Plasmonic Antennas and Arrays for Optical Imaging and
Sensing Applications
by
Yan Wang
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Edward S. Rogers Sr. Department of Electrical
and Computer Engineering
University of Toronto
Copyright c 2013 by Yan Wang
Abstract
Plasmonic Antennas and Arrays for Optical Imaging and Sensing Applications
Yan Wang
Doctor of Philosophy
Graduate Department of Edward S. Rogers Sr. Department of Electrical and Computer
Engineering
University of Toronto
2013
The optics and photonics development is currently driven towards nanometer scales.
However, diffraction imposes challenges for this development because it prevents confinement of light below a physical limit, commonly known as the diffraction limit. Several
implications of the diffraction limit include that conventional optical microscopes are
unable to resolve objects smaller than 250 nm, and photonic circuits have a physical
dimension on the order of the wavelength. Metals at optical frequencies display collective electron oscillations when excited by photon energy, giving rise to the surface
plasmon modes with subdiffractional modal profile at metal-dielectric interfaces. Therefore, metallo-dielectric structures are promising candidates for alleviating the obstacles
due to diffraction.
This thesis investigates a particular branch of plasmonic structures, namely plasmonic
antennas, for the purpose of optical imaging and sensing applications. Plasmonic antennas are known for their ability of dramatic near-field enhancement, as well as effective
coupling of free-space radiation with localized energy. Such properties are demonstrated
in this thesis through two particular applications. The first one is to utilize the interference of evanescent waves from an array of antennas to achieve near-field subdiffraction
focusing, also known as superfocusing, in both one and two dimensions. Such designs
could alleviate the tradeoffs in the current near-field scanning optical microscopy by
ii
improving the signal throughput and extending the imaging distance. The second application is to achieve more efficient radiation from single-emitters through coupling to a
highly directive leaky-wave antenna. In this case, the leaky-wave antenna demonstrates
the ability of enhancing the directivity over a very wide spectrum.
iii
Dedication
To my parents:
Lijun and Wenxia
iv
Acknowledgements
‘Knowledge is in the end based on acknowledgement.’
-Ludwig Wittgenstein
It was only through the guidance, help and support of the kind individuals and
organizations that this thesis is successful, to only some of whom it is possible to give
particular mention here.
First and foremost, I would like to express my sincere gratitude to my principal
supervisor, Prof. George V. Eleftheriades, who mentored me throughout my Master’s
and Ph.D studies. His extraordinary passion for research, tremendous insights for science,
and unparalleled patience for teaching are some of the most admirable characters that
one can possess as a researcher and an advisor. It is my blessing to be mentored by
such an individual, from whom I have learned valuable lessons both academically and
personally. I would also like to thank my co-supervisor, Prof. Amr S. Helmy, who guided
me patiently in an area of research that I was very unfamiliar with at the beginning. I am
tremendously grateful for his vast knowledge in the practical aspect of optical science,
and his constant reminder of concerning engineering designs with the real world. His
insights and attitude have had significant influence on my approach to scientific research.
Next, I would like to thank Prof. Costas D. Sarris, Prof. Sean V. Hum, Prof. Joyce Poon,
and Prof. Wai Tung Ng from the University of Toronto, and Prof. Filippo Capolino from
the University of California, Irvine, for being on my supervisory and examination committees. Their feedback and questions have helped me gaining valuable understanding
and additional perspectives of my work.
Throughout the entire duration of my graduate study, I have had the great fortune
of meeting the most wonderful labmates, who not only offer stimulating discussions concerning research problems, but also friendship and support that one often needs in a
relatively solitary research work. I am indebted to their aspiration of achieving excelv
lence, their passion for research as well as for beer. I cannot ask for a better group of
amazing individuals to share my interests, knowledge and perspectives of life.
I would also like to acknowledge the generous financial support provided by the Ontario Graduate Scholarship, the Ontario Graduate in Scholarship in Science and Technology, the Doctoral Completion Award of the University of Toronto, V. L. Henderson and
M. Bassett Research Fellowship in Electrical Engineering, Ewing Rae Graduate Scholarship in Electrical Engineering, and Frank Howard Guest Bursary in the Faculty of
Applied Science and Engineering, without which this work would not have been possible.
I would like to thank my mom, Lijun, for her unconditional love and support; as well
as my dad, Wenxia, for his encouragement and sharing his graduate school experience
with me. Last but not least, I would like to thank my boyfriend, Gerald, for sharing
the precious graduate school life experience together. It is a real blessing to have a likeminded companion who constantly helps me to become a better version of myself in all
aspects of life.
Yan Wang
March, 2013
Toronto, Canada
vi
Contents
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Overview of plasmonic antennas . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
Early developments and obstacles . . . . . . . . . . . . . . . . . .
3
1.2.2
Current developments . . . . . . . . . . . . . . . . . . . . . . . .
5
Thesis objective, scope and outline . . . . . . . . . . . . . . . . . . . . .
6
1.3
2 Background
2.1
2.2
10
Metals at optical frequencies . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.1
Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.2
Empirical data and analysis . . . . . . . . . . . . . . . . . . . . .
13
Wavelength scaling for plasmonic antennas . . . . . . . . . . . . . . . . .
15
2.2.1
Theoretical analysis for dipole antennas . . . . . . . . . . . . . . .
16
2.2.2
Numerical analysis for dipole and slot antennas . . . . . . . . . .
18
3 Plasmonic Antennas for Near-Field Superfocusing
3.1
3.2
24
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.1
Fundamental trade-offs in NSOM technologies . . . . . . . . . . .
25
3.1.2
Other superresolution techniques . . . . . . . . . . . . . . . . . .
29
3.1.3
Engineering optical near-field with plasmonic antennas . . . . . .
30
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
vii
3.3
3.2.1
Back-propagation method . . . . . . . . . . . . . . . . . . . . . .
33
3.2.2
Shifted-beam theory . . . . . . . . . . . . . . . . . . . . . . . . .
40
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3.1
Background signal . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.3.2
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.3.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4 Plasmonic Antennas for Near-Field Superfocusing (2D)
4.1
65
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.1.1
2D monopole-array probe at microwave frequencies . . . . . . . .
66
4.1.2
Plasmonic monopole antennas at optical frequencies . . . . . . . .
68
4.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.3.1
Aperture probe with a single monopole antenna . . . . . . . . . .
79
4.3.2
Aperture probe with a monopole antenna array . . . . . . . . . .
80
5 Plasmonic Antennas for Far-Field Sensing
5.1
91
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.1.1
Nanoscale emitters . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.1.2
Nanoemitter radiation without antennas . . . . . . . . . . . . . .
94
5.1.3
Improving nanoemitter radiation with antennas . . . . . . . . . .
96
5.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.3
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1
Plasmonic leaky-wave slot mode . . . . . . . . . . . . . . . . . . . 109
5.4.2
Antenna efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Summary and Future Work
119
viii
A Metals at Optical Frequencies
124
A.1 Analytical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.1.1 The Drude (free-electron) model . . . . . . . . . . . . . . . . . . . 124
A.1.2 The Lorentz (bound-electron) model . . . . . . . . . . . . . . . . 126
A.1.3 The Lorentz-Drude (LD) model . . . . . . . . . . . . . . . . . . . 127
A.2 Empirical data and parameterization . . . . . . . . . . . . . . . . . . . . 128
B Dispersion of the TM0 Surface-Mode for Rod-Antennas
132
C Parallel dipoles at air-dielectric interface
138
C.1 Radiation of an electric dipole . . . . . . . . . . . . . . . . . . . . . . . . 140
C.2 Radiation of a magnetic dipole . . . . . . . . . . . . . . . . . . . . . . . . 143
References
146
ix
List of Figures
2.1
Dispersion of the complex εr for Au, Ag and Al. . . . . . . . . . . . . . .
14
2.2
The geometry of a plasmonic nanorod antenna. . . . . . . . . . . . . . .
15
2.3
The effective wavelength dispersion of plasmonic rod antennas. . . . . . .
20
2.4
Resonance of a plasmonic rod antenna . . . . . . . . . . . . . . . . . . .
21
2.5
Resonance of a plasmonic slot antenna . . . . . . . . . . . . . . . . . . .
22
2.6
Comparison of the theoretical and simulated λeff dispersion. . . . . . . .
23
3.1
Scanning probe microscopy. . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2
Examples of NSOM probes . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3
Gaussian beam diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.4
Cutoff of the fiber mode(s) in a tapered probe . . . . . . . . . . . . . . .
28
3.5
Near-field plate schematics. . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.6
The schematic of the back-propagation system.
. . . . . . . . . . . . . .
34
3.7
An example of the target function. . . . . . . . . . . . . . . . . . . . . .
35
3.8
The spatial and spectral distributions of the transmission function. . . .
37
3.9
Comparison of the numerical and analytical transmission functions. . . .
38
3.10 The transmission function for achieving various focal lengths. . . . . . . .
39
3.11 The transmission functions for various Gaussian target beamwidths. . . .
39
3.12 An example of metascreen based on the shifted-beam theory. . . . . . . .
41
3.13 Comparison of the target beam and the 3-slot array radiation. . . . . . .
44
3.14 Comparison of scattered near-field. . . . . . . . . . . . . . . . . . . . . .
45
x
3.15 The FWHM for the target Gaussian beam is 0.12λ0 at z = 0.25λ0 . . . . .
46
3.16 Comparison of a single slot antenna and a 9-slot antenna array. . . . . .
48
3.17 Graphical illustration of the shifted-beam theory. . . . . . . . . . . . . .
48
3.18 Plasmonic metascreen operating at λ0 = 940 nm. . . . . . . . . . . . . . .
49
3.19 Tunneling of incident fields through the plasmonic screen. . . . . . . . . .
51
3.20 Slot antenna resonance with and without glass substrate . . . . . . . . .
52
3.21 Parametrization of the antenna lengths. . . . . . . . . . . . . . . . . . . .
54
3.22 Comparison of the signal strength at the nominal FWHM. . . . . . . . .
55
3.23 Electric field at the image plane (theory vs. simulation). . . . . . . . . .
56
3.24 FWHM of the beam along z (theoretical). . . . . . . . . . . . . . . . . .
57
3.25 FWHM of the beam along z (simulated). . . . . . . . . . . . . . . . . . .
58
3.26 The weights of the antenna-array (simulated). . . . . . . . . . . . . . . .
58
3.27 |Ex | and |Ey | components at the image plane . . . . . . . . . . . . . . . .
59
3.28 Sensitivity of FWHM and SLL. . . . . . . . . . . . . . . . . . . . . . . .
60
3.29 Wavelength dispersion of the FWHM beamwidth. . . . . . . . . . . . . .
61
3.30 Metascreens as optical probes . . . . . . . . . . . . . . . . . . . . . . . .
62
3.31 SEM images bowtie nano-antenna probes. . . . . . . . . . . . . . . . . .
63
3.32 Metascreens vs. traditional circular apertures. . . . . . . . . . . . . . . .
64
4.1
The microwave monopole-array probe. . . . . . . . . . . . . . . . . . . .
67
4.2
Aperture-probe-based monopole antenna. . . . . . . . . . . . . . . . . . .
68
4.3
Monopole antennas of various lengths. . . . . . . . . . . . . . . . . . . .
69
4.4
Monopole antenna schematic and model . . . . . . . . . . . . . . . . . .
69
4.5
Monopole excitation schemes. . . . . . . . . . . . . . . . . . . . . . . . .
70
4.6
TM surface mode field distribution. . . . . . . . . . . . . . . . . . . . . .
71
4.7
Aperture excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.8
A plasmonic monopole antenna excited by an aperture. . . . . . . . . . .
73
4.9
Monopole end-fire radiation vs. aperture size . . . . . . . . . . . . . . . .
74
xi
4.10 The end-fire radiation |Ez | of a monopole antenna. . . . . . . . . . . . .
76
4.11 Effect of increasing the number of satellite antennas. . . . . . . . . . . .
77
4.12 The diffracted beam (|Ez |) along z. . . . . . . . . . . . . . . . . . . . . .
78
4.13 |Ez | at the image plane of the single monopole configuration. . . . . . . .
81
4.14 2D near-field plasmonic monopole array as the NSOM probe. . . . . . . .
83
4.15 Contour diagrams of |Ez | at the image plane (z = 0.25λ). . . . . . . . . .
85
4.16 Radiation intensity at various imaging distances. . . . . . . . . . . . . . .
86
4.17 The field distribution of the monopole-array at the source plane. . . . . .
87
4.18 Comparison of the magnitude of electric field. . . . . . . . . . . . . . . .
89
4.19 E-field intensity at the image plane . . . . . . . . . . . . . . . . . . . . .
90
5.1
Exciting and collecting radiation from single nanoemitters. . . . . . . . .
93
5.2
Reciprocity theorem for an electric dipole on a substrate. . . . . . . . . .
94
5.3
The radiation pattern of an electric dipole on substrate. . . . . . . . . . .
95
5.4
An emitter coupled to a Yagi-Uda antenna placed on dielectric substrate.
97
5.5
The geometry of the leaky slot antenna. . . . . . . . . . . . . . . . . . .
99
5.6
The effective mode index of the infinite PEC leaky-slot. . . . . . . . . . . 102
5.7
The directivity of the PEC slot antenna. . . . . . . . . . . . . . . . . . . 103
5.8
H-plane directivity for an infinite PEC slot. . . . . . . . . . . . . . . . . 104
5.9
Radiation pattern for an infinite PEC slot of various slot width. . . . . . 105
5.10 A potential application of the plasmonic leaky-slot antenna . . . . . . . . 106
5.11 The effective mode index of the infinite plasmonic leaky-slot. . . . . . . . 109
5.12 Radiation pattern for the infinite PEC and plasmonic leaky-slots. . . . . 110
5.13 The leaky slot field distribution . . . . . . . . . . . . . . . . . . . . . . . 112
5.14 Power decay along the PLS antenna . . . . . . . . . . . . . . . . . . . . . 113
5.15 Power ratio: total, radiation and dissipation . . . . . . . . . . . . . . . . 115
5.16 H and E-plane directivity of the PLS antenna. . . . . . . . . . . . . . . . 116
5.17 Fabrication sensitivity analysis of the PLS antenna. . . . . . . . . . . . . 118
xii
A.1 The optical constants n and k of Ag. . . . . . . . . . . . . . . . . . . . . 129
A.2 LD model curving fitting for Ag. . . . . . . . . . . . . . . . . . . . . . . . 130
A.3 LD model curving fitting for Au. . . . . . . . . . . . . . . . . . . . . . . 130
A.4 LD model curving fitting for Al. . . . . . . . . . . . . . . . . . . . . . . . 131
B.1 The cylindrical waveguide configuration. . . . . . . . . . . . . . . . . . . 132
C.1 Incident wave from air to the dielectric. . . . . . . . . . . . . . . . . . . . 139
C.2 A parallel electric dipole in the plane of the air-dielectric interface. . . . . 140
C.3 The H-plane and E-plane of a parallel electirc dipole. . . . . . . . . . . . 140
C.4 The H and E-plane radiation of an electric dipole at the interface. . . . . 142
C.5 The E-plane and H-plane of a parallel magnetic dipole. . . . . . . . . . . 143
C.6 The E and H-plane radiation of a magnetic dipole at the interface. . . . . 145
xiii
List of Tables
3.1
Optimal weights for the 9-slot metascreen (theoretical predictions) . . . .
46
3.2
Achievable FWHMs for one and three slot antennas . . . . . . . . . . . .
57
3.3
Optimal weights for the 3-slot metascreen . . . . . . . . . . . . . . . . .
59
4.1
Optimal weights for the circularly-symmetric monopole antenna array . .
78
4.2
Optimal weights for the monopole array (theory and simulation) . . . . .
88
5.1
Wavelength dispersion of the leaky slot antenna (theory). . . . . . . . . . 105
xiv
List of Acronyms
DM
dichroic mirror
E-beam
electron beam
ENZ
epsilon near zero
FEM
finite element method
FIB
focused ion beam
FWHM
full width at half maximum
LD-model
Lorentz-Drude model
MoM
method of moment
NSOM/SNOM
near-field scanning optical microscopy/microscope
NRI
negative refractive index
PALM
photoactivated localization microscopy
PEC
perfect electric conductor
PLS
plasmonic leaky slot
PMMA
Poly(methyl methacrylate)
PSW
plasmonic slot waveguide
SEM
scanning electron microscopy/microscope
SiC
silicon carbide
SP
surface plasmon
SPP
surface plasmon polariton
xv
STED
stimulated emission depletion
STORM
stochastic optical reconstruction microscopy
TE
transverse electric
TEM
transverse electromagnetic
TM
transverse magnetic
TL
transmission line
xvi
Chapter 1
Introduction
1.1
Motivation
The study of plasmonics investigates the physical phenomena of light-metal interactions,
and is thus invaluable for developing metallic nanostructures for the purpose of manipulating light in the subwavelength regime. Traditionally, metals have not been highly used
for confining and transporting light due to their high reflectivity and absorption in the
visible and infrared spectra. Many dielectric and semiconductor materials, on the other
hand, are more popular choices for photonics applications because they are transparent
and have much lower losses. For example, silica has been the dominant material used to
manufacture optical fibers that transport light over long distances. Silicon, a material
that is prevalent in integrated electronic circuits, has also been the preferable choice for
optical interconnects that transfer data between microchips. However, the diffraction
limit severely constrains the possibility of controlling light on a subwavelength scale in
dielectrics. For instance, light cannot be confined in a spot that has a diameter smaller
than half of a wavelength in the respective media (λ/2n), where n is the refractive index. As a result, dielectric waveguides, switches and modulators cannot have physical
dimensions smaller than this limit. This size restriction has prevented the integration of
1
Chapter 1. Introduction
2
on-chip optical components with existing electronic circuits whose basic building blocks,
known as transistors, are already at the nanometer scale [1]. Another consequence of
the diffraction limit is that it defines the smallest feature size resolvable with conventional optical microscopes. Despite the improved quality of lens making, objects smaller
than 250 nm in the visible spectrum cannot be resolved because conventional optical microscopes rely on dielectric elements, such as lenses and dichroic filters, to redirect the
wavefront of the propagating spectrum of light. These diffractive elements are unable to
focus the evanescent waves with subwavelength resolution.
The study of surface plasmon polaritons (SPPs) presents new opportunities for circumventing the diffraction limit. SPPs, the hybrid collective electron-photon oscillations,
can be induced at the metal-dielectric interface at optical frequencies [2]. They form surface modes that are strongly confined to the interface and have a greater momentum
than the free-space propagating light of the same frequency. This gives rise to their subwavelength modal profile in the transverse plane, which is very attractive for guiding and
confining light below the diffraction limit. Although the fundamental physics of surface
plasmons have been understood for decades [3, 4], the physical implementations of many
plasmonic devices have been out of reach due to their challenging feature sizes. However, with recent breakthroughs in nanofabrication technologies, plasmonics has enjoyed
renewed prosperity in many areas of photonics [5]. Among them, slow-wave structures,
such as SPP waveguides, have proven their potential to solve the integration issue of onchip photonics and electronics [6–9]. The small modal size of SPPs has also been utilized
to achieve high-resolution images [10–12]. Additionally, transmission through subwavelength apertures can be strongly enhanced by coupling light through SPPs [13, 14]. This
enhancement defies the tremendous throughput loss predicted by Bethe’s classical theory [15], hence it is very useful for improving the signal strength in optical microscopy
and spectroscopy applications.
One branch of plasmonic research that has been gaining popularity in recent years is
Chapter 1. Introduction
3
plasmonic optical antennas [16]. At microwave and radiowave frequencies, the popularity
of antennas arises from their ability to efficiently link two very different length scales,
that is, the propagating radiation on the order of wavelength and the localized energy in
subwavelength circuits. This feature has enabled vast applications in telecommunications,
radars and satellite systems, etc. At optical frequencies, plasmonic nanostructures are
natural candidates for circumventing the diffraction limit because they can also effectively
link two different length scales. On one hand, the off-chip lasers and detectors work with
radiation on the micron-scales; on the other hand, the on-chip integrated circuits confine
energy on the order of a few tens of nanometers. Plasmonic antennas have the potential of
becoming the wireless communication front-end components at the nanometer scale [17].
Additionally, plasmonic antennas enable strong near-field enhancement, thus bring new
opportunities for advancing nano-optics beyond the original microwave inspirations. For
example, they can enhance several photophysical processes, which include light-emission
(e.g. light-emitting diodes (LEDs) and lasers [18]), light-collection (e.g. photovoltaics
[19]), as well as several other stimulated or spontaneous emission phenomena [11, 12, 20].
The discussions presented in this thesis focus on the optical microscopy and spectroscopy
applications in both near-field and far-field aspects, as such developments have become
the primary inspiration for this work.
1.2
Overview of plasmonic antennas
1.2.1
Early developments and obstacles
Unlike microwave antennas inspired by telecommunication, optical antennas stem from
near-field optical microscopy. In 1928, Synge was the first person who suggested the use
of nanoparticles to convert free-propagating optical radiation to localized fields, which
can be used for probing sample surfaces in the near-field [21]. The first experimental
evidence was demonstrated by Fischer and Pohl in 1988 [10], who imaged a metal film
Chapter 1. Introduction
4
with 320nm-diameter holes using a gold-coated polystyrene particle (which is now known
as the nanoshell [22]) and achieved roughly 50nm resolution. These early works illustrate
that although optical antennas exhibit substantial differences in terms of geometries compared to conventional microwave antennas, the ideas of utilizing the transducer nature
of antennas to convert free-propagating radiation to highly-confined localized energy are
sound.
Due to the frequency-invariant nature of fields and waves, the concept of adapting
existing microwave antenna designs for the purpose of nano-optics appeared very attractive. However, optical antennas have had a stunted development when compared to their
microwave counterparts, primarily due to feature size. More specifically, conventional
antennas have characteristic dimensions on the order of the wavelength. This leads to
their feature sizes in the centimeter range at radiowave and microwave frequencies, but
in the hundreds of nanometers in the visible spectrum. As a result, fabricating metallic
nanoantennas has not been possible until recent developments in high precision printing
techniques, such as focused-ion-beam (FIB) milling and electron-beam (E-beam) lithography. However, obtaining high consistency with low sensitivity is still challenging, even
at the current stage.
Besides nanofabrication challenges, there is also a substantial difference in terms of
the material properties of metals at the microwave and optical domains. Therefore, a
direct down-scaling of the antenna feature size is not feasible. Instead, careful studies of
the plasmon modes have to be carried out in order to determine the modal characteristics
of metallic antennas. Unfortunately, there have not been many investigations regarding
this aspect due to the complexity of most plasmonic nanostructures. Existing literature
covers only simple geometries such as spheres and cylindrical rods [11, 23].
Chapter 1. Introduction
1.2.2
5
Current developments
Current plasmonic antenna research is largely driven by the strong field enhancement
and localization of SPPs that enable accessing the important near-field information [24–
27]. For example, in near-field scanning optical microscopy (NSOM/SNOM), plasmonic
antennas are used as optical probes, which either illuminate in the near-field of the
samples or collect information from them. The probe is raster-scanned over the sample,
and the image is constructed by collecting optical information at every pixel. It has been
shown that these probes can achieve much higher resolution compared to conventional
NSOM aperture probes due to the small modal profile of highly localized SPPs [12, 20].
Additionally, the strong near-field intensity helps improve the signal level.
In addition to high-resolution microscopy, spectral analysis also benefits from plasmonic antennas. For example, two commonly collected radiative signals in spectroscopy
are fluorescence and Raman scattering. For fluorescence, a major benefit comes from
the increased excitation rate of the fluorescent dyes because plasmonic antennas enable
a stronger near-field illumination. The improvement in the fluorescence signal intensity
is proportional to the square of the local field enhancement factor (ESPP /Eincident ) [16].
The enhancement factor can be quite substantial close to the antenna surface. Raman
scattering can achieve even more impressive results. Unlike fluorescence, where the radiative energy is strongly red-shifted compared to the illumination, Raman scattering has a
much narrower emission spectrum which lies within the spectral width of the plasmonic
resonance. As a result, both excitation and emission are enhanced, which leads to an
improvement proportional to the fourth power of the local field enhancement factor [28].
More recently, it has been shown that far-field properties, such as the directivity and
polarization of the radiative energy, can be controlled through coupling localized energy
to plasmonic antennas [29–31]. The directivity and the polarization are dictated by the
radiation pattern of the antenna rather than the nanoscale emitter, much like in the case
of microwave antennas. Therefore, it enables small-angle collection of otherwise omni-
Chapter 1. Introduction
6
directionally scattered energy in the far-field and with prior knowledge of the polarization.
These characteristics help to improve the collection efficiency.
Plasmonic antennas have shown tremendous potential for optical microscopy, spectroscopy and other types of optical sensing applications. Continuing developments in this
field call for further analysis and understanding of the radiation and scattering characteristics of metallic nanostructures. Due to advancements in nanofabrication technologies,
new types of plasmonic antennas, and applications thereof, are expected to arise following
these efforts.
1.3
Thesis objective, scope and outline
The objective of this work is to continue current efforts in the development of plasmonic
antennas for optical microscopy and sensing applications. The contents of this thesis focus
primarily on theoretical analysis and simulation results. The designs introduced include
both near-field and far-field topologies. New antennas and antenna-arrays are proposed
for the general purpose of improving the resolution, sensitivity, as well as reducing the
complexity and cost of current technologies.
Chapter 2 provides the background information for this thesis. The first half focuses
on the theoretical analysis of the electric permittivity of metals at optical frequencies.
The theoretical models used for calculating the electric permittivity are derived from
classical kinetics. The Drude and Lorentz models are included to account for the freeand bound-electron contributions, respectively. They predict that the electric permittivity for metals, such as gold, silver and aluminium, is negative at optical frequencies.
Additionally, two sets of experimental data from different research groups are presented
to verify the theoretical models. The second half of the chapter outlines the effect of
negative permittivities, particularly on the surface plasmon modes supported in metallodielectric systems. It appears that the surface plasmons propagate at a much slower
Chapter 1. Introduction
7
speed compared to the free-space radiation. This implies that a direct down-scaling of
the antenna feature size from the microwave to the optical domain is not possible, and
that the antenna resonance can only be accurately predicted through studying the surface plasmon modes that it supports. The analytical derivation for a particular antenna
structure, the half-wavelength dipole antenna, is presented. The results show an almost
linear dispersion of the dipole resonance in the visible and near-infrared regions, with a
much smaller physical length than half of the excitation wavelength. According to the
duality principle, the complementary structure of the dipole, the slot antenna, would
have the same physical size at resonance. These results are validated through numerical
FEM simulations.
Chapter 3 begins with discussing the tradeoffs in the current NSOM technology,
which relies on extremely small apertures to acquire high-resolution. However, the power
transmitted through a small aperture decreases with decreasing aperture radius with a
factor of (r/λ0 )4 , as predicted by Bethe [15]. Therefore, high resolution can only be
attained at the expense of signal intensity. Another issue with the probes being used
is that imaging at longer distances is problematic. Light diffracts much faster for small
apertures, which leads to the tradeoff between the resolution and the imaging distance.
In order to overcome these tradeoffs, an approach based on manipulating the near-field
interference is presented, which in theory can achieve an arbitrary spot size at a desired
near-field distance. Given a specific subwavelength target function, a corresponding
transfer function can be derived with both the back-propagation method and shifted-beam
theory. In this thesis, attention is given to the latter method, since it not only renders
the transfer function, but also delivers a concrete ready-to-implement structure. Finally,
in order to demonstrate the shifted-beam theory, a near-field plate consisting of a linear
array of slot antennas is designed to achieve superfocusing in one dimension. Because
the slot antennas are separated by subwavelength distances, this near-field plate is also
named the metascreen. It is shown that the slots induce out-of-phase radiation between
Chapter 1. Introduction
8
the neighboring elements, which facilitate destructive interference for the evanescent
spectrum. Such interference can be controlled to achieve superfocused subdiffrational
beamwidth. The design presented is intended to operate at 940nm, but can in theory be
scaled for other wavelengths by changing the physical length of the slot antennas.
Chapter 4 extends the concept of shifted-beam theory from one dimension to two
dimensions, which enables practical raster-scans with superresolution, as in the case of
NSOM operations. It demonstrates the versatility of the optimization method employed
by the shifted-beam theory for multi-dimensional array design. It begins with the introduction of a near-field probe design operating at microwave frequencies, where a planar
monopole antenna array excited by a coaxial cable is utilized. This design enables the
near-field antenna radiation to be manipulated in a cylindrically symmetric fashion. The
issue of optical adaptation is then discussed, which states that plasmonic monopole antennas cannot be excited in similar ways as their microwave counterparts. A new excitation
scheme is proposed and used as the platform for implementing the optical probe. This
scheme is based on conventional NSOM aperture probes, and is hence compatible with
the existing near-field scanning setup. Finally, a planar rotationally-symmetric monopole
array is designed to demonstrate the 2D superfocusing, where in addition to the extended
imaging distance, the background signal is suppressed as well.
Chapter 5 presents a design that exploits far-field characteristics of a plasmonic leakywave slot antenna and demonstrates its potential for enhancing radiation from otherwise
inefficient nanoemitters. The chapter starts with the modal analysis of a small-width slot
sandwiched in two half-spaces. The analysis yields a complex propagation constant of
the fundamental mode, even in a lossless system. This indicates that the slot behaves as
a leaky-wave antenna, rather than a waveguiding structure. The directivity of the leakyslot antenna is calculated based on its propagation constant, which demonstrates a much
stronger angular confinement than that seen in the case of a nanoemitter radiating on its
own. Further, the dispersion of the leaky-slot mode is presented over a wide spectrum
Chapter 1. Introduction
9
in order to illustrate the ultrawideband performance of the design compared to existing
work. Finally, the efficiency and the fabrication tolerance of the design are presented.
Chapter 6 presents the summary of the results and findings, as well as several directions for extending to the work presented in this thesis.
Chapter 2
Background
This chapter aims to illustrate some of the most important differences between the optical
and microwave antennas. These include how metals interact with electromagnetic radiation in each respective spectrum, and the subsequent effect on the physical dimensions
of plasmonic antennas. Particular attention is given to the half-wavelength dipole antenna and its complementary structure, the slot antenna, as they will be used in designs
presented later in this thesis.
In order to understand the fundamental physics behind wave-metal interaction, a
theoretical analysis of the electric permittivity of metals is first introduced, since it is a
convenient and useful macroscopic interpretation of metallic properties. As will be shown
in Section 2.1.1, both classical Drude and Lorentz models are employed in order to describe the metallic response over a wide range of frequency. Section 2.1.2 presents several
empirical data sets for the refractive index of metals collected by various research groups.
These data sets illustrate the dispersion of εr (λ) in the visible and near-infrared regions
predicted by the Drude (free-electron) model, with additional interband transitions are
predicted by the Lorentz (bound-electron) model. The most important conclusion from
these results is that metals behave like dielectrics with negative complex electric permittivity at optical frequencies. One direct consequence from this conclusion is that the
10
Chapter 2. Background
11
scaling law of the physical dimension of antennas differs from convention. As discussed in
Section 2.2.1, an effective resonant wavelength is extracted theoretically to show that the
physical length tends to be much shorter for plasmonic antennas. Finally, these results
are compared against the ones obtained from numerical simulations in Section 2.2.2.
2.1
Metals at optical frequencies
2.1.1
Theoretical analysis
At microwave and radiowave frequencies, metals behave as good conductors that support
electric current. Their electric permittivity is largely imaginary, which leads to strong
attenuation of incident electromagnetic waves impinging upon them. However, the electromagnetic response of metals is substantially different at optical frequencies. Instead
of supporting current as in the case of good conductors, metals behave as lossy negative
0
00
dielectrics, whose electric permittivity is a complex quantity, i.e. εr = ε + jε . The
0
predominant negative real component ε gives rise to collective electron oscillations at
metal-dielectric interfaces, known as the surface plasmons. The negative imaginary part
00
ε indicates heat dissipation. At optical frequencies, metals become translucent towards
incident electromagnetic waves.
The frequency dispersion of the electric permittivity has been studied theoretically
from the perspective of classical kinetics, from which analytical models have been obtained [32]. The detailed derivations and properties of these models can be found in
Appendix A.1.1, only the results are presented here. The simplest and most commonly
used classical model, known as the Drude model, is considered adequate for modeling
metals at most optical frequencies [3, 4]. The Drude model describes the free-electron
motion that leads to the intraband absorption in metals, and has the form shown in
εr (ω) = 1 −
ωp2
ω 2 − jγω
(2.1)
Chapter 2. Background
where ωp =
q
ne2
mε0
12
is the plasma frequency, which is related to the electron density n
and the effective electron mass m of the metal. The average collision frequency of the
free-electron motion is represented by γ. The typical values of ωp for metals is in the
ultraviolet range, whereas the γ is in the infrared region. (Note that the time-harmonic
expressions in this thesis adapt the engineering convention of e jωt .)
The accuracy of the Drude model deteriorates at specific frequencies where the boundelectron contributions are no longer negligible. The bound-electron effect, known as the
interband absorption, can significantly alter the electric permittivity from that predicted
by the Drude model [32]. The interband effect is more accurately described with the
Lorentz model shown in
εr (ω) = 1 +
ωp2
ω02 − ω 2 + jγω
(2.2)
where ω0 is the resonance frequency of the Lorentz harmonic oscillator, at which metals
exhibit sudden and substantial increase of losses. Metals usually have several interband
resonance frequencies spanning over a wide spectrum. For gold, the most noticeable
one is at around 410 nm (f = 7.18 × 1014 Hz); and the one for aluminum is at around
804 nm (f = 3.7 × 1014 Hz) [33]. The total plasmonic effect at optical frequencies is
thus most accurately modeled by combining both the free-electron and bound-electron
contributions in the form of the Lorentz-Drude (LD) model shown in
bound
εr (ω) = εfree
(ω)
r (ω) + εr
2
2
X
ωpe
ωpn
=1− 2
+
2
ω − jγe ω
ω0n
− ω 2 + jγn ω
n
(2.3)
where the subscript e indicates the free-electron parameters and n refers to the nth
Lorentz oscillator. Since the interband transitions occur at several resonant frequencies,
the Lorentz multiple-oscillator model is employed in Eqn. 2.3, which consists of the sum
of several harmonic oscillators with different ωp , ω0 and γ.
Chapter 2. Background
2.1.2
13
Empirical data and analysis
The complex refractive index of noble metals collected by Johnson and Christy [34], and
from Palik’s Handbook of Optical Constants [35] are among the most popular data sets
for describing bulk metals at optical frequencies. Johnson and Christy measured the
optical constants based on the transmission and reflection experiments from 200 nm to
1800 nm. Conversely, the data in Palik ( [35]) consists of a collection of measurements at
different spectra by different groups, often with a variety of techniques. The particular
selection is based on experiments that employ high-purity samples, and with results that
show relatively good agreement at the junction wavelengths between different subsets.
From the complex refractive index ncomplex = n − jk presented in [34,35], the real and
imaginary components of the relative electric permittivity can be calculated from
0
= n2 − k 2
(2.4)
00
= −2nk
(2.5)
ε
ε
0
00
according to εr = ε + jε = (n − jk)2 . The results obtained from both [34] and [35] are
plotted in Fig. 2.1. In this case, the εr obtained from [34] consists of linearly interpolated
experimental data that reflects the original collection by a single group. Conversely, the
results obtained from [35] are interpolated with the LD model in order to account for
the disagreements at the junction frequencies between multiple data sets. The specific
parameters of the LD models are adapted from [33,36]. (The agreements between the LD
models and the empirical data can be found in Figs. A.2, A.3 and A.4 in Appendix A.2 for
silver, gold and aluminium, respectively.) Fig. 2.1 demonstrates similar results obtained
from [34] and [35] for gold and silver, whereas the εr for aluminium is only available
from [35]. It is also shown that aluminium has a much larger negative electric permittivity
when compared to gold and silver, which is due to a higher plasma frequency. As a result,
aluminium behaves closer to conductors in the visible and infrared spectra compared to
gold and silver. This property becomes important for the design in Chapter 5.
Chapter 2. Background
14
50
0
0
−10
−50
−20
εr (imaginary)
εr (real)
−100
−150
−200
−250
−300
−350
−400
−30
−40
−50
−60
Ag (JC)
Au (JC)
Ag (Palik)
Au (Palik)
Al (Palik)
500
−70
−80
1000
λ (nm)
0
1500
2000
−90
Ag (JC)
Au (JC)
Ag (Palik)
Au (Palik)
Al (Palik)
500
1000
λ (nm)
1500
2000
0
Figure 2.1: Dispersion diagram of the complex εr for Au, Ag and Al in the visible and
infrared domain.
Chapter 2. Background
2.2
15
Wavelength scaling for plasmonic antennas
The negative permittivity of metals leads to the formation of surface plasmon modes
in metallo-dielectric systems. As a result, the characteristic dimensions of plasmonic
antennas are significantly altered from the theoretical predictions used in classical antenna theory, which assumes perfect electric conductors (PECs). In microwave antenna
theory, antennas have dimensions on the order of the radiation wavelength, which scale
linearly with respect to frequency. For example, the ideal half-wavelength dipole antenna
made of PEC has a physical length of L ≈ 1/2λ when radiating at its resonance. (λ is
the wavelength of the radiation.) Unfortunately, this scaling law is only applicable at
low frequencies (e.g. microwaves), where the thin metal rods used for implementing the
dipole arms can be regarded as PECs and with infinitesimally small radius. At optical
frequencies, the incident waves are no longer perfectly reflected by the antenna arm due
to the complex εr with a predominant real component. In this case, not only is the PEC
assumption invalid, but the thickness of the antenna arm also becomes substantial with
respect to the wavelength. The resulting structure is essentially a plasmonic cylindrical
rod as illustrated in Fig. 2.2.
Figure 2.2: The geometry of a plasmonic nanorod antenna and the supported surface
mode. E represents the polarization of the external excitation, and k is the direction of
the plane wave propagation (adapted from [23] with permission. The original copyrighted
material can be found at http://prl.aps.org/abstract/PRL/v98/i26/e266802).
Chapter 2. Background
16
Another major difference between microwave and optical antennas is that microwave
antennas are usually connected to either the transmitter or the receiver circuit through
galvanic transmission lines because they support conduction currents. Plasmonic antennas, on the other hand, transfer energy through near-field coupling, that is, from/to a
local source that is able to excite/receive the energy carried by plasmonic oscillation.
When receiving energy from free-space radiation, as illustrated in Fig. 2.2, the plasmonic
rod antenna can be excited by an external electric field polarized along the longitudinal
direction of the rod. It exhibits a dipolar resonance similar to the conventional halfwavelength dipole antenna. Since the dominant surface plasmon mode has a much larger
k -vector than the free-space propagation, the plasmonic rod antenna can therefore be
characterized with a shorter effective wavelength whose value is a function of both the
permittivity εr and the antenna radius R.
2.2.1
Theoretical analysis for dipole antennas
This section is dedicated for theoretically deriving the effective wavelength (λeff ) scaling
properties for plasmonic dipole antenna with respect to both the wavelength and the
geometric variations of the antenna. These scaling properties are important for characterizing the physical dimensions of plasmonic rod antennas according to L ≈ 1/2λeff . As
demonstrated by Novotny in [23], the effective wavelength can be calculated from
λeff = λ0
k0
βz
− 4R
(2.6)
where λ0 is the free-space wavelength, k0 is the free-space wave number, and βz is the
propagation constant of the surface plasmon mode along the major axis of the rod.
Intuitively, the effective wavelength scales with the velocity factor (k0 /βz ) of the surface
mode, which can be calculated by analyzing the dominant lowest order TM0 surface mode
of a cylindrical negative-dielectric waveguide of infinite length [23]. The subtraction of
4R is an adjustment that accounts for the reactance of the rod ends. Therefore, the
Chapter 2. Background
17
task of finding the antenna characteristic dimension reduces to properly extrapolating
the propagation constant βz of the surface mode. The detailed derivation can be found
in Appendix B. Only the results are presented in this chapter.
From conventional dielectric waveguide theories [37], the cylindrical TM0 mode solution with βz can be found by solving the transcendental equation shown in
(2)
εd H1 (βρd R)
εm (λ) J1 (βρm R)
=
βρm R J0 (βρm R)
βρd R H0(2) (βρd R)
(2.7)
(2)
where Jn are the cylindrical Bessel functions and Hn are the cylindrical Hankel functions
of the second kind of order n, which are the solutions for the standing and traveling
cylindrical waves, respectively [23]. βρm and βρd are the transverse wave numbers in
the ρ-direction in metal and dielectric respectively, which are related to the propagation
p
p
constant βz through βρm = k02 εm − βz2 and βρd = k02 εd − βz2 .
Since the surface plasmon modes are evanescent in the transverse plane, this work
derives the transcendental equation in the form of the modified Bessel functions, which
is shown in
εd K1 (αd R)
εm (λ) I1 (αm R)
=−
(2.8)
αm R I0 (αm R)
αd R K0 (αd R)
p
p
where αm = βz2 − k02 εm and αd = βz2 − k02 εd are the transverse decay constants in
the ρ-direction in the metal and dielectric, respectively. Combining Eqn. 2.8 with the
constitutive relation
2
αm
− αd2 = k02 (εd − εm (λ))
(2.9)
the wavelength dispersion of βz can be solved.
These solutions lead to the effective wavelengths shown in Fig. 2.3, where the plasmonic rod antennas are made of either silver, gold or alumimium. Fig. 2.3 illustrates that
the effective wavelength scales almost linearly with respect to the free-space wavelength
in the visible and near-infrared spectra. The slope represents the velocity factor k0 /βz ,
which is much less than unity. This result is expected, given the surface plasmons are
slow waves. Rod antennas with various radii are analyzed to illustrate the geometrical
Chapter 2. Background
18
dependence of βz . In this analysis, the values of εm (λ) are based on the empirical data
in [34] for gold and silver, and [35] for aluminium, as presented in Fig. 2.1. The reason for
this choice is to allow fair comparison to the results presented in [23], which demonstrates
very similar linear scaling behavior with respect to the free-space wavelength.
2.2.2
Numerical analysis for dipole and slot antennas
In this section, numerical simulations are conducted in order to obtain more accurate
results of the effective wavelength. The previous theoretical analysis assumes infinitely
long cylindrical waveguides, where the edge effect at the antenna ends is taken care with
only a constant adjustment factor. Numerical analysis based on the boundary element
method has been conducted by Bryant et al. to study plasmonic rod antennas at optical
frequencies [38]. This study demonstrates that the dipolar plasmon resonance scales with
the excitation wavelength in a similar linear fashion as predicted by the modal analysis
yielding Eqn. 2.8.
In this work, both plasmonic rod and slot antennas are simulated using the commercial
software package Comsol Multiphysics R based on the finite element method (FEM).
This particular software is chosen because it allows for the flexibility of incorporating
either discrete experimental data or wavelength-dependent permittivity functions in the
plasmonic material parameters. The resonance is extracted by varying the antenna length
at a particular excitation wavelength. Under a plane wave illumination polarized along
the major axis of the dipole antenna (Fig. 2.4(a)), the fundamental resonance occurs
when the induced fields along the antenna arm achieve the strongest intensity. The
maximum E-field occurs at the two ends of the rod, whereas the maximum H-field is in
the middle as shown in Fig. 2.4(b). For the example of a silver rod antenna with radius
R = 20 nm, the physical resonance length is found to be 440 nm at λ0 = 1500 nm as
illustrated in Fig. 2.4(c). The resonance is found by evaluating |H| at 2 nm away from
the rod surface in the middle. On the other hand, for a slot antenna illuminated by a
Chapter 2. Background
19
plane wave polarized transverse to the slot length (Fig. 2.5(a)), similar resonance effect
occurs except the locations of the maximum E-field and H-field are switched as shown in
Fig. 2.5(b). In this case, the silver film is 40 nm thick and the slot is 40 nm wide, which
roughly approximates the complementary structure of the dipole antenna. The physical
resonance length of the slot antenna is found to be 450 nm at λ0 = 1500 nm as illustrated
in Fig. 2.5(c). This resonance is obtained by evaluating |E| in the middle of the slot. For
both rod and slot antennas, the electric permittivity of silver is −120 − j3.6 at 1500 nm,
which is adapted from [34].
Fig. 2.6 compares the simulated effective wavelengths (2Lresonance ) to the ones obtained
from the analytical Eqn. 2.8 when the free-space λ0 ranges from 400 nm to 1800 nm. The
simulations employ electric permittivity values based on linearly interpolated empirical
data in [34]. It can be seen that both methods demonstrate similar scaling behavior.
However, the analytical solution shows a steeper slope. It also illustrates that the effective
resonant wavelength is very similar for the rod and slot structures.
Chapter 2. Background
20
Ag
1600
λeff (nm)
1200
R = 20nm
R = 10nm
R = 5nm
800
400
0
400
600
800
1000
1200
1400
1600
1800
1200
1400
1600
1800
1000 1200
λ0 (nm)
1400
1600
1800
Au
1600
λeff (nm)
1200
R = 20nm
R = 10nm
R = 5nm
800
400
0
400
600
800
1000
Al
1600
λeff (nm)
1200
R = 20nm
R = 10nm
R = 5nm
800
400
0
400
600
800
Figure 2.3: The effective wavelength dispersion for plasmonic rod antennas at optical
frequencies. The radius R of the cylindrical rod ranges from 5 nm to 20 nm.
Chapter 2. Background
21
(a)
(b)
1.1
1
|H| (normalized)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
350
400
450
500
550
dipole antenna length (nm)
600
(c)
Figure 2.4: Resonance of a plasmonic rod antenna. (a) Schematic of a dipole antenna
illuminated by a plane wave (polarized along the antenna arm). (b) The electric and magnetic field intensity at resonance (Ldipole = 440 nm). (c) Extracting the dipole resonance
at λ0 = 1500 nm by varying the dipole length.
Chapter 2. Background
22
(a)
(b)
1.1
1
|E| (normalized)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
350
400
450
500
550
slot antenna length (nm)
600
(c)
Figure 2.5: Resonance of a plasmonic slot antenna. (a) Schematic of a slot antenna
illuminated by a plane wave (polarized transverse to the slot length). (b) The electric and
magnetic field intensity at resonance (Lslot = 450 nm). (c) Extracting the slot resonance
at λ0 = 1500 nm by varying the slot length.
Chapter 2. Background
23
1600
Analytical solution: Ag dipole (R=20nm)
FEM simulation: Ag dipole (R=20nm)
FEM simulation: Ag slot (W=40nm)
800
λ
eff
(nm)
1200
400
0
400
600
800
1000 1200
λ (nm)
1400
1600
1800
0
Figure 2.6: The comparison of the effective wavelength dispersion obtained from analytical solution of Eqn. 2.8 and FEM simulations.
Chapter 3
Plasmonic Antennas for Near-Field
Superfocusing
This chapter presents plasmonic antenna-array designs for the purpose of achieving subdiffraction focusing with applications in near-field scanning optical microscopy (NSOM/
SNOM). Section 3.1 presents with an overview of the fundamental trade-off between the
resolution and the working distance, i.e. the distance between the object and the imaging
apparatus. Both the importance and challenges to extend the working distance are discussed. Another trade-off between the resolution and the signal throughput is presented,
which arises from the current NSOM probe technology where extremely small apertures
are employed to overcome the diffraction limit. In Section 3.2, theories are presented to
illustrate the concept of antenna near-field interference and how it leads to superfocusing. Finally, Section 3.3 presents a specific implementation, which employs a linear slot
antenna-array to achieve one-dimensional superfocusing.
24
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
3.1
Overview
3.1.1
Fundamental trade-offs in NSOM technologies
25
Overcoming the diffraction limit has been one of most sought-after aspects of optical
microscopy. One propitious approach is by maintaining sufficient contribution of the
evanescent spectrum at the image location. These evanescent fields are essential for
achieving high resolution because they consist of large spatial frequency components
describing subwavelength details of the object of interest. Among many research efforts,
a popular and commercially available technique, known as the near-field scanning optical
microscopy, has proven effective for achieving subdiffractional resolution. This technique
is based on the scanning-probe methodology shown in Fig. 3.1, which illuminates/collects
evanescent fields directly through raster-scanning an optical probe over the sample surface
in its extreme near-field [39]. With this scheme, the imaging resolution is determined by
the size of the probe instead of the excitation wavelength of the light source.
Figure 3.1: Scanning probe microscopy. (Figure adapted from NanoScience Instruments
http://www.nanoscience.com/education/tech-overview.html with permission.)
The current NSOM instruments employ either the apertureless or aperture techniques.
In the apertureless setup, a large (diffraction limited) laser spot is focused onto a tiny tip
apex (Figs. 3.2(a)), and the scattered field from the sharp tip are utilized for obtaining
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
26
high resolution images. Although the achievable resolution can be very high (in the range
of 10 to 20 nm [40]), the strong background signals due to the illumination are problematic
for numerous applications. Conversely, aperture NSOM relies on tapered optical fiber
tips coated with a thin layer of metal (usually aluminium). Light is launched into the
fiber and scatters off the subwavelength aperture at the tip end (Fig. 3.2(b)). This
approach allows for achieving resolutions of 50 to 100 nm without any background [41].
Aperture NSOM is presently the most popular and extensively developed for the majority
of commercial instruments.
(a) An apertureless probe
Figure 3.2:
tions.
(b) An aperture probe
The (a) apertureless and (b) aperture probes for NSOM applica-
(The two figures are adapted from http://www.nanonics.co.il/products/
spm-probes-and-nanotools/nsom-probes/apertureless-nsom-probes.html
and [42], respectively, with permission.)
Although routinely achieving sub -100 nm resolution, NSOM requires the probe and
the object to be maintained within extremely close proximity (just a few nanometers) at
all times. Since large spatial components are generated only within close vicinity of the
tip end, probing farther away would severely diminish the resolution quality due to the
rapid decay of evanescent fields. In fact, the higher resolution generated by employing
smaller tips, the faster the focal spot size diffracts. This fundamental trade-off between
the focal spot size (i.e. the spatial resolution) and the working distance (i.e. the distance
between the probe tip and the sample) can be illustrated with the example of three
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
27
Gaussian beams with different beamwidths. The spatial distribution of a Gaussian beam
located at z = 0 is described by
2
− ρ2
Ex (ρ, z = 0) = E0 e ω0
(3.1)
where ω0 and ρ represent the Gaussian beamwidth and radial distance from the beam
center, respectively [43]. In this case, The diffracted Gaussian beam along z can be
described by
ω2
Ex (ρ, z = L) = E0 0
2
Z ∞
2
2
e−kρ ω0 /4 kρ J0 (kρ ρ)e−jkz z dkρ
(3.2)
0
Both Eqns. 3.1 and 3.2 are plotted in Fig. 3.3, which demonstrates the tradeoff between
the initial spot size and the rate at which the beam diffracts as the imaging distance
increases.
1
4
ω =λ
0
ω = λ/2
0
3
ω = λ/8
0
z/λ
0.6
x
|E |
0.8
2
0.4
1
0.2
0
−2
−1
0
ρ/λ
1
(a) Gaussian beams at z = 0
2
0
−4
−2
0
ρ/λ
2
4
(b) Diffracted beamwidths ω0 along z
Figure 3.3: The diffraction of three Gaussian beams along z with various initial
beamwidths.
Another tradeoff is for the aperture probe technology in particular, that is, high
resolution comes at the expense of high signal throughput loss. (The throughput or
transmission coefficient is defined as the power emitted by the aperture divided by the
power coupled into the taper region of the fiber [41].) Tapering optical fibers leads to
cutoff of the fiber mode(s) one after another until only the HE11 mode is left as shown
in Fig. 3.4. Eventually, the HE11 mode is also cut off and only evanescent tunneling
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
28
is present close to the tip end. Due to the small taper angle produced by the heating
and pulling process, this loss is quite significant. Furthermore, Bethe predicted that the
transmission coefficient for an ideal PEC subwavelength aperture decreases with a factor
of (r/λ0 )4 as the aperture radius r decreases [15], which further reduces the transmission
coefficient. Note that although the coefficients obtained by Bethe are not accurate for
practical aperture probes, it provides a good model for scaling the transmission coefficient
from one aperture size to another. For realistic aperture probes calculated by Novotny
et al. [44], an aperture with 100 nm diameter at λ0 = 488 nm is shown to only have a
transmission between 10−6 to 10−5 for a taper angle of 30◦ [41]. More importantly, this
power loss cannot be overcome by increasing the input power due to the low damage
threshold of the metal coating (≈ 10mW). As a result, small apertures (< 50 nm) is not
realistic for practical NSOM studies in the visible spectrum [41].
Figure 3.4: Cutoff of the fiber mode(s) in a tapered region of the aperture probe. The
cutoff diameters are obtained from [44]. (Figure adapted from [41] with permission.)
In conclusion, although there is no theoretical limit to the resolution that one can
achieve by shrinking the aperture diameter, the power throughput restricts practically
achievable resolution. NSOM aperture probes that can overcome the tradeoff between
the resolution and throughput is highly desirable. A longer working distance is also
advantageous in numerous situations. For example, when imaging sensitive specimens,
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
29
such as biological samples, the near-field information can be observed more accurately
owing to the reduced disturbance from the presence of the imaging probes. Additionally,
objects buried inside dielectrics cannot be imaged using contact-mode operation. Probing
further away without sacrificing the resolution is essential for obtaining those images.
Moreover, from the instrumentation point of view, probing at a longer distance could
reduce the overall complexity of the NSOM system since one of the most complicated
and costly components is the feedback system that prevents the probes from crashing
into the sample surface.
3.1.2
Other superresolution techniques
Since the proposal of the perfect lens by Sir Pendry in 2000 [45], the idea of overcoming
the diffraction limit utilizing negative-refractive-index (NRI) metamaterials has inspired
many research projects in both the microwave and optical regimes. Metamaterials are
known for their ability to support the growth of the evanescent waves and reverse the
phase for propagating waves. Therefore, a lens made of metamaterials can theoretically
ensure sufficient contribution of evanescent fields at the image location, which would
otherwise decay quickly in a conventional lens system. Ideally, a perfect lens that is able
to capture the entire evanescent spectrum would produce an exact image of the original.
However, practical issues regarding current physical implementations, most notably material losses, often prevent the entire spectrum at the image location to be reproduced
and leads to a broadened resolution. Although far from being “perfect”, these implementations demonstrate the ability of achieving subdiffractional resolution and are referred
to as the superlenses. The first superlens was successfully demonstrated experimentally
at microwave frequencies by Grbic and Eleftheriades in 2004 [46]. In this implementation, a planar negative-refractive-index transmission-line (NRI-TL) grid was utilized
to construct the metamaterial medium. A subsequent experiment at optical frequencies
was demonstrated by Fang et al. in 2005 using a thin slab of silver film with negative
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
30
permittivity [47]. Furthermore, subdiffraction imaging in free-space was demonstrated
utilizing a volumetric metamaterial at microwave frequencies in 2008 [48]. Although subdiffraction imaging has been achieved at various spectra, practical constraints strongly
hinder the performance of metamaterial-based superlenses. The significant attenuation
of the evanescent waves limits the imaging distance at which subdiffraction details can be
resolved. As a result, most of the experiments to date have only demonstrated a working
distance on the order of λ/10.
Some popular bio-imaging techniques, such as photoactivated localization microscopy
(PALM) [49], stochastic optical reconstruction microscopy (STORM) [50], and stimulated
emission depletion (STED) microscopy [51], have achieved superresolution utilizing farfield illumination and collection schemes based on fluorescent molecules. The first two
methods rely on stochastically exciting a subgroup of fluorophores at a time, and then
superimposing multiple images together in order to render the entire object. This means
that although subdiffractional objects can be resolved, the process is not simultaneous
and requires significant amount of post-processing. STED imaging creates a superfocused
spot through deactivating fluorophores in the surrounding region. All of these fluorescent
microscopy techniques require attaching fluorescent dyes to the samples, and therefore
cannot be generalized for most optical microscopy applications.
3.1.3
Engineering optical near-field with plasmonic antennas
NSOM still remains as one of the most practical superresolution techniques which can
be widely applied for various types of optical microscopy applications. Since diffraction
broadens the light scattered off the optical probe, it is intuitive to improve the probe
resolution through near-field diffraction engineering. A theoretical analysis regarding
manipulating the near-field diffraction has been proposed by Merlin in 2007 [52], and
subsequently demonstrated by Grbic and Merlin in 2008 [53]. This technique involves
constructing a patterned screen that interferes with incident electromagnetic waves, which
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
31
Figure 3.5: A schematic showing the near-field plate (z = 0) and focal plane (z = L).
The transmission and the target functions are illustrated for the near-field plate and the
focal plane, respectively. (Adapted from [54] with permission).
in turn induces convergence on the subwavelength scale in the near-field, such as the one
shown in Fig. 3.5. This screen is defined by Merlin and Grbic as a near-field plate,
which is a non-periodically patterned, planar structure that can focus electromagnetic
radiation to lines or spots of subwavelength dimension. This convergence phenomenon
is due to the near-field interference, also known as the radiationless interference. Hence
it is electrostatic or magnetostatic in nature. As a result, the spot size is not limited by
diffraction and, theoretically, can be made arbitrarily small. The convergence ability of
near-field plates is limited by the presence of the highest spatial evanescent component
at the focus. Therefore, the practical focal length L is usually constrained to be much
smaller than the wavelength, but could be much longer than the working distance of
conventional NSOM. These patterned screens can be seen as the near-field counterparts
of Fresnel zone plates, although the physics that governs their respective interference
mechanism is different. (Recall that traditional Fresnel zone plates focus the propagating
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
32
components, which leads to focal spot sizes limited by diffraction.)
Designing appropriate near-field plates that lead to superfocusing relies on backpropagating the desired imaging function from the focal plane to the transmission plane,
such that the field/current distribution of the interference pattern, i.e. the transmission
function, can be obtained. As discussed in [52], not all target functions (e.g. the ones
with singularities at the image plane) can be constructed using this approach. Some
examples of suitable image patterns, such as lines and Bessel functions with cylindrical
and azimuthal symmetries respectively, are proposed. In the initial experiment of Grbic
and Merlin at microwave frequencies [53], the radiationless interference effect was demonstrated using capacitive near-field plates that support appropriate current densities. The
superfocusing effect was achieved for working distances on the order of λ/10.
A related but alternative implementation has been proposed based on shifted-beam
theory [55]. The concept employs radiation emanating from spatially displaced antennas
(with subwavelength separation distance) to establish superfocused interference patterns.
The radiation intensity of the antenna array elements can be adjusted by varying their
lengths and widths, which leads to a discretized and weighted field distribution as the
transmission function. With properly designed weights based on the least-squares optimization, arbitrary waveforms can theoretically be synthesized in the near-field. Shiftedbeam theory establishes similar near-field interference phenomenon with an alternative,
simplified approach compared to the back-propagation method. It provides concrete antenna array structures for implementation, whereas the back-propagation method only
leads to transmission functions without suggesting specific implementations. Experimentally, one-dimensional superfocusing based on shifted-beam theory has been demonstrated at microwave frequencies for achieving a working distance as large as a quarter
wavelength in the broadside [55]. In this design, a linear antenna array consisting of three
slot antennas is designed to operate at 10 GHz (λ0 = 30 mm). The measured beamwidth
of the slot antenna array was reduced by a factor of 1.87 from that of a single slot antenna.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
33
With recent developments of plasmonic antennas, adapting antenna-array designs
from the microwave frequencies to the optical domain has become a viable approach for
achieving subdiffractional resolution for optical imaging. This is in contrast with the nearfield plate design in [53], which employs inter-digited capacitors that cannot be directly
scaled to the optical domain due to the lack of their optical-equivalents. Other designs
based on the near-field interference topology, such as the one utilizing high-order modes
of plasmonic waveguides [56], have also shown potentials for achieving one-dimensional
optical superfocusing. However, the configuration in [56] presents great implementation
challenges due to the requirement of constructing extremely narrow waveguides. In addition to manipulating the diffraction pattern, plasmonic antennas have also been shown to
enhance near-field signal strength. Therefore, incorporating resonant antenna structures
in the diffraction engineering designs further helps alleviate signal throughput issues. In
conclusion, the purpose of this work is to combine techniques used in microwave antennaarrays with those available from the domain of optical instrumentation and fabrication
in order to achieve superfocusing for NSOM applications. In the following section, the
theory of near-field interference is elucidated, followed by designs based on shifted-beam
theory that incorporate plasmonic slot antenna arrays operating at optical frequencies.
3.2
Theory
3.2.1
Back-propagation method
Obtaining field/current distributions at the transmission screen through back-propagating
the image spectrum has been proposed by both Merlin [52] and Eleftheriades [57]. Although with different inspirations, both present similar principles for constru cting desired
target waveforms from an incident wave using the interference pattern facilitated with
a near-field screen. The analysis begins with a desired spatial function at the image
plane, which contains ample evanescent fields and is defined everywhere. Its spectral
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
34
distribution is then obtained through the Fourier transform. The corresponding spectral
function at the transmission screen can thus be calculated by back-propagating the image
spectrum over the focal length. Note that as the image spectrum propagates backwards,
the propagating and the evanescent spectra experience phase reversal and exponential
amplitude growth, respectively. This is different from the optical phase conjugation effect
because in addition to the phase reversal of propagating waves, the amplitude growth
is necessary for the prevalent evanescent waves. Finally, the spatial distribution at the
transmission screen is obtained through the inverse Fourier transform.
Figure 3.6: The schematic of the back-propagation system.
The back-propagation method is best illustrated through an example, with the coordinate system shown in Fig. 3.6, where the target function of interest at the image
plane is a one-dimensional Gaussian beam with a subwavelength FWHM beamwidth
of 0.12λ0 . The analytical expressions of the spatial and spectral representations of the
target function are shown in
2
Efocal (x, z = L) = E0 e
− x2
ω0
x̂
√
e focal (kx , z = L) = E0 ω0 πe
E
(3.3)
ω2
− 40 kx2
x̂
(3.4)
respectively, where ω0 characterizes the Gaussian beamwidth and is related to its FWHM
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
35
through
ω0 =
FWHM
√
2 ln 2
(3.5)
Note that the Fourier and inverse Fourier transformations in this analysis assume the
engineering conventions shown in
Pe(kx ) =
Z ∞
p(x)ejkx x dx
(3.6)
Z ∞
(3.7)
−∞
1
p(x) =
2π
−∞
respectively, and the integral identity
Z ∞
e
−p2 u2 ±jqu
e
Pe(kx )e−jkx x dkx
du =
√
−∞
π − 4pq22
e
p
(3.8)
efocal (kx , z = L) (Eqn. 3.4) from Efocal (x, z = L) (Eqn. 3.3). For the
is used to derived E
example of a Gaussian beam with a FWHM beamwidth of 0.12λ0 , its spatial and spectral
distributions are illustrated in Figs. 3.7(a) and 3.7(b), respectively.
1
0.12
0.1
|Ex (kx, z=L)|
x
E (x, z=L)
0.8
0.6
0.12 λ0
0.4
0.08
0.06
0.04
0.2
0
−1
0.02
−0.5
0
x/λ
0.5
0
(a) Spatial distribution
1
0
−20
−10
0
kx/k0
10
20
(b) Spectral distribution
Figure 3.7: An example of the target function with subdiffractional spot size. (a) the
Gaussian spatial distribution with the FWHM of 0.12λ0 ; (b) the corresponding Gaussian
spectral distribution.
efocal (ky , z = L) is achieved by multiThe back-propagation of the image spectrum E
plying each plane-wave component with its corresponding back-propagation factor e jkz L ,
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
36
where L is the focal distance. The wave number kz along the direction of propagation
takes on different forms for the propagating and evanescent components, which are shown
in
kzprop =
q
k02 − kx2
q
evan
kz = −j kx2 − k02
(3.9)
(3.10)
respectively. Finally, the inverse Fourier transform is applied to the spectrum at the
transmission screen, and the spatial distribution of the transmission function Eaperture
can be related to the image spectrum as illustrated in
Z ∞
1
eaperture (kx ; z = 0)e−jkx x dkx
Eaperture (x; z = 0) =
E
2π −∞
Z ∞
1
efocal (kx ; z = L)ejkz L e−jkx x dkx
E
=
2π −∞
(3.11)
(3.12)
Given the previous Gaussian target function and a focal length L = 0.1λ0 , the spatial and
spectral distributions of the transmission function are illustrated in Fig. 3.8. Fig. 3.8(b)
shows that the back-propagated function is a cosine-modulated Gaussian, whose peak is
shifted from the zero spectral frequency towards both the positive and negative components of the modulation frequency. The spatial distribution in Fig. 3.8(a) thus shows
amplitude oscillation from positive to negative values with corresponding modulation
frequency. The oscillatory spatial distribution has also been demonstrated for the case
of a superfocused sinc target function by Merlin [52].
It is shown next that a closed-form expression can be formulated for the transmission
function in Eqn. 3.12 with reasonably good accuracy. In this case, it is assumed that
most of the energy is concentrated in the spectral regions where kz > |q| k0 , such that
the paraxial approximation kz ≈ −j|kx | can be applied. This is a reasonable assumption
when the evanescent spectrum dominates the target function, thus the contribution from
the propagating components can be neglected. With this assumption, the integral in
12
0.8
10
0.7
8
0.6
|Ex (kx, z=0)|
Ex (x, z=0)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
6
4
2
37
0.5
0.4
0.3
0
0.2
−2
0.1
−4
−1
−0.5
0
x/λ
0.5
1
−20
−10
0
k /k
0
10
20
x 0
(a) Spatial distribution
(b) Spectral distribution
Figure 3.8: The spatial and spectral distribution of the transmission function that facilitates the Gaussian target beam of 0.12λ0 FWHM at L = 0.1λ0 .
Eqn. 3.12 can thus be approximated as
E0 ω0
Eaperture (x; z = 0) ≈ √
2 π
Z −q
E0 ω0
√
2 π
2
−∞
+
=
2
ω0
e− 4 kx ej(jkx )L e−jkx x dkx
Z ∞
q
Z −q
e
ω2
− 40 kx2
2
ω0
e
j(−jkx )L −jkx x
e
2
e− 4 kx e−j(x−jL)kx dkx
−∞
+
Z ∞
q
dkx
e
ω2
− 40 kx2
e
−j(x+jL)kx
dkx
(3.13)
Since the modulation frequency is far away from the zero spectral frequency, the small
tails of the shifted Gaussian functions do not affect the overall integral significantly.
Hence the identity in Eqn. 3.8 can be applied to both integrals in Eqn. 3.13 as shown in
Z ∞
Z ∞
ω2
ω2
E0 ω0
− 40 kx2 −j(x+jL)kx
− 40 kx2 −j(x−jL)kx
e
dkx +
dkx
e
e
Eaperture (x; z = 0) ≈ √
e
2 π
−∞
−∞
!
√
√
2
(x+jL)2
−
E0 ω0 2 π − (x−jL)
π
2
2
2
e ω0 +
e ω0
= √
ω0
ω0
2 π
!
2
2
= E0
e
= 2E0 e
−
(x−jL)
2
ω0
L2 −x2
2
ω0
cos
+e
−
(x+jL)
2
ω0
2xL
ω02
(3.14)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
38
This result illustrates that the transmission pattern is a real and even function, consisting
of a Gaussian distribution modulated with a cosine function. The Gaussian distribution
2
2
shows an amplitude growth of 2eL /ω0 from the target function, while the beamwidth
remains as ω0 . The cosine function reveals that the spatial modulation frequency is
2L/ω02 . Therefore, it is concluded that applying back-propagation to the target Gaussian
function is to shift the Gaussian spectrum to higher spectral frequencies and with an
increase in the amplitude. The accuracy of the analytical formulation in Eqn. 3.14 is
compared with integrating Eqn. 3.12 numerically, and the results in Fig. 3.9 demonstrate
good agreement between the two.
numerical
analytical
12
10
Ex (x, z=0)
8
6
4
2
0
−2
−4
−0.4
−0.2
0
x/λ0
0.2
0.4
Figure 3.9: Comparison of the spatial transmission functions obtained through numerically integrating Eqn. 3.12 and from the analytical expression in Eqn. 3.14.
The effect of varying the focal length L and the FWHM of the target function are
illustrated in Figs. 3.10 and 3.11, respectively. It is shown that either increasing the focal
length or narrowing the target beamwidth shifts the Gaussian peak to a higher spectral
frequency. This is intuitive since larger spectral components are needed to facilitate these
more stringent superfocusing conditions. From the perspective of the spatial distribution,
larger amplitude oscillation with faster oscillating frequency is required to achieve a
narrower beamwidth or to maintain the same focus at a longer distance.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
L=0.08λ
L=0.10λ
L=0.12λ
25
20
1.6
|Ex (kx, z=0)|
x
1.8
1.4
15
E (x, z=0)
39
10
5
0
1.2
1
0.8
0.6
−5
0.4
−10
0.2
−0.3
−0.2
−0.1
0
x/λ
0.1
0.2
0.3
−20
−10
0
0
k /k
10
20
x 0
(a) Spatial distribution
(b) Spectral distribution
Figure 3.10: The transmission function for achieving the same Gaussian target function
(FWHM = 0.12λ) at various focal lengths.
30
20
1.4
|Ex (kx, z=0)|
1.2
10
x
E (x, z=0)
1.6
FWHM=0.10λ
FWHM=0.12λ
FWHM=0.14λ
0
1
0.8
0.6
0.4
0.2
−10
−0.3
−0.2
−0.1
0
x/λ0
0.1
0.2
(a) Spatial distribution
Figure 3.11:
0.3
−20
−10
0
kx/k0
10
20
(b) Spectral distribution
The transmission functions for achieving various Gaussian target
beamwidths at L = 0.1λ.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
3.2.2
40
Shifted-beam theory
The back-propagation method is an effective approach for obtaining transmission functions that facilitate near-field superfocusing. However, the method has several limitations. For instance, there are many restrictions on the target functions since not all
field distributions are suited for back-propagation. More importantly, such analysis only
presents the theoretical transmission functions without suggesting any concrete structures
that support them. This leads to the proposal of shifted-beam theory, where traditional
antenna-array theories describing the propagating wave interference in the far-field are
generalized to analyze the static evanescent wave interference in the near-field [55]. For
shifted-beam analysis, the design is based on concrete antenna elements and array configurations, and hence has the advantage of having readily available structures for physical
implementations. Additionally, this method employs the least-squares approximation,
which does not restrict the shape and form of the target function.
An example of such a design is the metascreen shown in Fig. 3.12, which consists of
a thin metallic sheet with a linear array of slot antennas of various lengths and widths.
A plane wave is incident from one side of the screen, and the scattered fields from the
slot antennas interfere on the other side in the near-field region. The slot patterns are
designed such that a subdiffractional focal spot can be formed at the image plane. Slot
antennas are good candidates for implementing metascreens due to the low background
signal since only the scattered fields from the slots contribute to the image.
In order to understand how shifted-beams emanating from an array of antennas lead
to near-field superfocusing, the difference between the proposed and traditional antenna
array representations are first discussed. In traditional antenna-array theory that analyzes far-field radiation patterns, the magnitude difference due to the spatial shift of an
antenna element is negligible. Hence a displacement by a distance d results only in a
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
41
Figure 3.12: An example of metascreen based on the shifted-beam theory. An array of
slot antennas separated by subwavelength distance are employed to facilitate near-field
interference at the image plane upon a plane wave illumination.
phase shift in the far-field, as described in
Ex,shifted = Ex (x, y, z)e−jkd cos θ
(3.15)
Consequently, the superposition due to an array of antenna elements with different phases
allows the total far-field pattern to be separated into an element factor and an array
factor [58]. In the near-field, however, both the magnitude and phase are affected, and
the radiation pattern is spatially shifted as shown in
Ex,shifted = Ex (x − d, y, z)
(3.16)
As a result, superposition based on those fields does not allow the separation of the
element and array factors; instead, the contributions from each radiator are summed
directly at the image plane in order to compute the total field due to the array. Therefore,
the total field has the form shown in
Etotal (x) =
X
N
wn En (x) ≈ f (x)
(3.17)
where En and wn are the field pattern and the corresponding weight of the nth radiating
element in the array. The separation has to be subwavelength in order to facilitate
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
42
interference among evanescent waves. With properly optimized weights, this sum could
best approximate the target function f (x). Therefore, the goal of finding the appropriate
transmission pattern based on the shifted-beam approach is to determine the optimized
weights.
Shifted-beam theory presented in this thesis employs the least-squares optimization
[59], which is an effective approach to approximate solutions of overdetermined systems,
i.e. sets of equations in which there are more equations than unknowns. For such a
system in
Aw = b
(3.18)
The solution of w minimizes the objective function kb − AwkL2 (i.e. the square of the
errors). In the metascreen case, A = [E1 [x], E2 [x], ..., En [x]] and b = f [x], where En [x]
and f [x] are column vectors representing the discretized fields in the image plane along
x. The column vector w contains the weights of the array elements. This problem has a
unique solution, provided that n columns of the matrix A are linearly independent. This
is the case for the shifted beams when assuming the antennas are linearly-independent
radiators, i.e. the mutual coupling effect between the antenna elements is not included.
A unique solution can be obtained by solving the normal equations in the matrix form
shown in
AH Aw = AH b
(3.19)
where AH is the Hermitian of the matrix A. The solution is shown in
wopt = (AH A)−1 AH b
(3.20)
Note that although shifted-beam theory is presented for analyzing near-field interference in this work, direct summation of individual shifted patterns without separating
into the element and space factors, as illustrated in Eqn. 3.17, is a generalized representation of the total field. In both cases, the individual antenna elements radiate as linearly
independent radiators, and the least-squares optimization can be applied to compute the
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
43
corresponding weights of the array elements. Therefore, shifted-beam theory is not restricted to near-field applications, but can also be applied for far-field radiation pattern
synthesis.
The design advantage of the shifted-beam theory can be illustrated through the example of achieving a superfocused Gaussian beam with a FWHM of 0.12λ at a distance
of λ/10 away from the transmission screen, which is similar to the example solved using
the back-propagation method in Section 3.2.1. The design of the particular metascreen
begins by calculating the radiated field pattern of individual slot antennas, where the
major axis of the slots is pointing in the y-direction as shown in Fig. 3.12. In this case,
the Ex component of the slot radiation at the image plane has the form illustrated in [58]
√2 2 2
Im z e−jk x +z +H /4
Im z e−jkR
=
(3.21)
Ex (x; z = L) =
j2π ρ2
j2π
x2 + z 2
where Im is the fictitious magnetic current in the slot, and whose strength depends on
the magnitude of the incident field. H is the length of the slot antenna, ρ is the distance
between the center of the slot-antenna and the image, and R is the distance between the
tip of the slot-antenna and the image. Note that Eqn. 3.21 is a closed-form solution that
contains both evanescent and propagating fields. By summing all the contributions of
spatially shifted slot antennas, the total field at the image location can be represented as
√
X
X Im z e−jk (x−nd)2 +z2 +H 2 /4
Ex,total (x; z = L) =
wn Ex,n =
wn
(3.22)
2 + z2
j2π
(x
−
nd)
N
N
According to the optimization method in Eqn. 3.20, for an antenna array with only three
elements, the optimized magnitude and phase at the transmission screen are [0.343, 1,
0.343] and [172◦ , 0◦ , 172◦ ], respectively. This result once again shows the (near) 180◦
phase offset between the neighboring elements that are separated with a subwavelength
distance.
The comparison between the target Gaussian beam and the one achieved with these
optimized weights is shown in Fig. 3.13. We can see that three elements are sufficient
to approximate the desired beamwidth when the image plane is distanced 0.1λ0 away
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
44
target function
3 slots
1
0.6
0.4
x
|E | (normalized)
0.8
0.2
0
−1
−0.5
0
x (λ0)
0.5
1
Figure 3.13: The FWHM is 0.12λ0 at z = 0.1λ0 for both the target Gaussian beam and
the radiation from the 3-slot array.
from the metascreen. The scattered field in the plane of propagation (xz -plane) can be
calculated for both the single slot antenna and the antenna array based on Eqns. 3.21 and
3.22, respectively. The results are shown in Figs. 3.14(a) and 3.14(b), where the electric
field magnitude is normalized with respect to the beam center. The FWHM of these
beams are illustrated in Fig. 3.14(c), which shows that the FWHM contour due to the
single-slot radiation spreads at a 45◦ angle with respect to the beam center. Note that
although the radiation from a single slot antenna is subdiffractional at z = 0.1λ0 , the
FWHM is 0.2λ0 , which is 67% larger than the target beamwidth. Conversely, a 3-slot
array with the previously shown optimized weights can achieve a FWHM 0.126λ0 , as
illustrated in Fig. 3.14.
The effect of increasing the number of elements in the array will now be discussed.
Theoretically, an infinite number of slots are necessary to produce the exact desired
target beam at any near-field locations. However, depending on the image plane distance
and the spot size of interest, a small number of slots can provide a remarkably close
approximation. For the previous example shown in Fig. 3.13, a Gaussian beam with a
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
0.2
0.2
1
1
0.15
z (λ0)
z (λ0)
0.15
0.1
0.05
0
−0.2
45
0.1
0.05
−0.1
0
x (λ0)
0.1
0.2
0
−0.2
0.1
−0.1
(a) 1 slot
0
x (λ0)
0.1
0.2
0.1
(b) 3 slots
0.2
z (λ0)
0.15
0.1
0.05
1 slot
3 slots
0
−0.2
−0.1
0
x (λ0)
0.1
0.2
(c) Contours of the FWHM
Figure 3.14: The comparison of scattered near-field |Ex | from a single slot antenna and
a 3-slot antenna array.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
n0
n±1
n±2
n±3
n±4
|wn |
1
0.764
0.373
0.136
0.031
wn
0◦
179◦
−1◦
176◦
−4◦
6
46
Table 3.1: Optimal weights for the 9-slot metascreen (theoretical predictions)
FWHM of 0.12λ0 can be produced using three elements when the image plane is 0.1λ0
away. However, if the same beamwidth is required at z = 0.25λ0 , the 3-slot array
is inadequate because it does not generate sufficiently large transverse spectral wave
numbers (kx ) from its transmission pattern. More specifically, it is only able to achieve
a FWHW of 0.22λ0 as illustrated in Fig. 3.15. A way to overcome this limitation is to
add more slots in the array, as shown in Fig. 3.15, with an example of a 9-slot array that
can achieve a FWHM of 0.14λ0 . For this case, the optimal weights for the slot array are
listed in Table 3.1. The option of maintaining the subwavelength spot size at increased
image distances by using more array elements provides a powerful design parameter for
achieving near-field superfocusing at longer imaging distances.
target function
3 slots
9 slots
1
0.6
0.4
x
|E | (normalized)
0.8
0.2
0
−1
−0.5
0
x (λ0)
0.5
1
Figure 3.15: The FWHM for the target Gaussian beam is 0.12λ0 at z = 0.25λ0 . The
FWHM is 0.14λ0 and 0.22λ0 for the 9-slot array and the 3-slot array, respectively.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
47
The scattered near-field Ex in the plane of propagation is shown in Figs. 3.16(a) and
3.16(b) for the single slot and the 9-slot array, respectively. It is illustrated that strong
radiation from multiple satellite elements leads to an ultra-narrow beamwidth at the
image plane of z = 0.25λ0 . However, this also implies that imaging at a distance much
shorter than the nominal image plane location is not feasible due to the high sidelobes
from the satellite elements. In this particular example, the sidelobes exceed half of the
main beam when the imaging distance is shorter than 0.15λ0 as shown in Fig. 3.16(c).
The high sidelobe level is undesirable because it triggers false detection signal in the
image plane.
As previously seen in the back-propagation analysis in Section 3.2.1, the fast oscillatory transmission function with 180o phase shift is the key feature that enables superfocusing in the near-field. This characteristic is also verified by the shifted-beam analysis
for both the 3-slot and the 9-slot antenna-arrays. It can be understood intuitively from
the illustration in Fig. 3.17. If the fields from all slot antennas are in phase as shown
in Fig. 3.17(a), the resulting radiation would produce a wider beamwidth compared to
just a single slot antenna due to static superposition. However, if the two satellite antennas radiate out of phase compared to the central element as shown in Fig. 3.17(b),
the resulting beam in Fig. 3.17(c) becomes much sharper.
In summary, shifted-beam theory offers a much easier alternative for constructing the
transmission screen for superfocusing applications when compared to the back-propagation
method. The spatially shifted antennas are treated as linearly independent radiators,
whose radiation pattern can be used as the basis functions. By properly weighting and
summing these basis functions, desired near-field distribution can be obtained. The discretized weights, i.e. the transmission pattern, can be calculated easily from Eqn. 3.20,
which is based on the least-squares optimization. Finally, the structure can be readily
implemented using linearly independent antenna elements. Shifted-beam theory is used
as the basis for the superfocusing design in this thesis.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
1
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
−0.5
0
x (λ )
0.5
1
0.3
z (λ0)
z (λ0)
0.3
48
0
−0.5
0.1
0
x (λ )
0
0.5
0.1
0
(a) 1 slot
(b) 9 slots
0.3
0.25
z (λ0)
0.2
0.15
0.1
0.05
0
−0.5
1 slot
9 slots
0
x (λ0)
0.5
(c) Contours of FWHM
Figure 3.16: The comparison of scattered near-field |Ex | from a single slot antenna and
a 9-slot antenna array.
Figure 3.17: Graphical illustration of the shifted-beam theory (adapted from [55] with
permission. http://prl.aps.org/abstract/PRL/v101/i11/e113901).
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
3.3
49
Results and discussion
In Section 3.2.2, shifted-beam theory predicts that a 3-slot antenna array can produce a superfocused beam that approximates a Gaussian function with subwavelength
beamwidth in the near-field. According to the effective wavelength analysis in Section 2.2,
plasmonic slot antennas can be designed to operate at optical frequencies to emulate the
radiation characteristics of their microwave counterparts. It will be shown that a plasmonic metascreen consisting of a linear array of slot antennas, such as the one shown in
Fig. 3.18, can be designed to achieve near-field superfocusing, where the target Gaussian
function has a FWHM beamwidth of 0.12λ at a focal length of 0.1λ away from the screen.
Figure 3.18: Schematic of the plasmonic metascreen operating at λ0 = 940 nm. The
gold film is 50 nm thick with εr = −36 − j2.5. The glass substrate has a refractive
index n = 1.513. The slot width w = 40 nm. The center-to-center separation distance d
between the slot antennas is 94 nm.
In this case, the plasmonic metascreen is designed using three slot antennas printed
on a gold film backed with glass substrate. It is illuminated by a plane wave from the
bottom at λ0 = 940 nm. The choice of the operating wavelength is to accommodate
the fabrication allowance of the FIB technology available, which can achieve a minimum
feature size of 40 nm. Technically, thicker films are favorable due to better shielding
capabilities. However, a large disparity between the slot width and thickness presents
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
50
fabrication challenges for obtaining uniformly shaped feature sizes for the entire thickness
of the film. The choice of 50 nm is to compromise fabrication accuracy and acceptable
screen opaqueness, which blocks more than 90% incident fields as shown in Section 3.3.1.
Glass substrate is used since it is the material employed in commercial optical fibers.
Finally, the separation distance of 0.1λ0 (d = 94 nm) is used for achieving the beamwidth
on the same scale. The desired weights are achieved by altering the length of the central
and satellite antennas, L1 and L2 , respectively.
3.3.1
Background signal
Although similar to microwave metascreens in principle, some characteristics are unique
for the optical implementation. A notable difference is that the plasmonic screen is not
completely opaque at optical frequencies. Depending on the thickness of the screen,
a certain percentage of the wave can leak through upon incidence, which leads to a
background signal. This background signal could limit the working distance since its
intensity becomes more comparable to the ones scattered from the slot antennas as the
working distance increases.
The fraction of leakage with respect to the incident field (|Ethrough |/|Ein |) is illustrated
in Fig. 3.19(a) for plasmonic screens of various thicknesses at λ = 940 nm. It is shown
that the leakage drops exponentially with the film thickness. At 940 nm, less than 10% of
the incident wave penetrates through a gold film that is 50 nm thick, but only 0.4% gets
through if thickness is more than 150 nm. The choice of the film thickness is important for
fabrication purposes, particularly when the slot feature size is relatively small. Therefore,
the results shown in Fig. 3.19(a) are helpful for choosing appropriate film thicknesses to
satisfy both fabrication and background signal intensity requirements. Fig. 3.19(b), on
the other hand, illustrates the wavelength dependence of the transmission when the film
thickness is fixed at 50 nm. It can be seen that the transmission is primarily determined
by the electric permittivity of the metal; therefore, although the film is optically-thicker
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
51
at shorter wavelengths, the smaller negative permittivity allows a much larger amount of
incident field to penetrate through. The peaks in gold and aluminium are related to the
interband absorption of the respective metals as discussed in Section 2.1.1. At 940 nm,
the transmitted field is 6.3%, 8.5% and 2.3% for silver, gold and aluminium, respectively.
The effect of the background signal on the proposed metascreen will be discussed in
Sections 3.3.3 and 3.3.4.
0.25
in
/E |
0.3
through
in
Ag
Au
Al
0.2
0.15
|E
/E |
0.3
|E
through
0.5
0.4
0.35
Ag
Au
Al
0.6
0.2
0.1
0.1
0.05
0
10
50
100
150
Film thickness (nm)
200
0
400
(a)
800
1200
λ (nm)
1600
(b)
Figure 3.19: (a) The fraction of the incident wave penetrating through the metallic screen
vs. the screen thickness at λ =940 nm. (b) The fraction of the incident wave penetrating
through a 50 nm-thick metallic screen vs. the wavelength.
3.3.2
Design
The physical length for a half-wavelength antenna has been obtained previously in Section 2.2. In the particular case of a gold slot antenna with a width of 40 nm and a
thickness of 50 nm, the resonance is found to be 243 nm when the metal screen is suspended in air. However, due to the presence of the glass substrate, this resonance length
shrinks from the previous result. This effect has been observed for dipole antennas on a
thick substrate at microwave frequencies, where the reduction factor is determined by the
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
propagation constant β, such that β ≈ k0
52
p
(1 + εr )/2 is averaged between the air and
the substrate electric permittivity [60]. Using this approximation, the reduction factor
β/k0 ≈ 1.282 for glass substrate, which implies that the expected resonance occurs at
190 nm for the plasmonic slot antenna. The results from numerical simulations using
Comsol Multiphysics are shown in Fig. 3.20, where the slot antenna exhibits resonance
at 200 nm. The central and satellite antenna lengths will be parameterized at around
this resonance.
22
20
with substrate
without substrate
|E
scattered
|
18
16
14
12
10
8
6
150
200
250
Slot antenna length (nm)
300
Figure 3.20: Comparison of the slot antenna resonance with and without the glass substrate.
It should be noted that shifted-beam theory assumes that each antenna in the array
radiates as a linearly independent element. However, mutual coupling can influence the
magnitude and phase of the radiation from the array [58], particularly when the antenna
elements are separated by subwavelength distances such as in the case of metascreens.
Therefore, the length of each slot antenna cannot be determined separately with the
goal of obtaining the expected optimal weight for individual elements. Instead, the array
of antennas are designed collectively such that the mutually coupled radiation leads to
the desired image function at the focal plane. This design principle holds for both the
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
53
microwave and optical implementations. The difficulty of predicting mutual coupling has
been a major obstacle which prevents designing metascreens with a large number of array
elements.
The design of plasmonic metascreens for a specific wavelength is obtained through
parametrization of the slot antenna lengths, such that particular combinations of L1 and
L2 lead to field distributions at the image plane that approximate the target function.
As shown in Fig. 3.21, for a set of L1 ’s around λeff /2 of 200 nm, L2 is varied in order to
alter the relative beam weights at the transmission screen. The resulting image functions
are analyzed from both perspectives of the FWHM beamwidth and the sidelobe level
(SLL). The combinations of L1 and L2 that lead to a FWHM smaller than the one of a
single slot antenna (i.e. below the black dashed line in Fig. 3.21(a)), as well as a SLL
lower than half of the main beam (i.e. below the black dashed line in Fig. 3.21(b)) are
considered as superfocused arrays.
For achieving the particular target Gaussian beamwidth of 0.12λ0 , a combination of
L1 = 240 nm and L2 = 125 nm is chosen in order to achieve the highest signal intensity
while satisfying the fabrication tolerance. The sensitivity analysis in Fig. 3.21(c) demonstrates the fluctuation of FWHM (the error bars) assuming ±10 nm fabrication tolerance
for the slot length. In this case, the nominal L2 (at the star of the error bar) for each
L1 curve is chosen such that it is shifted left by 10 nm from the one that produces the
minimum FWHM. This is the largest L2 that one can choose without being significantly
affected by the fabrication errors. (Note that at these nominal values, their corresponding
SLLs are also below half of the main beam, which satisfy the superfocusing requirement.)
This fabrication error limits the choice of L1 to be longer than 220 nm. Although choosing longer L1 helps to achieve higher resolution as demonstrated in Fig. 3.21(c), moving
further away from the resonance results in a more attenuated main beam signal strength
as shown in Fig. 3.22. Therefore, the choice of L1 = 240 nm and L2 = 125 nm presents
the highest signal intensity while stays within the fabrication allowance.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
0.5
L = 180nm
2
1
L = 220nm
L = 220nm
1
1
1.5
L = 240nm
1
1
L = 240nm
1
L = 260nm
0.3
1
L = 200nm
1
SLL
FWHM (λ0)
0.4
L = 180nm
L = 200nm
1
54
L = 260nm
1
1
0.2
0.5
0.1
100
120
140
160
L2 (nm)
180
0
100
200
120
(a) FWHM
140
160
L2 (nm)
180
200
(b) Sidelobe Level
0.25
0.8
0.7
0.5
SLL
FWHM (λ0)
0.6
0.2
0.4
0.3
0.15
0.2
0.1
0.1
100
110
120
130
L2 (nm)
140
(c) FWHM (zoomed in)
150
160
0
100
110
120
130
L2 (nm)
140
150
160
(d) Sidelobe level (zoomed in)
Figure 3.21: Parametrization of the antenna lengths and the effect on the image function.
(The black dashed line in (a) represents the FWHM due to a single resonant slot antenna
radiation, and the one in (b) indicates the maximum allowable sidelobe level, which is
50% of the main beam.)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
55
L1 = 200nm
1
L1 = 220nm
L1 = 240nm
0.8
|Ex|
L1 = 260nm
0.6
0.4
0.2
0
−0.2
−0.1
0
x (nm)
0.1
0.2
0.3
Figure 3.22: Comparison of the signal strength at the nominal FWHM for each L1 .
3.3.3
Results
This section compares the superfocusing performance of the metascreen predicted by
shifted-beam theory and the one from FEM simulations. For an image plane located
at 0.1λ0 as shown in Fig. 3.23(a), the field distributions of the single slot antenna and
the triple-slot antenna array are illustrated in Figs. 3.23(b) and 3.23(c), for the theoretical predictions and FEM simulations, respectively. The theoretical analysis assumes
ideal PEC half-wavelength slot antenna(s), while the simulated structure employs the
metascreen shown in Fig. 3.18, with L1 = 240 nm and L2 = 125 nm. Note that in this
work, the FWHM is defined with respect to the magnitude |Ex | (instead of the intensity)
in order to maintain consistency with previous relevant work in the microwave regime [55].
The corresponding achievable FWHMs are summarized in Table 3.2. It can be seen that
although the scattered field from a single slot antenna is subdiffractional (0.2λ0 ) at the
image plane, the FWHM is more than 1/3 larger than the desired beamwidth. However,
the three-slot antenna array is able to achieve the target beamwidth of 0.12λ0 in both
theory and simulations. Note that in Fig. 3.23(c), the leakage through the plasmonic
screen interferes with the scattered fields from the slot antennas, which leads to a higher
background signal. Finally,
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
56
(a) The relative position of metascreen and image plane
1
1
1 slot
3 slot
0.8
|Ex| (normalized)
|Ex| (normalized)
0.8
0.6
0.4
0.2
0
1 slot
3 slots
0.6
0.4
0.2
−0.2
−0.1
0
x (λ0)
0.1
0.2
(b) ideal PEC slot antennas (theory)
0
−0.2
−0.1
0
x (λ0)
0.1
0.2
(c) plasmonic slot antennas (simulation)
Figure 3.23: The scattered electric field at the image plane (z = 0.1λ0 = 94 nm). The
results are shown for both (b) theoretical analysis assuming PEC slot antenna(s) and
(c)FEM simulations based on the plasmonic metascreen. The blue dash-dotted line
represents the half maximum of |Ex |.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
1 slot
3 slots
theory
0.2λ0
0.12λ0
simulation
0.19λ0
0.12λ0
57
Table 3.2: Achievable FWHMs for one and three slot antennas
The FWHM of the diffracted light in the propagation plane (the xz -plane) is shown
in Figs. 3.24 and 3.25 from the theoretical prediction and the FEM simulation, respectively. Compared to the theoretical analysis where the slot antennas are assumed to be
infinitesimally narrow, the finite width (40 nm) of a plasmonic slot antenna widen the
beamwidth close to the transmission screen. Nonetheless, the superfocusing effect due to
the array interference predicted by shifted-beam theory is demonstrated.
0.2
1
0.2
z (λ0)
0.15
0
z (λ )
0.15
0.1
0.05
0
−0.2
1
0.1
0.05
−0.1
0
x (λ0)
0.1
0.2
(a) single slot antenna
0.1
0
−0.2
−0.1
0
x (λ0)
0.1
0.2
0.1
(b) triple-slot antenna array
Figure 3.24: The magnitude of Ex in the xz -plane from theoretical analysis. (The dashed
red line represents the FWHM of Ex .)
In order to verify the weights predicted by shifted-beam theory, the simulated field
distribution at the metascreen is shown in Fig. 3.26. The magnitude of the weights of
each slot antenna is obtained by integrating the electric field over the slot width. The
phases are taken at the slot center since they stay almost constant within the slot. As
shown in Table 3.3, the central and satellite antennas are nearly 180◦ out of phase.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
0.2
1
0.2
1
0.15
0
z (λ0)
0.15
z (λ )
58
0.1
0.05
0.1
0.05
0
−0.2
−0.1
0
x (λ )
0.1
0.2
0
−0.2
0.1
−0.1
0
x (λ )
0
0.1
0.1
0.2
0
(a) single slot antenna
(b) triple-slot antenna array
Figure 3.25: The magnitude of Ex in the xz -plane from FEM simulation. (The dashed
red line represents the FWHM of Ex .)
18
250
16
200
Ex (phase) [deg]
Ex (magnitude)
14
12
10
8
6
150
100
50
0
4
−50
2
0
−100
−0.2
−0.1
0
x (λ0)
(a) magnitude
0.1
0.2
−0.2
−0.1
0
x (λ0)
0.1
0.2
(b) phase
Figure 3.26: The magnitude and phase of the electric field at the metascreen surface
(z = 0). (The shadow regions represent the location and width of the three slot antennas.)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
w0
w±1
theoretical weights
16 0◦
0.346 172◦
simulated weights
16 0◦
0.466 167◦
59
Table 3.3: Optimal weights for the 3-slot metascreen
Finally, it is emphasized that the distructive interference effect only occurs in the Ex
component, hence the superfocusing effect is applicable for Ex only. This is illustrated in
Fig. 3.27, where Ey is also shown. (Note that Ez component is zero in this structure.)
Electric field (normalized)
1
|E |
x
|Ey|
0.8
0.6
0.4
0.2
0
−0.2
−0.1
0
x (λ)
0.1
0.2
Figure 3.27: |Ex | and |Ey | components at the image plane.
3.3.4
Discussion
The proposed metascreen is highly sensitive to geometrical variations. For the example
illustrated in Fig. 3.28, the central slot length (L1 ) is fixed at 240 nm and the satellite
(L2 ) is varied from 100 ∼ 200 nm. It can be seen that increasing L2 slightly from the
one that leads to minimum FWHM results in a sharp increase in the beamwidth. This
is because the SLL rises significantly during this transition, which exceeds 50% of or
even overcomes the field strength of the main beam. Additionally, superfocusing is very
0.5
2
0.4
1.5
SLL
FWHM (λ0)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
0.3
60
1
0.2
0.5
0.1
100
120
140
160
180
0
100
200
120
140
L (nm)
|Ex| (normalized)
L2 = 110nm
L2 = 130nm
1
1
0.5
0.5
0.5
0
0.5
0
−0.5
0.5
0
−0.5
L2 = 150nm
L2 = 144nm
|Ex| (normalized)
0
1
1
0.5
0.5
0.5
0
x (λ0)
200
0.5
0
−0.5
0
x (λ0)
0
0.5
L2 = 180nm
1
0
−0.5
180
L2 = 140nm
1
0
−0.5
160
L2 (nm)
2
0.5
0
−0.5
0
x (λ0)
0.5
Figure 3.28: The sensitivity of the FWHM and SLL with respect to the satellite antenna
length (L2 ) while the central slot antenna is kept constant (L1 = 240 nm). The image
beam profiles at individual sample L2 ’s are plotted to illustrate the relative amplitude of
the main beam and SLL.
narrowband due to the resonant slot antennas employed in the design. This effect is
demonstrated in Fig. 3.29, where the wavelength is varied ±200 nm from 940 nm for the
metascreen shown in Fig. 3.18 with L1 = 240 nm and L2 = 125 nm. The sharp change in
the beamwidth around the region of minimum achievable FWHM, as well as the effect
of the SLL are similar to the observation in the sensitivity analysis in Fig. 3.28. Finally,
this particular superfocusing design is effective only for the Ex component.
Metascreens are envisioned to be implemented at the end facet of traditional NSOM
probes to replace the circular subwavelength aperture as shown in Fig. 3.30. Tradition-
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
0.8
61
2
0.7
1.5
0.5
SLL
FWHM (λ0)
0.6
0.4
1
0.3
0.5
0.2
0.1
700
800
900
1000
λ0 (nm)
|Ex| (normalized)
λ0=750nm
800
900
λ0 (nm)
λ0=850nm
1
1
0.5
0.5
0.5
0
−0.5
0
0.5
0
−0.5
0
0.5
0
−0.5
λ0=950nm
1
1
0.5
0.5
0.5
0
x (λ0)
0.5
0
−0.5
0
x (λ0)
1100
0
0.5
λ0=1050nm
1
0
−0.5
1000
λ0=875nm
1
λ0=900nm
|Ex| (normalized)
0
700
1100
0.5
0
−0.5
0
x (λ0)
0.5
Figure 3.29: The wavelength dispersion of the FWHM and SLL around λ0 = 940 nm.
ally, apertures with various diameters are achieved by sectioning the taper region at
different locations, where the circular shape arises naturally from the heated and pulled
fiber. Recently, advanced techniques have been developed to carve apertures with different shapes, such as a bowtie [61]. As shown in Fig. 3.31, a large opaque screen can
be formed by sectioning off the tapered fiber at a large radius, followed by evaporating
metals onto the end facet. The thickness of the metal can be controlled to sub -100 nm.
Different metals, such as gold, can be evaporated instead of aluminium [62]. FIB milling
is then employed for directly writing the antenna patterns. Both [61] and [62] demonstrate comparable dimensions and material of the proposed metascreen, which suggests
fabrication of such a design on realistic NSOM fiber probe is possible.
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
62
Figure 3.30: Schematic of slot antennas and arrays at the end-facet of conventional
NSOM fiber probes.
Compared to traditional NSOM apertures, metascreens are able to alleviate the tradeoffs discussed in Section 3.1. Firstly, the resonant slot antenna(s) are shown to significantly enhance the near-field intensity in Fig. 3.32(a), where apertures of various sizes
and the slot antenna(s) are illuminated with a plane wave with the same magnitude
(|Ex | = 1). In all cases, the apertures and the slot antenna(s) are printed on a 50 nm
thick gold film on glass substrate. The choices of the aperture size are typical for commercial NSOM probes. The background signal due to the 50 nm gold film is much lower
than the signal level, hence does not pose problems for this design. More importantly,
metascreens based on shifted-beam theory presents a way to extend the benefits of high
optical resolution to longer working distances. This is illustrated in Fig. 3.32(b), where
small aperture size (e.g. 50 nm) provides high resolution, but the beam diffracts quickly.
This situation is dramatically improved for the slot antenna and the array. Extending
the working distance has been one of the most powerful capability of shifted-beam theory, which is in contrast of many current techniques that only improve the intensity and
resolution within a few nanometer distance [11, 12, 61, 62].
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
63
Figure 3.31: SEM images showing the different fabrication steps of a bowtie nano-antenna
probe. (a) A standard heat-pulled fiber coated with aluminium. (b) The coated fiber is
milled by FIB to obtain a 500–700 nm opening diameter. (c) The end facet of the tapered
fiber is coated again with aluminum. (d) The final step: a second FIB milling where the
bowtie nanoaperture is directly fabricated at the end-facet of the aluminum layer. (e and
f) Examples of two other bowtie probes showing gap regions of 50 nm and a length of
300 nm. The scale bars are 200 nm. (Figure adapted from [61] with permission.)
Chapter 3. Plasmonic Antennas for Near-Field Superfocusing
circular aperture (d=50nm)
circular aperture (d=100nm)
circular aperture (d=200nm)
1 slot
3 slots
background signal
30
Electric field intensity (dB)
64
20
10
0
−10
−20
0
0.05
0.1
0.15
0.2
0.25
z (λ0)
(a) |Ex | along z
0.25
0.2
0
z (λ )
0.15
0.1
0.05
0
−0.2
−0.1
0
x (λ0)
0.1
0.2
(b) FWHM of |Ex | along z
Figure 3.32: The comparison of the near-field for the single slot antenna, the 3-slot array
and circular apertures of various diameters.
Chapter 4
Plasmonic Antennas for Near-Field
Superfocusing (2D)
Using techniques developed in the field of metascreens, this chapter extends the shiftedbeam theory to two-dimensional applications. A new type of NSOM probe design is
introduced that achieves two-dimensional subdiffraction focusing at a quarter-wavelength
away from the probe. In Section 4.1, the design inspirations from the microwave nearfield imaging probe are introduced, followed by fabrication of monopole antennas at
optical frequencies. This work is the platform on which the optical antenna-array probe is
designed. In Section 4.2, the versatility of shifted-beam theory based on the least-squares
approximation is demonstrated by its ability to synthesize two-dimensional field patterns.
A circularly-symmetric design consisting of a planar array of monopole antennas is then
presented in Section 4.3. It is shown that the array elements radiate with different
amplitudes and phases, such that their near-field interactions lead to a smaller spot size
when compared to the single monopole antenna. Finally, several aspects regarding the
performance and characteristics of the near-field probe are discussed. The structure is
shown to enable superresolution focusing at a much longer distance than conventional
probes, while being compatible with the available instrumentation and measurement
65
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
66
setups. It could potentially be a viable candidate to alleviate the working distance issue
for the current NSOM technology. Although designed to operate in the visible frequency
range, this topology is scalable to other spectra as well.
4.1
Overview
4.1.1
2D monopole-array probe at microwave frequencies
In Section 3.3, shifted-beam theory is applied to design one-dimensional metascreens.
However, practical NSOM instrumentations demand confinement of light in two dimensions. The least-squares method can be applied to field patterns in multiple dimensions, thus shifted-beam theory can accommodate two-dimensional superfocusing designs. However, it is more challenging to find practical antennas that can be arranged
in a two-dimensional array such that the desired interelement spacing and the geometry
of the antennas can be accommodated to achieve planar symmetry. For example, slot
antennas (that utilize broadside radiation) are difficult to arrange in a two-dimensional
circularly-symmetric array due to the large disparity between the slot width and length.
Good candidates for a two-dimensional arrangement include half-wavelength dipoles and
quarter-wavelength monopole antennas on a ground plane, both utilizing the end-fire radiation. Although the far-field radiation of these antennas produces a null in the end-fire
direction, the near-field intensity is significant close to the antenna tip [58]. Note that the
term radiation usually refers to the propagating spectrum; however, it is generalized here
as the field emanating from an antenna, which includes both propagating and evanescent
fields.
One example of utilizing an array of monopole antennas for the purpose of nearfield superfocusing is the microwave near-field probe shown in Fig. 4.1 [63, 64]. This
rotationally-symmetric planar array of monopole antennas is implemented at 2.4 GHz.
As shown in Fig. 4.1(b), the array is excited through a coaxial feed at the central
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
67
Figure 4.1: The schematic of the monopole-array probe at microwave frequencies
(adapted from [64] with permission). (a) The top view: eight satellite monopoles surround a central monopole in a circular distribution. The finite ground plane is one
free-space wavelength in diameter. (b) The side view: only the central monopole is
connected to a coaxial feed.
monopole, whereas the satellite elements are excited via mutual coupling. Similar to
the one-dimensional metascreen in Chapter 3, the satellite and the central monopole
antennas are separated by a subwavelength distance (0.15λ0 in this case). The length
of the monopole antennas can be adjusted such that the central and satellite elements
radiate out of phase, resulting in destructive interference. The monopole array has been
shown, both in theory and experimentally, to significantly reduce the focal spot size when
compared to a single monopole radiation. Since the evanescent fields decay exponentially
in the near-field, the presence of the propagating waves becomes significant at a quarter
wavelength. The array probe is shown to suppress the large background signal caused
by the presence of the propagating waves, hence improving signal quality. In this work,
the focal length is chosen to be a quarter-wavelength because the spot size of a single
monopole probe roughly equals to the diffraction limit at this distance. Further confinement due to the array interference leads to subdiffractional spot sizes. This design is the
primary inspiration for the configuration of optical near-field probes presented herein.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
4.1.2
68
Plasmonic monopole antennas at optical frequencies
Recent work on fabricating a single plasmonic monopole antenna at the end-facet of a
conventional NSOM aperture probe, shown in Fig. 4.2, has been demonstrated at 514 nm
[12, 65]. The antenna length can be controlled (within 5 nm accuracy) to resonate at
different wavelengths, as shown in Fig. 4.3. Fig. 4.4(a) illustrates that a linearly polarized
electric field inside the optical fiber probe impinges on the subwavelength aperture, and
the scattered field at the aperture edge is used to excite the monopole antenna. A
simplified model is shown in Fig. 4.4(b), where the aluminium (εr = −31.3 − j8) cylinder
is placed next to an aperture on an infinitely large PEC screen that is 10 nm thick.
The incident wave (λ0 = 514 nm) is modeled as a plane wave with linear polarization.
This model has been shown to accurately predict the local field of an aperture probe
without the antenna [42], as well as the excitation and radiation when the antenna is
present [12, 65]. Hence it is adapted in this thesis for modeling the monopole antenna
array.
Figure 4.2: Aperture probe-based monopole antenna (SEM images): (a) viewed from a
52◦ angle and (b) side view (adapted from [12] with permission).
The work in [12,65] illustrates the achievable fabrication accuracy using FIB technology for nano cylindrical structures, and further presents an interesting excitation scheme
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
69
Figure 4.3: Tuning the antenna length to investigate its resonance. Length: 55 ± 5 nm,
70 ± 5 nm, 90 ± 5 nm, 105 ± 5 nm (adapted from [12] with permission).
Figure 4.4: The cross-sectional view of (a) the schematic and (b) the simplified model of
the monopole antenna at the end-facet of an fiber probe.
for monopole antennas. At microwave frequencies, the monopole antenna is formed by
extending the coaxial center pin by a quarter-wavelength, and attaching the ground plane
to the coax outer shell. The transverse electromagnetic (TEM) coaxial mode, excited by
the voltage potential difference between the center pin and the ground, has cylindrical
symmetry as illustrated in Fig. 4.5(a1). It generates currents flowing in the same direction
along the monopole antenna arm from all azimuthal directions as shown in Fig. 4.5(a2).
However, creating a voltage potential between the center pin and the ground cannot be
replicated at optical frequencies. If the same monopole structure is placed in an optical
fiber setup, it is instead illuminated by a linearly polarized plane wave. This illumination scheme can only excite high-order coaxial modes with odd azimuthal orders even
when the center pin and ground are PECs. As shown in the example in Fig. 4.5(b),
the x -polarized plane wave excites the TE11 coaxial mode. However, the symmetric
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
70
Figure 4.5: Excitation schemes of a PEC monopole antenna with different aperture fields:
(a) the TEM coaxial mode; (b) the high-order coaxial mode (TE11 ); (c) the circular
aperture mode (TE11 ). The modal profiles are obtained from FEM simulations using
Comsol Multiphysics.
electric field distribution along x leads to currents flowing in opposite directions in the
monopole antenna arm. As the result, the zero net current fails to excite the antenna.
For plasmonic coaxial-like waveguides at optical frequencies, several investigations have
been reported [66, 67], where only modes of odd azimuthal orders are observed. In the
experiments in [67], the coaxial-like waveguides are illuminated with linearly polarized
plane waves. However, placing the monopole antenna at the edge of a subwavelength
aperture can lead to effective excitation. When illuminated with a plane wave, the TE11
mode of the circular waveguide is excited (Fig. 4.5(c1)). This field distribution generates
desired currents in the antenna arm as shown in Fig. 4.5(c2).
For plasmonic monopole antennas at optical frequencies, this excitation scheme can be
understood from the perspective of mode coupling. The fundamental TM0 surface mode
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
71
Figure 4.6: Schematics of the field distribution for the the aperture excitation and the
TM0 surface mode of a plasmonic monopole antenna.
supported by the plasmonic rod antenna has cylindrical-symmetric field distributions in
the antenna arm, as shown in Fig. 4.6. The linearly polarized electric field inside the
optical fiber probe impinges on the aperture, which generates a significant amount of the
Ez component only at the aperture edge [68]. As shown in Fig. 4.6, Ez can effectively
couple to the TM0 surface mode. However, when the monopole antenna is placed in the
middle of the aperture, as in the form of an extended coaxial center pin, the excitation
fails because Ez vanishes at the aperture center. The Ez aperture field intensity without
the antenna, as well as the monopole antenna radiation when placed at the aperture edge
and center, are illustrated in Figs. 4.7(a), 4.7(b) and 4.7(c), respectively. The near-field
of the antenna tip exhibits a highly confined and intense spot when placed at the aperture
edge, whereas the one placed at the aperture center only consists of weak scattering from
the aperture. These results demonstrate that the mode coupling is only possible for edge
excitation.
Due to the non-uniformly distributed Ez at the aperture edge, as shown in Fig. 4.7(a),
this scheme leads to non-uniform excitation of the monopole antenna, depending on the
angle between the antenna and the polarization of the incident plane wave. For the
example shown in Fig. 4.8, where the illumination is polarized along the x -axis, the
monopole antenna is excited with maximum amplitude when it is aligned along this
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
72
150
0.06
100
0.05
y (nm)
50
0.04
0
0.03
−50
0.02
−100
−150
0.01
−100
0
x (nm)
100
(a) |Ez |2 at the aperture edge
−3
150
3.5
100
3
x 10
150
12
100
10
2.5
2
0
1.5
−50
1
−100
−150
0.5
−100
0
x (nm)
100
(b) |E|2 at the monopole tip
50
y (nm)
y (nm)
50
8
0
6
−50
4
−100
2
−150
−100
0
x (nm)
100
(c) |E|2 at the monopole tip
Figure 4.7: The excitation of the plasmonic monopole antenna through a subwavelength
(100 nm in diameter) aperture at 514 nm. The aluminium (εr = −31.3 − j8) cylinder
with a length of 70 nm and a radius of 20 nm is used to model the antenna. The aperture
is situated on a PEC screen of 10 nm thick. (a) |Ez |2 at 5nm below the aperture when
illuminated by a plane wave polarized in the x -direction; (b) |E|2 at the monopole antenna
apex when it is placed at the aperture edge, and (c) at the aperture center.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
73
direction. Furthermore, monopole antennas placed on opposite edges of the aperture
radiate 180◦ out of phase. Varying the aperture size also affects the monopole radiation
intensity because larger apertures generate stronger fields. This, along with changing the
monopole antenna length, provides several degrees of freedom for manipulating both the
magnitude and phase of the antenna end-fire radiation.
(a)
(b)
Figure 4.8: The end-fire radiation of a monopole antenna when the aperture is illuminated
with a plane wave polarized in the x -direction. (a) The monopole antenna is placed along
the aperture edge at various angles with respect to the polarization of illumination; (b)
The electric field is obtained at 1 nm below the apex of the antenna.
It is hereby recognized that this work presents a practical platform for implementing
monopole antenna-arrays in the visible wavelength regime. Since the design is based on
conventional NSOM aperture probes, it can be integrated easily with existing instrumentations. By combining the techniques used in microwave designs with those available for
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
74
9
8
7
z
|E |
6
5
4
3
2
1
0
60
80
100
120
140
160
180
200
Aperture diameter (nm)
Figure 4.9: Monopole end-fire radiation vs. aperture size (|Eexcitation | = 1[V/m]).
the domain of optical instruments and fabrication, a plasmonic monopole-array probe
is proposed to achieve two-dimensional superfocusing at a quarter-wavelength distance.
Previous work on two-dimensional superfocusing employing radiationless interference has
been reported at 1550 nm [69]. The achieved FWHM for the field intensity (|E|2 ) is
0.225λ0 , which translates to a FWHM for the field magnitude (|E|) of 0.474λ0 . It will be
shown that the proposed design can achieve much smaller spot (0.29λ0 ) when compared
to the existing design in [69]. Note that in this thesis, the FWHM is defined with respect
to the magnitude |E| (instead of the intensity) in order to maintain consistency with
previous relevant works in the microwave regime [63, 64]. However, the design in [69]
achieves two-dimensional focusing for dual polarized illumination, whereas the proposed
monopole antenna array only works for a single polarization.
4.2
Theory
The analytical solution of the end-fire radiation of a dipole antenna, or a monopole
antenna on an infinite ground plane, oriented along the z -axis is shown in Eqn. 4.1 [58],
k0 H e−jk0 r
η0 Ie e−jk0 R1 e−jk0 R2
+
− 2 cos
Ez = −j
4π
R1
R2
2
r
(4.1)
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
75
where H represents the antenna length. r, R1 and R2 are the distances from the center
and the two tips of the dipole antenna, respectively. They can be computed as follows:
p
p
x 2 + y 2 + z 2 = ρ2 + z 2
s
2 s
2
H
H
2
2
2
R1 = x + y + z −
= ρ + z−
2
2
s
s
2
2
H
H
2
2
2
R2 = x + y + z +
= ρ + z+
2
2
r=
(4.2)
(4.3)
(4.4)
This solution includes contributions from both the propagating and evanescent waves.
For the special cases when H has a value equal to odd integer multiples of half-wavelength,
the cosine term in Eqn. 4.1 vanishes. Additionally, Eqn. 4.1 demonstrates that although
the solution has a null in the far-field, the near-field intensity is significant in the vicinity
of the tips. At the desired image plane of z = λ0 /4, the magnitude of Ez is shown
in Fig. 4.10(a), where the red contour represents a FWHM of 2.2λ0 . Although the
beamwidth in Fig. 4.10(b) demonstrates subwavelength features due to the presence
of evanescent fields, the FWHM far exceeds the diffraction-limit because of the strong
background signal consisting of propagating waves.
As previously stated, shifted-beam theory can be extended to design a two-dimensional
planar array of monopole antennas. The relative weights of the antennas are once again
optimized using the least-squares approximation, such that a two-dimensional target
function can be achieved. In this case, the matrix A and vector b in Eqn. 3.18 are
modified to A = [E1 [x, y], E2 [x, y], ..., En [x, y]] and b = f [x, y] in order to account for
the two-dimensional field distributions. The optimized weights can then be found using
Eqn. 3.20. The application of this method is illustrated in the example of achieving a
subdiffractional target function at a quarter-wavelength distance. The target function is
shown in Fig. 4.11(a), consisting of a two-dimensional Gaussian field distribution with
a FWHM of 0.3λ0 . Due to the cylindrically symmetric nature of the target function, a
circular array is a natural choice of configuration for achieving such a field distribution.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
1.5
1
76
1
1
|Ez| (normalized)
0.8
0
y (λ )
0.5
0
−0.5
0.6
0.4
|E |
z
−1
−1.5
−1.5
half maximum of |E |
0.2
−1
−0.5
0
x (λ )
0.5
1
1.5
0.1
0
(a) |Ez | distribution in the xy-plane
0
−1.5
z
−1
−0.5
0
x (λ )
0.5
1
1.5
0
(b) |Ez | distribution along x-axis
Figure 4.10: The end-fire radiation |Ez | in the image plane at z = 0.25λ0 for a single
monopole antenna (analytical solution). The red contour/line represents the FWHM of
|Ez |.
The proposed monopole antenna array consists of a ring of satellite antennas surrounding
a central element. The effect of the number of satellite elements on the desired target
function is illustrated in Figs. 4.11(b)-(e). For a central-to-satellite separation of 0.2λ0 ,
a minimum of four satellite elements is required to obtain the desired field pattern, and
that the six-satellite configuration further improves the matching with the target function. Comparing these two results with Fig. 4.10(a), it is shown that the two-dimensional
antenna array is able to reduce the spot size of a single monopole antenna far below the
diffraction limit, while lowering the background signal. For both the four and six-satellite
configurations, the optimized weights is illustrated in Table 4.1. These weights confirm
the destructive interference condition between the central and satellite antennas, which
is necessary for reducing the spot size.
The diffracted beam along the z -direction for a single monopole antenna and the
antenna array with four-satellite monopoles are shown in Figs. 4.12(a) and 4.12(b), respectively. It can be seen that the radiation from a single monopole antenna diffracts
very quickly. At a quarter-wavelength distance, it has grown far beyond the diffraction
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
77
0.5
0.8
y (λ0)
0.6
0
0.4
0.2
−0.5
−0.5
0
x (λ )
0.5
0
(a) target function
0.5
0.5
1
1
0.8
0.6
0
y (λ0)
y (λ0)
0.8
0.6
0
0.4
0.4
0.2
0.2
−0.5
−0.5
0
x (λ0)
−0.5
−0.5
0.5
(b) 2 satellite antennas
0.8
0.6
0.6
0
(d) 4 satellite antennas
0.5
1
0.8
y (λ0)
y (λ0)
0.5
1
0
x (λ0)
0.5
(c) 3 satellite antennas
0.5
−0.5
−0.5
0
x (λ0)
0
0.4
0.4
0.2
0.2
−0.5
−0.5
0
x (λ0)
0.5
(e) 6 satellite antennas
Figure 4.11: Effect of increasing the number of satellite antennas on the achievable spot
size of |Ez |. (The blue dots represent the location of monopole antennas.)
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
wcentral
wsatellite
four-satellite configuration
16 0◦
0.3826 -6◦
six-satellite configuration
16 0◦
0.2556 -6◦
78
Table 4.1: Optimal weights for the circularly-symmetric monopole antenna array
limit. In contrast, the planar array with four satellite elements is able to maintain a
much more focused spot size. More specifically, the FWHM of the spot is 0.3λ0 , which
is very well within the diffraction limit.
0
1
0
0.1
0.1
0
z (λ0)
0.05
z (λ )
0.05
1
0.15
0.15
0.2
0.2
0.25
−0.5
−0.25
0
x (λ0)
0.25
0.5
0.1
0.25
−0.5
(a) single monopole antenna
−0.25
0
x (λ0)
0.25
0.5
0.1
(b) monopole array with four satellites
Figure 4.12: The comparison of the diffracted beam (|Ez |) along z for (a) a single
monopole antenna and (b) the antenna array that employs four satellites. (The red
contour represents the FWHM of the radiation.)
Although this particular target function naturally leads to a rotationally symmetric
array, it by no means imposes restrictions that such an array configuration is the only
choice for achieving the target function. The elements in the two-dimensional array can
be arranged in variety of ways to best suit the implementation convenience. For each
configuration, there is a particular weight distribution associated with it. It can also
employ a diverse collection of antenna structures other than slot, monopole and dipole
antennas.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
4.3
79
Results and Discussion
The single-monopole antenna excited by the NSOM probe aperture has shown great signal
intensity and resolution close to its tip. However, imaging at a distance of a quarterwavelength is problematic due to the significantly enlarged spot size. Section 4.2 has
presented the theoretical analysis of a monopole antenna array design that is capable of
achieving a FWHM beamwidth of 0.3λ0 (with respect to |Ez |) at the desired image plane.
This section begins with discussing the features that are unique to the single monopole
antenna on the end facet of a NSOM aperture probe, including the characteristics of the
antenna length, as well as the effect of aperture excitation on the signal resolution at the
image plane. A monopole antenna array is then designed at λ0 = 514 nm, with a scheme
that incorporates additional apertures for exciting the array. Finally, the performance of
the monopole array is evaluated through FEM simulations, and the results are contrasted
to the case of the single monopole antenna.
4.3.1
Aperture probe with a single monopole antenna
Although similar to microwave antennas in principle, the optical implementation exhibits
its own special features. Firstly, the antenna is physically much shorter than the conventional λ0 /4. This effect has been analyzed in Section 2.2. At λ0 = 514 nm, the resonance
occurs when the physical length of the aluminium monopole antenna is 70 nm (roughly
λ0 /7). Furthermore, due to practical limitations of nanofabrication, such as those imposed by FIB lithography, the resulting antenna radius is optically large at visible frequencies (approximately λ0 /25 at 514 nm) when compared to its microwave counterpart,
where it is usually less than a few hundredths of λ0 . As a result, the closed-form solution
of an ideal perfect-conducting monopole with infinitesimally thin arms seen in Eqn. 4.1
cannot accurately predict its end-fire radiation.
Numerical simulations using COMSOL Multiphysics R have been conducted to cal-
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
80
culate the field at z = 0.25λ0 . For this simulation, the simplified model in Fig. 4.4(b)
is applied. The plane wave excitation is at 514 nm with a linear polarization along the
x -axis. The subwavelength illuminating aperture with d = 100 nm is adapted from [65].
The radiation of a single plasmonic monopole at z = λ0 /4 is illustrated in Fig. 4.13(a),
showing that the field at the image plane is a combination of the monopole end-fire radiation and the scattered field from the illuminating aperture. Comparing the magnitude
of Ez in Fig. 4.13(b) along the y-axis with the analytical result shown in Fig. 4.10(b), the
simulation reveals that the plasmonic monopole antenna leads to a smaller FWHM than
that predicted by theory. The beamwidth at a quarter-wavelength away has a FWHM
beamwidth of only 0.48λ0 along the y-axis, whereas the one from Fig. 4.10(b) is 2.2λ0 .
Imaging with such an optical monopole in the extreme near-field, which is just a few
nanometers away from the antenna tip, exhibits several advantages compared to using
conventional aperture probes. Due to its small geometry, the near-field resolution at the
antenna apex is much higher than that achievable with the aperture alone. Additionally,
monopole antennas radiate strongly at resonance, which allows higher signal intensity
without sacrificing resolution. However, using this structure to illuminate beyond the
extreme near-field can be problematic. Aside from the rapidly increased spot size, the
field introduced by the aperture also interferes with the end-fire monopole radiation, thus
producing an unwanted sidelobe as shown in Fig. 4.13(a). This sidelobe, which effectively
acts as background signal, approaches the strength of the main beam as the imaging
distance is increased, hence severely reducing the resolving power along the direction
of the incident wave polarization (x -axis). As illustrated in Fig. 4.13(b), at a quarterwavelength away, the focus along the x -axis is blurred by the aperture interference.
4.3.2
Aperture probe with a monopole antenna array
The goal of designing a superfocused antenna array is to achieve opposing phases between
the central monopole and its satellites, such that the end-fire fields in the near-zone
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
1.5
81
1
1
y (λ)
0.5
0
−0.5
−1
−1.5
−1.5
−1
−0.5
0
0.5
1
1.5
0.1
x (λ)
(a)
1
z
|E | (normalized)
0.8
0.6
0.4
|Ez| along x−axis
0.2
|Ez| along y−axis
half maximum of |E |
z
0
−1
−0.5
0
x/λ (or y/λ)
0.5
1
(b)
Figure 4.13: |Ez | at the image plane (z = 0.25λ) for a plasmonic monopole antenna
excited by an aperture of 100 nm on a PEC screen of 10 nm thick. The model is based
on the one illustrated in Fig. 4.4(b), where the plane wave incidence that excites the
aperture is polarized along x -axis. Aluminum monopole antenna has a length of 70 nm
with εr = −31.8 − j8. (a) |Ez | in the transverse plane; (b) |Ez | in the transverse plane
along the x and y axes. (The red contour/line represents the half maximum of |Ez |).
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
82
interfere destructively to reduce the beamwidth of a single monopole antenna [63, 64].
The relative magnitude of the satellite radiation, along with their locations with respect to
the central monopole, determines the amount of beamwidth reduction. In general, a small
separation distance allows for achieving a small focal spot size. Given a fixed separation,
stronger satellite radiation can further reduce the main beamwidth, but at the expense of
increased sidelobe levels. At microwave frequencies, the out-of-phase radiation is achieved
by directly exciting the central monopole with a coaxial cable, and the satellite elements
are excited indirectly via mutual coupling, such as the one illustrated in Fig. 4.1. When
resonant monopoles are separated by subwavelength distances, the mutual coupling can
induce out-of-phase radiation [58]. For the design presented in [63, 64], the amplitude of
the satellite radiation can reach half of the main beam for a central-to-satellite separation
of 0.15λ0 . The corresponding focal spot size at a quarter-wavelength imaging distance is
shown to be 60% less than that of the single monopole.
The proposed plasmonic monopole-array probe designed for the visible spectrum is
shown in Fig. 4.14, which consists of five monopole antennas and three excitation apertures. The larger aperture (d1 ) used for exciting the central monopole antenna disrupts
the planar symmetry of the array. Besides introducing an asymmetrically distributed
background signal, this excitation scheme does not allow satellite monopoles to be placed
at the nominal distance of s = 0.15λ0 (77 nm) from all azimuthal directions of the central monopole. Therefore, the separation distance must be increased to about 0.193λ0
(100 nm) to allow for placing four satellite elements with circular symmetry, as illustrated in Fig. 4.14(b). The increased separation distance will increase the achievable
optical spot size. Additionally, at this separation distance, the satellite radiation induced by mutual coupling is only 1/5 of the main beam. Therefore, excitation for the
satellites through other means is necessary to enhance their radiation. Additional apertures are introduced to directly excite the satellite elements instead of relying on mutual
coupling alone. (The three-aperture configuration is envisioned to be implemented using
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
83
Figure 4.14: The proposed two-dimensional near-field monopole antenna array at the
end facet of a SNOM aperture probe. (a) The schematic of the probe configuration; (b)
The apertures and the circularly symmetric arrangement of the antenna array at the end
facet (transverse plane); (c) The respective locations of the source and image planes along
the longitudinal direction. The aperture diameters d1 = 100 nm and d2 = 70 nm. The
separation s = 100 nm. The antenna lengths L1 = 75 nm, L2 = 80 nm and L3 = 85 nm.
The permittivity of aluminium εr = −31.3 − j8 at λ0 = 514 nm.
FIB milling, similar to the process described in Fig. 3.31.) Finally, the amplitude and
phase of the excitation can be controlled through changing the position of the monopoles
along the aperture edge with respect to the polarization of the incident waves, as well as
adjusting the aperture size and the monopole lengths. In the proposed design, the effort
is focused on minimizing the number of additional apertures in order to minimize the
overall background signal. Therefore, only the two satellite monopoles on the right are
excited directly with smaller apertures (Fig. 4.14(b)), whereas the ones on the left are
located directly on the upper and lower edges of the large aperture that excites the central monopole. In this design, the smaller apertures have a diameter d2 = 70 nm, which
facilitates a weaker excitation for the satellites compared to the central radiator. The
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
84
left, central and right monopoles, all with a radius of 20 nm, have lengths of L2 = 80 nm,
L1 = 75 nm and L3 = 85 nm, respectively. The central radiator is designed to operate
closest to resonance to ensure maximum radiation. The satellite monopole lengths are
then adjusted to control the level of interference. Both the antenna lengths and aperture
sizes are physically implementable using current FIB technology [12].
It is demonstrated next in simulations that the design in Fig. 4.14 can lead to desired superfocusing. The simulations are conducted using Comsol Multiphysics, where
the model in Fig. 4.4(b) is adapted. The improvement of the monopole antenna array
over the single monopole antenna when imaging at a λ/4 distance is quantified using
two performance measures; the reduced FWHM of the focal spot size and the diminished aperture background signal interference. With respect to the reduced FWHM, the
contour diagram of |Ez | at the image plane is shown in Fig. 4.15(a). Compared to the
diffraction-limited beamwidth of the single monopole (blue), the smaller focal spot from
the monopole array (black) indicates that the FWHM of the array beamwidth is 0.29λ0
along the y-axis, which translates to a reduction of 40% of that associated with the single
monopole. The resolving power along the x -axis increases even more tremendously due to
the reduced interference from the aperture. Note that the monopole-array configuration
that produces the smallest focal spot size does not provide the strongest suppression of
the background singal. For the proposed design, a compromise has been made to achieve
a symmetric two-dimensional FWHM beamwith of 0.3λ0 from all azimuthal angles, while
maintaining a background intensity 20% below that of the main beam. It is instructive to
emphasize that the benefit of background signal reduction becomes increasingly important at longer imaging distances as shown in Fig. 4.16, which illustrates that the intensity
of the aperture background signal becomes more comparable to the main beam when the
imaging distance increases from 0.15λ0 to 0.25λ0 . Therefore, it is concluded that the proposed monopole antenna arrays enables higher SNR (lower background) in addition to
higher resolution (smaller spot size) compared to the single monopole antenna, especially
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
1
85
single monopole
monopole array
y (λ)
0.5
0
−0.5
−1
−1
−0.5
0
x (λ)
0.5
1
(a)
1
1
1
0.8
0.5
0.8
0.5
0.6
y (λ)
y (λ)
0.6
0
0
0.4
−0.5
−1
−1
0.2
−0.5
0
x (λ)
0.5
(b)
1
0
0.4
−0.5
−1
0.2
−1
−0.5
0
x (λ)
0.5
1
0
(c)
Figure 4.15: Contour diagrams of |Ez | at the image plane (z = 0.25λ). (a) Comparison
of the FWHM for the single monopole (blue) and the monopole array (black); (b) The
10% contour intervals for the single monopole antenna; (b) The 10% contour intervals
for the monopole antenna array.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
86
at longer working distances.
1
z
single monopole (|E |2)
1.5
0.5
y (λ)
0
−0.5
−1
−1.5
2
monopole array (|Ez| )
1.5
1
y (λ)
0.5
0
−0.5
−1
−1.5
−1.5 −1 −0.5
0
0.5
1
1.5
−1.5 −1 −0.5
0
0.5
1
1.5
x (λ)
x (λ)
−1.5 −1 −0.5 0 0.5 1 1.5
x (λ)
a) z = 0.15λ
b) z = 0.2λ
c) z = 0.25λ
Figure 4.16: The comparison of the normalized field intensity (|Ez |2 ) of the single
monopole and the monopole array at various imaging distances.
In order to verify shifted-beam theory, the electric field distribution in the sourceplane (z = 0) is shown in Fig. 4.17, which illustrates that the instantaneous electric
field of the central and satellite monopoles are nearly 180◦ out of phase. The normalized
weights of the antennas are shown in Table 4.2. In this case, the magnitudes deviate
from the theoretical predictions due to significant discrepancies between the plasmonic
and ideal monopole antennas, as well as the interference from the aperture excitation.
Nonetheless, the effect of out-of-phase radiation predicted by shifted-beam theory is
clearly demonstrated.
Finally, the near-field intensity of the proposed optical probe is compared to conventional aperture probes. As illustrated in Fig. 4.18, the magnitude of the electric field is
analyzed for the aperture, the single monopole, and the monopole array, with the same
excitation for all three cases. Note that for the conventional aperture probe, the working
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
0.5
87
0.6
0.25
y (λ)
0.3
0
0
−0.25
0
x (λ)
(a)
0.25
0.5
−0.3
180
0.75
135
0.5
90
0.25
45
0
0
−0.25
magnitude
phase
−0.5
−1
−0.5
Phase of Ez (deg)
−0.25
1
z
|E | (normalized)
−0.5
−0.5
−45
0
L/λ
(b)
0.5
−90
1
Figure 4.17: The field distribution (Ez ) at the source plane (z = 0). (a) The instantaneous electric field in the transverse plane; (b) The comparison of the magnitude and the
phase between the left, central and right elements along the dashed line in (a).
distance is measured from the center of the exiting facet of the aperture. For the single monopole and the monopole array probes, the working distance is from the apex of
the respective longest monopole antenna. The results indicate that the surface plasmon
resonance of the monopole antenna enhances the electric field around its apex, which
produces a higher field intensity compared to the illuminating aperture alone. Note that
the field amplification is more beneficial when imaging in closer proximity to the antenna
apex due to the fast divergence of evanescent fields. As illustrated in Fig. 4.18(a), the
electric field strength from the monopole end-fire radiation is stronger than the one trans-
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
wleft
wcentral
wright
theoretical prediction
0.386 -6◦
16 0◦
0.386 -6◦
simulated results
0.306 179◦
16 0◦
0.506 -165◦
88
Table 4.2: Optimal weights for the monopole array (theory and simulation)
mitted through the aperture probe within a working distance of 0.2λ. The field strengths
are still comparable at a working distance of 0.25λ.
Finally, it is emphasized that the distructive interference effect only occurs in the Ez
component, hence the superfocusing effect is applicable for Ez only. This is illustrated in
Fig. 4.19, where all three electric components and the total E-field intensity are shown.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
0
10
aperture
single monopole
monopole array
0
−0.25
z (λ)
5
−20
−0.5
−0.75
0
−1
−40
aperture
−0.5
0
0.5
1
0
−5
0
−0.25
z (λ)
E−Field Magnitude (dB)
89
−10
−20
−0.5
−0.75
−1
−15
−40
single monopole
−0.5
0
0.5
1
0
−20
z (λ)
−25
0
0
−0.25
−20
−0.5
−0.75
0.05 0.1 0.15 0.2 0.25
−z (λ)
(a)
−1
−40
monopole array
−0.5
0
x (λ)
(b)
0.5
1
Figure 4.18: The magnitude of electric field in the near-field of the single aperture, single
monopole and monopole array. (|Eexcitation | = 1[V/m] for all three cases). (a) The E-field
magnitude versus the working distance. (b) The magnitude (in dB scale) of the E-field
distribution in the xz -plane.
Chapter 4. Plasmonic Antennas for Near-Field Superfocusing (2D)
1
1
|E|2
x
0
0.6
0.5
0.5
y (λ)
0.6
0.7
|E |2
0.8
0.5
y (λ)
1
0.4
0
0.4
−0.5
0.3
0.2
−0.5
0.2
0.1
−1
−1
−0.5
0
x (λ)
0.5
−1
−1
1
−0.5
(a)
0
x (λ)
0.5
1
(b)
1
1
0.07
0.5
2
2
|E |
y
|E |
z
0.06
0.5
0.4
0.5
0.04
0
y (λ)
y (λ)
0.05
0.3
0
0.03
0.02
−0.5
0.2
−0.5
0.1
0.01
−1
−1
−0.5
0
x (λ)
(c)
0.5
1
−1
−1
−0.5
0
x (λ)
0.5
1
(d)
Figure 4.19: E-field intensity (normalized w.r.t. |E|2 ) at the image plane.
90
Chapter 5
Plasmonic Antennas for Far-Field
Sensing
This chapter presents a plasmonic leaky-wave antenna design for the purpose of achieving
directive far-field radiation over a wide spectrum (1200 nm to 2000 nm). The design leads
to efficient coupling from radiation to nanosize optics, such as subwavelength integrated
optical circuits, as well as single-molecule emitters. The chapter begins by analyzing the
radiation of nanoemitters, such as organic fluorescent dye molecules and semiconductor
quantum dots on a dielectric substrate. It is then shown that their far-field radiation
characteristics can be modified when coupled to plasmonic antennas. More specifically,
antennas with high directivity, such as leaky-wave antennas, are able to concentrate the
radiated energy within a much smaller solid angle [70]. The theory of the supported
leaky-modes is illustrated using the example of a microwave leaky-slot. The focus of the
discussion is on the ultrawideband performance of this antenna, as most existing freespace coupling schemes rely on resonant elements that are very narrowband. Finally, an
optical adaptation of the leaky-slot, the plasmonic leaky-wave antenna, is proposed and
its wideband performance is demonstrated.
91
Chapter 5. Plasmonic Antennas for Far-Field Sensing
5.1
Overview
5.1.1
Nanoscale emitters
92
Nanoscale emitters have found very wide applications in many scientific areas. For example, in biochemistry, by attaching them to an antibody, the fluorescent molecules serve
as labels for identifying the subject of study [71]. This technique has enabled many microscopy methods, from basic fluorescent confocal to subdiffractional techniques such as
PALM, STORM and STED imaging [49–51]. Quantum dots made of semiconductors can
be tailored to achieve different sizes, which enable artificially controlling their emission
spectra. These single-molecule based emitters and absorbers have proven very attractive
for light collection (i.e. photovoltaics) and emission (i.e. LEDs) applications.
The interaction between nanoscale emitters and light exhibits resonant characteristics
when the photon energy equals to the difference of discrete internal (electronic) energy
levels [43]. At optical frequencies, the light-particle interaction can often be approximated
with a two-level system. It involves excitation from the ground state to a higher energy
state, followed by radiative relaxation in the form of fluorescence, and some non-radiative
relaxation that dissipates as heat. The ratio of the radiative decay rate and the total
decay rate is defined as the quantum efficiency. Usually, the radiative energy experiences
a spectral shift from the incident wavelength towards the longer wavelengths known as
the Stoke shift. Each radiating particle can be modeled as an elementary electric dipole,
where maximum power transfer occurs when the external field is aligned with the dipole
moment of the nanoemitters [43].
The studies of the excitation and radiation of single emitters have traditionally been
done by embedding them in a thin polymer film on a substrate in order to fix the molecules
in space [43]. Such films can be produced by spin coating a solution that contains the
polymer (e.g. Poly(methyl methacrylate) PMMA) and the fluorescent molecules. The
density of the molecules can be controlled to obtain sparsely distributed single emitters.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
93
Such samples can then be studied using a setup such as the one illustrated in Fig. 5.1,
where different illumination methods are applied, i.e. either using a far-field focused
laser (as in case (3)) or a near-field local probe. The strategies based on local probes
can also be self-luminous as in the case of aperture probes (1) or externally excited
using an apertureless tip with irradiating laser (2). The fluorescence is collected from
the bottom of the sample through a high numerical aperture objective. Dichroic mirrors
(DM) and filters (F) are used to separate the excitation and the fluorescence based on
the Stoke shift. The following sections will illustrate the far-field radiation pattern of
single nanoemitters on dielectric substrate as stand alone radiators.
Figure 5.1: Schematic setup for exciting and collecting radiation from single nanoemitters
with different possible illumination geometries: (1) near-field aperture-probe illumination; (2) near-field apertureless-probe illumination; (3) far-field illumination (adapted
from
http://www.optics.rochester.edu/workgroups/novotny/courses/OPT463
with permission).
Chapter 5. Plasmonic Antennas for Far-Field Sensing
5.1.2
94
Nanoemitter radiation without antennas
An interesting approach for deriving the analytical expressions of the H and E-plane
radiation patterns has been proposed by Rutledge [60]. The analysis is based on the
reciprocity theorem, which can best be understood from the example shown in Fig 5.2.
In this case, an electric dipole with current Ie1 is placed on the substrate, which radiates
Figure 5.2: Reciprocity theorem for an electric dipole on a substrate.
field E1 in air. If there is a second electric dipole Ie2 in air that radiates field E2 at the
air-dielectric interface, then according to the reciprocity theorem, E1 = E2 if Ie1 = Ie2 .
Therefore, finding the electric field E1 radiated in air due to Ie1 becomes finding the
equivalent field E2 at the air-dielectric interface due to Ie2 . This is achieved from the
transmission and reflection analysis for a single interface. E2 can be found from
E2 = te21 Ea
(5.1)
where te21 is the transmission coefficient from air to the dielectric, and Ea is the field
of an elementary current in air. Similarly, the radiation pattern in the dielectric can be
calculated assuming the second dipole current is placed in the dielectric. In this case,
the tangential field is demonstrated in
E2 = te12 Ed
(5.2)
Chapter 5. Plasmonic Antennas for Far-Field Sensing
95
where te12 is the transmission coefficient from the dielectric to air, and Ed is the field of
a current element in the corresponding dielectric medium. The detailed derivation can
be found in Appendix C. Only the results are shown here in Fig. 5.3.
90
150
90
60
120
60
120
30
H−plane
E−plane
150
εr = 1
ε =1
1
180
0
180
r
330
240
0
ε =11.7
εr = 4
210
30
H−plane
E−plane
300
210
330
240
300
270
270
(a) Dipole on glass substrate
(b) Dipole on silicon substrate
Figure 5.3: The normalized radiation pattern of an electric dipole on substrate.
A notable problem with traditional collection schemes, where single molecules radiate
directly into dielectric substrate, is the angular confinement of their far-field radiation
pattern. Note that by fixing nanoemitters over a substrate, their dipolar radiation pattern
has been modified significantly compared to the one seen in a homogeneous medium. For
instance, the omni-directional pattern in the H-plane is now mostly concentrated in the
substrate. As shown in Figs. 5.3(a) and 5.3(b), the radiation patterns depend strongly
on the substrate permittivity. The ratio of the power distribution in dielectric and air
is approximately n3 : 1, where n = n2 /n1 . Additionally, the H-plane pattern has peaks
and the E-plane pattern has nulls at the critical angles. These radiation patterns also
indicate that the 3 dB beamwidth is wide and the energy splits into several lobes. As
a result, objective lenses with very high numerical apertures are necessary for collecting
single-molecule fluorescence. In this chapter, these patterns are used as the benchmarks
for evaluating the benefits of employing plasmonic antennas.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
5.1.3
96
Improving nanoemitter radiation with antennas
At microwave frequencies, antennas are used to efficiently couple energy from free-space
propagation to electronic circuits that are deeply subwavelength. These circuits serve
either as transmitters or receivers, which process information wirelessly through antennas. At optical frequencies, optical transmitters/receivers can consist of complicated
integrated photonic circuits, or of simple single fluorescent molecules.
The dimension of integrated plasmonic circuits is no longer confined by the diffraction limit. However, efficient coupling of optical energy into those deeply subwavelength
structures is still a major challenge. Present coupling techniques include both on-chip
and free-space schemes. The on-chip techniques rely on coupling with conventional bulky
dielectric waveguides [72–74], which not only increase the footprint beyond the diffraction limit, but also require additional consideration for coupling from free-space to these
bulky elements. Direct coupling from free-space has been proposed utilizing resonant
structures, such as dipole antennas and a chain of plasmonic spheres [75, 76]. Although
not relying on bulky dielectric waveguides, these structures operate within a very narrow bandwidth due to their resonant nature. Single molecules have dimensions of just
a few nanometers. Due to the large difference in size with the free-space propagation
to which they couple, those molecules neither radiate nor receive free-space illumination efficiently [77]. Plasmonic antennas present two major advantages for improving
nanoemitter radiation, the strong near-field intensity and the far-field directivity. The
former is known to enhance the excitation rate and to increase the quantum yield (i.e. the
ratio of emitted and absorbed photons) of emitters [16], which have been demonstrated
in experiments for plasmonic particles and rod antennas [11,12,78,79]. The near-field enhancement of plasmonic antennas has been discussed extensively in the previous chapters;
in this chapter, attention is focused on the far-field directivity improvement.
The direction and polarization of the radiated energy from nanoemitters can be controlled through coupling to antennas, such that the radiation pattern resembles the one of
Chapter 5. Plasmonic Antennas for Far-Field Sensing
97
the antenna instead of the emitter [16,29,77]. Therefore, it is possible to achieve radiation
patterns with significantly higher directivity when compared to single molecules radiating on their own. An experimental demonstration of such an improvement is shown by
Curto et al. [30], who couple a quantum dot radiating at 800 nm to a plasmonic Yagi-Uda
antenna. As shown in Fig. 5.4, the Yagi-Uda antenna consists of one driven element, one
reflector and three directors composed of alumimium rods. The quantum dot is end-fire
coupled to the driven element. Fig. 5.4(b) shows that the fluorescent emission is redirected into a single lobe, which resembles the far-field radiation pattern of the Yagi-Uda
antenna.
Figure 5.4: A y-oriented emitter coupled to a lossless Yagi-Uda antenna placed on a
dielectric substrate (εr = 2.25) (adapted from [80] with permission). (a) Snapshot of the
local field (xz -plane). (b) The angular directivity (linear scale).
Chapter 5. Plasmonic Antennas for Far-Field Sensing
98
Yagi-Uda antennas are examples from the family of traveling-wave antennas, which
are known for their high directivity. However, high directivity occurs only when the
elements are at resonance. In fact, most existing plasmonic antenna designs are based on
resonant elements [11,29,30,75,76], hence inevitably narrowband. A wideband structure
that enables directive radiation is highly desirable for illumination and signal collection
purposes. In this chapter, inspiration is drawn from a microwave design, which is a
leaky-wave slot antenna sandwiched between two dielectric media. It is shown that such
a structure exhibits ultrawideband directive radiation. The next section discusses the
theoretical aspect of the leaky-mode supported by this structure, and the effect of the
modal characteristics on the radiation pattern and the bandwidth performance. This
knowledge is very instructive for adapting the leaky-slot design for optical applications.
5.2
Theory
The inspiration for this work comes from Neto and Maci’s analysis for a perfect-electricconductor (PEC) slot placed between two semi-infinite homogeneous dielectric half-spaces
[81, 82], such as the one shown in Fig. 5.5. The PEC screen is assumed to have an
infinitesimal thickness and the slot width is very small compared to the wavelength (less
than 1/10λ0 ).
It is shown by Neto and Maci that according to the field equivalence principle, the
slot structure in Fig. 5.5 supports two longitudinal fictitious magnetic currents Im , above
and below the PEC screen, flowing in opposite directions when excited by an elementary
electric dipole polarized transverse to the slot [81]. This electric dipole excitation could
be from an radiating molecule, or nanosize photonic circuits. Its radiation pattern in
each half-space can thus be separately modeled with a surface magnetic current backed
by a conducting plane. For each half-space, the magnetic current distribution can be
assumed to be separable in terms of the space-dependence with respect to y and z, as
Chapter 5. Plasmonic Antennas for Far-Field Sensing
99
Figure 5.5: The schematic of the leaky slot antenna. The upper dielectric is silicon
(ε2 = 11.7) and the lower dielectric is air (ε1 = 1). An electric dipole is placed at z = 0
polarized transverse to the slot.
shown in
long
Im (y, z) = Itran
m (y) Im (z)
(5.3)
The transverse y-dependence has the form
Itran
m (y) = −
2
r
ws π
1
2 ẑ
1 − w2ys
(5.4)
The longitudinal magnetic current Ilong
m (z) is represented via its Fourier spectrum 1/D(kz )
as shown in
Ilong
m (z) =
1
2π
Z ∞
e−jkz z
dkz ẑ
−∞ D(kz )
(5.5)
where
q
q
2
1 X 2
ws
w
s
(2)
2
ki2 − kz2 × H0
ki2 − kz2
(k − kz ) J0
D(kz ) =
2k0 ς0 i=1 i
4
4
(5.6)
In Eqn. 5.6, k0 is the free-space wave number and ς0 is the characteristic impedance. The
(2)
index i corresponds to medium 1 and 2. J0 is the zero-order Bessel function and H0
is the zero-order Hankel function of the second kind. Both quasi-static and asymptotic
Chapter 5. Plasmonic Antennas for Far-Field Sensing
100
long
solutions of Im
are presented in
long
Im
(z) ≈
−jk0 ς0 ws ln(z)
2(k12 − k22 )
(for small z)
(5.7)
(for large z)
(5.8)
LW
long
Im
(z) ∼
e−jkz |z|
e−jk1 |z|
e−jk2 |z|
+
v
+
v
10
20
jD0 (kzLW )
(k1 z)2
(k2 z)2
where D0 (kz ) is the first-order derivative of D(kz ). The coefficients v10 and v20 are given
in Eqns. 29 and 30 in [81], respectively. The quasi-static solution (Eqn. 5.7) is for the
region very close to the source. The asymptotic solution in Eqn. 5.8 consists of the leaky
wave (the first term) and the lateral wave (the last two terms) contributions. Finally,
the radiation pattern in each medium is found by solving the vector potential using the
Green’s function of the respective medium and with the stationary phase method. It
is found that under the small-width approximation (ws < 0.1λ0 ), the leaky-wave phase
constant β LW can be approximated as
β
LW
≈
r
k12 + k22
2
(5.9)
which only depends on the averaged wave numbers of the two half-spaces. For the
air/silicon interface, β LW ≈ 2.52k0 . Hence the leaky slot mode is fast compared to
medium 2 (silicon), but slow compared to medium 1 (air), which leads to radiation
towards medium 2 only. On the other hand, the lateral waves have negligible effects on
the far-field radiation pattern [82].
In this thesis, the derivation of the far-field radiation pattern for the leaky slot structure is significantly simplified. Although less rigorous and accurate compared to the
solution in [82], it will be shown that the results are good approximations compared to
the one obtained from numerical simulations. Furthermore, this method can be used to
calculate the radiation pattern for plasmonic leaky slot at optical frequencies. In this
case, the lateral wave contribution is ignored in the magnetic current distribution. The
long
asymptotic solution of Im
is thus simplified as
−jkz
ẑ
Ilong
m (z) = Im0 e
(5.10)
Chapter 5. Plasmonic Antennas for Far-Field Sensing
101
where Im0 is a constant and k = β + jα is the complex propagation constant of the leaky
tran
mode. The current variation along the transverse direction Im
(y) is neglected and
approximated by a delta function instead. With this magnetic current distribution, the
H-plane far-field pattern can be calculated independently for each half-space, based on the
standard approach for traveling-wave antennas [58]. The E-plane exhibits unidirectional
radiation pattern (for each respective half-space) due to the magnetic current symmetry.
For the example of medium 2, the far-field magnetic field can thus be calculated according
to
Z l
k2 e−jk2 r
0
0
sin θ
2Im0 e−jkz e jk2 z cos θ dz 0
Hθ,2 = j
η 4πr
| 2 {z
} |0
{z
}
element factor
(5.11)
space factor
where the element factor represents the far-field due to a unit length infinitesimal magnetic current, and the space factor accounts for the phase offset due to the spatial current distribution. The factor 2 in the space factor is due to the image theory during
the application of the field equivalence principle. In this equation, k2 and η2 refer to
the propagation constant and the wave impedance of silicon, respectively; l is the slot
length and r is the far-field radiation distance. The angle θ is with respect to the z-axis.
A closed-form expression of the H-field can be obtained by evaluating the integral in
Eqn. 5.11, and the result is shown in
k2 l
k l
(cos θ − K)
2k2 Im0 l e−jk2 r
j 22 (cos θ−K) sin
2
sin θ e
Hθ,2 = j
k2 l
η2
4πr
(cos θ − K)
2
(5.12)
where K = k/k2 is the normalized complex propagation constant with respect to silicon.
Finally, the far-field power density can be obtained from
1
Pθ,2 = η2 |Hθ,2 |2
2
(5.13)
Similarly, the H-plane far-field radiation pattern in medium 1 is calculated from the same
magnetic current distribution (with opposite signs) along z, backed by a PEC screen.
Therefore, it yields the same expression as Eqn. 5.12 by replacing k2 and η2 with the
Chapter 5. Plasmonic Antennas for Far-Field Sensing
4
102
0
−0.1
3
2
α / k0
β / k0
−0.2
leaky−mode
n1 (air)
n2 (silicon)
−0.3
−0.4
1
−0.5
0
0.02
0.04
Ws/λ0
0.06
(a)
0.02
0.04
Ws/λ0
0.06
(b)
Figure 5.6: The normalized complex propagation constant (the effective mode index) of
the leaky mode for an infinite PEC slot placed between silicon and air. The slot width is
fixed at 46 nm, which corresponds to 0.03λ0 at 1550 nm. (a) normalized phase constant
β/k0 . (b) normalized leakage constant α/k0 .
corresponding parameters of medium 1. However, the field magnitude in the two media
are substantially different.
In order to compute the radiation pattern, the complex propagation constant k =
β + jα is obtained through numerical simulations using Comsol Multiphysics R . Twodimensional eigenmode analysis is conducted to obtain the slot mode indices, and the
results are illustrated in Figs. 5.6. In this case, the slot width ws is chosen to be 46 nm,
which corresponds to 0.03λ0 at the nominal wavelength of 1550 nm. The phase constant
β, illustrated in Fig. 5.6(a), is shown to be larger than the wave number in air but smaller
than that of silicon. This agrees with the findings in [82]. The analysis in Fig. 5.6(a)
represents a large wavelength range from 657 nm to 4600 nm. For this wide spectrum, the
normalized β is shown to stay relatively constant to the predicted value of 2.52 by Neto
and Maci. The leakage constant α, as illustrated in Fig. 5.6(b), exhibits more visible
variations. This is because the slot is wider for shorter wavelengths, hence leads to faster
leakage.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
103
The accuracy of the simplified solution is compared to numerical simulations using
ADS Momentum by Agilent Technologies, which is a software based on the method-ofmoments (MoM). ADS Momentum calculates the Green’s function for stratified media,
hence is able to compute the far-field radiation pattern due to a surface current at the
interface of two dielectric half-spaces. Comsol Multiphyscis R , on the other hand, is
unable to compute the far-field for heterogeneous half-spaces. The 3D directivity pattern
for the PEC slot with ws = 46 nm is shown in Fig. 5.7(a), where the transverse electricdipole excitation is located at z = 0. The slot length l is chosen to be 5λ0 in order to
approximate an infinite length. The corresponding H-plane radiation pattern is shown
in Fig. 5.7(b), where the beam angle θ is defined with respect to the z-axis.
14
12
7.5
10
5
8
7.5
5
10
0
0
5
2.5
0
5
6
4
2
10
z
−10
2.5
10
5
0
0
0
2.5
5
−5
7.5
θ
x
x
2.5
y
0
−5
7.5
10
−10
y
z
a)
b)
Figure 5.7: The simulated directivity of the PEC slot antenna placed between silicon and
air. The schematic of the structure is illustrated in Fig. 5.5. Both a) 3D directivity and
b) H-plane direcitivity (θ is the far-field beam angle) have linear scale.
By combining the expression in Eqn. 5.13 and the results of the complex propagation
constant in Fig. 5.6, the H-plane radiation pattern of the leaky-wave slot can be calculated. Furthermore, the directivity can be computed from just the H-plane radiation
pattern because of the symmetry in the E-plane. The H-plane directivity is illustrated in
Chapter 5. Plasmonic Antennas for Far-Field Sensing
104
Fig. 5.8, which demonstrates that the result obtained from the MoM simulation is similar
to the one obtained using the analytical expression in Eqn. 5.13. More specifically, the
maximum directivity and the beam angle obtained from the analytical solution are 13.4
and 40◦ respectively, and the ones from Momentum simulations are 14.2 and 40.2◦ , respectively. This result demonstrates that under the small width condition, the analytical
solution based on the simplified magnetic current distribution leads to reasonably good
approximation of the far-field radiation pattern and the directivity.
90
15
60
120
10
30
150
5
180
0
Momentum simulation
Analytical solution
210
330
240
300
270
Figure 5.8: The H-plane directivity for an infinite PEC slot placed between air and silicon
(linear scale).
The most unique and potentially very useful characteristic of this leaky-slot structure
is that the beam angle remains unchanged over a wide spectrum. This is because the
beam angle of leaky-wave antennas is determined by the phase constant β [70]. The
relationship in
β
≈
cos θ =
k2
r
ε1 + ε2
2ε2
(5.14)
demonstrates that the angle θ only depends on the electric permittivity of the two homogeneous half-spaces since β is averaged between the two wave numbers as previously
Chapter 5. Plasmonic Antennas for Far-Field Sensing
105
shown in Eqn. 5.9. In order to demonstrate the wideband performance of the leaky-wave
slot, the normalized directivity pattern is calculated when the wavelength is varied from
657 nm to 4600 nm, which corresponds to a slot width ws that ranges from 0.07λ0 to
0.01λ0 , respectively. The results shown in Fig. 5.9 illustrate a small variation in the
beam angle θ and a more noticeable variation in the 3dB beamwidth. The results are
summarized in Table 5.1.
90
120
60
1
150
30
0.5
180
0
W /λ = 0.01
s
0
W /λ = 0.03
210
s
330
0
W /λ = 0.07
s
0
240
300
270
Figure 5.9: The normalized H-plane directivity pattern for an infinite PEC slot placed
between air and silicon for various slot widths.
ws /λ0
λ0
θ
3dB beamwidth
0.01
4600 nm
41.6◦
13.1◦
0.03
1550 nm
40.2◦
18.6◦
0.07
460 nm
38.5◦
26.8◦
Table 5.1: Wavelength dispersion of the leaky slot antenna (theory).
Chapter 5. Plasmonic Antennas for Far-Field Sensing
5.3
106
Design
An optical adaptation of the leaky-slot antenna is proposed next for facilitating efficient
illumination and collection of single molecule emission. For the example illustrated in
Fig. 5.10, various single molecules that resonate at different wavelength can be illuminated
with a broadband signal. They radiate in the same direction towards the substrate,
where the signal can be collected for spectral analysis. This broadband property is very
attractive since most of the plasmonic antennas (such as Yagi-Uda antennas, nano-rods
and nano-spheres), as well as grating structures are very narrowband. for The electric
permittivity of metals is significantly different at optical frequencies compared to lower
frequencies, as discussed in Section 2.1. Therefore, it is expected that the propagation
characteristics of the leaky slot mode would deviate from theoretical analysis where the
metal is assumed to be PEC. Therefore, more accurate numerical analysis is required to
extract the propagation constant β and the attenuation constant α, from which the angle
of the leaky-wave radiation and beamwidth can be derived.
Figure 5.10: A potential application of the plasmonic leaky-slot antenna.
Before analyzing and designing the plasmonic leaky slot (PLS) placed between air and
silicon, it is instructive to first discuss the plasmonic slot waveguide (PSW) embedded in
Chapter 5. Plasmonic Antennas for Far-Field Sensing
107
air only, as this helps with gaining intuition regarding how the slot mode behaves in the
presence of an additional half-space. The fundamental mode of the plasmonic slot is a
coupled surface mode with additional field components along the longitudinal direction,
as opposed to the TEM mode in the PEC slot waveguide. This is commonly known as the
gap mode of negative-dielectric waveguides [72, 83]. The surface plasmon mode is known
to slow down significantly when operating close to the spectral region of the transverse
resonance condition (i.e. |εmetal | = εdielectric ) [7, 83, 84]; therefore, it is expected that the
propagation constant β increases.
The increased propagation constant could pose challenges in an PLS structure, where
either a superstrate or a substrate is present. For the example of the leaky-slot antenna,
if the slot mode becomes slower than the propagation in the dense medium, the energy
will be guided along the slot instead of being radiated. In fact, PSWs consisting of
two dielectric half-spaces have been reported previously [85], which have near identical
geometries as the proposed PLS antenna. Therefore, the key to designing operational
PLS antennas is to control the propagation constant, such that the slot mode is faster
than the radiation in the dense medium but slower than the one in air. In this design, it
is achieved by choosing the type of metals with appropriate electric permittivity at the
frequencies of interest.
For this purpose, aluminium has been chosen for implementing the PLS antenna.
As previously illustrated in Fig. 2.1(a), metals such as aluminium, gold and silver all
have a negative permittivity at optical frequencies. A negative permittivity with a larger
magnitude implies that the transverse resonance condition takes place at a shorter wavelength, which is the case for aluminium when compared to gold and silver. Therefore,
the fundamental mode (a forward even mode) in an aluminium slot propagates closer to
the light line in the spectral regime around 1550 nm. That is, the mode guided by an
aluminum slot is slower than that of a PEC slot, but faster than those of gold or silver.
For the previously reported PSW in [85] consisting of near identical geometries as the
Chapter 5. Plasmonic Antennas for Far-Field Sensing
108
proposed PLS antenna, silver is employed at 1550 nm to implement the waveguide. In
this section, it is shown that the aluminium slot printed on silicon substrate can support
fast slot modes that lead to highly directive radiation over a wide spectrum.
In order to analyze the radiation properties of the PLS antenna, 3D FEM simulations
using Comsol Multiphysics R are employed to extract the dispersion characteristics of
(β + jα)/k0 for the plasmonic slot mode. The results are plotted in Fig. 5.11. The results
for the PEC slot are also plotted for comparison. From this figure, three important
differences between the PEC leaky-wave antenna and the PLS antenna can be concluded.
First, the phase constant β of the PLS antenna cannot be accurately predicted by the
approximation in Eqn. 5.14 because it does not account for the wavelength dispersion of
the metal. Fig. 5.11(a) shows that besides having a larger phase constant β, the PLS
antenna does not possess a near-linear relationship between β and λ0 , such as the one
seen for a PEC leaky-wave slot. This leads to more visible beam-squinting, i.e. scanning
of the radiation beam angle with frequencies. Second, the PLS antenna has a lowerbound cutoff wavelength that is absent in its PEC counterpart. For the specific design
of a slot that is 46 nm thick and 46 nm wide on silicon, the cutoff wavelength is found to
be 1100 nm from Fig. 5.11(a). The effective mode index increases beyond the refractive
index of silicon (the dashed horizontal line) below 1100 nm, which indicates a guidedmode characteristic. Finally, the attenuation constant α in Fig. 5.11(b) consists of heat
dissipation in the metal in addition to the radiation loss. This additional loss mechanism
decreases the efficiency of the antenna since not all energy is converted to radiation. The
dissipation rate is more pronounced at shorter wavelengths because the mode penetrates
deeper into the metal. The efficiency of the PLS antenna due to these loss mechanisms
will be calculated analytically in Section 5.4.2.
The effects of these differences are reflected in the radiation patterns shown in Fig. 5.12.
In Fig. 5.12(a), the larger phase constant β of the PLS antenna leads to a smaller beam
angle compared to its PEC counterpart at the same wavelength. Additionally, as illus-
Chapter 5. Plasmonic Antennas for Far-Field Sensing
4
109
−0.3
PEC slot
plasmonic slot
n (silicon)
3.8
2
3.6
−0.35
α/k0
β/k0
3.4
3.2
−0.4
3
2.8
−0.45
PEC slot
plasmonic slot
2.6
2.4
1000
1200
1400
1600
λ (nm)
1800
2000
−0.5
1000
0
(a)
1200
1400
1600
λ (nm)
1800
2000
0
(b)
Figure 5.11: Comparison of the normalized complex propagation constant for the PEC
and plasmonic slot mode. For both cases, the slot is infinitely long and placed between
air and silicon. The aluminum slot is 46nm thick and 46nm wide. The PEC slot is
40 nm wide and infinitesimally thin. (a) normalized phase constant β/k0 ; (b) normalized
attenuation constant α/k0 .
trated in Fig. 5.12(b), the nonlinearity between β and λ0 results in the beam-squinting
when the wavelength is varied. The 3 dB beamwidth also increases visibly at shorter
wavelengths as shown in Fig. 5.12(c). This is due to the faster mode attenuation caused
by the additional metallic loss.
5.4
Discussion
5.4.1
Plasmonic leaky-wave slot mode
It is instructive to discuss the validity of the simplified theoretical approach outlined
in Section 5.2 for the case of plasmonic leaky-slot antennas. In Section 5.2, the field
equivalence principle is applied based on the assumption of PEC screen, which implies
that the tangential electric field only exists in the slot and is zero everywhere else along
the screen. This assumption is necessary for both the rigorous derivation in Neto and
Chapter 5. Plasmonic Antennas for Far-Field Sensing
120
90
1
110
45
60
0.5
150
θ (deg)
40
30
35
30
(b)
25
0
PEC slot
plasmonic slot
210
240
330
300
270
(a)
3dB Beamwidth (deg)
180
35
PEC slot
plasmonic slot
30
25
20
(c)
15
1200
1400
1600
λ0 (nm)
1800
2000
Figure 5.12: The comparison of the radiation pattern for an infinite PEC leaky slot and
a PLS antenna. a) The radiation pattern at the nominal wavelength of 1550 nm; b) The
beam angle variation (squinting) with wavelength; c) The 3 dB beamwidth fluctuation
with wavelength.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
111
Maci’s work and the approach in this thesis. Additionally, a TEM mode is supported in
the PEC slot, where only Ey is present. This leads to the magnetic current polarized in
the z-direction only according to Jm = −n̂ × E. Furthermore, it is also assumed that Im
along the transverse direction (y-axis) can be approximated with a delta function, and
its distribution along the longitudinal direction consists only the exponentially decaying
leaky-wave component.
The validity of approximating these conditions is demonstrated by comparing the
PEC and plasmonic (aluminium) leaky slot at λ0 = 1550 nm. Both structures are simulated using Comsol Multiphysics R under a delta current excitation at z = 0 polarized
transverse to the slot. The PEC slot is infinitesimally thin and has a gap width of 46 nm,
whereas the PLS has a slot width and thickness of 46 nm. Both field distributions along
the longitudinal and transverse directions are plotted in Fig. 5.13. In the longitudinal
direction, both structures exhibit exponential decay. The sharp increase close to z = 0 is
due to the dipole current excitation, which has been investigated in [81]. In the transverse
direction, the PEC slot only has Ey component within the gap width, whose distribution
closely matches with the analytical solution in Eqn. 5.4. Conversely, the PLS antenna
supports a quasi-TM mode, where all three components of the electric field are present.
However, Ey significantly dominates, which implies that the magnetic current is predominantly polarized in the z-direction, as assumed in Section 5.2. Finally, this quasi-TM gap
mode is a coupled surface mode, which penetrates into the metal. However, the overall
field profiles demonstrate strong concentration around the gap region, where the fields
outside of y ∈ (−0.05λ0 , 0.05λ0 ) is close to zero. This demonstrates that approximating
the transverse magnetic current distribution with a delta function, as well as applying
the field equivalence principle is valid.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
112
0
−10
0.8
−20
0.6
|Ey|
|E| (dB)
1
0.4
−30
0.2
−40
0
0
0.1
0.2
0.3
0.4
0.5
−0.2
−0.1
z (λ0)
(a) PEC slot (longitudinal)
0
y (λ0)
0.1
0.2
(b) PEC slot (transverse)
0
|Ey|
1
|E |
z
0.8
|Ex|
0.6
−20
|E|
|E| (dB)
−10
0.4
−30
0.2
0
−40
0
0.1
0.2
0.3
0.4
z (λ0)
(c) Al slot (longitudinal)
0.5
−0.2
−0.1
0
y (λ0)
0.1
0.2
(d) Al slot (transverse)
Figure 5.13: Comparison of the field distribution for the PEC and plasmonic (Al) leaky
slot antennas. The fields along the (a) longitudinal and (b) transverse directions of
the PEC leaky-slot antenna. The fields along the (a) longitudinal and (b) transverse
directions of the PLS antenna. The gray rectangular region in (b) and (d) indicates the
width and location of the slot.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
5.4.2
113
Antenna efficiency
Although theoretically it takes an infinitely long leaky-wave slot to radiate all the input
energy, the exponential decay of the slot mode along the z -direction suggests that a
PLS antenna with a short finite length could be very effective to approximate an infinite
slot. Due to the presence of the metallic loss, a portion of the energy is dissipated as
heat. The antenna efficiency, defined as ηeff = Prad (L)/Pin × 100, is calculated based
on the attenuation constant extracted from simulations for the lossy and the lossless
systems. For example, at the nominal wavelength λ0 = 1550 nm, the total and radiative
attenuation constants are αt = −0.370 k0 and αr = −0.207 k0 , respectively, which implies
that the attenuation constant due to dissipation is αd = −0.163 k0 .
In order to calculate Prad (L), the power distribution along the slot P (z) due to the
above attenuation constants are plotted in Fig. 5.14. In this case, the red solid line
100
Total attenuation
Radiative attenuation only
Dissipated attenuation only
90
80
60
in
P(z) / P (%)
70
50
40
30
20
10
0
0
∆z
0.5
1
1.5
z (λ )
0
Figure 5.14: Power decay along the PLS antenna due to the total, radiated and dissipated
attenuation constants, respectively.
represents the power remained in the lossy PLS, i.e. P (z) = Pin e2αt z ; and the blue
dashed line represents the power remained in the slot when the PLS is lossless, i.e.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
114
P (z) = Pin e2αr z . Next, the antenna length is discretized into L = N ∆z, where ∆z is
assumed to be infinitesimally small. Therefore, the total radiated power for a PLS of a
particular length L can be written as the sum of the radiative attenuation from each ∆z
as shown in
Prad (L) = lim
∆z→0
= lim
∆z→0
= lim
∆z→0
N
−1
X
n=0
N
−1
X
n=0
"
P (n∆z) 1 − e2αr ∆z
Pin e2αt n∆z 1 − e2αr ∆z
Pin 1 − e
−1
X
N
2αr ∆z
n=0
e
2αt ∆z n
#
(5.15)
Recognizing the summation in Eqn. 5.15 is the sum of a geometric series, the antenna
efficiency can then be written as
ηeff =
Prad (L)
× 100
Pin
"
= lim
∆z→0
#
2αt ∆z N
1
−
e
1 − e2αr ∆z
× 100
1 − e2αt ∆z
(5.16)
Applying L0 H ôpital0 s rule, the analytical expression for the antenna efficiency is obtained
ηeff = 1 − e2αt L
1 − 2αr
× 100
1 − 2αt
(5.17)
Similarly, the dissipated power percentage can be calculated from
1 − 2αd
Pdis (L)
× 100 = 1 − e2αt L
× 100
Pin
1 − 2αt
(5.18)
The results for the total, radiated and dissipated power percentage with respect to the
input are shown in Fig. 5.15, where the antenna efficiency can be extracted from the blue
dashed line. It is shown that it takes an antenna length of 0.5λ0 to lose 90% of the input
power; among this total loss, 50.5% is due to radiation and 39.8% due to dissipation.
Additionally, in order to lose 99% of the input power, the PLS antenna should be at least
one wavelength long. In this case, the radiation efficiency is 55.4% and the dissipation is
43.7%. Finally, a PLS with L = 1.5λ0 attenuates 99.9% of the input power, which can be
Chapter 5. Plasmonic Antennas for Far-Field Sensing
115
100
90
Ploss(L) / Pin (%)
80
70
60
50
40
30
Total loss
Radiation
Dissipation
20
10
0
0
0.5
1
1.5
L (λ )
0
Figure 5.15: The percentage of the total, radiative and dissipated power loss with respect
to the input.
regarded as equivalent of an infinite slot. Note that the unattenuated power is reflected
back towards the source, hence it is important to design a slot that can approximate an
infinite one.
The directivity patterns in the H and E planes are plotted in Figs. 5.16(a) and 5.16(b),
respectively, for a PLS antenna of three different lengths. It is shown that the 1.5λ0 PLS
produces a maximum directivity of 11.3 at a beam angle of 32◦ in the H-plane, with
a 3 dB beamwidth of 23.8◦ . The E-plane exhibits ominidirectional radiation within the
half-space of higher refractive index. These results approximate the ones of an infinite
slot.
It is instructive to note that we choose to analyze the spectrum ranging from 1200 nm
to 2000 nm due to the availability of data for the electric permittivity of aluminum. This
wavelength range represents a spectrum with 50% bandwidth centered at 1550nm. This
bandwidth is well beyond the definition of ultrawideband antennas. Additionally, it is
very important to point out that although the analysis is limited by an upper-bound
wavelength of 2000 nm due to the data availability for εr , this leaky-wave antenna could
Chapter 5. Plasmonic Antennas for Far-Field Sensing
90
116
12
120
60
10
8
6
150
30
4
2
180
0
L = 0.5λ0
L = 1.0λ
0
L = 1.5λ0
210
240
330
300
270
(a) H-plane
90
1.6
60
120
1.2
0.8
150
30
0.4
180
0
210
330
240
300
270
(b) E-plane
Figure 5.16: The H and E-plane directivity of the PLS antenna operating at 1550 nm for
three different antenna lengths (linear scale).
Chapter 5. Plasmonic Antennas for Far-Field Sensing
117
operate well beyond that limit at longer wavelengths, although it cannot overcome its
lower-bound cutoff. In fact, the PLS antenna will follow the behavior of its PEC leakywave counterpart closer at longer wavelengths because they are further away from the
plasma frequency. The slot width is optically smaller at longer wavelengths, which leads
to an even higher directivity. This trend can be seen from Figs. 5.12(b) and 5.12(c), where
both the beam angle and the 3 dB beamwidth of the PLS antenna approach its PEC
counterpart as the wavelength increases. This is a significant bandwidth enhancement
compared to existing traveling-wave antennas at optical frequencies. For example, the
leaky plasmonic-sphere chain only produces a directive beam within the range of 680 nm
to 740 nm, and the bandwidth for the optical Yagi-Uda antenna is only 150 nm centered
at around 800nm [30, 76].
Finally, the benefit of the proposed structure is discussed from the perspective of
practical implementations. The wideband behavior makes the PLS antenna less prone
to fabrication imperfections. This is because as the slot dimension changes with respect
to the frequency, the antenna’s radiation pattern stays relatively constant. One example
of the sensitivity analysis is illustrated in Fig. 5.17, where the thickness of the aluminum
is varied by 10% with respect to its nominal value of 46 nm. It is shown that this
10% variation does not change the radiation pattern much in terms of both the beam
angle and the 3 dB beamwidth. This is especially attractive for circumventing fabrication
imperfections at nanoscales.
Chapter 5. Plasmonic Antennas for Far-Field Sensing
118
90
120
1
60
0.8
0.6
150
30
0.4
0.2
180
0
41nm
46nm
51nm
210
240
330
300
270
Figure 5.17: The effect of ±10% deviation in the aluminum film thickness on the radiation
pattern (λ0 = 1550 nm).
Chapter 6
Summary and Future Work
The current trend of optics and photonics research is to move towards the nanoscale,
where structure dimensions are much smaller than the wavelength of the light with which
they interact with. The previous fundamental limitation, known as the diffraction limit,
no longer restricts the optical resolution that can be imaged, nor it hinders the size of
the photonic circuits that one can build. One approach that enables us to access the
subdiffraction modal size is to utilize surface plasmons supported in metallo-dielectric
systems. Among many surface-plasmon-based devices, plasmonic antennas present some
of the most promising opportunities to link localized energy with free-space propagation,
which leads to efficient observation using light, as well as manipulation of light in the
subdiffractional dimensions. To study the properties of optical antennas composed of
plasmonic infrastructures and adapt antenna designs from microwave frequencies for
optical applications have become the main subject of this thesis.
This thesis begins with the analysis of the properties of metals at optical frequencies,
which primarily discusses the dispersion of electric permittivity. These analyses reveal
that metals, such as gold, silver and aluminium, do not behave like good conductors in
the optical spectrum. Instead they exhibit negative electric permittivity. The theory
of negative εr (ω) is discussed from the perspective of classical kinetics, from which two
119
Chapter 6. Summary and Future Work
120
analytical models, namely the Drude and Lorentz models, are derived. Additionally,
several experimental data from various researchers are also presented and compared to
the analytical models.
Next, the effect of negative electric permittivity on the physical dimension of plasmonic antenna structures is discussed. For the example of a half-wavelength dipole
antenna, the problem is approached by solving the fundamental TM mode of a cylindrical waveguide, from which the dispersion of the mode index is extracted. It is concluded
that the cylindrical rod antenna exhibits dipolar resonance at half of the effective wavelength (λeff ), which scales nearly linearly with the free-space wavelength in the visible
and near-infrared regions. However, the slope is much smaller than one. This implies
that the physical length of the plasmonic dipole antenna is much shorter than the conventional half-wavelength. Additionally, the antenna radius can no longer be regarded as
infinitesimally narrow at optical frequencies, which affects the dispersion of the effective
wavelength. Finally, FEM simulations are utilized to verify the theoretical prediction
of λeff . The resonance wavelength for the slot antenna is also extracted, and the result
confirms the duality principles for complementary antenna structures.
With the ability to design resonant slot antennas in the optical regime, opportunities to manipulate an array of slot antennas is investigated. With the inspiration of
microwave metascreens, a linear array of plasmonic slot antennas are used to achieve
one-dimensional superfocusing beyond the extreme near-field. The fundamental concept
involved in this type of superfocusing technique is to manipulate the evanescent waves,
such that a desired subdiffraction spot size can be constructed owing to destructive interference. The theoretical approaches used to obtain the interference pattern at the
transmission screen encompass two methods, namely the back-propagation method and
the shifted-beam theory. Both designs start with the desired image function, which is
a Gaussian function with subdiffractional beamwidth, the spatial field distribution at
the transmission plane is then derived analytically. Both analyses show that a non-
Chapter 6. Summary and Future Work
121
periodic function with subwavelength spatial oscillation, where the phase offset between
the neighboring maxima and minima is 180◦ , is the key feature that leads to superfocusing. Finally, a plasmonic meta-screen consisting of a linear array of gold slot antennas on
glass substrate is designed, which demonstrates the one-dimensional superfocusing effect
at 940 nm, with a focal length at 0.1λ0 away from the transmission screen. The design
is shown to reduce the FWHM of a single slot antenna radiation by 25.5%. One useful
extension of the shifted-beam design is to incorporate additional satellite antennas, such
that a narrower beamwidth can be achieved at the same image plane location, or the
same beamwidth can be maintained when extending the imaging distance. The current
approach becomes cumbersome when a large number of parameters (i.e. the lengths of
all the slot antennas) need to be optimized. Therefore, it is instrumental to develop an
optimization method that is able to handle a large number of parameters efficiently.
Furthermore, a rotationally-symmetric monopole antenna-array is also designed to
achieve two-dimensional superfocusing at 514 nm. This design draws inspiration from a
microwave two-dimensional near-field probe. The difficulty of adapting such a design to
the optical domain, in particular the excitation of monopole antennas due to the lack
of optical equivalent coaxial structures, is discussed. A new excitation scheme based on
conventional NSOM aperture probes is discussed and shown capable of exciting monopole
antennas in the visible spectrum. This topology is adapted and modified for exciting the
monopole antenna array structure. The destructive interference phenomenon is observed
when the central and satellite array elements radiate out of phase, which again confirms
the prediction of the shifted-beam theory. It is shown that this near-field probe is able
to extend the focal length to a quarter wavelength away from the transmission screen.
It not only reduces the FWHM of a single monopole antenna radiation by almost 40%,
the destructive interference also helps to suppress the background signal tremendously.
These designs are potentially very attractive for near-field optical probes because of the
improved signal throughput, resolution, as well as imaging distance. Since the monopole
Chapter 6. Summary and Future Work
122
array is designed based on conventional NSOM probes, it is compatible with the existing
near-field scanning instrumentaion.
In addition to the near-field characteristics, far-field properties, such as the radiation
directivity of plasmonic antennas, are also exploited. In particular, a leaky-wave slot
antenna design, consisting of an optically narrow slot sandwiched between two dielectric
half-spaces is analyzed. It is shown that the fundamental slot mode has a mode index that
is the average between the refractive index of the two dielectrics. Therefore, the mode is
slow compared to lower index medium, but fast compared to the higher index one. As a
result, the slot serves as a leaky-wave antenna rather than a waveguide structure. The
dispersion of the slot mode index is analyzed over a very wide spectrum when a perfectelectric-conductor screen interfaces air and silicon, as well as when the PEC screen is
replaced with aluminium. The leaky-wave radiation pattern is calculated theoretically
based on the mode index under a small-width assumption. The result reveals a highly
directive radiation pattern, where the radiation angle is relatively constant over a very
wide spectrum. It is also shown that there is a cutoff for the plasmonic leaky-slot at
around 1100 nm, below which the fundamental mode becomes a guided mode. However,
an upper bound of the operating wavelength does not exist. The wideband characteristic
has been the most unique property of this plasmonic antenna design, which makes it very
attractive for interfacing nanoscaled localized energy and free-space propagation.
One interesting avenue for future exploration is to investigate the possibility of coupling light from air to the leaky-wave slot antenna, as this approach would enable illumination of nanoparticles and energy coupling with plasmonic circuits directly from
free-space. As discussed in Chapter 5, the leaky-mode has a propagation constant that
depends on the average of the electric permittivity of the two half-spaces. Therefore, the
key to enable free-space radiation is to employ a dielectric half-space with its permittivity
less than unity. Some natural materials behave as epsilon near zero (ENZ) media close to
their plasma frequencies, above which the permittivity is positive and smaller than unity
Chapter 6. Summary and Future Work
123
according to the Drude model in Eqn. 2.1. For example, polar dielectric materials, such
as silicon carbide (SiC), have their plasma frequency in the terahertz regime, which could
potentially be used as the dielectric half-space with εr < 1 at the optical and infrared
spectra [86]. Additionally, artificial dielectrics, also known as metamaterials, can be engineered at a desired frequency by embedding suitable inclusions in a host medium. One
implementation proposed by Brown in 1953 suggests that periodically arranged metallic
wires can be used to achieve such an artificial dielectric medium [87]. This concept was
proven analytically assuming the metals are PECs. This structure has recently been
revisited extensively for metamaterial implementations at various frequency ranges. As
suggested by Garcia et al. [88], a simple approximation of the effective permittivity can
be expressed as
ε = f εm + (1 − f )εd
(6.1)
where f is the metal volume filling factor, εm and εd represent the permittivities for
the metal and dielectric constituents, respectively. For natural noble metals, such as
gold and silver, the plasma frequency is in the ultraviolet region. Eqn. 6.1 indicates
that by changing the filling factor, it is possible to achieve new materials with plasma
frequencies much lower than pure metals. Extensive research regarding metamaterials
in the past decade has presented many possibilities to improve the present PLS antenna
design. However, it is instructive to investigate the dispersion properties of potential
metamaterials in the frequency range of interest, since many existing implementations
are narrowband.
Appendix A
Metals at Optical Frequencies
A.1
Analytical models
A.1.1
The Drude (free-electron) model
The simplest classical model that describes the optical properties of metals is derived
from the free-electron theory, known as the Drude model [32]. Metals, unlike insulators
and semiconductors, have either an energy band partially filled with electrons or a filled
band overlaps in energy with an empty band. The availability of vacant electron states in
the same energy band facilitates low-energy photon absorption, known as the intraband
absorption. the motion of electrons due to the intraband effect is therefore described as
the free-electron motion shown in
..
.
mx + bx = eE
(A.1)
where m is the effective electron mass, and the f rictional dissipation of the electron
energy due to ionic collisions is characterized with the constant b.
Assuming the electric field is a time harmonic oscillator with angular frequency ω,
Eqn. A.1 can be rewritten as
− ω 2 x + jγωx =
124
e
E
m
(A.2)
Appendix A. Metals at Optical Frequencies
125
where γ = mb represents the average collision frequency. The electron displancement is
then found by solving Eqn. A.2 and has the following form
x=−
e
E
m
ω 2 − jγω
(A.3)
In bulk metals, the macroscopic polarization P = np = nex, where n is the electron
density of the metal. From Eqn. A.3, we compute this polarization to be
where ωp =
q
ωp2
P =− 2
ε0 E
ω − jγω
(A.4)
ne2
is the plasma frequency. From the constitutive relation D = ε0 E + P ,
mε0
the relative electric permittivity of the free-electron model is shown to have the following
form:
ωp2
εr (ω) = 1 − 2
ω − jγω
(A.5)
The typical values of ωp for metals is on the order of 1015 rad/s (in the ultraviolet range),
whereas the collision frequency γ is about 1013 rad/s (in the infrared spectrum).
Inspecting the effective permittivity of metals at different frequency range can give us
insight into the characteristics of metals at various spectra. The discussion is divided in
three different ranges, which are the low frequency limit (ω γ, e.g. microwave), high
frequency limit (γ ω < ωp , e.g. optical), and the ultra high frequency limit (ω > ωp ,
e.g. ultraviolet).
At low frequencies (ω γ), such as the microwave spectrum, the electric permittivity
in Eqn. A.5 can be approximated as the following,
εr (ω) ≈ 1 − j
ωp2
ωp2
≈ −j
γω
γω
(A.6)
The approximation shows that εr is reduced down to a large imaginary number. In fact,
this imaginary εr explains the well-known skin effect in metals at microwave frequencies
because it is due to the complex propagation constant as shown
s
r
ωp2
√
|εr |
k = k0 εr = k0
(1 − j) ≈ k0
(1 − j)
2
2γω
(A.7)
Appendix A. Metals at Optical Frequencies
126
The large imaginary part of k indicates substantial attenuation for propagating waves
in bulk metals within a small penetration distance. Therefore, even a very thin layer of
metal can effectively shield microwaves.
At optical frequencies (γ ω < ωp ), the relative electric permittivity of metal is
negative but predominantly real as shown in
εr (ω) ≈ 1 −
ωp2
ω2
(A.8)
This leads to a purely imaginary k, which indicates that although propagation through
bulk metal leads to tremendous reflection, there is a possibility of longitudinal surfacewave propagation (such as the surface plasmons) under the transverse resonance condition.
At frequencies above the bulk plasma frequency ωp , εr becomes real and positive. This
leads to a real propagation constant k that allows the penetration of electromagnetic
waves through metals at extremely high frequencies (e.g. the X-rays). Most metals
become transparent and loose their shielding ability in this spectrum.
A.1.2
The Lorentz (bound-electron) model
In addition to the intraband absorption, some metals also exhibit significant interband
absorption [32]. In contrast to the intraband absorption, interband absorption happens
when high-energy photons excite electrons across the bandgap to a higher energy state.
Bandgaps commonly exist in nonconductive matrials where bound-electrons dominate.
Similar to the Drude model, the Lorentz model is also a classical microscopic representation of the electron motion, but modified to describe a harmonic oscillator for
bound-electrons. The equation of motion for such an oscillator is
..
.
mx + bx + Kx = eE
(A.9)
where the additional term Kx represents the “spring-like” restoring force due to atomic
attraction. The macroscopic optical constants can be derived similarly as for the Drude
Appendix A. Metals at Optical Frequencies
127
model in Section A.1.1. The relative electric permittivity is shown to have the following
form:
εr (ω) = 1 +
ωp2
ω02 − ω 2 + jγω
(A.10)
where ω02 = K
is the resonance frequency of the harmonic oscillator.
m
The frequency dependence of the electric permittivity is also analyzed at various
ω2
spectra assuming that γ ω0 . At low frequencies (ω ω0 ), εr ≈ 1 + ωp2 is a positive
0
real number. This leads to a real propagation vector k. At high frequencies (ω0 ω2
ω < ωp ), εr ≈ 1 − ωp2 , which is the same result as the free-electron model. The most
notable behavior of the Lorentz model happens when ω is of close vicinity of ω0 . In this
ω2
region, εr ≈ −j γωp is a large imaginary number similar to the low-frequency free-electron
model. This explains the sudden and substantial increase of losses around the resonance
frequencies. Metals usually have several resonance frequencies over a wide spectrum. For
gold, the most noticeable one is at around 410 nm (f = 7.18 × 1014 Hz); and the one for
alumnium is at around 804 nm (f = 3.7 × 1014 Hz) [33].
A.1.3
The Lorentz-Drude (LD) model
Since both the intraband and interband absorptions occur in metals, a composite dielectric function is used to account for polarizations due to both the free-electron and the
bound-electron contributions [32]. More specifically, it can be represented as a superposition of the Drude and the Lorentz models shown in Eqn. A.11. (The Lorentz model
contains multiple oscillators representing interband transitions at various frequencies.)
bound
εr (ω) = εfree
(ω)
r (ω) + εr
=1−
X
ωp2 n
ωp2 e
+
ω 2 − jγe ω
ω02 n − ω 2 + jγn ω
N
(A.11)
where the subscript e indicates the free-electron parameters and N is the number of
oscillators included in the Lorentz multiple-oscillator model.
Appendix A. Metals at Optical Frequencies
128
It is worth noting that the Lorentz multiple-oscillator model is sometimes represented
in its semi-quantum form instead of the classical representation [32,33]. More specifically,
X
fab ωp2 n
f0 ωp2 e
+
εr (ω) = 1 − 2
2
− ω 2 + jγn ω
ω − jγe ω
ω0ab
N
(A.12)
where ω0mn is the energy difference (divided by h̄) between the initial state a and excited
state b, and γn represents the probabilities of transition to all other quantum states. The
oscillator strengths fab represent the probability of excitating an electron from state a to
b. Despite that the parameters in the semi-quantum expressions take on very different
interpretation compared to their classical counterparts, the general forms of the two
expressions are very similar. This allows for analyzing the optical contants of metals
using the intuitive and much simpler classical approach with a good degree of accuracy
to the quantum effects. The parameters of the composite dielectric function are often
fitted to experimental data sets in order to extract the desired analytical models for the
metal of interest.
A.2
Empirical data and parameterization
In order to collect the optical response over a large spectrum, the empirical data for a
single metal often comprises measurements by a number of investigators with various
techniques. Due to many experimental complications, there is often a large spread in the
reported values, and the agreement at the spectrum junctions is also rare. The collections
presented in [35] are based on high-purity samples and relatively good agreement at the
junction wavelength between different data sets. One example of such a data set is shown
in Fig.A.1 for silver, which consists of four subsets of refractive indices collected from
different experiments at different wavelengths. Collections of the optical constant for
gold and aluminium are also presented in [35].
However, the experimental data shown in [35] only presents the original data. In order
to obtain a smooth dispersion function over several data sets, parametrized analytical
Appendix A. Metals at Optical Frequencies
129
2
10
0
n(ω) and k(ω)
10
n(ω) Hagemann
n(ω) Leveque
n(ω) Winsemius
n(ω) Dold
k(ω) Hagemann
k(ω) Leveque
k(ω) Winsemius
k(ω) Dold
−2
10
−4
10
−6
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
wavelength (µm)
Figure A.1: The optical constant of Ag collected from four experiments at various spectral
range [35]. The parameters n and k refer to the real and the imaginary component of
the complex refractive index.
models, such as the Lorentz-Drude model, are employed to fit the measurement data
[33, 36, 89]. For the interest of the visible and infared spectra, the Lorentz-Drude model
with parameters specified in [33] shows very good agreement with experimental data sets
as illustrated in Figs. A.2, A.3 and A.4.
Appendix A. Metals at Optical Frequencies
0
−100
−20
−200
−40
εr (imaginary)
0
εr (real)
−300
−400
−60
−80
−500
−100
−600
−120
−700
−140
−800
130
Leveque
Winsemius
Dold
LD model
−160
3
3
10
λ0 (nm)
10
λ0 (nm)
Figure A.2: The comparision of the complex εr of Ag between the experimental datasets
from [35] and the parameterized LD-model.
0
−100
−20
−200
−40
εr (imaginary)
0
εr (real)
−300
−400
−60
−80
−500
−100
−600
−120
−700
−140
−800
3
10
λ0 (nm)
−160
Theye
Dold
LD model
3
10
λ0 (nm)
Figure A.3: The comparision of the complex εr of Au between the experimental datasets
from [35] and the parameterized LD-model.
Appendix A. Metals at Optical Frequencies
0
131
0
−200
−100
−400
−200
εr (imaginary)
εr (real)
−600
−800
−1000
−1200
−300
−400
−500
−1400
−600
−1600
−700
−1800
−2000
3
10
λ0 (nm)
−800
Palik
LD model
3
10
λ0 (nm)
Figure A.4: The comparision of the complex εr of Al between the experimental datasets
from [35] and the parameterized LD-model.
Appendix B
Dispersion of the TM0 Surface-Mode
for Rod-Antennas
This appendix illustrates the derivation of the fundamental surface mode, the TM0 mode,
supported by the plasmonic rod-antenna shown in Fig. 2.2 in Section 2.2. This method
was adapted from [23], where the transcendental equation 2.7 was derived in the form
of propagating waves. In this Appendix, the expressions in the form of surface waves
(Eqn. 2.8) is also illustrated. The dispersion of λeff extracted using Eqn. 2.8 is used
to compare against the one shown in [23]. The rod antenna was treated as a negative
dielectric waveguide embedded in a positive dielectric space as shown in Fig. B.1, where
standard waveguide theories in cylindrical coordinate are applied.
Figure B.1: The cylindrical waveguide configuration.
132
Appendix B. Dispersion of the TM0 Surface-Mode for Rod-Antennas 133
According to the methods in [37], magnetic and electric vector potentials, A and F,
are introduced to facilitate the mode solution. For a TM mode to a given direction,
independent of the coordinate, it is sufficient to let the vector potential A have only a
component in the direction in which the fields are desired to be TM. In this case, we
assume the mode is transverse-magnetic in the z -direciton, i.e. TMz . Therefore, in the
cylindrical coordinate system, the vector potentials have the form illustrated in
A = Az (ρ, φ, z) b
az
F = 0
(B.1)
Electric and magnetic fields can thus be expressed as functions of Az only, as shown in
1 ∂ 2 Az
ωµε ∂ρ ∂z
1 1 ∂ 2 Az
= −j
ωµε ρ ∂φ ∂z
2
∂
1
2
+ β Az
= −j
ωµε ∂z 2
Eρ = −j
Eφ
Ez
1 1 ∂Az
µ ρ ∂φ
1 ∂Az
= −
µ ∂ρ
(B.2)
Hρ =
Hφ
(B.3)
Hz = 0
√
respectively, where β = k0 ε is the propagation constant in the respective medium.
Assuming the solution Az to the wave equation in a source-free region can be written
using the separation of variables as shown in
Az (ρ, φ, z) = f (ρ) g(φ) h(z)
(B.4)
the solutions for the mth mode in the metallic core and the surrounding positive dielectric
medium are expressed in
Azm = B Jm (βρm ρ) [C cos(mφ) + D sin(mφ)] e−jβz z
(B.5)
(2)
Azd = P Hm
(βρd ρ) [S cos(mφ) + T sin(mφ)] e−jβz z
(B.6)
Appendix B. Dispersion of the TM0 Surface-Mode for Rod-Antennas 134
(2)
respectively, where Jm and Hm are the mth -order Bessel function and Hankel function
of the 2nd kind, which represent the standing wave and the traveling wave solutions in
a cylindrical coordinate system, respectively. Additionally, we only consider the wave
propagating in the +z direction, which means only the solutions that contain e−jβz z
remain. The lower subscript m and d correspond to the metallic core and the positive
dielectric surrounding, respectively. The variables B, C, D, P , S, and T are constant
coefficients.
The constant coefficients, as well as βρ and βz can be found using the boundary
conditions at the metal and dielectric interface. That is, the tangential field components
are matched at ρ = R. Since the electric and magnetic field components in Eqns. B.2
and B.3 are not all independent from each other, only two tangential components are
needed to find the solution. In this case, we pick the tangential components Ez and Hφ
for the purpose of our derivation; however, other field combinations should also yield the
same result.
According to Eqns. B.2 and B.3, Ez and Hφ can be found from Az for both the metal
and the dielectric medium as shown in
1
2
−βz2 + βm
Azm
ωµεm
1
2
−βz2 + βm
B Jm (βρm ρ) [C cos(mφ) + D sin(mφ)] e−jβz z (B.7)
= −j
ωµεm
1
= −j
−βz2 + βd2 Azd
ωµεd
1
(2)
−βz2 + βd2 P Hm
(βρd ρ) [S cos(mφ) + T sin(mφ)] e−jβz z (B.8)
= −j
ωµεd
Ezm = −j
E zd
βρm
0
B Jm (βρm ρ) [C cos(mφ) + D sin(mφ)] e−jβz z
µ
βρ
(2)0
= − d P Hm
(βρd ρ) [S cos(mφ) + T sin(mφ)] e−jβz z
µ
H zm = −
(B.9)
Hzd
(B.10)
For the TM0 mode, the expressions in above equations containing the φ components
reduce down to a constant, which are absorbed into the coefficients B0 and P0 . Therefore,
Appendix B. Dispersion of the TM0 Surface-Mode for Rod-Antennas 135
for TM0 mode, the tangential field expressions can be further simplifed as
1
2
−βz2 + βm
B0 J0 (βρm ρ) e−jβz z
ωµεm
1
(2)
−βz2 + βd2 P0 H0 (βρd ρ) e−jβz z
= −j
ωµεd
Ezm = −j
(B.11)
E zd
(B.12)
βρm
0
B0 J0 (βρm ρ) e−jβz z
µ
βρ
(2)0
= − m P0 H0 (βρd ρ) e−jβz z
µ
Hzm = −
(B.13)
H zd
(B.14)
Since the boundary conditions imply that the tangential components are conserved
at ρ = R, Eqns. B.11 and B.12 yield the conservation of Ez , such that
1
1
(2)
2
−βz2 + βm
−βz2 + βd2 P0 H0 (βρd R)
B0 J0 (βρm R) =
εm
εd
(B.15)
and Eqns. B.13 and B.14 yield the conservation of Hφ , such that
(2)0
0
βρm B0 J0 (βρm R) = βρd P0 H0 (βρd R)
(B.16)
From the derivative properties of the Bessel functions of the first and second kinds in
m
Jm (x) − Jm+1 (x)
x
m
0
Ym (x) =
Ym (x) − Ym+1 (x)
x
0
(B.17)
Jm (x) =
0
(B.18)
(2)0
respectively, both J0 (βρm R) and H0 (βρd R) can be represented with respective higherorder functions. (Recall that Hankel functions are superpositions of Bessel functions, in
(2)
particular, Hm (x) = Jm (x) + jYm (x).) Therefore, Eqn. B.16 can be rewritten as
βρm B0 [−J1 (βρm R)] = βρd P0
h
(2)
−H1 (βρd R)
i
(B.19)
By taking the ratio of Eqns. B.19 and B.15, and normalizing the expressions with the
radius R, the transcendental equation is obtained in
(2)
εd H1 (βρd R)
εm (λ) J1 (βρm R)
=
βρm R J0 (βρm R)
βρd R H0(2) (βρd R)
(B.20)
Appendix B. Dispersion of the TM0 Surface-Mode for Rod-Antennas 136
where βρm and βρd are the transverse wavenumbers in the ρ-direction in metal and dielectric, respectively. Eqn. B.20 agrees with the finding in [23]. These two transverse
wavenumbers are not independent of each other since they are both related to the longitudinal propagation constant βz as in
q
βρm = k02 εm − βz2
q
βρd = k02 εd − βz2
(B.21)
(B.22)
For surface TM modes, the solution is better described by substituting βρ = −jα into
Eqn. B.20 since α represents the transverse decay constant of the surface modes. The
cylindrical Bessel function and Hankel function of the second kind can be related to the
modified Bessel function of the first and second kind respectively [37], according to
Jm (−jαR) = (−j)m Im (αR)
(2)
Hm
(−jαR) =
2 m+1
(j)
Km (αR)
π
(B.23)
where the modified Bessel function of the first and second kind of order m, Jm and Km ,
represent the evanescent waves in the −ρ and +ρ directions, respectively. Therefore, the
transcendental equation is obtained for the fundamental TM surface mode expressed in
term of evanescent wavenumbers as illustrated in
εd K1 (αd R)
εm (λ) I1 (αm R)
=−
αm R I0 (αm R)
αd R K0 (αd R)
(B.24)
where
q
αm = βz2 − k02 εm
q
αd = βz2 − k02 εd
(B.25)
(B.26)
Combining Eqns. B.24, B.25 and B.26, the solution of βz can be found numerically. The
method used in this thesis for solving the transcendental equation B.24 is a graphical
approach, where the left and right sides of the equation are plotted separately and the
Appendix B. Dispersion of the TM0 Surface-Mode for Rod-Antennas 137
intersection of the two graphs yields the solution for βz . This result can then be used in
Eqn. 2.6 to calculate the effective wavelength of the surface plasmon mode supported by
the rod antenna.
Appendix C
Parallel dipoles at air-dielectric
interface
This appendix applies the reciprocity theorem for obtaining the radiation pattern as proposed by Rutledge [60]. This method calculates the radiation pattern due to parallel
dipoles at the interface of two dielectric media. In [60], the radiation pattern due to an
electric dipole is given. In this appendix, the one due to a magnetic dipole is also derived. Both illustrate substantial difference from the dipoles embedded in a homogenous
medium.
In order to apply reciprocity theorem, the transmission coefficients of the E and H
fields need to be calculated. For the example of a plane wave that is incident from air (n1 )
to dielectric (n2 ), it can be decomposed into the s-polarized/TE wave and p-polarized/TM
wave as shown in Figs. C.1(a) and C.1(b), respectively. It will be demonstrated later
in Section C.1 and C.2 that for (either an electric or a magnetic) dipole oriented in the
plane of the air-dieletric interface, the TE-polarized transmission is used to calculate the
H-plane radiation pattern, whereas the TM-polarizated transmission is used to calculate
the E-plane radiation pattern.
For the TE and the TM-polarization shown in Fig. C.1, the transmission coefficients
138
Appendix C. Parallel dipoles at air-dielectric interface
(a) S-polarized/TE waves
139
(b) P-polarized/TM waves
Figure C.1: Incident waves from air to the dielectric half-space for TE and TM polarizations.
of the electric field are shown in
2n1 cos θ1
n1 cos θ1 + n2 cos θ2
2n1 cos θ1
=
n1 cos θ2 + n2 cos θ1
TE:
te21 =
(C.1)
TM:
te21
(C.2)
respectively. The results from dielectric to air can be easily obtained by swapping n1 and
n2 in the above expressions. Additionally, the transmission coefficients for the H-field
can be obtained using relations illustrated in
th21 = n te21
(C.3)
1
te12
n
(C.4)
th12 =
for both polarizations, where n = n2 /n1 is the refractive index ratio between dielectric
and air. In summary, there are four sets of transmission coefficients, each corresponding
to a particular field (E or H) with a specific polarization (TE or TM).
Appendix C. Parallel dipoles at air-dielectric interface
C.1
140
Radiation of an electric dipole
For an electric dipole oriented in the plane of the air-dielectric interface shown in Fig. C.2,
the current and field orientations in the H and the E-planes are illustrated in Figs. C.3(a)
Figure C.2: A parallel electric dipole in the plane of the air-dielectric interface.
and C.3(b), respectively. We can see that in order to obtain the H-plane radiation pattern,
(a) H-plane (TE-polarization)
(b) E-plane (TM-polarization)
Figure C.3: The H-plane and E-plane of an electirc dipole parallel to the plane of the
air-dielectric interface.
the transmission coefficients for the TE-polarization are used; on the other hand, the Eplane radiation pattern is obtained through analyzing the TM-polarized transmission.
The calculation of the radiation pattern is first done for the H-plane. As illustrated
Appendix C. Parallel dipoles at air-dielectric interface
141
in Fig. C.3(a), the power radiated in the air and the dielectric are shown in
1
1
|E1 |2 =
|te21 Ea |2
2η1
2η1
1
1
|E1 |2 =
|te12 Ed |2
=
2η2
2η2
P1 =
(C.5)
P2
(C.6)
respectively, where te21 and te12 correspond to the electric-field transmission for the TEpolarization. Ea and Ed , as shown in
jωµ0 Ie h −jn1 k0 r
e
4πr
jωµ0 Ie h −jn2 k0 r
=
e
4πr
Ea =
(C.7)
Ed
(C.8)
respectively, are the electric dipolar radiation in air and dielectric, respectively. Substituting Eqn. C.1 into Eqn. C.5 and C.6, we obtain the closed-form expressions of the
H-plane radiation pattern in both media can be calculated as
P1
P2
where A = η20
2
cos θ1
A
= n1
cos θ1 + n cos θ2
2
cos θ2
2
A
= n n2
n cos θ2 + cos θ1
(C.9)
(C.10)
ωµ0 Ie h 2
.
4πr
The E-plane radiation pattern can be derived from Fig. C.3(b). Compared to the
H-plane, the power radiated in the air and dielectric in the E-plane are illustrated in
P1
1
1
cos θ2
=
|E1 cos θ1 |2 =
Ea cos θ1
te21
2η1
2η1
cos θ1
2
1
1
cos θ1
|E1 cos θ2 |2 =
Ed cos θ2
te12
2η2
2η2
cos θ2
2
P2 =
=
1
|te21 Ea cos θ2 |2
2η1
(C.11)
=
1
|te12 Ed cos θ1 |2
2η2
(C.12)
respectively, where te21 and te12 correspond to the electric-field transmission for the TMθ2
E is used in order to obtain
polarization. Note that in Eqn. C.11, the expression te21 cos
cos θ1 a
the tangential electric field E2 . Additionally, E1 cos θ1 is used because only the transverse
electric field contributes to the power flow in the r-direction. Similar analogy holds for
Eqn. C.12. Finally, by substituting Eqn. C.2 into Eqn. C.11 and C.12, we obtain the
Appendix C. Parallel dipoles at air-dielectric interface
142
closed-form expressions of the E-plane radiation pattern in both media as shown in
P1
P2
2
cos θ1 cos θ2
= n1
A
cos θ2 + n cos θ1
2
cos θ1 cos θ2
2
= n n2
A
cos θ2 + n cos θ1
(C.13)
(C.14)
Comparing Eqn. C.9 to C.10 and Eqn. C.13 to C.14, it can be seen that the majority
of the power is in the dielectric for both polarizations. More specifically, the ratio of
the power distribution in dielectric and air is approximately n3 : 1. For an example of a
parallel electric dipole on a glass substrate, the far-field radiation patterns are illustrated
in Fig. C.4.
90
120
60
H−plane
E−plane
150
30
ε =1
r
180
0
ε =4
r
210
330
240
300
270
Figure C.4: The H-plane and the E-plane radiation pattern in both the air and the glass
substrate (ε2 = 4) for an electric dipole at the interface.
Appendix C. Parallel dipoles at air-dielectric interface
C.2
143
Radiation of a magnetic dipole
For a magnetic dipole oriented in the plane of the air-dielectric interface, the derivations
of the E-plane and H-plane radiation patterns are similar to those for the electric dipole,
with a few minor modifications, mostly due to the duality between the electric and
magnetic dipoles. The current and field orientations in the E and the H-planes are
illustrated in Figs. C.5(a) and C.5(b), respectively.
(a) E-plane (TM-polarization)
(b) H-plane (TE-polarization)
Figure C.5: The E-plane and H-plane of a magnetic dipole paralle to the plane of the
air-dielectric interface.
We start with the analysis in the E-plane. As illustrated in Fig. C.5(a), the power
radiated in the air and the dielectric can be obtained using
η1
η1
|H1 |2 =
|th21 Ha |2
2
2
η2
η2
2
|H1 | =
|th12 Hd |2
=
2
2
P1 =
(C.15)
P2
(C.16)
respectively, where where th21 and th12 correspond to the H-field transmission for the
TM-polarization. Ha and Hd , as shown in
Ha
Hd
jωn21 ε0 Im h −jn1 k0 r
e
=
4πr
jωn22 ε0 Im h −jn2 k0 r
=
e
4πr
(C.17)
(C.18)
Appendix C. Parallel dipoles at air-dielectric interface
144
are the magnetic dipolar radiation in air and dielectric, respectively. Note that unlike Ed
and Ea that have the same magnitude, Hd and Ha differ by a ratio of n2 .
By substituting the transmission coefficients for H in Eqn. C.3 and C.4 into Eqn. C.15
and C.16, we obtain the closed-form expressions of the E-plane radiation pattern in both
media as shown in
where B = 2η0
P1
n2
=
n1
P2
n4
=
n2
ωε0 Im h 2
.
4πr
cos θ1
n cos θ1 + cos θ2
cos θ2
n cos θ1 + cos θ2
2
2
B
(C.19)
B
(C.20)
Similarly, The H-plane radiation pattern can be derived from Fig. C.5(b), and their
expressions are shown in
η1
η1
cos θ2
|H1 cos θ1 |2 =
Ha cos θ1
th21
2
2
cos θ1
2
η2
η2
cos θ1
|H1 cos θ2 |2 =
=
Hd cos θ2
th12
2
2
cosθ2
2
P1 =
P2
=
η1
|te21 Ha cos θ2 |2
2
(C.21)
=
η2
|te12 Hd cos θ1 |2
2
(C.22)
After substituting th21 and th12 (of the H-plane) into the above equations, we obtain the
final expressions for the power in air and dielectric that are shown in
P1
n2
=
n1
P2
n4
=
n2
cos θ1 cos θ2
cos θ1 + n cos θ2
cos θ1 cos θ2
cos θ1 + n cos θ2
2
2
B
(C.23)
B
(C.24)
respectively.
Comparing Eqn. C.19 to C.20 and Eqn. C.23 to C.24, we can see that the majority of
the power is also in the dielectric for both polarizations. However, the ratio of the power
distribution in dielectric and air is approximately n : 1, which is much less compared to
the case of an electric dipole. For an example of a parallel magnetic dipole on a glass
substrate, the far-field radiation patterns are illustrated in Fig. C.6.
Appendix C. Parallel dipoles at air-dielectric interface
145
90
120
60
H−plane
E−plane
150
30
ε =1
r
180
0
ε =4
r
330
210
300
240
270
Figure C.6: The E-plane and the H-plane radiation pattern in both the air and the
substrate for a magnetic dipole.
References
[1] M. J. Kobrinsky, B. A. Block, J. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko,
F. Roberson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel
Technology Journal, vol. 8, no. 2, pp. 1535–864X, 2004.
[2] H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings.
Berlin: Springer-Verlag, 1988.
[3] E. N. Economou, “Surface plasmons in thin films,” Phys. Rev., vol. 182, no. 2,
pp. 539–554, 1969.
[4] J. J. Burke, G. I. Stegman, and T. Tamir, “Surface-polariton-like guided by thin,
lossy metal films,” Phys. Rev. B., vol. 33, no. 8, pp. 5186–5201, 1986.
[5] W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength
optics,” Nature, vol. 424, pp. 824–830, 2003.
[6] M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy
transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys.
Rev. B, vol. 62, pp. R16356–R16359, 2000.
[7] R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-range surface plasmon polaritons,” Opt. Express, vol. 13, no. 3, pp. 977–984, 2005.
146
REFERENCES
147
[8] R. Oulton, V. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic
waveguide for subwavelength confinement and long range propagation,” Nature Photon., vol. 2, pp. 496–500, 2007.
[9] M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Propagation characteristics
of hybrid modes supported by metal-low-high index waveguides and bends,” Opt.
Express, vol. 18, pp. 12971–12979, 2010.
[10] U. C. Fischer and D. W. Pohl, “Observation of single-particle plasmons by near-field
optical microscopy,” Phys. Rev. Lett., vol. 62, pp. 458–461, 1989.
[11] P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single
molecule fluorescence,” Phys. Rev. Lett., vol. 96, p. 113002, 2006.
[12] T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst,
“λ/4 resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano. Lett., vol. 7, pp. 28–33, 2007.
[13] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary
optical transmission through sub-wavelength hole arrays,” Nature, vol. 391, pp. 667–
669, 1998.
[14] H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. GarciaVidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science,
vol. 297, pp. 820–822, 2002.
[15] H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev., vol. 66, pp. 163–182,
1944.
[16] P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon.,
vol. 1, pp. 438–483, 2009.
REFERENCES
148
[17] A. Alù and N. Engheta, “Wireless at the nanoscale: Optical interconnects using
matched nanoantennas,” Phys. Rev. Lett., vol. 104, p. 213902, 2010.
[18] E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,”
Appl. Phys. Lett., vol. 89, p. 093120, 2006.
[19] K. R. C. A. Polman, “Plasmonic solar cells,” Opt. Express, vol. 16, pp. 21793–21800,
2008.
[20] A. Hartschuh, E. Sanchez, X. Xie, and L. Novotny, “High-resolution near-field raman
microscopy of single-walled carbon nanotubes,” Phys. Rev. Lett., vol. 90, p. 095503,
2003.
[21] L. Novotny, The history of near-field optics, vol. 50 of Progress in Optics, pp. 137–
184. Elsevier, 2007.
[22] J. Jackson, S. Westcott, L. Hirsch, J. West, and N. Halas, “Controlling the surface enhanced raman effect via the nanoshell geometry,” Appl. Phys. Lett., vol. 82,
pp. 257–259, 2003.
[23] L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.,
vol. 98, pp. 266802(1)–(4), 2007.
[24] J. R. Lakowicz, “Radiative decay engineering 5: metal-enhanced fluorescence and
plasmon emission,” Anal. Biochem., vol. 337, pp. 171–194, 2005.
[25] J. S. Biteen, N. Lewis, H. Atwater, H. Mertens, and A. Polman, “Spectral tuning
of plasmon-enhanced silicon quantum dot luminescence,” Appl. Phys. Lett., vol. 88,
p. 131109, 2006.
[26] F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement
of molecular fluorescence,” Nano Lett., vol. 7, p. 496501, 2007.
REFERENCES
149
[27] S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of singlemolecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys.
Rev. Lett., vol. 97, p. 017402, 2006.
[28] R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on
upconversion processes,” J. Appl. Phys., vol. 105, p. 033107, 2009.
[29] T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas
direct single-molecule emission,” Nature Photon., vol. 2, pp. 234–237, 2008.
[30] A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F.
van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,”
Science, vol. 329, pp. 930–933, 2010.
[31] I. Gryczynski, J. Malicka, Z. Gryczynski, and J. R. Lakowicz, “Radiative decay engineering 4. Experimental studies of surface plasmon-coupled directional emission,”
Anal. Biochem., vol. 324, pp. 170–182, 2004.
[32] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small
Particles. John Wiley & Sons, Inc., 1998.
[33] A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of
metallic films for vertical-cavity optoelectronic devices,” Appl. Opt., vol. 37, no. 22,
pp. 5271–5283, 1998.
[34] P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys.
Rev. B, vol. 6, no. 12, pp. 4370–4379, 1972.
[35] E. D. Palik, Handbook of Optical Constants of Solids, vol. I. Elsevier Inc., 1997.
[36] A. D. Rakić, “Algorithm for the determination of intrinsic optical constants of metal
films: application to aluminum,” Appl. Opt., vol. 34, no. 22, pp. 4755–4767, 1995.
150
REFERENCES
[37] C. A. Balanis, Advanced Engineering Electromagnetics. John Wiley & Sons, Inc.,
1989.
[38] G. W. Bryant, F. J. G. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances
of metallic nanoantennas,” Nano Lett., vol. 8, no. 2, pp. 631–636, 2008.
[39] E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and
surface modification beyond the diffraction limit,” Science, vol. 257, pp. 189–195,
1992.
[40] R.
Hillenbrand
and
F.
Keilmann,
“Material-specific
mapping
of
metal/semiconductor/dielectric nanosystems at 10 nm resolution by backscattering near-field optical microscopy,” Appl. Phys. Lett., vol. 80, no. 25, 2002.
[41] B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W.
Pohl, “Scanning near-field optical microscopy with aperture probes: Fundamentals
and applications,” J. Chem. Phys., vol. 112, no. 7761, 2000.
[42] J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, “High definition
aperture probes for near-field optical microscopy fabricated by focused ion beam
milling,” Appl. Phys. Lett., vol. 72, pp. 3115–3117, 1998.
[43] L. Novotny and B. Hecht, Principles of Nano-Optics. Cambridge University Press,
1st Edition, 2006.
[44] L. Novotny, , D. W. Pohl, and B. Hecht, “Scanning near-field optical probe with
ultrasmall spot size,” Opt. Lett., vol. 20, pp. 970–972, 1995.
[45] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85.
[46] A. Grbic and G. V. eleftheriades, “Overcoming the diffraction limit with a planar
left-handed transmission-line lens,” Phys. Rev. Lett., vol. 92, p. 117403, 2004.
REFERENCES
151
[47] N. Fang, H. Lee, C. Sun, and X. Zhang, “Subdiffraction-limited optical imaging with
a silver superlens,” Science, vol. 308, pp. 534–537, 2005.
[48] A. K. Iyer and G. V. Eleftheriades, “Mechanism of subdiffraction free-space imaing
using a transmission-line metamaterial superlens: an experimental verification,”
Phys. Rev. Lett., vol. 102, p. 173901, 2009.
[49] E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science, vol. 313, pp. 1642–1645,
2006.
[50] M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic
optical reconstruction microscopy (storm),” Nat. Methods, vol. 3, pp. 793–796, 2006.
[51] V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical
microscope,” Phys. Rev. Lett., vol. 94, pp. 143903–4, 2005.
[52] R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and
perfect focusing,” Science, vol. 317, pp. 927–929, 2007.
[53] A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with
patterned surfaces,” Science, vol. 320, pp. 511–513, 2008.
[54] M. F. Imani and A. Grbic, “An analytical investigation of near-field plates,” Metamaterials, vol. 4, pp. 104–111, 2010.
[55] L. Markley, A. M. H. Wong, Y. Wang, and G. V. Eleftheriades, “Spatially shifted
beam approach to subwavelength focusing,” Phys. Rev. Lett., vol. 101, p. 113901,
2008.
[56] R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless
interference of a plasmonic array,” Phys. Rev. Lett., vol. 102, p. 207402, 2009.
REFERENCES
152
[57] G. V. Eleftheriades and A. M. H. Wong, “Holograhy-inspired screens for subwavelength focusing in the near field,” IEEE Microw. Wireless Compon. Lett.,
vol. 18, pp. 236–238, 2008.
[58] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. John Wiley & Sons,
Inc., 2004.
[59] J. B. Fraleigh and R. A. Beauregard, Linear algebra 3rd Edition. Addison-Wesley,
1995.
[60] D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas
Propag., vol. AP-30, pp. 535–540, 1982.
[61] M. Mivelle, T. S. van Zanten, L. Neumann, N. F. van Hulst, and M. F. GarciaParajo, “Ultrabright bowtie nanoaperture antenna probes studied by single molecule
fluorescence,” Nano Lett., vol. 12, pp. 5972–5978, 2012.
[62] L. Neumann, Y. Pang, A. Houyou, M. L. Juan, R. Gordon, and N. F. van Hulst,
“Extraordinary optical transmission brightens near-field fiber probe,” Nano Lett.,
vol. 11, pp. 355–360, 2011.
[63] L. Markley and G. V. Eleftheriades, “Two-dimensional subwavelength-focused imaging using a near-field end-fire antenna-array probe,” IEEE Antennas Wireless
Propag. Lett., vol. 8, pp. 1025–1028, 2009.
[64] L. Markley and G. V. Eleftheriades, “Two-dimensional subwavelength-focused imaging using a near-field probe at a λ/4 working distance,” J. Appl. Phys., vol. 107,
p. 093102, 2010.
[65] T. H. Taminiau, F. B. Segerink, and N. F. van Hulst, “A monopole antenna at optical frequencies: single-molecule near-field measurements,” IEEE Trans. Antennas
Propag., vol. 55, pp. 3010–3017, 2007.
REFERENCES
153
[66] F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic
coaxial waveguides in the optical range: Role of the plasmonic modes,” Phy. Rev.
B, vol. 74, p. 205419, 2006.
[67] M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Plasmon dispersion in
coaxial waveguides from single-cavity optical transmission measurements,” Nano
Lett., vol. 9, pp. 2832–2837, 2009.
[68] T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst,
“Near-field driving of a optical monopole antenna,” J. Opt. A: Pure Appl. Opt,
vol. 9, pp. S315–S321, 2007.
[69] S. A. Hosseini, S. Campione, and F. Capolino, “Dual polarized near-field focusing plate for nearfield optical focusing in two dimensions,” Opt. Express, vol. 19,
pp. 24483–24498, 2011.
[70] J. L. Volakis, Antenna engineering handbook, 4th ed. McGraw-Hill, 2007.
[71] D. Coling and B. Kachar, “Principles and application of fluorescence microscopy,”
Curr. Protoc. Mol. Biol., vol. 44, pp. 14.10.1–14.10.11., 2001.
[72] P. Ginzburg, D. Arbel, and M. Orenstein, “Gap plasmon polariton structure for
very efficient microscale-to-nanoscale interfacing,” Opt. Lett., vol. 311, pp. 3288–
3290, 2006.
[73] G. Veronis and S. Fan, “Theoretical investigation of compact couplers between
dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic
waveguides,” Opt. Express, vol. 15, pp. 1211–1221, 2007.
[74] C. Lin, H. M. K. Wong, B. Lau, M. A. Swillam, and A. S. Helmy, “Efficient broadband energy transfer via momentum matching at hybrid junctions of guided-waves,”
Appl. Phys. Lett., vol. 101, p. 123115, 2012.
REFERENCES
154
[75] J. Wen, S. Romanov, and U. Peschel, “Excitation of plasmonic gap waveguides by
nanoantennas,” Opt. Express, vol. 17.
[76] X. X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive
radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B, vol. 82,
p. 144305, 2010.
[77] H. Giessen and M. Lippitz, “Directing light emission from quantum dots,” Science,
vol. 329, pp. 910–911, 2010.
[78] M. Pfeiffer, K. Lindfors, C. Wolpert, P. Atkinson, M. Benyoucef, A. Rastelli, O. G.
Schmidt, H. Giessen, and M. Lippitz, “Enhancing the optical excitation efficiency
of a single self-assembled quantum dot with a plasmonic nanoantenna,” Nano Lett.,
vol. 10, pp. 4555–4558, 2010.
[79] O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. G. Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,”
Nano Lett., vol. 7, pp. 2871–2875, 2007.
[80] T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation
and emission of single emitters by a nano-optical yagi-uda antenna,” Opt. Express,
vol. 16, pp. 10858–10866, 2008.
[81] A. Neto and S. Maci, “Green’s function for an infinite slot printed between two homogeneous dielectrics - Part I: magnetic currents,” IEEE Trans. Antennas Propag.,
vol. 51, pp. 1572–1581, 2003.
[82] S. Maci and A. Neto, “Green’s function for an infinite slot printed between two
homogeneous dielectrics - Part II: uniform asymptotic solution,” IEEE Trans. Antennas Propag., vol. 52, pp. 666–676, 2004.
REFERENCES
155
[83] Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional
coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express,
vol. 14, pp. 7279–7290, 2006.
[84] J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by
thin, lossy metal films,” Phys. Rev. B, vol. 33, pp. 55186–5201, 1986.
[85] G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot
in a thin metal film,” Opt. Lett., vol. 30, pp. 3359–3361, 2005.
[86] W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon
carbide,” Phys. Rev., vol. 113, pp. 127–132, 1959.
[87] J. Brown, “Mapping the plasmon resonances of metallic nanoantennas,” Proc. Inst.
Electr. Eng. – Part IV: Institution Monographs, vol. 100, pp. 51–62, 1953.
[88] N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band
gaps at the visible,” Appl. Phys. Lett., vol. 80, pp. 1449529–1440531, 2002.
[89] B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt.
Express, vol. 15, no. 3, pp. 1182–1190, 2007.