Guiding of electromagnetic energy in subwavelength periodic metal structures

Maier, Stefan

Guiding of electromagnetic energy in subwavelength periodic metal structures

Zusammenfassung

Inhaltsangabe:Abstract:
The ultimate miniaturization of optical devices requires structures that guide electromagnetic energy with a lateral confinement below the diffraction limit of light. In this thesis, the possibility of employing plasmon-polariton excitations in plasmon waveguides consisting of closely spaced metal nanoclusters for this purpose is examined. The feasibility of energy transport with mode sizes below the diffraction limit of visible light over distances of several hundred nanometers is demonstrated.
As a macroscopic analogue to plasmon waveguides, the transport of electromagnetic energy in the microwave regime along closely spaced centimeter-scale metal rods is examined. Full-field electrodynamic simulations show that information transport occurs at a group velocity of 0.65c for fabricated structures consisting of copper rods excited at 8 GHz. A variety of passive routing structures and an all-optical modulator are demonstrated.
The possibility of guiding electromagnetic energy at visible frequencies with mode sizes below the diffraction limit using plasmon waveguides is analyzed using a point-dipole model and finite-difference time-domain simulations. It is shown that energy transport occurs via near-field coupling between metal nanoparticles, which leads to coherent propagation of energy. For spherical gold particles in air, group velocities up to 0.06c are demonstrated, and a change in particle shape to spheroidal particles shows up to a threefold increase in group velocity. Pulses with transverse polarization are shown to propagate with negative phase velocities antiparallel to the energy flow.
Plasmon waveguides consisting of gold and silver nanoparticles were fabricated using electron beam lithography. The key parameters that govern the energy transport are determined for various interparticle spacings and particle chain lengths using far-field measurements of the collective plasmon modes. Spherical gold nanoparticles with a diameter of 50 nm and an interparticle spacing of 75 nm show an energy attenuation of 6 dB/30 nm. This loss can be reduced by one order of magnitude by a geometry change to spheroidal particles. Using the tip of a near-field optical microscope as a local excitation source and fluorescent nanospheres as detectors, experimental evidence for energy transport over a distance of 0.5 µm is presented for plasmon waveguides consisting of silver rods with a 3:1 aspect ratio.

Inhaltsverzeichnis:Table of […]

Leseprobe

Inhaltsverzeichnis

ID 6629

Maier, Stefan: Guiding of electromagnetic energy in subwavelength periodic metal

structures

Hamburg: Diplomica GmbH, 2003

Zugl.: Pasadena, Dissertation / Doktorarbeit, 2003

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iii

Acknowledgements

It is my great pleasure to thank the many people that have supported me during my

thesis work at Caltech and beyond.

Most of all, thanks is due to my advisor Harry Atwater, whose belief in the success of

this project never faltered during the last three years. I cannot recall a single day when he

was not willing to discuss obstacles and progress associated with my research, and his

constant support and encouragements have been exemplary.

Mark Brongersma deserves most credits for allowing a smooth transition from course

work into active research. During his postdoctoral work in our group, he started out the

initial investigations into the properties of ordered metal nanoparticle chains, and was of

great help in the planning of the first experiments. Most of the knowledge and skills I

acquired during this project I owe to him. The void he left after his far-too-early

disappearance to Stanford was filled by Pieter Kik, who has supported my research ever

since. Pieter's often intuitive insight into physical processes and his strong background in

optical experiments were of great help with this thesis.

I also had the great pleasure to work with a number of enthusiastic collaborators

outside of Caltech. Thanks to Sheffer Meltzer of the University of Southern California for

his willingness to contribute to this project with his invaluable experience as a surface

chemist. He assembled a great number of nanoparticle structures using his dedicated

atomic force micromanipulation system. During the last months of this work, Elad Harel

of the University of California in San Diego continued in Sheffer's footsteps.

For me, it was always a daunting task to conduct experiments whose success or

failure depend on the ordered arrangement of nanoscale structures and thus heavily on the

quality of the samples. But as it turned out there was no reason for worry since I

was working with the very best - Richard Muller, Paul Maker and Pierre Echternach of

the Jet Propulsion Laboratory in Pasadena. Thanks to them for their invaluable help with

electron beam lithography.

The experimental part of this work has benefited a lot from discussions about near-

field optical microscopy with Aaron Lewis of Nanonics Imaging Ltd. in Jerusalem. I am

very grateful for his aid in setting up a reliable and intricate near-field microscopy

station.

Funding for this work was mainly provided by the National Science Foundation

through the Center of the Science and Engineering of Materials at Caltech and the Air

Force Office of Scientific Research.

During my stay in Harry's group I was lucky to be surrounded by a number of great

fellow students. I especially want to thank Luke Sweatlock for a lot of good and fun

discussions and Robb Walters for providing a great office atmosphere.

I would never have had the pleasure of meeting all these fine people had it not been

for two advisors that especially influenced me during my studies at Technische

Universität München. Thanks to Fred Koch and Dmitri Kovalev for their training and

encouragement, and for the great times in the beer gardens.

And of course, nothing would have been and is possible without Mag.

Abstract

The miniaturization of optical devices to spatial dimensions akin to their electronic

device counterparts requires structures that guide electromagnetic energy with a lateral

confinement below the diffraction limit of light. This cannot be achieved using

conventional optical waveguides or photonic crystal defect waveguides. Thus, a size

mismatch between electronic and optical integrated devices exists and needs to be

overcome.

In this thesis, the possibility of employing plasmon-polariton excitations in "plasmon

waveguides" consisting of closely spaced metal nanoclusters with a subwavelength cross

section for the confinement and guiding of electromagnetic energy is examined both

theoretically and experimentally. The feasibility of energy transport with mode sizes

below the diffraction limit of visible light over distances of several hundred nanometers

is demonstrated.

As a macroscopic analogue to nanoscale plasmon waveguides, the transport of

electromagnetic energy in the microwave regime of the electromagnetic spectrum along

structures consisting of closely spaced centimeter-scale metal rods is investigated. The

dispersion relation for the propagation of electromagnetic waves is determined using full-

field electrodynamic simulations, showing that information transport occurs at a group

velocity of 0.65c for fabricated structures consisting of centimeter-scale copper rods

excited at 8 GHz (

= 3.7 cm). The electromagnetic energy is highly confined to the

arrays, and the propagation loss in a straight array is about 6 dB/16 cm. Routing of

energy around 90-degree corners is possible with a power loss of 3-4 dB, and tee

structures for the splitting of the energy flow and for the fabrication of an all-optical

modulator are investigated. Analogies to plasmon waveguides consisting of arrays

of nanometer-size metal clusters are discussed.

The possibility of guiding electromagnetic energy at visible frequencies with mode

sizes below the diffraction limit is analyzed using an analytical point-dipole model for

energy transfer in ordered one-dimensional arrays of closely spaced metal nanoparticles.

It is shown that such arrays can work as plasmon waveguides that guide electromagnetic

energy on the nanoscale. Energy transport in these arrays occurs via near-field coupling

between metal nanoparticles, which sets up plasmon modes. This coupling leads to

coherent propagation of energy with group velocities exceeding the saturation velocity of

electrons in semiconductor devices. The point-dipole model suggests the feasibility of

complex guiding geometries such as 90-degree corners and tee structures for the routing

of electromagnetic energy akin to the fabricated macroscopic guiding structures, and the

possibility of an all-optical modulator operating below the diffraction limit is suggested.

The interparticle coupling in plasmon waveguides is examined using finite-difference

time-domain (FDTD) simulations. Local excitations of plasmon waveguides show direct

evidence for optical pulse propagation below the diffraction limit of light with group

velocities up to 0.06c in plasmon waveguides consisting of arrays of spherical noble

metal nanoparticles in air. The calculated dispersion relation and group velocities

correlate well with predications from the simple point-dipole model. A change in particle

shape to spheroidal particles shows up to a threefold increase in group velocity for

structures that can be fabricated using electron beam lithography. Pulses with transverse

polarization are shown to propagate with negative phase velocities antiparallel to the

energy flow.

vii

Plasmon waveguides consisting of spherical and spheroidal gold and silver

nanoparticles were fabricated using electron beam lithography with lift-off on ITO coated

quartz slides. Far-field polarization spectroscopy reveals the existence of longitudinal and

transverse collective plasmon-polariton modes. Measurements of the polarization

dependent extinction confirm that the collective modes arise from near-field optical

interactions. The key parameters that govern the energy transport are determined for

various interparticle spacings and particle chain lengths using measurements of the

resonance frequencies of the collective plasmon modes. For spherical Au nanoparticles

with a diameter of 50 nm and an interparticle spacing of 75 nm, the energy attenuation of

the plasmon waveguide is 6 dB/30 nm. This loss can be reduced and the energy

attenuation length conversely increased by approximately one order of magnitude by

using spheroidal silver nanoparticles as building blocks of plasmon waveguides, which

show an enhanced interparticle coupling and a decreased plasmon damping.

Near-field optical microscopy allows for the local optical analysis and excitation of

plasmon waveguides. Using the tip of a near-field optical microscope as a local excitation

source and fluorescent polystyrene nanospheres as detectors, experimental evidence for

energy transport over a distance of about 0.5

µm is presented for plasmon waveguides

consisting of silver rods with a 3:1 aspect ratio and a center-to-center spacing of 80 nm.

Ways to further improve the efficiency of energy guiding in plasmon waveguides and

possible applications are discussed.

viii

Table of Contents

Chapter 1 Introduction ... 1

1.1 Towards nanoscale optical devices ... 1

1.2 Surface plasmons as a way to overcome the diffraction limit ... 2

1.3 Road map through this thesis ... 5

Chapter 2 Yagi waveguides ... 8

2.1 Introduction... 8

2.2 The dispersion of a Yagi waveguide ... 9

2.3 Guiding along linear and corner Yagi arrays: Experiments and simulations... 14

2.4 Towards active devices: A three-terminal modulator... 19

2.5 The link to nanoscale plasmon waveguides ... 22

2.6 Conclusions and outlook ... 23

Chapter 3 Going nanoscale: Point-dipole theory of plasmon waveguides ... 25

3.1 Plasmon resonances in small metal clusters ... 25

3.2 Near-field particle interactions in plasmon waveguides... 32

3.3 Routing and switching of electromagnetic energy in plasmon waveguides ... 37

3.4 Conclusions and limitations of the dipole model ... 39

Chapter 4 FDTD simulations of plasmon waveguides ... 41

4.1 Introduction... 41

4.2 Collective far-field excitation of plasmon waveguides ... 41

4.3 Locally excited plasmon waveguides ... 49

4.4 Tailoring of the guiding properties by particle design... 55

4.5 Conclusions and outlook ... 57

Chapter 5 Fabrication and far-field properties of plasmon waveguides ... 62

5.1 Introduction... 62

5.2 Fabrication of plasmon waveguides ... 63

5.3 Far-field characterization of interparticle coupling in plasmon waveguides ... 67

5.4 Conclusion and outlook: Decrease of waveguide loss by particle design... 75

Chapter 6 Local excitation of plasmon waveguides ... 79

6.1 Introduction... 79

6.2 Transmission NSOM analysis of plasmon waveguides: Facts and artifacts ... 83

6.3 Molecular fluorescence as a probe for localized electromagnetic fields ... 89

6.4 Local excitation and detection of energy transport in plasmon waveguides... 96

6.5 Conclusions and outlook ... 103

Chapter 7 Conclusions and outlook ... 109

Bibliography ... 116

Table of Figures

Figure 1-1 (color): Optical fibers and photonic crystals... 1

Figure 1-2 (color): Sketch of a plasmon waveguide network coupling two conventional

dielectric plane waveguides ... 5

Figure 1-3 (color): Artist's rendition of a plasmon waveguide ... 7

Figure 2-1 (color): Yagi antennas ... 8

Figure 2-2 (color): Geometry of a short 10-element Yagi array for the determination of

the dispersion relation using full-field electrodynamic simulations ... 10

Figure 2-3 (color): Influence of rod height on the guiding properties of Yagi arrays ... 10

Figure 2-4 (color): Transmission line model of a Yagi array ... 11

Figure 2-5: Dispersion relation for electromagnetic waves propagating on a linear Yagi

array consisting of 101 copper rods obtained by electromagnetic simulations... 13

Figure 2-6 (color): Top view of a fabricated 90-degree corner Yagi structure on

Styrofoam, showing a source and two probe dipoles ... 15

Figure 2-7 (color): Guiding energy along a linear Yagi array... 15

Figure 2-8 (color): Guiding energy along a corner Yagi array... 17

Figure 2-9 (color): Routing energy in a Yagi tee structure ... 18

Figure 2-10 (color): Modulation of energy in Yagi arrays ... 20

Figure 2-11 (color): Power modulation characteristic of a Yagi modulator... 21

Figure 3-1: Energy flux (Poynting vector) around a metal nanoparticle under plane wave

excitation at two frequencies... 26

Figure 3-2: Extinction of 30 nm Au colloids ... 27

Figure 3-3: Dipole resonance position for spheroids with different aspect ratios for

both the long- and the short-axis mode of excitation ... 28

Figure 3-4: Energy relaxation of a surface plasmon... 29

Figure 3-5: Plasmon decay time for Au (a) and Ag (b) nanoparticles... 31

Figure 3-6: Geometry of a plasmon waveguide consisting of a chain of noble metal

nanoparticles ... 34

Figure 3-7 (color): Dispersion relation of a plasmon waveguide.. ... 36

Figure 3-8 (color): Calculated power transmission coefficients

in plasmon waveguides

for a 90-degree corner and a tee structure ... 38

Figure 3-9 (color): A simulated nanoscale all-optical modulator... 39

Figure 4-1: Real (lower curves) and imaginary (upper curves) part of the dielectric

function of Au from the literature ... 42

Figure 4-2 (color): Determination of the single particle plasmon resonance frequency... 44

Figure 4-3 (color): Collective excitation of a plasmon waveguide... 46

Figure 4-4 (color): Collective resonance energies of nanoparticle chain arrays... 47

Figure 4-5 (color): Simulation of a locally excited plasmon waveguide ... 50

Figure 4-6 (color): Dispersion relation of plasmon waveguides obtained using FDTD

simulations ... 51

Figure 4-7 (color): Pulsed local excitation of plasmon waveguides... 52

Figure 4-8 (color): Energy decay during pulse propagation in plasmon waveguides... 53

Figure 4-9 (color): Negative phase velocity in plasmon waveguides... 54

Figure 4-10 (color): Pulse propagation through spheroidal plasmon waveguides... 56

Figure 4-11 (color): FDTD simulation of a 90-degree corner plasmon waveguide... 59

xii

Figure 4-12 (color): Interference on plasmon waveguides... 61

Figure 5-1 (color): Extinction coefficients of spherical Au (a) and Ag (b) clusters ... 63

Figure 5-2: Atomic force microscopy fabrication of plasmon waveguides (I) ... 64

Figure 5-3: Atomic force microscopy fabrication of plasmon waveguides (II)... 64

Figure 5-4: Schematic of the steps involved in an electron beam lithography fabrication

process of metal nanoparticles ... 65

Figure 5-5: Plasmon waveguides fabricated using electron beam lithography... 66

Figure 5-6: Far-field extinction spectrum of plasmon waveguides ... 69

Figure 5-7 (color): Collective resonance energy of versus interparticle spacing... 71

Figure 5-8 (color): Collective resonance energy of plasmon waveguides versus particle

chain length... 74

Figure 5-9 (color): Collective resonance energies of chains of spherical and spheroidal

particles... 77

Figure 5-10: Far-field extinction of rod shaped particles ... 78

Figure 6-1 (color): Dispersion relation of a plasmon waveguide consisting of spherical

nanoparticles ... 80

Figure 6-2: The optical tunneling effect... 81

Figure 6-3: Aperture-type NSOM geometries... 82

Figure 6-4 (color): The Nanonics NSOM-100 scanning head... 83

Figure 6-5 (color): Nanonics NSOM tip ... 84

Figure 6-6: Fragility of plasmon waveguides prevents contact mode scanning ... 85

Figure 6-7 (color): Topography (a) and transmitted light (b) NSOM image of plasmon

waveguides consisting of spherical Au nanoparticles ... 86

xiii

Figure 6-8 (color): Topography and transmitted NSOM image of solid Au

nanowires combined into a three-dimensional surface plot... 88

Figure 6-9 (color): Far-field spectrum of the fluorescence of a thin film of aggregated

nile-red fluorescent nanospheres obtained using a photoluminescence setup ... 91

Figure 6-10 (color): Fluorescent NSOM setup... 92

Figure 6-11 (color): Topography and fluorescent NSOM images of 100 nm polystyrene

nanospheres... 93

Figure 6-12 (color): Topography and fluorescent NSOM scans of 100 nm fluorescent

nile-red nanospheres ... 94

Figure 6-13 (color): Fluorescence near-field spectroscopy of plasmon waveguides... 95

Figure 6-14 (color): Far-field extinction spectrum of Ag nanoparticle chains and single

particles... 97

Figure 6-15 (color): Near-field optical microscopy excitation and energy transport

detection of plasmon waveguides ... 98

Figure 6-16 (color): Topography and fluorescent NSOM scans ... 100

Figure 6-17 (color): Evidence for energy transport in plasmon waveguides via the width

of the fluorescence intensity of fluorescent nanospheres ... 102

Figure 6-18: Placement of fluorescent nanospheres along plasmon waveguide structures

using the tip of an atomic force microscope... 105

Figure 6-19 (color): Polarization of NSOM excitations of plasmon waveguides... 107

Figure 7-1: A plasmon waveguide tee structure fabricated using electron beam

lithography ... 111

xiv

Figure 7-2 (color): Application of a plasmon waveguide as an energy collimator at

the end of a conventional dielectric waveguide... 114

Figure 7-3 (color): Application of a plasmon waveguide as an all-optical modulator

linking two resonant cavities... 114

Figure 7-4 (color): Plasmon waveguide as an optically tunable waveguide coupler... 115

Chapter 1

Introduction

1.1

Towards nanoscale optical devices

The miniaturization of optical devices to size dimensions akin to their electronic

counterparts is a major goal of current research efforts in optoelectronics, photonics and

semiconductor manufacturing. A high integration of optical components allowing the

fabrication of all-optical chips for computing and sensing requires both a confinement of

the guided optical modes to small dimensions and the ability to route energy around sharp

corners. Current technologies that are driving a revolution in the fabrication of integrated

optical components are planar waveguides, optical fibers and photonic crystals, which

can confine and guide electromagnetic energy in spatial dimensions in the micron and

sub-micron regime (Figure 1-1).

Figure 1-1 (color): Optical fibers and photonic crystals. a) Sketch of an optical fiber that

confines and guides light by total internal reflection (after [1]). b) Finite-difference time-

domain simulation of a guided light mode in a 90-degree corner photonic crystal defect

waveguide (after [2]).

Whereas waveguides based on the principle of total internal reflection such as

optical fibers do not allow for the guiding of light around sharp corners with a bending

radius considerably smaller than the wavelength of light

[1], bend engineering of defect

modes in photonic crystals has enabled the fabrication of defect waveguides with

complex guiding geometries [2, 3]. The further integration of active devices such as

defect mode lasers into photonic crystals [4] will ensure a prominent spot for this

technology in the creation of optical chips.

The size and density of optical devices employing these technologies is nonetheless

restricted by the diffraction limit

of light, which imposes a lower size limit of a few

hundred nanometers on the optical mode size. Thus, a size mismatch between highly

integrated electronic devices with lateral dimensions of a few tens of nanometers and

optical guiding components persists and needs to be overcome.

1.2

Surface plasmons as a way to overcome the diffraction limit

The diffraction limit for the guiding of electromagnetic energy can be overcome if the

optical mode is converted into a non-radiating mode that can be confined to lateral

dimensions smaller than the diffraction limit. Prominent examples that have been the

focus of intense research over the last couple of decades are surface plasmon-polaritons

in metals. Plasmons are coherent oscillations of the conduction electrons of the metal

against the static positive background of the metal ion cores. Whereas plasmons in bulk

metal do not couple to light fields, a two-dimensional metal surface can sustain plasmons

if excited by light either via evanescent prism coupling or the help of surface corrugations

to ensure momentum matching [5]. Such surface plasmons propagate as coherent electron

oscillations parallel to the metal surface and decay evanescently perpendicular to it.

Thus, the electromagnetic energy is confined to dimensions below the diffraction limit

perpendicular to the metal surface. Corrugations can further act as light-scattering centers

for surface plasmons, allowing for the fabrication of interesting optical devices such as an

all-optical transistor [6].

A further confinement of energy-guiding surface plasmon modes can be achieved

using metal nanowires instead of extended surfaces. In nanowires, the confinement of the

electrons in two dimensions leads to well-defined dipole surface plasmon resonances if

the lateral dimensions of the wire are much smaller than the wavelength of the exiting

light. Thus, the optical properties of metal nanowires can be optimized for particular

wavelengths of interest, and non-regular cross sections and coupling between closely

spaced nanowires allow a further tuning of the optical response [7, 8]. Indeed, the

propagation of electromagnetic energy has been demonstrated along noble metal stripes

with widths of a few microns [9] and along nanowires with subwavelength cross sections

[10, 11], and propagation lengths of a few microns have been found. Related to this

principle is the interesting idea of one-dimensional negative dielectric core waveguides,

where metals instead of dielectric materials are used as the core in a waveguide with a

cross section below the diffraction limit [12].

Another intriguing nanoscale system that can sustain surface plasmons are metal

nanoparticles, and their interaction with light has been the focus of intense research in

recent years [13, 14]. In metal nanoparticles, the three-dimensional confinement of the

electrons leads to well-defined surface plasmon resonances at specific frequencies. From

work on single noble metal nanoparticles, it is well established that light at the surface

plasmon resonance frequencies interacts strongly with metal particles and excites a

collective motion of the conduction electrons, or plasmon [15]. These resonance

frequencies are typically in the visible or infrared part of the spectrum for gold and silver

nanoparticles embedded in a variety of hosts. For particles with a diameter much smaller

than the wavelength

of the exciting light, plasmon excitations produce an oscillating

electric dipole field resulting in a resonantly enhanced non-propagating electromagnetic

near-field close to the particle surface.

Recently, it has been suggested that near-field interactions between closely spaced

metal nanoparticles in regular one-dimensional particle arrays can lead to the coherent

propagation of electromagnetic energy along the arrays with lateral mode sizes below the

diffraction limit [16]. For Au and Ag nanoparticles in air, energy decay lengths of a

couple of hundred nanometers have been predicted. Furthermore, it was suggested that

these so-called plasmon waveguides can guide electromagnetic energy around sharp

corners and tee structures, and an all-optical modulator based on interference operating

below the diffraction limit was proposed [17].

The use of ordered arrays of metal nanoparticles as plasmon waveguides is intriguing

from both a technology oriented and a fundamental point of view. Metal nanoparticles

can be fabricated using a wide arsenal of tools including electron beam lithography [18],

ion beam implantation [19], colloidal synthesis [20], and self-assembly [21]. This way,

metal nanoparticles of different shapes can be produced from a variety of materials, and

the synthesis of core-shell composite particles offers the incorporation of novel non-

linear materials for additional functionality. The fabrication of plasmon waveguides

consisting of such particles could enable the design of highly integrated optical

devices and logic elements operating below the diffraction limit (Figure 1-2).

Figure 1-2 (color): Sketch of a plasmon waveguide network coupling two conventional

dielectric plane waveguides.

From a fundamental point of view, a study of plasmon waveguides provides insight

into the nature of optical near-fields and electromagnetic light-matter interactions on the

nanoscale. The strong dipolar coupling between metal nanoparticles can also serve as a

model system for the study of other dipole-coupled energy transfer structures such as

quantum dot chains [22], magnetic nanoparticle arrays [23], and coupled-resonator

optical waveguides [24].

1.3

Road map through this thesis

In the following chapters, the physical properties of plasmon waveguides consisting

of ordered, closely spaced metal structures with subwavelength dimensions are analyzed

both theoretically and experimentally.

Chapter 2 presents experiments and full-field electrodynamic simulations on

macroscopic analogues to nanoscale plasmon waveguides [25-27]. The macroscopic

waveguides are based on centimeter-scale metal rods with a center-to-center spacing of a

few millimeters and operate in the microwave and RF regime of the electromagnetic

spectrum, akin to Yagi arrays used for the steering of RF beams. These so-called Yagi

waveguides can be excited using a dipole antenna and show a strong confinement of the

electromagnetic energy to the guiding structure. Linear arrays, 90-degree corner and tee

waveguides are characterized, and simulation results on an all-optical modulator are

presented.

The discussion of the properties of nanoscale metal nanoparticle plasmon waveguides

begins in Chapter 3 with the description of a point-dipole model of energy transfer in

plasmon waveguides consisting of spherical metal nanoparticles [17, 28]. The dispersion

relation and group velocity for energy transport are determined, and transmission

coefficients for network structures consisting of corners and tees are presented, as well as

the modulation characteristic of an all-optical switch in the point-dipole limit. The

important notion of plasmon damping is addressed and discussed rather extensively, to

offset some contradicting and confusing descriptions of the damping processes found in

the literature [29]. An experimentally accessible expression for the expected energy

attenuation of plasmon waveguides is derived.

Chapter 4 extends the analytical modeling of plasmon waveguides in Chapter 3 by

solving the full set of Maxwell's equations using finite-difference time-domain

simulations for extended metal nanoparticles modeled via a Drude model [30, 31].

Starting from a determination of the plasmon resonance frequencies of single noble metal

nanoparticles, interactions between nanoparticles located in each other's near-field are

analyzed using plane-wave excitations. This way, a realistic interparticle coupling

strength can be determined. Local excitation sources allow for direct determination of the

dispersion relation and characterization of optical pulse propagation in plasmon

waveguides. The optimization of the guiding properties of plasmon waveguides using

non-spherical particles is discussed.

The fabrication and far-field optical characterization of plasmon waveguides are

presented in Chapter 5. Plasmon waveguides consisting of spherical and rod-shaped Au

and Ag nanoparticles fabricated using electron beam lithography are analyzed using far-

field polarization spectroscopy [28, 32, 33]. Extinction measurements allow for the

determination of the interparticle coupling strength and bandwidth of energy transport,

thus yielding estimates for the group velocities and energy decay lengths of the fabricated

plasmon waveguides.

Chapter 6 presents a characterization of the guiding properties of plasmon

waveguides using a near-field optical microscope (NSOM) [34-36]. The tip of an

illumination-mode NSOM is used as a local excitation source, and fluorescent

polystyrene nanospheres are used as local detectors of energy transport. Evidence for

energy transport over distances of about 0.5

µm is presented for plasmon waveguides

consisting of rod-shaped Ag nanoparticles. Chapter 7 summarizes the work presented in

this thesis and gives an outlook on future work and applications of plasmon waveguides.

Figure 1-3 (color): Artist's rendition of a plasmon waveguide [37].

Chapter 2

Yagi waveguides

2.1 Introduction

As a large-scale analogue to nanoscale plasmon waveguides that operate in the visible

part of the electromagnetic spectrum, this chapter discusses the propagation of

electromagnetic energy along centimeter-scale periodic metal structures in the radio

frequency regime, similar to linear Yagi antenna arrays. Figure 2-1 shows a sketch of a 7-

element Yagi antenna array as it is typically used in radio frequency communications.

The antenna is driven by a dipole whose beam-profile is modified by the addition of

director and reflector rods to assure a directed emission and receiving of radiation [38].

h >

/ 2

/ 3

h <

/ 2

Figure 2-1 (color): Yagi antennas. a) Sketch of a linear Yagi antenna used in radio

frequency communication. The beam profile is shaped by the director and reflector rods.

b) Picture of a TV Yagi antenna.

A typical Yagi array (Figure 2-1 a) consists of a driven dipole source and a series of

equally spaced metal rods with a height h and an inter-rod spacing s. It is well known

experimentally [39] and theoretically [40, 41] that such arrays show guiding properties

for electromagnetic radiation as long as

, where

is the free-space

wavelength of the electromagnetic wave emitted by the source [38-41]. Under this

condition, the phase velocity of the guided wave is less than the free-space velocity

c. Conversely, for rods with

, the phase velocity is greater than c and the

waves are reflected in the backwards direction. Conventional Yagi antenna arrays are

aimed at radiating out electromagnetic energy into the far-field and have inter-rod

spacings on the order of

. Here we will describe structures with a significantly

smaller spacing of

, which allow for more efficient guiding of energy around sharp

corners. Smaller spacing between adjacent rods results in a slower phase velocity [40, 41]

and a stronger confinement of the electromagnetic waves to the structure due to increased

near-field coupling. The following sections will describe electromagnetic simulations and

experiments that examine energy propagation in straight arrays, 90-degree corners and

tee structures, and an all-optical modulator based on interference will be discussed. The

last part of this chapter will establish the link to nanoscale plasmon waveguides.

2.2

The dispersion of a Yagi waveguide

In order to determine the range of suitable heights and spacings for efficient Yagi

waveguides, full-field electrodynamic simulations of the 10-element Yagi array depicted

in Figure 2-2 were performed using an antenna simulation software (EZNEC v. 2.0). In

the simulation, the axes of the rods were chosen to be in the z-direction and the center of

the first rod (source) was located at (x,y,z) = (0,0,0).

Sweep

Oscillator

Cu rods

Sweep

Oscillator

Sweep

Oscillator

Cu rods

Figure 2-2 (color): Geometry of a short 10-element Yagi array for the determination of

the dispersion relation using full-field electrodynamic simulations.

The first rod was driven by a center-fed current source at 8.0 GHz (

=3.75 cm), and

the steady state electromagnetic field distribution around the structure was determined.

The heights h of the rods and the inter-rod spacings s were varied to explore their effect

on the guiding and dispersion of electromagnetic waves.

Figure 2-3 shows typical distributions of the absolute value of the z-component of the

electric field vector |E

| in the x-y plane for a linear Yagi array consisting of 10 copper

rods with an inter-rod spacing s = 0.05

(2 mm) for various rod heights h.

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.3

-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

1E-3

0.002

0.003

0.004

0.006

0.01

0.02

0.03

0.04

0.06

0.1

0.2

0.3

0.4

0.6

0.3

0.5

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.5

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.3

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.3

-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

1E-3

0.002

0.003

0.004

0.006

0.01

0.02

0.03

0.04

0.06

0.1

0.2

0.3

0.4

0.6

0.3

0.5

-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

1E-3

0.002

0.003

0.004

0.006

0.01

0.02

0.03

0.04

0.06

0.1

0.2

0.3

0.4

0.6

0.3

0.5

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.5

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.5

Figure 2-3 (color): Influence of rod height on the guiding properties of Yagi arrays. The

contour plots show the distribution of the absolute value of the z-component of the

electric field vector |E

| in the x-y plane for linear arrays of 10 rods with an inter-rod

spacing of 0.05

and variable heights h. The first rod is driven with a dipole current

source at 8 GHz (

= 3.75 cm). The color scale is exponential in the magnitude of |E

| and

spans about 3 orders of magnitude.

For rods small compared to the free-space wavelength (h

d 0.3

), the

absorption and scattering cross sections of the rods are too small to allow for efficient

guiding of electromagnetic energy along the array. For medium-sized rods

(0.3

d h < 0.5

), the cross sections are large enough to allow for efficient guiding of

the electromagnetic energy, and the field distribution along the array shows a standing-

wave type behavior due to reflection of the guided wave at the end of the structure. We

will call this rod size regime the transmission window of Yagi arrays for energy guiding.

For larger rods (h > 0.5

), the electromagnetic waves radiated by the source dipole

interfere destructively in the forward direction, causing an exponential decay of the

electric field along the rod structure and a concurrent back reflection. Thus,

electromagnetic energy can be guided along a Yagi array only in the transmission

window for rods with 0.3

d h < 0.5

The change in the guiding properties of Yagi arrays at a rod height h = 0.5

is due to

a change in the antenna impedance of a single rod at 0.5

and can be understood using a

loaded transmission line analysis [40, 42] depicted in Figure 2-4.

rod

lossless case:

Impedance of

unloaded line: Z

rod

lossless case:

Impedance of

unloaded line: Z

Figure 2-4 (color): Transmission line model of a Yagi array (after [40]).

The transmission line model approximates each rod as a load on an otherwise

unloaded transmission line of impedance Z [40]. Analysis of the loaded line's impedance

allows for the determination of the phase velocity of electromagnetic waves on the

transmission line. Assuming negligible losses inside the rods with an inter-rod spacing s,

the phase velocity v can be calculated from the shunt impedance iX and the impedance of

the unloaded line Z to

where c and

are the free-space velocity and wavelength of the guided electromagnetic

radiation. In order to allow for a guiding of electromagnetic radiation, the phase velocity

must be smaller than the free-space velocity of light. This requires rods with capacitive

antenna impedance X < 0. It is well known from antenna theory that this is the case for

rods with a height

. For larger rods, the antenna impedance becomes inductive,

resulting in a phase velocity greater than the free-space velocity of light and no guiding.

From the view point of optical theory, a phase velocity v < c implies a mean refractive

index of the Yagi array greater than one, leading to convergence of the source dipole

radiation along the structure analogous to a convex lens. The transmission line analysis

also suggests that for capacitive rods the phase velocity decreases with decreasing inter-

rod spacing s, leading to a stronger confinement of the electromagnetic energy along the

array. This was confirmed using full-field electromagnetic simulations of Yagi arrays

with various inter-rod spacings.

The dependence of the phase velocity of electromagnetic waves propagating on a

linear Yagi array on the rod geometry can be summarized by computing the dispersion

relation

(k) of the guided waves. Figure 2-5 shows this dispersion relation for a

linear Yagi array consisting of 101 copper rods as obtained by full-field electrodynamic

simulations.

300

600

900

1200

(GH

)

k (m

-1

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

of measurement:

=0.05

, h=0.37

No guiding due to insufficient

absorption and scattering cross

sections of the elements

(no resonant enhancement!)

No guiding due to destructive

interference in forward direction

Transmission Window

300

600

900

1200

(GH

)

k (m

-1

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

of measurement:

=0.05

, h=0.37

No guiding due to insufficient

absorption and scattering cross

sections of the elements

(no resonant enhancement!)

No guiding due to destructive

interference in forward direction

Transmission Window

Figure 2-5: Dispersion relation for electromagnetic waves propagating on a linear Yagi

array consisting of 101 copper rods obtained by electromagnetic simulations. The rods

have a diameter d of 1 mm, a height h of 1.4 cm, and were spaced by s = 2 mm apart.

The rods have a diameter d = 1 mm, height h = 1.4 cm and are spaced s = 2mm apart,

corresponding to the dimensions of the fabricated Yagi arrays described in section 2.3.

The first rod was driven at various frequencies from 6.0 to 9.5 GHz via a current source,

and the standing wave pattern along the structure was analyzed to determine the

corresponding wave vector k of the guided waves along the structure. The decrease in

slope with increasing wave vector k indicates a decrease of the group velocity of the

electromagnetic waves. Near the zone boundary at k =

the ratio

approaches

0.5, as predicted by linear antenna theory [40, 41]. At the excitation frequency of

8.0 GHz used in experiments on fabricated arrays the phase velocity is about 0.65c.

2.3

Guiding along linear and corner Yagi arrays: Experiments and

simulations

This section describes three fabricated Yagi waveguide structures consisting of arrays

of Cu rods, namely, a straight line, a 90-degree corner, and a tee structure. The rods have

a diameter of 1 mm (d = 0.002

), a height of 1.4 cm (h = 0.37

), and were spaced

2 mm apart (s = 0.05

). The rods were fixed in position using a Styrofoam construction

that showed negligible guiding properties as depicted in Figure 2-6. The first rod was a

center-fed dipole driven at 8.0 GHz using a HP 8620C sweep oscillator. The distribution

of the electromagnetic field along the array was studied by local power measurements

using a 1 cm long probe dipole and a HP 8472 crystal detector connected to a HP 415E

standing wave meter. These measurements were performed at a distance of 0.15

from

the array to avoid significant interference with the propagating electromagnetic waves.

None of the structures was terminated with impedance matching loads to obtain standing

wave patterns for physical analysis. The results were compared to full-field

electrodynamic simulations of the fabricated structures as discussed in Section 2.2.

Figure 2-6 (color): Top view of a fabricated 90-degree corner Yagi structure on

Styrofoam, showing a source and two probe dipoles (courtesy of Luke Sweatlock).

Figure 2-7 shows the distribution of the absolute value of the z-component of the

electric field vector |E

in the x-y plane for a linear array of 101 rods. The color scale is

exponential in the magnitude of |E

and spans three orders of magnitude. The inset shows

both the measured (squares) and simulated (thin solid line) power at a distance of 0.15

from the array, normalized to the power obtained at the position of the source.

-1

-2

-1

s o urc e

- 1 0

I/ I

(dB)

-1

-2

-1

s o urc e

- 1 0

I/ I

(dB)

Figure 2-7 (color): Guiding energy along a linear Yagi array. The plot shows the

distribution of the absolute value of the z-component of the electric field vector |E

| in the

x-y plane for a linear array of 101 rods. The rods are 1 mm in diameter, 1.4 cm long, and

spaced 2 mm apart. The color scale is exponential in the magnitude of |E

| and spans

about 3 orders of magnitude. The inset shows both the measured (squares) and simulated

(thin solid line) power at a distance of 0.15

from the array.

Most of the energy emitted by the source is lost due to radiation into the far-

field, as seen in the field plot and in the inset as an initial drop in the measured power

after rod number 1. Along the array a standing wave pattern is recognized, indicating

weak attenuation and strong reflection at the endpoints. The period of the standing wave

obtained by simulation and experiment are in excellent agreement. The discrepancies in

the measured and simulated powers can be attributed in part to the finite diameter of the

probe dipole, which causes an averaging-out of the sharp features seen in the simulation.

About 90% of the energy is confined within a distance of 0.05

from the array,

demonstrating strong guiding of the electromagnetic wave. The propagation loss

of the

electromagnetic wave and its reflection coefficient R can be estimated by assuming that

the standing wave pattern results form electromagnetic waves that travel back and forth

along the array. From such an analysis

6 dB/16 cm corresponding to 6dB/4

and

are determined. The remaining 25% of the power is radiated into the far-field at

the end of the array, as can be seen in the field plot. It is important to note that radiation

loss only occurs at the two ends of the linear array. More generally, radiation loss occurs

at each discontinuity in the structure where the guided electromagnetic wave changes

direction. For the design of more complex array structures it is important to quantify the

losses at such discontinuities. For this purpose, the transport around 90-degree corners is

investigated next.

Figure 2-8 shows the distribution of |E

in the x-y plane for a corner structure

consisting of two linear arrays of rods that meet at a 90-degree angle. The inset shows

both the measured and simulated power at a distance of 0.15

on the outside of the

corner, normalized to the power obtained at the position of the source.

- 1

- 2

- 1

s ourc e

- 1 0

-10

-20

I/ I

(dB)

-1 0

-2 0

Figure 2-8 (color): Guiding energy along a corner Yagi array. The plot shows the

distribution of |E

| in the x-y plane for a corner structure consisting of two linear arrays of

rods that make a 90-degree angle. The rods are 1 mm in diameter, 1.4 cm long, and

spaced 2 mm apart. The color scale is exponential in the magnitude of |E

| and spans

about 3 orders of magnitude. The inset shows both the measured (squares) and simulated

(thin solid line) power at a distance of 0.15

from the array.

As for the straight line, simulation and experiment are in reasonable agreement. As

expected, power is lost in turning the corner due to radiation into the far field. The ratio

of the power in the side arm to the power in the main arm is 3-4dB. This decrease in

power is due to radiation and back reflection at the corner. Similar results were found for

a tee structure (Figure 2-9) in which an electromagnetic wave was injected into the stem

and the power flow split into the two side arms. The ratio of the power in one side arm to

the power in the stem was about 8dB. This is roughly twice the power lost in the corner

structure.

Details

Seiten
Erscheinungsform: Originalausgabe
Jahr: 2003
ISBN (eBook): 9783832466299
ISBN (Paperback): 9783838666297
Dateigröße: 5 MB
Sprache: Englisch
Institution / Hochschule: California Institute of Technology (Caltech) – unbekannt
Schlagworte: wellenleiter nanooptik metallnanoteilchen plasmon optik

Autor

Stefan Maier (Autor:in)

Guiding of electromagnetic energy in subwavelength periodic metal structures

Zusammenfassung

Leseprobe

Inhaltsverzeichnis

Details

Autor

Stefan Maier (Autor:in)