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Bowtie and Bowtie Aperture Antenna at Microwave and Optical Frequency

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UNIVERSITY OF CALIFORNIA,
IRVINE
Bowtie and Bowtie Aperture Antenna at Microwave and Optical Frequency
THESIS
Submitted in partial satisfaction of the requirements
For the degree of
MASTER OF SCIENCE
in Electrical and Computer Engineering
By
RackGeun Kim
Thesis Committee:
Professor Filippo Capolino, Chair
Professor Ozdal Boyraz
Professor Regina Ragan
2013
UMI Number: 1541061
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INFORMATION TO ALL USERS
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a note will indicate the deletion.
UMI 1541061
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© 2013 RackGeun Kim
TABLE OF CONTENTS
Page
LIST OF FIGURES ............................................................................................................................................. iv
LIST OF TABLES...............................................................................................................................................vii
ACKNOWLEDGMENTS .................................................................................................................................viii
ABSTRACT OF THE THESIS ........................................................................................................................... ix
CHAPTER 1 INTRODUCTION ......................................................................................................................... 1
1.1 Motivations and Goals ...................................................................................................................................1
1.2 Classical Antennas vs. Optical Antennas .....................................................................................................2
1.3 Background and Historical survey ...............................................................................................................3
1.4 Scope of the Thesis .........................................................................................................................................6
CHAPTER 2 BACKGROUND OF BOWTIE ANTENNAS AT RADIO FREQUENCY ............................... 9
2.1 Introduction: Bowtie Antennas .....................................................................................................................9
2.2 Antenna Theory ...........................................................................................................................................11
2.3 Analysis on RF Bowtie Antennas ................................................................................................................16
2.4 Analysis on RF Bowtie Aperture Antennas ................................................................................................21
2.5 Summary: RF Antennas and Babinet’s Principle .....................................................................................25
CHAPTER 3 CHARACTERISTICS OF OPTICAL NANO-ANTENNAS ................................................... 27
3.1 Introduction..................................................................................................................................................27
3.2 Dispersive characteristics of Metals: Drude model and experimental data ...........................................29
3.3 Surface Plasmons (SPs) and Surface Plasmon Polaritons (SPPs)............................................................34
3.4 Field Enhancements for Surface Enhanced Raman Scattering (SERS) .................................................39
3.5 Summary: Analogy between the Radio Frequency and Optical Antennas .............................................40
CHAPTER 4 BOWTIE AND BOWTIE APERTURE ANTENNAS AT OPTICAL FREQUENCY ........... 41
4.1 Introduction..................................................................................................................................................41
4.2 Receiving Antennas: Analysis of Bowtie Optical Antennas ......................................................................43
4.2.1 Gap dependence ................................................................................................................................50
4.2.2 Resonance adjustments by the size ..................................................................................................51
4.2.3 Changes with Oblique Incidence .....................................................................................................54
4.3 Receiving Antennas: Analysis of Bowtie aperture Optical Antennas ......................................................56
4.3.1 Gap dependence ................................................................................................................................62
4.3.2 Resonance adjustments by the size ..................................................................................................65
4.3.3 Changes with Oblique Incidence .....................................................................................................67
4.4 Transmitting Antennas: Bowtie aperture Antenna with Hertzian Dipole...............................................69
4.5 Summary: Interesting Phenomena using Bowtie and Bowtie Aperture Antennas .................................73
CHAPTER 5 INTERACTION OF OPTICAL ANTENNAS AND MOLECULAR DIPOLES ................... 75
5.1 Introduction: Induced Dipole Moment of Molecules ................................................................................75
5.2 Green’s Function ..........................................................................................................................................76
5.3 Superposition in Electromagnetic Theory .................................................................................................78
5.4 Bowtie Aperture Optical Antenna and the Scattering Green’s Function ................................................79
5.5 Total Enhancement of the Induced Dipole Moment of a Molecule beside Optical Antenna .................81
5.6 Conclusion ....................................................................................................................................................84
ii
CHAPTER 6 CONCLUSION AND FUTURE APPLICATIONS ................................................................... 85
6.1 Various Applications ....................................................................................................................................85
6.2 Future work..................................................................................................................................................85
6.3 Conclusion ....................................................................................................................................................86
BIBLIOGRAPHY ............................................................................................................................................... 88
iii
LIST OF FIGURES
Page
Figure 1.1 Examples of antennas: (a) Yagi-Uda Antenna consists of an array of dipole antennas. (length
scale of meter) (b) Micro-strip Bowtie Antenna. (length scale of centimeter) (c) Nano particles act as an
Optical Antenna. (length scale of hundred nanometer) ................................................................................ 2
Figure 2.1 Transmitting and receiving antennas: (a) Transmitting model. The radiated power is
represented by the radiation resistance. (b) Receiving antenna where the generator (Voc) represents the
picked up signal as a voltage generator. ..................................................................................................... 11
Figure 2.2 (Top view) Detailed design of the bowtie antenna with a lumped port excitation polarized in yaxis in the gap (Conductivity of copper: 5.8X107 Siemens/m and FR-4 substrate r=4.4) ......................... 16
Figure 2.3 (a) Reflection coefficient S11 (in dB) of Bowtie Antenna operating at 2.4GHz. (b) Magnitude
of S11 and phase angle of S11. Zero-crossings occurred at 2.5GHz and 3.8GHz. ..................................... 18
Figure 2.4 (a) Input impedance versus frequency plot shows both real part and imaginary part of the
impedance of the bowtie antenna. The resonance is at 2.5GHz where the imaginary Z (reactance) is zero.
At 2.5GHz, the impedance is approximately 73Ω. (b) Gain versus frequency plot. Gain increases as the
frequency increases. At resonant frequency, the gain is -0.07dB ............................................................... 19
Figure 2.5 Farfield radiation of the bowtie antenna at 2.5GHz: (a) 2D plot shows the total E-field in the
polar plot as the radiation pattern (b) 3D plot of directivity in far-field. .................................................... 20
Figure 2.6 (Top view) Detailed design of bowtie aperture antenna with lumped port excitation polarized
in x-axis at the gap (Conductivity of copper:5.8 × 107 Siemans/m and FR-4 substrate εr = 4.4)........ 21
Figure 2.7 (a) Reflection coefficient S11 (in dB) of the bowtie aperture antenna operating at 4.65GHz... 22
Figure 2.8 (a) Input impedance versus frequency plot shows both the real part and imaginary part of the
impedance of the bowtie aperture antenna. At 2.25GHz, the impedance is approximately 1300Ω. At
4.65GHz, the impedance is 42Ω (b) Gain versus frequency plot. Gain increases as the frequency
increases. ..................................................................................................................................................... 23
Figure 2.9 Far-field radiation of the bowtie aperture antenna (y-axis polarization): (a) 2D plot shows the
total E-field in the polar plot as the radiation pattern (b) 3D plot of directivity in far-field. ...................... 24
Figure 3.1 Examples of plasma oscillation: (a) Localized nano-particle at plasma frequency. (b) Bulk
plasmons oscillation with the free electron density and effective mass. (c) Propagating surface plasmon
polaritons (SPPs) interacting with light. ..................................................................................................... 28
Figure 3.2 Comparison of Drude Model (theory) and experimental data [68]: (a) Experimental data (solid)
and theroretical Drude Model (dotted). Blue colored line represents the real part while red line represents
the imaginary part of the permittivity. (b) Difference in imaginary part of the Drude Model (Theory) and
experimental data (Exp) for wavelength range 200nm≤λ≤1200nm. ........................................................... 33
Figure 3.3 Surface plasmons (SPs) and surface plasmon polaritons (SPPs): collective oscillations at metal
and dielectric interface. ............................................................................................................................... 34
Figure 3.4 Surface plasmon dispersion curves for Au/ITO Substrate by using Drude Model and light line
where: It shows the radiative, quasi-bound, and bound surface plasmon dispersion relations for gold
indium tin oxide film. ................................................................................................................................. 38
Figure 4.1 Bowtie antenna structure in 3D and top view: (a) 3D view shows its general structure with
three materials (gold, ITO, silica) (b) 2D top view of the gold bowtie antenna that shows the substrate’s
dimensions (450x450nm2), gap (20nm), side length (80nm), and width (70nm) ....................................... 43
iv
Figure 4.2 Excitation and mesh view of the bowtie structure (a) Side view of the bowtie shows the
permittivity of the substrates. The excitation comes from the top and E-field is polarized along the x-axis
(b) Mesh configurations from the top view shows more tetrahedral concentrated in the gold and gap. (c)
Mesh view from the side of the bowtie antenna. ........................................................................................ 45
Figure 4.3 E-field distribution in the gap of the bowtie: (a) Normalized magnitude of E-field with respect
to incident wave on the substrate without the gold bowtie versus frequency plot shows the resonance
around 395THz (b) Normalized magnitude (with respect to its own maximum) of E-field distribution in
xy plane in the middle of the gold bowtie at 395 THz. ............................................................................... 47
Figure 4.4 Normalized E-field versus distance plot along the tip-to-tip line (tip-to-tip linear mid-line
distance is 20nm) in the gap at the resonance (395THz). Also, field map (normalized with respect to its
own maximum). .......................................................................................................................................... 48
Figure 4.5 Normalized voltage over the optical frequency range at the mid-point in the gap. (a) magnitude
of voltage (b) phase angle of voltage .......................................................................................................... 49
Figure 4.6 Normalized magnitude of E-field versus the optical frequency range at a mid-point in the gap
by varying the gap distance of 5nm, 12nm, 20 nm, 40nm, and 80nm ........................................................ 50
Figure 4.7 Normalized magnitude of E-field versus frequency at a mid point with the fixed gap (20nm) by
scaling the size of the bowtie depending on the length of 70nm, 87.5nm, and 105 nm (see table). ........... 52
Figure 4.8 Normalized magnitude of E-field versus frequency of the bowtie structure at the mid-point in a
fixed gap (20nm) for varying the incident angles of 0º, 15º, 30º, and 45º with respect to the normal
incidence: (a) Transverse Magnetic (TM) wave. (b) Transverse Electric (TE) wave. ................................ 54
Figure 4.9 Bowtie aperture optical antenna structure in 3D and top view: (a) 3D view shows its general
structure with three materials (gold, ITO, silica) (b) 2D top view of the gold bowtie antenna that shows
the substrate’s dimensions (450x450nm2), x-gap (20nm), y-gap (12nm), side length (80nm), and width
(70nm) ......................................................................................................................................................... 56
Figure 4.10 Excitation and mesh view of the bowtie aperture structure (a) Side view of the bowtie shows
the permittivity of the substrates. The excitation comes from the top and E-field is polarized in y-axis (b)
Mesh configurations from the top view shows more tetrahedral concentrated in the gold and gap. (c)
Mesh view from the side of the bowtie aperture antenna. .......................................................................... 58
Figure 4.11 Normalized magnitude of E-field of the bowtie aperture at the mid-point in the gap versus
frequency in THz ........................................................................................................................................ 59
Figure 4.12 Field distribution along the mid-line in the gap of the bowtie aperture at 195THz: (a)
Normalized E-field versus distance plot of mid-line in the gap. (b) 2D normalized field distribution in the
gap including the mesh view....................................................................................................................... 60
Figure 4.13 Normalized voltage over the optical frequency range in the gap: (a) Magnitude of normalized
voltage (b) Phase angle of the normalized voltage ..................................................................................... 61
Figure 4.14 Normalized magnitude of E-field versus infrared and optical frequency range at a mid-point
in the gap with varying y-gap distance by 5nm, 12nm, and 20 nm ............................................................ 62
Figure 4.15 Normalized magnitude of E-field versus infrared and optical frequency range at a mid-point
in the gap with varying x-gap distance by 20nm, 40nm, and 80 nm .......................................................... 63
Figure 4.16 Normalized magnitude of E-field versus frequency at a mid point with the fixed x-gap (20nm)
and y-gap(12nm) with scaling the sizes of the bowtie depending on the length of 70nm, 87.5nm, and 105
nm (see table). ............................................................................................................................................. 65
v
Figure 4.17 Normalized magnitude of E-field versus frequency of the bowtie aperture structure at the
mid-point with fixed gap with varying the incident angle 0º,15º, and 45º with respect to the normal
incidence: (a) Transverse Magnetic (TM) wave. (b) Transverse Electric (TE) wave. ................................ 67
Figure 4.18 Hertzian electric dipole coordinate settings: (a) Location of the Hertzian dipole (red arrow in
the middle of the gap). (b) Cartesian (x,y,z) and spherical (r,θ,φ) coordinates. ......................................... 69
Figure 4.19 Far-field radiation pattern (in dB) for Hertzian dipole with substrate and with bowtie aperture:
Hertzian dipole only (Red; inside) with the substrate (at -180º, maximum is 162dB). Hertzian dipole with
bowtie aperture antenna (Blue; outside) and substrate (at -180º maximum is 201dB). .............................. 71
vi
LIST OF TABLES
Page
Table 4.1 Bowtie optical antenna parameters and field enhancement results............................................. 53
Table 4.2 Bowtie aperture optical antenna parameters and field enhancement results with oblique
incidence ..................................................................................................................................................... 55
Table 4.3 Bowtie aperture optical antenna parameters and field enhancement results ............................... 67
Table 4.4 Bowtie optical antenna parameters and field enhancement results with oblique incidence ....... 68
Table 4.5 Hertzian dipole with substrate only case and bowtie aperture antenna case for theta angle of 0º
with the distance in y-gap of 5nm, 12nm, and 20nm .................................................................................. 71
vii
ACKNOWLEDGMENTS
I would first like to thank my advisor, Professor Filippo Capolino, who has given me the
guidance, insights, and support throughout my research. I would also like to thank my committee
members, Professor Ozdal Boyraz and Professor Regina Ragan, who gave me valuable advices. I
would like to thank Professor Syed Jafar for giving me the opportunity to work as his grader.
Most importantly, I want to thank those people who spent lots of time and effort to revise
this thesis. I truly appreciate Salvatore Campione who encouraged and guided me to the right
direction. I would also like to thank Faezah Tork Ladani for your effort and support. I feel very
thankful for Canar Guclu who was always supportive and giving me valuable comments. Moreover,
I thank Ananth Tamma for giving me invaluable assistance and advices.
Finally, I would like to thank my parents who always supported me with limitless love. To
Kyung Hyun whose support has always been my source of strength and inspiration. Your love and
support helped me to past the hard times.
It is great pleasure to thank everyone. I was able to finish this thesis with your help. Thank
you.
viii
ABSTRACT OF THE THESIS
Bowtie and Bowtie Aperture Antennas at Microwave and Optical Frequency
By
RackGeun Kim
Master of Science in Electrical and Computer Engineering
University of California, Irvine, 2013
Professor Filippo Capolino, Chair
The use of optical nano-antennas to circumvent the diffraction limit by amplifying the light
signal has been one of the goals in nano-plasmonics. This thesis exploits the idea of optical
antennas by investigating the bowtie and bowtie aperture structures to enhance the electric field
due to an incident signal. Due to the analogy of controlling electromagnetic waves with classical
antennas, the thesis first provides reviews and theoretical descriptions of classical antennas. We
then design and analyze bowtie and bowtie aperture antennas at microwaves. In the second half of
the thesis, we examine the important optical phenomena that accompanies with surface plasmons
in a metal-dielectric interface. By using the finite element method, we implement and characterize
the optical antennas in terms of field enhancement by gap dependence, structure dimensions, and
oblique incidence. As a result, bowtie and bowtie aperture optical antennas strongly enhance the
electric field in the gap at near-infrared and optical frequencies. Moreover, a new approach that
can explain the interaction of light scattering of a molecule and a nano-antenna is studied. Near
field interactions were analytically investigated by employing a Hertzian dipole excitation of the
nano-antenna and a dyadic Green’s function formulation. We finally compute the enhancement
factor and the required hypothetical polarizabiltiy that maximizes the field enhancement when a
newly defined critical condition is most nearly satisfied.
ix
CHAPTER 1 INTRODUCTION
1.1 Motivations and Goals
As modern technology advances, electronic, optical, and mechanical devices have been minimized
and their sizes vary even from tens of nanometers to a few microns with unique resonant
characteristics. Optical antennas are being studied and developed extensively with the growth of
optical devices and nanotechnology. This thesis exploits the fundamental idea of optical nanoantennas that can convert light into localized energy on a sub-wavelength scale. Although it takes
substantial effort and time to numerically calculate all the optical phenomena, studies on
interactions between light and antennas as well as optical properties can apparently make
breakthroughs.
Starting with classical antennas, this study will use the characteristics of propagating
electromagnetic radiation and localized fields usual of microwave antennas, for optical frequencies.
Since optical antennas play a critical role in the fields of nano-optics, spectroscopy, and
microscopy for biological research, the main objective is to explore and develop both theory and
applications of optical antennas. We will review literature history and theoretical background with
the properties in metal-dielectric interface. Then, we will demonstrate bowtie and the bowtie
aperture optical antennas with field enhancements and resonant characteristics. Near field
interactions between molecules and light with optical antennas will be examined and discussed.
1
1.2 Classical Antennas versus Optical Antennas
(a)
(b)
(c)
Figure 1.1 Examples of antennas: (a) Yagi-Uda Antenna consists of an array of dipole antennas. (length scale
of meter) (b) Micro-strip Bowtie Antenna. (length scale of centimeter) (c) Nano particles act as an Optical
Antenna. (length scale of hundred nanometer)
Antennas are the devices that convert electromagnetic radiation propagating in free space into
guided electromagnetic energy and vice versa. It can also be defined as a fundamental component
of any wireless communication systems and connects links between the transmitters and receivers.
Various designs of the antennas are employed for different purposes such as radios, televisions,
and smart phones. Figure 1.1 shows different types of antennas used depending on the application.
The Yagi-Uda antenna is commonly used for microwave applications and the dimensions range
from a few centimeters to meters. The bowtie antenna is widely used for wideband applications at
microwaves and the dimensions are typically in centimeters. The small snano-particles acting as
antennas are adopted for biomedical and optics researches and their dimensions are in the order of
a few hundred nanometers. Although their scales vary considerably, they all function similarly at
different frequencies. However, for nano-particles, there have been challenges due to optical
2
phenomena and high losses at visible frequency range. Light has been traditionally manipulated
and controlled by means of optical devices such as lenses, mirrors, and prisms. In optics, a
conventional microscope has a limitation of spatial resolution known as the diffraction limit which
is approximated by roughly a half of the wavelength depending on the incident angle. Since there
is a fundamental limit for any optical imaging system, different approaches have been made and
the concept of optical antennas has been applied to break the diffraction limit with the evanescent
and near fields produced by the nanostructures. Moreover, while the word antenna is often used as
a common name at microwaves, the term optical antenna explicitly describes antennas operating
at near infrared and optical frequencies. The objective of this thesis is to develop and analyze the
optical antennas with metal-dielectric interfaces consisting of the classical antenna theory.
1.3 Background and Historical survey
James Clerk Maxwell unified and formulated previous observations, experiments, equations, and
theories of electricity, magnetism, and optics into a set of profound equations known as Maxwell’s
Equations which were first published in 1861 [1]. In 1865, Maxwell proposed that light is also an
electromagnetic wave [2,3]. In 1886, the first antenna system and wireless transmission was
carried out by German physicist Heinrich Rudolph Hertz. Hertz conducted experiments with more
scientific methods to validate Maxwell’s equations. Hertz could successfully produce an electric
3
spark in the gap of a half wavelength dipole antenna. Guglielmo Marconi, an Italian inventor, sent
and received a wireless signal and letter information across the Atlantic Ocean by 1901. Although
Marconi had patent disputes with Nikola Tesla over the first wireless radio transmitter, Marconi’s
contributions to wireless radio communications had a huge impact. By World War II, antennas
had become ubiquitous and had transformed human lives with televisions and radio. Of all the
major inventions of the twentieth century, cellular phones have had a more profound impact on
people’s lives than any other electronic device. In 2011, according to the International
Telecommunication Union (ITU), global mobile cellular subscribers were nearly six billion while
the world population is about seven billion. In other words, a person carries more than one antenna
in the cell phones, laptops, or GPS. These applications of the antenna are mainly used at microwave
frequencies. The history of optics has started around 300 B.C. and Greek mathematician Euclid
was the one who studied and described the law of reflection and the geometrical optics. Later on,
a lot of efforts were made by many physicists, mathematicians, even philosophers. Those
inventions of glass lenses and microscopes have helped us study astronomy, biology, and
chemistry. Before we discuss recent advances of optical antennas, it is important to review their
history. In optics, lens has a diffraction limit which is roughly half of a wavelength of light. In the
early 1870s, Ernst Abbe presented the mathematical equation relating the best possible resolution
for two objects in light. According to Abbe’s equation in 1873, the achievable resolution with the
4
light is close to two hundred nanometers [4]. In 1928, Edward Hutchinson Synge developed an
idea of the use of a small aperture to image sub-wavelength resolution by localizing the light
radiation on the surface and it was theoretically possible to surpass the diffraction limit in optical
imaging with help of Albert Einstein [5-7]. However, due to the technical limitations, Synge’s idea
was forgotten until Bethe and Bouwkamp who theoretically investigated the wave propagation of
the a small aperture in a conducting plane in 1944 and 1950 [8,9]. Bethe’s study concentrated on
the effect of a small aperture in the cavity of microwave radiation. According to his demonstrations,
the electromagnetic radiation can be confined to a region that is smaller than the diffraction limit.
In 1956, John O’Keefe first developed a near-field microscope [10]. The first demonstration was
followed by Ash E. A. and Nichols G. with breaking the diffraction limit with microwave radiation
in 1972 [11]. In 1985, John Wessel mentioned the analogy between the classical antennas and the
gold particles or local microscopic light sources [12]. He also explained that the particle can serve
as a receiving antenna for the incoming electromagnetic field. The first experiment was followed
by Dieter W. Pohl and Ulrich C. Fischer in 1989 [13,21]. Their research in light microscopy was
focused on the technology known as near-field scanning optical microscopy (NSOM). Fortunately,
they not only discovered that Synge’s and Wessel’s proposals were feasible, but also carried out
multiple experiments for higher spatial resolution [13-24]. The first system of an NSOM was
developed by the Eric Betzig group at Bell Laboratories in the 1990s and the group proposed
5
improved methods and techniques [25-28]. Hans Kuhn also suggested near-field imaging for
molecules near the metal surface and carried out his idea and experiments [29-31]. In 1994, nearfield microscopy was first used to image the single florescent molecule [32,33]. With the
introduction of NSOM, different groups proposed the use of techniques such as tunneling
microscopy and the laser-irradiated metal tip of the tweezers for near-field enhancement and light
trapping [34]. During recent years, more creative techniques and various structures of optical
antennas were proposed and studied [34]. Some of the important studies are reviewed as we
progress.
1.4 Scope of the Thesis
The thesis describes the theory and applications of optical antennas that deal with the optical nearfield characteristics of metal nano-structures. This study will present basic constituents of optical
antennas and provide a qualitative overview of important characteristics. We first analyze bowtie
and bowtie aperture antennas at microwave frequencies to be familiarized with and to understand
the physics behind either antenna; we then extend the study to sub-wavelength scaled structures at
near infrared and visible range that may be employed for improving optical devices.
In Chapter 2, the background of electromagnetism and antennas is first reviewed. At microwave,
it introduces the theoretical analysis of the bowtie and its aperture antennas. The fundamental
6
definitions and standard parameters are measured to show the performance of the antenna, which
includes radiation patterns, directivity, radiation properties, bandwidth, efficiencies, polarizations,
and input impedances. Moreover, the idea of Babinet’s principle for complementary structures is
reviewed and discussed for radio frequency applications.
In Chapter 3, the physical properties of optical nano-antennas are presented. We also review
primary properties including the electric conductivity of the metal and its dispersive characteristics.
Collective oscillations of free electrons in the medium are approximated by classical models of
Drude and Lorentz. These approximations are also compared with the experimental data by Palik.
For optical properties, we will discuss the surface plasmons and surface plasmon polaritons on
metal-dielectric interfaces considering the dimensions and materials. Using the plasmon resonance
characteristics, one of the techniques called Surface Enhanced Raman Scattering (SERS) to detect
single molecules on the metal surface is studied. There is also a short discussion of the Finite
Element Method (FEM), which was used to simulate the optical antennas. Eventually, we will
compare and contrast physical characteristics of antennas at radio and optical frequencies.
In Chapter 4, we demonstrate and analyze the characteristics of bowtie and bowtie aperture optical
plasmonic nano-antennas. We will discuss how we design and model bowtie and bowtie aperture
structures using the commercial finite element method in HFSS by Ansys Inc. The resonant
characteristics of the optical antennas are examined and optical field enhancement and localization
7
in a deep sub-wavelength region of the antennas is studied extensively with numerous simulations
using the near field perspective and plasmonic effects. Moreover, the transmitting antenna concept
is applied with Hertzian dipole exciations.
In Chapter 5, we investigate the interactions of scattered fields of the antenna, molecular
polarizability, and dipole moment with the dyadic Green’s function and Hertzian Dipole
excitations.
In Chapter 6, the conclusion and various applications of nano-technologies are discussed. We
briefly mention also the techniques for sub-wavelength spectroscopy, photovoltaic effect, and
sensors. We will also provide discussions of future work using the bowtie and bowtie aperture
optical antennas.
8
CHAPTER 2 BACKGROUND OF BOWTIE ANTENNAS AT RADIO FREQUENCY
2.1 Introduction: Bowtie Antennas
The classical antenna theory is essential for understanding the concept of the optical antennas.
Therefore, in this chapter, the basic microwave antenna theory is introduced. The antenna theory
can be applied to various types and shapes of antennas such as wire, aperture, micro-strip, array,
reflector, lens, and so forth, eventually including the bowtie antennas under study in this thesis.
For infinitesimally thin wire antennas, we can easily approximate the radiation properties by using
the far-field dipole approximations. Because the bandwidth characteristic of the antenna is very
sensitive to frequency, wider bandwidth antennas were desirable. Wide bandwidth characteristic
requires that the antenna has nearly a constant input resistance and negligible reactance, as a
consequence, the reflection at the antenna ports is fairly low.
Wide band antennas find
application in areas where reception from a wide range of frequency channels is crucial. There
have been challenges and methods to find more practical geometries of wider band characteristics.
The biconical antenna, as a wide-band antenna solution, represents a simple geometry and the
characteristics can be approximated by using transmission line analogy by Schelkunoff [37].
Biconical antennas were employed and used for many years but the geometrical structure was
expensive to build and impractical to use. Therefore, triangular sheet and bow-tie antennas known
to have wider bandwidth characteristics were introduced and investigated by Brown and
9
Woodward [38]. The resistance and reactance fluctuate more in triangular sheet and bowtie
antennas than biconical antennas. However, the bowtie antenna is a balanced antenna which has
low cross-polarizations. The property of broader bandwidth when compared to the regular wire
dipole had led to further developments such as broadband arrays and Sierpinski fractal patches
[36].
The micro-strip antenna was invented by Bob Munson in 1972 [39,40]. Later, various types of
micro-strip antennas have been developed such as patch antennas and PIFA [36]. Having a very
low profile, light weight, low fabrication cost, and small volume were advantageous for portable
technologies and micro-strip antennas have been the basic radiation unit for most of the devices.
The performance of bowtie antennas with varying lengths and angles at microwave range were
examined in previous studies [41-43] leading to ground rules for bowtie antenna design.
Employing finite element method (FEM), the characteristics of the bowtie and the bowtie aperture
antennas at microwave are going to be studied in this chapter as a means of providing the first
insight to their radiation performances.
10
2.2 Antenna Theory
(b)
(a)
ZA
XA
RLoss
Vs
RRad
Voc
ZL
Figure 2.1 Transmitting and receiving antennas: (a) Transmitting model. The radiated power is represented by
the radiation resistance. (b) Receiving antenna where the generator (Voc) represents the picked up signal as a
voltage generator.
Figure 2.1 shows the equivalent circuit models for the transmitting and receiving antennas.
The
transmitting antenna model in Figure 2.1 (a) shows the radiation resistance which represents the
radiated power while its source Vs is supplying the power. The receiving antenna model in Figure
2.1 (b) shows the voltage generator (Voc) which represents the picked up signal. Without
considering the load ZL , the impedance ZA on the receiving mode is equal to the sum of the
impedances on the transmitting mode, i.e. ZA = (R Rad + R Loss ) + jXA . Here, starting from basic
antenna theory, I provide the derivations for important equations to understand the concept of
antennas. At first, the time averaged Poynting vector calculations are going to be reviewed. For
more details, readers can find the general information in the third edition of Antenna Theory by C.
11
A. Balanis [36]. To build bottom-up discussions, we use the Maxwell’s equations to describe the
time dependent currents [1]. Assuming the time-harmonic convention ejωt , the current density is
given by
∇ ∙ () = jωρ().
(2.1)
where ρ is the charge density at the location .
Also, under Lorentz guage the magnetic vector potential, , can be written using the electric scalar
potential, Φ, as
∇ ∙ () = jωμεΦ(),
(2.2)
where μ = μ0 μr , ε = ε0 εr .
Therefore, the Helmholtz equations become
2π 2
[∇2 + ( ) ] () = −μ(),
λ0
[∇2 + (
And hence both
2π 2
ρ()
) ] Φ() = −
.
λ0
ε
(2.3)
(2.4)
equations obtain the form
[∇2 + (
2π 2
) ] f() = −g().
λ0
(2.5)
To derive the scalar Green’s function, G0 (,  ′ ), we replace g() with a single point source using
Dirac-delta function. Then, we have
2π 2
[∇2 + ( ) ] G0 (,  ′ ) = −δ( −  ′ ).
λ0
(2.6)
where  denotes the location of the field point and  ′ designates the location of the point source.
12
Using the scalar Green’s function, G0 (,  ′ ), the magnetic vector potential and the electric scalar
potential can be written in terms of current density as
() = μ ∫ ( ′ ) G0 (,  ′ )dV ′ ,
(2.7)
V
Φ() =
1
∫ ρ( ′ ) G0 (,  ′ )dV ′ .
ε
(2.8)
V
The physical solution in free space using (2.6) is then
′
G0 (, 
′)
e−jk|− |
=
.
4π| −  ′ |
(2.9)
Going back to the Maxwell’s equations [1-3], we have
() = ∇ × (),
(2.10)
() = −∇Φ() − jω().
(2.11)
where E denotes the electric field and B denotes the magnetic induction.
Until now, the formulation established the link between the time-harmonic electric field and the
currents as its source. For dipole antennas, its current density can be easily derived by assuming
the infinitesimally thin wire and feed line.
In Antenna theory, power and energy associated with electromagnetic fields are important
radiation properties. The averaged Poynting vector represents the flux of electromagnetic energy,
and at any point in space it is defined in [36] as
1
ave = Re( ×  ∗ ).
2
13
(2.12)
where E denotes the electric field and H denotes the magnetic field. The total power radiated from
a volume can be calculated by integrating the Poynting vector over the surface enclosing the
volume as follows
1
Ρ = ∯ Re( ×  ∗ ) ∙ d.
2
(2.13)
S
In the far zone, fields decay as 1/r and thus the pointing vector assume 1/r2 dependence. The
definition of the radiation intensity is the radiated power over solid angle in the far-field
represented as
U(θ, φ) = r 2 WRad .
(2.14)
Note that the radiation intensity has no dependence on the distance, r, while it bears dependence
on the angular position in the space. The integration of the radiation intensity over a certain solid
angle thus leads to the mathematical expression for the radiated power in the angular region
denoted by the solid angle, Ω, given as
2π
π
PRad = ∯ U ∙ dΩ = ∫ ∫ U sin θ dθdϕ .
0
Ω
(2.15)
0
By using the radiation intensity, one can define directivity which is a measure of the antennas
ability to resolve a source’s angular position in space. Directivity of an antenna under test is defined
as its radiation intensity divided by the radiation intensity of an isotropic antenna that radiates the
same amount of power as the antenna under test. The divisor can be also referred to as the average
14
radiation intensity, found by the total power divided by 4π, the solid angle corresponding to a
surface of a whole sphere. Then directivity of an antenna can be written as
D(θ, φ) =
4πU(θ, φ)
.
PRad
(2.16)
For lossless isotropic source, the gain G(θ, φ) is expressed as
G(θ, φ) =
4πU(θ, φ) 4πU(θ, φ) PRad
PRad
=
= D(θ, φ)
,
Pin
PRad
Pin
Pin
(2.17)
where Pin = PRad + PLoss and PRad = ℯPin .
Thus the radiation efficiency ℯ takes the following representations,
ℯ=
G(θ, φ)
R Rad
=
.
D(θ, φ) R Loss + R Rad
(2.18)
The open-circuit voltage Voc of the receiving antenna is related to the incident electric field Einc ,
by defining the effective height or length by
heff =
Voc
.
Einc
(2.19)
As a receiving antenna, a certain amount of power is extracted by the antenna and delivered to a
load for detection of the incoming signal. However, the current induced on the antenna by the
incoming electric field results in re-radiation as scattered fields at the same time. This causes the
mismatch between the total electric field from the incident wave and the electric field on the
antenna. Since the incident power is expressed in terms of electric field Pinc = |E|2 /2η where
μ
η = √ ε , the relationship between the total power delivered by incident wave and the received
power PR is written as
15
PR = ℯPRad = Ae Pinc ,
(2.20)
where Ae is the effective area and ℯ is the radiation efficiency. Moreover, the maximum
effective area Aem can be expressed as Aem = eap Ap where Ap is the physical area and eap
is the aperture efficiency.
2.3 Analysis on RF Bowtie Antennas
FR-4 Substrate (1.66 mm thick)
z
y
We = 30 mm
Gap = 1 mm
x
L/2
W = 10 mm Copper (35 μm thick)
Wc = 10 mm
Port
S = 37.4 mm
Le = 60 mm
Figure 2.2 (Top view) Detailed design of the bowtie antenna with a lumped port excitation polarized in y-axis
in the gap (Conductivity of copper: 5.8X107 Siemens/m and FR-4 substrate r=4.4)
The bowtie antenna with a lumped port excitation operating at 2.5GHz is designed as shown in
Figure 2.2. 35μm thick copper is used assuming the standard printed circuit board (PCB utilizing
FR-4 substrate with the relative permittivity of 4.4 and the physical parameters provided in Figure
16
2.2.). Copper is one of the mostly used metals for antennas due to the high conductivity, 5.8X107
Siemens/m, combined with ductility. At the wavelength 120mm (frequency of 2.5GHz), the length
of the copper S is set close to a quarter wavelengths (≈30mm) as shown in Figure 2.2. The
conventional and modified bowtie structured micro-strip antennas at microwave have been
characterized for wide bandwidth [41-43], here the main objective of this chapter is only to make
comparisons and contrasts between the bowtie and bowtie aperture antenna designs by the means
of input impedance characteristics, radiation patterns, bandwidth, and polarization. The antenna is
characterized by means of full wave simulations via HFSS 14 that utilizes finite-element-method
in phasor domain. The design procedure of a micro-strip bowtie antenna is similar to the design of
rectangular micro-strip antenna. The resonant frequency of a bowtie structured antenna’s
parameters, the dominant TM10 mode, can be approximated by Transmission line model as follows
from [42-43]:
fr ≅
1
1.152
c
1.152
=
≅ 2.5 GHz,
2L√εeff √μ0 ε0 R t
2L√εeff R t
L (W + 2∆L)(Wc + 2∆L)
≅ 0.965,
2 (W + 2∆L)(S + 2∆L)
W
(εeff + 0.3) ( i + 0.262) h
h
∆L = 0.412
≅ 11.3 mm,
Wi
(εeff − 0.258) ( + 0.813)
h
Rt =
(2.21)
(2.22)
(2.23)
1
εeff
εr + 1 εr − 1
h −2
=
+
[1 + 12 ] ≅ 3.475,
2
2
W
W + Wc
Wi =
≅ 6 mm.
2
17
(2.24)
(2.25)
where the thickness, relative, and effective permittivity of the substrate are denoted by h, εr and
εeff, respectively.
(b)
1
0
-10dB Bandwidth
Zero Crossings
20
0.8
-10 (2.25GHz,-10dB)
Mag of S11
dB(S(1,1))
-5
(2.71GHz,-10dB)
0
0.6
-20
0.4
-40
0.2
-60
-15
2.50GHz
1
2
3
Frequency [GHz]
4
5
40
0
0
Phase of S11
(a)
1
3.80 GHz
2
3
4
Frequency [GHz]
5
-80
6
Figure 2.3 (a) Reflection coefficient S11 (in dB) of Bowtie Antenna operating at 2.4GHz. (b) Magnitude of S11
and phase angle of S11. Zero-crossings occurred at 2.5GHz and 3.8GHz.
As shown in Figure 2.3, the reflection coefficient S11 determines the range of operation frequency
of the antenna. The phase of S11 exhibits two resonances at 2.5GHz and 3.8GHz. The bowtie
antenna is designed to operate at 2.5GHz and as a result, and it is apparent that most of the signal
is radiated at 2.5GHz. The reflection coefficient of -10dB designates 10% of reflected power at the
input terminal of the antenna; therefore, more than 90% of the signal power is transmitted in the
frequency range of operation. From 2.25GHz to 2.71GHz, the reflection coefficient is within the
range of operation. The bowtie antenna provides 460MHz of bandwidth and the resonant
18
frequency is 2.5GHz. Therefore, the percent bandwidth of operation of the bowtie antenna is 19.2%
which performs better in terms of bandwidth than rectangular micro-strip patch antennas.
(a)
(b)
500
Re(z) = 235 Ω
2
300
200
0
Re(z) = 73Ω
100
Gain [dB]
Impedance(Z) [Ohms]
400
4
Real(Z)
Imag(Z)
Im(z) = 0Ω
0
-100
-200
-2
-4
-6
-300
-400
-500
0
(2.5GHz,-0.07dB)
2.50GHz
1
-8
3.80 GHz
2
3
4
Frequency [GHz]
5
-10
0
6
1
2
3
4
Frequency [GHz]
5
6
Figure 2.4 (a) Input impedance versus frequency plot shows both real part and imaginary part of the impedance
of the bowtie antenna. The resonance is at 2.5GHz where the imaginary Z (reactance) is zero. At 2.5GHz, the
impedance is approximately 73Ω. (b) Gain versus frequency plot. Gain increases as the frequency increases. At
resonant frequency, the gain is -0.07dB
In Figure 2.4 (a), the input impedance versus frequency plot shows that the input impedance is
approximately 73Ω at 2.5GHz and 235Ω at 3.8GHz, respectively, while the reactance reaches zero.
Since the signal is fed from 50Ω port, there are reflections due to the mismatch with 73Ω. Gain
versus frequency plot is shown in Figure 2.4 (b). Although there are ways to match the input
impedances, here we try not to include the impact of feeding line when making a comparison. Gain
of the bowtie antenna increases at higher frequencies and the gain is 0.98 or negative 0.07 dB at
19
2.5GHz where S11 is the lowest. Since this bowtie antenna is designed to have a quarter wave
antenna characteristics, the gain is small, close to 0dB, i.e. unity. At microwave, the bowtie
structured antennas can be scaled to operate at different frequencies by varying the sizes of the gap,
length and width. At the same time, tradeoffs such as resonant frequency, bandwidth, gain, and
efficiency must be considered.
Radiation Pattern 5
Bowtie
ANSOFT
Curve Info
(a)
dB(rETotal)
Setup2 : Sweep
Freq='2.4GHz' Phi='0deg'
(b)
0
-30
dB(rETotal)_1
Setup2 : Sweep
Freq='2.4GHz' Phi='90deg'
30
13.00
6.00
-60
(dB)
60
-1.00
-8.00
-90
90
H-plane
E-plane
-120
-150
120
150
-180
Figure 2.5 Farfield radiation of the bowtie antenna at 2.5GHz: (a) 2D plot shows the total E-field in the polar
plot as the radiation pattern (b) 3D plot of directivity in far-field.
Figure 2.5 shows the far-field radiation pattern in 2D and 3D at 2.5 GHz. As we closely look at the
far-field, the radiation pattern is omni-directional which has a similar characteristic to the dipole
antenna. In Figure 2.5 (a), the pattern shows that the magnetic field is in H-plane is almost constant.
20
Figure 2.5 (b) displays the total directivity (in dB) in the far-field. General characteristics of
transmitting bowtie antennas are discussed. However, according to the reciprocity, the radiation
and receiving patterns of transmitting and receiving antennas are identical.
2.4 Analysis on RF Bowtie Aperture Antennas
z
y
We = 30 mm
x
W = 10 mm
Gap = 1 mm L/2
FR-4 Substrate
(1.66 mm thick)
Wc = 10 mm
Port
S = 37.4 mm
Copper (35 μm thick)
Le = 60 mm
Figure 2.6 (Top view) Detailed design of bowtie aperture antenna with lumped port excitation polarized in xaxis at the gap (Conductivity of copper:.  ×  / and FR-4 substrate  = . )
The aperture version of the bowtie micro-strip antenna is designed by using the exact same
parameters with lumped port excitation at the gap as shown in Figure 2.6. Since there are no
analytic formulations for design, as a starting point the identical shape of bowtie in Section 2.3 is
subtracted from the copper film of 35µm thickness and in addition a gap in the center is formed
21
by removing the metal strip as an excitation point. Apparently this is not exactly a complementary
structure because the feeding port at the center is not metalized, left as a gap, for both of the designs.
The lumped port is applied in the gap polarized in x-direction as in Figure 2.6. Also, the gap has
the same dimensions of 1mm in length and 2mm in width as the previous bowtie antenna design.
The physical difference between the bowtie and bowtie aperture is that the bowtie aperture
structure has more copper metal surface area and 90º angular rotation in polarization. Thus, the
bowtie aperture antenna has the lumped port excitation polarized in the y-axis.
0
1
200
160
120
80
-10dB Bandwidth
-10
0.8
(3.80GHz,-10dB)
(5.25GHz,-10dB)
-15
Mag of S11
dB(S(1,1))
-5
0.6
40
0
-40
0.4
0.2
-20
2.25GHz
4.65GHz
-25
-80
-120
-160
Zero Crossings
2
4
6
Frequency [GHz]
0
0
8
1
4.65GHz
2 3 4 5 6
Frequency [GHz]
7
Phase of S11
(b)
(a)
-200
8
Figure 2.7 (a) Reflection coefficient S11 (in dB) of the bowtie aperture antenna operating at 4.65GHz.
(b) Magnitude of S11 and phase angle of S11. Resonances occurred at 2.25GHz, 4.65GHz and 8GHz.
There are three resonances occurred at 2.25GHz, 4.65GHz, and 8GHz as shown in Figure 2.7 (b).
22
Figure 2.7 (a) shows that the antenna has the resonant frequency at 4.65 GHz. The -10dB
bandwidth is 1.45 GHz ranging from 3.8GHz to 5.25GHz leaves 1.45GHz centered at 4.65GHz
and the percent bandwidth of operation is 32%. Compared to the bowtie structure, the bowtie
aperture antenna exhibited a higher bandwidth and the resonance frequency is shifted from 2.5GHz
to 4.65GHz. The minimum reflection coefficient reached -23dB.
1600
1400
1200
1000
800
600
400
200
0
-200
-400
-600
-800
0
(b)
10
Real(Z)
Imag(Z)
8
(4.65GHz,6.87dB)
6
Re(z) = 1300Ω
4
Im(z) = 0Ω
Re(z) = 42 Ω
Gain [dB]
Impedance(Z) [Ohms]
(a)
2
0
-2
-4
-6
2.25GHz
1
-8
4.65GHz
2
3
4
5
6
Frequency [GHz]
-10
0
7
1
2
3
4
5
Frequency [GHz]
6
7
Figure 2.8 (a) Input impedance versus frequency plot shows both the real part and imaginary part of the
impedance of the bowtie aperture antenna. At 2.25GHz, the impedance is approximately 1300Ω. At 4.65GHz,
the impedance is 42Ω (b) Gain versus frequency plot. Gain increases as the frequency increases.
Figure 2.8 (a) shows the input impedance versus frequency plot of the bowtie aperture structure.
When the zero reactance occurred at 2.25GHz and 4.65GHz, the resistances of the input impedance
were 1300Ω and 42Ω, respectively. Since the signal is fed by a 50Ω port, there are very little
23
reflections due to the mismatch. According to the gain versus frequency plot in Figure 2.8 (b), the
gain was increased at higher frequencies and the gain reaches the highest value of 4.80 or 6.87 dB
at 4.65GHz. Compared to the bowtie antenna, the bowtie aperture antenna had much higher gain
at the resonance. However, the gain can be changed with matching conditions. The interesting
phenomenon is that the input impedance abruptly increased at around 2.25GHz, which is close to
the resonance of the bowtie antenna in Section 2.3. Although the bowtie aperture is not exactly
complementary of the bowtie structure, it might imply that we can exploit this relationship to
design the complementary and aperture structures.
Radiation Pattern 2
Comp1
ANSOFT
Curve Info
(a)
(b)
0
-30
30
dB(rETotal)
Setup1 : LastAdaptive
Freq='2.4GHz' Phi='0deg'
dB(rETotal)_1
Setup1 : LastAdaptive
Freq='2.4GHz' Phi='90deg'
8.00
1.00
-60
(dB)
60
-6.00
-13.00
-90
90
H-plane
E-plane
-120
-150
120
150
-180
Figure 2.9 Far-field radiation of the bowtie aperture antenna (y-axis polarization): (a) 2D plot shows the total
E-field in the polar plot as the radiation pattern (b) 3D plot of directivity in far-field.
24
In the far-field plot, the directional radiation pattern is omni-directional which is similar to the
bowtie structure. However, higher radiation is concentrated along z direction. In Figure 2.9 (a), the
pattern shows that the field is that the field in E-plane is almost constant due to the polarization
difference compared to Figure 2.5. Figure 2.9 (b) displays the total directivity in the far-field.
General characteristics of transmitting bowtie aperture antennas are discussed and by introducing
the Babinet’s principle, the comparisons and contrasts are discussed in the chapter summary.
2.5 Summary: RF Antennas and Babinet’s Principle
In conclusion, we have reviewed the classical antenna theory and discussed the essential
characteristics of the bowtie and bowtie aperture structures. The micro-strip type of bowtie and
bowtie aperture antennas were designed and simulated. Comparisons and contrasts between the
bowtie and bowtie aperture antenna designs were made by the means of input impedance
characteristics, radiation patterns, bandwidth, and polarizations. To compare the input impedance,
Babinet’s principle was considered. Babinet’s principle in optics states that the sum of the
transmitted wave on the screen with a small aperture and the wave behind the screen is equal to
the wave under the condition of no screen. For antenna applications, Babinet's principle can be
used for formulating a strong relationship in the gain and input impedance only if the screen is a
perfect conducting and indefinitely thin plane [44]. Later, H.G. Booker extended the idea by
25
introducing the polarization and practical conducting screens for microwave applications [45].
However, even before we apply the limitations mentioned above, it is important to stress that the
bowtie and bowtie aperture antennas are not exact complementary to each other. The
complementary shape of the bowtie would be the bowtie aperture whose feed point would be filled
with copper instead of a gap. Therefore, the Babinet’s principle could not be considered to compare
the impedance characteristics of bowtie and bowtie aperture structures.
It is important to note that RF antennas are usually fed from the wires so that their input impedances
are essential characteristics to determine the performance. On the other hand, the optical antennas
are not fed by wires due to high losses and metal reflections. Optical antennas usually receive the
incident wave at a certain angles and the enhanced field is scattered outwards. In next chapters, we
discuss the optical properties and eventually introduce the optical antennas.
26
CHAPTER 3 CHARACTERISTICS OF OPTICAL NANO-ANTENNAS
3.1 Introduction
Many modern technological advancements ranging from microcopy to optical communications
are possible due to precise control over light. Dielectric materials are conventionally used to build
optical components due to their transparency and low losses. Conversely, metals have a high
absorption and large reflectivity. However, interesting optical properties can be observed at the
metal-dielectric interface and is the subject of the field of Plasmonics. Plasmonics is the study of
the behavior of coupled electromagnetic and electron wave oscillations that occurs at a metaldielectric interface. In this chapter, we introduce the fundamental properties and phenomena
behind plasmonics. Historically, the first quantitative studies of light scattering from spherical gold
colloidal particles were performed in 1908 by Gustav Mie [50]. Subsequently, in 1957, Rufus
Ritchie theoretically predicted the existence of surface plasmons on thin metallic films [51]. In
1988, Thomas Ebbesen demonstrated an enhancement in transmitted light intensity from a
subwavelength hole in a silver sheet at 730nm wavelength [46,47] and further experiments with
different geometries were performed by Henri Lezec [47-49].
27
(a)
(b)
(c)
k
Figure 3.1 Examples of plasma oscillation: (a) Localized nano-particle at plasma frequency. (b) Bulk plasmons
oscillation with the free electron density and effective mass. (c) Propagating surface plasmon polaritons (SPPs)
interacting with light.
Plasmons can be classified as bulk or surface plasmons and depending on the shape of the metaldielectric interface, surface plasmons can be further viewed as localized or propagating surface
plasmons as detailed schematically in Figure 3.1. In Figure 3.1 (a) describing a localized surface
Plasmon resonance, an external electric field is imposed on the metal nano-particle and displaces
the free conduction electrons. The resulting positive ions create a restoring field such that the
electrons collectively oscillate at resonance. Figure 3.1 (b) shows the bulk plasmon oscillations
which are associated with purely transverse electromagnetic fields. Figure 3.1 (c) shows confined
propagating surface plasmons at the planar interface between metal and dielectric layers. Surface
plasmons and enhanced transmission of light through a subwavelength hole have pioneered the
field of plasmonics. In this thesis, we focus on the field enhancements due to sub-wavelength
confinement of light in nano-antennas. Optical nano-antennas convert propagating light into
confined fields and help concentrating radiation in a small region. Exploiting the characteristics of
28
optical antennas has enabled applications including fluorescence and single molecular sensing
using Surface Enhanced Raman Spectroscopy (SERS) [52-55], sub-wavelength lithography [56],
color filtering and imaging [57], energy harvesting for photovoltaic devices [58], magnetic
recording [59], higher harmonic generation [60], and more efficient organic light emitting diodes
(OLED) [61,62]. In order to understand optical antennas qualitatively, it is important to understand
the interaction between nano-antennas and light. Starting from the dispersive characteristics of
metals at optical frequencies, this chapter is devoted to provide a qualitative understanding of
optical antennas.
3.2 Dispersive characteristics of Metals: Drude model and experimental data
The resonance condition of monopole and dipole antennas is determined by the physical length of
the antennas. A monopole antenna receives or transmits a wireless signal most efficiently when it
is exactly quarter wavelength long. A dipole antenna can be resonantly excited when its length is
half of the free space wavelength. The Fundamental principle of optical antennas is the same as
antennas at microwave regime except the dispersive metal permittivity, high losses in the metal
and presence of surface Plasmon polaritons. At optical frequencies, metals no longer conduct
electrons without restraint since there is a strong frequency dependency in conductivity and
therefore the skin depth plays an important role. Skin depth is defined as the depth of penetration
29
of the electromagnetic wave in conductors or metals. If a metal has an infinite thickness, current
flows only in a certain depth of the metal. In most radio frequency applications, the skin effect is
negligible when compared to the metal thickness of the antenna. The skin depth of metals in the
visible region goes tens of nanometers depending on the conductivity of that particular metal.
Therefore, skin effect needs to be considered when designing an optical antenna due to the high
losses and because, depending on the frequency, most nano-antennas use metal thickness of the
order of few tens to 100 nm.
At optical frequencies, the conductivity of frequently used metals such as gold (Au), silver (Ag),
copper (Cu), and aluminum (Al) decreases with increasing frequency. Therefore, metals can be
treated as lossy dielectrics with complex values of permittivity and with negative real part of
permittivity. The electrical conduction characteristics of metals at optical frequencies can be
represented by the Drude Model. The optical properties of metals were first explained by Paul
Drude in the context of conductivity. The characteristics of metals at optical frequencies can be
approximated by the Drude Model while assuming that the free-electrons are not bound to a
particular nucleus. Paul Drude proposed the Drude model of electrical conductivity in 1900 [66,67]
and he explained the validity of Ohm’s law. The classical Drude model gives a good estimate of
conductivity in metals. From the Ohm’s law expressions,
30
(ω) = [
σ0
] (ω),
jω
1+(γ)
(3.1)
Where, ω is the angular frequency of the light wave, γ is the collision frequency, and σ0
denotes the static conductivity.
For an applied oscillating field, we have
σ(ω) =
σ0
.
jω
1+(γ)
(3.2)
As described by the model, at very high frequencies, the conductivity becomes purely imaginary,
i.e., 90˚ out of phase with the applied field. Considering an ideally free electron and linear,
homogeneous, isotropic media, the permittivity model of the classical Drude model [63] can be
derived as
εr (ω) = 1 −
ωp 2
ωp 2
ωp 2 γ
=
(1
−
)
−
j
(
),
ω2 − jωγ
ω2 + γ2
ω3 + ωγ2
ωp 2 = γσ0 c 2 μ0 ,
(3.3)
(3.4)
where, ωp is the plasma frequency, ω is the angular frequency of the light wave, γ is the
collision frequency where γ = 1/τ and τ is the mean free time between collisions. However, this
is modified by Arnold Sommerfeld [37] with quasi-free-electron model as
εr (ω) = ε∞ −
ωp 2
ωp 2
ωp 2 γ
=
(ε
−
)
−
j
(
).
0
ω2 − jωγ
ω2 + γ2
ω3 + ωγ2
(3.5)
For gold metals, the Drude-Sommerfeld model parameters ε∞ = 9.5, ωp = 1.36 × 1016 rad/s,
and γ = 1.05 × 1014 rad/s were used in [65-67].
31
The Drude-Sommerfeld model assumed quasi-free electrons in the metal. When compared to the
experimentally measured complex permittivity data for gold [68], the theoretical model diverges
from experimental data after a certain wavelength as plotted in Figure 3.2 (a). To convert the
refractive index to complex permittivity,
ε(ω) = ε′ (ω) − jε′′ (ω)
(3.6)
where ε′ is the real permittivity and ε′′ is the imaginary permittivity. We express
ε′ = Re(nc 2 ) = (n2 − k 2 ),
(3.7)
ε′′ = −Im(nc 2 ) = 2nk,
(3.8)
and we define nc = n − jk, and we define ‘n’ as the refractive index and ‘k’ as extinction coefficient.
The complex permittivity can be defined in terms of a loss tangent,
ε = ε′ (1 − j tan δ),
(3.9)
ε′′
tan δ = ′ .
ε
(3.10)
where, the loss tangent is defined as
32
(b)
(a)
300
5
Drude: Imag ()
Palik: Imag()
Solid: Palik
Dotted: Drude
200
Permittivity ()
Permittivity ()
100
0
-100
-200
0
-5
-300
-400
-500
Real ( )
Imag ( )
500 1000 1500 2000 2500 3000
Wavelength [nm]
-10
200
400
600
800 1000
Wavelength [nm]
1200
Figure 3.2 Comparison of Drude Model (theory) and experimental data [68]: (a) Experimental data (solid) and
theroretical Drude Model (dotted). Blue colored line represents the real part while red line represents the
imaginary part of the permittivity. (b) Difference in imaginary part of the Drude Model (Theory) and
experimental data (Exp) for wavelength range 200nm≤λ≤1200nm.
When comparing the theoretical Drude Model with the experimental data, Figure 3.2(a) shows that
the real part of gold’s permittivity has a good agreement in the wavelength range from 500nm to
1.2μm. At higher wavelengths, the disagreement increases in both real and imaginary components
because in reality, free electrons condition no longer applies. Figure 3.2(b) plots only the imaginary
component of permittivity from 200 nm to 1200 nm and shows the large mismatch from 200nm to
700nm caused by inter-band transitions in the metal [35]. Therefore, instead of using the classical
Drude-Sommerfeld model, experimental data [68] for gold was used in the frequency range from
1THz to 800THz with interpolation in all the simulation data presented in this thesis. With
33
increasing wavelength, the permittivity of metals like gold becomes close to negative infinity and
the therefore metals can be treated as perfect conductors at radio frequencies. But at optical
frequencies, gold has negative numbers for its real component of permittivity. The characteristics
of optical antennas are predominantly dictated by the conductivity of electrons in metals.
3.3 Surface Plasmons (SPs) and Surface Plasmon Polaritons (SPPs)
z
Dielectric
δd
x
δm
Ez
Indium Tin Oxide [ε = 2.47]
H
Metal
Figure 3.3 Surface plasmons (SPs) and surface plasmon polaritons (SPPs): collective oscillations at metal and
dielectric interface.
As discussed in the previous section, surface plasmons are caused by the electron density
oscillations at the interface of two materials. In the case of metal-dielectric interface, the fixed
positive ions in the metal cause the electrons to oscillate by interacting with electromagnetic
waves. The surface plasmon resonance (SPR) is similar to the bulk plasmon resonance. As surface
34
plasmons are coupled with the light, the resulting excitation is called the surface plasmon
polaritons, which propagates along a surface of the metal. When the charge density was applied at
the metal-dielectric surface with the incident wave, the field is the maximum at the interface and
the evanescent field quickly decays away from the interface as shown in Figure 3.3. Optical
antennas exploit this phenomenon and have been used for spectroscopic measurements by surface
enhancement. For bulk plasmons considering metal and dielectric surface, the Maxwell curl
equations are expressed as
∇ × i = −jωε0 εi i ,
(3.11)
∇ × i = −jωμ0 i ,
(3.12)
where i = d (d denotes dielectric) with z > 0 and i = m (m denotes metal) with z < 0.
For a TE mode, no surface mode exists since we want wave propagation in the direction of x axis
with evanescent decaying in z-axis.
For a TM mode,
i = (Exi , 0, Ezi ),
(3.13)
i = (0, Hyi , 0),
(3.14)
∂i
.
∂
(3.15)
and are related by
∇ × i = εi
Where,
35
i = (0, Hyi , 0)exp{−j(k xi x + k zi z − ωt)}
(3.16)
i = (Exi , 0, Ezi ) exp{−j(k xi x + k zi z − ωt)} .
(3.17)
and
Solving the curl equation,
(
∂Hzi ∂Hyi ∂Hxi ∂Hzi ∂Hyi ∂Hxi
−
,
−
,
−
) = (jk zi Hyi , 0, −jk xi Hyi ) = (jωεi Exi , 0, jωεi Ezi ). (3.18)
∂y
∂z ∂z
∂x ∂y
∂y
Assuming continuous boundary conditions for Hyd = Hym and Exd = Exm ,
k zm k zd
=
,
εm
εd
(3.19)
where k zi ≡ k zm = k zd .
The wavevector (k) components satisfy below,
ω 2
k 2 = k x 2 + k zi 2 = εi ( ) .
c
(3.20)
Since k sp ≡ k x ,
2
k sp
ω
= √εi ( ) − k zi 2 .
c
(3.21)
Therefore, a complex wavenumber kx of surface plasmon dispersion for bulk media is expressed
by using (3.20, 3.21)
kx = k′x + jk′′x =
In z-axis,
36
ω
εm εd
.
√
c εm + εd
(3.22)
kzi = k′zi + jk′′zi =
ω
ε2
√ i
,
c εm + εd
(3.23)
where
εm = ε′m − jε′′
m.
(3.24)
For a bound surface plasmon mode, k x is purely real and k zi is imaginary. Then, ε′m < −εd .
The real and imaginary components can be expressed as follows assuming εm ′′ ≪ |εm ′ |,
k ′x
ω
ε′m εd
ω ε′m εd
≅ √ ′
, k ′′
≅
−
(
)
c εm + εd x
c ε′m + εd
3/2
(
ε′′
m
).
2(ε′m )2
(3.25)
For surface plasmon polariton (SPP) wavelength,
λSPP ≅
2π
=
k ′x
λ
ε′ ε
√ ′m d
εm + εd
,
(3.26)
where, λ is the wavelength of light excitation.
The propagation length L is expressed as,
L=
1
|2k′′x |
.
(3.27)
Therefore, equation 3.22 gives the dispersion relation which determines the SPP wavelength and
propagation length.
37
5
Energy [eV]
Plasmon
4 Polariton
Light line
(Photon)
damping
3
2
Surface Plasmon
1
0
0
0.1
-1
0.2
0.3
kx [nm ]
Figure 3.4 Surface plasmon dispersion curves for Au/ITO Substrate by using Drude Model and light line where:
It shows the radiative, quasi-bound, and bound surface plasmon dispersion relations for gold indium tin oxide
film.
Figure 3.4 shows that if we compare dispersion of light in air with that of the surface plasmon
polaritons, the wavevector k does not couple with surface plasmon modes at far-field. Using
higher index medium is one of the techniques to couple light to surface plasmon polaritons. For
example the use of SiO2 prisms can enable the light excitation; however, the thickness of SiO2 is
strongly related with the incident angle [69]. We show Figure 3.4 to give a formal discussion of
the aspects and nature of plasmon and surface plasmon modes at metal-dielectric interface. Bulk
38
plasmon oscillations do not directly relate the idea of field enhancements by optical antennas since
they cannot interact or couple with light.
3.4 Field Enhancements for Surface Enhanced Raman Scattering (SERS)
Raman scattering was discovered by Sir C.V. Raman in 1922 [70,71] and is one of the most
important effects in analyzing the behavior of light interaction with matter. When photons are
scattered from an atom or molecule, the majority of scattered photons have the same energy as the
incident photons. This phenomenon is called Rayleigh scattering [72] and it represents the electric
polarizability of the particle. At the same time, a small number of scattered photons lose or gain
energy thereby having a different frequency than the incident photon.
This effect is called Raman
effect and scientists exploited this scattering behavior in chemical spectroscopy the in the 1970’s.
The surface enhanced Raman scattering was first discovered by Fleischmann in 1977 [73]. Also,
theoretical studies predicted the enhancement in metal clusters in 1980 [74]. Around the same time,
advances in the field of scanning near-field optical microscopy pioneered the field of microscopy
as well as spectroscopy [52-55]. Thus, the new era of near-field optics successfully began and over
the past years, new experimental techniques and developments had been proposed.
39
3.5 Summary: Analogy between the Radio Frequency and Optical Antennas
After reviewing the basic optical properties, it is now easier to differentiate regular microwave
antennas and optical nano-antennas. At optical frequencies, especially in the visible region, surface
plasmons and polaritons play a major role because the collective excitation of the electrons in the
metal propagates and couples light with nano-structures. Therefore, it is a plausible idea that an
aperture in the bulk metal could exhibit higher intensity in a sub-wavelength scaled region.
Moreover, the interaction between light and nano-structures could result in strong scattering,
absorption, and field enhancement at resonance. As derived in the previous sections, the resonance
in the nano-structures is not the same as the microwave regime since surface plasmons occur. The
finite conductivity of metals is also a significant factor that has effects on the oscillation of charges.
We reviewed the major aspects of optical properties and more qualitative analysis and bottom-up
discussions were delivered in this chapter.
40
CHAPTER 4 BOWTIE AND BOWTIE APERTURE ANTENNAS AT OPTICAL
FREQUENCY
4.1 Introduction
In optics, a precise control of light has led to studies of structures that deal with near field and
evanescent fields in deep sub-wavelength regions of space. After acknowledging the analogy with
the classical antenna concept [12], dimer structures such as dipole and bowtie antennas recently
gained much attention. Because of the lightening rod effect [35], bowtie structures have the ability
to confine large electric fields from two sharpened metal points with a radius of curvature smaller
than the incident wavelength. Prior research investigated the optical properties of bowtie shaped
antennas in the optical frequency range [75-92]. Some studies experimentally proved the gap is
the main factor that leads to enhanced E-field in bowtie antennas [76,79,89]. The near-field
confinement and enhancement due to capacitive coupling in a bowtie structure was reviewed in
[86]. Previous studies of material dependence and high transmissions of bowtie apertures can be
found in [93-96]. Further studies of bowtie aperture structure can be found in [107-111].
Since these noble bowtie and bowtie aperture structures exhibit unique properties, it was necessary
to explore the characteristics of propagating electromagnetic radiation and localized fields. One of
the major physical differences between microwave and optical antennas is that optical antennas do
not have the feeding wire. Although it might be possible to use fiber optic cables for transmitting
41
enhanced light signal by optical antennas, it is hard to imagine that metal wires with high losses
can be fed to optical antennas. Due to these limitations, optical antennas are currently used only
for spectroscopic purposes. However, the study of input impedance concept in an optical nanoantenna can be found in [88].
As discussed in Chapter 3, surface plasmons are found by the electron density oscillations between
the (metal and dielectric interface. When coupled to electromagnetic waves, the metal and
dielectric surface exhibits the collective oscillations in a localized field. Control and manipulation
of light and electromagnetic fields by optical antennas can make improvements in different
applications, such as Surface Enhanced Raman Spectroscopy (SERS) [52-55] and the Near-field
Scanning Optical Microscopy (NSOM) [35,102]. Therefore, the bowtie and bowtie aperture
antennas at optical frequencies are thoroughly examined in this chapter by looking at field
distribution, oblique incidence illumination, gap dependence, and thickness dependence. In
particular, the resonant characteristics of the optical antennas are analyzed and optical localized
field enhancement is studied extensively with numerous simulations. A quantitative approach by
controlling and manipulating the structures of bowtie and bowtie aperture is performed.
42
4.2 Receiving Antennas: Analysis of Bowtie Optical Antennas
(a)
(b)
Figure 4.1 Bowtie antenna structure in 3D and top view: (a) 3D view shows its general structure with three
materials (gold, ITO, silica) (b) 2D top view of the gold bowtie antenna that shows the substrate’s dimensions
(450x450nm2), gap (20nm), side length (80nm), and width (70nm)
The optical properties including the permittivity of the metal are implemented on the simulations
employing the experimental results from Palik [68] for gold. Gold is widely used in
nanotechnology because of the resistance to surface oxidation which results in a greater longevity.
Figure 4.1 represents the general structure and design parameters of the bowtie and the substrate.
Similar to the micro-strip antenna at the radio frequency counterpart, two triangular structured gold
metals, which form a structure called bowtie, are placed on the dielectric surface. The thickness of
the gold is 20nm and the side of the bowtie is 80nm. The radius of curvature at the edge of the
bowtie shape is chosen to be 9nm to minimize the errors from the mesh calculations for finite43
element method (FEM) when compared to the sharp edges. The gap between the triangles is 20nm
and is varied to observe the changes in field-enhancement in the middle of the gap. In Figure 4.1,
Indium Tin Oxide, whose thickness is 50nm and refractive index is 1.95, is placed below the gold
nano-structure. Indium Tin Oxide (ITO) is composed of indium (III) oxide and tin (IV) and it is
generally used at optical frequencies because it reduces the charging effect during the electron
beam lithography (EBL). Moreover, its electrical conductivity and optical transparency are
exceptional because most transparent conducting films lose its transparency as it gets more
conductive. Using the electron beam lithography (EBL), the ITO film can be deposited on the
substrate, fused silica. Fused silica is one form of silicon dioxide (SiO2) which has a high purity
of silica in amorphous or non-crystalline form which means that it has a high melting point about
2000˚C. The typical usage of fused silica was to make optical fibers and semiconductors because
of its thermal stability. The purpose of getting very high field enhancements in the gap might cause
a lot of heat from the light and this is why fused silica was chosen to be the substrate. The substrate
is placed below the Indium Tin Oxide. The thickness of fused silica is 500nm and its refractive
index is 1.47.
44
(a)
(b)
(c)
Side view
Figure 4.2 Excitation and mesh view of the bowtie structure (a) Side view of the bowtie shows the permittivity
of the substrates. The excitation comes from the top and E-field is polarized along the x-axis (b) Mesh
configurations from the top view shows more tetrahedral concentrated in the gold and gap. (c) Mesh view from
the side of the bowtie antenna.
Figure 4.2 (a) represents the side view and the dielectric’s permittivites for ITO as 3.8 and Fused
Silica as 2.16. In HFSS simulations, the Floquet ports excitation method is applied together with
the master and slave boundaries conditions. This in turns models a 2D periodic set of bowties. For
simplicity of calculations, we assume that bowties are arrayed in a square lattice, with a period of
45
450nm, large enough so that arrays unit cells are weakly coupled to insignificantly affect the
maximum field between the bowties. Therefore, the Floquet port is primarily enables the simulated
structure to be limited to a unit cell. From mesh view in Figure 4.2 (b), the tetrahedrons are used
for integration. Highest concentration of mesh is in the gap and gold metal because field variations
are there expected, lower concentration in the substrate.
The incident plane wave is propagating toward negative z direction with TEM mode excitation
with the electric field along the x-axis. Therefore, the bowtie optical antennas are polarized along
the x-axis. Infrared and visible frequency ranging from 200THz to 500THz excitation signals are
applied on the bowtie nano-structure with substrate. Moreover, the same signal is applied without
the bowtie structure (i.e., only the multilayered environment is present) and the electric field E0 at
the point of interest is used for normalization purposes.
46
(a)
(b)
45
Field at the point
40
35
|E|/|E0|
30
25
20
15
y
10
5
0
250
300
350
400
450
Frequency [THz]
z
500
x
Figure 4.3 E-field distribution in the gap of the bowtie: (a) Normalized magnitude of E-field with respect to
incident wave on the substrate without the gold bowtie versus frequency plot shows the resonance around
395THz (b) Normalized magnitude (with respect to its own maximum) of E-field distribution in xy plane in the
middle of the gold bowtie at 395 THz.
Our interest is in the gap effects to field enhancement values because the edge of each triangular
structure concentrates the field near the metal resulting in higher field confinement in the gap.
Figure 4.3 (a) shows the normalized magnitude of E-field versus frequency in the middle of the
gap as shown in Figure 4.3 (b). It shows that the normalized magnitude of the electric field has a
resonant peak at 395THz because the magnitude of the electric field and oscillation is maximum
at resonance. The normalized magnitude of E-field is 40 times larger than the incident field at the
resonant frequency and drops quickly over the frequency. Figure 4.3 (b) shows the normalized
47
magnitude of E-field distribution in xy plane in the gap at 395THz. The magnitude of E-field is
higher near the gold metal and is mainly concentrated around the gap.
80
Field along the tip-to-tip line
|E|/|E0|
60
40
20
0
0
5
10
dist(nm)
15
20
Figure 4.4 Normalized E-field versus distance plot along the tip-to-tip line (tip-to-tip linear mid-line distance is
20nm) in the gap at the resonance (395THz). Also, field map (normalized with respect to its own maximum).
Figure 4.4 clearly demonstrates that the normalized magnitude of the electric field is minimum at
the mid-point of the line and as it approaches to the metal, the normalized magnitude of the electric
field becomes higher and hits the maximum. Since the confined field is higher near the gold metal,
we are going to reduce distance of the gap and observe the behavior in the next section.
48
Using the antenna theory, the confined field in the gap of the gold metal can be seen as a voltage
source. To find the voltage drop in the gap of the bowties, we integrate the electric field along the
tip-to-tip line as
A
∆Φ = ΦA − ΦB = − ∫  ∙ dl.
(4.1)
B
By performing the integration of the real and imaginary part of the electric field, the change in
magnitude and phase of the voltage is calculated and reported in Figure 4.5.
(a)
(b)
50
200
Resonance
Phase Angle [degrees]
|V|/|V0|
40
30
20
10
0
250
0
Zero Crossing
-100
395THz
395THz
300
350
400
Frequency [THz]
100
-200
250
450
300
350
400
Frequency [THz]
450
Figure 4.5 Normalized voltage over the optical frequency range at the mid-point in the gap. (a) magnitude of
voltage (b) phase angle of voltage
49
The normalized magnitude of voltage is plotted in Figure 4.5 (a) and its shape of the graph is
almost same as the normalized magnitude of E-field. Figure 4.5 (b) shows that the phase angle
change had a zero crossing at resonance.
4.2.1 Gap dependence
Although previously analyzed [76,79,89], we report and confirm here that the gap largely affects
the field enhancement properties of bowtie structures.
200
gap=5nm
gap=12nm
gap=20nm
gap=40nm
gap=80nm
180
160
|E|/|E0|
140
120
gap
70 nm
100
80
60
40
20
0
250
300
350 400 450
Frequency [THz]
500
Figure 4.6 Normalized magnitude of E-field versus the optical frequency range at a mid-point in the gap by
varying the gap distance of 5nm, 12nm, 20 nm, 40nm, and 80nm
50
In Figure 4.6, the plot demonstrates the gap dependence of the bowtie optical antenna at the mid
point in the gap by plotting the normalized magnitude of E-field versus the frequency in terahertz.
The highest normalized magnitude of electric field at resonance of 370THz was close to 170 when
the gap in the x-axis is 5nm. The lowest normalized magnitude of electric field at resonance
frequency of 405THz was close to 6 when the gap in the x-axis is 80nm. Varying the gap changed
the electric field confinement significantly. As the gap in the x-axis increases, the magnitude of
the electric field decreases and vice versa. Using the circuit resonance analogy, increasing the gap
between the two metals can make it less capacitive. Conversely, if two metals are placed closely,
it becomes more capacitive thus justifying the observed blue shift.
4.2.2 Resonance adjustments by the size
Another factor that changes the field confinement is the bowtie size. Varying the size of the bowtie
structure with respect to the length can also help on achieving even higher local field enhancement.
51
length=70nm
length=87.5nm
length=105nm
60
50
width gap=20nm
length
|E|/|E0|
40
30
Scaling
20
10
0
200
250
300 350 400
Frequency [THz]
450
Width
Length
80nm
70nm
100nm
87.5nm
120nm
105nm
500
Figure 4.7 Normalized magnitude of E-field versus frequency at a mid point with the fixed gap (20nm) by
scaling the size of the bowtie depending on the length of 70nm, 87.5nm, and 105 nm (see table).
The sizes of the bowtie were scaled depending on its length and plotted with fixed gap distance at
20nm in Figure 4.7. As shown above, the bigger the bowtie is, the higher the electric field
enhancement is, though the variation of the maximum value is not dramatic (from about 40 to
about 43). In particular, the difference was not as large as the one notices for the gap dependence
as shown in Figure 4.6. Besides, there was a larger difference in the resonance. When the length
was 70nm, the resonance was at 395THz. However, when the length was 105nm, the resonance
dropped to 335THz. The resonance is red-shifted for increasing bowtie length. The following
52
Table 4.1 summarizes the gap, and size dependence results. Note that the local field intensity
enhancement factor can be calculated by
2
Menhanced
|E|
=(
) .
|E0 |
(4.2)
and scattered field of Raman Scattering is often approximated by |E|4 as
4
Senhanced ≅
2
Menhanced
|E|
=(
) .
|E0 |
(4.3)
where E0 is the incident field with the substrate and E is scattered field with the metal. This is
also reported in Table 4.1 and shows that bowtie structures yield low detection limits generating
108-109 fold signal enhancements for (chem/bio) molecular detection in surface enhanced Raman
scattering (SERS) measurements that provide a molecular fingerprint of analyte molecules of
interest as recently shown in [113,114] for nanosphere clusters. Large control is observable by
structural parameters of the bowtie antenna.
Table 4.1 Bowtie optical antenna parameters and field enhancement results
Length
Gap
Thickness
Resonant Freq
Wavelength
[nm]
[nm]
[nm]
[THz]
[nm]
70
5
20
370
811
170
.×8
70
12
20
390
769
70
.×7
70
20
20
395
759
40
.×6
87.5
20
20
370
811
47
.×6
105
20
20
335
896
44
.×6
70
40
20
400
750
15
.×4
70
80
20
405
741
6
.×3
53
||/||
Raman Enhancement
(||/||)
4.2.3 Changes with Oblique Incidence
(a)
(b)
35
30
25
20
15
15
10
5
5
350
400
450
Frequency [THz]
500
θ
k
20
10
0
300
TE(0deg)
TE(15deg)
TE(30deg)
TE(45deg)
30
|E|/|E0|
|E|/|E0|
25
35
TM(0deg)
TM(15deg)
TM(30deg)
TM(45deg)
0
300
z
y
350
400
450
Frequency [THz]
x
500
Figure 4.8 Normalized magnitude of E-field versus frequency of the bowtie structure at the mid-point in a fixed
gap (20nm) for varying the incident angles of 0º, 15º, 30º, and 45º with respect to the normal incidence: (a)
Transverse Magnetic (TM) wave. (b) Transverse Electric (TE) wave.
In Figure 4.9, the normalized magnitudes of E-field versus frequency at the mid-point for oblique
incidence cases for both TM and TE waves are plotted. For oblique incidence, we want to avoid
the presence of higher order modes in the substrate. We thus look for the angles for which they
appear at a given frequency. It can be derived from
k 2 − k x,n 2 = 0.
(4.4)
where n is the integer. By assuming k as a free space wavenumber, it becomes
2πn 2
) = 0,
d
2πn
k = (k sin θ −
),
d
k 2 − (k x 2 −
54
(4.5)
(4.6)
where d denotes the length of the period (substrate). Then,
2πn
= k(sin θ − 1).
d
(4.7)
As we solve for θ by using k = 2π/λ,
nλ
θ = sin−1 ( + 1) .
d
(4.8)
nλ
θ = sin−1 ( + √εr ) .
d
(4.9)
With the substrate, it is expressed as
With the period of 450nm as in Figure 4.1, higher order modes would appear in the frequency
range analyzed in Figure 4.9. Therefore, we decreased the size of the substrate to 250nm by 250nm.
When λ is 600nm at 500THz, we get θ ≅ 68˚. Therefore, for the angles analyzed in Figure 4.9,
we can understand the pattern of oblique incidence cases without the effects of the higher order
modes.
Table 4.2 Bowtie aperture optical antenna parameters and field enhancement results with oblique incidence
Raman
Incident
Length
x-gap
Resonant
Wavelength
Angle (º)
[nm]
[nm]
Freq [THz]
[nm]
TM
0
70
20
405
741
23.0
2.80×105
TM
15
70
20
405
741
20.4
1.73×105
TM
30
70
20
405
741
16.4
7.23×104
TM
45
70
20
405
741
13.6
3.42×104
TE
0
70
20
405
741
23.0
2.80×105
TE
15
70
20
405
741
22.9
2.75×105
TE
30
70
20
405
741
22.3
2.47×105
TE
45
70
20
405
741
21.1
1.98×105
Wave
55
|E|/|E0|
Enhancement
(|E|/|E0|)4
The resonance was fixed at 405THz and the magnitude changed accordingly with the incident
angle as shown in Table 4.2. The resonance of TM wave at 45º was 405THz but the normalized
field was decreased to 13.6 while the normalized field of the TE wave at 45º was 21.1. Therefore,
the TE wave is not as sensitive as TM wave for the field enhancements. Although the incident
angle was varied, we could still observe the resonant frequency and field enhancements at the midpoint in the gap.
4.3 Receiving Antennas: Analysis of Bowtie aperture Optical Antennas
(a)
(b)
Au
Radius of curvature: 9 nm
x gap: 20 nm
80 nm
ITO
y
z
70 nm
450 nm
y gap: 12 nm
x
450 nm
Figure 4.9 Bowtie aperture optical antenna structure in 3D and top view: (a) 3D view shows its general structure
with three materials (gold, ITO, silica) (b) 2D top view of the gold bowtie antenna that shows the substrate’s
dimensions (450x450nm2), x-gap (20nm), y-gap (12nm), side length (80nm), and width (70nm)
56
The bowtie aperture structure is shown in Figure 4.10 for the bowtie aperture shape: the gold is
deposited outside of the bowtie shape and the gap is 20nm (x-gap) by 12nm (y-gap). The entire
bowtie shape was substracted from the gold metal excluding the gap. Again, Palik’s experimental
data for gold was applied in bowtie aperture antenna simulations. The thicknesses for gold, ITO,
and fused silica remained the same as in Figure. 4.1.
57
(a)
(b)
y
z
x
(c)
Side view
Figure 4.10 Excitation and mesh view of the bowtie aperture structure (a) Side view of the bowtie shows the
permittivity of the substrates. The excitation comes from the top and E-field is polarized in y-axis (b) Mesh
configurations from the top view shows more tetrahedral concentrated in the gold and gap. (c) Mesh view from
the side of the bowtie aperture antenna.
The bowtie aperture structure should be excited with an illumination that has a 90˚ phase rotation
in the xy plane with respect to the bowtie in Section 4.2 and thus the electric field is here polarized
58
along the y-axis. Figure 4.11 (b) represents the mesh configurations of the structure. Tetrahedrons
are evenly distributed over the gold metal and they are more concentrated in the gap.
100
x-gap=20-nm
Mid-Point
80
70 nm
|E|/|E0|
y-gap=12nm
60
40
20
0
50
100 150 200 250 300
Frequency [THz]
350
Figure 4.11 Normalized magnitude of E-field of the bowtie aperture at the mid-point in the gap versus frequency
in THz
Figure 4.12 shows the normalized magnitude of E-field versus frequency plot at the mid point in
the gap. The normalized magnitude of E-field goes up to 80 at the resonance 195THz and drops
quickly over the frequency. When compared to bowtie structure of 12nm gap (≈70), the bowtie
aperture structure has slightly more normalized field magnitude.
Figure 4.13 represents the normalized magnitude of E-field distribution in xy plane in the gap at
195THz. The bowtie structure exhibited the behavior of higher normalized field near the metal. In
constrast to bowtie structure, the field distribution looks almost constant in the gap as seen in
59
Figure 4.13 (b). This is a very interesting phenomenon which can be applied to single molecule
spectroscopic applications as a larger area with constant large fields is available for molecules to
attach to. To clarify, the normalized magnitude of the electric field at the resonance is plotted
versus frequency along the linear mid-line in the middle of the gap in Figure 4.13 (a).
(b)
(a)
100
Field along the line
80
|E|/|E0|
12nm
60
20nm
40
1
20
0
0
0.5
2
4
6
8
dist(nm)
10
12
y
z
x
0.1
Figure 4.12 Field distribution along the mid-line in the gap of the bowtie aperture at 195THz: (a) Normalized
E-field versus distance plot of mid-line in the gap. (b) 2D normalized field distribution in the gap including the
mesh view.
Figure 4.13 clearly presents that the normalized magnitude of the electric field is almost constant
(≈80) at the resonance of 195THz in the gap. As mentioned above, this behavior might give more
control for various applications. By using equation (4.1), the voltage drop along the line (y-axis)
of the bowtie aperture is calculated and tabulated.
60
(a)
(b)
100
Resonance
200
Phase Angle [degrees]
|V|/|V0|
80
60
40
20
150
100
50
0
-50
Zero Crossing
-100
-150
-200
0
195THz
195THz
150
200
Frequency [THz]
150
200
Frequency [THz]
250
250
Figure 4.13 Normalized voltage over the optical frequency range in the gap: (a) Magnitude of normalized
voltage (b) Phase angle of the normalized voltage
The normalized magnitude of voltage is plotted in Figure 4.14 (a) and the shape of the plot is
almost the same as the normalized magnitude of E-field in Figure 4.12, considering from 100THz
to 270THz. The phase angle versus the frequency is also plotted in Figure 4.14 (b) and the phase
angle has a zero crossing at resonance. We can view this as the voltage source and the enhaced
field produced by the bowtie aperture is proportional to the normalized voltage.
61
4.3.1 Gap dependence
To perform the gap dependence of bowtie aperture antenna in y-axis, we vary the distance of the
y-gap with a fixed distance of 20nm in x-axis.
200
180
160
y-gap=5nm
y-gap=12nm
y-gap=20nm
|E|/|E0|
140
x-gap=20nm
70 nm
y-gap
120
100
80
60
40
20
0
50
100 150 200 250 300
Frequency [THz]
350
Figure 4.14 Normalized magnitude of E-field versus infrared and optical frequency range at a mid-point in the
gap with varying y-gap distance by 5nm, 12nm, and 20 nm
In Figure 4.15, the plot demonstrates the gap dependence of the bowtie aperture optical antenna at
the mid-point in the gap by plotting the normalized magnitude of E-field versus frequency in
terahertz. The highest normalized magnitude of electric field at resonance of 175THz was close to
183 when the y-gap is 5nm. The lowest normalized magnitude of electric field at resonance
62
frequency of 210THz was 45 when y-gap is 20nm. Varying the y-gap changed the electric field
confinement significantly. As the gap in the y-axis increases, the magnitude of the electric field
decreases and vice versa. Using the circuit resonance analogy, increasing the gap between the two
metals can make it less capacitive. Conversely, if two metals are placed closely, it becomes more
capacitive, justifying the resonance blue shift. Behaviors of resonance shifteness and changes in
E-field confinement of both bowtie and bowtie aperture structures share the same aspects of gap
dependence in the polarized axis. Now, we fix the y-gap and vary the x-gap while looking at the
resonance and normalized E-field.
100
|E|/|E0|
80
x-gap=20nm
x-gap=40nm
x-gap=80nm
x-gap
70 nm
y-gap=12nm
60
40
20
0
50
100 150 200 250 300
Frequency [THz]
350
Figure 4.15 Normalized magnitude of E-field versus infrared and optical frequency range at a mid-point in the
gap with varying x-gap distance by 20nm, 40nm, and 80 nm
63
Figure 4.16 demonstrates x-gap dependence of the bowtie aperture antenna at the mid-point in the
gap from plotting the normalized magnitude of E-field versus the infrard and optical frequency in
terahertz. The highest normalized magnitude of electric field at resonance of 195THz was close to
78 when the x-gap is 20nm. The lowest normalized magnitude of electric field at resonance
frequency of 140THz was observed at 60 when x-gap is 80nm. Table 4.3 presents the results of
gap dependence of the bowtie aperture optical antenna. Varying the x-gap changed the electric
field confinement moderately since it was only down to 60 when the x-gap was 80nm. Comparing
the result to bowtie structure in Figure 4.6, it shows a significant difference because normalized
magnitude of E-field of the bowtie structure with x-gap of 80nm was only 6. As the x-gap of the
bowtie aperture increases, the resonance frequency has decreased or red-shifted. This gives an
extra control of resonance and the E-field confinement of the bowtie aperture optical antenna. By
controlling the distances of both x-gap and y-gap, we can manipulate the frequency of operation
as well as high field enhanacements. In addition, we observed almost constant field over the gap
at the resonant frequencies for various gap values. It means that the bowtie aperture structure can
have more space for molecules than bowtie with almost a constant field.
64
4.3.2 Resonance adjustments by the size
As seen in the section 4.2.2, bowtie size affects field confinement in the bowtie structure. Likewise,
varying the size of the bowtie aperture shape with respect to the length might also achieve a higher
local field enhancement, and this is inspected in this section.
120
length=70nm
length=87.5nm
length=105nm
100
|E|/|E0|
width
length
80
60
y-gap
12nm
Scaling
40
20
0
50
x-gap=20nm
100
150 200
250
Frequency [THz]
Width
Length
80nm
70nm
100nm
87.5nm
120nm
105nm
300
Figure 4.16 Normalized magnitude of E-field versus frequency at a mid point with the fixed x-gap (20nm) and
y-gap(12nm) with scaling the sizes of the bowtie depending on the length of 70nm, 87.5nm, and 105 nm (see
table).
The sizes of the bowtie aperture were scaled depending on its length and width and plotted with a
fixed x-gap(20nm) and y-gap(12nm) in Figure 4.17. As expected, the bigger the bowtie aperture
is, the higher the electric field enhancement is. However, the difference was not as large as the gap
65
dependence as shown in Figure 4.16, which affects more the resonance location rather than the
enhancement value. When the length was 70nm, the resonance was at 195THz. However, when
the length was 105nm, the resonance dropped to 145THz. The resonance was to be red-shifted as
the length is increased. The following Table 4.3 summarizes the gap and size dependance of the
bowtie aperture optical antenna.
66
Table 4.3 Bowtie aperture optical antenna parameters and field enhancement results
Raman
Lengt
x-gap
y-gap
Thickness
Resonance
Wavelengt
h [nm]
[nm]
[nm]
[nm]
Freq [THz]
h [nm]
70
20
5
20
175
1714
183
.×9
70
20
12
20
195
1538
78
.×7
70
20
20
20
210
1429
45
.×6
70
40
12
20
175
1714
70
.×7
70
80
12
20
140
2143
60
.×7
87.5
20
12
20
170
1763
83
4.75×07
105
20
12
20
150
1998
88
6.00×7
Enhancemen
||/||
t (||/||)
From the results, we presented that it is possible to control the resonance and field enhancements
with the size and gap distance of the bowtie aperture antenna.
4.3.3 Changes with Oblique Incidence
(a)
(b)
80
70
60
50
40
40
30
20
20
10
10
150
200
250
Frequency [THz]
300
k θ
50
30
0
100
TE(0deg)
TE(15deg)
TE(30deg)
TE(45deg)
70
|E|/|E0|
|E|/|E0|
60
80
TM(0deg)
TM(15deg)
TM(30deg)
TM(45deg)
0
100
z
y
150
200
250
Frequency [THz]
x
300
Figure 4.17 Normalized magnitude of E-field versus frequency of the bowtie aperture structure at the midpoint with fixed gap with varying the incident angle 0º,15º, and 45º with respect to the normal incidence: (a)
Transverse Magnetic (TM) wave. (b) Transverse Electric (TE) wave.
67
In Figure 4.19, the normalized magnitude of E-field versus frequency of the bowtie aperture
structure at the mid-point is plotted. With the period of 450nm as in Figure 4.10, higher order
modes would appear in the frequency range analyzed in Figure 4.19. Therefore, we decreased the
size of the substrate to 350nm by 350nm. Using equation (4.10), we can find the exact angle of
grating lobe. When λ is 857nm at 350THz, we get θ ≅ 78˚ by using the equation (4.10).
Therefore, we can understand the pattern of oblique incidence cases without the effects of the
higher modes as shown in Figure 4.19.
Table 4.4 Bowtie optical antenna parameters and field enhancement results with oblique incidence
Wav
Incident
Length
x-gap
y-gap
Resonance
Wavelength
Normalized
Enhanceme
e
Angle (º)
[nm]
[nm]
[nm]
Freq [THz]
[nm]
|E|
nt (|E|/|E|)
TM
0
70
20
12
200
1500
58.5
1.17×107
TM
15
70
20
12
200
1500
54.1
8.57×106
TM
30
70
20
12
200
1500
45.6
4.32×106
TM
45
70
20
12
200
1500
38.6
2.22×106
TE
0
70
20
12
200
1500
58.5
1.17×107
TE
15
70
20
12
200
1500
58.0
1.13×107
TE
30
70
20
12
200
1500
56.7
1.03×107
TE
45
70
20
12
200
1500
54.2
8.63×106
As shown in Table 4.4, the resonance frequency stayed the same and the magnitude change was
occurred over the entire frequency range. For TM0º, the resonance was at 200THz and the
normalized field was 47.5. At 45º, the normalized field reaches 30 at 195THz. For TE As a result,
68
the magnitude of field in bowtie aperture optical antennas were greatly affected by the angle of
incidence as well as bowtie antennas shown in Figure 4.8.
4.4 Transmitting Antennas: Bowtie aperture Antenna with Hertzian Dipole
(b)
(a)
z
r
Hertzian Dipole
20nm
y
z
12nm
x
x
l/2
θ
l/2
φ
y
Figure 4.18 Hertzian electric dipole coordinate settings: (a) Location of the Hertzian dipole (red arrow in the
middle of the gap). (b) Cartesian (x,y,z) and spherical (r,θ,φ) coordinates.
Reciprocity states that the properties of transmitting and receiving antennas are identical. Using
the reciprocity theorem in electromagnetics, the model of incident wave coming from the top
represents the receiving antenna while the excitation from the gap represents the transmitting
antenna. We place a Hertzian dipole in the gap to look at far-field of the bowtie aperture optical
antenna as if it is transmitting a signal. As shown in Figure 4.20 (a), the radiated fields of a Hertzian
1
jη
electric dipole are defined as  = μ ∇ ×  and  = − k ∇ × . The fields radiated by this dipole
69
are outlined in Sec. 9.2 in [115] or in [116], and reported here as
() =
where v =
c
√ εr
vk2
4π
(̂ × )
e−jkr
r
1
(1 + jkr),
(4.10)
., and
1
() = 4πε {k 2 (̂ × ) × ̂
e−jkr
r
1
jk
+ [3̂(̂ ∙ ) − ] (r3 + r2 ) e−jkr }.
(4.11)
with  the electric dipole moment. Note that the electric vector lies in the plane defined by ̂
and . As seen in Figure 4.20 (b), one nanometer radius Hertzian dipole is inserted in the middle
of the gap of the bowtie aperture optical antenna when x-gap is 20nm and y-gap is 12nm. Instead
of applying the periodic boundaries, we use the single bowtie aperture antenna with Hertzian
dipole to transmit a signal. The Hertzian dipole wave excitation (I∙Dipole Length=1A∙m) is applied
in the y-axis in HFSS because of the polarization of the bowtie aperture. To look at the far-field,
the perfect matched layer is placed at a distance of a quarter wavelengths outside of the bowtie
aperture structure.
70
0
-30
198.00
30
186.00
-60
60
174.00
162.00
-90
90
-120
120
-150
150
-180
Hertzian Dipole
With
Optical Antenna
Figure 4.19 Far-field radiation pattern (in dB) for Hertzian dipole with substrate and with bowtie aperture:
Hertzian dipole only (Red; inside) with the substrate (at -180º, maximum is 162dB). Hertzian dipole with bowtie
aperture antenna (Blue; outside) and substrate (at -180º maximum is 201dB).
The radiation pattern at resonance is plotted in Figure 4.21. The resonance of the bowtie aperture
antenna while receiving was at 195THz. Likewise, for transmitting antenna, the bowtie aperture
structure radiated the highest amount of field at 195THz. Far-field pattern gives an implication that
the radiation is directed to both upwards and downwards. It also implies that the field confinement
can be found from the top (+z-axis) and the bottom (-z-axis). Thus, the maximum E-field at farfield is at 0 and -180º. The difference of two plots in Table 4.5 is calculated to show how the
reciprocity applies at theta 0º.
Table 4.5 Hertzian dipole with substrate only case and bowtie aperture antenna case for theta angle of 0º with
71
the distance in y-gap of 5nm, 12nm, and 20nm
y-gap
Resonance
Substrate only
Including Bowtie
Increased
[nm]
[THz]
[dB]
aperture [dB]
Field [dB]
5
175
159.36
205.40
46.03
200.24
12
195
160.65
199.45
38.81
87.15
20
210
161.65
195.47
33.82
49.11
Normalized Field
Table 4.5 shows the results of the increased field with the theta angle of 0º at the resonance.
Looking at the far-fields, higher field increase is observed, in agreement with the increase in the
normalized magnitude of E-field in receiving mode in Figure 4.15. For the bowtie aperture antenna
with 12nm y-gap, the excitation produced by the Hertzian dipole has increased the magnitude of
the field by 38.81dB or 87.15. The percent differences for three cases of receiving and transmitting
modes were less than 10 percent. Since we used the Hertzian dipole excitation for the transmitting
mode, there were mismatches of excitation field and received field. However, the result gives an
implication that reciprocity is still valid for the bowtie aperture antennas. When y-gap is 20nm, the
typical values of normalized magnitude of E-field are ranging from 40 to 50 for different theta
angles. When the gap decreased to 10nm, the number went from 70 to 85. Also, when it is down
to 5nm, it ranged from 165 to 200. In summary, when we compared these results with Table 4.3,
the result coincides within the range of 10 percent errors.
72
4.5 Summary: Interesting Phenomena using Bowtie and Bowtie Aperture Antennas
Table 4.1 and Table 4.3 summarize the resonance characteristic and normalized magnitude of
electric field as well as enhancements for both bowtie and bowtie aperture optical antennas. Table
4.2 and Table 4.4 show the results of oblique incidence cases. Moreover, Table 4.5 proves the
reciprocity of transmitting and receiving antenna models. The metal-dielectric interface exhibited
the very strong oscillation and both bowtie and bowtie aperture offered very high field
enhancements at the resonance. For bowtie structured optical antennas, we have confirmed the
blue-shift and decrease in the E-field enhancements as the gap increased. At the same time, the
bowtie aperture optical antennas exhibited very interesting characteristics. When its y-gap was
increased with polarization in y-axis, its behavior was the same as the bowtie structure. However,
when x-gap was increased with polarization in y-axis, it showed the characteristics of red-shiftness
and slight decrease in the field enhancements. More importantly, interesting observation was that
the field distribution in the middle gap of the bowtie aperture structure was almost constant. For
oblique incidence cases, both of structures exhibited consistent behavior of having the same
resonance frequency and lower field enhancements with higher angles. Even at incidence angles
as large as 45 degrees, the optical antennas were found to resonate at the same frequency but there
was a slight decrease in the field enhancement. Moreover, we developed the transmitting optical
antenna model by using the Hertzian dipole. We observed and confirmed that the bowtie aperture
73
antenna increased the field by 46 dB from the transmitting case as shown in Table 4.5. Overall,
constant field distribution for bowtie aperture structure with varying x-gap is a quite impressive
result because it provides more area but almost the same enhancement for the spectroscopy dealing
with the molecules and polarizability.
74
CHAPTER 5 INTERACTION OF OPTICAL ANTENNAS AND MOLECULAR DIPOLES
5.1 Introduction: Induced Dipole Moment of Molecules
One of the important applications of optical nano-antennas is enhancing fluorescence and Raman
scattering of molecules. The objective of this chapter is to show a path to determine the best electric
dipole polarizability of molecules for the strongest enhancement. Polarizability is a measure of
redistribution of charges or ability to be polarized when a molecule is exposed to an external field
[112]. When a molecule is exposed to an electric field, it exhibits an electric dipole moment which
is a measure of non-uniform distributions of charges among the molecules. In general, such
response of the molecule to the electric field is directional dependent and the molecule has a greater
tendency to get polarized in one direction to the other. Therefore, the dipole moment of the
molecule can be expressed as
 =  ∙ loc ().
(5.1)
where Eloc () is the electric field acting on the molecule located at  and  is the tensor of
molecular polarizability. Polarizability of molecules has the SI units of C ∙ m2 ∙ V −1 or Fm2 . In
matrix notation using Cartesian coordinates, it appears as
αxx
px
[py ] = [αyx
αzx
pz
αxy
αyy
αzy
αxz
Ex
αyz ] ∙ [Ey ].
αzz
Ez
(5.2)
In many applications including molecular spectroscopy, achieving the maximum induced dipole
75
moment of the molecule is the primary goal. To gain a strong dipole moment, we need to use either
an intense electric field or highly polarizable molecule. Since molecules have a very small
polarizability due to their small dimensions, we need to increase the local electric field intensity
to induce a significant dipole moment in the molecule. Throughout this thesis, we have shown that
when optical nano-antenna is illuminated by a plane wave, the electric field is localized in the
vicinity of the optical antenna. Therefore, optical antennas can benefit excitations of the molecules
located in the hot spots of the electric field produced by the nano structures. However, if the
molecules are located very close to the optical antenna, the scattering of the incident wave is not
only the contributing factor to the local electric field enhancement. But also the interaction of the
molecule with the optical antenna contributes to the total electric field. In this chapter, we study
this interaction called self-coupling [112] and investigate when self-coupling significantly
enhances the excitation of a molecule beside our optical antenna. For this purpose, we use Green’s
function theory in calculating the electromagnetic field. Then, we numerically extract Green’s
function for one of the structures discussed in this thesis, and calculate the total electric field at the
location of the molecule resulted from all the mechanisms.
5.2 Green’s Function
As briefly mentioned in Chapter 2, in electromagnetic field theory, Green’s function related to
76
electric field in a simple form is the electric field produced by a point source (an ideal dipole) in a
linear space. Therefore, the electric field generated by any distribution of source can be calculated
through convolution of the source with the Green’s function of the medium. This is in analogy
with a linear time domain network, whose output to any input signal can be calculated by
convolution of the impulse response of the system with the input signal. However, since the source
and the produced electromagnetic field are vector quantities, the most general function describing
the electric field due to a point source is not simply a scalar Green’s function but is a dyadic Green’s
function e (,  ′ ) which relates the electric field at the location  generated by a dipole source
located at  ′ , regarding the specific direction of the source and the field as vector quantities.
Dyadic Green’s function is a matrix that gives the vector of electric field when a dot product
operation with the vector of the dipole source is calculated. Therefore, knowing the electric dyadic
Green’s function, the electric field produced by a current source is calculated as
() = −jω ∭ e (,  ′ ) ∙ ( ′ )dV ′ .
(5.3)
where the prime sign denotes the source parameters  ′ and dV ′ , location and differential volume
occupied by the source respectively. If the source is an ideal dipole moment and modeled as a
Dirac-Delta current, the volume integration reduces to
() = ω2 μe (,  ′ ) ∙ ( ′ ),
where we use the relationship
77
(5.4)
( ′ ) =
( ′ ) ∙ V ( ′ )l
=
,
jω
jω
(5.5)
where l and V denotes to the infinitesimally small length and the volume occupied by the dipole
respectively. If we use more compact notation by (,  ′ ) = ω2 μe (,  ′ ), then (5.4) becomes
() = (,  ′ ) ∙ ( ′ ).
(5.6)
In Cartesian coordinates, it appears as
Gxx
Ex
[Ey ] = [Gyx
Ez
Gzx
Gxy
Gyy
Gzy
Gxz
px
Gyz ] ∙ [py ].
pz
Gzz
(5.7)
The equation (5.7) shows the concept of dyadic Green’s function in electromagnetic field theory,
where define how each component of the point source contributes to any components of the electric
field .
5.3 Superposition in Electromagnetic Theory
z
p
r
r
y
x
Figure 5.1 Different scattering mechanisms contributing to the local electric field.
78
Assume that the space where we are interested in calculating the electric field contains an electric
dipole and the optical antenna. The total electric field is not only composed of the incident field
due to the dipole, but also the scattered field due to the optical antenna. This scattered field is
indeed produced indirectly by the current source since the incident electric field by the dipole
induces charge displacement or current on the optical antenna. For such cases in a linear medium,
superposition can be used to determine the electric field at the location  by summation of two
terms, as shown in Figure 5.1. The first term is the direct incidence of the dipole source in free
space, Ef, and the second term is the field scattered by the antenna due to the dipole source, Esc.
Then, we have
loc () = f () + sc () = 0 (,  ′ ) ∙ ( ′ ) + sc (,  ′ ) ∙ ( ′ ),
(5.8)
where 0 is the free space dyadic Green’s function and sc is the scattering dyadic Green’s
function which results in only the scattered field.
5.4 Bowtie Aperture Optical Antenna and the Scattering Green’s Function
As we mentioned in the introduction, optical nano-antennas can enhance inducement of a strong
dipole moment in the molecules in two ways. In previous chapters, we clearly showed how the
nano-antenna enhances the plane wave incidence in close vicinity of the structure. In this section,
we calculate the second contribution. For this purpose, we numerically calculate the scattering
79
Green’s function at the location of the molecule for the optical antenna, here a bowtie aperture
structure.
20nm
x
z
12nm
y
Figure 5.2 Bowtie aperture optical antenna with the Hertzian dipole excitation (red sphere)
The general settings are almost the same as the transmitting antenna case in section 4.5. As shown
previously, the bowtie aperture structure can only enhance an x polarized plane wave of incidence.
Therefore, the most scattering occurs in x polarized fields and the significant component of the
sc (,
scattering Green’s function to be considered here is Gxx
). Here,  is at the center of the
sc (,
structure where the molecular dipole will be located. To calculate Gxx
), we put a Hertzian
dipole source along x axis and approximate the x component of the scattered electric field at the
location of the dipole by its value very close to the dipole. Because of the singularity of the radiated
field by the dipole at its own location, we cannot evaluate the exact field at the dipole location.
The scattered field at  very close to the dipole is determined as
sc (,
Exsc () = Extot () − Exf () = Gxx
′) ∙ px (′),
80
(5.9)
where Exf () is the field produced by the Hertzian dipole at  ≈ ′ in free space. Since px ()
sc (,
can be represented as (5.4), Gxx
) can be calculated by
sc (,
Gxx
) =
Exsc () Exsc () jωExsc ()
=
=
,
Iℓ
px ()
Iℓ
jω
(5.10)
where Iℓ is set to 1Am using the Hertzian dipole wave excitation in HFSS.
5.5 Total Enhancement of the Induced Dipole Moment of a Molecule beside Optical Antenna
Now, we are ready to combine the previous sections and calculate the total induced dipole moment
of a molecule located beside an optical nano-antenna. In Figure 5.2, the bowtie aperture optical
antenna is illuminated by a plane wave whose electric field, E inc (), is polarized along x. Using
the superposition, the total electric field at the center of the structure, r, is contributed from 3 terms:
(i) the electric field due to direct incidence of the plane wave, Exinc (); (ii) the scattered field by
the bowtie aperture nano-antenna due to the incident wave, Exsc,inc (); and, (iii) the field due to
the self-coupling which is the field that molecule scatters and receives back through the rescattering by the nano-antenna, Exsc,m (). Therefore, the total field is
Extot () = Exinc () + Exsc,inc () + Exsc,m ().
(5.11)
If we combine the contribution of the processes in (i) and (ii) as Ex1 (), we can write the equation
as
81
Ex1 () = Exinc () + Exsc,inc (),
(5.12)
Extot () = Ex1 () + Exsc,m ().
(5.13)
sc (,
Extot () = Ex1 () + Gxx
) ∙ px (),
(5.14)
Using (5.9), it becomes
where both Extot and px are unknown. However, they are related by
px = αm ∙ Extot (),
(5.15)
where αm is a scalar molecular polarizability.
If we combine these two equations together, we can express Extot () as
Extot () =
Ex1 ()
.
sc (, )
1 − αm Gxx
(5.16)
sc (,
We can define the term χ = 1/{1 − αm Gxx
)} which is dependent on the polarizability of the
molecule and self-coupling through the optical antenna. Ideally, one can induce a very high
enhancement when the denominator becomes zero, hence χ → ∞ . Therefore, for a specific
plasmonic scattering structure, we can calculate the polarizability, named as critical polarizability,
which results in such large enhancement [122]. Such polarizability is calculated and reported in
Figure 5.3 for the bowtie apperture ideally but we may also be able to get close to these values in
practice for other structures.
82
x 10
1.5
2
2
real(c) [Fm ]
5
(b)
-35
imag(c) [Fm ]
(a)
4.5
4
3.5
100
150
200
250
300
Frequency [THz]
x 10
1
0.5
0
100
350
-35
150
200
250
300
Frequency [THz]
350
Figure 5.3(a) Real part of the critical polarizability of the molecule modeled as a Hertzian dipole. (b)
Imaginary part of the critical polarizability.
Figure 5.3 shows the real and imaginary part of the critical polarizability of the molecules for the
specific setup of the bowtie aperture, which would ideally produce infinite enhancement. It
demonstrates that the imaginary part of polarizability is positive. Therefore, the molecule should
be an active molecule to counterbalance the losses in the plasmonic nano-antenna. Besides, the
real part of the critical polarizability is of the order of -35 Fm2 which is too high and not achievable
in practical molecules. Although this result shows that the bowtie aperture structure is not the best
example to take the advantage of self-coupling in exciting molecules, this discussion explains how
self-coupling can have the extra effect in the enhancement of the electric field and hence the
induced dipole moment of molecules.
83
5.6 Conclusion
In this chapter, we have investigated the scattered fields created by a single molecule located at
the center of the gap of the bowtie aperture optical antenna. We have first assumed that the
molecule is located very close to the optical antenna and we observed the near field interaction of
the molecule with the optical nano-antenna which produces the strong field enhancement in the
gap. By employing Green’s function, we computed the critical polarizability and the enhancement
factor χ. This new approach allows us to clearly explain the behavior of light scattering of
molecules near nano-antennas. In particular it explains how molecules excitation can be enhanced
sc (,
due to self-coupling, when the critical condition αm Gxx
) = 1 is most nearly satisfied.
84
CHAPTER 6 CONCLUSION AND FUTURE APPLICATIONS
6.1 Various Applications
Bowtie and bowtie aperture optical nano-antennas have the ability to enhance the infrared and light
signals very effectively. In particular, the bowtie aperture structure has a potential to develop as
optical devices and nanotechnology. It is possible to accelerate the growth of applications such as
Surface Enhanced Raman Spectroscopy (SERS) [52-55], Tip-Enhanced Raman Spectroscopy
(TERS) [100-103], NSOM and Tip-enhanced Near-field Optical Microscopy (TENOM), subwavelength lithography [56], color filtering and imaging [57], energy harvesting for photovoltaic
devices [58], magnetic recording [59], higher harmonic generation [60], and efficient Organic
Light Emitting Diodes (OLED) [61,62].
6.2 Future work
We faced challenges in analyzing the optical nano-antennas and also thought that more
improvements should be made by developing the theory. Dealing with light interactions with
different materials and being able to fully understand natural phenomena are some of the most
significant avenues for future research. The concept of optical nano-antennas can be applied to
enhancing the performance of optical devices as well as electric devices. Bowtie aperture optical
antennas can be fabricated using the electron beam lithography for the further discussions of nearly
85
constant field distribution, tunable resonance properties, and oblique incidences. The experimental
results will verify these characteristics of theoretical simulations. It is also crucial to develop
comprehensive equations using circuit theory analysis for optical antennas [97-99], including the
ohmic losses and plasmonic behaviors. Another future objective is to achieve limitless spatial
resolution by using the evanescent and near fields created by optical antennas.
6.3 Conclusion
Bowtie antennas and bowtie aperture antennas have been extensively analyzed and studied by
exploiting the optical properties and behaviors of plasmonic nano-structures. Bowtie and bowtie
aperture optical antennas, which formed by two 70nm length gold triangles, have been excited and
showed the resonance shift behavior and field enhancements varying the distance of the gap.
Moreover, topics to understand the optical properties and characterizations of RF electronics and
plasmonics have been reviewed. Starting from the antennas at microwaves, the study explored the
analogy between antennas at the microwaves and optical frequencies. We have studied the bowtie
and bowtie aperture antennas at microwaves to gain confidence in properties using antenna theory.
For optical antennas, bowtie and bowtie aperture nano-structures have created a strongly enhanced
field in the gap much smaller than the wavelength. This high enhancement of the electromagnetic
field enabled the detection of Surface Enhanced Raman Scattering (SERS) for single molecules.
86
More importantly, interesting behaviors such as constant field distribution in the gap and gap
dependence with different axes of polarization of bowtie aperture optical antennas have been
observed. The study not only confirms the resonance shifts and field enhancement of the plasmonic
nano-structures but also it carries out interesting analysis on both bowtie and bowtie aperture
antennas. Table 4.1, Table 4.2, Table 4.3, Table 4.4, and Table 4.5 summarize the resonance
characteristic, oblique incidence, and normalized magnitude of the electric field as well as
enhancements. We have also applied antenna theory to model a transmitting antenna by using a
Hertzian dipole excitation. The idea of using a dipole extended our research in near-field
interaction with molecular polarizability and we carried out the behavior of light scattering of
molecules near nano-antennas.
87
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