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Energy transport and plasmon dispersion in linear arrays of metal nanoparticlesCompaijen, Paul Jasper

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Energy Transport and Plasmon Dispersion

in Linear Arrays of Metal Nanoparticles

Zernike Institute PhD series 2016-07ISSN: 1570-1530

ISBN: 978-90-367-8644-7 (printed version)ISBN: 978-90-367-8643-0 (electronic version)

The work described in this thesis was performed at the Zernike Institute forAdvanced Materials of the Rijksuniversiteit Groningen.

Cover design by Jasper Bosch

Printed by Grafimedia, University of Groningen

Copyright c© 2016 Paul Jasper Compaijen

This thesis is part of NanoNextNL, a micro and nanotechnology innovation con-sortium of the Government of the Netherlands and 130 partners from academiaand industry. More information on www.nanonextnl.nl.

Energy Transport and PlasmonDispersion in Linear Arrays of

Metal Nanoparticles

Proefschrift

ter verkrijging van de graad van doctor aan deRijksuniversiteit Groningen

op gezag van derector magnificus prof. dr. E. Sterken

en volgens besluit van het College voor Promoties.

De openbare verdediging zal plaatsvinden op

vrijdag 11 maart 2016 om 14.30 uur

door

Paul Jasper Compaijen

geboren op 25 maart 1988te Elburg

PromotorProf. dr. J. Knoester

CopromotorDr. V.A. Malyshev

BeoordelingscommissieProf. dr. ir. C.H. van der WalProf. dr. A. PolmanProf. dr. O.J.F. Martin

Contents

1 Introduction 1

1.1 Diffraction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Modern plasmonics . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Sub-diffraction waveguides . . . . . . . . . . . . . . . . . . 9

1.3.2 Optical antennas . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Surface-mediated light transmission in metal nanoparticle chains 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 System Set Up and Formalism . . . . . . . . . . . . . . . . . . . 17

2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Isolated chain . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 Perfect reflector versus non-perfect reflector . . . . . . . . 24

2.3.3 Influence of chain-interface separation . . . . . . . . . . . 28

2.4 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 30

3 Engineering plasmon dispersion relations: hybrid nanoparticlechain - substrate plasmon polaritons 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Contents

4 Elliptically polarized modes for the unidirectional excitation ofsurface plasmon polaritons 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Electromagnetic pulse propagation through a chain of metalnanoparticles 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . 745.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4.1 Frequency domain . . . . . . . . . . . . . . . . . . . . . . 775.4.2 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . 835.4.3 Discussion of the time domain results . . . . . . . . . . . 89

5.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 92

Samenvatting 93

Dankwoord 101

Publication list 103

Bibliography 120

Chapter 1

Introduction

In 1965 Gorden E. Moore published the observation that the number of tran-sistors in an integrated circuit doubled approximately every two years and heprojected that this line of growth would continue for at least the next decade [1].Revisiting this statement after 10 years, it was shown that his expectation wasaccurate and it became better known as Moore’s law. In fact, rather then justan observation, it had become a road map for the long-term planning withinthe semiconductor industry. Therefore, Moore’s law actually is a self-fulfillingprophecy rather than a law, and great efforts and investments are made to keepup with the desired line of growth.

Increasing the number of transistors in a circuit, while keeping the dimen-sions similar, requires the fabrication of smaller transistors. The strong drivefor miniaturization induced by Moore’s law, has been a great impulse for thedevelopment of new fabrication techniques. Currently, the smallest features oncomputer chips are as small as 14 nanometers [2]. Importantly, apart from thetechnological benefit of the fabrication of such small structures, it was realizedthat the nanometer length scale has a plethora of interesting physics to offer.

When physical phenomena at the nanoscale are studied and explored, ofcourse optics cannot be neglected. The ability to control light at the nanoscaleopens up a wide range of possibilities, from studying the absorption and emis-sion of a single molecule, to using light as an information carrier on sub-microncomputer chips.

A beautiful example from nature showing the interaction of light with ananostructured material, is the presence of so-called structural coloring in but-terfly wings and peacock feathers. Figure 1.0.1 shows the nanostructures on the

2 Introduction

Figure 1.0.1: a) Morpho menelaus butterfly, b) TEM image of a cross section ofthe wing showing lamellae with sub-wavelength periodicity. Image reproducedfrom [3]

wings of the Morpho butterfly, which is famous for the fact that its luminouscolor seems to change when it is viewed under different angles, a phenomenoncalled iridescence [3]. Instead of using different materials to create different col-ors, the pigments are made of the same materials, however each having a veryspecific periodicity. The period is smaller than or similar to the wavelength,and therefore light experiences an effective structure with a specific, frequencydependent, refractive index. This technique for the manipulation of light is nowwidely applied in the field of photonic crystals and metamaterials. However, ma-nipulating light far into the nanometer domain is challenging, due to the presenceof the diffraction limit: light cannot be focused to a size of roughly half of itswavelength. In optics this limit is around 200 nm and, therefore, in order totruly do nano-optics and match with current day nano-electronics, techniquesare needed to shrink or even bypass the diffraction limit.

1.1 Diffraction limit

When we investigate something under a microscope, we look at the light thathas scattered of the object that we are studying. By making use of a systemof lenses, the image of the object can be enlarged and focused. Looking attwo objects that are moving closer together, there will be, of course, also twoseparate objects in the image, moving closer together. However, at some pointthe two objects will appear to be one big object, which indicates that the distancebetween the two objects is smaller than the microscope can resolve. Although inpractice one can usually improve on this limit by replacing the lens system or the

1.1 Diffraction limit 3

CCD-sensor, there appears to be a fundamental limit of how small the distancebetween two objects can be in order to still resolve to separate objects. This isthe so-called diffraction limit. Physically, the origin of the diffraction limit canbe easily explained from the famous uncertainty relation [4], which states that∆E ·∆t ≥ ~/2. In free-space, the dispersion relation of light is given by ω = c|k|,where |k| is the length of the corresponding wavevector

|k| =√k2x + k2

y + k2z . (1.1.1)

Considering the x-direction and using E = ~ω and c∆t = ∆x, we can write

~∆kx ·∆x ≥ ~/2⇒ ∆x ≥ 1/2∆kx. (1.1.2)

In order words, the uncertainty in x position is inversely proportional to the un-certainty in the x-component of the wavevector. In a homogeneous environmentthe maximum value of ∆kx is the total length of the wavevector |k|, which isequal to 2π/λ. Therefore, by this reasoning, in free-space the lower limit of theconfinement of light is λ/4π.

According to the above, the spread in position can be decreased if kx is madelarger than |k|. Then, to satisfy Eq. 1.1.2, one of the other two wavevector com-ponents has to be purely imaginary. In this case, assuming that for examplekz is purely imaginary, the wave equation states that exp[ikzz] = exp[−|kz|z],which implies that the amplitude is exponentially decaying in the +z directionand exponentially growing in the −z direction. The latter contribution is un-physical and therefore a purely imaginary wavevector component is not allowedin free-space. However, in an inhomogeneous system the boundary conditionsmay be such that an imaginary wavevector component only yields exponentiallydecaying solutions. In such a system, in that particular direction, light can beconfined to arbitrarily small dimensions. An example of a system where this ispossible, is the interface between a metal and a dielectric. To a first approxima-tion, a metal can be treated as a free electron gas, with a certain correspondingplasma frequency that depends on a.o. the electron density of the metal. Forfrequencies smaller than the plasma frequency of the metal, its permittivity εmwill be negative. The wavevector of light within the medium km, is related tothe free-space wavevector k0, as km =

√εmk0. Applying the fact that εm < 0

in the boundary conditions at the interface, we can find solutions that are expo-nentially decaying solutions both into the metal and into the dielectric, i.e. they

4 Introduction

Figure 1.2.1: Fourth-century roman chalice exposed in the British Museumin London. The cup is green under ambient lighting conditions (left), whenilluminated from inside it becomes translucent red (right). [Image courtesy:The Guardian]

are localized to the metal-dielectric interface. The exponential decay lengths canbe much smaller than the wavelength of light, yielding a localization well belowthe diffraction limit. These types of electromagnetic waves are called surfaceplasmons.

1.2 Plasmonics

1.2.1 Historical overview

Plasmonics is the study of plasmons, collective oscillations of the free electronsof a metal. Although bulk plasmons can exist, for applications in nanosciencethe interactions at the surface are far more important. Therefore, in this thesis,when the word plasmon it used, it is implied that it concerns a surface plasmon,rather than a bulk plasmon.

Most probably the first historic report of a plasmon based phenomenon is thebeautiful Lycurgus cup, a fourth-century roman chalice. As shown in Fig. 1.2.1,the cup appears green under ambient lighting conditions, but it becomes translu-cent red when it is illuminated from the inside. Obviously the Romans were notaware of the existence of plasmons and for centuries scientists kept wonderingabout this strange phenomenon. Analysis of the glass revealed small quantitiesof silver and gold nanoparticles and in 1857 Michael Faraday showed with a bril-

1.2 Plasmonics 5

liant experiment that the color of metallic colloidal solutions depends on the sizeand the materials of the particles [5]. Theoretical understanding of these resultscame in 1908 when Gustav Mie composed a theory with which it is possible toascribe these observations to electromagnetic resonances of the nanoparticles,the Localized Surface Plasmon resonance (LSP) [6]. Physically, it is the freeelectron cloud of the metal nanoparticle that is moving along with the oscilla-tions of the electromagnetic field impinging on it. The curved surface of theparticle effectively exerts a restoring force on the electron cloud, giving rise to aresonance. This implies that for frequencies close to the resonance frequency theoptical response, i.e. the amplitude of the electron cloud, will be much largerthan away from the resonance. Additionally, since the electron cloud is chargedand accelerating charges radiate, both the field inside and outside the particlewill be greatly amplified when the LSP is excited. It is because of the strongoptical response and the large local field enhancement that the LSP resonanceof metal nanoparticles has been the subject for a large amount of studies.

Figure 1.2.2a shows a contour plot of the electric field when the LSP isexcited in a gold nanopshere, with a diameter of 60 nm and that is embeddedin air. It can be clearly seen that the amplitude of the electric field close tothe sphere can be much larger than that of the external excitation. Also, theelectric field pattern closely resembles that of an oscillating dipole. In fact, forspheres much smaller than the wavelength of the light, only the dipole LSPwill be excited. Therefore, to simulate the optical response of a nanoparticle,it is not always necessary to employ the full Mie theory, but often a simpledipole approximation is sufficient. Using such an approximation, the scatteringefficiency, i.e. the scattering cross section divided by the particle’s geometricalcross section, is evaluated and shown in Fig. 1.2.2b. The strong dependence ofthe scattering efficiency on the wavelength, or frequency, is obvious from thesharp resonance around 480 nm. At resonance the particle will strongly absorband scatter the incident light. Now we can find the explanation for the dichroısmthat is observed in the Lycurgus cup. The metal alloy clusters were found tohave a resonance frequency in the blue part of the spectrum. Under ambientlighting the color of the chalice that we perceive is dominated by reflection, i.e.backscattering. The strong blue contribution of the metal clusters makes thecup’s appearance greenish. However, when the light source is inside the cup,we only observe the transmitted light. Since the clusters absorb blue light, thiscolor will be absent from the transmission spectrum and the light that reaches

6 Introduction

400 450 500 550 6000

1

2

3

4

5

Sca

tterin

g ef

ficie

ncy

wavelength (nm)

-100

Figure 1.2.2: a) The Localized Surface Plasmon resonance of a gold nanopar-ticle with a diameter of 60 nm embedded in air. The particle is excited with aplane wave coming from the left with an amplitude of E0. The electric field isnormalized with respect to E0. Clearly seen is the local field enhancement, thefield close to the particle can be significantly larger than the external excitation[Source: juluribk.com]. b) Plot of the scattering efficiency, i.e. the scatteringcross section divided by the geometrical cross section, for the gold nanosphereas a function of the wavelength of the excitation.

1.2 Plasmonics 7

our eyes is predominantly red.

Surprisingly, the first famous plasmonic studies where conducted more than50 years after the publication of Mie and where not carried out on metal nanopar-ticles, but on metallic substrates. In contrast to the LSP of a metal nanoparticle,which is localized to the particle and therefore non-propagating, the plasmonexcitation of a metallic substrate is able to propagate along the interface be-tween the metal and a surrounding dielectric. For these types of excitations,the oscillation of the electrons in the metal is strongly coupled to the light inthe dielectric and therefore these plasmons are referred to as Surface PlasmonPolaritons (SPPs), see Fig. 1.2.3a. They can only exist for transverse magneticpolarization and, as explained at the end of Sec. 1.1, are characterized by electricfield decaying exponentially away from the interface, both in the metal and inthe dielectric (Fig. 1.2.3b). Many of the first studies, performed by among otherRitchie, Nishikawa, and Bennet, revolved around the modeling and measurementof the dispersion relation of the SPPs on various metallic substrates [7–10]. Anexample of a characteristic SPP dispersion curve is shown in Figure 1.2.3c. Sincethese excitations are polaritons, the dispersion relation shows an avoided cross-ing with the dispersion of light, resulting in two branches, one lying above thelight line and one below. The latter branch is referred to as the SPP dispersionrelation. The fact that the SPP dispersion lies below the light line, indicatesthat all wavevectors of the mode are larger than those of light for the same fre-quency. This implies that the SPPs cannot couple to free-space radiation andspecial phase-matching techniques are needed to excite the SPPs. Reversely, thisalso implies that the SPPs will not lose their energy to radiation and thereforethey can propagate along the metal-dielectric interface for a substantial distance.When instead of a smooth substrate a rough substrate is considered, the SPPscan radiate locally, giving rise to so-called hotspots. The huge field intensities atthose hotspots are shown to be very interesting for the enhancement of nonlineareffects such as Second Harmonic Generation and Surface Enhancement RamanScattering. In 1988 the comprehensive textbook ”Surface Plasmons on Smoothand Rough Surfaces and on Gratings” written by Heinz Raether appeared [11].It is this book that summarizes most previously obtained knowledge and formeda solid basis on which many of the later studies were built.

8 Introduction

Figure 1.2.3: a) Schematic representation of the electric and magnetic fieldscorresponding to an SPP excitation on the interface between a metal and adielectric. b) The electric field decays exponentially away from the interface,with decay constants δd in the dielectric and δm in the metal. c) Characteristicdispersion relation for SPPs. At low frequencies the modes behave like light, athigher frequencies the dispersion starts to deviate more from the light line anda more plasmonic character is obtained. Reproduced from [12].

1.2.2 Modern plasmonics

Around the turn of the 21st century a view brilliant insights preluded the hugeboost of the field of plasmonics. It was realized that manipulating and propa-gating light below the diffraction limit was not just a nice novelty, but that itmight be a necessity in order to keep on improving the performance of electronicdevices. Starting with the pioneering work of Quinten and coworkers [13], andfollowed by the very careful investigations performed by the Atwater’s group atCaltech [14–17], it was shown that a chain of metal nanoparticles can be usedas a sub-diffraction transmission line of electromagnetic energy in the visibleregime. These classic papers were the main reason why many researchers be-came interested in plasmonics. Interestingly, at the same time as the demandfrom industry grew, the techniques to fabricate the devices just became availableand many theoretical concepts still had to be developed. Therefore, the growthpotential of the research field was immense, and scientists and engineers frommany different research areas joined in. Apart from sub-diffraction waveguiding,many other interesting discoveries were made, e.g., negative refraction, whichallows the construction of a perfect lens and even an optical cloak; extraordi-

1.3 Applications 9

nary optical transmission, the observation that much more light is transmittedthrough a sub-wavelength aperture in a metallic substrate than would be ex-pected purely from geometry; and plasmon excitations in graphene with a verystrong confinement and which show little dissipation.

1.3 Applications

Ever since the beginning of the 21st century, research in plasmonics has beenmotivated by the potential applications. Over the past decade, researchers haveshown to be very creative and many application areas have been proposed, someof which can only exist using plasmons, others where the use of plasmons is amere improvement of already known devices. Since the first publications ap-peared, numerous review articles have been published, describing the generalprinciples of nano-optics, the elementary theory, and the various fabrication tech-niques [12, 18–26]. Although most envisioned applications stem from only twoproperties of plasmons, namely the sub-wavelength confinement and the stronglocal field enhancement, they do span quite a large variety of possibilities. Somepromising application areas are: sub-diffraction waveguides [12–26], optical nano-antennas [27–32], sensing [33–37], solar cells [38–40], plasmon lasers [41–43] andcancer treatment [44–46]. The first two of the mentioned applications will bediscussed in some more detail below, since they are closely related to the researchpresented in this thesis.

1.3.1 Sub-diffraction waveguides

As was mentioned earlier, the biggest drive for the field of plasmonics was thepromise of sub-diffraction optical communication. The high frequencies of lightgive rise to a very large bandwidth, which is a huge benefit compared to con-ventional electronic communication. In addition, also the signal velocities aremuch larger in optical waveguides. Quinten and coworkers realized that a chainof MNPs could be applied as a sub-diffraction waveguide [13]. In the followingyears many studies on this subject were performed. The first studies mainlyfocused on theory and simulations [14–16], later the results were also experi-mentally verified [17]. The dispersion relations for these systems, also known asplasmonics arrays, revealed that the collective plasmonic modes of the systemare in fact Surface Plasmon Polaritons, similar to the plasmon excitations of a

10 Introduction

a) b)

Figure 1.3.1: a) Artist impression of a chain of metal nanoparticles, correspond-ing to the work of Maier et al. [17]. b) Illustration of the hybrid plasmonicwaveguide proposed by Oulton et al. [47], comprising a dielectric nanowire anda metallic substrate.

metallic substrate [48–50]. Also for the array the waveguiding modes lie belowthe light line, implying that they cannot lose their energy to radiation, all energyis guided along the chain. Physically this can be explained from interference,for large wavevectors the electromagnetic far-fields generated by the individ-ual particles will destructively interfere. In contrast to a metallic substrate, nophase-matching conditions are needed to excite the SPPs of a plasmonic array:due to the discrete nature of the system they can be excited by simply excitinga small part of the array. Other advantages of plasmonic arrays are the inher-ent one dimensional nature, which is important for signal transmission and theamount of design parameters, the operation frequency and wavelength stronglydepend on the material, size, shape and spacing of the particles and the envi-ronment of the array. Theoretically, plasmonic arrays are very interesting studyobjects. Simple as they seem, many nontrivial phenomena have been observed,such as localization [49, 51], negative phase velocity [16] and a complex modestructure [52, 53]. Although these systems were first introduced as waveguides,it turned out that their efficiencies are relatively low, and that the arrays aredifficult to fabricate. Therefore, many other geometries have been explored forthe design of sub-diffraction optical waveguides. An overview of different possi-ble waveguide geometries is given in [25]. Particle arrays are still considered tobe interesting for antenna and spectroscopy applications.

1.3 Applications 11

Among the other explored plasmonic waveguide geometries are metallic nano-wires [47,54,55], and 1D structures on metallic substrates, such as wegdes [56,57],grooves [58, 59] and channels [60]. A problem that is commonly encountered inthe design of plasmonics systems, is the trade-off between propagation and con-finement: if the excitation is strongly confined this often implies that most of theenergy is localized in the metal, and since metals suffer from ohmic losses, theexcitation will experience more dissipation. To circumvent this difficulty, partic-ularly promising are the so-called hybrid systems, where one type of plasmonicsystem is combined with another plasmonic or even with a dielectric counter-part [47,61,62].

1.3.2 Optical antennas

a) b)

Figure 1.3.2: a) Schematic illustration of the function of the antenna. b) Artistimpression of a nano-antenna corresponding to the work of Curto et al. [30].

Optical antennas are in fact the sub-micron wavelength analogues of the well-known radio wave antennas, and they can be thought of as devices that convertfreely propagating optical radiation into localized energy, and vice versa [28].Looking back in time, the technological and social benefits that arose from in-vention of radio wave antennas are obvious, in particular the development ofwireless communication. Due to the scale invariance of Maxwell’s equations,many of the concepts of radio waves antennas can be straightforwardly trans-lated to nanometer length scales. However, in general, antennas are much smallerthan the wavelength at which they operate, on the order of λ/100. Therefore,the dimensions of optical antennas are very small and the fabrication of antennasthat operate in the visible spectrum has been a bottleneck for its development

12 Introduction

for a long time. Only since the past decade the field really started to grow.

Typically an optical antenna would be applied to enhance the signal trans-mission from an emitter to a receiver. The presence of the antenna can benefitin two different ways, it can extract more signal out of the emitter and it candirect the transmission towards the receiver. As was discussed earlier, the opticalcross section of a metal nanoparticle dramatically increases when it is excitedresonantly. Using metallic nanostructures as building blocks for optical anten-nas can therefore greatly enhance, for example, the amount of energy that canbe extracted from the far field, or the emission rate of an emitter close to theantenna. The first is very interesting for applications in photodetection [63]and photovoltaics [39], the latter is useful for the creation of very bright sin-gle photon sources [64]. Directing the radiation pattern of an antenna can beachieved by considering arrays of nanoparticles. By carefully tuning the particlesizes and the inter-particle spacing, interference can be used to achieve a highdirectionality. A well-known design for this, originating from radio antennas,are so-called Yagi-Uda antennas [29, 65]. It is not only the directional radiationof antennas that is interesting, an increasing amount of effort is spent in usingnano-antennas to unidirectionally couple an excitation into guided modes of awaveguide [62,66]. In particular the use of polarization in tuning the directivityof such antennas lately has received a great deal of attention [67–69]. A detailedreview on the basics and applications of optical antennas is written by Novotnyand Van Hulst [31].

1.4 This thesis

The research presented in this thesis focuses on the calculation of the opticalresponse of hybrid plasmonic systems, in particular a one-dimensional metalnanoparticle chain hybridized with a metallic substrate. As was discussed inSec. 1.3, the applications for these types of systems are various, ranging from sub-diffraction waveguides to optical nano-antennas and plasmon enhanced sensors.Although these types of systems have been the subject of a large amount ofstudies, many important questions are still unanswered, such as: what is thenature of the interaction between a chain of metal nanoparticles and a layeredstructure, and how does such a structure affect the optical properties of thechain?

In this thesis, we address these questions by constructing a theoretical frame-

1.4 This thesis 13

work based on the point dipole approximation. It is known that this model,simple as it is, allows for a fast and accurate calculation of systems of metalnanoparticles, because the particle-particle and particle-substrate interactionsbecome simple analytical expressions. Additionally, being able to write down theinteraction in the coupled system, makes it possible to find the collective modesof the system, which forms the basis for understanding the optical response of thesystem. Therefore, this model serves as an excellent tool to explore the opticalproperties of linear chains of metal nanoparticles in a layered structure.

Although the system under consideration was first proposed as a sub-diffractionwaveguide, it was soon realized that it possesses a plethora of interesting physics.Therefore, with the theoretical framework at hand, we will not only study thewaveguiding properties of the system, but also investigate it from a more funda-mental perspective.

1.4.1 Outline

In Chapter 2 the efficiency of signal transmission through a chain of metalnanoparticles localized above a specific substrate is studied. This work wasinspired on the promise of the use of hybrid plasmonic structures and the re-sults obtained by Evlyukhin and Bozhevolnyi in 2006 [61], in which the obtainedvery long plasmon propagation lengths in a chain of gold nanoparticles situatedabove a golden substrate. Additionally, when applying plasmonic arrays eitherin devices or in an experimental set-up, it will always be embedded in a layeredsurrounding. Therefore, it is crucial to study the influence of a substrate on thetransmission efficiency of the chain. In this chapter we show that the effect of asubstrate on the transmission efficiency of a plasmonic array is non-trivial andthere is a complex interplay between multiple transmission channel. Neverthe-less, we give a complete description of these effects and show that it is possibleto enhance the efficiency of a plasmonic array by placing it in close to a silversubstrate.

The work presented in Chapter 3 describes the interaction between the plas-monic modes of a chain of metal nanoparticles and the surface plasmon polari-tons of a metallic substrate. We show that the modes of the chain and thesubstrate can indeed hybridize into new chain-substrate plasmons. The amountof hybridization is very sensitive to the material properties of the chain and thesubstrate. This insight provides a powerful tool to engineer the hybrid modes ofthe system, and thus, its optical properties.

14 Introduction

Chapter 4 is based on the interesting phenomenon of the unidirectional ex-citation of Surface Plasmon Polaritons on a metallic substrate. It was shown inRefs. [70, 71] that it is possible to excite SPPs on a metallic substrate with aspecific propagation direction using an out-of-plane circularly polarized dipole.This effect can be explained from the fact that for SPPs spin and propagationdirection are coupled. Due to angular momentum matching, a circularly polar-ized dipole will preferentially excite SPPs with a given spin and thus, a specificdirectionality. All setups described so far, rely on the external excitation to pro-vide the angular momentum. In this chapter, we show that the hybrid modesof an asymmetric nanoparticle dimer above a metallic substrate are ellipticallypolarized. Therefore, the chirality is inherent and thus, using this system, unidi-rectional excitation of SPPs can be achieved regardless of the external excitation.

The final Chapter, 5, deals with the time-dependent signal transmissionthrough a plasmonic array in a homogeneous environment. Most studies on plas-monic waveguides are performed under continuous wave excitation. However, inorder to use the waveguide for communication, it is of crucial importance toinvestigate the time-dependent properties of the waveguide. To that extent, westudied the transmission of a Gaussian pulse through a plasmonic array as afunction of time. Surprisingly, we found that not one, but two signals are ex-cited simultaneously. In addition to the expected signal formed by the collectivemodes of array, also an optical precursor is obtained. The latter propagates withthe speed of light in the embedding medium, and does not feel dispersion nordissipation. We show that, because of this, the precursor is of great importancein long plasmonic arrays.

1.4.2 Framework

All calculations were performed by applying the point dipole approximation forthe metal nanoparticles and using a Green’s tensor approach to calculate thecoupling between the particles in the chain and between the particles and thesubstrate. Naturally, great care was taken regarding the validity of these approx-imations. The calculations were carried out using an in-house developed codewithin the environment of Matlab. Each chapter clearly states the importantequations and parameters, and, if necessary, provides some derivations of theformalism. For the development of the programs extensive use was made of thetextbooks cited in [4, 72–75].

Chapter 2

Surface-mediated lighttransmission in metalnanoparticle chains

We study theoretically the efficiency of the transmission of optical signals througha linear chain consisting of identical and equidistantly spaced silver metal nanopar-ticles. Two situations are compared: the transmission efficiency through an iso-lated chain and through a chain in close proximity of a reflecting substrate. TheOhmic and radiative losses in each nanoparticle strongly affect the transmissionefficiency of an isolated chain and suppress it to large extent. It is shown that thepresence of a reflecting interface may enhance the guiding properties of the array.The reason for this is the energy exchange between the surface plasmon polari-tons (SPPs) of the array and the substrate. We focus on the dependence of thetransmission efficiency on the frequency and polarization of the incoming light,as well as on the influence of the array-interface spacing. Sometimes the effectof these parameters turns out to be counter-intuitive, reflecting a complicatedinterplay of several transmission channels.

This chapter is based on P. J. Compaijen, V. A. Malyshev and J. Knoester, Physical ReviewB, vol. 87, p. 205437, 2013

16 Surface-mediated light transmission in metal nanoparticle chains

2.1 Introduction

Metal NanoParticles (MNPs) are known to have scattering cross sections thatcan exceed the MNP’s geometrical cross section by more than an order of mag-nitude, if the excitation is close to the localized surface plasmon resonance of theparticle [72,75]. This property makes MNPs attractive building blocks for nano-optics. The strong near-field enhancement and light scattering of the nanopar-ticles have led to a number of interesting applications in spectroscopy [76] andnano-antennas [28, 29, 31, 32, 77]. An important application, which will be thefocus of this work, is the possibility to guide and propagate an optical excitationthrough a chain of metal nanoparticles. This application was first proposed byQuinten et al. [13] in 1998, and since then this system has attracted a great dealof attention. Guiding, bending, splitting [14,53,78,79], and localization [51,80] ofthe optical excitations of the chain, as well as the time dependent properties [16]have been studied carefully. Important insight into guiding properties of thesesystems was gained after computing the dispersion relations [14,48–50,52,81,82],both for finite and infinite chains. From these studies, it became clear that far-field interactions are very important [48] and that the excitations in the chainare of a Surface Plasmon Polariton (SPP) nature [49].

It is well known that the radiative properties of emitters change if theirlocal environment is modified. The lifetime of a molecule in the proximity of ametal or dielectric substrate can change drastically, depending on its orientationand distance to the substrate [83–85]. Similarly, one can expect changes inthe guiding properties of the MNP chain. For small enough array-substratespacings, not only the radiative but also the non-radiative decay properties willbe altered [86]. It has been shown that under some conditions, SPP modes onthe interface of the substrate can be excited and guided along the array [61].

In this chapter, we modify the local environment of an array of MNPs bypositioning the array close to a (partially) reflective substrate. We discuss theeffect of the substrate on the guiding properties of the array. Understanding thisinfluence is important: in any nano-scale application or experiment, there willbe a reflecting interface in the proximity of the array. From a more fundamentalpoint of view, studying this system is interesting because it allows one to tunethe interactions between the MNPs [87], and, to a certain extent, the radiativelosses of the particles. We consider and compare three different cases: an isolatedarray, an array close to a perfectly reflecting substrate and an array close to a

2.2 System Set Up and Formalism 17

real substrate (silver, in particular). We study how the array’s efficiency totransmit an optical signal depends on the frequency and polarization of theexcitation, as well as on the array-substrate spacing. The MNPs are modeledin the point-dipole approximation, making use of a generalized Drude model fortheir permittivity. The coupling between the MNPs in free space is calculatedusing the full, retarded Green’s tensor for a homogeneous medium. The presenceof the interface is taken into account by constructing a Green’s tensor for thescattered field, within the framework of Sommerfeld’s treatment for the field thatis reflected from the interface.

The chapter is organized as follows. In the next section, we describe thesystem set-up and the mathematical formalism. In Sec. 2.3, the results of calcu-lations of the transmission efficiency (T ) of the array are presented for the threedifferent system choices mentioned above. The dependence of the T on the exci-tation polarization, excitation frequency, and on the array-substrate spacing arediscussed. Finally, in Sec. 2.4 we summarize.

2.2 System Set Up and Formalism

We consider a linear array of N identical spherical nanoparticles, embedded ina medium with permittivity ε1, positioned parallel and at a distance h from asubstrate with permittivity ε2. The nanoparticles have radius a and are equallyspaced with center-to-center distance d (see Fig. 2.2.1). In the calculations,we treat each MNP as a point dipole and describe the interaction between theparticles using the retarded dipole-dipole interactions. The point-dipole approx-imation is accurate if the variation of the field over the particle is small (a� λ,λ being the excitation wavelength), and the inequality d > 3a holds [88]. Weassume a continuous wave (CW) excitation of only the leftmost particle.

The amplitude of the dipole moment p induced in an MNP, subject to anelectric field of amplitude E, can be characterized by the frequency dependentMNP polarizability α(ω),

p = ε1αE , (2.2.1)

where we omitted the argument ω in α for the sake of simplicity. Please note thatalso the time dependence is removed from the above equation, we will focus onthe amplitudes of steady-state solutions only. In the point-dipole approximation,


Figure 2.2.1: Schematics of the system under consideration: a linear array ofidentical and equidistantly spaced spherical MNPs with radius a and center-to-center distance d. The array is embedded in a medium with permittivity ε1 andlocated at a distance h from a substrate with permittivity ε2. Only the leftmostparticle is considered to be excited. The x-axis is parallel to the array and thez-axis is perpendicular to the interface.

α is given by

1

α=

1

α(0)− k2

1

r− 2i

3k3

1 . (2.2.2)

Here, α(0) is the so-called bare polarizability, which can be derived from electro-statics, and k1 = (ω/c)

√ε1 is the wave vector of light in the host medium. The

k1-dependent terms are corrections due to the depolarization field generated in-side the nanosphere [89]. The k2

1-term describes the spatial dispersion correction,whereas the k3

1-term accounts for radiation damping. For particles with radii ofa few tens of nanometers, it is sufficient to use just these first two corrections,so higher order terms can be safely neglected. In the quasi-static approximation(a� λ) the bare polarizability reads [72,73]

α(0)(ω) =ε(ω)− ε1ε(ω) + 2ε1

a3 , (2.2.3)

where ε(ω) indicates the permittivity of the bulk metal. Note that ε stronglydepends on the frequency ω and can be negative in a certain frequency range.The poles of α(0)(ω), i.e. the solutions to the equation Re[ε] = −2ε1, correspondto the localized surface plasmon resonance (see e.g. Ref. [75]).

2.2 System Set Up and Formalism 19

Figure 2.2.2: Illustration of the orientation of the induced image dipoles. Fora dipole polarized perpendicular to the interface, the image dipole will have thesame polarization. For a dipole polarized parallel to the interface, the imagedipole will have anti-parallel polarization.

In a chain, the MNPs will couple to each other due to electromagnetic inter-actions. The electromagnetic field produced by an oscillating dipole embeddedin a homogeneous medium, can be written in the terms of the Green’s tensor ofa homogeneous medium. In Cartesian coordinates, this tensor is given by [4]

GH(r, r0) =eik1R

R

[(1 +

ik1R− 1

k21R

2

)I +

3− 3ik1R− k21R

2

k21R

2

RR

R2

]. (2.2.4)

In this equation, R represents the distance from the source r0 to the detectionpoint r. The corresponding electric field is E = ε−1

1 k21GH(r, r0)p. To account for

the presence of a substrate, we introduce a Green’s tensor GS which describesthe field reflected from the interface. To do this, we make use of the work done bySommerfeld on radio-antennas close to the earth, which can be directly appliedto radiating dipole positioned close to an interface. There is extensive literatureabout this method [74,90]. As an example, we will only present the zz-componentof GS (see Fig. 2.2.1 for the definition of the x, y and z-directions):

GSzz(r, r0) =[1 +

ik1R′ − 1

k21R′2 +

3− 3ik1R′ − k2

1R′2

k21R′2

(z + h)2

R′2

]eikR′R′

− 2i

k21

∫ ∞0

J0(kρρ)k3ρ

k1z

ε1k2,z

ε1k2z + ε2k1zeik1z(z+h)dkρ .

(2.2.5)

This equation gives the z-component of the electric field in medium 1 producedby a z-polarized oscillating dipole in the same medium, located at a height h


above the interface with medium 2. R′ is the distance from the image dipole inmedium 2 to the detection point r, i.e. R′ = [x2+y2+(z+h)2]1/2, ρ = (x2+y2)1/2,ki denotes the wavenumber in medium i, kρ = (k2

x + k2y)

1/2 is the in-plane wave-

vector, and kiz = (k2i −kρ)1/2, is the component of the wave-vector perpendicular

to the interface in medium i. The function J0 is the zeroth order Bessel function.The integrals of GS are evaluated along an appropriate integration path usinga Gauss-Kronrod quadrature. A detailed description of this method is given inRef. [90].

From comparison of Eqs. 2.2.4 and 2.2.5, it becomes clear that the firstpart of Eq. 2.2.5 represents the field produced by a free space dipole located atz = −h (image dipole, see for illustration Fig. 2.2.2), and therefore, it can beidentified as the field scattered by a perfectly reflecting interface. In the caseof a perfect reflector, we have ε2 → ∞, so that, indeed, the second term inEq. 2.2.5 will vanish. Thus, the integral in Eq. 2.2.5 can be interpreted as acorrection to a perfect reflector, and includes effects coming from the excitationof surface modes. For kρ < k1, the reflected waves are propagating away fromthe interface into medium 1, whereas for kρ > k1, the z-component of the wavevector, k1,z, is imaginary and therefore these waves will be surface waves, boundto the interface. A special type of surface wave, the so-called Surface PlasmonPolariton (SPP), occurs when kρ is such that the denominator ε1k2z + ε2k1z = 0,i.e. kρ = k0(ε1ε2/(ε1 + ε2))1/2. In the case of an interface between a metal and adielectric, the conditions ε1ε2 < 0 and |ε2| > |ε1| can be satisfied, which impliesthat kρ is real and kiz is imaginary, i.e., the mode propagates along the interfaceand is bound to it.

With this knowledge in mind, we can set up a system of coupled equationsfor the dipole moment pm of each particle m as

1

ε1

∑m

[ 1

αδnmI− k2

1

(GH(rn, rm) + GS(rn, rm)

)]pm = En . (2.2.6)

Here, GH(rn, rn) should be taken equal to zero 1. Using this equation, thedipole moment of each particle can be calculated for a given input electric fieldof amplitude E0.

1Note that Im[GH(rn, rn)], which describes the radiation damping of the nth particle, isalready taken into account in the polarizability in Eq. 2.2.2

2.3 Results and Discussion 21

2.3 Results and Discussion

In this section, we present and discuss the transmission of optical excitationsthrough a chain of 20 silver nanospheres, choosing a particular set of parame-ters: radius a = 25 nm, center-to-center distance d = 75 nm and distance fromthe chain axis to the substrate h = 50 nm. The point-dipole approximation forthis particular parameters has been verified using a Boundary Element Method(BEM) calculation2 and the validity of this approximation with respect to en-ergy conservation is carefully studied in Ref. [91]. To quantify the transmission,we calculate the transmission efficiency T , which we define as the ratio of themodulus squared of the dipole moments of the rightmost (last) to the leftmost(first) particle,

T =||pN ||2

||p1||2. (2.3.1)

Remember that only the leftmost MNP is driven by the incoming field. This isincorporated by setting En = δn1E0 in Eq. 2.2.6. The permittivity of the silverMNPs and substrate is described with a generalized Drude model:

ε(ω) = 5.45− 0.73ω2p

ω2 + iωγ, (2.3.2)

where ωp = 17.2 fs−1 and γ = 0.0835 fs−1. Equation 2.3.2 provides a goodfit to experimental data in the relevant frequency region [92]. Three differentgeometries of the system will be considered: an isolated chain, a chain in theproximity of a perfect reflector, and a chain close to a silver substrate. We alsoare interested in different excitation geometries: x-polarized, y-polarized, andz-polarized. The results obtained for the T at a fixed array-substrate separationare presented in Fig. 2.3.1 as a function of the wavelength in medium 1.

2.3.1 Isolated chain

First, we consider the T of an isolated chain for different excitation polariza-tions (dashed-dotted curves in all panels of Fig. 2.3.1). As is seen, the T s fory- and z-polarized excitations (transversal) are equal to each other, whereas inboth cases the T is much lower than that for x-polarization (longitudinal). The

2Private communication with J. Munarriz


|p20

|2 /|p1|

2|p

20|2 /|p

1|2

|p20

|2 /|p1|

2

(a) x excitation

(b) y excitation

(c) z excitation

Substrate:NonePerfect reflector

Silver

350 375 400 425 450 475 500Wavelength (nm)

0.070

0.060

0.050

0.040

0.030

0.020

0.010

0

0.006

0.005

0.004

0.003

0.002

0.001

0

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0

λ

Figure 2.3.1: Caption on the following page


Figure 2.3.1: Transmission efficiency T , Eq. 2.3.1, for a chain of 20 identicalspherical silver MNPs (radius a = 25 nm, center-to-center distance d = 75 nm)for three different systems: in red (dashed-dotted) - an isolated chain, in blue(dashed) - a chain above a perfectly reflecting substrate (height h = 50 nm),and in green (solid) - a chain above a silver substrate (height h = 50 nm). Thepolarization of the excitation is denoted in the legend. The black vertical dottedline indicates the plasmon resonance of a single particle (366 nm).

reason for this is that for an inter-particle separation of 75 nm, near field interac-tions are dominant. This interaction is twice as large for longitudinally orienteddipoles as compared to transversally oriented ones, which gives rise to a higherT . For dipoles oriented longitudinal to the chain axis, the radiation is directedoutward of the chain. For long excitation wavelengths more of these dipoleswill be oscillating in phase [48], and therefore more energy will be radiated out,instead of transmitted along the array. This explains the absence of the long-wavelength tail, which is seen for transverse excitation. This tail originates fromthe asymmetry of the polarizability of a single MNP.

Although in the rest of this chapter we limit our discussion to a chain ofN = 20 particles and an inter-particle spacing of d = 75 nm, the effects of chang-ing these parameters can be easily argued from understanding the properties ofretarded dipole-dipole interactions. Increasing d will reduce the inter-particlecoupling strength and, as a result, decrease T . The coupling of dipoles with apolarization perpendicular to the chain (y or z) contains terms depending onR−1, R−2 and R−3, whereas for dipoles with a polarization parallel to the chain(x) this only contains the R−2 and R−3 terms. Therefore the decrease in Twill be stronger for parallel polarization. Adding more particles to the chain,while leaving d unchanged, implies adding more loss channels (both radiativeand Ohmic), and therefore will decrease the signal at the last particle.

The T relates the dipole moment of the last particle of the chain to the first.Thus, increasing the number of particles while keeping the inter-particle spacingunchanged, will give rise to a lower T , because more losses are introduced in thechain.

The position of the T maximum can be explained by calculating the disper-sion relations of the system. As discussed in Ref. [49] the mode with the longestpropagation length is expected to be the mode for which the product of group


velocity and the lifetime is maximized. The frequency of this mode correspondsexactly to the T maximum. For an isolated chain, there is one mode dominatingthe T , but this is not necessarily the case for a chain over a reflector, as we willsee below.

2.3.2 Perfect reflector versus non-perfect reflector

When considering the signal transmission through a chain in the proximity ofa reflector, it is worth noting that the presence of a reflecting interface notnecessarily enhances the T of the chain. In particular, when the MNPs have apolarization parallel to the interface, the T decreases as compared to an isolatedchain of the same length. The reason is that the dipole moment of an MNP,together with its image, forms an effective quadrupole (see Fig. 2.2.2, right), thusreducing the effective electromagnetic interaction between neighboring MNPs. Inthe case of z-excitation, the dipole moment of an MNP and its image forms anenlarged dipole (see Fig. 2.2.2, left), which leads to a stronger coupling betweenthe MNPs, and, as a result, in an increased T as compared to an isolated chain.For both parallel and perpendicular polarization, the interaction with the imagegives rise to a red-shift of the MNP’s plasmon resonance. A detailed study ofthese effects for a single MNP can be found in Ref. [93].

The T spectra for a chain over a perfectly reflecting substrate (dashed curvesin Fig. 2.3.1) show two peaks for x- and z-polarized excitation, instead of a singleone, as in the case of an isolated chain (see Figs. 2.3.1a and 2.3.1c). Inspectingthe directivity of dipole radiation reveals that due to the presence of a reflector, az-polarized dipole can excite in the neighboring particle an x-polarized dipole andvise versa (see Fig. 2.3.2). Thus, whereas x-,y- and z-polarizations are completelydecoupled for an isolated chain, x- and z-polarizations now form coupled modes.One of the peaks originates mainly from x-polarized dipoles and the other oneresults mainly from z-polarized dipoles. This fact is supported by the observationthat, in the case of a chain over a perfectly reflecting substrate, the two peaksoccur at the same wavelengths for both polarizations. For excitation along the x-axis, the left peak is much more intense than the right one, indicating a dominantcontribution to the T of the x-polarized propagating modes. The right (muchsmaller) peak comes from the interface-mediated coupling to the z-polarizedmodes. The situation is reversed when the excitation is z-polarized.

With or without a reflecting interface, y-polarized dipoles will only exciteoscillations of neighboring MNPs with the same polarization. This gives rise to


Log

|E|

Silver substrate

-2

-1

0

1

2

3

z

xy

Figure 2.3.2: Maps of the x-component of the electric field produced by a z-polarized metal nanoparticle. Upper plot – an isolated particle. Lower plot– a particle located 50 nm above a silver substrate. Dashed circles indicatea neighboring nanoparticle. The lower plot shows that in the presence of areflecting interface, there will be a coupling between x- and z-polarization. Inthe absence of a reflector this coupling is identical to zero in the point-dipolemodel. The above figure shows that this approximation is valid if the particlesare small enough and not too closely spaced. Note that the absolute value ofthe field is plotted on a logarithmic scale. The white stripes correspond to zeroelectric field.

a single peak in the T for an isolated chain and for a chain over a perfect reflector.The reason for the weak transmission in the case of a perfectly reflecting substrateis that the reflected field has opposite phase as compared to the field comingdirectly from the MNP. The destructive interference of these two contributionsresults in a weaker interaction along the chain and therefore in a lower T .

As was noticed in Sec. 2.2, a silver substrate differs from a perfect reflector


350 375 400 425 450 475 500

Wavelength (nm)

0.5

1

1.5

2

2.5

3

3.5

|E12,Ag|/|E

12,PR|

λ

Exx

Eyy

Ezz

Figure 2.3.3: The ratio |E12,Ag|/|E12,PR| of the electric field magnitudes pro-duced by the first (1) particle of the chain in the position of the neighboring one(2) for a chain at a height of h = 50 nm above a silver substrate (Ag) and abovea perfect reflector (PR), as a function of the wavelength for different excitationpolarizations. The center-to-center distance between the particles is d = 75 nm.

because not all radiation is reflected from the former; some part of it is transmit-ted into the silver or excites surface modes, like SPPs. At first glance, it mightcome as a surprise that the T of a chain of MNPs close to a non-perfect reflector,like silver, can be higher than in the presence of a perfectly reflecting substrate.The physics of this counterintuitive result can be understood from the couplingof the chain excitations to the SPP-modes of the substrate. Due to the fact thatthese modes are localized on the interface, they give rise to strong fields closeto the interface, and can therefore enhance the coupling between neighboringMNPs. Figure 2.3.3 shows the ratio of the electric field magnitudes |E12,Ag| and|E12,PR| produced by the first particle of the chain in the position of the secondone for a silver substrate (Ag) and a perfect reflector (PR), respectively, as afunction of frequency of wavelength for different excitation polarizations. Notethat this shows the relative electric fields, not the actual field strengths. Fromthis figure, it can be clearly seen that over almost the whole range of wavelengths,the field produced in a neighboring MNP above a silver substrate is larger thanfor the case of a perfectly reflecting substrate. Only for x-polarization at small


wavelengths the perfect reflector gives rise to a stronger coupling, which explainsthe slight decrease of the T for the left peak under x-polarized excitation in thecase of the silver substrate (see Fig. 2.3.1a).

Due to the bound nature of the SPP modes of the substrate, they have a largerwave-vector than light of the same frequency has. Therefore, to excite thesemodes with MNPs, phase-matching conditions dictate that the chain-substratespacing should be smaller that the actual wavelength, i.e., the substrate shouldbe positioned within the near-field region of the MNP. The reason for this isthat the near field of oscillating dipoles contain the required high wave-vectorcontributions. Since the amplitudes of SPP modes decay exponentially awayfrom the interface, a strong influence of these modes on the T is only expectedfor small chain-interface separations. It has been shown that, for the geometryunder consideration, SPP modes can be guided over the interface, along a chainof MNPs [61] and that these surface modes can propagate for long distances [76].In the case of a silver substrate, the transmission of optical signals through theMNP chain will be due to the propagation of collective chain-substrate modes.For substrate-SPP modes, it is well known that the propagation length increasesfor larger wavelengths [76], and therefore also an increased T and a redshiftof the transmission maximum is expected. Dipoles oscillating perpendicular toa metal substrate are known to couple stronger to SPP modes than dipolesoriented parallel, because the latter will excite both s- and p-polarized surfacewaves, whereas the former will only excite surface waves with p-polarization ofwhich the SPP is one. Fig. 2.3.1 shows that in the presence of a silver substrate,the z-polarized peaks of the T increase and shift to the red, as compared with aperfectly reflecting substrate.

Interestingly, also a broadening of the transmission spectrum is observed.Broadening is often associated with a decreased lifetime, for example due to thepresence of an extra decay channel. In fact the presence of a silver interface can beconsidered as a decay source for an MNP, because exciting substrate-SPPs is anextra channel to lose its excitation. Calculation of the effective polarizability ofan MNP above a silver substrate, indeed shows a broadening. A broader spectralresponse of a single MNP will give rise to a broader transmission efficiency aswell, which implies that a larger range of frequencies can be transmitted alongan array of MNPs. This is interesting from the point of view of applications.


2.3.3 Influence of chain-interface separation

a) x−Excitation

b) y−Excitation

|p20

|2 /|p1|

2|p

20|2 /|p

1|2

|p20

|2 /|p1|

2

c) z−Excitation

Wavelength (nm)λ350 370 390 410 430 450

0.050

0.040

0.030

0.020

0.010

0

0.006

0.005

0.004

0.003

0.002

0.001

0

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0

h=50 nmh=60 nmh=70 nmh=100 nm

h=150 nm

Figure 2.3.4: Caption on the following page.


Figure 2.3.4: Transmission efficiency for a chain of 20 identical spherical silverMNPs (radius a = 25 nm, center-to-center distance d = 75 nm) for differentheights h above a silver substrate. The values of h are indicated in the figurelegend. The black, vertical, dotted line indicates the plasmon resonance of asingle particle (366 nm).

For the system under y-excitation, one observes in Fig. 2.3.1 an interestingdifference between the T spectra for a perfectly reflecting substrate and a (real)silver one: in the latter case, the spectrum has an extra peak. To understandthe origin of this peak, we performed a study of the T dependence on the chain-interface separation h (see Fig. 2.3.4). It is seen from Fig. 2.3.4b that, uponincreasing h, the right peak approaches the one for an isolated chain, suggestingthat both have the same physical origin. This is not surprising: for larger h, theinteractions between the MNPs is much larger than the MNP substrate couplingand dominates the T . The red shift of this peak upon reducing h results fromthe formation of effective dipole-image quadrupoles. Since this formation leadsto weaker electromagnetic forces along the chain, it also reduces the magnitudeof the peak.

By contrast, the left peak increases in magnitude upon decreasing h, hintingtowards a strong contribution of SPP modes. A careful examination of the peakbehavior shows that, for small h, the peak position exactly matches the singleparticle plasmon resonance in the presence of the silver substrate. This impliesthat in this case, the MNP-substrate coupling is much stronger than the inter-particle interactions. Therefore, the physical origin of the right peak derives fromthe decay of the plasmon excitation of the leftmost MNP into SPP modes of thesubstrate; these modes are then guided along the chain and excite the otherparticles. A detailed study of these SPP-mediated interactions can be found inRef. [85].

Taking a closer look at the T spectra for x- and z-excitation, we see thatin this case the silver substrate also gives rise to an additional, small and red-shifted peak as compared with a perfect reflector. The main difference between aperfect reflector and a silver substrate in this frequency domain is the possibilityto excite surface modes. Therefore, these extra peaks can also be attributed tothe SPPs contribution. Here, the peak position does not match the single par-ticle resonance, because the inter-particle interaction still is quite strong. Since


the transmission can occur both via x- and z-polarized modes, there are severalcompeting channels. The SPP channel is expected to become weaker when in-creasing the chain-substrate spacing h. However, for x-excitation, increasing halso reduces the coupling to z-polarized modes, thereby lowering the z-polarizedT . Because of that, the SPP channel becomes more important. As a result, thecorresponding peak grows upon increasing h. Naturally, for even larger spacings,this peak will decrease again, since the coupling to SPPs decays exponentiallyaway from the interface and can only occur if the substrate is within the near-zone region of the MNP field.

2.4 Summary and Concluding Remarks

We studied theoretically the transmission efficiency of visible light through alinear chain consisting of equidistantly spaced identical silver Metal NanoParti-cles (MNPs) under CW excitation of only the leftmost particle. Three differentarrangements of the system where considered: an isolated chain, a chain in prox-imity to a perfect reflector, and a chain close to a real reflector (silver substrate).We also considered different geometries of excitation: x-polarized (along thechain axis), y-polarized (perpendicular to the chain axis, parallel to the sub-strate), and z-polarized (perpendicular to the chain axis and the substrate). Wefound a complicated dependence of the transmission efficiency on the polariza-tion and provided simple explanations of the peculiarities observed, making useof the dipole-image picture and coupling of the MNP chain to Surface PlasmonPolariton (SPP) modes.

Surprisingly, for z-polarization the silver substrate leads to a much largertransmission efficiency than a perfect reflector, additionally giving rise to a widerspectral range of the transmission. We attribute this effect to the efficient en-ergy exchange between the substrate-SPPs and the chain-SPPs. The former havelarger propagation lengths than the latter, and therefore provide better condi-tions for the light transmission. In the case of x-polarization, the transmissionefficiency of both systems are comparable.

We also addressed the dependence of the transmission efficiency and its spec-trum on the array-interface spacing (for a silver substrate) and found a compli-cated behavior, depending on the excitation polarization, which, nevertheless,has a transparent physical explanation.

To conclude, we point out that, similar to the case of an isolated chain [49],

2.4 Summary and Concluding Remarks 31

the dispersion relation of the collective electromagnetic excitations of the MNPchain in the presence of a reflecting interface is of great importance to furtherunderstand the energy exchange between the chain and the substrate and thepossibility to guide light below the diffraction limit. This is the topic of ongoingresearch.

Chapter 3

Engineering plasmon dispersionrelations: hybrid nanoparticlechain - substrate plasmon po-laritons

We consider the dispersion relations of the optical excitations in a chain of silvernanoparticles situated above a metal substrate and show that they are hybridplasmon polaritons, composed of localized surface plasmons and surface plasmonpolaritons. We demonstrate a strong dependence of the system’s optical proper-ties on the plasma frequency of the substrate and that choosing the appropriateplasma frequency allows one to engineer the modes to have a very high, verylow or even negative group velocity. For the latter, Poynting vector calculationsreveal opposite phase and energy propagation. We expect that our results willcontribute to the design of nano-optical devices with specific transport proper-ties.

3.1 Introduction

Over the past decade there has been considerable effort in understanding theoptical properties of nano-structured materials and designing sub-wavelength

This chapter is based on P. J. Compaijen, V. A. Malyshev and J. Knoester, Optics Express,vol. 23, no. 3, p. 2280, 2015

34Engineering plasmon dispersion relations: hybrid nanoparticle chain

- substrate plasmon polaritons

optical waveguides [12,21,25,26,63]. In these developments plasmons, collectivefree electron oscillations in metals, have proven to be of particular importance,since they allow localization and guiding of optical signals well below the diffrac-tion limit. Usually, plasmons are divided in two different classes: LocalizedSurface Plasmons (LSPs) and Surface Plasmons Polaritons (SPPs), the formerbeing non-propagating excitations in metal nanoparticles (MNPs) embedded in adielectric environment, whereas the latter generally refer to propagating surfacewaves bound to the interface between a metal and a dielectric. Both of thesemodes have been studied in detail and are well understood [4, 75].

Plasmons resonances of MNPs are very sensitive to their environment, whichmakes them highly valuable for sensor applications [94–97]. Furthermore, theycan couple strongly to nearby plasmonic nanoparticles, yielding collective plas-monic modes, a property which is often exploited in designing sub-wavelengthoptical waveguides [13, 16, 17, 50, 53] and nano-antennas [28, 29, 31, 32, 98]. Sub-strate SPP modes can only be excited if the appropriate phase matching condi-tions are met. A common method to supply the extra momentum is to make useof the electromagnetic near field of a local emitter. Therefore, the LSP mode ofan MNP can excite and couple to the SPP mode of a metal substrate. Thesecollective modes have been described very clearly and elegantly in the frameworkof plasmon hybridization [99,100]. During the past years, the exploration of theproperties of these modes has been the subject of several studies. It was shownthat the optical properties of an MNP above a metal substrate are strongly de-pendent on the polarization and frequency of the excitation and their relativeposition [93, 101, 102]. Furthermore, recent time-dependent studies revealed therelevance of different decay channels [103].

A thoroughly studied example of a plasmonic waveguide is a linear chainof closely spaced MNPs. Such chains can support propagating plasmon modesand have already been used to demonstrate guiding, splitting, bending [14, 53,61,79,104–106] and localization [51,80] of optical excitations, all well below thediffraction limit. The chain modes can be described as coupled LSPs [16, 48],and actually, for chains longer than the resonance wavelength in the surroundingmedium, represent polariton modes [49]. The polaritons result from the strongcoupling of the LSPs to electromagnetic radiation and will arise when the effectsof retardation are important [107], otherwise the excitations are referred to asstatic chain modes.

Both substrate and chain SPPs have attracted considerable attention over

3.2 Formalism 35

the past years because they are promising candidates for applications in nano-optical waveguides, antennas and sensors. Many nano-optical devices includecoupled MNPs as well as metal substrates, meaning that generally both typesof SPPs will be present. Understanding and exploiting the full functionality ofthese compound devices requires a deep understanding of how these two modesinteract with each other. The coupling between both types of SPPs has beenstudied recently by several groups, considering linear chains [108, 109] and twodimensional nanoparticle arrays [110,111] above metal substrates. These systemwere investigated both experimentally [111] and theoretically [108, 109, 111]. Ithas been shown that the coupling between the chain and substrate plasmonmodes is strong, and that the dispersive properties of hybrid modes can besignificantly different from the individual, uncoupled modes.

In this chapter, we study the dispersion relations of the SPP modes of acompound system: a one dimensional chain of silver nanospheres and a metalsubstrate. In contrast to previous work, in which only one particular metalsubstrate has been studied [108], we show that the plasma frequency of themetal substrate plays a dominant role in determining the optical properties ofthis compound system. Choosing the appropriate plasma frequency, i.e., byselecting a specific substrate, the hybrid polaritons in this system can have veryhigh, almost zero, or even negative group velocities. Furthermore, the dispersionrelations of the compound system turn out to be highly polarization dependent.These insights open up a new possibility to engineer the optical response ofcoupled plasmonic systems. We present the dispersion relations for a wide rangeof plasma frequencies and discuss them with the aid of plasmon hybridizationand dipole-dipole interactions. Finally, we use real material parameters to givean example of a system for which the negative index modes can be observed, andcalculate the Poynting vector for one of these modes. In the conclusion, we willdiscuss the relevance of our findings for the development of nano-optical devices.

3.2 Formalism

We consider a one dimensional chain of identical and equally spaced silver nanosphereswith radius a = 25 nm, center-to-center spacing d = 75 nm, embedded in glass(ε1 = 2.25) at a height of h = 50 nm above a metal substrate, as shown inFig. 3.2.1. Each nanosphere is treated as a point dipole. The MNP-MNP andMNP-substrate interactions are described using dyadic Green’s functions which



a

d

h

Medium 2Medium 1

z

xy

Figure 3.2.1: A schematic illustration of the considered system: a linear chainof equally spaced, identical silver nanospheres, embedded in medium 1, situatedabove and parallel to a metallic substrate (medium 2).

include retardation effects. For the configuration considered here, the dipole ap-proximation provides an accurate description of the system at hand [91,112]. Asto including the contribution of higher order multipole effects, see Ref. [113].

Within the dipole-approximation, the dipole moment induced in the MNPby an electric field with amplitude E, is p = ε1αE. Here, ε1 is the dielectricconstant of the host medium, whereas the optical properties of the MNPs aredescribed by the so-called polarizability α. For spheres of radius much smallerthan the wavelength in the host medium, α is given by

1

α(ω)=

1

α(0)(ω)− k2

1

a− 2i

3k3

1 , (3.2.1)

where α(0) is the bare polarizability, as is derived from electrostatics [72,73]. Thelast two terms, depending on the wavevector of light in medium 1, k1 =

√ε1ω/c,

correct α(0) for spatial dispersion and radiative damping respectively: we aredealing with non-static fields [89]. In terms of the frequency dependent permit-tivity of the sphere, ε(ω) and the radius of the sphere a, the bare polarizabilityhas the form

α(0)(ω) =ε(ω)− ε1ε(ω) + 2ε1

a3 . (3.2.2)

Since we consider silver nanoparticles, we use a generalized Drude responsefunction which is fitted to experimental data [92], ε(ω) = 5.45 − 0.73 ω2

p/(ω2 +

iωγ), with plasma frequency ωp = 17.2 rad fs−1 and damping coefficient γ =0.0835 fs−1.

3.2 Formalism 37

The dipole moment induced in one of the MNPs is proportional to the electricfield acting on that particle. This field consists of the external excitation fieldEext, the field produced by all the other particles in the chain and the fieldreflected from the substrate. The dipole moment of each MNP in a chain of Nparticles can be found by solving the following system of equations

1

ε1

∑m

[ 1

αδnmI− k2

1(G0(ω; rn, rm) + Grefl(ω; rn, rm))]pm = Eext

n . (3.2.3)

Here Eexti and pi are 3 × 1 vectors, representing x,y, and z components of the

polarization, and I is the 3 × 3 unit tensor. G0(ω; rn, rm) is the 3 × 3 Green’stensor describing the electric field produced at rn, by a unit dipole (oscillatingwith frequency ω) that is located at rm. In a homogeneous environment

G0(ω; rn, rm) =[I +∇∇k2

1

]exp(ik1|rn − rm|)|rn − rm|

. (3.2.4)

Grefl is the Green’s tensor that describes the field that is reflected from the sub-strate. The tensor components can be derived by expanding the spherical wavesof the dipole field into plane waves with the appropriate Fresnel coefficients.This procedure, first applied by Sommerfeld, has been extensively studied inliterature. For the expressions used in this research, we refer to Refs. [4, 74, 90].The optical properties of the substrate enter Grefl through the frequency de-pendent permittivity of the metal, for which we will assume a standard Drudeform: ε2(ω) = 1 − ω2

p/(ω2 + iωγ). Throughout this chapter, we will consider

different values for the plasma frequency, keeping the damping coefficient fixedat γ = 0.0835 fs−1, the value for silver. The properties of the plasmon mode ofthe substrate are contained in the Fresnel coefficient for p-polarized (TM) light.

In order to calculate the dispersion relation for the system under consider-ation, we need to find the eigenmodes of the set of coupled equations (3.2.3).These modes are obtained by solving the homogeneous system, i.e., setting theexternal excitation field Eext

m = 0. Since both radiation and ohmic damping areaccounted for, losses can be quite significant and therefore, Eq. (3.2.3) will onlyhave solutions at complex frequencies, commonly referred to as normal modes fre-quencies. The imaginary parts of these frequencies represent the mode quality.Diagonalizing this system of equations, evaluated at the obtained frequencies,will give the corresponding modes and from this a dispersion relation can beconstructed [48].



Although this method is exact and the only correct method if a short chain ofnanoparticles is considered, it suffers from a major difficulty: the Green’s tensorshave to be evaluated at complex frequencies, which is numerically very challeng-ing, especially, for the Sommerfeld integrals contained in Grefl. A workaround forthis problem is to consider the limit of an infinite chain and apply periodic bound-ary conditions, as is done in the so-called eigendecomposition method [114]. Ithas been shown that for a chain of MNPs this limit can be reached for chains asshort as 10 particles [79]. This condition is easily met in practice for nanoparti-cle waveguides. Furthermore, studying polariton-polariton interactions requiresa system that is much longer than the wavelength.

In the eigendecomposition method, Bloch’s theorem is used to write pm =p exp (iqmd) and Eext

n = Eext exp (iqnd), where q is the quasi-wavevector of theBloch mode and d the spacing between the nanoparticles. Substituting this inEq. (3.2.3) reduces the system to three equations, one for each component ofthe polarization. For an infinite chain in the geometry under consideration, thisresults in [108]

1

ε1

[ 1

αI− k2

1

∞∑m=−∞

(G0(ω; 0x,mdx) + Grefl(ω; 0x,mdx))eiqmd]p = Eext . (3.2.5)

The part of this equation that is between brackets, can be thought of as aninverted generalized polarizability of the system, α. Now, instead of finding thecomplex roots of this equation, calculation of the imaginary part of α as a func-tion of the real frequency ω and the wavevector q gives dispersive properties ofthe plasmon modes of the system, as well as the quality of these modes [114].Physically, this approach corresponds to the calculation of an absorption spec-trum, where one is not only able to tune the frequency of the excitation, butalso the wavevector. Here, in contrast with the exact method, the mode quality,or lifetime, of the mode is reflected in the width of the spectrum: the wider themode, the shorter the lifetime.

3.3 Results

Before discussing the dispersion relations of the compound system, it is insightfulto investigate the dispersion relations of the chain and substrate surface plas-mons, assuming them uncoupled. These relations have been published before

3.3 Results 39

and are discussed in detail in e.g. Refs. [48,49,115]. In order for this work to bereasonably self-contained, we briefly discuss the most important results.

As was mentioned in the introductory part, plasmon polaritons result fromthe strong coupling between plasmons and electromagnetic radiation, which im-plies a characteristic anti-crossing of the plasmon dispersion branch with thelight line. In the MNP chain, this concerns the mixing of the LSPs with elec-tromagnetic radiation. For the metal substrate it is in fact the non-propagatingsurface plasmon (SP), i.e. the large wave vector limit of the SPP, which is mixingwith radiation. This gives rise to well-known substrate SPP.

This is illustrated in Fig. 3.3.1, where the solutions to Eq. (3.2.3) are shownfor different polarizations, setting Grefl = 0 (uncoupled situation). In this case,the chain-substrate interaction is zero, and therefore, it is not necessary to com-pute the Green’s tensor of the metal substrate and the exact method of findingthe complex roots of Eq. (3.2.3) can be easily applied. In Fig. 3.3.1 only the realpart of the frequency is depicted, the imaginary part, i.e. the mode quality, isnot shown.

Figure 3.3.1(a) displays the completely uncoupled situation, excluding retar-dation: the quasistatic longitudinal and transverse chain modes (green squaresand red circles, respectively), the light line (black dashed) and the non-propagatingsurface plasmon of the substrate ωsp = ωp(1 + ε1)−1/2 (black dotted). In the ge-ometry under consideration (see Fig. 3.2.1), longitudinal modes consist of dipoleswith x-polarization, while transverse modes contain y- or z-polarized dipoles.For an isolated chain these two transverse modes are degenerate. Comparingthe longitudinal and the transverse modes, we see from the slope of the disper-sion that the inter-particle interactions for the two modes have opposite signs.From the energy difference between the high and low wavevector modes, we candeduce that longitudinal dipoles are interacting stronger than transverse ones.This is exactly what would be predicted from the near field interaction of pointdipoles [73]. The right plot shows the dispersion relations when retardation isincluded, implying the plasmon modes are interacting with the light line. Theeffect of radiative interaction on the chain plasmons can be seen by compar-ing Figs. 3.3.1(a) and 3.3.1(b). It is clear that an avoided crossing with thelight line occurs only for the chain modes with a polarization perpendicular tothe chain axis (y- and z-polarization), indicating a strong coupling of the staticchain modes to electromagnetic radiation. The reason for this is that a dipoleemits mainly along the direction perpendicular to the dipole orientation. The



0 0.01 0.02 0.03 0.043

3.5

4

4.5

5

Re

[ω]

(ra

d/f

s)

0 0.01 0.02 0.03 0.04

a) b)

wavevector q (nm−1

)

Transverse

Longitudinalω

sp

Light line

SPP

Figure 3.3.1: Dispersion relations for non-interacting chain and substrate plas-mons are shown, excluding (left panel) and including (right panel) retardation.These relations are obtained by solving Eq. (3.2.3) with N = 20, r = 25, d = 75,h = 50 nm, and ε1 = 2.25 (glass). The substrate is described by a Drude modelwith ωp = 6.00 rad/fs and γ = 0.0835 rad/fs. ωsp is the non-propagating surfaceplasmon (SP) of the substrate, given by ωp(1 + ε1)−1/2. The wavevector q runsfrom 0 to π/d, the edge of the first Brillouin zone.

branch of the substrate SPP is added manually, using the well-known dispersionrelation kspp = [ε1ε2(ω)/(ε1 + ε2(ω))]1/2ω/c. From the figure it is clear that thisbranch arises from the anti-crossing of the non-propagating surface plasmon ofthe substrate and the light line.

We now turn to the discussion of the dispersion relation for an MNP chainclose to a metallic substrate. If the distance between the MNP-chain and thesubstrate is smaller than the wavelength, the near-field of the MNPs can exciteSPPs on the substrate and hence, will couple the chain and substrate plasmons,which are shown uncoupled in Fig. 3.3.1(b). In this case, the dispersion relationsare easier to calculate using an eigendecomposition, i.e. by solving Eq. 3.2.5[108, 114]. To compare the two different methods, Fig. 3.3.2(a) also shows thedispersion relations for the isolated chain. It is clearly seen that they are verysimilar to the ones shown in Fig. 3.3.1. In addition, the dispersion relations ofFig. 3.3.2 also show the quality of the modes. For wavevectors q to the left of the

3.3 Results 41

3.0

3.5

4.0

4.5

5.0

3.5

4.5

5.5

Log[Im[α]]

ω(r

ad/fs

)

3.0

3.5

4.0

4.5

5.0ω

p=6.00

ωp=7.40

0 0.01 0.02 0.03 0.043.0

3.5

4.0

4.5

5.0

wavevector q (nm−1)0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04

Chain over a substrate:

Chain over a substrate:

Isolated chain Isolated MNP

x y,z

mainly x mainly z

mainly x mainly z

y

y

a)

b)

c)

rad/fs

rad/fs

AS

S

S

AS

S

S

S

S

ASAS

AS AS

Figure 3.3.2: Dispersion relations of a chain of silver MNPs, with r = 25, d = 75,h = 50 nm, and ε1 = 2.25, calculated using the eigendecomposition method.Plotted is Im[α] on a logarithmic scale as a function of the frequency ω and quasi-wavevector q. The left column of b) and c) shows the result for x-polarizationfor two specific plasma frequencies, the middle column for y-polarization and theright column for z-polarization. Animations presenting these dispersion relationsfor a wide range of plasma frequencies for x, y and z-polarization can be found inthe supplementary material of [116]. (Caption continued on the following page)



Figure 3.3.2: (continued) The solid white line gives the substrate surface plas-mon polariton (SPP) dispersion. The steep dashed and dotted horizontal linesrepresent the light line and the substrate surface plasmon (SP), respectively. Theripples that occur close to the light line for the transverse mode of the isolatedchain, result from the fact that a finite, but very long, chain is used for the cal-culations. Close to the light line the slowly decaying radiative interactions arevery important and effects of the finiteness of chain can be seen. The labels AS,S, || and ⊥ refer to the sign of the coupling and the polarization of the hybridpolaritons, explained in more detail in Fig. 3.3.3. The wavevector q runs from 0to π/d, the edge of the first Brillouin zone.

Ene

rgy

AS

S

S

AS

Figure 3.3.3: Hybridization diagram for the plasmons of an MNP and a metalsubstrate. The black and blurred arrows indicate the polarization of the inducedcharge distributions in the MNP and substrate, respectively. It is shown thatthe LSP of the MNP hybridizes with the SP of the substrate into a symmetric(S) and anti-symmetric (AS) mode. The sign of the interaction is different forpolarization parallel (||) or perpendicular (⊥) to the substrate.

light line, we see that the modes have much higher losses than those to the right ofthis line. The reason is that the chain modes will only radiate if their wavevectoris smaller than that of light in the surrounding medium. As a reference for the

3.3 Results 43

strength and spectral position of the resonance, the polarizability of an isolated,single MNP is plotted as well. Figures 3.3.2(b) and 3.3.2(c) display the dispersionrelations for the chain at a height of 50 nm above a substrate characterized bya plasma frequency of ωp = 6.00 and ωp = 7.40 rad/fs, respectively. Animationspresenting the dispersion relations for a wide range of plasma frequencies can befound in the supplementary material of [116]. In these animations the plasmafrequency of the substrate is stepwise increased, such that the substrate plasmonmode is scanned through the plasmon resonance of the chain.

One has to be careful when the interaction between the chain and the sub-strate is taken into account. Not only will the longitudinal and transverse chainmodes mix with the modes of the substrate, the latter also introduces a couplingbetween the x- and z-polarized chain modes, lifting the degeneracy between thetransverse chain modes. Although the x- and z-polarized modes are coupled, itis still possible to separate them because in each branch the nanoparticles willhave a dominant polarization along one direction, denoted by ’mainly x’ and’mainly z’ in the figures. The coupling between x- and z-polarized is mediatedby the substrate and therefore it has a phase difference with respect to the ’di-rect’ MNP-MNP interaction. This implies that for large wavevectors it is moredifficult to distinguish between the x- and z-polarized modes due to the strongdifference in phase between neighboring particles. A consistent way to distin-guish between the different modes is by separating them by the phase of thedipole moment rather than by magnitude. This subtlety only occurs at the edgeof the Brillouin zone, for smaller wavevectors both methods of separation givethe same results. Therefore the modes can still be referred to as ’mainly x’ or’mainly z’.

As is well known, two interacting modes will hybridize into two new modes,a symmetric (S) and an anti-symmetric (AS) mode. This is shown in Fig. 3.3.3,where the hybridization scheme for an MNP above a metal substrate is given[99, 100]. The arrows indicate the polarization of the charge distributions thatare induced in the MNP and the substrate, respectively. From dipole-dipoleinteractions, one can calculate that the interaction between the MNP and thesubstrate is stronger when the polarization is perpendicular to the substrate.Furthermore, the sign of the interaction is opposite for parallel and perpendicularpolarizations. This explains the interchange of the S and AS modes, as well asthe energy separation between them.

The interaction strength between the two modes does not only depend on the



polarization, but also on their spectral separation. In Fig. 3.3.2(b) the disper-sion relations are shown for the chain close to a substrate with plasma frequencyωp = 6.00 rad/fs. For this ωp, the surface plasmon frequency ωsp is spectrally faraway from the chain mode, and therefore, the interaction is small. Comparingthese plots with those of the isolated chain, we see that the upper branches (S)of the dispersion relations for x- and y-polarization are indeed very similar tothose of the isolated chain, in particular for y-polarization. For x-polarization theresemblance is smaller because in addition to the mixing of the chain and sub-strate modes, also the x- and z-polarized modes are coupled. As was mentionedbefore, distinguishing both polarizations is not so clear for large wavevectors.Comparing Figs. 3.3.2(a) and 3.3.2(b) carefully, reveals that, for the ’mainly x’configuration, the part of the upper branch that is below the lightline actually isvery similar to the transverse branch of the isolated chain. Similarly, the largewavevector part of the upper ’mainly z’ polarized branch is comparable to thelongitudinal mode of the isolated chain.

The lower frequency branch is always positioned below ωsp, between ωp andωsp no fields can penetrate the metal substrate. Due to the large spectral separa-tion in Fig. 3.3.2(b), the lower branch (AS) has very low intensity. Interestingly,for x-polarization the lower branch results from the hybridization of the chainSPP with the substrate SP, rather than the SPP. This shows that, similar to theisolated chain, the radiative interaction is not important for this polarization.The lower branch for y-polarization (AS) does show influence of the substrateSPP mode: once this branch has been crossed, the mode intensity goes to zero.This is explained by the fact that substrate SPPs only exist for TM polarization,and therefore, the SPPs excited by y-polarized dipoles, will not propagate alongthe chain, but instead perpendicular to it, channeling the energy away from thechain.

Contrary to x- and y-polarizations, the presence of the substrate stronglyalters the dispersion relation for z-polarization. From this we can conclude thatfor z-polarization the interaction with the substrate is more important, as is alsopredicted from dipole-dipole interactions (see Fig. 3.3.3). The upper branch forthis polarization is the AS mode, for which the interaction between the particles ispartially canceled by the contribution from the substrate. This weak interactiongives rise to the almost flat dispersion that is observed. The opposite occursfor the lower branch (S): the surface contribution is in phase with the directcontribution, yielding a strong interaction and a steep dispersion. This mode is

3.3 Results 45

strongly related to the substrate SPP mode (white solid line), only with a smallredshift due to the symmetric coupling with the chain.

Figure 3.3.2(c) shows the dispersion relations for the same system as before,but now with ωp = 7.40 rad/fs and hence, a smaller spectral separation. Thefeatures of the dispersion relations are very similar to those discussed above.However, since the substrate is now spectrally closer to the chain, the modesof the former carry more intensity. For x-polarization, we can now see a veryclear anti-crossing between the chain SPPs and substrate SP modes, indicatingstrong hybridization of the chain-substrate modes. Again, a very strong influ-ence of the substrate SPPs can be found for z-polarization, especially, at lowerfrequencies. Here, there are modes with a very steep dispersion, thus with a highgroup velocity, and a high mode quality, i.e., low losses. Therefore, these modeshave a long propagation length, that is promising for sub-wavelength guidingapplications [49,112]. Interestingly, for z-polarization no qualitative difference isobserved as a function of the plasma frequency, eventhough the particle-substrateinteraction is the strongest for this polarization. The reason for this counterin-tuitive effect lies in the fact that the large interaction results in a large splitting.In this case the influence of the plasma frequency is less prominent, because theSP of the substrate is located in the gap between the two branches. Thereforeno crossing or mixing with this line occurs and no qualitative difference arises.

The low frequency branches of the x-polarized dispersion relations have an-other interesting feature: the slope of these modes is negative over a wide rangeof wavevectors. For lossless media, a negative slope is associated with a negativegroup and energy velocity, indicating that the direction of energy flux is oppositeto the phase advance. These type of modes are commonly referred to as negativeindex modes [117, 118]. The animations in the supplementary material of [116]show that, for a large spectral overlap between the substrate surface plasmonand the chain SPPs, the upper and lower branch repel each other. The lower,anti-symmetric branch is pushed down by the upper, symmetric branch.

So far we have only considered artificial metal substrates, characterized bya certain plasma frequency ωp and a fixed damping coefficient γ = 0.0835 fs−1.To observe the mode mixing effects that are presented in this chapter, the mostimportant parameter is the spectral overlap between the chain and substrateplasmon modes. This implies that depending on the size and material of theMNPs, different substrates can be selected. As an illustration, we will show thatthe negative index mode that is observed for x-polarization in Fig. 3.3.2(c), is



wavevector q (nm−1)

ω(r

ad/fs

)

x−polarization platinum

a) b)

−0.04 −0.02 0 0.02 0.043

3.5

4

4.5

5

3.5

4

4.5

5

5.5

6

y position (nm)

zpo

sitio

n(n

m)

−50 0 50−50

0

50

100

150

−50 0 50−1

−0.5

0

0.5

1

Poy

ntin

g ve

ctor

(arb

. uni

ts)

MNP MNP

substrate substrate

A

B

A B

Dispersion Poynting vector

Figure 3.3.4: a) Similar to Fig. 3.3.2, but now for a Platinum substrate (ωp =7.81 rad/fs, γ = 0.1051 1/fs [119]). The full first Brillouin zone is shown for thex-polarized mode. b) The x-component of the Poynting vector (i.e. parallel tothe chain axis) is plotted over a surface perpendicular to the chain axis, bisectingthe chain between two neighboring particles. Mode A corresponds to a point onthe lower branch with a negative wavevector, mode B to a point on the upperbranch with a positive wavevector (also indicated in a) ). The integrated value ofthe energy flux of mode B is about 4 times larger than that of mode A, and bothvalues are positive. Mode A has opposite phase and energy flux, and therefore,it is a negative index mode.

also present when a realistic platinum substrate is considered. The permittivityof platinum is fitted with a simple Drude model, yielding ωp = 7.81 rad/fs andγ = 0.1051 fs−1 [119]. Figure 3.3.4(a) shows the dispersion relation of the x-polarized modes for a chain of silver MNPs close to a platinum substrate. Dueto the somewhat larger damping coefficient for platinum the mode quality is a bitsmaller as compared to Fig. 3.3.2(c), but the key features are still there. To takea closer look at the direction of the phase and the energy flow in the negativeindex modes, the full first Brillouin zone is plotted. A dispersion relation showsthe wavevector and energy of all the modes in the system, however not all thesemodes can be accessed by optical excitation. Due to causality the energy fluxof a mode must be in the same direction as its excitation. Therefore, insteadof considering negative group velocity (i.e. negative energy flux) and positive

3.3 Results 47

phase, one has to consider positive group velocity and negative phase. Thus theexcited mode will be on the left side of the dispersion diagram, instead of on theright side.

In a system in which losses are present there is not a straightforward con-nection between group and energy velocity anymore [118]. The clearest way toexamine the direction of the energy flux is to explicitly calculate the directionof the Poynting vector with respect to the phase advance. In the Green’s tensorformalism applied here, this vector can be easily calculated using [4]

S =1

2Re[E×H∗], with

E =k2i

εiGp, H = −iω[∇× G]p. (3.3.1)

In the above equation G is the electromagnetic Green’s tensor for a layeredmedium and ki and εi are representing the wavevector of light and the dielectricconstant in the ith medium, respectively.

Figure 3.3.4(b) displays the x-component of the Poynting vector, i.e., thecomponent parallel to the chain axis, calculated through a plane perpendicularto the chain axis, bisecting the chain in between two particles. In general theenergy flux can be in any direction. Since the flux directed along the y or z-axis implies energy flowing out of the chain, only the x-component of the fluxis relevant for studying energy propagation in this system. The total energythat is propagating in the x-direction can be obtained by integrating the x-component of the Poynting vector over the defined plane. The energy flux hasbeen calculated for two specific modes, one with negative phase (q = −0.02451/nm) and positive group velocity, the other with positive phase (q = 0.02451/nm) and positive group velocity. Furthermore, the former is on the lowerbranch, whereas the latter is on the upper. Note that for this calculation it isassumed that it is possible to excite these modes exclusively. The position ofthe modes on the dispersion relation is indicated with the letters A and B. Asexpected, the integrated value of the energy flux is positive for both modes. Thisimplies that mode A indeed is a negative index mode with opposite energy fluxand phase advance. The total energy flux of mode B is about 4 times as largeas that of mode A. This is supported by the observation that mode B has aslarger group velocity and a higher mode quality. Taking a closer look at thecolor profile of the Poynting vector, one can see that it resembles those in a



Metal-Dielectric-Metal (MDM) waveguide, if the substrate and MNP chain areconsidered as the metal cladding layers, see Refs. [118,120–122]. Comparing theprofile of the two selected modes in between the substrate and the MNP, modeA has a continuous blue coloring, whereas mode B has two red areas separatedby a white space. The white area represents a zero in the energy flux, similarto a node in that can be observed in higher order modes. This matches withthe fact that mode B lies on the upper, high energy branch of the dispersionrelation.

Interestingly, mode B has a negative energy flux in the dielectric gap betweenthe chain and the substrate, even though the integrated energy flux is positive.In MDM waveguides, the negative contribution is usually found inside the metallayers. The fact that it occurs here in the dielectric, opens up the possibility tostudy this phenomenon in more detail, f.e. by placing small emitters in the gapbetween the chain and the substrate.

3.4 Summary

We found that the presence of a metal substrate strongly alters the dispersionrelations of a chain of metal nanoparticles. The properties of the resulting hy-brid chain-substrate SPPs depend to a large extent on the spectral overlap ofthe plasmon modes of the chain and the substrate. In particular, for a givennanoparticle chain, choosing the appropriate plasma frequency, i.e., selectingthe appropriate substrate, results in a device in which the plasmons will eitherhave a very high, very low, or negative group velocity. The latter is confirmedby considering a realistic device with a platinum substrate. Poynting vectorcalculations reveal that the negative group velocity corresponds to anti-parallelenergy flux and phase advance and thereby confirm that these modes have a neg-ative refractive index. In addition, the optical properties of the system stronglydepend on the polarization and the frequency of the excitation. Our findingsshow that this simple system gives rise to a wide range of physical phenomena,all of which can be explained very well by considering interacting dipoles andthe plasmon hybridization model. The model which we have applied is an easytool to calculate the dispersion relations and can easily be extended to dipoleemitters embedded in more general layered media, e.g. Metal-Dielectric-Metalor Dielectric-Metal-Dielectric geometries.

Chapter 4

Elliptically polarized modes forthe unidirectional excitation ofsurface plasmon polaritons

We propose a new method for the directional excitation of surface plasmon po-laritons by a metal nanoparticle antenna, based on the elliptical polarizationof the normal modes of the antenna when it is in close proximity to a metallicsubstrate. The proposed theoretical model allows for the full characterization ofthe modes, giving the dipole configuration, frequency and lifetime. As a proof ofprinciple, we have performed calculations for a dimer antenna and we report thatsurface plasmon polaritons can be excited in a given direction with an intensityof more than two orders of magnitude larger than in the opposite direction. Fur-thermore, using the fact that the response to any excitation can be written as asuperposition of the normal modes, we show that this directionality can easilybe accessed by exciting the system with a local source or a plane wave. Lastly,exploiting the interference between the normal modes, the directionality can beswitched for a specific excitation. We envision the proposed mechanism to be avery useful tool for the design of antennas in layered media.

P. J. Compaijen, V. A. Malyshev and J. Knoester, Optics Express, accepted

50Elliptically polarized modes for the unidirectional excitation of

surface plasmon polaritons

4.1 Introduction

Plasmons hold great promise for the field of nanophotonics, since these electro-magnetic waves can be confined well below the diffraction limit and can propagateover distances of several tens of microns [12,21,25,26,63]. These properties makeplasmonic structures the ideal platform for studying single emitters, as well asfor designing nanophotonic circuitry [22,29,47,123–126].

Of particular importance are Surface Plasmon Polaritons (SPPs), propagat-ing surface waves which can be excited on the interface between a dielectric anda metal. Since these are guided modes, they cannot be excited by direct illumi-nation, but some phase-matching techniques are needed. A common approach isto use a local emitter, for example the plasmon resonance of a metal nanoparticle(MNP), as a near-field source for the excitation of the guided modes. The pres-ence of the substrate will also affect the optical response of the MNP, the plasmonof the MNP will mix with the modes of the substrate. A very clear description ofthis phenomena can be given in the framework of plasmon hybridization [99,100].For a single MNP close to a substrate, resonance frequency shifts and high fieldintensities in the gap have been observed, as well as the interplay between thelocalized plasmon of the MNP and the propagating SPP [70, 71, 101, 103, 127].When a chain of MNPs close to a metallic substrate is considered, the hybridmodes of the system are shown to have interesting properties, such as guiding,bending and negative phase velocity [61,87,109,112,115,116].

For applications in nano-photonic circuitry and sensing, it is desired to be ableto excite guided modes with a preferred direction. An important step towardsachieving this goal was taken by hybridizing a plasmonic antenna with a dielectricwaveguide. The antenna couples the free-space radiation directionally to theconfined modes of the waveguide and vice versa [62]. Plasmonic antennas canbe created from collections of metal nanoparticles (MNPs) and can be designedto have specific properties, like localization and directivity [28, 29, 31, 32, 51, 77,80, 98]. Most commonly used are the so-called phased array antennas, in whichthe directionality is obtained by engineering the constructive and destructiveinterference between the antenna elements. Based on this approach, antennashave been created that allow for polarization-dependent, directional excitationof SPPs [66, 67]. Alternatively, grooves in a metallic substrate have been usedas antenna elements and directionality is obtained due to the interference ofthe electric field scattered from separate grooves [128–130], or even due to the

4.1 Introduction 51

interference of the modes of a single, structured groove [131].

In Refs. [70, 71, 132], it was reported that directional excitation of guidedmodes can even be obtained from a single metal nanoparticle, provided its in-duced dipole moment is out-of-plane circularly polarized. In particular, thesereferences have studied the directional excitation of SPPs by circularly polarizedMNP above a metal substrate. The phase difference between the dipole momentparallel and perpendicular to the waveguide gives rise to constructive interferencein one direction, and destructive inference in the opposing direction. Optimizingthe system can even lead to a complete cancellation of SPPs in a particular direc-tion [70]. The first experimental realization of the directional excitation of SPPsby using circularly polarized light was achieved using a nanoslit rather than ametal nanoparticle [133]. Later, the effect was demonstrated for a single opticalnano-antenna by exciting it with a polarization tailored beam [132]. Directionalemission from circularly polarized dipoles can also be explained in the contextof angular momentum matching. For surface polaritons, spin and propagationdirection are coupled, and therefore a circularly polarized dipole will preferen-tially excite SPPs with one particular propagation direction [134, 135]. Overthe last few years, there has been a great effort in getting a better fundamentalunderstanding of the important physics, see e.g. the discussion in [136, 137],and finding new implementations of this phenomenon [68, 69, 138, 139]. A clearoverview of recent progress is given in [140].

In order to achieve a circularly, or in general elliptically, polarized dipole,the external excitation needs to be elliptically polarized. In practice, this willlimit the applicability of this method and, furthermore, it can be complicatedto excite the system with a specific elliptically polarized beam when the systemis embedded in a layered structure. In this chapter, we will demonstrate anew mechanism for obtaining directional near-field excitation of SPPs, basedon the elliptical polarization of the plasmonic modes of coupled MNPs above ametallic substrate. The method for finding the collective modes of the hybridsystem is presented and, as a proof of principle, we will discuss the results for anasymmetric dimer above a metallic substrate and show that it contains normalmodes which couple directionally to SPPs. Since this effect originates from themodes of the system, the directionality is inherent, i.e. it does not rely onelliptically polarized excitation.



4.2 Formalism

The calculations presented in this chapter are based on a coupled dipoles ap-proach, i.e., the MNPs are treated as point dipole scatterers and the couplingbetween the MNPs is taken into account using dipole-dipole interactions, includ-ing retardation effects. The interaction with the substrate is calculated from therecipe prescribed by Sommerfeld. For a dimer of MNPs, which is the system ofinterest, the geometry of the system under consideration is shown in Fig. 4.2.1.The radii of the particles are given by a1 and a2 respectively, the center-to-centerspacing between the particles is given by d, and the dimer is situated at a heighth from the substrate. The formalism presented below may easily be generalizedto arrays of an arbitrary number of MNPs placed parallel to the substrate. Westress that all calculations have been performed under conditions for which thepoint dipole approximation is known to be valid [88, 112], i.e. the particle radiiare much smaller than the wavelength, and the spacing and height satisfy d ≥ 3aand h ≥ 2a. Therefore, the dipole model provides an excellent framework to per-form the calculations, without obscuring the important physical phenomena thatare necessary for a deep understanding.

In order to understand the collective resonances of the hybrid system, itis useful to first consider only a single isolated spherical MNP. Assuming theparticle is much smaller than the wavelength, the dipole moment induced in theMNP will be p = ε1αE, where ε1 is the permittivity of the dielectric (medium 1),E is the electric field exciting the particle, and α(ω) is the frequency dependentpolarizability of the MNP, given by

1

α(ω)=εMNP (ω) + 2ε1εMNP (ω) + ε1

1

a3− k2

1

a− 2i

3k3

1. (4.2.1)

The first term of this equation is the polarizability as derived from electro-statics, with εMNP representing the permittivity of the MNP. The second andthird terms are dynamical corrections taking into account spatial dispersion andradiation damping [81, 89]. The wavevector in the surrounding medium is de-fined as k1 =

√ε1ω/c. Generally, since the MNP is a metallic particle, it has

a Drude-type permittivity and it is important to take into account the ohmiclosses. Calculating |α|2 from Eq. (4.2.1) as a function of ω shows a Lorentzian-like response with a strong peak and well-defined width. The peak is the plasmonresonance frequency of the metal nanoparticle and the width is due to both the

4.2 Formalism 53

Figure 4.2.1: A schematic illustration of the considered system: a dimer ofsilver nanospheres with different sizes, embedded in medium 1, located above andparallel to a metallic substrate (medium 2). a1 and a2 are the radii of particle1 and 2, respectively, d is the center-to-center spacing between the MNPs and his the distance from the dimer to the substrate. Throughout this work, we willconsider the metallic substrate to be silver and the embedding medium to beglass.

ohmic and the radiation losses. The resonance in fact corresponds to exciting thenormal mode of the MNP, which has a corresponding complex normal mode fre-quency. The real part of this frequency corresponds to the peak position, whilethe imaginary part is associated with the Full Width Half Maximum of the peak.The normal mode frequency corresponds to the singularity of the polarizability,i.e., the frequency for which the dipole moment of the MNP becomes infinite.

In a dimer, each MNP does not only experience the monochromatic externalfield Eext, but also the field generated by the other particle. Within the Green’stensor formalism, the electric field at position r generated by a point dipole p′

located at r′ and oscillating with frequency ω is given by

E(r) =k2

1

ε1G(ω, r, r′)p′, (4.2.2)

where G(ω, r, r′) is the 3 by 3 electromagnetic Greens tensor. Hence, for adimer the dipoles of the MNPs are governed by the following system of coupledequations



1

ε1

(1

α1(ω) I −k21G(ω, r1, r2)

−k21G(ω, r2, r1) 1

α2(ω) I

)(p1

p2

)=

(Eext

1

Eext2 ,

)(4.2.3)

or, in short, Mp = E, where M is a 6x6 matrix, and p and E are 1x6 vectors.For the general case of N particles the system will be of dimensions 3Nx3N .This equation allows one to calculate the dipole moments induced in the MNPsgiven a specific excitation, or reversely, the electric field produced for a given setof dipoles. Mathematically, this system is very similar to a set of driven coupledharmonic oscillators. Physically, one gains a great deal of insight into thesesystems by calculating the so-called quasinormal modes, and their correspondingcomplex frequencies. In this work, we will refer to these modes simply as normalmodes, however all losses are properly taken into account. The real part ofthe frequency corresponds to the oscillation frequency of the dipoles and theimaginary part is the inverse lifetime of the mode. To find the normal modesof such a system, the homogeneous equation has to be solved, by setting E =0. This implies that the complex frequencies corresponding to det(M) = 0have to be found and subsequently the eigenvalue problem has to be solved foreach of those frequencies. This procedure is carefully described in, for example[48, 50, 52, 81]. In this study, the positions of the roots of the determinant werefirst estimated by calculating |det(M)| on the complex frequency plane, whichwere then used as an input for the build-in rootfinder of Matlab R2015a.

Since losses are inherent to the system, the normal modes will in generalnot be orthogonal. However, they do form a complete set and therefore it is stillpossible to expand any solution of p as a superposition of the normal modes. Thiscan be achieved by constructing a so-called bi-orthogonal system (see AppendixA).

It is important to note that calculating the normal modes for a given systembecomes numerically more difficult for increasing system size. The reason liesin the calculation of the interaction at complex frequency which diverges atlarge distances, because the electric field at large distances diverges. Assuming atime-dependence of exp[−iωt], implies that the normal mode frequencies have tosatisfy Im[ω] < 0 in order to obtain modes that are decaying in time. Insertingsuch a frequency in a propagating wave exp[ikr] = exp[iωr/c] gives a wave thatis growing as it propagates. It is important to realize that this does not violatecausality, in reality there is a wavefront r/c = t at which the decay in time

4.2 Formalism 55

and the increase in distance exactly cancel and no interactions can occur beforethe wavefront arrives. For infinitely long chain of MNPs this difficulty can becircumvented by assuming Bloch modes and using analytical continuation tocalculate the dipole sums forming the interaction [49].

Needless to say, the normal modes will be determined to a large extent bythe detailed form of G, which in turn depends strongly on the environment. Fora dipole in a homogeneous medium, the tensor is often referred to as GH and isdefined in Cartesian coordinates as [4]

GH(ω; r, r′) =[I +∇∇k2

1

]exp(ik1|r− r′|)|r− r′|

. (4.2.4)

From the above equation it is obvious that if r − r′ is parallel to one of thecoordinate axes, the tensor will be diagonal, i.e. Gij = 0 if i 6= j, hence the threedifferent polarizations are completely decoupled. For a linear chain of dipoles ina homogeneous medium this will always be the case and therefore the modes ofsuch a chain will always be linearly polarized.

In general, any inhomogeneity that is introduced into the system will breakits symmetry and therefore gives rise to a cross-coupling between the differ-ent polarizations. In particular, for the set-up considered in this chapter (seeFig. 4.2.1), the presence of the substrate introduces a coupling between x andz-polarized modes. In this case, there are two possible interaction paths betweenthe particles, a direct one, mediated by free-space photons, and an indirect path,comprising contributions both from reflections and SPPs at the substrate. Ingeneral, the two interactions will have a different polarization and phase, re-sulting in an elliptically polarized field at the particle position, and hence, inelliptically polarized dipoles, with a polarization oscillating in the xz-plane.

To characterize the phase difference and the resulting polarization, one needsto calculate the Green’s tensor for an oscillating dipole located above a metallicsubstrate. A well-known approach for this was developed by Sommerfeld, and isbased on expanding the dipole field into a product of cylindrical waves and planewaves for which the interaction with the substrate is simply given by Fresnel re-flection [4, 74, 90]. In particular, the Fresnel coefficient for TM polarized lightwill describe the coupling to the SPP of the substrate. The outlined procedureresults in contour integrals that have to be calculated numerically. Extra careneeds to be taken when calculating the normal modes for this system, since theintegrals then have to be evaluated for complex frequency, so that the conver-



gence is very sensitive to the choice of the integration path and the branch cuts.This is explained in more detail in Appendix B.

4.3 Results and discussion

To illustrate the above we have performed calculations for a dimer of silvernanospheres above a silver substrate (see Fig. 4.2.1). To describe the frequencydependent permittivity of silver a generalized Drude model is used [92],

εMNP = ε2 = ε(ω) = 5.45− 0.73ω2p

(ω2 + iωγ), (4.3.1)

with plasma frequency ωp = 17.2 rad fs−1 and damping coefficient γ =0.0835 fs−1. The surrounding medium is chosen to be glass with ε1 = 2.25.

In order to make this chapter reasonably self-contained, we will first brieflydiscuss the optical response of a single silver MNP above a silver substrate. InFig. 4.3.1(a), the dispersion relation of the SPP on the silver substrate is given.The surface plasmon frequency, ωsp, and the resonance frequency of the MNP,ωMNP , are indicated by the dotted lines. It is important to note that the SPPresults from hybridization of the surface plasmon of the metal and the photonsin the dielectric. From the SPP dispersion relation it can be seen that theMNP plasmon resonance has an appropriate frequency to couple to the SPPsof the substrate. Figure 4.3.1(b) gives the optical response, i.e. the modulussquared of the dipole moment of a silver MNP situated at h = 50 nm above thesubstrate. The particle is excited by a stationary z-polarized electric field withangular frequency ω. In this graph, two peaks can be distinguished: one closeto the MNP resonance, the other close to the surface plasmon of the substrate.The relatively large frequency spacing between both resonances implies thatthe hybrid MNP-substrate modes will be closely related to their non-interactingconstituents. In Fig. 4.3.1(c), the square of the real part of the z-componentof the electric field is shown on a logarithmic scale, when the MNP is excitedresonantly. The high intensity close to the substrate is a clear indication thatindeed SPPs are excited.

We will now turn to the modes of a symmetric MNP dimer above a silversubstrate. A detailed description of these modes can be found in Ref. [141],including o.a. the dependence on the particle size and inter-particle spacing.In Fig. 4.3.2 it is shown how the well-known modes of a dimer, consisting of

4.3 Results and discussion 57

0.010 0.020 0.030 0.040

6

5

4

3

2

3 4 53.5 4.5 5.50

0.5

1

1.5

2

2.5

SPPLight

ωsp

ωMNP

wavevector (nm-1)

Fre

quen

cy (

rad/

fs)

Frequency (rad/fs)

log[Re[Ez]

2]

(arb

. uni

ts)

a)

b)

0

100

200z (n

m)

300

400

500

-500 0 500x (nm)

-2

-6

-10

-14

c)

Figure 4.3.1: (a) SPP dispersion on the interface between silver and glass. Alsoindicated are: (i) the resonance frequency ωMNP of a silver MNP with a radiusof a = 25 nm, (ii) the frequency of the surface plasmon resonance ωsp, (iii) thelightline in glass. (Caption continued on the following page.)



Figure 4.3.1: (continued) b) The modulus squared of the dipole moment of asilver MNP located in glass at h = 50 nm above the glass-silver interface, inducedby a stationary z-polarized electric field of angular frequency ω. c) Logarithmicplot of the square of the real part of the z-component of the electric field thatis produced by the MNP is plotted in the xz-plane (y = 0) for an excitationfrequency of ω = 4 rad/fs (highest peak in panel b). The system’s geometry isthe same as in panel b).

Symmetric MNP dimer in free space Symmetric MNP dimer above silver substrate

ω (rad/fs)dipolesω (rad/fs) dipoles

Figure 4.3.2: The modes of a dimer comprising two equal-size silver MNPsembedded in glass are shown, with (right) and without (left) the presence ofa silver substrate, using a = 25, d = 90 and h = 50 nm. The black arrowindicates the dipole moment, while the dashed line represents the contour tracedout by the dipole as a function of time. For each mode the complex normal modefrequency ω in units of rad/fs is also given.

two equal particles, are changed when a silver substrate is present. The modesdepicted here were found by solving Eq. (4.2.3) for E = 0 under the givenconditions. Generally, the presence of the substrate will give rise to three effects:the cross-coupling of longitudinal and transversal modes, shifts of the resonancefrequencies due to the interaction with the reflected field, and, to a smaller extent,the hybridization of the dimer modes with the surface plasmon of the substrate.


In Fig. 4.3.2, the cross-coupling between the longitudinal and transversal modesis obvious from the tilted dipoles. Depending on the phase difference between thetwo polarizations, the amplitudes of the dipoles trace a straight line (no phasedifference) or, in general, an ellipse. Comparing the left and the right panelof Fig. 4.3.2 it becomes clear that not only the dipole moments are altered,but also the resonance frequencies have changed. These shifts result from thechange in the effective inter-particle coupling due to the field that is reflectedfrom the substrate. There is also a contribution from the hybridization of thedimer modes with the surface plasmon. However, due to the large frequencyseparation between the isolated dimer and the surface plasmon, the effect isvery small. Therefore, we can assume that modes that are depicted here mainlyoriginate from the dimer. It is important to note that the system contains manymore normal modes, which occur at higher frequency and mainly correspond tothe substrate.

As expected, some of the modes of the MNP dimer above the substrate,shown in Fig. 4.3.2, indeed contain elliptically polarized dipoles. According to[70,71], dipoles with such a polarization can excite SPPs in a preferred direction.However, due to the symmetry of the geometry considered here, both MNPscause fields of opposite directionality, yielding a net non-directional response.This effect can be easily overcome by lifting the symmetry of the system. To thisend, we consider a dimer consisting of two silver nanospheres with different sizes.Other approaches may be using particles of different shape or different material.Figure 4.3.3 shows the normal modes and the corresponding frequencies for adimer consisting of silver nanospheres with sizes of a1 = 15 nm and a2 = 25 nm,respectively. As can be seen from this figure, the asymmetry results in modesthat are mainly localized on one of the particles and show strong asymmetrywith respect to x = 0. Therefore, the optical response of the system will beasymmetric and directional excitation of SPPs on the substrate can be achieved.

To illustrate the directional response of this system, Fig. 4.3.4 shows theradiation profiles of the asymmetric dimer above a silver substrate. The dipolesof the MNPs are assumed to oscillate according to mode 4 of Fig. 4.3.3, i.e., onlythis mode is considered to be excited. In Fig. 4.3.4(a) the square of the real partof the z-component of the electric field is plotted. The bright spots show theposition of the dimer and the coupling of the dimer to the SPP for this modeis seen from the high intensity wave that is propagating on the substrate. Itis clear from the figure that the excitation of the SPP by the dimer is strongly



-500 0 500

0

-1

-2

-3

-4

x (nm)

log[

Inte

nsity

]

a)

b)

ω (rad/fs) dipoles

mode 1

mode 2

mode 3

mode 4

Figure 4.3.3: a) As in the right panel of Fig. 4.3.2, but now for an asymmetricdimer, with particle sizes a1 = 15 and a2 = 25 nm, respectively. The asymmetryand ellipticity of the modes are clearly seen from the arrows representing thedipole moments. b) The intensity of the electric field generated as a function ofx along the line y = 0, z = 5 nm by each of the modes depicted in a). The modesare normalized and the intensity is scaled in such a way that the highest peakin mode 4 equals 1. All modes radiate more in the -x direction and this effect isthe strongest for mode 4.


asymmetric. This is further illustrated in Fig. 4.3.4(b) in which the normalizedintensity of the electric field is plotted on a surface that is parallel to and at aheight of z = 5 nm above the substrate. As can be seen, the strong asymmetrydoes not only occur close to the dimer axis: over a width of about 100 nmtransverse to the dipole axis there is a large intensity contrast visible. Theintensity contrast along the x-axis is characterized by a factor of 140 betweenx = +500 and x = −500 nm. It should be pointed out that the radiation profileof an asymmetric dimer in free space also is directional, however, much lesspronounced. In absence of the substrate, the most directional normal mode hasan intensity contrast along the same line of only a factor of 5.

So far we have shown that the considered system has quasi-normal modeswhich can directionally excite SPPs. It is an inherent property of the system,since those modes exist without any external excitation. Therefore, no externalsource of elliptically polarized light is needed. However, in order to use thismechanism to directionally excite guided modes, an important question arises:how to supply energy to the modes that are responsible for the directionality?In the present situation, a strong intensity contrast of two orders of magnitudeis only seen for mode 4 (see Fig. 4.3.3(b)) and therefore, it is crucial to beable to couple most of the excitation to this specific mode. As was mentionedearlier, the response of the system to any excitation, can be written in terms ofa superposition of the system’s normal modes (see Appendix A). The prefactorsin the superposition will depend on the frequency and the polarization of thegiven excitation. Due to the presence of both Ohmic and radiative losses, thespectral dependence of each mode is relatively broad and, because of that, thereis a significant overlap between the modes.

Furthermore, the fact that the modes are not orthogonal makes it less straight-forward to use polarization to select one particular mode: even if the providedexcitation is such that the polarization and phase of the electric field acting atthe location of each particle match exactly to one of the modes, still a superpo-sition of multiple modes will be excited. Here “match exactly” means that theelectric field at each particle has a direction parallel to the mode’s dipole on thatparticle, while the ratio of the electric field amplitudes and their relative phasesat both particles equal the amplitude ratio and phase difference for the mode’sdipoles on both particles. However, it is worth noting that there are techniquesavailable to optimize the excitation of one particular mode, even under theseconditions. Let us denote the set of non-orthogonal modes vi, where i denotes



z (n

m)

x (nm)

a)

b)

log[Re[E

z ] 2]

-500 0 500

0

100

200

300

400

500-2

-6

-10

-14

-500 0 500-500

0

500

-1

-2

-3

x (nm)

y (n

m)

log[Intensity]

0

-4

Figure 4.3.4: The radiation profiles of the asymmetric dimer are plotted. Inboth cases, the field produced by mode 4 of Fig. 4.3.3 is shown. a) The squareof the real part of the z-component of the electric field in the xz-plane. b) Thenormalized electric field intensity profile on a plane parallel to the xy-plane, at aheight of z = 5 nm above the substrate. Both pictures show a strong asymmetryalong the x-axis, with an intensity contrast of a factor of 140 between x = +500and x = −500 nm along the line y = 0 nm.

the mode index and v is a complex vector with length 4, corresponding to thex and z polarization of both particles. Then, a corresponding bi-orthogonal setvi can be constructed [142], which has the property that vivj = δi,j , where δi,j


-500 0 500

0

-1

-2

-3

x (nm)

log[

Inte

nsity

]Excitation 1

Excitation 2

Figure 4.3.5: The normalized intensity of the field radiated by the asymmetricdimer as a response to two different excitations. 1 - z-polarized excitation on thesmaller MNP only, with ω = 4.50 rad/fs; 2 - both particles excited by an equalz-polarized electric field at both particles, with ω = 4.85 rad/fs. The intensity iscalculated along a line parallel to and at a height of 5 nm above the substrate,and with y = 0 nm.

is the Kronecker delta. Exciting the system along one of the vectors vi (i.e.choosing the electric field values at both particles such that their orientations,phases, and amplitude ratio matches vi), implies that the excitation will onlycouple to mode vi. Therefore, to excite only a single mode the excitation shouldnot be matched to the mode itself, but to its counterpart in the bi-orthogonalset. In order to achieve such an excitation, beam shaping techniques with whichboth amplitude and polarization of the input electric field can be altered, mightbe needed [143,144]. Additionally, it was shown in Ref. [145] that chirped pulsescan be used to selectively excite metal nanoparticles with different sizes if theyare positioned above a metallic ground plane. However, since the directionalityis an inherent property of the system, even standard excitations, such as nearfield excitation or plane wave excitation yield a significant directionality. As isseen from the mode configurations in Fig. 4.3.3(a), the modes are mainly local-



ized on one of the particles. This implies that the modes are very sensitive to alocal excitation. From Fig. 4.3.3(b) one can recognize that mode 4 will couplestrongly to a z-polarized CW excitation at the smaller MNP, with a frequencyof ω = 4.50 rad/fs. The response of the dimer to this particular excitation(Excitation 1), is plotted as the blue solid line in Fig. 4.3.5. Even though a com-plex superposition of the modes is excited, still a strongly directional responseof more than one order of magnitude is observed. In practice, such excitationconditions can be easily achieved by making use of near field excitation with anoptical fiber, or electronically with cathodoluminescence [77, 146, 147]. Interest-ingly, since the modes in a superposition have different phases and amplitudes,interference between the normal modes can occur, which can lead to counterin-tuitive phenomena. An example is depicted for Excitation 2 in Fig. 4.3.5, theresponse of the system to an equal z-polarized CW electric field at both MNPswith a frequency of ω = 4.85 rad/fs. The response shows a preferred emissionin the +x direction, whereas all modes (see Fig. 4.3.3(b)) show a directionalityin the −x direction. Calculating the radiation of each of the modes separately,revealed that modes 2 and 4 are destructively interfering in the −x direction,whereas there is constructive interference in the +x direction, yielding a netdirectional response in the +x direction.

4.4 Summary

We have provided a method which can be used to find the normal modes of acollection of metal nanoparticles above a metallic substrate. The modes of suchsystem are shown to be elliptically polarized. Using the fact that elliptically po-larized dipoles can directionally excite surface plasmon polaritons on the metallicsubstrate, we have designed a system for which this directionality is an inherentproperty: it is embedded in the normal modes. For the system under consid-eration, modes are found which can have an efficiency of SPP excitation in onedirection which is more than two orders of magnitude higher than in the oppo-site direction. Although a complicated excitation setup is needed to exclusivelyexcite the mode with the highest directionality, it is shown that even with a localor plane wave excitation an efficient directional response with a contrast of oneorder of magnitude can be achieved. The asymmetric dimer above a metallicsubstrate is the simplest class of systems in which this effect inherently embed-ded. It will be present for an asymmetric chain of MNPs, independently of the

4.4 Summary 65

chain length. The contrast is expected to be higher for larger systems, becausethe radiation transverse to the chain direction can be canceled by destructiveinterference, as is known for phased array and Yagi-Uda antennas [31, 32, 77].Since the proposed mechanism is based on the interference between the directinteraction of the MNPs and the interaction via the substrate, the system canbe tuned and optimized by varying the inter-particle spacing, the distance to thesubstrate, the type of material, and the shape of the particles. In addition, thismethod can be straightforwardly extended to the directional excitation of theguided modes of dielectric waveguides. We envision the proposed mechanism tobe a very useful tool in the design of directional couplers and antennas in layeredmedia.

Appendix A: Non-orthogonal modes

In the presence of losses, the quasi-normal modes of the system will not beorthogonal in general. However, they do form a complete set and therefore, thatthey span the whole space. This implies that it is possible to write any dipolevector p as a superposition of the normal modes, i.e.

p = a1v1 + a2v2, (4.4.1)

where v1,2 are the normal modes, and a1,2 the corresponding coefficients. In orderto find the coefficients for the situation when v1 and v2 are not orthogonal, butare linearly independent, one can construct a so-called bi-orthogonal system.This is a related system consisting of the non-orthogonal vectors v1 and v2 withthe property that vivj = δi,j , where δi,j is the Kronecker delta. Such a systemcan easily be found by taking the inverse of the matrix of which the columnsare the normal modes. The row vectors of the inverse will form a set which isorthogonal to the normal modes. Therefore, the coefficients ai can be easily foundby calculating vip and the dipole vector p can be written as a superposition ofthe normal modes of the system.



Appendix B: Calculation of the Sommerfeld integralsfor complex frequencies

The interaction via the metallic substrate can be calculated using the methodproposed by Sommerfeld. This procedure has been studied and described ex-tensively in the literature. A very clear derivation of all the relevant equationscan be found in, e.g. [4, 74]. The basic ingredient in the method is the so-calledSommerfeld identity,

eik1R

R= i

∫ ∞0

dkρkρk1,z

J0(kρρ)eik1,z |z|, k1,z =√k2

1 − k2ρ. (4.4.2)

This states that one can write the spherical wave as a product of cylindricalwaves in the ρ direction and plane waves in the z direction, summed over allin-plane wave numbers. In this equation R is the distance between the sourceand the point of measurement, ρ = (x2 + y2)1/2 the distance parallel to thesubstrate, k1 the wavevector in the medium, and kρ and k1,z the wavevectorsparallel and perpendicular to the substrate, respectively. Note that the effect ofthe substrate is not in the equation yet, it is only an expansion of the field inwaves propagating parallel and perpendicular to the xy-plane.

Since only the plane waves will encounter the substrate, the influence of thesubstrate only has to be taken into account for those and has the well knownform of the Fresnel reflection coefficients

rs =µ2k1,z − µ1k2,z

µ2k1,z + µ1k2,z, rp =

ε2k1,z − ε1k2,z

ε2k1,z + ε1k2,z, ki,z =

√k2i − k2

ρ, (4.4.3)

where εi and µi are the permittivity and permeability of medium i. Furthermore,ki represents the wavevector in medium i and ki,z is the wavevector normal to theinterface pointing into medium i. Now the Green’s tensor for the interaction withthe substrate can be constructed by splitting Eq. (4.4.2) for s and p polarizationand multiplying it with the corresponding reflection coefficient [4]. As an examplethe zz-component of the tensor will be

Grefzz = i

∫ ∞0

k3ρ

k21k1,z

(ε2k1,z − ε1k2,z

ε2k1,z + ε1k2,z

)eik1,z(z+h)dkρ (4.4.4)

Taking a close look at Eq. (4.4.4), one notices that there are two double-valued functions present, k1,z and k2,z. In order to ensure convergence, it is

4.4 Summary 67

0 0.02 0.04 0.06 0.08 0.100

-0.005

-0.01

-0.015

-0.02

-0.025

-0.03

-0.035

-0.04

Branch cut

Silver substrate

Branch cut

GlassSPP singularity

Integration path

Figure 4.4.1: A typical form of the integration path. The dotted lines representthe branch cuts corresponding to the glass medium and the silver substrate. Thecircle indicates the position of the singularity of rp, i.e. the SPP. The frequencyconsidered here is ω = 4− 0.6i.

important that the proper Riemann sheet is selected, i.e. Im[ki,z] > 0. Boththese functions give rise to a branch point at ki and a corresponding branch cut.Furthermore, in the case of a metal substrate, the reflection coefficient rp willhave a singularity at kρ = (ε1ε2/(ε1 + ε2))1/2, the surface plasmon polariton.

A very fast and stable method to numerically evaluate these type of integralswas proposed by Paulus et al. [90]. For real frequencies, both the branch pointsand the singularity will lie in the first quadrant, and, being functions of ω,their exact position will depend on ω. Close to these points the integrand willhave rapid oscillations which make the numerical evaluation complicated. Theproposed method is to deform the integration path into the fourth quadrant,staying away from the difficulties. This yields a rapid and accurate calculationof the field.

It should be noted that the original Sommerfeld integration along the realaxis does not converge anymore when complex frequencies are considered: thebranch cut corresponding to medium 1 is such that the correct value of the squareroot k1,z should be below the cut. A valid integration path is found empirically



by beginning with the standard Sommerfeld formulation and slowly increasingthe imaginary part of the frequency. The integration path was found to bepushed down by the singularity and the branch cut, into the fourth quadrant.For accurate and fast evaluation, the integration path has to be chosen verycarefully for each frequency in such a manner that the branch cuts and thesingularity are avoided, while not moving too far away from the real axis, sincethis leads to an exponential growth of the Bessel functions.

An example of an integration path is shown in Fig. 4.4.1. The branch cutsarise from the double-valued functions ki,z for both media. The branch pointscorresponding to the glass medium, ±k1, are connected through Re[kρ]→∞, and±k2, i.e. the branch points corresponding to the silver substrate, are connectedthrough Im[kρ] → ∞. The circle indicates the position of the singularity ofrp, which is corresponding to the surface plasmon polariton of the glass-silverinterface.

The integration path has been verified by comparing the obtained normalmode frequencies to spectroscopic calculations with real frequencies. The posi-tion and width of the peaks in the optical response of the system are correctlydescribed by the real and imaginary parts of the normal mode frequencies.

Chapter 5

Electromagnetic pulse propaga-tion through a chain of metalnanoparticlesPrecursor and normal mode contributions

We theoretically studied the propagation of an optical pulse with a Gaussiantime-envelop through a chain of identical and equidistantly spaced metal nanopar-ticles. Surprisingly, for a single excitation, we obtained two signals travelingthrough the array, each with its own characteristics: (i) - the collective plasmonsignal propagating with the group velocity and damping exponentially, accord-ing to the system’s dispersion relations, and (ii) - a signal propagating with thespeed of light in the background medium, which drops off geometrically with R−1

or R−2, depending on the polarization. The latter signal has not been obtainedbefore and can, in fact, be considered as a precursor. In addition, we showed thateven though this signal has lateral dimensions which are much larger than thewavelength, the field profile close to the chain axis does not change as function ofthe propagation distance, indicating that this part of the fast signal is confinedto the array.

P. J. Compaijen, V. A. Malyshev and J. Knoester, To be submitted

70Electromagnetic pulse propagation through a chain of metal

nanoparticles

5.1 Introduction

Recent advances in fabrication techniques have led to an impulse in the appli-cation of nanotechnology, in particular in the development of nano-electronicchips. Miniaturizing electronics and improving its performance, requires moreinformation transport through smaller interconnects. A significant improvementin performance is expected when the communication is mediated by light, ratherthan by electrons. However, since the size of traditional optical devices is limitedby diffraction, there is a large size-mismatch between the electronic and opticalcomponents. Potentially, this problem can be solved using plasmonics, wherethe energy is stored in the collective excitations of the free electrons of a metallicstructure - plasmons [4, 12, 21, 75]. For such optical excitations it is known thatthey can be confined well below the diffraction limit of the light. Dependingon the considered geometry, these properties can be exploited for the design ofnano-scale optical waveguides, antennas and sensors.

A well-known example of a plasmonic waveguide is a chain of metal nanopar-ticles (MNPs), also known as a plasmonic array. Excitation of one of the MNPswill induce a so-called Localized Surface Plasmon (LSP), which in turn will cou-ple to the neighboring particles in the chain. In this way, an optical signal canpropagate through the chain, maintaining the strong energy confinement prop-erties of the LSP. This system was first introduced by Quinten et al. [13] andsince then it has been extensively studied and discussed in the literature. Thewaveguiding properties of the plasmons of such a chain have been investigatedin detail, considering both the frequency dependence and the propagation dis-tance [14,16,53,78,79,81,82,104,105,112]. Furthermore, due to the large opticalcross section associated with the LSP, it has become clear that this system canalso be used as a nano-antenna, either to localize the excitation [51, 80] or toradiate with a large directionality [28, 29, 31, 32, 77]. The environment of thechain plays an important role in the interactions between the particles in thechain and, therefore, the optical response of the chain is significantly alteredwhen it is embedded in a layered environment [61,108,112,114,116]. The opticalproperties of the system at hand can be summarized very clearly in the disper-sion relations, from which information about the velocity and the lifetime of theplasmonic modes can be derived [48–50,52,81,82,116,148].

Although plasmonic arrays have been subjected to a large amount of studies,most of these focused on the steady-state response of the system. For applications

5.2 Formalism 71

in communication, it is of crucial importance to also understand time dependentbehavior. Therefore, we investigate the propagation of an optical pulse throughsuch a plasmonic waveguide.

Metals are known to be strongly dispersive and dissipative at optical frequen-cies, and consequently, the propagation velocity and damping of the pulse canbe highly dependent on the frequencies comprising the pulse [16,82,149]. Whendispersion and dissipation are small, it is evident that a signal will propagateat the group velocity vg = dω/dq. However, both conditions are not necessarilymet in plasmonic arrays, where the dispersion relations have steep regions anddissipation is a necessary condition to have sub-wavelength confinement [150].

In this chapter, we simulate the propagation of a Gaussian optical pulsethrough a long chain of MNPs. Surprisingly, we obtain two propagating sig-nals rather then one, each with its own characteristics. In the next section, weintroduce the mathematical model that is used for the calculations. Then, inSection 5.3, the dispersion relations are derived and discussed. Section 5.4 firsttreats the frequency domain solutions, followed by the time-dependent propa-gation simulations. The obtained results are discussed and analyzed using ananalytical solution based on the Green’s function of the system. Additionally, thelateral dimensions of the electric field of the pulses is calculated and discussed inview of confinement. Finally, in Section 5.5, we summarize the obtained results.

5.2 Formalism

The plasmonic waveguide that we consider in this chapter is a linear array ofidentical and equidistantly spaced, spherical MNPs. The array is oriented alongthe x-axis and the center of the first MNP coincides with the origin. The systemis depicted schematically in Fig. 5.2.1, in which the center-to-center distancebetween the MNPs is given by d = 75 nm and their radii by a = 25 nm. Thearray is embedded in a glass matrix with a permittivity of εb = 2.25. Theplasmonic array is chosen to consist of N = 4000 particles. Note that the chainis much longer than necessary for any practical implementations. However, sincewe are interested in the propagation of an optical pulse through the array, a longchain enables us to find the amplitude and velocity of the signal, without thecomplication of reflections due to the end of the chain. To study the propagationof an optical pulse through this system, we assume that only the first MNP ofthe array is excited. In practice this can be achieved with, e.g. a tapered


nanoparticles

optical fiber as is used, for example, in near-field optical microscopy [4]. Theexcitation is chosen to be an oscillating electric field with a carrier frequency ofω0 and an amplitude that has a Gaussian time-envelope. To probe the effects ofdispersion on the pulse propagation, it is important that the temporal width ofthe excitation is relatively long, so the frequency spectrum of the pulse is narrowcompared with the variation of dispersion relation.

x

z

d

εb

ε(ω)

Figure 5.2.1: Schematics of the system under consideration: a linear arrayof identical and equidistantly spaced spherical MNPs with radius a = 25 nmand center-to-center distance d = 75 nm. The array is embedded in a mediumwith permittivity εb = 2.25. Only the first particle is considered to be excited.Throughout this work we will consider a chain consisting of N = 4000 MNPs.

The frequency dependent response of MNPs is well understood, and therefore,we will first calculate the optical response in the frequency domain. The MNPsare much smaller than the wavelength and the inter-particle spacing satisfiesd ≤ 3a, therefore it suffices to describe the MNPs as oscillating point dipoles [88].An external field of Eexp[−iωt] induces a dipole moment of pexp[−iωt], andthe amplitudes satisfy the following relation p = εbα(ω)E, where α(ω) is thefrequency dependent polarizability of the metal, which for spherical MNPs canbe written as

1

α(ω)=ε(ω) + 2εbε(ω)− εb

1

a3−k2b

a− 2i

3k3b . (5.2.1)

In this equation, ε(ω) is the permittivity of the material of which the MNPsare composed and kb =

√εbω/c = ω/v is the wavevector in the background

medium, where c and v are the velocities of light in vacuum and the back-ground medium, respectively. The first term in Eq. 5.2.1 is the electrostaticpolarizability, the second and third terms account for spatial dispersion andradiation damping, respectively. In this chapter, the MNPs are considered

5.2 Formalism 73

to be silver with a permittivity given by a generalized Drude model ε(ω) =5.45− 0.73(16.2 · 1016)2/(ω2 + 0.0835iω) [92].

The induced dipole moment of the MNP will oscillate along with the appliedexternal field and therefore it will also generate an electric field. The electricfield at position r produced by an oscillating point dipole with an amplitude ofp′ located at r′ can be written in terms of the Green’s tensor G as

E(r) =k2b

εbG(r, r′)p′, (5.2.2)

where the ω-dependence is dropped. Taking into account that the dipole momentof an MNP is proportional to the total experienced field, which is the sum of theexternal field and the fields that are produced by all the other MNPs, we canwrite the following equation of motion for the amplitudes of the coupled systemin the frequency domain

1

εb

∑m 6=n

[ 1

αδnmI− k2

bG(rn, rm)]pm = Eext

n . (5.2.3)

The exact form of the Green’s tensor is, of course, strongly dependent onthe environment of the dipoles. In the present situation, we consider a chain ofMNPs in an otherwise homogeneous background with permittivity εb. In thiscase the Green’s tensor is simply given by [4]

G(rn, rm) =[I +∇∇k2b

]eikb|rn−rm||rn − rm|

. (5.2.4)

From the symmetry of the system (see Fig. 5.2.1), it is obvious that onlytwo independent polarizations will exist: longitudinal (||), when the dipole mo-ments are oriented along the chain axis, and transversal (⊥), with the dipolemoments are oriented perpendicular to the chain axis. Therefore, only the diag-onal elements of the Green’s tensor will be nonzero and we can simplify Eq. 5.2.4to

G||(xn, xm) = 2

(− i

kb|xn − xm|2+

1

k2b |xn − xm|3

)eikb|xn−xm|

G⊥(xn, xm) =

(1

|xn − xm|+

i

kb|xn − xm|2− 1

k2b |xn − xm|3

)eikb|xn−xm|,

(5.2.5)


nanoparticles

where we have introduced G|| = Gxx and G⊥ = Gyy = Gzz.

The optical response of the array to a given external excitation can now easilybe found by inserting Eq. 5.2.5 into Eq. 5.2.3 and solving the latter for pn. Notethat this is the solution in the frequency-domain and that the external excitationis considered to be continuous wave.

In order to investigate the propagation of a pulse, we have to find the time-domain solution. The pulse we will consider in this chapter has a Gaussianenvelope with a standard deviation of ∆t, a carrier frequency of ω0 and is centeredaround t = t0, i.e.,

E(t) = E0e−iω0te

−(t−t0∆t

)2

. (5.2.6)

The time-domain response can be obtained by taking the inverse Fourier trans-form of the frequency domain response. To this extent, we first have to take theFourier transform of the above defined Gaussian, which will again be Gaussian

E(ω) =E0∆t√

2ei(ω−ω0)t0e

−(ω−ω02/∆t

)2

. (5.2.7)

We now substitute E(ω) into Eq. 5.2.3 and solve for the dipole moment perfrequency component pn(ω). Then the time-dependent response pn(t) can easilybe found by taking the inverse Fourier transform, i.e.

pn(t) =1√2π

∫ ∞−∞

pn(ω)e−iωtdω. (5.2.8)

5.3 Dispersion relations

Resonant excitation of an MNP will excite a plasmon in the particle, and there-fore the collective, quasinormal modes of a chain of MNPs will also be plasmons,which are coherently shared by a number of MNPs. Since it is expected that thepropagation of energy through the chain is mediated by the collective modes ofthe system, it is useful to calculate the properties of these modes. The way tofind these modes is to solve Eq. 5.2.3 assuming E = 0: the quasinormal modesof the system exist without the presence of an external source. There are severalschemes available that can be used to find the modes, however, in general, it isnot a straightforward task. For short arrays the summation in Eq. 5.2.3 can becarried out directly and the modes are easily found using the method proposed

5.3 Dispersion relations 75

by, e.g., Weber et al. [48] Due to poor convergence of the summation it becomesincreasingly difficult to apply this method when longer arrays of MNPs are con-sidered. Fortunately, the infinite chain approximation has been shown to bealready accurate for relatively short chains [79] and in this case the summationscan be written down in terms of polylogarithms, which can be evaluated withvery fast convergence using analytical continuation [49,50,151].

For an infinite, periodic, chain with an inter-particle spacing d, we can writethe collective modes of the system as Bloch modes, i.e. pm ∝ pexp[iqmd].Inserting this in Eq. 5.2.3 with E = 0 yields

1

εb

1

α−∑m 6=n

k2bGβ(nd,md)eiq(m−n)d

p = 0, (5.3.1)

where the Green’s tensor is replaced by Gβ, the scalar interaction for eitherlongitudinal or transversal polarization, i.e., β = || or ⊥, respectively. The sum-mation over m is the so-called dipole sum, Sβ(kb, q). Splitting this summationin two parts corresponding to m < n and m > n, i.e. forward and backwardinteraction, it can be rewritten using polylogarithms [152], giving

S||(kb, q) =1

d3

(−2ikbd

[Li2(z+) + Li2(z−)

]+ 2

[Li3(z+) + Li3(z−)

])S⊥(kb, q) =

1

d3

(k2bd

2[Li1(z+) + Li1(z−)

]+ ikbd

[Li2(z+) + Li2(z−)

]+[Li3(z+) + Li3(z−)

] ).

(5.3.2)

The argument z± = exp[i(kb ± q)d], where z− and z+ correspond to the for-ward and backward interactions, respectively, and Lis(z) =

∑∞n=1 z

n/ns is thepolylogarithm of order s.

Now, the dispersion relation, ω(q), can be obtained by numerically solvingthe following equation:

1

α(ω)− Sβ(kb, q) = 0. (5.3.3)

Figure 5.3.1 shows the thus obtained dispersion relations for an infinite lineararray of silver nanospheres with parameters as in Fig. 5.2.1. The upper and lowerpanel show the real and imaginary parts of the normal mode frequencies, respec-tively. A time-dependence of exp[−iωt] is assumed, and therefore the real part


nanoparticles

4.6

4.4

4.2

4.0

3.8

3.6

0.5

0.4

0.3

0.2

0.1

0

Re[

ω] (

rad/

fs)

-Im

[ω] (

rad/

fs)

wavevector q (nm-1)

a)

b)

0 0.01 0.02 0.03 0.04

Light lineLongitudinalTransversal

Figure 5.3.1: The dispersion relations of an infinite chain of silver sphericalMNPs of radius 25 nm and equidistant spacing of 75 nm. a) Real part of the fre-quency versus the wavevector for longitudinally (green) and transversally (red)polarized plasmons. The black dotted line indicates the dispersion of light inglass. b) The imaginary part of the frequency versus the wavevector. For refer-ence purposes: the plasmon resonance of an individual MNP is ω = 4.14− 0.19irad/fs. Note that the wavevector q runs from 0 to the first Brillouin zone edgeat π/d


of the frequency refers to the oscillation frequency, whereas the imaginary partgives rise to exponential damping in time. In fact, the lifetime of the modes isdetermined by 1/|Im[ω]|. From Fig. 5.3.1a one can see a clear difference betweenthe longitudinal and transversal chain modes. Firstly, at q = 0 the longitudinalmode reaches a minimum that lies below the plasmon resonance of a single MNP,whereas the transversal mode reaches a maximum, lying above the single parti-cle resonance. This reflects the fact that the sign of the near-field dipole-dipolecoupling is opposite when comparing the modes. This is also clearly seen fromEq. 5.2.5. Secondly, a very important difference between both polarizations isthe anti-crossing with the light line that is only observed for the transversal case,indicative of a plasmon polariton. The reason for this is the fact that light is atransversal wave and therefore couples strongly to transversally polarized plas-mons propagating along the array, giving rise to the avoided crossing at q = kb.This coupling cannot occur for longitudinal plasmons. This is also reflected inthe fact that the radiative interaction term, proportional to 1/R, is absent forthe longitudinal interaction, whereas it is present for transversal polarization.Looking at Fig. 5.3.1b, a large reduction of the losses is observed when the lightline, kb = ω/v, is crossed. Within the light cone, q < kb, the modes of the chainsuffer from both ohmic and radiative losses. However, when q > kb, the modescannot couple to free-space radiation anymore and only ohmic damping will bepresent. These are the so-called guided modes of the system and the ones ofinterest when pulse propagation is considered.

From the dispersion relations, one can derive important properties for theenergy transport through such a system. Each mode can propagate at its ownphase velocity, defined as vp = Re[ω]/q. When a collection of modes is excited,a wavepacket is formed and this will propagate with the group velocity. Specif-ically, it is expected that a pulse with a central frequency of ω0, therefore willpropagate with a velocity of dRe[ω]/dq|ω0 .

5.4 Results and Discussion

5.4.1 Frequency domain

In Sec. 5.2, the polarizability α(ω) of a single MNP and the interaction betweentwo MNPs G(rn, rm) were given in the frequency domain. The optical responseof the chain can now be easily obtained by inserting these quantities in Eq. 5.2.3


nanoparticles

-1

-3

-5

-7

-9

Log[

|pn(

ω)|

/|max

(p1(

ω)|

]

-1

-3

-5

Log[

|pn(

ω)|

/|max

(p1(

ω)|

]

Transversal

Longitudinal

ω (rad/fs)

-72 3 4 5 6

1000

2000

3000

4000

0

part

icle

num

ber

n

1000

2000

3000

4000

0

part

icle

num

ber

n

Figure 5.4.1: The frequency domain characterization of the energy transportthrough a plasmonic array. The absolute values of the dipole moments pn areplotted on a logarithmic scale. The dipole moment of each particle is normalizedwith respect to the maximum dipole moment of the first particle. Only thefirst particle is excited and the amplitude of the external field is the same forall frequencies. The chain comprises 4000 silver nanospheres with radii a = 25nm and spacing d = 75 nm. The maximum dipole moment of the last particleoccurs at ω = 4.12 rad/fs for longitudinal polarization, and at ω = 3.46 rad/fsfor transversal polarization.


0 1 2 3 4

0

-2

-4

-6

log[n]

log[max(|pn/p 1

|)]

Longitudinal

n-2.13

Transversal

n-0.95

Figure 5.4.2: Double logarithmic plot of the maximum value of the dipole mo-ment for each particle obtained from Fig. 5.4.1, as a function of the positionalong the chain n = x/d. The dipole moments are normalized with respect tothe maximum value of the first dipole. It is clear that after an initial exponentialdecay, the decrease becomes linear, indicating a power law decay along the chainaxis. Using a linear fitting, it is determined that the decay is proportional tox−2.13 for longitudinal polarization, and x−0.95 for transversal polarization.

and solving for the dipole moments. The frequency domain response of the sys-tem is depicted in Fig. 5.4.1 for a chain consisting of N = 4000 silver nanosphereswith radii a = 25 nm and interparticle spacing d = 75 nm. The excitation isconsidered to act only on the first particle of the chain and with the same am-plitude for all frequencies. Both for longitudinal and transversal polarization, itis clearly seen that the dipole moments are large close to the beginning of thechain, but gradually decrease further down the chain. Figure 5.4.1 shows thatmore towards the end of the array, efficient energy transmission occurs only in anarrow frequency interval. Furthermore, while at the beginning of the chain thedipole moments are larger for longitudinal polarization, more towards the end ofthe chain the response is more intense for transversal polarization.

To appreciate this effect better, a graph, showing the maximum values of thecontour plots of Fig. 5.4.1 as a function of the distance along the chain, is given


nanoparticles

in Fig. 5.4.2. From this figure, it can be clearly observed that indeed for thefirst part of the chain longitudinal polarized transport is more efficient than thetransversal one, but after about 130 particles transversal polarization dominates.Interestingly, whereas all collective modes damp exponentially due to dissipation,the line of maximum energy transport along the chain follows a power law atlarge distances, i.e. p ∝ xs. From the slope of the decay, one can determinethe value of s, which turns out to be s ≈ −0.95 for transversal polarizationand s ≈ −2.13 for longitudinal polarization. Furthermore, the frequency atwhich the dipole amplitude is maximized, matches the frequency at which thedispersion relations cross the light line. Even though Fig. 5.4.1 was calculated bynumerically solving Eq. 5.2.3, in order to interpret this surprising effect at longdistance, it is useful to consider the analytical solution to this equation. If onlya single particle is excited in an infinite chain, Eq. 5.2.3 essentially representsthe Green’s function for polarization, and can be solved by taking the Fouriertransform [81,153]

Pβ(xn, 0) =d

2πεb

∫ πd

−πd

eiqxndq

1/α(ω)− Sβ(kb, q). (5.4.1)

Here, Pβ(xn, 0) is the dipole moment that is induced in the particle situatedat xn, when only the particle at x = 0 is excited. The integration runs overthe real axis from q = −π/d to q = +π/d, which are the edges of the firstBrillouin zone. To find the different contributions to the integral, we deformthe integration path into the complex plane. In the present case, we can movethe path into the upper half-plane, to ensure the convergence of the integral forpositive values of xn. Taking a close look at the integrand in Eq. 5.4.1, we observethat for the contour deformation we have to take into account the poles, given by1/α(ω)−Sβ(kb, q) = 0, and logarithmic branch cuts at q = ±kb originating fromthe polylogarithms in the dipole sum S. The deformed contour is illustrated inFig. 5.4.3.

Since the integrand is the same in the Brillouin zone edges, the contributionscoming from the left and right vertical parts of the integration contour willcancel. In addition, the integrand is exponentially decaying as a function of theimaginary part of q. Therefore, the contribution of the upper part of the contourwill tend to zero as long as we move the path far enough away from the real axis.Interestingly, this shows that there are two contributions to the dipole momentof the particle at xn, coming from the pole and from the branch cut. The poles


Figure 5.4.3: Illustration of the deformed integration path for Eq. 5.4.1. Thecrosses indicate the positions of the poles and the dashed lines depict the loga-rithmic branch cuts.

of the integrand in Eq. 5.4.1 correspond exactly to the collective modes of thesystem. The above statement therefore implies that the response of the systemcannot be simply written as a superposition of the modes, the contribution ofthe branch cut also has to be taken into account.

The contribution of the pole P pβ can easily be calculated using the ResidueTheorem. Denoting the position of the pole by q = qp, we find

P pβ (xn, 0) =d

iεb

eiqpxn

S′β(kb, qp), (5.4.2)

where S′ denotes the derivative of S with respect to q. Due to the presenceof dissipation, we know that the poles will occur at complex values of q and,therefore, it is easy to see that the contribution coming from the collective modesof the system will decay exponentially as a function of xn with a decay constantof Im[qp].

The origin of the power law decay of the signal for large xn thus has to comefrom the contribution of the integration around the branch cut, P bcβ . Since theintegrand vanishes exponentially according to exp[−Im[q]xn], the contributionsof both vertical paths along the cut will be zero for large xn. Therefore, at largedistances, we expect to only see the contribution from the integration over thereal axis from kb − ε to kb + ε, for arbitrary small ε, i.e.,


nanoparticles

P bcβ (xn, 0) = limε→0

d

2πεb

∫ kb+ε

kb−ε

eiqxndq

1/α(ω)− Sβ(kb, q). (5.4.3)

To find the asymptotic behavior of this integral for large xn, we can substitute

eiqxn =1

ixn

∂

∂qeiqxn (5.4.4)

in Eq. 5.4.3 and integrate by parts twice to obtain

P bcβ (xn, 0) = limε→0

d

2πεb

( 1

ixn

eiqxx

1/α− Sβ

∣∣∣kb+εkb−ε

+1

x2n

eiqxxS′β

(1/α− Sβ)2

∣∣∣kb+εkb−ε

− 1

x2n

∫ kb+ε

kb−ε

eiqxn [(1/α− Sβ)S′′β − 2(S′β)2]dq

(1/α− Sβ)3

).

(5.4.5)

Note that the dependence of Sβ on kb and q is suppressed in Eq. 5.4.5.

The branch cut originates from the polylogarithms in the dipole sum Sβ andtherefore, it is important to understand the behavior of these functions close tothe cut. The arguments of the polylogarithms are z+ and z−, which reduce toz+ = exp[2ikb] and z− = exp[∓iε] at the integration boundaries. This impliesthat the branch cut at q = +kb is only due to the polylogarithms containingz−, i.e the forward interaction. One can show that limε→0

[Lis(e

+iε)− Lis(e−iε)

]equals 0 for s = 2 and 3, but will be nonzero for s = 1. For transversal polariza-tion, polylogarithms with s = 1, 2 and 3 occur. Therefore, none of the terms inEq. 5.4.5 vanish, indicating that the large distance response of the system decaysas 1/x, in close agreement to what was observed in Fig. 5.4.2. For longitudinalpolarization, there are only second and third order polylogarithms in the dipolesum, and therefore S||(kb, kb + ε) = S||(kb, kb− ε) as ε goes to zero, implying thatthe first term in Eq. 5.4.5 disappears. The second term of Eq. 5.4.5 contains thefirst derivative of S with respect to q. Using the fact that the derivative of apolylogarithm with respect to the argument decreases its order by 1, we deducethat S′|| contains a polylogarithm of the first order and therefore, the secondterm does not vanish for arbitrary small ε. Thus, the large distance responsefor longitudinal polarization decays as 1/x2, in close agreement to what wasalso observed in Fig. 5.4.2. Additionally, from the above analysis it is easy tounderstand why the maximum transmission occurs at the frequency where the


dispersion relation crosses the light line. The crossing point means that Re[qp]and kb are equal, i.e. the pole lies very close to the branch point, which impliesthat the denominators in Eq. 5.4.5 are small, yielding a large response.

5.4.2 Time domain

As was mentioned before, the time-dependent propagation can easily be obtainedfrom the frequency domain solution with the aid of the inverse Fourier transform.The Gaussian pulse with a carrier frequency of ω0 has a Gaussian spectral dis-tribution centered around ω0, as is given by Eqs. 5.2.6 and 5.2.7. To obtain thefrequency domain solution depicted in Fig. 5.4.1, the amplitude of the excitationwas put equal for all frequencies. Since the external excitation enters linearlyin the equations, the response to a Gaussian excitation can be obtained by sim-ply multiplying the solution with a Gaussian spectral distribution. Numericallythis is advantageous, since the time-consuming frequency domain response onlyhas to be calculated once and consequently, the frequency domain response ofthe chain to a Gaussian excitation for different values of ω0 can be generatedvery fast. The inverse Fourier transform was performed using the build-in FastFourier Tranform (FFT) function of Matlab R2015a. To obtain a sufficientlyhigh resolution in the time-domain, the frequency grid that is used had a stepsize of 0.001 rad/fs. Furthermore, the considered pulse was chosen relativelylong compared to the oscillation period of the normal modes, with a width of∆t = 80 fs. This implies that the spectral distribution has width of ∆ω = 0.025rad/fs, which is narrow enough to probe the influence of different ω0.

An example of the time-dependent pulse propagation through the consideredsystem is given in Fig. 5.4.4, where the absolute value of the dipole moment,|pn(t)|, is plotted as a function of the particle number and time. For longitu-dinal polarization, the carrier frequency ω0 is 4.3 rad/fs, while for transversalpolarization ω0 = 3.8 rad/fs. Note that these values are just chosen for illus-tration purposes and that calculations have been performed for a wide range ofω0. The propagation velocity can be determined from the slope of the contours:the steeper the contours, the larger the velocity. For both polarizations, one canclearly distinguish two different signals: a fast, slowly decaying signal over thediagonal, and a slower, more lossy signal. In the top right corner the reflectionof the fast signal at the end of the chain can be seen. Using Brillouin’s definitionof signal velocity vs, the slope of the contours was determined by tracing themaximum value of the signal as a function of time [154]. The velocity of the fast


nanoparticles

0 500 1000 1500 2000

-2

-6

-10

-14

-18

1000

2000

3000

4000

0

time (ps)

Par

ticle

num

ber

Log[

|pn(

t)|/|

p1(

t0)|

]

1000

2000

3000

4000

0

Par

ticle

num

ber

-2

-6

-10

-14

-18

Log[

|pn(

t)|/|

p1(

t0)|

]

-22

Longitudinal ω0 = 4.3 rad/fs

Transversal ω0 = 3.8 rad/fs

~~

~~~

Figure 5.4.4: The time-dependent response of a plasmonic array consistingof 4000 silver nanospheres with a radius of a = 25 nm and a center-to-centerspacing of 75 nm. The absolute value of the dipole moment —pn(t)| is plottedas a function of the particle number and time. The first particle of the chain isexcited by a Gaussian pulse with a width of ∆t = 80 fs and a central frequency ofω0. The dipole moments are normalized to the maximum value of the first dipole.For longitudinal polarization ω0 = 4.3 rad/fs and for transversal polarizationω0 = 3.8 rad/fs.

signal is found to be exactly equal to the velocity of light in the embedding glass


matrix, 200 nm/fs, and interestingly, is independent of the carrier frequency ofthe exciting pulse, ω0. In contrast, the velocity of the slower signal turns out tobe strongly dependent on ω0, and as will be shown, it is this signal that matchesthe guided plasmons of the nanoparticle chain.

The observation of two signals propagating at different velocities, hints to-wards the two different contributions that were found in the frequency domainsimulations in the Sec. 5.4.1. Since we have obtained analytical expressionsfor both contributions, by taking the Fourier transform we can find the veloci-ties with which these signals are propagating. The integrals are complicated toevaluate exactly, however, in order to obtain the propagation velocities, a fullcalculation is not necessary. The Fourier transform of the contribution comingfrom the branch cut is proportional to

P bcβ (xn, 0) ∝∫eikb(ω)xne−iωtdω, (5.4.6)

which is essentially a superposition of plane waves exp[i(kb(ω)xn − ωt)] withdifferent frequencies. The functional dependence of kb on ω is well known andis kb = ω/v. Therefore, it is evident that this contribution propagates with thevelocity v, i.e. the speed of light in the surrounding medium, independentlyof the excitation frequency. Thus, the fast propagating signal arises from theintegration along the logarithmic branch cut.

Similarly, for the contribution coming from the pole of Eq. 5.4.1, i.e. thecollective plasmons of the chain, we can write

P pβ (xn, 0) ∝∫eiqp(ω)xne−iωtdω. (5.4.7)

Although we do not have a simple analytical expression for qp(ω), to a first ap-

proximation, we can write qp(ω) = qp,0 +dqpdω

∣∣ω0

(ω − ω0). Inserting this in the

above equation, we find that the signal is propagating according to exp[i(dqpdω

∣∣ω0xn−

ωt)], i.e. the propagation velocity is dωdqp

∣∣ω0

. Within this approximation, this isequal to the group velocity vg at the carrier frequency ω0. Therefore, the col-lective plasmons of the chain propagate with the group velocity, which can bederived from the dispersion relation.

To verify the above result, Fig. 5.4.5 shows both the group velocity, derivedfrom the dispersion relation, and the signal velocity vs of the slower signal,which is obtained by tracing the maximum in the time-dependent propagation


nanoparticles

3.6 3.8 4.0 4.2

vg

vs

ω0 (rad/fs)

ω0 (rad/fs)

velo

city

(nm

/fs)

velo

city

(nm

/fs)

Transversal

Longitudinal

4.4 4.6

2.2 2.6 3.0 3.4 3.8 4.2

0

20

40

60

80

100

120

140

0

50

100

150

200

250

0.0205 0.0207

4.13

4.11

Figure 5.4.5: A comparison of the group velocity vg (blue solid), calculated fromthe dispersion relation, and the signal velocity vs (red circles), obtained from thetime-dependent propagation shown in Fig. 5.4.4. The inset shows a close-up ofthe crossing point of the dispersion curve for longitudinal polarization with thelight line.

simulations. A very close match between both velocities is observed, confirmingthe hypothesis. Close to the light line the comparison becomes worse. This isbecause near the light line the changes in the dispersion relation are large, and


therefore, taking only the first order in the Taylor series is not sufficient, buthigher order terms have to be taken into account.

For longitudinal polarization at high frequencies we still observe a signal,whereas according to the dispersion relation no modes exist for this frequency.However, due to the width of the excitation and the broadening of the modesarising from dissipation, there is still sufficient overlap to excite the modes at theedge of the first Brillouin zone. Surprisingly, for longitudinal polarization, rela-tively large group and signal velocities are recorded, much larger then expectedfrom looking at the slope of the dispersion curve in Fig. 5.3.1. As is shown in theinset of Fig. 5.4.5, in which a close-up of the light line crossing for longitudinalpolarization is plotted, the dispersion relation coincides with the light line fora small selection of wavevectors. This implies that for a very narrow frequencyinterval, the plasmons will actually propagate with the velocity of light. Thisis the reason for the peak around ω0 = 4.1 rad/fs for longitudinal polarization.The above result might seem counterintuitive, because radiative interaction onlyexists for transversal polarization. However, in the interaction there is also thenon-static intermediate-field term, proportional to R−2, that results from retar-dation and gives rise to a cusp in a narrow region around q = kb [155,156].

In addition to the signal velocities, we can also extract information aboutthe signal damping from the time-dependent simulations. Tracing the heightof the peaks as a function of distance or time, one can deduce the propagationlength or the lifetime of the signals. Doing this for the fast signal, the samepower law decay as observed in the frequency domain calculations was obtained,confirming that this signal is indeed originating from the branch cut. For theslower signal, we deduced the lifetime from the simulations as a function of thecarrier frequency. The result is compared with the lifetime that was obtainedfrom the imaginary part of the dispersion relation in Fig. 5.3.1. The comparisonis shown in Fig. 5.4.6. It is obvious that the decay constant exactly matches withIm[ω], another confirmation that the slower signal is indeed due to the collectiveplasmons. It is important to stress that, even though realistic dissipation wastaken into account, the group velocity provides a good fit to the obtained signalvelocities. Even for simulations with twice the realistic amount of ohmic losses(not shown), the match between both velocities still is very good. This indicatesthat it is not the amount of dissipation that is important, but its variation withrespect to the wavevector or the frequency, i.e. the changes in the imaginarypart of the dispersion relation. Close to the light line, the variation in both the


nanoparticles

Transversal

0.01 0.02 0.03 0.04wavevector q (nm-1)

0.01

0.02

0.03

0

-Im

[ω] (

rad/

fs)

0.02 0.03 0.04wavevector q (nm-1)

0.02

0

0.04

0.06

0.08

0.1

-Im

[ω] (

rad/

fs)

Longitudinal

Figure 5.4.6: The dashed line represents the imaginary part of the normal modefrequencies as a function of the wavevector. The red circles show the decayconstants that are obtained by tracing the maximum amplitude of the slowersignal in the pulse propagation simulations for different modulation frequencies.Please note that 1/Im[ω] is equal to the lifetime of the corresponding normalmode.

real and imaginary part of the dispersion relation is large, indicating that in thisregion the group velocity is not a good approximation.


5.4.3 Discussion of the time domain results

Two different signals are found to propagate along the plasmonic array, one trav-eling with the speed of light in the embedding medium and decaying as a powerlaw, the other propagating according to the group velocity of the collective plas-mons of the chain and decaying exponentially. The latter signal is shown to arisefrom the pole of the response function, proving that it is in fact the plasmonicnormal mode of the chain. We have shown that the fast signal originates fromthe branch cut of the integrand in Eq. 5.4.1. Both the high propagation velocityand the slow geometrical decay, make this signal very interesting. The questionarises how to interpret the branch cut contribution physically. Note that sincethe branch cut originates from the dipole sum, the fast signal is a collective effectof the chain and not simply the electric field that the first dipole generates ateach particle in the chain. In fact, as was shown in the previous section (belowEq. 5.4.5), the contribution at the branch cut is solely due to the sum of theforward interacting contributions. Therefore, the fast signal can be thought ofas a forward scattered signal. Since the electric field generated by an oscillatingdipole propagates with the speed of light, it is understandable that such a signalshould always be present. The intensity of such a signal will be the highest if allforward scattered contributions are adding up in phase. This condition is satis-fied exactly when the wavevector of the plasmonic array matches the wavevectorof light, as was observed in Sec. 5.4.1.

The signal that propagates with the speed of light and decays geometricallyrather than exponentially, as would be expected in a dissipative medium, in factis a so-called forerunner or precursor [154]. We stress however that the condi-tions under which we found this signal, differ substantially from those of thewell-known Sommerfeld precursor. The standard formulation of the precursorproblem consists of an incoming pulse with a broad spectrum, which propa-gates through a weakly dispersive medium, far from resonance. In this case, thehigh frequency components propagate with the phase velocity, i.e. the speedof light, forming the precursor, whereas the central spectral components travelwith the group velocity and form the collective mode response. The physicalinterpretation of the precursor is the fact that the medium dipoles can not reactimmediately on the external action: they need some time to rise. Therefore, theSommerfeld precursor actually is the incoming field which propagates with thespeed of light, feeling neither dispersion nor dissipation.

It is important to realize that the system we considered here is quite different


nanoparticles

from those where precursors are commonly studied. First of all, the systemunder study is a discrete medium, with a geometry such that the expansionparameter kd ≈ 1. Secondly, it is a highly dispersive medium and because ofthat plasmon polaritons are formed, resulting in an anti-crossing at the light linewhere the plasmon-light interaction is very strong. Furthermore, in our setup,the external excitation only acts on the first particle of the chain and is thereforenot propagating through the system. Thus, the precursor we obtained is in factthe secondary field produced by the MNPs, which travels with the speed of lightthrough the chain. Normally, this contribution is overshadowed by the moreintense external excitation.

In Eq. 5.4.5 it was shown that the amplitude of the precursor obtains amaximum for those frequencies where the dispersion relation crosses the lightline. The reason for this is the following. Since the precursor propagates with thespeed of light, its wavevector is given by the dispersion relation of light, i.e. thelight line. The dispersion relation of the collective plasmons of the chain describesthe frequency and wavevector corresponding to the natural resonances of thesystem. When the excitation frequency is such that the dispersion relation of thechain crosses the light line, the wavevector with which the precursor propagatesis close to a collective mode of the system and, therefore, it can be enhanced bythe collective properties of the system.

Signal transmission through plasmonic arrays is studied mostly in the contextof sub-diffraction waveguides. It is known that for such systems there exists atrade-off between confinement and losses [150]. Therefore, in order to have atruly sub-wavelength confinement, one has to deal with a smaller propagationlength. In the present case, the plasmon signal is dominating the transmissionin the first part on the array, but after a certain distance the decay shows apower law behavior, indicating that the forward scattered signal has the highestintensity. Due to the slow decay, it can propagate for remarkable distances.Naturally, the question arises if this signal is also, in some sense, confined tothe array. To answer this question, the electric field along a plane orientedperpendicular to the chain axis was calculated. Figure 5.4.7 shows the electricfield components corresponding to the fast, forward scattered, signal as shownin Fig. 5.4.4. The field plotted along two lines parallel to the y-axis, one locatedin between particles n = 2000 and n = 2001, and the other between n = 3000and n = 3001, is depicted. The main figures show the fields far away from thechain, the insets display the fields near to the chain. Note that the units on the


n=2000n=3000

|Ex/

E0|

|Ez/

E0|

n=2000n=3000

|Ex/

E0|

|Ez/

E0|

0

2

4

6

8

10

12

14

0

1

2

3

4

5

6

7

8x 10-3

x 10-3

y (μm)

Longitudinal ω0 = 4.3 rad/fs

Transversal ω0 = 3.8 rad/fs

y (nm)

y (nm)

n=2000

40 80 1200 160 200

4 x 10-3

0 250 500

1 x 10-3

0 250 500

Figure 5.4.7: The electric field profile of the forward scattered signal is calculatedalong the line parallel to the y-axis situated at two positions in the chain: betweenthe particles n = 2000 and n = 2001, i.e., the line (x2000+d/2, y, 0), and similarlybetween particles n = 3000 and n = 3001. The excitation is the same as inFig. 5.4.4 and the electric field is normalized with respect to the amplitude ofthe external excitation at ω0, E0. The main figures give the response far awayfrom the chain and the insets show the response close to the chain axis. Notethat in the inset the distance is given in nm, rather than in µm as in the mainfigure.

horizontal axis are in µm and nm, respectively. The figure clearly shows that,far away from the chain, the field intensity decreases and broadens, exactly asexpected for a signal that decays geometrically. Interestingly, this behavior is not


nanoparticles

observed close to the chain. Even though the amplitude of the field decreases, itdoes not spread laterally. From this we can conclude that this part of the signalis confined to the chain.

5.5 Summary and Outlook

We have investigated theoretically the propagation of a Gaussian-envelope elec-tromagnetic pulse through an array of equidistantly spaced spherical metal nanopar-ticles and have found that there are two different contributions to the opticalresponse of a plasmonic array, one traveling with the group velocity and decayingexponentially, and the other propagating with the velocity of light and decayinggeometrically. The former signal was shown to arise from the excitation of thenormal modes, i.e. the collective plasmons of the chain of MNPs. The lattersignal has not been obtained in previous studies and cannot be deduced fromthe normal modes of the system. This contribution was interpreted as the sumof all the forward scattered electric fields. It acquires a maximum in intensitywhen the frequency matches the crossing point of the dispersion relations of thecollective plasmons and light, i.e. when the forward scattered fields add up inphase.

Furthermore, the fast propagating signal is, in fact, a precursor or forerunner.The system considered here is strongly dispersive and the collective modes of thesystem are close to the light line crossing point. Therefore, the precursor and thecollective modes can both be excited by applying a pulse with a relatively narrowspectrum. Since only the first particle of the chain is excited, the precursor weobtained is a secondary effect, which is otherwise overshadowed by the moreintense precursor signal originating from the front of the external pulse, as is thecase for the Sommerfeld precursor.

Lastly, by calculating the electric field profile along a line perpendicular tothe chain axis, the lateral dimensions of the forward scattered signal was stud-ied. It was shown that, surprisingly, even though the signal does not have sub-wavelength dimensions, close to the chain axis the profile does not change as afunction of the propagation distance, indicating that this part of the signal isconfined to the array.

Samenvatting

Het bestuderen en manipuleren van licht speelt al vele eeuwen eenbelangrijke rol in wetenschappelijke en technologische ontwikkeling.Met de opkomst van de nanotechnologie, ontstond ook de wens omlicht op de nanometerschaal te kunnen beınvloeden. Dit biedt onderandere de mogelijkheid om de communicatie binnenin en tussen com-puterchips enorm te versnellen. Het blijkt echter dat de traditioneletechnieken om licht te sturen, met spiegels en lenzen, niet langer wer-ken voor deze kleine afmetingen. Nano-optica is het vakgebied datzich bezig houdt met het ontwikkelen van technieken om ook op dezelengteschaal licht te kunnen manipuleren. Een belangrijke bouwsteenin dit vakgebied is een keten van metalen nanodeeltjes. Dit systeembiedt namelijk de mogelijkheid om optische signalen te kunnen gelei-den op de nanometerschaal. De onderliggende fysica van dit procesblijkt echter niet triviaal, alsook de invloed van externe factoren. Indit proefschrift hebben wij een theoretisch model ontwikkeld om be-rekeningen te kunnen doen aan ketens van metalen nanodeeltjes ende belangrijke natuurkundige aspecten van dit systeem te kunnen ex-traheren.

Sub-diffractie optica

Een bekend voorbeeld van de manipulatie van licht is de optische microscoop.Met een microscoop is het mogelijk om voorwerpen te bekijken die zo klein zijndat we ze met ons blote oog niet kunnen zien. In de meeste gevallen zit er eenlamp onder het preparaat en als we door de microscoop kijken zien we feitelijkhet licht dat verstrooid wordt door het object dat we willen bestuderen. Eenslimme samenstelling van lenzen maakt het mogelijk om het beeld van een ob-

94 Samenvatting

ject te vergroten en te focusseren. Het probleem van traditionele optica kaneenvoudig geıllustreerd worden aan de hand van het volgende voorbeeld. Steldat we door een microscoop naar twee voorwerpen kijken en de afstand tussendeze voorwerpen steeds iets kleiner maken. Op het beeld dat de microscoopvormt zien we dan twee vergrootte objecten die steeds dichter bij elkaar komen.Op een gegeven moment zal de afstand tussen de voorwerpen zo klein zijn datwe geen twee afzonderlijke objecten meer waarnemen, maar dat het een grootobject lijkt. Dit betekent dat de afstand tussen de voorwerpen kleiner is danhet oplossend vermogen (de resolutie) van de microscoop. Meestal kan de reso-lutie verhoogd worden door een sterkere lens te gebruiken. Echter, er blijkt eenfundamentele grens te zijn aan de resolutie van een optische microscoop, en dusaan de afstand tussen de twee voorwerpen waarbij ze nog te onderscheiden zijn.De reden hiervoor is de verstrooiing van licht rond de objecten. Een verstrooi-ingspatroon heeft een bepaalde minimale omvang en als de patronen van beideobjecten voldoende overlappen, kunnen ze niet meer onderscheiden worden. Ditgeeft een fundamentele limiet aan de resolutie van een optische microscoop, dezogenaamde diffractielimiet.

Voor zichtbaar licht is de diffractielimiet van de orde van 100 nanometer, datwil zeggen 100 miljardste van een meter, oftewel een duizendste van de diktevan een haar! Hoewel het dus om een ontzettend kleine afstand gaat, is dezenog steeds groot in vergelijking met de hedendaagse nanotechnologie. De klein-ste structuur op een moderne computerchip is slechts 14 nanometer. Afgelopenjaren is de snelheid van computers toegenomen door steeds meer en kleineretransistoren op een chip te plaatsen. Tegenwoordig is de limiterende factor voorde snelheid van computers de communicatiesnelheid tussen en binnenin com-puterchips en om de volgende stap te kunnen maken moet die communicatieverbeterd worden. Potentieel zou dit probleem opgelost kunnen worden doorgebruik te maken van optische communicatie, zoals bijvoorbeeld gebeurd metglasvezelkabels voor breedband internet. Echter, om dit te kunnen bewerkstelli-gen moet er een alternatief gezocht worden voor glasvezelkabels, met afmetingendie vergelijkbaar zijn met die van elektronische componenten. Dit betekent dater technieken nodig zijn waarmee licht gemanipuleerd kan worden op een leng-teschaal ver beneden de golflengte van het licht.

Interessant genoeg geeft nanotechnologie ons niet alleen een limiet voor hetgebruik van traditionele optische technieken, maar biedt het ook fascinerendenieuwe mogelijkheden om licht te manipuleren. Door materialen op de nanome-

Samenvatting 95

terschaal op te bouwen en samenstellen, kunnen materialen gemaakt worden metoptische eigenschappen die voorheen niet mogelijk waren, bijvoorbeeld een nega-tieve brekingsindex. Deze ’nieuwe’ materialen worden ook wel metamaterialengenoemd. Belangrijke bouwstenen van deze materialen zijn metalen oppervlak-ken en metalen nanodeeltjes. De optische reactie van dit soort structuren kanzeer sterk zijn door de aanwezigheid van zogenaamde oppervlakte plasmonen.Dit zijn resonante, collectieve oscillaties van de vrije elektronen van het metaal.Deze resonanties hebben typisch een frequentie in de buurt van zichtbaar lichten kunnen worden aangeslagen op het oppervlak van een metaal.

Een bekend voorbeeld van een materiaal met een sterke plasmonische opti-sche respons is een metalen nanodeeltje. Als het deeltje wordt beschenen metlicht dat een frequentie heeft die dicht bij de frequentie van de plasmonresonantieligt, wordt veel energie van het licht geabsorbeerd en omgezet in een oscillatiebe-weging van de elektronen in het deeltje. De diameter van metalen nanodeeltjesligt typisch tussen de 20 en 100 nm, veel kleiner dan de golflengte. Door gebruikte maken van metalen nanodeeltjes kan licht dus worden omgezet in iets dat metdezelfde hoge frequentie oscilleert, maar een omvang heeft die veel kleiner is degolflengte van het licht. Dit soort nanodeeltjes hebben niet alleen een sterke wis-selwerking met licht, ze hebben ook een sterke interactie met elkaar. Dit betekentdat het plasmon van het ene deeltje, een plasmon in een nabijgelegen deeltje kanaanslaan. Als we een keten maken van metalen nanodeeltjes en het eerste deeltjeaanstralen, dan kan er dus een plasmonsignaal door de keten gaan lopen. Hetvoordeel van de hoge frequentie van licht blijft behouden, maar de omvang wordtniet langer beperkt door de diffractielimiet. Daardoor is een plasmonische keteneen potentiele kandidaat voor een nanometerschaal alternatief voor een optischefiber. Dit soort ketens vormen het onderwerp van dit proefschrift.

Theoretische beschrijving

Om goed te kunnen begrijpen hoe een plasmonische keten een optisch signaal kangeleiden en welke parameters daarbij van invloed zijn, is het van belang om eendegelijke theoretische beschrijving van het systeem te kunnen geven. Aangezienplasmonica volledig door Maxwells vergelijkingen beschreven kan worden, ligthet voor de hand om een commercieel softwarepakket te gebruiken die dezevergelijkingen voor elke geometrie kan oplossen. Echter, de grote variatie inde sterkte van het elektrische veld en de sterk frequentie afhankelijke respons

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van metalen nanodeeltjes, maken het erg moeilijk om nauwkeurige resultaten tebehalen. Daarnaast is het vaak lastig om de belangrijke natuurkundige aspectente extraheren uit de grote hoeveelheid data die dit type simulaties genereert.

Om deze reden hebben wij er voor gekozen om gebruik te maken van een sterkvereenvoudigde beschrijving van het systeem: de punt-dipool benadering. In ditmodel wordt elk nanodeeltje slechts beschreven door een dipool met een bepaalddipoolmoment, dat wil zeggen, een plus en min lading op een bepaalde afstandvan elkaar. De grootte van het dipoolmoment is frequentie-afhankelijk en heefteen maximum op de plasmonresonantie. Uiteraard is grote zorg besteed aan hetcontroleren van de validiteit van deze benadering. Binnen de punt-dipool bena-dering kan de koppeling tussen verschillende nanodeeltjes eenvoudig beschrevenworden met behulp van dipool-dipoolinteracties. Daarnaast is het ook mogelijkom de invloed van de omgeving van de deeltjes mee te nemen, bijvoorbeeld dievan het materiaal waardoor ze omgeven worden, maar ook de invloed van eenoppervlak of een gelaagde structuur. Dit is erg belangrijk omdat in werkelijk-heid de nanodeeltjes altijd op een oppervlak moeten rusten en het is bekend datde aanwezigheid van een oppervlak een significante invloed heeft op de optischeeigenschappen van nanodeeltjes. Deze invloed is bijzonder sterk in het gevalvan een metalen oppervlak. De reden hiervoor is niet alleen dat een metalenoppervlak een spiegelende werking heeft, en het deeltje dus interactie heeft metzichzelf via de spiegel, maar vooral omdat op het metalen oppervlak ook plas-monen aangeslagen kunnen worden. Het plasmon van een metalen nanodeeltjezal dan koppelen met het plasmon van het oppervlak, met als gevolg een sterkeverandering van de optische eigenschappen van het nanodeeltje.

Wanneer een aantal metalen nanodeeltjes sterk met elkaar koppelt, wat hetgeval is in een plasmonische keten, dan worden de optische eigenschappen vande keten niet meer bepaald door de som van de individuele plasmonresonanties,maar gaat het om de collectieve plasmonresonanties van het hele systeem. Elkvan deze resonantiefrequenties correspondeert met een bepaalde faserelatie tus-sen de deeltjes, bijvoorbeeld alle dipolen oscilleren in-fase, de ene helft oscilleertmet een tegengestelde fase als de andere helft, etc. Dit worden de collectievetoestanden van de keten genoemd. Het uitrekenen van de resonantiefrequen-ties voor al deze toestanden resulteert in een zogenaamde dispersierelatie engeeft een volledige beschrijving van de optische eigenschappen van de keten. Uiteen dispersierelatie kan men niet alleen afgeleiden welke toestanden aangeslagenworden door een bepaalde frequentie, maar ook hoe snel de excitatie zich zal

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voortbewegen en hoe snel hij zijn energie zal verliezen.

Dit proefschrift

Zoals de titel van het proefschrift aangeeft hebben wij ons gericht op het beschrij-ven van “energietransport en plasmondispersie in lineaire ketens van metalennanodeeltjes”. In de bovenstaande secties is uitgelegd dat een plasmon-excitatiezich door een keten van metalen nanodeeltjes kan voortbewegen. Dit betekentdat de energie van het plasmon, dat wil zeggen de bewegingsenergie van deelektronen en de potentiele energie van de nanodeeltjes, worden getransporteerdlangs te keten. Niet alle collectieve toestanden van de keten zijn even geschiktom gebruikt te worden voor energietransport: voor bepaalde frequenties zal hettransport heel efficient verlopen, terwijl voor andere nagenoeg geen energie heteinde van de keten bereikt. Dit wordt veroorzaakt door de dispersie van de col-lectieve plasmontoestanden en daarom is het van belang om de dispersierelatiesvan het systeem goed te begrijpen.

In hoofdstuk 2 van dit proefschrift hebben we de efficientie van het transportvan een optisch signaal door een plasmonisch keten bestudeerd. In het bijzonderhebben we gekeken naar de invloed van een oppervlak onder de keten. In demeeste theoretische beschrijvingen zijn plasmonische ketens in de vrije ruimtebeschouwd, terwijl in een experiment de deeltjes altijd op een oppervlak moetenrusten. Een belangrijk verlieskanaal in een plasmonische keten is straling, dedeeltjes koppelen niet alleen met elkaar maar zullen ook een elektrisch veld pro-duceren in de richting loodrecht op de keten. Om deze reden hebben we gekekennaar de invloed van een spiegelend oppervlak onder de keten, zodat een gedeeltevan de stralingsverliezen gereflecteerd wordt en terug komt bij de keten. Hetblijkt dat dit inderdaad een positieve invloed op de efficientie heeft. Echter, deefficientie gaat nog verder omhoog als een zilveren oppervlak gebruikt wordt. Ditis verrassend omdat zilver zeker geen perfecte spiegel is en dus behoorlijk veelelektrisch veld zal absorberen in plaats van reflecteren. De reden voor de toe-name in efficientie is dat op het zilveren oppervlak ook een plasmon resonantiekan worden aangeslagen. De plasmonen in de keten van nanodeeltjes koppelenmet het plasmon van het zilveren oppervlak en zorgen samen voor een efficientenergietransport.

In hoofdstuk 3 is de koppeling tussen de plasmonen van een keten van meta-len nanodeeltjes en die van een metalen oppervlak verder bestudeerd. Door het

98 Samenvatting

uitrekenen van de dispersierelaties van het systeem, konden we de menging vande plasmonen van de keten en van het oppervlak bestuderen als functie van deplasmafrequentie van het metalen oppervlak. Hoe kleiner het verschil tussen deplasmonresonanties van het oppervlak en van de keten, des te sterker de mengingtussen beide toestanden. Het combineren van het concept van plasmon hybridi-satie en de symmetriebreking die het oppervlak induceert, maakt het mogelijkom gekoppelde toestanden helder te karakteriseren. In het geval van sterke kop-peling tussen keten en oppervlak, kunnen voor dezelfde geometrie toestandengevonden met een erg hoge groepssnelheid, een groepssnelheid van bijna nul, enzelfs een negatieve groepssnelheid. Voor het laatste geval laten we aan de handvan Poynting vector berekeningen zien dat het hier optische toestand gaat meteen negatieve brekingsindex.

Zowel in hoofdstuk 2 als in hoofdstuk 3 tonen we aan dat, voor een plasmo-nische keten boven een oppervlak, de toestanden met een polarisatie langs deketen gekoppeld zijn met de toestanden waarbij de polarisatie loodrecht op hetoppervlak staat. In hoofdstuk 4 maken we van dit feit gebruik om een systeem zote ontwerpen dat de plasmonische toestanden elliptisch gepolariseerd zijn. Dit isinteressant omdat elliptisch gepolariseerde dipolen in staat zijn om alleen plas-monen op het oppervlak aan te slaan die in een bepaalde richting propageren.In eerdere onderzoeken werd het nanodeeltje altijd elliptisch gepolariseerd doorhet aan te stralen met een bundel met de gewenste polarisatie. Het voordeel vanhet systeem dat wij voorstellen is dat de ellipticiteit een natuurlijke eigenschapvan het systeem is. Het directioneel aanslaan van plasmonen op het oppervlakis daardoor onafhankelijk van de externe bron.

Tot slot, in hoofdstuk 5, beschouwen we een zeer lange plasmonische ketenbestaande uit 4000 nanodeeltjes in de vrije ruimte. Opnieuw hebben we gekekennaar de efficientie van energietransport door de keten. Echter, in dit geval nietals functie van de frequentie, maar als functie van de tijd. Door het eerstedeeltje van de keten aan te slaan met een korte laserpuls en de propagatie van hetplasmon te volgen over de tijd, hebben we bepaald wat de snelheid is waarmee hetplasmon door keten beweegt. Het blijkt dat voor elke excitatie twee verschillendesignalen door keten lopen. In de eerste plaats het signaal dat gevormd wordtdoor de collectieve plasmonen van de keten, met de daarbij behorende snelheid endemping, maar ook een signaal dat met de lichtsnelheid door de keten beweegten zich uitbreidt in de ruimte, zonder te dempen. Dit laatste signaal is eenzogenaamde precursor, een signaal dat wordt gevormd door verstrooiing rond de

Samenvatting 99

deeltjes, nog voordat het plasmon aangeslagen is. Doordat de verstrooiing eeninstantaan effect is, kan het zich voortplanten met de snelheid van het licht enzonder te dempen, het beweegt zo snel dat de elektronen van de nanodeeltjes hetsignaal niet kunnen volgen. De intensiteit van de precursor is laag, maar verderopin de keten speelt hij een dominante rol doordat hij zijn energie veel minder snelverliest dan het collectieve plasmon signaal. Voor lange plasmonische ketensmoet daarom zeker rekening gehouden worden met dit signaal. Daarnaast is het,gezien zijn hoge snelheid en geringe verval, de moeite waard om te verkennen ofdit signaal als informatiedrager gebruikt kan worden binnen de nano-optica.

100 Samenvatting

Dankwoord

Dit proefschrift had niet tot stand kunnen komen zonder de onmisbare hulp vaneen aantal mensen. Daarom wil ik graag de mensen bedanken die, hetzij opprofessioneel of persoonlijk vlak, een belangrijke bijdrage hebben geleverd.

In de eerste plaats wil ik natuurlijk Jasper Knoester bedanken voor de afgelopenjaren. Bedankt voor het vertrouwen en de vrijheid die je me hebt gegeven inmijn onderzoek. Ik heb altijd erg genoten van onze discussies en heb veel mogenleren van je natuurkundige inzicht en je scherpe blik. Bedankt dat je, ondanks jedrukke agenda, zoveel tijd vrij weet te maken voor wetenschap, in het bijzondertijdens de afgelopen maanden waarin de meeste communicatie verliep via tele-foongesprekken in het weekend of ‘s avonds laat. Ik waardeer het erg dat je nietalleen wetenschappelijke begeleiding geeft, maar ook persoonlijk betrokken bent.

Victor, I very much enjoyed our interactions over the past years. The physicalintuition and critical mindset that I learned from you will very valuable for therest of my career. It was an honor to work with you!

Uiteraard wil ik ook Niels en Roel bedanken voor de fijne tijd. De eindeloze flau-wekul, goede gesprekken en uiteraard de vaste koffiemomentjes zorgden ervoordat ik altijd met veel plezier naar de universiteit kwam. Het is een eer om nogeen keer als ‘de drie musketiers’ ten strijde te kunnen trekken!

It was a pleasure and a valuable experience to be part of the Theory of CondensedMatter group and I want to thank all the members and former members forall the nice interactions: Jasper, Maxim, Bernhard, Thomas, Marwa, Anka,Ana, Arunesh, Alessio, Maria, Tenzin, Bintoro, Niels, Roel, Andrey, Erik, Bas,Wijnand, Frank, Dennis, Anna Stradomska, Chungwen and Santanu.

During my PhD I had the pleasure to co-supervise both bachelor and masterprojects. I’d like to thank Niels Jelsma, Maarten Blok, Xabier Inchausti andArsalan Torke Ghashghaee for the great time and their contributions to thisthesis.

102 Dankwoord

I would also like to thank Andrey Malyshev for hosting me at UniversidadComplutense de Madrid, introducing me to the wonderful field of plasmonicsand for the fruitful discussion that we have had over the past years. A spe-cial thanks to the group members Javier, Angela, Belen and Irene for the nicefriendships and making me feel welcome. Javi, thank you very much for helpingme to get started with my research. I am really happy for you and Angela andwish both of you all the best. I hope we will stay in touch!Bernhard, bedankt voor de vele interessante en leerzame gesprekken. Fijn datjij altijd bereid was om te hulp te schieten als wij even niet meer wisten hoewe verder moesten. Ik hoop dat we goed contact mogen blijven houden en detijd kunnen vinden om een aantal van die openstaande problemen te kunnenaanpakken.Natuurlijk wil ik ook Iris en Annelien bedanken. Jullie zijn de steun en toeverlaatvan de groep. Bedankt dat jullie altijd klaar stonden om te helpen of om evengezellig te praten (en voor de altijd gevulde pot met snoep).Tijdens mijn verblijf in Groningen heb ik het geluk gehad om niet alleen eenprettige werkomgeving gehad te hebben, maar vooral ook een groep goede vrien-den om me heen die voor de nodige ontspanning zorgden. In het bijzonder wilik graag Jasper en Vincent bedanken voor de vele mooie, gezellige avonden enavonturen. Ik hoop dat we elkaar veel zullen blijven zien en spreken, ook nu deafstand wat groter geworden is. Kees, bedankt voor de goede vriendschap diewe al zoveel jaar hebben en die hopelijk nog vele jaren blijft bestaan. Hilbert,bedankt voor introductie bij Nedap. Leuk dat we nu niet meer als studiegenoten,maar als collega‘s samen mogen werken. Jelle, bedankt voor gezellige filmavon-den en de goede gesprekken over wetenschap, geloof en maatschappij. Verderwil ik natuurlijk Els, Pieter, Esther, Arend, Gerrit, Wieke, Izra, Jelle, Ester enAdri bedanken voor de vele gezellige weekendjes en de wintersportvakanties.Uiteraard had dit onderzoek niet tot stand kunnen komen zonder de eindelozesteun van mijn ouders. Bedankt dat jullie altijd klaar staan om te helpen en mijsteunen in de keuzes die ik maak.Het meest dankbaar ben ik Marriet. Bedankt voor alle liefde die je me geeft enhet vertrouwen dat je in me hebt. In het bijzonder tijdens het afgelopen jaarwaarin we, ondanks de drukte van het afronden van dit proefschrift, het verhuizenen het starten van nieuwe banen, vooral samen hebben kunnen genieten van onzekleine, lieve Elin. Jullie laten me zien hoe mooi het leven is!

Publication list

• P.J. Compaijen, V.A. Malyshev, and J. Knoester, “Electromagnetic pulsepropagation through a chain of metal nanoparticles: precursor and normalmode contributions”, To be submitted.

• P.J. Compaijen, V.A. Malyshev, and J. Knoester, “Elliptically polarizedmodes for the unidirectional excitation of surface plasmon polaritons”, Toappear in Optics Express

• P.J. Compaijen, V.A. Malyshev, and J. Knoester, “Engineering plasmondispersion relations: hybrid nanoparticle chain -substrate plasmon polari-tons”, Optics Express, vol. 23, no. 3, p. 2280, 2015

• P.J. Compaijen, V.A. Malyshev, and J. Knoester, “Substrate-mediatedsub-diffraction guiding of optical signals through a linear chain of metalnanoparticles: Polarization dependence and the role of the dispersion rela-tion”, 7th International Conference on Advanced Electromagnetic Materi-als in Microwaves and Optics, 247, 2013.

• P.J. Compaijen, V.A. Malyshev, and J. Knoester, “Surface-mediated lighttransmission in metal nanoparticle chains”, Physical Review B, vol. 87, p.205347, 2013

104 Publication list

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