two different BNA (having different geometrical parameters).

Chapter four is reserved to present the NL part of this thesis, the EO effect. In that chapter we will present a new FDTD self-consistent code used to calculate the optical response of a structure when applying a static electric field. We also discuss different assumptions to estimate the variation of the refractive index and all methods will be compared for various structures. The relationship between the optical field confinement inside the structure and the EO effect will also be discussed in the chapter.

A general conclusion and future perspectives are presented in chapter five.

Chapter 2

Numerical tools and scientific problematic

Introduction

This chapter deals with the numerical method that will be used to model the light-matter interaction in the context of nano-optics domain. More precisely, we will be interested in simulating efficient linear and non-linear phenomena that can take place within nanostructures due to the confinement of the electromagnetic field inside such structures as it will be demonstrated in the next two chapters respectively. Therefore, in the current chapter we will present the property of our FDTD code that conduct to the modeling of plasmonic structure. In addition, we will discuss different dispersion models, in order to well describe the optical properties of metals over a wide range of wavelength. On the other hand, a nonlinear FDTD code will also be described which will allow the modeling of the electro-optic (EO) effect of nano-structures.

Various numerical methods were introduced to estimate the optical response of different structures. Those methods are based on solving Maxwell's equations and are divided into two groups: methods based on the resolution of Maxwell's equations in the frequency domain such as the plane wave expansion (PWE) method [START_REF] Guo | Simple plane wave implementation for photonic crystal calculations[END_REF][START_REF] Shi | Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers[END_REF], the transfer-matrix method (TMM) [START_REF] Soukoulis | éditeur. Photonic Band Gaps and Localization[END_REF][START_REF] Charalambos | Siapkas : General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference[END_REF] and the Rigorous coupled-wave analysis (RCWA) [START_REF] Mg Moharam | Rigorous coupled-wave analysis of grating diffractionŮe-mode polarization and losses[END_REF][START_REF] Lalanne | Highly improved convergence of the coupled-wave method for TM polarization[END_REF], or in the spatial domain as the finite difference time domain (FDTD) method [START_REF] Kunz | The finite difference time domain method for electromagnetics[END_REF][START_REF] Mekis | Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides[END_REF][START_REF] Taflove | Computational Electrodynamics: The Finitedifference Time-domain Method[END_REF] and the finite elements method (FEM) [START_REF] Shlager | A survey of the finite-difference time-domain literature[END_REF][START_REF] Zschiedrich | Advanced finite element method for nano-resonators[END_REF][START_REF] Pomplun | Adaptive finite element method for simulation of optical nano structures[END_REF], etc. The choice of a particular method depends on its domain of validity. In our work we are interested in using the Finite-Difference Time-Domain Method (FDTD). The main advantage of using the FDTD method in our work is that, by launching one calculation we can study the optical response of a defined structure in a wide frequency band using Fourier transformation to obtain the optical response as a function of the wavelength. By using FDTD method we can simulate the linear and nonlinear optical responses of the structure that we wish to study. We note that some restrictions on the structure geometry (the optical field confinement factror inside the structure) will be discussed on chapter 4.

Finite-Difference Time-Domain Method (FDTD) is a numerical tool first introduced by Yee [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media[END_REF] in 1966, in order to numerically solve the curl Maxwell's equations, consequently to study the propagation of electromagnetic fields, in time and space domain. Despite its simplicity and elegancy, Yee algorithm, has not received the sufficient interest immediately; High computational cost (memory and time consuming simulations) in addition to some inherent limits were the main reasons behind the lack of interest. In 1975 Taflov

and Brodwin [3] reported the correct numerical stability criteria of the Yee algorithm.

However the developments of the FDTD method and solving the inherent limits have continued through years. Many researchers contributed to the evolution of the FDTD such as

Mur [START_REF] Mur | Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations[END_REF] who published in 1981 the numerically stable, second-order accurate, absorbing boundary condition, Berenger [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] and many others researchers [START_REF] Raymond | Fdtd for n th-order dispersive media[END_REF][START_REF] Sacks | A perfectly matched anisotropic absorber for use as an absorbing boundary condition[END_REF][START_REF] David | Piecewise linear recursive convolution for dispersive media using fdtd[END_REF][START_REF] Roden | Convolutional pml (cpml): An efficient fdtd implementation of the cfs-pml for arbitrary media[END_REF].

Now we can say that the FDTD is the most used method to simulate the electromagnetic fields.

FDTD has been used to numerically solve the Maxwell's curl equations for various structures and in different, linear and nonlinear, media. Solving the optical response in a linear medium using the FDTD has attracted the interest of many researchers and many studies, discuss various types of structures and show a high accordance between FDTD results and experimental ones. However, various phenomena could take place while studying the optical response of a nonlinear medium, depending on the characteristics of medium and the existence of some external parameters that might affect the medium properties. When talking about nonlinear media the widely know effects are: the Kerr effect which is related to third order susceptibility of the medium, the second harmonic generation and the EO effect related to the second order susceptibility.

In 1992, Goorjian et al [START_REF] Peter | Direct time integration of maxwell's equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons[END_REF] presented a new algorithm that permits the direct time integration of Maxwell's equations in 2-D for material media having linear and nonlinear instantaneous and Lorentz-dispersive effects in the electric polarization. The nonlinear modeling takes into account such quantum effects as the Kerr and Raman interactions.

Also in 1992, Zheng and Chen [START_REF] Zheng | Transient analysis of microstrip lines with ferrite substrate by extended fd-td method[END_REF] published their work presenting an FDTD technique to simulate the transient signal propagating in a nonlinear, assumed to be Kerr-type, substrate. However, Tran [START_REF] Tran | Photonic-band-structure calculation of material possessing Kerr nonlinearity[END_REF] has developed an alternative FDTD scheme for Kerr nonlinearity when the electric field is obtained from the flux density via the solution of a cubic equation.

On the other hand, second harmonic generation has also attracted many researchers to simulate the effect using the FDTD method. In 2000, Xu et al [START_REF] Xu | Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide[END_REF] used the FDTD with the tight binding approximation to analyze coupled resonators waveguide and they investigated the SHG process in the coupled resonators. SHG is a known phenomena that occurs in a non-centrosymetric materials, however, various studies and models have been proposed to investigate the SHG in a centrosymetric material due to the broken symmetry at the interface. Other studies have investigated the SHG in metallic structures.

An algorithm for nonlinear FDTD was proposed by Barakat et al [START_REF] Barakat | Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate[END_REF] to estimate the enhancement of the SHG in a metallo-dielectric structure. The authors proposed to calculate the optical response of the structure for each wavelength separately in order to bypass the convolution product that appears in the polarization equation.

Problematic

In our study we are interested in presenting a new FDTD algorithm able to estimate the optical response of a nonlinear medium. More specifically we are interested in studying the optical response coming from the EO effect (Pockels effect). Pockels effect can only occur in materials that are noncentro-symmetric as it is described by a second-order nonlinear susceptibility. EO effect is the linear variation of the refractive index of a material as a function of the strength of the applied electric field. Therefore, the Pockels effects as well as the all other linear and non-linear effects, appear in Maxwell's equations due to the polarization term. Thus, due to medium property, the light polarization may show linear and non-linear terms. The general equation of the light polarization is given as:

⃗ P = ⃗ P L + ⃗ P N L = ε 0 (χ (1) ⃗ E + χ (2) ⃗ E ⃗ E + χ (3) ⃗ E ⃗ E ⃗ E + ...) (2.1)
Where χ (n) is the nth order nonlinear susceptibility. χ (1) E describes the linear response of the material. χ (2) ⃗ E ⃗ E defines the SHG, sum and the difference frequency generation and the linear EO effect by changing the second E to be the static electric field

χ (2) ⃗ E ⃗ E 0 ; χ (3) ⃗ E ⃗ E ⃗
E is both four-wave mixing and the third harmonic generation. In our work we are interested in studying the variation of the material refractive index as a function of the applied external electric field (the EO effect). Thus, we limit the study to the second order nonlinear susceptibility χ (2) . Consequently, the nonlinear polarization will be expressed as:

⃗ P N L (ω) = ε 0 χ (2) (ω) ⃗ E(ω) ⃗ E 0 (2.2)
where ⃗ E 0 denotes the applied external electric field.

Therefore, In our study we will be interested in studying the cases of structure fabricated by using the Lithium Niobate (LN) material. LN, which belongs to the calss of trigonal crystal symetry, is going to be used as the nonlinear material, it is one of the best known crystals used for various linear and non-linear optical applications [START_REF] Roussey | Experimental and theoretical characterization of a lithium niobate photonic crystal[END_REF]. Consequently, its refractive index variation with respect to the external applied electric field is linked to the EO tensor as following:

                           ∆(1/n 2 ) 1 ∆(1/n 2 ) 2 ∆(1/n 2 ) 3 ∆(1/n 2 ) 4 ∆(1/n 2 ) 5 ∆(1/n 2 ) 6                            =                            0 -r 22 r 13 0 r 22 r 13 0 0 r 33 0 -r 51 0 r 51 0 0 -r 22 0 0                                     E 0x E 0y E 0z          (2.3)
where r ij are the elements of the EO tensor and E 0 are the external applied electric field elements. Therefore, for LN, there is only four distinctive non zero elements of the EO tensor and r 33 (r 33 = 33 pm/V ) is the largest elements among them [START_REF] Shen | The Principles of Nonlinear Optics[END_REF].

Assuming that the largest electric field components is measured with the r 33 whcih is along the z crystalline direction. By neglecting the other small EO elements contributions and according to the Pockels effect, the refractive index variation can be simplified [START_REF] Boyd | Nonlinear optics[END_REF] as:

∆n = - 1 2 .n 3 .r 33 .E 0 (2.4)
Therefore the implementation of the classic Pockels effect is not complicated in the FDTD code. A linear FDTD study is conducted in order to verify the classical Pockels effect in a bulk LN, we have calculated the phase velocity of a monochromatic wavelength propagating in LN while applying different external electric fields. The results of this study are presented in chapter 4.

On the other hand, when a structure is presented in a bulk noncentro-symetric material and due to the optical electric field confinement that take place in the structure, the intrinsic non-linear optical property of the material are modified according to the degree of non-linearity. Thus, the second order susceptibility tensor χ (2) becomes dependent on the field confinement inside the structure [START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF][START_REF] Lu | Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut thin film[END_REF] through the relation:

χ (2) mod = χ (2) bulk × f 2 (2.5)
where f is the field factor that was defined as [START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF][START_REF] Lu | Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut thin film[END_REF]: the ratio of the total electromagnetic field calculated inside the structure to the electromagnetic field calculated over the same region but in the absnece of the structure (in the bulk). f is given by the equation:

f (ω) = E structure (ω) E bulk (ω) (2.6)
This definition is valid for both electromagnetic (EM) and Electrostatic fields and leads to an average value of the field factor. By introducing the modified χ (2) in equation 2.2, the Pockels effect becomes non-linear due to the term |E(ω)| 2 × E(ω):

⃗ P N L (ω) = ε 0 χ (2) f 2 ⃗ E(ω) ⃗ E 0 (2.7)
Nevertheless, as it will be discussed in chapter 5, different methods for the determination of the field factor (f) can be applied. In all of them, the field factor depends on the EM wave frequency and thus its temporal variations that is needed to be implemented in the FDTD algorithm, is obtained by finding the Fourier transform of equation 2.7 :

⃗ P N L (t) = ε 0 ∫ ∫ ∞ 0 ∫ χ(2) b (t 1 , t 2 , t 3 )f op (t-t 1 )f op (t-t 2 ) ⃗ E(t-t 3 ) ⃗ E 0 dt 1 dt 2 dt 3 (2.8)
However, equation 2.8 presents a triple convolution product which is very hard to calculate numerically. Therefore, to bypass this difficulty in our FDTD code, different assumptions will be studied and some assumptions are applied to simplify the calculation.

In the coming parts of this chapter we will present both linear and non-linear algorithms of the FDTD method. In addition, due to the presence of dispersive materials in our structures, such as metals, a brief description of different methods to simulate this dispersion with respect to the frequency range will be given. As well, we will present the stability criteria for both algorithms and the concept of boundary conditions. We end this chapter by presenting the susceptibility tensor χ (2) of nonlinear material, i.e., the LN.

FDTD principle

Yee algorithm [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media[END_REF] consists of discretizing Maxwell's equations in space and time domains. Maxwell's equations for the electric and magnetic fields are given by:

E E E E E H H H H x x x x E E E E E H H H H x x x x t = 0 t = 0.5 Dt t = Dt t = 1.5 Dt x = 0 x = Dx x = 2Dx x = 3Dx x = 4Dx
∂ ⃗ B ∂t = -∇ × ⃗ E (2.9) ∂ ⃗ D ∂t = ∇ × ⃗ H (2.10)
Where, as usual, ⃗ E is the electric field vector, ⃗ H is the magnetic field vector, ⃗ D is the displacement field vector and ⃗ B denote the magnetic flux density vector. In practice, the number of fields is reduced to two by the constitutive relations,

⃗ D = ε 0 ↔ ε r ⃗ E + ⃗ P (2.11) ⃗ B = µ 0 ↔ µ r ⃗ H (2.12)
Where, µ 0 and ε 0 are the magnetic permeability and dielectric permittivity of vacuum, respectively. The dimensionless quantities ↔ µ r and ↔ ε r are the relative permeability and permittivity tensors of the medium, respectively.

By developing Maxwell's equations to fit the approximation given by Yee's algorithm, in the case of linear, homogenous, isotropic and non-dispersive material, we get: In the case of LN, the non-linear polarization term should be taken into account in the equation relating D to the electric field,

H n+1/2 x (i, j, k) = H n-1/2 x (i, j, k) + ∆t µ∆z (E n y (i, j, k) -E n y (i, j, k -1)) - ∆t µ∆y (E n z (i, j, k) -E n z (i, j -1, k)) (2.13) E n+1 x (i, j, k) = E n x (i, j, k) + ∆t ε∆y (H n+1/2 z (i, j + 1, k) -H n+1/2 z (i, j, k)) - ∆t ε∆z (H n+1/2 y (i, j, k + 1) -H n+1/2 y (i, j, k)) (2.
⃗ D(ω) = ε 0 ε r ⃗ E(ω) + ⃗ P (ω) (2.15)
Assuming an isotropic material ( ↔ ε r = ε r and using equation 2.2 and 2.15 the electro- magnetic field becomes:

H n+1/2 x (i, j, k) = H n-1/2 x (i, j, k)+ ∆t µ(i, j, k)∆z [ E n y (i, j, k) -E n y (i, j, k -1) ] - ∆t µ(i, j, k)∆y [E n z (i, j, k) -E n z (i, j -1, k)] (2.16) H n+1/2 y (i, j, k) = H n-1/2 y (i, j, k)+ ∆t µ(i, j, k)∆x [(E n z (i, j, k) -E n z (i, j, k -1))] - ∆t µ(i, j, k)∆z [(E n x (i, j, k) -E n x (i, j -1, k))} (2.17) H n+1/2 z (i, j, k) = H n-1/2 z (i, j, k)+ ∆t µ(i, j, k)∆y [E n x (i, j, k) -E n x (i, j, k -1))] - ∆t µ(i, j, k)∆x [ E n y (i, j, k) -E n y (i, j -1, k) ] (2.18) E n+1 x (i, j, k) = E n x (i, j, k)+ ∆t ε(i, j, k)∆y [ H n+1/2 z (i, j + 1, k) -H n+1/2 z (i, j, k) ] - ∆t ε(i, j, k)∆z [ H n+1/2 y (i, j, k + 1) -H n+1/2 y (i, j, k) ] -(P n+1 x (i, j, k) -P n x (i, j, k)) (2.19) E n+1 y (i, j, k) = E n y (i, j, k)+ ∆t ε(i, j, k)∆z [ H n+1/2 x (i, j + 1, k) -H n+1/2 x (i, j, k) ] - ∆t ε(i, j, k)∆x [ H n+1/2 z (i, j, k + 1) -H n+1/2 z (i, j, k) ] -(P n+1 y (i, j, k) -P n y (i, j, k)) (2.20) E n+1 z (i, j, k) = E n z (i, j, k)+ ∆t ε(i, j, k)∆x [ H n+1/2 y (i, j + 1, k) -H n+1/2 y (i, j, k) ] - ∆t ε(i, j, k)∆y [ H n+1/2 x (i, j, k + 1) -H n+1/2 x (i, j, k) ] -(P n+1 z (i, j, k) -P n z (i, j, k)) (2.

FDTD stability criteria

Spatial and temporal meshing parameters should be well defined in the FDTD code in order to have accurate simulation results.

Spatial stability condition

The first parameter is the maximum cell dimension in the spatial step ∆x which should be short enough to well describe the desired structure. It should also be well defined to overcome any numerical dispersion coming from the transition between the analytical form of Maxwell's equations and the numerical approximation (central finit difference). A convergence numerical study shows that a spatial step of λ min /20 of order of magnitude is needed to get a maximum error of 0.1 on the phase velocity for an oblique wave propagating at 45 o from the grid axis [START_REF] Taflove | Computational Electrodynamics: The Finitedifference Time-domain Method[END_REF]. λ min is defined as the smallest wavelength propagating in the grid so then:

∆x ≤ λ min 20

(2.22)

Temporal stability condition

In addition to the spatial step condition, a temporal step ∆T condition should be well considered. This criterion was given by Taflov and Brodwin in 1975 [3] to overcome the numerical instability. Once the spatial step is defined then the temporal step condition is given by the Courant-Friedrichs-Lewy [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF] criterion:

∆t ≤ 1 c ( 1 ∆x 2 + 1 ∆y 2 + 1 ∆z 2 ) 1/2 (2.23)
Where c represents the maximum velocity of the wave in a medium, generally taken to be the speed of light in vacuum. We have to note that while studying the linear response of a nano-structure we usually choose a time step close to the stability limit. Taking a shorter time step increases the simulation cost (time for each simulation) and didn't enhance the results accuracy. On the contrary, for a non-linear study we will see in chapter 4 that taking a smaller time step can bypass a diverging problem since ∆t is dependent on the 2. Numerical tools and scientific problematic field factor temporal variation and not only on the spatial step.

Dispersion

The existence of metals in our plasmonic structures (nano-antenna) requires the integration of analytical models to well describe its electromagnetic dispersion properties:

ε = f (ω) (2.24)
Three main models of dispersion: Drude, Drude-lorentz and Drude critical point, are widely used depending on the metal nature and the wavelength range of interest.

Drude model

Drude model was proposed three years after the discovery of the electron by J. Thomson in 1897. In his theory, Drude considered the metal electrons as a gas of negative charged particles traversing a uniform positively charged medium [START_REF] Kittel | Physique de l'état solide: cours et problèmes[END_REF] and then he suggestet applying the kinetic theory of gas to the electrons. However, Drude model is based on three main hypotheses:

-The interaction between electrons and other particles is only considered during the collision, otherwise the interaction is negligible.

-Collisions are considered to be instantaneous events that modify the velocity of an electron.

-An electron experiences a collision with a probability of 1/T; Where T is independent of the electrons position and is known as relaxation time, collision time, or mean free time. In Drude model, T is assumed to be independent of the electron position and is independent of time.

Based on these assumptions, the dielectric permittivity of metals according to the Drude model is given as:

ε D (ω) = 1 - ω 2 p ω(ω + iγ D ) (2.25)
where γ is the characteristic collision frequency of the free electron gas and ω 2 p = ne 2 /(ε 0 m) is the plasma frequency of the free electron gas, which depends on the density n of the electrons in the metal and on the mass, m, of the electron.

The main advantage of the Drude model comes from its simplicity. In the case of silver and aluminum, adapting the values of ω p and that of γ, the model is sufficient and allows us to correctly understand the physical phenomenon in the visible range. Whereas, in the case of gold, it is impossible to find a unique value for ω p and for γ to enable the use of this model over all the visible range. To overcome this issue, a Lorentzian part must be added to the Drude model.

Drude-Lorentz model

This model is an extended form of the Drude model by adding the Lorentzian term. It is based on the assumption that the movement between electrons and nucleus is harmonic.

In addition by adding this part, the model becomes able to describe the permittivity of metals in a wider frequency band.

ε r (ω) = 1 - ω 2 P ω(ω + iγ) - ∆ε.Ω 2 l (ω 2 -Ω 2 l ) + iωΓ l (2.26)
Ω l and Γ l represent, respectively, the pulsation free oscillation and the friction coefficient associated with the Lorentz terms. While ∆ε is a scalar factor between the two parts of the model.

Drude-critical points model

Describing the optical properties of some metals (e.g. Au) are more difficult to represent in the visible/near-UV region with the Drude or Drude-Lorentz model. The reason for that is the more important role played by inter-band transitions [START_REF] Peter | Fundamentals of semiconductors: physics and materials properties[END_REF] in the violet/near-UV region (Au has at least two inter-band transitions). Describing this later with a Drude-Lorentz model [START_REF] Novotny | Principles of nano-optics[END_REF] requires more Lorentzian terms and makes the model complicated with many unphysical parameters. For that, a combination of Drude and two Critical-Point (CP) model was proposed by Etchegoin et al [START_REF] Pablo G Etchegoin | An analytic model for the optical properties of gold[END_REF]. Following this approach, the frequency-dependent optical properties of Au in the visible/near-UV range can be very well described by an analytic formula:

ε D2CP (ω) = Ω 2 D Ω 2 + iωγ D + p=2 ∑ p=1 G p (ω) (2.27) G p (ω) = A p Ω P ( ε ipϕ Ω P -ω -ipΓ + ε -ipϕ Ω P + ω + ipΓ ) (2.28)
In equation 2.27 the symbols represent: A P amplitude; ϕ the phase; Ω D energy of the gap; and γ D broadening. Note that a comparative study between the Drude-Critical points and Drude-Lorentz model was achieved by Vial and Laroche [START_REF] Vial | Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method[END_REF] and they demonstrate that the Drude-critical points is more realistic to describe the dielectric function of Gold and Silver.

These three dispersion analytical models were integrated into the FDTD algorithm: Drude model by simply applying a Fourier transform of the constitutive equation [START_REF] Rose | Direct time integration of maxwell's equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses[END_REF][START_REF] Parent | Theoretical study of transient phenomena in near-field optics[END_REF]. The Drude-Lorentz and Drude critical point models were adapted to the FDTD using a recursive convolution scheme [START_REF] Vial | Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method[END_REF][START_REF] Vial | Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method[END_REF].

Boundary Conditions

As we have described in the previous sections, FDTD method consists of creating a simulation window in which we define the structure that we need to study. However the simulation window should be well defined and the method could not consider the case of an infinite window. In our work we will consider two kinds of boundary conditions depending on the structure that we are dealing with:

-In chapter 3, while looking into the optical response of a Bowtie nano-antenna BNA we will use a full 3D-FDTD code and thus the perfect matched layer (PML) will be applied on the six bounds of the simulation window in order to avoid any reflection on the window limits.

-In chapter 4, we will be looking into the refractive index variation of a multiple periodic structure. Therefore, a periodic 3D-FDTD code will be used and one periode of the structure is defined and due to the periodicity condition the simulation will infinitely reproduce the structure. In this code we will use two types of boundary conditions: the PML on the top and at the bottom sides while the periodic boundary conditions (PBC) will be applied on the lateral bounds where the periodicity of the structure is applied.

Perfectly Matched Layer PML

Perfectly matched layer (PML) boundary condition was proposed by Berenger in 1994 [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF]. PML technique relies on the principle of impedance matching at the interface between two media having the same refractive index. The PML is based on surrounding the simulation window by a perfect absorbant medium which has nonzero electric σ e and magnetic σ m conductivities (see figure 2.3). These two conductivities are in reality dependent on each other through the impedance matching condition:

σ e ε = σ m µ (2.29)
Unfortunately due to the high absorption in the PML, a parasitical reflection can be generated back into the structure. Those reflections come from the fact that the spatial discretization is not adapted to the variation of the electric field. To bypass this problem, the conductivity is considered to increase as we go further in the PML (polynomial behavior).

Therefore using the PML techniques, all propagating light outgoing from the simulation window is absorbed by the PML and consequently there is no reflected light to the interior (10 -6 to 10 -7 reflection coefficient). We have to note that the PML is not able to absorb the evanescent wave. Thus, it is necessary to leave enough space between the structure and the PML region. A distance equal to λ max /2 is fairly enough to eliminate any effect due to the reflection of the evanescent wave, where λ max is the maximum value of the wavelength range in which we are interested.

Periodic Boundary Conditions (PBC)

For a periodic structure (e.g. photonic crystal), using a periodic 3D-FDTD code with the periodic boundary conditions [START_REF] Ko | Implementation of floquet boundary condition in fdtd for fss analysis[END_REF][START_REF] Harms | Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for fss structures[END_REF] consumes time required by the simulations.

In addition it makes the simulation similar to an infinite structure case. The periodic boundary conditions are applied along the periodicity directions while PML is still considered in the other directions (non-periodic) (see figure 2.4). PBC is obtained through the Floquet-Bloch theorem which considers that the electromagnetic fields ⃗ E and ⃗ B at the boundary cell are the same as the electromagnetic fields at the first cell with a phase term difference (e ik x .p ) .

⃗ E(x

= p x , y, t) = ⃗ E(x = 0, y, t)e ik x .p x (2.30a) ⃗ E(x, y = p y , t) = ⃗ E(x, y = 0, t)e ik y .p y (2.30b) ⃗ H(x = p x , y, t) = ⃗ H(x = 0, y, t)e ik x .p x (2.30c) ⃗ H(x, y = p y , t) = ⃗ H(x, y = 0, t)e ik y .p y (2.30d)
We note that when using the periodic 3D-FDTD code, we only deal with the normal incidence. In the case of the normal incidence the phase shift term becomes equal to zero and thus it is easy to be integrated in the FDTD algorithm. Otherwise, in the case of an incident angle different from zero the phase shift term becomes frequency-dependent and it will be difficult to implement in the FDTD code [START_REF] Belkhir | Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: Application to the study of a radar dome[END_REF].

PBC

Second order susceptibility tensor of LN

Pockels effects, as we have seen in the previous part (see 2.2), is related to the second order susceptibility tensor. However, this tensor is a physical characteristic of the material in use, in other words, to its point group. Second order susceptibility tensor χ (2) , is a third rank tensor with 27 components χ

(2) ijk in Cartesian coordinate system. Since the dielectric permeability tensor is real and symmetric, then the EO tensor must also be symmetric in its last two indices χ

(2) ijk = χ

(2) ikj . The third rank tensor can be represented through a contracted tensor χ ij , a two dimensional matrix with 18 different elements r ih out of the initial 27 elements.

h =                                 
Using this EO tensor matrix, EO polarization of equation 2.2 will be given as:

         P x P y P z          = 2ε 0         
                                    E x E 0x E y E 0y E z E 0z E y E 0z + E z E 0y E x E 0z + E z E 0x E y E 0x + E x E 0y                            (2.32)
In our case, we are interested in the EO effect of a LN photonic crystal. However, LN has the point symmetry group 3m and thereby the second order susceptibility has an anisotropic behavior. Nonzero components of the EO coefficients of crystals of LN are r 22 = r 21 = r 16 = 3 pm/V , r 31 = r 32 = r 15 = r 24 = 5 pm/V and r 33 = 33 pm/V . Therefore, the EO polarization corresponding to the crystal in use is given as:

         P x P y P z          = 2ε 0          0 0 0 0 r 31 -r 22
-r 22 r 22 0 r 31 0 0 r 31 r 31 r 33 0 0 0

                                    E x E 0x E y E 0y E z E 0z E y E 0z + E z E 0y E x E 0z + E z E 0x E y E 0x + E x E 0y                            (2.33)
From the equation 2.33 and the nonzero coefficients we can see that an electric field component along the optical axis (Z axis) is needed to exploit the biggest EO coefficient r 33 . Therefore, the equation 2.33 leads to the three components of the EO polarization vector:

P x = 2ε 0 (r 31 (E x E 0z + E z E 0x ) + r 22 (E y E 0x + E x E 0y ))
(2.34a)

P y = 2ε 0 (-r 22 E x E 0x + r 22 E y E 0y + r 31 (E y E 0z + E z E 0y )) (2.34b) P z = 2ε 0 (r 31 E x E 0x + r 31 E y E 0y + r 33 E z E 0z ) (2.34c)
We can conclude that when the static electric field and the optical electric field, are parallel to the crystal axis of LN, the EO polarization term becomes more efficient.

By the way, the other 2 components r 22 and r 31 can be neglected in comparison with r 33 and thus Pz reduces to:

P z = 2ε 0 r 33 E z E 0z (2.35)
As expected, the EO polarization is proportional to the electric field meaning that it is a linear effect. Its implementation in the FDTD algorithm is very easy through the direct integration of the P expression into the constitutive relation. Consequently, the E z equation given by equation 2.21 becomes:

E n+1 z (i, j, k) = E n z (i, j, k) + ∆t ε∆x [ H n+1/2 y (i, j + 1, k) -H n+1/2 y (i, j, k) ] - ∆t ε∆y [ H n+1/2 x (i, j, k + 1) -H n+1/2 x (i, j, k) ] -2ε 0 r 33 E 0z [ E n z (i, j, k) -E n-1 z (i, j, k) ] (2.36)
In this chapter we have presented the numerical concept of FDTD in addition to the theory for solving Maxwell's equations by using the Yee algorithm [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media[END_REF]. The Maxwell's equations were presented in two ways to be able to consider both linear and EO effects of different structures. Different methods to describe the metal dispersion were presented as well, henceforth the simple Drude model will be used. In addition, we have presented the FDTD criteria and the boundary conditions. Theoretical results for linear and EO effects will be shown in chapters 3 and 4, respectively.

Chapter 3

Resonance sensitivity of a fibered Bowtie Nano-Aperture

Introduction

In this chapter we are interested in studying the optical sensitivity of the Bowtie Nano-Aperture BNA to the distance D separating it from a given substrate [START_REF] Elie M Atie | High optical resonance sensitivity to its environment of a fibered bowtie nano-aperture antenna[END_REF]. The study shows the sensitivity of the Bowtie nano-aperture to the distance D, and consequently introduce the Bowtie nano-aperture as a promising tool for tip to sample control alternative to mechanical system used nowadays. We start by presenting a full numerical study achieved using a home made 3D-FDTD code, followed by an experimental study in order to verify the theoretical results. In the theoretical study, we consider the collection mode (CM) of the BNA executed by illuminating the tip with a plane wave from the substrate part and the emission mode (EM) by injecting light from the fiber side. Theoretical study will discuss the optical response of the BNA to the distance separating it from a high (Indium Phosphide InP) and low (Glass) refractive index substrate. Although we limit the experimental study [START_REF] Elie M Atie | Remote optical sensing on the nanometer scale with a bowtie aperture nano-antenna on a fiber tip of scanning near-field optical microscopy[END_REF] to the collection mode and only consider the case of high refractive index substrate.

Theoretical study

Structure description

As we have mentioned above, in our work we are interested in studying the resonance properties of a BNA engraved at the apex of a fiber metal coated tip [START_REF] Mathieu Mivelle | Bowtie nano-aperture as interface between near-fields and a single-mode fiber[END_REF]. The latter

Antenna in vacuum

Firstly, we started our work by conducting a study, in which we discuss the resonance of the BNAT in vacuum (simulation window filled by a medium with refractive index n = 1

and without the presence of a substrate). In figure 3.2 we present the results corresponding to the antenna operating in collection mode. Figure 3.2(a) shows the near-field normalized spectrum recorded at the gap center of the BNA with a maximum denoting the resonance excitation at λ = 1118.5 nm. The latter corresponds to the cut-off wavelength of the fundamental mode of the infinitely-long waveguide having a bowtie-shape section [1].

Normalization in that case is the division of intensity collected at the BNA gap center divided by the intensity collected at the same point without the presence of the BNA. We note that for the BNA operating in the emission mode the same resonance spectrum is obtained.

Light polarization is a key parameter affecting the resonance of the BNA which could be ON or OFF depending on the light polarization direction. When the electric field is whereas the antenna is off resonance when the illumination is turned by 90 o (see figure

3.2(d)). As we can clearly see by a comparison of both figures, the light enhancement is

negligible when the BNA is off resonance. The case of resonant BNA will be considered in all coming results. As well, figure 3.2(b) demonstrates that the high confinement of the electric field takes place in the BNA gap and not in all the opened area (see figure 3.3). In fact, at the cutoff wavelength, the group velocity of the guided mode tends to zero while the associated phase velocity tends to infinity. Consequently, this slow light mode is at the origin of the electric field enhancement in the gap zone [1].

A cross-section of the intensity map (figure 3.2(b)) made along the tip axis is presented in figure 3.2(c). From this figure we can estimate the distance limit. This latter corresponds to the maximum distance between the tip apex and an object for which the resonance properties of the BNA is not affected. In fact, this distance d can be estimated as the one for which the intensity of the electric field decreases by a factor of 1/e 2 . For both emission and collection modes the calculations demonstrate that d = 38nm.

Let us mention that in the following, the resonance wavelength (RW) of the BNAT can be determined differently according to the calculated signal: near-field signal in the gap zone (detector at a single grid cell) or collected energy inside the fiber (energy flux through a section of the fiber). Thus, we have defined two different RW, namely a near-field RW (RW N F ) and a far-field one (RW F F ). A relation between the far and near field resonance wavelengths in the case of simple geometries could be determined due to the formula given in [START_REF] Menzel | The spectral shift between near-and far-field resonances of optical nano-antennas[END_REF]. As well, we have defined the normalized near field χ N F and far field χ F F enhancement factor results. Near field enhancement factor χ N F results are defined as the energy calculated at BNA gap center divided by the energy calculated at the same point without the BNAT. While the far field enhancement factor χ F F are defined as the Poynting vector flux through an internal transverse section of the tip calculated far from its apex in order to eliminate possible surface plasmon waves propagating at the inner and/or the outer metallic walls of the tip. In fact, even if these waves are excited by the BNA, they cannot significantly contribute to the collected far-field power due to their absorption by the metal during their propagation. A small diffused part of light can affect the detected signal when the plasmon waves encounter small protrusions (metallic or dielectric surface irregularities) during propagation.

Collection mode

In figure 3.4 we present the near field resonance spectra of the BNA for different tipto-substrate distances. it's only ∆λ = 26 nm in the case of the glass substrate for a distance variation from D = 0 nm to D = 5 nm, denoting the high sensitivity of the BNA to the substrate optical index.

Resonance Wavelength variation

To well understand the resonance sensitivity of the BNA against the substrate we have started by studying the resonance wavelength variation as a function of the distance D when this latter varies from 0 nm to 300 nm. of the distance in near field and far field, respectively. In both figures, the blue solid line corresponds to the RW variation when the BNA is placed in front of a high index substrate (InP) while the red solid line corresponds to the case of a glass substrate. We notice that the RW variation proves a prompt decline with respect to D. As we have concluded earlier from figure 3.4, RW presents a much important blue shift in the case of a high optical index substrate (i.e. InP). In fact, when the BNA is placed in front of an InP substrate the near field RW goes from λ = 1808 nm for D = 0 nm to λ = 1319 nm for D = 10 nm while for a similar distance variation of BNA in front of a glass substrate the RW varies from λ = 1233 nm to λ = 1190 nm. RW shows the same variation for both N F and F F results, the variation is remarkable between D = 0 nm and D = 100 nm, while for higher distance, λ tends to reach the RW of the BNA when operating in vacuum. However, a significant blue-shift, between the resonance wavelength obtained in far field and near field resonance wavelength (see figure 3.6), occurs due to the Rayleigh scattering effect (power in 1/λ 4 ). In fact, the BNA acts as a dipole that diffuses light inside the fiber. Its near-field spectrum density is then different from the diffused far-field power because the diffused energy depends on the frequency.

Enhancement factor variation

As discussed earlier, the BNA exhibits two optical properties: the resonance wavelength and the enhancement factor. The RW was studied in the previous section, and in the same manner a study concerning the enhancement factor variation as a function of distance D will be conducted in this section. We notice that, in both near field and far field, the enhancement factor variation in the case of InP substrate shows a more remarkable oscillation than the one detected in the case of the glass substrate. In fact this oscillation denotes the efficiency of the interference waves propagating back and forth between the tip and the substrate.

Enhancement factor variation presented in figure 3.7 corresponds to the enhancement factor at the resonance wavelength which in its turn depends on the distance variation. Figure 3.8 gives us a better understanding of the variation of the enhancement factor being a function of the distance by choosing three different operating wavelengths. We can clearly see that, depending on each wavelength and for the same BNA, the variation of χ presents different behaviors: decreasing (black line corresponds to λ = 1100 nm), increasing (red line corresponds to λ = 1700 nm) and presenting a maxima at a given distance (blue line corresponds to λ = 1500 nm). Consequently, we can design the BNA to operate in a way that the collected power by the fiber shows a maximum at a desired distance which may correspond for example to the feedback set point of the shear force signal usually used in SNOM experience. By the way, these results open the way towards an alternative nano-positioning technique and toward a BNA designed for nano-sensing application.

Optical index sensitivity

As we have demonstrated earlier in this chapter the sensitivity of the BNA, either in terms of resonance wavelength or enhancement factor, is dependent on the optical index of the substrate. In addition, we concluded that the BNA presents a higher sensitivity in front of a high optical index substrate.

To well understand the influence of the substrate on the BNA optical properties, a study showing the response of the BNA facing a substrate with optical index going from n = 1 (vacuum medium) to n = 3.5 was conducted. Results shown in figure 3.9

represent the ones collected in far field. In fact, based on the results found in the previous sections, the far field and the near field outcomes, do not show a significant difference except the blue shift in term of resonance wavelength due to the Rayleigh scattering effect. In figure 3.9(a) we present the RW variation, while in figure 3.9(b) we present the enhancement factor variation as a function of the optical index n. We clearly notice that the optical index of the substrate is a critical parameter which highly influences the optical properties of the antenna. Consequently, the BNA should be optimized depending on the desired application.

longitudinal sensitivity of the Bowtie nano-aperture has been demonstrated and the antenna response is highly related to the distance separating it from the substrate. However, to well understand the sensitivity of the antenna a study was completed in order to estimate the lateral sensitivity of the Bowtie nano-aperture. Calculated results show that the antenna displays sub-wavelength lateral sensitivity as shown in figure 3.10. In this figure we consider the case of a BNA facing a silicon (Si) substrate in which we integrate a small piece of glass (depth h = 100 nm), thus the BNA will be considered as facing a substrate with an index step (see figure 3.10(a)). In our simulations we have considered three different distances separating the BNA from the substrate D = 5 nm 3.10(b), D = 50 nm 3.10(c) and D = 150 nm 3.10(d). The BNA is then moved laterally to figure out the optical response variation of the BNA with respect to the distance x from the Glass step. The sensitivity of the BNA is higher for small distances as expected. In fact, when the BNA is close to the substrate, the variation of the medium effective optical index as a function of x will be more important which induces a higher optical sensitivity.

Emission mode

After studying the optical response of the BNA when operating in collection mode, we will further adduce the optical response of a BNA operating in emission mode. Emission mode consists of injecting light inside the fiber, however theoretically the Gaussian beam is injected from the end of the metallic fibered tip. The goal of this section is that it clearly verifies the independence of the BNA resonance sensitivity from the operating mode. The optical response, and consequently the resonance sensitivity, only dependent on the geometrical parameters and the on the medium overall refractive index.

Resonance wavelength and enhancement factor variation

In figure 3.11 we presents the optical response sensitivity of the BNA to the distance variations. from λ = 1932 nm to λ = 1365 nm for a distance variation from D = 0 nm to D = 10 nm. However, for a similar distance variation when the BNA is placed in front of glass substrate the RW variation goes from λ = 1224 nm to λ = 1171 nm. On the other hand, the enhancement factor shows a higher intensity than that recorded in collection mode (see figure 3.5) and this is due to the non-leakage of light in the simulation window. We have to note that the normalization in the emission mode is obtained by dividing the results collected at the BNA gap center by the results collected at the same point by removing the metal from the tip, which is at a distance equal to the half-thickness of the metal from the polymer tip.

Due to the similarities in the results for both collection and emission modes we stop the comparison between both modes and assume that all results would be produced with an acceptable difference between both of them. In the following, we will present an experimental study of a BNA operating in collection mode, the study will prove the benefit of using such a system as a remote sensing or as a distance controller.

Bowtie antenna

Resonance sensitivity is not only a characteristics related only to the BNA. In fact, any resonant structure should present the same sensitivity phenomena and its optical response will varies as a function of the relative index of the medium. In order to verify this idea we will consider a Bowtie nano-antenna and discuss its sensitivity in term of resonance wavelength and enhancement field in the gap center. We consider a BA with lateral width L = 220 nm and with a gap dimension G = 25 nm. The BA is suspended in air and separated by a distance D from an InP substrate. See figure 3.12.

Bowtie antenna sensitivity

Resonance sensitivity of the BA is discussed as a function of the distance D separating it from an InP substrate. In figure 3.13(a) we present the resonance wavelength variation while in 3.13(b) we present the resonance intensity variation. We can notice that for a distance variation of 10 nm from D = 0 nm to D = 10 nm RW goes from 1815 nm to 1122 nm which is bigger than the variation calculated for the BNA in previous section.

However in figure 3.14 we present the intensity variation calculated for three different wavelength we can see that similarly to the case of BNA the BA intensity variation is dependent on the injected light wavelength. For λ = 1100 nm the intensity present variation present a maxima for D = 160 nm while for λ = 1500 nm the maxima is calculated for D = 5 nm and for λ = 1700 nm the intensity is decreasing when the distance increase.

Experimental study

An experimental study was conducted in order to verify the capability of using a BNA as a remote sensor, in other words to achieve an experimental verification of the high optical sensitivity to the distance variation. The study was conducted thanks to our colleague Ali EL ETER who was working under the supervision of Thierry GROSJEAN. In the following parts we will describe the different parts of the experimental study starting by the fabrication process, then the experimental setup and finally by presenting our results.

Fabrication process

The fabrication of the metallic coated fiber tip involves in several steps. Firstly, a polymer tip is produced at the cleaved facet of an optical fiber, then coating the polymer tip with a metal (aluminum) layer and finally the aperture is engraved at the top of the tip.

Tip fabrication

The fabrication of the polymer tip is achieved by LOVALITE company 1 . This startup company is a leader in polymer tip fabrication at the end of an optical fiber. The process involves a drop of polymerizable solution (pentaerythritol triacrylate (polymer) + eosin (photo initiator)) that is deposited at the cleaved end of the fiber. After that, and in order to create a conical-shape polymer tip, laser light must be injected from the other side of the fiber [START_REF] Bachelot | Integration of micrometer-sized polymer elements at the end of optical fibers by free-radical photopolymerization[END_REF]. The illuminated part of the solution would be polymerized in a conical shape. Then, the tip is immersed in an alcoholic solution which eliminates the non-polymerized solution.

Metal coating

Once the polymer tip is created, a metallic (aluminum) layer should cover the tip to allow the confinement of light and eliminate any leakage of light outside the tip. This part is achieved at MIMENTO technology center by using the evaporation techniques. We have to note that to increase the thermal resistance of the metal layer a titanium layer of 5 nm is deposited before the aluminum layer. The titanium layer has no remarkable effects on the optical response of the BNA.

Etching

The final step of fabricating the BNA at the top of a metallic coated tip is achieved using a Focal Ion Beam (dual Beam FEI Helios 600i with a Raith Elphy Multibeam attachment)to engrave the metal by respecting the geometrical parameters of the BNA. This step is also achieved at the MIMENTO technical center. Figure 3.15 shows a BNA engraved at the tip apex.

Experimental set-up

Nano-antenna is approached to a Si flat surface (commercial wafer) and studied optically with the experimental set-up shown in Figure 3.17 proximity of the surface of the Si wafer. The Si sample is mounted onto a calibrated piezo stage (PI company) to be accurately moved away from the tip along the longitudinal direction (perpendicular to the sample surface). The BNA on tip is used in collection mode. Laser light at 1550 nm (4mW ) is slightly focused within a lens onto the tip apex through the Si wafer. The fiber output is coupled to an InGaAS photodiode (Thorlabs), and a synchronous detection scheme is used to enhance signal-to-noise ratio. The output voltage is directly monitored with a numerical oscilloscope connected to a computer. The BNA is resonantly excited by aligning the polarization direction of the incident optical waves along the BNA's polarization axis, i.e., the symmetry axis of the BNA that passes through each metal triangle's tip.

Experimental results

The We have previously demonstrated through the theoretical part that BNA1 has a resonance wavelength of 1500 nm when it is placed at D = 5 nm from the sample we will see later that the BNA2 presents a resonance wavelength of 1850 nm for the same distance.

Polarization

We have mentioned in the previous parts the effect of light polarization on the BNA optical response. Due to its importance, as it indicates the status of the BNA (ON resonance or OFF resonance), we started the experimental study by presenting the polarization diagram see figure 3.17. In fact a weakly focused linearly polarized beam is projected onto the BNA and thus the signal collected by the tip (signal only come from the BNA resonance) is measured as a function of the incident polarization direction. We notice

Intensity variation

In figure 3.18 we present the intensity variation as a function of the distance separating the BNA from the sample. We remark that for BNA1 3.18(a) the intensity collected decreases as we increase the distance, whereas for BNA2 3. 46 mV.nm -1 , -19 mV.nm -1 and -4 mV.nm -1 for the same distances. Since the millivolt range is beyond the noise level of the current highly sensitive optical detection devices, one can expect with BNA-on-tips distance control schemes to achieve a sub-nanometer accuracy along the longitudinal direction perpendicular to the sample surface. The advantage of using a BNA for distance control comes from the fact that, in case of shear force and atomic force microscopy, transverse spatial resolution can be of the order of the angstrom, thus the spatial sensing depth is limited to a few nanometers.

Whereas in the case of a BNA we can expect distance regulation to be at least two orders of magnitude higher with subwavelength sensing lateral area see figure 3.10.

Conclusion

In summary, we have presented a complete theoretical study demonstrating the high resonance sensitivity, of a BNA engraved at the apex of a metallic fibered tip and placed at a distance D from a given substrate, to the distance variation. Theoretical study considered the cases of two substrates, high (InP) and low (Glass) refractive index, and the two possible functional modes (emission and collection) of the BNA. Also we have numerically discussed the optical response of a BNA as a function of the substrate refractive index and

Modeling of the enhanced Pockels effect by FDTD

dimensional Photonic Crystal (PhC) by using a 3D periodic FDTD code to evaluate the optical response. All cases will be compared with results that we obtained while using the others methods.

Validation of the classical Pockels effect

We will start our work by simply validating our FDTD algorithm through the simulation of the EO effect in the case of a conventional Pockels effect. For that we will consider a simulation window filled with homogeneous LN material and we will use a 1D FDTD code. LN is going to be used as the nonlinear material due to its important physical characteristics [START_REF] Volk | Lithium niobate: defects, photorefraction and ferroelectric switching[END_REF].

As described previously, the NL effect is introduced into the FDTD algorithm by considering the NL polarizability that leads to an additional current term ∂(εχ (2) • ⃗ E ⃗ E 0 ) ∂t in the Maxwell-Ampre equation. The main objective is then to validate the modification of the LN optical index according to the relation [START_REF] Boyd | Nonlinear optics[END_REF]:

∆n = - 1 2 .r 33 .n 3 e .E 0 (4.1)
Where n e = 2.138 is the refractive index of LN [START_REF] Sellmeier | Zur erklärung der abnormen farbenfolge im spectrum einiger substanzen[END_REF], r 33 = 33.10 -12 m/V is the largest element of the second order susceptibility tensor of LN [START_REF] Shen | The Principles of Nonlinear Optics[END_REF] and E 0 is the applied electric field.

To this end we consider a LN simulation window subjected to an homogeneous external electric field with various strength. A monochromatic plane wave (λ = 1000 nm) is then injected inside the window. This wave is linearly polarized along the optical axis (OZ) of the LN crystal. Consequently, for an external electric field parallel to this axis, only the z-component of the polarizability is non zero. We only need to inject this value P z = 2ε 0 r 33 E z E 0 into equation 2.21. Due to the small value of r 33 , we need to consider a very large propagation distance in order to be able to measure the index variation through the modification of the phase velocity of the wave. Then, this variation of the optical refractive index as a function of the applied electric field could be compared to its analytical value mentioned above.

In our simulation we have considered three different strength of the external electric field with E 0 = 1.10 7 V /m, E 0 = 1.10 8 V /m and E 0 = 1.10 9 V /m respectively.

The calculation of the phase velocity of each wave while applying the electric field leads us to deduce the corresponding index of refraction of the medium and thus to the variation of this index as a function of E 0 . In the first step we calculate the wavelength of the wave propagating along the window, note that in the FDTD code we inject a monochromatic plane wave with λ = 1000 nm. In table 4.1 we show the wavelength of the propagating beam (λ p ) for the corresponding E 0 . λ p is calculated by multiplying the period (the difference between two crests of the graph representing the spatial electric field variation along the propagation axis) by the medium refractive index n e .

strength of the electric field λ p 0 1000 10 7 999. 30 10 8 992. 45 10 9 932.16

∆n = λ period -n e (4.
2)

The results showing the variation of the refractive index with respect to the external electric field are given in table 4.2.

E 0 ∆(n) FDTD ∆(n) Boyd 0 0 0 10 7
1.612.10 -3 1.6125.10 -3 10 8

1.612.10 -2 1.6125.10 -2 10 9

1.612.10 -1 1.6125.10 -1 Table 4.2: Variation of the refractive index of the bulk as a function of the applied electric field, results calculated using equation 4.2.

As mentioned above, we have to note that in order to measure the variation of the refractive index in an accurate way we needed to perform the simulation in a very large simulation window. In fact, the variation of the period is so small while applying the external electric field, we had to measure more than 1 period. Our simulation window has 40000 spatial steps along the Z-axis (propagation axis), uniform spatial step ∆(z) = 0.5 nm and the calculation was achieved for nt = 7 × 10 4 where nt is the total number of time steps (δt = ∆(z)/2c ; c is the velocity of light in vacuum). This correspond to an interaction real time of 8.33 × 10 -19 s .

These results confirm that the Pockels effect, in its conventional form (bulk materials), is accurately modeled by the FDTD (see table 4.2) when the polarizability term ⃗ P is taken into account in the updated equations 2.19, 2.20 and 2.21. Nonetheless, when assuming that the intrinsic NL properties of the material are modified by electromagnetic confinement, the resolution of Maxwell's equations becomes very difficult due to the dispersion properties induced by the optical response of the nano-structures. The next section is devoted to the presentation of the precise problematic of the FDTD algorithm, in this case and in the four different methods that were proposed to solve it, including our original self-consistent method.

Enhanced Pockels effect modeling

Factual background

As mentioned previously, the EO effect which is a linear effect in the case of bulky media, becomes non-linear when assuming an intrinsic modification of the susceptibility tensor induced by the light confinement. This main assumption that was introduced by [START_REF] Razzari | Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals[END_REF] and experimentally verified (in a certain configuration) since 2005 by Roussey et al. in refs [START_REF] Roussey | Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons[END_REF] and H. Lu [START_REF] Lu | Integrated temperature sensor based on an enhanced pyroelectric photonic crystal[END_REF] can be expressed through:

χ(2) mod (x, y, z, ω) = χ(2) bulk (x, y, z, ω)f 2 op (x, y, z, ω) (4.3)
Where

χ(2)
mod is the modified second order susceptibility of the medium induced by

the light confinement, χ (2)
bulk is the initial second order susceptibility (in bulk where no field confinement exists) and f op is the optical electric field factor given in equation 2.6.

This last equation is at the origin of the non-linearity due to the term f 2 op (x, y, z, ω) that is, in addition, multiplied by ⃗ E(x, y, z, ω) in the expression of ⃗ P which then becomes:

⃗ P (x, y, z, ω) = ε 0 χ(2) bulk (x, y, z, ω)f 2 op (x, y, z, ω) ⃗ E(x, y, z, ω) ⃗ E 0 (4.4)
The implementation of this equation into the FDTD requires to express it in the time space through its Fourier transform which can be writen as:

⃗ P (t) = ε 0 ∫ ∫ ∞ 0 ∫ χ(2) b (t 1 , t 2 , t 3 )f op (t -t 1 )f op (t -t 2 ) ⃗ E(t -t 3 ) ⃗ E 0 dt 1 dt 2 dt 3 (4.5)
As we known, the numerical calculation of this triple convolution is a very hard task due to the fact that all six EM field components must be stored at all the computational window nodes and for all time steps. Consequently, some assumptions are needed for its implementation into the FDTD algorithm.

General assumptions

The first simplification concerns the anisotropy of LN (linear term χ(1) ). In fact, LN, which is the material of interest in our case, exhibits a quite large birefringence of ∆n = 7.3 × 10 -2 at λ = 1550 nm 1 [83]. Nonetheless, we will assume LN as isotropic with an average optical index of n = 2.138 [START_REF] Weis | Lithium niobate: summary of physical properties and crystal structure[END_REF][START_REF] Schlarb | Refractive indices of lithium niobate as a function of wavelength and composition[END_REF]. This approximation will be justified in the following through the comparison of numerical FDTD results demonstrating barely differences between isotropic and anisotropic simulations [START_REF] Qiu | Guided resonances on lithium niobate for extremely small electric field detection investigated by accurate sensitivity analysis[END_REF] for the spectral range of interest.

The second assumption that is commonly adopted by theoreticians is to suppose that the material is non dispersive. Consequently, both χ b are considered as invariant function of the EM wavelength. Generally, this approximation is justified whenever we consider a narrow spectral response. For LN, the optical index (n

= √ 1 + χ (1) ) is
ordinarily given by a Sellmeier equation [START_REF] Volk | Lithium niobate: defects, photorefraction and ferroelectric switching[END_REF] while there are no precise data for the second order susceptibility except at some fixed values of the wavelength (λ = 633 nm and λ = 1550 nm [START_REF] Sellmeier | Zur erklärung der abnormen farbenfolge im spectrum einiger substanzen[END_REF][START_REF] Yariv | Optical waves in crystal propagation and control of laser radiation[END_REF]).

Finally, we want to emphasize the fact that, in general, all the three components of the polarizability should be taken into account by the FDTD algorithm. In fact and contrary to the commonly studied configuration in bulk optics where a linearly polarized guided mode inside an LN waveguide is considered, a nano-patterned structure will induce the depolarization of the EM field through the diffraction phenomenon. Consequently, even if experimental data show that the EO coefficient r 33 is at least 3 times larger than the other coefficients (see equation 2.33), the amplitude of the diffracted electric field can counterbalance this weakness and contribute efficiently to the polarizability.

Moreover, this is still valid even if the electrostatic field is aligned along the crystal axis

( ⃗ E 0 = E 0 ⃗ e Z ).
More rigorously, the latter field also undergoes depolarization by the presence of a nano-structure but this phenomenon will be neglected in the following 2 structure and will only be discused for the infinit PhC structure (see figure 4.11) .

Taking into account this last remark, and using the expression of χ as a function of the r ij coefficients, equations 2.34a can be written as:

P x;nl (x, y, z, ω) = 2ϵ 0 r b 31 f 2 op (x, y, z, ω)E x (x, y, z, ω)E 0z (x, y, z) (4.6a) P y;nl (x, y, z, ω) = 2ϵ 0 r b 31 f 2 op (x, y, z, ω)E y (x, y, z, ω)E 0z (x, y, z) (4.6b) P z;nl (x, y, z, ω) = 2ϵ 0 r b 33 f 2 op (x, y, z, ω)E z (x, y, z, ω)E 0z (x, y, z) (4.6c)
To implement these equations into the FDTD algorithm, different assumptions for the calculation of the f op term will be adopted leading to different methods that will be detailed and discussed in the next section.

Calculation of the optical field factor f op

Estimating the variation of the optical index as a function of the applied electric field was performed in different studies and four different assumptions were used. However, all these methods are based on calculating the variation of the refractive index over the whole medium and then included in an FDTD simulation. On the other hand, the variation of the optical index is calculated through the formula:

∆n(x, y, z, ω) = - 1 2 .n 3 e .r bulk 33 .f 2 op (x, y, z, ω).f el (x, y, z).E 0 (4.7)
Where f el represents the electro-static field factor and f op is the optical factor given as the confinement of the optical electric field in the structure divided by the confinement of the optical electric field in the bulk as shown below.

f op (x, y, z, ω) = ||E structure (x, y, z, ω)|| ||E bulk (x, y, z, ω)|| (4.8)
The difference between the four methods comes from the way the optical field factor are calculated and consequently the variation of the optical index. Note that the electrostatic field factor is considered to be calculated in the same way as the optical field factor but without the contribution of the wavelength.

-In the first approximation, the transmission spectrum of the structure with E 0 = 0 V /m is calculated using one FDTD simulation and Fourier transform is done to pass from time domain to frequency domain. The average of the optical field confinement in the structure is then calculated at the nearest resonance wavelength

|| ⃗ E structure (x, y, z, λ res )|| due to the fact that the confinement is maximum at the resonance. The same steps are repeated for a bulk substrate of LN and the optical field confinement in the bulk is calculated at the same wavelength

|| ⃗ E bulk (x, y, z, λ res )||. Consequently, the optical field factor is calculated by dividing the two average terms:

f Avg_λ R op = f op (λ res ) = ∫ ∫ ∫ || ⃗ E structure (x, y, z, λ res )||dxdydz || ⃗ E bulk (x, y, z, λ res )||dxdydz (4.9)
-The second approximation focuses on calculating the confinement of the field factor at each point of the structure at the resonance wavelength. Thus the field factor for both structures || ⃗ E structure || and || ⃗ E bulk || are calculated for each spatial position. Note that the field confinement inside the bulk could be taken as a constant since the light is homogenous and so is the substrate.

f point_λ R op = f op (x, y, z, λ res ) = || ⃗ E structure (x, y, z, λ res )|| || ⃗ E bulk (x, y, z, λ res )|| (4.10)
-In the third method the calculation of the optical field factor would be done for all wavelengths, separately. In fact there is no reason to consider a large f op away from the resonance wavelength. Thus we will calculate the average of the field confinement inside the structure for each wavelength || ⃗ E structure (x, y, z, λ)||.

The same procedure is applied to calculate the field confinement in the bulk LN

|| ⃗ E bulk (x, y, z, λ)|| f Avg_λ op = f op (λ) = ∫ ∫ ∫ || ⃗ E structure (x, y, z, λ)||dxdydz || ⃗ E bulk (x, y, z, λ)||dxdydz (4.11)

Modeling of the enhanced Pockels effect by FDTD

-The fourth approximation is similar to the second one whereas the difference comes from the fact that the field confinement will be calculated in each spatial cell and for all wavelengths, separately. Thus, each spatial cell will have its own || ⃗ E structure (x, y, z, λ)||, however, the || ⃗ E bulk (x, y, z, λ)|| could be taken as a constant throughout the bulk and depending on the wavelength only.

f point_λ op = f op (x, y, z, λ) = || ⃗ E structure (x, y, z, λ)|| || ⃗ E bulk (x, y, z, λ)|| (4.12)
In addition to the just cited four methods, we will introduce another (self-consistent)

one in which we estimate the variation of the optical index of the medium and the corresponding optical response. However, the variation of the optical index in this method is not achieved directly but through the modification of the second order susceptibility tensor elements. The relation between the second order susceptibility tensor elements and the refractive index of the medium is given by M. Izdebski et al. [START_REF] Izdebski | On relationships between electrooptic coefficients for impermeability and nonlinear electric susceptibilities[END_REF] as:

χ ijk = n 4 r ijk (4.13)
in which n is the ordinary refractive index of the medium (LN). After that the modified tensor elements will be introduced in the equation to calculate the nonlinear polarization:

⃗ P i (x, y, z, ω) = 2.ϵ 0 . ∑ ij χmodified ijk (x, y, z, ω) ⃗ E opt (x, y, z, ω). ⃗ E 0 (4.14)
Therefore, in the self-consistent method the optical field factor in the presence of the substrate is calculated at each time step of the simulation and for each spatial position.

The modification of the optical field factor leads to a modification for the susceptibility tensor elements and consequently to the polarization of the propagating light. We note that the light polarization is also depending on the strength of the applied external electric field.

Comparative study and results

As we have mentioned earlier three types of structures will be studied in the coming sections. We begin with the simplest form of PhC, the Bragg reflector, then we consider the case of two Bragg gratings separated by a cavity, and finally we will present a more complicated form which is a 2 D PhC exhibiting a Fano resonance.

Bragg reflector

In the first step the case of a simple 1-D PhC, Bragg reflector, will be treated. A Bragg reflector is a structure composed from multiple layers of alternating media with different refractive index. When the wavelength of the propagating beam in the Bragg is close to four times the optical thickness of the layers, each layer boundary causes a partial reflection which combine through constructive interferences to act as a high quality reflector. The reflectivity efficiency and the band gap width are dependent on the number of the grating periods and on the difference between the refractive indices of the media. each layer having a thickness of 200 nm. Furthermore, a 3D-FDTD simulation was initiated in order to obtain the resonance spectrum of this Bragg. In our simulation, we apply the PBC along the X-axis and Y-axis and the PML is applied along the Z direction, thus, the structure is infinitely repeated along X and Y. Furthermore, a uniform spatial step is defined along the three direction with ∆x = ∆y = ∆z = 20 nm.

The study shows that under these criteria a reflection of 100% occurred for wavelength between λ = 1070 nm and λ = 1600 nm.The transmission spectrum is presented in figure 4.1(b). As we can see 10 periods are enough to get almost 100% reflectivity (0% transmission). Vertical edges of the band gap are enough abrupt to allow possible modulation of the transmitted signal.

Bragg with applied static field

On the same structure considered earlier, we now apply an external static electric field.

and then we study the variation of the refractive index in the LN. The variation of the refractive index of LN layers will induce a shift in the transmission spectrum of the Bragg reflector. The position of the band gap will be modified as well as the sharpness of the band gap edges. In figure 4.2 we present the transmission spectrum of the Bragg reflector as a function of the applied electric field. We notice that the band gap is more red shifted when the strength of the electric field is greater. The external electric field is assumed to be applied using two electrodes placed in parallel with the Brag direction and different voltages were applied. The results collected using the self consistent method

for different E 0 are presented in figure 4.2. The red curve represents the transmission spectrum corresponding to an applied electric field E 0 = 10 6 V /m, the green curve corresponds to E 0 = 10 7 V /m and the black one to E 0 = 10 8 V /m. In addition the blue curve represents the spectral response in the absence of any external field in order to compare all the results. We noticed that the slope of the curve corresponding to the higher band gap edge is inversely proportional to the strength of the applied field (there is a decrease when the applied electric field become stronger). However, the lower band gap edge curve does not show an important modification, it is red shifted for stronger electric fields (shifted in the same way as the band gap), while it almost conserves the same slope.

The variation of the high band edge (wavelength at the maximum intensity) versus the strength of the applied static electric field is represented in the inset of figure 4.3. We can see that the variation is not linear and this is due to the confinement of the optical electric field.

The difference between the two band gap limits response under the application of an We note that for E higher than E = 10 8 V /m the calculation using the self consistent method diverges. On the other hand, the same calculations using the other methods have all given results that are quite similar, thus, using the simplest method for a structure that does not present a high field confinement is good enough, and no need to complicate the method.

Comparative study between all methods

A comparative study between various methods to calculate the optical response as a function of the refractive index variation was conducted. In the comparative study we have only considered the application of an external electric field with E = 10 7 V /m, and we assume that for all other EO field the comparison will be similar. The results are shown in figure 4.4, in which, the blue curve shows the optical response in absence of the external field, the black curve corresponds to the self consistent method, and the red represents the optical response while applying the average_resonance method, the green curve is for point_resonance method, the dark blue curve corresponds to the average_lambda method and finally the magenta curve describes the response for the point_lambda method. Note that in figure 4.4 we focus on the high band edge to represent the discrepancy between the results collected from the various methods.

As we can see from these results the difference between outcomes of each method is not very significant. All the methods show the same property for the optical response as a function of the applied electric field, and that is the intensity at the maximum and the slope of the edge both decrease. Therefore, each method presents a red shift different from the others and the behavior is somewhat different. However, the self consistent results are located around the middle of the other results and it seems to look as an average result between the results of the other methods.

Two Bragg separated by a cavity

In this section, we are going to study, in the same way as in the previous section, the case of two Bragg reflector separated by a LN cavity. The structure presents a pronounced resonance with an important quality factor. Thus the electro-magnetic confinements becomes larger allowing a large modification of the optical index.

Optimizing the Bragg cavity geometry

Similarly to the Bragg reflector used in the previous section, the Bragg cavity is considered to be the combination of an LN layer separated by a vacuum layer of equal thicknesses.

The scheme of the structure is represented in figure 4.5(a) in which the optical response is expected to be directly dependent on the number of layers and the thickness of the cavity. The quality factor of the peak presented in the band gap is related to the number of layers constituting the Bragg however its position is related to the refractive index and thickness of the cavity.

In our case we have chosen each Bragg reflector formed with 5 LN layers. The cavity is then symmetrical and has a thickness of D = 330 nm. The 5 layers were sufficient in order to get a peak with an important quality factor (Q) and relatively a important electromagnetic field confinement. In fact, the quality factor is related to the field factor; field factor is bigger for high quality factor; thus we were looking for an intermediate quality factor to optimize the optical response of the structure and also the simulation time (simulation time is proportional to the confinement EM field factor). From figure 4.5 (b) we see that the optical response of the structure shows a band gap between λ = 1070 nm and λ = 1670 nm. Furthermore, we can see that a resonance occurred in the band gap due to the presence of the cavity and the resonance peak of this anomaly is at λ = 1400 nm with a quality factor Q = 63.

The simulation was achieved using a periodic 3D-FDTD, the periodicity was applied along the x and y direction, thus, the structure and the light propagation were along the Z-axis. Therefore, the PBC is applied along the X and Y direction and the PML at the upper and down interfaces. The simulation window is spatially discretized with uniform meshing along the three directions with ∆x = ∆y = ∆z = 10 nm.

Bragg cavity with applied static field

Similarly to the study conducted in the case of one Bragg, we will apply different voltages to our structure and we calculate the optical response corresponding to each case. In figure 4.6 we present the optical response of the structure while applying different voltages using two electrodes placed in parallel to the structure.

In figure 4.6 the blue curve corresponds to an applied external electric field E = 0 V /m, the red curve shows the resonance spectrum corresponding to an applied electric field E = 10 6 V /m, the green curve corresponds to E = 10 7 V /m, and the magenta curve presents the optical response while applying an E = 10 8 V /m. We note that for an E = 10 5 V /m very small variation of the resonance shift occurred.

From the results we can see that the variation of the applied external electric field has induced a modification in the position of the gap peak and especially on its shape.

The resonance wavelength was red shifted when applying an external electric field and the shift is proportional to the electric field strength (see the inset of figure 4.6), however, the intensity at the resonance is lower for stronger electric fields. On the other hand, the shape of the resonance peak shows an anomaly, which in our opinion comes from the fact that in our self consistent assumption, the modification of the second order tensor, consequently the refractive index, on each spatial cell is dependent on the confinement of the optical electric field in the cell itself. Therefore, the shift induced by the external electric field is limited by the fact that the initial resonance of the structure is sharp, which induced a steep descent on the left side of the resonance peak. The steep descent is located at the left side of the peak because the index of refraction is increasing when applying the external electric field and hence the resonance wavelength is red shifted. We note, that the non-linearity presented by the structure due to the external electric field is of type Duffing [START_REF] Kovacic | The Duffing equation: nonlinear oscillators and their behaviour[END_REF].

Comparative study between different methods for Bragg cavity

In this section we present the comparative study between the different methods cited above. The comparison will be conducted by considering the case of the green curve on figure 4.6 which refers to an applied external electric field of E 0 = 10 7 V /m. The variation of the index of refraction is calculated separately by applying the same external static electric field. This variation is then obtained for each method under the same conditions stated previously and then implemented in the FDTD simulation. Collected results are presented in figure 4.7.

In this figure the blue curve shows the optical response in the absence of an external field, the black curve corresponds to the self consistent method, the red represents the optical response while applying the Average_resonance method, the green curve is for point_resonance method, the dark blue curve corresponds to the average_lambda method and finally the magenta curve describes the response for the point_lambda method.

Using the average_lambda and the average_resonance methods have conducted to almost the same result (difference of 1 nm), however, the self consistent method and the point_lambda results present almost 2 nm resonance wavelength difference (similar to the Bragg reflector results). The point_resonance method presents a resonance wavelength so far from the other methods (as the electro-magnetic field confinement increase the resonance presented by this method become farther than the resonance calculated using the other methods).

Similarly to the case of the Bragg study, we did not collect the same result using the different methods. However the difference between the results corresponding to each method has become more important. Furthermore, the results collected by the self consistent method are the only ones showing a steep on the left side of the band gap peak and that is due to the variation of the refractive index during the simulation, which is not existing in the other methods. This steep descent appeared on the left side of the band gap peak since the refractive index is increasing when we apply an external electric field and consequently the peak is red shifted.

Infinite PhC

In this last part of the chapter we will study a more complicated structure. We choose a PhC that presents a Fano resonance in order to make a comparative study with a well known structure that was experimentally studied in our team. The PhC in use is a square The simulation study is achieved using a periodic 3D-FDTD code, therefore, a computational window of 2a × a in the XY plane is needed. The structure is illuminated by a plan wave traveling along the Z-axis (normal incidence). PBC is applied along the X and Y direction, thus, the simulation consider an infinite structure along these two axes. However, the PML is applied along the Z-axis to eliminate the parasite reflections from the window edges. Spatial step is defined to be uniform along the X and Y with δx = δy = 30 nm and non-uniform along the Z-axis where the smaller step δz = 35 nm is applied around the structure and the bigger step ∆z = 45 nm is applied in the remaining parts of the window; Non-uniform meshing is applied to well describe the structure thickness and to reduce the calculation time. On the other hand, In figure 4.8(b) we show the optical response of the PhC for a wavelength between 1400 nm and 1700 nm when no external electric field is applied. From the graph we see that the transmission spectrum shows a Fano deep resonance for λ res = 1492 nm with a quality factor Q = 290. We note that in the study achieved by QUI et al [START_REF] Qiu | Guided resonances on lithium niobate for extremely small electric field detection investigated by accurate sensitivity analysis[END_REF] the authors discussed the effect of different geometric parameters on the resonance wavelength position. In our work, we only chose this specific parameter in order to make a comparison with experimental results for different electric field strength.

2D PhC with applied static field

The influence of an applied static electric field on the refractive index of the medium, consequently, on the optical response of the PhC is discussed in this section.

In figure 4.9, we represent the optical response of the PhC, collected by using the self consistent method, when applying two different electric field E 0 = 3 × 10 6 V /m and E 0 = 8.5 × 10 6 V /m. The self consistent method shows a discontinuity between λ = 1504 nm and λ = 1510 nm for E 0 = 3×10 6 V /m and between λ = 1514 nm and λ = 1518 nm for E 0 = 8.5 × 10 6 V /m (around the expected resonance wavelength).

This discontinuity comes in fact from the divergence of the simulations when injecting a beam in that wavelength range. For those wavelengths, where the confinement is optimal, the field factor at some cells becomes very high due to the high confinement of light.

Therefore, the refractive index, which is updated at each time step of the simulation, undergoes a high variation at consecutive time steps which leads to the divergence of the simulation see figure 4.10.

Effect of the electro-static field factor

In the previous parts of this study we have only consider the effect of the optical field factor and neglecting the contribution of the electro-static field factor (the previous structure were not presenting a complex geometry). However, in the PhC crystal and as we can see from

Comparative study case of PhC

In this section we are going to compare the results collected by the self consistent method with the 4 other ones. The study was achieved by considering a static electric field strength E 0 = 8.5 × 10 6 V /m (red curve of figure 4.9) and the results are presented in figure 4.12. The optical response of the PhC while using the average_lambda method shows a deep resonance at λ = 1509 nm (dark blue curve of figure 4.12), however the point_lambda method gives a resonance at λ = 1529 nm (magenta curve of figure 4.12). On the other hand, the average_resonance (red curve) and the point_resonance (green curve) shows respectively, a resonance at λ = 1586 nm and λ = 1703 nm. As we can see, these results show somehow a similarity to the results collected by the other methods. Result collected by the self consistent method and the one by the point_lambda method, present a small difference (about 5 nm) in term of the resonance wavelength. This difference as we can see has become larger while going from a Bragg reflector structure to the current structure, due to the difference if the confinement factor.

As well, the point_resonance method continue to presents a resonance wavelength so far from the all other methods.

Comparison with experimental results

An experimental study of this structure has also been conducted by (Wentao QIU Abdullaye NDAO) and the results obtained show that when heating the structure the resonance shifts with respect to the temperature in a quasi linear form (see figure 4.13). A relationship between the Pyro-electric effect and the EO effect is given through the relationship defining the static electric field in terms of the temperature: Where ε 0 and ε r are the vacuum the LN and relative electrostatic dielectric constants (ε r = 28.7), and ρ is the pyro-electric coefficient (ρ = -6×10-5 Cm -2 K -1 ). From wavelength of ∆λ = 50 nm is recorded. This result is comparable to red curve in figure 4.9 as they both correspond to E 0 = 8.5 × 10 6 V /m. By comparing both results we can see that the shift recorded experimentally is more significant than the shifts collected using the average_lambda and point_lambda methods.

In the theoretical results, the EO effect is the only effect that modifies the optical response of the structure, however, in the experiment multiple effects can affect the optical response such as the mechanical stress which comes from heating the structure, and due to this heat process the geometry of the structure could be modified and consequently the optical response would also be modified.

Effect of time step on the divergence

The divergence of simulations when using the self consistent method requires more intensive studies in order to bypass it. For that, we have studied the influence of the time step ∆t at the diverging point. In the main code we were using the FDTD stability condition to define the time step (see equation 2.23), thus we require to minimize the time step in a way that the difference of ∆n at a consecutive time steps remains acceptable. Therefore, we choose to study the case of an applied E 0 = 8.5×10 6 V /M and for λ = 1517 nm which is expected to be the resonance wavelength and we decrease the time step by a factor of 2 in order to conserve reasonable simulation time. The variation of the diverging point have presented a slightly nonlinear variation, unfortunately, we cannot count on it to bypass the divergence. High confinement of the optical field in the structure leads to a variation of the refractive index of the medium in an unphysical way (see figure 4.10).

Various ways could be used to overcome this limitation. For example by minimizing the spatial step size. However it is worth noting that in the results that we presented, we have only taken into account the optical field factor and assuming a static electric field that has only one component along the axis of propagation. Introducing the other components and the electro-static field factor will be another challenge with more complicated diverging sources. This approach will not be discussed in this work, nevertheless, it constitutes a new challenge to be undertaken.

Conclusion

This chapter was reserved to the study of the EO effect of various structures. Various methods have been used to study the effect of an applied electric field on the optical response of the structure and a new method has been presented in order to estimate the variation of the refractive index in a self consistent way.

The beginning was by validating the conventional Pockels effect in bulk where no confinement of optical electric field is present. In order to validate the variation of the refractive index of the medium as a function of the applied electric field, we have calculated the phase velocity of the propagating beam. The results collected and presented in table 4.2 show a high accordance between analytical and numerical results.

In addition, a study discussing the case of a 1D PhC has been conducted. In the study we have considered two different structures: a simple Bragg reflector with 10 layers and a Bragg cavity structure described in figure 4.5. The results obtained show a modification of the optical response of the structure which comes from the variation of the refractive index. A comparison between the results corresponding to each structure shows that in the case of a Bragg reflector results obtained using the various methods are more close than the results in the case of the Bragg cavity. In addition, the self-consistent method diverges for a lower applied electric field in the case of the Bragg cavity. The difference between the two structures comes from the fact that the field is more confined in the case of the Bragg cavity which leads to a higher field factor and consequently to a higher variation of the refractive index at consecutive time steps.

In the last part of this chapter we discussed a more complicated structure by considering a 2D PhC. As in the case of the previous structure a comparison study between different methods have demonstrated that the higher is the field confinement in the struc-

Chapter 5 Conclusion and Perspectives

To end our this thesis we will present a general conclusion for the studies that we have presented. In addition, some perspectives for future studies are suggested.

Conclusion

In our work we have discussed two different optical aspects: the linear optical response of a BNA and the electro-optic effect of 1D and 2D PhC. The confinement of light in the structure was the reason behind both effects and their proposed applications. In the BNA case, the dependency of the volume mode and the optical properties of the structure to the medium effective index were the reason behind the high sensitivity of the BNA which paved the way toward a new alternative nano-positioning technique. However, the optical confinement of light in the PhC in addition to an external static electric field are the reason of the variation of the medium refractive index, consequently, to the appearance of the EO effect.

Sensitivity of the BNA

In the first section of the thesis, we have presented the ability of using a BNA, engraved at the apex of a metal-coated tip, as an optical technique of nano-positioning based on the monitoring of diffraction-free plasmon resonance, offering an alternative to nanomechanical and diffraction-limited optical interference-based devices. In our work we have conducted a full theoretical study and an experimental study in order to validate our theoretical results. In the theoretical study we have considered the emission mode and the collection mode of the antenna, as well we have discussed the cases of two different substrates (high and low refractive index) and a view of the resonance wavelength variation of the BNA with respect to the refractive index of the substrate placed at a distance D.

Therefore, a 3D-FDTD code, was used to complete the numerical simulation and we note that we implemented a linear Drude dispersion model to well describe the metal properties and we used the perfectly matching layer to eliminate any reflection of light from the simulation window interfaces.

On the other hand, in the experimental study we have discussed the case of a BNA operating only in collection mode. The study discusses the case of two different BNA (BNA1 and BNA2), with different geometrical parameters, when they are placed in front of a high refractive index substrate (silicon n = 3.4). Experimental results present a high accordance with the one recorded in the theoretical study. BNA1 presented experimentally a sensitivity of about -119 mV.nm -1 , -41 mV.nm -1 , -15 mV.nm -1 and -5 mV.nm -1 at tip-to-sample distances of 15 nm, 30 nm, 100 nm and 200 nm, respectively. However, BNA2 experimental curves of Fig. 3.18(c) show local tangential slopes of about 233 mV.nm -1 , 46 mV.nm -1 , -19 mV.nm -1 and -4 mV.nm -1

for the same distances. Thus, in the case of a BNA we can expect distance regulation to be at least two orders of magnitude higher than the sensitivity corresponding to shear force and atomic force microscopy techniques.

We conclude that the BNA sensitivity is dependent on various parameters, such as the antenna geometry, the refractive index of the sample in front and the separating distance between the antenna and the sample. The sensitivity of the antenna is higher for small distances, because the confinement mode localized in the gap of the antenna is more affected by the displacement of the substrate. In other words, the confinement mode of the BNA localized at the gap center is at the origin of the sensitivity phenomena measured in our study, thus, the sensitivity of the BNA is proportional to the light confinement in the vicinity of the BNA.

Electro-optic effect

In the second part of our work, we were concerned with studying the EO effect of different types of PhC going from the simplest (Bragg reflector) to a more complicated PhC (holes engraved in LN substrate). In this study we were looking to present a new algorithm in which we estimate the variation of the medium refractive index directly in the code and at each time step.

We have proposed an algorithm in which the variation of the refractive index of the medium is instantaneous, the variation of the index is described by the variation of the second order susceptibility tensor elements. Results of this method were compared with the results collected with 4 different other methods and to the results of the experimental study for the case of the 2D PhC. The study demonstrates that the application of the self consistent method is limited by various diverging sources and the efficiency of its results could vary depending on the structure in use (optical field factor). In the case of a Bragg reflector the applied electric field is limited to be less than E 0 = 10 9 V /m however, it is smaller than that in the case of a Bragg gratting with cavity and much smaller in the case of 2D PhC. The limit of the applied electric field is related to the variation of the refractive index between two consecutive time steps.

On the other hand, results collected by the other 4 methods have as well presented a dissimilarity for different structures. In these methods the variation of the refractive index is calculated separately and implemented in the FDTD code thus the refractive index of the medium does not change during the simulation. The results collected by those methods and the one corresponding to the self consistent method were quite similar for the case of Bragg reflector in which the confinement of light is smaller than the others two structures. Whereas the difference between the results collected by the various methods became larger in the case of the Bragg reflector with cavity structure and even more so for the 2D PhC. This dissimilarity is due to the difference in the light confinement factor for each structure. For high light confinement the optical field factor become higher and consequently ∆n becomes bigger.

The light confinement is at the origin of various optical phenomena. In our study we have demonstrated that this confinement plays the key role behind the high sensitivity of the BNA to a sample placed at a distance D in front of it. As well, this confinement is one of the main reasons behind the variation of the medium refractive index when this medium is subjected to an applied external electric field (EO effect).

Perspective

Modeling the EO effect of an optical structure remains an important goal to achieve. As we have show in our experiments, an important difference between the results collected by the experimental study and the theoretical one still exists even while using different methods. This difference is proportional to the confinement of light in the structure as well to the strength of the applied static electric field. Various physical effects could appear during the experimental study which are not defined in our simulation code. These could be incorporated in future work by:

-Studying the mechanical stress coming from heating the sample and looking into the existence of other physical effects in order to be introduced in our FDTD code, which then allows a faithful study of the electro-optics effect that could be dependent on other phenomena in addition to its dependency on the strength of the electric field and light confinement.

-Bypassing the diverging source of the self consistent method coming from the high value of the optical field factor and thus making the difference between the susceptibility tensor elements at two consecutive time steps become very large.

-Modeling the BNA in order to enhance its lateral and vertical sensitivity to the distance from a sample. However many other studies could take place to benefit from the high confinement of light in a small region (gap zone).

Publication

List of publication

4. 5 . 3 . 5 32 1. 8

 535328 figure is the zoom of the cavity region. . . . . . . . . . . . . . . . . . 33

2. 2 67 3. 2 69 3. 4 71 3. 5 72 3. 6 73 3. 7 75 3. 8

 2672694715726737758 Space time chart of the Yee algorithm for a 1D wave propagation example showing the use of central differences for space derivatives and leapfrog for the time. Examples the three components with dashed circle are required to calculate the component with the same color with solid circle. . . . . 2.3 Scheme representing the simulation window of a full 3D-FDTD code, (a) show the PML at different sides of the simulation window as well as on the sides intersection zone (blue zone), (b) shows the needed conductivities of three different zones (circled area on figure (a)). . . . . . . . . . . . . . 2.4 Scheme representing the simulation window of a periodic 3D-FDTD code. We see the PML at the top and bottom sides of the simulation window while the PBC is applied on the sides. . . . . . . . . . . . . . . . . . LIST OF FIGURES 3.1 Bowtie shaped antenna (width L = 215 nm, gap dimension G = 45 nm) engraved at the end face of a fiber tip coated with 105 nm aluminum layer, and placed at a distance D from the substrate. . . . . . Optical response of BNA operating in vacuum and operating in collection mode (a)Near-field spectrum detected at the gap center,(c) and(d) present the distribution of the forth root of the electric field amplitude in the YZ plane through the center of the gap when the BNA is ON and OFF resonance respectively (b) Variations of the intensity in front of the tip along its axis (BNA ON resonance), d = 38 nm corresponds to the decay-length. 68 3.3 Electric field distribution in XY plan taken at the middle of the BNA . . Resonance spectra of the BNA versus the distance separating it from: (a) an InP substrate, (b) a glass substrate. . . . . . . . . . . . . . . . . . Resonance wavelength variation as a function of the distance, (a) results collected in Near-Field, (b) results collected in Far-Field. . . . . . . . . Difference between Near-field and Far-Field resonance wavelength. . . . . Enhancement factor variation as a function of the distance. (a) Results collected in Near-Field (b) Results collected in Far-Field. . . . . . . . . Enhancement factor for three wavelength black line corresponds to λ = 1100 nm, blue line for λ = 1500 nm and red line for λ = 1700 nm.

76 3. 9 ( 107 4. 4 110 4. 6 111 4. 7 115 4. 9

 7691074110611171159 a),(b) Show respectively the variation in far field, of the resonance wavelength and the enhancement factor, as a function of the substrate optical index for D = 0 nm (black curve) and D = 10 nm (red curve). . . . 77 3.10 Resonance wavelength variations as a function of the distance (x) between the BNA center and an index step (glass-Si with h = 100 nm) for three different values of D, (a) schematic of the supposed configuration. results for BNA at 5 nm, 50 nm and 150 nm in (b), (c) and (d) respectively.3.11 Resonance wavelength and enhancement factor variation as a function of the distance from an InP substrate (blue curve) and glass substrate (red curve), respectively. Results are collected in Near Field. . . . . . . . . . 3.12 Schematic of a bowtie antenna with gap G = 25 nm placed at a distance D from a substrate with optical index n. . . . . . . . . . . . . . . . . 3.13 Optical response of Bowtie antenna as a function of the distance. (a) Resonance wavelength variation (b) Enhancement factor variation. Results are collected in Near Field. . . . . . . . . . . . . . . . . . . . . . . . 3.14 Resonance wavelength and enhancement factor variation as a function of the distance fro an InP substrate (blue curve) and glass substrate (red curve), respectively. results are collected in Near Field. . . . . . . . . . 3.15 SEM micrograph of a BNA on a SNOM fiber tip (top view). . . . . . . 3.16 Scheme of the experimental set up. . . . . . . . . . . . . . . . . . . . 3.17 Experimental and ideal polarization diagrams of the BNA-on-tip. . . . . 3.18 Experimental (blue point line) and theoretical (red solid line) result, presenting the variation of the intensity as a function of the distance by injecting a monochromatic light with λ = 1500 nm, (a) for the first BNA and, (b) for the second BNA. . . . . . . . . . . . . . . . . . . . . . . 4.1 In our study we used a 10 layer of LN Bragg reflector, (a) Schematic of the Bragg reflector, (b) the resonance spectrum . . . . . . . . . . . . . . . 4.2 Transmission spectrum of the Bragg while applying an external electric field, results are collected using the self consistent method. The inset represent the variation of the wavelength as a function of E 0 . . . . . . . . 4.3 (a) Distribution of the electric field inside the Bragg reflector for λ = 1043 nm, (b) cross section of the intensity variation along the structure (blue line on (a)), (c) Distribution of the electric field inside the Bragg reflector for λ = 1643 nm, (d) cross section of the intensity variation along the structure (green line on (c)). . . . . . . . . . . . . . . . . . Resonance spectrum of the Bragg while applying an external electric field E 0 = 10 7 V /m, all methods are used in order to compare the different response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 A Bragg cavity structure composed of 5 LN layers from each side, (a) schematic of the structure, (b) resonance spectrum. . . . . . . . . . . . Transmission spectrum of the Bragg cavity while applying an external electric field. Results are collected using the self consistent method. The inset represents the variation of the wavelength as a function of E 0 . . . . . . Transmission spectrum of the Bragg cavity while applying an external electric field E 0 = 1.10 7 V /m, all methods are used in order to compare the different response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.8 2D PhC composed by holes engraved in a LN substrate, (a) schematic of the structure in which t is the thickness of the LN substrate p x is the PhC period along the X-axis, p y the period along the Y-axis and s is the displacement of the second hole column, (b) the optical response of the structure when illuminated by a plane wave with λ = 1000 nm. . . . . The optical response of the PhC when applying different external electric field and by using (a) the self consistent method. The gray zone correspond to the wavelength where the simulation have diverged. . . . . . . . . . 117 4.10 Distribution of the field factor all over the PhC, (a) 4 time steps before divergence, (b) 3 time steps before divergence, (c) 2 time steps before divergence and (d) 1 time step before divergence. . . . . . . . . . . . . . 118 LIST OF FIGURES 4.11 (a) Distribution of the static electric field along the structure, (b) optical response by taking into account the contribution of f el (missing points correspond to a diverging wavelength). . . . . . . . . . . . . . . . . . 4.12 The optical response by using different optical field calculation methods and considering an electric field E 0 = 8.5 × 10 6 V /m. . . . . . . . . 4.13 Experimental result showing the resonance wavelength as a function of the PhC temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1 Introduction 1 . 1

 111 Research backgroundEvolution of technology, especially in terms of fabrication techniques and tools, were the reason behind the appearance of optical structures at the nano-meter scale. The capacity of fabricating such very small structures has opened the way towards a new scientific field: Nano-optics. In definition, nano-optics is the study of the behavior of light on the nanometer scale, and of the interaction of nanometer-scale objects with light.

Figure 1 . 1 :

 11 Figure 1.1: Figure showing different geometrical shapes of nano-antennas (a) nano-rod, (b) dimer, (c) Bowtie, (d) diabolo and (e) BNA.

Figure 1 . 2 :

 12 Figure 1.2: Difference in terms of electric field confinement between single and paired nano-structures.

Figure 1 . 3 :Figure 1 . 4 :

 1314 Figure 1.3: (a) Schematic of the BNA, (b) response of the Bowtie aperture for various incidence angles, (c) and (d) wavelength dependence of the maximum |E| 2 value for Bowtie aperture on gold and silver films, respectively.

Figure 1 . 5 :

 15 Figure 1.5: (a) AFM images of a lithography made on a photo-resist using the BNA (b) cross-sectional scan of the pattern taken from the position illustrated in the inset. Inset is the zoom-in image of the square area in (a).

Figure 1 . 6 :

 16 Figure 1.6: Demonstration of the 3D optical trapping of a 0.5 micrometer latex bead with a BNA on fiber tip, (a-c) the polarization of the in-fiber illumination (λ = 1064 nm) is parallel to the polarization axis of the BNA: a particle is trapped at the tip apex by the BNA resonantly excited (time range from 0 to 15 seconds), (d-f) the input polarization is turned by 90 o : the BNA is off-resonance. The initially trapped particle leaves the tip apex.

1 .

 1 Introduction modulation of light through the enhancement of different NL effects each depending on the material used and or the domain of interest. Mach Zehnder structures have been used

Figure 1 . 7 :

 17 Figure 1.7: (a)Measured transmission of the PhC for different applied external voltages(0, 40, and 80 V). Inset: Measured transmission of a different LN substrate with the same geometry, where the central peak has been observed.(b) Wavelength shift measured for a λ = 1300 nm at 0 V as a function of external positive and negative voltages. (c)

Figure 1 . 8 :

 18 Figure 1.8: Normalized transmission spectra of the PhC cavity structure for three different applied voltage values (DC = 0 V, 10 V and 20 V) and the inset figure is the zoom of the cavity region.

Figure 1 . 9 :Figure 1 .

 191 Figure 1.9: (a) Fabry-Perot resonance peak obtained by 3D-FDTD simulation of the device and the experimentally measured transmission, (b) experimentally measured transmission spectra of the peak as a function of the temperature, (c) 3D-FDTD simulation transmission spectra of the peak as a function of the temperature

Figure 1 .

 1 Figure 1.11: (a) Sketch of PhC fabricated on thin film LN presenting the different geometrical parameter. (b) FDTD calculation result of the normalized transmission spectra for a = 630 nm, r = 230 nm, t = 700 nm and varying s. (c) Numerically calculated plot showing the resonance wavelength as a function of the E-field (the inset show the zoom view of the first few data points) for infinite PhC air bridged structure with a = 630 nm, r = 230 nm, t = 700 nm and varying s.

  the second chapter of this thesis.

Figure 2 . 1 :

 21 Figure 2.1: Elemental Yee cell, in which we represent the components of the electromagnetic field. Yee cell is defined by the spatial steps ∆x, ∆y and ∆z.

  figure 2.2) relation between them by approximating the spatial and temporal derivatives by their central finite difference. For example the electric field at a time (n + 1)∆t is a function of the electric field at (n)∆t and of the magnetic field at (n + 1/2)∆t see equation 2.14.

Figure 2 . 2 :

 22 Figure 2.2: Space time chart of the Yee algorithm for a 1D wave propagation example showing the use of central differences for space derivatives and leapfrog for the time. Examples the three components with dashed circle are required to calculate the component with the same color with solid circle.

Figure 2 . 3 :

 23 Figure 2.3: Scheme representing the simulation window of a full 3D-FDTD code, (a) show the PML at different sides of the simulation window as well as on the sides intersection zone (blue zone), (b) shows the needed conductivities of three different zones (circled area on figure (a)).

Figure 2 . 4 :

 24 Figure 2.4: Scheme representing the simulation window of a periodic 3D-FDTD code. We see the PML at the top and bottom sides of the simulation window while the PBC is applied on the sides.

r 11 r 12 r 13 r 14 r 15 r 16 r 21 r 22 r 23 r 24 r 25 r 26 r

 1626 31 r 32 r 33 r 34 r 35 r 36

Figure 3 . 1 :

 31 Figure 3.1: Bowtie shaped antenna (width L = 215 nm, gap dimension G = 45 nm) engraved at the end face of a fiber tip coated with 105 nm aluminum layer, and placed at a distance D from the substrate.

Figure 3 . 2 :

 32 Figure 3.2: Optical response of BNA operating in vacuum and operating in collection mode (a)Near-field spectrum detected at the gap center,(c) and(d) present the distribution of the forth root of the electric field amplitude in the YZ plane through the center of the gap when the BNA is ON and OFF resonance respectively (b) Variations of the intensity in front of the tip along its axis (BNA ON resonance), d = 38 nm corresponds to the decay-length.

Figure 3 . 3 :

 33 Figure 3.3: Electric field distribution in XY plan taken at the middle of the BNA .

Figures 3. 4

 4

Figure 3 . 4 :

 34 Figure 3.4: Resonance spectra of the BNA versus the distance separating it from: (a) an InP substrate, (b) a glass substrate.

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Resonance wavelength variation as a function of the distance, (a) results collected in Near-Field, (b) results collected in Far-Field.

Figures 3. 5

 5 Figures 3.5(a) and (b) represent the resonance wavelength variations as a function

  Figures 3.5(a) and (b) represent the resonance wavelength variations as a function

  Figures 3.7

  (a) and (b) show the enhancement factor as a function of the distance D in near field (χ N F ) and far field (χ F F ), respectively.

Figure 3 . 7 :

 37 Figure 3.7: Enhancement factor variation as a function of the distance. (a) Results collected in Near-Field (b) Results collected in Far-Field.

3 .Figure 3 . 8 :

 338 Figure 3.8: Enhancement factor for three wavelength black line corresponds to λ = 1100 nm, blue line for λ = 1500 nm and red line for λ = 1700 nm.

Figure 3 . 9 :Figure 3 . 10 :

 39310 Figure 3.9: (a),(b) Show respectively the variation in far field, of the resonance wavelength and the enhancement factor, as a function of the substrate optical index for D = 0 nm (black curve) and D = 10 nm (red curve).

Figures 3. 11

 11

  (a) and (b) present respectively, near field resonance wavelength variation and enhancement factor (at the resonance) of the BNA. Results clearly align with those recorded in collection mode: the RW variation when the BNA is placed in front of a high optical refractive index is significantly greater than that recorded in front of a low refractive index substrate. For an InP substrate the near field RW varies 3. Resonance sensitivity of a fibered Bowtie Nano-Aperture

Figure 3 . 11 :Figure 3 . 12 :

 311312 Figure 3.11: Resonance wavelength and enhancement factor variation as a function of the distance from an InP substrate (blue curve) and glass substrate (red curve), respectively. Results are collected in Near Field.

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: Optical response of Bowtie antenna as a function of the distance. (a) Resonance wavelength variation (b) Enhancement factor variation. Results are collected in Near Field.

  . The BNAs on fiber tip are mounted onto a tuning fork and placed into a SNOM head from NT-MDT company (NTEGRA System). A shear-force detection based feedback loop is used to position the tip in close

Figure 3 .

 3 Figure 3.15: SEM micrograph of a BNA on a SNOM fiber tip (top view).

Figure 3 . 16 :

 316 Photodiode

Figure 3 . 17 :

 317 Figure 3.17: Experimental and ideal polarization diagrams of the BNA-on-tip.

  18(b) the intensity increases as we vary the distance from D = 20 nm to D = 45 nm, however it decreases as D increases. The point for which the intensity presents a maximum is dependent on multiple parameter: geometric parameters of the BNA, optical index of the substrate as well as the operating wavelength. The closest attempted distance is about 20 nm due to experimental limits as going closer would break our tip, consequently the maximum presented on figure 3.8 couldn't be verified experimentally.By calculating the tangential slopes of the experimental curves, we figure out the voltage variation at different points of the curves, which is generally used in the nanopositioning techniques in scanning probe microscopies. BNA1 presents about -119 mV.nm -1 , -41 mV.nm -1 , -15 mV.nm -1 and -5 mV.nm -1 at tip-to-sample distances of 15 nm, 30 nm, 100 nm and 200 nm, respectively. However, BNA2 experimental curves of Fig.3.18(b) show local tangential slopes about 233 mV.nm-1 ,

Figure 3 . 18 :

 318 Figure 3.18: Experimental (blue point line) and theoretical (red solid line) result, presenting the variation of the intensity as a function of the distance by injecting a monochromatic light with λ = 1500 nm, (a) for the first BNA and, (b) for the second BNA.

Figure 4 . 1 :

 41 Figure 4.1: In our study we used a 10 layer of LN Bragg reflector, (a) Schematic of the Bragg reflector, (b) the resonance spectrum

Figure 4 . 2 :

 42 Figure 4.2: Transmission spectrum of the Bragg while applying an external electric field, results are collected using the self consistent method. The inset represent the variation of the wavelength as a function of E 0 .

Figure 4 . 3 :

 43 Figure 4.3: (a) Distribution of the electric field inside the Bragg reflector for λ = 1043 nm, (b) cross section of the intensity variation along the structure (blue line on (a)), (c) Distribution of the electric field inside the Bragg reflector for λ = 1643 nm, (d) cross section of the intensity variation along the structure (green line on (c)).

Figure 4 . 4 :

 44 Figure 4.4: Resonance spectrum of the Bragg while applying an external electric field E 0 = 10 7 V /m, all methods are used in order to compare the different response.

Figure 4 . 5 :

 45 Figure 4.5: A Bragg cavity structure composed of 5 LN layers from each side, (a) schematic of the structure, (b) resonance spectrum.

Figure 4 . 6 :

 46 Figure 4.6: Transmission spectrum of the Bragg cavity while applying an external electric field. Results are collected using the self consistent method. The inset represents the variation of the wavelength as a function of E 0 .

Figure 4 . 7 :

 47 Figure 4.7: Transmission spectrum of the Bragg cavity while applying an external electric field E 0 = 1.10 7 V /m, all methods are used in order to compare the different response.

  lattice of air holes milled in LN as shown in figure4.8. In our model we consider an infinite PhC. The PhC has a period p y along the Y direction and p x = 2p y along the X direction. In addition a shift s is applied to the hole position in every two columns of air holes as schematically shown in figure4.8(a). This structure was studied extensively by Wentao QIU during her PhD studies and based on her study we have chosen specific geometrical parameters to get resonance near the telecome wavelength. The PhC has a period a = 630 nm, s = 30 nm and a thickness t = 700 nm, and the holes have a radius r = 230 nm.

Figure 4 . 8 :

 48 Figure 4.8: 2D PhC composed by holes engraved in a LN substrate, (a) schematic of the structure in which t is the thickness of the LN substrate p x is the PhC period along the X-axis, p y the period along the Y-axis and s is the displacement of the second hole column, (b) the optical response of the structure when illuminated by a plane wave with λ = 1000 nm.

  figure 4.11(a) the distribution of the external electric field is not homogenous along the structure. The optical response of the structure, for both E 0 , is red shifted relatively

Figure 4 . 9 :

 49 Figure 4.9: The optical response of the PhC when applying different external electric field and by using (a) the self consistent method. The gray zone correspond to the wavelength where the simulation have diverged.

Figure 4 . 10 :

 410 Figure 4.10: Distribution of the field factor all over the PhC, (a) 4 time steps before divergence, (b) 3 time steps before divergence, (c) 2 time steps before divergence and (d) 1 time step before divergence.

Figure 4 .

 4 Figure 4.11: (a) Distribution of the static electric field along the structure, (b) optical response by taking into account the contribution of f el (missing points correspond to a diverging wavelength).

4 .

 4 Modeling of the enhanced Pockels effect by FDTD transmission E

Figure 4 . 12 :

 412 Figure 4.12: The optical response by using different optical field calculation methods and considering an electric field E 0 = 8.5 × 10 6 V /m.

Figure 4 . 13 :

 413 Figure 4.13: Experimental result showing the resonance wavelength as a function of the PhC temperature.

equation 4 .

 4 [START_REF] Sendur | Near-field radiation of bow-tie antennas and apertures at optical frequencies[END_REF] we can calculate that a temperature variation ∆T = 36 o C corresponds to a static field E 0 = 8.5 × 10 6 V /M . From figure4.13 for T = 60 o C a resonance

Paper 1 Paper 3 Paper 4 Paper 5

 1345 Etude numrique de la rsonance dŠune nano-antenne optique ouverture papillon Elie M. Atie, Ali El Eter, Thierry Grosjean, Tony Tannous et Fadi I. Baida Lebanese sciences journal Paper 2 Fiber-integrated optical nano-tweezer based on a bowtie-aperture nano-antenna at the apex of a SNOM tip El Eter, Ali; Hameed, Nyha M; Baida, Fadi I; Salut, Roland; Filiatre, Claudine; Nedeljkovic, Dusan; Atie, Elie; Bole, Samuel; Grosjean, Thierry Optics Express, Vol. 22 Issue 8, pp.10072-10080 (2014) Light funneling from a photonic crystal laser cavity to a nano-antenna: overcoming the diffraction limit in optical energy transfer down to the nanoscale Mivelle, Mathieu; Viktorovitch, Pierre; Baida, Fadi I; El Eter, Ali; Xie, Zhihua; Vo, Than-Phong; Atie, Elie; Burr, Geoffrey W; Nedeljkovic, Dusan; Rauch, Jean-Yves; Callard, Sgolne; Grosjean, Thierry Optics Express, Vol. 22 Issue 12, pp.15075-15087 (2014) High optical resonance sensitivity to its environmentof a fibred Bowtie Nano-aperture Antenna Elie M. Atie, Tony Tannous, Thierry Grosjean and Fadi I. Baida Applied Physics B, Vol 120 Issue 4, pp. 581-586 (2015) Remote optical sensing on the nanometer scale with a bowtie aperture nano-antenna on a fiber tip of scanning near-field optical microscopy Elie M. Atie, Zhihua Xie, Ali El Eter, Roland Salut, Dusan Nedeljkovic, Tony Tannous, Fadi I. Baida, and Thierry Grosjean Appl. Phys. Lett. Vol 106, 151104 (2015)

Table of contents

 of

Table 4 .

 4 1: This table presents the calculated wavelength propagating in Bulk when an electric field E 0 is applied.

Consequently, from the results of table 4.1 we can calculate the variation of the refractive index with respect to E 0 . The calculation of ∆n is done through the relation:

http ://www.lovalite.com

http://www.lambdaphoto.co.uk/pdfs/Inrad_datasheet_LNB.pdf

Modeling of the enhanced Pockels effect by FDTD ture the more different are the results corresponding to each method, and the divergence in the case of the self consistent method appears now for a lower static field.

Acknowledgment

Firstly, I would like to express my sincere gratitude to my advisors Prof. Fadi BAIDA and Dr. Tony TANNOUS for the continuous support of my Ph.D study and related research, for their patience, motivation, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisors and mentors for my Ph.D study. A special thanks goes to Prof. Fadi BAIDA for all the time he gives to me for all our discussion, I could not imagine or express how my work would be without your supervision.

Besides my advisors, I would like to thank the rest of my thesis committee: Prof. Fouad EL HAJ HASSAN, Prof. Brahim GUIZAL, Prof. Ayman AL FALOU and Prof. Wehbe FARAH, for their insightful comments and encouragement, but also for the discussion which incited me to widen my research from various perspectives.

My sincere thanks also goes to Dr. Jihad ATTIEH, who provided me an opportunity to join the university of Balamand and was always supporting me during my trip. Without they precious support it would not be possible to conduct this research and even to start my Ph.D. I thank my fellow colleagues in FEMTO-ST and at Balamand for the stimulating discussions, for all the fun we have had in the last four years. I will always remember our enjoyable moments.

Last but not the least, I would like also to thank, my parents without them and their support It wouldn't be easy to be what I am, my sisters for their deep support and for Acknowledgment all the grammar corrections they did and a special thanks goes to my special one Diala for her support, encouragement and being the shining spot when the life goes to be dark. To all of them I say "the best is yet to come".

Chapter 4

Modeling of the enhanced Pockels effect by FDTD

Introduction

This chapter is reserved to the study conducted and the steps we implemented in order to build an original FDTD code that was able to simulate the nonlinear optical response coming from the electro-optic (EO) effect in a self-consistent way. In this chapter we will present, in addition to the steps we have taken to develop the code and the assumptions taken into account, a comparison between these results and the ones obtained by using four other different ways to estimate the variation of the refractive index of the medium. The beginning will be by determining the optical index variation in Lithium Niobate (LN) due to the application of an external electric field, conventional Pockels effect, thus we prove the ability of using FDTD to study the Pockels effect. Then we will show the case of a Bragg mirror using a one-dimensional FDTD code as well as a the case of a combination of 2 Bragg separated by a cavity. In addition, we will study the case of two