# Antennes Gap Plasmon: Une Source de Plasmon Polariton de Surface

Cam Nhung Vu

► To cite this version:

Cam Nhung Vu. Antennes Gap Plasmon: Une Source de Plasmon Polariton de Surface. Optique / photonique. Université Claude Bernard Lyon 1, 2022. Français. NNT: 2022LYO1089. tel-03891996

HAL Id: tel-03891996

https://theses.hal.science/tel-03891996

Submitted on 9 Dec 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

N° d'ordre NNT : 2022LYO1089

## THÈSE de DOCTORAT DE L'UNIVERSITÉ DE LYON

Opérée au sein de :

l'Université Claude Bernard Lyon 1

### Ecole Doctorale ED52

Physique et Astrophysique

**Discipline : Physique**

Soutenue publiquement le 04/11/2022, par :

Nhung-Cam VU

# Opening-gap Plasmonic Antennas : A Surface Plasmon Polariton Source

Devant le jury composé de :

DEL FATTI Natalia, Pr., Université Claude Bernard Lyon 1

BOUHELIER Alexandre, Dr., Directeur de Recherche CNRS

BRAMATI Alberto, Pr., Sorbonne Université

DELEPORTE Emmanuelle, Pr., Ecole Normale Supérieure Paris-Saclay

LAVERDANT Julien, MCF, Université Claude Bernard Lyon 1

Présidente

Rapporteur

Rapporteur

Examinatrice

Directeur de thèse
UNIVERSITE CLAUDE BERNARD - LYON 1

Président de l'Université

Président du Conseil Académique

Vice-président du Conseil d'Administration

Vice-président du Conseil Formation et Vie Universitaire

Vice-président de la Commission Recherche

Directeur Général des Services

M. le Professeur Frédéric FLEURY

M. le Professeur Hamda BEN HADID

M. le Professeur Didier REVEL

M. le Professeur Philippe CHEVALIER

M. Fabrice VALLÉE

M. Alain HELLEU

COMPOSANTES SANTE

Faculté de Médecine Lyon Est – Claude Bernard

Faculté de Médecine et de Maïeutique Lyon Sud – Charles Mérieux

Faculté d'Odontologie

Institut des Sciences Pharmaceutiques et Biologiques

Institut des Sciences et Techniques de la Réadaptation

Département de formation et Centre de Recherche en Biologie Humaine

Directeur : M. le Professeur J. ETIENNE

Directeur : Mme la Professeure C. BURILLON

Directeur : M. le Professeur D. BOURGEOIS

Directeur : Mme la Professeure C. VINCIGUERRA

Directeur : M. X. PERROT

Directeur : Mme la Professeure A-M. SCHOTT

COMPOSANTES ET DEPARTEMENTS DE SCIENCES ET TECHNOLOGIE

Faculté des Sciences et Technologies

Département Biologie

Département Chimie Biochimie

Département GEP

Département Informatique

Département Mathématiques

Département Mécanique

Département Physique

UFR Sciences et Techniques des Activités Physiques et Sportives

Observatoire des Sciences de l'Univers de Lyon

Polytech Lyon

Ecole Supérieure de Chimie Physique Electronique

Institut Universitaire de Technologie de Lyon 1

Ecole Supérieure du Professorat et de l'Education

Institut de Science Financière et d'Assurances

Directeur : M. F. DE MARCHI

Directeur : M. le Professeur F. THEVENARD

Directeur : Mme C. FELIX

Directeur : M. Hassan HAMMOURI

Directeur : M. le Professeur S. AKKOUCHE

Directeur : M. le Professeur G. TOMANOV

Directeur : M. le Professeur H. BEN HADID

Directeur : M. le Professeur J-C PLENET

Directeur : M. Y.VANPOULLE

Directeur : M. B. GUIDERDONI

Directeur : M. le Professeur E.PERRIN

Directeur : M. G. PIGNAULT

Directeur : M. le Professeur C. VITON

Directeur : M. le Professeur A. MOUGNIOTTE

Directeur : M. N. LEBOISNE
Abstract

Nanopatch antennas based on metallic nanoparticles-on-mirror configuration offer great ad-
vantages as a potential source for launching surface plasmon polaritons (SPPs). In this
work, we propose a plasmonic platform consisting of gold bipyramid nanoparticles (AuBPs)
deposited over a gold thin film. A sub-10nm TiO2 gap is introduced between the two metallic
structures to prevent conductive contact. The synthesized AuBPs have the ability to support
a strong localized surface plasmon resonance along its longitudinal axis which behaves like a
radiating dipole. When the nanoparticles are brought to interact with the gold mirror, each
AuBP resembles a nanoscale nanopatch antenna with an “opening-mouth”, which allows for
effectively launching SPPs over the metallic surface around the bipyramid. In the terms of
the experiment, we employ a leakage radiation microscope working in the dark-field config-
uration to study the scattering of the antenna and also excite and collect the SPPs. Here,
we highlight the importance of the orientation effect in the optical response of the AuBPs.
A rich mode structure ranging from the visible to the near-infrared regime is opened by
tilting AuBP with one tip touching the substrate, whereas other orientations only display a
single dipolar longitudinal mode. The plasmon modes exhibit both radiative nature and are
confined within the gap. The COMSOL calculation confirmed that the plasmon gap mode
behaves similarly to the Fabry-Pérot resonances. Finally, the bipyramid nanopatch is em-
ployed to optically excite SPPs, showing that the efficiency of launching SPPs is resonantly
enhanced at the plasmon resonance wavelength of the mode. Especially, in the comparison
to the plasmon generated by scattering from a dielectric bead nanoparticle of silica on the
same gold substrate, the bipyramid nanopatch induces a greater SPPs efficiency by around
1 order of magnitude. These characteristics of the AuBP nanopatch can pave the way for
electrically driven plasmon antennas, for which many applications such as energy harvesting
or quantum information, are beneficial.
Résumé

Les antennes nanopatch basées sur une configuration de nanoparticules métalliques sur miroir
offrent de grands avantages en tant que source potentielle pour lancer des plasmons polaritons
de surface (SPP). Dans ce travail, nous proposons une plateforme plasmonique constituée de
nanoparticules en forme de bipyramide d'or (AuBP) déposées sur un film mince d'or. Une
couche de TiO2 inférieure à 10 nm est introduite entre les deux structures métalliques pour
empêcher le contact conducteur. Les AuBPs synthétisées ont la capacité de supporter une
forte résonance plasmonique de surface localisée le long de son axe longitudinal qui se com-
porte comme un dipôle rayonnant. Lorsque les nanoparticules sont amenées à interagir avec
le miroir d'or, chaque AuBP ressemble à une antenne nanopatch à l'échelle nanométrique
avec une "ouverture", qui permet de lancer efficacement des SPP sur la surface métallique
autour de la bipyramide. Dans nos expériences, nous utilisons un microscope à fruites radia-
tives fonctionnant dans la configuration de fond noir pour étudier la diffusion de l'antenne et
également exciter et collecter les SPP. Ici, nous soulignons l'importance de l'effet d'orientation
dans la réponse optique des AuBPs. Une structure de mode riche allant du régime visible
au proche infrarouge apparaît en inclinant l'AuBP quand la pointe touche le substrat, tandis
que d'autres orientations n'affichent qu'un seul mode longitudinal dipolaire. Le calcul COM-
 SOL a confirmé que le mode gap plasmon se comporte de manière similaire aux résonances
de Fabry-Pérot. Enfin, le nanopatch bipyramide est utilisé pour exciter optiquement les
SPP, montrant que l'efficacité du lancement des SPP est améliorée à la longueur d'onde
de résonance du mode. En particulier, en comparaison avec le plasmon généré par diffu-
sion à partir d'une nanoparticule de billes diélectriques de silice sur le même substrat d'or,
le nanopatch bipyramide possède une plus grande efficacité des SPP d'environ 1 ordre de
grandeur. Ces caractéristiques du nanopatch AuBP peuvent ouvrir la voie à des antennes
à plasmon à commande électrique, pour lesquelles de nombreuses applications telles que la
récupération d'énergie ou l'information quantique, sont bénéfiques.
Acknowledgements

First and foremost, I would like to express my gratitude to my thesis director Julien Laverdant for his tremendous guidance and open discussion. Thanks for giving me the freedom to be right and to be wrong, plenty of opportunities to strengthen my abilities, and sincere advice on any topic.

I would also like to thank members of my thesis committee: Alexandre Bouhelier, Bramati Alberto, Natalia Del Fatti, and Emmanuelle Deleporte, who have accepted to participate in my jury. Thqnk you also for all your remarks and corrections after the defence, particularly to Alexandre and Bramati for taking time to review my work with many insights of the work.

I am also grateful to the members and Ph.D students of the Agrégats et Nanostruc-
tures (AgNano) team: Emmanuel Cottatin, Christophe Bonnet, Jean Lermé, Michel Pellarin,
Marie-Ange Lebeault, Michel Broyer, Mathias Hillenkamp, Sylvain Hermelin, Elise Camus,
and Floriane Perrier. I really enjoy the time to be a part of the team for the past three years.

This Thesis would have been completed without the fruits of the collaboration with Frédéric Lerouge from ENS Lyon; Michel Pellarin, Remy Fulcrand, and Emmanuel Cottatin from iLM; Laurent Coolen, Agnès Maitre, and Zakarya Ouzit from INSP (Paris); Stephanie Buil and Bruno Bérini from GEMaC (UVSQ). I deeply thank all of you for your always generous support.

Thanks to my sisters, Ngà and Hanh, who have shared with me both joy and sadness
from the early dates of our Ph.D. in France. Thanks to my friend, Hong, for her endless
positive energy and lovely distractions from writing.

Big thanks to my friends and family in Vietnam, always love me and support me uncon-
ditionally. They are my motivation to be better every passing day.

Lastly, sincerely thanks to myself of the last 5 year.
Introduction

Plasmonic antennas are analogs of radio antennas that transfer electromagnetic energy from free space to localized fields, and vice versa. The attractiveness of plasmonic nanostruc- tures is the ability to concentrate light into deep sub-wavelength volumes and offer relative broad optical resonances while plasmons can propagate over micrometer distances. Plasmonic nanoantennas are pervasive in various domains of modern technologies, ranging from sub- nanometer resolution optical imaging, quantum information processing, energy harvesting, and conversion, to biological sensing [1]. For instance, recent advances in nanofabrication al- low to couple a single emitter into metallic nanostructures, thus increasing the spontaneous emission rate of the emitter, making it becomes a reliable ultrafast photon source [2]. A promising route towards using advantageously plasmonics to construct optically thick but physically very thin photovoltaic absorbers for solar energy harvesting [1]. To bring plas- monic antennas into the practical realm, it is critical to optimize the energy transfer between a localized source and the free-space radiation. Several challenging requirements must be met, including large local density of optical states (LDOS) enhancement, good photon-radiation efficiency, and the ability to control the emission direction.

A single metallic nanoparticle with sharp corners can strongly confine light to volume
with dimensions in the range of 10-100nm [3]. Even more tightly confined fields – on the
scale of sub-10nm – can be achieved by coupling nanoparticle plasmons together in a dimer
configuration. However, a major challenge is to separate individual nanostructures precisely
and reproducibly in the subnanometer scale. An alternative promising route toward control-
ling plasmonic gaps is provided by nanopatch antennas in a nanoparticle-on-mirror (NPoM)
configuration, where a metallic nanoparticle is separated from a metallic film by a thin di-
electric layer, which behaves similarly to a dimer of nanoparticles [4]. The attractiveness
of the plasmonic nanopatch or NPoM is that large numbers of plasmonic nanogaps can be
fabricated in a controllable and reproducible manner, and thus extend their uses in various
practical applications.

Generally, the optical response of the nanopatch based on nanoparticle-on-mirror geometry is strongly modified by the exact geometry of the overlying particle, especially by the bottom facet that contacts the underlying metallic film. Nanospheres and nanocubes are the most extensively used and studied to produce the gap plasmon for the patch antenna, thanks to their symmetrical geometry. However, when it moves to less symmetrical particles, more degrees of freedom are required to be controlled. In the framework of this Thesis, we aim to study a new kind of nanopatch antenna made from gold bipyramid nanoparticles (AuBPs) placed separately above a thin gold film, focusing on investigating one of the most crucial degrees of freedom induced by the elongated shape of particle: its 3D orientation. The optical responses of a single AuBP have been well studied in the team previously [5], showing that that particle has the ability to support strongly localized surface plasmon resonances along its longitudinal axis and allow to tingly confine field around its sharp tip ends. In our work, we harvest these distinctive characteristics of the individual AuBPs and couple them to the metallic film, from which a rich optical response hybridized between confined gap plasmon and radiating localized plasmon is opened. This structure is then employed to optically excite surface plasmon polaritons propagating on the gold film.

In Chapter 1, we set out the basic concepts of surface plasmon polaritons. We begin this chapter with fundamental discussions of the interaction of light and metals, where their ability to confine light below the diffraction limit is highlighted. Subsequently, both propagating and localized surface plasmon at the planar and confined metallic structures is presented, which provides a profound understanding of the plasmon hybridization in the following chapter.

In Chapter 2, we investigate numerically the surface plasmon hybridization in alternative
multilayers of dielectric/metal by using a dipole as a probe. We start by showing how we can
use the classical approach to study the spontaneous emission of the dipole. The optical modes
supported by the multilayer metallic structures are then identified from the spatial spectrum
of the dissipated power of the dipole. In this Chapter, we mainly focus on a practical long-
range surface plasmon device and alternative N-stacks of metallic and insulator thin film.
Excitation and collection of surface plasmon can be done in the far field. In Chapter 3, we present the experimental methods that are employed to investigate the optical responses of the nanopatch. In which, a dark-field microscope equipped with Fourier-plane filtering to measure scattering spectra of the nanoparticles. Besides, the leakage radiation microscope is applied to efficiently collect the SPPs.

Owning to the elongated geometry, the random deposition of bipyramid nanoparticles on a flat substrate can induce many orientations of the particle, which as will be shown later results in significantly different optical responses. Chapter 4 is therefore dedicated to presenting the measurement of the 3D orientation of the bipyramids on the glass substrate. The orientation is measured by polarization analysis of the longitudinal localized surface plasmon resonance of the bipyramid, which behaves like a linear dipole. We demonstrate that the bipyramids display two main orientations: lying horizontal to the surface or tilting with a tip touching the surface.

In Chapter 5, we introduce the opening-gap nanopatch antenna which consists of bipyrami- mids deposited separately on the gold film by a dielectric spacer. We prove explicitly that tilting bipyramids on the surface can enhance the interaction strength of the particles and the underlying substrate, from which various optical modes are opened with interesting be- haviors: radiative and confined inside the gap like Fabry-Pérot resonances. The nature of these plasmon modes is identified with the support of the COMSOL simulation.

The SPPs can offer great potential for various photonic applications thanks to their ability to concentrate light in space in time. In Chapter 6, we therefore employ scattering from the bipyramid nanopatch antenna to optically launch SPPs. Besides fundamental properties of the SPPs such as propagation length or dispersion are well characterized, we also measure the SPP emission efficiency of the antenna and seek to comparison to a dielectric nanobead. The physical mechanisms involved in generating SPPs also will be discussed.

The results in this Thesis demonstrate the potential of the opening-gap plasmon antennas based on gold bipyramids to confine light to the nanoscale and convert it into desired output channel of SPPs. Such properties may be utilized as a flexible platform for surface energy transfers over a long distance and in a very short time, which can be applied in perspective to various practical fields such as sensing, photovoltaics, and optoelectronic devices.
Résumé substantiel en français

Les antennes plasmoniques sont analogues aux antennes radio qui transfèrent l'énergie électromagnétique de l'espace libre vers des champs localisés, et vice versa. L'attrait des nanostructures plasmoniques est la capacité de concentrer la lumière dans des volumes profonds de dimensions sub-longueur d'onde et d'offrir des résonances optiques relativement larges tandis que les plasmons peuvent se propager sur des distances micrométriques. Les nanoantennes plasmoniques sont omniprésentes dans divers domaines des technologies modernes, allant de l'imagerie optique à résolution inférieure au nanomètre, du traitement de l'information quantique, de la récupération et de la conversion d'énergie, à la détection biologique [1]. Par exemple, les progrès récents en nanofabrication permettent de coupler un seul émetteur dans des nanostructures métalliques, augmentant ainsi le taux d'émission spontanée de l'émetteur, ce qui en fait une source de photons ultra-rapide fiable [2]. Une voie prometteuse est l'utilisation avantageuse de la plasmonique pour construire des absorbeurs photovoltaïques optiquement épais mais physiquement très fins pour la récupération de l'énergie solaire [1]. Pour amener les antennes plasmoniques dans le domaine pratique, il est essentiel d'optimiser le transfert d'énergie entre une source localisée et le rayonnement en espace libre. Plusieurs exigences difficiles doivent être satisfaites, notamment une amélioration de la grande densité locale d'états optiques (LDOS), une bonne efficacité du rayonnement photonique et la capacité de contrôler la direction d'émission.

Une seule nanoparticule métallique aux angles vifs peut fortement confiner la lumière dans un volume dont les dimensions sont comprises entre 10 et 100 nm [3]. Des champs encore plus étroitement confinés - à l'échelle inférieure à 10 nm - peuvent être obtenus en couplant des plasmons de nanoparticules ensemble dans une configuration dimère. Cependant, un défi majeur consiste à séparer les nanostructures individuelles de manière précise
et reproductible dans le régime subnanométrique. Une voie alternative prometteuse vers le contrôle des lacunes plasmoniques est fournie par les antennes nanopatch dans une configuration de type nanoparticules sur miroir (NPoM), où une nanoparticule métallique est séparée d'un film métallique par une fine couche diélectrique, qui se comporte de manière similaire à un dimère de nanoparticules [4]. L'attrait du nanopatch plasmonique ou NPoM est qu'un grand nombre de nanogaps plasmoniques peuvent être fabriqués de manière contrôlable et reproductible, et ainsi étendre leurs utilisations dans diverses applications pratiques.

Généralement, la réponse optique du nanopatch basée sur la géométrie des nanoparticules sur miroir est fortement modifiée par la géométrie exacte de la base géométrique de la nanoparticule, en particulier par la facette inférieure qui entre en interaction avec le film métallique sous-jacent. Les nanosphères et les nanocubes sont les plus largement utilisés et étudiés pour produire le plasmon gap pour l'antenne patch, grâce à leur géométrie symétrique. Cependant, lorsqu'il se déplace vers des particules moins symétriques, plus de degrés de liberté doivent être contrôlés. Dans le cadre de cette thèse, nous visons à étudier un nouveau type d'antenne nanopatch fabriquée à partir de nano-bipyramides d'or (AuBP) placées séparément au-dessus d'un mince film d'or, en nous concentrant sur l'étude de l'un des degrés de liberté les plus cruciaux induits par la forme allongée de particule : son orientation 3D. Les réponses optiques d'une seule AuBP ont été bien étudiées dans l'équipe précédemment [5], montrant que cette particule a la capacité de supporter des résonances de plasmon de surface fortement localisées le long de son axe longitudinal et de permettre de confiner le champ autour de ses extrémités pointues. Dans notre travail, nous utilisons ces caractéristiques distinctives des AuBP individuels en les couplant au film métallique, à partir duquel une réponse optique riche hybride entre le plasmon à espace confiné et le plasmon localisé rayonnant est attendue. Cette structure est ensuite utilisée pour exciter optiquement des plasmons polaritons de surface se propageant sur le film d'or.

Dans le chapitre 1, nous exposons les concepts de base des polaritons des plasmons de surface. Nous commençons ce chapitre par des discussions fondamentales sur l'interaction de la lumière et des métaux, où leur capacité à confiner la lumière en dessous de la limite de diffraction est mise en évidence. Par la suite, le plasmon de surface se propageant et localisé au niveau des structures métalliques planes et confinées est présenté, ce qui nous permet de
fournir une compréhension approfondie de l'hybridation du plasmon dans le chapitre suivant.

Dans le chapitre 2, nous étudions numériquement l'hybridation des plasmons de surface dans des multicouches alternatives de diélectrique/métal en utilisant un dipôle comme sonde. Nous commençons par montrer comment nous pouvons utiliser l'approche classique pour étudier l'émission spontanée du dipôle. Les modes optiques supportés par les structures métalliques multicouches sont alors identifiés à partir du spectre spatial de la puissance dissipée du dipôle. Dans ce chapitre, nous nous concentrons principalement sur un dispositif pratique à plasmons de surface à longue portée et sur des piles N alternatives de couches minces métalliques et isolantes.

L'excitation et la collection des plasmons de surface peuvent se faire en champ lointain. Dans le chapitre 3, nous présentons les méthodes expérimentales utilisées pour étudier les réponses optiques du nanopatch. Un microscope à champ noir utilisant la manipulation du plan de Fourier a été mis en place pour mesurer les spectres de diffusion des nanoparticules. En outre, le microscope à fuite radiative nous permet pour collecter efficacement les SPP.

En raison de leur géométrie, le dépôt aléatoire des bipyramides sur un substrat plat peut induire de nombreuses orientations des particules, ce qui, comme on le verra plus loin, se traduit par des réponses optiques significativement différentes. Le chapitre 4 est donc con- sacré à la présentation de la mesure de l'orientation 3D des bipyramides à l'échelle individuelle sur substrat de verre. L'orientation est mesurée par une analyse fine de la polarisation diffusée par la résonance plasmon longitudinale de la bipyramide. Cette réponse peut être modélisée comme un dipôle linéaire. Nous démontrons que les bipyramides affichent deux orientations principales : couché horizontalement à la surface ou inclinée avec une pointe touchant la surface.

Dans le chapitre 5, nous introduisons l'antenne nanopatch à ouverture de gap formée par des bipyramides déposées séparément d'un film d'or par une fine couche diélectrique. Nous prouvons explicitement que l'inclinaison des bipyramides sur la surface peut améliorer la force d'interaction des particules avec le substrat sous-jacent, à partir de laquelle différents modes optiques sont créés avec des comportements intéressants : radiatifs et confinés à l'intérieur de l'entrefer comme les résonances de Fabry-Pérot. La nature de ces modes plasmons est identifiée à l'aide de la simulation COMSOL.
Les SPP peuvent offrir un grand potentiel pour diverses applications photoniques grâce à leur capacité à concentrer la lumière dans l'espace et dans le temps. Dans le chapitre 6, nous utilisons donc la diffusion de l'antenne nanopatch bipyramide pour lancer optiquement les SPPs. Outre les propriétés fondamentales des SPP telles que la longueur de propagation ou la dispersion sont bien caractérisées, nous mesurons également l'efficacité d'émission SPP de l'antenne et cherchons à la comparer à une nanobille diélectrique de silice. Les mécanismes physiques impliqués dans la génération des SPP seront également discutés.

Les résultats de cette thèse démontrent le potentiel des antennes à plasmon à ouverture de gap basées sur des bipyramides d'or pour confiner la lumière à l'échelle nanométrique et la convertir en canal de sortie souhaité de plasmon de surface. De telles propriétés peuvent être utilisées comme plate-forme flexible pour les transferts d'énergie de surface sur une longue distance et en très peu de temps, qui peuvent être appliquées en perspective à divers domaines pratiques tels que la spectroscopie, la détection, le photovoltaïque et les dispositifs optoélectroniques.
# Contents

<table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>1</td><td>Surface Plasmon Polaritons</td><td>15</td></tr><tr><td>1.1</td><td>Metals and Light: Material perspective</td><td>16</td></tr><tr><td>1.1.1</td><td>Dielectric function of bulk metals</td><td>16</td></tr><tr><td>1.1.2</td><td>Light confinement in metals</td><td>19</td></tr><tr><td>1.1.3</td><td>The loss in metal</td><td>22</td></tr><tr><td>1.2</td><td>Surface plasmon polaritons at a single interface</td><td>24</td></tr><tr><td>1.3</td><td>Plasmon hybridization in planar structures</td><td>30</td></tr><tr><td>1.4</td><td>Localized surface plasmon of metallic nanoparticles</td><td>33</td></tr><tr><td>1.4.1</td><td>Scattering, absorption and extinction</td><td>34</td></tr><tr><td>1.4.2</td><td>Quasi-static approximation</td><td>38</td></tr><tr><td>1.4.3</td><td>Mie theory</td><td>41</td></tr><tr><td>2</td><td>Coupling a dipole to multilayer surface plasmon devices</td><td>45</td></tr><tr><td>2.1</td><td>Spontaneous emission of an emitter</td><td>46</td></tr><tr><td>2.2</td><td>Spontaneous emission of a dipole near planar interfaces</td><td>51</td></tr><tr><td>2.3</td><td>Coupling a dipole to fundamental planar plasmonic structures</td><td>53</td></tr><tr><td>2.4</td><td>Coupling a single dipole to a long-range surface plasmon device</td><td>59</td></tr><tr><td>2.5</td><td>Surface plasmon hybridized dressed states</td><td>64</td></tr><tr><td>3</td><td>Experimental methods for Surface Plasmons</td><td>70</td></tr><tr><td>3.1</td><td>Investigation of the optical response of metallic nanoparticles</td><td>71</td></tr><tr><td>3.2</td><td>Dark-field scattering polarimetry</td><td>73</td></tr><tr><td>3.2.1</td><td>Dark-field imaging by Fourier-plane filtering</td><td>73</td></tr><tr><td>3.2.2</td><td>Single metallic nanoparticle spectroscopy</td><td>77</td></tr><tr><td>3.2.3</td><td>Scattering polarimetry</td><td>79</td></tr><tr><td>3.3</td><td>Exciting and imaging propagating surface plasmon</td><td>80</td></tr><tr><td>3.4</td><td>Launching SPPs by leakage radiation microscopy</td><td>84</td></tr><tr><td>3.5</td><td>Launching SPPs by scattering from a point source</td><td>87</td></tr><tr><td>4</td><td>Measuring the 3D Orientation of Single Gold Bipyramid Nanoparticle</td><td>89</td></tr><tr><td>4.1</td><td>Properties of Gold Bipyramid nanoparticles</td><td>90</td></tr><tr><td>4.2</td><td>Scattering polarimetry for orientation measurement</td><td>93</td></tr><tr><td>4.3</td><td>Orientation of the AuBPs on the glass substrate</td><td>97</td></tr><tr><td>4.4</td><td>Role of collection numerical aperture</td><td>99</td></tr><tr><td>4.5</td><td>Conclusion</td><td>104</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td><b>5 Opening-gap nanopatch antennas</b></td>
      <td><b>105</b></td>
    </tr>
    <tr>
      <td>5.1 Introduction to opening-gap nanopatch antennas</td>
      <td>106</td>
    </tr>
    <tr>
      <td>5.2 Characterization methods</td>
      <td>113</td>
    </tr>
    <tr>
      <td>5.3 Gold bipyramids on a high refractive index substrate</td>
      <td>116</td>
    </tr>
    <tr>
      <td>5.4 Gold bipyramids on the gold film</td>
      <td>122</td>
    </tr>
    <tr>
      <td>5.4.1 Scattering spectra</td>
      <td>122</td>
    </tr>
    <tr>
      <td>5.4.2 Physical formation of the modes</td>
      <td>126</td>
    </tr>
    <tr>
      <td>5.5 The complexity of the AuBP nanopatch antenna</td>
      <td>129</td>
    </tr>
    <tr>
      <td>5.6 Conclusion</td>
      <td>131</td>
    </tr>
    <tr>
      <td><b>6 Optically launching of Surface Plasmon Polaritons by Bipyramid Nanopatch Antennas</b></td>
      <td><b>133</b></td>
    </tr>
    <tr>
      <td>6.1 SPP emission spectrum</td>
      <td>134</td>
    </tr>
    <tr>
      <td>6.2 Spatial and angular emission of SPPs</td>
      <td>136</td>
    </tr>
    <tr>
      <td>6.3 SPP emission efficiency</td>
      <td>139</td>
    </tr>
    <tr>
      <td>6.4 Role of nanopatch antenna in SPP emission</td>
      <td>144</td>
    </tr>
    <tr>
      <td>6.5 Conclusion</td>
      <td>148</td>
    </tr>
    <tr>
      <td><b>Conclusion and Perspectives</b></td>
      <td><b>149</b></td>
    </tr>
  </tbody>
</table>
# Chapter 1

## Surface Plasmon Polaritons

As the ultimate gold of this work is to investigate the optical response of a plasmonic structure that hybrids between the nanoparticle and metallic film, it is necessary to begin with the properties of the individuals. This Chapter thus aims to provide the underlying theory of light-metal interaction. It begins with a basic description of the optical response of bulk metal, discussing both pros and cons of using metallic material for light confinement. From there, the propagating surface plasmon polaritons (SPPs) at a single dielectric/metal interface and their hybridization in the three-multilayer structures will be considered. On the other hand, the confinement of the surface plasmon in the nanoparticles of the finite dimensions yields distinctive optical spectra through the phenomenon of localized surface plasmon resonance (LSPR). The quasi-static approximation and Mie theory will be discussed to introduce the basic theory of localized surface plasmon resonance.
# Contents

<table><tr><td>1.1</td><td>Metals and Light: Material perspective</td><td>16</td></tr><tr><td>1.1.1</td><td>Dielectric function of bulk metals</td><td>16</td></tr><tr><td>1.1.2</td><td>Light confinement in metals</td><td>19</td></tr><tr><td>1.1.3</td><td>The loss in metal</td><td>22</td></tr><tr><td>1.2</td><td>Surface plasmon polaritons at a single interface</td><td>24</td></tr><tr><td>1.3</td><td>Plasmon hybridization in planar structures</td><td>30</td></tr><tr><td>1.4</td><td>Localized surface plasmon of metallic nanoparticles</td><td>33</td></tr><tr><td>1.4.1</td><td>Scattering, absorption and extinction</td><td>34</td></tr><tr><td>1.4.2</td><td>Quasi-static approximation.</td><td>38</td></tr><tr><td>1.4.3</td><td>Mie theory.</td><td>41</td></tr></table>

## 1.1 Metals and Light: Material perspective

### 1.1.1 Dielectric function of bulk metals

The optical response of the metal can be described by its dielectric functions, which reflect its electronic structure. We follow Drude [6] in making a simple model studying the collective response of metal in interaction with light. Classically, the metal is constructed from a lattice of stationary periodic ions with conduction electrons freely moving around. The free electrons of density $N_e$ (number of electrons per unit volume in the undisturbed state, $N_e \sim 10^{-23} \text{ cm}^{-3}$ [7]) are treated as classical particles, which have a given mass $m_e$, elementary charge $e$, and moving with velocity $v$. The Drude model solves the motion of the free electrons induced by the applied electromagnetic (EM) field with harmonic time dependence $\vec{E}(t) = \vec{E}_0 e^{-i\omega t}$
using Newton's equation. The model approximates that the electrons are not subjected to magnetic forces and electric fields from other electrons (i.e., no restoring force or potential energy is zero). During the movement of the electrons under the EM field, they are damped via collisions with other particles (electrons, ions, defects) at an average rate of $\gamma_0$. The equation of motion for one electron can be expressed as:

$$m_e \frac{d^2 \vec{x}}{dt^2} = -e \vec{E}(t) - m_e \gamma_0 \frac{d \vec{x}}{dt} \qquad (1.1)$$

As the driving field is the harmonic oscillation and without restoring force, the electron also oscillates at the same frequency $\omega$ as the incident field; its position is defined as $\vec{x}(t) = \vec{x}_0 e^{-i\omega t}$. Replacing the exponential expression of the $\vec{x}(t)$ and $\vec{E}(t)$ into the equation 1.1, we found the position of the electron displaced by the incident field as:

$$\vec{x}(t) = \frac{e}{m_e(\omega^2 + i\omega\gamma_0)} \vec{E}(t) \qquad (1.2)$$

For one electron, its charge displacement induces a dipole with a moment $\vec{p} = -e\vec{x}$. In the case of many electrons with the number density $N_e$, all of them will contribute to the macroscopic polarization $\vec{P}$ of the metal and are given by

$$\vec{P} = N_e \vec{p} = - \frac{N_e e^2}{m_e (\omega^2 + i\omega\gamma_0)} \vec{E}(t) \qquad (1.3)$$

To find the dielectric function of the material, we recall the basic definitions of the macroscopic polarization $\vec{P}$ and electric displacement $\vec{D}$ of a linear, isotropic, and nonmagnetic medium: $\vec{P} = \varepsilon_0 \chi \vec{E}$ and $\vec{D} = \varepsilon_0 \vec{E} + \vec{P} = \varepsilon_0 \varepsilon(\omega) \vec{E}$, with $\varepsilon(\omega) = \chi + 1$ is the relative dielectric function of the metal and $\chi$ is the dielectric susceptibility, $\varepsilon_0$ is the electric permittivity<fnref n="1" /> of vacuum. We identify the Drude dielectric function of the bulk metal as

$$\varepsilon^D(\omega) = 1 - \frac{\omega_p^2}{\omega^2 + i\omega\gamma_0} \qquad (1.4)$$

where $\omega_p^2 = \frac{N_e e^2}{\varepsilon_0 m_e}$ is the plasma frequency – a natural frequency of a local charge density

<footnote n="1">¹$\varepsilon_0 \approx 8.854 \times 10^{-12} \text{F/m}$</footnote>
(plasma electrons) that oscillates collectively in response to a fluctuation [8], which is as shown depending on the electron density. From the complex expression of the dielectric function $\epsilon_r(\omega)$, we derive its real and imaginary part in the condition of small damping $\gamma_0 \ll \omega$ as

$$ 
\begin{aligned}
Re[\epsilon^D(\omega)] &= 1 - \frac{\omega_p^2}{\omega^2 + \gamma_0^2} \approx 1 - \frac{\omega_p^2}{\omega^2} \\
Im[\epsilon^D(\omega)] &= \frac{\gamma_0 \omega_p^2}{\omega(\omega^2 + \gamma_0^2)} \approx \frac{\omega_p^2}{\omega^3} \gamma_0
\end{aligned} 
\quad (1.5) $$

## Behavior of the Drude dielectric function

Roughly, in the case of simple metals, Drude model can successfully interpret the optical properties of metals such as

* For $\omega < \omega_p$ and is very close to $\omega_p$: the real part of $\epsilon^D(\omega)$ is predominant and negative, resulting in a complex wavevector of light in metal $k = \pm i\sqrt{|\epsilon^D|}\omega/c$. This means that the light is evanescent inside metal and cannot go through that is strongly reflected at the interface. This also explains why gold and silver are good optical reflectors in the visible to mid-infrared regions since their plasma wavelength is $\lambda_p = 2\pi c/\omega_p \approx 200$nm.

* For $\omega = \omega_p$: the real part of $\epsilon^D(\omega)$ is equal zero. This means that all the electrons oscillate in phase throughout the metal propagation length. This oscillation is known as plasma oscillation.

* For $\omega > \omega_p$: the $\epsilon^D(\omega)$ is almost real and positive. Therefore, the light can propagate through the metal, making it being transparent.

## Dielectric function of noble metals

The description of the optical response of metals by Drude model is only valid for photon energies below interband transition of metal, which will be discussed in more detail later. Intuitively, Drude model is not adequate at the large frequency ($\omega \gg \omega_p$) because electrons are not completely free anymore, i.e. their energy band structure is not perfectly a parabolic [10]. Figure 1.1 displays a comparison between the dielectric function of gold and silver calculated by the Drude model and those measured by Johnson and Christy [11]. As can be seen in the imaginary part of $\epsilon^D(\omega)$, the validity of the Drude model curve breaks down at the occurrence of interband transitions: $E_{IB}$ (gold) ~ 2eV (650nm) and $E_{IB}$ (silver) ~ 4eV
<figure><img src="image_3.png" /><figcaption>Figure 1.1: Real ($ε_1$) and imaginary ($ε_2$) parts of dielectric function of gold and silver. Solid lines are the Drude model and squares are experimental measurement of Johnson and Christy. $ħω_{IB}$ is the interband transition threshold. Source: E.Cottantin et al. [9]</figcaption></figure>

(330nm). Therefore, to calculate the dielectric function of the noble metals at energy above $E_{IB}$, we need to take the interband transition into account. The dielectric function then has an additional term $ε^{IB}$ [6, 9]:

$$ε = ε^D + ε^{IB} - 1 \qquad (1.6)$$

Here, $ε^{IB}$ is related to interband transition, which mostly accounted of the contribution of the d-core electrons [6].

### 1.1.2 Light confinement in metals

Light emission plays an instrumental role in ant areas where advanced technological applications are desired. In general, the interaction of light with single emitters, such as molecules or nanocrystals, is inefficient because of the dimension mismatch between the point-like character of such emitters and the limit-diffraction light beam. Light confinement at the nanoscale is thus the fundamental role of nanophotonics.
From a material perspective, there are two strategies for the light localization using: (i) high-index dielectrics or (ii) negative-permittivity materials such as metals or doped semi-conductors. Using the high-index dielectric materials allows for reducing the wavelength to $\lambda/n$, with $n \sim 3$ (Si, Ge, GaP,..) [12, 13], which is usually prevented by the limit of diffraction (this might be broken down by the nanoscale dielectric discontinuities [14, 13]). On the other hand, metals allow light concentration beyond the diffraction limit by coupling electromagnetic fields to the large momentum of the conduction electrons. This diffraction limit beating also can be interpreted from an energy conversation point-of-view based on the work of J.Khurgin [15], as illustrated in Figure 1.2.

Here, the light oscillates inside the finite dimension materials which is considered as an optical resonator. For dielectrics with refractive index $n$ (Figure 1.2 a and b), depending on the resonator' size $a$, the electric or magnetic fields might have different harmonic spatial dependence

$$ 
\begin{align} 
\vec{E} &= \vec{E}_0 e^{-i\omega t} e^{i\frac{2\pi}{\lambda}z}, \quad \vec{H} = \vec{H}_0 e^{-i\omega t} e^{i\frac{2\pi}{\lambda}z} \quad (a > \frac{\lambda}{2n}) \tag{1.7} \\ 
\vec{E} &= \vec{E}_0 e^{-i\omega t} e^{i\frac{\pi}{a}z}, \quad \vec{H} = \vec{H}_0 e^{-i\omega t} e^{i\frac{\pi}{a}z} \quad (a < \frac{\lambda}{2n}) 
\end{align} 
$$

For the resonator size beyond the limit of diffraction ($a > \frac{\lambda}{2n}$), from the two first Maxwell's equations $\nabla \vec{H} = \varepsilon \partial \vec{E} / \partial t$ and $\nabla \times \vec{E} = -\mu \partial \vec{H} / \partial t$, we can find a similar relation between the magnetic and electric fields as $H = \sqrt{\varepsilon/\mu} E$, resulting in an equal average electric field and magnetic energies: $u_E = \frac{\varepsilon E^2}{2} = u_H = \frac{\mu H^2}{2}$. This means that the energy inside the resonator converts between the electric energy $u_E$ and magnetic energy $u_H$, which is analogous to the energy transformation between the kinetic energy and potential energy in a mechanical harmonic oscillator for “self-sustaining oscillation”.

For the dielectric resonator with sub-wavelength size, resolving the first Maxwell's equation gives $H = \frac{a}{\lambda/2n} \sqrt{\varepsilon/\mu} E$, which is much smaller than $E$ due to $a \ll \lambda$. Subsequently, the magnetic energy is also much smaller than electric energy $u_H \ll u_E$, the self-oscillation cannot be maintained, hence the energy is dissipated outside the resonator as expected from the diffraction limit. Besides, since the magnetic field is so small, the second Maxwell's equation requires that the electric field must be irrotational, resembling an electrostatic field at a sub-wavelength regime.
<figure><img src="image_4.png" /><figcaption>Figure 1.2: Energy balance in photonic and plasmonic structures. (a) Optical resonator size $a > λ/2n$, energy converts between electric $u_E$ and magnetic energy $u_H$ in order to self sustain oscillation. Below the limit of diffraction: (b) magnetic energy of dielectric resonator is too small, the energy is dissipated outside the resonator; (c) metallic resonator of sub-wavelength dimension can beat the diffraction limit by storing energy in the kinetic energy of free carriers. Source: J. Khurgin [15].</figcaption></figure>

In contrast, the metallic resonator can store the dissipated energy below the diffraction limit in the form of the kinetic energy of the free electrons, which can be quantified by the Drude model (Figure 1.2c). In response to the applied EM field, the movement of electrons introduces an internal current density $\vec{j} = -N_e e \vec{v}$ which delivers a power $P_j = 1/2 Re(\vec{j} \cdot \vec{E}^*)$. This power is associated with the rate change of the kinetic energy of the free electrons $^2$. Assuming the damping is small $γ_0 \ll ω$ we can find the average kinetic energy $u_K =$

<footnote n="2">²the potential energy is zero</footnote>
$1/2mv^2 \approx \frac{1}{2}\varepsilon_0 \frac{\omega_p^2}{\omega^2} E^2$. Therefore, at a certain frequency $\omega$, the $u_K$ can compensate for the small magnetic energy and make a balanced exchange with the electric energy $u_E$. In other words, in metals, the energy coherently oscillates between the electric and kinetic energy of free electrons, and thus allow the light to be stored its electric energy at below limit of diffraction.

For example, Figure 1.3 shows state-of-art optical cavities made from dielectric or metalic of different structures used to boost light-matter interaction (accelerating the spontaneous emission rate of emitters for example). Comparisons between cavities are based on two figure-of-merits: (i) the time of the photon is trapped inside the cavity characterized by the quality factor $Q = \omega/\Delta\omega$ ($\Delta\omega$ is the spectral width of the resonance) and (ii) the optical mode volume $V_{eff}$ allows to confine the light. A combination of a large quality factor and a small volume mode is thus essential for enhancing light-matter interaction. The dielectric cavities can offer huge $Q$ values ($10^3 - 10^8$) by trapping photons traveling back and forth in between Bragg mirrors of the Fabry-Pérot cavity [16, 17], or moving circularly inside the whispering gallery resonator (spheres or toroids) [18, 19], or making its slowdown in the dielectric band gap of photonic crystals [20]. In turn, the reduction of mode volume is prevented by the diffraction limits. As a counterpart, the metallic cavities allow localized the field into deep-subwavelength volume near the tips of elongated nanoparticles [21] or inside the gap structures [22, 23]. However, this success comes at a price of a low-quality factor due to strong losses.

### 1.1.3 The loss in metal

As discussed above, free electrons in metals can drag photons below the diffraction limit by storing their electric field energy in the kinetic energy, which however can get lost via dissipative damping of electron motion. Consider a simplified energy band structure that results from the outer elections of metals in Figure 1.4, from which the hybrid $ns-p$ band is separated spatially from the $(n-1)d$ band.

The $ns-p$ band is only partially filled (conduction band), all below the Fermi level of energy $E_F$ – the highest energy that electrons can occupy at 0K. By absorbing photons, the electrons from the unoccupied states inside $ns-p$ band can be promoted into the unoc-
<figure><img src="image_5.png" /><figcaption>Figure 1.3: Cavities for enhanced light-emitter interaction. Upper row: dielectric microcavities provide high quality factor *Q* but usually, they can only focus light in the scale of wavelengths. Lower row: plasmonic resonators/cavities hold the modest value of *Q* but the light is confined to the deep sub-wavelength volume *V*<sub>eff</sub>. Adapted from [24, 18, 20, 21, 23, 22]</figcaption></figure>

cupied region above the Fermi level and move almost freely throughout the crystal. Such
optical transitions are called intraband transitions and take place in the visible and near-IR
range, which will be the most interesting in the works that follow. However, because electron
momentum is much larger than that of photons, the absorption loss during the intraband
transitions is thus unavoidable due to the momentum conservation. The absorption can be
assisted by several processes such as (i) phonons accompanied by the generation of “hot”
electrons and holes (i.e. charge carriers with significant additional energy before they ther-
malize via scattering [15]), (ii) electron-electron scattering that many carriers might involve in
dissipating photon energy, and (iii) direct absorption induced by large momentum of tightly
confined plasmon mode when the particle size *a* ≪ λ/2 (Landau damping). In addition,
if a sufficiently energetic photon is absorbed, electrons from the fully filled (n − 1)d band
<figure><img src="image_6.png" /><figcaption>Figure 1.4: Optical transitions of the noble metals with schematic dispersion relation of (n-1)d band and ns-p band. Intraband transitions take place inside ns-p band, whereas the interband transitions between the (n-1)d band and conduction band requires an energy threshold $\hbar\Omega_{IB}$. Adapted from J. Khurgin [15].</figcaption></figure>

(valence band) can be excited to the unoccupied levels of the ns-p band. This band-to-band transition is a so-called interband transition that requires a photon’s energy larger than the energy threshold $E_{IB} = \hbar\Omega_{IB}$. For example, the interband transition becomes significant around 650nm ($E_{IB} \sim 2$eV) for gold and 300nm ($E_{IB} \sim 4$eV) for silver [9].

In conclusion, loss in metals is an unavoidable process, that can affect the performance of metallic devices in applications that ask for high efficiency of emissions such as single-photon light sources, solar cells, or sensing. However, the generation of the hot carriers assisted absorption can be harvested for productive purposes such as acceleration of chemical reactions [25, 26].

## 1.2 Surface plasmon polaritons at a single interface

Above we have considered the optical response of bulk metals, noticing that the bulk plasmons (only exist at $\varepsilon(\omega_p)=0$) are naturally longitudinal waves that cannot be excited to the transverse field of light. However, when it comes to a dielectric/metallic interface, the light
waves can couple to the conductive electrons forming a hybrid surface mode. In this interaction, the free electrons near the surface oscillate collectively in resonance with the light wave, which are termed surface plasmon polaritons (SPPs) reflecting their hybrid nature. The SPPs are confined below the diffraction limit in the direction perpendicular to the interface while propagating in a long distance of micrometric scale along the surface). This fascinating characteristic of SPPs was first highlighted by Wood in 1902 [27].

The aim of this section is therefore to characterize the SPPs in terms of their dispersion relation (i.e., the frequency-dependent surface plasmon wavevector) and spatial profile, starting by identifying the conditions from which the surface plasmon modes can exist. These can be deduced by solving Maxwell's equations under the appropriate boundary conditions at the interface.

<figure><img src="image_7.png" /><figcaption>Figure 1.5: Schematic representation of a surface plasmon propagating along the dielectric interface/metal. $\varepsilon_d$ and $\varepsilon_m(\omega)$ are the relative dielectric function of the dielectric and metal, respectively.</figcaption></figure>

We consider a simple geometry that can support SPPs: a flat dielectric/metal interface, as schematized in Figure 1.5. A dielectric of negligible with positive dielectric constant $\varepsilon_d > 0$ is located in $z > 0$ and a metal with a complex dielectric function $\varepsilon_m(\omega)$ is located in $z < 0$. The interface is set in a plane at $z = 0$. We simplify the problem to one-dimension surface wave, i.e., the waves propagate along the x-axis ($\partial/\partial x = ik_{\parallel}$), and no spatial variation in the y-axis ($\partial/\partial y = 0$). Here, $k_{\parallel} = \sqrt{k_x^2 + k_y^2}$ is the in-plane wavevector, which is necessarily conserved across the interface and therefore

$$k_{\parallel}^2 + k_{j,z}^2 = \varepsilon_j k_0^2 \qquad (1.8)$$
where $j = d, m$ denotes for the dielectric and metal media, $k_j, z$ is normal component of the wavevector and $k_0$ is the wavevector of light in the vacuum.

## Conditions to sustain SPPs at the interface

In order to find the condition that excitation can couple to SPPs, we seek the solutions of the Maxwell's equations for a homogeneous medium in the absence of the external charges and current density, yielding to the wave equation of the electric field [7]:

$$ \nabla^2 \vec{E}_j + k_0^2 \varepsilon_j \vec{E}_j = 0 \qquad (1.9) $$

Naturally, a similar equation exists for the magnetic field $\vec{H}$

we derive the fields in tow sets of polarization: transverse magnetic mode (p-polarization) and transverse electric mode (s-polarization), and then apply the boundary conditions at the interface.

* s-polarization excitation: only components $H_x, H_z$, and $E_y$ are nonzero.

The electric and magnetic fields in each half-space medium can be expressed for s-polarization as

$$ \vec{E}_j = e^{-i(\omega t - k_{||}x)} e^{ik_{j,z}z} E_{j,y} \vec{e}_y, \quad \vec{H}_j = e^{-i(\omega t - k_{||}x)} e^{ik_{j,z}z} (H_{j,x} \vec{e}_x + H_{j,z} \vec{e}_z) \qquad (1.10) $$

The one-dimension propagating surface wave problem allows to simplify the Maxwell equation $\nabla \times \vec{E} = -\frac{\partial \vec{E}}{\partial t}$ into- the relation $\frac{\partial E_y}{\partial z} = -i\omega\mu_0 H_x$. On the other hand, the continuity at the interface of the in-plane components of the electric and magnetic fields requires:

$$ E_{d,y}|_{z=0} = E_{m,y}|_{z=0}, \quad \frac{\partial E_{d,y}}{\partial z}|_{z=0} = \frac{\partial E_{m,y}}{\partial z}|_{z=0} \qquad (1.11) $$

which are respectively equivalent to conditions

$$ E_{d,y} = E_{m,y}, \quad k_{d,z} E_{d,y} = k_{m,z} E_{m,y} \qquad (1.12) $$

These conditions are not fulfilled, excepted for $E_{d,y} = E_{m,y} = 0$. Therefore, the surface plasmon at the interface cannot be excited by the s-polarization
* p-polarization excitation: only components $E_x$, $E_z$, and $H_y$ are nonzero. The electric and magnetic fields in each half-space medium can be expressed for the p-polarization as

$$ \vec{E}_j = e^{-i(\omega t - k_{||}x)} e^{ik_{j,z}z} (E_{j,x}\vec{e}_x + E_{j,z}\vec{e}_z), \quad \vec{H}_j = e^{-i(\omega t - k_{||}x)} e^{ik_{j,z}z} H_{j,y}\vec{e}_y \quad (1.13) $$

In the absence of the external charges in each medium, we have $\nabla \cdot \vec{D} = 0$: $k_{||}E_{j,x} + k_{j,z}E_{j,z} = 0$. On the other hand, the continuity of the in-plane component of $\vec{E}$ and normal component of $\vec{D}$ imposes two boundary conditions at $z = 0$: $E_{d,x} = E_{m,x}$ and $\varepsilon_d E_{d,z} = \varepsilon_m E_{m,z}$. Combining these three equations, we obtain the final condition for the p-polarization excitation

$$ \frac{k_{d,z}}{\varepsilon_d} = \frac{k_{m,z}}{\varepsilon_m} \quad (1.14) $$

This condition can be fulfilled by the two media have opposite signs of their relative permittivity such as dielectric/metal or dielectric/doped semiconductor. In terms of the excitation, we can conclude that the SPPs at the interface are essential to be excited by the p-polarization.

## Dispersion relation of surface plasmon polaritons

By introducing the relation 1.14 into the equation $k_{||}^2 + k_{j,z}^2 = \varepsilon_j k_0^2$, we can obtain the dispersion relation of SPPs, which is expressed by a function

$$ k_{SPP} = \frac{\omega}{c} \sqrt{\frac{\varepsilon_d \varepsilon_m}{\varepsilon_d + \varepsilon_m}} \quad (1.15) $$

For limit of low damping (implying $Im(\varepsilon_m) = 0$), we can plot the dispersion relation between angular frequency $\omega$ and the real part of $k_{||}$, as shown in Figure. We can see that for small $k_{||}$ corresponding to low frequencies (wavelength in mid-infrared and higher), the dispersion curve of SPPs is close to the respective light line of the dielectric $\omega = \frac{c}{\sqrt{\varepsilon_d}}k_0$. When $k_{||}$ increases, the SPPs dispersion is more separated from the light line and it asymptotes to a frequency $\omega_{SP} = \frac{\omega_p}{\sqrt{1+\varepsilon_d}}$ as $k_{||} \to \infty$. The $\omega_{SP}$ is called the surface plasmon frequency. Besides, the SPP dispersion is nearly flat for very large $k_{||}$, the group velocity is thus $v_g = \partial\omega/\partial k \to 0$, which means that the photons are dragged to be slowdown at the interface and the mode
<figure><img src="image_8.png" /><figcaption>Figure 1.6: **SPP dispersion relation.** The dispersion of SPP (red curve) is below the light cone, i.e., the propagating regimes, of the light line (black line).</figcaption></figure>

thus acquires electrostatic character [7].

The most important feature of the SP dispersion relation is that for a given frequency, the dispersion curve of SPPs is always under the respective light line of the dielectric. In other words, the momentum of the SPPs $\hbar k_{SPP}$ is always greater than that of a free-space photon at the same energy, $\hbar k_{inc}$ ($k_{inc}$ is the free-space wavevector). This implies that the SPPs at the interface cannot be directly excited from free-space waves of the corresponding dielectric medium. Instead, special phase-matching techniques are required to provide additional momentum if the light is to be used to generate SPPs. There are three common optical techniques have been used to increase the momentum: (i) prism coupling [28, 29], (ii) grating coupling [30] (periodic configuration is patterned in the metal’s surface), or (iii) near-field coupling induced by local scatters (i.e., particle or holes) [31, 32]. In this work, the techniques used for the excitation of SPPs will be discussed in more detail in Chapter 3.

## Propagation length of SPPs

Owing to the losses arising from the absorption of the metals, the SPPs are gradually damped during propagating over the interface. The attenuation of the plasmon intensity along the x-axis is linked to imaginary part of $k_{SPP}$ as

$$I_{SPP} = I_0 e^{-2Im(k_{SPP})x} \qquad (1.16)$$

The propagation length is thus defined as the distance at which the plasmon intensity decayed
<figure><img src="image_9.png" /><figcaption>Figure 1.7: Propagation of SPPs (a) attenuation of plasmon intensity along the surface and (b) evanescence of field inside the media along the perpendicular z-direction.</figcaption></figure>

to $1/e$ of its initial value (see Figure 1.7a) and can be given by

$$L_{SPP} = \frac{1}{2Im(k_{SPP})} \qquad (1.17)$$

Here, the imaginary part of $k_{SPP}$ can be calculated by introduce the complex $\varepsilon_m(\omega)$ of metals into equation 1.15. Therefore, the propagation length of SPPs is mainly dictated by the loss in metals which is responsible by the imaginary part of its permittivity. For example, in the visible regime, the smooth surface of silver can support long propagation lengths in the range of 10-100µm due to its low losses, which is 1 order of magnitude greater than relative absorbing metal such as aluminium [33].

In terms of time domain, we also can define the decay time of SPPs as $\tau_{SPP} = \frac{L_{SPP}}{v_g} = \frac{2\pi}{\omega_{SPP}}$, with $\omega_{SPP}$ is the imaginary part of the SPP frequency, which can be deduced from the complex $k_{SPP}$. The decay time of a dielectric/metal interface is usually in the order of 10-100fs [34, 35], which can propel their use in various applications.

## Extension of field inside the media

The characteristic of the surface waves requires the electric field to be strongly confined near the surface and evanescent decay perpendicularly away from it, which is expressed by the term $e^{-|k_{j,z}||z|}$ in the field equation. The reciprocal values $\delta_j = 1/|k_{j,z}|$, which corresponds to the z-distance at which $|E_z|$ decayed by $1/E$ (see Figure 1.7b), can determine the spatial extension or the confinement of the fields perpendicular to the surface. This is so-called evanescent decay length.
From the conversion relation of the in-plane wavevector $k_{\parallel}^2 + k_{j,z}^2 = \epsilon_j k_0^2$, we can deduce the decay lengths of SPPs inside the dielectric and respectively are:

$$ \delta_d = \frac{\lambda}{2\pi} \sqrt{\frac{|\epsilon_d + Re(\epsilon_m)|}{\epsilon_d^2}} \qquad (1.18) $$

$$ \delta_m = \frac{\lambda}{2\pi} \sqrt{\frac{|\epsilon_d + Re(\epsilon_m)|}{\epsilon_m^2}} $$

Because of $|\epsilon_m| > |\epsilon_d|$, the field penetrates more inside the dielectric half-space. Typically, the decay length $\delta_d$ into the dielectric is in the order of half wavelength, whereas the decay length into metal is associated with the skin depth at room temperature $\delta_m \sim 25nm$. [7].

## 1.3 Plasmon hybridization in planar structures

In the previous section, we have seen that a single dielectric/ metal interface can support surface plasmon which propagates along the interface while remains confined with a perpendicular component field near the surface. When the SPPs from two individual interfaces are brought near together (separated distance should be smaller than attenuation length of SPPs $\delta$ inside the overlapping medium), they can couple and results in hybrid plasmon modes. Typically, the planar structures that support the hybrid plasmon modes are based on two fundamental structures: (i) insulator-metal-insulator (IMI) layers or (ii) metal-insulator-metal (MIM) layers. The goal of this section is therefore to show how the SPPs of each single interface interact and to give the main characteristics of the new modes that arise from the SPPs coupling.

We consider three-layer structures consisting of a thin slab of thickness $t$ sandwiched between two infinite cladding made from the same material, as shown in Figure 1.8. The relative dielectric functions of the cladding and slab respectively are $\epsilon_s$ and $\epsilon_c$, which necessarily have opposite signs in order to support SPPs. Depending on the structure, the cladding is made from metal, and the slab is made from a dielectric (MIM structure) or vice versa (IMI structure).

The geometry of the three-layer system in Figure 1.8 is analog to a waveguide, which
<figure><img src="image_10.png" /><figcaption>Figure 1.8: Schematic of a three-layer system that supports hybridization of SPPs
A thin slab of dielectric function $\varepsilon_s$ and thickness t is embedded between two similar cladding media of dielectric function $\varepsilon_c$.</figcaption></figure>

might exhibit oscillatory modes for the sufficient thickness of the slab. Here we are only interested in the lowest-order bound modes. The electric and magnetic field components of the p-polarization that are non-oscillatory in the z-direction and propagate along the x-direction are written as

$$ 
\begin{aligned}
E_x &= E_x^0(z)e^{-i(\omega t - k_{||}x)}, \quad E_z = E_z^0(z)e^{-i(\omega t - k_{||}x)} \\
H_y &= H_y^0(z)e^{-i(\omega t - k_{||}x)}
\end{aligned}
\qquad (1.19) $$

Here, $k_{||}^2 = k_{j,z}^2 + \varepsilon_j k_0^2$ is the in-plane component of wavevector, $k_z$ is the associated normal component; $j = s, c$ denotes for the slab and cladding media. From the boundary conditions that require $H_y$ and $E_x$ at upper and lower interfaces ($z = \pm \frac{t}{2}$) are continuous, two dispersion relations and associated field components can be given by solutions of

$$ \tanh(k_{s,z}\frac{t}{2}) = -\frac{k_{c,z}\varepsilon_s}{k_{s,z}\varepsilon_c}, \quad E_z^0(z) = E_z^0(-z) \propto \frac{k_{||}}{\omega\varepsilon_s}\cosh(k_{s,z}z) \qquad (1.20) $$

$$ \tanh(k_{s,z}\frac{t}{2}) = -\frac{k_{s,z}\varepsilon_c}{k_{c,z}\varepsilon_s}, \quad E_z^0(z) = -E_z^0(-z) \propto \frac{k_{||}}{\omega\varepsilon_s}\sinh(k_{s,z}z) \qquad (1.21) $$

For the solutions of equation 1.20, the field component $E_z(z)$ is an even function that exhibits a symmetry profile along the z-axis. On the contrary, for the solutions of equation 1.21, the $E_z(z)$ is an odd function and the field profile is anti-symmetric along the z-axis. Respectively, they are called symmetric mode and anti-symmetric mode. Now we can apply
these dispersion relations to describe the coupled SPP modes of IMI and MIM structures.

<figure><img src="image_11.png" /><figcaption>Figure 1.9: **Plasmon hybrid modes in the IMI and MIM structure.** Spatial profile of the field component $Re[E_z(z)]$ and dispersion relations of the coupled SPPs in (a) IMI structure and (b) MIM structure. Adapted from P.Berini 2006 [36]</figcaption></figure>

In the IMI structure, SPPs on either side of the metal slab can hybridize, resulting in $E_z$-symmetric and anti-symmetric modes, Figure 1.9a illustrates the field profile and the associated dispersion relation of the hybrid mode of the IMI structure. Here, the spacing between two dispersion curves depends on the thickness of the metallic slab. For a large slab thickness, the surface plasmon of each interface barely interact and the dispersion of two hybrid modes can converge into the single SPP dispersion. The hybridization can take place when the individual SPPs overlap and become significant when the metal thickness is smaller than its skin depth ($t < 20$nm) [6]. At a given frequency, the in-plane wavevector $k_{||}$ of the anti-symmetric mode ($k_{SR}$) is always greater than that of the symmetric mode ($k_{LR}$). In the other words, the anti-symmetric mode has tighter confinement inside the slab than the symmetric one. This also can be seen by an examination of the component field. As the slab thickness vanishes, while the anti-symmetric mode increases its confinement due
to the field null inside the metal, the symmetric mode evolves into a plane wave supported by the surrounding homogeneous dielectric. Since the better confinement implies a lower propagation length, the anti-symmetric mode is called short-range surface plasmon (SRSP) and the symmetric mode is called long-range surface plasmon (LRSP). Depending on the wavelength or the materials considered, the propagation length of the LRSP can take place in the order of a few hundred micrometers [37]. Therefore, the LRSP in the IMI structure can be employed to assist the information or energy over a long distance.

For the MIM geometry, the SPPs on either side of the dielectric medium similarly hybridize inside the slab, as shown in Figure 1.9b. However, for the insulator thicknesses of interest, only the fundamental symmetric mode (or long-range plasmon) is supported that has no cut-off for vanishing slab thickness, unlike the anti-symmetric mode that is cut-off at roughly $\lambda/2$ [38]. We refer to the symmetric mode of this structure as the MIM plasmon gap mode. The most striking feature of the MIM plasmon gap is that it has a very large $k_\|$ , usually in the 1 order of magnitude as compared to the SPP at a single interface, which can be achieved even for excitation well below surface plasmon frequency. Therefore, the metallic nanostructures based on the MIM geometry are a platform for deep-subwavelength confinement of light with mode volumes smaller than $\sim 10^{-6}(\lambda/n)^3$ [38, 22].

To conclude, we have shown that the hybridization of SPPs in the planar system allows extending the properties of the individual SPPs in the terms of propagation length and light confinement. Further investigation of these two structures will be done by the analytical calculation in Chapters 2 and 5.

1.4 Localized surface plasmon of metallic nanoparticles

Previously, we have seen that the light can couple to electron plasma of metal at a planar dielectric/metal interface, yielding propagating surface plasmon. In this section, we will see that when the interface is reduced into nanoscale dimension of the metallic particle, the free electron cloud can oscillate resonantly with electromagnetic field, resulting in non-propagating surface plasmon. This resonance is called the *localized surface plasmon* resonance (LSPR). We will begin with an overview problem of optical responses, including scattering,
absorption and extinction, of an arbitrary object. Then, appropriate analytical treatments
will be discussed, depending on the size of the particle, to show how this give raise to the
phenomenon of the LSPR.

1.4.1 Scattering, absorption and extinction

Generally, when light interacts with an object, it can be scattered or absorbed by the object. While absorption corresponds to the energy dissipation (heating) in the object, scattering is an elastic interaction, i.e. the object re-emits the electromagnetic field without changing the wavelength. The total energy that an object takes from the incident light, thus it is attributed to scattering and absorption, is called extinction. Usually, these optical responses of the object are characterized by optical cross-sections. This section aims to define these optical cross-sections. Let us consider a general situation where an arbitrary scatter object of relative permittivity $\varepsilon(\omega)$ with a finite size is surrounded by a homogeneous transparent medium and is excited by an incoming plane wave $\vec{E}_{inc}$, as shown in Figure 1.8.

<figure><img src="image_12.png" /><figcaption>Figure 1.10: Schematic diagram of optical responses of an arbitrary particle under the plane wave excitation. The dotted ring presents an imaginary surface with surface normal vector $\vec{n}$, which may be used to calculate the energy flux in the far field.</figcaption></figure>

Scattering cross-section

It is clear that not all the photons but only a finite number of photons of the incoming
light are scattered by the object. Therefore, the probability that leads to the scattering
is proportional to an effective area, so-called scattering cross-section. The scattering cross-
section is defined as the ratio between the average scattered power $P_{sc}$ and the intensity of the incident field $I_{inc}$

$$ \sigma_{sc} = \frac{P_{sc}}{I_{inc}} = \frac{1}{|\langle \vec{S}_{inc} \rangle|} \iint_A \langle \vec{S}_{sc} \rangle \cdot \vec{n} dA \quad (1.22) $$

where $\langle \vec{S}_{inc} \rangle$ and $\langle \vec{S}_{sc} \rangle$ respectively are the time-average Poynting vector of the incident and scattered field. Here, the EM energy flux of scattering through an imaginary surface A far from the object (Figure 1.8) can be calculated as $P_{sc} = \iint_A \langle \vec{S}_{sc} \rangle \cdot \vec{n} dA$, with $\vec{n}$ is an outward-facing normal vector. It is important to note that the scattering cross-section has unit of area but it does not represent a geometrical cross-section of the object. Instead, the physical meaning of scattering cross-section is related to the energy of the incident light that has been lost in scattering.

## Absorption cross-section

Since the object has a permittivity, it can be considered as a perturbation of the background medium [39]. Therefore, the total fields around the object can be represented as a sum of the incident fields and scattered fields:

$$ \begin{aligned} \vec{E}_{tot} &= \vec{E}_{inc} + \vec{E}_{sc} \\ \vec{H}_{tot} &= \vec{H}_{inc} + \vec{H}_{sc} \end{aligned} \quad (1.23) $$

As the medium is non-absorbing, the total EM energy flux inward the imaginary surface around the object will be equal to the absorbed power

$$ P_{abs} = - \iint_A \langle \vec{S}_{tot} \rangle \cdot \vec{n} dA \quad (1.24) $$

The minus thus said that the direction of the time-averaged Poynting vector of the total field $\langle \vec{S}_{tot} \rangle$ (inward) is opposite to $\vec{n}$ (outward). In the analogy to scattering, we can define absorption cross-section as

$$ \sigma_{abs} = \frac{P_{abs}}{I_{inc}} = - \frac{1}{|\langle \vec{S}_{inc} \rangle|} \iint_A \langle \vec{S}_{tot} \rangle \cdot \vec{n} dA \quad (1.25) $$

## Extinction cross-section

The time-averaged Poynting vector of the total field $\langle \vec{S}_{tot} \rangle$ can be formulated as a function
of the fields as

$$ \langle \vec{S}_{tot} \rangle = \frac{1}{2} \mathrm{Re}(\vec{E}_{tot} \times \vec{H}_{tot}^*) \\ = \underbrace{\frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{inc}^*)}_{\langle \vec{S}_{inc} \rangle} + \underbrace{\frac{1}{2} \mathrm{Re}(\vec{E}_{sc} \times \vec{H}_{sc}^*)}_{\langle \vec{S}_{sc} \rangle} + \underbrace{\frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*)}_{?} \quad (1.26) $$

Here, the first and second terms of equation 1.26 are equal to the time-avarage Poynting vectors of incident and scattered fields $\langle \vec{S}_{inc} \rangle$ and $\langle \vec{S}_{sc} \rangle$, respectively. We now want to find the physical meaning of the last cross term. Applying the area integration for both sides of equation 1.26, we have

$$ \iint_A \langle \vec{S}_{tot} \rangle . \vec{n} dA = \iint_A \langle \vec{S}_{inc} \rangle . \vec{n} dA + \iint_A \langle \vec{S}_{sc} \rangle . \vec{n} dA + \iint_A \frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*) . \vec{n} dA \\ - P_{abs} = 0 + P_{sc} + \iint_A \frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*) . \vec{n} dA \quad (1.27) $$

Here $\iint_A \langle \vec{S}_{inc} \rangle . \vec{n} dA = 0$ is the result of the conservation of the incoming and out-coming energies through the imaginary surface. We can express the cross term as

$$ - \iint_A \frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*) . \vec{n} dA = P_{sc} + P_{abs} = P_{ext} \quad (1.28) $$

which as shown, is equal to the sum of the scattered power and the absorbed power. Therefore this term is extinct power $P_{ext} = P_{sc} + P_{abs}$, which tells that the object can take some energy from the incoming field and then distributes it into scattering and absorption. The energy that the incident field has lost is called extinction. Besides, from equation 1.28, we can interpret the extinction as an interference between the incident and scattered fields. Now, we can formulate the extinction cross-section as

$$ \sigma_{ext} = \frac{P_{ext}}{I_{inc}} = \frac{1}{|\langle \vec{S}_{inc} \rangle|} \iint_A \frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*) . \vec{n} dA \quad (1.29) $$

Subsequently, we also have $\sigma_{ext} = \sigma_{sc} + \sigma_{abs}$.

**For numerical calculation:** The absorption cross-section can be calculated by applying the Poynting's theorem for the harmonic field in a homogeneous medium $\nabla(\vec{E}_{tot} \times \vec{H}_{tot}^*) =$
$-\vec{j}^*.\vec{E}_{tot}$, with $\vec{j}$ is the current density that the incident field induces in the particle. Making a volume integration of this equation yields to the absorbed power

$$P_{abs} = \frac{1}{2} \iiint_V Re(\vec{j}^*.\vec{E}_{tot})dV \qquad (1.30)$$

Similarly, we also can express the extincted power in the term of volume integral of the current density by applying the reciprocity theorem $\nabla(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*) = -\vec{j}^*.\vec{E}_{inc}$, we obtain

$$P_{ext} = \frac{1}{2} \iiint_V Re(\vec{j}^*.\vec{E}_{inc})dV \qquad (1.31)$$

The two expressions 1.30 and 1.31 of the powers of absorption and extinction would be very useful for numerical calculations.

## Optical responses of metallic nanoparticles: Problem with size

Now we will use the optical cross-sections discussed above to characterize the optical response of metallic nanoparticles. Roughly, because of the small dimension, the EM field can penetrate significantly inside the nanoparticles, however, the penetration can be different depending on the particle's size. Figure 1.11 considers two spherical metallic nanoparticles of different radii, which are surrounded by a non-absorbing medium and submitted to an incident field.

<figure><img src="image_13.png" /><figcaption>Figure 1.11: Schematic diagram of optical responses of an arbitrary metallic particle under the plane wave excitation. Depending on the particle's size, the external field induce a dipole or multipole in the particles. Source: PhD thesis of [5]</figcaption></figure>

For a vanishing small particle of diameter is much smaller than the wavelength of light in the surrounding medium ($a \ll \lambda$), the field is homogeneous over the particle volume. Therefore, the oscillating EM field can be simplified to an electrostatic field and the optical
responses of the small particle now can be analyzed using the quasi-static approximation.

On the other hand, for a large particle of diameter in the order of wavelength ($a \sim \lambda$), it can feel the spatial variation of the field, resulting in a more complex surface charge distribution. In this case, a rigorous theory is required to study the optical responses of the particle. This theory was developed completely by Mie in 1908 [40]. The discussion in the following sections will be separated based on the two regimes of the particle's dimension.

### 1.4.2 Quasi-static approximation

In this section, we aim to consider the interaction of EM waves with a small metallic nanoparticle of characteristic dimension is much smaller than the wavelength. Using the quasi-static approximation, we will see that the absorption and scattering cross-sections can be resonantly enhanced at the LSPR.

<figure><img src="image_14.png" /><figcaption>Figure 1.12: Schematic diagram of a small spherical metallic nanoparticle subjected to an EM plane wave. The particle of radius $a \ll \lambda$ is approximated to a dipole of moment $\vec{p}$.</figcaption></figure>

We consider the most convenient geometry for an analytical treatment: a spherical particle of the radius $a \ll \lambda$ is submitted to a plane-wave illumination $\vec{E}_{inc}(x, t) = E_0 e^{-i(\omega t - kx)} \cdot \vec{e}_z$, surrounding by an isotropic, non-absorbing medium with dielectric constant $\varepsilon_d$. The dielectric function of the particle in response to the applied field is described by a complex number $\varepsilon_m(\omega)$. The sketch of the situation is illustrated in Figure 1.12.

The strategy to find $\varepsilon_m(\omega)$ is first solving the Laplace equation in the absence of free
charges for potential $\nabla^2 V = 0$, from which we can calculate the electric field $\vec{E} = -\nabla V$ inside and outside the particle. The boundary conditions at the interface of the particle are also applied for both the potential and normal components of the displacement field. In the following, we will present the main results as the consequences of the quasi-static approximation:

**Electrostatic field of excitation:** the spatial variation of the incident field over the particle volume can be ignored. Therefore, the harmonically oscillating field can be simplified to an electrostatic field $E_{inc}(t) = E_0e^{-i\omega t}$.

**Induce dipole surface plasmon:** free electron cloud of the metallic particle oscillates in response to the applied EM field, yielding an oscillating induced dipole of moment $\vec{p}(t) = \epsilon_0\epsilon_d\alpha\vec{E}_{inc}$ with polarizability $\alpha$ can be given by:

$$ \alpha = 4\pi a^3 \frac{\epsilon_m(\omega) - \epsilon_d}{\epsilon_m(\omega) + 2\epsilon_d} \quad (1.32) $$

We can see that the polarizability $\alpha$ of the particle is resonantly enhanced at the vanishing of $\epsilon_m(\omega) + 2\epsilon_d$. For the case of small $Im[\epsilon_m(\omega)]$, the resonance condition simplifies to $Re[\epsilon_m(\omega)] = -2\epsilon_d$. For particles made from Drude metal with dielectric function negligible damping $Re[\epsilon_m(\omega)] \approx 1 - \omega_p^2/\omega^2$, the value of $\alpha$ can approach maximum at a frequency

$$ \omega_{LSPR} \approx \frac{\omega_p}{\sqrt{\epsilon_m(\omega) + 2\epsilon_d}} \quad (1.33) $$

When the frequency of the applied field is $\omega_{LSPR}$, the electron cloud in the metallic nanoparticle oscillates collectively and results in resonance that enhances the polarizability. This resonance is called the localized surface plasmon that can appear in the optical spectra. We can see that the resonance frequency $\omega_{LSPR}$ depends on the dielectric function of the environment $\epsilon_d$, which makes metallic nanoparticles employed for optical sensing. Besides, $\omega_{LSPR}$ is not affected by the particle's size as a consequence of the quasi-static approximation.

**Field enhancement:** As a consequence of the quasi-static approximation, if we look at the particle from the outside, we could see it as a dipole located in the electrostatic field. Therefore, a total field outside the particle can be interpreted as a superposition of the incident field and the near field of the dipole ($kr \ll 1$), where its electric field is $\sim 1/r^3$.
The electric field outside the particle can be determined from the associated potentials via $\vec{E} = -\nabla V(\vec{r})$ and is formulated as

$$ \vec{E}_{out} = \vec{E}_{0} + \frac{3\vec{e}_{r}(\vec{e}_{r} \cdot \vec{p}) - \vec{p}}{4\pi\varepsilon_{0}\varepsilon_{d}} \cdot \frac{1}{r^{3}} \quad (1.34) $$

Here we see that the resonance in $\alpha$ at $\omega_{LSPR}$ also implies resonant enhancement in the dipolar field. For example, if we consider the $\vec{E}_{out}$ on the surface of a particle $r = a$ and at around the induced dipolar axis (i.e., $\vec{E}_{out}$ is strongest), we can obtain a maximum field enhancement at resonance ($Re[\varepsilon_{m}(\omega)] = -2\varepsilon_{d}$) as

$$ \frac{E_{res}}{E_{0}} \approx 1 + \frac{3Re[\varepsilon_{m}(\omega)]}{Im[\varepsilon_{m}(\omega)]} \quad (1.35) $$

For example, if we take a gold nanoparticle illuminated by $\lambda = 650$nm which has $\varepsilon_{m} = -11 + 1i$ [11], the field at the poles along the axis of dipolar surface plasmon enhances by $\sim 30$ times as compared to the field $E_{0}$ of the surrounding medium. Since the resonance is also associated with the field enhancement, the metallic particle can act as a local refractive index sensor, which is the most sensitive within the near-field region of order 10-50nm.

**Optical cross-section enhancement:** As the particle is approximated to a dipole, its scattered power $P_{sc}$ can be calculated from the Poynting vector associated with its far-field radiation, i.e., retaining only the term $1/r$ of the dipole field [39]. For the absorption, the mean power absorbed by the particle is responsible by the imaginary part of the polarizability $\alpha$ via expression $P_{abs} = \frac{1}{2}E_{0}^{2}\omega\varepsilon_{d}Im(\alpha)$. On the other hand, the particle is subjected to an incident plane wave, which has an average incident irradiance $I_{inc} = \frac{1}{20}\sqrt{\varepsilon_{d}cE_{0}^{2}}$. Recalling the definition of the scattering and absorption cross-sections in equations 1.22 and 1.25, respectively, we obtain

$$ \begin{aligned} \sigma_{sc} &= \frac{P_{sc}}{I_{inc}} = \frac{1}{6\pi}\varepsilon_{d}^{2}(\frac{2\pi}{\lambda})^{4}|\alpha|^{2} \\ \sigma_{abs} &= \frac{P_{abs}}{I_{inc}} = \sqrt{\varepsilon_{d}}(\frac{2\pi}{\lambda})Im(\alpha) \end{aligned} \quad (1.36) $$

Besides, from the scattered far-field of the dipole, we can determine the interference with the incident field, from which the extinction cross-section can be calculated [40], resulting in $\sigma_{ext} \approx \sigma_{abs}$.
The dependence of the optical cross-sections on the polarizability $\alpha$ implies that they exhibit the same resonance condition at LSPR frequency $\omega_{LSPR}$. Noticing that $\alpha$ scales with $a^3$ (equation 1.32) or $V$, with $v$ is the volume of the spherical particle, we can see that $\sigma_{sc} \sim V^2$ and $\sigma_{ext} = \sigma_{abs} \sim V$. When the size of the particle decreases, it is clear that the extinction is dominated by absorption. In the other words, the small plasmonic particles are good absorbers and bad scatters. Besides, the $\sigma_{sc}$ is proportional to $1/\lambda^4$, which corresponds to the Rayleigh scattering.

### 1.4.3 Mie theory

We have shown that the resonantly absorbing and scattering light field of vanishingly small particles is well-predicted by the quasi-static approximation. In response to the EM field that approximates constant throughout the particle, the electrons oscillate collectively respected to the fixed ion, forming a dipolar mode. However, for a particle of dimensions in the order of wavelength, it feels the inhomogeneous spatial variation of the driving field. A more complex surface charge distribution encloses the particle, which can induce multipolar modes. In this situation, the dipolar or quasi-static approximation is no longer valid, and a rigorous electrodynamic treatment is required. Mie in 1908 developed a complete theory that gives an appropriate analytic solution to scattering and absorption problems for a sphere of arbitrary radius embedded in a homogeneous non-absorbing medium.

A detailed description of the Mie theory can be found in the book of Bohren and Huffman [41]. Generally speaking, the theory still follows the main strategy of calculating the optical cross-sections of the object: applying the boundary conditions for fields inside and outside the particle, which are also necessarily satisfied Maxwell's equations. To address the higher-order modes, the field outside the particle, including incident and scattered fields, and the internal field are expanded into a series of the two basic vector functions $\vec{M}$ and $\vec{N}$. These functions have all the required properties of an electromagnetic field in a homogeneous medium: solenoidal fields ($\nabla \cdot \vec{\psi}$) and satisfy the vector Helmholtz wave equation ($\nabla^2 \vec{\psi} + k^2 \vec{\psi} = 0$), where $\vec{\psi}$ stands for electric field $\vec{E}$ and magnetic field $\vec{H}$. Therefore, $\vec{M}$ and $\vec{N}$ are referred to the vector spherical harmonics, which can be constructed from the scalar function $\psi$ in
spherical coordinates as

$$ 
\begin{aligned}
\vec{M} &= \nabla \times (\vec{r}\psi) \\
\vec{N} &= \frac{1}{k}(\nabla \times \vec{M})
\end{aligned}
\qquad (1.37) $$

where $\vec{r}$ is the radius vector. These expressions are necessary to satisfy the vector Helmholtz wave equation and the problem of finding solutions to the field equations can simplify the finding solutions to the scalar wave equation $\psi$. By resolving the scalar Helmholtz equation in the spherical coordinates, we can obtain solutions to a scalar function

$$ 
\begin{aligned}
\psi_{elm} &= P_l^m(\cos\theta)z_l(kr) \cos m\phi \\
\psi_{olm} &= P_l^m(\cos\theta)z_l(kr) \sin m\phi
\end{aligned}
\qquad (1.38) $$

where notations $e$ and $o$ stand for even and odd due to the presence of cosine terms. The $P_l^m(\cos\theta)$ is associated with *Legendre polynomials*. The $z_l(kr)$ are the spherical *Bessel functions* of 1st, 2nd, or 3rd kind, each involves in the expansions of incident, internal, or scattered fields, depending on whether it either converges or diverges at the origin. Indices $m$ and $l$ are integers, $l \ge 1$ and $-l \le m \le +l$, which determine the form of each vector spherical harmonics $\vec{M}_{lm}$ and $\vec{N}_{lm}$. Here, $l$ corresponds to the order of polarity of the different resonant modes, e.g. dipolar ($l=1$), quadrupolar ($l=2$), and $m$ is the azimuth number. The fields now can be developed to an arbitrary order from a linear combination of these basis functions.

Returning to the scattering and absorption problem, while the incident field is already known (e.g. plane wave), the expansions of the scattered field and the internal field contains four unknown coefficients. These expansion coefficients can be found by applying the boundary conditions at the interface of the sphere: the tangential components of the vector fields across the surface boundaries are preserved. Finally, the extinction and scattering cross-section can be derived by calculating the flux of the Poynting vector over an imaginary surface enclosing the particle. The extinction cross-section is expressed as

$$ 
\begin{aligned}
\sigma_{ext} &= \frac{1}{|\langle \vec{S}_{inc} \rangle|} \iint_A \frac{1}{2} \mathrm{Re}(\vec{E}_{inc} \times \vec{H}_{sc}^* + \vec{E}_{sc} \times \vec{H}_{inc}^*).\vec{n}dA \\
\sigma_{ext} &= \frac{2\pi}{k^2} \sum_{l=1}^{\infty} (2l + 1) \mathrm{Re}(a_l + b_l)
\end{aligned}
\qquad (1.39) $$
and scattering cross-section is given by

$$ \sigma_{sc} = \frac{1}{|\langle \vec{S}_{inc} \rangle|} \iint_A \langle \vec{S}_{sc} \rangle \cdot \vec{n} dA \qquad (1.40) \\ \sigma_{sc} = \frac{2\pi}{k^2} \sum_{l=1}^{\infty} (2l + 1)(|a_l|^2 + |b_l|^2) $$

Here, the $a_l$ and $b_l$ are called Mie coefficients (the expressions can be found in [41]). The cross-section is given by the sum, this means that the energy scatters through independent modes corresponding to $a_l$ and $b_l$, with $l$ defining the order of the polarity of the modes. When the particle size is much smaller than the wavelength, the contribution of all modes beyond the lowest order may be negligible. The cross-sections remain only dipolar term, which recovers the quasi-static approximation with corrections.

<figure><img src="image_15.png" /><figcaption>Figure 1.13: Optical cross-sections of silver nanoparticles of various radii $\phi$ in water calculated by Mie theory. Extinction is presented bold black line, absorption is presented in grey line, and scattering is shown in thin black line. Here, D labels for dipolar mode and Q labels for quadrupolar mode. Adapted from [9].</figcaption></figure>

For example, the Figure 1.13 presents the results of the Mie theory for 4 silver spheres with radii of 15nm, 30nm, 60nm, and 120 nm, surrounded by water (n = 1.33) [9]. We can see that for small sizes, extinction is dominated by absorption and exhibits only the dipolar
mode. When the radius increases, the scattering contributes significantly to the extinction and the resonance is very different from the one expected in the quasi-static approximation: resonance of dipolar mode is red-shifted and quadrupolar mode appears at a lower wavelength.

The Mie theory can be applied to spherical core/shell structure and extended to several spherical nanoparticles in an interaction excited by an incident plane wave. For particles with shapes that cannot be approximated by spheres or ellipsoids, numerical methods are employed to calculate the optical cross-sections.
# Chapter 2

## Coupling a dipole to multilayer surface plasmon devices

Hybridization of surface plasmon in the multilayer structures allows for enhancing the performance of single-surface plasmons, such as extending the propagation length to around 1 order of magnitude or deep-subwavelength confinement of light. In this Chapter, we investigate numerically the surface plasmon hybridized inside the planar multilayer structure by employing a dipole as a probe. The first section will be dedicated to presenting how we can use the classical approach to study the spontaneous emission of a dipole. Then by calculating the spatial spectrum of the power dissipated by a buried inside the plasmonic structure, we can identify and construct the optical modes that are supported by the structure. Moreover, figures of merit were deduced, including the Purcell factor and coupling efficiency to each identified mode. We mainly focus our study on two multilayer structures: (i) a realistic insulator/metal/insulator (IMI) device and (ii) alternative N-stacks of metallic and insulator thin films. For the IMI device, we found that the structure can support many modes including the long-range surface plasmon of a few ten µm propagation length and waveguide modes. The coupling efficiency to the modes can be controlled by the position and orientation of the dipole. For the N-stack device, we show that when N is large, the plasmon modes evolve toward an energy continuum in a manner similar to the energy bands in a solid.
# Contents

<table><tr><td>2.1 Spontaneous emission of an emitter</td><td>46</td></tr><tr><td>2.2 Spontaneous emission of a dipole near planar interfaces</td><td>51</td></tr><tr><td>2.3 Coupling a dipole to fundamental planar plasmonic structures</td><td>53</td></tr><tr><td>2.4 Coupling a single dipole to a long-range surface plasmon device</td><td>59</td></tr><tr><td>2.5 Surface plasmon hybridized dressed states</td><td>64</td></tr></table>

## 2.1 Spontaneous emission of an emitter

### Quantum description: a two-level system

Proper treatment of the interaction of light and matter (molecules or other emitters) requires rigorous theoretical treatment of quantum electrodynamics (QED), in which both electronic states of an emitter and electromagnetic field are quantized. An emitter is described as a two-level system: ground state $|g\rangle$ of energy $E_g$ and excited state $|e\rangle$ of energy $E_e$. Interacting with a quantized radiation field promotes electrons from the ground state to the excited state, subsequently relaxing into the fundamental level after a certain time, accompanied by the emission of a photon of energy $\hbar\omega_0 = E_e - E_g$. The spontaneous emission is thus defined as the emission of a photon by a system previously brought to an excited state and occurs independently of any external field. This process is schematized in Figure 2.1a.

As the relaxation is spontaneous, the probability that the recombination takes place is an exponential decrease of time $e^{-\Gamma t}$, with $\Gamma$ being the decay rate of spontaneous emission; which is inverse proportional to the lifetime of the excited state of the emitter. Before Purcell's
<figure><img src="image_16.png" /><figcaption>Figure 2.1: **Spontaneous emission: Quantum description.** An emitter is described as a two-level system: (a) the relaxation in a homogeneous medium and (b) coupling to local environment defined by its local density of state $\rho(\omega_0)$.</figcaption></figure>

analysis in 1946 [42], the rate of spontaneous emission was thought to be the unchangeable property of atoms or emitters, which is expressed in vacuum as [43]:

$$ \Gamma_0^q = \frac{\omega_0^3 \mu_{ge}^2}{3\pi_0 \hbar c^3} \qquad (2.1) $$

where $\hbar\omega_0$ is the difference between ground and excited energies and $\mu_{ge}$ is the dipole matrix element between the two optical transition levels.

It has been realized later that the spontaneous emission can be modified by interaction between the emitter and its electromagnetic environment. The modified rate is usually expressed by Fermi's golden rule [44] and is rewritten as:

$$ \Gamma^q = \frac{2\omega_0}{3\hbar\mu_{ge}^2} \rho(\omega_0) \qquad (2.2) $$

Here, the decay rate depends on two factors: the internal structure of the emitter represented by the transition dipole moment $\mu_{ge}$ and the electromagnetic environment, which is described by the local density of states (LDOS) $\rho(\omega_0)$ at frequency $\omega_0$. The LDOS is the number of electromagnetic modes, per unit frequency and per unit volume. In other words, the presence of the local environment opens new channels that are available for the emitter to radiate into, and thus modify the spontaneous emission rate through its LDOS (Figure 2.1b). The idea of enhancement of spontaneous emission was first proposed by Purcell [42], through coupling an atom to a cavity. The ratio between the modified and free-space emission rates
has thus come to be known as the Purcell factor:

$$F_P = \frac{\Gamma^q}{\Gamma_0^q} \qquad (2.3)$$

Classical treatment: a harmonic oscillator

The classical picture of spontaneous emission can be described by treating an emitter as a harmonically oscillating dipole. In terms of energy, a harmonic oscillator can be equivalent to a system that has an infinite number of equidistant levels. As the dipole oscillates, it dissipates its energy (determined by integrating the Poynting vector over a surface enclosing the emitter) into electromagnetic radiation. The spontaneous emission rate is now equivalent to the damping rate of the harmonic oscillator, which is defined as the rate at which the energy decayed to 1/e of its initial value. In an arbitrary environment, the classical spontaneous emission rate can be found by resolving the equation of motion:

$$\frac{\partial^2 \vec{\mu}}{\partial t^2} + \Gamma_0^{cl} \frac{\partial \vec{\mu}}{\partial t} + \omega_0^2 \vec{\mu} = \frac{e^2}{m} \vec{E}_{loc}(\vec{r}_0) \qquad (2.4)$$

where $e$ and $m$ respectively are the charge and mass of electron, $\vec{\mu}$ is the dipolar moment of the dipolar emitter [39]. The term on the right of the equation is related to the driving force $\vec{F} = -e\vec{E}_{loc}(\vec{r}_0)$, and induced the local field $\vec{E}_{loc}(\vec{r}_0)$ that arrives back to the dipole (at position $\vec{r}_0$) after being scattered by the inhomogeneous environment. For example, this field can be reflecting field from an interface if the emitter is placed in proximity to this interface.

In the free space, this backscattered field is absent and the equation is now the equation of motion for an undriven harmonic oscillator, oscillating at the natural frequency $\omega_0$ with a damping rate $\Gamma_0^{cl}$. The spontaneous emission rate of the emitter in the free space is now equal to $\Gamma_0^{cl}$, which is given by:

$$\Gamma_0^{cl} = \frac{e^2 \omega_0^2}{6\pi_0 m c^3} \qquad (2.5)$$

With the presence of an inhomogeneous environment, both the emission rate $\Gamma_0^{cl}$ and the oscillating frequency $\omega_0$ are modified by self-interacting with its backscattered field. These can be found (as we will present in detail later) by resolving equation 2.4.
<figure><img src="image_17.png" /><figcaption>Figure 2.2: **Spontaneous emission: Classical approach.** An emitter is described as a harmonic oscillator of the infinite equidistant energy levels. The damping rate can be modified by the homogeneous medium induced the backscattered field of the medium.</figcaption></figure>

## Connecting both pictures

We stress that the decay rate calculated from the harmonic oscillator system is not as rigorously accurate as the one obtained by quantum treatment. Yet, it is possible to account for the spontaneous emission modification of the decay rate $\Gamma^{cl}/\Gamma_0^{cl}$, as we will show later, this gives the same result as the Purcell factor determined by the quantum treatment, under certain conditions. So the question is in which conditions that the classical theory still works properly for the calculation of normalized decay rate? This happens in the situation, in which an emitter is a two-level system, and reacts as a harmonic oscillator. When the two-level system is driven, some electrons distributed at the ground state will be promoted in the excited state. This happens similarly for the harmonic oscillator, except for after being excited, the electrons can immediately go to above-excited states, as the harmonic system contains an infinite number of equidistant energy levels. Thus, the two-level system and the harmonic oscillator could be identical at a weak excitation, from which we can neglect the excitation in higher levels of the harmonic oscillator. This requires the interaction with $\vec{E}_{loc}(\vec{r}_0)$ to be weak, resulting in the shift of frequency being small compared to $\omega_0$
and both terms of $\vec{\mu}$ and $\vec{E}_{loc}(\vec{r}_0)$ have a harmonic time dependence $e^{-i\omega t}$, with is new resonance frequency. Replacing these trial solutions for dipole moment and the driving field into equation 2.4, we obtain:

$$ -\omega^2 - i\Gamma_0\omega + \omega_0^2 = \frac{e^2}{m\mu_0}E_{r,loc} \quad (2.6) $$

where $\mu_0$ is the amplitude of dipolar moment $\vec{\mu}$ and $E_{r,loc}$ is the complex amplitude of the component field along dipole axis $\vec{u}_r$ of $\vec{E}_{loc}(\vec{r}_0)$. The solution of $\omega$ in equation 2.6 is a complex number represented as $\omega = Re(\omega) + Im(\omega)$, in which the real part defines the frequency shift as $\Delta\omega = Re(\omega) - \omega_0$ and the imaginary part determines the total decay rate as $\Gamma = -2Im(\omega)$. As the excitation is weak, the frequency shift should be small compared to its natural frequency: $\Delta\omega \ll \omega_0$. Besides, we assume that the dipolar oscillator has $\Gamma \ll \omega$ (i.e. the oscillation amplitude remains unchanged over period of oscillation), so that later the quadratic terms of $\Gamma^{cl}$ can be neglected. Applying these assumptions into equation 2.6, we obtain the solution:

$$ \begin{aligned} \Delta\omega &= -\frac{\Gamma_0^2}{8\omega_0} - \frac{1}{2}\frac{e^2}{\mu_0\omega_0 m}Re(E_{r,loc}) \\ \Gamma^{cl} &= \Gamma_0^{cl} + \frac{e^2}{\mu_0\omega_0 m}Im(E_{r,loc}) \end{aligned} \quad (2.7) $$

These two solution means that the radiation of a dipole in an inhomogeneous medium leads to the modification of the total decay rate and the shift of resonance frequency, which are respectively responsible for the imaginary and real parts of the backscattered field. Now we can determine the normalized spontaneous emission rate classically as:

$$ \frac{\Gamma^{cl}}{\Gamma_0^{cl}} = 1 + \frac{6\pi_0 c^3}{\mu_0 \omega_0^3} Im(E_{r,loc}) \quad (2.8) $$

Following [39], the scattered field induces by dipole, at the location of the dipole itself $\vec{r} = \vec{r}_0$, can be represented by the Green's function as $E_{r,loc}(\vec{r}_0) = G(\vec{r}_0, \vec{r}_0)$ (projected along the dipole axis). On other hand, the LDOS can be calculated from the imaginary part of the Green's function: $\rho(\omega_0) \propto Im[G(\vec{r}_0, \vec{r}_0)]$, from which the modification of spontaneous emission rate calculated by the quantum treatment (i.e., Fermi's golden rule) and the classical
self-interaction are equivalent:

$$F_P = \frac{\Gamma^q}{\Gamma_0^q} = \frac{\Gamma^{cl}}{\Gamma_0^{cl}} \qquad (2.9)$$

The derivation of the connection between the LDOS and the imaginary part of Green's function can be found in the book [39] of Lukas Novotny and Bert Hecht. In this chapter, now we can use the classical treatment for spontaneous emission of emitter, that latter will be deployed for studying the plasmonic structures.

## 2.2 Spontaneous emission of a dipole near planar interfaces

The expression of the Purcell factor $F_P = \frac{\Gamma^q}{\Gamma_0^q} = \frac{\Gamma^{cl}}{\Gamma_0^{cl}}$ suggests that we can use the classical approach to study the dynamics of spontaneous emission of a dipole. To serve the purpose of studying the planar plasmonic structure, we first study the spontaneous emission of a dipole near planar interfaces. Lukosz [45, 46, 47] first determined an analytical expression of $\frac{\Gamma^{cl}}{\Gamma_0^{cl}}$ for a dipole in the vicinity of interfaces by a classical near-field calculation. The near-field emission of the dipole is displayed as a superposition of plane waves and evanescent waves of varying in-plane wavevector $k_\|$. As the decay rate of the dipole is proportional to the total power it radiates

$$P_{tot} = -\frac{dW_{dipole}}{dt} = -\Gamma W_{dipole} \qquad (2.10)$$

where $W_{dipole}$ is the dipole's energy, the calculation of $\Gamma^{cl}/\Gamma_0^{cl}$ (or the Purcell factor) is equivalent to the calculation of the normalized total power $P/P_0$ radiated by the dipole (where $P_0$ is the power radiated by the dipole in vacuum). To effectively excite the surface plasmon modes that will be discussed later, we employ a dipole that is perpendicular to the interface. Following [46], we express the normalized dissipated power as:

$$(\frac{P_{tot}}{P_0})_\perp = \frac{3}{2} Re(\int_0^\infty (1+c^p) \frac{u^3}{\sqrt{1-u^2}} du) \qquad (2.11)$$

where $u = \frac{k_\|}{k_0} = \sqrt{1 - (k_{z,i}/k_0)^2}$ is the normalized in-plane wavevector, with $k_0 = \omega/c$ is the wavevector in vacuum and $k_{z,i}$ represents the vertical component of the wavevector in
the medium $i$ where the dipole is located.

The expression 2.11 can be interpreted as the interference of the field of the dipole in an infinite medium and the reflected field of the nearby interfaces. The latter is represented by the complex field amplitude $c^p$, which is determined by the boundary conditions for p-polarized fields. Considering two distinguished situations: a dipole near a semi-infinite layer ($c_{\downarrow}^p$, Figure 2.3a) or a dipole is buried in a layered medium consisting of multilayer ($c_{\downarrow\uparrow}^p$, Figure 2.3b); the complex field amplitudes are respectively given by:

$$c_{\downarrow}^p = r_{i,\downarrow}^p(u) \cdot e^{2ik_{z,i}d_{\downarrow}} \qquad (2.12)$$

and

$$c_{\downarrow\uparrow}^p = \frac{2r_{i,\downarrow}^p(u)r_{i,\uparrow}^p(u)e^{2ik_{z,i}(d_{\downarrow}+d_{\uparrow})} + r_{i,\uparrow}^p e^{2ik_{z,i}d_{\uparrow}} + r_{i,\downarrow}^p e^{2ik_{z,i}d_{\downarrow}}}{1 - r_{i,\downarrow}^p(u)r_{i,\uparrow}^p(u)e^{2ik_{z,i}(d_{\downarrow}+d_{\uparrow})}} \qquad (2.13)$$

<figure><img src="image_18.png" /><figcaption>Figure 2.3: **Two situation of a dipole near interfaces.** Radiating of a dipole near (a) a single interface: direct radiation of dipole can interfere with its reflection on the interface and (b) in between two interfaces: multiple interactions are induced by the two interfaces.</figcaption></figure>

The indices $\downarrow, \uparrow$ respectively, denote upward and downward media. For multilayer structure, $r_{i,\downarrow}^p$ and $r_{i,\uparrow}^p$ correspond to the Fresnel reflection (p-polarization) coefficients calculated by the transfer matrix (see Ph.D. these of Kelvin [48]). The quantities $d_{\downarrow}$ and $d_{\uparrow}$ are the distance from the dipole to the below interface and upper interface. The geometry is schematized in Figure 2.3. In the case of a dipole close to the planar interface (Figure 2.3a), it only feels the reflection of the surface. For a dipole embedded in a layered medium (Figure 2.3b), the fields emitted by the dipole are initially reflected by the interfaces $i, \uparrow$ and $i, \downarrow$ and are then reflected back and forth between the two interfaces. This leads to the multiple beam-interference factors appearing as the denominator in the expression 2.13 of $c_{\downarrow,\uparrow}^p$.
The integral in the expression 2.11 represents the normalized power dissipated by the dipole in the directions of the optical modes whose component in-plane wavevector is $u$. The values of $u < 1$ correspond to radiative emission, and the values $u > 1$ are associated with near-field emission. By separating the integral spatially in different domains of $u$, the optical modes of the system can be identified. We, therefore, employ a single vertical dipole as a local probe to study plasmonic structures. In the following, from the calculation of the dissipated power presented above, we extract three important physical quantities for the characterization of the plasmonic system:

* the spatial distribution of the dissipated power $\alpha_{\perp}$ that allows identifying the contribution of the component optical modes,

* the coupling efficiency to the modes $\beta^m$,

* the Purcell factor $F_P$.

## 2.3 Coupling a dipole to fundamental planar plasmonic structures

The layered structure with subsequent lateral patterning are based around three fundamental structures: (i) insulator/ metal (IM), (ii) insulator/metal/insulator (IMI), or (iii) metal/insulator/metal (MIM). In this Chapter, we first employ a dipole to investigate the optical modes supported by these planar plasmonic structures. Figure 2.4 illustrates the schematic of each structure. For calculation, the insulator and the metallic layers are assumed to be air and gold ($n_{Au} = 0.15 + 3.32i$ at $\lambda = 650$nm [11]), respectively. To effectively excite the optical modes of the surface [45], we place a vertical dipole emitting at $\lambda = 650$nm in the air layer at a distance of $d_{\downarrow} = 15$nm. The gap in the IMI and MIM structures has an equal thickness $g = 2d_{\downarrow} = 30$nm. We stress that the main goal of this work is to probe in a given configuration the Purcell factor and the optical modes, the choice of the configuration's parameters is not optimal.
<figure><img src="image_19.png" /><figcaption>Figure 2.4: Coupling a dipole to surface plasmon. (a) single insulator/dielectric (IM) interface. (b) insulator-metal-insulator (IMI) structure. (c) metal-insulator-metal (MIM) structure. Second row: Calculation of spatial spectrum of the power dissipated by the dipole. Third row: Calculation of coupling efficiency to the modes as a function of the dipole’s distance (IM and IMI) and decay rate enhancement of the MIM modes as a function of the gap thickness.</figcaption></figure>

### Identification of optical modes supported by the structure

To determine the optical modes supported by the structure, we calculate the spatial spectrum of the power dissipated by the dipole (i.e., the dissipated power as a function of the normalized in-plane wavevector $u$), which is given by:

$$ \alpha_{\perp}(u) = \frac{d\left(\frac{P_{tot}}{P_0}\right)_{\perp}}{du} \qquad (2.14) $$

The calculation of $\alpha_{\perp}$ is presented in Figure 2.4 (second row). The presence of the surface plasmon in each structure is filled in red, which superimposes to a broad “background” containing both values of $u$ less than 1 (far field) and greater than 1 (near field). For
$0 < u < 1$ (white region), the processes associated with photon radiation the free-space. The mode corresponds to values of $u$ above surface plasmon modes (grey region) referred to as lossy surface waves (LSWs) or quenching. This transfers the dissipated power of the dipole to the electron-hole pairs of metal.

For the IM structure, we observe a peak at $u = 1.1$ (red region), which is associated with the wavevector of SPPs propagating along the air/ gold interface ($k_{SPP}/k_0 = 1.1$). In the case of IMI structure, the spatial spectrum of $\alpha_{\perp}$ features two distinctive peaks in the near-field regime ($u > 1$), which results from the SPP the hybridization of SPPs bound to each interface. The mode at smaller $u$ has a narrower width (which means fewer losses or longer propagation length) and is called long-range surface plasmon (LRSP). The other one holds an opposite feature and is referred to as short-range surface plasmon (SRSP). In terms of the field, these two modes are also usually referred symmetric (LRSP) and anti-symmetric (SRSP) gap modes. The mode hybridization in the IMI structure is very sensitive to the surrounding dielectric optical indices. If the difference between the optical index of the upper and lower insulators are very different (10%), the SPPs of each interface will be decoupled, and only the independent SPPs will remain [49]. In the MIM structure, the SPPs on either side of the insulator medium similarly hybridize to form symmetric gap mode (the anti-symmetric mode is cut-off at roughly $\lambda/2n$ [38, 50]), which features as a monomode with a broad width on the spatial spectrum of $\alpha_{\perp}$.

For comparison with the two other structures, the gap MIM has the largest $k_{\|}$ and spectra full-width at half-maximum (FWHM), which means the strongest field confinement. However, the confinement of the gap MIM is accompanied by short propagation length. The propagation length of the mode is determined by the imaginary part of the $k_{\|}$ as $L = 1/(2Im(k_{\|}))$, which in turn is related to the FWHM of the plasmon peaks as $Im(k_{\|}) = k_0 \cdot \text{FWHM}$. From this, we deduce the propagation lengths of 9µm (SPPs), 2µm (SRSP), 44µm (LRSP), and 10nm (gap MIM).

## Coupling efficiency to the mode

In proximity to the metallic surface, the emission process of a dipole is mainly a competition between the surface plasmon modes and quenching (LSWs). We define the $\beta$-factor, which gives the fraction of the total dissipated power funneled into one mode by integrating
the spatial spectrum over the spatial region yields to the mode. It can be represented as

$$ \beta = \frac{P_m}{P_{tot}} = \frac{\Gamma_m}{\Gamma_{tot}} = \frac{\int_{u \in modes} \alpha_{\perp}(u) du}{\int_{0}^{+\infty} \alpha_{\perp}(u) du} \quad (2.15) $$

This fraction is equivalent to the normalized decay rate of dipole emission into the mode. Figure 2.4 (third row) shows the β-factor of surface plasmon modes and quenching calculated as a function of the normalized separation $d_{\downarrow}/\lambda$ (for IM and IMI) or the insulator gap thickness $g$ (for MIM). When the $g$ changes, we note that the dipole is always placed in the middle of the gap as $d_{\downarrow} = g/2$.

In the IM and IMI structures, there are optimal distances yield to a maximum of β-factor of plasmon modes: $\beta_{SPP}^{max} = 80\%$ at $d_{\downarrow} = 0.3\lambda$, $\beta_{SRSP}^{max} = 55\%$ at $d_{\downarrow} = 0.25\lambda$, and $\beta_{LRSP}^{max} = 55\%$ at $d_{\downarrow} = 0.5\lambda$. At these optimal distances, the plasmon modes largely overcome the quenching (blue curve, Figure 2.4). Above this distance, the radiative is the dominant channel. The quenching tends to be localized near the metallic surface, which has the coupling efficiency scales as $d_{\downarrow}^{-3}$ as $d_{\downarrow} \to 0$. This scaling is understood from the local and static nature of the quenching. Below 10nm, the quenching is the dominant decay channel of the dipole.

For the MIM structure, we prefer to present the normalized decay rate of the modes $\Gamma_m/\Gamma_0$ rather than their coupling efficiency because the structure does not exhibit an optimal distance where the coupling efficiency to the mode reach maximum, as seen in the IM and IMI structures. Indeed, the gap mode and the quenching have identical separation distance $d_{\downarrow}^{-3}$ scaling as $d_{\downarrow}$ vanished (Figure 2.4). We note that the dissipated power into quenching of MIM gap (or quenching rate) is ~2-3 times higher than that of the dipole near a single interface of IM or IMI structure. Interestingly, we find that the coupling to plasmon gap mode overcomes the quenching despite the very proximity to the metal ($d_{\downarrow} < 10$nm). This is attributed to the slowdown of the gap plasmon mode (group velocity decrease) and thus strong field confinement when the gap thickness is reduced [51, 52].

## The Purcell factor

The Purcell factor defines the total decay rate enhancement (as compared to vacuum decay rate $\Gamma_0$) that can be calculated by integrating the spatial spectrum of $\alpha_{\perp}$ over all values of $u$,
$$F_P = \frac{\Gamma}{\Gamma_0} = \int_0^{+\infty} \alpha_{\perp}(u) du \qquad (2.16)$$

<figure><img src="image_20.png" /><figcaption>Figure 2.5: **The Purcell factor.** (a) Purcell factor of the dipole in the IM and IMI structure as a function of the dipole distance. Two gap thicknesses of the IMI structure are considered: $g_M = 3$nm and $g_M = 30$nm. (b) Similar to (a) but focusing on smaller distance and compare to the MIM structure. Only the IMI of $g_M = 3$nm is considered.</figcaption></figure>

Figure 2.5 a shows the calculation of $F_P$ as a function of the normalized separation distance $d_{\downarrow}/\lambda$ of the IM and IMI structure. We can see that the Purcell factor of the IMI with the gold slab thickness $g_M = 30$nm (blue line) is nearly identical to that of the single air/gold interface (black line). When the dipole is far from the interface, the $F_P$ slightly oscillates around unity ($\Gamma = \Gamma_0$) due to the interference between the far-field radiation of the dipole and its reflected field from the surface. As $d_{\downarrow} < 10$nm, the Purcell factor of these two structures increases rapidly; however, this is dominantly due to the quenching as shown in Figure 2.4.

In Figure 2.5 b, we also calculate the Purcell factor of the MIM structure as a function of gold thickness $g = 2d_{\downarrow}$, focusing on the thicknesses of interest $g < 20$nm. We note that dielectric gap thickness at a 10nm scale is possible to achieve by state-of-art lithography or self-assembled monolayer of molecules ($g < 2$nm) [53, 54]. Here we find that the $F_P$ is also significantly enhanced as the gap thickness reduces ($F_P \approx 10^5$ at $g = 2$nm) and is more prominent by around 2 orders of magnitude than that of IM or IMI ($g_M = 30$nm) structures. Moreover, as proved previously, the large $F_P$ of the MIM structure is predominantly driven by
the gap plasmon mode but not the quenching, even at very small $d_{\downarrow}$. It is thus apparent that the MIM nanogaps can provide extreme light confinement inside the dielectric gap that its excitation can compete with the quenching, thus propelling their use in various applications related to enhancing spontaneous emissions such as single photon sources or sensing.

### The IMI structure with extremely thin metallic gap thickness

Above we have considered the IMI structure with gold thickness $g_M = 30$nm, which shows no significant difference in Purcell factor as compared to a single IM interface. In fact, above the gold's skin depth ($> 20$nm), the $F_P$ of the IMI structure is not understood to depend on the gap thickness, except that the plasmon mode hybridization is loose. Here, we are interested in calculating the Purcell factor of an IMI structure with a very small thickness, $g_M = 3$nm for example (the thinnest metal sheets have reached), shown as the blue dashed line in Figure 2.5. This geometry has been recently studied in graphene plasmonics [55] or nanoparticle-on-foil system [56], which allow tightly confined light and can be used as molecular sensing. Here we find that in the far-field regime ($d_{\downarrow}$ is large, Figure 2.5a), the $F_P$ is nearly unaffected by the presence of very thin gold film thickness, which shows no significant difference from that of a dipole in a vacuum. In contrast, in the near-field regime (Figure 2.5b), the $F_P$ of 3nm-gap-IMI is largely enhanced by 2 orders of magnitude compared to 30nm-gap-IMI, which is an asymptote to the huge $F_P$ of the MIM structure for $10 < d_{\downarrow} < 20$nm. This is primarily contributed by the strong field confinement of the as-symmetry IMI mode (SRSP) as its field is null inside the metal gap [56]. Besides, in terms of the in-plane wavevector ($k_{\|}$), when $g_M$ reduces, $k_{\|}$ of LRSP mode remains unchanged and very close to the $k_{\|}$ of SPPs on a single gold interface. In contrast, $k_{\|}$ of SRSP largely increases by $\sim 10$ times, making it possible to be confined inside the extremely thin gold gap. When $d_{\downarrow} < 10$nm, the Purcell factor curve of the 3nm-gold-IMI structure converges toward the Purcell factor curve of IM, and the 30nm-gold-IMI structure owing to the dominance of the quenching in this regime.
## 2.4 Coupling a single dipole to a long-range surface plasmon device

Studying the plasmon modes hybridized inside the IMI structure above reveals several promising advantages of the long-range surface plasmon modes: very few losses compared to the SPPs at the single interface as long as millimeter range propagation length can be reached [37], and the tolerance on the structure symmetry is a few percent (achievable experiment). The LRSPs are thus the potential for many applications and have been proposed for optoelectronic devices [57], energy transfer [58], and biological sensors [59]. In a recent paper, we experimentally demonstrated a strong coupling between the LRSP and J-aggregate dyes [37]. We demonstrated that this device enhances the coupling and, depending on the position of the dyes, new modes can be excited.

The motivation of this section is to present a new study of the “more realistic” LRSP device [60, 37] in the weak coupling regime. Using the same dipolar approach above, we focus on the identification and reconstruction of the different modes supported by the device. Besides, the coupling efficiency to the mode is extracted depending on the dipole’s position and orientation. Finally, we present an optimization procedure as a guideline for future optoelectronic devices.

The device is presented in Figure 2.6a. We set the same gold slab of 30nm between two finite glass layers denominated substrate and superstrate (index 1.5, thicknesses 750nm) and we place the whole structure on a prism of index 1.8. The dipole is placed at $d_{\downarrow} = 50$nm from the gold layer inside the superstrate. This configuration is also known as Otto or leakage configuration [29, 61]. The chosen parameters here serve the purpose of demonstrating the system, while their influence will be discussed later.

### Identification and reconstruction profile of optical modes supported by the device

Figure 2.6b shows the spatial spectrum of $\alpha_{\perp}$ of the dipole as a function of u. We can see that the device supports the plasmonic modes (LRSP and SRSP) located at $u > 1.5$. Furthermore, new modes appear for $u \in [1; 1.5]$, which belongs to the glass side of index 1.5. The inset presents the Purcell factor as a function of the distance from the gold layer. No difference is
<figure><img src="image_21.png" /><figcaption>Figure 2.6: **A realistic long-range surface plasmon configuration.** (a) Schematic of the structure. (b) Plot $\alpha_{\perp}$ as function of the in-plane wavevector $u$. Inset: Corresponding Purcell factor.</figcaption></figure>

observed with the case of IMI structure in Figure 2.5a in the near-field regime ($d_{\downarrow} = 50$nm) but a new structure appears in the far-field distance (red arrow). These new modes are thus propagative, belong to the glass, and will be attributed to wave-guided-modes (WGMs). In Figure 2.7, we present a procedure to identify, optimize and reconstruct each mode’s profile.

The identification of the optical modes is done by comparing the intensity reflection coefficient ($R_p, R_s$) as function of the normalized in-plane wavevector $u$ in p and s polarizations with the calculated $\alpha_{\perp}$. These latter three curves are plotted in Figure 2.7a. It is important to note that for reflectivity calculations, the wavevectors are limited by the index of the prism ($u < 1.8$). We see that the modes in the $\alpha_{\perp}$ curve also appear in $R_p$ at the same positions, resulting from the vertical dipole near an interface mostly radiating p-polarized field inside a higher index medium [39]. On the other hand, if we set a horizontal dipole, the dissipated power would have been distributed in both p and s modes (the role of the dipole orientation will be examined at the end of this section). The mode at $\alpha_{\perp}$ ($u = 1.6$) appears only in p-polarization, while it is absent in s-polarization. This dependence on the polarization confirms that this mode has plasmonic nature, which we attribute to LSRP mode. Moreover, we observe that the SRSP is located at $\alpha_{\perp}$ ($u > 1.6$), thus making it impossible to detect in this configuration. For $1 < u < 1.5$, many other modes are present in both polarizations. These modes are attributed to wave-guided modes (WGMs).

In Figure 2.7b, we calculated the coupling efficiency $\beta$ of the vertical dipole within the
<figure><img src="image_22.png" /><figcaption>Figure 2.7: (a) Dissipated power and reflectivity simulated as a function of $u$. (b) Coupling efficiency to the different modes as a function of the vertical distance in the glass. (c) Mode profiles $|\vec{e}_n(r_0)|^2$.</figcaption></figure>

superstrate as a function of the separation distance to the gold. First, we note that the coupling efficiency of the LRSP and the SRSP does not change in this device as compared to the ideal structure IMI with insulator sides made from glass. The finite size of the structure (750nm of glass thickness) does not disturb these modes. Secondly, we see that the coupling efficiency to the guided modes can be: $\beta_{WGM}^{max} = 40\%$, which is obtained deeply inside the glass layer. We have one and two maxima for WGM1 and WGM2 respectively which seem to reproduce the spatial shape of the modes. One important feature is that the number of modes depends on the position of the dipole inside the glass slab. For example, for a dipole position at $d_{\downarrow} = 0.3d_{sup}$, three modes are excited with 25% of total power dissipated to WGM2, 15% to the WGM1, and 15% to LRSP.

To complete the description of the modes and to display the energy propagation within the structure, we reconstruct the mode profiles. It has been shown that in the case of discrete
modes, the LDOS can be formulated through a summation. This calculation is then weighted by the amplitude of the eigenmodes at the dipole position $r_0$ as [62]:

$$ \rho(r_0, \omega) = \sum_n |\vec{e}_n(r_0)|^2 \delta(\omega = \omega_n) \qquad (2.17) $$

We have reconstructed the mode profile $|\vec{e}_n(r_0)|^2$ by first selecting the mode wavevector $u$ and then moving the position of the dipole. Figure 2.7c presents the profile structure of the four modes independently. We observe the profile shape of the plasmon with an exponential decrease in the dielectric surrounding the gold. In the case of the WGM, we observe the number of nodes, in the upper glass, according to the order of the mode. We notice that the WGM profiles are not completely symmetric in the superstrate because of the as-symmetry in terms of the optical index of the surrounding medium (i.e., the device around the gold film).

## Optimization of coupling efficiency

Until now, only a vertical dipole has been set to clarify the discussion and to optimize the coupling to plasmonic modes. Using complementary reflectivity calculations, we have identified all the modes. In Figure 2.7a, we also clearly see that the waveguided modes also occur for s-polarization which may be excited with a horizontal dipole. Depending on the purpose of various applications, it may be more interesting to mostly couple to the LRSP or to both LRSP and WGMs in a defined proportion.

For all these reasons, we introduce a new contrast parameter $C$ between the LRSP and the WGM. This factor of merit highlights the tolerance and optimization of the coupling within the desired mode.

$$ C = \frac{\beta_{LRSP} - \beta_{WGM}}{\beta_{LRSP} + \beta_{WGM}} \qquad (2.18) $$

The coupling efficiencies for WGMs are integrated for $u \in [1; 1.5]$. Figure 2.8 displays the behavior of $C$ with the most important parameters which are the gold distance and the dipole orientation. We retrieve the previous result which is a better coupling to the LRSP with a vertical dipole at short distances. In this case, the range of angle tolerance is around 20°. A horizontal dipole ($\theta = 90°$) will barely couple to the LRSP (only at very short distances).
<figure><img src="image_23.png" /><figcaption>Figure 2.8: Contrast map as function of the vertical distance $d_↓$ in the superstrate and with the dipole orientation $θ$. The dash lines point-out the delimitation between each region. Inset: schematic of the dipole orientation.</figcaption></figure>

There is an intermediate range (for $C = 0$), where the dipole will couple to each mode with equivalent efficiency, this region is marked with the white dash. This calculation is only valid for one dipole. In the case of a monolayer of random independent emitters, we have to average our calculation over all orientations $θ$.

**Conclusion:** To conclude this section, we have investigated the dipole coupling into a Long-Range plasmonic device. In a symmetric configuration where a gold slab is inserted in-between two identical mediums, the dipole may radiate into a Long-Range plasmon. The dissipated power is redistributed into the Long Range as well as in a Short Range plasmon. The coupling efficiency to the LRSP is around 57 and the propagation length is around 44m for wavelength of 650nm. In an even more realistic configuration, where the mediums have finite sizes, waveguided modes may also appear in the dielectric layers without modifying the properties of the LRSP. According to the orientation of the dipole, the coupling efficiency may be optimized and balanced between the LRSP and the WGM. One interesting feature is that these channels may be independent (in wavevector) or coupled (excited by the same dipole) depending on the optimization. This configuration may be interesting for multimode integrated devices such as single photons on chip, quantum entanglement or for optical communications. More details of this part can be found in [49].
## 2.5 Surface plasmon hybridized dressed states

Multiple-hybridization between metallic layers inside plasmonic nanocavities displays properties similar to atomic-dressed states. In this framework, we propose to numerically and analytically investigate the case of a multilayered structure composed of N stacks of metallic (M) and insulator (I) thin films. We also use a vertical dipole as an optical probe buried inside the structure to excite the optical modes. To understand the multiple-plasmon hybridization of the structure, we begin with a layered structure with a small number of metallic slabs ($N = 3; 4$). When the number of blocks is increasing, we will show that the resulting modes evolve toward an energy continuum in a manner similar to the energy bands in a solid.

### Surface plasmon modes of alternative insulator/metal multilayer: begin with 3 and 4 metallic layers

In the following, we discuss about the construction of the plasmonic modes due to multi-hybridization by first analyzing coupling between 3 and 4 metallic layers. These configurations are labeled MIMIM and MIMIMIM. In Figure 2.9, the dissipated power is calculated and the profile of each mode is extracted for each geometry. All layer thicknesses are set to 20nm.

In the case of 3 metallic layers, we observe two modes noted M1 and M2 in Figure 2.9a. We clearly observe different widths in $u$ for these modes which is a fingerprint of their losses. In Figure 2.9b, the M1 mode, with smaller losses, is rather located near the outer limit of the structure, whereas the M2 mode, which has more losses, is propagating close to the central metallic layer.

In the case of 4 metallic layers, we observe the appearance of three modes with different widths. The profile of these new modes displays similar features as those presented in Figure 2.9a and b. The first mode M1, with the narrowest width, is more likely located in the central insulator. Its profile increases in the vicinity of the external metallic layers. By contrast, the modes (M2 and M3) with broader widths tend to localize closer to the central metallic layer. In particular, the intermediate M2 mode is vanishing inside the middle of the structure. Considering the cases of 3 and 4 metallic layer structures, we generally observe
<figure><img src="image_24.png" /><figcaption>Figure 2.9: **Three and four multilayers.** Dissipated power as a function of the wavevector $u$ for (a) MIMIM and (c) MIMIMIM structures. (b,d) Corresponding spatial profiles of each selected mode. The dipole is placed at 10nm from the metal in the left insulator (I) layer displayed (b,d). The finite layer thicknesses are set to 20nm.</figcaption></figure>

that the modes which present more losses are those that are more closely located near the central metallic layer. In both cases, the SPP modes at lower wavevector display fewer losses, propagate farther, and increase systematically near the external metallic layers.

## Multiple-plasmon hybridization between metallic layers

To understand the creation of these new coupled modes, one of the insulator (I) layers is replaced by a dielectric with a variable refractive index $n$, while the other insulators are air. On the basis of our previous analysis, changing the optical index is expected to detune the respective energies between the modes to decouple them [63, 64]. This results in a characteristic anti-crossing pattern. For this analysis, we assume that the independent modes are those of the MIM structure. The evolution of the independent MIM mode where the index of dielectric $n$ is changed is denoted MnM and the results are presented in Figure 2.10.

We observe the appearance of 2 and 3 branches for the case of 3 and 4 metallic layers
<figure><img src="image_25.png" /><figcaption>Figure 2.10: Construction of plasmonic modes in the case of structures with 3 and 4 metallic layers and a dielectric layer of variable refractive index. (a,d) Schemes of the devices. The position of the dipole is shown by a red arrow and located at 10nm from the metallic surface. (b,e) Evolution of $\alpha_{\perp}$ as function of $n$ and $u$ of the associated structures. The dashed lines are the positions of the independent modes ($u_{MIM}$ and $u_{MnM}(n)$) and the continuous lines are the calculated eigenmodes of the interaction matrix. (c,f) Schematic diagram of the mode construction for 3 and 4 metallic layers respectively.</figcaption></figure>

in the anti-crossing maps (Figure 2.10b, e). For example, for 4 metallic layers, a simple matrix formalism, grounded on physical considerations, can be introduced to reproduce the simulations. This case is based on the interaction between three coupled MIM structures. The interaction matrix can be written as:

$$u_{MIMIMnM} = \begin{pmatrix} \tilde{u}_{MIM} & V_{12} & V_{13} \\ V_{12} & \tilde{u}_{MIM} & V_{23} \\ V_{13} & V_{23} & \tilde{u}_{MIM}(n) \end{pmatrix} \quad (2.19)$$

In matrix $u_{MIMIMnM}$, we insert the wavevectors of the SPP for the MIM structure $\tilde{u}_{MIM}$. This value can be slightly corrected due to the finite size of the layer. The SPP wavevector of the MnM structure is noted as $\tilde{u}_{MIM}(n)$. These two values are extracted from simulations. We also define $V_{ij}$ as the interaction strength between $i$ and $j$ layers, in order to reproduce the numerical simulation. In Figure 2.10b, e, we present, in black continuous lines, the calculated eigenmodes from the matrix. These results are superimposed on the simulations for the cases of 3 and 4 metallic layers. We found that only the first neighbor’s MIM structures are interacting and that the coupling strength between neighbors does not depend on the overall structure (namely 3 or 4 metallic layers). For the case of 4 metallic layers, the independent mode of the MIM structure is used twice in the matrix. As these 2 modes are degenerate
(wavevector $\tilde{u}_{MIM}$), only one vertical dash line appears in Figure 2.10e. In the case of 4 layers of equation 2.19, we deduce $V_{12} = V_{23} = V = 0.5$, $V_{13} = 0$. The wavevector of the MIM plasmon of the single IM interface is corrected by 0.25, so that $\tilde{u}_{MIM} = u_{MIM} + 0.25$. Finally, we also found that the correction to MnM is $\tilde{u}_{MnM} = u_{MnM}(n) + 0.3$. The most important result is that, in the case of successive layers with the same thickness and index, we always have only first neighbors coupling with the same coupling $V = 0.5$. Finally, it is worthwhile to note that the regime of strong coupling can be achieved when the ratio of the coupling strength to their intrinsic losses is higher than 1 [64]. In our case, the coupling strength is $2V = 1$ and the losses, given by the width of the independent MIM mode, is $\Delta u = 0.1$. The strong regime criterion is, therefore, $V/\Delta u \gg 1$.

## Surface plasmon hybridization: N layers

In the following, we analyze the evolution of these hybridized modes as a function of the number of metallic layers. All dielectric layers are set to air to maximize the hybridization coupling strength. We note $N_b$ as the number of metallic layers so that the overall structure possesses ($N_b - 1$) air layers. For example, the MIM structure is obtained for $N_b = 2$. The thicknesses of both finite gold and air layers are set to 20nm. The dipole is placed in the center of the global structure (similar to Figure 2.10). If the center is air, the dipole is placed in the center and if it is metallic, the dipole is set 10nm above. The results are presented in Figure 2.11.

In Figure 2.11a, we see that the number of plasmonic modes increases with the number of layers. It is important to note that in our simulations, the dipole probes the Local Density of Optical States (LDOS). We see that for the case of 8 gold layers, the amplitudes of the modes are different. The distribution of the nodes of the various modes in the structure are different, so that the dipole may not be able to excite all the modes with the same efficiency. Nevertheless, when the number of layers is very large, all the modes merge to form a continuum in a similar way as the energy bands in a solid. The simulated colormap in Figure 2.11b illustrates this evolution. We see that when Nb is increasing, multiple hybridization occur and we observe the coalescence of all modes. In Figure 2.11a and b, we also detect two asymptotic limits of the wavevector values that will be noted $u_{limit} = u_{max}$ or $u_{min}$.
<figure><img src="image_26.png" /><figcaption>Figure 2.11: **Evolution of the dissipated power $\alpha$ as function of the number of metallic layers $N_b$.** (a) Dissipated power $\alpha$ as a function of the in-plane wavevector $u$ for 3 different numbers $N_b$ (3, 8 and 200) and (b), as a colormap with $u$ and $N_b$ from 2 (MIM scheme) to 40.</figcaption></figure>

In Figure 2.11b, we find them close to $u_{max} = 3.4$ and $u_{min} = 1.5$. These asymptotic values are dependent on the thickness of the gold or the air layers as illustrated in Figure 2.12. As expected, the asymptotic evolution ($N_b \to \infty$) of the wavevector $u$ tends to the initial values of building blocks (MIM or IMI) presented in Fig.1. When increasing the thicknesses ($e_M$ or $e_I$), the building blocks forming the overall structure become actually decoupled. When the dielectric layer thickness $e_I$ is infinite, the global structure tends to an IMI scheme with long-range ($LR_{IMI}$) and short-range ($SR_{IMI}$) plasmon. In the opposite case, when the metallic layer thickness $e_M$ is very high, the global structure adopts the behavior of a MIM structure with one plasmon mode ($SPP_{MIM}$).

**Conclusion:** To conclude this section, we have studied multiple hybridization between MIM structures. Hybridization occur at a small distance between adjacent MIM blocks with
<figure><img src="image_27.png" /><figcaption>Figure 2.12: Evolution of asymptotic bounds $u_{limit}$ of the mode continuum. (a) $u_{limit}$ as function of the metallic thickness $e_M$ (air layers thickness kept to 20nm) and (b), as function of the air thickness $e_I$ (gold layers of 20nm thick).</figcaption></figure>

a coupling strength V. To model this coupling, we used a strong coupling matrix formalism and proposed a schematic diagram in Figure 2.10c, f. We found that the coupling strength between adjacent layers is independent of the number of layers but is directly related to the thicknesses and the indices of the layers, as found for Bragg mirrors. Increasing the number of layers causes a multi-hybridization process and the plasmonic modes merge to construct a continuum similar to the formation of energy bands in a solid. Our result is also valid for the opposite case where the outside layers are made of insulators. The development of this work has many potential applications in plasmonic for architecture design in optoelectronics. More details of this section can be found in [65].
# Chapter 3

## Experimental methods for Surface Plasmons

We introduce optical techniques used in this Thesis for the investigation of localized surface plasmons (LSPR) in single metallic nanoparticles and de-localized surface plasmons (SPPs) at planar interfaces. The polarization-resolved scattering of the particles is determined by dark-field imaging. We employ Fourier-plane filtering to achieve nearly background-free images. For the planar interfaces, the wavevector-matching condition is necessarily satisfied to launch propagating SPPs, in contrast to LSPR, where its plasmon resonances can be excited by direct light illumination. We present here techniques for increasing the wavevector component of excitation light by means of evanescent waves or scattering from a point source. In terms of detection, the SPPs propagating on the interface are collected by leakage radiation microscopy. The SPPs are characterized by inspecting both Fourier-plane images (k-space) and real-space images. In combination, these experimental methods will be applied to study LSPR of gold bipyramid nanoparticles and gold film in the following chapters.
# Contents

<table><tr><td>3.1</td><td>Investigation of the optical response of metallic nanoparticles</td><td>71</td></tr><tr><td>3.2</td><td>Dark-field scattering polarimetry</td><td>73</td></tr><tr><td>3.2.1</td><td>Dark-field imaging by Fourier-plane filtering</td><td>73</td></tr><tr><td>3.2.2</td><td>Single metallic nanoparticle spectroscopy</td><td>77</td></tr><tr><td>3.2.3</td><td>Scattering polarimetry</td><td>79</td></tr><tr><td>3.3</td><td>Exciting and imaging propagating surface plasmon</td><td>80</td></tr><tr><td>3.4</td><td>Launching SPPs by leakage radiation microscopy</td><td>84</td></tr><tr><td>3.5</td><td>Launching SPPs by scattering from a point source</td><td>87</td></tr></table>

## 3.1 Investigation of the optical response of metallic nanoparticles

The study of localized surface plasmon resonances is not only necessary for the fundamental understanding of the properties of nanoparticles but also for practical applications such as emission enhancement [66, 67] and sensing [68]. The scattering of metallic nanoparticles at a single level can be investigated by means of methods based on near-field and far-field characterizations. Near-field optical microscopy provides information about the optical response, including the local field distribution and the spectrum of plasmon modes [69, 70]. One strategy is to employ a local probe, e.g., fiber tip with an aperture (~100nm), for investigation of the evanescent field of the particle. Because of the complex tip-particle interactions, the near-field methods require a large effort to interpret the results obtained in the far-field and
<figure><img src="image_28.png" /><figcaption>Figure 3.1: **Observation of single gold nanoparticles by dark-field imaging.** (a) Dark-field image together with SEM images. (b) scattering spectra. Source: Kuwata et al., 2003 [71].</figcaption></figure>

thus being limited in spectroscopy and sensing applications.

In dark-field microscopy, scattered light is selectively collected by filtering the incident light away from the detector. Thus, the particles appear in dark-field images as bright spots, induced by scattering, over a dark background. As apposed to near-field microscopy, single-nanoparticle observation in dark-field imaging is only achieved for well-separated nanoparticles. The nearly background-free measurement is usually achieved by using a condenser to construct the illumination in an annular beam and scattered light is collected in a cone of a smaller aperture [71, 72, 59]. Another approach is to locally probe the particles by evanescent waves from a total internal reflection illumination of an interface and scattering collection takes place in the free space of the smaller optical index medium [73]. Due to this simple operation and relatively fast acquisition time, dark field spectroscopy has been widely performed to study scattering of various nanoparticles of different geometry and in different environment. Figure 3.1 shows an example of a dark-field image and corresponding scattering spectra of individual gold nanoparticles [71]. Even under the diffraction limit, the gold nanoparticles of various sizes and shapes enable to distinguish based on their spot color
or spectrum resonance, defined by the resonance wavelength of the localized surface plasmon.

It is worth noting that dark-field spectroscopy or scattering-based techniques are not suitable for very small nano-objects, e.g., few tens nanometers for silver and gold nanospheres, surrounded by a background of scatters. This is due to the fact that the scattering cross-section scales rapidly with the square of particle volume $V^2$. Therefore, the scattering of the particles is overwhelmed by the background scattering (low signal-to-noise ratio). On the other hand, absorption is the dominant optical response in this small-size regime, and thus absorption-based techniques are alternatives for sub-10nm nanoparticle detection [74, 75].

In our work, the gold bipyramid nanoparticles (AuBPs) has relatively small volume (equivalent to a sphere of radius ~ 25 nm). However, the bipyramids are able to produce a strong scattering cross-section under resonant excitation due to the large oscillation strength of plasmons along the main axis of the particles (i.e., longitudinal localized surface plasmon resonances). Therefore, we employ dark-field microscopy to measure the scattering spectrum and analyze the polarization of the bipyramids.

## 3.2 Dark-field scattering polarimetry

### 3.2.1 Dark-field imaging by Fourier-plane filtering

In order to achieve nearly background-free images for dark-field imaging, our strategy is to discriminate angularly the incident and scattered light by Fourier-plane filtering. Let us first present the principle of the Fourier-plane manipulations. This technique is not only employed to support the measurement of the scattering of localized surface plasmon of the gold nanoparticle but also to characterize the propagating surface plasmon at the planar interfaces.

**What is Fourier-plane imaging?**

Mathematically, Fourier transform is a function that allows transforming between two reciprocal domains, e.g., the time-temporal frequency or space-spatial frequency domains, by multiplying the input function by an exponential that is linear in phase and making integration. In optics, the two-dimensional Fourier transform of light diffracted from an object can be made by propagating the light at a very large distance or just simply using a
<figure><img src="image_29.png" /><figcaption>Figure 3.2: Fourier transform of light by a lens. Spherical wave produced from one point source at position $(x, y)$ on the object is transformed to plane waves, each moving in a direction $(\theta_x, \theta_y)$. Imaging the back focal plane (Fourier-plane) of the lens gives angular distribution of diffracted light.</figcaption></figure>

converging lens. The Fourier transform in the latter case is depicted in Figure 3.2.

The field distribution in the back focal plane (BFP) of the lens ($\tilde{E}(k_x, k_y)$) can be interpreted as a superposition of plane waves or Fourier transform of the diffracted field $E(x, y)$ from the object and is expressed as:

$$ \tilde{E}(k_x, k_y, z = f) = \iint E(x, y) \cdot e^{-i(k_x x + k_y y)} dxdy \quad (3.1) $$

We can see that the lens makes Fourier transform light waves from the real space $(x, y)$ to Fourier space defined by wavevectors $(k_x, k_y)$ or angles $(\theta_x, \theta_y)$ with $\theta_x \approx -x/f$ and $\theta_y \approx -y/f$ (paraxial approximation). If a screen is placed at the BFP, we would image the spatial frequency spectrum (or angular distribution) of diffracted light from the object, the so-called radiation pattern. The Fourier-plane imaging is thus extensively used in nano-optics to access the angular spectrum of propagating radiation. For example, this technique has been used to probe the single fluorescence molecule orientation [76, 77] or measure the directionality of plasmonic antennas [78].

In our work, we image the radiation pattern in the BFP of an inverted oil immersion high numerical aperture objective (Nikon 60x, NA = 1.49). For a light-emitting object deposited on the glass substrate, the objective can collect all radiation confined inside azimuth angle $\theta_{max} \sim 80^{\circ}$. Figure 3.3a depicts an arbitrary radiated beam is refracted at the surface of the
<figure><img src="image_30.png" /><figcaption>Figure 3.3: **Fourier-plane imaging by a high numerical aperture objective.** (a) Scheme of a radiation collected and projected on the BFP of the objective. (b) Emission diagram of an aggregation of CdSe/CdS nanocrystals in the vicinity to the glass substrate is recorded by an oil immersion objective.</figcaption></figure>

objective and then projected onto the BFP. The spherical angular coordinates $(\theta, \phi)$ of this radiation can be obtained from the polar coordinates $(n \sin \theta, \phi)$ in the BFP, where $n \sim 1.5$ is the refractive index of glass.

The resulting intensity distribution on the Fourier-plane image is related to the radiation pattern by:

$$I_{FP}(k_x, k_y) = I(\theta, \phi) \cdot \frac{1}{\cos \theta} \qquad (3.2)$$

where $I_{FP} \sim |\tilde{E}(k_x, k_y)|^2$ and $k_x/k_0 = n \sin \theta \cos \phi$, $k_y/k_0 = n \sin \theta \sin \phi$. The refracting surface of the objective is approximated as a reference sphere and the apodization factor $1/\cos \theta$ is introduced in order to converge the energy along each radiation path [39].

As an example, we employ the oil immersion objective to measure the emission diagram of an aggregation of CdSe/CdS nanocrystal deposited on a glass substrate (Figure 3.3b). We refer to the radial coordinate $n \sin \theta$ as the “in-plane wavevector” $k_{||} = \sqrt{k_x^2 + k_y^2}$ of the radiation normalized by $k_0$. In the Fourier-plane image, the radius of the upper bound (red circle) corresponds to the maximum collection of the objective (NA = 1.49). A bright circle at $k_{||}/k_0 = 1$ is associated with the critical angle of the total internal reflection (TIR) at the glass/air interface: $\theta_c = \sin^{-1}(1/n) = 41.8^\circ$. The intensity of the radiation pattern peaks at the critical angle is interpreted as the evanescent components of the fluorescence. This field can couple to the far-field radiation beyond the critical angle ($\theta > \theta_c$) when the particle is
placed near the glass substrate [39].

## Dark-field imaging by Fourier-plane filtering

Now let us return to our main work where we make manipulations in the Fourier plane to obtain scattering from AuBPs. For a small bipyramid particle size, the localized surface plasmon resonance of AuBPs behaves like a linear radiating dipole. Because of the presence of the interface, the scattering is mostly redirected towards a denser index medium at angles beyond the critical angle of TIR. Therefore, our strategy is to excite the particle in the air and collect its forward scattering in the glass. Figure 3.4 presents the main idea of the measurement: the particle is illuminated in a small aperture and collection takes place in glass by a high numerical aperture objective. The collection consists of both transmitted excitation and forward

scattering that mostly distributes at larger angles. The low spatial frequency of the excitation is filtered out of detection by using a beam stop installed in the BFP (or Fourier plane) of the objective. Finally, only scattering of the particle is effectively collected.

In the detection, only a fraction of the total scattered power $P_{sca} = I_{inc} \cdot \sigma_{sca}$ is collected and can be parameterized as:

<figure><img src="image_31.png" /><figcaption>Figure 3.4: Dark-field imaging by Fourier-plane filtering.</figcaption></figure>

$$P_{sca}^{D} = I_{inc} \cdot \iint_{D} \frac{\partial \sigma_{scat}}{\partial \Omega} d\Omega \qquad (3.3)$$

where $I_{inc}$ is the incident intensity and $\sigma_{scat}$ is the scattering cross-section of the nanoparticle that will be enhanced at LSPR wavelength. The power $P_{sca}^D$ depends on the collection angle of the detector ($D$) and the radiation pattern of the scattering ($\frac{\partial \sigma_{scat}}{\partial \Omega}$). In which the scattering pattern can be modified by the local environment. For example, in the case of AuBPs placed near an air/glass interface, the scattering is transmitted or reflected by the interface, and thus the scattered power in each medium depends on the Fresnel coefficients.
### 3.2.2 Single metallic nanoparticle spectroscopy

Usually, the polarization-resolved fluorescence of single nanoemitters (e.g., dye molecules or nanocrystals) is measured by counting all fluorescent power detected on an avalanche photodiode. For metallic nanoparticles, scattering spectrum is necessarily recorded in order to identify the optical mode for the polarization analysis. The experimental setup for single particle measurements is depicted in Figure 3.5. In general, two excitation channels are possible: from the top (transmission configuration) and/or from the back (reflection configuration) of the collection objective, while there are three detection modes: dark-field imaging, spectroscopy, and Fourier-plane imaging, respectively, all for single particles deposited on a glass substrate.

For the dark-field imaging, we utilize the transmission configuration. The particles are excited by the beam from a fiber-coupled white lamp focused by a 0.25-NA objective. Then, both the scattered light and the direct illumination are collected in transmission by the oil immersion objective (NA = 1.49). The matching refractive indices of the oil and the glass substrate ensure the acquisition of the scattering in a homogeneous medium of index $n \sim 1.5$. As mentioned previously, a beam stop needs to be installed in the Fourier plane of the collection objective in order to block the transmitted excitation. However, it is difficult to access the primary Fourier plane (F.P, Figure 3.5a) because it usually lies inside the objective's body. Therefore, a confocal system is employed to make an image of F.P at position $F' \cdot P'$ (Figure 3.5a). The size of the beam stop is optimized at NA = 0.45 to cut off the low spatial frequency of the direct transmission and retain only the scattered light. It is supported by a thin and homogeneous glass coverslip (thickness $\sim 130\mu m$) so that the phase delay between spatial frequencies can be negligible. In addition, a diaphragm is integrated into the beam stop in order to control the collection numerical aperture. The signal is then recorded by an imaging spectrometer (Andor, Kimera 193i model), i.e., images are captured at the zero order of the grating. Different lenses are performed to switch between the real space ($L_r$) and the Fourier space ($L_F$).

#### Manipulation of the beam stop in the Fourier plane

In order to achieve a nearly background-free image, the position of the beam stop in the secondary Fourier plane $F' \cdot P'$ is adjusted in live-mode imaging. For this, both excitation
<figure><img src="image_32.png" /><figcaption>**Figure 3.5:** Dark-field imaging and polarization analysis. (a) Schematic of the experimental setup: F.P: primary Fourier plane of the objective, $F'$.P': image of the F.P, BS: beam stop, $\lambda/2$: achromatic half-wave plate, P: fixed polarizer, $L_r$: lens for real-space imaging, $L_F$: lens for Fourier-plane imaging. (b) and (c) Fourier-plane images and associating real-space images before and after, respectively, the beam stop is introduced to filter the excitation. The bright disk in (b) is associated with the excitation beam.</figcaption></figure>
channels from the top and from the back of the oil objective are performed. The recorded images are shown in Figure 3.5b and c. Angular distribution of the two excitation channels can be observed in the Fourier-plane image (Figure 3.5b, left). The bright disk in the center (radius $k_\parallel/k_0 = 0.25$) comes from the direct transmission of the white-lamp incidence, while the rest is the light reflected by the glass substrate when the oil objective is illuminated from the back. We can see a bright two-dimensional ring as the result of the TIR of light beams exceeding the critical angle $\theta_c = 41.8^\circ$ ($k_\parallel/k_0 = n \cdot \sin \theta_c = 1$) of the glass/air interface. In this case, the full back focal plane of the objective is illuminated and thus, the radius of Fourier-plane image is limited by $k_\parallel/k_0 = NA = 1.49$. In the corresponding real-space image of the bright field (Figure 3.5b, right), we only see the direct image of the excitation beam (induced by the core of the fiber) but not the AuBPs due to the overexposure. However, when the beam stop is introduced in the center of the Fourier-plane image to block the excitation (red-dotted circle, Figure 3.5c, left), we can see that AuBPs appear in the real-space image (Figure 3.5c, right) as well-separated bright spots on the dark background. The particles are then selected individually and sent to the spectrometer. The obtained spectra have a good signal-to-noise ratio because evanescent components of scattering are effectively collected.

### 3.2.3 Scattering polarimetry

The polarization analysis is then performed by the combination of an achromatic half-wave plate $\lambda/2$ rotating with an angle $\alpha/2$ and a fixed polarizer plate (see Figure 3.5a). The role of the half-wave plate and the polarizer is equivalent to a single rotating polarizer of analysis angle $\alpha$, however, maintaining a polarization state to enter the spectrometer. Because the polarization analysis will be employed to deduce the 3D orientation of AuBPs later, the polarization devices are necessarily placed in the detection after the oil objective but not in the excitation (polarization analysis of excitation only gives the information about the in-plane angle [79, 80]). It is worth noting that the beam splitter is removed and the dielectric mirrors are alternated by the silver mirrors (if used) to avoid depolarization. A calibration test has been conducted and showed that the depolarization of the overall experimental system is below $3^\circ$.

For polarization analysis, the scattering spectra are simultaneously recorded for each
<figure><img src="image_33.png" /><figcaption>Figure 3.6: **Polarization analysis of scattering.** Scattering spectrum of a single particle is recorded for each analysis angle $\alpha$. The detected intensities at resonance (blue dots) are plotted as a function of $\alpha$ following the Malus’s curve (red line).</figcaption></figure>

rotating angle of the half-wave plate. For instance, Figure 3.6 (left) shows three scattering spectra of a single AuBP respectively recorded for $\alpha = 0, \alpha = 60^\circ$, and $\alpha = 90^\circ$. We can see that all the spectra peak at wavelength $\lambda = 670\,nm$ associated with the longitudinal LSPR of the AuBP. The intensity at resonance is maximum when the polarizer is in the direction of the longitudinal axis of AuBP ($E_\|\|$) and is minimum in the orthogonal direction ($E_\perp$). For others, the detected intensities oscillate between the maximum and minimum values following the curve of Malus’law (red line, Figure 3.6 right).

## 3.3 Exciting and imaging propagating surface plasmon

### Excitation of propagating surface plasmon polaritons (SPPs)

In order to excite propagating surface plasmon, both energy and momentum conservation are necessarily accomplished. However, for a given energy $h\omega$, the wavevector of SPPs $k_{SP}$ on a dielectric/metal interface is always greater than the wavevector of light $k$ in the dielectric. This can be seen in the SPPs dispersion relation (see Chapter 1) where the SPP dispersion curve lies under the light line (given by $\omega = kc$). Therefore, SPPs excitation by light propagating in free space is not possible unless its in-plane wave vector is increased. There are several ways to achieve the wavevector-matching, including both local SPPs excitation by near-field optical microscopy [81] and macroscopic excitation using a prism or grating
coupling [29, 28, 30], or using spontaneous emission of fluorophores [82, 83]. In this Thesis,
the propagating surface plasmon is excited by means of evanescent waves and scattering from
a point source.

<figure><img src="image_34.png" /><figcaption>Figure 3.7: **Generation of propagating SPPs.** (a) SPPs of an air/gold interface is gener-
ated by evanescent field from glass/gold interface. The incident angle of the excitation beam
needs to be higher than critical angle of air/glass interface: θ<sub>SP</sub> > θ<sub>c</sub>. (b) SPPs excitation
by scattering from a small particle (a ≪ λ<sub>SP</sub>) on the gold surface</figcaption></figure>

SPPs excitation by a Kretschmann configuration: An approach is to excite SPPs of a planar dielectric/ metal interface by means of evanescent waves from another higher refractive index medium that is brought into contact with the metal surface. This is usually known as the Kretschmann configuration [28]. Figure 3.7a shows an example scheme: a thin gold film is sandwiched between air and a glass substrate (n ~ 1.5). The wavevector of any light beam propagating inside the glass is always greater than that in the air by a factor of refractive index n (k = k₀·n). Thus, the SPPs at the air/gold interface can be excited by light beams from glass traveling under incident angles θ<sub>SP</sub>, corresponding to in-plane wavevector equal the real part of the SPPs wavevector: k'<sub>SP</sub> = k₀·n · sin θ<sub>SP</sub>. The incident angle θ<sub>SP</sub> is necessarily greater than the critical angle θ<sub>c</sub> = sin<sup>-1</sup>(1/n) of the TIR at the air/glass interface. In addition, an effective SPPs excitation is only achieved for critical thicknesses of the metal film (e.g., ~ 50nm for gold) from which the evanescent wave created at the glass/metal interface can penetrate through the metal layer.

SPPs exciation by near field: The surface plasmon can also be generated by scattered
light from a point source whose characteristic dimensions are smaller than the wavelength
(a ≪ λ), e.g., surface defects, particles on the surface [82] or structured holes in the metal film
[32, 84]. The electromagnetic field can be localized in the near-field regime of the scattered
sources where wavevectors extend beyond the light cone of the propagating wave. Therefore, it is possible to excite the SPPs if the point source is near the interface (Figure 3.7b). Besides, the directivity of the SPPs launching on the interface is defined by the radiation pattern of the point source.

## Imaging propagating SPPs by leakage radiation microscopy

The SPPs field propagating along the air/metal interface does not couple to free waves in the air but can be leaked through the metal film and damped into radiation in the glass substrate. This can be interpreted by inspecting the dispersion relations. Figure 3.8 plots the dispersion relations of SPPs and light propagating in free spaces of air ($k_0 = \omega/c$) and a substrate ($k = n.\omega/c$). We can see that the SPP dispersion curve of metal/air interface is always located in the forbidden region of the air light line, and therefore SPPs do not radiate to the free space in the air. However, the radiation loss of SPP into the substrate can appear in a region of the dispersion curve that lies in between the light lines of air and substrate, the so-called “leakage radiation” (Figure 3.8).

<figure><img src="image_35.png" /><figcaption>Figure 3.8: Dispersion relation of SPPs at a metal/air interface. Region of leakage radiation locates in between light lines of air and substrate. Source: S. Maier [7]</figcaption></figure>

The leakage radiation of SPPs can be collected by using the prism (Kretschmann configuration) or the oil-immersion objective (the index of oil is taken very close to that of the glass). In this Thesis, we employ a high numerical aperture objective (NA = 1.49) to investigate the leakage SPPs. There are several advantages of using the objective lens than the prism. Besides allowing tightly focused beams and higher resolution to visualize SPP propagation,
<figure><img src="image_36.png" /><figcaption>Figure 3.9: SPPs by leakage radiation microscopy. (a) Schematic of excitation and collection of SPPs by the oil-immersion objective. (b) A measured Fourier plane where the excited SPPs show their footprint as a dark circle of radius $k'_{SP}$. (c) Dispersion of selected direction on the Fourier-plane (white lines in (b)). (d) Intensity profile of a vertical line cut on the 2D map of (b) at 700nm. In-plane wavevector $k_{||} = \sqrt{k_x^2 + k_y^2}$</figcaption></figure>

a single acquisition of the SPPs dispersion is possible to obtain in the BFP of the objective that encodes information about the angular distribution of the radiation. On the contrary, the measurement of the SPPs dispersion by the prism in the Kretschmann configuration must be made repeatedly for each excitation angle. In the following, we will use the oil objective to investigate the SPPs launched by the two described techniques using evanescent waves and scattering.
### 3.4 Launching SPPs by leakage radiation microscopy

Figure 3.9 depicts the excitation and collection of SPPs propagating over an air/gold interface by leakage radiation microscopy. The 45nm-gold is deposited on the glass substrate (thickness ~ 130µm) and is brought into contact with the oil immersion objective. The objective's back focal plane is completely filled by a white lamp illumination (reflection configuration). For clarity, we only depict the excitation on one side of the objective (Figure 3.9a). For a given wavelength, an associated wavevector fulfilled the wavevector matching condition (red ray) will be responsible for SPPs excitation, whereas others will be reflected or transmitted through the gold film. Therefore, the partial excitation light that excited SPPs will show up as a minimum in the reflectivity (grey ray, Figure 3.9a). The oil objective with NA = 1.49 contains all desired angles to excite SPPs corresponding to the white lamp spectrum. All the information related to the angular distribution is encoded inside the Fourier-plane image of the objective in Figure 3.9b. The Fourier-plane image is filled by the reflection of the white lamp on the gold/glass substrate. The dark circle ring is the result of the partial reflection of light after exciting SPPs at the air/gold interface and has a radius proportional to the real part of SPPs wavevector $k'_{SP} = k_0.n.sin\theta_{SP}$.

#### Measuring the SPPs dispersion relation

Each point in the Fourier-plane image encodes information about radiation angles ($\theta, \phi$) and wavelength $\lambda$. In order to obtain the SPPs dispersion $k_{SP}(\lambda)$, we make an image of the Fourier plane onto a vertical entrance slit of the spectrometer and close the slit (width ~ 100µm) to select one direction of the reflection (white lines Figure 3.9b). An obtained spectra image is shown in Figure 3.9c. This is a two-dimensional map of reflected intensities $I(k_\parallel/k_0, \lambda)$. The SPP dispersion can be seen as a dark blue trace above the light line of the air $k_\parallel = k_0$ that retrieves all the features of the typical SPP dispersion of a dielectric/metal interface. In the near infrared wavelength region, the SPP propagation constant $k'_{SP}$ is asymptotically approached the light line and bends away when the wavelengths decrease. Besides, the width of the SPP dispersion (which is proportional to the imaginary part of the SPP wavevector $k''_{SP}$) is broadened when the wavelength is getting smaller, especially where the interband transitions take place (below 550nm).
Identifying SPPs from the reflectivity

We now want to examine the properties SPPs by taking a closer look at the dispersion relation. We select one wavelength on the SPPs dispersion map, $\lambda = 700nm$ for example, and plot the reflected intensity as a function of the in-plane wavevector $k_\parallel/k_0$ in Figure 3.9d. The SPPs is characterized by a dip in reflectivity. As mentioned earlier, this missing light can be interpreted as a conversion to the SPPs which carry away the energy along the interface so that it cannot reach the spectrometer. Therefore, the full width at half maximum $\Gamma_{SP}$ of the dip is contributed by all the losses of the SPPs excited at the air/ gold interface and can be expressed as:

$$ \Gamma_{SP} = \Gamma_{abs} + \Gamma_{LK} + \Gamma_{scat} \qquad (3.4) $$

where $\Gamma_{abs}$ is Ohmic damping caused by the absorption of metal; $\Gamma_{LK}$ is the leakage radiation into glass substrate that depends on thickness of metallic film; and $\Gamma_{scat}$ is damping caused by the surface roughness of the metallic surface. If the surface roughness is neglected ($\Gamma_{scat} = 0$), the reflection coefficient in the region of the resonance can be approximated as a Lorentzian function

$$ R = 1 - \frac{4\Gamma_{LK}\Gamma_{abs}}{[k_{SP} - (k_{SP}^{(0)} + \Delta k_{SP})]^2 + (\Gamma_{abs} + \Gamma_{LK})^2} \qquad (3.5) $$

The formula is obtained from [7] where the calculation is based on the Fresnel coefficients. In which, $k_{SP}^{(0)}$ is the wavevector of SPPs propagating along the interface of two infinite media air/metallic. This is not equal, but is shifted by $\Delta k_{SP}$ from the wavevector $k_{SP}$ of SPPs of the air/metal/glass system due to the finite thickness of metallic film and the leakage radiation of SPPs into glass substrate. From the formula, it is apparent that a zero reflectivity can be achieved for a critical metal film thickness that allows $\Gamma_{LK} = \Gamma_{abs}$. At this thickness, there is also a perfect destructive interference between the reflected light and the leakage radiation of SPPs [7, 39].

Usually, the complex value of $k_{SP}$ can be obtained by fitting the reflected curve with the Lorentzian function. In our experimental situation, this is not straightforward because the reflected curve of SPPs does not really have a well-defined shape due to a low efficiency of SPP excitation. This is because of the fact that only small proportion of the white-lamp incidence is responsible for SPPs excitation when the full back focal plane of the objective is
illuminated. A more efficient way to excite SPPs can be achieved by using an off-axis laser beam to locally excite SPPs around angles associated with $k_{SP}$. The highly focused laser beam also allows for an excitation in a diffraction-limited spot area and thus the propagation length of SPPs can be measured directly in the real-space image. Moreover, the excitation of a continuum of SPPs at different wavelengths is possible by using a white-light continuum laser. The SPPs dispersion appears as a rainbow jet in the Fourier-plane image [85].

### Measuring dielectric film thickness by SPP dispersion

As shown above, leakage radiation microscopy using white lamp for excitation is not efficient and accurate to determine characteristics of SPPs at given excited wavelength (e.g., decay length, wavevector). However, because the measurement allows to retrieve the dispersion of SPPs, it can be useful for tracing the change of optical index of the near environment: a thin dielectric film deposited on gold film for example.

<figure><img src="image_37.png" /><figcaption>Figure 3.10: Dispersion relation of SPPs at different interfaces. (a) air ($\epsilon_d = 1$)/45nm gold film and (b) air/4nm-TiO$_2$ ($\epsilon_{TiO2} = 2.4$)/45nm gold film. $\epsilon_{eff}$ is an effective index of air and thin TiO$_2$. White lines is the simulation of SPP dispersion where dotted lines is simulated for $t = 3$nm and $t = 5$nm.</figcaption></figure>

Figure 3.10 shows SPP dispersion measured by leakage radiation microscope for two kinds of films: (a) bare gold film in air and (b) 4nm-TiO$_2$ deposited on gold film. We can see that the dispersion curve of the latter is shifted to higher $k_\|\|$ as compared to the bare gold film. This happens at all the wavelengths of experimental window; however, it is more significant in the lower wavelengths. Besides, the centre (or the dip where reflection is minimum) of the
measured dispersion is well-followed the simulated curves (solid white lines). Changing the TiO₂ film thickness by *t* ± 1 nm already shows the visible shift of the SPP dispersion (dotted white lines). Therefore, this measurement allows us to determine nanometric-scale thickness of thin films with 1nm of precision. Here, the thin film induces the changing of the optical index at interface, which leads to shift SPP dispersion following $k_{SP} = k_0 \sqrt{(\varepsilon_d \varepsilon_m)/(\varepsilon_d + \varepsilon_m)}$. In the case of air/gold interface, $\varepsilon_d$ is equal to the refractive index of air $\varepsilon_d = 1$. When thin TiO₂ film is deposited, now the optical index of above surface can be considered as a mixing of air and TiO₂, which is called effective index $\varepsilon_{eff}$. Therefore, now it is similar a single interface between a dielectric medium of $\varepsilon_d = \varepsilon_{eff}$ and gold. This is still true as long as the TiO₂ thickness is sufficient smaller than the evanescent decay length of SPPs in the air $t < \delta_d$. The shift of SPP dispersion presented here is also the mechanism behind sensing applications based on surface plasmon.

## 3.5 Launching SPPs by scattering from a point source

The increase of the free-space wavevector to excite SPPs can be achieved by means of scattering from a subwavelength point source such as defects or structured ridge on the metallic surface. For example, we use non-absorbing silica bead nanoparticles to excite SPPs of the air/gold interface. The beads have an average diameter of 150nm and refractive index ~ 1.5, and are deposited separately on the 45nm-gold film. A simplified experimental scheme is drawn in Figure 3.11a where the bead is excited by a focused laser beam ($\lambda = 650$ nm) and the leakage radiation is collected by the oil immersion objective. The beam stop is also installed in the Fourier plane to block the transmitted laser beam.

When the bead is excited by the laser, any scattered light satisfies the wavevector-matching condition is responsible for generation of SPPs. The SPPs are then leaked into glass substrate and appears in the Fourier-plane image as a bright circle. Figure 3.11b show an example of a Fourier-plane image which is measured by the experimental scheme on the left. Here, the radius of the red ring is proportional to $k'_{SP} = k_0 n \sin \theta_{SP}$, which is just above $k_{||} = k_0$. Because only partial scattering involves the SPPs excitation, the Fourier-plane image can be interpreted as a superposition of the SPPs leakage radiation and the
<figure><img src="image_38.png" /><figcaption>Figure 3.11: Launching SPPs by scattering from a nanobead. (a) Schematic of experimental setup. The particle is excited by a 650nm-laser. LR is the leakage of SPPs. Black disk is a beam stop, which is used to block the transmission of the laser (b) A measured Fourier plane where the leakage SPPs show their footprint as a bright red circle of radius $k'_{SP}$.</figcaption></figure>

background scattering. Here, we can see that the contribution of SPPs is dominant due to its confinement characteristic, while the background scattering is negligible.
# Chapter 4

## Measuring the 3D Orientation of Single Gold Bipyramid Nanoparticle

Similarly to the elongated nanoparticles, the field enhancement around the gold bipyramid nanoparticles (AuBPs) also displays high anisotropy. A simulation shows that this enhancement is strongest near end tips along the longitudinal axis of the AuBPs and is about 25 times greater than a sphere [5]. This is so-called longitudinal surface plasmon resonance (LSPR) which is mostly sensitive to the local environment at the vicinity of the tips. Thus, in the case of a AuBP on a high index or an absorbing substrate, e.g., a gold mirror, how the particle lies on the surface plays an important role. Furthermore, the knowledge of the nanoparticle orientation is required to maximize the desired properties for many applications. Among them, we can cite dipolar interactions and directional scattering. For instance, color routing has been achieved by manipulating the scattering diagram of two interacting metallic nanoparticles [86, 87]. Another wide range of examples may be found in controlling the spontaneous emission of single emitters with nanoparticles by tailoring their local electromagnetic environment with applications in nanoantennas [88, 66], light-emitting devices, Raman spectroscopy, and biosensors. It is then crucial that the emitter and the metallic nanoparticles are well aligned in order to maximize their coupling.

In this Chapter, the 3D orientation of a single AuBP is determined by plasmon-resonant scattering polarimetry. The AuBPs are deposited on a glass substrate in order to avoid any significant orientation-induced effects such as an LSPR shift or the appearance of new modes. Here, we develop a technique to precisely control the collection angle by Fourier-plane filtering of the scattering. The crucial role of the collection numerical aperture is also studied and discussed thoroughly.
# Contents

<table><tr><td>4.1</td><td>Properties of Gold Bipyramid nanoparticles</td><td>90</td></tr><tr><td>4.2</td><td>Scattering polarimetry for orientation measurement</td><td>93</td></tr><tr><td>4.3</td><td>Orientation of the AuBPs on the glass substrate</td><td>97</td></tr><tr><td>4.4</td><td>Role of collection numerical aperture</td><td>99</td></tr><tr><td>4.5</td><td>Conclusion</td><td>104</td></tr></table>

## 4.1 Properties of Gold Bipyramid nanoparticles

The synthesis procedure of the AuBPs used in this work has been described previously [89, 90]. Briefly, the nanoparticles are produced by a colloidal seeded growth method. The bipyramid anisotropic growth is controlled by adjusting the amount of gold salt and the nature of the reductant, surfactant and structural agent. Eventually, this method produces water-soluble AuBPs stabilized by a bilayer of cetyltrimethylammonium bromide (CTAB) that prevents the particles from aggregation.

The morphology of the bipyramids has been studied by a transmission electron microscope (TEM) (Fig 4.1). The AuBPs with sizes deduced from the TEM images have an average length L=70nm and width W= 25nm with rounded tips and edges (rounding radius r= 5nm). However, the real sizes might be different from these measured values by 5 - 10% depending on the orientations of the AuBPs which is due to the fact that the TEM gives 2D projections of the particles. Moreover, the precise shape of the AuBPs is also observed in 3D electron tomography, a technique that reconstructs the volume of particles from a series of 2D projection TEM images recorded at different angles. The reconstruction from the 3D-electron-tomography images (Fig 4.1c) shows that the AuBPs are twisted at the middle with 5 triangular facets on each pyramidal half.
<figure><img src="image_39.png" /><figcaption>Figure 4.1: **Morphology of the AuBPs.** (a) TEM image of the AuBPs. (b) Distributions of the length *L* and width *W* measured on the TEM images, *r* denotes for a rounding radius of the tips green(c) 3D shape of a AuBP reconstructed from tomographic TEM.</figcaption></figure>

The optical response of the AuBPs is studied by both ensemble measurement and at the
individual level. The UV-visible spectrum of the bipyramid suspensions (Fig4.2) shows two
distinct peaks: (i) a broad band around 500-600 nm associated with the transverse LSPR and
(ii) a more intense and narrow band at the near infra-red region (750-1050 nm) attributed to
longitudinal LSPR of the AuBPs. The bi-products (spheres or nanorods) might contribute
to these resonances, but only a small proportion (around 5% [89]). For different sample
preparations in Fig 4.2, the wavelength at which the longitudinal LSPR occurs can be tuned
by changing the size and aspect ratio of the AuBPs. However, controlling the aspect ratio is
more favorable because increasing the size of a particle leads to broadening of the resonance
that is mostly caused by multipolar excitations [89].

At a single particle level, the scattering spectrum of a AuBP on a glass substrate is stim-
ulated by COMSOL and measured by dark-field spectroscopy (Fig 4.3). In the calculation,
the particle is excited by an unpolarized plane wave in a normal incidence. We can see
that the simulated spectrum is dominated by an intense band centered at 740nm associated
with the longitudinal LSPR. The charge distribution of this mode (inset, Fig4.3a) shows the
strong localization around the tips of the bipyramid, which resembles an induced dipole. On
<figure><img src="image_40.png" /><figcaption>Figure 4.2: **Optical study on ensembles of biyramids:** UV-visible spectra of different suspensions of the AuBPs. Source: Navarro et al [90].</figcaption></figure>

the contrary, the quadrupolar mode is also excited at around 580 nm, however, this mode weakly scattered light because of its zero net dipole moment. Interestingly, a small resonance that arises at 650 nm is the result of hybridization between the radiating dipolar and the non-radiating quadrupolar modes. This interaction is mediated by the substrate and therefore, only appears when the AuBPs have a facet contact with the surface. In addition, the hybridization is more significant if the particles are deposited on the higher optical substrate [91, 92]. At the single particle level, the transverse LSPR yields virtually non-contribution to the scattering cross-section due to its very weak dipolar moment and strongly damping by the interband transitions (below 550 nm) [11, 9].

The dark-field spectroscopy <fnref n="1" /> is performed to retrieve alsmot the forward-scattering of each individual AuBP and a scattering spectrum is shown in Fig 4.3b. We can see that only the longitudinal LSPR shows up as a well-defined line while the other modes are vanished due to the too weak signal-to-noise ratio. The spectrum follows very well the Lorentzian fit (red curve) centered at 700nm with narrow full width at a half-maximum (FWHM) of 40nm. This value is equivalent to a resonator with quality factor $Q \sim 20$, suggesting that the AuBPs can be used as plasmonic probes with a good $Q$ factor that is very beneficial for sensing applications [5, 75].

<footnote n="1">¹see Chapter 3</footnote>
<figure><img src="image_41.png" /><figcaption>Figure 4.3: **Optical studies of the AuBPs at a single level.** (a) Numerical simulations of scattering spectrum of a AuBP on the glass substrate excited by a unpolarized plane wave. Inset: charge distribution associated with the main resonance (720nm).(b) Scattering spectrum measured by the dark-field spectroscopy (black dots) and fitted with the Lorentzian function.</figcaption></figure>

## 4.2 Scattering polarimetry for orientation measurement

In general, the orientation of a single nanoparticle is obtained experimentally from their radiation pattern or by polarization analysis. First, we briefly review the principles of the conventional methods for the determination of orientation, with possible applications.

### Methods of emission orientation determination

**Fourier-plane imaging:** the emission of the nano-object is directly encoded in the radia-
tion pattern that can be accessed by imaging the intensity distribution in the objective's
back focal plane (Fourier plane). This method has been applied to measure the orien-
tation of various materials, including nanocrystals [93, 94], fluorescent molecules [77],
or gold nanoparticles [59, 92]. Besides orientation, the emission Fourier plane image is
able to discriminate the dipole dimensions of the emitter [76]. However, this method
usually requires very sensitive detectors and a large number of photons to obtain a
sufficient signal-to-noise ratio.

**Defocused imaging:** by just a slight displacement of an objective out of its focal plane, the defocused patterns on the recorded images can be used to retrieve the orientations of the object [95, 96]. This method offers a very convenient implementation but its precision is very sensitive to the alignment and aberrations of the experiment setup.
**Focused orientation and position imaging (FOPI):** the core idea is to place the particles on an absorbing substrate, e.g. gold, to enhance the vertical dipolar component of the object inducing a focused doughnut-shape image [97]. The 3D orientations are extracted from this image without angular degeneracy.

**Anisotropy measurement:** determine the in-plane orientation by using a polarization beam splitter to separate optical responses of nano-objects into two orthogonal polarization directions to measure the optical anisotropy. This method is commonly used to measure the orientation of dynamic nanoparticles in biological solutions [98]. Besides, it also allows for determining the dipole dimension of nano-objects.

**Emission polarization analysis:** emission polarization is analyzed by continuously rotating a polarizer placed on the emission path. The 3D orientations are extracted from the detected intensity as a function of rotating angles. This method allows for determining the orientation of the emitter in situations, e.g. in proximity to a gold film, for which the radiation pattern-based techniques are not sufficiently sensitive [79, 76].

## Theoretical framework of scattering polarization analysis

In our study, we determine the orientations of a single AuBP by polarization analysis of its longitudinal LSPR scattering. While the scattered light is effectively collected by Fourier-plane filtering in the dark-field imaging (see Chapter 3), the measurement and the analytical model of the polarization analysis are suggested from the emission polarization analysis [79, 99]. This technique has been successfully applied to determine the 3D orientation of a permanent dipole inside semiconductor nanocrystals [76] or fluorescent molecules [100]. In our work, we are also able to use this method to measure orientations of the induced dipole associated with the longitudinal LSPR, which is equivalent to the orientation of the AuBPs. In the following, we present a brief description of the theoretical modeling used in this study to interpret the data and extract the 3D orientation of a AuBP. A more comprehensive calculation of the different environments can be found in this Ph.D. thesis [80].

To illustrate the relation between the orientation of a dipole and its polarization, we consider the simple case of a linear dipole radiating in a homogeneous medium. The 3D orientation is defined by an in-plane angle $\Phi$ and an out-of-plane angle $\Theta$ as shown in Fig4.4a.
The emission of the dipole is collected by an objective and then analyzed by a polarizer that continuously rotates with an analysis angle $α$. The radiation of the dipole in each direction is characterized by its emission diagram, which spreads mostly in the transverse directions to the dipole axis and there is no radiation at all in the direction of the dipole. We can see in Fig4.4 that the collected emission by the objective is strongly polarized for the horizontal dipole $Θ = 90°$ (opposite for the vertical dipole $Θ = 0°$).

<figure><img src="image_42.png" /><figcaption>Figure 4.4: Schematic of a simulated situation of a dipole radiating in a homogeneous medium. (a) 3D orientation of the dipole is defined by angles ($Θ, Φ$). Emission diagram of a horizontal dipole (b) and a vertical dipole (c) is collected by an objective and analyzed by a rotating polarizer ($α$ is the analysis angle).</figcaption></figure>

The model calculates the sum of the emission polarization along each propagation direction that have taken into account the apodization factor of the objective. Ultimately, the in-plane dipole orientation $Φ$ can be extracted from the dependence of the polarized intensity $I$ as a function of the analysis angle $α$:

$$I(\alpha) = I_{min} + (I_{max} - I_{min}) \cdot \cos^2(\Phi - \alpha), \quad (4.1)$$

This equation shows that the emission intensity is partially polarized, with a Malus component in the square-cosine function. On the other hand, the out-of-plane orientation $Θ$ defines the contrast between the maximum and minimum values of the detected intensities
and therefore can be deduced from the measured degree of polarization (DOP) $\delta$:

$$ \delta = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \qquad (4.2) $$

For example, vertical and horizontal dipoles induces strongest polarization and depolarization, respectively, and thus, $\delta$ is maximum for $\Theta = 90^{\circ}$and is minimum for $\Theta = 0$ (Fig 4.4).

It is important to note that $I(\alpha)$ and $\delta$ are parameterized with the following values:

* The dipolar dimension of the nanoparticles, e.g. 1D dipole - a linear radiating dipole (nanorod [101]), 2D dipole - an incoherent sum of two orthogonal 1D dipoles (dot-in-plate [76]), or 3D dipole - an incoherent sum of three 1D dipoles (isotropic emitters).

* The collection numerical aperture of the objective.

* The environment close to the dipole, e.g. optical index or interface.

In the case of the AuBPs, the simulated situation is depicted in Fig 4.5. The emitter is located in the air near a glass interface (refractive index $n \sim 1.5$) described as a semi-infinite medium. Given the emitter's length $L = 70$ nm (average size of the AuBPs deduced from the TEM images), the scattering AuBP is modeled as a point-like dipole located at a distance $z = (70/2) \cdot \cos\Theta$ (nm) from the glass surface. The emission of the dipole is collected by an oil immersion objective (NA = 1.49), a minor part of the collected light is cut off by a beam stop (NAbs = 0.45) placed at the Fourier-plane of the objective, and then analyzed by a polarizer that is continuously rotated with an analysis angle $\alpha$.

For this situation, the calculation has to take into account the transmission and redirection of the emission at the air-glass interface. Besides, the sums of the emission inside a solid angle of collection is limited by the beam stop ($\theta_{min} = asin(NA_{bs}/n)$) and the objective numerical aperture ($\theta_{max} = asin(1.49/n)$).
<figure><img src="image_43.png" /><figcaption>Figure 4.5: **Schematic of a simulated situation of associated with the AuBPs.** The bipyramid is modeled as a linear dipole at a vicinity to a glass substrate. Off-axis angle $\theta$ related to the collection angle. BS denotes for a beam stop.</figcaption></figure>

## 4.3 Orientation of the AuBPs on the glass substrate

In this section, we apply the described scattering polarization analysis to measure the orientations of the AuBPs. The sample is prepared by drop-casting the AuBPs solution on a clean glass substrate without any specific surface treatment or controlling the orientation. The concentration of the AuBPs has been adjusted to display well-separated nanoparticles (around 1 AuBP per $50\mu m^2$). The experimental configuration of the dark-field imaging and polarization analysis has been depicted in the Chapter 3. In the following, the procedure to measure the 3D orientations of an individual AuBP is presented first, then this measurement of the out-of-plane angle is repeated for other AuBPs to obtain the orientation distribution.

### Determination of the 3D orientation of a single AuBP

We plot on Fig 4.6a the dependence of the scattering intensity on the analysis angle $\alpha$. The measured intensity oscillates between $I_{max}$ and $I_{min}$ (red dots) with a square-cosine dependence in good agreement with Eq 4.1. This fit (blue line) gives $\Phi = 30^\circ$ and $\delta = 0.72$.

Alternatively, the values $I_{max}$ and $I_{min}$ can be obtained directly by looking for the maximum and minimum of the scattering spectrum. Such spectra of a single AuBP are shown in Fig 4.6b. For this AuBP, we have a resonance at 670nm with a narrow FWHM of 40nm. This resonance is attributed to the longitudinal LSRP mode. Using Eq 4.2, we calculate that $\delta = 0.7$ for this particle, which is different by only 3% from $\delta$ extracted from the Malus dependence in Fig 4.6a. Therefore, for our following measurements, $\delta$ will be simply deter-
<figure><img src="image_44.png" /><figcaption>Figure 4.6: Polarimetric measurement of an isolated AuBP. (a) The intensity of the scattering spectra as a function of the analysis angle $\alpha$. The red dots are the experimental measurements. The blue line corresponds to the fit with Eq 4.1. (b) Scattering spectra for two orthogonal polarizations corresponding to maximum ($I_{max}$) and minimum ($I_{min}$) intensities. (c) Theoretical dependence of $\delta$ on the dipole orientation $\Theta$. The dashed line represents the experimental $\delta$ extracted from (b).</figcaption></figure>

mined by taking $I_{max}$ and $I_{min}$ on the scattering spectrum. Once $\delta$ is known, $\Theta$ is determined from the theoretical dependence $\delta(\Theta)$ [77, 80] as presented in Fig 4.6c that gives $\Theta = 75^{\circ}$.

## Statistical orientation $\Theta$ of the AuBPs

In the above section, we gave a description about the procedure to measure 3D orientation for one bipyramid by scattering polarization analysis. In the following, we present a statistical measurement of the out-of-plane angle $\theta$. This angle is the important parameter that might lead to different interaction between the AuBPs and the substrate.

We perform the same measurement of $\delta$ for a selection of 30 AuBPs. Fig 4.7a presents the result as a distribution of $\delta$ between 0.5 – 0.8. By using the theoretical curve of $\delta(\Theta)$ reported previously, we deduce out-of-plane orientation $\Theta$ and report its histogram in Fig 4.7b. These values might be directly linked to the specific geometry of the AuBPs. We, therefore, calculate a geometrical orientation

$$ \Theta_{contact} = \frac{\pi}{2} - tan^{-1} \frac{W - 2r}{L - 2r} \qquad (4.3) $$

which is associated with the cases from which a facet of the AuBPs touches the surface, with the widths $W$, the lengths $L$, and the rounding radius $r$ deduced from the TEM images, and plot the distribution in Fig 4.7c. Both angular histograms display a peak at around $75^{\circ}$, but the extension of the wings on either side of this peak is larger by $15^{\circ}$ for the polarization
<figure><img src="image_45.png" /><figcaption>Figure 4.7: Orientation distribution of the deposited AuBPs on the glass substrate. (a) Histograms of measured $\delta$ and (b) deduced orientations $\Theta$, for 30 AuBPs by the polarization analysis. (c) Illustration of orientations when a AuBP is horizontal ($\Theta = 90^\circ$) or touches the surface ($\Theta_{contact}$), and histogram of $\Theta_{contact}$ is calculated from the TEM images.</figcaption></figure>

analysis. Especially, many AuBPs are measured depositing parallel to the substrate ($\Theta = 90^\circ$). This orientation might be induced by the stabilizer surrounding the AuBPs (CTAB, ~2nm [89]) or the surface roughness of the glass (~3nm, measured by profilometer). However, these side factors do not introduce any unpredictable orientations as we did not find any particle with $\Theta < 60^\circ$. These angles are observed in the previous study [5, 102] on the AFM image, but for the case of overlapping of two AuBPs. However, the dark-field scattering spectrum allows us to identify whether the object is a single AuBP or an aggregation.

## 4.4 Role of collection numerical aperture

The previous theoretical analysis [99] stated the important role of the collection numerical aperture (NA) in the interpretation of experimental values of $\delta$. Fig 4.8 demonstrates how
the curve $\delta(\Theta)$ is significant varied when different NA objective is used. This is due to the fact that the emission of a 1D dipole into a specific direction is always polarized and thus, different collection NA results in different values of $\delta$. In this section, the relation between $\delta$ and collection NA is investigated experimentally, providing deep insights to the polarization analysis: (i) another way to measure orientation with higher certainty, and (ii) the choice of an optimal collection objective for the measurement.

<figure><img src="image_46.png" /><figcaption>Figure 4.8: Theoretical calculation of $\delta(\Theta)$ for 1D dipole with different collection NA (ON: overture numerique), calculated in a situation when the dipole is located in the air, near a glass surface, and without presence of a beam stop. Source: PhD thesis of C. Lethiec [80]</figcaption></figure>

## The dependence of $\delta$ on the collection NA

To change the collection NA, we set a diaphragm at the secondary Fourier-plane position together with the beam stop (see Chapter 3) as shown in Fig (Chap 3). We, therefore, study $\delta$ as a function of NA for $NA \in [0.45; 1.49]$. Fig 4.9a presents the determination of $\delta$ for 5 values of NA on the same AuBP. For each diaphragm radius (corresponding to a specific NA), we measured $\delta$ as previously explained (see Fig 4.6). Because, the decrease of the collection NA means fewer emission is collected, we also measure the evolution of the scattering maximum intensity $I_{max}$ with NA. The experimental results agree well with a theoretical fit (solid lines). These calculations have taken into account the presence of the beam stop at $NA = 0.45$. If we omit the presence of the beam stop and integrate over all collected angles down to $\theta = 0$, only a slight difference is observed (dash line in Fig 4.9b) because most of the scattering occurs above the critical reflection angle $\theta_c = \sin^{-1}(1/n)$. This confirms that the polarization analysis protocol is not fundamentally modified by working in the dark-field
<figure><img src="image_47.png" /><figcaption>Figure 4.9: **Evolution of the polarization measurement as a function of the objective numerical aperture.** (a) $\delta$ and normalized maximum intensity $I_{max}$. Red dots are experimental measurements on the same single AuBP. The lines show two cases of theoretical calculations: NA $\in$ [0.45; 1.49] with a 0.45-NA beam stop (solid lines) and NA $\in$ [0; 1.49] without any beam stop (dashed lines). The blue zone corresponds to radiation directions above the critical angle. (b) Simulated scattering diagram and its projection on the Fourier plane.</figcaption></figure>

mode.

During the opening of the diaphragm, $\delta$ drops by 30%, especially near the critical region defined by $NA_c = n \sin \theta_c = 1$. This trend can be understood by correlating to the emission diagram. For that purpose, we calculated the radiation pattern in the Fourier plane of a dipole near a glass substrate and in the same experimental situation as for the AuBPs (Fig 4.9b). The center of the Fourier plane corresponds to a narrow solid angle of collection where the direction of the polarization is only linked to the projection of the dipole in the plane of the substrate resulting in $\delta \approx 1$. When collection NA increases, the radiated field is no longer parallel to the dipole, and different radiation directions show different field polarization, so that $\delta$, averaged over all collected angles, is lowered. This effect is stronger at large NA because the scattered field is radiated mostly at high angles, as the evanescent waves of the dipole near-field in air couple to the light propagating modes in glass [39] (blue zone, Fig 4.9a).

**Validation of dipole model and extraction of $\Theta$ from the curve $\delta(NA)$**