AN ABSTRACT OF THE THESIS OF
Nicholas Anthony Kuhta for the degree of Doctor of Philosophy in Physics presented on
July 9, 2012.
Title: The Optical Properties of Multi-Scale Plasmonic Structures
and Their Applications in Optical Characterization and Imaging
Abstract approved:
Viktor A. Podolskiy
The optical response of metallic structures is dominated by the dynamics of their
free electron plasma. Plasmonics, the area of optics specializing in the electromagnetic
behavior of heterogeneous structures with metallic inclusions, is undergoing rapid development, fueled in part by recent progress in experimental fabrication techniques and novel
theoretical approaches. In this thesis I outline the behavior of four plasmonic material
systems, and discuss the underlying physics that governs their optical response. First, the
anomalous optical properties of solution-derived percolation films are explained using scaling theory. Second, a novel technique is developed to characterize the optics of amorphous
nanolaminates, leading to the creation of a meta-material with anisotropic (hyperbolic)
dispersion. The properties of such materials can be tuned by adjusting their composition.
Third, the electrodynamics of vertically aligned multi-walled carbon nanotubes is derived
through the development of a spectroscopic terahertz transmission ellipsometry algorithm.
Lastly, a new diffraction based imaging structure based on metallic gratings is presented
to have resolution capabilities which far outperform the diffraction limit.
c
⃝
Copyright by Nicholas Anthony Kuhta
July 9, 2012
All Rights Reserved
The Optical Properties of Multi-Scale Plasmonic Structures and Their Applications in
Optical Characterization and Imaging
by
Nicholas Anthony Kuhta
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented July 9, 2012
Commencement June 2013
Doctor of Philosophy thesis of Nicholas Anthony Kuhta presented on July 9, 2012
APPROVED:
Major Professor, representing Physics
Chair of the Department of Physics
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon State
University libraries. My signature below authorizes release of my thesis to any reader
upon request.
Nicholas Anthony Kuhta, Author
ACKNOWLEDGEMENTS
To my wife Heather, without whose unwavering love, selflessness and forgiveness, this PhD
would not have been possible. This work is also dedicated to my son, Elliot, and to the
memory of my father, both of whom make me strive to be a better man each day of my
life. I would also like to thank my loving family and friends for their kindness and support
over the last years.
TABLE OF CONTENTS
Page
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.
A Review Of Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2.
Surface Plasmon Polariton Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.
Dispersion Equation and Poynting Vector for Non-Magnetic Uniaxial Anisotropic
Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.
Transfer Matrix Method Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Transfer Matrix Method For Uniaxial Media . . . . . . . . . . . . . . . . . . . .
1.4.2 General N-Layer Recursive Fresnel Coefficients . . . . . . . . . . . . . . . . . .
1.5.
7
10
14
Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. ASYMMETRIC REFLECTANCE AND CLUSTER SPATIAL EFFECTS IN
SILVER PERCOLATION FILMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.
Percolation Film Synthesis and Characterization . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.
Reflection, Transmission, and Absorption of Random Percolation Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Generalized Ohm’s Law for Asymmetric Structures . . . . . . . . . . . . . .
2.3.2 Scaling Theory Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
28
2.4.
Deriving the Necessary Conditions for Nonzero ∆R . . . . . . . . . . . . . . . . . . . . . 31
2.5.
Comparing Scaling Theory with Experimental Results . . . . . . . . . . . . . . . . . . 32
2.6.
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3. OPTICAL PROPERTIES OF AMORPHOUS NANOLAYERS . . . . . . . . . . . . . . . 36
3.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.
Nanofabrication and Material Characterizaton . . . . . . . . . . . . . . . . . . . . . . . . . . 37
TABLE OF CONTENTS (Continued)
Page
3.3.
Bulk Optical and Electrical Properties of Amorphous Metallic and Dielectric Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.
Effective Anisotropic Medium - Dispersion Engineering . . . . . . . . . . . . . . . . . 44
3.5.
Necessary Conditions for Non-Magnetic Negative Refraction . . . . . . . . . . . . 47
3.6.
Layer Thickness Verification using Effective Medium Error Analysis . . . . . 49
3.7.
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4. TERAHERTZ ELLIPSOMETRY OF VERTICALLY ALLIGNED MULTI-WALLED
CARBON NANOTUBES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.
Carbon Nanotube Fabrication and Characterization. . . . . . . . . . . . . . . . . . . . . 53
4.3.
Time-Averaged Transmitted Terahertz Power - Bolometer Measurements 56
4.4.
Terahertz Ellipsometry Theoretical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Optical Path Lengths in Uniaxial Material Systems . . . . . . . . . . . . . .
4.5.
62
Time Domain Spectroscopy Comparison and Concluding Remarks . . . . . . 64
5. SUBWAVELENGTH IMAGING RECONSTUCTION USING A RIGOROUSLY
COUPLED WAVE ANALYSIS ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2.
Transmission and Reflection Coefficients in General Periodic Medium . . . 68
5.3.
Multiple Slit Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4.
Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.
Diffraction Based Imaging Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6. DISSERTATION CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
TABLE OF CONTENTS (Continued)
Page
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
A
Optical properties of Horizontally Aligned Carbon Nanotube Dipole Antennae Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B
Scaling Theory Fortran Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
LIST OF FIGURES
Figure
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4
Page
General planar material schematic showing the difference between positive and negative refraction. When negative refraction occurs light is
transmitted on the same side of the optical axis as the incident light as
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Geometrical schematic for a surface plasmon polariton wave which is
propagating in the z-direction and exponentially decaying in the x-direction.
Note that the decay lengths depend on the material cladding, and are
asymmetric in general. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
General N-layer planar thin-film layered structure with transmission and
reflection coefficients for each material region. TM incident electric field
with amplitude E0 is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Transmission and reflection amplitudes for all waves passing through a
three layer thin-film system. Coefficients rij and tij represent standard
Fresnel amplitude coefficients between materials with ni and nj . Note
that normal incidence is assumed and that the angles of rays are drawn
only to distinguish between different paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
General layered structure composed of a silver percolation film clad by
air to the left and glass to the right. Incident light may come from either
the air or substrate side as shown. Both air and glass regions are taken
to be semi-infinite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Scanning electron micrograph of a chemically deposited silver film with
metal filling fraction p ≃ 0.52. The scale bar is 500nm. . . . . . . . . . . . . . . . . . .
22
Measured reflectance (red diamonds), transmittance (blue boxes), and
absorbance (green circles) as function of incident wavelength for measured
metal filling fraction p ≃ 0.52. Solid lines represent the results of scaling
theory calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Schematic of a metal-dielectric percolation film on a glass substrate. At
the far left the first region is vacuum, the center grey region with thickness d is a composite medium composed of silver and vacuum, and the
right region is a glass dielectric substrate. Dashed vertical lines represent reference planes, not physical objects, used in the implementation
of GOL as a fitting parameter. Light is incident from both directions as
indicated by the solid and dashed arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
LIST OF FIGURES (Continued)
Figure
2.5
2.6
2.7
3.1
3.2
3.3
Page
Reflectance (red long-dashed), transmittance (black short-dashed), and
absorbance (blue solid) through our percolation film as a function of
surface coverage fraction for a 10cm reference plane GOL system. The
percolation threshold is assumed to be pc = 0.5, and light is incident
from the air side of the film. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Points represent the measured change in reflectance (∆R = R1 − R2 )
for various incident wavelengths. Black circles 500nm, green triangles
600nm, red boxes 700nm. Corresponding colored solid lines (black solid
500nm, green short-dashed 600nm, red long-dashed 700nm) represent
the results of scaling theory reflectance calculations. The inset shows
the change in reflectance over the entire surface coverage range. Note
that the 2D scaling model fails for large metal concentrations, where
the three-dimensional structure of the composite dominates the optical
response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Points represent the measured (a) reflectance, (c) transmittance, and (e)
loss from the air side as a function surface coverage fraction. Connecting
lines are a guide for the eye. Calculated (b) reflectance, (d) transmittance, and (f) loss when the correlation length parameter is ξ0 = 2nm
(solid line), ξ0 = 5nm (dashed line), and ξ0 = 10nm (dotted line). For
all graphs the incident wavelength is 700nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
TEM image of a 10 bilayer TiAl3 -AlPO system. Dark regions represent TiAl3 (metal) and light regions represent AlPO (dielectric). Vector
directions are marked as referenced in the body text. . . . . . . . . . . . . . . . . . . . .
38
Amorphous material characterization. (a) Wide-angle image of laminated structure containing both amorphous metals (TiAl3 ,ZrCuAlNi)
with AlPO dielectric layers interspersed showing no crystalline spots.
Darkened zone indicates where the included high-resolution image (b)
was taken. (b) High-resolution TEM image of amorphous metal / AlPO
nanolaminate with constituent layers labeled. (c)Electron diffraction
taken from the high-resolution image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
TM and TE polarized reflectance from an optically thick 200 nm TiAl3
film as a function of wavelength. Solid black lines represent theoretical
reflectance results and points are measured data for 20◦ (red), 45◦ (green),
70◦ (blue), and 80◦ (brown) angles of incidence. It should be noted that
spurious data points near 900nm are related to the Xe lamp spectrum,
and not material resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
LIST OF FIGURES (Continued)
Figure
3.4
3.5
3.6
3.7
Page
TM and TE polarized reflectance from an optically thick 284 nm ZrCuAlNi film as a function of wavelength. Solid black lines represent theoretical reflectance results and points are measured data for 20◦ (red),
45◦ (green), 70◦ (blue), and 80◦ (brown) angles of incidence. . . . . . . . . . . . .
41
Real part of the dielectric response as a function of wavelength for bulk
dielectric AlPO (blue), and amorphous metals TiAl3 (black) and Zr-CuAl-Ni (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Imaginary component of the dielectric function for bulk amorphous metals TiAl3 (black) and Zr-Cu-Al-Ni (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Real part of the effective anisotropic dielectric constant for the 4.7nm
TiAl3 - 11.3nm AlPO (black) and the 8nm ZrCuAlNi - 8nm AlPO (red)
composites. Yellow shaded regions represent the spectral regions where
the composites have hyperbolic dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.8
TM and TE polarized reflectance from 10 bilayers of 8nm ZrCuAlNi and
8 nm AlPO as a function of wavelength. Solid lines represent theoretical
effective medium theory reflectance results and points are measured data
for 20◦ (red), 45◦ (green), 70◦ (blue), and 80◦ (brown) angles of incidence. 46
3.9
TM and TE polarized reflectance from 10 bilayers of 4.7nm TiAl3 and
11.3 nm AlPO as a function of wavelength. Solid lines represent theoretical effective medium theory reflectance results and points are measured
data for 20◦ (red), 45◦ (green), 70◦ (blue), and 80◦ (brown) angles of
incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.10 Normalized reflectance error for both TiAl3 -ALPO and ZrCuAlNi-ALPO
10 bilayer systems at 20◦ and 45◦ incidence for different theoretical MetalDielectric thickness ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.1
4.2
(a) SEM image of the vertically aligned MWCNT on a Silicon substrate
for d=21.5µm. (b) Ellipsometry characterization schematic for THz
transmission measurements: linearly polarized (s and p polarization),
broadband THz pulses are incident on the given sample at some incident
angle. The two THz detection schemes are: (i) time-averaged integrated
power spectrum Si:Bolometer measurements and (ii) THz time-domain
spectroscopy measured using electro-optic sampling. . . . . . . . . . . . . . . . . . . . . .
54
Spectrally integrated THz power transmitted through the CNT samples
vs. the incident angle for (a) p-polarization and (b) s-polarization. The
solid black lines represents the theoretical transmission for a bare Silicon
substrate (n=3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
LIST OF FIGURES (Continued)
Figure
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.1
5.2
5.3
5.4
Page
Ray diagram for temporally separated pulses traveling through the layered CNT-substrate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
(a) Real and (b) imaginary parts of the refractive index for all CNT films
in the THz regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
(a) Real and (b) imaginary parts of the terahertz spectrum for pulses in
air. The spectrum is found by taking the Fourier transform of the time
dependent THz air pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Ray diagram for light moving through a planar slab (thickness d) at some
incident angle. Dark lines with arrows represent the path taken by the
⃗ ....................................................
Poynting vector (S).
62
Ray Diagram for the CNT-Silicon multilayered thin-film system. Solid
lines represent the path taken by light pulses moving through the layers.
64
P-polarization THz waveforms transmitted through the CNT samples for
incident angles between 0 and 60◦ (a-d) experiment and (e-h) theory. . . . .
65
S-polarization THz waveforms transmitted through the CNT samples for
incident angles between 0 and 60◦ (a-d) experiment and (e-h) theory. . . . .
66
General schematic of a three layer system where layers 1 and 3 are air,
and layer 2 is a metallic grating with spatial period Λ and thickness
L. An incident plane waves illuminates an object located z0 from the
grating. That signal is then transformed into diffraction modes whose
spacing depends on the grating period. Measurement of the diffracted
object waves are taken along the far-field measurement plane (z = zf ). . .
68
Linear least squares fitting for trigonometric spectral basis functions.
Field recovery (with truncated kx spectrum) (a) and spectral comparison
(b) for a three object system with slit widths (λ0 /8, λ0 /4, λ0 /2). Note
that the trigonometric spectral fit fails for large kx . . . . . . . . . . . . . . . . . . . . . .
76
Linear least squares fitting for bessel function spectral basis functions.
Field recovery (with truncated kx spectrum) (a) and spectral comparison
(b) for a three object system with slit widths (λ0 /8, λ0 /4, λ0 /2). Again
the bessel spectral fit fails for large kx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Linear least squares fitting for pixel source array spectral basis. Field
recovery (without any truncation) (a) and spectral comparison (b) for a
three object system with slit widths (λ0 /8, λ0 /4, λ0 /2). . . . . . . . . . . . . . . . . . .
77
LIST OF FIGURES (Continued)
Figure
5.5
5.6
5.7
A1
Page
Standard deviation of the recovered spectrum as a function of the object
grating separation distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Standard deviation of the recovered spectrum as a function of the number
of grating modes used in the imaging algorithm. . . . . . . . . . . . . . . . . . . . . . . . .
79
Standard deviation of the recovered spectrum as a function the maximum
kx spectrum summing value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Horizontally aligned carbon nanotube arrays with horizontal spacing lx
and vertical spacing ly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
THE OPTICAL PROPERTIES OF MULTI-SCALE PLASMONIC
STRUCTURES AND THEIR APPLICATIONS IN OPTICAL
CHARACTERIZATION AND IMAGING
1.
1.1.
INTRODUCTION
A Review Of Metamaterials
Advances in nanofabrication techniques and theoretical knowledge are driving the
innovation of new material systems which display unique and useful optical and electrical
properties. Fundamentally new physics has been found, such as nonlocal optical response
which deviates from conventional dispersion theory.[1, 2, 3, 4, 5] In this work I will focus on
the field of metamaterials, which is the study of material composites that create effective
optical responses which are different than their constituent materials, and uncommon in
nature.
Although intensive metamaterials research is relatively recent, scientists first proposed such systems 40 years ago. In 1968 Veselago proposed that negative refraction
will occur in a material with simultaneously negative electric permittivity and magnetic
permeability.[6] The problem with applying Veselago’s theory is that there are no known
natural materials which display these properties, therefore this work was not used for over
30 years until scientists and engineers discovered how to make composite materials which
displayed the correct effective properties. In 2001 researchers were able to experimentally
verify negative index of refraction in the microwave frequency regime using arrays of copper split-ring resonators.[7] Split-ring resonators are metal loops which produce effective
2
magnetism from current loops that create a non-zero electric field curl. Many advances
have been made in furthering the theoretical understanding and design of split-ring resonator arrays.[8, 9, 10, 11, 12] The ultimate goal of negative refraction materials is to
fabricate planar lenses which are free from abberations, and can amplify evanescent waves
to produce resolution which outperforms the diffraction limit. In practice, the planar
negative index superlens system is not physically viable due to inherent material loss.[13]
Air
Material
positive
refraction
optical axis
negative
refraction
incident radiation
FIGURE 1.1: General planar material schematic showing the difference between positive
and negative refraction. When negative refraction occurs light is transmitted on the same
side of the optical axis as the incident light as shown.
As nanofabrication techniques evolved to pattern smaller structures, negative refraction was achieved at higher frequencies. By 2005 researchers were able to produce negative
refraction at wavelength values of 1.5µm using periodic arrays of gold nanorods.[14] Negative refraction itself seems impossible due to the conservation of momentum parallel to
an interface that is prescribed by Snell’s Law, but closer examination reveals that there
is no broken momentum conservation due to the negative phase velocity of light waves
moving in an isotropic negative index media. Dual configuration transmission line networks can also create effective negative refraction and focusing with large bandwidths
ranging from megahertz to tens of gigahertz.[15] By exploiting hyperbolic anisotropic dispersion in metal-dielectric layers, and in cylindrical nanorod arrays embedded in dielectric
3
films, researchers have shown that negative refraction is possible even without magnetic
materials.[16, 17, 18]
Negative refraction is only one of many interesting properties that is currently being explored. Arrays of gold helical structures have been fabricated to create broadband
circular polarizers.[19] Due to the coupling of light to plasmons, extraordinary optical
transmission through subwavelength hole arrays has been achieved, and furthers the potential for local nanoscale light sources.[20] In 2006, a radial array of copper split-ring
resonators successfully cloaked the scattering of microwaves from an solid metal cylinder
placed in the middle of the rings.[21] Within a year theory progressed to demonstrate
how to extend macroscopic cloaking to the visible spectrum by using arrays of radially
oriented spheroidal silver wires embedded in a dielectric host.[22] While cloaking is an
abstract and exciting realization of a topic that used to be classified as science fiction,
many more practical ideas are playing a critical role in the development of modern optical
technology. Circuits with light on the nanoscale have been classified and compared to
analogous electrical circuits.[23] Optical response in non-magnetic materials is encoded in
the dielectric constant, ϵ, which describes how charges within a system are polarized given
an incident electric field. Nanoparticle circuit elements with dielectric response, Re(ϵ) > 0,
are equivalent to capacitors at optical frequencies, metallic response, Re(ϵ) < 0, creates
effective inductors, and lossy dielectrics, Im(ϵ) ̸= 0, respond as conventional resistors. By
using light-based systems to perform signal processing, the field of photonics has evolved
to compete with technology that has been traditionally dominated by electronics. Integrating silicon photonics and CMOS transistor technology is a very broad research goal,
with groups successfully developing efficient electro-optic modulators, and fiber-waveguide
couplers.[24]
The field of plasmonics, which is the study of optical response in metallic systems,
is currently gaining wide interest by researchers in many disciplines. Due to the tunability
4
of plasmonic resonances through geometric design, metallic nanostructures are emerging
as ideal candidates for many waveguiding and sensing applications.[25, 26] One beautiful
example of the power of sensing is shown by a work which characterizes DNA bound to
different gold nanoparticles, which leads to a wide range of solution colors based on the
size and shape of the gold nanoparticles.[27] Homogeneous nucleation (seeded) and heterogeneous nucleation methods have produced incredible shape control of colloidal metal
nanoparticles, and greatly increase the functionality and selectivity of catalysts, plasmonic
sensors, and spectroscopy applications.[28] Plasmonics is also playing an increasing role in
enhancing new biophysics experiments. Enhanced fluorophore-plasmon interactions may
result in ultrabright nanoprobes which do not photobleach, measurement of distances
which are inaccessible using fluorescence resonance energy transfer (FRET) technology,
create localized multiphoton probes, selectively enhance emission at desired wavelengths,
and improve wide-field subwavelength optical microscopy.[29]
Focusing and confining light to nanometer scale dimensions is an inherent feature
of plasmonic systems which is continually exploited to develop new devices.[30, 31, 32, 33]
One major technological gap which is difficult to overcome is the integration of modern
optical and electronics devices. Electronics components are on the nanometer scale, where
as optical devices are on the micron scale and larger. Classical diffraction theory limits
the focus of light using conventional dielectric lenses to approximately the wavelength of
incident light (∼ λ/2), and is therefore incompatible with the size scale of cutting edge
electronic circuits. Metallic structures with rectangular and conical shapes have been used
to build light concentrators and resonators, which leads to highly localized field sources for
nonlinear applications, and C-shaped apertures which drastically increase the transmitted
power through subwavelength apertures.[34] In 2006, researchers designed a silver-based
plasmonic superlens nanolithography method which resolved spatial features which were
four times smaller than the illumination wavelength.[35] By coating gold nanodisk arrays
5
with redox-controllable molecules, researchers showed that active control of plasmonic
response is achievable through surface chemistry resonances which can be changed through
exposure to different chemicals.[36] Enhanced absorption in plasmonic systems is being
used to redesign solar cells by reducing the physical thickness of photovoltaic absorber
layers.[37]
1.2.
Surface Plasmon Polariton Dispersion
Surface plasmon polaritons (SPP) are charge waves which are confined to an interface between a metal and dielectric material. Understanding resonant coupling to SPP
waves is critical for the design of plasmonic devices. The SPP waves propagate along the
surface and exponentially decay away from the interface as evanescent waves. To calculate
the surface plasmon polariton dispersion relationship first consider surface waves which
are propagating in the z-direction, and decaying away from the x = 0 boundary as shown
below in Fig.(1.2).
x
ε1
z
ε2
SPP Propagation
y
FIGURE 1.2: Geometrical schematic for a surface plasmon polariton wave which is propagating in the z-direction and exponentially decaying in the x-direction. Note that the
decay lengths depend on the material cladding, and are asymmetric in general.
6
To derive the required resonance condition for TM SPP waves we write the magnetic
field in each material region as
⃗1 = H0 exp (ikz z − κ1 x − ωt) ŷ
H
(1.1)
⃗2 = H0 exp (ikz z + κ2 x − ωt) ŷ,
H
(1.2)
√
(j)
where the decay constants κj = ikx =
2
kz2 − ϵj ωc2 , and the different exponential signs
ensure fields decay away from x = 0. Using Maxwell’s equations we can relate the electric
field which is parallel to the x-interface to the parallel magnetic field
−iω
c ϵEz
=
∂Hy
∂x .
Taking the spatial derivative we arrive at,
(1)
κ1 Hy
(2)
κ2 Hy
=
(1)
iω
c ϵ1 Ez
(1.3)
(2)
= − iω
c ϵ2 Ez ,
(1.4)
and assuming that electric and magnetic fields which are parallel to the x-interface are
continuous at x = 0 the standing wave condition for SPP is revealed.
κ2
−κ1
=
ϵ1
ϵ2
(1.5)
When Eq.(1.5) is true, then a SPP will be formed on the interface between materials 1
and 2. Plugging the definition of κ back into the SPP standing wave condition yields
√
−ϵ2
ω2
kz2 − ϵ1 2 = ϵ1
c
√
kz2 − ϵ2
ω2
,
c2
(1.6)
and algebraic manipulation yields the SPP dispersion relationship, which gives the wavelength of the SPP surface mode.
kz =
2π
λSP P
ω
=
c
√
ϵ1 ϵ2
ϵ1 + ϵ2
(1.7)
7
1.3.
Dispersion Equation and Poynting Vector for Non-Magnetic Uniaxial Anisotropic Media
The optical properties for uniaxial anisotropic materials are derived directly from
Maxwell’s equations.[38, 39, 40, 41] Assuming there is no free charge or free current density
Maxwell’s equations in a non-magnetic (µ = 1) materials are
⃗ =0
∇·D
(1.8)
⃗ =0
∇·H
⃗
⃗ = −1 ∂ H
∇×E
c ∂t
⃗
⃗ = 1 ∂D .
∇×H
c ∂t
(1.9)
(1.10)
(1.11)
⃗ = ϵ̂E,
⃗ where
The displacement field in a material is related to the electric field by D
ϵ̂ is a matrix describing the material polarization in different directions given an applied
electric field. Given a ẑ primary propagation direction (optical axis), the displacement
vector is related to the electric field by the following diagonal matrix in uniaxial materials.

 ϵxy 0 0

ϵ̂ = 
 0 ϵxy 0

0
0 ϵz






(1.12)
Note that ϵxy is not an off-diagonal matrix term, and represents dielectric response in the
xy-plane. Using a (∇×) operator on the Maxwell’s curl equations produces the standard
wave equation for all fields. In cartesian coordinates the wave equation can be solved by
a linear combination of plane waves. Therefore, if we assume that fields have plane wave
[
]
form exp i(⃗k · r − ωt) , the gradient operator is proportional to the momentum vector,
∇ = i⃗k = i (kx , ky , kz ), and Maxwell’s equations are written in the following convenient
form.
8
⃗k · D
⃗ =0
(1.13)
⃗k · H
⃗ =0
(1.14)
⃗k × E
⃗ = ωH
⃗
c
⃗k × H
⃗ = −ω ϵ̂E
⃗
c
(1.15)
(1.16)
The general dispersion master equation is found by applying a (⃗k×) operator to the
modified Maxwell’s equations, and solving by using vector identities.
(
)
2
⃗k × ⃗k × E
⃗ = −ω ϵ̂E
⃗
⃗ = ω ⃗k × H
c
c2
(1.17)
Using a vector identity turns Eq.(1.17) into the dispersion master equation,
(
)
2
⃗k ⃗k · E
⃗ − k2 E
⃗ = −ω ϵ̂E,
⃗
c2
(1.18)
which has the following matrix form.






k2
−
kx2
−
2
ϵxy ωc2
−kx ky

−kx ky
k 2 − ky2 − ϵxy
−kx kz
−ky kz
−kx kz
ω2
c2
−ky kz
2
k 2 − kz2 − ϵz ωc2

  Ex 


 E  = 0
 y 


Ez
(1.19)
The dispersion equation is found by requiring that the matrix equation above has a nontrivial solution, which happens when the determinant is equal to zero. After taking the
determinant of the dispersion matrix equation and factoring one is left with the following
necessary condition for a non-trivial solution.
)
)(
c2 k 2 − ϵxy ω 2 c2 ω 2 (−ϵz kz2 − ϵxy (kx2 + ky2 )) + ϵxy ϵz ω 4 = 0
{z
}|
|
{z
}
(
TE
(1.20)
TM
Therefore, the dispersion equation for TE waves in uniaxial media is given by the spherical
equation
9
k 2 = kx2 + ky2 + kz2 = ϵxy
ω2
,
c2
(1.21)
and the dispersion equation for TM waves is given by the elliptical or hyperbolic equation
kx2 + ky2
k2
ω2
+ z = 2.
ϵz
ϵxy
c
(1.22)
Now we will derive the Poynting vector for uniaxial crystals, which describes the
magnitude and direction of energy flow within the material. Again assuming a ẑ optical
⃗ = (0, Ey , 0), and using Maxwell’s equations
axis the TE electric field waves are defined as E
⃗ =
the corresponding magnetic field would be H
c
ω
(−kz Ey , 0, kx Ey ). The Poynting vector
is found by taking the vector product of electric and magnetic fields.
(
)
⃗= c E
⃗ ×H
⃗
S
4π
(1.23)
Plugging in the defined fields above yields the Poynting vector for TE polarized waves in
uniaxial crystals.
c2
S⃗TE =
|Ey |2 ⃗k
4πω
(1.24)
Note that the energy and momentum are coincident for TE waves in uniaxial crystals.
TM waves are calculated using the same formalism. First we define the magnetic field,
⃗ = (0, Hy , 0), and use Maxwell’s equations to find the corresponding TM electric field
H
(
)
⃗ = ckz Hy , 0, −ckx Hy . Plugging these fields into Eq.(1.23) yields the Poynting vector
E
ωϵxy
ωϵz
for TM waves in a uniaxial medium.
⃗ =
STM
c2
|Hy |2
4πω
(
kx
kz
, 0,
ϵz
ϵxy
)
(1.25)
Note that the Poynting vector and momentum direction are not the same for TM waves
in uniaxial material. Utilizing the conservation of parallel (kx ) momentum (Snell’s Law),
when ϵz < 0 and ϵxy > 0 is achieved in uniaxial materials, negative refraction of TM
10
wave energy (Sx < 0) occurs. Note that when negative refraction of TM energy occurs in
uniaxial materials the momentum undergoes conventional positive refraction.
1.4.
1.4.1
Transfer Matrix Method Algorithm
Transfer Matrix Method For Uniaxial Media
The transfer matrix method (TMM) is one of the most useful analysis tools available
for analyzing any general layered thin-film system.[42, 43] Here a general transfer matrix
is derived to calculate the angle-dependent transmission and reflection coefficients of any
monochromatic linearly polarized light waves moving through uniaxial anisotropic media.
This general matrix is also simplified to model isotropic materials and normal incidence
measurements, and is used extensively in upcoming chapters. By employing field continuity boundary conditions to Maxwell’s equations, fields in every region of a multi-layered
film are completely deterministic.
x
z
.....
a1(+)
a2(+)
a3(+)
(-)
1
(-)
2
(-)
3
aN-1(+)
aN(+)
aN-1(-)
aN(-)
.....
E0
a
a
a
θ1
z1
z2
z3
zN-2
zN-1
FIGURE 1.3: General N-layer planar thin-film layered structure with transmission and
reflection coefficients for each material region. TM incident electric field with amplitude
E0 is shown.
⃗ = (Ex , 0, Ez ), and H
⃗ = (0, Ey , 0). By choosing
First consider TM polarized plane waves, E
the total electric field as the primary field, the layer dependent electric field can be written
11
as a linear combination of forward and backward moving plane waves.
(±)
E⃗j = aj exp [i(kx x ± kz,j z − ωt)] (cos(θj )x̂, 0, sin(θj )ẑ)
(1.26)
a±
j represents the amplitude coefficients for forward and backward moving waves. If we
assume that the dielectric response in all layers is anisotropic, with permittivity components parallel and perpendicular to the layer interfaces being different (uniaxial crystal),
then we can write the permittivity matrix as

 ϵxy 0 0

ϵ̂ = 
 0 ϵxy 0

0
0 ϵz



,


(1.27)
and referring to Sec.(1.3.) the dispersion equation for TM waves in the j th material is
kx2 + ky2
(j)
+
ϵz
2
kz,j
(j)
ϵxy
=
ω2
,
c2
(1.28)
and


θj = tan−1 
√


sin(θ1 )

) .
(
(j)
sin2 (θ1 )
ϵxy 1 − (j)
(1.29)
ϵz
⃗ =
Using Maxwell’s equation, ∇ × H
iω ⃗
c ϵ̂E,
we can relate the parallel magnetic field in
each layer to the parallel electric field within the layer.
(j)
Hy(j) =
ωϵxy
(j)
ckz
Ex(j)
(1.30)
To find the transfer matrix consider the boundary conditions which enforce the continuity
of electric and magnetic fields which are parallel to the j th material boundary located at
zj .
12
[
]
[
]
(j)
(j)
(j+1)
(j+1)
(+)
(−)
(+)
(−)
cos(θj ) aj eikz zj + aj e−ikz zj = cos(θj+1 ) aj+1 eikz zj + aj+1 e−ikz zj
(1.31)
(j)
] cos(θ )ϵ(j+1) [
]
cos(θj )ϵxy [ (+) ikz(j) zj
j+1 xy
(−) −ikz(j) zj
(+) ikz(j+1) zj
(−) −ikz(j+1) zj
a
e
−
a
e
=
a
e
−
a
e
j
j
j+1
j+1
(j)
(j+1)
kz
kz
(1.32)
After algebraic manipulation we arrive at the general transfer matrix for TM polarized
waves in non-magnetic uniaxial media.




(−)
aj+1
(+)
aj+1


κj )ϕ−
j

(−)
aj
(1 −

 (1 +


=β


(+)
−
(1 − κj )/ϕ+
(1
+
κ
)/ϕ
a
j
j
j
j
κj )ϕ+
j
(1.33)
where the TM parameters are
β=
1
2
sec θj+1 cos θj
(1.34)
(j) (j+1)
ϵxy kz
(j+1) (j)
ϵxy kz
κj =
{ (
) }
(j+1)
(j)
ω
ϕ±
k
±
k
=
exp
i
zj .
z
z
j
c
The transfer matrix for TE polarized waves is found using exactly the same formalism.
⃗ = (0, Ey , 0) and H
⃗ = (Hx , 0, Hz ).
First assume for TE waves the fields have the form E
⃗ =
By using the Maxwell’s equation ∇ × E
⃗
−1 ∂ H
c ∂t
we can relate the parallel magnetic field
in a layer to the corresponding parallel electric field. Note that again we are choosing the
total electric field as the primary field.
(j)
Hx(j) =
−ckz
Ey(j)
ω
(1.35)
Just as before the TE transfer matrix is found by considering the boundary conditions
which enforce the continuity of electric and magnetic fields which are parallel to the j th
material boundary located at zj .
13
[
] [
]
(j)
(j)
(j+1)
(j+1)
(+)
(−)
(+)
(−)
aj eikz zj + aj e−ikz zj = aj+1 eikz zj + aj+1 e−ikz zj
[
]
[
]
(j)
(j)
(j+1)
(j+1)
(+)
(−)
(+)
(−)
kz(j) aj eikz zj − aj e−ikz zj = kz(j+1) aj+1 eikz zj − aj+1 e−ikz zj
(1.36)
(1.37)
Rearrangement yields the general TE transfer matrix given the primary total electric field.




(−)
aj+1
(+)
aj+1



κj )ϕ−
j
(−)
aj
(1 −



 (1 +


=β
(+)
+
−
aj
(1 − κj )/ϕj (1 + κj )/ϕj
κj )ϕ+
j
(1.38)
recall that the TE dispersion equation is shown in Eq.(1.21). The polarization matrix
parameters are
β=
1
2
(1.39)
(j)
z
κj = k(j+1)
kz
{ (
) }
(j+1)
(j)
ω
ϕ±
=
exp
i
k
±
k
zj .
z
z
j
c
The interface transfer matrices above can be used to calculate the total transfer
matrix, and total transmission and reflection coefficients. To calculate the total transfer matrix consider the transfer matrix at the first interface a⃗2 = Tˆ12 a⃗1 , and the second
interface a⃗3 = Tˆ23 a⃗2 . Combining these two equations we arrive at the necessary transformation from the first to third material regions, a⃗3 = Tˆ23 Tˆ12 a⃗1 . Extrapolating this to N
layers reveals that the total transfer matrix is the product of individual transfer matrices
ˆ ...Tˆ23 Tˆ12 . To calculate the reflection and
starting from the last interface, Tˆtot = TN −1,N
transmission coefficients consider the following simplified transfer matrix form.




(−)
a2
(+)
a2



(−)
a1
  t11 t12  

=


(+)
t21 t22
a1
This matrix equation is equivalent to the following set of linear equations.
(1.40)
14
(−)
(+)
(−)
(1.41)
(−)
(+)
(+)
(1.42)
a1 t11 + a1 t12 = a2
a1 t21 + a1 t22 = a2
To solve these matrix equations for transmission and reflection coefficients we have to
(−)
assume that the reflection in the last layer is zero (a2
= 0), and we have to represent
(+)
the amplitude coefficients as a fraction of the incident amplitude (a1 ), which must be
known. The transmission and reflection amplitude coefficients are a ratio of transfer
matrix elements.
(−)
r=
a1
t=
a2
(+)
a1
=
−t12
t11
(1.43)
=
det T̂
t11
(1.44)
(+)
(+)
a1
Note that these amplitude coefficients represent the fraction of primary field which is
transmitted and reflected. Recall that we chose the total electric field in this work as the
primary field for both TE and TM polarizations. If a particular experiment was set up to
measure only a component of the electric field, then we could repeat this formalism again
by choosing the measured field component as the primary field.
1.4.2
General N-Layer Recursive Fresnel Coefficients
One significant drawback of using the transfer matrix algorithm is the numerical
error which occurs for significantly thick layered systems. In this section additional formalism is developed to derive a recursive relationship that models the optical properties of multi-layered structures. This method, along with other eigenmode expansion
algorithms[44], have the advantage of isolating phase information, and removing the layer
interface locations which cause numerical error. First, consider multiple reflections inside
a three layered thin-film system.
15
n1
n2
.
.
.
n3
.
.
.
.
.
.
t23r21r23r21r23t12exp(5ik2d2)
t21r23r21r23t12exp(4ik2d2)
t23r21r23t12exp(3ik2d2)
t21r23t12exp(2ik2d2)
t12exp(ik2d2)
r12
1
d2
FIGURE 1.4: Transmission and reflection amplitudes for all waves passing through a
three layer thin-film system. Coefficients rij and tij represent standard Fresnel amplitude
coefficients between materials with ni and nj . Note that normal incidence is assumed and
that the angles of rays are drawn only to distinguish between different paths.
The total transmission (reflection) in the three layered system above is calculated
by summing all of the waves which exit in the third (first) material regions.
rtotal = r12 + t21 r23 t12 e2ik2 d2 + t21 r23 r21 r23 t12 e4ik2 d2 + ...
(1.45)
ttotal = t12 eik2 d2 + t23 r21 r23 t12 e3ik2 d2 + t23 r21 r23 r21 r23 t12 e5ik2 d2 + ...
To solve the transmission and reflection sums the geometric series,
∞
∑
(1.46)
xn =
n=0
1
, is
1−x
used. First rewrite Eq.(1.45) in the following suggestive form
[
rtotal = r12 + t12 r23 t21 e
2ik2 d2
1 + r12 r23 e
2ik2 d2
(
+ r12 r23 e
2ik2 d2
)2
(
+ ... + r12 r23 e
2ik2 d2
)n ]
.
(1.47)
Using the geometric series solution and the Fresnel identities r12 = −r21 and t12 t21 +r212 = 1
we arrive at the compact solution for the total reflection.[45]
16
r = r13 =
r12 + r23 e2ik2 d2
1 + r12 r23 e2ik2 d2
(1.48)
Similarly, the total transmission has the following compact solution.
t = t13 =
t12 t23 eik2 d2
1 + r12 r23 e2ik2 d2
(1.49)
Calculating the total transmission and reflection from a system with many more
layers is exceedingly difficult if one attempts to trace all the possible paths that a light ray
may take. Thankfully, the solution is much easier if one is to consider the extension of the
functional forms of Eqns.(1.48,1.49) above. Note that another commonly used method for
calculating the optical properties of multi-layered films is the amplitude coefficient transfer
matrix method, which is equivalent to this algorithm. Extending the results above, the
total reflection from a five layer system is
r15 =
r12 + r25 e2ik2 d2
,
1 + r12 r25 e2ik2 d2
(1.50)
which looks the same as Eq.(1.48) except regions 3 through 5 are considered to be one
material. Looking at Eq.(1.50) we still do not know what the r25 coefficient is, but again
one can consider a material which starts at region 2 and ends at region 5 using the same
formalism.
r25 =
r23 + r35 e2ik3 d3
1 + r23 r35 e2ik3 d3
(1.51)
Finally, we finish the recursion problem by calculating the last unknown coefficient r35 ,
which has a simple three layer solution.
r35 =
r34 + r45 e2ik4 d4
1 + r34 r45 e2ik4 d4
(1.52)
The multi-layered transmission coefficients are calculated using exactly the same formalism. Now we can extend this reasoning to an arbitrary number of layers. The general
17
Fresnel amplitude coefficients between any arbitrary layer j and any other arbitrary layer
n (assuming n > j+2) are given by the following recursion relationships.
1.5.
rj,n =
rj,j+1 + rj+1,n e2ikj+1 dj+1
1 + rj,j+1 rj+1,n e2ikj+1 dj+1
(1.53)
tj,n =
tj,j+1 tj+1,n eikj+1 dj+1
1 + rj,j+1 rj+1,n e2ikj+1 dj+1
(1.54)
Dissertation Outline
In this dissertation I will be presenting research pertaining to four different plasmonic metamaterial systems. In Chapter(2.) we will explore the optical properties of
unique solution-derived silver percolation films, which were fabricated and measured by
research collaborators in the department of Physics at the University of Oregon.[46] Reflectance, transmittance, and absorbance are measured and modeled as a function of
surface filling fraction, and laser wavelength. Around the percolation threshold, when
approximately 60 percent of the film’s surface is covered with silver, there is anomalous
optical response. This interesting behavior is explained using a scaling theory model, and
accounting for the fabrication specific particle sizes in our experiments. In Chapter(3.)
we discuss the optical characterization of an amorphous nanolaminate materials platform,
which was developed by researchers in the Electrical Engineering and Chemistry departments at Oregon State University. The material system is composed of alternating layers
of amorphous metals and dielectrics. Effective medium structures were successfully fabricated and shown to display hyperbolic dispersion. Future dispersion engineering can
be achieved by expanding the materials set.[47] In Chapter(4.) we develop an ellipsometry algorithm that successfully provides physical insight into the terahertz anisotropic
dielectric response of vertically grown multi-walled carbon nanotubes.[48] Using an in-
18
dependent damped-driven oscillator dielectric function model, electron-defect scattering
times were approximated. It is believed that we are the first researchers to probe the interwall CNT scattering dynamics. Lastly, in Chapter(5.) we will explain a subwavelength
imaging algorithm based on a diffraction-based plasmonic material structure. This new
diffraction-based imaging system has the ability to reduce the pixel size in modern digital
cameras, and has the potential to outperform the resolution limitations of refraction based
imaging systems.[49]
19
2.
2.1.
ASYMMETRIC REFLECTANCE AND CLUSTER SPATIAL
EFFECTS IN SILVER PERCOLATION FILMS
Introduction
This chapter is an extension of our previously published work.[46] Research into the
optics of semicontinuous metal-dielectric films has been enjoying a sustained interest due
to a unique combination of novel physics and the practical applications offered by such
composites. It has been demonstrated, both theoretically and experimentally, that the
electromagnetic (EM) response of these structures is dominated by a non-trivial interplay
between Anderson-localized and delocalized surface plasmon polaritons.[50, 51, 52] This
results in unusual optical properties that include: greatly enhanced absorption, giant
intensity fluctuations of local EM fields, giant local chiral response, and strongly enhanced
optical nonlinearities.[53, 54, 55, 56, 57, 58, 59] In a related context, it has been recently
shown that the reflectance of semicontinuous silver nanocomposites, chemically deposited
on glass substrates, strongly depends on the direction of incident light.[60] In particular,
the reflectance of such a system irradiated from the substrate-film interface side can differ
by as much as 15% from its reflectance given film-air side incidence (see Fig.2.1). Moreover,
this large asymmetry in reflectance has been found to be extremely broadband, spanning
most of the visible frequency spectrum. For comparison, the reflectance asymmetry of thin
continuous silver films does not exceed 3% when measured over the same range of optical
frequencies, and does not exhibit any broadband characteristics. It has been suggested
that the origin of this large broadband asymmetry is in the enhanced optical absorbance
which is often seen in percolation-type systems. Here we develop a quantitative description
of the observed phenomenon.
The geometry of the system described in this chapter is shown in Figure 2.1. We
approximate the silver percolation film as a uniform material with thickness d [in our
20
calculations d = 50nm (see section 2.3)]. The structure of the film is characterized by the
surface metal filling fraction p ranging from p = 0 for a bare glass substrate to p = 1 for
a substrate which is fully covered with metal. At the critical value p = pc , known as the
percolation threshold, the dc conductivity response of the entire random metal-dielectric
composite undergoes an insulator-conductor phase transition.[61] The unique optical properties of our films are manifested particularly in the vicinity of the percolation threshold,
and we therefore adopt the conventional description of the response as function of the
parameter p − pc for the purpose of both modeling and data analysis. All theoretical and
experimental results in this work use pc = 0.6, which correlates well with two dimensional
site percolation on a square lattice (pc ≃ 0.593).[61]
FIGURE 2.1: General layered structure composed of a silver percolation film clad by air
to the left and glass to the right. Incident light may come from either the air or substrate
side as shown. Both air and glass regions are taken to be semi-infinite.
As mentioned above, the reflectance R1 of the composite film, measured using light
impinging from the air-metal interface, strongly differs from R2 - the reflectance measured
when light is incident from the substrate-film side. Since the transmittance of our system,
as the transmittance of any non-chiral homogeneous film is symmetric (i.e. T1 = T2 )[62,
21
63], the asymmetry in reflectance ∆R ≡ R1 −R2 directly reflects the asymmetry in losses.1
As we show below, in contrast to vacuum-deposited percolation films, (i) ∆R as well as the
computed combined losses exhibit a local minimum at p ≃ pc , (ii) ∆R exhibits broadband
response in the vicinity of p − pc ≃ ±0.05 , (iii) the reflectance exhibits a local maximum
in the vicinity of p ≃ pc , and (iv) the transmittance exhibits a local minimum near p ≃ pc .
2.2.
Percolation Film Synthesis and Characterization
Semi-continuous silver films with controllable filling fractions were deposited on
glass microscope slides using a modified Tollen’s reaction as described previously.[60, 64]
The amount of silver deposited on the substrates was controlled by monitoring deposition
times, with reactions ranging between 1-6 hours. These chemically deposited films appear
as highly disordered polycrystalline aggregates, with large grain-size distributions. In
addition, we note the non-uniform coating of the substrates by the metal, resulting in
highly discontinuous morphologies.
Normal incidence optical reflectance and transmittance spectra were collected using
a spectroscopic optical microscopy setup.[60] The spectral response of the film shown in
Fig.2.2 is depicted in Fig.2.3.
2.3.
Reflection, Transmission, and Absorption of Random Percolation
Composites
Many metal-dielectric composite systems are described by conventional effective
medium techniques (EMTs)[65, 66, 67, 68] by representing the composite as an effec1
Since the model we use here utilizes a uniform smooth film of known thickness, standard boundary
conditions allow only specular reflection to occur. We therefore employ the common approach which does
not distinguish between specular and diffuse loss mechanisms, lumping them together into a generalized
combined loss.
22
500 nm
FIGURE 2.2: Scanning electron micrograph of a chemically deposited silver film with
metal filling fraction p ≃ 0.52. The scale bar is 500nm.
tive homogeneous layer which successfully models the system’s average optical properties.
However, it is known that the optical properties of these films close to the percolation
threshold cannot be adequately described by EMTs.[69, 70] The reason for the consistent
failure of EMTs in this case is two-fold. First, although the dimensions of the components in percolation films are much smaller than the free-space wavelength, the optical
properties of the composites are dominated by the dynamics of resonant clusters that
can be comparable in size to the wavelength. Second, as result of a dc metal-dielectric
phase transition, the effective parameters of the percolation films in the vicinity of pc become scale-dependent and therefore cannot be described by quasi-static effective medium
models. Although some percolation films have been successfully described in terms of
Generalized Ohm’s Law (GOL)[71, 72, 73], straightforward extensions of GOL formalism
to our system are not consistent with our experimental observations.
2.3.1
Generalized Ohm’s Law for Asymmetric Structures
As stated above GOL[71, 72, 73] does not adequately describe the current optical
properties shown by the silver percolation films synthesized with the Tollen’s reaction.
Here we clarify this statement and present the comprehensive derivation of GOL formalism
23
FIGURE 2.3: Measured reflectance (red diamonds), transmittance (blue boxes), and absorbance (green circles) as function of incident wavelength for measured metal filling fraction p ≃ 0.52. Solid lines represent the results of scaling theory calculations.
in asymmetric structures. One primary advantage of GOL as compared to many effective
medium theories is that it avoids implementing the quasistatic field approximation, and
can therefore be used on much larger systems. The interactions between generalized
electric (magnetic) currents, jE (jH ), define the physical picture for GOL. Defining four
generalized optical conductivities (s, m, g1 , g2 ) the generalized current equations take the
following form.
jE = sE0 + g1 [ẑ × H0 ]
(2.1)
jH = mH0 + g2 [ẑ × E0 ]
(2.2)
The E0 and H0 terms in Eq.(2.1) represent electric and magnetic fields at the reference planes, located at a distance L0 from the percolation film as shown in Fig(2.4). In
traditional effective medium modeling many systems are considered to be purely twodimensional, and effective properties are modeled by averaging constituent materials
within the film. Imaginary reference planes in GOL are located on each side of the film,
24
Vacuum
Left
Incidence
x
z
Percolation
Film
Glass
Substrate
Rleft
Tleft
Tright
Rright
L0
d
Right
Incidence
L0
FIGURE 2.4: Schematic of a metal-dielectric percolation film on a glass substrate. At
the far left the first region is vacuum, the center grey region with thickness d is a composite medium composed of silver and vacuum, and the right region is a glass dielectric
substrate. Dashed vertical lines represent reference planes, not physical objects, used in
the implementation of GOL as a fitting parameter. Light is incident from both directions
as indicated by the solid and dashed arrows.
and the reference plane electric and magnetic fields are related to effective currents within
the film through boundary conditions due to the linearity of Maxwell’s equations. By
moving the boundary conditions away from the physical film interface 3D optical properties are not excluded from the GOL model, where the film fields are not assumed to
by curl-free and z-independent. In the limit where the film grains are disk-shaped and
the inhomogeneity scale D is much smaller than the wavelength λ, the fields on the reference plane are two dimensional and therefore curl-free to order D/λ.[73] We consider a
planar system consisting of an effectively two-dimensional (2D) random Ag-Vacuum layer
which is surrounded by vacuum on one side, and a glass substrate on the other side. The
morphology of such films is completely characterized by the surface metal coverage p. To
solve for the GOL generalized conductivities first assume that spatial field distributions
are linear functions of the reference fields as shown below.
25
E(z) = a(z)E0 − c(z)H0
H(z) = b(z)H0 + d(z)E0
(2.3)
⃗ = (0, Hy , 0) and
TEM polarization is used exclusively for the current model, with H
√
⃗ = (Ex , 0, 0). By introducing the layer impedance χj = ϵj /µj , a plane wave expansion
E
is used as a basis for the field solutions in each region.
⃗ j = c1 eikj z + c2 e−ikj z
E
⃗ j = c1 χj eikj z − c2 χj e−ikj z
H
(2.4)
All amplitude coefficients in Eq.(2.3) can be solved piecewise for each material region.
By adding up the generalized current across the system the general GOL conductivity
coefficients are derived,
∫
d/2+L0
s=
−d/2−L0
∫
a(z)σE dz
(2.5a)
b(z)σH dz
(2.5b)
c(z)σE dz
(2.5c)
d(z)σH dz
(2.5d)
d/2+L0
m=
−d/2−L0
∫
g1 =
∫
g2 =
d/2+L0
−d/2−L0
d/2+L0
−d/2−L0
where σE = (−iωϵ/4π), and σH = (iµω/4π). Note that (µ = 1) for all current simulations.
For layered systems with inversion symmetry there is a corresponding asymmetry between
gyrotropic conductivities (g1 ̸= g2 ).
To calculate the reflection of left incident light from a layered structure using GOL
formalism, first consider the plane wave expansion shown in Eq.(2.4) at the location z = 0.
26
The reference fields can be written as a linear combination of incident and reflected waves
as shown below.
E0 = Ei + Er
H0 = χ1 Ei − χ1 Er
(2.6)
By considering Maxwell’s equations in the following differential form,
dE(z)
4π
4π
=
σH H(z) =
j
dz
c
c E
−dH(z)
4π
4π
=
σE E(z) =
j ,
dz
c
c H
(2.7)
and defining the vacuum reference fields as (E0 , H0 ), and the glass reference fields as
(E1 , H1 ), Eq.(2.7) is simplified to the following useful form.
4π
(mH0 + g2 E0 )
c
4π
H0 = H1 +
(sE0 − g1 H0 )
c
E0 = E1 −
(2.8)
Using the relationship (χ3 E1 = H1 ) along with Eqs.(2.6,2.8), the total reflection for left
incident light rl = (Er /Ei ) is obtained.
rl =
c (χ1 − χ3 ) − 4π (s − χ1 g1 + χ3 g2 + χ1 χ3 m)
c (χ1 + χ3 ) + 4π (s + χ1 g1 + χ3 g2 − χ1 χ3 m)
(2.9)
Similar physical reasoning leads to all other GOL reflection and transmission coefficients
shown below.
27
c (χ3 − χ1 ) + 4π (χ3 g2 − χ1 χ3 m − s − χ1 g1 )
c (χ3 + χ1 ) + 4π (χ3 g2 − χ1 χ3 m + s + χ1 g1 )
(
)
2χ1 (c + 4πg1 )(c + 4πg2 ) + 16π 2 sm
tl =
c [c(χ1 + χ3 ) + 4π (s + χ1 g1 + (g2 − χ1 m)χ3 )]
2χ3 c
tr =
c (χ1 + χ3 ) + 4π (χ3 g2 − χ1 χ3 m + χ1 g1 )
rr =
(2.10a)
(2.10b)
(2.10c)
1.0
R,T,A
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
p (surface metal filling fraction)
1.0
FIGURE 2.5: Reflectance (red long-dashed), transmittance (black short-dashed), and
absorbance (blue solid) through our percolation film as a function of surface coverage
fraction for a 10cm reference plane GOL system. The percolation threshold is assumed to
be pc = 0.5, and light is incident from the air side of the film.
In the limit of small reference plane distance (L0 ≪ λ) the GOL and Transfer Matrix
transmission and reflection coefficients become equivalent and may be used interchangeably. Effective Medium Theory models commonly employ a weighted averaging scheme to
homogeneous amplitude coefficients to calculate the average optical properties of a twocomponent film. Composite weighted averaging models do not verify our experimental
measurements, particularly near the percolation threshold, therefore Scaling Theory must
be employed to describe the insulator-metal phase transition properly. You will see below
that Fig.(2.5) does not accurately represent the optical response of our percolation films.
28
2.3.2
Scaling Theory Formalism
The only technique that explicitly accounts for the dc conductivity phase transition
of percolation systems is known as scaling theory.[69, 70, 74, 75, 76] In this technique, the
conductivity of the film is assumed to be explicitly dependent on the size of the cluster
L over which it is measured. The average conductivity of a metal-dielectric composite
mixture of length L is modeled by the following function,
[
]
σaverage = σm L−µ/ν F (σd /σm )L(µ+s)/ν ,
(2.11)
where µ is the dc conductivity critical exponent, s is the capacitance critical exponent,
and ν determines the correlation length scaling behavior. F [x] = C1 + C2 x for metallic
conductivity, and F [x] = C3 x + C4 x2 for dielectric conductivity. Using the definitions
above the average conductivity of a conductive cluster of size L is given by
C1 σdc
σm (L) =
1 + ω2τ 2
(
L
ξ0
)−µ/ν
[
C1 σdc ωτ
+i
1 + ω2τ 2
(
L
ξ0
)−µ/ν
(
− C2 ωC0
L
ξ0
)s/ν ]
,
(2.12)
and that of a dielectric (insulating) cluster is
[
( )
( )
( )s/ν ]
C3 ω 2 C02 L (µ+2s)/ν
C3 ω 2 C02 ωτ L (µ+2s)/ν
L
σd (L) =
+i
− C4 ωC0
.
σdc
ξ0
σdc
ξ0
ξ0
(2.13)
The expressions above explicitly assume that the conductivity of the conductive
component of the film is given by the Drude model,
σ1 =
σdc
,
1 − iωτ
(2.14)
where σdc is the dc conductivity, τ is the electron relaxation time, and ω is the angular
frequency of the incident light. The ac response of a dielectric film component is equivalent
to that of a capacitor,
σ2 = −iωC0 ,
(2.15)
29
where C0 is the average capacitance between neighboring metal clusters. The parameters
σdc , τ, ξ0 , C0 and C1 . . . C4 coefficients are uniquely determined by the composition and
micro-geometry of the percolation film. The critical exponents for 2D percolating films
are µ = s = 1.3 and ν = 4/3.[69, 70, 77] For p ≪ pc percolation films are governed by
dielectric conductivity, which is dominated by the capacitance coefficients C3 and C4 . For
p ≫ pc metallic conductivity (governed by C1 , C2 ) dominates the optical properties of the
system.
Despite the scale-dependence on the microscopic and mesoscopic levels, the percolation film appears homogeneous when conductivity is measured over a significantly large
area. The transition from the scale-dependent to the homogeneous dc response occurs at
the scale known as the correlation length, ξ, that characterizes the typical cluster size.
One can define a correlation function g(r) which represents the probability that a site at
distance r away from an occupied (metallic) site is also occupied, and belongs to the same
cluster. Given a correlation function the correlation length is defined as
∑
r2 g(r)
r
ξ2 = ∑
.
(2.16)
g(r)
r
In the vicinity of the percolation threshold, the correlation length diverges as
p − pc −ν
.
ξ = ξ0 pc (2.17)
The constant ξ0 represents the smallest metal cluster size, which occurs at p → 0.
At finite frequencies, the oscillatory motion of electrons within conducting clusters
leads to the length scale correction of the homogeneous response of the system,


 B0 ξ0 (λ0 /2πξ0 )1/(2+θ) ,
L(λ0 ) = min

 ξ(p)
(2.18)
30
where θ = 0.79 is related to the fractal dimension of the film, the fitting parameter
B0 = 4.0, and the free space wavelength is given by λ0 .[69, 70]
The ac conductivity of the percolation films, calculated using the expressions above
can be directly related to an effective film index, which along with the film thickness
d can be used to determine the macroscopic optical properties of the film, including R
and T . In our calculations, we use the technique introduced in [69]. In this approach, the
optical properties of the film are calculated as a weighted average of conductive (dielectric)
film contributions, where the average conductivities are given by Eqs.(2.12) and (2.13)
respectively to yield
∫
T =
∫
Ri =
∞
[f Tσ (zσm ) + (1 − f )Tσ (zσd )] P (z)dz
(2.19)
[f Ri,σ (zσm ) + (1 − f )Ri,σ (zσd )]P (z)dz
(2.20)
0
∞
0
where the parameter
[
) ( )1/ν ]
(
L
1
p − pc
,
f=
1+
2
pc
ξ0
(2.21)
is the metal occupation probability. For small surface metal concentrations,
[
[
( )−1/ν ]
( )−1/ν ]
L
p < pc 1 − ξ0
, the occupation probability f → 0. When p > pc 1 + ξL0
the occupation probability f → 1. For intermediate surface metal concentrations centered
( )−1/ν
at p = pc with full-width ∆p = 2pc ξL0
, the occupation probability varies linearly
as a function of p from unoccupied (f = 0) to occupied (f = 1). As shown by the
above inequalities, the range of metal surface coverage values for which scaled metal
and dielectric optical properties are averaged depends non-trivially on the correlation
length, applied frequency, and film geometry. The function P (z) gives the distribution
of the conductivities of conductive [dielectric] clusters around their mean values given by
Eq.(2.12) [Eq.(2.13)]. Following Ref.[69, 78] we assume that P (z) is adequately described
by a log-normal distribution function with standard deviation of σsd = 0.3. Integrating
31
over all scaled conductivities averages out the length dependent optical conductivity and
allows for percolation films to be modeled by the contributions from planar homogeneous
constituent layers.
The homogeneous-layer optical properties are given by, [45]
2
√
4nf ns Φ
Tσ = 2
(1 + nf )(nf + ns ) + (1 − nf )(nf − ns )Φ (1 − nf )(nf + ns ) + (nf − ns )(1 + nf )Φ2 2
R1,σ = (1 + nf )(nf + ns ) + (1 − nf )(nf − ns )Φ2 (nf − ns )(1 + nf ) + (1 − nf )(nf + ns )Φ2 2
R2,σ = (1 + nf )(nf + ns ) + (1 − nf )(nf − ns )Φ2 where the glass substrate index is ns = 1.5166, the effective film index is nf =
(2.22)
(2.23)
(2.24)
√
1 + 4πiσ/ω,
and the phase parameter is Φ = exp(i ωc nf d).
As noted, all critical exponents in the expressions above are universal for all 2D
percolating networks, while the parameters C0 . . . C4 , σdc , τ , and ξ0 are unique for a given
percolation film.[69, 70] In our calculations, we use σdc = 2.574 × 1017 sec−1 , frequency
dependent relaxation time [79] 1/τ = 1/τ0 + βω 2 , where τ0 = 3.0fs, β = 0.2fs, C0 = 0.5,
C1 = C2 = 0.046, C3 = 0.028, C4 = 0.055, and ξ0 = 2nm.
2.4.
Deriving the Necessary Conditions for Nonzero ∆R
One might assume, correctly in fact, that the change in reflectance ∆R = R1 −
R2 (see Fig.2.1) in a three layer film is caused by broken inversion symmetry due to
the interior reflections being equivalent within the central material region regardless of
incidence direction, and the larger impedance mismatch on one side. In this section, we
will prove this assumption quantitatively and show an additional necessary condition that
is required to obtain nonzero ∆R.
Starting with the previously derived three layer total reflectance equations for left
32
and right incidence, and consider their difference.
r12 + r23 e2ik2 d2 2
R1 = |r13 |2 = 1 + r12 r23 e2ik2 d2 (2.25)
r32 + r21 e2ik2 d2 2 r23 + r12 e2ik2 d2 2
R2 = |r31 | = =
1 + r12 r23 e2ik2 d2 1 + r12 r23 e2ik2 d2 (2.26)
r12 + r23 e2ik2 d2 2 r23 + r12 e2ik2 d2 2
−
∆R = R1 − R2 = 1 + r12 r23 e2ik2 d2 1 + r12 r23 e2ik2 d2 (2.27)
2
Using the substitution β = 2k2 d2 along with complex conjugate formalism we arrive at
the following equivalent formula for the difference in reflectance.
∆R =
(r∗12 r23 − r12 r∗23 )eiβ + (r12 r∗23 − r∗12 r23 )e−iβ
2
|1 + r12 r23 eiβ |
(2.28)
By visual inspection one can see that ∆R = 0 when r∗12 r23 = r12 r∗23 . If the first and
third regions are the same material (symmetric), then r23 = r21 = −r21 , and there is no
change in reflectance. Additionally, if r12 and r23 are purely real there is no change in
reflectance. Therefore the necessary conditions for nonzero ∆R is a system with broken
inversion symmetry which contains loss.
2.5.
Comparing Scaling Theory with Experimental Results
A comparison of the experimentally obtained spectral response of the silver films
with the predictions of scaling theory is shown in Fig.(2.3). It is seen that both the
broadband nature of the reflectance asymmetry and its non-monotonic behavior near the
percolation threshold are are well reproduced by the theoretical model, as demonstrated
in Fig.(2.6). We note however, that the model fails for the large metal concentrations p →
1[60, 64] where the structure of the composite becomes substantially three-dimensional
and cannot be treated as a thin homogeneous film. To further illustrate the robustness
33
of the presented technique we show in Fig.(2.7) a comparison of experimentally measured
values of R1 and T1 , as well as the losses (computed as A1 = 1 − R1 − T1 ,) with our
theoretical model. As mentioned before, both theoretical and experimental results clearly
show that despite a strong reflectance asymmetry, the transmittance of the films remains
symmetric. Therefore the asymmetry in reflectance is directly related to the asymmetry
in losses (∆R = R1 − R2 = A2 − A1 ).[60]
0.20
0.2
0.1
ΔR
0.15
0.0
-0.1
ΔR
0.10
-0.2
-0.6
-0.4
-0.2
0.0
p-pc
0.2
0.4
0.05
0.00
-0.6
-0.4
-0.2
0.0
0.2
p-pc
FIGURE 2.6: Points represent the measured change in reflectance (∆R = R1 − R2 )
for various incident wavelengths. Black circles 500nm, green triangles 600nm, red boxes
700nm. Corresponding colored solid lines (black solid 500nm, green short-dashed 600nm,
red long-dashed 700nm) represent the results of scaling theory reflectance calculations.
The inset shows the change in reflectance over the entire surface coverage range. Note
that the 2D scaling model fails for large metal concentrations, where the three-dimensional
structure of the composite dominates the optical response.
We now examine the loss or, alternately the reflectance, in more detail. We note
that several previous experiments as well as theoretical models have observed absorption
maxima in the vicinity of p = pc . In contrast, our experimental data clearly demonstrate a
local minimum in losses near the percolation threshold, as seen in Fig.2.7(e). Additionally,
we observe a local maximum in reflectance and a local minimum in transmittance near
p = pc , as seen in Fig.2.7(a,c).
We suggest that this anomalous behavior stems from the dramatically reduced correlation length in our solution-derived percolation films. The smallest metallic particle
34
FIGURE 2.7: Points represent the measured (a) reflectance, (c) transmittance, and (e)
loss from the air side as a function surface coverage fraction. Connecting lines are a guide
for the eye. Calculated (b) reflectance, (d) transmittance, and (f) loss when the correlation
length parameter is ξ0 = 2nm (solid line), ξ0 = 5nm (dashed line), and ξ0 = 10nm (dotted
line). For all graphs the incident wavelength is 700nm.
size produced in our experiments is on the order of 2 nm, in contrast to 10 nm reported
in Ref.[70]. In addition, the optical response of solution-derived metals is typically affected by the reduced electron mean free path [80], which may further reduce the effective
particle size, described by the parameter ξ0 .
Figure 2.7 demonstrates the evolution of optical properties of percolation systems
when the correlation length is reduced, corresponding to a change in ξ0 from 10 nm to 2
nm. It is clearly seen that at ξ0 ≃ 2nm the absorption reaches a local minimum in the
vicinity of p = pc , while at ξ0 = 10nm, the system recovers the absorption maximum at
p = pc , as observed in previous studies.
The dramatic difference between correlation lengths in our solution-derived films
[60, 64] and other vacuum-deposited counterparts [71, 69, 81] is consistent with the dif-
35
ference in fabrication technique: while thermal deposition under vacuum typically yields
uniform metal films with almost perfect Ohmic contacts between adjacent grains, solutionbased deposition is routinely associated with quantitatively weaker contacts between conducting grains. The latter results in films with reduced electron mean-free-paths and lower
correlation lengths.
2.6.
Conclusion
We have developed an analytical description of the phenomenon of broadband asymmetric reflection in percolation composites. The developed technique, based on scaling
theory, is not only capable of describing the spectral response of our films, but also explains that the reduced correlation length in our solution-derived composites is the primary
cause of the experimentally observed anomalous optical properties near the percolation
threshold. Our work demonstrates that the correlation length is an important factor that
fundamentally affects the optical properties of percolation composites in the vicinity of
the percolation threshold.
36
3.
3.1.
OPTICAL PROPERTIES OF AMORPHOUS NANOLAYERS
Introduction
Many optical and electrical properties are derived from the natural periodicity of
material structures. By engineering periodicity through material geometry, scientists aim
to create devices which have tunable optical and electrical responses.[82] Due to advances
in nanofabrication and theoretical understanding, the fields of plasmonics and metamaterials have recently seen many examples of geometric structures that have properties
which are distinctly different from the constituent materials.[83] One such example is the
split-ring resonator, which is the optical analog of the LC tank in electronics.[84, 85, 86]
By changing the spatial dimensions and material composition of split-ring resonators, one
can tune the permittivity and permeability of resulting devices.
The goal of engineering metamaterial devices is not only to create fundamentally
new behavior, but to outperform conventional device limitations such as the diffraction
limit. Perhaps the most intriguing examples of this are cloaking devices, which have the
potential of making objects invisible.[87, 88, 89] Another recently proposed possibility is
the optical black hole, which absorbs all incident electromagnetic radiation.[90, 91] Surface
plasmons, which are waves that are confined to the surface between a metal and dielectric,
can be focused and controlled on the nanoscale far beyond the diffraction limit, and create
extraordinary transmission in subwavelength optical devices.[92, 93, 94, 95, 96] Self similar
nanospheres have been proposed to superfocus electromagnetic radiation and allow for
the manipulation of single molecules.[97] Due to the ability to tune plasma resonance
using geometry, nanoplasmonic structures display many different colors based on their
size and shape, and are ideal for sensor applications.[98, 99] Solar cell technology and
energy storage methods can be made more efficient by engineering electrical and optical
37
response in nanomaterials.[100]
In this work a new type of metamaterial comprised of completely amorphous constituents is presented. There is short-range chemical bonding order in these materials, and
the long-range periodicity is a result of the engineered layered geometry. The drawbacks of
amorphous metals are high material losses and comparatively low electrical conductivity.
These drawbacks are offset by the ability to fabricate smooth and contiguous thin films
at single-nanometer thickness scales.
3.2.
Nanofabrication and Material Characterizaton
All experimental work in this chapter was performed by William Cowell and Christopher Knutson from the departments of Electrical Engineering and Chemistry at Oregon State University. Metal targets with chemical compositions of Zr40 Cu35 Al15 Ni10
and Ti25 Al75 (TiAl3 ), were sputtered via DC magnetron onto thermally oxidized silicon.
A new class of aqueous, solution-deposited, amorphous dielectric films utilize dissolved
metal oxides in solution as well as the surface tension of water to produce ultra-smooth
surfaces. This method allows for low-temperature processing of oxide dielectrics from
water.[101, 102, 103, 104] A solution of aluminum oxide phosphate (AlPO) precursor was
prepared as previously described by Meyers et. al. [101] and spin coated onto the amorphous metal. Dielectric-layer thicknesses are determined either by the concentration of the
solution, or by utilizing multiple coating steps. The sputtering/spin coating was repeated
until the desired number of bilayers was achieved.
Sputtered amorphous metals and solution-processed dielectrics have been proven to
be a low-cost, and highly efficient materials set for producing nanolaminated metal/insulator
systems.[105] When a single metallic element is sputtered it will typically result in a polycrystaline layer. It is difficult to achieve an amorphous single element metallic thin film.
38
FIGURE 3.1: TEM image of a 10 bilayer TiAl3 -AlPO system. Dark regions represent
TiAl3 (metal) and light regions represent AlPO (dielectric). Vector directions are marked
as referenced in the body text.
Conversely, when an alloy with appropriate chemical complexity, in terms of atomic radius and crystal structure is chosen (ie., quite varying radii and differing crystal structures), the resultant film will be amorphous. While amorphous metals provide an excellent
method of creating ultra-smooth, ultra-thin metallic films, they are typically confined to
low-temperatures due to the propensity of materials to crystallize.
Polarized optical reflectance measurements were carried out on a spectroscopic ellipsometer (SE) with a Xe arc lamp source whose spectrum ranged from 300 nm to 1500
nm. Data was collected at angles of incidence ranging from 20◦ to 80◦ .
3.3.
Bulk Optical and Electrical Properties of Amorphous Metallic and
Dielectric Glass
Bulk (in this case thick film) measurement and characterization are essential steps
in engineering nanostructures. Later in this contribution, individual material dielectric
responses are combined to analyze composite thin-film systems. While the primary focus
of this work is the high-frequency optical response of our amorphous metamaterials, the
DC resistivity of the amorphous metals was also analyzed, and in contrast to the optical
response, the DC resistivity is approximately equal for bulk ZrCuAlNi and TiAl3 . Using
39
(a)
(c)
(b)
FIGURE 3.2: Amorphous material characterization. (a) Wide-angle image of laminated
structure containing both amorphous metals (TiAl3 ,ZrCuAlNi) with AlPO dielectric layers
interspersed showing no crystalline spots. Darkened zone indicates where the included
high-resolution image (b) was taken. (b) High-resolution TEM image of amorphous metal
/ AlPO nanolaminate with constituent layers labeled. (c)Electron diffraction taken from
the high-resolution image.
4-point probe measurements of sheet resistance carried out on multiple film thicknesses
ranging from 10-300nm the DC resistivity of amorphous ZrCuAlNi was 191.5 µΩ·cm, and
TiAl3 was 191.8 µΩ·cm. The stability of the measured resistivity values over the given
length scales is a strong indication that amorphous metals have continuous morphology to
much smaller lengths than their polycrystalline counterparts. As expected, these amorphous metals are approximately 2 orders of magnitude less conductive than noble metals
due to weak electron localization[106].
Reflectance measurements from a spectroscopic ellipsometer are used to characterize all materials in this work. The primary advantage of ellipsometry measurements is the
large angular and frequency measurement space available for characterization and mate⃗
rial parameter data fitting. The Jones vector, J=(R
TM ,RTE ), is measured from surface
reflectance for TE and TM polarized incident laser radiation from 300nm to 1500nm.
Referring to Fig.(I) the z-direction, which is normal to layer interfaces, is defined as the
40
RTM
20º
0.6
0.5
45º
0.4
70º
0.3
80º
0.2
RTE
400
600
1.0
800
1000 1200 1400
800
1000 1200 1400
λ(nm)
80º
0.9
70º
0.8
45º
0.7
0.6
20º
400
600
λ(nm)
FIGURE 3.3: TM and TE polarized reflectance from an optically thick 200 nm TiAl3 film
as a function of wavelength. Solid black lines represent theoretical reflectance results and
points are measured data for 20◦ (red), 45◦ (green), 70◦ (blue), and 80◦ (brown) angles of
incidence. It should be noted that spurious data points near 900nm are related to the Xe
lamp spectrum, and not material resonance.
propagation direction, thus for TM polarization there is no Hz field, and for TE polarization there in no Ez field. For all angles and incident wavelength values there was negligible
coupling between different polarization reflectance states.
Optical reflectance from bulk metal samples into air is calculated with single interface Fresnel coefficients,
RTM
RTE
√
ϵ 1 + cos(2θ) − √2ϵ + cos(2θ) − 1 2
√
= √
ϵ 1 + cos(2θ) + 2ϵ + cos(2θ) − 1 √
1 + cos(2θ) − √2ϵ + cos(2θ) − 1 2
√
= √
,
1 + cos(2θ) + 2ϵ + cos(2θ) − 1 (3.1)
(3.2)
41
RTM
20º
0.6
0.5
45º
0.4
70º
0.3
0.2
80º
400
600
800
1000 1200 1400
λ(nm)
RTE
80º
0.9
0.8
70º
0.7
45º
0.6
0.5
20º
0.4
400
600
800
1000 1200 1400
λ(nm)
FIGURE 3.4: TM and TE polarized reflectance from an optically thick 284 nm ZrCuAlNi
film as a function of wavelength. Solid black lines represent theoretical reflectance results
and points are measured data for 20◦ (red), 45◦ (green), 70◦ (blue), and 80◦ (brown)
angles of incidence.
where θ is the angle of incidence (and reflection), and ϵ is the dielectric function of the
amorphous materials. Note that all materials in this work are non-magnetic (µ = 1, n =
√
ϵ). To extract the frequency dependent complex dielectric material response 13 unique
TM polarized angular reflectance measurements were taken (at each wavelength) to minimize the difference between measured and modeled bulk reflectance using nonlinear least
squares fitting. Numerical dielectric results were independent of the fitting algorithm and
epsilon starting locations. Note that the anomalous peaks near 900 nm in the reflectance
spectrum is due to lamp fluctuations, not material resonance. The results were also verified by the scanning spectroscopic ellipsometer software model. The dielectric response
for each individual bulk material is isotropic, in contrast to the anisotropic composite
42
behavior that will be discussed in the next section.
In general, for both amorphous metal samples TM reflectance decreases, and TE
reflectance increases as the incident angle increases. TM polarization reflectance shows an
interesting change in functional behavior by changing the slope for high incident angles.
Theoretical curves tend to overestimate TE reflectance of ZrCuAlNi for large angles.
Scanning ellipsometry measurements of these optically thick films of AlPO, TiAl3 and
ZrCuAlNi were used to extract the dielectric functions. Results are shown in Fig.(3.5).
The AlPO shows data that is typical for a transparent material.
Re(є)
2
AlPO
0
-2
TiAl3
-4
-6
Zr-Cu-Al-Ni
400
600
800
1000 1200 1400
λ(nm)
FIGURE 3.5: Real part of the dielectric response as a function of wavelength for bulk
dielectric AlPO (blue), and amorphous metals TiAl3 (black) and Zr-Cu-Al-Ni (red).
As expected, the real part of both amorphous metal dielectric constants are negative, representing the decaying behavior of light inside of metallic materials due to the
fast rearrangement of free charge. Dielectric constants which are more negative correspond to materials which are more conductive, with shorter corresponding decay lengths.
Unlike the DC conductivity, the frequency dependence of our amorphous metals uniquely
depends on the atomic speciation. As shown in Fig.(3.5) the conductivity of amorphous
ZrCuAlNi decreases as a function of frequency (Re(ϵ) becomes less negative), which is consistent with a conventional Drude-model metallic response. In contrast, the conductivity
of TiAl3 increases as a function of frequency (Re(ϵ) becomes more negative). The phys-
43
ical mechanism responsible for this effect in TiAl3 is believed to be caused by interband
transitions to higher mobility conduction states.
Im(є)
50
40
TiAl3
30
20
Zr-Cu-Al-Ni
10
400
600
800
1000 1200 1400
λ(nm)
FIGURE 3.6: Imaginary component of the dielectric function for bulk amorphous metals
TiAl3 (black) and Zr-Cu-Al-Ni (red).
The AlPO loss, Im(ϵ), is not shown because it is always at least three orders of
magnitude smaller than the metal loss, and does not play a significant role in the measured
optical response. In realistic plasmonic systems loss usually plays a primary constraining
role, and these metals are no exception. The loss in these amorphous multi-component
metals limits the transmission device design capabilities to very thin layers. Attempting
to significantly decrease the loss in these metals is a fundamentally difficult task due to
the requirement of lattice purification, which is directly counter to the disordered nature
of the films. Therefore, the realistic applications for these systems should take advantage
of large material loss. Plasmonic systems can provide improved efficiency in solar cells
by increasing light absorption in thin-film and organic systems.[107, 108] Measuring the
number of carriers and understanding carrier dynamics in different types of amorphous
metallic glass is ongoing work.
44
3.4.
Effective Anisotropic Medium - Dispersion Engineering
By fabricating alternating layers of metal and dielectric materials with bilayer thicknesses much smaller than the wavelength of light (quasistatic), one is able to create composites which display effective average anisotropic dielectric response. It is important to
note that obtaining planar quasistatic metal-dielectric structures is not an easy task in the
visible regime using polycrystalline metals as it is difficult to deposit them in a very thin
and contiguous embodiment. The amorphous nature of these films plays a crucial role in
allowing for reliable fabrication of contiguous metal-dielectric layers which have bilayer
thickness of 16nm. Studies as to how small continuous bilayers may be fabricated are
ongoing, however early results indicate that the bilayer thicknesses reported in this contribution are not the minimum. Note that minimizing layer thicknesses is not advantageous
beyond the point where conventional Drude dispersion breaks down.
Maxwell-Garnett and Bruggeman effective medium theory techniques have been well
established for calculating the average dielectric response in systems with spherical, and
elliptical material inclusions.[65, 66] Here there are planar metal-dielectric layers, so the
average dielectric response is calculated by taking the spatial average of the displacement
and electric field ϵave. =
<D>
<E>
across a bilayer.
(
ϵxy =
< ϵE >
=
<E>
dm ϵm E0 +dd ϵd E0
dm +dd
(
(
ϵz =
dm E0 +dd E0
dm +dd
dm D0 +dd D0
dm +dd
)
)
=
dm ϵm + dd ϵd
dm + dd
(3.3)
)
<D>
ϵ ϵ (d + dd )
)= m d m
=(
d
D
/ϵ
+d
D
/ϵ
m 0 m
d 0 d
< D/ϵ >
dm ϵd + dd ϵm
(3.4)
dm +dd
The metal and dielectric layer thicknesses are given by dm and dd , the quasistatic
field amplitudes are E0 and D0 , and the metal and dielectric layer complex permittivities
are given by ϵm and ϵd . Fabricating thin metallic layers is necessary for any transmissive
applications for these films, but it is not ideal to create non-local optical response by
45
confining electrons to a distance that is smaller than the distance an electron moving the
fermi velocity travels at optical frequencies, which is on the order of 1-2nm.[109] When
materials display anisotropic dispersion, uniaxial response in this case, the dispersion
equation for TM is shown in Eq.(1.22), where the xy and z directions are parallel and
perpendicular, respectively, to layer interfaces as shown on Fig.(3.1).
6
ZrCuAlNi-Re(єz)
Re(єEMT)
4
TiAl3-Re(єz)
2
TiAl3-Re(єxy)
0
-2
-4
ZrCuAlNi-Re(єxy)
400
600
800 1000
λ(nm)
1200
1400
FIGURE 3.7: Real part of the effective anisotropic dielectric constant for the 4.7nm TiAl3 11.3nm AlPO (black) and the 8nm ZrCuAlNi - 8nm AlPO (red) composites. Yellow shaded
regions represent the spectral regions where the composites have hyperbolic dispersion.
If the sign of either anisotropic dielectric constant becomes negative, then the resulting material displays hyperbolic dispersion as seen in Eq.(1.22). Negative refraction
occurs when Re(ϵz ) < 0 and Re(ϵxy ) > 0, which has been the most ideal way to fabricate
negative refraction structures in non-magnetic materials without using material patterning [16, 17]. Here, negative refraction is not observed due to the high metal loss and
small ALPO dielectric constant, however an alternative type of hyperbolic material where
Re(ϵz ) > 0 and Re(ϵxy ) < 0 is observed. We also demonstrate a plausible mechanism for
creating negative refraction by utilizing novel, low-cost production materials and methods.
The shaded regions in Fig.(3.7) show the spectral regions where these layered amorphous
metamaterials display hyperbolic dispersion. When these materials are hyperbolic one
expects positive refraction of transmitted wave energy and negative phase velocity. Direct
46
RTM
0.5
0.4
20º
80º
45º
0.3
0.2
70º
0.1
400
600
800
1000 1200 1400
λ(nm)
RTE
80º
0.9
0.8
70º
0.7
0.6
45º
20º
0.5
0.4
400
600
800
1000 1200 1400
λ(nm)
FIGURE 3.8: TM and TE polarized reflectance from 10 bilayers of 8nm ZrCuAlNi and 8
nm AlPO as a function of wavelength. Solid lines represent theoretical effective medium
theory reflectance results and points are measured data for 20◦ (red), 45◦ (green), 70◦
(blue), and 80◦ (brown) angles of incidence.
measurements of negative phase velocity are not possible for the current systems in use
due to signal to noise constraints from the high losses in these amorphous metals.
As shown in Figs.(3.8,3.9) an effective medium dielectric response model, shown by
the solid lines, represents the measured reflectance from the metal-dielectric layers, shown
by the data points, quite well for all angles of incidence and measured frequencies.
One of the goals in creating effective medium structures is to engineer the dielectric response in fabricated composites, particularly when the optical response is not readily
available in bulk materials. Ideally, by creating a suitable set of metal targets for sputtering, and having various dielectric glasses which can be deposited in the fashion described
here, an entire metamaterial production platform could be created using the techniques
47
RTM
0.5
20º
0.4
45º
0.3
80º
0.2
70º
RTE
400
600
800
1000 1200 1400
λ(nm)
80º
0.8
70º
0.6
45º
0.4
20º
0.2
400
600
800
1000 1200 1400
λ(nm)
FIGURE 3.9: TM and TE polarized reflectance from 10 bilayers of 4.7nm TiAl3 and 11.3
nm AlPO as a function of wavelength. Solid lines represent theoretical effective medium
theory reflectance results and points are measured data for 20◦ (red), 45◦ (green), 70◦
(blue), and 80◦ (brown) angles of incidence.
presented. This materials platform could be used to engineer material dispersion for a
number of applications including: biological sensing, improved solar-cell efficiency, optical
filters, and anisotropic thermal conduction.
3.5.
Necessary Conditions for Non-Magnetic Negative Refraction
In the quasistatic limit, where layer thicknesses are much smaller than the incident
wavelength, the effective average dielectric constant for planar layered structures is given
by Eqs.(3.3,3.4). The material dielectric constants (ϵm , ϵd ) shown above are complex
values, making the average anisotropic dielectric response functions complex. In order to
48
find the conditions which must be satisfied for negative refraction we will consider the real
part of the effective epsilons, which does include all complex material values. Start by
introducing the following notation for the real and imaginary material dielectric constants.
ϵm = ϵ′m + iϵ′′m
(3.5)
ϵd = ϵ′d + iϵ′′d
(3.6)
Plugging the above definitions into the effective anisotropic dielectric equations yields,
dm (ϵ′m + iϵ′′m ) + dd (ϵ′d + iϵ′′d )
dd + dm
(3.7)
(ϵ′m + iϵ′′m )(ϵ′d + iϵ′′d ) (dd + dm )
.
(ϵ′m + iϵ′′m )dd + (ϵ′d + iϵ′′d )dm
(3.8)
ϵxy =
ϵz =
With knowledge of the anisotropic Poynting vector the real part of Eqns.(3.7,3.8) is
needed, which in our system corresponds to the propagating waves in our system. Recall
that for negative refraction to occur that Re(ϵz ) < 0, and Re(ϵxy ) > 0, those are the
conditions where negative refraction of wave energy occurs. Without any approximation
the real average dielectric constants are
Re(ϵxy ) =
Re(ϵz ) =
dm ϵ′m + dd ϵ′d
dd + dm
(3.9)
(ϵ′m ϵ′d − ϵ′′m ϵ′′d )(ϵ′m dd + ϵ′d dm )dbl + (ϵ′′m ϵ′d + ϵ′m ϵ′′d )(ϵ′′m dd + ϵ′′d dm )dbl
,
(ϵ′m dd + ϵ′d dm )2 + (ϵ′′m dd + ϵ′′d dm )2
(3.10)
where dbl is the bilayer thickness dbl = dd + dm . Eqn.(3.9) is quite easy to visualize, by
inspection we can say that the ϵxy condition for negative refraction is,
Re(ϵxy ) > 0
when
dm ϵ′m > −dd ϵ′d
or
−ϵ′m dm
< 1.
ϵ′d dd
(3.11)
49
Note that ϵ′m in Eqn.(3.11,3.12,3.13) is negative, and all other constants are positive.
Using algebra Eqn.(3.10) can be reduced to find the ϵz condition for negative refraction.
′′2
′
′2
′′2
Re(ϵz ) < 0 when ϵ′m dm (ϵ′2
d + ϵd ) < −ϵd dd (ϵm + ϵm )
or
′′2
−ϵ′m dm (ϵ′2
d + ϵd )
>1
′′2
ϵ′d dd (ϵ′2
m + ϵm )
(3.12)
In order for true negative refraction to occur, meaning negative refraction of the Poynting
vector inside the planar layers, both inequalities in Eqns.(3.11,3.12) must be simultaneously true. Combining all inequalities yields the following condition for negative refraction.
′′2
(ϵ′2
−ϵ′m dm
m + ϵm )
<
<1
′′2
ϵ′d dd
(ϵ′2
d + ϵd )
(3.13)
Negative refraction will occur when Eqn.(3.13) is satisfied.
3.6.
Layer Thickness Verification using Effective Medium Error Analysis
Effective medium modeling is a useful metrology tool for verifying the metal and
dielectric layer thicknesses within nanolaminates. First define some Metal:Dielectric ratio
as the ratio of metal to dielectric thickness,
dm
dd .
Experimentally, this ratio is determined
by deposition rates and sputtering times for the metals, and well as surface tension and
spin time for the dielectrics. Theoretically, one can vary the layer thicknesses until the
error between the measured and modeled reflectance is minimized. This is how to verify
the layer thicknesses.
Normalized Error =
∑
λ=300−1500nm
Rmeasured − Rmodel Rmeasured
(3.14)
More precisely the normalized reflectance error is defined as the sum of the normalized difference in reflectance over the measured frequency spectrum as show in Eq.(3.14).
50
50
ZrCuAlNi 20
45
Normalized Error (%)
ZrCuAlNi 45
40
35
TiAl3 20
30
TiAl3 45
25
20
15
10
Metal/Dielectric Ra!o
5
0
0
0.5
1
1.5
2
2.5
3
3.5
FIGURE 3.10: Normalized reflectance error for both TiAl3 -ALPO and ZrCuAlNi-ALPO
10 bilayer systems at 20◦ and 45◦ incidence for different theoretical Metal-Dielectric thickness ratios.
Fig.(3.10) shows the normalized reflection error as a function of the metal dielectric ratio
( ddmd ) for 20◦ and 45◦ incident light. The ZrCuAlNi-ALPO bilayer system shows a metal
dielectric ratio of 1, meaning the metal and dielectric thicknesses are equal (8nm in this
case). The TiAl3 -ALPO bilayer system shows a metal dielectric ratio of approximately
0.5, meaning there is roughly half as much metal as dielectric (4.7nm metal and 11.3nm
dielectric). Both these metal-dielectric ratios were verified using TEM micrographs. Ultimately another independent measurement is helpful to obtain the actual thickness of
at least one layer, but once one thickness is known this reflectance error measurement
scheme is accurate in producing the correct metal to dielectric ratios at single nanometer
length scales. Note that this error analysis will become dielectric rich for large angles of
incidence due to the increased path length through lossy metals.
3.7.
Conclusion
The amorphous nature of these materials makes them close to ideal systems, with
extremely smooth interfaces and uniform field distributions. While the DC electrical
51
resistivity of the two amorphous metal samples are approximately equal, the optical dielectric response is completely unique for each bulk amorphous metal. The Re(ϵ) of
amorphous ZrCuAlNi decreases as a function of frequency from 300nm to 1500nm, and
the Re(ϵ) of amorphous TiAl3 increases with frequency. Combining these bulk amorphous
metal-dielectric films creates effective composites with hyperbolic dispersion in the optical
spectrum.
Expanding the materials set enhances the possibility of engineering the dispersion
of composites through material choice and thickness. As with many plasmonic systems
material loss plays a primary governing role, and realistic applications for the current
materials should take advantage of loss. Modern processor interconnect switching times
are limited by the RC time constant which is caused by the dielectric material between
electrodes. Designing metamaterials to have a broadband dielectric constant near zero
would alleviate this fundamental limitation and create faster computers. Additional technologies which could benefit from the current materials includes tunneling diodes, optical
filters, thermo-electric devices, and tunable mirrors.
52
4.
4.1.
TERAHERTZ ELLIPSOMETRY OF VERTICALLY ALLIGNED
MULTI-WALLED CARBON NANOTUBES
Introduction
Carbon nanotubes (CNTs) have exceptional electrical and optical properties which
have inspired unique applications in nanoscale optoelectronics.[110, 111, 112] The electrodynamics of CNT at terahertz (THz) frequencies are of considerable interest not only
for fundamental materials research, but also for practical high speed electronics and biosensing applications.[113, 114, 115, 116] A CNT can behave as a metal or a semiconductor
depending on the chirality of the rolled up carbon sheet.[117] For plasmonic applications
and nano-antennae design metallic CNT response is ideal, however controlling chirality
(metallic CNT concentration) is a difficult task experimentally, and on average only thirty
percent of CNT are metallic. Previous THz studies of CNT thin-films show strong absorption of broadband THz radiation, demonstrating their metallic nature.[118, 119, 120] The
extreme aspect ratio, and resonant motion of electrons between adjacent carbon atoms in
single-walled CNTs (SWCNT) leads to strong absorption anisotropy in aligned SWCNT
thin-films.[121, 122] Broadband THz polarizers can be achieved by exploiting the strong
anisotropic nature of SWCNT thin-films.[123, 124, 125]
In contrast to previous works, we present the optical characterization of densely
packed vertically aligned multi-walled CNTs (MWCNT) using THz ellipsometry. A nanoforrest of vertical MWCNTs is an ideal black material in the visible and infrared spectral
regimes, absorbing light completely at all angles, therefore THz spectroscopy is an ideal
candidate for probing the electronic characteristics of these thin-films.[126, 127] Our vertically aligned MWCNT show near perfect blackness by visual inspection. In the THz
spectral regime absorption is strong, but nowhere near perfect, thus transmission measurements can be utilized. By measuring the transmission of THz pulses through the
53
vertically aligned MWCNT the dielectric response along the primary CNT growth axis
(z-direction) and horizontal direction (xy-plane) is extracted. The dielectric response
shows unexpectedly weak anisotropic behavior, with stronger conduction in the xy-plane
than expected. The primary mechanism for the high conductivity in the horizontal direction is believed to be intershell (within one MWCNT) transport.[128] As later results
will show, this study directly probes the electron scattering environment in the vertically
aligned MWCNT, particularly horizontally within one shell, which is believed to be an
experimental precedent.
4.2.
Carbon Nanotube Fabrication and Characterization
The CNT samples were prepared by experimental collaborators (Aixtron UK) using
low-pressure chemical vapor deposition (LP-CVD) with 2-nm Fe on 10-nm Al2 O3 catalyst on high-resistivity Si substrates. Individual CNTs are multi-walled, semi-metallic
conductors.[129] Four samples were fabricated with varying thickness: 0 µm, 21.5 µm, 62.5
µm, and 132 µm. For consistency the 0 µm (blank) sample had the same metal catalyst
as the other CNT samples. Although the catalyst does undergo some geometrical change
during the CNT synthesis reaction it was determined that the catalyst has no effect on
our THz measurements, and thus further analysis of the true nature of the blank before
and after CNT synthesis is not necessary.
Fig.4.1(a) shows the SEM image of the 21.5 µm thick MWCNT film. The films are well
aligned perpendicularly to the substrate interface. Angle resolved transmission of broadband THz pulses were measured using free-space THz time-domain spectroscopy (TDS) by
Yun-Shik Lee’s terahertz research group at Oregon State University. THz measurements
are an ideal non-destructive probe for local carrier dynamics of metallic thin films.[130, 131]
Fig.4.1(b) illustrates the measurement schematic where the THz field is oriented paral-
54
;ĂͿ
;ďͿ
‫ݖ‬
‫ݕ‬
ƉͲƉŽů
‫ݔ‬
;ŝͿŽůŽŵĞƚĞƌ
ĚсϮϭ͘ϱµŵ
θ
ƐͲƉŽů
;ŝŝͿKƐĂŵƉůŝŶŐ
FIGURE 4.1: (a) SEM image of the vertically aligned MWCNT on a Silicon substrate
for d=21.5µm. (b) Ellipsometry characterization schematic for THz transmission measurements: linearly polarized (s and p polarization), broadband THz pulses are incident
on the given sample at some incident angle. The two THz detection schemes are: (i)
time-averaged integrated power spectrum Si:Bolometer measurements and (ii) THz timedomain spectroscopy measured using electro-optic sampling.
lel (perpendicular) to the plane of incidence for s-polarization (p-polarization). Timeaveraged transmitted power measurements were obtained using a liquid helium cooled
Si:Bolometer. Time resolved electric field waveforms were obtained using THz TDS under
N2 purge with electro-optic sampling of a ZnTe crystal.
Fig.4.2 shows the spectrally integrated THz transmitted power through the four
samples with varying CNT lengths for s and p polarization as a function of the incident
angle (θ). Transmission through the bare Si substrate (n=3.42) is shown by solid black
lines, and transmission through the blank containing a metal catalyst layer (with no CNTs)
is shown by the red dots, indicating that the THz response to the catalyst layer is negligible. The s-polarized transmission, which depends only on the xy-conductivity because
the electric field is only in the xy-plane, decreases with increased incident angle (or effective thickness). The most notable feature in Fig.4.2 is that the p-polarized transmission
undergoes a pronounced change in curvature, which increases with CNT thickness. The
typical isotropic dielectric (lossless) angular behavior is similar to the blank p-polarized
55
1.0
1.0
;ĂͿ
ƉͲƉŽů
ĂƌĞ^ŝƚŚĞŽƌLJ
ൌ Ͳ
0.6
ĂƌĞ^ŝƚŚĞŽƌLJ
0.8
Transmission
Transmission
0.8
;ďͿ
ƐͲƉŽů
ʹͳǤͷ
0.4
͸ʹǤͷ
0.6
ൌ Ͳ
ʹͳǤͷ
0.4
͸ʹǤͷ
0.2
0.2
ͳ͵ʹ
ͳ͵ʹ
0.0
0.0
0
10
20
30
40
50
Angle (degree)
60
0
10
20
30
40
50
60
Angle (degree)
FIGURE 4.2: Spectrally integrated THz power transmitted through the CNT samples
vs. the incident angle for (a) p-polarization and (b) s-polarization. The solid black lines
represents the theoretical transmission for a bare Silicon substrate (n=3.42).
trends, the transmission increases as a function of angle until Brewster’s angle (typically
70◦ ). The three CNT samples show striking differences in integrated power as a function
of incident angle, when d= 21.5µm the transmitted power increases in an isotropic (conventional) fashion, when d= 62.5µm the transmitted power remains relatively constant,
and when d= 132µm the transmitted power decreases as a function of angle (similar to
s-polarization). The strong deviation from conventional p-polarized transmitted power
angular response can be justified by lossy material and anisotropic dielectric response. A
detailed THz ellipsometry spectral analysis confirms that the angle-dependent trends are
caused by dielectric anisotropy.
56
4.3.
Time-Averaged Transmitted Terahertz Power - Bolometer Measurements
Due to temporally separated THz pulses the transmitted power in the carbon nanotube (CNT) thin-film systems does not depend on the silicon thickness. Experimentally
this is evident by the lack of interference oscillations when the transmittance is plotted
as a function of angle. To calculate the transmitted power, only the phase information in
the CNT layer is considered.
(1) Air
(2) CNT
(3) Silicon
(4) Air
.
.
.
.
.
.
t13(r34r31)2t34
t13r34r31t34
t13r34
1
t13t34
t13
FIGURE 4.3: Ray diagram for temporally separated pulses traveling through the layered
CNT-substrate system.
Referring to the figure above, consider incident light of magnitude 1 passing through a
CNT layer where the total transmission coefficient (including phase) through the CNT
layer is encompassed in t13 . The total CNT transmitted ray is then traversing back and
forth in the silicon substrate layer without acquiring any silicon phase information before
it is ejected in the last air region. The total transmitted power in this temporally separated
system is found by summing up the individual powers of each exiting wave.
2
Ttot = |ttot |2 = |t13 t34 |2 + |t13 r34 r31 t34 |2 + t13 (r34 r31 )2 t34 + ... + |t13 (r34 r31 )n t34 |2 (4.1)
Rewriting the transmission sum in a convenient geometric series form, we arrive at the
57
total transmission power coefficient as measured by the bolometer.
Ttot =
∞
∑
|t13 t34 |2 (|r34 r31 |2 )n =
n=0
4.4.
|t13 t34 |2
1 − |r34 r31 |2
(4.2)
Terahertz Ellipsometry Theoretical Algorithm
As described in Sec.(1.4.1), to model the polarization dependent transmission through
uniaxial anisotropic layers, we use Maxwell’s equations along with the continuity of electric and magnetic fields parallel to interface boundaries to obtain a transfer matrix for the
(±)
amplitude coefficients aj
representing the total electric-field amplitudes for forward (+)
and backward (−) moving monochromatic waves in the j-th material region:




(−)
aj+1
(+)
aj+1

κj )ϕ−
j


(−)
aj
(1 −

 (1 +


=β


(+)
+
−
(1 − κj )/ϕj (1 + κj )/ϕj
aj
κj )ϕ+
j
(4.3)
where the polarization dependent parameters are
1
1
βs = , βp = sec θj+1 cos θj ,
2
2
(j)
(j) (j+1)
kz
ϵxy kz
κsj = (j+1) , κpj = (j+1) (j) ,
kz
ϵxy kz
{ ω(
) }
±
(j+1)
(j)
ϕj,s = exp i
kz,s
± kz,s
zj ,
c
) }
{ ω(
(j+1)
(j)
ϕ±
kz,p
± kz,p
zj .
j,p = exp i
c
(4.4)
with the dispersion relations
(p-pol)
(s-pol)
k2
ω2
kx2
+ z = 2,
εz
εxy
c
ω2
kx2 + kz2 = εxy 2 .
c
(4.5)
The transfer matrix describes the coupling between fields in the j and j + 1 material
regions, which meet at interface position zj . Although there are multiple transmitted
58
exit pulses due to the internal reflections within the Si substrates, they are temporally
separated and do not interfere. Only the Fourier spectrum of the first transmitted pulse
is used to obtain the THz response. Using Eq.(4.3) above, the transmission coefficient for
the first exit pulse is t = tAir−CN T −Si · tSi−Air , which contains all CNT phase information.
The vertical CNT film is modeled as a planar uniaxial dielectric material with
polarization that is governed by independent damped-driven oscillator dynamics. Consider
⃗ = E⃗0 e−iωt , the electron response to this monochromatic wave
an applied electric field E
is given by the general damped-driven oscillator differential equation,
mr̈ + mΓṙ + mω02 r = −eE0 e−iωt
(4.6)
where m is the effective mass of the electron, Γ is the damping parameter which is proportional to the electron velocity and electron scattering rate, and ω0 is the resonant
frequency. By using the ansatz r = r0 e−iωt the oscillation amplitude of the electron is
determined.
r0 =
m
(
ω2
eE0
)
+ iΓω − ω02
(4.7)
The electron velocity is found by differentiating the electron oscillation amplitude function,
v=
dr
dt
= −iωr. Given the volume density of electrons, N, the electron current density is
found by J⃗ = N e⃗v .
J=
iωN e2 E0
= σE0
ω 2 + iΓω − ω02
(4.8)
Next consider Maxwell’s equations in the following form,
[
(
)]
ω2
iσ
2
⃗ = 0,
∇ + 2 1+
E
c
ϵ0 ω
which implies that ϵ = 1 +
iσ
ϵ0 ω .
(4.9)
Using this definition for ϵ along with Eq.(4.8) allows us
to write the conventional damped-driven oscillator dielectric response model.
59
(
ϵ = ϵ0 1 −
N e2
ϵ0 m(ω 2 + iΓω − ω02 )
The numerator term is combined together using
N e2
ϵ0 m
)
(4.10)
= b2 . The general damped-driven
oscillator model is generalized for modeling the dielectric response of the vertical CNT
thin-films. It is assumed that the dielectric response of our thin-film samples has the form
ϵα = ϵ∞
α −
b2α
, α = xy, z
ω 2 + iωΓα − ωα2
(4.11)
where ϵ∞
α is the high frequency permittivity limit, bα is proportional to the oscillator
strength (or plasma frequency for metals), ω = 2πν is the applied angular frequency,
ωα = 2πνα is the resonant angular frequency, and damping parameters Γα , dictates the
electron scattering rates.
ϵ∞
xy
Γxy (THz)
bxy (THz)
νxy (THz)
1.20±0.003
339±106
40±6
2.2±0.4
ϵ∞
z
Γz (THz)
bz (THz)
νz (THz)
1.2±0.2
229±149
51±20
0.0±0.01
TABLE 4.1: Uniaxial dielectric function parameters. Averaged results from 2000 independent Nelder-Mead search algorithm starting locations and their corresponding standard
deviation.
Figure 4.4 shows the real and imaginary components of nxy and nz spectra. The
anisotropic nature of the THz properties of the V-MWCNTs is evident, yet the ratio of
the z-axis conductivity to the xy-axis conductivity (σz /σxy ∼
= 2.3, which is nearly constant
over the broad spectral range, 0.4-1.6 THz) is significantly smaller than that of a SWCNT.
The ratio of the V-MWCNTs is even smaller than that of graphite, σz /σxy ∼
= 4.2 [132].
The relatively weak anisotropy of the V-MWCNT samples indicates that THz fields can
readily induce electron transport between neighboring shells.
60
1.4
1.7
;ĂͿ
1.0
Im(n)
Re(n)
1.5
‡ሺ݊௭ ሻ
1.4
0.6
1.2
0.4
‡ሺ݊௫௬ ሻ
0.6
0.8
1.0
1.2
Frequency (THz)
1.4
1.6
ሺ݊௭ ሻ
0.8
1.3
1.1
0.4
;ďͿ
1.2
1.6
0.2
0.4
ሺ݊௫௬ ሻ
0.6
0.8
1.0
1.2
1.4
1.6
Frequency (THz)
FIGURE 4.4: (a) Real and (b) imaginary parts of the refractive index for all CNT films
in the THz regime.
The oscillator parameters were extracted by minimizing the difference between the
measured and modeled blank-normalized transmitted intensity spectrum using a NelderMead nonlinear least squares algorithm [133, 134]. In principle, one could minimize the
difference between the measured and modeled TDS signals directly, but there are two
fundamental difficulties when using this formalism. The first difficultly is that the TDS
functions themselves are very large due to the Fourier spectral sum, and using them for
computations is very time consuming. Additionally, there is an arbitrary phase relationship when producing time dependent pulse trains using discrete Fourier transforms, so
one would need to temporally calibrate the model to the measured pulses. Using the
intensity spectrum formalism CNT lengths were fit simultaneously. First, normal incidence data was used to extract ϵxy parameters, then ϵz parameters were extracted using
both p-polarized experimental data and the ϵxy result. This process was performed over
the FWHM of the incident electric-field spectrum (0.4-1.6 THz). The results for the oscillator parameters are listed in Table 4.1. The z-axis parameter νz = 0 indicates that
CNT-axis conduction is due purely to free charge carriers, while the nonvanishing xy-axis
resonant frequency (νxy = 2.2 THz) implies that intershell conduction is not Drude-like,
but undergoes shallow potential barriers. Using the oscillator parameters, and assuming
a Fermi-velocity along the MWCNT axis, vF = 8 × 105 m/s [135], we estimate the average
61
electron scattering mean free path in the z-direction to be 3.5±1.4 nm. This is comparable to the typical scattering length in metals at room temperature. The Fermi-velocity
in the xy-direction is not known, but should be less than 8 × 105 m/s due to the weaker
coupling between electron orbitals in different shells of the MWCNT. Using this upper
bound on radial velocity, we predict that the average electron scattering mean free path
in the xy-direction is less than 2.4±0.6 nm, much less than the typical MWCNT diameter (10–20 nm). These estimates indicate that Drude-like conduction (with anisotropic
scattering parameters) can be expected, which is consistent with the parameters in Table
4.1.
The theoretical transmission spectra, ttot (θ, ν), is used along with the incident THz
electric-field spectrum in air, a(ν), to perform an inverse Fourier transform to model the
time-domain THz pulses,
e t) = Re
E(θ,
[
∑
]
a(ν)ttot (θ, ν)ei(kz0 −ωt)
(4.12)
ν
where ttot (θ, ν) contains all the optical path length phase information for the Air-CNT-SiAir system, and z0 is the measurement position. Theory results are shown in Figs. 4.8(eh) and 4.9(e-h) and are consistent with experimental data [Figs. 4.8(a-d) and 4.9(a-d)].
Re[a(ν)] (a.u)
Im[a(ν)] (a.u)
(a)
(b)
0.02
0.02
0.01
0.01
1
2
3
4
ν (THz)
1
-0.01
-0.01
-0.02
-0.02
2
3
4
ν (THz)
FIGURE 4.5: (a) Real and (b) imaginary parts of the terahertz spectrum for pulses in
air. The spectrum is found by taking the Fourier transform of the time dependent THz
air pulse.
Originally the experimental goal was to fabricate and measure THz response in
62
arrays of horizontally aligned CNTs which would lay flat on a substrate. While the
experiments were achieved with some success, there was never a large enough density
of CNTs in the horizontal direction to see a strong modulation of the THz signal. The
polarizability model for such a horizontal CNT array system is shown in Appendix(A1).
4.4.1
Optical Path Lengths in Uniaxial Material Systems
The total transmission function ttot (θ, ν), which is derived using the total transfer
matrix, already contains all phase information. For specific devices or experiments, one
might want to engineer a particular phase shift, therefore the analytical expression for the
total optical path length for this anisotropic system will be derived.
Air
Path
dsec(θ1)
θ1
dsec(
θ2 )
θ2
Incident
Light
y
x
θ1
Material
Path
x
Th
in-
Film
z
d
FIGURE 4.6: Ray diagram for light moving through a planar slab (thickness d) at some
incident angle. Dark lines with arrows represent the path taken by the Poynting vector
⃗
(S).
Before moving on to the anisotropic system, it is useful to consider light rays moving
through a simple isotropic material. Such a material is shown in Fig.(4.6), where the
sample is turned to mimic the THz experiments. The top horizontal line represents the
unimpeded path a wave moving through air would take, and the dark lines represent the
63
actual path a light wave takes moving through the isotropic material. Ultimately the
difference in distance between the air and material path is needed in order to define some
phase shift with respect to an air pulse. Referring to Fig.(4.6), straight forward geometry
reveals the distance x = d sin(θ1 ) [tan(θ1 ) − tan(θ2 )], and y = d cos(θ1 ) [tan(θ1 ) − tan(θ2 )].
∑
[(ki di )material − (ki di )air ],
By looking at the difference in optical path lengths, ∆Φ =
i
we can define the total phase difference between the waves moving through the materials
with respect to the same wave moving through air.
∆Φ = k2 d sec(θ2 ) + k1 d sin(θ1 ) [tan(θ1 ) − tan(θ2 )] − k1 d sec(θ1 )
(4.13)
The incident angle θ1 is given and the transmitted angle θ2 is found using Snell’s Law,
and the dispersion equation for isotropic material defines the momentum, ki = ni ωc . The
transition to anisotropic materials is just a matter of adapting Eq.(4.13) to an anisotropic
system, namely differences would occur in the effective index of the thin-film, and the
transmitted angle. For uniaxial anisotropic materials the effective index neff for the film
region is replace by an effective index.[136]
1
nef f
(
=
sin2 (θ1 ) cos2 (θ1 )
+
n2z
n2xy
)1/2
(4.14)
Additionally, the transmitted angle changes in anisotropic materials for TM waves.




sin(θ1 )

θi = tan−1 
)

 √ (i) (
sin2 (θ1 )
ϵxy 1 − (i)
(4.15)
ϵz
To calculate the relative phase shift for the CNT thin-film system we need to consider
an anisotropic CNT layer on a silicon substrate as shown in Fig.(4.4.1).
Using the same formalism as above, the relative phase shift for the CNT-Silicon
thin-film system is
64
Air
Path
θ1
Incident
Light
h
θ2
θ3
Material
Path
CNT
(ani
s
otro
d
pic)
Si-S
2
x
ubs
trate
z
d
3
FIGURE 4.7: Ray Diagram for the CNT-Silicon multilayered thin-film system. Solid lines
represent the path taken by light pulses moving through the layers.
∆Φ =
ω
[nef f d2 sec(θ2 ) + n3 d3 sec(θ3 ) + h sin(θ1 ) − (d2 + d3 ) sec(θ1 )] ,
c
(4.16)
where the parameter h = (d2 + d3 ) tan(θ1 ) − d2 tan(θ2 ) − d3 tan(θ3 ), nef f is given by
Eq.(4.14), and the region dependent transmission angle is given by Eq.(4.15). It is important to note that our transmission calculations employ the transfer matrix method and do
not have any path-length corrections because phase information is already encoded in the
amplitude coefficients. This spectral transfer matrix method formalism was independently
verified by theoretical collaborator Sandeep Inampudi at U. Mass. Lowell.
4.5.
Time Domain Spectroscopy Comparison and Concluding Remarks
To gain more insight into the vertical and horizontal carrier dynamics of the VMWCNT films, time-resolved THz ellipsometry was used to obtain a time-dependent
65
transmission function for both s- and p-polarization, ts,p (t, θ). Figures 4.8(a-d) and 4.9(ad) show the directly transmitted waveforms with p- and s-polarization through each CNT
sample at θ = 0, 10, 20, 30, 40, 50, 60◦ , measured by THz-TDS. The incident-angle dependence of each transmitted waveform has been normalized to the relative power transmission to remain consistent with the power transmission measurements shown in Fig 4.2. A
Fourier transform of the THz-TDS data yields the transmission spectrum, ts,p (ν, θ), which
is compared to a uniaxial Drude-Lorentz model.
ൌ ʹͳǤͷ ൌ ͸ʹǤͷ ൌ ͳ͵ʹ
ൌ Ͳ
ETHz(t) (a.u.)
džƉĞƌŝŵĞŶƚ
;ĂͿ
θ
;ďͿ
;ĐͿ
;ĚͿ
;ĨͿ
;ŐͿ
;ŚͿ
4 5 6
Time (ps)
4 5 6
Time (ps)
4 5 6
Time (ps)
ETHz(t) (a.u.)
;ĞͿ
dŚĞŽƌLJ
o
0
o
10
o
20
o
30
o
40
o
50
o
60
4 5 6
Time (ps)
FIGURE 4.8: P-polarization THz waveforms transmitted through the CNT samples for
incident angles between 0 and 60◦ (a-d) experiment and (e-h) theory.
The experiment and theory results show that (i) the THz response along the z-axis is
stronger than that of the xy-plane, yet the anisotropy is much weaker compared with that
of an isolated, metallic SWCNT, (ii) strong absorption in the horizontal direction indicates
that charge carriers transport between adjacent shells, and (iii) the z-axis THz response
of MWCNTs is not overwhelmingly metallic in contrast to that of SWCNTs [123, 124,
125]. Intershell charge transports instigate scattering sites within the multi-shell structure,
reducing the effective scattering length dramatically along the z-direction and introducing
66
ൌ ʹͳǤͷ ൌ ͸ʹǤͷ ൌ ͳ͵ʹ
ൌ Ͳ
ETHz(t) (a.u.)
džƉĞƌŝŵĞŶƚ
;ĂͿ
θ
o
0
o
10
o
20
o
30
o
40
o
50
o
60
;ĐͿ
;ĚͿ
;ĨͿ
;ŐͿ
;ŚͿ
4 5 6
Time (ps)
4 5 6
Time (ps)
4 5 6
Time (ps)
ETHz(t) (a.u.)
dŚĞŽƌLJ
;ĞͿ
;ďͿ
4 5 6
Time (ps)
FIGURE 4.9: S-polarization THz waveforms transmitted through the CNT samples for
incident angles between 0 and 60◦ (a-d) experiment and (e-h) theory.
a significant decrease in absorption.
In conclusion, time-resolved THz transmission ellipsometry reveals the anisotropic
carrier dynamics in vertically aligned MWCNTs. The conductivity along the z-axis is
larger than the xy-plane, but they are similar orders of magnitude. The considerably
strong THz response along the xy-plane indicates that charge carrier transport occurs between neighboring shells in MWCNTs, also creating a non-negligible reduction in absorption along the length of the nanotubes. The THz ellipsometry algorithm will also be useful
to understand carrier dynamics in other nanomaterials consisting of novel two-dimensional
conductors such as multilayer graphene, where transport aniosotropy is expected, yet is
hard to measure with conventional electrode techniques.
67
5.
5.1.
SUBWAVELENGTH IMAGING RECONSTUCTION USING A
RIGOROUSLY COUPLED WAVE ANALYSIS ALGORITHM
Introduction
Conventional imaging systems are composed of lenses that are shaped in order to
direct beams of light through refraction to the sought location.[137] This is commonly
shown in the ray picture, where Snell’s Law designates how much light waves bend when
moving between materials with a different index of refraction. Light rays only represent
a signal’s propagating spectrum, which does not contain subwavelength spatial information. Subwavelength spatial information is encoded in the evanescent spectrum, which is
commonly lost due to the decaying nature of evanescent waves. This loss is the fundamental mechanism for the diffraction limit. While many thoughtful techniques have been
developed[138, 139, 140] to overcome imaging resolution limitations, the fundamental difficulties with refraction based optics remain difficult to overcome. Metamaterials, and
plasmonic systems in particular, are becoming increasingly strong candidates for outperforming conventional imaging techniques[141, 142], but required field manipulation within
a material device is a primary challenge for free space experimental research.
Near field scanning optical microscopy (NSOM) solves the problem of losing evanescent waves by placing a microscope near the radiation source in order to detect exponentially decaying waves.[143, 144, 145] The drawbacks to NSOM are the relatively slow
data acquisition requirements and costly experimental apparatus. Structured illumination spectroscopy utilizes transmission through a wavelength size grating to outperform
the diffraction limit by a factor of 2.[146, 147]
This work proposes another diffraction based imaging scheme in which angle dependent plane waves are incident upon a set of objects which are placed close to a subwavelength metallic grating. Each object will have some inherent field pattern which is
68
diffracted into different grating modes and transmitted into the far-field for measurement.
One primary benefit of the diffraction based imaging system is that increments of momentum may be added to evanescent waves such that they can be shifted into the propagating
spectrum, and recovered in the far-field. Therefore, this system provides a platform to
outperform conventional resolution limits. Given full knowledge of the grating transmission function, a series of subwavelength slit objects, represented by an unknown Fourier
wavevector spectrum, can be recovered through far-field measurements and computer processing.
5.2.
Transmission and Reflection Coefficients in General Periodic Medium
Rigorous Coupled Wave Analysis (RCWA)[148] generalizes Maxwell’s equations to
describe waves which propagate through layers of periodic medium, such as a diffraction
grating.
Far-Field Measurement Plane
z=zf
Diffraction
Modes
3
2
Λ
z=L
.....
.....
z=0
z0
z
Object
1
x
θ
Incident
plane wave
FIGURE 5.1: General schematic of a three layer system where layers 1 and 3 are air,
and layer 2 is a metallic grating with spatial period Λ and thickness L. An incident plane
waves illuminates an object located z0 from the grating. That signal is then transformed
into diffraction modes whose spacing depends on the grating period. Measurement of the
diffracted object waves are taken along the far-field measurement plane (z = zf ).
69
⃗ = (Ex , 0, Ez ) and H
⃗ =
Consider TM polarized monochromatic plane waves, E
(0, Hy , 0), which are incident upon the object-grating system shown in Fig.(5.1). Incident
plane waves will have the form E, H ∝ exp(iωt−i⃗k·⃗r), where the jth transverse wavevector
mode is given by,
kxj = kx0 + jkΛ ,
(5.1)
where kx0 denotes the incident transverse momentum prescribed by the incident angle,
kΛ =
2π
Λ
is a unit of grating momentum, and the integer value j spans the number of
grating modes. The propagating wavevector kzj is related to kxj through the dispersion
relationship, which is this case is merely isotropic dispersion. The periodic dielectric
response in the grating region is represented by the following Fourier series.
∞
∑
ϵ2 (x) =
ϵ̂j ei(jkΛ x)
(5.2)
j=−∞
The dielectric coefficients are found using conventional Fourier analysis.
1
ϵ̂j =
Λ
∫
Λ
ϵ2 (x)e−i(jkΛ x)
(5.3)
0
All of the simulations will employ the binary grating model defined by an air region
centered about x = 0, and both metal and air region thickness (L) are half of the grating
spatial period, which is Λ = 2λ0 /3. Referring to the Fourier series in Eq.(5.3) the dielectric
coefficients for an air-centered binary grating when j ̸= 0 are
ϵ̂j =
2 (ϵair − ϵmetal ) sin
jkΛ Λ
(
jkΛ L
2
)
,
(5.4)
and when j = 0 the grating coefficient is a weighted average of constituent dielectric
contributions ϵ̂j=0 =
ϵair +ϵmetal
.
2
system will have the form
Diffracted TM polarized fields in a general periodic
70
∞
∑
⃗ =
E
⃗ =
H
j=−∞
∞
∑
[Exj (z)x̂ + Ezj (z)ẑ] e(−ikxj x)
(5.5)
Hyj (z)e(−ikxj x) ŷ
(5.6)
j=−∞
Recovering field amplitude coefficients is done through the manipulation of Maxwell’s
equations while considering the material periodicity. First consider the following curl equations in nonmagnetic material.
⃗
⃗ = −1 ∂ H
∇×E
c ∂t
⃗
⃗ = 1 ∂D
∇×H
c ∂t
(5.7)
(5.8)
Solving Maxwell’s equations for TM waves yields the following set of matrix equations.
⃗ xj (z)
∂E
⃗ zj (z) = −iω H
⃗ yj (z)
+ ikxj E
∂z
c
⃗ yj (z)
∂H
iω ⃗
−
= D
xj (z)
∂z
c
⃗ yj (z) = iω D
⃗ zj (z)
−ikxj H
c
(5.9)
(5.10)
(5.11)
Due to the periodic nature of ϵ the displacement field in Eq.(5.9) is represented by the
following sums.
⃗ =
D
[
∞
∑
j=−∞
]
∞
∑
⃗ j e−ikxj x
ϵ̂n eikΛ nx E
(5.12)
n=−∞
By setting m = j − n we can write the displacement field in the following useful form,
⃗ =
D
∞
∑
j,m=−∞
⃗ j e−ikxm x .
ϵ̂j−m E
(5.13)
71
Now using the displacement field formalism above along with Eqns.(5.9) the master matrix
equation (for parallel fields) of the RCWA algorithm is shown below.




∂Hy
∂z
∂Ex
∂z


 
=
−iω
c ϵ̂
0
ic
ω kxj ϵ̂j−m kxm
−
iω ˆ
c I
0

  Hy 


Ex
(5.14)
Eq.(5.14) is solved using conventional linear system differential equation formalism[149],
where the solutions are written as a linear combination of eigen-solutions of the master
matrix,




N
∑
⃗
 Hy,n (z) 
(±)  Hy,jn  qn z −ikxj x
,
Cj 
=
e e

⃗ x,jn
Ex,n (z)
E
j=−N
(5.15)
⃗ y,jn and E
⃗ x,jn together are the eigenvectors of the master matrix, qn are the
where H
corresponding eigenvalues, n ranges from 1 to 2(2N+1) where N is the number of grating
(±)
modes, and Cj
are the amplitude coefficients of transmitted and reflected waves in each
material region. Parallel field continuity at the front and back grating interface is used to
find the amplitude coefficients in each material region. First, consider the continuity of
the Hy field component at the front grating interface (z=0).
N
∑
[δj0 + rj ] e−ikxj x =
j=−N
N
∑

∑

2(2N +1)

⃗ y,jn  e−ikxj x
Cn H
(5.16)
n=1
j=−N
⃗ y,jn is only the upper 2N+1 elements of the eigenvector, rj is the reflection
Note that H
amplitude coefficient, and the delta function δj0 represents the incident wave which does
not have any added grating momentum. From Maxwell’s Equations we can write the Ex
field as a function of Hy as follows, Ex =
ic ∂Hy
ϵω ∂z ,
therefore the continuity of Ex at the first
grating boundary (z=0) is given by the following expression.
(1)
N
∑
kzj c
j=−N
ϵ(1) ω
(δj0 − rj ) e−ikxj x =
N
∑
j=−N

∑
2(2N +1)

n=1

⃗ x,jn  e−ikxj x
Cn E
(5.17)
72
Now consider the continuity of the Hy field component at the back grating interface, when
z = L.
N
∑

∑

2(2N +1)

j=−N
⃗ y,jn eqn L  e−ikxj x =
Cn H
n=1
N
∑
tj e−ikxj x
(5.18)
j=−N
Likewise, the Ex field continuity condition may be written as
(3)
N [
N
]
∑
∑
kzj c −ik x
⃗ x,jn eqn L e−ikxj x =
t e xj .
Cn E
(3) ω j
ϵ
j=−N
j=−N
(5.19)
The boundary condition sums above are solved by assuming that each term in the j
sums must be equal, which simplifies the problem to the following set of linear equations
that can be solved for the transmission and reflection coefficients.
∑
2(2N +1)
δj0 + rj =
⃗ y,jn
Cn H
(5.20)
n=1
(1)
kzj c
ϵ(1) ω
∑
2(2N +1)
(δj0 − rj ) =
∑
⃗ x,jn
Cn E
(5.21)
n=1
2(2N +1)
⃗ y,jn eqn L = tj
Cn H
(5.22)
n=1
∑
(3)
2(2N +1)
n=1
⃗ x,jn eqn L =
Cn E
kzj c
ϵ(3) ω
tj
(5.23)
By eliminating rj and tj from the linear equations above you arrive at the Cn coefficient
matrix.

(


⃗
⃗
(
)
H
+
E
y,jn
x,jn



 [( (3) )
 C
]
=
n


kzj c
q
L
⃗ y,jn − E
⃗ x,jn e n
H
ϵ(3) ω
(1)
kzj c
ϵ(1) ω
)
(1)
2kz,0 c
ϵ(1) ω



(5.24)
0
Now the matrix above is inverted to calculate Cn coefficients, and the modal reflection
and transmission coefficients are calculated using the following equations.
73
∑
2(2N +1)
rj =
⃗ y,jn − δj0
Cn H
(5.25)
⃗ y,jn eqn L
Cn H
(5.26)
n=1
∑
2(2N +1)
tj =
n=1
5.3.
Multiple Slit Imaging
An ideal plane wave in free space is completely determined by a single unique
wavevector which describes the wave’s momentum direction. As plane waves interacts
with objects the single unique wavevector (momentum) is scattered into a continuous
spectrum of wavevector values, a(kx ). When the diffracting object is smaller then larger
transverse wavevector (kx ) values are needed to describe the resulting scattered wave.
To test the diffraction based imaging system, a series of subwavelength slit objects are
illuminating with angle dependent plane waves, the slit signals are transmitted through
the metallic grating, and the resulting fields are measured in the far-field along a line
which is parallel to the grating. The far-field signal is a complicated pattern which must
be unscrambled to recover the slit object array. Using what we know about the grating
transmission function, and applying spectral concepts from Fourier optics, the field which
is transmitted from the object-grating system and measured in the far-field (z = zf ) can
be written in the following form.
Hy (x) =
∞
∑
−kx =∞
dkx a(kx − kx0 )
∑
(+)
⃗ y,jn C (+) eλn
H
j
zf −ikxj x
e
(5.27)
n,j
All of the terms in the (j,n) sum in Eq.(5.27) represent the grating transmission function,
which was discussed in detail in Sec.(5.2.). The object wavevector spectrum a(kx − kx0 ) is
transformed by grating the transmission function to produce the measured far-field pattern. As shown below, the unknown object wavevector spectrum is the function which is
74
ultimately recovered from the measured far-field data. Note that the kx0 variable represents
the incident angle, and is zero for normal incidence. Recall that any periodic function can
be represented by the conventional integral shown below
1
f (x) =
2π
∫
∞
a(kx )eikx x dkx ,
−∞
(5.28)
where a(kx ) is the wavevector spectrum of the spatial function f(x). In order to produce a
plane wave a delta function spectrum, a(kx − kx0 ) = δ(kx − kx0 ), is used. Consider a single
slit with width d which is centered about x = 0 through which a plane wave with incident
angle kx0 is passed, the wavevector spectrum for such a single slit is calculated below.
a(kx −
kx0 )
1
=
2π
∫
d
2
f (x)e
−ikx x
− d2
1
dx =
2π
∫
d
2
− d2
e
−i(kx −kx0 )x
[
]
d
0 d
sinc (kx − kx )
dx =
2π
2
(5.29)
Extrapolating this formalism to multiple slits with different widths which are located
at z = z0 results in the general multiple slit wavevector spectrum shown below,
Nobj
a(kx −
kx0 )
=
∑
k=1
]
]
[
[
1
exp iXk (kx − kx0 ) + ikz z0 sin (kx − kx0 )wk ,
0
π(kx − kx )
(5.30)
where k is summed over the number of slit objects, wk is the k th slit width, and Xk is the
is the k th slit half-width. The general multiple slit wavevector spectrum is required for
calculating the field which is measured in the far-field, but the ultimate goal is to use only
the far-field measurements along with an imaging algorithm to recover a set of unknown
objects placed before the grating.
Referring back to measured field expression in Eq.(5.27) we assume that the multiple
object wavevector spectrum a(kx ) is unknown, and is to be determined by minimizing the
difference between measured and modeled transmission data. Measurements are taken
over multiple xi values along the far-field line at z = zf for a set of incident angles, then
75
the unknown wavevector spectrum is found by minimizing the error (S) below.
S=
∑
|Hdata (xi , θi ) − Hmodel (xi , θi )|2
(5.31)
xi ,θi
The subwavelength nature of unknown objects is reflected in the high kx spectral contribution. Referring to the analytical expression for the finite size slit wavevector spectrum,
note the trigonometric functional form. First the unknown object wavevector spectrum is
modeled as a sum of trigonometric functions with unknown coefficients (aj , bj ).
a(kx ) =
M
∑
(
aj cos
j=0
2πjkx
λx
)
+
M
∑
(
bj sin
j=1
2πjkx
λx
)
(5.32)
Given the functional form of Eq.(5.30) the sinusoidal basis should model the multi-slit
system spectrum effectively. The λx variable in Eq.(5.32) is taken to be the span of kx
sum values in Eq.(5.27). Given the linear unknown coefficients the minimum of the S error
function can be found analytically using linear least squares fitting. Once the unknown
spectrum function is known the recovered objects are calculated using the equation below.
Hobj =
∑
a(kx ) exp(−ikx x)
(5.33)
kx
Further system details for Fig.(5.2) are as follows: 11 grating modes, grating period
Λ = 2λ/3, grating thickness L = λ0 /10, metal dielectric constant ϵm = −100 − 0.1i,
Max(kx ) = 75 (but Fig.(5.2)(a) is truncated to 19), with dkx = 0.06, incident angles
kx0 = (−40, −20, 0, 20, 40)dkx , object-grating distance z0 = λ/40, measurement plane distance zf = 7λ0 , 250 x-measurements between ±7λ0 , and M = 20 unknown trigonometric
coefficients. It is shown in Fig.(5.2)(b) the spectral fit fails for large kx , thus the field
recovery calculation must be done with a truncated spectrum. The truncated portion of
the spectrum is close to noise levels in this system, therefore the trigonometric basis is
not the ideal candidate for the unknown object spectrum.
76
1.5
0.02
Hz Recovered Image
Hz Object
(a)
|a(kx)| − Linear Fit
|a(kx)| − source
(b)
0.015
Hz
| a(kx) |
1
0.01
0.5
0.005
0
−1
−0.5
0
x/λ0
0.5
0
−100
1
−50
0
kx
50
100
FIGURE 5.2: Linear least squares fitting for trigonometric spectral basis functions. Field
recovery (with truncated kx spectrum) (a) and spectral comparison (b) for a three object
system with slit widths (λ0 /8, λ0 /4, λ0 /2). Note that the trigonometric spectral fit fails
for large kx .
Next a bessel function basis was used for the unknown object spectrum to take
advantage of the decaying nature of bessel functions for large numerical arguments.
a(kx ) =
M
∑
m=0
(
a m Jm
2πmkx
λx
)
(5.34)
These are bessel functions of the first kind, with λx again being defined by the span of kx
values in the field sum. All other system parameters are the same for the trigonometric
and bessel basis simulations.
As seen in Fig.(5.3) the bessel function spectral basis is significantly worse that the
trigonometric basis for the same system parameters, and the decaying nature of bessel
functions does not stop the spectral fitting from blowing up at large kx . Even the truncated spectrum produces a highly oscillatory field pattern, therefore a fundamentally new
spectral basis model must be employed.
Moving away from the functional unknown object spectral function, we employ a
pixel method which creates an array of sources on the object plane.
77
7
6
Hz Recovered Image
Hz Object
0.2
(a)
|a(kx)| − Linear Fit
|a(kx)| − source
0.15
5
4
| a(kx) |
Hz
(b)
3
2
0.1
0.05
1
0
−1
−0.5
0
x/λ0
0.5
0
−100
1
−50
0
kx
50
100
FIGURE 5.3: Linear least squares fitting for bessel function spectral basis functions. Field
recovery (with truncated kx spectrum) (a) and spectral comparison (b) for a three object
system with slit widths (λ0 /8, λ0 /4, λ0 /2). Again the bessel spectral fit fails for large kx .
a(kx ) =
M
∑
1
px (j) exp [i(kx xj + kz z0 )]
2Max(kx )
(5.35)
j=−M
The source array points are located at xj , and px (j) represents the contribution to the
source located at xj . Finding the numerical contributions to a series of rectangular source
points will recover the unknown series of objects.
2
0.02
Hz Recovered Image
Hz Object
(a)
(b)
0.015
| a(kx) |
Hz
1.5
1
0.5
0
−1
|a(kx)| − Linear Fit
|a(kx)| − source
0.01
0.005
−0.5
0
x/λ0
0.5
1
0
−150 −100 −50
0
kx
50
100
150
FIGURE 5.4: Linear least squares fitting for pixel source array spectral basis. Field
recovery (without any truncation) (a) and spectral comparison (b) for a three object
system with slit widths (λ0 /8, λ0 /4, λ0 /2).
Further system details for Fig.(5.4) are as follows: 15 grating modes, grating pe-
78
riod Λ = 2λ/3, grating thickness L = λ0 /10, metal dielectric constant ϵm = −100 −
0.1i, Max(kx ) = 100 (no truncation required), with dkx = 0.134, incident angles kx0 =
(−40, −20, 0, 20, 40)dkx , object-grating distance z0 = λ/40, measurement plane distance
zf = 7λ0 , 50 x-measurements between ±7λ0 , and 1500 pixel sources. Fig.(5.4) clearly
demonstrates that the pixel array source method outperforms other functional spectral
methods, and solves the diabolical large kx spectral runaway problem.
5.4.
Stability Analysis
This section will discuss the primary imaging algorithm parameters which can be
adjusted to optimize the unknown wavevector spectrum recovery. The statistical numerical
result which will be used to determine the quality of imaging given a set of parameters
is the standard deviation of the recovered spectrum (when compared to the exact known
multiple slit spectrum). For all statistical analysis system parameters are the same as
Fig.(5.4), and one variable at a time is adjusted.
√
σ=
2
1 ∑ (exact)
(recovered) a(k
)
−
a(k
)
x
x
Nkx
(5.36)
kx
The recovered wavevector spectrum standard deviation is plotted below as a function of
the object-grating separation distance.
Subwavelength imaging requires the objects to be placed close enough to the grating
so that the evanescent wave amplitudes are significantly large before momentum transformation shifts then into the propagating spectrum. As seen in Fig.(5.5) the spectrum standard deviation minimizes when the series of slits are placed at approximately z0 /λ0 = 0.03.
Fig.(5.6) shows the spectrum standard deviation as a function of the number of
grating modes. According to statistical analysis of the three object system 50 grating
modes minimizes the fit spectrum error. In practice, using many modes increases the
79
σ
0.030
0.028
0.026
0.024
0.02
0.04
0.06
0.08
0.10
z0/λ0
FIGURE 5.5: Standard deviation of the recovered spectrum as a function of the object
grating separation distance.
calculation time and memory significantly, therefore operating in the first plateau region
where mode numbers are greater than 15 is reasonable.
σ
0.026
0.025
0.024
0.023
10
20
30
40
50
60
70
Grating Modes
FIGURE 5.6: Standard deviation of the recovered spectrum as a function of the number
of grating modes used in the imaging algorithm.
As expected a large transverse wavevector is necessary for recovering subwavelength objects, which is verified by Fig.(5.7) where the spectrum standard deviation decreases as a
(max)
function of kx
.
Although many parameters can be adjusted in this diffraction based imaging algorithm, the error analysis of three critically important parameters are shown in this section.
To successfully recover arrays of subwavelength slit objects one must use a sufficient number of grating modes, and spectral width, while placing the objects close enough to the
80
σ
0.04
0.03
0.02
0.01
k (max)
10
20
30
40 x
FIGURE 5.7: Standard deviation of the recovered spectrum as a function the maximum
kx spectrum summing value.
grating.
5.5.
Diffraction Based Imaging Conclusions
This work demonstrates the viability of a diffraction based imaging system composed
of a subwavelength metallic grating. The RCWA grating transmission function is used
to model signals coming from a series of subwavelength objects. Imaging recovery is
done using an unknown pixel basis wavevector spectrum and minimizing the difference
between far-field transmission measurements and theory. Stability analysis shows that the
imaging algorithm has a higher performance when objects are placed close to the grating,
a sufficient number of grating modes are used, and a large enough spectrum is used for
representing optical fields. Ultimately, this technique creates a new imaging paradigm,
and outperforms the diffraction limit by an order of magnitude.
81
6.
DISSERTATION CONCLUSIONS
Plasmonic metamaterials are increasingly promising candidates for bridging the spatial gap between optics and electronics, creating new sensor systems, enhancing lightmatter interactions, exploring biophysics, advancing waveguide systems, and developing
spectroscopy applications. This work has explored four plasmonic material systems to
gain insight into the fundamental interactions that light has with metallic structures.
Solution-derived silver percolation film composites deposited on glass were shown
to display giant asymmetry in reflectance. Scaling Theory successfully accounted for the
spectral optical response in these films, as well as the experimentally observed anomalies
near the percolation threshold. In a multi-disciplinary collaboration, a bulk amorphous
materials set was optically characterized. The bulk amorphous dielectric response was
then used to engineer planar layered effective optical structures which displayed hyperbolic
dispersion. By expanding metrology algorithms to include an entire spectrum of incident
light a spectroscopic terahertz transmission ellipsometry algorithm was invented. This
algorithm was then used to classify the anisotropic electrodynamics of an array of vertically
grown multi-walled carbon nanotubes. The optical response between the carbon nanotube
walls was much higher than expected. Finally, a diffraction based imaging system was
created to invent a new way to overcome the drawbacks of conventional refraction based
imaging. A subwavelength metallic grating and computer post-processing were used to
image complex objects with spatial resolution that outperforms the diffraction limit by
an order of magnitude.
82
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90
APPENDICES
91
A
Optical properties of Horizontally Aligned Carbon Nanotube Dipole
Antennae Arrays
In this appendix we will give a detailed description of the derivation of the optical
properties of horizontally aligned carbon nanotube arrays using a dipole approximation.
First we extend previous formalism[150] to calculate the transmission and reflection coefficients through a layer of periodic dipoles that have an effective polarizability per unit
area (located at z = 0). To calculate the optical reflection and transmission from such a
system first consider the displacement field wave equation.
(
)
1 ∂2 ⃗
2
∇ − 2 2 D=0
c ∂t
(A.1)
⃗ = E
⃗ + 4π P⃗ , and assuming monochromatic normal incidence plane waves the
Using D
wave equation simplifies to
(
)
ω2 ⃗
ω2
2
∇z + 2 E = −4π 2 P⃗ .
c
c
(A.2)
Next we can define the polarization of the dipole sheet as
⃗
P⃗ = ηαδ(z)E,
(A.3)
where α is the effective polarizability of one particle, and η is the number of particles per
unit area. The electric field wave equation becomes
)
(
ω2
ω2 ⃗
2
⃗
∇z + 2 E = −4π 2 ηαδ(z)E.
c
c
(A.4)
By requiring tangential electric field continuity at z=0, and integrating Eq.(A.4) about
z=0 we arrive at the following boundary condition equations.
92
E(z = 0+ ) = E(z = 0− )
∂
∂z E(z
= 0− ) −
∂
∂z E(z
(A.5)
2
= 0+ ) = 4π ωc2 ηαE(z = 0)
(A.6)
Consider the case when the dipole layer is on a substrate, which is the real physical
scenario which one would measure. Assuming the dipole layer exists at z=0 the boundary
equation simplify to
t=r+1
(A.7)
2
ik1 (1 − r) − ik2 t = 4π ωc2 ηα(1 + r),
(A.8)
where r and t are the reflection and transmission amplitude coefficients, and k2 describes
the momentum in the substrate region. Solving the simultaneous equations above gives
the amplitude coefficients for the dipole layer on substrate system.
r=
k1 − k2 + α ‡
k1 + k2 − α ‡
(A.9)
t=
2k1
k1 + k2 − α‡
(A.10)
The momentum wavevector ki = ni ωc , and the parameter α‡ = 4πi ωc2 ηα. Note that when
2
α‡ = 0 the conventional Fresnel amplitude coefficients are recovered.
To calculate the effective polarizability of a 2D array of CNT dipole emitters consider
the array shown in Fig.(A1), where each dipole has cartesian coordinate (xi , yi ). The total
electric field from the collection of dipoles is
E⃗dip (x, y) =
∑
[3(⃗
p · r̂)r̂ − p⃗]
ij
4πϵ0 [(x − xi )2 + (y − yj )2 ]3/2
.
(A.11)
93
ly
lx
E
θ
H
FIGURE A1: Horizontally aligned carbon nanotube arrays with horizontal spacing lx and
vertical spacing ly .
Assuming the polarizability is exclusively in the vertical (y) direction, and p0 is the dipole
moment of a single CNT the total dipole electric field which is in the direction of the local
dipole moments becomes
⃗ y = p0
E
[
N
N
∑
∑
i=−N j=−N
2ly2 j 2 − lx2 i2
(lx2 i2 + ly2 j 2 )5/2
]
.
(A.12)
It’s convenient to define the sum parameter S,
S(a) =
∞
∑
i2
,
2 + a2 j 2 )5/2
(i
i,j=1
(A.13)
and the zeta function,
σ=
∞
∑
1
.
i3
i=1
By summing over an infinite domain Eq.(A.12) becomes
(A.14)
94
[
Ey = 4p0
]
2S (lx /ly ) S (ly /lx )
σ
σ
−
+
−
.
ly3 2lx3
ly3
lx3
(A.15)
Assuming an incident applied field magnitude of E0 which is aligned with the CNT axis,
and an effective polarizability of a single CNT of α0 we arrive at the effective polarizability
(αef f ) for the infinite 2D CNT array shown in Fig.(A1).
αef f
[
(
])
2S (lx /ly ) S (ly /lx )
σ
σ
= α0 E0 + 4p0 3 − 3 +
−
ly
2lx
ly3
lx3
(A.16)
95
B
Scaling Theory Fortran Source Code
Below is the primary block of Fortran 90 source code for the percolation films
project. An Intel Visual Fortran Compiler and IMSL and MKL libraries are required to
compile this code.
!
!
Percolation Films calculation
(c)
2011 Nicholas A. Kuhta
include ’link_f90_static.h’
!
Load the IMSL Library
use linear_operators
implicit none
integer i,j
real*8 sigDC,tau0,beta,c,C0,B
real*8 A1,A2,A3,A4
real*8 mu,sigma,nu,s,theta
real*8 pc,d,CorLength
real*8 val1,val2,val3,test(4)
complex*16 c1
real*8 pi
complex*16 ci
integer Nlam,nP
real*8 lamArray(10),RTAlam(21),R1Array(21),T1Array(21),A1Array(21),ptest
real*8 R1p(601),T1p(601),A1p(601),lamP,sdev
real*8 pArray(601),pStart,pStop,pStep
96
!
DeltaR array DeltaR(lam,p-pc)
real*8, allocatable :: DeltaR(:,:)
parameter (ci=(0d0,1d0), pi=3.141592653589793d0)
!
dc conductivity in (sec^-1)
sigDC=2.574e17
!
characteristic relaxation time
tau0=3.0e-15
beta=2.0e-16
!
speed of light(m/s)
c=3.0e8
!
single link capacitance
C0=0.5
!
Correlation Length
CorLength=2.0e-9
!
B scalar in L function
B=4.0
!
Conductivity parameters
A1=0.046
A2=0.046
A3=0.028
A4=0.055
!
Critical exponents
97
mu=1.3
!
dc conductivity exponent
s=1.3
!
capacitance exponent (and superconductivity)
nu=4.0/3.0
theta=0.79
!
!
!
correlation length scaling
related to RMS distance for random walk on 2D fractal
percolation threshold
pc=0.6
!
metal thickness
d=50.0e-9
!
log-normal standard deviation
sdev=0.3
!
lamda data from 400-850nm in 50nm steps
Nlam=10
do i=1,Nlam
lamArray(i)=350e-9+50e-9*i
end do
!
p data points
nP=601
pStart=0.0
pStop=1.0
pStep=(pStop-pStart)/nP
allocate (DeltaR(Nlam,nP))
!
(p-pc) Array
98
do j=1,nP
pArray(j)=(pStart+(j-1)*pStep)
end do
do i=1,21
RTAlam(i)=500e-9+(i-1)*15e-9
end do
contains
!
relaxation time as a function of frequency
function tau(w)
implicit none
real*8 tau,w
tau=1/((1/tau0)+beta*w**2)
end function
!
metal epsilon as a function of conductivity and frequecy
function epsM(sigma,w)
implicit none
complex*16 epsM,sigma
real*8 w
epsM=1.0+(4*pi*ci*sigma)/w
end function
99
!
Correlation Length Function
function xi(p,xi0)
implicit none
real*8 xi,p,xi0
xi=xi0*abs((p-pc)/pc)**(-nu)
end function
!
distance traveled in the film
function Lfun(w,p,xi0)
implicit none
real*8 Lfun,p,xi0,w
real*8 lam,Lomega
lam=2*pi*c/w
Lomega=B*xi0*(lam/(2*pi*xi0))**(1/(2+theta))
if (xi(p,xi0).lt.Lomega) then
Lfun=xi(p,xi0)
else
Lfun=Lomega
end if
end function
!
Average Metal and Dielectric Scaled Conductivities
function sigmaM(w,p,xi0)
implicit none
real*8
w,p,xi0,L,cval
100
complex*16
sigmaM
L=Lfun(w,p,xi0)
sigmaM=(A1*sigDC*(L/xi0)**(-mu/nu))/(1+w**2*tau(w)**2)+&
ci*((A1*sigDC*w*tau(w)*(L/xi0)**(-mu/nu))/(1+w**2*tau(w)**2)-&
A2*C0*w*(L/xi0)**(s/nu))
end function
function sigmaD(w,p,xi0)
implicit none
real*8
w,p,xi0,L
complex*16
sigmaD
L=Lfun(w,p,xi0)
sigmaD=(A3*w**2*C0**2*(L/xi0)**((mu+2*s)/nu))/sigDC+&
ci*((A3*w**2*C0**2*w*tau(w)*(L/xi0)**((mu+2*s)/nu))/sigDC-&
A4*C0*w*(L/xi0)**(s/nu))
end function
function fFun(w,p,xi0)
implicit none
real*8
w,p,xi0,fFun
fFun=0.5*((p-pc)/pc*(Lfun(w,p,xi0)/xi0)**(1/nu)+1)
end function
!
log-normal probability distribution function
function lognormal(mu2,sig,x)
101
implicit none
real*8
lognormal,mu2,sig,x
!normal
!lognormal=exp(-(x-mu2)**2/(2*sig**2))/(sqrt(2*pi)*sig)
!lognormal
lognormal=exp(-(log(x)-mu2)**2/(2*sig**2))/(sqrt(2*pi)*x*sig)
end function
!
Normal incidence TMM routine for metal on substrate
function R1(d1,w,kz1,eps2)
implicit none
real*8
d1,w,eps1,eps3,kz1,kz3,R1
complex*16
T12(2,2),T23(2,2),TLeft(2,2)
complex*16
k12,k23,kz2,eps2
!,ci
eps1=1.0
eps3=2.3
kz2=sqrt(eps2)*kz1
kz3=sqrt(eps3)*kz1
k12=kz2*eps1/(kz1*eps2)
k23=kz3*eps2/(kz2*eps3)
T12(1,1)=0.5*(1+k12)*exp(-ci*(kz2-kz1)*d1)
T12(1,2)=0.5*(1-k12)*exp(-ci*(kz2+kz1)*d1)
T12(2,1)=0.5*(1-k12)*exp(ci*(kz2+kz1)*d1)
T12(2,2)=0.5*(1+k12)*exp(ci*(kz2-kz1)*d1)
T23(1,1)=0.5*(1+k23)
T23(1,2)=0.5*(1-k23)
102
T23(2,1)=0.5*(1-k23)
T23(2,2)=0.5*(1+k23)
!
total transfer matrix
TLeft = T23 .x. T12
!
RTarray(RLeft,RRight,TLeft,TRight)
R1=abs(-TLeft(1,2)/TLeft(1,1))**2
end function
function R1Scaled(w,p,xi0)
implicit none
integer j
real*8
f,w,p,xi0,a,b,n,h
real*8
sig0,mu0,sig2,mu2,sig
real*8
R1Scaled,prob
real*8
RTm1,RTm2,RTm3,RTm4
real*8
RTd1,RTd2,RTd3,RTd4
!
!,lam,xifunc,Lfunc
control the varience and mean value to sig0 and mu0 respectively
sig0=sdev
mu0=1.0
sig2=log(sig0**2/mu0**2+1)
mu2=log(mu0)-sig2/2
sig=sqrt(sig2)
f=fFun(w,p,xi0)
103
!
conductivity integration limits
a=1e-1
b=10.0
n=100
h=(b-a)/n
RTm1=0.0
RTm2=0.0
RTm3=0.0
RTm4=0.0
RTd1=0.0
RTd2=0.0
RTd3=0.0
RTd4=0.0
! metal terms (simpson integration)
RTm1=R1(d,w,w/c,epsM(a*sigmaM(w,p,xi0),w))*lognormal(mu2,sig,a)
do j=1,n/2-1
RTm2=2*R1(d,w,w/c,epsM((a+2*j*h)*sigmaM(w,p,xi0),w))*&
lognormal(mu2,sig,a+2*j*h)+RTm2
end do
do j=1,n/2
RTm3=4*R1(d,w,w/c,epsM((a+(2*j-1)*h)*sigmaM(w,p,xi0),w))*&
lognormal(mu2,sig,a+(2*j-1)*h)+RTm3
end do
RTm4=R1(d,w,w/c,epsM(b*sigmaM(w,p,xi0),w))*lognormal(mu2,sig,b)
! dielectric terms (simpson integration)
104
RTd1=R1(d,w,w/c,epsM(a*sigmaD(w,p,xi0),w))*lognormal(mu2,sig,a)
do j=1,n/2-1
RTd2=2*R1(d,w,w/c,epsM((a+2*j*h)*sigmaD(w,p,xi0),w))*&
lognormal(mu2,sig,a+2*j*h)+RTd2
end do
do j=1,n/2
RTd3=4*R1(d,w,w/c,epsM((a+(2*j-1)*h)*sigmaD(w,p,xi0),w))*&
lognormal(mu2,sig,a+(2*j-1)*h)+RTd3
end do
RTd4=R1(d,w,w/c,epsM(b*sigmaD(w,p,xi0),w))*lognormal(mu2,sig,b)
R1Scaled=f*(h/3)*(RTm1+RTm2+RTm3+RTm4)+(1-f)*(h/3)*(RTd1+RTd2+RTd3+RTd4)
! simplified solution
!R1Scaled=f*R1(d,w,w/c,epsM(sigmaD(w,p,xi0),w))+(1-f)*&
R1(d,w,w/c,epsM(sigmaD(w,p,xi0),w))
end function
function T1(d1,w,kz1,eps2)
implicit none
real*8
d1,w,eps1,eps3,kz1,kz3,T1
complex*16
T12(2,2),T23(2,2),TLeft(2,2)
complex*16
k12,k23,kz2,eps2
eps1=1.0
eps3=2.3
kz2=sqrt(eps2)*kz1
105
kz3=sqrt(eps3)*kz1
k12=kz2*eps1/(kz1*eps2)
k23=kz3*eps2/(kz2*eps3)
T12(1,1)=0.5*(1+k12)*exp(-ci*(kz2-kz1)*d1)
T12(1,2)=0.5*(1-k12)*exp(-ci*(kz2+kz1)*d1)
T12(2,1)=0.5*(1-k12)*exp(ci*(kz2+kz1)*d1)
T12(2,2)=0.5*(1+k12)*exp(ci*(kz2-kz1)*d1)
T23(1,1)=0.5*(1+k23)
T23(1,2)=0.5*(1-k23)
T23(2,1)=0.5*(1-k23)
T23(2,2)=0.5*(1+k23)
!
total transfer matrix
TLeft = T23 .x. T12
!
RTarray(RLeft,RRight,TLeft,TRight)
T1=sqrt(eps3)*abs(det(TLeft)/TLeft(1,1))**2
end function
function T1Scaled(w,p,xi0)
implicit none
integer j
real*8
f,w,p,xi0,a,b,n,h
real*8
sig0,mu0,sig2,mu2,sig
real*8
T1Scaled
106
real*8
RTm1,RTm2,RTm3,RTm4
real*8
RTd1,RTd2,RTd3,RTd4
!
control the varience and mean value to sig0 and mu0 respectively
sig0=sdev
mu0=1.0
sig2=log(sig0**2/mu0**2+1)
mu2=log(mu0)-sig2/2
sig=sqrt(sig2)
f=fFun(w,p,xi0)
!
conductivity integration limits
a=1e-1
b=10.0
n=100
h=(b-a)/n
RTm1=0.0
RTm2=0.0
RTm3=0.0
RTm4=0.0
RTd1=0.0
RTd2=0.0
RTd3=0.0
RTd4=0.0
!!!
SOMBMF--(HLK=ILY)
! metal terms (simpson integration)
RTm1=T1(d,w,w/c,epsM(a*sigmaM(w,p,xi0),w))*lognormal(mu2,sig,a)
107
do j=1,n/2-1
RTm2=2*T1(d,w,w/c,epsM((a+2*j*h)*sigmaM(w,p,xi0),w))*&
lognormal(mu2,sig,a+2*j*h)+RTm2
end do
do j=1,n/2
RTm3=4*T1(d,w,w/c,epsM((a+(2*j-1)*h)*sigmaM(w,p,xi0),w))*&
lognormal(mu2,sig,a+(2*j-1)*h)+RTm3
end do
RTm4=T1(d,w,w/c,epsM(b*sigmaM(w,p,xi0),w))*lognormal(mu2,sig,b)
! dielectric terms (simpson integration)
RTd1=T1(d,w,w/c,epsM(a*sigmaD(w,p,xi0),w))*lognormal(mu2,sig,a)
do j=1,n/2-1
RTd2=2*T1(d,w,w/c,epsM((a+2*j*h)*sigmaD(w,p,xi0),w))*&
lognormal(mu2,sig,a+2*j*h)+RTd2
end do
do j=1,n/2
RTd3=4*T1(d,w,w/c,epsM((a+(2*j-1)*h)*sigmaD(w,p,xi0),w))*&
lognormal(mu2,sig,a+(2*j-1)*h)+RTd3
end do
RTd4=T1(d,w,w/c,epsM(b*sigmaD(w,p,xi0),w))*lognormal(mu2,sig,b)
T1Scaled=f*(h/3)*(RTm1+RTm2+RTm3+RTm4)+(1-f)*(h/3)*(RTd1+RTd2+RTd3+RTd4)
! simplified solution
108
!T1Scaled=f*T1(d,w,w/c,epsM(sigmaD(w,p,xi0),w))+(1-f)*&
T1(d,w,w/c,epsM(sigmaD(w,p,xi0),w))
end function
!
test simpson method
function simpson(a,b,n)
implicit none
integer j,n
real*8
a,b,term1,term2,term3,term4,h,simpson
term1=0.0
term2=0.0
term3=0.0
term4=0.0
h=(b-a)/n
! metal terms (simpson integration)
term1=sin(a)
do j=1,n/2-1
term2=2*sin(a+2*j*h)+term2
end do
do j=1,n/2
term3=4*sin(a+(2*j-1)*h)+term3
end do
109
term4=sin(b)
simpson=(h/3)*(term1+term2+term3+term4)
end function
end