First of all, I would like to thank my thesis supervisor, Prof. Fadi BAIDA, who enabled me to apply for and be awarded this thesis subject, which was totally in adequation with my aspirations. I would like to thank him for his warm welcome, the time spent together and the sharing of his expertise on a daily basis. Thanks also to his confidence, he was a precious help in the most delicate moments. I would also like to thank everyone in the P. M. Duffieux Optics Department for their warm welcome and team spirit, and the SPIM Doctoral School, presented by its director Ms. Thérèse LEBLOIS, who played a very effective and essential role in the success of this thesis, as well as the secretariat presented by Ms. Alika ROSSETTI.

Of course, I can't forget to salute my current work team, FC'INNOV, introduced by its General Director, Ms. Tatiana LOCATELLI, who has been extremely flexible in helping me complete the administrative procedures required to register my current work as an optical engineer. My most sincere thanks go to my supervisor, Dr. Jassem SAFIOUI, who believed in me and pushed me beyond my limits. Thanks to all the members of my team: Christelle, Nebal, Mihaela, Lucie, Elmehdi, Rabah, Florent, Ammar, Hauriann, Xavier, .. . I would like also to acknowledge the financial support of my PhD studies and my second family who supported me financially also : They know themselves! iii This journey has been marked by challenges, growth and moments of inspiration. I owe a deep debt of gratitude to my family, my leader "Fayssal", my heaven "Rola", my brother "Alaa" and my sister "Walaa", for their constant love and encouragement. To my wife, "Janjoun", your unwavering support and understanding during late-night research sessions have been my source of strength.

I would like to thank my colleagues for their patience and for always being there to cheer me up: Ayman H., Mostafa, Vage, Lucas, Ayman, Souhaila, Ali Nouho, Abder, Valéria, Kirthanaa, .. . I would especially like to thank my uncles, Dr Hassan and Rateb.

Obtaining a PhD is no mean feat, and is often accompanied by moments of humor. I'd like to thank the department's coffee machine for its endless cups of motivation, my trusty laptop and desktop for their unfailing resilience, and the supercomputing center "Mesocentre de Franche Comté" for offering to perform the intensive simulations.

GENERAL INTRODUCTION

The history of the nano-emitter, as quantum dots (QDs), dates back to the 1980s, when researchers began exploring the electronic properties of ultrasmall semiconductor crystals. In 1984, Louis E. Brus at Bell Labs synthesized the first CdSe QDs [Brus, 1984], which were only a few nanometers in size.

These early QDs exhibited a strong quantum confinement effect, which led to their unique optical properties, including tunable emission.

The first demonstration of QDs emission occurred in the mid-1980s, when researchers used photoluminescence spectroscopy to measure the emission spectrum of individual CdSe QDs [START_REF] Rossetti | Excited electronic states and optical spectra of ZnS and CdS crystallites in the 15 to 50 Å size range: Evolution from molecular to bulk semiconducting properties[END_REF]. Over the next decade, researchers made significant progress in understanding the electronic and optical properties of QDs and in developing new synthetic methods to control their size, shape, and composition.

One of the major challenges in QDs emission is the divergence of the emitted light. Due to their small size, QDs typically emit light in all directions, resulting in a wide divergence angle. This divergence can limit the efficiency of QDbased devices, particularly for applications that require collimated light, such as displays and lighting.

Since the 1990s, researchers began exploring ways to control the divergence of QD emission [START_REF] Temmyo | Quantum disk lasers with self-organized dot-like active regions[END_REF], Benisty et al., 1999]. One approach was to place the QDs inside a waveguide or cavity structure [START_REF] Rao | Single quantum-dot purcell factor and β factor in a photonic crystal waveguide[END_REF], Kim et al., 2018, Chang et al., 2022], which could help to direct the emitted light. Two additional approaches for controlling the divergence of QDs emission are using micro-or nano-lenses to collimate the emitted light [START_REF] Thual | Micro-lens on polarization maintaining fibre for coupling with 1.55 µm quantum dot devices[END_REF], Northeast et al., 2021], and placing the QDs near a planar interface or metallic or/and dielectric nanoparticles (NPs) to enhance the light emission in a specific direction [START_REF] Yamamoto | Quantum state control in semiconductor pnjunctions (II): controlled spontaneous emission in quantum well microcavity lasers[END_REF][START_REF] Alaeian | [END_REF], Wang et al., 2019, Xu et al., 2021].

1 CONTENTS More recently, researchers have developed new techniques for controlling the divergence of QDs emission, including the use of plasmonic and photonic nanostructures [START_REF] Lodahl | Interfacing single photons and single quantum dots with photonic nanostructures[END_REF], Li et al., 2017, Liu et al., 2018, Park et al., 2020, Qiu et al., 2023]. Plasmonic nanostructures can be used to enhance the local electromagnetic (EM) field around the QDs, which could be accompanied by directional emission. Photonic nanostructures, such as photonic crystals, can be used to modify the dispersion properties of the QDs emission, leading to a narrower emission angle.

In most cases, the interactions occur between nano-emitters such as QDs and resonant or non-resonant nanosystems. However, our study focuses on a slightly more complex system, involving nanostructures other than QDs, such as a photonic nanocavity and a nanoparticle.

Our study is motivated by a previous study in our team, which aimed to modify the resonance quality factor of a photonic crystal cavity (PCC) by coupling it with a metallic nanoantenna, while maintaining resonance. It is important to note that these two phenomena are closely related, as the coupling of two nanoresonators inevitably leads to changes in the wavelengths of the resonances as well as in the resonance quality factor.

Our current study focuses on quantitative rather than qualitative aspects. We investigate how the far-field scattered intensity can be modified and controlled, taking into account the interaction between two distinct nanostructures. It is important to emphasize that this study is almost a continuation of the previous study, and will allow to better understand the interactions between nanostructures.

Therefore, in this thesis, we are interested in both enhancing the emission of light extracted from a PCC, and controlling its emission properties by coupling it with a resonant dielectric NP.

In the next chapter (chapter 1), we will present a state of the art of the effect of the surrounding environment on the emission properties of various nanoemitters (such as silicon QDs, core-shell CdSe/ZnS QDs, perovskite nanocrystals, ...). We will discuss several studies that consider different types of structures (resonant and non-resonant) coupled to such nano-emitters in order to control the intensity and the direction of their emissions.

As it is known, the numerical study of such a multi-dimensional structure is very complex to realize by a pure analytical method, so we need a numerical method to do it. Therefore, this work has a theoretical/numerical aspect.

It is realized by enormous simulations based on a rigorous method named "Finite Difference Time Domain (FDTD)", which is already developed and widely used in our institute to model the light-matter interaction in nanooptics (method detailed in chapter 2).

In chapter 3, we will develop in details the FDTD algorithm in the frame of the integration and the extension of the TFSF technique. This technique consists in determining the scattering and absorption cross sections for different types of resonant scatterers (metallic or dielectric) studied individually or when coupled to other resonant scatterers or to a non-resonant structure such as a plane interface (dielectric or metallic).

In chapter 4, which is the core of this work, we will study the coupling, in terms of emission efficiency and directivity, of a photonic crystal cavity (PCC) coupled to a resonant dielectric nanoparticle (NP). Our objective is to demonstrate that it is possible to control the light emission from behavior by simply modifying the spatial position of the NP. A different FDTD algorithm has been used to simulate the optical response of the latter nanostructure (i.e. PCC) when it is studied alone (without coupling) and when it is coupled to the NP.

In this part, we will see how the spatial position of the NP with respect to the PCC to which it is coupled, can impact the intensity and the direction of the light extracted by the whole coupled system (NP+PCC) in the far field. The determination of the radiation patterns, either from each resonator individually or from the whole coupled system, has been performed by exploiting the results obtained by the FDTD method and by using the angular spectrum (or plane wave) expansion (method detailed in chapter 2).

Finally, the manuscript is ended by a general conclusion summarizing the most important results and highlighting key points and perspectives arising from this work.

MENT

The coupling between a nano-emitter, such as a quantum dot (QD), and its surrounding environment is a critical element in determining the emission properties of the QD. The emission properties of the QD are influenced by the density of states (DOS) of the surrounding materials, and the coupling strength between the QD and the surrounding media. The latter can be classified into two categories: radiative and non-radiative coupling. Radiative coupling is the interaction between the QD and the electromagnetic (EM) field of the environment, and it leads to the emission of light [Drexhage, 1970, Jiang et al., 1998[START_REF] Parent | Fluorescence lifetime of a molecule near a corrugated interface: application to near-field microscopy[END_REF], Pelton, 2015, Chikkaraddy et al., 2016]. Non-radiative coupling, on the other hand, is the interaction between the QD and non-radiative modes, such as phonons or other charge carriers, which can result in energy loss through nonradiative recombination processes [START_REF] Jiang | Radiative and non-radiative inter-subband transition in self assembled quantum dots[END_REF], Barnes et al., 1999, Miyazaki et al., 2011, Murphy et al., 2017, Fan et al., 2021].

To optimize the coupling between QDs and its surrounding environment, various techniques have been developed, including embedding QDs in specific materials with tailored DOS, using plasmonic or photonic structures to enhance radiative coupling, and utilizing surface chemistry to modify the surface states of the QD and the surrounding materials.

These techniques can be implemented in a wide range of applications, such as light-emitting diodes (LEDs) [START_REF] Demir | Quantum dot integrated leds using photonic and excitonic color conversion[END_REF], Du et al., 2021], solar cells [START_REF] Nozik | Colloidal quantum dots of III-V semiconductors[END_REF], Lee et al., 2008, Semonin et al., 2012, Abdelhameed et al., 2021, Rasal et al., 2022], and quantum information processing [START_REF] Imamog | Quantum information processing using quantum dot spins and cavity QED[END_REF], Kiraz et al., 2004, Flamini et al., 2018, Kagan et al., 2020, Paudel et al., 2022].

In the following, we will discuss different types of coupling between a nanoemitter and its surrounding environment by seeing its effect on the emission properties of the nano-emitter.

1.2/ COUPLING EFFECT BETWEEN A NANO-EMITTER AND METAL-LIC STRUCTURES

On the one hand, the coupling effect between a nano-emitter and metallic structures has been an area of significant interest in the field of nanophotonics. When a nano-emitter, such as a quantum dot (QD) or a molecule, is brought into close proximity to a metallic structure, such as a metallic film, a nanoparticle or a nanowire, the interaction between the two can result in enhanced light-matter interactions. This can lead to phenomena such as increased emission rates, modified emission spectra, and reduced lifetimes of the emitters.

1.2.1/ COUPLING EFFECT BETWEEN A NANO-EMITTER AND A METALLIC PLANAR

INTERFACE

The use of metallic planar interfaces to control the divergence of QDs emission has been a topic of research for several decades [START_REF] Chance | Molecular fluorescence and energy transfer near interfaces[END_REF], Moskovits, 1985, Hellen et al., 1987, Girard et al., 1995].

One of the first theoretical studies was carried out by [START_REF] Chance | Molecular fluorescence and energy transfer near interfaces[END_REF].

Their study consisted in approaching electric dipoles oriented parallel and perpendicular to a plane metallic surface and to study the changes on their life time and their quantum yields. In particular, this approach was performed near a silver surface of index n Ag = 0.06 + 4.11*i (for all wavelengths). They verified that the lifetime of the emitter varies strongly according to its orienta-tion with respect to the surface and its distance to this one. Knowing that the emission of the dipole is related to its lifetime, we understand that it will also vary according to these criteria.

In the early 2000s, researchers began to consider the use of metallic planar interfaces for controlling the emission from QDs. Various metallic structures, such as films, have been utilized to control the emission of an ensemble of QDs by coupling them [START_REF] Kulakovich | Enhanced luminescence of CdSe quantum dots on gold colloids[END_REF], Chen et al., 2003, Gryczynski et al., 2005, Okamoto et al., 2006].

For example, in 2006 [START_REF] Okamoto | Surface-plasmon enhanced bright emission from CdSe quantum-dot nanocrystals[END_REF], Okamoto and colleagues successfully achieved bright light emission from CdSe QDs by utilizing the surface-plasmon (SP) coupling technique (see figure 1.1a,b). They observed a significant 23-fold increase in photoluminescence (PL) intensity (see figure 1.1c) and a two-fold increase in PL decay rates when the QDs were placed on evaporated gold films (see figure 1.1e). However, this enhancement was not observed in CdSe cores with ZnS shells (ZnS/CdSe) (see figures 1.1d,f).

The difference in results between the two systems can be explained using the SP dispersion diagram and considering the SP coupling mechanism. Furthermore, several other studies have gone beyond controlling the intensity of the QDs emission to control its direction [START_REF] Benner | Angular emission profiles of dye molecules excited by surface plasmon waves at a metal surface[END_REF], Lakowicz et al., 2003, Gryczynski et al., 2005, Cao et al., 2012, Liu et al., 2012].

For example, in 2012, Liu and colleagues examined the behavior of CdTe QDs in surface plasmon-coupled emission (SPCE) and utilized this phenomenon to create a sensor for detecting Hg(II) ions [START_REF] Liu | Directional surface plasmon-coupled emission of CdTe quantum dots and its application in Hg (II) sensing[END_REF]. To do this, they attached CdTe QDs synthesized in aqueous solution to a 50 nmthick Au film through a process called layer-by-layer (LBL) assembly (see fig- (from [START_REF] Okamoto | Surface-plasmon enhanced bright emission from CdSe quantum-dot nanocrystals[END_REF])

More recently, a study was published in 2022 by Wang & al. on almost the same topic [START_REF] Wang | Surface plasmon coupled directional emission for integrated fluorescence-Raman biodetection: A proof-of-concept study[END_REF]. The authors in this study utilized a method that allowed Raman spectra, which are typically obscured by strong fluorescence signals, to be identified by taking measurements at different angles. To demonstrate the effectiveness of their approach, they created dualmode nano-composites for testing (see figures 1.3a,b). By using surface plasmon-coupled emission (SPCE), the fluorescence (SPCF) signal was pre- (from [START_REF] Liu | Directional surface plasmon-coupled emission of CdTe quantum dots and its application in Hg (II) sensing[END_REF])

dominantly observed at an angle of 46 • while the Raman (SPCR) signal was predominantly observed at 50 • (see figures 1.3c-f). Due to the unique directional radiation properties of SPCE, the efficiency of signal collection was greatly improved, resulting in an almost 5-fold increase in intensity compared to conventional fluorescence methods and a 14-fold increase compared to Raman methods (see figures 1.3g,h).

1.2.2/ COUPLING EFFECT BETWEEN A NANO-EMITTER AND A METALLIC NANOS-

TRUCTURE

The history of controlling the divergence of QD emission by metallic nanostructures goes back several decades [START_REF] Tanaka | Stimulated emission from optically pumped GaN quantum dots[END_REF], (from [START_REF] Wang | Surface plasmon coupled directional emission for integrated fluorescence-Raman biodetection: A proof-of-concept study[END_REF]) [START_REF] Fujisawa | Spontaneous emission spectrum in double quantum dot devices[END_REF]], [START_REF] Song | Large enhancement of fluorescence efficiency from CdSe/ZnS quantum dots induced by resonant coupling to spatially controlled surface plasmons[END_REF], Brolo et al., 2006, Fedutik et al., 2007, Bakker et al., 2008].

One of the earliest studies on this topic was published in 2007 by Fedutik & al. in Physical Review Letters [START_REF] Fedutik | Exciton-plasmon-photon conversion in plasmonic nanostructures[END_REF]. In this study, the authors demonstrated the use of a plasmonic nanocavity structure to enhance the emission properties of CdSe QDs. They were able to experimentally create a 1D plasmonic nanocavity using an Ag nanowire functionalized with CdSe nanocrystals on top of a SiO 2 shell of 15 nm thickness. The CdSe nanocrystals can be excited optically and have three channels for energy relaxation, namely photoluminescence into free space, dipole-dipole interaction with the damped mirror dipole (known as "lossy surface waves"), and excitation of surface plasmons. This excitation can result in plasmon-photon conversion, which can occur with low efficiency due to surface roughness and structural discontinuities, such as the Ag-nanowire ends, but can also occur with high efficiency.

Another approach is considered by [START_REF] Hoang | Ultrafast spontaneous emission source using plasmonic nanoantennas[END_REF],

where they have successfully demonstrated a highly efficient and ultrafast source of spontaneous emission using a hybrid structure of single plasmonic nanopatch antennas (NPAs) coupled to colloidal quantum dots (see figure 1.4). This structure consists of a Ag nanocube on top of a gold film, separated by a 1 nm polyelectrolyte spacer layer and a sparse layer of ≈ 6 nm diameter CdSe/ZnS QDs. The results showed an increase in the spontaneous emission rate of over 90 GHz, with a factor of 880 enhancement and a 2300-fold increase in the total fluorescence intensity, indicating a high radiative quantum efficiency of around 50%. This approach provides a promising solution for nanophotonics based on ultrafast spontaneous emission by tuning the nanopatch antenna geometry from the visible to the near-infrared.

In addition, there have been many other studies investigating the use of plasmonic nanostructures for controlling the direction of QDs emission [Vasa, 2003, Podolskiy et al., 2005, Taminiau et al., 2008, Curto et al., 2010].

For example, in the latter reference [START_REF] Curto | Unidirectional emission of a quantum dot coupled to a nanoantenna[END_REF] More recently, in a 2022, a study was performed by Li and colleagues [START_REF] Li | Directional radiation enhancement of nanowire quantum dots based on line-array plasmonic antenna coupling[END_REF] aimed to enhance the spontaneous emission of a single photon in the vertical direction for on-chip photonic integrated circuits. The authors propose using a line-array plasmonic antenna with specific periodicity to On the other hand, the coupling effect of a nano-emitter to a dielectric structure can have also significant impact on the emission properties of the emitter, including its radiation pattern, spectral properties, and efficiency. The dielectric structure can be a photonic crystal, a nanocavity, or a waveguide, among others, and its interaction with the nano-emitter can be either weak or strong, depending on the size, geometry, and composition of the structure.

1.3.1/ COUPLING EFFECT BETWEEN A NANO-EMITTER AND A DIELECTRIC PLA-NAR INTERFACE

The coupling effect between a nano-emitter and a dielectric planar interface has been a topic of interest in the field of nanophotonics. The main objective is to investigate and manipulate the emissions within a thin film. This can be achieved by modifying the photonic mode density (PMD) of the surrounding media and by identifying non-radiative decay mechanisms on dielectric and semiconductor substrates [Barnes, 1998].

One early example of this phenomenon was observed in 1988 by [START_REF] Shu | A study of fluorescent intensity emitted by Eu 3+ ions near dielectric interfaces[END_REF], where the emission intensity of Eu 3+ ions above an indium oxide surface was investigated . It was observed that the fluorescence was significantly quenched when the spacer thickness was reduced from 100 to 2 nm. The distance dependence of the quenching effect was measured, and it was found that the commercially important transparent conductor was responsible for the tenfold decrease in fluorescence.

Later, in 1995, [START_REF] Snoeks | Measuring and modifying the spontaneous emission rate of Erbium near an interface[END_REF] investigated the lifetime of Er 3+ ions. These ions were embedded just beneath the surface of a glass slide, and it was observed that the lifetime was influenced by the refractive index of the liquid covering the slide's surface.

More recently, in 2009 [START_REF] Pomozzi | Ensemble measurement of the orientation-dependent variations in chromophore lifetimes near a dielectric interface[END_REF], a technique has been created that enables the measurement of the angle between excitation and emission dipole moments, as well as the radiative and nonradiative decay rates of organic dyes. This method involves polarization-resolved excitation and detection of molecular fluorescence in the proximity of a dielectric interface.

By exploiting the distinct photonic mode densities for dipoles parallel and perpendicular to the interface, these parameters can be accurately determined. In order to create a multilayer structure with precise positioning of chromophores, the authors embedded a layer containing covalently attached chromophores within a multilayer film (see figure 1.7a). In figures 1.7(b,d), the authors show the field transmission coefficients, which are the local-field components normalized by the electrical field amplitude of the incident plane wave. The fields were evaluated for a three-layer system consisting of fused (from [START_REF] Pomozzi | Ensemble measurement of the orientation-dependent variations in chromophore lifetimes near a dielectric interface[END_REF] Additionally, in 2020, Xu and colleagues demonstrated the strong coupling of a single quantum emitter to a dielectric substrate, which consists of hybrid low-refractive-index/high-refractive-index membranes at room temperature [START_REF] Xu | Strong coupling of single quantum dots with low-refractive-index/high-refractive-index materials at room temperature[END_REF]. They studied the photoluminescence (PL) spectra of colloidal quantum dots that were placed on the surface of a SiO 2 /Si material.

They varied the collection angles and also measured the reflection spectra (RL) of the SiO 2 /Si material to compare it with the PL spectra. During their analysis, they observed that the PL spectra showed a splitting pattern which they interpreted as a result of the strong coupling between the colloidal quantum dots and the SiO 2 /Si material.

From the point of view of the control of the QDs emission directivity, we will only mention a recent study published by two teams at the University of Paris-Saclay [START_REF] Au | High directional radiation of single photon emission in a dielectric antenna[END_REF]. In this study, the authors demonstrated, both theoretically and experimentally, the generation of high directivity radiation from QDs by integrating them into a polymer based dielectric antenna. From this citation, we are just interested in highlighting the far field radiation angular behavior of a spherical QD emitter having two orthogonal transition dipoles (out-of-plane "OP" and in-plane "IP") and deposited on a dielectric film (SU-8, MicroChem) with a thickness of 2 µm on a glass substrate (n S U-8 = 1.6, n glass = 1.52) (see figure 1.8).

Figure 1.8: Schematic of a single QD in the dielectric film (left). The orientation of a QD in 3D is random and can be described by a bright plane that contains two isotropic orthogonal dipoles, with a c-axis that is perpendicular to the bright plane (middle). Simulated angular radiation patterns in far-field of a QD on a dielectric film with two orthogonal transition dipoles IP and OP (right). (from [START_REF] Au | High directional radiation of single photon emission in a dielectric antenna[END_REF])

1.3.2/ COUPLING EFFECT BETWEEN A NANO-EMITTER AND A DIELECTRIC

NANOPARTICLE

Being a relatively new area of study, there have not been many investigations on the use of dielectric nanoparticles (NPs) for regulating the divergence of quantum dot (QD) emission [START_REF] Gérard | Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere[END_REF], Evlyukhin et al., 2010, Bleuse et al., 2011, Rolly et al., 2012].

One of the earliest studies in this area was published in 2009 by Gerard & al. in Journal of the Optical Society of America B [START_REF] Gérard | Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere[END_REF]. In this study, the authors experimentally and theorically demonstrated the use Since there, a significant improvement has been achieved in this area of coupling emitters with nanoresonators to control the emission of light at the subwavelength scale with high efficiency [START_REF] Devilez | Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission[END_REF], Rolly et al., 2012, Staude et al., 2013, Staude et al., 2015, Au et al., 2019, Suárez et al., 2020, Fang et al., 2021]. Recently, [START_REF] Colom | Enhanced purcell factor for nanoantennas supporting interfering resonances[END_REF] were published in Physical [START_REF] Gaponenko | Electrons and photons in mesoscopic structures: Quantum dots in a photonic crystal[END_REF]. In this study, the authors demonstrated that the emission properties of QDs could be controlled by placing them into a PC structure.

Since then, there have been many other studies investigating the use of photonic nanostructures to manipulate the divergence of QD emission [START_REF] Lee | Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab[END_REF], Hermann et al., 2002, Irman, 2003, (from [START_REF] Colom | Enhanced purcell factor for nanoantennas supporting interfering resonances[END_REF]) [START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF], Li et al., 2010, Yan et al., 2014]. For example, Ogawa & al. [START_REF] Ogawa | Control of light emission by 3D photonic crystals[END_REF] used a 3D woodpile PC to control the emission of light , while [START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF]] used a 3D inverse opal PC to control the dynamics of the spontaneous emission from QDs and [START_REF] Englund | Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal[END_REF] used 2D-PCC to modify the rate of the spontaneous emission from a single QDs (see figure 1.12).

More recently, several researchers have been interested in studying the lightmatter interaction in the strong coupling regime in order to control light emission from QDs. For example, [START_REF] Rigal | Probing disorder and mode localization in photonic crystal cavities using site-controlled quantum dots[END_REF] were able to probe the photonic modes of a L 33 and L 61 PC cavity by using single or multiple site-controlled pyramidal QDs and they demonstrated this will impact their disorder-induced distortions (figure 1.13).

Furthermore, [START_REF] Felici | Broadband enhancement of light-matter interaction in photonic crystal cavities integrating site-controlled quantum dots[END_REF] were able to integrate single, site-controlled Ga(AsN)/Ga(AsN):H quantum dots (QDs) within a photonic crystal (PhC) cavity. They then studied the properties of these QD-PhC cavity systems by using photon correlation techniques, which provided clear A SEM image of the structure (on bottom right). [START_REF] Ogawa | Control of light emission by 3D photonic crystals[END_REF]. (b) 3D inverse opal PCs. A SEM image of a surface of the structure. [START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF]. (c) 2D-PCC. Electric field intensity distribution in the fundamental mode of the cavity, calculated by using the FDTD method (at left) and (at right) a SEM image of the structure. ( from [START_REF] Englund | Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal[END_REF]).

evidence of single-photon emission, and time-resolved microphotoluminescence spectroscopy. To gain a deeper understanding of the dynamics of the integrated nanodevices, they developed a system of rate equations that accounts for all the major processes involved in the capture, relaxation, and recombination of carriers inside and outside the QD. By comparing the experimental results with the solutions obtained from this system, they were able to track the evolution of the relevant recombination rates for varying energy detuning, ∆E, between the QD and the PhC cavity. When the QD exciton transition was nearly resonant with the cavity mode, a large enhancement of the spontaneous emission rate (more than 10-fold) was observed, in agreement with the predictions of Jaynes-Cummings (JC) theory [START_REF] Jaynes | Comparison of quantum and semiclassical radiation theories with application to the beam maser[END_REF].

However, for intermediate detunings (∆E ≈ 1.5-3.5 meV), the observed enhancement was significantly greater than that predicted by JC theory. This was due to the important role played by acoustic phonons in mediating the QD-PhC cavity coupling in a solid-state environment (see figure 1.14). Some researchers have coupled these nano-emitters to a dielectric substrate by integrating them inside the substrate or depositing them on the substrate, while others have placed them in close proximity to a metallic substrate in order to enhance or attenuate the intensity of their photoluminescence.

Others were able to demonstrate an increase in the Purcell factor, enhance- Furthermore, some are interested in controlling the emission of light from nano-emitters by embedding them in a two-dimensional photonic crystal cavity that may control the emission of light from site-controlled nano-emitters embedded within.

In our team, they are interested in studying the coupling effect between two resonant structures, where QDs are integrated into one of these structures to excite its resonance mode. The coupling of a Bowtie aperture nano-antenna etched at the end of a metallized SNOM probe with a photonic crystal cavity (named CL7) [START_REF] Baida | Double-way spectral tunability for the control of optical nanocavity resonance[END_REF] was studied since 2015 as a function of the vertical distance "d" between the BNA and the cavity, but also as a function of the BNA's orientation along the orthogonal "x" and "y" axes. By calculating the near-field spectrum at 15 nm above the cavity center, it was found two opposite behaviors: a significantly redshift for a high value of "d" and a low blueshift for d = 30 nm. This means that two different phenomena are involved: the BNA resonance, which tends to increase the volume of the mode by expanding the light distribution within its gap (dielectric resonance), and the metal tip effect or magnetic effect, which compresses the PC cavity mode (blue shift). This interpretation has been confirmed by the results of figure 1.15 f, obtained for the BNA off-resonance, which show only blue shifts, regardless of the distance between the tip and the cavity.

Consequently, despite these advances, no study has proposed to control light emission from nano-emitters coupled to two resonant structures simultaneously. Thus, the problematic of this thesis is the following: Is it possible to control the emission of nano-emitters embedded in a photonic crystal cavity by coupling it, spectrally and geometrically, to a dielectric NP that exhibits a dipole mode resonance? What are the advantages of this particular and new coupling mode compared to previous ones?

2.1/ INTRODUCTION

Since the aspect of my thesis is purely theoretical, more particularly it is numerical, this chapter will describe the different computational methods used in this study in order to reach our objectives, either by Mie's theory or by the Finite Difference Time Domain (FDTD) method. Thus, let us quickly introduce the principle of the interaction between electromagnetic (EM) fields and a nanoparticle. This principle is based on the phenomenon of light-matter interaction in which an incident EM wave defined by its fields

( ⃗ E inc (⃗ r, t); ⃗ H inc (⃗ r, t))
propagates in a defined medium (which we will name "medium 1") and illuminates a nanoparticle immersed in this same medium.

Under the effect of the incident illumination, an EM fields ( ⃗ E 2 (⃗ r, t); ⃗ H 2 (⃗ r, t))

will appear within the particle ("medium 2"). Moreover, the interaction between the particle and the incident radiation will produce an EM field scattered by the particle in the medium 1 (see Fig. 2.1); this field, which we will note ( ⃗ E sca (⃗ r, t); ⃗ H sca (⃗ r, t)), will be superimposed on the incident field ( ⃗ E inc (⃗ r, t); ⃗ H inc (⃗ r, t)) in the medium 1, so that the total EM field in the medium 1 will be: currents, which are written:

⃗ E tot (⃗ r, t) = ⃗ E inc (⃗ r, t) + ⃗ E sca (⃗ r, t) and ⃗ H tot (⃗ r, t) = ⃗ H inc (⃗ r, t) + ⃗ H sca (⃗ r, t
                           ⃗ rot ⃗ E = -∂ ⃗ B ∂t ⃗ rot ⃗ H = ∂ ⃗ D ∂t div ⃗ D = 0 div ⃗ B = 0 (2.1)
Where, ⃗ B = µ ⃗ H and ⃗ D = ε ⃗ E with ε and µ denote respectively the relative dielectric permittivity and the relative magnetic permeability of the medium.

This set of equations is accompanied by continuity conditions at the interface between the particle and the host medium; these obviously concern the total field in the two media, i.e. the ( ⃗ E 2 (⃗ r, t); ⃗ H 2 (⃗ r, t)) field in the particle and the

( ⃗ E 1 (⃗ r, t); ⃗ H 1 (⃗ r, t)
) field in the surrounding medium, and as no magnetic material is considered (i.e. µ = µ 0 ), so that all components of ⃗ H (i.e. H tangential and H normal ) are continuous, while the tangential component of the electric field E tangential and the normal component of electric displacement vector D normal on both sides of the interface are continuous. Unfortunately, when the shape of the particle has no particular geometrical symmetry, it is impossible to find analytical solutions to these continuity equations, and only numerical simulations allow to evaluate them. In the case of a spherical particle, Mie's theory is used to solve these equations while in the more general case of any geometrical shape particle, only numerical methods such as finite difference methods or finite element methods "FEM" could be used to this end. In the following, we will describe both methods, first the Mie method and then the FDTD method.

2.2/ MIE THEORY

The name "Mie theory" refers to the analytical solution of the EM problem described above (i.e. determination of the EM fields ( ⃗ E 2 (⃗ r, t); ⃗ H 2 (⃗ r, t)) inside the particle and ( ⃗ E 1 (⃗ r, t); ⃗ H 1 (⃗ r, t)) outside it) when the particle is spherical. This theory is named after the German physicist Gustav Mie, who published his calculations in 1908 [Mie, 1908] while studying the absorption and scattering properties in the visible of aqueous suspensions of colloidal gold particles.

From the analytical solution of the EM problem of Mie, we will deduce the expressions of the effective extinction and scattering cross sections. As we will see later, the mathematical expressions of Mie are quite complex. In order to lighten the notations, we have therefore decided to write our analytical developments without the temporal harmonic term e (-iωt) which will be omitted and implied.

2.2.1/ ANALYTICAL SOLUTION OF THE EM PROBLEM OF MIE

The solution is reached through a rather lengthy mathematical process, which is well documented in the references [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF] and [START_REF] Hulst | Light scattering by small particles[END_REF]. Here, we will simply give the analytical expression for the scattered EM field ( ⃗ E sca (⃗ r, t); ⃗ H sca (⃗ r, t)), since this result is sufficient to express some of the radiation-particle interaction quantities, namely the extinction and scattering powers and cross sections.

The spherical particle, of radius a, is assumed to be illuminated by a plane wave having a propagation direction ⃗ k and polarization ⃗ p (⃗ p ⊥ ⃗ k); the incident electric field ⃗ E inc (⃗ r) has therefore the expression :

⃗ E inc (⃗ r) = E 0 ⃗ p e i ⃗ k.⃗ r (2.2)
The geometric symmetry presented by the particle naturally leads to move to a spherical coordinate system (r,θ, ϕ), in which the two angles θ and ϕ are defined by θ= ( ⃗ k,⃗ r), ( 0 ≤ θ ≤ π ) on the one hand, and by ϕ = angle defined by ⃗ p and the projection of ⃗ r in the plane orthogonal to ⃗ k (0 ≤ ϕ ≤ 2 π ) on the other.

In this coordinate system, the scattered EM field ( ⃗ E sca (⃗ r, θ, ϕ) ; ⃗ H sca (⃗ r, θ, ϕ)) from Mie's calculations has the following analytical expression:

                         ⃗ E sca (⃗ r, θ, ϕ) = +∞ n=1 E n (-b n ⃗ M h2 o1n (⃗ r, θ, ϕ) + ia n ⃗ N h2 e1n (⃗ r, θ, ϕ)) ⃗ H sca (⃗ r, θ, ϕ) = ∇ ∧ ⃗ E sca (⃗ r, θ, ϕ) iωµ 2 = k 2 iωµ 2 +∞ n=1 E n (-b n ⃗ M h2 o1n (⃗ r, θ, ϕ) + ia n ⃗ N h2 e1n (⃗ r, θ, ϕ)) (2.3)
This formulation requires some explanation:

• The coefficients E n (n ≥ 1) are given by E n = i n E 0 2n+1 n(n+1)

• The two series a n and b n , commonly referred to as the Mie series, have rather complex expressions that simplify slightly in the special case where the particle and the host medium have identical magnetic permeabilities. The coefficients a n and b n (n ≥ 1) can then be calculated using the following relations:

                   a n = mΨ n (mx)Ψ ′ n (x)-Ψ n (x)Ψ ′ n (mx) mΨ n (mx)ξ ′ n (x)-ξ n (x)Ψ ′ n (mx) b n = mΨ n (mx)Ψ ′ n (x)-mΨ n (x)Ψ ′ n (mx) mΨ n (mx)ξ ′ n (x)-mξ n (x)Ψ ′ n (mx)
(2.4) in which :

• The functions Ψ n (ρ) and ξ n (ρ) are the Ricatti-Bessel functions defined respectively by :

Ψ n (ρ) = ρ j n (ρ) and ξ n (ρ) = ρh n (ρ) (2.5)
The notations j n and h n referring respectively to the spherical Bessel and Hankel functions (see [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF] for more details).

• The quantity x, defined by x = k 1 a , is a dimensionless parameter commonly called size parameter related to the host medium (medium 1); x is real because the host medium is assumed to be non-absorbing.

• The quantity m, defined by m = n 2 n 1 = k 2 k 1 , is also a dimensionless parameter and represents the complex optical index contrast between the particle and the host medium; the product mx is equal to k 2 a , size parameter related to the particle (medium 2) and possibly a complex quantity if the particle is made of a material with absorption.

• Finally, the functions ⃗ M h2 o1n (⃗ r, θ, ϕ); ⃗ M h2 e1n (⃗ r, θ, ϕ); ⃗ N h2 o1n (⃗ r, θ, ϕ); ⃗ N h2 e1n (⃗ r, θ, ϕ) are qualified as vector spherical harmonics (see [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF] for more details).

⃗ M h2

e1n (⃗ r, θ, ϕ) = -h n (k 2 .r)

P 1 n (cosθ) sinθ sinϕ ⃗ e θ -h n (k 2 .r) dP 1 n (cosθ) sinθ cosϕ ⃗ e ϕ (2.6) ⃗ M h2 o1n (⃗ r, θ, ϕ) = h n (k 2 .r) P 1 n (cosθ) sinθ cosϕ ⃗ e θ -h n (k 2 .r) dP 1 n (cosθ) sinθ sinϕ ⃗ e ϕ (2.7) ⃗ N h2 e1n (⃗ r, θ, ϕ) = n(n + 1) h n (k 2 .r) k 2 .r P 1 n (cosθ)cosϕ ⃗ e r + 1 k 2 .r d(r.h n (k 2 .r)) dr dP 1 n (cosθ) dθ cosϕ ⃗ e θ - 1 k 2 .r d(r.h n (k 2 .r)) dr dP 1 n (cosθ) sinθ sinϕ ⃗ e ϕ (2.8) ⃗ N h2 o1n (⃗ r, θ, ϕ) = n(n + 1) h n (k 2 .r) k 2 .r P 1 n (cosθ)sinϕ ⃗ e r + 1 k 2 .r d(r.h n (k 2 .r)) dr dP 1 n (cosθ) dθ sinϕ ⃗ e θ - 1 k 2 .r d(r.h n (k 2 .r)) dr dP 1 n (cosθ) sinθ cosϕ ⃗ e ϕ
(2.9) in which the notation P 1 n refers to the associated Legendre functions of the first kind and (⃗ e r ,⃗ e θ ,⃗ e ϕ ) denotes the local direct orthonormal basis associated with the coordinates (r,θ,ϕ).

2.2.2/ EXPRESSIONS OF THE EXTINCTION AND SCATTERING CROSS SECTIONS

Knowing now the analytical expression (Eq. 2.3) of the EM field E sca (⃗ r, θ, ϕ) and H sca (⃗ r, θ, ϕ) scattered by the spherical particle, we can calculate the extinction power Ψ P ext ( ⃗ k, ⃗ p) and scattered one Ψ P sca ( ⃗ k, ⃗ p). The expression of the ⃗ E sca (⃗ r, θ, ϕ) field in the far field is obtained from (Eq.

2.3) and using the following two equivalences:

h n (ρ) ∼ (-i) n e iρ iρ and h ′ n (ρ) ∼ (-i) n e iρ ρ , when ρ → ∞ (2.10)
All calculations done, we obtain :

⃗ E sca (⃗ r, θ, ϕ) ∼ E 0 e (-ik 2 r) -ik 2 r [ S 2 (cos θ) cos ϕ ⃗ e θ -S 1 (cos θ) cos ϕ ⃗ e ϕ ], when ⃗ r → ∞ (2.11)
Expression in which we have introduced the functions S 1 and S 2 defined by :

                     S 1 (cos θ) = +∞ n=1 2n+1 n(n+1) [a n π n (cos θ) + b n τ n (cos θ)] S 2 (cos θ) = +∞ n=1 2n+1 n(n+1) [a n τ n (cos θ) + b n π n (cos θ)]
(2.12)

The functions π n and τ n being themselves defined by :

π n (cos θ) = P 1 n (cos θ) sin θ and π n (cos θ) = dP 1 n (cos θ) dθ (2.13)
At this point, the expression of the extinction power (referring to [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]), and knowing that π n (1) = τ n (1) = n(n+1)

2

, will be:

Ψ p ext ( ⃗ k, ⃗ p) = πε 2 c 2 E 2 0 k 2 2 +∞ n=1 (2n + 1)Re(a n + b n ) (2.14)
Which shows no dependence of Ψ p ext ( ⃗ k, ⃗ p) either on the direction of propagation ⃗ k or on the direction of polarization ⃗ p; this result was, after all, expected, given the particular geometric symmetries exhibited by a spherical particle.

The expression (Eq. 2.9) is therefore also that of the power Ψ np ext extinguished by the particle in non-polarized (np) illumination, whatever the direction of incidence of this illumination, so that, dividing Ψ np ext first by the incident surface power

Ψ inc = k 2 E 2 0 2µ
2 ω and then by the geometric cross section πa 2 of the particle, we successively reach the expressions of the effective extinction cross section of the particle C np ext and of its associated efficiency Q np ext :

C np ext = 2π k 2 2 +∞ n=1 (2n + 1)Re(a n + b n ) and Q np ext = 2 x 2 +∞ n=1 (2n + 1)Re(a n + b n ) (2.15)
The diffused power has the expression [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]:

Ψ p sca ( ⃗ k, ⃗ p) = E 2 0 2ωµ 2 k 2 4π [|S 2 (cos θ)| 2 cos 2 ϕ + |S 1 (cos θ)| 2 sin 2 ϕ] dΩ (2.16)
Which depends neither on ⃗ k nor on ⃗ p but only on the size parameter x and the complex optical index contrast m; this power is therefore also the power Ψ sca scattered by the particle in non-polarized illumination, whatever the direction of incidence of this illumination. Continuing the calculation of the integral (Eq.

2.16), and making use of special orthogonality properties of the functions π 2 and τ 2 , we arrive at reducing the expression of Ψ np sca to the following simplified formula:

Ψ np sca = πε 2 c 2 E 2 0 k 2 2 +∞ n=1 (2n + 1)[|a n | 2 + |b n | 2 ]
(2.17)

So that the expressions for the effective scattering cross section of the particle C np sca and its associated efficiency Q np sca are written :

C np sca = 2π k 2 2 +∞ n=1 (2n + 1)[|a n | 2 + |b n | 2 ] and Q np sca = 2 x 2 +∞ n=1 (2n + 1)[|a n | 2 + |b n | 2 ]
(2.18) series, and should be programmed with care. In chapter 4 of their book [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF], BOHREN and HUFFMAN present a review of the algo-rithmic means available in the 1980's to numerically compute the scattering coefficients a n and b n as well as the extinction efficiency Q np ext and scattering efficiency Q np sca . In general, the larger the size parameter x, or the larger its product by the modulus of the complex optical index contrast m, the slower the series converge: thus the study of the scattering by visible light by a 1 mm diameter water drop (x and |m|x of the order of 10 4 ) would require the computation of several tens of thousands of coefficients a n and b n for correct estimates of the efficiencies Q np ext and Q np sca . Here, we have developed a Matlab code where the functions are predefined which makes the implementation of the code very easy. It is tested with two nanoparticles suspended in air, one is metallic (Gold) with radius R NP = 90nm and the other is dielectric (Diamond) with radius R NP = 160nm and refractive index n = 2.417 as shown in Fig. 2.2. This code will be used to determine the properties of our spherical nanoparticles in chapter 4.

2.3/ FINITE DIFFERENCES TIME DOMAIN (FDTD)

In this paragraph, we will explain the used numerical method of calculation, namely the method of finite differences in space and time, the FDTD (Finite Difference in Time Domain). This method is based on the algorithm proposed by K. S. Yee in 1966 [Yee, 1966]. From Maxwell's equations that govern the propagation of EM waves in any medium, the FDTD allows to approximate the solutions, for structures in one, two and three dimen-sions, dispersion curves, transmission, reflection and extinction coefficients and field maps based on a development in finite differences of the partial derivatives involved in the equations in question. These finite differences are based on a discretization in space (for the spatial derivatives) and another one in time (for the temporal derivatives), which discretizations must be well chosen by verifying the spatial convergence criterion which ensures the convergence of the unknown functions sought [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]. The FDTD is based on spatial discretization of the structure under study through parallelepiped cells in the Cartesian coordinates. This kind of meshes is not well suitable to model structures with curved surfaces. Thus, in order to minimize an eventual stair-casing problem, several techniques were developed in the literature [START_REF] Dey | A modified locally-conformal FDTD algorithm for modeling 3-D perfectly conducting objects[END_REF], Dey et al., 1999, Yu et al., 2000, Nilavalan et al., 2008]. Nonetheless, in the case of a purely dielectric material without absorption, the averaging technique seems to be quite powerful. In this case, the dielectric constant of each cell comprising two media of different refractive index is calculated as the average value of the dielectric constant over the cell volume. We have tested this technique and demonstrated that it allows a very good agreement with Mie's theory.

2.3.1/ MAXWELL'S EQUATIONS

FDTD method is based on the direct solution of Maxwell-Faraday and Maxwell-Ampère rotational equations. These can be written as follows:

⃗ rot ( ⃗ E) = - ∂ ⃗ B ∂t (2.19) ⃗ rot ( ⃗ H) = ∂ ⃗ D ∂t + ⃗ j (2.20)
These equations are completed by the two constitutive equations that link the EM response of a given medium to the applied field. These equations are given by: ⃗ B = µ ⃗ H and ⃗ D = ε ⃗ E where ε and µ denote respectively the dielectric permittivity and the magnetic permeability of the medium which is considered as a homogeneous (a medium whose composition is the same at all points), isotropic (a medium whose properties are the same in all directions) and non-dispersive (ε is independent of the frequency of the wave propagating in the medium). In these equations, ⃗ B and ⃗ D are therefore given by:

⃗ B = µ r µ 0 ⃗ H ; ⃗ D = ε r ε 0 ⃗ E (2.21)
Where µ r and µ 0 are respectively the relative permeability and the free space permeability, and ε r and ε 0 are respectively the relative permittivity and the free space permittivity. Note that Gauss's laws for electric and magnetic inductions are implicit in the FDTD algorithm. Recall that in a 3D-Cartesian frame of origin (O, x, y, z), Maxwell's differential equations lead to 2x3 scalar equations linking the three components of the magnetic and electric excitation to the applied EM field ( ⃗ E, ⃗ B) :

                   ∂H x ∂t = 1 µ ( ∂E y ∂z -∂E z ∂y ) ∂H y ∂t = 1 µ ( ∂E z ∂x -∂E x ∂z ) ∂H z ∂t = 1 µ ( ∂E x ∂y - ∂E y ∂x ) (2.22)                    ∂E x ∂t = 1 ε ( ∂H z ∂y - ∂H y ∂z ) ∂E y ∂t = 1 ε ( ∂H x ∂z -∂H z ∂x ) ∂E z ∂t = 1 ε ( ∂H y ∂x -∂H x ∂y )
(2.23)

The numerical solution of Maxwell's equations requires the discretization of the space and time according to the scheme of Yee [Yee, 1966] which uses the centered differences to express the partial derivatives that appear in equations 2.22 and 2.23. Thus, suppose that f (x, y, z, t) represents a component of the electric field ⃗ E or of the magnetic field ⃗ H. The function evaluated at node (i, j, k) and at time n is written :

f (i∆x, j∆y, k∆z, n∆t) = f n (i, j, k) (2.24)
The approximation of the first derivative f by the centered differences is written as follows:

∂ f n (i, j, k) ∂t = f n+ 1 2 (i, j, k) -f n-1 2 (i, j, k) ∆t + O(∆t) (2.25) ∂ f n (i, j, k) ∂x = f n (i + 1 2 , j, k) -f n (i -1 2 , j, k) ∆x + O(∆x) (2.26)
Where:

∆x, ∆y, ∆z correspond to the spatial cell dimensions along the x, y and z directions respectively and ∆t is the temporal step. This is equivalent to replacing the continuous Euclidean space by a grating of points located at the nodes of a three-dimensional grid. If the axes ( ⃗ Ox), ( ⃗ Oy) and ( ⃗ Oz) are orthogonal, which is the case here, the three-dimensional grid consists on rectangular parallelepipedic cells. Each cell is then marked by the triplet (i, j, k) meaning that the cell origin is located at coordinates (x = i∆x, y = j∆y, z = k∆z) from the frame origin. This manner of numbering is illustrated in figure 2.3a.

The indices i, j and k are bounded, i.e.:

                   i ∈ [1, i max ] j ∈ [1, j max ] k ∈ [1, k max ] (2.27)
This obviously corresponds to working with a finite calculation volume.

2.3.2/ YEE'S ALGORITHM

Yee proposed to use such a spatial discretization in terms of cells but assigning to each of them the 6 components of the EM field in an intercalated way. This means that no two components are ever placed at the same location (see Fig. 2.3b). Therefore, each component of a field is surrounded by the components of the other field which form a loop (integration path for Gauss's and Ampere's laws) used to calculate the rotational. By the way, we can directly apply the formulas of the centered finite differences to calculate the derivatives of each component [Yee, 1966] cell by cell function of the components of the neighboring cells. This mesh fills the space with loops of fields that interpenetrate and create the contours along which Faraday's and Ampere's laws of induction will apply. It also immediately reveals the local and integral forms of Maxwell's equations.

This simplicity and this adequacy of the cells with the intrinsic properties of the EM fields will be extremely useful to define the singularities and the boundary conditions of the model. Yee also proposes to shift the calculation of the fields ( ⃗ E and ⃗ H) by a half step in time called "leap-frog time schema" (Fig. 2.4) for the field :

• ⃗ E is given at times: (n -1 2 )∆t, (n + 1 2 )∆t, (n + 3 2 )∆t ... etc., and are placed in the middle of the cell edges,

• ⃗ H at times: n∆t, (n + 1)∆t, (n + 2)∆t ... etc. and are placed in the middle of the cell faces.

Thus in a cell (i, j, k) and at a time t, we have :

                                           H n x (i, j + 1 2 , k + 1 2 ) = H x {i∆x, ( j + 1 2 )∆y, (k + 1 2 )∆z, n∆t} H n y (i + 1 2 , j, k + 1 2 ) = H y {(i + 1 2 )∆x, j∆y, (k + 1 2 )∆z, n∆t} H n z (i + 1 2 , j + 1 2 , k) = H z {(i + 1 2 )∆x, ( j + 1 2 )∆y, k∆z, n∆t} E n x (i + 1 2 , j, k) = E x {(i + 1 2 )∆x, j∆y, k∆z, n∆t} E n y (i, j + 1 2 , k) = E y {i∆x, ( j + 1 2 )∆y, k∆z, n∆t} E n z (i, j, k + 1 2 ) = E z {i∆x, j∆y, (k + 1 2 )∆z, n∆t}
(2.28) This algorithm has been very successful. Since then, many advances have been made to improve the calculation of EM fields with FDTD. We now apply the above concepts and notation to perform a numerical approximation of Maxwell's rotational equations in three dimensions, given by equations (2.22) to (2.23). For example, consider the forth equation from (2.28): Considering E x at the point (i + 1 2 )∆x, j∆y, k∆z as it is shown in Fig. 2.3 at time (n∆t), we can write the equation with a "leap-frog" schema as follows:

⃗ E n+ 1 2 (i + 1 2 , j, k) = ⃗ E n-1 2 (i + 1 2 , j, k) + ∆t ε ∇ ∧ ⃗ H n (2.29)
By applying the spatial centered FDTD scheme (see Fig. 2.5), we get the expression of each component of the field ⃗ E, i.e. E x , E y and E z , for time

(n + 1 2 )
given by:

E n+ 1 2 x (i + 1 2 , j, k) = E n-1 2 x (i + 1 2 , j, k) + ∆t ε 0 ε r (i, j, k) H n z (i + 1 2 , j + 1 2 , k) -H n z (i + 1 2 , j -1 2 , k) ∆y - H n y (i + 1 2 , j, k + 1 2 ) -H n y (i + 1 2 , j, k -1 2 ) ∆z (2.30) E n+ 1 2 y (i, j + 1 2 , k) = E n-1 2 x (i, j + 1 2 , k) + ∆t ε 0 ε r (i, j, k) H n x (i, j + 1 2 , k + 1 2 ) -H n x (i, j + 1 2 , k -1 2 ) ∆z - H n z (i + 1 2 , j + 1 2 , k) -H n z (i -1 2 , j + 1 2 , k) ∆x (2.31) E n+ 1 2 z (i, j, k + 1 2 ) = E n-1 2 z (i, j, k + 1 2 ) + ∆t ε 0 ε r (i, j, k) H n y (i + 1 2 , j, k + 1 2 ) -H n y (i -1 2 , j, k + 1 2 ) ∆x - H n x (i, j + 1 2 , k + 1 2 ) -H n x (i, j -1 2 , k + 1 2 ) ∆y (2.32)
And by analogy we can derive the finite difference equations for the rest of the equations for the magnetic field:

H n+1 x (i, j + 1 2 , k + 1 2 ) = H n x (i, j + 1 2 , k + 1 2 ) + ∆t µ 0 µ r (i, j, k) E n+ 1 2 y (i, j + 1 2 , k + 1) -E n+ 1 2 y (i, j + 1 2 , k) ∆z - E n+ 1 2 z (i, j + 1, k + 1 2 ) -E n+ 1 2 z (i, j, k + 1 2 ) ∆y (2.33) H n+1 y (i + 1 2 , j, k + 1 2 ) = H n y (i + 1 2 , j, k + 1 2 ) + ∆t µ 0 µ r (i, j, k) E n+ 1 2 z (i + 1, j, k + 1 2 ) -E n+ 1 2 z (i, j, k + 1 2 ) ∆x - E n+ 1 2 x (i + 1 2 , j, k + 1) -E n+ 1 2 x (i + 1 2 , j, k) ∆z (2.34) H n+1 z (i + 1 2 , j + 1 2 , k) = H n z (i + 1 2 , j + 1 2 , k) + ∆t µ 0 µ r (i, j, k) E n+ 1 2 x (i + 1 2 , j + 1, k) -E n+ 1 2 x (i + 1 2 , j, k) ∆y - E n+ 1 2 y (i + 1, j + 1 2 , k) -E n+ 1 2 y (i, j + 1 2 , k) ∆x (2.35)

2.3.3/ NUMERICAL CONSTRAINTS OF THE FDTD

Numerical computational methods are normally governed by "limitations" to be accurate. On the one hand, one of these limitations is the criterion of convergence or numerical stability (δ < min(λ) 16 ) which is ensured if the propagation speed of an EM wave in the mesh grid is bounded; the EM wave must therefore travel at a speed lower than the maximum physical speed actually allowed by the medium (v max ). On the other hand, the definition of a finite discretization domain is in contradiction with the real situation in the study of radiating structures in "open" geometry where the spatial domain of the calculated EM fields is unlimited in one or more directions. In addition to the limited memory space, absolutely no computer can store an unlimited amount of data, so it is necessary to find an artifice to simulate the continuity of the propagation outside the limits of the mesh. In general, two families of solutions exist:

• Techniques that express the field at the boundary only as a function of the field already computed inside the studied domain (this is the case of the absorbing boundary conditions of Mur [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF], Mur, 1981].

• Techniques that add a layer around the calculation window, not necessarily physical, whose impedance is matched to that of the free space, but which does not cause any reflection and which absorbs almost any EM field propagating there (Perfectly Matched Layers "PML" conditions [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF], Berenger, 1994, Berenger, 1996]).

2.3.3.1/ STABILITY CRITERION

The Yee scheme is subject to a stability condition [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF] fixing the time step from the initial discretization of the simulation space. The stability problems of explicit numerical methods have been analyzed in detail by Courant, Friedrich and Levy [Courant et al., 1967], as well as by Von Neumann, using a rigorous mathematical approach. This analysis shows that explicit schemes are stable under a so-called "CFL" condition (for Courant, Friedrich and Levy) applied to the FDTD method in the case of a regular mesh [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]:

∆t ≤ 1 c. 1 ∆x 2 + 1 ∆y 2 + 1 ∆z 2 (2.36)
where c denotes the light celerity.

In the case of a uniform mesh (∆x = ∆y = ∆z = ∆), the CFL criterion reduces to:

∆t ≤ ( 1 c . ∆ √ 3 ), in 3D (2.37) ∆t ≤ ( 1 c . ∆ √ 2 ), in 2D (2.38)
However, it is possible to overcome the restrictive assumptions of the regular mesh and the homogeneity of the environment that allow the previous result to be obtained with the generalized criterion:

∆t ≤ min( 1 v m . 1 ∆x 2 m + 1 ∆y 2 m + 1 ∆z 2 m ), with: v m = 1 µ.ε (2.39)
The minimization on the parameter "m" represents a minimization on all the elementary meshes of the computational volume. The time step is then imposed by the dimensions of the smallest cell of the mesh (∆x m , ∆y m , ∆z m ). It follows from this stability condition that the temporal step must be sufficient to describe the propagation of the wave from a node to the nearest node distant by one spatial step.

2.3.3.2/ ABSORBING BOUNDARY CONDITIONS (ABC)

The limitation of the memory space of the computational resources impose to restrict the dimension (volume) of the calculation window. All the fields, located at the edges of the calculation window represented in Fig. 2.3a, cannot be calculated with the classical FDTD equations. A first solution would be to let them equal to zero. Then, these edges act as electric mirrors reflecting all the outgoing field into the calculation window. All these nonphysical reflections disturb the modeling of the EM response inside the calculation window and could lead to many parasitical artifacts. The objective is therefore to use a special algorithm for these edge components to minimize these spurious reflections. It is easy to understand that the use of effective boundary conditions has been the major difficulty in the implementation of the FDTD technique. Several methods exist with different philosophies [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]] but we will briefly present only two with their advantages and drawbacks.

2.3.3.3/ MUR'S CONDITION

The absorbing boundary technique of Mur [Mur, 1981] is based on the theory of Engquist and Majda applicable in a cartesian FDTD grid [START_REF] Engquist | Absorbing boundary conditions for numerical simulation of waves[END_REF]. This consists in applying to the fields arriving at the the solution of the propagation equation in a privileged direction to the fields arriving at the privileged direction. Let us consider the propagation equation:

∂ 2 U ∂x 2 + ∂ 2 U ∂y 2 + ∂ 2 U ∂z 2 - 1 c 2 . ∂ 2 U ∂t 2 = 0 (2.40)
where U is the scalar component of the ⃗ E or ⃗ H field. A partial differential operator L is then introduced such that:

L ≡ ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 - 1 c 2 . ∂ 2 ∂t 2 (2.41)
The wave equation 2.40 is then written :

LU = 0 (2.42)
The factorization of the operator L allows to write:

LU = L + L -U = 0 (2.43)
The two operators L + and L -are defined by:

L + ≡ ∂ ∂x + 1 c . ∂ ∂t . √ 1 -S 2 (2.44) L -≡ ∂ ∂x - 1 c . ∂ ∂t . √ 1 -S 2 (2.45)
With:

S 2 ≡ ∂ 2 ∂y 2 + ∂ 2 ∂z 2 1 c 2 . ∂ 2 ∂t 2 (2.46)
We thus obtain the two solutions for each direction corresponding to the two directions L -U = 0 and L + U = 0. Placing on the edges x = 0 and x = d, the outgoing and incoming solutions respectively must verify:

L -U = 0, for x=0 (2.47) L + U = 0, for x=d (2.48)
They reflect the analytical absorption boundary conditions for a wave propagating within the spatial domain in the Ox direction. A similar factorization is obviously possible in y = 0 and y = d and z = 0 and z = d.

For their numerical implementation, the objective is to linearize the square roots in order to be able to apply a defined difference scheme to the operators. This linearization can, under certain conditions, be done by a Taylor expansion at first or second order. This is the case for a plane wave around the normal incidence where we can consider that the derivative along y and z is small compared to the time derivative. The square root term is then written:

√ 1 -S 2 ≈ 1 - 1 2 .S 2 (2.49)
This second order approximation implies that the outgoing and incoming solutions, applied to the tangential components of the field at x = 0 and x = d respectively, are written as follows:

∂ 2 U ∂x∂t - 1 c . ∂ 2 U ∂t 2 + c 2 . ∂ 2 U ∂y 2 + c 2 . ∂ 2 U ∂z 2 = 0, for x=0 (2.50) ∂ 2 U ∂x∂t + 1 c . ∂ 2 U ∂t 2 - c 2 . ∂ 2 U ∂y 2 - c 2 . ∂ 2 U ∂z 2 = 0, for x=d (2.51)
By analogy, we obtain the differential equations for the other absorbing boundaries of the grid:

∂ 2 U ∂y∂t - 1 c . ∂ 2 U ∂t 2 + c 2 . ∂ 2 U ∂x 2 + c 2 . ∂ 2 U ∂z 2 = 0, for y=0 (2.52) ∂ 2 U ∂y∂t + 1 c . ∂ 2 U ∂t 2 - c 2 . ∂ 2 U ∂x 2 - c 2 . ∂ 2 U ∂z 2 = 0, for y=d (2.53) ∂ 2 U ∂z∂t - 1 c . ∂ 2 U ∂t 2 + c 2 . ∂ 2 U ∂x 2 + c 2 . ∂ 2 U ∂y 2 = 0, for z=0 (2.54) ∂ 2 U ∂z∂t + 1 c . ∂ 2 U ∂t 2 - c 2 . ∂ 2 U ∂x 2 - c 2 . ∂ 2 U ∂y 2 = 0, for z=d (2.55)
These equations correspond to the Mur conditions and are discretized by the nested difference method according to the Mur scheme. The discretization according to the Mur scheme is detailed in [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]]. We will only present the result of the discretization of the x = 0 boundary absorption condition for the U component of the field in the case of a uniform mesh grid with constant spatial pitch in all the three directions (∆x = ∆y = ∆z = ∆) which is written:

U n+1 0, j,k = -U n-1 1, j,k + c.∆t -∆ c.∆t + ∆ (U n+1 1, j,k + U n-1 0, j,k ) + 2.∆ c.∆t + ∆ (U n 0, j,k + U n 1, j,k )+ (c.∆t) 2 2.∆(c.∆t + ∆) ×                    (U n 0, j+1,k -4U n 0, j,k + U n 0, j-1,k + U n 1, j+1,k -4U n 1, j,k + U n 1, j-1,k + U n 0, j,k+1 + U n 0, j,k-1 + U n 1, j,k+1 + U n 1, j,k-1 ) (2.56)
In conclusion, we can say about the Mur's conditions that :

1) They are strictly valid only for waves arriving at normal incidence at the boundary of the domain. Parasitic reflections will appear for incidences that deviate from them. In our case, since we are going to be at normal incidence only (1D), we considered only the first term and we neglected the second term :

U n+1 0, j,k = -U n-1 1, j,k + c.∆t -∆ c.∆t + ∆ (U n+1 1, j,k + U n-1 0, j,k ) (2.57)
2) They lower the spurious reflections in the FDTD space to a global level, of the order of 1% to 5%, sufficiently low for the simulations not to be altered.

3) They are practically not applicable to the corners of the computational domain without the expansion of the equation (2.49) being reduced to first order ( √ 1 -S 2 ≈ 1). The solution to the limit x = 0 is then written as follows:

∂ 2 U ∂x∂t - 1 c . ∂ 2 U ∂t 2 = 0 (2.58) 2.3.3.4/ PERFECTLY MATCHED LAYERS (PML) CONDITIONS
The application of absorbing boundary conditions has evolved rapidly with the advent of perfectly matched absorbing layers (PML) developed in the 1990s by Berenger (2D [Berenger, 1994] and 3D [Berenger, 1996] cases), and which are considered today as the most efficient absorbing conditions.

Their performance is largely due to the variation in the properties of the layers as one moves away from the calculation area/PML interface. In addition, and in contrast to wall conditions, they allow for higher levels of reflection attenuation by adjusting their thickness. In fact, the implementation of such absorption conditions is perfectly integrated in an FDTD scheme. The use of PMLs is based on the impedance matching condition at the interface between two media of the same index but one of which is absorbing (of equivalent electrical conductivity σ and non-zero magnetic conductivity σ * ). This condition is expressed by:

σ ε = σ * µ (2.59)
In this case, the wave is not reflected at the interface between the two media and is attenuated in the PML zone. This impedance adaptation is only possible at normal incidence because a reflection at the interface between the two media appears as soon as we move away from it. Bérenger gets around this problem by decomposing the wave propagating in the PML medium along two main axes. The projection along the normal axis is equivalent to considering a plane wave at normal incidence which satisfies the condition 2.59 and which will therefore not be reflected at the interface between the two media.

The decomposition along the tangential axis describes a plane wave at grazing incidence for which no absorption appears and which consequently does not undergo any reflection (see Fig. 2.6). The decomposition of the fields into subcomponents on the two main axes is accompanied by the definition of virtual conductivities (σ x , σ * x , σ y , σ * y , σ z , σ * z ) for each direction, respecting the condition 2.59 and allowing the cancellation of reflections on each edge of the domain. The introduction of virtual conductivities and EM subcomponents in Maxwell's equations gives [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]:

µ 0 ∂H xy ∂t + σ * y H xy = - ∂(E zx + E zy ) ∂y (2.60) µ 0 ∂H xz ∂t + σ * z H xz = ∂(E yx + E yz ) ∂z (2.61) µ 0 ∂H yz ∂t + σ * z H yz = - ∂(E xy + E yz ) ∂z (2.62) µ 0 ∂H yx ∂t + σ * x H yx = ∂(E zx + E zy ) ∂x (2.63) Figure 2.6: Operation of a PML medium. µ 0 ∂H zx ∂t + σ * x H zx = - ∂(E yx + E yz ) ∂x (2.64) µ 0 ∂H zy ∂t + σ * y H zy = ∂(E xy + E xz ) ∂y (2.65) ε ∂E xy ∂t + σ y E xy = ∂(H zx + H zy ) ∂y (2.66) ε ∂E xz ∂t + σ z E xz = - ∂(H yx + H yz ) ∂z (2.67) ε ∂E yz ∂t + σ z E yz = ∂(H xy + H yz ) ∂z (2.68) ε ∂E yx ∂t + σ x E yx = - ∂(H zx + H zy ) ∂x (2.69) ε ∂E zx ∂t + σ x E zx = ∂(H yx + H yz ) ∂x (2.70) ε ∂E zy ∂t + σ y E zy = - ∂(H xy + H xz ) ∂y (2.71)
we note that if the virtual conductivities are all zero, we would recover the equations of propagation in free-space.

The solution of these equations in the PML medium is done by spatial and temporal discretization at the central difference. Take for example equation 2.60 where the update of the H xy component at time t = (n + 1 2 )∆t is written:

H n+1 xy (i, j + 1 2 , k + 1 2 ) = 2µ 0 -∆t.σ * y 2µ 0 + ∆t.σ * y H n xy (i, j + 1 2 , k + 1 2 ) - 2.∆t ∆y(2µ 0 + ∆t.σ * y ) × (E n+ 1 2 zx (i, j + 1, k + 1 2 ) -E n+ 1 2 zx (i, j, k + 1 2 )) + (E n+ 1 2 zy (i, j + 1, k + 1 2 ) -E n+ 1 2 zy (i, j, k + 1 2 ))
(2.72)

The other 11 equations are discretized in the same way.

Despite this decomposition, a residual reflection remains because of the discontinuity induced, at the vacuum/PML interface, by the spatial discretization. It is all the more important as the direction of incidence of the wave moves away from the normal. The abrupt variations of conductivities at this interface degrade the absorption performances. This effect is however reduced by imposing a power law variation to the absorption in the PML layers [START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF]:

σ(ρ) = σ max ( ρ e ) p (2.73)
where (ρ = x, y, z) and σ max is the maximum conductivity reached when the depth inside the PML is equal to the total depth e of this layer. The order of the growth equation p (also called the PML order) is, in most cases, chosen between 2 and 5. A perfect Mirror condition is imposed at the external PML boundary without any consequent energy reflections in the calculation domain.

Berenger PMLs in their basic version, have one important limitation, which is that they do not absorb evanescent waves. The absorbing layers must therefore be placed at a minimum distance from the structure to be studied (at least equal to λ 2 ) in order to avoid spurious reflections of this type of wave.

The other major difficulty of this type of absorbing conditions is its cost in memory and calculation time, especially in 3D. Indeed, the biaxial medium leads to an increase in the number of field components to store and the number of operations to perform at each time step. In many cases (large or very resonant structures), the reduction of the memory space and the computation time still remain a priority compared to the reduction of the reflections on the edges of the domain.

2.3.4/ DISPERSION MODELS

The metals occupy a very important place in the optical devices especially those which involve plasmonic resonances such as certain photonic crystals [START_REF] Sigalas | Metallic photonic band-gap materials[END_REF], Fan et al., 1996]. It is therefore important in our case to properly describe the optical properties of the metals. As a first approximation, a metal can be considered as an isotropic medium having an electric constant ε, a magnetic permeability µ and a conductivity σ. Maxwell's equations are written as follow:

µ ∂H ∂t + ⃗ ∇ ∧ ⃗ E = 0 (2.74) -ε ∂E ∂t + ⃗ ∇ ∧ ⃗ H = ⃗ J (2.75)
⃗ J denotes the current density vector related to the electric field by : ⃗

J = σ ⃗ E.
For a monochromatic plane wave of EM field vectors:

⃗ E = ⃗ E 0 exp(-iωt), ⃗ H = ⃗ H 0 exp(-iωt)
, the wave equation is written:

∆ 2 ⃗ E + k 2 ⃗ E = 0 (2.76)
The complex electric constant is defined:

ε = ε + i σ ω (2.77)
In reality the electric constant and the conductivity depend on the frequency.

The optical response of a metal is directly related to the properties of the electrons and the atomic crystal lattice. A modification of the electronic distribution is thus translated by that of the optical properties. Three electronic models are involved in photon/matter interactions and are among the origins of optical properties: Drude, Drude-Lorentz (DL) [START_REF] Lee | Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications[END_REF] and Drude-Critical Point (DCP) [START_REF] Vial | Comparison of gold and silver dispersion laws suitable for fdtd simulations[END_REF]. The two first models allow a good description of the dispersion properties only on small spectral inter-vals. As we will see below, the FDTD method requires an analytical form of the electric constant of the metal in order to describe correctly its dispersion properties. Thus, we have used a more sophisticated model that was recently proposed by Vial, A. and T. Laroche [START_REF] Vial | Comparison of gold and silver dispersion laws suitable for fdtd simulations[END_REF]. This model consists of a Drude term and two critical point terms (n = 2) and can satisfactorily represent the electrical permittivity of metals over a wide range of frequencies [START_REF] Etchegoin | Erratum:"an analytic model for the optical properties of gold[END_REF]. From a computational point of view, the DCP model is advantageous compared to the DL model since the former requires only a smaller number of terms. Since then, the DCP model has been used to model the electrical permittivity of metals such as gold, silver and others with a good accuracy [START_REF] Vial | Comparison of gold and silver dispersion laws suitable for fdtd simulations[END_REF].

The dispersive DCP model expresses interband transitions as asymmetric line shapes with critical point terms instead of Lorentzian terms [START_REF] Etchegoin | Erratum:"an analytic model for the optical properties of gold[END_REF]. The relative electrical permittivity according to the Drude model can be written as:

ε DL (ω) = ε ∞ + χ D (ω) = ε ∞ - ω 2 D ω 2 + iγω (2.78)
Where ε ∞ is the relative electrical permittivity at infinite frequency, χ D (ω) the Drude susceptibility, and ω D is the frequency of the Drude pole and γ the inverse of the relaxation time of the pole.

In the case of the Drude-Lorentz model the dielectric permittivity is written:

ε DL (ω) = ε ∞ + χ D (ω) - ∆εΩ 2 l ω 2 -Ω 2 l + iωΓ l (2.79)
Where Ω l and Γ l are, respectively, the oscillator strength and the spectral width of the Lorentz oscillators, and ∆ε is a weighting coefficient.

Furthermore, the relative electrical permittivity according to the DCP model can be written as:

ε DCP (ω) = ε ∞ + χ D (ω) + n p=1 χ p (ω) (2.80)
Where the susceptibility at the critical point is expressed as:

χ p (ω) = A p Ω p ( e iϕ p Ω p -ω -iΓ p + e -iϕ p Ω p + ω + iΓ p ) (2.81)
Where A p is the amplitude, h Ω p is the energy gap, ϕ p is the phase, and Γ p is the pole expansion. The time dependence is written according to the convention e (-iωt) . Notice that in our study we considered only the first two terms (n=2), and here is an example of the parameters of a noble metal "Gold" expressed in the table 2.1 using the experimental data of Palik [Palik, 1985].

Moreover, we tested these models by calculating the values of the dielectric constants of gold and compared them with the experimental values measured by Palik. Figure 2.7 represents the spectra of real and imaginary parts of the dielectric constants of gold as a function of the wavelength. Figure 2.7(a)

shows the spectra of real values (ε ′ ) and figure 2.7(b) shows the spectra of imaginary values (ε ′′ ).

Gold A 1 0.26698 A 2 3.0834 ε ∞ 1.11683 ϕ 1 (rad) -1.2371 ϕ 2 (rad) -1.0968 ω D (

2.4/ ANGULAR RADIATION SPECTRUM

We will see that in free space, the knowledge of two components of the EM field is sufficient to know the entire field. Let us suppose that the longitudinal components E x and E y of the EM field are detected at a plane P perpendicular to the axis of propagation of the incident wave z positioned at D z = z 0 (see Fig. 2.8). The objective is to obtain the other four components of the EM field.

First of all, we decompose the plane wave spectrum of E x and E y into an angular spectrum [Fabbro, 1999]. We then obtain the spectrum of these two components in the dual space:

Ẽx (k x , k y , z 0 , λ) = +∞ -∞ E x (x, y, z 0 , λ) e -i(k x .x+k y .y) dx.dy (2.82) Ẽy (k x , k y , z 0 , λ) = +∞ -∞ E y (x, y, z 0 , λ) e -i(k x .x+k y .y) dx.dy (2.83)
With:

• k x is the wave vector along the x axis,

• k y is the wave vector along the y axis.

Thus the field distribution at the plane P can be described as a superposition of plane waves of direction ⃗ k such that:

⃗ k =               k x k y k z               such as: k 2 0 = k 2 x + k 2 y + k 2 z (2.84)
k z being defined differently depending on the sign of the wave vector k 0 , thus:

k z =            k 2 0 -k 2 x -k 2 y for k 2 0 ≥ k 2 x + k 2 y
, which corresponds to the propagating waves,

i k 2 x + k 2 y -k 2 0 for k 2 0 ≤ k 2 x + k 2
y , corresponding to the evanescent waves.

(2.85)

Each plane wave propagates along the direction ⃗ k and is written as follows:

⃗ E prop =                    E prop x = Ẽx (k x , k y ) e i(k x .x+k y .y+k z .z) E prop y = Ẽy (k x , k y ) e i(k x .x+k y .y+k z .z) E prop z (2.86)
The component along z of the considered plane wave, E z , can be found thanks to the relation of divergence of the field :

∇. ⃗ E prop = 0 ⇐⇒ δ x E prop x + δ y E prop y + δ z E prop z = 0 (2.87)
By deriving the expressions for the E prop

x and E prop y components from (2.86) into the equation (2.87), we obtain:

δ z E prop z = -ik x E prop x -ik y E prop y (2.88)
The relation which must satisfy E prop z is obtained by integrating the previous equation with respect to z:

E prop z = - k x E prop x + k y E prop y k z = - k x Ẽ prop x + k y Ẽ prop y k z e i(k x .x+k y .y+k z .z) (2.89)
The magnetic field can now be known from the electric field through the equation Maxwell-Faraday equation written in the harmonic regime in a medium without charge nor current:

-→ rot ⃗ E prop = jωµ 0 ⃗ H prop ⇐⇒ ⃗ H prop = - i ωµ 0               δ x δ y δ z               ∧               E prop x E prop y E prop z               (2.90)
By developing the rotational operator, we obtain the following relations for each of the components of the magnetic field:

⃗ H prop =               H prop x H prop y H prop z               = 1 ωµ 0               k z E prop y -k y E prop z k x E prop z -k z E prop x k y E prop x -k x E prop y               (2.91)
The field is then recomposed as the sum of all plane waves:

⃗ E = 1 4π 2 +∞ -∞                Ẽx Ẽy - (k x Ẽx +k y Ẽy ) k z                .e i(k x .x+k y .y+k z .z) dk x dk y (2.92) ⃗ H = 1 4π 2 ωµ 0 +∞ -∞               k z Ẽy -k y Ẽz k x Ẽz + k z k x (k z Ẽy -k y Ẽz ) - k y k x (k y Ẽy -k z Ẽz ) -k x Ẽy              
.e i(k x .x+k y .y+k z .z) dk x dk y

(2.93)

We can conclude that in free space the knowledge of two components of the EM field at the abscissa z 0 is sufficient to determine it completely whatever z ≥ z 0 .

Moreover, the angular spectra of the radiation can be calculated as a function of the Euler angles (i.e. θ and ϕ) thanks to the knowledge of the electric field distributed on a plane P ( -→ E P (x, y, z 0 , λ)). Thus, considering the PWE theory, the determination of the electric field associated to the propagation of each plane wave along the direction (θ, ϕ) is as follows:

-

→ E (θ, ϕ, z 0 , λ) = +∞ -∞ -→ E P (x, y, z 0 , λ)e (-i - → k . - → r ) dx dy = +∞ -∞
-→ E P (x, y, z 0 , λ)e (-ik x .x-ik y .y-ik z .z 0 ) dx dy

(2.94) With:

• - → k =                                            k x (θ, ϕ, λ) = 2πn λ sinθcosϕ k y (θ, ϕ, λ) = 2πn λ sinθsinϕ
, are the coordinates of the considered wave vec

k z (θ, ϕ, λ) = ( 2πn λ ) 2 -k 2 y -k 2 x
• n is the refractive index of the medium, After the determination of the three components of the angular spectrum, we calculate the far field intensity by considering the asymptotic limit given in [START_REF] Sherman | Asymptotic approximations to angular-spectrum representations[END_REF], Stamnes, 1986] by:

I(ϕ, θ, λ) = lim z→∞ || - → E (ϕ, θ, z, λ) || 2 ∝ || -i λ cosθ - → E (ϕ, θ, z 0 , λ) || 2 (2.95)

2.5/ CONCLUSION

In conclusion, we were able to present the different types of computational methods used in this thesis to achieve our objectives. Thanks to the Mie method, we are able to determine the optical properties (i.e. scattering, absorption and extinction cross sections, or even the spectra of electric or magnetic multipolar resonances) for any spherical micro-nanoparticle immersed in a homogeneous medium despite its radius or nature. Fortunately, a technique, called TFSF, has solved the limitation of determining the optical properties of a structure more complicated than a spherical particle. It will demonstrate that these properties can be determined for non-spherical particles suspended in a homogeneous medium or coupled to a dioptre or a photonic crystal. It is essential to mention that a further chapter is dedicated to the development of this technique with its extension in the case of a dielectric/metallic nanoparticle coupled with a planar substrate (for more details, see chapter 3). This last technique is a branch of the FDTD method which is based on the spatial and temporal discretization of the computational domain by considering the algorithm proposed by Yee. Finally, we have exploited the results obtained by the FDTD method to determine the angular spectrum of the radiation using the modal development method. This method is based on the decomposition of the near field into plane waves and aims to determine the amplitudes and directions of propagation of these plane waves in the far field. or the finite element method (FEM).

In our study, we will consider general cases where structures we will model by FDTD (i.g. either a substrate or a photonic crystal coupled to a NP), then we only focus on the case of an technique that based on the FDTD method to resolve this problem.

This technique named "Total-Field Scattered-Field" (TFSF). It is known by this name because it is based on a spatial division of the calculation domain into two areas: the area of total field (TF) that contains the scatterers (i.e.

NPs) and where the calculated EM field consists on the incident and scattered fields, and a second area, that encompasses the first one and where only the scattered field (SF) is calculated by subtracting the incident field over all the external interface of the TF area (see figure 3.1a). We can also distinguish two different cases: the first one is the simple case defined by an NP having an arbitrary geometry immersed in a homogeneous medium (here, we have considered a spherical NP immersed in air, see the figure 3.1a), while the second case includes the presence of a substrate in the vicinity of such an NP (see figure 3.1b). The code in the second case is improved by sub-59 tracting not only the incident field but also the reflected one on the substrate interface [START_REF] Markovich | Magnetic dipole radiation tailored by substrates: numerical investigation[END_REF][START_REF] Taflove | Computational Electrodynamics, the Finite-Difference Time-Domain Method[END_REF], as we will see in the following sections.

This technique, detailed in [START_REF] Merewether | On implementing a numeric Huygen's source scheme in a finite difference program to illuminate scattering bodies[END_REF], provides in the computational domain uniform plane waves with variable polarization and time dependence. The total electric and magnetic fields E tot and H tot are decomposed as follows:

E tot = E inc + E sca ; H tot = H inc + H sca (3.1)
The incidence fields E inc and H inc are defined at each cell of the calculation domain, but scattered-wave fields E sca and H sca are initially unknown. The TFSF technique allows to determine them. Therefore, by using this technique, we will be able to determine the optical responses (scattering, absorption and extinction) of different types of NPs (either spherical NPs or with any geometry such as Diabolo Antenna "DA") immersed in an homogeneous medium (as air or water) [Taflove et al., 2005c].

On the other hand, let us remember that our study deals with the coupling between a resonant NP and a PC implemented in an InP layer. Thus, it is necessary to first study the effect of the presence of such a substrate on the optical behavior of the NP. This chapter describes this technique which is based on the calculation of the scattered and the absorbed fields by correcting the FDTD update equations, and then shows the development of these equations in case of the presence of a planar substrate.

3.2/ TFSF IN 1D CASE

To explain the principle of this transformation in a simple way, we will first present it in the 1D problem, and generalize it to the case of 3D structures in the next section. In case, the incident field is necessarily a plane wave and is assumed to be polarized along the z axis and to propagate in the x direction (see figure 3.2). The boundaries between the total field and the scattered field are located at the points x = i a and x = i b . A scatterer is placed entirely in the TF area.

Let us first consider the first boundary at x = i a . As shown on figure 3.2, the electric field E tot z (i a ) and the magnetic field H scat y (i a -1 2 ) are located on either side of this boundary. Both require special treatment because the classical Yee's scheme does not allow them to be calculated directly because the total field E tot z (i a ) should be calculated taken into account, not only the scattered one H scat y (i a -1 2 ) but also the incident magnetic field at the same spatial position H inc y (i a -1 2 ). We exactly have the same constraint for the updating equation of the magnetic field H scat y (i a -1 2 ). Therefore, we understand why the conventional update equations become not consistent at the border between the two areas, as shown below in the equations (3.2 and 3.3). This implies that these fields should be corrected separately by adding or subtracting the incident one.

E z, tot | n+ 1 2 i a = E z, tot | n-1 2 i a - ∆t ε i a ∆x (H y, tot | n i a + 1 2 -H y, scat | n i a -1 2 ) (3.2) H y,scat | n+1 i a -1 2 = H y,scat | n i a -1 2 + ∆t µ a ∆x (E z,tot | n+ 1 2 i a -E z,scat | n+ 1 2 i a -1 ) (3.3)
Figure 3.2: A scheme representing the scatterer in the TF region and the spatial distribution of the electric and magnetic fields, in the 1D case. These fields are indicated by acronyms relative to the regions in which they are found: "tot" refers to the field existing in the TF region and "scat" refers to the field existing in the SF region.

More precisely, to obtain a consistent update of these fields, we must subtract the incident magnetic field from the total one while we must to add the incident electric field as following:

E z,tot | n+ 1 2 i a = E z,tot | n-1 2 i a - ∆t ε i a ∆x (H y,tot | n i a + 1 2 -H y,tot | n i a -1 2 ) - ∆t ε i a ∆x H y,inc | n i a -1 2 (3.4) H y,scat | n+1 i a -1 2 = H y,scat | n i a -1 2 + ∆t µ a ∆x (E z,scat | n+ 1 2 i a -E z,scat | n+ 1 2 i a -1 ) + ∆t µ a ∆x E z,inc | n+ 1 2 i a (3.5)
Similarly, the TFSF equations can be determined for the other boundary (at

x = i b ), as follows: E z,tot | n+ 1 2 i b = E z,tot | n-1 2 i b - ∆t ε i b ∆x (H y,tot | n i b + 1 2 -H y,tot | n i b -1 2 ) - ∆t ε i b ∆x H y,inc | n i b + 1 2 (3.6) H y,scat | n+1 i b -1 2 = H y,scat | n i b -1 2 + ∆t µ b ∆x (E z,scat | n+ 1 2 i b -E z,scat | n+ 1 2 i b -1 ) + ∆t µ b ∆x E z,inc | n+ 1 2 i b (3.7)
These equations are valid only in the case where the scatterer (nanoparticle)

is the only structure illuminated by the incident beam. As mentioned before, the resonance properties of a NP greatly depends on the host medium [Atie et al., 2015a, Atie et al., 2015b] so that they will be affected by the presence of a substrate for example. In such case, both incident and reflected EM fields should be taken into account in the TFSF boundary equations. Let us consider a specific case where the particle is located in medium (1) and placed in front of a planar substrate (medium 2) such presented in figure 3.3.

The TFSF boundaries are now located in two different media. The correction of the updated TFSF equations must take into account this constraint meaning two different expressions of the total field for each boundary. This leads to the following equations for the total fields:

E (1) tot = E inc + E re f l + E scat ; H (1) tot = H inc + H re f l + H scat (3.8) E (2) tot = E tr + E scat ; H (2) tot = H tr + H scat (3.9)
Where E re f l and H re f l are the reflected electric and magnetic fields, respectively, and the indices (1) and ( 2) are marked on the fields to specify the total field for each medium. Therefore, using the equations (3.8 and 3.9) into the equations (3.4 and 3.5), we get the following correction equations in medium 1:

E z,tot | n+ 1 2 i a = E z,tot | n-1 2 i a - ∆t ε i a ∆x (H y,tot | n i a + 1 2 -H y,tot | n i a -1 2 ) - ∆t ε i a ∆x (H inc y | n i a -1 2 + H re f l y | n i a -1 2 ) (3.10) H y,scat | n+1 i a -1 2 = H y,scat | n i a -1 2 + ∆t µ a ∆x (E z,scat | n+ 1 2 i a -E z,scat | n+ 1 2 i a -1 )+ ∆t µ i a ∆x (E inc z | n+ 1 2 i a +E re f l z | n+ 1 2 i a ) (3.11)
Contrarily, for equations of the second boundary at x = i b (3.6 and 3.7), it is necessary to take into account the transmitted "tr" fields (E tr or H tr ) through the interface by using the equations of (3.9) as following:

E z,tot | n+ 1 2 i b = E z,tot | n-1 2 i b - ∆t ε i b ∆x (H y,tot | n i b + 1 2 -H y,tot | n i b -1 2 ) - ∆t ε i b ∆x H tr y | n i b + 1 2 (3.12) H y,scat | n+1 i b -1 2 = H y,scat | n i b -1 2 + ∆t µ b ∆x (E z,scat | n+ 1 2 i b -E z,scat | n+ 1 2 i b -1 ) + ∆t µ i b ∆x E tr z | n+ 1 2 i b (3.13)
It should be noted that the integration of these equations in the FDTD algorithm requires a parallel calculation of the incident and reflected fields.

3.3/ TFSF IN 3D CASE

The real structure is 3D, so that we need to perform 3D simulation. The main task of this section is to determine the update of the EM equations at the TFSF boundaries that are composed now of 6 planes of a box separating the TF from the SF (see figure 3.4). These boundaries are defined by nodes whose coordinates are (i a , j a , k a , i b , j b and k b ).

A plane wave propagating along the z-axis and linearly polarized along the y-axis, where the two components of the incident field (E inc y and H inc x ) are non-zero, is considered. Note that any other polarization (linear, elliptical or circular) can be considered but only in the case of an incident plane wave.

An illumination by any beam (Gaussian for example) could be taken into account but requires a 3D calculation to determine the incident and reflected E and H components. By analogy with the previous case (1D), to correct the TFSF boundary equations, the updated equations of E and H must add or subtract the incident field. Notice that only the tangential components of E and H are involved in the correction on the TFSF boundary planes, so each cell on these planes contains only two components of the E and H field to be corrected. To do this, the following equations present the corrections on the six planes of the box that corresponds to the boundary between the TF and SF zones according to the notations in figure 3.4 and taking into account the interleaving between the spatial positions of E and H :

• For i ∈ [i a , i b ] and k ∈ [k a , k b ] (xz -plane)
, E z at j a and E z at j b can be written as :

E z (i, j a , k) = E z (i, j a , k) + ∆t ε j a (i, k) ∆ H inc x ( j a - 1 2 ) (3.14) E z (i, j b , k) = E z (i, j b , k) - ∆t ε j b (i, k) ∆ H inc x ( j b + 1 2 ) (3.15)
• For i ∈ [i a , i b ] and j ∈ [ j a , j b ] (xyplane ), E y at k a and E y at k b can be written as :

E y (i, j, k a ) = E y (i, j, k a ) - ∆t ε k a (i, j) ∆ H inc x (k a - 1 2 ) (3.16) E y (i, j, k b ) = E y (i, j, k b ) + ∆t ε k b (i, j) ∆ H inc x (k b + 1 2 ) (3.17) • For i ∈ [i a , i b ] and j ∈ [ j a , j b ] (xy -plane ), H x at k a -1 2 and H x at k b + 1 2
can be written as :

H x (i, j, k a - 1 2 ) = H x (i, j, k a - 1 2 ) - ∆t ∆ E inc y (k a ) (3.18) H x (i, j, k b + 1 2 ) = H x (i, j, k b + 1 2 ) + ∆t ∆ E inc y (k b ) (3.19) • For j ∈ [ j a , j b ] and k ∈ [k a , k b ] (yz -plane ), H y at i a -1 2 and H y at i b + 1 2
can be written as :

H y (i a - 1 2 , j, k) = H y (i a - 1 2 , j, k) + ∆t ∆ E inc y (i a ) (3.20) H y (i b + 1 2 , j, k) = H y (i b + 1 2 , j, k) - ∆t ∆ E inc y (i b ) (3.21)
Due to the implicated components in the two remaining planes (xz and yz) are always null, no correction of the EM field is necessary.

As in the case of (1D), we consider a semi-infinite planar substrate which divides the computational domain into two homogeneous media of different refractive indices. The xz and yz planes of the TFSF boundaries are perpendicular to the interface separating the two media and penetrate simultaneously in these two media while the two xy planes are each in a homogeneous medium. Therefore, the correction of the equations depends on the medium where the wave is injected and the medium from which the wave will be transmitted, that is to say that the correction on the planes (addition or subtraction of fields) should take into account the position of the cells. For the cells located in the medium of incidence, the incident and reflected fields will be taken into account while for the cells of the planes located in the medium of transmission, only the transmitted field is considered. For example, we consider that the medium of injection of the plane wave is the one where the xy plane at k = k b (the top plane according to figure 3.4b), then the correction of the equations on such a plane as well as the parts of the xz and yz planes penetrated in this medium is done by subtractions and additions of the incident "inc" and reflected "re f l" electric and magnetic fields; whereas the correction of the equations in the xy plane defined in the medium where the wave is transmitted as well as the other parts of two vertical planes (xz and yz) which are in this last medium is done by subtractions and additions of the transmitted "tr" electric and magnetic fields. The equations then become as follows:

• For i ∈ [i a , i b ] and k ∈ [k a , k b ] (xz -plane)
, E z at j a and E z at j b can be written as :

E z (i, j a , k) = E z (i, j a , k) + ∆t ε inc/tr j a (i, k) ∆ (H inc/tr x ( j a - 1 2 ) + H re f l x ( j a - 1 2 )) (3.22) E z (i, j b , k) = E z (i, j b , k) - ∆t ε inc/tr j b (i, k) ∆ (H inc/tr x ( j b + 1 2 ) + H re f l x ( j b + 1 2 )) (3.23) • For i ∈ [i a , i b
] and j ∈ [ j a , j b ] (xyplane), E y at k a and E y at k b can be written as :

E y (i, j, k a ) = E y (i, j, k a ) - ∆t ε tr k a (i, j) ∆ H tr x (k a - 1 2 ) (3.24) E y (i, j, k b ) = E y (i, j, k b ) + ∆t ε inc k b (i, j) ∆ (H inc x (k b + 1 2 ) + H re f l x (k b + 1 2 )) (3.25) • For i ∈ [i a , i b ] and j ∈ [ j a , j b ] (xy -plane ), H x at k a -1 2 and H x at k b + 1 2
can be written as :

H x (i, j, k a - 1 2 ) = H x (i, j, k a - 1 2 ) - ∆t ∆ E tr y (k a ) (3.26) H x (i, j, k b + 1 2 ) = H x (i, j, k b + 1 2 ) + ∆t ∆ (E inc y (k b ) + E re f l y (k b )) (3.27) • For j ∈ [ j a , j b ] and k ∈ [k a , k b ] (yz -plane ), H y at i a -1 2 and H y at i b + 1 2
can be written as :

H y (i a - 1 2 , j, k) = H y (i a - 1 2 , j, k) + ∆t ∆ (E inc/tr y (i a ) + E re f l y (i a )) (3.28) H y (i b + 1 2 , j, k) = H y (i b + 1 2 , j, k) - ∆t ∆ (E inc/tr y (i b ) + E re f l y (i b )) (3.29)
Otherwise, to calculate the effective cross sections, it is not necessary to know the EM field inside the particle. It is sufficient to calculate the flux of the Poynting vector through a closed surface that encompasses the particle.

For this, the fields are recorded on the planes of two additional boxes, one (surface S1) located in the TF zone for absorption and the other (surface S2) in the SF zone for scattering. Then, the calculation of the effective absorption and scattering cross section is deduced using these relations:

σ abs = P abs P in (3.30)
Where P abs = S 1 -→ Π abs • -→ n dS 1 , is the total power absorbed by the system (in W), and σ scat = P scat P in (3.31)

Where • P scat = S 2 - → Π scat • - → n dS 2 ,
is the total power scattered by the system (in W),

• P inc is the incident power received by the geometric section of the particle,

• -→ Π i , such as (i = "abs" or "scat"), is the Poynting vector defined by

- → Π = - → E ∧ - → H * 2 , with - → H * is the conjugation of - → H.
Therefore, the effective extinction cross section is easily accessible by summing the two contributions:

σ ext = σ scat + σ abs (3.32)
Thus, the calculation of effective cross sections is possible whatever the nature and geometric shape of the particle. However, the object must be described as well as possible, especially for objects with complex shapes. To In the follow, we will present various applications of this technique. First, we will show the optical behaviors of several NPs immersed in an homogeneous medium and having a different refractive indices and geometries (either spherical or not). Second, we will bring these NPs closer to different types of substrates (i.e. having different refractive indices) in order to study the influence of such media on the NP properties. These studies could allow us to understand the evolution of the NP's properties when the latter is coupled to a more complex structure than a substrate (i.e. coupled to a photonic crystal).

3.4/ APPLICATIONS

After the validation of our TFSF code in two simple passive configurations, we plan to study the effect of the presence of substrate, having different refractive indices, on the optical properties of NP (the spectral position and the intensity of the NP's resonance) when the latter is placed near the substrate.

Therefore, in this section we will numerically investigate the characteristics of dielectric and metallic resonant NPs with different geometries, either spherical or "butterfly" shaped (called Diabolo antenna (DA)). Two calculation methods are used to characterize spherical NPs which are immersed alone in a homogeneous medium (here it is air): the Mie theory and the TFSF technique. However, only the TFSF technique is considered in the study of the optical properties of DA (non-spherical NP) or of those which are spherical when each one is coupled to another structure to design a more complex system.

3.4.1/ STUDY OF SPHERICAL NPS

To understand the variation of the optical behavior of an NP when its environment is modified (such as the presence of a substrate or a photonic crystal in its vicinity), it was first necessary to study its optical behavior when it is alone and suspended in a homogeneous medium. Thus, two spherical resonant NPs were chosen for this study. One is considered as a silicon NP of radius R = 150 nm while the other is considered as a gold NP of radius R = 90 nm. The silicon NP presents several multipolar modes while the metallic one presents just one, probably plasmonic, mode.

As mentioned in chapter 2, the Mie theory allows us to calculate separately the contribution of each resonance mode that appears on the spectrum. We can thus identify the nature of different resonances of our NP. In case of a silicon NP, the figure 3.7a only present its three higher order of resonant modes: Using Mie's theory, we have reproduced all these spectra (those in dotted lines) which strongly match the spectra calculated using the TFSF technique.

Indeed, this comparison helps, one time again, to validate our FDTD codes which will be used in all the following simulations. Fortunately, we see a fairly good agreement between the two results (see the Appendix A for more information). wave at the same wavelength of the excited mode and polarized along the

x-axis and propagates along the z > 0 axis.

Referring to figures 3.7a, we see that the scattering efficiencies of the MQ and MD modes are higher than the ED one of the silicon NP. Thus, and as expected, we can see that the electric and magnetic field at the MQ and MD resonances are more exalted than those of the ED one. Notice that the peak of the QM mode is the narrowest, which means that the lifetime of the photons trapped in the NP at resonance is the longest compared to the other excited modes, and then its electric and magnetic field intensities are the highest. and air in (b)) for the two cases of incident field polarization. We find that these quantities (i.e. efficiencies) depend significantly on the direction of polarization of the incident field and the medium surrounding the DA. Regardless of the medium in which the DA is immersed, the transition from the "Off-resonance" to the "On-resonance" case leads to a red shift of the DA resonance wavelength. For example, in the case of the water as surrounding medium, the wavelength for the DA in "Off-resonance" (DA-off) case (at λ = 875 nm) shifts towards the red region when passing towards the DA in "On-resonance" (DA-on) case (at λ = 1458 nm). This shift of the resonance wavelength to longer wavelengths (in the "DA-on" case) is due to the fact that the volume of the mode, which results from the confinement of the electric field to the corners, is larger than that in the "DA-off" case. By the way, comparing these spectra according to the medium surrounding the DA, there is a redshift of the spectral position of each case resonance by a factor (named by "S f ") identical to the value of the ratio between the refractive indices of two media (S f = n water n air = 1.315). It can be seen that the transition from a surrounding medium of low index to another of higher index leads to a decrease in the extinction, absorption and scattering efficiencies of the DA.

We also present the distributions of the electric field intensity in figure 3.11(a) and magnetic field intensity in figure 3.11(b) at the resonance wavelength in the DA-on case when the DA is immersed in water. The electric field presents localized confinements at the corners due to the accumulation of charges at the corners (antenna effect). The magnetic field, on the other hand, is exalted at the center due to the oscillating current between the two metallic parts through the thin metallic junction at the center. These oscillations are directly related to the direction of the incident electric field (polarization) which induces them. In the case where the incident electric field is polarized perpendicular to the axis of the DA (figure 3.9(c)), the magnetic field is no longer exalted, only the electric field is exalted at the pointed extremities.

3.5/ NP-TO-SUBSTRATE COUPLING

As shown in the previous section, the optical response of the NP is highly dependent on the surrounding medium. This section have as objective to numerically study a coupling system by using an adapted TFSF code.

This study is able to better understand the influence of the presence of a planar substrate on the optical behavior of these various NPs individually [Atie et al., 2015a, Atie et al., 2015b]. In the following, we will consider the same configurations discussed in the previous section (spherical NPs and the gold DA) but this time when each NP is placed individually near a planar substrate. The below results were obtained using an adapted TFSF-FDTD code taken into account the presence of the substrate. The illumination consists of a plane wave polarized along the x-axis and incident from the NP side perpendicularly to the air or water-substrate interface (see schematic of the system in figure 3.1b). for the ED mode and by about 12 nm for the MD mode while the MQ mode is slightly shifted towards the blue region by about 2 nm. This coupling also has an effect on the value of the scattering efficiency of each mode. We can see that the scattering efficiency of the ED mode is increased by a factor of 2 and that of the MQ mode is decreased by a factor of 2, compared to the case of NPs alone without coupling. The MD scattering efficiency in this configuration has maintained its value but it has become larger than that of the case studied in figure 3.7a. A periodic modification of the σ s along the z-direction occurs due to the presence of the interference pattern between the incident field and the small one reflected by the NP itself. This leads to a small modulation of the effective scattering cross section over all the considered spectral range with a peculiar behavior in the near field where the coupling involves evanescent waves that are diffracted by the NP. Therefore, in a first approach, we can consider that the NP is in an interference pattern with fringes parallel to the interface and visibility close to unity. Notice that, at such a distance D z = 0 nm, the spectral position of each resonance mode is more shifted in the same direction comparing to the coupling case between a such NP and a InP substrate. So by comparing to the case of the self-suspended NP (see figures 3.7a), the scattering efficiency of each mode is increased differently, such that : the MQ mode is slightly increased by a factor of about 1.1 times, the MD mode is increased by a factor of about 1.5 times while the ED one is the most increased by a factor of about 5 times.

This increase of scattering efficiency of the NP resonance modes is accom-panied by an enhancement of the electric and magnetic field intensity (see figures 3.12(b-d)).

As expected, this increase is accompanied by a large shift in the spectral position of the ED mode where the redshift is up to 85 nm. In same way, the MD mode have a large redshift in its spectral position (up to 32 nm) as well as a blueshift (up to 10 nm) in the spectral position of the MQ mode.

In the same way as in figure 3.12, we presented in figures 3.12(b-d) the normalized electric and magnetic field intensity distributions of the NP which exhibits a MQ, ED and MD resonances, in the yz planes that passing by the center of the NP when the latter is in contact with PEC substrate.

3.5.3/ GOLD DA TO DIELECTRIC SUBSTRATES COUPLING

In this section, we considered the DA in the configuration of the DA-on immersed in water and placed close to a flat dielectric substrate. The whole system (DA+substrate) is illuminated from the water side by a plane wave that propagates perpendicularly to the water-substrate interface along z < 0 and linearly polarized along the y axis. In order to study the impact of the presence of a flat substrate on the optical response of the DA, we vertically moved the latter close to two dielectric substrates, one of which has a high refractive index (such as InP with n = 3.17) and the other has a low refractive index (such as glass with n = 1.45).

3.5.3.1/ GOLD DA TO INP SUBSTRATE COUPLING

We will first investigate the impact of the optical response of the DA when coupled to an InP substrate by moving the DA away from the substrate. water. This increase is accompanied by a large shift of the spectral position towards the long wavelength region (a maximum red shift reaches ≈ 72 nm)

when the DA is deposited on the substrate (D z = 0 nm), and another towards the short wavelength region (a maximum blue shift reaches ≈ 13 nm) when the DA is at D z ∈ [150; 360] nm above the substrate for a resonance wavelength of λ = 1445.62 nm. distribution of the electric and magnetic field intensities of the DA, in the xy plane when the DA placed at such a distance, can be seen respectively.

As expected, the magnetic field intensity (shown in Fig. 3.15d) is slightly exalted by a factor of ≈ 1.16 (from a maximum normalized intensity in the DA case alone of 12.8 a.u. to 15 a.u. in case of coupling) while the electric field intensity (shown in figure 3.15c) is about 1.08 (from a normalized intensity maximum in the DA case alone of in case of coupling). This probably explains the missing spectral shift at the coupling between a low index substrate and a resonant NP, even if the latter is placed at a short distance near the substrate.

3.6/ CONCLUSION

In conclusion, we have developed in this chapter a computational technique, named TFSF which is based on the FDTD method using homemade codes, in order to calculate the effective extinction, absorption and scattering cross sections of different types of NPs. The results obtained by this technique are validated by comparing them with the results calculated using the rigorous Mie theory method. This comparison was done exclusively on NPs with a spherical geometry in order to validate our TFSF code.

In addition, an extension on the TFSF method was applied for a purpose to be used to determine the optical behavior of various categories of NPs (either spherical or "butterfly" shape, either dielectric or metallic) by approaching the NP individually to flat interface substrate with different refractive index values inluding a perfectly electric conductor. This extension of calculation method solves the influence of another medium of different refractive index (i.g. a substrate plane or a photonic crystal) placed in the vicinity of the scatterer.

This type of coupling sheds light on the impact of the coupling with a non resonant structure such as a flat substrate shwoing that it can impact the optical response of a resonant NP. In this chapter, we present a numerical study to determine the angular radiation spectrum of light extracted from a PCC coupled to a dielectric NP. This coupling with such a NP, which exhibits either an electric dipole resonance (ED) or a magnetic dipole resonance (MD) at the same resonance wavelength of the PCC mode, could be responsible for transferring more energy outwards and also for controlling the direction of the light extracted from this system (PCC-NP) according to the NO position close to the cavity. Therefore, this chapter will be divided into three main sections. The first one consists in presenting each resonator individually, addressing their geometrical and optical properties. The second section deals with the effect of the presence of a substrate -made of the same material as the PC -on the optical behavior of the NP when the latter is placed close to the substrate. Finally, a third section is dedicated to discuss the results of such a near-field coupling between the PCC and an NP. Thanks to a detailed description of the properties of each element of the considered system in each of the subsections 4.2.1 and 4.2.2 as well as a study of the influence of the properties of the NP as a function 85 of the distance that separates it from a substrate made of the same material as the PCC (section 4.3), we were able to predict the significant positions of the NP along the three directions (x, y and z) in the case of its coupling with the PCC 4.2.

4.2/ STUDY OF THE PROPOSED CONFIGURATION

In the following, we will present the properties of the proposed system. As mentioned above, the later is composed of a spherical NP coupled to a PCC.

Thus, we will first study the optical properties of each one individually and, second, their coupling. Since the cavity mode of the considered PCC occurs for a well given value of the wavelength, we need to design the NP to have either an electric dipole (ED) or magnetic dipole (MD) resonance at the same wavelength in order to study the coupling phenomena between the resonances of the excited modes of the PCC and the NP. Let us mention that the calculations are performed with Mie's theory in the case of self-suspended spherical NPs, while the developed TFSF-FDTD algorithm in chapter 3 is used in the case of NPs in the vicinity of an InP substrate (which is the same material in which the PCC has been fabricated).

4.2.1/ PRESENTATION OF THE PHOTONIC CRYSTAL

The considered PCC was proposed since 2003 by [START_REF] Monat | Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane[END_REF] to exhibit a cavity mode in the NIR spectral range. It consists on a 2D hexagonal photonic crystal composed of air hole cylinders of radius r = 105 nm with a lattice constant a = 420 nm. The holes are engraved into an InP layer of 300 nm-thickness deposited on a silica (SiO 2 ) layer of 1 µm-thickness. These two materials are assumed to be non-dispersive in the considered spectral range and having the following refractive indices:

n InP = 3.17 FDTD codes include perfectly matched layers (PML) as absorbing boundary conditions. As the structure has a finite dimension, the computational window (including the PML) has a total size of 544 × 720 × 216 cells along the x, y and z axes, respectively. Note that we need to perform two separate calculations for each configuration, the first calculation allows us to determine the transmitted or reflected field spectra in the impulse regime (which allows us to determine the spectral position of the resonance modes), while the second allows us to determine the evolution of the electric and magnetic fields in the steady state (when the structure is resonating at its own resonance wavelength).

We can see that the size of the window is quite large, for this we used a powerful computer of the University of Franche-Comté (Mesocentre) to perform all the calculations. Although the numerical simulations were performed by FDTD codes which consider parallelizing the calculations, the calculation of a single spectrum takes at least 8 days. Moreover, each calculation takes a rather large memory space of about 33 GB.

The near-field spectrum calculated 15 nm above the PCC center is shown in figure 4.1b.

The cavity mode occurs at the wavelength λ mode 0 = 1602.92 nm with a quality factor of Q 0 = 4035 as experimentally measured in ref. [START_REF] Grosjean | Extraordinary blueshift of a photonic crystal nanocavity by reducing its mode volume with an opaque microtip[END_REF]. The fundamental mode of the PCC is excited by taking into account a point source (dipole) radiating over a broad spectral range and embedded in the PCC so that its direction and location correspond to the excitation of the fundamental mode of the cavity. The dipole source can simulate a quantum emitter coupled to the cavity [START_REF] Novotny | Principles of nanooptics[END_REF].

We limited our spectrum analysis to a tiny spectral range centered on the resonance wavelength of the cavity's fundamental mode since this coupling causes the excitation of various cavity modes. First, we can see that, at the cavity center, the E y -component is maximum while the E x -and E z -components are zero. This is the reason we chose to excite this mode numerically by placing a point source at the center of the cavity and fixed its polarization to be along the y-direction. Second, it is clear that the electric field components are spatially distributed differently along the major axis of the cavity i.e. a possible coupling to a NP should take into account the position of the latter as we will see later.

The magnetic field components (H x , H y and H z ) are also given in figure 4.2d-f, respectively. As expected and because the cavity mode is an in-plane mode, H z is predominant but its value vanishes at the cavity centre similarly to H y while H x is maximum. It may be mentioned here that, in the vertical direction (Oz), the cavity mode is centred on the InP layer and extends vertically through it. The far-field emission properties of the PCC will be detailed later in the context of its coupling with the NP.

4.2.2/ PRESENTATION OF THE NPS

In this section, we will numerically investigate the characteristics of dielectric resonant NPs with spherical geometries. In the context of this thesis work, we limit ourselves to spherical particles made of purely dielectric materials.

As mentioned in section 4.2, the considered spherical NP is designed so that it exhibits an electric dipole (ED) or a magnetic dipole (MD) resonance close to the wavelength resonance of the PCC mode. Thus, knowing the operation wavelength to be λ mode 0 = 1602.92 nm, two solutions can be envisaged : (i)

Fixing the radius R of the NP and changing its refractive index n p , or (ii) fixing the index and varying its radius. Nonetheless, if we want the coupling of the NP with the PCC to be local, we should consider the size of the NP smaller than the size of an electric or magnetic field lobe of the PCC mode (see figure 4.3a). This limits the choice to solution (i). From the distributions shown in

4.3/ NP-TO-INP SUBSTRATE COUPLING

In this section, we present the influence on the NP optical response when coupled to a dielectric substrate. We will consider the same optical and geometrical properties of the NP as those discussed in the previous section but in this time an InP substrate is placed in front of the dielectric NP individually.

The choice of an InP substrate come from the fact that it corresponds to the material composing the PCC to be coupled to the NPs. It is therefore necessary to study first the coupling with the substrate itself before taking into account the presence of the PCC, in order to determine the influence of the presence of such a substrate (without the cavity) on the optical behavior of the NP. The results were obtained using an TFSF-FDTD code taken into account the presence of the substrate. The illumination consists of a plane wave polarized along the x-axis and propagates from the substrate perpendicularly to the air-InP interface (see schematic of the system in figure 4.6(a)).

As mentioned in the section 4.2.1, we model all the structures studied in this chapter (either PC, substrate or NP) with a uniform mesh size of 15 × 15 × 15 nm 3 . Since the substrate is considered as a finite structure with a given dimension, we perform the following coupling studies using a FDTD code including PML with 10 cells thickness and surrounding the structure from all directions. To perform a study of one of the configurations, we need at least one computational window (including PML) with a dimension of 160×160×250 cells along the x, y and z axes, respectively. Each calculation requires 6.3 GB of memory space and approximately 8.33 hours of computation time. Note that each curve presented in the following sections (4.3.1 and 4.3.2) concerns an approach curve that requires the calculation of 54 spectra corresponding to 54 positions of the NP.

4.3.1/ CASE OF THE ED EXCITATION

In this paragraph, we consider the NP with R p = 220 nm having a refractive index of n p = 4.85 to be coupled with a InP substrate in order to study the effect of this coupling as according to the substrate-NP distance (D z ). In particular, we seek in this section to see the spectral position evolution of the excited ED mode of this NP when it is placed near an InP substrate.

Therefore, the figure 4.6(a) shows the 3D schematic of a spherical NP placed at D z above a flat InP substrate. Let notice that, at this distance, the intensity of the electric field is enhanced by a factor of ∼ 1.3 times compared to the case of the self-suspended NP (see figures 4.5(b,c)). This increase is accompanied by a large shift in the spectral position of the ED mode (up to 22 nm) when approaching the NP to the substrate, revealing the possibility of a tunable control of the PCC resonance wavelength when coupled to the same NP as it will be shown in the following.

4.3.2/ CASE OF THE MD EXCITATION

In this section, we will study the coupling effect between a substrate InP and the NP which excites the MD mode at such a wavelength by maintaining its geometry and changing its refractive index to n p = 3.48 (refractive index of silicon in the studied spectral range). This increase is accompanied by a large shift in the spectral position of the MD mode (up to 24 nm) but this time the shift is toward to the blue re-gion. This approach reveals the possibility of a tunable control of the PCC resonance wavelength to the blue region when coupled to a dielectric resonant NP contrarily to what was obtained in previous studies in our team using a metallic nano-antenna as shown in the refs. [START_REF] Grosjean | Extraordinary blueshift of a photonic crystal nanocavity by reducing its mode volume with an opaque microtip[END_REF], Baida et al., 2015].

4.3.3/ CONCLUSION

As expected, a slightly small quasi periodic modification of σ s along the zdirection occurs due to the presence of the interference pattern between the incident field and the small one reflected by the NP itself. This leads to a small modulation of the effective scattering cross section over all the considered spectral range with a peculiar behavior in the near field where the coupling involves evanescent waves that are diffracted by the NP. We can notice that the oscillations appearing in the two curves are not synchronous even far from the substrate. This can be attributed to a geometrical effect of the resonance contrary to that of a point dipole (i.e. the spherical NP has a volume contrary to the point dipole).

4.4/ STUDY OF THE PCC AND ED-EXCITED NP COUPLING AND

DISCUSSION

Let us describe the approach considered in order to achieve the main objective of this study that consists on demonstrating how the presence of NP can modify the pattern of light scattered by the cavity. Unfortunately, we have almost infinite possibilities for the spatial position of the NP with respect to the PCC.

Consequently, we have made the choice to start from the center of the PCC, where the electric field intensity is maximum, and to move the NP along three perpendicular axis parallel to the z, x and y-directions respectively.

Indeed, we begin the study by moving the NP along the z-direction due to the fact that the coupling efficiency is more sensitive to the NP-to-PCC distance.

In a second step, we will move the NP along the two lateral axes (the x and y axes) at a fixed distance above the PCC.

From a numerical point of view and as mentioned in the section 4.2.1, all the simulations of the studied system (PCC+NP) are performed by the same FDTD code that was used in case of the isolated PCC (see section 4.2.1).

Thus, we are dealing with a FDTD code has a computational window with a total dimension (inluding the PML) of 544 × 720 × 216 cells along the three perpendicular axes x, y and z respectively. The calculation window is meshed by uniform cells which each one has a volume of 15 × 15 × 15 nm 3 .

Similarly to the study of the isolated PCC, we also need to perform two separate calculations for each configuration. The first calculation allows us to determine the near-field intensity spectrum as well as the superstrate and substrate electrical intensity spectra (those propagating vertically straight ahead) normalized to those of the isolated PCC (without coupling) in the impulse regime, while the second one provides the spatial distributions of the electric and magnetic fields in the steady state (CW regime).

The considered system (PCC+NP) is illuminated by taking into account a point source (dipole) radiating over a broad spectral range and embedded in the PCC at 45 nm above the substrate-PC interface and x-polarized. Nevertheless, it is important to note that in the CW regime, we consider the same type of source with a fixed central wavelength (λ mode

0

) for all configurations.

Note that the total number of performed calculations is 2 × (28 + 28 + 25) = 162 calculations which requires nearly 8×162 = 1296 cumulative days of numerical simulations and generates over 80GB of data.

Then in the following, the coupling results between PCC and ED-excited NP (the NP made in material with n p = 4.85) will be discussed in two separate parts. The first one consists in determining the near field spectra, as well as the transmitted spectra in both z>0 (superstrate) and z<0 (substrate) directions of the whole system (PCC-NP) while the second consists in determining the whole radiation pattern (in 3D and 2D views) as a function of the NP position. As mentioned before, we begin the study by calculating the spectral behavior of the whole system resonance when the NP moves along the z-direction.

Therefore, we calculated the evolution of the normalized electric field intensities of the whole system (PCC+NP) compared to that of the isolated PCC for:

• Normalized near-field "NF " intensity defined by the electric intensity recorded at z d = 15 nm above the center of the PCC, given by the following formula :

NF = || - → E d (0, 0, z d , λ)|| 2 sys || - → E d (0, 0, z d , λ)|| 2 PCC (4.1)
• Normalized far-field intensity of the wave propagating straight to the superstrate region along the z > 0, given by the following formula :

I superstrate = || +∞ -∞ - → E super (x, y, z super , λ) dx.dy || 2 sys || +∞ -∞ - → E super (x, y, z super , λ) dx.dy || 2 PCC (4.2)
• Normalized far-field intensity of the wave propagating straight to the substrate region along the z < 0, given by the following formula :

I substrate = || +∞ -∞ - → E sub (x, y, z sub , λ) dx.dy || 2 sys || +∞ -∞ - → E sub (x, y, z sub , λ) dx.dy || 2 PCC (4.3)
Where :

• || - → E || is the modulus (norm) of the electric field vector - → E . • - → E d (0, 0, z d , λ
) is the electric field vector recorded by a point detector "d" placed at a distance "z d = 15 nm" above the center of the PCC (0,0,0) (red point in figure 4.8).

• -→ E super (0, 0, z super , λ) is the electric field vector recorded in an xy-plane at "z super " (dashed red line in figure 4.8).

• -→ E sub (0, 0, z sub , λ) is the electric field vector recorded in an xy-plane at "z sub " (dashed blue line in figure 4.8).

Note that "z sub " is fixed at 30 nm below the PC for all configurations, while "z super " varies according to the vertical position of the NP, such that z super = D z + 2 × R p + 30 nm. In the isolated PCC case, z super is equal to 30 nm. Therefore, we present in figures 4.9(a-c) the evolution of: (a) "NF ", (b) "I superstrate ", and (c) "I substrate " as a function of the vertical position (D z ) of the NP. For all cases of coupling between the PCC and the NP when the latter moves away from the PCC along the z axis, the NP is placed above the central point (0,0,0) that corresponds to a maximum electrical intensity of the isolated cavity (see figure 4.3a).

Figures 4.9(d-f) present the corresponding results when the NP moves along the x-direction parallel to the PCC major axis starting from its center while figures 4.9(g-i) correspond to a y-displacement of the NP along the perpendicular direction (PCC minor axis). In the last two cases (x and y displacements) cases, the NP vertical position was fixed to the D z = 45 nm.

Let's begin by interpreting the curves obtained when we move the NP vertically along z > 0 (figures 4.9(a-c)). As expected, a red shift of the resonance wavelength occurs for small distances accompanied by an enhance- This is the signature of the great dominance of the PCC resonance even if the coupling is critical; predictable through the great value of its quality factor compared to that of the ED resonance of the NP alone.

To go further, we have also studied the influence of the lateral position of the NP on the PCC resonance behavior. As can be seen in figure 4.3(a), a strong modulation of the intensities is obtained as a function of the position of the NP depending on whether it is above a maximum or minimum of the electric intensity of the cavity mode.

Moreover, we see on figures 4.9(e,f) that the minima of the transmitted and reflected intensities occur simultaneously and correspond to a total extinction for D x = 0.5 a = 210 nm and D x = 1.5 a = 630 nm. The same phenomenon also occurs when the NP moves along the y direction for D y = 0.5 a = 210 nm (see figures 4.9(h,i)), due to the presence of a single total extinction in this direction (see figure 4.3a). This means that the energy is no longer scattered in the direction perpendicular to the structure (along the z-axis) but probably in other directions which remain to be determined. In addition, the same figures show that the maxima of transmission and reflection also occur simultaneously meaning that the energy flow is probably mainly directed along the vertical direction.

From this study, we can simply conclude that the behavior of the scattered signal along the z-axis appears to be directly related to the distribution of the electric field generated by the PCC at resonance. When the NP is in contact with a maximum electric field, it radiates more energy along the two directions parallel to the z-axis. Nevertheless, to better understand this scattering phenomenon, it is important to determine the total angular radiation pattern and not only in one direction.

As we see in figure 4.9, it is almost certain that each configuration has its own resonance wavelength. Thus, due to experimental restrictions, assuming that we are going to use a monochromatic laser at the resonance wavelength of the isolated PCC, it will be very difficult to adjust the resonance wavelength for each configuration. Consequently, we limit our studies of determining the total angular radiation pattern to selecting a single wavelength (λ mode 0 ) for all configurations.

4.4.2/ EMISSION PROPERTIES AS A FUNCTION OF THE NP POSITION

In order to provide a clear and complete explanation on the modification of the emission by the NP-to-PCC coupling, we have calculated the whole angular radiation pattern of the light emitted by the isolated PCC as well as in the presence of the NP when it move :

First, along the z direction above the PCC center (section 4.4.2.1), second, along the major axis of the PCC (x-axis, section 4.4.2.2), and third, along the minor axis of the PCC (y-axis, section 4.4.2.3). The lateral displacement is considered in a plane located at D z = 45 nm above the PC. .11: 2D view of the calculated total radiation angular spectra in air (z > 0) and in the substrate side (z < 0) followed by a 3D presentation (bottom of the same column), respectively. (a) corresponds to the case of the isolated PCC, (b) shows the case the PCC-NP system when the NP is deposited on the PCC at its center (0,0,0), (c) shows the case the PCC-NP system when the NP is placed 45 nm above the center of the PCC (0,0,45), and (d) shows the case the PCC-NP system when the NP is placed 630 nm above the center of the PCC (0,0,630).

In figure 4.11a, we see that the angular spectra of the light emitted by the isolated PCC form, in the air, two main lobes with azimuth angles ϕ = 0 Moreover, in order to quantify the effect of the coupling on the total intensity extracted from the PCC, we define the enhancement factor τ as the total radiated intensity calculated in the presence of the NP divided by the same quantity for the isolated PCC. Out of curiosity, we calculated the far field intensity in some particular directions. -

→ Π = - → E ∧ - → H =                E y .H * z -E z .H y E z .H * x -E x .H z E x .H * y -E y .H x                (4.4)
On the other hand, when the NP is placed far from the PCC, the quadrupole shape disappears completely and we find the distribution of each element when isolated (see lower figures 4.15a and 4.15b for the PCC and figures 4.5c and f for the NP).

In order to complete this study and to better explain the effect due to the diffusion phenomena of the light extracted from the PCC through the NP, that could be seen as an "optical funneling effect", we present in figures 4.18 and 4.17 the distributions of the Poynting vectors in the xOz and yOz planes, respectively.

The optical funneling effect is based on the near-field coupling between two optical resonators whose Q-factors are strongly unbalanced. The PCC (the high Q-factor micro-resonator) acts as a micron-scale light accumulator for the NP (the low Q-factor nano-resonator) that acts as a nanoscale loss channel for the PCC [START_REF] Mivelle | Light funneling from a photonic crystal laser cavity to a nanoantenna: overcoming the diffraction limit in optical energy transfer down to the nanoscale[END_REF]. Therefore, it is clear that the efficiency of this funneling effect depends on the NP vertical position so that it reaches its maximum for the critical coupling, here at D z = 120 nm.

As presented for the field distributions, figures 4.17 and 4.18 show the spatial distribution of the Poynting vector in the two same planes xOz and yOz respectively.

Let mention that to better visualize these vector distributions, and because of the strong EM confinement in the very near-field zone, we have presented the Poynting vector in the far zones (white arrows) and its square root in the near zone (green arrows). that this "donut" shape will be reproduced in the air along the major axis of the cavity (x-axis) when the NP is placed above an electric intensity minimum. In figure 4.20, we also present the angular radiation pattern considering different configurations when the NP is placed also further from the PCC center. According to the results presented in figures 4.19 and 4.20, we are interested to quantify the effect of coupling on the total intensity extracted from the whole system by defining the enhancement factor τ. In this last configuration, we can observe that the electric field intensity has recovered approximately its value without the NP. This could be further argued that approaching the NP to the edge of the cavity leads to a strong perturbation of the condition imposed in the isolated PCC case (i.e., the displacement of two holes from the CP to the edge of the cavity), and thus this increases the emission of the light extracted by the PCC.

In addition, we can see that the intensity distributions sinusoidally oscillate between a maximum (when the NP is placed at D x = m × a = positions that correspond to an intensity maximum) and a minimum (when the NP is placed On the other hand, four lobes of electric intensity appear in side the NP. Their It is clear that the electric field intensity oscillates in the same way as the τ curve, where it is minimal for small distances (i.e., the second "D y = 0 nm" and third "D y = 120 nm" configurations from the bottom), and then it begins to recover as the distance (D y ) increases.

According to the top configuration which corresponds to a distance (D y = 630 nm), the NP is completely outside the PCC but the electric field intensity

has not yet recovered its value without the NP. Therefore, this confirms the gradually increasing oscillation of the curve presented in figure 4.26(a).

GENERAL CONCLUSION AND

PERSPECTIVES

In this thesis, we successfully modeled and numerically investigated a spectrally tunable coupling system between two optical nano-resonators. Our goal was to study the emission properties of light extracted from a photonic crystal nanocavity (PCC) by geometrically coupling it to a spherical dielectric nanoparticle (NP).

Such a nanocavity is generally excited using nano-emitters (such as quantum dots) integrated inside and radiating coherently at the wavelength of the nanocavity resonance mode. After a state of the art on the coupling between two nanoresonators and the way to optimize the light extraction, we presented the numerical tools necessary for this type of modeling, namely the FDTD method as well as the extension of a very specific technique to the FDTD which is the TFSF technique. The latter allows to subtract the incident field from the total (radiated + incident) field and that also to compute the effective scattering, absorption and extinction cross sections of the NP.

The tools that we have developed in this framework allow to study any coupling between nanoresonators, for example a QD with a Yagi-Uda type nanoantenna.

These numerical tools were used to determine the optical properties of each resonator. The next step was to consider the coupling between them in view of determinig the influence of the spatial position of one with respect to the other on the emission properties of the whole structure. The coupling study shows that a red shift of the PCC resonance wavelength (up to ≈ 2.5 nm) occurs when the NP is at a distance smaller than 300 nm from the PCC.

Moreover, this study shows that when the NP is at 120 nm, we get a critical coupling such that the extracted energy by emission is 3.4 larger than for the 133 isolated PCC. By studying the radiation behavior in the far field, we found that we can extract light almost in particular directions according to the position of the NP.

As a perspective, it would be interesting to study the coupling with other couple, or more, of optical nanoresonators in view of obtaining better results.

A possible approach would be to replace the current NP by another one which presents a magnetic dipole, an electric or magnetic quadrupole at the same resonance wavelength as the PCC. It should be noted that the computational time required to study a single configuration (8 days) was the main problem that led to limit the study to a 3-axis displacement for the NP and not in a volume. This could be solved by considering smaller cavities such as CL5, CL3 and why not a CL1. The second method used in this paper is the so-called "Total Field -Scat-177 its extension in the case of a dielectric/metallic nanoparticle coupled with different type of plane substrate 3.

Contrary to Mie's theory, the FDTD method does not allow to determine the nature of different resonances that appear on the spectrum. Nevertheless, calculating the spatial distribution of the electromagnetic field at the structure at the resonance wavelength gives important information about the nature of the resonance. For particles of any geometric shape, one can also calculate the effective cross sections in the same way as for spherical particles.

Document generated with L AT E X and: the L AT E X style for PhD Thesis created by S. Galland -http://www.multiagent.fr/ThesisStyle the tex-upmethodology package suite -http://www.arakhne.org/tex-upmethodology/ écol e doct oral e sci ences pour l 'i ngéni eur et mi crotechni ques

Université Bourgogne Franche-Comté 32, avenue de l'Observatoire 25000 Besançon, France Title: Study of the coupling between frequency-tuned optical nanoresonators for the control of their emission properties: modeling Keywords: Far-field emission control, nano-rasonator coupling, tuned spectral coupling, plane wave spectrum, radiation pattern, FDTD, TFSF, Mie theory.

Abstract:

The thesis focuses on the determination of the optical response of dielectric and/or metallic nanoparticles of different sizes immersed in a transparent homogeneous medium, and on the study of their coupling in two regimes: weak coupling with a planar substrate and strong coupling with a photonic crystal cavity (PCC) as a function of their spatial positions. To study the coupling, the properties of each element, i.e., the spherical NP and the CL7 PCC, were described and the optical properties of the entire system were determined.

The study showed that the light extracted from the PCC could be controlled by coupling it to an NP whose electrical dipole resonance matches the resonance wavelength of the PCC. Technical codes and simulations were developed for this study, and Mie theory was used to design the NP. Another study was conducted to show the behavior of light extracted from the same PCC when coupled to a dielectric NP with a magnetic dipole resonance at the same resonance wavelength as the PCC.

Titre : Etude du couplage entre nanorésonateurs optiques accordés en fréquence pour le contrôle de leurs propriétés d'émission: modélisation Mots-clés : Contrôle de l'émission en champ lointain, couplage nano-rasonateur, couplage spectral accordé, spectre d'ondes planes, diagramme de rayonnement, FDTD, TFSF, théorie de Mie.

Résumé :

La thèse porte sur la détermination de la réponse optique de nanoparticules diélectriques et/ou métalliques de différentes tailles immergées dans un milieu homogène transparent, et sur l'étude de leur couplage dans deux régimes : couplage faible avec un substrat plan et couplage fort avec une cavité à cristal photonique (PCC) en fonction de leurs positions spatiales.

CONTENTS1

  State of the Art 1.1 Introduction to the coupling effect . . . . . . . . . . . . . . . . . 1.2 Nano-emitter coupled to metallic structures . . . . . . . . . . . 1.2.1 Nano-emitter coupled to a metallic planar interface . . . 1.2.2 Nano-emitter coupled to a metallic nanostructure . . . . 1.3 Nano-emitter coupled to dielectric structures . . . . . . . . . . 1.3.1 Nano-emitter coupled to a dielectric planar interface . . 1.3.2 Nano-emitter coupled to a dielectric nanoparticle . . . . 1.3.3 Nano-emitter coupled to a photonic crystal . . . . . . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Calculation Methods 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Analytical solution of the EM problem of Mie . . . . . . . 2.2.2 Expressions of the extinction and scattering cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Numerical solutions of the EM problem of Mie . . . . . . 2.3 Finite Differences Time Domain (FDTD) . . . . . . . . . . . . . 2.3.1 Maxwell's Equations . . . . . . . . . . . . . . . . . . . . 2.3.2 Yee's algorithm . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Numerical constraints of the FDTD . . . . . . . . . . . . v

  GENERAL INTRODUCTION ON THE COUPLING EFFECT BE-TWEEN A NANO-EMITTER AND ITS SURROUNDING ENVIRON-

Figure 1

 1 figure 1.2e), as predicted by theoretical calculations.

Figure 1

 1 Figure 1.2: (a) Schematic of the steps for the preparation of CdTe QD multilayers by LBL assembly on gold film. (b) Configuration of the reverse Kretschmann sample for SPCE observation of CdTe QDs. (c) The SPCE to FS intensity ratio for CdTe QD multilayers. (d) Angular distribution of CdTe QD SPCE. (e) Spectra of emission of CdTe QDs observed at SPCE angle (red line) and at FS conditions (black line). The inset represents the spectra QDs SPCE with p-polarization (green line) and s-polarization (blue line).(from[START_REF] Liu | Directional surface plasmon-coupled emission of CdTe quantum dots and its application in Hg (II) sensing[END_REF]

Figure 1

 1 Figure 1.3: (a) Schematic of the steps for the preparation of Ag/PATP @ SiO2/CdTe composite nanoprobe. (b) Schematic of the steps for the construction of sandwich immune system using the probe to simultaneously measure dual mode fluorescence (SPCF) and Raman (SPCR) signals. (c) and (d) present SPCF and SPCR spectra at different detection angles for the dual-mode fluorescence-Raman probe captured by human IgG specifically bound to an antibody modified silver substrate. (e) and (f) represent the angular distribution diagram for SPCF and SPCR, respectively. Comparison of fluorescence (g) and Raman (h) spectra acquired separately. The sample is a dual-mode fluorescent and Raman probe captured by a human IgG specifically bound to an antibody-modified silver substrate. (from [Wang et al., 2022])

  , Curto & al. demonstrated a method for achieving one-way emission of light from a single emitter by connecting it to a Yagi-Uda antenna that has been fabricated at a nanoscale level. The structure of the fabricated five-element Yagi-Uda antenna consisting of a feed element, one reflector, and three directors. By placing a quantum dot close to the antenna, it is possible to use it to activate the antenna's resonant feed element (see figure 1.5). This results in the emission of polarized and highly focused light from the quantum dot in a specific direction, which can be controlled by adjusting the dimensions of the antenna. These findings demonstrate the potential of using optical antennas to facilitate energy communication among nano-emitters.

Figure 1

 1 Figure 1.4: (a) 3D illustration of the sample structure with a representation of the simulated directional radiation pattern from the NPA. (b) Cross-sectional schematic of the NPA. (c) TEM image of a NPA and QDs (scale bar = 50 nm). Simulated spatial distributions of (d) Purcell factor (spontaneous emission rate enhancement) and (e) quantum radiative efficiency for a vertically oriented QD dipole located in the space between the gold film and the Ag nanocube. (f) Enhanced fluorescence intensity of QDs as a function of average incident laser power for: QDs on a glass slide, QDs on Au film and coupled to individual NPAs. (g) Histogram shows the fluorescence enhancement factor distribution of the 11 measured NPAs. (from[START_REF] Hoang | Ultrafast spontaneous emission source using plasmonic nanoantennas[END_REF]

Figure 1

 1 Figure 1.5: (a) SEM image of the fabricated Yagi-Uda antenna, where a QD is attached to one end of the feeder inside the red square. (b) Comparison of SEM and confocal scanning luminescence (CSL) microscopy images of three QD-driven antennas. (c)Luminescence intensity versus time in both polarizations for one of the antennas shown in (b), showing the blinking of a single QD. (d) CSL images of QDs on 60 nm gold reference squares (left), half-wave (λ/2) dipole antennas (middle), and YU145 antennas (right), respectively. Different colors indicate variations in the degree of linear polarization (scale bar = 2 mm). (e) Radiation pattern ( distribution of intensity in the back focal plane of the objective) of an individual structure shown in (d), indicating the critical angle (θ C ) and the numerical aperture angle (θ NA ). For YU antenna, a high-pass filter of 830 nm was used. (f) Angular radiation diagram for the YU antenna (black) with the theoretical prediction (red). (from [Curto et al., 2010])

Figure 1 . 6 :

 16 Figure 1.6: From left to right :(a) Schema of the NWQD on the substrate; Schematic diagram of the energy bands for the QD integrated into a nanowire; The distribution of the electric field (V/m); Radiation pattern in far-field (V/m) of a single NWQD with a diameter d = 10 nm deposited on the substrate. (b) Schematic of the configuration AS; The distribution of the electric field (V/m) of the configuration AS; Radiation pattern in far-field (V/m) of the configuration AS with h Au = 560 nm. (c) Schematic of the configuration SS; The distribution of the electric field (V/m) of the configuration SS; Radiation pattern in far-field (V/m) of the SS with h Au = 560 nm.The dipole moment of the QD is oriented along the axis of the bowtie antenna; the value of the gap equal 15 nm. The origin of the coordinate system is at the center of the QD. (from[START_REF] Li | Directional radiation enhancement of nanowire quantum dots based on line-array plasmonic antenna coupling[END_REF]

CHAPTER 1 .Figure 1

 11 Figure 1.7: (a) Schematic of the structure geometry. (b) and (d) present the coefficients of the field transmission as a function of θ. (d) Transmission coefficients as a function of d above when θ ex =0.687 is set fixed. (e) show the radiative decay rates as a function of d above .(from[START_REF] Pomozzi | Ensemble measurement of the orientation-dependent variations in chromophore lifetimes near a dielectric interface[END_REF]

  of a dielectric microsphere to control the emission directionality of a single fluorescent molecule. They found that the interaction between a single fluorescent molecule and the specific electromagnetic distribution by the dielectric microsphere allows to direct the light emitted at high incidences toward the optical axis. The microsphere allows for a collection efficiency of up to 60% while also increasing the excitation intensity received by the molecule by a factor of 2.2. Combining these two effects can boost the amount of collected fluorescence photons by a factor of up to 5. In figure 1.9a, the experimental setup for confocal single molecule detection enhanced by a single microsphere is schematically shown. In figure 1.9b shows the electric field intensity distribution near a dielectric microsphere (diameter 2 µm, refractive index 1.59) illuminated with a tightly focused Gaussian beam at λ=633 nm with 1.2 NA (logarithmic scale). The refractive indices of the outer medium and glass slide are 1.33 and 1.5, respectively. The angular distribution of the fluorescence intensity of a molecule located at 100 nm above a 2 µm polystyrene microsphere in water (red curves) is shown for a dipole oriented along the Z axis (see figure 1.9c) and averaged over all orientations (see figure 1.9d). The emission of the dipole without the microsphere is presented by non-filled curves.

Figure 1

 1 Figure 1.9: (a) Schematic of the experimental setup. (b) Calculated distribution of the electric field intensity when the QD is placed near a dielectric microsphere. (c) and (d)show the angular radiation patterns of the fluorescence intensity of a molecule located at 100 nm above a 2µm polystyrene microsphere in water (red curve) for a dipole pointed along the z axis and averaged over all orientations, respectively. The non-filled curves correspond to the dipole emission without the microsphere. (from[START_REF] Gérard | Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere[END_REF]

Figure 1 .

 1 Figure 1.11: Modal decomposition of θ-dependent radiation patterns for a dipole localized on the symmetry axis of a nanodisk in case without (a to d) and with (e to h) a substrate. The green (black) solid curve represents the far-field modal energy flux resolved in angle s 1 (s 2 ) corresponding to the low-Q (high-Q) mode, respectively. The red dashed curve corresponds to the total energy flux s tot . The wavelengths of the dipole emission correspond to the different Γ markers highlighted in figure 1.10 which are reproduced in the lower right corner of each radiation diagram. (from [Colom et al., 2022])

Figure 1

 1 Figure 1.12: (a) 3D woodpile PCs. A schematic of the structure (on top). A schematic of the InGaAsP multiple quantum wells (MQWs) used as the emitters (on bottom left).A SEM image of the structure (on bottom right).[START_REF] Ogawa | Control of light emission by 3D photonic crystals[END_REF]. (b) 3D inverse opal PCs. A SEM image of a surface of the structure.[START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF]. (c) 2D-PCC. Electric field intensity distribution in the fundamental mode of the cavity, calculated by using the FDTD method (at left) and (at right) a SEM image of the structure. ( from[START_REF] Englund | Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal[END_REF]).

Figure 1

 1 Figure 1.13: (a) SEM image of an L 33 PhC cavity. The red and green small triangles correspond to examples of multiple and single site-controlled QD configurations, respectively. (b) computed 2D FDM mode intensity patterns of an L 33 PhC cavity for different cases : the lowest energies (M 0 , M 1 , M 2 ), and the higher one (M 9 ). The geometrical parameters are : a=225 nm, r= 61 nm, refractive index of membrane is n 2D =3.13. (c to e) The structure of a L 33 PhC cavity modes excited by 16 integrated QDs. (c) Spectrum of the photoluminescence (PL). (d) Optical field with spectral resolution (excitation power: P=500 µW, T=10 K). (e) 2D FDM calculated modes M 1 in an unperturbed structure (green curves) compared to the measured envelopes of the observed modes in (d) (blue curves).(These modes have been integrated along the y-direction). (from[START_REF] Rigal | Probing disorder and mode localization in photonic crystal cavities using site-controlled quantum dots[END_REF]

AFigure 1

 1 Figure 1.14: A: Schematics of the fabrication steps necessary to deterministically integrate a single Ga(AsN)/Ga(AsN):H QD into a PhC cavity (elements shown in the schematics are not to scale). (a) SEM image of an circular HSQ masks array (The inset shows a zoomed view of a single HSQ mask). (b-d) SEM images of a typical fabricated L 3 PhC cavity. (e) Spectrum of micro-photoluminescence of a QD-PhC device. (from[START_REF] Felici | Broadband enhancement of light-matter interaction in photonic crystal cavities integrating site-controlled quantum dots[END_REF]

Figure 1 .

 1 Figure 1.15: (a) Schema of the BNA etched at the apex of the SNOM tip of radius R. BNA enhancement factor measured at 15 nm in front of its apex and given as the ratio of the intensity with the BNA to the same calculated intensity without metal. The inset of (b) illustrates the distribution of intensity (logarithmic scale) in the yOz longitudinal plane to highlight the confinement of light that occurs in the gap area. The proposed CL7 cavity is shown in (c) with vertical and horizontal sections along the structure. Calculated near-field spectrum at the center of the cavity is shown in (d), where fundamental mode resonance occurs at λ R 0 = 1602.94 nm with a Q-factor of 1590. Calculated near-field spectra at 15 nm above the PC for two different BNA-on-tip polarizations: in (d) the BNA-on resonance and in (f) the BNA-off resonance. (from[START_REF] Baida | Double-way spectral tunability for the control of optical nanocavity resonance[END_REF]

Figure 2

 2 Figure 2.2: Scattering cross-section spectrum, calculated by Mie's method using a Matlab code, of a diamond NP of radius (R NP = 160 nm) (a) and a gold NP of radius (R NP = 90 nm) (b) as a function of wavelength, by specifying the different types of multipolar resonances (ED: Electric Dipole, MD: Magnetic Dipole, EQ: Electric Quadripole, MQ: Magnetic Quadripole and MO: Magnetic Octopole). (c) and (d) represent the scattering cross-section spectra of diamond and gold NPs, respectively, as a function of their radius.

Figure 2

 2 Figure 2.3: (a) Spatial discretization of the 3D computational domain and (b) the Yee's cell of coordinate (i,j,k) with the distribution of the E and H field components in this cell.

Figure 2 . 4 :

 24 Figure 2.4: Principle of temporal discretization in finite differences called "Leap Frog" schema.

Figure 2 . 5 :

 25 Figure 2.5: Spatial distributions of the fields in a Yee cell for calculation of (a) the E n+ 1 2 x

Figure 2

 2 Figure 2.7: Comparison between experimentally calculated values of gold dielectric constants and those using the Drude and DCP models as a function of wavelength. (a) represents the real values (ε ′ ) and (b) the imaginary values(ε ′′ ) of gold dielectric constants.

Figure 2

 2 Figure 2.8: (a) Projection of the tangential wave vector ( ⃗ k P ) of the incident plane wave field onto the xy plane. (b) The 3D views of the intensity of electric field distribution on the P-plane. Green circles show the position of the photonic crystal air holes.

  Several methods are used to calculate the scattered field of non-spherical particles, such as the T-matrix method and the Discrete Dipole Approximation (DDA) method, as well as numerical methods such as the FDTD method

Figure 3

 3 Figure 3.1: (a) The NP self-suspended in air and surrounded by the TF box (red box) and the SF box (blue box).(b) The NP is suspended in medium 1 at a distance D z above an substrate (medium 2) and surrounded by the TF and SF boxes, which are considered to penetrate into the substrate. The three green arrows represent the incident plane wave that illuminates our structure.

Figure 3

 3 Figure 3.3: A scheme representing the scatterer located in the medium 1 at distance D x from the second medium in the TF region, as well as the spatial distribution of the electric and magnetic fields into the both media (1D case).

Figure 3

 3 Figure 3.4: 3D-schematic of TF and SF planes with the TFSF boundary nodes (i a , j a , k a , i b , j b and k b ) that used in the correction. (a) represents a scatterer immersed alone in a homogeneous medium and surrounded by the TFSF boundary box. (b) A substrate is placed near the scatterer in which the TF and SF planes are considered to pass through it.

  Figures 3.5a and 3.5b illustrate respectively the incident wave at the beginning of its propagation in the TF zone and at the moment when it totally illuminates the NP before the scattered light reaches the SF zone. Figure3.5c shows that the light scattered by the NP appears on the outer side of the

Figure 3 . 5 :

 35 Figure 3.5: Temporal evolution of the electric field intensity scattered by a silicon NP immersed in air. The dotted green lines correspond to the TFSF boundary.

  the magnetic dipolar mode (MD) achieved at λ = 1086.29 nm, the electric dipolar mode (ED) achieved at λ = 847.58 nm and the magnetic quadrupolar mode (MQ) achieved at λ = 752.42 nm. The other NP which is metallic has only one mode which is the electric dipole mode (ED) achieved at wavelength λ = 589.25 nm (see figure 3.7b). These resonance wavelength values are obtained by calculating the extinction spectrum of each NP by using TFSF technique. As expected, the dielectric NP is non-absorbing unlike the metallic NP. Thus, the imaginary part of the refractive index is neglected in the calculations, and thus the effective absorption cross section is zero. Therefore, only the scattering spectrum is presented in figure 3.7a while the three spectra (scattering, absorption and extinction) corresponding to the gold NP are presented in figure 3.7b. Notice that a critical Drude model (DCP) is used to describe the dispersive properties of the metal.

Figure 3 .

 3 Figure 3.8 shows the spatial distribution of the electric and magnetic field intensity recorded in the plane yz passing through the center of NP. Figure 3.8a (the first column on the left)corresponding to the distribution of the electric and magnetic intensity of the gold NP that excited at the wavelength of ED mode (λ = 589.25 nm), respectively.

Figures 3

 3 Figures 3.8(b-d) corresponding to the distribution of the electric and magnetic intensity of the silicon NP when it is excited at the wavelength of : the MQ mode (in the second column (b)), the ED mode (in the third column (c)) and the MD mode (in the last column on the right (d)), respectively. Such a study is performed in the continuous wavelength (CW) regime, by injecting a plane

Figure 3 . 8 :

 38 Figure 3.8: Spatial distributions of the electric and magnetic field intensities recorded in the yz plane at several excitation modes corresponding to each NP. NPs are immersed in air and illuminated by a plane wave linearly polarized along the x-axis and propagating along z >0 axis. Yellow dotted lines define the TFSF boundary while the white solid line defines the edge of the silicon NP.

  3.4.2/ STUDY OF THE DIABOLO ANTENNAIn this paragraph we consider a Diabolo Antenna (DA) which is supposed to be made of gold. The DA is described in the FDTD algorithm with a uniform mesh in the three directions (∆x = ∆y = ∆z = 5 nm). The geometrical parameters of the DA (given in the legend of figure3.9) are chosen to have a resonance around the wavelength λ = 1458 nm[Hameed, 2016]. Notice that a critical Drude model (DCP) is used to describe the dispersive properties of the metal.

Figure 3 .

 3 Figure 3.10 shows the spectra of the extinction, absorption, and scattering efficiencies of the DA as a function of the surrounding medium (water in (a)

Figure 3

 3 Figure 3.9: (a) 3D representation of the DA. It is characterized by a side length D = 135 nm, a side junction G = 25 nm and a thickness e = 20 nm. The DA is immersed in two different media: in water with refractive index n m = 1.315 or in air with refractive index n m = 1. It is illuminated at normal incidence (along z-axis) by a linearly polarized plane wave (along x-or y-axis). Two distinct configurations of the DA are considered: (b) "On resonance" when the incident field is polarized along the axis of the DA and (c) "Off resonance" when the incident field is polarized perpendicular to the axis of the DA.

Figure 3 .

 3 Figure 3.10: Scattering (green lines), absorption (red lines) and extinction (black lines) cross section spectra corresponding to a gold Diabolo antenna (DA) immersed in water (a) (n m = 1, 315) and in air (b) (n m = 1). Two different configurations of DA are considered: the solid lines correspond to a DA in case of on-resonance while the dotted lines correspond to a DA in case of off-resonance.

  Figure 3.11: (a) and (b) represent the spatial distribution of the intensity of the electric and magnetic fields respectively in the (xy) plane at the resonance wavelength of the gold DA immersed in water. Green dashed lines determine the shape of the DA in xy-cut.

  3.5.1/ SILICON NP TO INP SUBSTRATE COUPLINGIn this paragraph, we consider a silicon NP with radius R = 150 nm and refractive index n p = 3.48 coupled to an InP substrate in order to study the effect of this coupling on the NP optical behavior as a function of the substrate-NP distance. Figure 3.12a shows the variations of the scattering cross section (σ s ) of the silicon NP as a function of D z for a spectral range including the highest electric (ED) and magnetic (MD) dipole modes as well as the highest magnetic quadrupole (MQ) mode. The scattering efficiency spectra oscillate in such a way that two maxima of a specific mode are separated by a distance equal to λ mode 2 . This distance is defined as the result of the constructive and destructive interference induced by the substrate. The scattering efficiency spectrum (σ s ) corresponding to the NP when deposited on the InP substrate (D z = 0 nm) is shown at the bottom of figure 3.12a. Comparing with figure 3.7a and at this substrate-NP distance, the spectral position of the electric and magnetic dipole modes is shifted towards the red region by about 15 nm

Figure 3 .

 3 Figure 3.12: (a) Variations of scattering efficiency (σ s ) spectra of a silicon NP (with R = 150 nm) coupled with an InP substrate (n = 3.17) as a function of the NP-substrate distance D z . A scattering efficiency spectrum of the NP, when it is placed at 0 nm above the substrate, is shown in an insert located at the bottom in column (a). (b-d) represent the electric and magnetic field intensity distributions recorded in the yz plane for the three excitation modes of the NP at D z = 0 nm. The incident plane wave propagates from the NP side to the air-substrate interface. The dotted yellow frame represents the TFSF boundary.

Figures 3 .

 3  present the normalized electric and magnetic field intensity distributions of the three excitation modes recorded in the yz vertical plane that passes through the center of the NP, when the latter is at D z = 0 nm. The illumination wavelength considered in each case corresponds to the one that excites each mode in the case of the NP without coupling. Such a consideration aims at understanding the effect of coupling on the optical behavior of the NP. Notice that at this distance, the electric and magnetic field intensities of the ED mode are enhanced respectively by a factor of ∼ 3.85 and ∼ 2.42 times

Figure 3 .

 3 Figure 3.13a shows the variations of the scattering cross section (σ s ) of the NP coupled to a PEC substrate as a function of D z for a spectral range including the three highest orders of the electric and magnetic multipole modes. As can be seen in Figure 3.13a, when the position of the NP corresponds to a dark fringe -for example at D z = 300 nm for λ ≈ 905.64 nm -it is then almost not electrically illuminated and thus the ED disappears completely and only the MD and MQ remain possible to excite (see figure 3.13a).

  figure 3.13a where we have superimposed the fringe system (green and blue lines) on the response of the NP (scattering efficiency).The previous interpretation does not take into account the coupling of the

Figure 3 .

 3 Figure 3.13: (a) Variations of scattering efficiency (σ s ) spectra of a silicon NP (with R = 150 nm) coupled with a PEC substrate as a function of the NP-substrate distance D z . A scattering efficiency spectrum of the NP, when it is placed at 0 nm above the substrate, is shown in an insert located at the bottom in column (a). (b-d) represent the electric and magnetic field intensity distributions recorded in the yz plane for the three excitation modes of the NP at D z = 0 nm. The incident plane wave propagates from the NP side to the air-substrate interface. The dotted yellow frame represents the TFSF boundary.

Figure 3 .Figure 3

 33 Figure 3.14a shows the evolution of the DA efficiency as a function of the increase of the DA-substrate distance to reach maximum efficiencies when the DA is at a distance D z = 250 nm. At such a distance, the extinction, absorption and diffusion efficiencies of DA reach the following values respectively: 74.38, 49.49 and 25.09. These values demonstrate an increase of a factor of ≈ 3.6 times greater than those obtained when the DA is suspended alone in

Figures 3 .

 3 Figures 3.14b and 3.14d respectively show, in the polarization plane (yz plane) and in the xy plane, the magnetic field intensity distributions of the DA placed at a distance D z = 250 nm.Figure 3.14(c) shows the electric field

  Figure 3.15: (a) Normalized effective extinction, absorption and scattering cross section spectra as well as their wavelengths, corresponding to a gold DA immersed in water and coupled with a glass substrate (refractive index n = 1.45 nm), as a function of the DA-Substrate distance D z , for which the incident wave is from the water side. (b,d) are the magnetic field intensity distributions recorded in the yz and xy planes. (c) show the distribution of the electric field intensity recorded in xy plane at a wavelength (λ DA = 1458 nm) for D z = 270 nm. The green dotted frame determines the DA shape. A solid green line defines the water-substrate interface.

  and n S iO 2 = 1.45. The optical cavity is composed of seven missed and aligned air holes (then called CL7) as shown in figure 4.1a. Note that the two air holes at the extremities of the cavity are offset 75 nm towards the outside of the cavity in order to enhance the quality factor of the cavity mode resonance keeping it inside the photonic bandgap of the photonic crystal dispersion diagram.The modeling of the structures (PC or NP) was done with a uniform mesh of 15 × 15 × 15 nm 3 . All numerical simulations were performed using FDTD codes. In order to accurately describe the small features of the structure, especially those with curved shapes, we used an averaging technique to calculate the average value between two existing dielectric constants inside a single cell. This technique is used to almost fold the staircase effect induced by spatial discretization by dividing the "hybrid" cell (the one containing two different dielectric constants) about forty cells and then averaging them[Al-Aridhee, 2016].

  Figure 4.1: Schematic of the considered PCC in (a). The inset shows the structure in xy plane. Near-field intensity spectrum calculated 15nm in air above the center of the PCC in (b).

Figures 4 .

 4 Figures 4.2a-c present the amplitude distributions of the 3 components of the electric field E x , E y and E z respectively.

Figure 4

 4 Figure 4.3(a,b) present the total intensity distribution of the electric and magnetic field recorded in the xy-plane that is placed at 15 nm above the PCC in air, respectively. In order to clearly present the intensity distribution of

Figure 4

 4 Figure 4.2: (a) to (c) are the amplitude distributions of the three components of the electric field recorded at 15nm above the PCC in air in xy plane, while (d) to (f) are the amplitude distributions of the three components of the magnetic field recorded at 15nm above the PCC in air in the same plane. The green circles show the position of the air holes of the photonic crystal in the xy plane.

Figure 4

 4 Figure 4.3: (a,b) present the total-intensity distributions of the electric and magnetic field recorded in xy plane at 15nm above the PCC in air, respectively. (c,d) present the intensity distributions of the electric field (logarithmic scale) at resonance in vertical planes (yz and xz) passing by the PCC center, respectively. (e,f) present the intensity distributions of the magnetic field (logarithmic scale) at resonance in vertical planes (yz and xz) passing by the PCC center, respectively. The green contours show the position of the air holes of the photonic crystal and its edges in the 3 orthogonal planes.

Figure 4 .

 4 figures 4.2a-c and 4.3a, we estimate the size of this lobe to be about 440 nm (see the scale bar in figure 4.2a). Therefore, we will vary its index so that it resonates at the wavelength of the PCC (λ mode 0

Figure 4 . 5 :

 45 Figure 4.5: Spatial distributions of the electric and magnetic field intensities (logarithmic scale) recorded in the xy, xz and yz planes at the ED and MD excitation of two spherical NP. The parameters of the NPs are: n p = 4.85 in (a-f), n p = 3.48 in (g-l) and R p = 220 nm in all cases. NPs are immersed in air and illuminated by a plane wave linearly polarized along the x-axis and propagating along z>0 axis. Black arrows present the electric and magnetic field vectors in each planes and green dotted circle defines the edges of NP.

Figures 4

 4 Figures 4.5(a-c) show the electric intensity distributions of the excited NP at the ED wavelength, while figures 4.5(d-f) show the magnetic intensity distributions of the same excited NP. In the figures (4.5(d,e)) one can see a double intensity lobe where the magnetic field migrates from one to the other, as well as in the figure 4.5f where the magnetic field rotates around the NP center inside the NP, which validates the existence of an ED perpendicular to the yz plane (ED along the x axis) [Kruk et al., 2017]. The spatial distributions of the electric and magnetic field intensities of an NP excited at the MD wavelength are also shown in figures 4.5(g-i) and figures 4.5(j-l), respectively. By analogy with figures 4.5(d-f), figure 4.5h shows that the electric field rotates around the center of the NP in the xz plane, which therefore validates the existence of a MD perpendicular to this plane (MD along the y axis). Moreover, distributions of the electric intensity in the form of tow-lobes appear in the other orthogonal planes in figures 4.5(g,i).

  Figure 4.6(b) shows the variations of the values of the normalized effective scattering cross section of the NP (σ s , which is represented by the dashed red line) and its corresponding spectral position (λ ED , which is represented by the solid blue line) as functions of D z . We can see that the curves show the appearance of oscillations due to interference between the incident field and that reflected by the NP.

Figures 4

 4 Figures4.6(c,d) present the normalized electric field intensity distributions in the two vertical planes (xz and yz) that cross the center of the NP, when the latter is at D z = 45 nm above the substrate for a ED excitation (λ = 1624nm).

Figure 4

 4 Figure 4.6: (a) Schematic of the considered configuration. (b) Variations of the ED spectral position (λ ED ) and the corresponding scattering efficiency (σ s ) of a NP (refractive index n = 4.85 nm) coupled with an InP substrate (n = 3.17), as a function of the NPsubstrate distance D z , for which the incident wave is from the substrate side. (c,d) are the electric field intensity distributions recorded in the xz and yz plane at the ED excitation for D z = 45 nm. The dotted yellow rectangle represents the TFSF boundary.

Figure 4

 4 Figure 4.7(a) shows the variations of the values of the normalized effective scattering cross section of the NP (σ s , which is represented by the dashed red line) and its corresponding spectral position (λ MD , which is represented by the solid blue line) as functions of D z . In figure 4.7(a), we can see that the maximum of the normalized effective scattering cross section (σ s ) of the MD excitation is achieved when the NP is placed at 225 nm above the substrate. One also notes the same phenomenon of appearance of oscillations of the curves due to the interference between the incident field and that reflected by the NP. In the same way, we present in figures 4.7(c,d) the normalized electric field intensity distributions of the NP at its MD resonance, in the two vertical planes (xz and yz) passing by the NP center for D z = 225 nm where the maximum of scattering efficiency (σ s ) is reached. Notice that, at such a distance, the scattering efficiency (σ s ) is slightly decreased compared to the case of the self-suspended NP (see figures 4.4(c)). Despite this decrease of scattering efficiency (σ s ), the intensity of the electric field is enhanced compared to the case of self suspended NP (see figures 4.5(h,i)).

Figure 4

 4 Figure 4.7: (a) Variations of the MD spectral position (λ MD ) and the corresponding scattering efficiency (σ s ) of a silicon NP (refractive index n = 3.48 nm) coupled with an InP substrate (n = 3.17), as a function of the NP-substrate distance D z , for which the incident wave is from the substrate side. (b,c) represent the electric field intensity distributions recorded in the xz and yz plane at the ED excitation for D z = 225 nm. The dotted yellow frame represents the TFSF boundary.

4. 4 .

 4 STUDY OF THE PCC AND ED-EXCITED NP COUPLING AND DISCUSSION 101 4.4.1/ SPECTRAL STUDY OF THE COUPLING

Figure 4 . 8 :

 48 Figure 4.8: Schematic illustrating the principle of detection of the system propagating wave.

Figure 4 . 9 :

 49 Figure 4.9: Normalized near-field, superstrate (transmission) and substrate (reflection) far-field curves of the PCC coupled to the NP (n p = 4.85, R p = 220 nm) as a function of the NP's positions along the (x,y,z) axes. The dashed green line shows the resonance wavelength of the fundamental mode of the unperturbed PCC, indicated by λ mode 0 .

4. 4 .

 4 STUDY OF THE PCC AND ED-EXCITED NP COUPLING AND DISCUSSION 105

Figure 4

 4 Figure 4.10: Schematic of the PC presented in the xOz (a) and xOy (b) planes, as well as Euler angles (θ,ϕ) and the wavelength vector -→ k (solid red arrow) and its projected -→ k p (dashed red arrow) on the xOy plane.

  • and 180 • along the main axis of the PCC (x-axis) and directed outward from the vertical axis (z-axis) along almost θ = 31 • , while three lobes are formed in the substrate side where two of these lobes pointing outward from the vertical axis along almost θ = 42.7 • and the third is directed vertically (θ = 0 • ). We can see that the lobes extend symmetrically in ϕ between -30 • and +30 • in the superstrate region (z > 0). The obtained angular spectra in figures 4.11(b,c) clearly demonstrate that the scattered light by the PCC-NP system is mainly directed in the vertical axis in superstrate (air) and substrate regions. While in figure 4.11d (at D z = 630 nm) the scattered light, in the air, shows three main lobes where two of them are 4.4. STUDY OF THE PCC AND ED-EXCITED NP COUPLING AND DISCUSSION 107 directed outward from the vertical axis (z-axis) along almost θ = 34 • , and the third one is directed vertically (θ = 0 • ). From the substrate side, two lobes are formed along x-axis and pointing outward from the vertical axis along almost θ = 44 • .

  Figure 4.12(a) gives the values of τ for different NP positions along the vertical axis. As the figure4.12(a) illustrated, the maximum of the extracted intensity is obtained for D z = 120 nm. This configuration will be discussed later.

Figure 4

 4 Figure 4.12: (a) Normalized total intensity (τ) emitted by the system PCC-NP as a function of the vertical position of the NP. (b) Emitted intensity by the system recorded in three directions in the both emission region (solid lines correspond to the emitted intensity in air while dashed lines correspond to the emitted intensity in the substrate) as a function of the vertical position of the NP.

  Figure 4.12(b) shows the variation of the intensity of the scattered light by the system along three directions in both the superstrate and the substrate sides as function of the vertical position of the NP. Therefore, these directions are defined by specific Euler angles (ϕ,θ) such that: • (0 • ,0 • ) presented by blue line curves that illustrate the evolution of the intensity emitted on the vertical direction (solid line for z>0 while dotted line for z<0), • (0 • ,42 • ) presented by red curves giving the evolution of the intensity emitted in xOz plane and pointing outward from the vertical axis along almost θ = 42 • (solid line in air and dotted one in substrate), and, • (90 • ,42 • ) presented by yellow curves that give the evolution of the intensity emitted in yOz plane and pointing outward from the z-axis along almost θ = 42 • . Referring to figure 4.12b, we can see that the intensities emitted in the air are more important than those emitted in the substrate (solid curves compared to the dashed ones). The blue curves indicate that the majority of the intensities are emitted in the right direction in both areas. The solid red and yellow curves, which show the emitted intensities along the xOz and yOz planes respectively and point outward from the vertical axis along almost θ = 42 • , explain the extension of the intensity lobe when the PCC-NP distance increases up to reach a maximum of extension when the NP placed 120 nm above the center of PCC. This latter PCC-NP distance corresponds to a critical coupling between the two resonators allowing a maximum extraction of the energy stored in the cavity. Therefore, we present in figure 4.13(a) the 4π sr radiation diagram (3D view) of the PCC-NP system when the NP is at 120 nm above the PCC center, while in figure 4.13(b) we present the 2D views calculated in air (z > 0) and in the substrate side (z < 0) of this configuration. In figure 4.13a, we see that the 3D angular spectrum of the light emitted by the system, in air, begins to generate two main lobes located in the xOz plane and directed outward from the vertical axis (z-axis) along almost θ = 19 • , while four lobes are formed in the substrate side where two of them at θ = 11.77 • and the two others are directed along almost θ = 38.5 • .To better visualize the PCC-to-NP coupling, we present the distributions of the electric and magnetic field intensities (in logarithmic scale) in the 3 orthogonal planes for some specific configurations. This study aims to understand more the effect of the vertical position of the NP on the behavior of the energy extracted from the whole system (PCC-NP).

Figure 4 .Figure 4

 44 Figure 4.14a shows the spatial distributions of the electric field intensity (in

Figure 4 .

 4 Figure 4.14: Intensity distributions of the electric (a) and magnetic (b) fields (in logarithmic scale) recorded on the xOy plane at 15nm above the PCC in air. The considered configurations from bottom to top are : isolated PCC, NP deposited on the PCC center, NP at 45 nm above the PCC center, NP at 120 nm above the PCC center and NP at 630 nm above the PCC center. The green circles show the position of the air holes of the photonic crystal in xOy plane.

Figure 4 .

 4 Figure 4.16: (a) presents the electric intensity distributions (in logarithmic scale) recorded in the yOz plane passing through the PCC center. (b) presents the magnetic intensity distributions (in logarithmic scale) recorded in the same plane. 5 different cases are considered: isolated PCC, NP deposited on the PCC center, NP at 45 nm above the PCC center, NP at 120 nm above the PCC center, NP at 630 nm above the PCC center. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.17: Electric intensity distributions (in logarithmic scale) recorded in the xOz plane passing through the center of the PCC, superimposed by the Poynting vectors (green and white vectors). 5 different cases are considered (from bottom to top) : isolated PCC, NP deposited on the PCC center (D z = 0 nm), NP at 45 nm above the PCC center, NP at 120 nm above the PCC center, NP at 630 nm above the PCC center. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.18: Electric intensity distributions (in logarithmic scale) recorded in the yOz plane passing through the center of the PCC, superimposed by the Poynting vectors (green and white arrows). 5 different cases are considered (from bottom to top) : isolated PCC, NP deposited on the PCC center (D z = 0 nm), NP at 45 nm above the PCC center, NP at 120 nm above the PCC center, NP at 630 nm above the PCC center. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.19: (a) Top-view of the PCC schema showing the three specific positions of the NP marked by 3 colored dots. 2D view of the radiation angular patterns calculated in air (z > 0) and in the substrate side (z < 0) are presented in (b) for the PCC-NP system when the NP is placed 45 nm above the PCC at (210,0,45), in (c) for the PCC-NP system when the NP placed 45 nm above the PCC at (420,0,45) and in (d) for the PCC-NP system when the NP placed 45 nm above the PCC at (630,0,45). The 3D view of the total (4π sr) radiation angular patterns are shown below the 2D views corresponding to each case.

Figure 4 .

 4 Figure 4.20: (a) Top-view of the PCC schema showing the three specific positions of the NP marked by 3 colored dots. 2D view of the radiation angular patterns calculated in air (z > 0) and in the substrate side (z < 0) are presented for the PCC-NP system when the NP is placed at : (840,0,45) in (b), (1020,0,45) in (c) and (1260,0,45) in (d). The 3D view of the total (4π sr) radiation angular patterns are shown below the 2D views corresponding to each case.

  figure 4.19(c)), the angular radiation pattern obtained clearly demonstrates that the energy scattered by the PCC-NP system is mainly directed along the vertical axis in the supertrate (air) and reflection (substrate) regions. In air, the lobe of the emitted light is found to be nearly extended with two lobes that are directed at θ = 29.5 • from the vertical axis and located symmetrically to the x-axis along an azimuthal angle ϕ = 48.1 • .In figure4.20(c), the NP is placed above the third intensity minimum at (1020,0,45). In this case, the extracted light scatters again in a quasi-isotropic shape with extinction in the vertical direction leading to a "donut" shape and directed outward from the vertical axis (z-axis) along almost θ = 42 • and θ = 32.5 • , while a lobe is formed on the substrate side in the xOz plane (major axis plane) which is pointed outward from the vertical axis along almost θ = 31 • in the opposite direction of the NP position (along x<0).In figure4.20(d), we placed the NP at a distance equal to D x = 3×a = 1260 nm away the center of the PCC. Such a position corresponds to the position of the third intensity maximum (see figure4.3(a)). At this position of the NP, the energy scattered by the system, in the air, takes on a new pattern other than that obtained when the NP is placed above an intensity maximum such that a major lobe is directed outward from the vertical axis along almost θ = 48.5 • in the x<0 direction, whereas on the substrate side, almost all the energy is concentrated in a lobe extended from the vertical axis to the direction pointed outward from the vertical axis along almost θ = 44 • in the opposite direction of the NP position (i.e with an azimuthal angle ϕ = 180 • ). This new pattern could be due to the fact that the NP approaches the edge of the cavity and thus simulates the role of a light reflector. This hypothesis remains to be elucidated.

  Figure 4.21(a) gives the values of τ for different positions of the NP along the x-axis. We can see that the curve of τ oscillates in a sinusoidal way with an increase of amplitude as the NP moves laterally towards the outside of the cavity. This increase in amplitude can be explained on the basis of figure 4.21(b) which presents the curves describing the variation of intensity emitted by the system in different directions in the both regions (air and substrate). All the curves, with the exception of the solid blue one, reach a high amplitude in the interval of the lateral distance D x = [1000 nm 1200 nm ]; this indicates that for such large distances, the light extracted by the system is oriented in specific directions corresponding to these curves. Moreover, let us recall that the isolated cavity is initially designed in such a way that the light is trapped inside. For this, we recall that the holes of the PC on the edge of the cavity have been moved in such a way as to ensure maximum trapping (high Q-factor at resonance) (see section 4.2.1). Consequently, the fact of approaching the NP of this zone will greatly disturb this condition and thus leads to an increase in the emission.

Figure 4 .

 4 Figure 4.21: (a) The total normalized intensity extracted by the PCC-NP system (τ) as a function of the position of the NP along the x-axis. (b) Emitted intensity by the system recorded in three directions in the both emission region (solid lines correspond to the emitted intensity in air while dashed lines correspond to the emitted intensity in the substrate) as a function of the lateral position of the NP along the x-axis.

Figure 4 .

 4 Figure 4.22: (a) presents the intensity distributions of the electric field (in logarithmic scale) recorded on the xy-plane at 15nm above the PCC in air. (b) presents the intensity distributions of the magnetic field (in logarithmic scale) recorded at 15nm above the PCC in air. The different configurations are presented from bottom to top when the position of the NP increases in 210 nm steps from the center of the PCC as indicated in the middle column. The green circles show the position of the air holes of the PC in xOy-plane.

Figure 4 .

 4 Figure 4.22(a) shows the spatial distributions of the electric field intensity (in

  at D x = (p + 1 2 ) × a = positions that correspond to an intensity minimum), such that m = 0 ; 1 ; 2 or 3 and p = 0 ; 1 or 2. Similar to figure 4.22(a), we present in figure 4.22(b) the spatial distributions of the magnetic field intensity recorded in a xOy plane placed 15 nm above the PCC for the same configurations considered above. The same behavior of the intensity distributions were reproduced when the PCC is coupled to the NP (i.e a sinusoidal oscillation). Another rather interesting way of presenting these results is to show the distributions of electrical and magnetic intensities in the xOz plane (figure 4.23). This allows us to get an overview on the coupling effect due to the presence of NP near the PCC by following the amplitude and direction of the light emitted by the system as well as the verification of the existence of the light either into the NP or into the PCC. In figures 4.23(a,b), we show the spatial distributions of the electric and magnetic field intensities for the same configurations as mentioned in figure 4.22, respectively. As expected, the spatial distributions of the intensities of both electric and magnetic fields in the PCC gradually reach their maximum values (the values in case of isolated PCC) when the NP sufficiently moves away from the PCC center.

Figure 4 .

 4 Figure 4.23: (a) presents the electric intensity distributions (in logarithmic scale) recorded in the xOz-plane passing through the PCC center. (b) presents the magnetic intensity distributions (in logarithmic scale) recorded in the same plane. Cases from bottom to top are: isolated PCC, NP at 45 nm above the PCC center, NP at D x = 210 nm, NP at D x = 420 nm, NP at D x = 630 nm, NP at D x = 840 nm, NP at D x = 1020 nm and NP at D x = 1260 nm. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.24: Electric intensity distributions (in logarithmic scale) recorded in the xOz plane passing through the center of the PCC, superimposed by the Poynting vectors (green and white arrows). 8 different cases of the NP's position are considered as indicated on the top of each sub-figure. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.25: (a) Top-view of the PCC schema showing the three specific positions of the NP marked by 3 colored dots. 2D view of the radiation angular patterns calculated in air (z > 0) and in the substrate side (z < 0) are presented in (b)for the PCC-NP system when the NP is placed 45 nm above the PCC at (0,210,45), in (c) for the PCC-NP system when the NP placed 45 nm above the PCC at (0,420,45) and in (d) for the PCC-NP system when the NP placed 45 nm above the PCC at (0,630,45). The 3D view of the total (4π sr) radiation angular spectra are shown below the 2D spectra corresponding to each case.

Figure 4 .

 4 Figure 4.26: (a) Total normalized intensity extracted by the PCC-NP system (τ) as a function of the position of the NP along the y-axis. (b) Emitted intensity by the system recorded in three directions in the both emission region (solid lines correspond to the emitted intensity in air while dashed lines correspond to the emitted intensity in the substrate) as a function of the lateral position of the NP.

Figure 4 .

 4 Figure 4.27: (a) presents the intensity distributions of the electric field (in logarithmic scale) recorded on the xOy plane at 15nm above the PCC in air. (b) presents the intensity distributions of the magnetic field (in logarithmic scale) recorded at 15nm above the PCC in air. The different configurations are presented from bottom to top as follow : isolated PCC, NP at 45 nm above the PCC center, NP at D y = 120 nm, NP at D y = 240 nm, NP at D y = 300 nm, NP at D y = 420 nm and NP at D y = 630 nm. In all previous configurations, the NP is placed at D z = 45 nm above the PCC. The green circles show the position of the air holes of the photonic crystal in xy-plane.

Figure 4 .

 4 Figure 4.27(a) shows the spatial distributions of the electric field intensity (in logarithmic scale) recorded in a xOy plane placed at 15 nm above PCC for the different configurations indicated in the middle column in the same figure.

  figure 4.27, respectively. As in figure 4.27, the spatial distributions of the intensities of both electric and magnetic fields in the PCC gradually reach their maximum values (the values in case of isolated PCC) when the NP sufficiently moves away from the PCC. Finally, to highlight and validate more the radiation pattern variations, we once again calculated the Poynting vectors and superimposed them on the electric intensity distributions (presented in logarithmic scale) of the system recorded in the yOz plane for the different cases considered in figure 4.28 (see figure 4.29). Due to the high light confinement near the cavity, we again visualized the Poynting vectors at two different scales, so that the white arrows are larger than the green ones by a square factor. As expected, the figure 4.29 clearly shows the distribution of the Poynting vectors so that they well describes the shape of the angular radiation patterns presented in figure 4.25 (see the direction and length of the white arrows above the NP and those below the PC in the substrate region).

4. 4 .

 4 Figure 4.28: (a) presents the electric intensity distributions (in logarithmic scale) recorded in the yOz plane passing through the PCC center. (b) presents the magnetic intensity distributions (in logarithmic scale) recorded in the same plane. Cases from bottom to top are: isolated PCC, NP at 45 nm above the PCC center, NP at D y = 120 nm, NP at D y = 240 nm, NP at D y = 300 nm, NP at D y = 420 nm and NP at D y = 630 nm. The green circle show the edges of the NP above the PC that presented by green rectangles.

Figure 4 .

 4 Figure 4.29: Electric intensity distributions (in logarithmic scale) recorded in the yOz plane passing through the center of the PCC, superimposed by the Poynting vectors (green and white arrows). 5 different cases are considered (from bottom to top): the first is for the isolated PCC and the remaining four lateral positions are those where the NP is placed 45 nm above the PCC for a spacing of 210 nm outwards from the center of the PCC. The green circle show the edges of the NP above the PC that presented by green rectangles.

1. 3 Figure

 3 Figure A.1: (a) The NP self-suspended in air and surrounded by the TF box (red box) and the SF box (blue box). (b) Relative difference between the wavelength (blue curve) and the normalized scattering cross section (red curve) calculated by the FDTD code and Mie's theory as a function of the relative spatial sampling (λ/δ). (c) The NP is suspended in air at a distance of D z above an InP substrate and surrounded by the TF and SF boxes, which are penetrated into the substrate.

  ).

	Figure 2.1: Incident, internal (i.e. within the particle) and total EM fields.
	At that point, we are faced with the problem of determining the expressions
	of the two unknown EM fields, namely the ( ⃗ E 2 (⃗ r, t); ⃗ H 2 (⃗ r, t)) fields in the par-
	ticle and the ( ⃗ E sca (⃗ r, t); ⃗ H sca (⃗ r, t)) fields in the host medium. These two fields
	each verify Maxwell's four equations without the terms of free charges and
	27

  10 15 rad/s) 13.202 Ω 1 (10 15 rad/s) 3.8711 Ω 2 (10 15 rad/s) 4.1684

γ (10 15 rad/s) 0.10805 Γ 1 (10 15 rad/s) 0.44642 Γ 2 (10 15 rad/s) 2.3555 Table 2.1: Parameters for the DCP model to fit the dielectric functions of a noble metal (Gold) over the wavelength range where λ ∈ [200 ; 2000] nm (experimental data from Palik)

  Pour étudier le couplage, les propriétés de chaque élément, à savoir la NP sphérique et la PCC CL7, ont été décrites et les propriétés optiques du système entier ont été déterminées. L'étude a montré que la lumière extraite du PCC pouvait être contrôlée en la couplant à une NP dont la résonance dipolaire électrique correspond à la longueur d'onde de résonance du PCC. Des codes techniques et des simulations ont été développés pour cette étude, et la théorie de Mie a été utilisée pour concevoir la NP. Une autre étude a été menée pour montrer le comportement de la lumière extraite du même PCC lorsqu'elle est couplée à une NP diélectrique dont la résonance dipolaire magnétique se situe à la même longueur d'onde de résonance que le PCC.

ACKNOWLEDGEMENTS

I would like to thank all the people who have contributed to the success of my thesis: my thesis supervisor, my wife, my family and my colleagues.

In figures 4.15(a,b), we show the spatial distributions of the electric and magnetic field intensities (in logarithmic scale) in xOz plane for the same configurations as mentioned in figure 4.14, respectively. As in figure 4.14, the spatial distributions of the intensities of both electric and magnetic fields in the PCC gradually reach their maximum values (the values in case of isolated PCC) when the NP sufficiently moves away from the PCC.

On the other hand, in the figure 4.15, especially when the NP is placed at small distances near the PCC (at least when D z ≤ 120 nm), we can clearly see that the NP remains excited and that the EM field distribution inside resembles that of a quadrupole mode. The origin of this phenomenon remains to be elucidated.

We also present, respectively, in figures 4. 16(a,b) of the structure under study through parallelepiped cells in the Cartesian coordinates. This kind of meshes is not well suitable to model structures with curved surfaces. Thus, in order to minimize an eventual stair-casing problem, several techniques were developed in the literature [START_REF] Dey | A modified locally-conformal FDTD algorithm for modeling 3-D perfectly conducting objects[END_REF], Dey et al., 1999, Yu et al., 2000, Nilavalan et al., 2008].

Nonetheless, in the case of a purely dielectric material without absorption, the averaging technique seems to be quite powerful. In the case, the dielectric constant of each cell comprising two media of different index is calculated as the average value of the dielectric constant over the cell volume. We have tested this technique and demonstrated that it allows a very good agreement with Mie's theory. At the same time, we have also verified our TFSF code.

As its can be seen from figure A.1b, the convergence of the method depends greatly on the FDTD cell size and is only of 100% if the cell size tends to zero. That is why we set the cell size at 15 × 15 × 15 nm 3 corresponding to a deviation from the Mie theory of only 0.8% in the case of a spherical NP of radius 220 nm and a refractive index of 4.85 (see Fig. A.1b). In addition, our FDTD home-made code includes a non-uniform mesh that allows us to accurately describe the fine details of the structure.

As for the TFSF technique, it is based on a spatial division of the calculation window into two areas: the area of total field (TF) that contains the scatterers and where the calculated electromagnetic field consists on the incident and diffracted fields, and a second area, that encompasses the first one and where only the scattered field (SF) is calculated by subtracting the incident field over all the external interface of the TF area. We can then distinguish two different cases: the first one is the simple case defined by an NP having an arbitrary geometry immersed in a homogeneous medium (here, we have considered a spherical NP immersed in air, see the fig.A.1a), while the second case includes the presence of a substrate in the vicinity of such an NP (see fig.A.1c). The code in the second case is improved by subtracting not only the incident field but also the reflected one on the substrate interface [START_REF] Markovich | Magnetic dipole radiation tailored by substrates: numerical investigation[END_REF], Taflove et al., 2005a]. It is essential to mention that a further chapter is more dedicated to the development of this method with