Avril Alicia Meza Olivo

Alicia Meza Olivo

Dra Karina

Garay Palmett

Dr Rafael Salas Montiel

Emission de nanoémetteurs exaltée par plasmon de surface en photonique silicium : vers des sources

Keywords: Photons-Émission, Microscopie en champ proche, Optique intégrée, Nanophotonique, Plasmons, Optique quantique Single-photon emission, near-field microscopy, integrated optics, nanophotonics, plasmonics, quantum optics photon emission -integrated opticsnanophotonics -plasmons (physics) -quantum optics -near-field microscopy

Le développement de sources intégrées de photon unique efficaces dans les longueurs d'ondes des télécommunications est en train de jouer un rôle majeur pour le développement des technologies quantiques. Ce projet de recherche doctoral montre une contribution au développement de ces sources en proposant et démontrant expérimentalement la réalisation d'une paire de dispositifs nanophotoniques intégrés complémentaires. Deux principes fondamentaux de l'optique électromagnétique ont été utilisés pour leur conception, le théorème de réciprocité et le principe de Babinet. Les résultats montrent que les structures intégrées permettent un confinement extrême de la lumière et par conséquence, un renforcement des champs électrique et magnétique. Réciproquement, l'émission spontanée d'émetteurs électriques et magnétiques localisés à la place des champs confinés est accélérée sélectivement dans les modes du champ par effet Purcell électrique et magnétique. Les dispositifs complémentaires ont été fabriqués à l'aide de techniques avancées de lithographie électronique et caractérisés expérimentalement en champ lointain, via des mesures de durée de vie, ainsi qu'en champ proche à l'aide d'un microscope optique en champ proche. Ces mesures représentent une première étape de validation expérimentale vers la réalisation de sources intégrées de photon unique à base d'émetteurs quantiques. Les perspectives des travaux portent sur la réalisation de sources intégrées déterministes et efficaces et les combiner avec de circuits photonique sur silicium plus complexes.

iv Resumen de la tesis que presenta Avril Alicia Meza Olivo como requisito parcial para la obtención del grado de Doctor en Ciencias en Óptica.

Emisión de nanoemisores exaltada por plasmones de superficie en fotónica de silicio: hacia la implementación de fuentes integradas

Resumen aprobado por:

Dra. Karina Garay Palmett

Codirector de Tesis

Dr. Rafael Salas Montiel

Codirector de Tesis Esta investigación se realizó bajo el marco del programa de doble grado, celebrado entre el Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE) y la Universidad de Tecnología de Troyes (UTT). Durante los cuatro años de este doctorado, tanto las actividades curriculares como de investigación se distribuyeron equitativamente entre ambas instituciones. La fabricación de fuentes de fotones integradas, altamente eficientes, operando en la banda de las telecomunicaciones, es una creciente área de investigación hacia el desarrollo de tecnologías de información cuántica. En este proyecto, se propone y se demuestra experimentalmente el principio de operación de un conjunto de dispositivos nanofotónicos integrados, como una contribución hacia el desarrollo de la nanofotónica cuántica integrada. El diseño de los dispositivos está basado en los conceptos físicos fundamentales de teorema de reciprocidad y principio de Babinet. Los resultados muestran que las estructuras integradas confinan y exaltan los campos eléctricos y magnéticos y, recíprocamente, modifican la densidad local de estados por efecto Purcell; incrementando así la tasa de decaimiento de las transiciones eléctricas y magnéticas de emisores colocados en la cercanía de las estructuras diseñadas. Dichas estructuras complementarias se fabricaron por litografía de haz de electrones y se caracterizaron experimentalmente, en campo lejano, con mediciones de tiempos de vida y en campo cercano, mediante microscopía óptica de barrido en campo cecano (NSOM por sus siglas en inglés). Las medidas NSOM representan el primer paso hacia la validación experimental de fuentes de fotones individuales integradas, basadas en emisores cuánticos. El trabajo futuro involucra la utilización de estos dispositivos como base para la implementación de fuentes de fotones individuales integradas en un chip, con las siguientes características: determinísticas, altamente eficientes, con ancho de banda amplio en la ventana de las comunicaciones ópticas e integrados en fotónica de silicio para su transporte a largas distancias, con bajas pérdidas; por tanto, adecuadas para su uso en aplicaciones de procesamiento cuántico de información.

Palabras Clave: Emisión de fotones, microscopía de campo cercano, óptica integrada, nanofotónica, plasmónica, óptica cuántica. Because the essential for a family is not to live together, but to be united... I love you more.

vi "I am always doing that which I cannot do, in order that I may learn how to do it." -Pablo Picasso "Science offers the boldest metaphysics of the age. It is a thoroughly human construct, driven by the faith that if we dream, press to discover, explain, and dream again, thereby plunging repeatedly into new terrain, the world will somehow become clearer and we will grasp the true strangeness of the universe. And the strangeness will all prove to be connected, and make sense."

-Edward O. Wilson

"A photon is the future baby of a pregnant atom. A proper environment should surround it to happily exist. Nanophotonics provides him the best possible option." -Avril Meza vii Al Consejo Nacional de Ciencia y Tecnología (CONACyT) por brindarme el apoyo económico para realizar mis estudios de doctorado (y previamente de maestría), a través de la asignación de becas nacionales y de beca mixta, en la modalidad de programas de doble titulación. No. de becario: 233924. Así como a Campus France por la beca complementaria CR LN1415J. También por su apoyo económico para llevar a cabo este proyecto, agradezco al Laboratorio Nacional de Materia Cuántica: Materia Ultrafría e Información Cuántica (LANMAC). No. de proyecto: 293471.

Al Centro de Investigación Científica y de Educación Superior de Ensenada en México y a la Universidad de Tecnología de Troyes en Francia; a todos los investigadores, profesores, personal administrativo y de servicios generales que ahí laboran; por su compromiso constante hacia la sociedad, en la continua labor de formar recursos humanos de alta calidad científica y profesional, pero sobre todo, de alta calidad humana.

Gracias a mis padres y hermanas por su amor y apoyo incondicional. Por enseñarme a amar la vida con su ejemplo de amor, superación, trabajo y dedicación constantes.

Es gracias a ustedes que soy todo lo que soy y que amo todo lo que hago. Gracias por construir mis alas para volar. Agradezco también el apoyo de mis tíos, tías, primos y primas. Gracias por su cariño, por estar siempre al pendiente de mi.

A mis amigos y compañeros de viaje, literalmente, aquí y en China; de Francia y de cada rincón del mundo que he tenido la fortuna de conocer. Son los abrazos, las risas compartidas y las complicidades formadas, grandes alimentos para mi alma y la fuerza que me permite seguir en los momentos más inciertos.

A mis codirectores de tesis, la Dra. Karina Garay Palmett y el Dr. Rafael Salas Montiel, por su confianza, apoyo, consejos y paciencia infinita. Por su dedicación y esmero viii en mi formación académica y profesional. Gracias también por cada momento compartido fuera de los laboratorios y/o las aulas; además de su calidad como científicos, admiro infinitamente su calidad humana. Son y serán siempre un modelo a seguir en muchos ámbitos de mi vida.

Gracias a mis sinodales y miembros de comité de tesis doctoral por su tiempo, sus cuestionamientos, por las discusiones entabladas en cada oportunidad. Su contribución en mi aprendizaje ha sido fundamental. Agradezco también a los doctores Roger Cudney, Raúl Rangel, Kevin O'Donnell, Pedro Negrete y Eugenio Méndez por el esmero y pasión con la que me compartieron sus conocimientos, así como por las gratas conversaciones y enseñanzas. A los doctores Israel Rocha y Pedro Negrete por su gran apoyo como coordinadores del Posgrado de Óptica. A Dolores Sarracino, Citlali Romero, Isabelle Leclercq y Pascale Denis por su apoyo y esmero en labores administrativas. A todos mis compañeros de clases y al grupo de trabajo por la continua retroalimentación que, igualmente, es parte fundamental del aprendizaje.

Finalmente, porque para lograr la obtención de un doctorado no solo se requieren conocimientos científicos, sino una gran resistencia mental; y porque no todo el camino fué sencillo, sino todo lo contrario; agradezco infinitamente a Francisco, Josslyn, Lila y Dalia, colegas y grandes amigos. Gracias por las discusiones profesionales y filosóficas, pero sobre todo, gracias por adoptarme como un miembro de su familia. A doña Rebeca y doña Rosy, por alimentarme y cuidarme con todo su cariño, como a una hija adoptiva. Y, por contribuir en gran medida a mi transformación personal, muchas gracias a Marcia y a Pedro por la incansable búsqueda, por todas las experiencias y aprendizaje compartidos. Son y serán siempre mis grandes compañeros de viaje.

Gracias a la vida, que me ha dado tanto...

Chapter 1. Introduction

Optics is the field of study historically involving the generation, propagation, and detection of light. Over the last seventy years, developments such as the invention of the laser, the fabrication of low-loss optical fibers, and the introduction of semiconductor optical devices, promoted an increasing renewed importance of this field in modern technologies. As these developments have evolved, different disciplines to describe the new technologies have emerged, among which photonics reflects the increasing role that semiconductor materials and devices play in optical systems. In the same way as electronics involves the control of electric charge flow (in vacuum or in matter); photonics involves the control of photons (in free space or in matter). The term photonics also reflects the importance of the photon nature of light in the description of many optical devices. Moreover, the increasing pace of development has been characterized by an striking progress in miniaturization and integration of optical components; up to the point that, in recent years, the science of controlling the propagation, absorption and emission of light at the nanometer scale (10 -9 meters) has become an effervescent field of study, which has coined the term nanophotonics.

In the context of light emission, an ideal single-photon source, able to generate photons on demand, represents the ultimate control of the photon generation process, i.e., single photons that can be generated within short time intervals with a deterministic time interval between successive photon generation events. Such a source has the potential of enabling many new applications in the field of photonics and quantum information technology [START_REF] Lounis | Single photon sources[END_REF]. This is particularly true for quantum cryptography [START_REF] Ekert | Quantum cryptography based on Bell's theorem[END_REF], which exploits the fundamental principles of quantum mechanics to provide unconditional security for communication. Possible other applications are optical quantum computing [START_REF] O'brien | Optical quantum computing[END_REF], ultrasensitive metrology [START_REF] Giovannetti | Quantum metrology[END_REF], random number generation [START_REF] Ma | Quantum random number generation[END_REF], and quantum teleportation [START_REF] Bouwmeester | Experimental quantum teleportation[END_REF].

The requirements on the properties of single-photon sources depend on their specific application. Desirable properties for all sources are a high and constant internal quantum efficiency, low multiphoton emission probability, a high emission efficiency into a single mode, a low jitter (i.e. temporal uncertainty in the emission of the photons), and high emission rates.

To this day, macroscopic sources based on parametric downconversion [START_REF] Burnham | Observation of simultaneity in parametric production of optical photon pairs[END_REF][START_REF] Hong | Experimental realization of a localized one-photon state[END_REF], which generate pairs of correlated photons (twin photons) in free space at high speed rates, are used in an overwhelming majority of quantum-optics experiments. When a short laser pulse is sent into a nonlinear crystal, it generates pairs of signal and idler photons, which are highly correlated in space and time. Provided the probability of generating two pairs at the same time remains negligible and that the two-photon state is factorable i.e. the continuum variable (spatial and spectral) degrees of freedom are decorrelated; such photon pairs can be used as sources of heralded single photons.

However, recent progress in the optical detection, characterization and manipulation of single nano-objects makes quantum emitters an attractive alternative as a straightforward method to develop non-classical light sources, delivering one individual photon at a time. This property called antibunching of the emitted photons, ensures that the probability of having more than one photon at the same time remains negligible. The implementation of deterministic single-photon sources is achievable using single quantum emitters, in the form of individual atoms or molecules, or electron-hole pairs in a quantum dot (QD) with a suitable radiative transition between their quantum levels. In this case, the photons are generated by spontaneous emission, when a transition from an excited to the ground state causes a single-photon to occupy one or more spatial-temporal modes. Provided that the quantum states (as a function of time) can be controlled, and that the emitted photons are efficiently coupled to well-defined spatial modes, single quantum emitters represent a viable approach for the realization of temporarily spaced (antibunched) single-photon sources.

For the development of quantum information and communication technologies, it is desirable that the single-photon sources could be efficiently integrated into nanophotonic devices compatible with current silicon (Si) photonic platforms [START_REF] Soref | The past, present, and future of silicon photonics[END_REF]. Furthermore, the emission wavelength of the source should be one that minimizes optical losses in the telecommunications transmission window, e.g., 1.3 μm and 1.5 μm for long distance propagation, and maximizes the photo-detection efficiency.

On the other hand, up to date, the control over the generation of single photons by spontaneous emission has been mainly focused to the investigation of the spontaneous decay of electric dipole (ED) transitions, because the strength of ED transitions in typical optical quantum sources is orders of magnitude greater than that of magnetic dipole (MD) transitions [START_REF] Landau | En: Electrodynamics of continuous media[END_REF]. In fact, this difference is the reason why the permeability of natural materials is close to 1 in the visible range [START_REF] Merlin | Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism[END_REF], and it also determines the difficulties of detecting and controlling the magnetic field at optical frequencies [START_REF] Burresi | Probing the magnetic field of light at optical frequencies[END_REF]. Nevertheless, certain quantum emitters, such as rare-earth ions [START_REF] Carnall | Spectral intensities of the trivalent lanthanides and actinides in solution. II. Pm3+, Sm3+, Eu3+, Gd3+, Tb3+, Dy3+, and Ho3+[END_REF][START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF] and semiconductor QDs (Zurita-Sánchez y [START_REF] Zurita-Sánchez | Multipolar interband absorption in a semiconductor quantum dot. II. Magnetic dipole enhancement[END_REF], possess MD transitions whose strength is comparable or even greater than the competing ED ones. For instance, quantifying the ED and MD transitions around the 1.55 μm spectral line of trivalent erbium (Er 3+ ) could improve the design of optical amplifiers by reducing spontaneous emission noise (Digonnet, 2001;[START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF]. Coupling of lanthanide ions and QDs to various nanophotonic systems including bulk materials [START_REF] Deloach | Evaluation of absorption and emission properties of Yb 3+ doped crystals for laser applications[END_REF], planar structures [START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF][START_REF] Karaveli | Spectral tuning by selective enhancement of electric and magnetic dipole emission[END_REF], dielectric [START_REF] Shi | A new dielectric metamaterial building block with a strong magnetic response in the Sub-1.5-Micrometer Region: Silicon Colloid Nanocavities[END_REF][START_REF] Sanz-Paz | Enhancing magnetic light emission with all-dielectric optical nanoantennas[END_REF] and plasmonic nanoantennas [START_REF] Hussain | Enhancing Eu 3+ magnetic dipole emission by resonant plasmonic nanostructures[END_REF][START_REF] Feng | Controlling magnetic dipole transition with magnetic plasmonic structures[END_REF], and metamaterials [START_REF] Simovski | Wire metamaterials: physics and applications[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF] have been proposed. Here, advances in nanofabrication techniques, along with the increasing study of magnetic quantum emitters have stimulated the investigation of the magnetic side of spontaneous emission. This work is devoted to explore the integration of single quantum emitters that possess electric and magnetic dipole transitions into Si nanophtonic devices, towards the physical implementation of on-chip, deterministic, and highly efficient single-photon sources that are compatible with current and future optical platforms for the development of quantum information technologies.

Problem and Proposal

To produce single photons on demand using quantum emitters, it is necessary to achieve three basic tasks: 1) to isolate a single emitter, 2) to excite it, and 3) to efficiently collect and direct the emitted photons into the optical system to be used. Indeed, an essential requirement for implementing highly efficient single-photon sources is to optimize the extraction, collection, and control of the emitted radiation to be strongly coupled to well-defined spatial-temporal modes in order to feed integrated photonics circuitry. In this sense, the generation of strong and coherent coupling between individual quantum emitters and photons is the most challenging step to achieve, because of the reasons explained below.

In general, the coupling between atoms, molecules etc., and optical radiation is, by nature, poor because of the significant size mismatch between atomic wavefunctions (matter) and the wavelength of the emitted radiation (optical modes) [START_REF] Barnes | Solid-state single photon sources: light collection strategies[END_REF]. Synthetic atoms such as QDs have larger sizes, but there is still a big mismatch.

The problem caused by this mismatch is that we wish to generate not just any random photon; we want to generate a single optical mode, typically that of an optical fiber or in our case, a Si waveguide. Therefore, it is necessary to alter the modes into which emission may take place by adjusting the local photonic environment, the so-called local density of optical states, LDOS. Such a modification of the environment may be effective by recalling a simplified version of the Fermi's Golden rule, which governs the probability of transitions from an excited state |e〉 to a ground state |g〉

 eg ∝ |M eg | 2 ρ ω eg .
(

where  eg is the rate of spontaneous emission (i.e. the probability of emission), M eg is the matrix element that connects the excited and ground states of the emitter via the radiative decay channel (usually an electric dipole transition) and ρ(ω eg ) is the LDOS at the emission frequency ω eg i.e. the total available optical modes at ω eg into which the emitted photon could decay. Fermi's Golden rule as expressed in Eq. 1.1, states that the spontaneous emission decay rate is proportional to such photonic mode density.

For the case of emission in free-space, all directions of emission as well as an infinite number of polarization states are allowed, and there will be a poor match into a single mode of the desired optical system. Then, physical solutions to this problem will be to engineer ways in which the physical (photonic) environment may be manipulated to adjust the modes into which spontaneous emission may take place. Such manipulation of the LDOS is at the heart of cavity quantum electrodynamics (CQED). Consequently, common physical solutions to the collection problem are referred to as CQED approaches. Details on the coupling of quantum emitters to cavities will be given in chapter 3.

For the moment, it is worth noting that these CQED approaches are possible due to the fact that spontaneous emission is not an intrinsic property of an isolated emitter, but rather a property of the coupled system of the emitter and the electromagnetic mo-des in its environment. Indeed, the possibility of modifying the fundamental process of spontaneous emission is at the heart of this doctoral thesis as well.

Background

The essential fact that the spontaneous emission rate can be increased, decreased or even suppressed by varying the electromagnetic environment of a quantum source was discovered by E. M. [START_REF] Purcell | Spontaneous emission probabilities at radio frequencies[END_REF][START_REF] Purcell | Spontaneous emission probabilities at radio frequencies[END_REF]. Since then, the modification of the spontaneous emission rate of a quantum source induced by its interaction with the environment has been referred to as the Purcell effect.

Purcell suggested that nuclear magnetic transition rates could be increased by placing atoms inside a resonant cavity. The ratio between the modified emission (the atoms inside the cavity) relative to free-space emission rates became known as the Purcell factor. For a cavity whose fundamental mode is resonant with the transition frequency, for a dipole aligned with the polarization of this cavity mode and located at the position of maximum field in this mode, and for an emitter linewidth that is narrow compared to the cavity linewidth, the Purcell factor: .2) turns out to be proportional to Q, the quality factor of the cavity and to 1/ V eƒ ƒ , the effective volume of the resonant mode. Therefore, a significant increase in the emission rate requires an optical resonator that confines light down to small dimensions (small V eƒ ƒ ) and that stores it there for a long time (high Q). These two requirements are to some extent contradictory since tighter confinement means higher losses. Therefore, research into emission modification is nowadays directed either towards the development of resonant cavities with higher Q or towards the efficient confinement of the effective volume of the available modes. Because the use of cavities imposes a restriction on the bandwidth and the size of devices, an alternative strategy is to use an interface to bridge the size gap. Confining the light field to small effective volumes enables stronger coupling to the emitter. In this sense, plasmonic modes can be squeezed into volumes far below the diffraction limit, and therefore provide an excellent interface between single photons and emitters [START_REF] Marquier | Revisiting quantum optics with surface plasmons and plasmonic resonators[END_REF].

F P ∝ Q V eƒ ƒ . ( 1
Although the Purcell effect was originally discussed in the context of nuclear mag-netic resonance, it applies to the broad range of quantum sources, that demonstrate spontaneous decay from the excited state [START_REF] Krasnok | An antenna model for the Purcell effect[END_REF][START_REF] Pelton | Modified spontaneous emission in nanophotonic structures[END_REF], including atomic or molecular electric and magnetic transitions, quantum dots (QDs) [START_REF] Laucht | Broadband Purcell enhanced emission dynamics of quantum dots in linear photonic crystal waveguides[END_REF], quantum wells [START_REF] Okamoto | Surface-plasmon-enhanced light emitters based on InGaN quantum wells[END_REF], and defect centers in diamond nanocrystals [START_REF] Santori | Nanophotonics for quantum optics using nitrogen-vacancy centers in diamond[END_REF].

For instance, at communications frequencies, typical quantum sources exhibit slow spontaneous emission decay rates, with lifetimes of few microseconds. Then, to increase their emission rates, and consequently their LDOS by Purcell effect, semiconductor

QDs have been integrated into dielectric resonators [START_REF] Humer | Integration, photostability and spontaneous emission rate enhancement of colloidal PbS nanocrystals for Sibased photonics at telecom wavelengths[END_REF], photonic crystal cavities [START_REF] Fushman | Coupling of PbS quantum dots to photonic crystal cavities at room temperature[END_REF], plasmonic metamaterials [START_REF] Tanaka | Multifold enhancement of quantum dot luminescence in plasmonic metamaterials[END_REF] and nanoantennas [START_REF] Akselrod | Efficient nanosecond photoluminescence from infrared PbS quantum dots coupled to plasmonic nanoantennas[END_REF], achieving Purcell factors up to 13, 30, 100, and 1100, respectively. All these cases represent great advances in the manipulation of spontaneous emission rates, however some issues remain to be addressed yet. The use of resonant structures limits the spectral range of the source and provides a poor collection of the emitted light into guided media, such as optical waveguides and optical fibers. Additionally, in dielectric cavities, even though exhibiting high quality factors, the enhancement of the Purcell factor is restricted due to the diffraction-limited photonic modes (with significant effective mode volumes). Furthermore, in all the previous examples, the emitted photons are radiated to free space and the given Purcell factors represent a total enhancement of the spontaneous emission rate (i.e. enhancement of emission in all possible directions), but in all cases, additional coupling steps are required for the photons to be used in integrated optical systems. Then, the corresponding Purcell factors when the emission happens to be coupled into specific spatial modes will be lower than those presented, a consideration that is often neglected.

In order to increase both the achievable bandwidth delivered by the source and the coupling efficiency into well-defined modes; and because of the reasons that will become clear throughout this manuscript, we propose the use of non resonant structures composed by plasmonic waveguides that support surface plasmon polaritons (SPPs) integrated on top of a Si waveguide. Such hybrid plasmonic-photonic nanodevices will exploit the ability of SPPs to extremely confine the electromagnetic field far beyond the diffraction limit, together with the benefit of long-range distance propagation of radiation into well-defined electromagnetic modes provided by Si photonics wavegui-des.

This hybrid approach has become possible over the last two decades due to the rapid development of nanoscale fabrication techniques, and also due to the availability of theory tools in the form of powerful electromagnetic simulation methods, making both Si photonics and plasmonics very active research fields for the manipulation of light at the nanoscale, and allowing them to become a powerful tool to enhance the emission properties of nanoemitters, as described in the next section.

Motivation

Surface plasmon polaritons are surface electromagnetic waves, which result from the coupling of electromagnetic excitations (radiation) to charge density waves (matter), propagating along metal-dielectric interfaces [START_REF] Barnes | Surface plasmon subwavelength optics[END_REF][START_REF] Tame | Quantum plasmonics[END_REF]. The electromagnetic field in such surface waves is confined to the near vicinity of the interface, to volumes far below the diffraction limit (Gramotnev and Bozhevolnyi, 2010), therefore providing an excellent interface between individual photons and quantum emitters.

Although it is well known that SPPs undergo high losses by absorption as they propagate along the metal, potentially limiting their use in real-world applications, these limitations are not fundamental if SPP devices are combined with dielectric waveguides. In this case, the hybrid plasmonic-photonic platform provides the benefits of high confinement of plasmonics, together with the benefits of Si photonics for longer distances propagation. In this sense, the role of SPPs in controlling light-matter interactions at the quantum level, opens up a new frontier for the realization of controlled quantum devices including controllable and efficient single-photon sources and ultra-compact circuitry at the nanoscale.

The challenge in the development of hybrid photonic-plasmonic platforms, is the mismatch between the spatial distribution of the optical modes in each individual device. As mentioned, the optical modes in plasmonic devices can be confined far below the diffraction limit, meanwhile in Si photonics devices the optical modes are propagated at around λ/ 2n where n is the material refractive index. In this context, an effective approach to bridge the size of the modes in dielectric waveguides to that of the plasmonic structures is through the so-called directional coupling (Salas-Montiel y Blaize, 2015). This method has been used in integrated optics to transfer the energy between two waveguides. In the coupling zone there are usually two modes that propagate and interfere since their propagation constants are different, leading to an energy exchange between the waveguides. This directional coupling mechanism is the approach used in this work, and will be detailed in subsequents chapters.

Finally, because the light is well confined in the nanophotonic structures that we propose here, the optical near field will be used to characterize them. Optical near field has proven to be very useful for the imaging of the light propagation in Si photonics [START_REF] Salas-Montiel | Quantitative analysis and near-field observation of strong coupling between plasmonic nanogap and silicon waveguides[END_REF].

In summary, despite the increasing progress towards the control of the spontaneous emission of quantum emitters in order to realize efficient single-photon sources, challenges still remain such as the need of practical platforms capable of integrating high rates of spontaneous emission, high directionality and coupling to waveguide structures for transmission and broadband operation.

In general, this thesis is an attempt to decrease the gap towards the practical implementation and use of single-photon sources integrated in nanophotonic devices.

In

Hypothesis

"By Purcell effect, it is possible to enhance and to control the rate and direction of the spontaneous emission of electric and magnetic dipole emitters, modifying the electric and magnetic local density of states into which the emission transitions could decay. This can be achieved by the design of non resonant nanometric plasmonic structures integrated in Si photonic waveguides, thus opening the possibility to implement highly efficient, broadband, integrated single-photon sources, operating in the optical telecommunications window."

Objectives of the doctoral research project

To understand the physical mechanisms that enable both high Purcell Factors, and efficient coupling between quantum emitters and plasmonic modes supported by nonresonant plasmonic structures integrated on silicon photonic waveguides, towards the realization of controllable, highly efficient, deterministic and broadband single-photon sources, operating in the near infrared optical spectrum, thus suitable for their use in quantum information technologies.

Particular objectives

1. To understand the coupling mechanisms that allow efficient energy transfer between the photonic modes supported by a Si waveguide and the plasmonic modes supported by metallic nanostructures in order to achieve maximum electromagnetic field confinement and minimum effective mode volume.

2. To design plasmonic geometries able to enhance and confine the electric and magnetic fields.

Organization of the manuscript

The theoretical foundations to understand and to manipulate the light-matter interactions at the nanometer scale are given in chapter 2. Here, electromagnetism is described as an eigenvalue problem from which all the information relative to optical systems can be obtained by linear algebra and symmetry properties. Additionally, the fundamental concepts of Green's function, reciprocity theorem and Babinet's principle are explained as fundamental tools to understand the relationship between electric and magnetic fields and the sources that originate them. Nanophotonics and plasmonics are then presented as the fields devoted to modify the light-matter interactions by changing the electric dielectric constant of materials as a function of the frequency, at subwavelength and subdiffraction scales.

In chapter 3, the properties of single photons emitted by single quantum emitters are discussed. The spontaneous emission process is modeled by a two-level system, whose radiative properties can be altered depending on the electromagnetic environment which the emitter resides in. Here the concept of local density of states emerges together with its interpretation in classical terms, via the Green's tensor and the power radiated by a classical oscillator. The computational methods to model the light-matter interactions are also presented.

Based on all the above, the design, numerical analysis, fabrication, and experimental validation of a set of complementary integrated devices capable of extremely confining the electric and magnetic fields, and by reciprocity, promoting the enhancement of the electric and magnetic LDOS of electric and magnetic dipole emitters by Purcell effect at telecommunication frequencies are detailed in chapter 4.

The general conclusion is given in chapter 5, and a comprehensive summary of the thesis in French is presented in chapter 6.

Chapter 2. Theoretical foundations for the spontaneous emission enhancement in complex structures

Introduction

In this chapter, the theoretical framework to understand and modify light-matter interactions at the nanoscale are presented. Specifically we want to modify and enhance the spontaneous emission of ED and MD quantum emitters in a controlled manner.

It should be mentioned that to rigorously understand light-matter interactions at the nanometer scale, quantum electrodynamics (QED) is required. A good understanding of this subject can be found in textbooks such as [START_REF] Cohen-Tannoudji | Photons and atoms: introduction to quantum electrodynamics[END_REF][START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Craig | Molecular quantum electrodynamics[END_REF]. Here we invoke important results derived from QED when needed. However, nanostructures are usually too complex to be rigorously solved in this way. Therefore, the treatment in this manuscript will be presented in a classical description; not only because it is often more intuitive, but because the results obtained in this way have been validated [START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Baranov | Modifying magnetic dipole spontaneous emission with nanophotonic structures[END_REF][START_REF] Carminati | Electromagnetic density of states in complex plasmonic systems[END_REF], and as long as there are no experimental contradictions, it constitutes a convenient approach.

Classical electromagnetics in complex photonic and plasmonic systems

To understand the physical mechanisms allowing the control of light-matter interactions at the nanoscale, let's first recall the behavior of light within macroscopic media.

All of macroscopic electromagnetism is governed by the four macroscopic Maxwell equations. In SI units, they are:

.

∇ • D = ρ, ∇ • B = 0, ∇ × E = - ∂B ∂t , ∇ × H = J + ∂D ∂t , (2.1)
where, respectively, E and H are the macroscopic electric and magnetic fields, D and B are the displacement and magnetic induction fields and ρ and J, are the free charge and current densities. If these equations are restricted to the case of 'simple' optical materials, i.e. a composite of regions of homogeneous material where the propagation of light is a function of the position vector r, in which the structure does not vary with time, and there are no free charges or currents, then ρ = 0 and J = 0.

The interaction of electromagnetic waves with matter is determined by the appropriate constitutive relations. In general, the components D  of the displacement field D are related to the components E  of the electric field E by a power series:

D  / ϵ 0 = j ϵ j Ej + j,k χ jk E j E k + O E 3 , (2.2)
with ϵ 0 ≈ 8.854 × 10 -12 Farad/m being the vacuum permittivity. However, for many dielectric materials, the following approximations can be made:

1. The field strengths are assumed to be small enough to stay in the linear regime, so that χ jk and all higher-order terms can be neglected.

2. The material is assumed macroscopic and isotropic, so that E(r, ω) and D(r, ω)

are related by ϵ 0 multiplied by a scalar dielectric function ϵ (r, ω), also called the relative permittivity.

3. Any explicit frequency dependence (material dispersion) of the dielectric constant is ignored. Instead, the value of the dielectric constant appropriate to the frequency range of the physical system under consideration is chosen.

4. Only transparent materials are considered, which means that ϵ(r) is purely real and positive.

Assuming these four approximations to be valid, we have: .3) where μ 0 = 4π × 10 -7 Henry/m is the vacuum permeability, but for most materials at optical frequencies the relative magnetic permeability μ (r) is very close to unity, so that B = μ 0 H. In such the case, ϵ is the square of the refractive index n. In general, n = ϵμ. With all these assumptions, the Maxwell equations become:

D (r) = ϵ 0 ϵ (r) E (r) B (r) = μ 0 μ (r) H (r) , ( 2
∇ • [ϵ (r) E (r, t)] = 0, ∇ • H (r, t) = 0, ∇ × E (r, t) = -μ 0 ∂H (r, t) ∂t , ∇ × H (r, t) = ϵ 0 ϵ (r) ∂E (r, t) ∂t .
(2.4)

In general, both E and H are complicated functions of time and space. However, since Maxwell's equations are linear, the time and spatial dependences can be separated by expanding the fields into a set of harmonic functions. This approach represents no big limitation, since by Fourier analysis, any solution can be built with an appropriate combination of these harmonic functions. Therefore, we now analyze the restrictions that Maxwell's equations impose on fields that vary sinusoidally (harmonically) with time. These are often referred to as harmonic modes, or simply as modes or states of the system. Also, for mathematical convenience, complex fields will be used, from which the real part will then be taken to obtain the physical fields. In this way, a harmonic mode can be written as a spatial pattern (or complex amplitude, or mode profile)

times a complex exponential:

E (r, t) = E (r) e -ωt , H (r, t) = H (r) e -ωt .
(2.5)

To find the mode profiles for a given frequency, the above equations are inserted into the two divergence Maxwell's equations, leading to the conditions:

∇ • [ϵ (r) E (r)] = 0, ∇ • H (r) = 0, (2.6)
which have a simple physical interpretation: there are no point sources or sinks of displacement and magnetic fields in the medium. Additionally, the fields are made of transverse electromagnetic waves.

The two curl equations relate E (r) to H (r):

∇ × E (r) = ωμ 0 H (r) , ∇ × H (r) = -ωϵ 0 ϵ (r) E (r) .
(2.7)

These equations can be decoupled in the following way:

1. Divide ∇ × H (r) by ϵ (r) and then take the curl:

∇ × 1 ϵ (r) ∇ × H (r) = -ωϵ 0 ∇ × E (r) .
2. Use ∇ × E (r) to eliminate E (r):

∇ × 1 ϵ (r) ∇ × H (r) = ω 2 ϵ 0 μ 0 H (r) .
3. Finally, the constants ϵ 0 and μ 0 can be combined to yield the vacuum speed of light, c = 1/ ϵ 0 μ 0 . The result is an equation entirely in H (r) :

∇ × 1 ϵ (r) ∇ × H (r) = ω c 2 H (r) . (2.8)
Equation (2.8) is referred to as the master equation. Together with the divergence equations (2.6), it contains all the information about H (r). Then, a common strategy to describe the propagation of light through complex media is: for a given structure ϵ (r), solve the master equation to find the modes H (r) and the corresponding frequencies, subject to the transversality requirement. Then use the curl equations (2.42) to find E (r): .9) This procedure guarantees that E satisfies the transversality condition, because the divergence of a curl is always zero. Thus, only one transversality constraint needs to be imposed, rather than two. The reason why the problem is formulated in terms of H (r) and not E (r) is merely one of mathematical convenience (detailed in section 2.2.1). However, H can also be found from E as:

E (r) =  ωϵ 0 ϵ (r) ∇ × H (r) . ( 2
H (r) = -  ωμ 0 ∇ × E (r) .
(2.10)

Electrodynamics as an eigenvalue problem

The meaning of the master equation, Eq. (2.8), is the following: perform a series of operations on a function H (r), and if H (r) is an available electromagnetic mode, the result will be a constant times the original function H (r). This situation arises often in mathematical physics, and is called an eigenvalue problem. If the result of an operation on a function is just the function itself, multiplied by some constant, then the function is called an eigenfunction or eigenvector of that operator, and the multiplicative constant is called the eigenvalue.

In this case, the left side of the master equation will be identified as the operator Θ acting on H (r): .11) with Θ the differential operator taking the curl, then dividing by ϵ (r), and taking the curl again:

ΘH (r) = ω c 2 H (r) , ( 2
Θ ∇ × 1 ϵ (r)
∇× .

(2.12)

The eigenvectors (also called eigenmodes) H (r) are the spatial patterns of the harmonic modes, and the eigenvalues (ω/ c) 2 are proportional to the squared frequencies of those modes. It is important to notice that the operator Θ is a linear operator. Therefore, any linear combination of solutions is itself a solution; if H 1 (r) and H 2 (r) are both solutions of Eq. (2.11) with the same frequency ω, then so is αH 1 (r) + βH 2 (r),

where α and β are constants. For this reason, two field patterns that differ only by an overall multiplier are considered to be the same mode.

The utility of solving electrodynamic problems as an eigenvalue problem is that the solutions of the system can be analyzed in terms of modes which, based on the properties explained below, will provide us information about the behavior of the system in a simple way. Additionally, because of the possible linear combinations, the system will have infinite number or solutions (modes). The problem does not require a source to be solved (as opposite to the standard linear problem), since the eigenmodes exist with no driving.

This operator notation is analog to quantum mechanics, in which an eigenvalue equation is obtained by operating on the wave function with the Hamiltonian operator.

The eigenfunctions of the Hamiltonian exhibit certain key properties, which will also hold for this formulation of electromagnetism: the eigenfunctions have real eigenvalues, they are orthogonal, and they can be classified by their symmetry properties.

These properties are briefly reviewed in the next section; a deeper description of each of them can be found for example in [START_REF] Shankar | Principles of quantum mechanics[END_REF].

Here, it is enough to recall that these properties rely on the fact that the main operator is a special type of linear operator known as a Hermitian operator. The definition of an Hermitian operator is based on the inner product of two vector fields F (r) and .13) where '*' denotes complex conjugation. From this definition of the inner product, it can be seen that:

G (r): (F, G) d 3 rF * (r) • G (r) , ( 2
(F, G) = (G, F) * ,
(F, F) is always real and positive, even if F itself is complex. In fact, if F (r) is a harmonic mode, it can always be set such that (F, F) = 1 as follows:

Given F' (r) with (F',F') = 1, F (r) can be defined as: .14) Because of the linearity property, F (r) and F' (r) are indeed the same mode, since they differ only by an overall multiplier, but now (F, F) = 1. F (r) has been normalized.

F (r) = F (r) (F , F ) . ( 2
Normalized modes are very useful to get information about the spatial distributions of the fields. If however, the physical energy of the field is of interest and not just its spatial profile, the overall multiplier should be taken into account.

Based on these properties of the inner product, a Hermitian operator  is defined such that: (2.15) for any vector fields F (r) and G (r). That is, it does not matter which function is operated upon before taking the inner product. Not all operators are Hermitian, but Θ is. The demonstration can be found in several in textbooks like [START_REF] Joannopoulos | Photonic crystals: molding the flow of light[END_REF].

F, G = F, G ,

General properties of harmonic modes

Since Θ is Hermitian, if H (r) is an eigenvector of Θ, its eigenvalues (ω/ c) 2 are real numbers.

ω 2 is always nonnegative for ϵ > 0. The Θ operator is said to be positive semidefinite.

Any two harmonic modes H 1 (r) and H 2 (r) with different frequencies ω 1 and ω 2 are said to be orthogonal modes: the Hermiticity of Θ forces H 1 and H 2 to have inner product equal to zero:

(H 1 , H 2 ) = 0.
The electromagnetic energy stored in a harmonic mode can be found through the electromagnetic variational theorem -not derived here- [START_REF] Joannopoulos | Photonic crystals: molding the flow of light[END_REF], in analogy to the variational principle of quantum mechanics. The timeaveraged physical energy can be separated into a contribution from the electric field, and a contribution from the magnetic field:

 E ≡ ϵ 0 4 d 3 rϵ (r) |E (r)| 2 ,  H ≡ μ 0 4 d 3 r|H (r)| 2 .
(2.16)

But in a harmonic mode, the physical energy is periodically exchanged between the electric and magnetic fields, and it can be shown that  E =  H . Roughly, a mode tends to concentrate its electric-field energy in regions of high dielectric constant, and to remain orthogonal to the modes with lower frequencies.

The rate of energy transport is given by the Poynting vector, S: 2.17) where ℜ denotes the real part. This is the time-average flux of electromagnetic energy in the direction of S, per unit time and per unit area, for a time-harmonic field. The component of S in a given direction is the light intensity. And the radio of the energy flux to the energy density defines the velocity of energy transport [START_REF] Jackson | Classical electrodynamics[END_REF].

S ≡ 1 2 ℜ{E * × H}, (
As pointed out before, the master equation (2.8) was expressed in terms of H for mathematical convenience. The reason is the following:

In terms of E, the master equation yields to: .18) Because there are operators on both sides, this equation is referred to as a generalized eigenproblem. We can divide by ϵ(r), however, the operator in the left side would be no longer Hermitian, therefore the properties of the harmonic modes would be slightly more difficult to analyze.

∇ × ∇ × E (r) = ω c 2 ϵ (r) E (r) . ( 2

Spectrum of an electromagnetic system

It is also convenient to analyze how the Hermiticity of an operator influences the continuity or discreteness of the spectrum produced by a given system.

The spectrum of an electromagnetic system consists of the totality of all its eigenvalues ω. The available frequencies are determined by the spatial domain of the mode profiles H (r) (or E). If the fields are spatially bounded, as in a waveguide for example, the frequencies ω form a discrete set. Otherwise they can form a single continuous range, a set of continuous ranges, or a combination of continuous ranges and discrete sets (for a combination of localized and extended modes).

Indeed the discreteness of the frequency spectrum is a general property for many

Hermitian eigenvalue problems since it is a consequence of the orthogonality of the modes [START_REF] Courant | Application of the calculus of variations to eigenvalue problems[END_REF]. Suppose a continuous range of eigenvalues, so that the frequency ω can be varied continuously to get some eigenmode H ω (r) for each ω.

As ω is continuously changing, the field H ω will change continuously as well, so that for an arbitrarily small change δω there is a correspondingly small change δH. If there are two spatially bounded modes H and H + δH that are arbitrarily similar, their inner product will be (H, H + δH) = (H, H) + (H, δH), where the first term is positive and the second term is arbitrarily small when integrated over a finite domain, i.e. a system with spatially bounded modes. Thus, the inner product = 0 and the continuous spectrum is therefore incompatible with the required orthogonality of the modes, unless the modes are of unbounded spatial extent.

Electromagnetic modes in different system's symmetries

It is useful to study the symmetries of electromagnetic systems because they allow to make general statements about the system's behavior. For example, we can simply determine the allowed modes in a system given its symmetry properties.

In this section, only two kinds of symmetries present in electromagnetic systems are studied, since those will be used in the design of our devices: the inversion symmetry and the continuous translational symmetry.

Inversion symmetry

Suppose the two-dimensional metal cavity shown in Fig. 1. It has the following symmetry: if inverted about its center, its shape is exactly the same. In this case, if a particular pattern H 1 (r) is a mode with frequency ω of the cavity, then the mode Then α should be +1 or -1. r). This is called an even mode. In the case that α = -1, then H 1 (r) = -H 1 (-r), which is called an odd mode. Now, it is useful to define the inversion symmetry in terms of operators. Suppose  is a matrix operator that inverts vectors, so that a = -a. To invert a vector field f (r), both the vector f and its argument r must be inverted: Ô f (r) = f (r). If a system has inversion symmetry, then: .19) this means that it is the same to operate a system by Θ or to first invert its coordinates, then operate it by Θ and then invert it back again. Equation (2.19) can be rewritten as Ô Θ -Θ Ô = 0. From which it is said that the operators Θ and Ô commute.

H 2 = H 1 (-
If α = 1, H 1 (r) = H 1 (-
Θ = Ô-1  Θ Ô. ( 2
In general, a commutator of two operators  and B is defined as: .20) And it has been shown that Θ and Ô commute: Θ, Ô = 0.

Â, B Â B -B Â. ( 2
Generally speaking, whenever two operators commute, it is possible to construct simultaneous eigenfunctions of both operators. This is convenient since the eigenfunctions and eigenvalues of Θ are not easy to determine, but if Θ commutes with a simple symmetry operator Ŝ, the symmetry properties can be used to determine a set of eigenfunctions of Θ. In the case of inversion symmetry, the Θ eigenfunctions are classified as either odd or even, as explained before.

Continuous translational symmetry

Let's now consider the propagation of light through a system that does not vary if we translate everything through some displacement d in a certain direction. It is then said that this system has continuous translation symmetry. For each d, a translation operator Td can be defined such that it shifts the argument by d. If our electromagnetic system is translational invariant; then Td ϵ (r) = ϵ (rd) = ϵ (r). Or equivalently, Td , Θ = 0. The modes of Θ can now be classified according to how they behave under the symmetry Td .

For example, for a system with continuous translational symmetry in the z direction, it can be easily verified that the functional form e kz is an eigenfunction of any translation operator in the z direction.

Tdẑ e kz = e k(z-d) = e -kd e kz .

(2.21)

The corresponding eigenvalue is e -kd and by linearity, any eigenfunction of Td for all d = dẑ must be proportional to e kz for some k (k must be a real number in an infinity system where the modes are required to have bounded amplitudes at infinity).

In fact, an homogeneous medium is a system with continuous translational symmetry in all three directions: ϵ(r) is a constant (ϵ = 1 for free-space). The modes must then have the form: (2.22) where H 0 is any constant vector. These are plane waves, polarized in the direction of Another way to state the meaning of translational symmetry is that the components of the wave vector k along the symmetry directions are conserved quantities. If a field pattern starts with a particular eigenvalue e -kd of Td , then it will have that eigenvalue at all the future times.

H k (r) = H 0 e k•r ,
H 0 .

Refraction at a single planar interface

Consider now another translational invariant system, such as a flat single interface between two dielectric materials of semi-infinite extent (an infinite half space is a region in space that is bounded at only one edge, it extends to infinity on all other sides) and with refractive index n 1 = ϵ 1 and n 2 = ϵ 2 as depicted in Fig. 2. This is a system with continuous translational symmetry along the interface (in the y-direction).

The refraction of a light ray is usually obtained from the Snell's law: n 1 sin θ  = n 2 sin θ t , where θ  is the angle that the incident ray makes with the normal to the interface. If θ  > sin -1 (n 2 / n 1 ), then the law would give sin θ t > 1, for which there is no real solution; the interpretation is that the ray is totally reflected. The critical angle θ c = sin -1 (n 2 / n 1 ) exists only for n 2 > n 1 , thus total internal reflection (TIR) occurs only within the higher-index medium. In fact, Snell's law is the combination of two conservation laws: Conservation of frequency ω. From the linearity and time-invariance properties of Maxwell equations. Note, however, that the wavelength is different in medium 1 and 2.

k || n 1 n 2 z x y x 1 || y x 2 y || x ||
Conservation of the component of k that is parallel to the interface, k . From the continuous translational symmetry along the interface. The conservation of k is the only way for the phase fronts to match (exist), at all times, everywhere along the interface.

From the geometry of the problem (see Fig. 2) k = |k| sin θ  , where  = 1, 2; and with the dispersion relation |k| = nω/ c, Snell's law is obtained by equating k in both sides of the interface. Using the translational symmetry to understand this problem will allow us to generalize beyond the ray-optics regime, which is only valid on length scales much larger than the wavelength of light.

Let's now analyze the electromagnetic modes of this system, i.e. the allowed frequencies ω versus the wave vectors k (both of them conserved quantities). The normalized dispersion curve for an air-glass interface (ϵ 1 = 1, ϵ 2 = 11.4) is plotted in Fig.

3.

First, consider the modes that do not suffer TIR into the glass, these modes are not confined and they extend into the air and out to infinity. Far away from the glass they are superpositions of free-space plane waves with ω = c|k| = c k 2  + k 2 . For a given value of k , there will be modes with every possible frequency greater than ω = ck (the light line) because k  can take any value. Thus, the spectrum for every frequency above the light line is continuous, as marked in Fig. 3. The modes above the light line are real solutions to the Snell's law. However, the presence of glass introduces new solutions that will lie below the light line. These solutions can take only discrete values of ω, therefore the spectrum below the light line will be discrete. The explanation is as follows:

Since n gss > n r , the modes will have higher |k| in the glass than in air (k = nω/ c).

At some point (for light traveling from glass to air with θ  ≥ θ c ), k will become too long to be conserved across the interface (or alternatively, the spatial variations at the interface will be to high to be conserved -as k increases, λ decreases); and as a consequence, TIR will exist. In the glass, the total reflected fields will interfere with each other and only those interfering constructively will remain, they will form a set of discrete solutions, called the guided modes.

Below the light line, the only possible solutions in air are those with imaginary k  = ± k 2ω 2 / c 2 , which correspond to fields that decay exponentially away from the glass, they are localized in the vicinity of the glass. In the air, these are called evanescent fields. A summary of the possibilities of k conservation for a TE mode (see section 2.2.3 about mode's polarization) of this system is presented in Fig. 4. Notice that through the analysis of the modes, nothing has been said about their amplitudes, polarization or phase changes, neither for the reflected nor refracted mo-des. This is so, since the symmetry of the problem does not provide any information regarding those dynamic properties. Usually, they are given in terms of the Fresnel coefficients (see for example in [START_REF] Jackson | Classical electrodynamics[END_REF]). Fresnel coefficients are nothing but the application of appropriate boundary conditions:

n 1 <n 2 n 1 >n 2 i < c n 1 n 2 k || k || n 1 n 2 k || k || k || k || n 1 n 2 n 1 >n 2 i > c 0 0 -1 1 -0.
The component of the electric E and the magnetic H fields that is parallel or tangential to the boundary between two media is conserved.

E ,1 = E ,2 ; H ,1 = H ,2 .
(2.23)

The component of the displacement D and the magnetic induction B fields that is perpendicular or normal to the boundary between two media is conserved.

D ⊥,1 = D ⊥,2 ; B ⊥,1 = B ⊥,2 .
(

From this refraction problem we have seen that:

The incident field (plane wave) is scattered (split) by an object with ϵ(r).

Translation invariance provides parallel momentum conservation. This is the principle of guiding light in waveguides, for example in optical fibers.

Figure 5a).

TIR implies evanescent fields, which can be used in microscopy. Figure 5b).

Boundary conditions determine the amplitudes.

More generally, from this problem, we can see that the spatial distribution of matter i.e. ϵ(r), μ(r) controls the behavior of light fields.

Indeed, nanophotonics is about controlling the light that happens to be very different from a plane wave by engineering such specific material properties (ϵ,μ) in space and over length scales << λ 0 = 2πc/ ω, the vacuum wavelength; recalling that the refractive index and consequently the electric permittivity and magnetic permeability are not absolute constants of materials, but depend on frequency (the dispersion phenomenon). In the following sections, the magnetic permeability will be considered as μ = 1, since it is the case for ordinary materials at optical frequencies. Then we will focus to the understanding of the dielectric constant ϵ.

Microscopic description of the dielectric constant

In dielectric materials, the electrical charges are strongly attached to specific atoms or molecules. When an external electric field is incident to the material, individual microscopic displacements of charges will occur [Fig. 6a)]. Such microscopic displacements of charges, in all the atoms or molecules together, will produce a macroscopic induced polarization in the medium. (sketched in Fig. 6b). To describe the macroscopic response of a linear material we use:

D = ϵ 0 E + P. (2.25)
Expressing the constitutive relation [Eq. (2.3)] in terms of the macroscopic dielectric constant (or relative permittivity) and susceptibility of the material: Then, the derivation of the microscopic ϵ can be summarized as follows:

D = ϵ 0 ϵE = ϵ 0 χE, ( 2
1. The atomic motion is described by comparing bound charges to a mass on a spring, according to the Lorentz osillator model [START_REF] Born | Chapter XI -Rigorous diffraction theory[END_REF].

∂ 2 r ∂t 2 + γ ∂r ∂t + ω 2 0 r = - e m e E, (2.29)
where e and m e are the charge and mass of the electron, respectively. γ is a damping term that represents internal collisions in the solid and radiation emitted by the electron (any accelerating charge emits radiation). It is the simplest possible type of damping, being linearly proportional to the velocity of the mass. The constant ω 0 is the resonant frequency (also called natural frequency or fundamental frequency) of an undamped spring-like oscillator. This frequency basically represents the spring force yielding to the electron's displacement and it is derived from Hooke's law as ω 0 = k/ m, being k the spring constant. E and the displacement r are considered to vary harmonically over time: E = Ee -ωt , and

r = re -ωt .
2. This equation can be solved for r in the frequency domain by Fourier transformation:

r(ω) = - e m e E(ω) ω 2 0 -ω 2 -ωγ .
(2.30)

3. The electric moment of the charge displaced by r is then obtained as μ = -er:

μ(ω) = e 2
m e E(ω) .31) 4. The statistical volume-averaged dipole moment defines the material polarization as: .32) with N the atoms density.

ω 2 0 -ω 2 -ωγ . ( 2
P(ω) = 1 V μ  (ω) = N〈μ(ω)〉 = ϵ 0 χ(ω)E(ω). ( 2
5. Using the two right-hand terms from the above equation, the susceptibility is: .33) where the plasma frequency is defined as: .34) 6. Finally, as χ = 1 + ϵ, we have: .35) Equation (2.35) corresponds to the microscopic frequency-dependent response of dielectric materials to an incident electric field. But, what about metals?

χ(ω) = ω 2 p ω 2 0 -ω 2 -ωγ . ( 2
ω 2 p = Ne 2 ϵ 0 m e . ( 2
ϵ(ω) = 1 + ω 2 p ω 2 0 -ω 2 -ωγ . ( 2
In 1900, Paul Drude proposed the Drude model for electrical conduction, the result of applying kinetic theory to electrons in a solid. In metals, the electrons are not bound to the nuclei. These delocalized electrons constitute which is known as the "cloud of electrons" that flows freely around the lattice of nuclei, and are the one thing that allows metals to conduct electricity. Consider a particular change in the Lorentz oscillator model: in this metallic bond, if the electrons are not bound, then there is no analogous of a restoring "spring" force. Hence, the equivalent spring constant associated with is k = 0, yielding to ω 0 = 0. Nevertheless, there is still a damping term, mostly due to the collisions within the electron cloud and with the nuclei. Setting ω 0 = 0 in Eq.

(2.35), the model then becomes: .36) The definition of ω p in Eq. (2.34) remains the same, except that N is now interpreted as electron density N e , and m e is the effective mass of the electron. This is the so-called Drude-Lorentz model for metals. Although updated models exist for metals, which take advantage of quantum theory and Boltzmann statistics, the Drude model is very useful to develop accurate intuitions regarding the optical behavior of metals.

ϵ(ω) = 1 - ω 2 p ω 2 + ωγ . ( 2
As observed from Eqs. (2.35) and (2.36), the dielectric permittivity is a complex function of frequency. By further analyzing the real and imaginary parts of this function at optical frequencies (analyses found in many textbooks as the previously cited), it follows that:

Dielectrics have real ϵ > 1, corresponding to bound charges.

The imaginary part of ϵ signifies loss. In metals, this represents Ohmic resistance.

A perfect metal has real ϵ = -∞.

Real metals have real ϵ < 0 up to the ultraviolet and visible frequencies (i.e.

ω < ω p ).

According to the Lorentz-Drude model, the free electrons forming an ionic cloud or 'plasma' explain the reflectivity of metals at low frequencies. The metal is shielded from external fields by surface charges, and the permittivity represents how quickly the charges can move to shield fields. For frequencies higher than ω p , loss vanishes and metals become transparent (the fields move faster than the shielding plasma cloud). For example, for gold, the charges oscillate at a plasma frequency of 2,185 THz.

Then, electromagnetic waves with wavelength above λ p ≈ 140 nm, cannot propagate through it.

Another fundamental concept necessary to understand the manipulation of light is the classical Green's function, which relates an electric or magnetic field to the source that originates it. As will be seen later, the Green's function will allow us to relate the quantum-mechanical process of spontaneous emission with a classical description of an oscillating dipole source.

The total Green's function

For a closed non-absorbing medium, the electric Green's function G(r, r 0 , ω) is defined as the solution to the inhomogeneous wave equation: .37) with  being the unit tensor, together with appropriate boundary conditions on the surface of the closed cavity (the boundary conditions are the same as for the electric field, see section 2.2.1.1). The Green's function defined in this way has a simple physical interpretation: for a monochromatic point electric dipole (ED) source with dipole moment, d, located at a position r 0 , the electric field radiated at a point r is: .38) In terms of the eigenmodes of the system, the Green's function takes the form [START_REF] Carminati | Electromagnetic density of states in complex plasmonic systems[END_REF]:

(∇ × ∇×) -ϵ (r) ω 2 c 2 G (r, r 0 , ω) = δ (r -r 0 ) , ( 2
E (r) = μ 0 ω 2 G (r, r 0 , ω) d. ( 2
G (r, r 0 , ω) = k c 2 e * k (r 0 ) ⊗ e k (r) ω 2 k -ω 2 , ( 2.39)
Here, the summation runs over all the eigenmodes e k of the system with eigenfrequencies ω k . The eigenmodes e k are the solutions to the generalized eigenvalue problem detailed in Eq. (2.18) i.e. the homogeneous wave equation, normalized by the condition d 3 rϵ(r)e  (r) • e j (r) = δ j . A relationship between the classical Green's function to the local density of states, which is defined in terms of the quantummechanical description of the spontaneous emission rate will be derived in section 3.3.1. Indeed, the relationship between these quantum-mechanical and classical quantities is at the center of this work.

Now that we have reviewed the basic interactions of electromagnetic waves with matter at optical frequencies, we can start to manipulate it by engineering different devices. We proceed with the description of light propagation through waveguides.

Trapping light in photonic waveguides: confinement limited by diffraction

We have seen from the refraction problem in section 2.2.1.1 that it is possible to confine (trap) light into devices by TIR. In general, the structures used to confine the propagation of waves to a single path are called waveguides. There are some generalities regarding any kind of waveguide:

Waveguides support an infinite number of discrete modes.

The modes have a constant amplitude profile that just accumulates phase as it propagates.

Modes have cutoff frequencies, below which they are not supported and decay very quickly.

As pointed out before, a guided mode exists if:

In the high index medium, the component of the wavevector perpendicular to the interface is k 2 ⊥ > 0.

In the low index medium, k 2 ⊥ < 0 (exponential field).

And we have seen, that according to the inversion symmetry of a system, the modes can be classified as even or odd modes. Another way to classify them is according to the orientation of linearly polarized waves relative to specific devices. In this manner, the guided modes are classified as (see sketch in Fig. 7):

TE: Transverse electric mode/perpendicular/s. The electric field is polarized perpendicular to the plane of incidence.

TM: Transverse magnetic mode/parallel/p. The electric field is polarized parallel to the plane of incidence.

For example, a plane wave propagating in free-space is said to be a TEM (transverse electromagnetic) wave . It means that the component of both the electric and Plane wave in free space a) b) magnetic field in the propagation direction of the wave is zero (the electromagnetic field is transverse to k). Similarly, in a TE mode, the component of the electric field in the propagation direction is zero, whereas for a TM mode, the component of the magnetic field in the propagation direction is zero.

In general, solving a waveguide (i.e. finding the allowed modes supported by a waveguide and their parameters) is rather complicated, therefore computational techniques, such as the ones described in section 3.5 are commonly used. Nevertheless, it should be pointed out that, as before, the guided waves should follow the k conservation principle, and they must satisfy boundary conditions at every interface. A formal waveguide analysis can be found in [START_REF] Snyder | Optical waveguide theory[END_REF]. Here, the general steps are given (illustrated in Fig. 8): 1. Draw the waveguide.

2. Assume a form of the solution. Consider that in regions outside the waveguide the fields must decay exponentially (dielectric) or vanish (metallic). In general, a mode in a waveguide with propagation direction along the x-axis has the following mathematical form:

E (, y, z) = E 0 (y, z) e -β ;
H (, y, z) = H 0 (y, z) e -β .

( 2.40) where E 0 and H 0 are complex amplitudes that represent the spatial distribution of the modes, e -β is the term representing the accumulation of phase in the  direction, and β is a phase constant. See Fig. 8.

3. Substitute solutions based on the symmetry (geometry) of the waveguide.

4. Manipulate equations into the governing wave equation. For the eigenvalue problem, this is the master equation (2.8).

5. Solve the master equation in each homogeneous region of the waveguide.

6. Connect the solutions in each region using boundary conditions.

7. Obtain the overall field solution.

8. Use the field solution to calculate the waveguide parameters such as the profile of the fields and the phase constant β.

The confinement of light into waveguides is a very common procedure in order to transport optical information. In an overwhelming majority of applications today, dielectric 'glass' photonics are the guiding devices (i.e. optical fibers), able to guide light with spatial size around > (λ/ 2) 2 . Metallic devices such as films, ridges, and grooves, on the other hand, are able to decrease the spatial distribution of light up to (λ 0 / 100) 2 . This difference among dielectric and metallic guides is due to boundary conditions, as exemplified in the slab (planar) waveguides shown in Fig. 9.

Indeed, over the last two-three decades, the basic principle of confining and guiding light has enabled not only low-loss transportation of light, but a wide range of applications based on the control and manipulation of light at the micrometer scale.

This is at the heart of Integrated Optics technologies [START_REF] Hunsperger | Integrated optics: theory and technology[END_REF]. In particular, beyond the fiber optics industry, there is now significant interest at integrating photonic components onto a silicon chip (because of reasons explained below). This recent field of research has coined the name of Silicon Photonics.

Silicon Photonics waveguides

The aim of Si photonics is to integrate all the optical components onto a single (or a few) chips, taking advantage of the Si dominance in microelectronics for many decades, which is predicted to continue for the forseeable future [START_REF] Paul | Silicon photonics: a bright future?[END_REF]; thereby reducing cost and also potentially increasing the performance of systems for a given application.

With the idea of creating optical waveguides to propagate light on chip-scale packages, scientists started to use silica (SO 2 ), which is an easily available material. The technology came to be known as silica on silicon (S-SO 2 ) or silicon on insulator (SOI),

where the silicon (high refractive index of ≈ 3.5) was embedded within silica (lower refractive index of 1.4). The fabrication techniques for silicon were well established (thanks to electronic chips) and at the same time, silicon was compatible with other CMOS techniques, which helped boost research into silicon photonics technology. Additionally, efficient coupling with standard optical fibers is readily obtained using either grating [START_REF] Roelkens | High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit[END_REF] or inverted taper couplers [START_REF] Almeida | Nanotaper for compact mode conversion[END_REF]. Figure 10 shows examples of common Si waveguides, among which single-mode silicon ridge waveguides with cross section of 220 nm by 400 nm are today a standard in silicon photonics [START_REF] Delacour | Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics[END_REF].

Optical confinement in dielectric waveguides

Up to date, dielectric waveguides, including Si photonic waveguides, are widely used for low-loss transport of light (i.e. control over the propagation of light is achieved). However, the control over absorption or emission of individual photons is not possible by the use of regular dielectric optics. The reason of this limitation is that the achievable spatial confinement in dielectric waveguides cannot go below (λ/ 2).

So, for light in the telecommunications window (λ = 1550nm), for example, dielectric waveguides can achieve a maximum confinement of about ≈ 750nm. If we consider the size of an atom as the Bohr radius, of about 1 Angstrom, or ≈ 0.1 nm, or the size of an ensemble of atoms such as in a quantum dot to be of ≈ 1 -10 nm, then there is a three orders of magnitude size mismatch between k photon vs. k tom , and individual light-matter interactions cannot be controlled. The question is then, how to confine the light further to make it interact with individual light emitters and receivers?

As will be shown, negative ϵ (r) are required: i.e. metals. This is the reason why we will use plasmonics (optics in metals) to achieve sub-diffraction confinement. This will be further explored in section 2.4.1. For the moment, the physical concepts of electromagnetic reciprocity and Babinet's principle are introduced. These principles play a fundamental role in the methodology towards the design of the devices that we propose in this thesis.

Reciprocity and Babinet's principles for electromagnetic sources and fields

In this section, Babinet's principle is presented to understand the relationship between electric and magnetic fields produced by complementary screens; together with the electromagnetism reciprocity theorem which relates electromagnetic fields to the sources that originate them.

Babinet´s principle for electromagnetic fields

In scalar diffraction theory, Babinet's principle relates the diffraction pattern of a screen to that of its complementary one. The geometry of a simple problem illustrating this principle is schematically shown in Fig. 11. The diffraction screens S 1 and S 2 are complementary in a sense such that S = S 1 + S 2 . In the absence of either screen [Fig. 11c)], a given source producing a field ψ in Region I will propagate to Region II without perturbation. When the screen S 1 is placed [Fig. 11a)], a perturbation ψ 1 will be created in Region II. Similarly, if now S 2 is separating regions I and II [Fig. 11b)],

the perturbation field ψ 2 will be generated. The relationship between the fields ψ 1 and ψ 2 will be given by: ψ = ψ 1 + ψ 2 . In the case that ψ represents an incident plane wave, Babinet's principle states that the diffraction patterns are the same for both screens away from the incident direction (θ = 0). This is:

From Fig. 11c, ψ = 0 for θ = 0.

Then

ψ 1 = -ψ 2 .
Thus

 1 =  2 .
This result is valid for scalar fields (i.e. independent of polarization). However, a rigorous vectorial statement of Babinet's principle for electromagnetic fields can be made for thin, perfectly conducting, plane screen S and its complementary screen S c .

A required condition to satisfy the full vectorial Babinet's principle, is that S c must be illuminated by the complementary source fields: E c 0 (r) = cB 0 (r) and B c 0 (r) = -E 0 (r)/ c, such that the complementary source fields have opposite polarization characteristics relative to the fields incident to S, E 0 (r) and B 0 (r). Given this condition, Babinet's principle states that the total fields (incident + scattered) behind the screen satisfy [START_REF] Jackson | Classical electrodynamics[END_REF]:

I 1 | 1 | 2 I 2 | 2 | 2
E(r) = E 0 (r) + cB c (r), B(r) = B 0 (r) -E c (r)/ c. (2.41)
To illustrate the vectorial Babinet's principle, consider the triangular screen (original screen S O ) and it's complementary screen (S C , with a triangular aperture) in Fig. 12.

The behavior of E due to S O is the same as B due to S C , with rotated polarization.

The aim of using the vectorial Babinet's principle is to deduce unknown magnetic field distributions based on known electric field distributions for a given set of complementary structures. Recall that, in general, this principle holds in the case of thin, perfectly conducting, planar screens in free-space. However, the integrated plasmonic structures that we will deal with do not satisfy these conditions: they have finite size as well as finite conductivity, and they are not located in free-space but coupled to a waveguide. Therefore, in chapter 4 we investigate its applicability to integrated plasmonic structures on a Si waveguide configuration, considering that the principle has been already studied in the case of finite thickness and conductivity [START_REF] Koo | Extraordinary magnetic field enhancement with metallic nanowire: role of surface impedance in Babinet's principle for sub-skin-depth regime[END_REF], leading to a successful demonstration.

Together with Babinet's principle, it is convenient to introduce another fundamental property of classical electromagnetic systems that will serve for design purposes in this work. This is the well-known electromagnetism reciprocity theorem, which provides a deeper and direct understanding of the behavior of the harmonic fields generated by an optical system, in response to oscillating current source distributions located at various points.

Electromagnetism reciprocity theorem for sources

Electromagnetic reciprocity is a property of wave propagation and scattering that relates time-harmonic electric current densities (sources) and their resulting electromagnetic fields in time-invariant linear media. Basically, it states that, under certain constraints, the relationship between an oscillating current and its resulting electric/magnetic field is unchanged if the position of the source and the position where the field is measured are interchanged. Indeed, the Green's tensor of the wave equation in the presence of a scatterer satisfies reciprocity. In addition, this property holds whatever the distance between the scatterer and the observation points. The reciprocity principle and its application to current dipole sources is fundamental for the design of the nanophotonic device proposed in this work, as detailed in chapter 4.

The derivations of the reciprocity relations in electromagnetism are based on the use of an integral theorem that is due to Lorentz [START_REF] Nieto-Vesperinas | Scattering and diffraction in physical optics[END_REF], and it is usually applied without sources near the scatterer, which implies dealing with sourcefree fields, namely, fields without evanescent components. However, for the purposes of this work, evanescent waves need to be considered since a waveguide configuration is employed. Thus, in this section, the Lorentz's reciprocity theorem with sources present at a finite distance from the scatterer is detailed.

Electric and magnetic dipole sources

Consider two different situations as depicted in Fig. 13. In a first situation, V 1 is a source volume with a current density J 1 (r) radiating at a frequency ω. E 1 (r) and

a)

scatterer The fields in each situation satisfy the Maxwell's curl equations:

J 1 (r) b) V 2 scatterer J 2 (r) V 1 E 1 (r) H 1 (r) E 2 (r) H 2 (r) ε r, ) μ r, ) ε r, ) μ r, )
∇ × E k (r) = ωB k (r), ∇ × H k (r) = J k (r) -ωD k (r), (2.42)
where k = 1, 2. The scatterer might be an inhomogeneous and anisotropic medium, as long as the linear constitutive relations hold:

D k (r) = ϵ 0 ↔ ϵ (r, ω) • E k (r), B k (r) = μ 0 ↔ μ (r, ω) • H k (r). (2.43)
Carrying out the same procedure as in reference [START_REF] Landau | En: Electrodynamics of continuous media[END_REF], the following scalar products, for each point r, are taken:

H 2 • ∇ × E 1 , E 2 • ∇ × H 1 , -H 1 • ∇ × E 2 , -E 1 • ∇ × H 2 ,
and adding them all together:

(H 2 • ∇ × E 1 -E 1 • ∇ × H 2 ) + (E 2 • ∇ × H 1 -H 1 • ∇ × E 2 ) ,
yields to:

= ω (B 1 • H 2 -H 1 • B 2 ) -ω (D 1 • E 2 -E 1 • D 2 ) + (J 1 • E 2 -J 2 • E 1 ) . (2.44)
Substituting the constitutive relations in each scalar product:

B 1 • H 2 = μ 0 ↔ μ • H 1 • H 2 , H 1 • B 2 = H 1 • μ 0 ↔ μ • H 2 , D 1 • E 2 = ϵ 0 ↔ ϵ • E 1 • E 2 , E 1 • D 2 = E 1 • ϵ 0 ↔ ϵ • E 2 .
Provided the tensors ↔ ϵ (r) and ↔ μ (r, ) are symmetrical, i.e.:

↔ ϵ (r) = [ ↔ ϵ (r)] T and ↔ μ (r) = [ ↔ μ (r)] T , then B 1 • H 2 = H 1 • B 2 and D 1 • E 2 = E 1 • D 2 .
Also, by vector identities the left-hand side in the equality (2.44) can be written as ∇

• (E 1 × H 2 -E 2 × H 1 )
. Under these conditions, we have:

∇ • (E 1 × H 2 -E 2 × H 1 ) = J 1 • E 2 -J 2 • E 1 . (2.45)
Integrating this equation over all space; the term in the left-hand side can be transformed into a surface integral over a sphere whose radius tends to infinity, and gives zero. Then: .46) which constitutes the reciprocity theorem with sources. The integrals are taken only over the volumes V 1 and V 2 respectively, since J 1 and J 2 are zero elsewhere.

V 1 J 1 (r) • E 2 (r)d 3 r = V 2 J 2 (r) • E 1 (r)d 3 r, ( 2
In the particular case of ED sources, the current densities are given by J k (r) = -ωd k δ (rr k ), where d k represents ED moments. After substitution into Eq. (2.46) we have:

d 1 • E 2 (r 1 ) = d 2 • E 1 (r 2 ) , (2.47)
where E 2 (r 1 ) andE 1 (r 2 ) represent the fields due to each of the two sources at the position of the other. The reciprocity theorem in this form (i.e. for dipole sources) states that the component of the electric field in the direction of the polarization of the source is unchanged when the positions of source and detector are interchanged.

Following the same procedure but taking into account higher-order transitions in the multipolar expansion of the source, the reciprocity theorem for magnetic dipole sources is [START_REF] Landau | En: Electrodynamics of continuous media[END_REF]: .48) where m k are the magnetic dipole moments.

B 2 (r 1 ) • m 1 = B 1 (r 2 ) • m 2 , ( 2
Reciprocity is a fundamental property that will provide us with a powerful tool to design complex photonic systems. It holds whenever the electric permittivity and the magnetic permeability at ω, are symmetric 3×3 matrices (symmetric rank-2 tensors), including the common case where they are scalars (for isotropic media), of course.

They need not be real, therefore materials with losses are included, such as conductors with finite conductivity σ (which is included in ϵ via ϵ = 1 + σ ωϵ 0

). The symmetry condition is almost always satisfied for linear materials. For nonlinear media, no reciprocity theorem generally holds. Reciprocity does not generally apply for time-varying (active) media either; for example, when ϵ is modulated in time by some external process. (In both of these cases, the frequency ω is not generally a conserved quantity).

Finally, in this chapter, the concepts of surface waves and evanescent fields are presented; which will allow us to understand the propagation of electromagnetic waves along interfaces between dielectric and metallic media.

Surface waves and evanescent fields

A surface wave can be thought as a kind of guided mode, which is confined to the interface between two different materials comprising two infinite half spaces. The field decays exponentially away from the interface (evanescent fields); but, along the interface, it is free to propagate without decaying (surface waves).

Surface waves and evanescent fields are consequence of the translational symmetry along a planar interface explained in the refraction problem of section 2.2.1.1.

According to this translational invariance, the component of the wavevector which is parallel to the interface, k , should be the same in both materials; but if such a component is too long to be conserved in one of them, the field may not propagate into it. Under this condition, evanescent fields confined to the low-index material along the propagation direction will be created. These fields decay exponentially away from the interface and they do not transport energy since no oscillations are present. However, in the direction parallel to the interface, oscillations transporting energy do exist. Figure 14, illustrates the resulting surface waves for different values of θ  at the same interface between air, ϵ = 1, and glass with ϵ gss = 11.4, discussed in section 2.2.1.1.

As observed, the larger the θ  above the critical angle θ c , the higher the spatial frequency of the surface waves (an the smaller the evanescent fields).

Several types of surface waves exist, but the one of interest to control light-matter interactions at the nanoscale, must be one including a material with ϵ (r) < 0 (like metals below the plasma frequency). For this reason, a particular kind of surface wave relevant to this work is introduced, the so-called surface plasmon polariton (SPP). To describe the concept of SPPs, the familiar principle of k will be used, and it will be shown that a SPP is indeed a kind of surface wave produced at dielectric-metallic interfaces as a consequence of the same k-matching condition presented before.

Near field microscopy principle

Before analyzing the conditions and characteristics of surface plasmon polaritons, let's first briefly introduce the principle of operation of near field microscopy since it is strongly related to the generation of evanescent fields and surface waves as a result of the k matching condition.

Indeed, near-field optics refers to the study of non-propagating inhomogeneous (evanescent) fields and their interaction with matter. Optical near-fields are localized to the source region of optical radiation or to the surfaces of materials interacting with free radiation (secondary sources). In near-fields, the optical energy is localized to length scales smaller than the diffraction limit of roughly λ/ 2 (see section 2.4.3).

To illustrate the simplest near-field interaction, consider the same planar interface between two infinite half-spaces, where ϵ 1 > ϵ 2 , thus producing TIR and the generation of evanescent fields in the material with ϵ 2 .

As depicted in Fig. 15 from images 1 to 6, if a medium with higher refractive index approaches closer and closer to the evanescent field, up to distances below the wavelength (and even below λ/ 2), it may no longer be cutoff and become a propagating wave that contains the information of the field traveling in the material ϵ 1 . In this way, by approaching a high-index material (i.e. a tip of a microscope) to the evanescent field localized at the the interface of a waveguide, for example, it is possible to recover information regarding the electromagnetic field that is propagating confined to the waveguide. This phenomenon is also called frustrated total internal reflection or electromagnetic tunneling, in analogy to electron tunneling through thin insulators.

Further details about the near-field scanning optical microscopy (NSOM) technique will be presented in chapter 4.

Surface plasmon polaritons

Surface plasmons polaritons are surface waves that propagate, specifically, along interfaces between a material with positive dielectric constant (a dielectric) and a material with negative dielectric constant (a metal). As sketched in Fig. 16 , an SPP is a coupled state resulting when an electromagnetic field propagating in the dielectric medium couples to electron charge density waves present in the metallic surface. In this special kind of surface wave, the field decays exponentially (there are evanescent The system in Fig. 16 exhibits continuous translation symmetry along the interface between two infinite half-spaces with ϵ 1 (dielectric) and ϵ 2 (metal), respectively. We start with the master equation (2.8), but the two media will be analyzed independently as homogeneous materials, in such a way that ϵ will be constant in each medium, then:

1 ϵ  (∇ × ∇ × H  ) = ω c 2 H  ,
where  = 1, 2. From Eq. (2.9), we have:

E  =  ωϵ 0 ϵ  ∇ × H  .
From the geometry of the problem and boundary conditions we have:

1. Because the interface comprises two infinite half-spaces in the z-plane, every derivative along y vanishes: ∂ ∂y = 0.

2. If a wave is to propagate along the surface of a metal, the electric field must be polarized normal to the surface. Otherwise, boundary conditions will require it to be zero. According to Fig. 7b), this configuration corresponds to a TM mode:

H  = 0, E  = 0;
the electric field is polarized parallel to the plane of incidence (E z = 0) and the magnetic field perpendicular to it (H y = 0). Notice that this should always be the case for a SPP, they are always TM modes.

With these considerations:

∇ × H  = x - ∂H y ∂z + ŷ - ∂H z ∂ + ẑ ∂H y ∂ , ∇ × ∇ × H  = x ∂ 2 H z ∂z∂ -ŷ ∂ 2 H y ∂ 2 + ∂ 2 H y ∂z 2 + ẑ ∂ 2 H z ∂ 2 = ϵ  ω c 2 xH  + ŷH y + ẑH z  .
Now, if the wave is a surface wave, it must be confined to the interface. This can only happen if the field decays exponentially away from the interface (in the z direction).

And, from the k matching condition, the component k  should be the same along the interface in both materials, and only k z will differ. A general solution for the magnetic field is then assumed in the form:

H  (z) = H y, e -β e -ϰ  |z| , with      z > 0, if  = 1 z < 0, if  = 2 ;
where k  = β and k z, = ϰ  , with ϰ real. Substituting in the master equation for H y, :

β 2 -ϰ 2  H y, = ϵ  ω c 2 H y, ;
from which the general dispersion relation for the  th medium is obtained as:

β 2 -ϰ 2  = ϵ  k 2 0 , ϰ 2  = β 2 -k 2 0 ϵ  , (2.49) with k 2 0 = ω 2 / c 2 .
The dispersion relation has the same form in both media because ∂ 2 H y, / ∂ 2 is negative for  = 1, 2 and ∂ 2 H y, / ∂z 2 is positive, independent on the medium. This implies that k z, is imaginary for  = 1, 2, representing evanescent decay in both materials.

In the dielectric, E  will be:

E ,1 = - ϰ 1 ωϵ 0 ϵ 1 H y,1 ,
and in the metal:

E ,2 = ϰ 2 ωϵ 0 ϵ 2 H y,2 .
Applying boundary conditions, E  and H y are both components tangential to the boundary and must be conserved, therefore E ,1 = E ,2 and H y,1 = H y,2 Then:

- ϰ 1 ϵ 1 = ϰ 2 ϵ 2 .
(2.50) Equation ( 2.50) is known as the existence condition of a SPP. From this, we have that in order for a SPP to exist:

ϵ 2 = - ϰ 2 ϰ 1 ϵ 1 . (2.51)
To have exponential decay (evanescent field) in both sides of the interface, ϰ 1 and ϰ 2 should be positive quantities. This implies that ϵ 1 and ϵ 2 must have opposite signs to support a SPP. The dispersion relation in terms of the propagation constant β is:

β = k 0 ϵ 1 ϵ 2 ϵ 1 + ϵ 2 . (2.52)
Recall that Eq. (2.36) gives ϵ (ω) for metals, according to the Lorentz-Drude model.

Indeed, this equation can be simplified even further. In a plasma, the electrons in the "cloud" are far enough from each other and from the ionized nuclei that they do not collide, statistically speaking (i.e., the plasma has a very small value of mean free collision time). Then, the only loss of energy would be due to re-radiation, but then that re-radiation would be absorbed by some other dipole and re-re-radiated, "ad infinitum". For this reason, the damping term γ is null. The simplified permittivity equation for plasmas is then just:

ϵ 2 (ω) = 1 - ω 2 p ω 2 .
(2.53)

An expression for the surface plasma frequency can be derived, by substituting Eq.

(2.53) into the dispersion relation, letting ω = ω sp and taking the limit when β → ∞:

ω sp = ω p 1 + ϵ 1 .
(2.54) lectric materials with ϵ 1 > ϵ 2 , therefore generating an evanescent field in the direction perpendicular to the interface. For a certain frequency ω, this evanescent field would couple to ω spp , thus creating a SPP at the interface between the materials with ϵ 2 and ϵ(ω), respectively. Figure 18a).

2. Kretschmann configuration. The concept is the same, except that in this case the positions of ϵ 2 and ϵ(ω) are exchanged for which the layer with ϵ(ω) (usually a metal) should be very thin so that the evanescent field is able to penetrate the skin and to couple to the bottom material (ϵ 2 ). Figure 18b).

3. Grating coupler configuration. In a diffraction grating, a wave vector incident to it is expanded into high-order modes, each of them with a higher k than the previous one. Then, a high-order spatial harmonic (usually the second order) produces a high spatial frequency that matches the propagation constant of the surface wave. If the lower material has negative ϵ, this surface wave will be a SPP. Figure 18c).

4. Evanescent coupling configuration. Although all the configurations explained above make use of evanescent fields generated by TIR, this configuration refers specifically to those evanescent waves created around a waveguide. The evanescent field outside the core of a waveguide has a high spatial frequency that is cutoff by the cladding materials. Once again, such spatial frequency will match the one supported at the interface between a material with negative ϵ and the cladding to generate a SPP. Figure 18d).

It is interesting to note that in the Otto, Kretschmann and grating coupling configurations, reflectance measurements on metallic films are possible, which exhibit a characteristic reflection dip beyond the critical angle of total internal reflection. The reflection dip is consistent with thin film theory, showing that it is associated to the generation of an evanescent wave (and consequently, of a SPP) because there is no transmitted light.

For the design of the hybrid photonic-plasmonic device proposed in this work, the evanescent coupling configuration is used to excite SPPs.

dielectric 1 >0 2 (< 1 ) (ω)<0 a) dielectric ε 2 >0 ε 1 >0 ε(ω)<0 dielectric ε(ω)<0 dielectric ε 1 >0 ε 2 (<ε 1 ) dielectric b) c) d) ε(ω)<0

Optical confinement limit: subwavelength versus subdiffraction

The field associated with SPPs can be highly confined to both subwavelength and subdiffraction limits, in contrast to other systems, as sketched in Fig. 19. For simplicity this will be shown by analyzing 2D media but the same applies for 3D materials.

It has been previously stated that the dispersion relation satisfy the equation of a

circumference of radius k, k 2  + k 2 z = k 2 = k 2 0 ϵ
, with k 0 = 2π/ λ 0 being λ 0 the free-space wavelength. In the k-space, by Fourier transformation, the spatial spread of a wave in direction  can be obtained as Δ = 2π/ Δk  , where  = , z.

Bulk light (homogeneous medium). Let's first consider light traveling in an homogeneous medium. In this case, the circumference does not change and the maximum spread of the wavenumber is given by the diameter of the circle i.e.

Δk  = 2k, leading to the diffraction limit:

Δ = 2π Δk = λ 0 2n .
Although diffraction-limited, subwavelength confinement (compared to λ 0 ) can be achieved in materials with high n.

2D Light. At the interface between two materials with positive dielectric functions, for which ϵ 1 > ϵ 2 , when TIR takes place, it has been shown that

k 2  + k 2 z,1 = k 2 0 ϵ 1 and k 2  -ϰ 2 z,2 = k 2 0 ϵ 2 .
Since k  should match in both materials, the system is again diffracted-limited. In the medium with ϵ 2 , the evanescent field is considered to vanish at an amplitude 1/ e, for a decay distance δ D = 1/ ϰ z ≥ (λ 0 / 2n 1 ). Which again can be subwavelength confined for large n 1 .

Surface plasmon polariton. In the case of SPPs, by replacing the ϵ 2 material with one that has a negative dielectric function, it is possible to 'break' the diffraction limit. Here, noble metals such as gold can be used, where the effective response of the electrons at the surface to the coupled field can be described by the Lorentz-Drude dielectric function, ϵ 2 (ω), which is negative for frequencies below the plasma frequency. In this negative regime, as found for the SPPs,

k 2  -ϰ 2 z,1 = k 2 0 ϵ 1 , and k 2  -ϰ 2 z,2 = k 2 0 ϵ 2 (ω);
where both k z,1 and k z,2 have become imaginary. Although k 2  must match across the boundary, its value is no longer limited, which in principle enables confinement to arbitrary spatial extent in the y plane. However, SPPs can no longer exist for frequencies above ω spp introduced in the previous section. The confinement limit is given by [START_REF] Tame | Quantum plasmonics[END_REF]:

Δ ≥ λ 0 2n 1 - ϵ 1 ϵ 2 (ω)
The amount of the confinement depends on the materials and geometry, with nanowires and channel waveguides providing larger field confinement.

The ability of SPPs to confine and guide their coupled light field within regions far below the diffraction limit, using materials with negative ϵ such as metals, superconductors, and graphene, is one of their major strengths (in this work, the choice is to use metallic nanoantennas). By confining light using SPPs it is possible to significantly alter the dynamics of light-matter interactions. However, since SPPs suffer from big losses due to absorption in metals, the proposal in this thesis is to use hybrid dielectricplasmonic platforms, thus taking advantage of the low-loss transport characteristics of Silicon (dielectric) waveguides, together with the extreme confinement provided by SPPs.

Conclusion

In this chapter, the fundamental concepts of classical electromagnetism that will allow us to manipulate the light-matter interactions in complex photonic and plasmonic systems were studied.

By analyzing optical systems as an eigenvalue problem, it was shown that the symmetry of the system, together with the dielectric description of the materials as a function of the frequency, provide the foundations towards high confinement of electromagnetic fields. We have seen that, by engineering the material properties (ϵ,μ) in space and over length scales below the vacuum wavelength, nanophotonics provides a way to controlling light, first up to subwavelength limits by means of photonic waveguides and even further to subdiffraction regions by SPPs. Additionally, the Green's function, the reciprocity theorem and the Babinet's principle were presented as valuable tools to relate electric and magnetic fields with the sources originating them. In particular, the case of dipole sources was analyzed.

Chapter 3. Quantum emitters coupled to complex plasmonic and photonic structures

Introduction

In this chapter, we analyze the properties of single photons emitted by quantum sources and their modification when the emitters are coupled to complex structures. To this end, the single emitters are modeled as two-level systems (TLS). The modification of the dynamics of the emitted photons is detailed, together with a description of how to understand the quantum-mechanical process of spontaneous emission in classical terms. The computational methods to model the interaction between the emitters and complex photonic and plasmonic structres are also discussed.

The two-level system model

The single-photon sources considered here are based on spontaneous emission. According to quantum electrodynamics (QED, [START_REF] Loudon | The quantum theory of light[END_REF]), the spontaneous emission from an emitter in free-space is due to vacuum fluctuations. The simplest (but effective) model of a single emitter, which reflects many phenomena inherent to the real optical emitters, is a TLS interacting with an infinite number of field modes, each mode characterized by its wavevector k. In the following subsections, the key properties of single-photons emitted by a TLS and their interaction with its surrounding environment are presented.

Properties of single photons emitted by quantum emitters

The properties of single photons are generally discussed in terms of a given single mode. To be used in practice, however, photons have to leave the source and should rather be described by wavepackets spread over several modes, each mode characterized by its wavevector k. Such wavepackets are characterized by different parameters, which can vary from source to source depending on its nature. The parameters relevant to this work (schematically shown in Fig. 20), will provide information that can be classified into three broad categories as follows:

1. Space: Once that the photon have left the source, where does it go, with what spatial distribution, and with what polarization?

2. Time: When will the photon appear?

3. Matter: Which frequency/wavelength will the photon have?

The parameters containing this information are (where the super-script indicates which from the previous categories they belong to):

Spatial mode 1 : For telecommunications, computation or interference applications, photons have to be collected and collimated into a given spatial mode (spatial distribution of the electric and magnetic fields). Unless the emitter undergoes spatial motion or reorientations, a solid-state source based on a quantum emitter will, in general, emit all its photons in the same spatial mode. Mobile emitters such as atoms may deliver different spatial wavepackets at different times or may couple with variable efficiency to the fixed spatial mode of a cavity. For an efficient utilization, the spatial mode of the photon at the output of the source must be matched to those of all subsequent optical components, in particular that of the detector.

Spectrum 3 : The desirable spectrum of a single-photon source also depends on the application. In the case of optical telecommunication systems, the propagation of photons should favor the three main transmission windows of silica fibers, around 840 nm, 1320 nm and 1550 nm. To fully exploit the quantum properties of a single-photon source, the detector must be a photon-counter with a high detection efficiency in the near-infrared domain. Photon-counting devices based on III-V semiconductors are now available in the near-IR window. In this work, the wavelength is chosen around λ 0 = 1550 nm.

Polarization state 1,3 : It is determined by the microscopic nature of the emitter and by how it is coupled to the emission mode. Each k-vector possesses two linearly independent polarization states. For example, for a linear dipole emitter such as a single molecule, the polarization pattern of the emitted photons depends on the orientation of the dipole moment. If the dipole lies in the focal plane of the collection objective, the polarization is close (but not identical) to linear, parallel to the dipole, for all photons across the whole spatial mode. If the dipole is perpendicular to the focal plane, the polarization in the collected mode is radial.

Emission lifetime 2,3 : In the TLS, the ground |g〉 and excited |e〉 energy states are separated by the transition energy ℏω eg . The excited state of an individual quantum system (an atom, a molecule or a quantum dot) is created in a rapid process, and the subsequent emission follows on a longer timescale. Therefore, the spontaneous emission lifetime eventually limits the rate at which the single photons can be emitted,  0 , thus representing an important feature of the source. The emission lifetime in vacuum τ 0 is a characteristic of the emitter. It is determined by the transition frequency, ℏω eg , and by the transition dipole moment matrix element, d eg = 〈e|qr|g〉, which governs the probability of transitions between the ground and excited states, (i.e. it selects the available optical transitions by the standard selection rules). Calculation of this process requires accounting for the interaction of a TLS with the continuum of electromagnetic modes of the freespace. In this way, it was demonstrated that [START_REF] Weisskopf | Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie[END_REF]:

τ -1 0 =  0 = ω 3 eg 3πℏϵ 0 c 3 |d eg | 2 , (3.1)
where ℏ is the reduced Planck's constant and ϵ 0 and c are the vacuum permittivity and speed of light, respectively. The emission lifetime, and consequently the spontaneous emission rate can be considerably altered if the object is placed in a nanostructured environment, which may amplify or reduce vacuum fluctuations at the emission frequency, thereby 'forcing' or 'blocking' spontaneous emission. This was the phenomenon originally described by Edward Purcell while studying nuclear magnetic moment transitions at radio frequencies [START_REF] Purcell | Spontaneous emission probabilities at radio frequencies[END_REF].

ω 1 ω 2 ω 3 ω k ħω ω 1 ω 2 ω k ω 3 τ -1 = Γ λ = 2πω eg

Single-photon emission in homogeneous media

Purcell found that the spontaneous emission probability of the TLS coupled to a resonant electrical circuit was increased (and thus the relaxation time was reduced), relative to the uncoupled system (i.e. the TLS in free-space), by a factor:

F P = 3 4π 2 λ 3 eg Q V , ( 3.2)
where λ eg = 2πc/ ω eg , Q is the quality factor of the resonator and V is a characteristic volume of the system. Furthermore, Purcell also claimed that for a non-resonant circuit,

F P ∝ λ 3 / V.
In his formulation, the emitter is supposed to stand at the point r m , where the electric-field is maximum.

The expression in Eq. ( 3.2) was named the Purcell factor, from which it can be observed that F P ∝ (Q, 1/ V). Thus, it is possible to increase the Purcell factor either by: 1) the use of resonators with high Q or by 2) the reduction of the characteristic volume V.

This was a fundamental discovery applicable not only to magnetic transitions and not only at radio frequencies. In general: the spontaneous emission of an emitter is strongly dependent on the electromagnetic environment it resides in. It is a property of the coupled system of the emitter and the available electromagnetic modes surrounding it. Since then, the dependence of the spontaneous emission rate on the environment was called the Purcell effect and it was experimentally demonstrated at optical frequencies by the study of the interaction of light with monomolecular dye layers [START_REF] Drexhage | IV Interaction of light with monomolecular dye layers[END_REF]. Indeed, the electrical resonator used by Purcell at radio waves corresponds to a single-mode resonant cavity at optical frequencies; where Q is defined as the ratio of the resonance frequency to the cavity damping rate:

Q = ω 0 /  c ,
and the characteristic volume pointed out by Purcell turns out to be the effective volume of the resonant mode of the cavity, V eƒ ƒ . After these demonstrations, Cavity Quantum Electrodynamics (CQED) was then born to study the interaction of emitters with tailored electromagnetic fields [START_REF] Haroche | Fundamental systems in quantum optics[END_REF].

Local density of states (LDOS) and the Green's function

We have seen that the spontaneous emission decay rate of an emitter changes when it is inside a cavity, relative to that in free-space. As detailed in this section, the reason of this change is that the walls of the cavity will limit the number of available electromagnetic modes ω K into which the photon can couple. In free-space, the available modes are infinite but this is not the case in the presence of a different environment.

When a TLS is placed inside a closed, non-absorbing cavity, the spontaneous emission rate is given by the Fermi's golden rule, derived from quantum mechanics firstorder perturbation theory (Scully y Zubairy, 1997):

 = πω eg ℏϵ 0 |d eg | 2 ρ n r 0 , ω eg , ( 3.3)
where ρ n r 0 , ω eg is the partial local density of states (partial LDOS) of the electromagnetic field at the TLS position r 0 and it is defined as:

ρ n r 0 , ω eg = k n • e k (r 0 ) ⊗ e * k (r 0 ) • n δ ω k -ω eg (3.4)
Here, the summation runs over all the eigenmodes e k of the system with eigenfrequencies ω k . The eigenmodes e k are the solutions to the generalized eigenvalue problem detailed in Eq. (2.18) i.e. the homogeneous wave equation, normalized by the condition d 3 rϵ(r)e  (r) • e j (r) = δ j . The unit vector n points in the direction of the TLS dipole moment d eg .

To understand the meaning of the partial LDOS, the concepts of density of states (DOS), and local density of states (LDOS) should also be introduced. Consider a molecule, atom, quantum dot, etc., as located at a fixed position, r 0 and oriented along a fixed direction, n. The total photonic density of available modes (DOS) at a frequency ω eg just counts the number of eigenmodes in an infinitely small frequency range:

ρ ω eg = 1 V k δ ω k -ω eg . DOS (3.5)
This is a global quantity that characterizes the spectral density of eigenmodes of the medium as a whole. It does not take into account the position and orientation of the TLS. In practice, however, the total available optical states have to be weighted by how well the orientation and position of the TLS match to them. Then, a local quantity, the LDOS, ρ r 0 , ω eg , that takes into account the position of the TLS can be introduced through a summation weighted by the amplitude of the eigenmodes at the point r 0 :

ρ r 0 , ω eg = k |e k (r 0 ) | 2 δ ω k -ω eg . LDOS (3.6)
Finally, the partial LDOS is a projection of the LDOS in the direction of the orientation of the TLS. It accounts for both the position and orientation of the TLS:

ρ n r 0 , ω eg = k |n • e k (r 0 ) | 2 δ ω k -ω eg . partial LDOS (3.7)
Indeed, the full LDOS in Eq. (3.6), in terms of the partial LDOS:

ρ r 0 , ω eg = n=n  ,n y ,n z ρ n r 0 , ω eg , ( 3.8)
describes the density of states at the position r 0 , summed up over the three directions n  , n y , and n z of the TLS. Now, going back to the Fermi's golden rule in Eq. ( 3.3), it can be observed that by varying the partial LDOS (and therefore the LDOS), the spontaneous emission rate of the TLS can be efficiently modified.

Notice that the delta-function in the expression for the LDOS, Eq. (3.6), implies an integration over a finite distribution of final frequencies, and even for a single final frequency, the apparent singularity introduced through δ ω kω eg is compensated by the normal modes whose magnitude tends to zero for a sufficiently large mode volume. In any case, it is convenient to get rid of these singularities by representing ρ r 0 , ω eg in terms of the electric Green's function instead of normal modes [START_REF] Novotny | Principles of nano-optics[END_REF], emphasizing the fact that it is a classical concept relating a timeharmonic current density to the electric field that it produces.

A relationship between the total Green's function (from section 2.2.2) to the quantum-mechanical description of the LDOS derived from the Fermi's golden rule (Eq. 3.

3), can be established in a phenomenological way (i.e. as long as the experiments do not contradict the theory), by replacing the TLS quantum emitter located at position r 0 by the classic monochromatic point ED that defines the Green's function (Eq. (2.38)). The replacement is as follows:

quantum TLS → classical ED

ω eg → ω, d eg → d, r 0 → r.
(3.9)

The substitution of d by d eg includes a factor of two since only positive frequencies are handled in the emission problem (in quantum mechanics positive and negative are treated specifically), while the classical treatment involves both positive and negative frequencies. This simple procedure allows now to represent the partial LDOS as:

ρ n r 0 , ω eg = 2ω eg πc 2 n • m G r 0 , r 0 , ω eg • n , (3.10)
and the LDOS as:

ρ r 0 , ω eg = 2ω eg πc 2 m Tr G r 0 , r 0 , ω eg , (3.11) where m refers to the imaginary part and Tr denotes the trace of a tensor.

Single-photon emission in inhomogeneous photonic and plasmonic structures

The LDOS calculated, or measured in this way, is valid for a point r 0 lying in vacuum, although this point might be at close proximity from a material surface, including the surface of an absorbing metal [START_REF] Carminati | Electromagnetic density of states in complex plasmonic systems[END_REF]. Under these conditions, the imaginary part of the Green's function at r = r 0 is non-singular and the LDOS in Eq.

(3.11) is a well-defined quantity [START_REF] Guérin | Singularity of the dyadic Green's function for heterogeneous dielectrics[END_REF]. Indeed, even though the source point r 0 is located in vacuum, the Green's function G (r, r 0 , ω) can describe an arbitrary environment surrounding the emitter. Therefore, Eq. (3.11) will be used as a definition of the LDOS in an arbitrary environment, where only the imaginary part of the Green's function will change depending on the system. This definition does not assume any particular set of eigenmodes. It should be mentioned, however, that the Fermi's golden rule in Eq. (3.3) gives the correct solution to the spontaneous emission problem when the interaction between the emitter and the electromagnetic modes inside the cavity is weak enough. This scenario corresponds to a dynamics in which the system does not remember its evolution and results in the exponential decay. Otherwise, the emitter may demonstrate non-exponential decay. The strength of the interaction is classified as follows.

TLS coupled to a cavity: weak and strong coupling regimes

From CQED, it was found that for a resonant cavity with a single-mode field, the emitter-cavity interaction is described by the Jaynes-Cummings Hamiltonian [START_REF] Jaynes | Comparison of quantum and semiclassical radiation theories with application to the beam maser[END_REF]; from which the strength of the interaction between the optical transition of the emitter and the resonant mode of the cavity is characterized by the coupling rate g (Fig. 21):

g = d eg ℏ ℏω eg 2ϵ 0 V eƒ ƒ . ( 3.12)
As for the Purcell factor, the emitter is considered to be in a region of maximum electric-field inside the cavity. Depending on the comparison of g with the damping rates of both the emitter and the cavity (,  c ), CQED can be split into two regimes [START_REF] Walls | Quantum optics[END_REF]:

strong coupling (or high Q-cavity) regime, for g  c , , and weak coupling (or low Q-cavity) regime, for g  c , .

In the strong coupling case [Fig. 21a)], the emitted photons are stored in the cavity long enough so that the emitter is coherently coupled to the cavity field and spontaneous emission becomes reversible. The presence of one excitation induces 'vacuum'

Rabi oscillations [START_REF] Majumdar | All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity[END_REF] between the two coupled states: 1) excited emitter with empty cavity, and 2) ground-state emitter with one photon in the cavity.

In the weak coupling regime [Fig. 21b)], the emitted photons are very rapidly and irreversibly dissipated by the cavity walls, thus the emitter excitation is irreversibly lost to the continuum of all available photon states, including the free-modes outside the cavity (leak modes) and those inside the cavity (cavity modes). In fact, the presence of the cavity walls changes the field distribution around the TLS and alters the emitter's spontaneous emission rates. The transition frequencies are also modified by the coupling to the cavity walls.

Generalized electric and magnetic Purcell factors

As long as the TLS-cavity interaction belongs to the weak coupling regime, we can use the previous definitions of the partial and total LDOS in terms of the electric Green's function, and the spontaneous decay rate can be expressed as (by substituting

Eq. (3.10) in Eq. (3.3)):

 = 2ω 2 eg ℏϵ 0 c 2 |d eg | 2 n • m G r 0 , r 0 , ω eg • n . (3.13)
In the particular case of a dipole placed in free-space, the imaginary part of the free-space dyadic Green's function, G 0 r, r 0 , ω eg , at r = r 0 is simply [START_REF] Carminati | Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle[END_REF]:

m G 0 r 0 , r 0 , ω eg = k 0 6π . (3.14) leading to:

 0 = ω 3 eg 3πℏϵ 0 c 3 |d eg | 2 . (3.15)
Using Eqs. (3.13) and (3.15), it is possible (and convenient) to characterize the spontaneous emission rate by a dimensionless quantity: .16) and in terms of the partial LDOS:

F P =   0 = 6πc ω eg n • m G r 0 , r 0 , ω eg • n . ( 3
F P =   0 = 3π 2 c 3 ω 2 eg ρ n r 0 , ω eg . ( 3.17)
In both expressions, the right-hand side of the equations can be understood as a generalized electric Purcell factor F P . It should be noticed that the F P does not depend on the TLS dipole moment magnitude and it is only defined by the electromagnetic properties of the environment, which are reflected in the imaginary part of the Green's tensor. This expression applies for any system, including open absorbing media that are usually encountered in plasmonics. In the particular case of a single mode cavity, it leads to the historical Purcell factor (Eq. (3.2)).

All the previous definitions correspond to quantities for an electric dipole quantum emitter, i.e. electric Green function, electric Purcell factor, electric LDOS, and so on.

These may be mapped in a straightforward way to the description of the spontaneous emission from a magnetic dipole emitter, as briefly reviewed below.

The spontaneous emission rate  (m) of a TLS with the MD transition moment m eg is given by the Fermi's rule: (3.18) where ρ n (m) (r 0 , ω eg ) is the magnetic partial LDOS for the given environment:

 (m) = πω eg ℏ μ 0 |m eg | 2 ρ n (m) r 0 , ω eg ,
ρ n (m) r 0 , ω eg = 2ω eg πc 2 n • m G (m) r 0 , r 0 , ω eg • n, (3.19)
where G (m) r 0 , r 0 , ω eg is the Green's tensor of a magnetic dipole, which connects the magnetic field at the position r to the magnetic dipole located at r 0 via H (r) = k 2 0 G m r, r 0 , ω eg m, with k 0 = ω eg / c. Here, r = r 0 . Using Eq. (3.19), the expression relating the MD spontaneous emission rate to the magnetic Green's tensor is:

 (m) = 2μ 0 ω 2 eg ℏc 2 |m eg | 2 n • m G (m) r 0 , r 0 , ω eg • n.
(3.20)

Finally, the magnetic Purcell factor is defined as the enhancement of a MD emitter decay rate with respect to the vacuum value. Since the electric and magnetic LDOS are equal in free space due to the symmetry of Maxwell's equations,

 0,(m) = ω 3 eg 3πℏc 3 μ 0 |m eg | 2 ; (3.21)
the generalized magnetic Purcell factor is then:

F P,(m) = 6πc ω eg n • m G (m) r 0 , r 0 , ω eg • n. (3.22)
As for the case of an ED, expression (3.22) may be obtained from a classical argument by considering the work P (m) performed by the magnetic field of a classical magnetic dipole m on the oscillating magnetic current: .23) In this project, the MD LDOS enhancement is explored making use of an adaptation of the vectorial Babinet's principle.

P (m) = ω eg 2 m [m * • B (r 0 )] = μ 0 c 2 ω 3 eg 2 |m| 2 n • m G (m) r 0 , r 0 , ω eg • n. ( 3

Power radiated by a TLS

We have seen that it is possible to calculate the spontaneous emission rate of a quantum emitter given the knowledge of a classical characteristic -the Green's tensor G(r, r 0 , ω). Indeed, this relationship [expressed in Eq. (3.13)], offers an interpretation of the Purcell effect in classical terms: the acceleration of spontaneous emission from an ED emitter can be understood as the enhancement of work done by the electric field of a dipole on the oscillating electric current. The work per unit time performed by the electric field E of a monochromatic ED source which oscillates at ω, de -ωt , and which is located at r 0 , (i.e. the power radiated by a classical oscillator) is given by: .24) where the electric field can be once again expressed in terms of the Green's function (as in Eq. (2.38)). Then, using the same phenomenological approach given by the substitutions in the expression (3.9), the power emitted by an ED can be then expressed as:

P = ω 2 m [d * • E (r 0 )] , ( 3
P = μ 0 ω 3 eg 2 |d eg | 2 n • m G r 0 , r 0 , ω eg • n . (3.25)
Dividing this quantity by the power P 0 = ω 4 eg 12πϵ 0 c 3 |d| 2 emitted by the same dipole into free-space, an expression identical to the generalized Purcell factor derived in Eq. (3.16) is obtained.

Thus, another relevant aspect of the LDOS introduced in terms of Green's functions is that it allows to study the quantum spontaneous emission process by means of the classical radiation of an electric dipole. Based on the previous analyses, we can now establish the relations which will be the basis of the numerical and experimental techniques that were used in order to design and evaluate the performance of the nanophotonic devices proposed in this work. These relations are:

F P = τ 0 τ =   0 = ρ ρ 0 = P P 0 . (3.26)
The generalized Purcell factor is equal to the ratios between the spontaneous decay rate, local density of states, and radiated power in a given environment relative to those in free-space. It is reciprocal to the ratio between the emission lifetime in the environment relative to that in free-space.

Quantum Yield and β-factor

In either case, for ED and MD emitters, the total decay rate  accounts for both radiative and non-radiative processes. The fraction of energy emitted into photons is defined as the quantum yield:

η =  rd  rd +  nonrd +  nt ; (3.27)
where  nonrd is the decay rate due to the electromagnetic losses in the environment and  nt is the rate of the intrinsic nonradiative decay that occurs even when an isolated emitter is placed in free space.

In most practical cases, the enhancement of the emission radiated to the far field is of specific interest, thus an appropriate figure of merit is the radiative Purcell factor

F rd P =  rd /  0 .
As detailed in the previous section, the radiative decay rate  rd can be calculated by integrating the Poynting vector over a surface enclosing the dipole and its environment. In terms of the quantum yield, the radiative Purcell factor is

F rd P = ηF P .
Additionally, the fraction of the light emitted into a specific spatial mode, ω k , of the system can be obtained. It is characterized by the spontaneous emission coupling efficiency, called the β-factor, given by: (3.28) from which the Coupled Purcell factor is defined as:

β k =  k  rd ,
F P,k =  k  0 = β k F rd P . (3.29)
From these definitions, one way to characterize the Purcell effect, and consequently the modification of the LDOS by different structures, is summarized as follows (schematically shown in Fig. 22):

1. A specific position and orientation of the dipole is chosen.

2. Calculation of the reference power P 0 by integration of the Poynting vector over a surface enclosing the dipole lying in the homogeneous medium with ϵ = 1. Figure 22a). 3.5. Computational electromagnetic modeling: electric and magnetic dipoles coupled to photonic and plasmonic structures Because Maxwell's equations are practically exact and the relevant material properties are well known, photonic systems are specially suitable for computation. Indeed, quantitative theoretical predictions can be made from first principles without any questionable simplifications or assumptions. Since the results have consistently agreed with experiments, it is possible and preferable to optimize the design of complex photonic systems on a computer and then manufacture them. This is exactly the approach followed here.

Many standard numerical techniques to solve partial differential equations have been applied to electrodynamics, each of them having its own particular strengths and weaknesses. Here the most important methods are summarized.

In general, there are three types of problems in computational electromagnetics:

1. Frequency-domain eigenvalue problems: The dispersion relation ω(k) and the associated fields are found by expressing the problem as a finite matrix eigenproblem A = ω 2 B and applying linear algebra to find a few of the eigenvectors  and eigenvalues ω 2 . Alternatively, the eigenproblem can be formulated at a fixed ω for the wavevector k along a single uniform direction as a generalized

Hermitian eigenproblem with eigenvalue k.

2. Frequency-domain responses: Given a current density J (x) e -ωt at a fixed frequency ω, the resulting fields are found by expressing the problem as a finite matrix equation A = b, and applying linear algebra to solve for .

Time-domain simulations:

The fields E (x, t) and H (x, t) propagating in time are simulated, usually starting with some time-dependent current source J (x, t).

Another way to classify numerical methods for partial differential equations is by the approach used to discretize to a finite number, N, the infinite number of unknowns (the fields at every point of space, for example). Four important discretization schemes are:

Finite differences: represent unknown functions ƒ () by their values ƒ n ≈ ƒ (nΔ)

at discrete points n on a grid, and their derivatives by differences on the grid.

Finite elements: divide the space into a set of finite geometric elements and represent the unknown functions by simple approximations defined on each element (i.e. low-degree polynomials).

Spectral methods: represent unknown functions as a series expansion in a complete basis set of smooth functions, truncating the series to have a finite number of terms. The most common is the Fourier series, also called a plane wave method in two or three dimensions. When the boundary conditions are not periodic other basis functions such as Chebyshev polynomials can be used.

Boundary-element methods: only the boundaries between homogeneous regions are discretized, in contrast to discretization of all the space. The homogeneous regions are treated analytically. The discretization may employ a finite element or spectral basis. The multipole method, transfer-matrix method and the coupledwave methods are essentially boundary-element methods with specialized spectral basis.

Among the above, the simplest methods to implement and analyze are those that operate on a uniform grid: the finite difference methods and the spectral method with a plane wave basis.

A general comparison between frequency-domain and time-domain methods is that in the former one, only a snapshot of the field can be observed after a "long" time has passed, the transient response cannot be observed. In time-domain methods, however, the fields' evolution over time can be observed and thus the transient response of the system can be recorded.

Unlike solvers for a frequency-domain response, time-domain methods can compute the response of a linear system at many frequencies with a single computation by taking the Fourier transform of the response to a short pulse. For example, a short pulse (which has a broad bandwidth) can be sent into a device, and the resulting fields To develop this work, a method that simulate the full time-dependent Maxwell equations, propagating the fields in both space and time was employed. This is the finitedifference time-domain method, or FDTD, which is by far the most common technique for time-domain simulations. It is described below.

E (t)

Finite-difference time domain (FDTD) methods

As As illustrated in Fig. 23, in the Yee grid, the field components are placed in physically different locations, this implies that they may reside in different materials even if they are in the same unit cell. Also, the components will be out of phase and they are also staggered in time. The consequences and convenience of using a Yee grid are:

The system is divergence-free, the physical boundary conditions are naturally satisfied, and the Maxwell's curl equations are approximated in an elegant arrangement (see the 3D-Yee grid in Fig. 23). The benefits and drawbacks of using FDTD methods are summarized in table 1.

Further theoretical and implementation details about FDTD are well described in textbooks such as [START_REF] Taflove | Computational electrodynamics: the finitedifference time-domain method[END_REF].

Since the nanophotonic devices proposed in this manuscript are complex coupled systems, weakly resonant, and broadband; the FDTD method represents more advantages than disadvantages to our needs. Therefore, these are the kind of numerical calculations performed in the rest of this work. The commercial available FDTD si- 2. Excellent for broadband and/or transient simulations.

2. Slow for small devices.

3. Accurate, robust, and mature method: Sources of error are well understood and it is a proven method in many fields. A lot of literature is available.

3. Very inefficient for highly resonant devices.

4. Naturally handles nonlinear behavior. Directly handles nonlinearities due to nonlinear materials or incorporation of circuit elements.

4. Require a long time to resolve a sharp spectral feature.

5. Great for learning electromagnetics. Field animations and direct simulation of Maxwell's equations make FDTD a great learning tool.

5. If high spatial resolution is required, high temporal resolution is required as well in order to maintain numerical stability.

mulator of Lumerical Solutions©, was employed. The numerical calculations were, in part, performed with the resources of the High Performance Computing (HPC) Center ROMEO, in France, Grand Est region.

The implementation of the model to evaluate the interaction between quantum emitters and the complex structures of interest in this thesis is based on the following single and coupled mode expansion analysis.

Power radiated by electric and magnetic dipole sources: mode expansion analysis

In regions where a system is invariant along the propagation direction (i.e. rectangular waveguides), the electric and magnetic fields can be expanded into the normal modes as: (3.30) where  k and b k are the constant expansion coefficients for the forward and backward guided modes, respectively, and e k , h k , are the kth normal eigenmodes. The third term in the equations correspond to the non-guided modes of the system (i.e. the radiated modes into free-space). Because the eigenmodes are orthogonal, the expansion coefficients can be calculated as:

E(r, ω) = k  k e k (r, ω) + k b k e k (r, ω) + k E rd dk. H(r, ω) = k  k h k (r, ω) + k b k h k (r, ω) + k H rd dk,
 k + b k = 〈E|e k 〉 〈e k |e k 〉 .  k -b k = 〈e k |H〉 〈e k |e k 〉 , ( 3.31)
where, for any vectors f, and g, 〈f|g〉 is the bra-ket notation of their inner product defined in section 2.2.1. Equations (3.31) are obtained by a projection of the total electric and magnetic field to the kth eigenmode, using the orthogonal property.

The expansion coefficients can then be calculated as:

 k = 1 4N k [〈E|e k 〉 + 〈e k |E〉]. b k = 1 4N k [〈E|e k 〉 -〈e k |E〉], (3.32)
where

N k = 1 2 (e k × h k ) • n  dA =〈e k |e k 〉
is the normalization factor of the kth eigenmode along the propagation direction n  .

The power carried by the kth forward eigenmode is: .33) and by the kth backward eigenmode: .34) By this expansion, each individual kth mode supported by each independent wa-veguide can be identified. The effective refractive index n eƒ ƒ of each mode can be calculated as: .35) where  x pk , λ x pk , and k  pk ≡ β  pk are, respectively, the components parallel to the propagation direction of the velocity, wavelength and propagation constant of the kth mode with p polarization (TE or TM). The effective index gives the ratio of the velocity of light in vacuum to the phase velocity of a mode in the guided media for a given polarization in the direction of propagation in a guiding structure (in our case, along the  direction).

P k = k | k | 2 N k , ( 3
P k = k |b k | 2 N k . ( 3
n eƒ ƒ pk = c  x pk = λ 0 ν λ x pk ν = 2πk 0 2πk x pk = β 0 β x pk , ( 3
The values of n eƒ ƒ for each mode were numerically obtained using the commercial software of Lumerical©, called Mode solutions.

Mode expansion along the coupling region

The electric and magnetic fields can be further expanded into the modes along the hybrid region (i.e. the plasmonic structure over the Si photonic waveguide). However, in the coupling region, the expansion coefficients are not constant scalars and should account for the propagation and mode coupling with the explicit dependence on the propagation axis ( -s) as:

E(r, ω) = k A k ()e k (r, ω) + k B k ()e k (r, ω) + k E rd dk. H(r, ω) = k A k ()h k (r, ω) + k B k ()h k (r, ω) + k H rd dk.
(3.36)

The expansion coefficients are given by the following general coupled mode equation:

dA k () d = β k A k () +  k ϰ k k A k (). (3.37) with the coupling coefficient ϰ k k = ω 2 〈e k |Δε|e k 〉 -ω 2 〈e ,k |Δε|e ,k 〉.
Here e  is the longitudinal x-component of the mode fields and β k is the propagation constant of the kth mode. For two coupled modes considered in the coupling region, the two-mode coupled equations are:

dA 1 () d = β 1 A 1 () + ϰ 12 A 2 (). dA 2 () d = β 2 A 2 () + ϰ 21 A 1 ().
(3.38)

The solutions can be found with initial condition A 1 (0) = 1 and A 2 (0) = 0 (i.e. just before starting the hybrid region the photonic mode does exist and the plasmonic mode does not) as:

A 1 () = (cos  +  Δ  sin )e ϕ . A 2 () =  ϰ 12  sin()e ϕ .
(3.39)

The energy is exchanged between the two modes with a coupling length

 c = π 2|| , ( 3.40)
where  = Δ 2 + ϰ 12 ϰ 21 and Δ = 1 2 (γ 2γ 1 ), with γ 1 and γ 2 the propagation constants of the coupled modes.

Conclusion

Up to now, the properties of single-photons have been presented along with their interactions with the available electromagnetic modes of a given environment. It was shown that such interactions can be inferred by classical arguments via the Green's tensor describing the sources producing electromagnetic fields. The computational method of FDTD to solve complex photonic and plasmonic systems was described, together with the model to evaluate them, based on single modal analysis and coupledmode theory.

In the case of the plasmonic nanoantenna [Fig. 24c)], the triangular (tapered) shape is a common configuration that favors the confinement of the electromagnetic fields at its apex. For the moment, the only parameter to establish is its thickness=30 nm, which is a suitable value to support SPPs in gold (it is given by the skin depth in gold, also at λ = 1550 nm). After analyzing each waveguide independently, the modes supported by the hybrid structure will be calculated [Fig. 24d)]. It is important to notice that the tapered (triangular) geometry of the Au nanoantenna is not translational invariant along the propagation direction , thus the spatial cross section profile (over the plane yz) at each nanotaper width will be different. To simplify the analysis, the metallic waveguide is simulated as a rectangular waveguide with variable width. This approach provides continuous translational symmetry to the system and it can be then easily solved.

To solve each waveguide (i.e. find the supported modes) independently, the following procedure is needed.

Single-mode silicon waveguides at communication frequencies

According to the geometry of the problem [see Fig. 24)], the waveguide has inversion symmetry relative to both y-and z-axes and we know that the thickness is fixed at 220 nm. Choosing y-as the symmetry axis, the effective refractive index n eƒ ƒ was obtained for the first three symmetric (even) and anti-symmetric (odd) modes relative to this axis, at λ = 1550 nm and for a width range from 0 < dth < 1 μm, the results are plotted in Fig. 25. The refractive index of S was taken as n S = 3.48 and for the substrate, SiO 2 , it was taken as n sb = 1.46. As observed, over a width range ≈ 250 < dth < 500 nm, only the fundamental symmetric S 0 [Fig. 25a)] and anti-symmetric A 0 [Fig. 25b)] modes correspond to guided-modes (i.e. n eƒ ƒ > n sb ), in contrast to the higher order modes S 1 , S 2 , A 1 , and A 2 . Plotting only S 0 and A 0 over this range width [Fig. 25c)], we can observe that at a width of 500 nm, A 0 has a higher n eƒ ƒ of ≈ 2.4 and S 0 of ≈ 1.7. The group velocity of each mode was calculated, and it can be seen [Fig. 25d)], that both modes are well confined at a width of 500 nm, since they propagate at less than a quarter of the speed of light in vacuum, 0.25c.

Also from the geometry of the system, since  is the propagation direction, for the TE-polarized mode, the electric field must be oriented along the y -s, E y , and for

Plasmonic waveguide for the electric field confinement

Following the same methodology, the dispersion curve [Fig. 27a)] and the group velocity [Fig. 27b)] of the supported modes of a rectangular plasmonic waveguide placed over a SiO 2 substrate were calculated for a width range of 0 <  < 500 nm.

In this case, only one symmetric mode is confined (guided) and we know that every plasmonic mode should be a TM-polarized mode in order to satisfy boundary conditions (see section 2.4.1).

As observed, the mode begins to confine for a width  < 100 nm, for which n eƒ ƒ starts to increase asymptotically. The intensity of the spatial transverse profile of the mode for a width of the plasmonic waveguide of  = 60 nm is plotted in the inset of Fig. 27a). In the same way, the drastic reduction of the group velocity for  < 100 nm, up to 10 % of the speed of light in vacuum (0.1c) indicates a temporal confinement of the plasmonic mode. A width of 300 nm was chosen as the base of the Au nanotaper, where 0 nm represents its apex, as sketched in the inset of Fig. 27b). From the previous analyses, the dimensions of the Si waveguide have been established to 220 nm (thick) by 500 nm (height). The base width of the triangular nanotaper has been set to  = 300 nm. To find the appropriate length L of the nanotaper, we need to study the hybrid device, that is the plasmonic waveguide placed over the Si-photonic waveguide. As they will be in close proximity, an energy transfer process between the waveguides will occur, this is:

Consider first the purely photonic Si waveguide. The propagating guided-modes have an evanescent field which may excite the plasmonic mode of the Au nanotaper by evanescent coupling (see section 2.4.2) at the base of the nanotaper.

Once the plasmonic mode is excited, the modes of the Si waveguide will start to interfere with it along the coupling region. At some distances there will be constructive interference meanwhile for some other distances the interference will be destructive, such that for a given distance L a maximum energy will be transferred to the plasmonic nanotaper.

To find this length, the coupling among the modes will be analyzed as follows.

By these procedure, the coupled modes of the photonic-plasmonic structure were numerically calculated, using the same mode solver as before. Once again, the plasmonic nanotaper was simulated for a variable width from 0 <  < 500 nm with a constant width of the Si waveguide of 500 nm, as sketched in Fig. 28a. As observed in Fig. 28b), there are four guided coupled-modes. From basic electromagnetic theory, we know that modes of perpendicular polarization cannot interfere with each other. Since the plasmonic modes are always TM-polarized, the only photonic mode able to interfere with them is the TM 0 mode of the Si waveguide. After carefully analyzing the symmetry of each coupled mode, and their spatial distributions [Fig. 28c)], we have:

The H TM 0 mode is the result of the interaction between TM 0 (Si waveguide) and S 0 (plasmonic waveguide).

The H TE 0 mode is basically TE 0 (Si waveguide) barely interacting with other fields.

The H TM 1 mode is the result of the interaction between TM 0 (Si waveguide) and A 0 (plasmonic waveguide, not shown before).

The H TE 1 mode is basically TE 1 (Si waveguide, the second symmetric mode plotted in Fig. 25a) slightly interacting with other fields.

From these, we see that H TM 0 is the only mode strongly confined towards the apex of the nanotaper (i.e. its effective index increases asymptotically to infinity). This is also shown in Fig. 28d), where the group velocity of each mode is plotted as a function of the width of the nanotaper, where H TM 0 exhibits a strong temporal confinement with a group velocity of 5 % the speed of light in vacuum at the nanotaper apex. With this information, now it is possible to determine the length for which a maximum energy transfer from the Si waveguide to the plasmonic nanotaper will occur (at a distance of half the coupling length of Eq. (3.40)).

Determination of the coupling length

Once the hybrid modes are excited, those with TM-like polarization H TM 0 and H TM 1 will undergo a vertical energy transfer process (bouncing up and down) along the coupling region. And similarly for the H TE 0 and H TE 1 modes. However, only the interaction between the H TM -like modes results in a maximum energy confinement into the plasmonic Au nanotaper, where the H TM 1 has efficiently transferred all of its energy into the H TM 0 mode. The concept of the energy transfer process is schematically shown in Fig. 29a. In terms of the eigenvalue of each mode (i.e. the effective index), the coupling length for a specific width is defined as: According to Eq. (4.1), the coupling length as a function of  is plotted in Fig. 29b) (solid black curve). Although this  c was calculated for structures where the plasmonic waveguides are rectangular (i.e. a fixed width at a time), we can estimate the  c of the tapered structures by taking the geometrical average. For a triangular taper, the average corresponds to about 1/ 2 of its maximum width,  = 300 nm (width at the base of the triangle). Then, for / 2 = 150 nm, we have:

 c () = λ Re{n eƒ ƒ (H TM 0 ()) -n eƒ ƒ (H TM 1 ())} . ( 4
 c  2 = 1550nm 2.5721 -1.4541 = 1386 nm. (4.2)
The energy is efficiently coupled into the plasmonic structure at half the coupling length, this is around ≈ 700 nm (dashed black line). Indeed, a length L = 800 nm was chosen for simulation and fabrication of the plasmonic nanotaper. As observed in Fig. 29c), for a length of the nanotaper L = 800 nm, the calculated power is gradually transferred from the Si waveguide to the plasmonic one along the propagation direction, reaching a maximum transference at the nanotaper apex. In 29d), electric field profile in the Si waveguide, along the propagation direction (x-axis), decreases gradually over the coupling region, whereas the electric field in the Au nanotaper increases up to its maximum value (≈ 2) at the nanotaper apex, i.e.  = 0.5μm.

Confinement of the electric field by the hybrid device

From the previous sections, the principle of operation can be described as follows (schematically shown in Fig. 30). The hybrid device consists of a plasmonic Au nanotaper of base width  = 300 nm, length L = 800 nm, and 30 nm thick, placed on top of a Si waveguide of 500 × 220 nm. Both waveguides are separated by a 20 nm thick SiO 2 layer that favors the energy transfer process (Fig. 30a). In order to strongly confine the electric field at the apex of the Au nanotaper, the Si waveguide must be illuminated with the TM 0 mode. This photonic mode propagates towards the base of the plasmonic Au nanotaper where it efficiently excites the hybrid photonic-plasmonic modes H TM 0 and H TM 1 . Along the overlapping area, a mode coupling process occurs that allows the transition of energy between the Si and plasmonic waveguides. Because the width  of plasmonic waveguide decreases gradually, a spatial and temporal confinement of the electric field at its apex occurs. The effective plasmonic mode volume of the electric field is given by:

V eff = ϵ(r)|E(r)| 2 dV max(ϵ(r)|E(r)| 2 ) . (4.3)
After being numerically evaluated, the effective mode volume of the electric field for the H TM 0 mode at the nanotaper apex exhibits subdiffraction confinement, as expected due to the excitation of SPPs. The numerical value is given and discussed in chapter 5.

Plasmonic waveguide for the magnetic field confinement: the application of the Babinet´s principle

According to the vectorial Babinet's principle (section 2.3.1), it is possible to deduce the magnetic field distribution produced by a screen, based on the known electric field distribution of its complementary screen as long as the fields incident to the complementary screen have rotated polarization characteristics relative to those illuminating the original screen, this is: For the hybrid complementary structure the Si waveguide is the same as for the original device. The plasmonic nanotaper has the same dimensions as well, except that now the materials air-gold are interchanged. In this case, we have a gold rectangular screen 30 nm thick, with a triangular hollow of base width  = 300 nm, and a length L = 800 nm. Between each end of the nanotaper and each end of the Au rectangle there are 100 nm. A 20 nm thick layer of SiO 2 is placed between the Si waveguide and the Au nanostructure.

E c 0 (r) = cB 0 (r). B c 0 (r) = -E 0 (r)/ c,
From Eq. (2.41), we have that the total electric field behind the original structure (i.e. above the metallic nanotaper) is related to the magnetic field incident to the complementary structure by:

E(r) -cB c (r) = E 0 (r). (4.5)
Analyzing for each  = , y, z field component, we have: Before verifying Eq. (4.6) for our set of complementary nanostructures, it should be noticed that this expression is valid when the screens are in free-space, which is not our case. In order to account for the guided configuration, the intrinsic impedance of a given mode of interest should be considered.

E  E 0 i - c E 0 i B c  = 1. E  E 0 i + B c  B c 0 i = 1, from which: B c  B c 0 i = - E  E 0 i -1 . ( 4
From the requirement of rotated polarizations, we have that if the original structure is illuminated by the TM 0 mode of the Si waveguide, then the complementary structure should be illuminated by the TE 0 . We also know that H = E/ Z. Considering the amplitude of the incident electric field in the TM 0 mode equal to 1, it would be useful to normalize the amplitude of the incident magnetic field in the TE 0 mode as well, allowing for easy H field enhancement measurement; this is:

H TE 0 = E TE 0 Z TE 0 , 1 = E TE 0 Z TE 0 , E TE 0 = Z TE 0 . (4.7)

Determination of Z TE 0

The impedance concept describes propagating modes in waveguides provided we consider the transverse fields. By matching this transverse impedance, the reflectance can be minimized when a waveguide is fed by an external input or when a single waveguide is split into multiple guides. If  is the propagation direction, and the transverse fields E y and H z were constant through the transverse cross section, the simple definition of the characteristic impedance for the TE-polarized mode would be

Z TE = E y / H z .
However, in a guided configuration the transverse fields are spatial functions of the transverse coordinates (y,z) and hence the ratio E y / H z depends on the cross-sectional position and is not unique. In this situation, the characteristic impedance of a specific pth polarized mode can be calculated as [START_REF] Biswas | Impedance of photonic crystals and photonic crystal waveguides[END_REF]: In this way, Eq. (4.6) becomes:

Z p ≡ U p S p =  p
H c  H c TE 0 ,i = - E  E TM 0 ,i -1 . (4.9)
Which provides the relations between the electric and magnetic enhancement field distributions of the original and complementary screens, respectively.

Finally, it should be remarked that, in general, the vectorial Babinet's principle is valid for perfectly conductive and infinitely thin planar screens, which again is not the case of the hybrid devices proposed here. Under these conditions, Babinet's principle can only be expected to hold approximatively.

To validate the principle, the full vectorial 3D finite-difference time-domain (FDTD) method, implemented by Lumerical©FDTD Solutions, was used as follows (depicted in (Fig. 32).

The original structure is injected with the photonic forward-propagating TM 0 mo- The enhancement of the electric and magnetic field distributions resulting from applying Babinet's principle to the set of complementary structures that we propose are presented and discussed in chapter 4. However, the enhancement field distributions are not the only way to validate the principle. Indeed, if Babinet's principle hold, the complementary structure should behave completely analogous to the original device, except that all of the fields should exhibit a 90 degree rotated polarization. To verify this, the same methodology to describe the original hybrid device was followed to describe the complementary structure.

Principle of operation of the complementary structure

Thanks to the application of Babinet's principle, the behavior of the magnetic fields due to the presence of the complementary structure is similar to that of the electric fields due to the presence of the original structure.

In the complementary structure, the incident field injected to the Si photonic waveguide corresponds to its fundamental TE 0 . At the beginning of the hybrid region, To verify this complementary behavior, the same approach as for the original case was followed. This is schematically shown in Fig. 33: first the photonic and plasmonic waveguides are analyzed independently and then the hybrid coupling region is solved.

In this case, the Si photonic waveguide is the same as for the original device, supporting only the fundamental TE 0 and TM 0 modes for a width of 500 nm, at λ = 1550 nm.

The dispersion relations, group velocities, and mode transverse profiles (at  = 60 nm) for the single-mode and coupled-mode fields were calculated, as well as the coupling length curve between the H C TE 0 and the H C As observed, from Figs. 34a)-34b), although the symmetric mode S C 0 has and effective index slightly above n sb , only the asymmetric plasmonic mode A C 0 is strongly confined in space and time as the width of the nanotaper decreases. This is shown by the asymptotically increase of its n eƒ ƒ for  < 100 nm, as well as by the drastic reduction of its group velocity in the same width range, reaching a value up to 10 % of c.

Since the analysis for the Si photonic waveguide is the same as before, the (constant) effective indexes and group velocities of the fundamental TE 0 and TM 0 modes are also plotted. As insets in these figures, the transverse mode profiles at  = 60 nm and the top view of the plasmonic complementary waveguide are respectively shown.

When analyzing the coupling region we have, as expected from Babinet's principle, that the hybrid mode H C TE 0 is the only one achieving spatial and temporal confinement.

Its dispersion curve, group velocity and spatial profile are shown in Figs. 34c), 34d and 34f), respectively.

Finally, the fields that contribute to the energy transfer process in the complementary device are the coupled H C TE 0 and H C TE 1 modes. From which, following the same procedure as for the original structure, the coupling length as a function of the nanotaper width was obtained [Fig. 34e)], and the averaged value is calculated as: ding Babinet's principle, we will keep the same nanotaper length as for the original screen (i.e. L C = 800 nm). And as plotted in Fig. 34f), the profile of the calculated power along the propagation direction does indicate energy transfer from the Si waveguide to the plasmonic one at this L. To evaluate the magnetic field confinement by the complementary device, the effective volume of the coupled mode can be numerically implemented as (the resulting value is discussed in the following section):

 C c (150) =
V c eff = |H c (r)| 2 dV 2 |H c (r)| 4 dV . (4.11)
The design of the complementary device through Babinet's principle has a simple motivation: we want to design a set of nanostructures able to modify the spontaneous decay rate of the electric and magnetic transitions of suitable quantum emitters. Up to now, we have shown that indeed the complementary structures achieve strong spatial-temporal confinement for a given mode. The purpose now is to exploit this fact to modify the electric and magnetic LDOS of ED and MD emitters by Purcell effect.

To this end, the electromagnetism reciprocity theorem presented in section 2.3.2 is applied as described below.

Electric and magnetic Purcell effect: the application of the electric and magnetic reciprocity theorem

Following the Lorentz' reciprocity theorem with sources, if a dipole source is placed around the apex of the plasmonic nanotaper, the following process will occur:

because of the strong confinement of either the electric or magnetic field of a given plasmonic mode, an ED or MD placed around the apex will predominantly radiate and couple into the plasmonic mode which, in turn, will propagate through the overlapping (photonic-plasmonic) region until finally coupling into a Si photonic mode. This process is characterized by a change in the spontaneous emission rate of the dipoles by Purcell effect due to the electric or magnetic LDOS enhancement. As detailed in chapter 2, all the figures of merit characterizing this effect are functions of the position, frequency, and orientation of the dipoles. They account for a particular mode field excited by the emission of the dipoles, among radiative modes and forward and backward guided modes.

To verify the reciprocity property, we need to evaluate the performance of the complementary devices when the source and the detection points are interchanged, giving raise to the following situations:

1. The direct scenario corresponds to the previous cases already studied, the source is given by the fundamental TM 0 (TE 0 ) of the Si waveguide which propagates in the forward direction ( > 0) from the base to the apex of the original (complementary) plasmonic nanotaper and produces an electric (magnetic) field confinement which is measured around the apex of the nanotaper.

2. In the reciprocal scenario [Fig. 35)], the source will be an ED (MD) positioned around the nanotaper apex of the original (complementary) device. In principle, the emission of the dipole will excite the plasmonic mode which will propagate in the backwards direction ( < 0) from the apex to the base of the nanotaper, and finally it would be able to transfer energy to the Si waveguide, measured in the TM 0 (TE 0 ) photonic mode.

dipole source Both of these situations were analyzed by propagating light through the integrated structures. Similarly to the case of Babinet's principle, the full vectorial 3D finitedifference time-domain (FDTD) method implemented by Lumerical©FDTD Solutions, was used as follows (illustrated in Fig. 36).

To evaluate the field confinement by the original (complementary) structure, it was illuminated by the forward-propagating TM 0 (TE 0 ) fundamental mode. Then, the electric (magnetic) field profile in the frequency domain was collected from simulation results, at λ = 1550 nm, across some spatial region within an output plane located 10 nm above the metallic nanostructure, i.e. Z 0 = 10 nm [Fig. 36a)].

In the reciprocal situation for the original (complementary) structure, an ED (MD)

source was located at different positions, one at a time, lying in the Z 0 plane (10 nm above the plasmonic nanotaper), and the power carried by the backwardpropagating TM 0 (TE 0 ) mode of the Si waveguide was taken as the output. In this case, expansion monitors were implemented to calculate the power carried by a particular forward/backward kth mode. It should be noticed, as sketched in Fig. 36b), that to determine the power emitted by the source, a monitor box surrounding the device must be also implemented. In this way, the Purcell factor, quantum yield and β-factor can be then calculated following the methodology presented in section 3. The corresponding results for each complementary and reciprocal scenario are presented and discussed in the following section.

Up to now, a set of complementary devices capable to confine the electric and magnetic fields was designed, based on the application of the Babinet's principle. Indeed, we expect that, by reciprocity, these set of nanostructures are able to enhance the electric and magnetic LDOS when ED and MD are positioned in close proximity to the devices. In the following, we analyze the validity of this approach.

Numerical analysis of the plasmonic waveguides on silicon photonics based on FDTD methods

We have proposed a methodology to design a set of complementary devices based on Babinet's principle, and assuming that the reciprocity theorem will allow them to enhance the corresponding electric and magnetic LDOS. Here, the applicability of the principles is analyzed.

Qualitative demonstration of the Babinet's and reciprocity principles

To verify Babinet's principle, the length of the plasmonic nanotaper was fixed at The distributions of the total electric and magnetic near-field enhancements are shown in Fig. 38a) and Fig. 38c), respectively. It can be observed that the near-field distribution of the electric field exhibits a maximum enhancement at the nanotaper From this, it can be confirmed that even though the metallic layers are not perfectly conducting thin screens, the distributions of the electric and magnetic field near-field enhancements in the original and complementary structures are equivalent, hence Babinet's principle qualitatively holds.

To verify the reciprocity property, the original structure is illuminated with the forward-propagating photonic TM 0 mode and the electric field enhancement represents the output, collected across the Z 0 plane. In the reciprocal situation, the source corresponds to an ED with emission peak at λ 0 = 1550 nm located at every point across the Z 0 plane, and the power collected at the backward-propagating TM 0 mode is obtained at the output. The same setup was then applied to the complementary device: it is excited by the forward-propagating TE 0 mode and the output corresponds to the magnetic field enhancement, collected across the Z 0 plane. In the reciprocal situation, the source is a MD with emission peak at λ 0 = 1550 nm located at every point across the Z 0 plane, whereas the power carried by the backward-propagating TE 0 mode is calculated at the output.

The distributions of the direct case are those already discussed in the demonstration of Babinet's principle [Figs. 38a) and 38c)], where the electric and magnetic fields were enhanced at a position 10nm above the nanotaper apex by factors of 9 an 4, respectively. In the reciprocal situation, we have that when the ED is placed 10 nm above the nanotaper apex, the power collected into the TM 0 mode is enhanced by a factor of 36 [Fig. 38b)]. The same applies to the complementary structure, in this case the power collected into the TE 0 exhibits an enhancement of 2 [Fig. 38d)].

From chapter 3, we know that ratio P/ P 0 actually represents the modification of the LDOS, which in turns gives us the modification of the spontaneous emission rate of the dipole relative to free space. In addition, we know that the LDOS enhancement is polarization-dependent; therefore, the maps presented in Figs. 38b) and 38d) were obtained by averaging the results produced by dipoles oriented along the , y and z directions.

Since the power is measured into each specific guided mode at the output, the factors of 36 and 2 represent the electric and magnetic LDOS produced by the original and complementary structures, respectively. This means that the spontaneous emission rate of the ED, collected into the TM 0 is being enhanced F

x,y,z TM 0 = 36 times; and the spontaneous emission rate of the MD is being enhanced by a factor F x,y,z TE 0 = 2, when collected into the TE 0 , at the output of the Si waveguide.

The enhancement of the spontaneous emission rate of the dipoles is a consequence of the modification of the available photonic states due to the presence of the hybrid plasmonic-photonic device. In particular, we have seen that this modification of the LDOS is the result of the strong electromagnetic field confinement achieved by the plasmonic nanostructures, which is characterized by the extreme reduction of the effective mode volume in each device. Following Eqs. (6.18) and (6.19), the effective mode volume of the electric field produced by the original structure at the nanotaper apex was calculated to be V eff = 0.0017(λ/ 2n) 3 , which is less than 0.2 % the diffraction limit. In the case of the complementary device, the magnetic mode volume of V c eff = 0.0044(λ/ 2n) 3 was obtained, being less than 0.5 % the diffraction limit.

The profiles at y = 0 (i.e. the nanotaper apex) of the results are plotted in Fig. 39.

The apex of both nanotapers (top views shown as insets) are located at ( = 0.5, y = 0)μm. As observed, in either case the maximum values are obtained at this position.

Additionally, it is clearly seen that the spatial profile distributions are similar point to point.

Once that Babinet's principle and the reciprocity theorem were proven to hold, we proceeded to characterize the LDOS modification produced by both the original and the complementary devices. Indeed the electric and magnetic LDOS can reach different enhancements depending on the dipole wavelength, the length of the plasmonic nanotaper, and the dipole orientations.

Electric and magnetic LDOS enhancement: influence of the geometry and dipole orientation

To evaluate the LDOS enhancement, a parametric study was performed based on FDTD calculations, varying the orientation ( = , y, z) and position of the dipoles, as well as the length of the plasmonic nanotaper from 300 nm to 1300 nm, over a wavelength range from 1400 nm to 1600 nm. This is the wavelength range of interest since we are considering the use of the structures across the S-and C-bands of the telecommunication wavelength (S-band: 1460 nm -1530 nm, C-band: 1530 nm -1565 nm), thus able to transmit in the near infrared, with the peak centered at λ = 1550 nm.

To begin with, the dependence on the position was analyzed for several locations of the dipoles. The results presented here correspond to the maximum values reached for a fixed position of the dipoles (given below). Additionally the dipoles are unpolarized (randomly oriented), for which the average over the , y and z orientations of the dipoles was performed.

In Fig. 40, maps of the total electric and magnetic LDOS, the guided-mode LDOS, the quantum yields, and the β-factors, are plotted as a function of the wavelength of the source and the length of the nanotaper.

As it can be observed, the behavior of each figure of merit is approximately constant over the emission wavelength, as expected from the non-resonant behavior of the devices. On the other hand, strong variations are present as a function of the nanotaper length mainly due to the vertical energy transfer process between the coupled modes along the overlapping region. The results in Fig. 40 correspond to a specific position of the ED (MD), for which the electric (magnetic) LDOS enhancement was at its maximum. In the original (complementary) structure, the ED (MD) is located at  = +5 nm( C = -5 nm), and y = 0 nm from the nanotaper apex, at Z = 10 nm above the SiO 2 transfer layer (insets in Fig. 40). The Z position should not be confused with the position Z 0 = 10nm above the metallic structures used in the previous studies. In fact here, the dipoles are closer to the plasmonic nanostructures and thus the LDOS enhancement can reach higher values compared to those obtained before.

Purcell factor and guided Purcell factor

From this parametric study, the original structure exhibits a maximum electric Purcell factor 272 [Fig. 40a)] versus a maximum magnetic factor of 90 produced by the complementary structure [Fig. 40b)]. The guided-mode or coupled Purcell factor, which represents the spontaneous emission enhancement coupled into one of the fundamental modes at the output of the Si waveguide reaches a maximum value of 77 in the case of the original structure [Fig. 40c)], and of 21 in the complementary device [Fig. 40d)]. The results are summarized in table 2. According to the reciprocity theorem, the electric LDOS enhancement measured at the TM 0 mode in the original structure, should follow the same behavior as the electric field enhancement produced when the device is injected by the TM 0 mode (i.e. when the input and the output are interchanged). This means that if the structure achieved a maximum field enhancement at L = 800 nm then, the LDOS enhancement should be found at the same length. Similarly, the maximum magnetic LDOS enhancement The left-hand side scale corresponds to the electric (magnetic) field enhancement (red curve) and the right-hand scale measures the total (black curve) Purcell factor, and the guided-mode Purcell factor coupled into the TM 0 (TE 0 ) mode in the backward direction. As observed, all the curves exhibit oscillations along the nanotaper length due to the energy transfer process (constructive and destructive interference), as we already knew. Notice that in both structures, the field enhancement and the guided Purcell factors follow the same oscillatory behavior, with their maximum and minimum peak values at the same nanotaper length, as predicted by the reciprocity theorem.

On the other hand, observe that the total Purcell factor exhibits its maximum value at a longer L. Remember that the total Purcell factor considers the enhancement due to every radiative and non-radiative process, including the emission into free-space, into the guided modes in the forward and backward directions, and also the increasing of losses due to absorption in the metal. Whereas the guided-mode Purcell factors accounts only for the contribution of the guided-mode in the backward direction. In particular, it might be the case that at longer nanotaper length, the contributions of the non-radiative transitions due to losses and of the forward guided-modes become stronger.

Quantum yield and β-factor

Also from the parametric study, we have that, in both cases, for the original and the complementary devices, the quantum yield reaches a maximum value of 0.62 [Figs.

40e)-f)]. The radiative emission of the ED is coupled into the TM 0 with a maximum efficiency given by a β-factor of 0.4, whereas the radiative emission of the MD is coupled into the TE 0 with a β-factor of 0.2. The results are summarized in table 3. Observe that, although these values correspond to randomly polarized electric and magnetic dipoles, analyzed over different nanotaper lengths and different wavelengths, very interesting results emerge. From the application of Babinet's principle, the electric field enhancement in the original structure was about twice the magnetic field enhancement in the complementary structure (≈ 9 :≈ 4.5). Now, a factor of three Based on the promising numerical results, we then proceeded to fabricate and characterize the set of complementary plasmonic structures, as follows.

Fabrication of the nanostructures

The integrated hybrid devices were fabricated by standard electron beam lithography (sketched in Fig. 44), evaporation deposition, and lift-off process. It should be mentioned that the Si photonic waveguides were already fabricated at the beginning of the project. Thus, the process described here involves the fabrication of the complementary plasmonic Au nanostructures, placed on top of the Si waveguides, as follows.

To begin with, the samples should undergo a cleaning process consisting of an ultrasonic bath of acetone over 1 min. After that, the sample is immersed in an ethanol and isopropanol bath. The sample is then dried with compressed air.

Prior to the electron beam lithography process, the 20 nm thick layer of SiO 2 was deposited over the Si waveguides by thermal evaporation. Recall that this layer is devoted to favor the energy transfer process between the photonic and plasmonic After this process, the exposed regions are removed, thus creating a 'hole' with the shape defined by the mask.

Once that the exposed PMMA resist has been removed, the metallic layer can be deposited on the substrate. Since the metal to be deposited is gold, a 3 nm to 5 nm thick layer of chrome was first deposited, otherwise the metallic material cannot be attached to the SiO 2 film. Then the 30 nm thick layer of gold was thermally evaporated onto the wafer.

After the metal deposition, the remaining (unexposed) PMMA resist must be dissolved by immersion of the sample in an appropriate solvent (we used acetone), over 1-2 days. This process is called lift off. In this way, the metal sticking to the resist looses "floor" so that only the metal attached to the SiO 2 layer remains.

Several samples of the original and complementary plasmonic structures were fabricated, over different substrates as well as over the Si waveguides to integrate the hybrid photonic-plasmonic devices. Scanning electron microscopy (SEM) images were taken and some of them are presented in the following subsections.

Integrated complementary structures on silicon photonic waveguides

The plasmonic structures were also integrated onto the Si waveguides of interest. In Fig. 47, SEM images of a set of the complementary structures are provided. follows.

Choice of the quantum emitters at telecom frequencies: PbS quantum dots

Since we are interested in the integration of nanophotonic devices that operate at the telecommunication frequency band, the choice of the quantum emitters is rather limited compared to the available options in the visible window. Additionally, in order to explore the magnetic LDOS enhancement, it is desired that the emitters exhibit not only ED transitions, but strong MD transitions as well.

Rare-earth ions and semiconductor QDs are quantum emitters that possess both,

Characterization of the integrated plasmonic structures on silicon waveguides

The characterization of the plasmonic structures consisted of far-field and near-field measurements. In the case of the plasmonic structures fabricated over glass and silicon substrates (nonguided media), we were able to characterize the fluorescence lifetime of the PbS quantum dots by time-resolved plotoluminiscence (PL) measurements. In the case of the plasmonic structures integrated on the Si waveguides, the observations correspond to near-field scanning optical microscopy (NSOM) images.

Time-resolved PL measurement of PbS QDs in nonguided configuration

The measurement of the fluorescence lifetime decays of quantum emitters provides a powerful experimental tool to characterize the LDOS enhancement produced by a given electromagnetic environment. To record the fluorescence lifetime of PbS quantum dots in different optical environments, the Time Correlated Single Photon Counting technique (TCSPC) was used. As shown in Fig. 49a), a pulsed laser is incident to the sample, which triggers a timer at t=0. The laser pulse then excites the PbS quantum dots, and the photon emitted after some time Δt is detected by a single-photon detector, which in our case corresponds to a single-photon avalanche photodiode (SPAD).

The signal produced at the output of the detector stops the timer. The period of time between the laser pulse and the SPAD output pulse is then recorded. This process is repeated over and over again at a frequency given by the laser repetition rate. The collected data is then statistically analyzed to characterize the exponential decay rate of the emitters. The experimental data is finally fitted to an exponential curve e -t/ τ , with τ being the fluorescence lifetime of interest.

A mandatory requirement imposed by this method in order for the measurements to be valid, is that the probability that the laser pulse excites more than one fluorescence photon remains negligible. As a general thumb-rule, this requirement is covered by assuring that the counts rate of the fluorescence signal keeps between 2 % and 5 % of the laser repetition rate.

The experimental setup is shown in Figs. 49b)-c). The pump signal is a pulsed laser 112 of 12 picoseconds width and repetition rate of 200kHz. The wavelength of incident the laser is λ = 520 nm, which need to be blocked by the use of a suitable filter in order to avoid nonlinearities in the detector. We used the ID210 infrared single-photon detector, which has a detection window in the near infrared between 900 < λ < 1800 nm. The time correlation between the input and the output pulses was carried out by the use of the Pico-Quant HydraHarp 400© correlator. The different environments for which the PbS quantum dots fluorescence lifetime were measured are:

1. PbS quantum dots in toluene solution. (i.e. homogeneous medium).

2. PbS quantum dots spin-coated in close proximity to the original plasmonic nanotaper. In contrast to the hybrid photonic-plasmonic integrated structures, the plasmonic nanotaper is placed directly over a glass substrate, i.e. the Si waveguide is not present.

3. PbS quantum dots spin-coated in close proximity to the complementary plasmonic nanotaper. Same as for the previous point, the plasmonic antenna is placed directly over a glass substrate, in absence of the Si waveguide.

The obtained curves are plotted in Fig. 50

Imaging of the electric near-field in the set of complementary structures

To characterize the electric field enhancement produced by each structure, the NSOM technique was employed since it provides sub-wavelength resolution for the observation of electromagnetic fields in nanostructured systems. This imaging technique allows to break the far-field resolution limit by exploiting the properties of evanescent waves, as explained in section 2.4. When a high refractive index nanotip probe is placed in close proximity to the structure of interest, it is able to convert the evanescent field bounded to the surface into scattered field, which is then collected, and detected in the far field. In particular, to characterize the plasmonic structures, two different NSOM configurations were used: aperture and apertureless (or scattering) NSOM in transmision.

Plasmonic structures on glass: commercial aperture NSOM

The aperture NSOM in transmision is a very common configuration in which the excitation laser light is focused through an aperture with a diameter smaller than the excitation wavelength, resulting in an evanescent field (or near-field) on the far side of the aperture (Fig. 51). When the sample is scanned at a small distance below the aperture, the optical resolution of transmitted light is limited only by the diameter of the aperture. The optical resolution attainable is in the range of 60 -100 nm. The optical image is generated by scanning the sample's surface point-by-point and lineby-line.

The particular NSOM employed to analyze the plasmonic structures on glass substrates is the WITec alpha300 S. It uses micro-fabricated cantilever (the base where the nanotips are placed) sensors and a customized near-field objective with an aperture The NSOM images were collected at the excitation wavelength λ = 520 nm, rather than at the fluorescence emission spectrum. This is because the full microscope system is optimized for detecting light in the visible range, and the collection in the NIR spectrum is highly challenging. In this sense, the future work involves the optimization of the optical components in the NIR frequency range. However, even from these images, interesting results are present. The dark regions in Figs. 52e)-f) correspond to the metallic structures, where absorption takes place. From this, we expect that, for PbS QDs positioned over these samples, fluorescence photons will be emitted, and maps similar to the simulation results could be obtained. Additionally, the perturbations over the illuminated regions are indication of radiative modes propagating through the substrates.

The plasmonic structures integrated on the Si waveguides were then characterized by means of the aperturless NSOM described below.

Plasmonic structures on silicon waveguides: homemade apertureless NSOM

The principle of operation of the apertureless (or scattering) s-NSOM is sketched in Fig. 53. A sample is placed under an Atomic Force Micorscope (AFM) in tapping mode, which measures the topography of the sample. The nanotip of the AFM scatters the evanescent wave into free-space, producing a perturbation in the transmitted field, which is coupled to a single-mode optical fiber. The transmitted signal is measured by an Avalanche Photo-Diode (APD) that transforms the optical field into an electrical signal. Since the scattered and transmitted fields are modulated by the frequency of oscillation of the nanotip, the field of interest must be demodulated from the oscillation frequency by means of a lock-in amplifier. In this way, the intensity of the optical field propagating in the sample is obtained. The configuration is schemed in Fig. 53c). In particular, this NSOM configuration counts with an integrated optics stage, which allows the direct measurement of guided structures. However, it should be pointed out that during the characterization of the samples, the instrument experienced some problems related to the head of the AFM microscope (where the tip is located), resulting in rather ghost images. In addition, this instrument allows only measurements of electric near-field distributions, since specially designed NSOM probes would be needed to measure the magnetic near-field distributions [START_REF] Ernandes | Fresnel reflection in fiber optics. Reflection -EX-FO animated glossary of Fiber Optics[END_REF]. The achieved NSOM observations with this setup are presented below.

To demonstrate the confinement and enhancement of the electric and magnetic fields, NSOM images were obtained for the samples shown in Figs. 54a-b). The length of the fabricated original and complementary structures are L = 865±5 nm and L c = 740±5 nm with an apex radius of R = 12±5 nm .

The samples were then characterized by the scattering NSOM setup in transmission also described above, which allows to measure the electric (and only the electric) near-field distributions over the integrated plasmonic structures. The magnetic field distribution can not be measured with this NSOM configuration, since specially designed NSOM probes would be needed to this end [START_REF] Ernandes | Fresnel reflection in fiber optics. Reflection -EX-FO animated glossary of Fiber Optics[END_REF].

By the use of a polarizer at the input of the Si waveguide, in the original structure the silicon photonic TM 0 mode was excited; whereas in the complementary structure, the TE 0 was the excited mode. The topography (AFM images) of the devices [Figs. Although the NSOM maps present ghost images due to AFM head problems, the structure clearly presents confinement and enhancement of the electric field at the apex. In the complementary structure, the electric field is confined inside the structure.

To verify these experimental observations, the electric field distributions were calculated and collected at different distances above the metallic surface of the complementary integrated plasmonic structures, modeling the distance between the tip of the NSOM and the metallic screens. The maps shown in Fig. 55 correspond to a distance of 20nm.

The simulated results confirms that the excitation of the TM 0 mode into the original structure produces an electric field distribution that is confined to the apex of the plasmonic structure with an enhancement factor of 5 (Fig. 55a). For the complementary structure, the electric field is confined to the apex but at the interior edges, where an enhancement factor of 1.2 is at its maximum. The spatial resolution of the NSOM measurement does not allow to measure this value experimentally. However, the field distribution confirms the expected behavior when the device is excited by the TE 0 mode. To model the excitation of unpolarized modes, we simulated the excitation of a combination of ( 50 lation (electric field enhancement) and the experimental (ϵ|E| 2 ) results for the original structure is shown in Fig. 56, where agreement between the calculated and experimental spatial distribution is observed.

To carry out this comparison, it should be taken into account that the simulated electric field enhancement (EF) is defined as the ratio of the amplitude of the electric field at the apex of the structure to the amplitude of the incident field, that is, the amplitude of the TM 0 mode. However, since the amplitude of the incident field cannot be experimentally obtained (because of the guided configuration), a quantitative analysis is difficult to achieve. An estimation of the experimental EF can be obtained from the NSOM measurements, by defining the enhancement factor as

EF * = |E pe | 2 / |E pe,0 | 2 ,
where |E pe | 2 is the NSOM signal at the apex and |E pe,0 | 2 is the minimum value in lated EF. A reason for the difference between the experimentally and simulated EF is that the experimental distance between the AFM tip and the surface of the sample is not precisely known and the field at the apex of the structure exponentially decreases along the normal direction. To appreciate this effect, the profile at two different distances from the surface, i.e. at 20 nm and 30 nm are plotted. As observed, although the experimental curve exhibits broadening, the spatial distributions of the profiles follow the same pattern.

At this point, it should be mentioned that experimental measurements of the Purcell effect could not be obtained, because it was not possible to couple the PbS QDs over the set of complementary structures. However, the fluorescence lifetimes presented in section 4.5.1 still hold as a proof-of-principle characterization.

Conclusion

The results presented in this chapter show that both reciprocity and Babinet's principles hold for the set of complementary integrated plasmonic structures. The set of complementary plasmonic structures in nonguided media were characterized in the far-field by time-resolved PL measurements, where significant modification of the fluorescence lifetime were obtained, as an indication of LDOS enhancemenet. Finally, the plasmonic structures integrated on Si waveguides were characterized in the nearfield by NSOM measurements, exhibiting electric field confinements which are in good agreement with the calculations.

Chapter 5. General conclusion

In this work, the spontaneous emission of ED and MD emitters was analyzed in terms of the local density of states and the appropriate figures of merit to characterize the processes were defined and evaluated. From this analysis, it was shown that the modification of the LDOS in an arbitrary electromagnetic environment can increase the emission decay rate, characterized by a generalized total Purcell factor, but more importantly the radiative processes were also studied in terms of the guided-mode

LDOS, which provides relevant information regarding what fraction of the enhanced spontaneous emission is coupled into a specific guided mode of interest.

The manuscript was then focused to the understanding of the properties of electromagnetic systems that allow to modify the local density of states in an efficient way, and we saw that this can be achieved by either 1) the use of resonant cavities with high Q factor, or by 2) strong confinement of the optical fields to extremely small effective mode volumes. In our case, the choice was to decrease the effective mode volume because the use of resonant cavities not only limit the bandwidth of the devices, but also their physical sizes, which represent an important drawback towards the integration of on-chip optical platforms.

To drastically reduce the effective volume of optical fields, a plasmonic nanoantenna supporting surface plasmon polaritons was designed, because of their ability to confine the electromagnetic fields to regions far below the diffraction limit. However, SPPs suffer from big losses due to absorption in the metal and thus the light cannot propagate over long distances. For this reason, the plasmonic nanoantenna was vertically integrated on top of a Si photonic waveguide, which provides low-loss transport and is thus suitable for long distance propagation.

The methodology to design and characterize the integrated photonic-plasmonic platform consisted of the analysis of each individual structure separately, from which the single-mode fields supported by each of them were found. Then, the hybrid photonicplasmonic device was integrated and analyzed following the coupled-mode theory. Because of the geometry of the device and thanks to the vertical coupling configuration, the integrated structure promotes an adiabatic energy transfer process of the optical modes, bouncing up and down between the Si waveguide and the plasmonic nanoan-123 tenna, which results into a strong spatial and temporal confinement of the electric field at the apex of the plasmonic nanostructure. From numerical calculations, the effective mode volume was obtained to be less than 0.2 % of the diffraction limit at λ = 1.55μm and the corresponding electric field enhancement, measured 10 nm above the apex of the plasmonic nanotaper, was found to be |E|/ |E 0 | = 9.

Next, the full vectorial Babinet's principle was used to design a complementary hybrid device, which promotes a magnetic near-field enhancement that exhibits a similar spatial distribution to the electric near-field enhancement produced by the original device. The complementary structure achieved a maximum magnetic field enhancement of |H|/ |H 0 | = 4.3 and a confinement of the magnetic field up to less than 0.5 % of the diffraction limit. It is important to mention that the rigorous vectorial derivation of Babinet's principle considers the ideal case of infinitely thin complementary screens in free-space, whose conductivity is also required to be infinite. Since the devices studied here do not satisfy these requirements, an adaption of this principle was developed, and its applicability to our guided-configuration was explored, leading to successful demonstration.

After demonstrating the electric and magnetic field enhancements achieved by the set of complementary structures, the Lorentz' reciprocity theorem with dipole sources was applied in order to modify the LDOS of electric and magnetic dipole emitters. The Finally, the integrated photonic-plasmonic platforms were fabricated and experimental observations based on NSOM and TCSPC techniques served to confirm the principle of operation of the structures.

In summary, a set of complementary nanophotonics devices was theoretically studied, numerically analyzed, fabricated and characterized. The results obtained show that both the reciprocity and Babinet's principles qualitatively hold for the original and complementary integrated plasmonic structures on silicon waveguides at telecommunication frequencies, even though the structures do not strictly satisfy the conditions imposed by Babinet's principle.

By the use of these principles, the integrated plasmonic structures on silicon photonic waveguides were proven to confine and enhance the electric and magnetic fields and reciprocally, to modify the electric and magnetic local density of states and thus, to enhance the emission of electric and magnetic dipoles.

Given the fact that the s-NSOM employed for the experimental characterization is a very unique homemade instrument, with the capability of measuring the near-field directly from guiding structures due to its integrated optics stage, the observations are challenging and further work is required. These NSOM measurements correspond to the first step towards the experimental validation of the principles. However the theoretical results presented still provide evidence of the enhancement of the spontaneous emission decay rates of electric and magnetic dipole emitters.

The configurations presented here are suitable for several scientific applications.

To begin with, the enhancement and confinement of electromagnetic radiation to na-nometer scale is already a valuable feature in fields such as spectroscopy, medical applications and quantum information. Moreover, the realization of on-chip nanofocusing devices compatible with silicon photonics platform adds a key functionality and provides opportunities in sensing, trapping, local heat generation, nonlinear optics, and on-chip signal processing among others. In addition, due to the underlying mechanisms promoting the energy transfer process form a propagating guided mode to a trapped plasmonic mode at the apex of the plasmonic nanotaper, both the original and complementary devices exhibit a mode selectivity, which provides opportunities for multiplexing different functionalities by using different polarizations in the same structure. Regarding the enhancement of the spontaneous emission rate of ED quantum emitters, it is a desirable feature for efficient laser operation, fluorescent microscopy, nanoscale imaging and sprectroscopy, biological studies, and of course, for the design of efficient single-photon sources. In the context of the enhancement of the magnetic Purcell factor, it is an increasing field of study that constitutes an emerging possibility to empower the development of nanooptics tools and facilitate the development of efficient magnetic field optical probes, which could be used as spectroscopic tools for measurement of the magnetic optical fields at the nanoscale, allowing, for example, to distinguis different multipolar contributions into the total fluorescence from a given quantum emitter and/or to fully characterize the optical modes into which the emitter radiates.

In particular, future perspectives of the work developed here involves exploring the use of these devices as the basis for the implementation of on-chip integrated singlephoton sources with the following features:

1. Deterministic, With all these characteristics, the hybrid devices proposed here constitute a suitable platform for quantum information processing applications in which the electric and magnetic dipole transitions of the emitters could be independently enhanced.

Chapter 6. Résumé de la thèse en français

Introduction

Pour le développement des technologies de l'information et de la communication quantiques, il est souhaitable que les sources à photon unique puissent être intégrées à des dispositifs nanophotoniques compatibles avec les plateformes photoniques au silicium (Si) actuelles [START_REF] Soref | The past, present, and future of silicon photonics[END_REF]. En plus, la longueur d'onde d'émission de la source devrait être celle qui minimise les pertes optiques dans la fenêtre de transmission des télécommunications, par exemple entre 1.3 μm et 1.5 μm, pour la propagation sur de longues distances et, aux mêmes temps, celle qui optimise l'efficacité de la photodétection.

Jusqu'à présent, le contrôle de la génération de photons uniques par émission spontanée a été principalement axé sur la recherche de la décroissance spontanée des transitions dipolaires électriques (ED), car la force des transitions ED dans des sources quantiques optiques typiques est supérieure que celle des transitions dipolaires (MD) magnétiques [START_REF] Landau | En: Electrodynamics of continuous media[END_REF]. Cependant, certains émetteurs quantiques, tels que les ions de terres rares [START_REF] Carnall | Spectral intensities of the trivalent lanthanides and actinides in solution. II. Pm3+, Sm3+, Eu3+, Gd3+, Tb3+, Dy3+, and Ho3+[END_REF][START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF] et les points quantiques de semi-conducteurs (Zurita-Sánchez y [START_REF] Zurita-Sánchez | Multipolar interband absorption in a semiconductor quantum dot. II. Magnetic dipole enhancement[END_REF], possèdent des transitions MD dont la force est comparable ou supérieure à celles électriques. Le couplage des ions lanthanides et des points quantiques à divers systèmes nanophotoniques, notamment des matériaux en vrac [START_REF] Deloach | Evaluation of absorption and emission properties of Yb 3+ doped crystals for laser applications[END_REF], des structures planaires [START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF][START_REF] Karaveli | Spectral tuning by selective enhancement of electric and magnetic dipole emission[END_REF],des nanoantennas diélectriques [START_REF] Shi | A new dielectric metamaterial building block with a strong magnetic response in the Sub-1.5-Micrometer Region: Silicon Colloid Nanocavities[END_REF][START_REF] Sanz-Paz | Enhancing magnetic light emission with all-dielectric optical nanoantennas[END_REF] et plasmoniques [START_REF] Hussain | Enhancing Eu 3+ magnetic dipole emission by resonant plasmonic nanostructures[END_REF][START_REF] Feng | Controlling magnetic dipole transition with magnetic plasmonic structures[END_REF] et des métamatériaux a été proposé [START_REF] Simovski | Wire metamaterials: physics and applications[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF]. Ou les progrès des techniques de nanofabrication, ainsi que l'étude croissante des émetteurs quantiques magnétiques ont stimulé l'investigation du côté magnétique de l'émission spontanée.

Ce travail prétend explorer l'intégration des émetteurs quantiques avec des transitions dipolaires électriques et magnétiques dans des dispositifs nanophoniques en silicium, vers le développement de sources à photon unique, déterministes et hautement efficaces, compatibles avec les plateformes optiques actuelles et futures pour le développement des technologies de l'information quantique.

Concepts préliminares

La nanophotonique consiste à contrôler la lumière en créant de propriétés optiques spécifiques en la matière (ϵ, μ), dans l'espace et sur les échelles de la longueur d'onde du vide; en rappelant que l'indice de réfraction et, par conséquent, la permittivité électrique et la perméabilité magnétique ne sont pas des constantes absolues des matériaux, mais dépendent de la fréquence (le phénomène de dispersion). Dans les sections suivantes, la perméabilité magnétique sera considérée comme μ = 1, comme c'est le cas pour les matériaux ordinaires à des fréquences optiques.

La fonction totale de Green

Pour un milieu fermé non absorbant, la fonction de Green électrique G(r, r 0 , ω) est définie comme la solution à l'équation d'onde non homogène:

(∇ × ∇×) -ϵ (r) ω 2 c 2 G (r, r 0 , ω) = δ (r -r 0 ) , ( 6.1)
avec  le tenseur unitaire, et avec les conditions aux limites appropriées à la surface de la cavité fermée. La fonction de Green a une interprétation physique simple: pour une source dipolaire électrique ponctuelle (ED), monochromatique, avec un moment dipolaire d, situé à une position r 0 , le champ électrique rayonné en un point r est:

E (r) = μ 0 ω 2 G (r, r 0 , ω) d. (6.2)
En termes de modes propres du système, la fonction de Green prend la forme (Car-minati et al., 2015): [START_REF] Jackson | Classical electrodynamics[END_REF]. Une exaltation du champ magnétique, composant par composant, sur l'écran complémentaire peut être obtenue à partir des composants de l'exaltation du champ électrique de l'écran d'origine comme: .4) où  = , y, z. Les structures plasmoniques intégrées que nous proposons ici ne satisfont pas les conditions d'application du principe de Babinet (c'est-à-dire des écrans minces plans parfaitement conducteurs dans une étendue infinie dans l'espace libre).

G (r, r 0 , ω) = k c 2 e * k (r 0 ) ⊗ e k (r) ω 2 k -ω 2 , ( 6
B C  (r) B C 0 (r) = 1 - E  (r) E 0 (r) . ( 6
Cependant, son applicabilité à l'épaisseur et à la conductivité finies a été discutée [START_REF] Koo | Extraordinary magnetic field enhancement with metallic nanowire: role of surface impedance in Babinet's principle for sub-skin-depth regime[END_REF], avec une démonstration satisfaisante dans le régime de sous-profondeur de peau. Ici, nous considérons sa validité dans une configuration de guide d'ondes.

Pour la structure d'origine, nous considérons le champ incident comme le mode photonique TM 0 du guide d'onde en Si et du mode TE 0 pour la structure complémentaire.

Nous allons utiliser le principe de Babinet vectoriel pour déduire des distributions de champs magnétiques inconnues à partir des distributions de champs électriques connues pour un ensemble donné de structures complémentaires.

Théorème de réciprocité électromagnétique

Pour les sources de courant dipolaires électriques, avec les densités de courant données par J k (r) = -iωp k δ(rr k ), où k = 1, 2, et p k sont les moments dipolaires électriques, une dérivation formelle du théorème de réciprocité conduit à [START_REF] Jones | The theory of electromagnetism[END_REF][START_REF] Carminati | Reciprocity of evanescent electromagnetic waves[END_REF]: .5) Ou encore, en remplaçant les moments ED par les moments MD m 1 et m 2 , conduit à la condition de réciprocité pour les champs magnétiques [START_REF] Zangwill | Modern electrodynamics[END_REF][START_REF] Prat-Camps | Circumventing magnetic reciprocity: a diode for magnetic fields[END_REF]: Pour analyser les performances des dispositifs hybrides, l'analyse modale suivante a été réalisée.

p 1 • E 2 (r 1 ) = p 2 • E 1 (r 2 ) . ( 6
B c 2 (r 1 ) • m 1 = B c 1 (r 2 ) • m 2 . ( 6
β = k 0 ϵ 1 ϵ 2 ϵ 1 + ϵ 2 . ( 6
E(r, ω) =    e  (r, ω) +  b  e  (r, ω) + k E rd dk. H(r, ω) =    h  (r, ω) +  b  h  (r, ω) + k H rd dk. ( 6
| h j 〉= 1 2 ( e  × h j ) •   dA = N  δ ,j
, les coefficients de l'expansion peuvent être calculés comme: .9) où N  = 1 2 (e  × h j ) •   dA =〈e  |e  〉 est le facteur de normalisation du th eigenmode. Les éqs. 6.9 ont été obtenues par la pré-opération du champ électrique et magnétique total avec le th mode propre et avec l'utilisation de la propriété d'orthogonalité.

  + b  = 〈E|e  〉 〈e  |e  〉 .   -b  = 〈e  |H〉 〈e  |e  〉 . ( 6
Les coefficients de l'expansion peuvent alors être calculés: 6.10) La puissance porté par le  mode propre en avant est: .11) et pare le  mode propre en arrière: comme: (6.13) Les coefficients de l'expansion sont donnés par l'équation suivante pour modes couplés: En termes de valeur propre de chaque mode (c'est-à-dire l'indice effectif), la longueur de couplage pour une largeur spécifique est définie par: .18) pour le champ électrique dans la structure d'origine et .19) pour le champ magnétique dans la structure complémentaire. Ils représentent un champ de mode particulier excité par l'émission des dipôles, des modes radiatifs, des modes photoniques en avant et en arrière.

  = 1 4N  [〈E| e  〉 + 〈 e  |E〉]. b  = 1 4N  [〈E| e  〉 -〈 e  |E〉]. (
P  =  |  | 2 N  . ( 6
P  =  |b  | 2 N  . ( 6
E(r, ω) =  A  () e  (r, ω) +  B  () e  (r, ω) + k E rd dk. H(r, ω) =  A  () h  (r, ω) +  B  () h  (r, ω) + k H rd dk.
dA  () d = β  A  () +  j A j ()κ j . ( 6
d = β 1 A 1 () + A 2 ()κ 12 . dA 2 () d = β 2 A 2 () + A 1 ()κ 21 . ( 6
 c (W) = λ Re(n eƒ ƒ (H TM,0 (W)) -n eƒ ƒ (H TM,1 (W))) . ( 6
V eff = ε|E(r)| 2 dV max(ε|E(r)| 2 ) , ( 6
V c eff = |B c (r)| 2 dV 2 |B c (r)| 4 dV , ( 6
Le facteur de Purcell peut être défini comme: .20) où    est la somme des émissions radiative et non radiative des dipôles placés à r 0 , orientés le long des axes  = , y, z et couplés dans le mode guidé  ou, en l'absence d'indice, dans les modes radiatifs.  0 est l'émission du même dipôle dans l'air. L'efficacité quantique est définie comme:

F   (r 0 , ω) =    (r 0 , ω)  0 (ω) = ρ   (r 0 , ω) ρ 0 (ω) , ( 6
η  (r 0 , ω) =  ,r (r 0 , ω)   (r 0 , ω) = ρ ,r (r 0 , ω) ρ  (r 0 , ω) , ( 6.21)
où  ,r est le taux de décroissance des émissions radiatives. Le facteur β est l'efficacité de couplage du rayonnement dans le mode  du guide d'ondes en Si: (r 0 , ω) . On sait que le LDOS est proportionnel à la puissance rayonnée par un dipôle tel que [START_REF] Carminati | Electromagnetic density of states in complex plasmonic systems[END_REF]: 5). Si une orientation spécifique est considérée, pour la structure originale, un ED orienté vers l'axe  présente un facteur de Purcell maximal de F  = 770, efficacité quantique de η  = 0.9, facteur de Purcell guidé de F  une dispersion de taille réduite (≈ 5 %) et efficacité quantique élevée jusqu'à 80 % peuvent être produits [START_REF] Maikov | Photoluminescence properties of PbSe/PbS core-shell quantum dots[END_REF]. Parmi ces options, nous avons choisi les QD PbS. Nous avons mesuré le spectre de photoluminescence, qui est tracé à la Fig. 67a). La durée de vie de fluorescence des QD PbS en solution (toluène), mesurée par le fabricant (Mesolight ©) était τ = 1.365 μs. La décroissance exponentielle est illustrée à la Fig. 67b).

β   =  ,r  (r 0 , ω)    (r 0 , ω) = ρ ,r  (r 0 , ω) ρ   (r 0 , ω) . ( 6
ρ   (r 0 , ω) = 4 π ε(r 0 )P   (r 0 , ω) , ( 6
La caractérisation des structures plasmoniques a consisté en des mesures en champ lointain et en champ proche. Dans le cas des structures plasmoniques fabriquées sur des substrats de verre et de silicium (structures non-guidées), nous avons pu caracté-

v

  DedicationTo my parents Fernando and Maru, my sisters Claudia and Miriam, baby Fer and Mimi... So far away in the distance, but so close in our hearts.

  photon sources integrated in Si photonics platforms, for both of the research groups involved in this study: the 1) Nonlinear Interactions and Quantum Optics laboratory (LI-NOC, in Spanish) from the Department of Optics in CICESE, and the 2) Nanophotonics topic from the Light, Nanomaterials and Nanotechnology (L2n) team of the Institute Charles Delaunay in UTT. This doctoral research project is based upon the following hypothesis.

3 .

 3 To understand the emission properties of electric and magnetic dipole emitters located near plasmonic nanostructures as a function of its position, orientation, and frequency. 4. To fabricate the structures by means of electron beam lithographic methods. 5. To characterize the nanophotonic devices based on the near-field scanning optical microscopy (NSOM) imaging of the light propagation along the fabricated structures.

Figure 1 :

 1 Figure 1: Metallic cavity with inversion symmetry. a) Even mode: H(r) = H(-r). b) Odd mode: H(r) = -H(-r).

Figure 2 :

 2 Figure 2: Conservation of k . The conservation of the parallel component of the wavevector along the interface determines the directions of the allowed refracted wavevectors.

Figure 3 :

 3 Figure 3: Dispersion relation for a planar air-glass interface.

Figure 4 :

 4 Figure 4: k conservation in an air-glass interface for a TE mode. a) Light traveling from air to glass: k is conserved for any θ  . For light traveling from glass to air, k can be conserved if b) θ  < θ c , but if c) θ  > θ c , TIR takes place and evanescent fields decaying exponentially away from the interface are created.

Figure 5 :

 5 Figure 5: Total internal reflection. a) A guided mode in optical fibers, adapted from (EXFO-Tube, 2011) and (Lohmann, 2010). b) Evanescent-tail microscopy, adapted from (Khan et al., 2013).

Figure 6 :

 6 Figure 6: Polarization in dielectrics. a)Microscopic and b) macroscopic pictures.

Figure 7 :

 7 Figure 7: TEM, TE and TM modes. a)Plane waves in free-space have both the electric and magnetic field polarized in a plane transversal to k. b) TM and TE modes relative to a device, the arrows represent linearly polarized electric fields.

Figure 8 :

 8 Figure 8: General form of waveguides solutions. The modes have constant amplitudes that propagate with an accumulation of phase. This allows to analyze only the cross section, thus reducing the 3D to a 2D problem. Reproduced from (Rumpf, 2018).

Figure 9 :

 9 Figure 9: Dielectric vs metallic planar (slab) waveguides.

Figure 10 :

 10 Figure 10: Examples of Si waveguides. (left) Slab waveguide. (right) Ridge waveguide.

Figure 11 :

 11 Figure 11: Scalar Babinet's principle applied to the diffraction of complementary single slits.

Figure 12 :

 12 Figure 12: Vectorial Babinet's principle applied to a triangular metallic screen (left) and its complement (right).

H 1

 1 (r) are the fields created by this source in the presence of a scatterer described by its constitutive tensors ↔ ϵ (r, ω) and ↔ μ (r, ω) [Fig. 13a)]. In a second situation, V 2 is a source volume with a current density J 2 (r) radiating at the same frequency ω. E 2 (r) and H 2 (r) are the fields created by this source in the presence of the same scatterer [Fig. 13b)].

Figure 13 :

 13 Figure 13: Geometry considered in the derivation of Lorentz's reciprocity theorem with sources. Adapted from (Nieto-Vesperinas, 2006)

Figure 14 :

 14 Figure 14: Surface waves generation. As θ  increases, the surface waves vary rapidly, transporting energy along the direction parallel to the interface, meanwhile evanescent fields decay away from the interface more rapidly along the propagation direction.

Figure 15 :

 15 Figure 15: Principle of near field microscopy. If a high-index material (i.e. a tip of a microscope) approaches closer and closer (from 1 to 6) to the evanescent field, up to distances smaller than λ (in the near-field), the total reflected electromagnetic field can be converted into propagating waves.

Figure 16 :

 16 Figure 16: SPPs. Surface waves propagating along a metal-dielectric boundary. Reproduced from (Rumpf, 2018).

Figure 18 :

 18 Figure 18: Excitation of SPPs. a) Otto, b) Kretschmann, c) grating coupler and d) evanescent coupling configurations.

Figure 19 :

 19 Figure 19: Subwavelength vs. subdiffraction confinement. Subwavelenght confinement can be achieved in a) homogeneous media and at b) interfaces between two dielectrics, whereas the subdriffaction limit can only be achieved by c) SPPs at the interface between a positive and a negative ϵ.

Figure 20 :

 20 Figure 20: Single photons emitted by spontaneous emission in free-space. A two-level system (left) is excited by the absorption of an energy ℏω eg , which is determined by the allowed transitions in the material. After the time τ 0 , the system decays to ground state by the emission of a photon with λ eg . The photon can then be coupled into any of the infinite available modes ω k , where the sub-index k designates simultaneously the direction of the k-vector and the polarization state (right). Each k-vector possesses two linearly independent polarization states. Figure adapted from (Novotny y Hecht, 2006)

Figure 21 :

 21 Figure 21: Weak and strong interactions. a) Strong coupling regime, the emitted photon will be reabsorbed before it leaves the cavity, the spontaneous emission process is reversible. The probability of finding the TLS in the excited state oscillates from 1 to 0. In the b) weak coupling regime, the emitted photon leaves the resonator (after some reflections), without being reabsorbed and the spontaneous emission process is irreversible. The probability of finding the TLS in the excited stated decays exponentially.

3 .

 3 Calculation of P ner-ƒ ed by integrating the Poynting vector over a surface enclosing a small volume surrounding only the dipole, when it is located in close proximity to the inhomogeneous and/or open and/or lossy environment with ϵ(r, ω).

Figure 22b). 4 . 7 . 8 .

 478 Figure 22b).

Figure 22 :

 22 Figure 22: Characterization of the Purcell effect. a) The reference power P 0 is measured when the dipole is in an homogeneous medium, usually with ϵ = 1. b) The near-and far-field averaged powers are then obtained when the emitter is in the vicinity of the environment with ϵ (r, ω) to determine the figures of merit. These can be calculated for specific orientations n of the dipole and for specific modes k of the system.

  and H (t) at the output are Fourier-transformed to yield E (ω) and H (ω), from which the energy flux is obtained at each ω. In all cases absorbing layers are used to simulate open boundaries (such as the perfectly matched layer, PML).

Figure 23 :

 23 Figure 23: Schematic of 1D, 2D, and 3D Yee grids. The different components of each vector correspond to different locations on the grid cell. Reproduced from (Rumpf, 2018).

Figure 24 :

 24 Figure 24: Methodology to design the hybrid photonic-plasmonic platform. a) The structures will be vertically integrated with a 20 nm thick SiO 2 layer between them, which favors the energy transfer from one waveguide to the other. The b) silicon and the c) plasmonic waveguides are first designed independently and then d) solved together.

Figure 25 :

 25 Figure 25: Symmetric and antisymmetric modes of the Si waveguide at λ = 1550 nm.

Figure 26 :

 26 Figure 26: Spatial distribution of the fundamental TE 0 (up) and TM 0 (down) modes at λ = 1550 nm. Scale bar is 500 nm.

Figure 27 :

 27 Figure 27: Only one symmetric mode is confined at λ = 1550 nm.

Figure 28 :

 28 Figure 28: Coupled modes supported by the a) hybrid region at λ = 1550 nm. The b) dispersion curves, c) spatial profiles calculated for  = 60 nm, and the d) group velocities as a function of the nanotaper width.

Figure 29 :

 29 Figure 29: a) Energy transfer process along the hybrid region. b) Coupling length as a function of . c) Profile of the power transference from the Si waveguide to the Au nanotaper along the propagation direction, for L = 800 nm. d)Electric field profile along the propagation direction in the Si waveguide (blue) and in the Au nanotaper (orange) for a length L=800nm.

Figure 30 :

 30 Figure 30: Principle of operation of the photonic-plasmonic hybrid device.

  E 0 (r) and B 0 (r) are the fields incident to the original screen and E c 0 (r), B c 0 (r) are the complementary incident fields. Using this principle, we propose the design of a new hybrid device which is complementary to the original device presented in the previous sections, the complementary devices are shown in Fig. 31.

Figure 31 :

 31 Figure 31: Original and complementary hybrid devices.

. 6 )

 6 Equation(4.6) tells us that the enhancement of the magnetic field produced by the complementary structure is, component by component, numerically equal to minus the enhancement of the electric filed produced by the original structure, differing by one unity.

  de. The electric field enhancement |E|/ |E 0 | distribution in the frequency domain is then collected from simulation results, at λ = 1550 nm, across some spatial region within an output plane located 10 nm above the metallic nanostructure, i.e. Z 0 = 10 nm. The complementary structure is injected with the photonic forward-propagating TE 0 mode. The magnetic field enhancement |H C |/ |H C 0 | distribution in the frequency domain is then collected from simulation results, at λ = 1550 nm, across Z 0 = 10 nm.

Figure 32 :

 32 Figure 32: Numerical implementation of Babinet's principle.

1 .

 1 photonic TE 0 mode, meanwhile the purely-photonic TM 0 mode excites two modes with TM-like polarization, H C TM 0 and H C TM 1 . At half the coupling length  c , the H C TE 1 will have

Figure 33 :

 33 Figure 33: Methodology to design the complementary hybrid platform. a) The structures are vertically integrated with a 20 nm thick SiO 2 layer between them. The solutions found for the b) silicon waveguide remain unchanged. c) The complementary plasmonic waveguide is solved independently and then d) the hybrid structure is simulated.

Figure 34 :

 34 Figure 34: Behavior of the complementary hybrid platform. The a) dispersion relation and b) group velocity curves for the single plasmonic mode fields were calculated, followed by the c) dispersion and d) group velocity curves of the coupled modes along the hybrid region. e) The coupling length between the H C TE -like polarized modes as well as the f) profile of the power transference from the Si waveguide to the plasmonic one along the propagation direction for L = 800 nm. g) The coupled-mode profiles were also obtained. In summary, the behavior of the complementary device is similar to the original one, but exhibiting rotated polarization features.

Figure 35 :

 35 Figure 35: Verification of Reciprocity theorem. If an ED (MD) is placed around the apex of the original (complementary) plasmonic nanotaper, by reciprocity, the emitted light should couple to the plasmon modes, propagate in the backwards direction and finally couple into one of the fundamental modes of the Si waveguide.

Figure 36 :

 36 Figure 36: Numerical implementation of the reciprocity theorem. a) Direct scenario. The corresponding fundamental mode is injected to the Si waveguide in the forward direction and the field enhancement is collected at the plane Z 0 = 10 nm. b) In the reciprocal scenario, the input is and ED or MD placed across Z 0 and the output is collected at the forward and backward fundamental mode by field expansion monitors.

L

  Figs. 37e)-f) and vice versa.

Figure 37 :

 37 Figure 37: Real part of the electric field component distributions for the original Au nanotaper. a) E x /E 0 , b) E y /E 0 , and c) E z /E 0 , and of the magnetic field component distributions for its complementary screen d) H C x /H C 0 , e) H C y /H C 0 , and f) H C z /H C 0 , at Z 0 =10 nm. Each component, point to point, exhibits similar values and opposite sign.

Figure 38 :

 38 Figure 38: Validation of reciprocity and Babinet's principles by the set of plasmonic structures integrated on silicon photonics. (a) Electric and (c) magnetic near-field distributions for the original and complementary structure. The electric and magnetic fields are confined at the apex of the nanotaper. (b) Electric LDOS into the backward TM 0 mode for an ED and (d) magnetic LDOS into the backward TE 0 mode for a MD scanned across the y-plane, 10 nm above the metallic surface. λ = 1550 nm in all cases.

Figure 39 :

 39 Figure 39: Spatial profiles at the apex of the nanotaper (y=0). a) Results of reciprocity. b) Results of Babinet's principle. The spatial distributions are equivalent point to point.

F

  . 40a)-d) that the maximum values (brighter regions in the maps) are exhibited at the same emission wavelength, λ = 1.6 μm in all cases. However, the maximum values for the total Purcell factors [Figs. 40a)-b)] correspond to a longer nanotaper (L ≈ 1.3 μm), compared to the guided Purcell factors [Figs. 40c)-d)], with their maximum at (L ≈ 800 nm). The reason why the guided Purcell factor is found at this length is due to reciprocity, as follows.

Figure 40 :

 40 Figure 40: Electric and magnetic LDOS enhancement by Purcell effect of randomly-polarized dipoles, placed at 5 nm in front of the apex, as a function of the wavelength emission and the length of the metallic nanotaper. (a) Electric and (b) magnetic Purcell factors collected over the all 4π srad. Quantum yields of the (c) electric and (d) magnetic dipoles. Guided Purcell factors collected into the (e) TM 0 and (f) TE 0 modes of the silicon waveguide. β-factor into the (g) TM 0 and (h) TE 0 modes.

  measured at the TE 0 mode of the complementary structure should correspond to the maximum magnetic field enhancement when the device is injected by the TE 0 mode at the input of the Si waveguide. In fact, the reciprocal behavior was analyzed as a function of the length of the nanotapers. The total and guided Purcell factors compared to the field enhancement produced by each structure at λ = 1.55μm are plotted in Fig.41, for the length range between 0.3 μm < L < 1.3 μm.

Figure 41 :

 41 Figure 41: Total (black color) and guided (blue color) Purcell factors at λ = 1.55μm as a function of the nanotaper length for the a) original and b) complementary structures. In both plots, the red curves follow the left-hand side scale, and the rest follow the right-hand side one. The reciprocity property imposes that the red and blue curves in both figures should follow the same qualitative behavior, which is indeed the case.

( 3 .

 3 272:90) is obtained regarding the electric versus the magnetic total Purcell factors, and a bit bigger for the guided Purcell factors (77:21). However, in both cases, the maximum quantum yields are the same. Moreover, the maximum β-factor for the For the complementary device, the MD oriented along the z-axis exhibits the maximum values: F z = 201, η z = 0.87, F z

Figure 43 :

 43 Figure 43: Electric and magnetic partial LDOS enhancement by Purcell effect of polarized dipoles, placed at 5 nm in front of the apex, for L=800 nm.

Figure 44 :

 44 Figure 44: Fabrication process. After a) PMMA deposition, the b) mask of the desired nanostructures is designed. c) The electron beam will then irradiate the selected regions. d) Development process to eliminate the exposed PMMA. e) Evaporation deposition of the metallic material. f)Lift off process to remove the non exposed PMMA.

4. 4 . 1 .FigFigure 45 :

 4145 Fig. 45).

Figure 46 :

 46 Figure 46: SEM images of the original (left) and complementary (right) plasmonic waveguides fabricated over a glass substrate. Scale bars are 1 μm (top), 500 nm (center) and 100 nm (bottom).

Figure 47 :

 47 Figure 47: Original and complementary plasmonic nanotapers fabricated onto an integrated "Y"beam-splitter made up of Si waveguides by electron beam lithography.

Figure 49 :

 49 Figure 49: a) Principle of operation of the TCSPC technique to measure fluorescence lifetimes. b) Schematic representation of the setup employed. c) Photography of an implemented experimental setup.

  and the fluorescence lifetimes for each structure are summarized in table 4. As observed, The fluorescence lifetime of the PbS quantum dots decays faster when they are in close proximity to the plasmonic structures relative to solution. From the experimental values of the lifetimes, it is possible to calculate the corresponding experimental radiative Purcell factors as F rd = τ 0 τ , where τ 0 is the fluorescence lifetime in solution and τ is the fluorescence lifetime in the given environment. The results are also shown in table 4.

Figure 50 :

 50 Figure 50: Fluorescence lifetime decay of PbS quantum dots in different environments.

Figure 51 :

 51 Figure 51: Principle of operation of aperture NSOM. Taken from reference (WITec-Resources, 2018)

Figure 52 :

 52 Figure 52: Characterization of the plasmonic structures over glass substrate. SEM images of the a) original and b) complementary structures. AFM topography images of the c) original and d) complementary structures and NSOM images of the e) original and f) complementary plasmonic nanostructures.

Figure 53 :

 53 Figure 53: Principle of operation of the NSOM technique. a) The evanescent field confined to the guided structure b) can be converted into propagating waves radiating to the far-field by approaching a nanotip with high refractive index to the surface of the sample, resulting in a perturbation of the transmitted and scattered optical field. c) The transmitted field is coupled into an optical fiber and then converted into an electrical signal by an APD. This signal is demodulated with a lock-in amplifier using the frequency of oscillation of the nanotip as reference.

  54c)-d)] an their optical NSOM signals [Figs. 54e)-f)] are recorded at the output as the Si tip (tip with nominal radius of 20 nm) raster scans the sample.

  %) TE and (50 %) TM modes (Figs. 55(c)-55(d)), from which no enhancement is observed.A comparison of the profiles at y = 0 (the tip of the nanotaper) between the simu-

Figure 54 :

 54 Figure 54: Experimental optical near-field results. SEM image of the (a) original and (b) complementary structure. Scale bar is 200 nm. Topography image of the (c) original and (d) complementary structures. NSOM images of the electric field density at λ = 1550 nm on the (e) original and (f) complementary integrated structure. Scale bar is 1 μm.

Fig. 56 .

 56 Fig. 56. The experimental EF is about EF * = 2.4. This value is about half the simu-

Figure 55 :

 55 Figure 55: Simulated FDTD results. Electric field distributions 20 nm above the surface of the metal for the original and complementary integrated plasmonic structure on Si waveguide. (a) Original and (b) complementary structure illuminated with a TM mode. (c) Original and (d) complementary structure illuminated with 50/50 TE/TM modes. (e) Original and (f) complementary structure illuminated with a TE mode. Scale bar is 500 nm.

Figure 56 :

 56 Figure 56: Simulated vs. Experimental EF. Profile distributions at the position of the apex (y = 0) of the simulated electric field enhancement at a) 20 nm and b) 30 nm above the Au nanotaper. And c) the experimental NSOM raw signal (ϵ|E| 2 ).

=

  respective processes were characterized by the calculation of the total and guidedmode Purcell factors, along with their corresponding quantum yield and β-factor efficiencies. This characterization was performed by considering multiple positions and orientations of the dipoles, over a nanotaper length ranging between 0.3 < L < 1.3 μm and over a source spectrum of 1.4 < λ < 1.6 μm. For the ED, a maximum total Purcell factor of F x = 769 was found for a dipole oriented along the -direction with quantum yield η x ≈ 0.9. The radiative spontaneous emission was coupled into the TM 0 mode at the output of the Si waveguide with a coupling efficiency of β x ≈ 30 %, resulting in a guided-mode Purcell factor of F x TM 0 220. The MD, exhibited a maximum total Purcell factor of F z = 201 for a dipole oriented along the z-direction with quantum yield η z ≈ 0.9. The radiative spontaneous emission was coupled into the TE 0 mode at the output of the Si waveguide with a coupling efficiency of β z ≈ 30 %, resulting in a guided-mode Purcell factor of F z TE 0 =51. 124 Notice that the maximum values obtained in each case, correspond to dipole sources with rotated polarizations. In addition, both devices exhibit approximatively the same values of quantum yield and coupling efficiencies. The difference in the absolute values for the Purcell factors can thus be related to the fact that the effective mode volume undergoes a stronger confinement in the original structure compared to the complementary one. These results not only corroborate the enhancement of ED and MD emission by the nanophotonic devices proposed, but they also reflect a perfect reciprocal behavior as well as perfect complementarity provided by the Babinet's principle.

2. highly efficient, 3 .

 3 with broadband operation in the telecommunications window, 4. integrated on Si photonics for low-loss transport over long distances.

6. 1 . 1 .

 11 Objectif du projet de recherche doctoraleComprendre les mécanismes physiques qui permettent des facteurs de Purcell élevés ainsi qu'un couplage efficace entre les émetteurs quantiques et les modes plasmoniques supportés par des structures plasmoniques non résonantes, intégrées sur des guides d'onde photoniques en silicium, vers la réalisation des sources à photon unique contrôlables, hautement efficaces et déterministes, fonctionnant dans le spectre optique proche infrarouge, convenant donc à leur utilisation dans les technologies de l'information quantique.

6. 2 . 3 .

 23 Principe de Babinet vectoriel pour les champs électromagnétiques Étant donné un écran mince planaire parfaitement conducteur infinie S et son écran complémentaire S c éclairé par une onde complémentaire présentant des caractéristiques de polarisation opposées E c 0 (r) = cB 0 (r) et B c 0 (r) = -E 0 (r)/ c, le principe vectoriel de Babinet indique que les champs totaux derrière l'écran satisfont: E(r) -cB c (r) = E 0 (r) et B(r) -E c (r)/ c = B 0 (r), où E 0 (r) et B 0 (r) sont les champs d'incident sur l'écran d'origine S

. 6 ) 6 . 2 . 5 .

 6625 Le théorème de réciprocité sous cette forme stipule que la composante des champs électriques ou magnétiques dans la direction d'orientation de la source ne change pas lorsque les positions de la source et des champs sont interchangées. Polaritons de plasmons de surface Les polaritons de plasmons de surface sont des états couplés résultant lorsqu'un champ électromagnétique qui se propage dans un milieu diélectrique se couple aux ondes de densité de charge d'électrons présentes dans une surface métallique. Dans ce type particulier d'onde de surface, le champ diminue de manière exponentielle (champs évanescent) dans les deux matériaux constituant l'interface. Afin d'avoir une décroissance exponentielle (champ évanescent) des deux côtés de l'interface, ϵ 1 et ϵ 2 doivent avoir des signes opposés. La relation de dispersion d'un SPP en termes de constante de propagation β est:

. 7 )

 7 Fig. 57. Sur les surfaces métalliques planes, les SPP sont non radiatives. Ils se propagent le long de la surface et leur longueur d'onde à une fréquence donnée ω est inférieure à celle de l'espace libre (sous la ligne de lumière). À la fréquence du plasma

Figure 57 : 6 . 3 .Figure 58 :

 576358 Figure 57: Relation de dispersion SPP.

. 12 ) 6 . 3 . 1 . 2 .

 126312 Dans le cas d'un dipôle orienté le long des axes  = , y, z, la puissance rayonnée est appelée P   (r 0 , ω), où r 0 est la position du dipôle et  donne l'orientation du dipôle. Pour analyser les modes propres des structures originales et complémentaires, nous commençons par tracer le TM 0 et TE 0 modes guidés purement photoniques dans le guide d'ondes en Si pour une largeur constante de 500 nm (Fig. 59(d)). Ensuite, pour trouver les modes propres des structures plasmoniques (c'est-à-dire originales et complémentaires), nous avons simulé des guides d'ondes rectangulaires purement plasmoniques pour 0 < W < 500 nm, comme indiqué dans les Figs. 59(c) et 59(e). A ces dimensions, la structure plasmonique d'origine ne supporte que le mode symétrique fondamental (S 0 ) à λ = 1550 nm. Tandis que la structure complémentaire supporte les modes fondamentaux symétrique (S C 0 ) et asymétrique (A C 0 ). Cela se voit dans les relations de dispersion montrées en Figs. 59(a) et 59(b), où la partie réelle de l'indice effectif des modes respectifs est supérieure à 1,46, l'indice de réfraction du substrat. Pour la structure plasmonique originale, l'indice effectif du mode S 0 augmente asymptotiquement à l'infini lorsque W diminue en dessous de 80 nm, indiquant un confinement élevé du mode. Réciproquement, pour le cas complémentaire, le mode A C 0 est celui qui présente cet confinement, alors que l'indice effectif de S C 0 reste constant. En plus, les deux modes S 0 et A C 0 modes présentent un confinement temporel. Ceci est montré dans les Figs. 59(g) et 59(h), où la vitesse de groupe de chaque mode est tracée. Comme observé, la vitesse de groupe décroît asymptotiquement à 0 pour W < 80 nm. Puissance transportée par les modes couplés Les champs électriques et magnétiques peuvent être exprimés dans les modes de la région hybride (c'est-à-dire la structure plasmonique sur le guide d'ondes photoniques en Si). Cependant, dans la région de couplage, les coefficients de l'expansion ne sont plus des constantes scalaires et devraient prendre en compte la propagation et le couplage des modes avec la dépendance explicite à l'axe de propagation (axe x)

Figure 59 :

 59 Figure 59: Analyse des modes des champs monomodes. Relation de dispersion de la structure plasmonique (a) originale et (b) complémentaire en fonction de la largeur des guides d'ondes plasmoniques. La largeur du guide d'onde photonique est constante W = 500 nm. (c) Structure plasmoniques originale, (d) structures purement photonique et (e) structure plasmonique complémentaire. (f) Distribution transversale de l'intensité de chaque mode propre. Vitesse de groupe des modes en fonction de la largeur du nanotaper pour les structures (g) originale et (h) complémentaire. V g tend vers 0, lorsque W tend vers 0.

. 15 )Figure 60 :

 1560 Figure 60: Analyse des modes pour les champs couplés. Relation de dispersion des structures photoniques plasmoniques hybrides (a) originale et (b) complémentaire en fonction de la largeur des guides d'ondes plasmoniques. La largeur du guide d'ondes photonique est constante W = 500 nm. (c) Structures originale et (d) complémentaire simulées. e) Distribution transversale de l'intensité de chaque mode couplé. Vitesse de groupe des modes en fonction de la largeur du nanotaper pour les structures (f) originale et (g) complémentaire. Distribution de puissance calculée pour les structures (h) originale et (i) complémentaire.

  Dans les deux cas, les structures originale et complémentaire, la relation de dispersion (Figs. 60(a) et 60(b)) et les courbes de vitesse de groupe (Figs. 60(f) et 60(g)) en fonction de W, démontrent le confinement spatial et temporel des modes H TM,0 et H rmC TE,0 , respectivement. Leur indice effectif augmente asymptotiquement à l'infini, tandis que leur vitesse de groupe diminue asymptotiquement à 0 pour W < 50 nm.

6. 4 .Figure 61 :

 461 Figure 61: Théorème de réciprocité et principe de Babinet dans une structure plasmonique intégrée sur la photonique sur silicium. (a) Distributions électriques et (c) magnétiques de champ proche pour la structure originale et complémentaire. Les champs électriques et magnétiques sont confinés au sommet du nanotaper. (b) LDOS électrique dans le mode TM 0 en arrière pour un ED et (d) LDOS magnétique dans le mode TE 0 en arrière pour un MD balayé sur le plan y, à 10 nm au-dessus de la surface métallique. λ = 1550 nm dans tous les cas.

  0 , ω) est la puissance rayonnée par le dipôle. L'équation(6.23) nous permet d'obtenir les valeurs de mérite et d'évaluer la modification de la LDOS à l'aide de calculs numériques. Dans tous les cas, pour la vérification de la réciprocité et des principes de Babinet et pour le calcul de la puissance rayonnée par des dipôles électriques et magnétiques, nous avons utilisé la méthode FDTD (3D-Difference Time-Domain) entièrement vectorielle et implémentée par Lumerical FDTD Solutions. Nous avons utilisé des moniteurs d'expansion pour calculer les émissions couplées à un mode particulier de  pour les scénarios réciproques. Pour évaluer ces modifications de la LDOS, une étude paramétrique a été réalisée en faisant varier l'orientation ( = , y, z) et la position des dipôles, ainsi que la longueur du nanotaper plasmonique de 300 nm à 1300 nm, sur une longueur d'onde allant de 1400 nm à 1600 nm. C'est la gamme de longueurs d'ondes d'intérêt puisque nous envisageons l'utilisation des structures sur les bandes S et C de la longueur d'onde des télécommunications (bande S: 1460 nm -1530 nm, bande C: 1530 nm -1565 nm), donc capable de transmettre dans le proche infrarouge, avec le pic centré à λ = 1550 nm. Dans la Fig. 63, les cartes des LDOS électrique et magnétique (Figs. 63(a)-63(b)), les efficacités quantiques (Figs. 63(c)-63(d)), les LDOS guidées (Figs. 63(e)-63(f)), et les facteurs β (Figs. 63(g)-63(h)), sont tracés en fonction de la longueur d'onde et de la longueur du nanotaper. Comme observé, le comportement est approximativement constant sur les fréquences d'émission comme prévu par le comportement non résonnant des dispositifs. D'autre part, de fortes variations sont présentes en fonction de la longueur du nanotaper principalement en raison du processus de transfert d'énergie vertical entre les modes couplés au le long de la région de recouvrement. Les résultats de la Fig. 63 correspondent à une position spécifique du dipôle dans la structure originale (complémentaire): l'ED (MD) est situé à  0 = +5 nm ( c 0 = -5 nm), y 0 = 0 nm et z 0 = 0 nm de l'apex du nanotaper (montrés à la Fig. 63), pour un dipôle avec orientation aléatoire, c'est-à-dire une moyenne sur les trois orientations des dipôles (Table

Figure 63 :Table 5 :F 6 .Figure 64 : 6 . 6 . 2 .

 635664662 Figure 63: Exaltation électrique et magnétique de LDOS par effet Purcell des dipôles avec polarisation aléatoire, placés à 5 nm devant de l'apex, en fonction de l'émission en longueur d'onde et de la longueur du nanotaper métallique. (a) Facteurs de Purcell électriques et (b) magnétiques collectés sur les 4 π srad. Efficacités quantiques des dipôles (c) électriques et (d) des dipôles magnétiques. Facteurs de Purcell guidés collectés dans les modes (e) TM 0 et (f) TE 0 de la guide d'onde en silicium. Facteur β dans les modes (g) TM 0 et (h) TE 0 .

Figure 65 :Figure 66 :

 6566 Figure 65: Images au MEB des guides d'ondes plasmoniques originaux (à gauche) et complémentaires (à droite) fabriqués sur un substrat de verre. Les barres d'échelle sont 1 μm (en haut), 500 nm (centre) et 100 nm (en bas).

Figure 67 : 6 . 6 . 3 . 3 .

 676633 Figure 67: a) Spectre d'émission de photoluminescence. b) Durée de vie de la fluorescence de PbS CQDs.

Figure 68 :Table 6 : 4 . 6 . 6 . 4 . 1 .

 68646641 Figure 68: Durée de vie de fluorescence de points quantiques de PbS dans des différents environnements.

Figure 69 :Figure 70 :

 6970 Figure 69: 4 288/5000 Caractérisation des structures plasmoniques sur un substrat de verre. Images au MEB des structures a) originales et b) complémentaires. AFM images de topographie des structures c) originales et d) complémentaires et images NSOM des nanostructures plasmoniques e) originales et et f) complémentaires.

Figure 71 :

 71 Figure 71: Résultats expérimentaux en champ proche. Image MEB de la structure (a) originale et (b) complémentaire. La barre d'échelle est 200 nm. Image topographique des structures (c) originales et (d) complémentaires. Images NSOM de la densité de champ électrique à λ = 1550 nm sur la structure intégrée (e) originale et (f) complémentaire La barre d'échelle est 1 μ m.

  a time-domain method, FDTD is more intuitive than other techniques, it works by creating a "movie" of the fields flowing through a device. FDTD divides space and time into a staggered grid (usually uniform) of discrete points and approximates the derivatives (∇× and ∂/ ∂t) of the Maxwell equations by finite differences. The propaga-

tion in time uses a scheme where the E fields at time t are computed from the E fields at time t -Δt. Similarly, the H fields at time t + Δt/ 2, are computed from the H fields at t -Δt/ 2. In this way, the E and H field patterns are matched through time, offset by a time step Δt/ 2. Such methods employ a "Yee" grid in which the different components of each vector are associated with different locations on the grid cell.

Table 1 : Advantages and disadvantages of using FDTD methods.

 1

	Advantages	Disadvantages
	1. Excellent for large scale simulations. It	1. Structured grid does not efficiently re-
	is said that numerical complexity scales	present curved surfaces.
	linearly with problem size. Typically, met-
	hods scale exponentially. FDTD methods
	are easily parallelized.

Table 2 : Maximum values of the total and guided Purcell factors for randomly-polarized dipo- les. ED (MD) placed 5 nm in front(back) of the apex.

 2

Table 3 : Maximum values of the quantum yield and β-factor for randomly-polarized dipoles. ED (MD) placed 5 nm in front(back) of the apex.

 3

		η ,y,z β-factor	,y,z k
	ED	0.6	0.4 (k=TM 0 )
	MD	0.6	0.2 (k=TE 0 )

Table 4 : Fluorescence lifetimes and radiative Purcell factors of PbS quantum dots in different environments.

 4

		τ (μs)	F rd
	Solution	0.98	1
	Original plasmonic structure	0.12	8.32
	Complementary plasmonic structure	0.06	15.75

  .3) Ici, la sommation inclut tous les modes propres e k du système avec les fréquences propres ω k . Les modes propres e k sont les solutions à l'équation d'onde homogène, Un guide d'onde peux être représenté par un système constitué de trois matériaux avec comme indice de réfraction respectifs n 1 , n 2 , et n 3 , où n 2 > n 1 , n 3 . Le matériel d'indice n 2 est le coeur du guide d'onde, tandis que les matériaux d'indices n 1 et n 3 constituent la gaine du guide. Quand une onde plane entre dans le coeur avec un angle inférieur à l'angle de réflexion totale interne, la lumière est réfléchie par le gaine et se propage le long du coeur du guide. Pour chaque réflexion, l'onde a un changement de phase de π rad. Lorsque l'onde est reflèchie deux fois, le changement global de phase est 2π et l'onde est reproduit identiquement. Les champs avec ces propriétés sont appelés modes guidés.

	En fonction de l'orientation des ondes polarisées linéairement relatifs à des dispo-
	sitifs spécifiques, les modes guidés sont classés en:
	TE: Mode électrique transversal / perpendiculaire / s. Le champ électrique est
	polarisé perpendiculairement au plan d'incidence.
	normalisée par la condition d 3 rϵ(r)e  (r) • e j (r) = δ j . TM: Mode magnétique transverse / parallèle / p. Le champ électrique est polarisé
	6.2.2. Guides d'ondes photoniques: confinement limité par diffraction parallèlement au plan d'incidence.
	Certaines généralités concernant tout type de guide d'onde sont:
	Les guides d'ondes supportent un nombre infini de modes discrets.
	Les modes ont un profil d'amplitude constant qui n'accumule que la phase pen-
	dant la propagation.
	Les modes ont des fréquences de coupure en dessous desquelles ils ne peuvent
	plus se propager.
	Un mode guidé existe si:
	Dans le milieu à indice élevé, la composante du vecteur d'onde perpendiculaire à
	l'interface est k 2 ⊥ > 0.
	Dans le milieu d'indice bas, k 2 ⊥ < 0 (champ évanescent).

6.3.1. Analyse des modes des nanostructures photoniques-plasmoniques 6.3.1.1. Puissance transportée par un champ monomode

	Dans les régions du dispositif nanophotonique où le guide d'onde est invariant en
	la direction de propagation (c'est-à-dire la guide d'onde Si sans structure plasmoni-
	que), les champs électriques et magnétiques peuvent être exprimés dans les modes
	normaux comme:

  .14) avec le coefficient de couplage κ j = ω 2 〈 e j |Δε| e  〉-ω 2 〈e ,j |Δε|e , 〉. Ici e  est la composante longitudinale en x des champs et β  est la constante de propagation du mode . Pour deux modes couplés considérés dans la région de couplage, les équations couplées

	sont les suivantes:	dA 1 ()

  vertical (rebondissant) dans la région de couplage. Et un processus similaire se produit pour les modes H TE,0 et H TE,1 . Cependant, à la moitié de la longueur de couplage  c , seule l'interaction entre les modes de type H TM provoque un confinement maximal de l'énergie dans le nanotaper plasmonique, où H TM,1 a efficacement transféré l'ensemble son énergie dans le mode H TM,0 . La structure complémentaire hybride présente un comportement similaire, mais avec des caractéristiques de polarisation en rotation. En effet, à W = 300 nm, deux modes hybrides à polarisation de type TE, H C TE,0 et H C TE,1 sont excités par le mode purement photonique TE 0 , tandis que le mode purement photonique TM 0 excite deux modes avec une polarisation de type TM, H TM,0 C et H C TM,1 . À la moitié de la longueur de l'accouplement  c , le mode H C TE,1 a efficacement transféré toute son énergie vers le mode H C TE,0 , qui se traduit par un confinement maximal de l'énergie dans le nanotaper complémentaire plasmonique.

  il excite efficacement les modes hybrides photonique-plasmonique H TM 0 et H TM 1 ou H TE 0 et H TE 1 . En la zone hybride, un processus de couplage des modes se produit qui permet la transition entre les guides d'ondes en Si et plasmoniques.

	Comme W diminue progressivement, il se produit un confinement spatial et temporel
	des champs électriques ou magnétiques à son sommet. Le volume effectif en mode
	plasmonique est donné par:
	.17)
	Figs. 60(h)-60(i) montrent la distribution des profils de puissance calculés en la
	direction de propagation pour les structures originale et complémentaire. On peut-
	it observer que la puissance est transférée de la guide d'onde en Si aux nanotaper
	plasmoniques, atteignant une valeur maximale au sommet du nanotaper, pour une
	longueur L = 800 nm.

Le principe de fonctionnement peut être décrit comme suit: le guide d'ondes en Si est éclairé avec le mode TM 0 (l'originale) ou TE 0 (le complémentaire) selon le principe de Babinet. Le mode photonique se propage vers la base du guide d'onde nanotaper plasmonique où

Acknowledgements

This work was conducted under the Agreement for double doctoral degree program between Centro de Investigación Científica y de Educación Superior de Ensenada (CI-CESE) and Université de Technologie de Troyes (UTT). In particular, the co-directors of this thesis are with the Nonlinear

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The dispersion relation illustrating the SPP's characteristics analyzed in this section is shown in Fig. 17. In summary, on planar metal surfaces, SPPs are nonradiative. They propagate along the surface and their wavelength at a given frequency ω is shorter than the one in free-space (below the light line). At the surface plasma frequency, and in the ideal case of no damping, the wavelength of the SPP goes to zero (β goes to ∞) and the associated field becomes localized to the very surface of the metal, i.e. the decay length of the evanescent wave goes to zero.

Excitation of SPPs

Based on the concepts developed in the previous sections, in general, the excitation (creation) of a SPP follows from the evanescent field generated by total internal reflection. In particular, Fig. 18 shows four different configurations that take advantage of this concept to create SPPs:

1. Otto configuration. A surface wave is created at the interface between two die- In this chapter, the methods and materials used for the design, fabrication and characterization of the hybrid nanophotonic-plasmonic platforms are detailed. Because of the reasons explained in previous chapters, our proposal is to control the spontaneous emission rate of ED and MD quantum emitters by combining a Si photonic waveguide and a plasmonic antenna, able to support SPPs. This is a complex coupled system and although its behavior follows the same concepts presented before, the solutions of the composed system are not obtainable in an analytical way. Instead, they are calculated by means of powerful computational techniques available for this task. In particular, based on the FDTD method described in the previous chapter. The devices are then fabricated by electron beam lithography and, finally, characterized by far-field as well as by near-field measurements.

Design and modal analysis of complementary plasmonic waveguides on silicon photonics

The idea is to exploit the features of plasmonic waveguides that support SPPs to strongly confine the electromagnetic field; together with the low-loss transport and integration capabilities of a Si waveguide. To this end, a plasmonic gold (Au) nanoantenna will be vertically integrated over a Si waveguide, since the vertical coupling configuration has proven to efficiently promote an energy transfer process between both structures [START_REF] Luo | Compact on-chip plasmonic light concentration based on a hybrid photonic-plasmonic structure[END_REF].

The methodology to design the devices is sketched in Fig. 24. To begin with, common configurations for each independent waveguide are employed as the basis of the hybrid device [Fig. 24a)]. For the Si waveguide [Fig. 24b)], we want a configuration that supports only the fundamental TE 0 and TM 0 modes. Since the commercial available Si wafers have a standard thickness of 220 nm, this will be a fixed parameter. The width will be then calculated such that only the modes of interest are supported at λ = 1550 nm. In Fig. 42 the behaviour of the quantum yield and the β-factor is also analyzed as a function of the length of the nanotaper. The left-hand side scale corresponds to the electric (magnetic) field enhancement (red curve) and the right-hand scale measures the quantum yield (curve in pink), and the β-factor (green curve), of the original (complementary) device. These figures of merit represent the quantum efficiency and the coupling-mode efficiency of each device, respectively. Recall also that, by definition, these two figures of merit are inversely proportional (i.e. η ∝  rd meanwhile βƒ ctor ∝ 1/  rd ). Therefore, a trade-off between the two quantities is required for an optimum performance of the devices. We desire a majority of the emission of the ED (MD) to be radiative (high quantum yield), but we also need that most of the radiative emission get coupled into the specific guided mode of interest.

The previous results correspond to randomly polarized dipoles (averaged over , y, and z orientations) whose emission was analyzed over a nanotaper length 0.

Electron beam lithography

The substrate is spin-coated with a thin layer of a resist, which chemically reacts under the exposure to the electron beam.

In our case, the Poly(methyl methacrylate) or PMMA was the electron-sensitive resist of choice.

The PMMA (with concentration of 30g/L) was spin-coated with speed of 4000 rpm and acceleration of 4000 rpm 2 over 30 s, in order to provide a PMMA thickness around 200 nm.

After the PMMA deposition, the sample must be annealed in an oven at 160°C over 3 hours.

Before the electron beam exposure, an ESPACER deposition is usually carried out. The ESPACER is a resist used to avoid the charges diffusion over the surface of the PMMA and hence, to obtain a better penetration of the electrons. The ESPACER must be spincoated few minutes before the exposure, with a speed of 3000 rpm over 30 s.

The electron beam exposure was then performed by the use of the Raith eLiNE electron beam microscope, in which a maximum resolution of 20 nm can be achieved at the surface of the PMMA. A mask should be designed in order to expose only the desired region where the Au film will be deposited.

Under exposure to an electron beam, the PMMA undergoes a chemical reaction so that the exposed areas can be easily dissolved by a specific lithography solvent. This process is called development (in analogy with development of photographic films).

The ESPACER is first removed by a water bath over ≈15 s.

Next, a MIBK:IPA bath of 1 min is provided to remove the exposed PMMA.

Finally, an isopropanol bath of ≈ 15 s is given to clean the sample.

emission in the near infrared (NIR) spectrum, and MD transitions whose strength is comparable to the competing ED ones. Indeed, the calibration of some NIR detectors and several setup alignments used in this thesis were performed with the aid of an erbium (Er 3+ ) doped glass substrate, whose emission around λ = 1.55 μm is very strong. However, for single-photon emission, we decided to use commercial available (of relative low-cost) suspended colloidal QDs (CQDs).

In particular, the electronic and optical properties of lead-salt (PbS, PbSe) CQDs structures as a function of temperature are well understood. In fact, recently, high quality air-stable PbS and PbSe, with low size dispersion (≈ 5 %) and high quantum efficiency (QE) up to 80 % can be produced [START_REF] Maikov | Photoluminescence properties of PbSe/PbS core-shell quantum dots[END_REF]. Among these options, PbS QDs were chosen. We measured the photoluminiscence spectrum, which is plotted in Fig. 48a) In the following section, the characterization of the plasmonic structures in nonguided media (over glass and Si substrates) and integrated on the Si waveguides are presented.

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