Coupled Plasmonic Systems and Devices: A

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Coupled Plasmonic Systems and Devices: Applications in Visible Metamaterials, Nanophotonic Circuits, and CMOS Imaging
Citation
Burgos, Stanley P.
(2013)
Coupled Plasmonic Systems and Devices: Applications in Visible Metamaterials, Nanophotonic Circuits, and CMOS Imaging.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/N2JK-5318.
Abstract
With the size of transistors approaching the sub-nanometer scale and Si-based photonics pinned at the micrometer scale due to the diffraction limit of light, we are unable to easily integrate the high transfer speeds of this comparably bulky technology with the increasingly smaller architecture of state-of-the-art processors. However, we find that we can bridge the gap between these two technologies by directly coupling electrons to photons through the use of dispersive metals in optics. Doing so allows us to access the surface electromagnetic wave excitations that arise at a metal/dielectric interface, a feature which both confines and enhances light in subwavelength dimensions - two promising characteristics for the development of integrated chip technology. This platform is known as plasmonics, and it allows us to design a broad range of complex metal/dielectric systems, all having different nanophotonic responses, but all originating from our ability to engineer the system surface plasmon resonances and interactions. In this thesis, we demonstrate how plasmonics can be used to develop coupled metal-dielectric systems to function as tunable plasmonic hole array color filters for CMOS image sensing, visible metamaterials composed of coupled negative-index plasmonic coaxial waveguides, and programmable plasmonic waveguide network systems to serve as color routers and logic devices at telecommunication wavelengths.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
plasmonics; nanophotonics; metamaterials; color filters; waveguides; negative index; hole arrays
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Thesis Committee:
Atwater, Harry Albert (chair)
Painter, Oskar J.
Vahala, Kerry J.
Schwab, Keith C.
Defense Date:
21 May 2013
Record Number:
CaltechTHESIS:06042013-150337959
Persistent URL:
DOI:
10.7907/N2JK-5318
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
7835
Collection:
CaltechTHESIS
Deposited By:
Stanley Burgos
Deposited On:
22 Sep 2015 20:39
Last Modified:
08 Nov 2023 00:12
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Coupled Plasmonic Systems and Devices:
Applications in Visible Metamaterials,
Nanophotonic Circuits, and CMOS Imaging

Thesis by

Stanley P. Burgos
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2013
(Defended May 21, 2013)

2013

Stanley P. Burgos

iii

Acknowledgements
I have been fortunate enough to have come across several important people throughout my studies, to whom I would like to convey my deepest appreciation. First on the
list is my advisor, Prof. Harry Atwater, for all his support throughout my career as a
student at Caltech. Harry has been extremely instrumental in determining the outcome of my studies, having provided me with invaluable advice and encouragement,
which have helped shape both my academic and personal life. His dedication, enthusiasm, and positive outlook have definitely helped me aspire throughout my PhD. I
am extremely grateful for him having opened the doors to the research playground
that is his lab, and for allowing me the freedom to pursue my research interests. I
feel extremely fortunate to have had the opportunity to be a part of his lab.
I would also like to thank Profs. Kerry Vahala and Oskar Painter for their invaluable support, guidance, and advice, spanning from my undergraduate to graduate
studies at Caltech. It is with the utmost appreciation that I thank them for their
willingness to help even in the most inconvenient of circumstances. I likewise thank
Professor Keith Schwab for his advice and perspective; he was definitely instrumental
in helping me decide my post-doctoral career path. Similarly, I would like to thank my
undergraduate professors at Occidental College: Profs. Daniel Snowden-Ifft, George
Schmiedeshoff, and Alec Schramm, who were not only excellent professors, but also
great advisors and people whom I could always turn to for help. They have positively
helped shape the person and researcher I am today. Also, having had the privilege
of doing research under the guidance of Prof. Snowden-Ifft, I thank him for being
a great advisor on an exciting project; his dedication and guidance has definitely
helped me aspire to be the best researcher I can be. Equally, I would like to thank

iv
Profs. Scott Funkhouser and Adrian Hightower for their guidance and support while
at Oxy. Professor Hightower’s advice was instrumental in my attending Caltech,
‘surviving’ graduate school, and embarking on the next stage of my research career.
As I started in Harry’s group, I was also lucky enough to have worked with
Prof. Jennifer Dionne, Dr. Luke Sweatlock, Dr. Carrie Hoffmann, and Dr. Brendan
Kayes (all students at the time). Luke, Carrie, and Brendan all facilitated my start
with plasmonics and with the Atwater group – Luke helped me get started with computational plasmonics, and Carrie and Brendan helped me get started with electron
microscopy. Beyond that, their guidance was absolutely critical in the development
of my research career. Also, I had the pleasure of working with Jennifer, who has
been one of my role models throughout my graduate career; her perspective and excitement for the theory and application of plasmonics definitely left an impression on
me, which has helped lead me to where I am today. I thank her for her guidance and
support, as well for teaching me the FIB.
During my graduate studies, I also had the privilege of working with a lot of
unique individuals, many of whom I am honored to call my friends. One of those
individuals is Dr. Eyal Feigenbaum, who is one of the most intelligent, enthusiastic,
and compassionate individuals I know. Eyal, with his critical perspective and genuine
excitement for research, has helped me aspire to become a better individual. Similarly,
I was extremely fortunate to have shared an office with Prof. Jeremy Munday who
showed me that you can be a great researcher while building your own resonator
guitar! Also, I had the privilege of overlapping time in the group with Profs. Marina
Leite and Deirdre O’Carrol, with whom I shared many an afternoon espresso. Having
met them definitely made my studies at Caltech more enjoyable. Similarly, I would
also like to thank Prof. Domenico Pacifici for being a good friend, a great mentor,
and, additionally, a good gym buddy! Likewise, I would like to thank Prof. Koray
Aydin for his advice and support. I was also lucky enough to have overlapped with
Dr. Henri Lezec, a person whose mastery of both the field of plasmonics as well as the
FIB helped me not only to get started in the field, but also to aspire throughout my
PhD. Similarly, I would like to thank Dr. Michael Kelzemberg and Dr. Rob Walters for

their help and support as I started in the Atwater group. Mikes computer knowledge
definitely came in handy throughout the computational parts of this thesis.
I would also like to acknowledge Dr. Min Seok Jang, with whom I shared not
only an office but also many discussions on our perspectives on research and life.
Min’s seemingly effortless intelligence and appreciation for leisure time make him
one of the most interesting individuals I’ve had the pleasure of meeting at Caltech.
I was also fortunate to have worked with Dr. Ryan Briggs. Ryan helped me get
started in KNI and he was kind and talented enough to help fabricate the integrated
RGWN network investigated in Chapter 10. Also, toward the end of my studies, I
was fortunate to have worked with Dr. Howard Lee, who helped me complete part of
the experiments described in Chapter 10. Howards attention to detail and willingness
to help was absolutely critical in the completion . I also thank Dr. Ragip Pala for his
advice, perspective, kindness, and willingness to help. Equally, I thank Dr. Jonathan
Gradidier, Dr. Ruzan Sokhoyan, Dennis Callahan, Seokmin Jeon, and Seyoon Kim for
being good friends and people I could to turn to for advice and meaningful discussions.
I would also like to thank several Atwater people for making my stay at Caltech
more enjoyable: Nick Batara, Jeff Bosco, Ana Brown, Chris Chen, Naomi Coronel, Michael Deceglie, Carissa Eisler (take care of Bebe), Jim Fakonas, Vivian Ferry,
Dagny Fleischman, Cristofer Flowers , Yousif Kelaita, Lise Lahourcade, Krista Langeland, Andrew Leenheer, Josue Lopez, Gerald Miller, Prineha Narang, Georgia Papadakis, Morgan Putnam, Imogen Pryce, Bryce Sadtler, Faisal Tajdar, Raymond
Weitekamp, and Kelsey Whitesell.
Working in the Atwater group has also allowed me to work in several international
collaborations that have led me to meet some truly amazing individuals. I had the
pleasure of working with Prof. Albert Polman and his students Dr. Rene DeWaele,
Ruben Maas, and Marie van de Haar. Albert was kind enough to let me work in his
lab at AMOLF in the Netherlands, where I had the pleasure of working with Ruben
and Marie in work related to Part 1 of this thesis. Similarly, I had the privilege of
working with Rene while he was a visiting student in the Atwater group as I started
my PhD. I was lucky to have shared an office with him and there found that we

vi
shared many of the same perspectives on research and life, making for a fun and
fruitful collaboration (described in Part 1). Rene definitely helped me get started
with research and aided in setting the course of my thesis. I would also like to thank
Profs. Ewold Verhagen and Femius Koenderink for their guidance and support as
well as Piero Spinelli, Rutger Thijssen, Toon Coenen, Claire van Lare, Jorik van de
Groep, and Benjamin Brenny for welcoming me to their lab and showing me around
Amsterdam.
Similarly, I have had the privilege of working with Prof. Ulf Peschel and his students Arian Kriesh and Daniel Ploβ. Arian came from Prof. Peschels lab to work
in the Atwater group for a couple of months in 2010, and it was then that I learned
about the amazingly motivated and intelligent person that he is. Then, about a year
later, Prof. Peschel was kind enough to invite me to work in his lab in Germany.
There I met Daniel Ploβ, Sarina Wunderlich, and Sabina Dobmann, some of the
most friendly and intelligent people I’ve had the pleasure of meeting. I would also
like to thank Hannes Pfeif, Alois Regensburger,Vincent Schultheiβ, and Ali Mahdavi
for welcoming me to their lab and showing me around Germany. Also, I would like to
thank Prof. Peter Banzer, Thomas Bauer, and Uwe Mick for their hospitality during
my visit to Germany.
I also had the privilege of working with Dr. Sozo Yokogawa, a researcher from
Sony who visited our group for a year in 2010, looking for ways to use plasmonics
in combination with Sony’s CMOS IS technology. The work that came out of that
collaboration is discussed in Part 2 of this thesis. Sozo is one of the nicest, most
hardworking, and intelligent individuals that I know, and I feel fortunate to have
worked with him.
I would also like to thank several people, without whom many of the devices presented in this thesis would not be possible: Bophan Chhim, Guy de Rose, Melissa
Melendes, Nils Asplund, Ali Ghaffari, and Jim Lacy. I would also like to thank the
amazing administrative support both inside and outside of the Atwater group, consisting of April Niedholdt, Lyra Haas, Jennifer Blankenship, Tiffany Kimoto, Christy
Jenstead, Connie Rodriguez, Rosalie Rowe, and Eleonora Vorobief. April was amaz-

vii
ing at helping Harry manage a group as large as ours; her positive energy and support
definitely reflected in the group dynamics. Similarly, Jennifer and Tiffany did a great
job at filling in her shoes – I thank them for putting up with me and for making my
last years at Caltech more enjoyable.
And of course, this thesis would not be possible without the support of my family.
I would like to thank Vanessa for all her support while I worked at completing my
studies. Similarly, I would like to thank all her family for the support they have
provided me throughout all these years. And lastly, I would like to thank my mom,
grandma, and family for always believing in me. This thesis is as much an accomplishment for all of them as it is for me.
I would finally like to thank the NSF Graduate Fellowship program for helping
me get started on my PhD career. Similarly, I would like to thank everyone else who
made my stay at Caltech more enjoyable and successful, namely the staff of the Financial Aid and Registrars Office, as well as that of Brown Gym, the Health Center,
and Chandler. These people are the unsung heroes of Caltech, working behind the
scenes to make for a smooth infrastructure so that we can do the science that we love.
Lastly, I would like to thank Oxy and Caltech for giving me such a great experience
in going from a 3/2 Physics undergraduate at Occidental to a graduate student at
Caltech. If I could do it all again, I probably wouldn’t change a thing, for it is those
experiences, both positive and negative alike, that have led me to where I am today.

Stanley P. Burgos
May 2013
Pasadena, CA

viii

Abstract
With the size of transistors approaching the sub-nanometer scale and Si-based photonics pinned at the micrometer scale due to the diffraction limit of light, we are unable to
easily integrate the high transfer speeds of this comparably bulky technology with the
increasingly smaller architecture of state-of-the-art processors. However, we find that
we can bridge the gap between these two technologies by directly coupling electrons
to photons through the use of dispersive metals in optics. Doing so allows us to access
the surface electromagnetic wave excitations that arise at a metal/dielectric interface,
a feature which both confines and enhances light in subwavelength dimensions – two
promising characteristics for the development of integrated chip technology. This
platform is known as plasmonics, and it allows us to design a broad range of complex metal/dielectric systems, all having different nanophotonic responses, but all
originating from our ability to engineer the system surface plasmon resonances and
interactions. In this thesis, we demonstrate how plasmonics can be used to develop
coupled metal-dielectric systems to function as tunable plasmonic hole array color
filters for CMOS image sensing, visible metamaterials composed of coupled negativeindex plasmonic coaxial waveguides, and programmable plasmonic waveguide network
systems to serve as color routers and logic devices at telecommunication wavelengths.
The first part of this thesis is dedicated to studying the coaxial metal-dielectricmetal (MDM) waveguide configuration as it applies to coupled negative index waveguide metamaterials and ultra-small mode-volume nanocavities. We begin by presenting transmission measurements done in combination with an analytic and numeric
study to experimentally determine the dispersion relation of the positive index coaxial
mode, demonstrating how their facet-end reflection coefficients can be tuned to engi-

ix
neer ultra-small mode-volume mode cavities. This study is followed by a theoretical
investigation of the negative index modes in MDM coaxial waveguides, along with a
study of the effect of geometry and materials on the configuration. We then move
on to coupled waveguide geometries, in which we present a new type of metamaterial design, consisting of coupled negative index coaxial waveguides, demonstrating
how these photonic material composites can be engineered to operate with refractive indices that are continually tunable from negative to positive values at visible
frequencies.
In the second part of this thesis, we explore the optical response of another type
of coupled system, consisting of hole arrays, demonstrating, based on a hole-pair
scattering model, that the transmission of large-size hole arrays is determined by
local (2nd nearest neighbor) rather than long-range order. Furthermore, using this
model, we find that the peak transmission efficiency of hole arrays reach ∼ 90% that
of an infinite array at ∼ 6×6 µm2 - the smallest size array showing near-infinite array
transmission properties. We substantiate these findings with a set of experiments in
which we investigate the response of hole arrays in terms of spatial color cross-talk,
random defects, and array size. Finally, we demonstrate their performance as efficient
color filters by integrating them onto a CMOS image sensor and analyzing the quality
of their high-resolution full-color images.
Finally, in the last section of this thesis, we investigate Resonant Guided Wave
Networks (RGWNs), another new type of artificial photonic material concept based
on the interaction of closed-loop wave resonances in waveguide networks. We describe their building blocks in detail and demonstrate how they can be engineered
to form high-Q plasmonic resonators as well as di- and tri-chroic routers operating
at telecommunication wavelengths. We demonstrate the concept of RGWNs in a
hybrid Si-photonic/plasmonic experimental platform, using NSOM measurements to
demonstrate efficiency coupling from SOI ridge waveguides to subwavelength channel plasmon polariton (CPP) networks. We furthermore demonstrate ultracompact
4-way equal power splitters, the basic element of an RGWN, and 2×2 plasmonic resonators operating as a logic device – the first demonstration of a truly subwavelength

integrated plasmonic circuit.

Contents
Acknowledgements

iii

Abstract

viii

1 Introduction
1.1

Perspective on Metal Optics . . . . . . . . . . . . . . . . . . . . . . .

1.2

Scope of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1

Coaxial Plasmonic Waveguides and Metamaterials . . . . . . .

1.2.2

Hole Array Color Filters and Plasmonic CMOS Image Sensing

1.2.3

Resonant Guided Wave Networks and Hybrid Plasmo-Photonic
Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Introduction to Metal Optics

2.1

Optical Properties of Metals . . . . . . . . . . . . . . . . . . . . . . .

2.2

The Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Surface Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Coaxial Plasmonic Waveguides and Metamaterials

3 Plasmon Dispersion in Coaxial Waveguides from Single-Cavity
Optical Transmission Measurements

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Measuring the Dispersion of a Coaxial Plasmonic Waveguide . . . . .

xii
3.3

Comparing Experiment with Theory . . . . . . . . . . . . . . . . . .

3.4

The Phase Shift at Reflection . . . . . . . . . . . . . . . . . . . . . .

3.5

A More Comprehensive Study . . . . . . . . . . . . . . . . . . . . . .

3.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Negative Refractive Index in Coaxial Plasmon Waveguides

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Anlaytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Coaxial Waveguide Dispersion . . . . . . . . . . . . . . . . . . . . . .

4.4

Conditions for Achieving a Negative Mode Index . . . . . . . . . . . .

4.5

Effect of Materials and Geometry . . . . . . . . . . . . . . . . . . . .

4.6

Visualizing the Negative Index Mode . . . . . . . . . . . . . . . . . .

4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 A Single-Layer Wide-Angle Negative-Index Metamaterial

at Visible Frequencies

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Isolated Coaxial Waveguide Dispersion . . . . . . . . . . . . . . . . .

5.3

NIM Slab Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

Pitch Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Parameter Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

Wedge Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hole Array Color Filters

6 Plasmonic Color Filters for CMOS Image Sensor Applications

68
69

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Comparing with FDTD . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

Size Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii
6.5

Spatial Crosstalk . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6

Random Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

Nearest Neighbor Model . . . . . . . . . . . . . . . . . . . . . . . . .

6.8

Size-Corrected Transmission Efficiencies

. . . . . . . . . . . . . . . .

6.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Color Imaging via Integrated Plasmonic Color Filters on a CMOS
Image Sensor

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3

Fabrication and Integration . . . . . . . . . . . . . . . . . . . . . . .

7.4

Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.5

Transmission Measurements . . . . . . . . . . . . . . . . . . . . . . .

7.6

Transmission Simulations . . . . . . . . . . . . . . . . . . . . . . . . .

7.7

Color Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.8

Image Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9

Measuring the Color Quality . . . . . . . . . . . . . . . . . . . . . . .

7.10 Angle Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Scattering-Absorption Nearest-Neighbor Model Description of Hole
Arrays
8.1

102

Nearest Neighbor Scattering-Absorption
Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.2

Extracting the Hole-Pair Scattering-Absorption Coefficients . . . . . . 105

8.3

Number of Contributing Nearest Neighbors . . . . . . . . . . . . . . . 106

8.4

Setting the Periodicity of the Array . . . . . . . . . . . . . . . . . . . 108

8.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xiv

III

Resonant Guided Wave Networks

9 Resonant Guided Wave Networks

113
114

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.2

Plasmonic RGWN Components . . . . . . . . . . . . . . . . . . . . . 116

9.3

Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.4

Tailoring the Optical Properties of Artificial Materials . . . . . . . . . 122

9.5

Programming the Optical Properties of a Network . . . . . . . . . . . 124

9.6

Multi-Chroic Filters using RGWNs . . . . . . . . . . . . . . . . . . . 128

9.7

Possible Implementations . . . . . . . . . . . . . . . . . . . . . . . . . 133

9.8

Conclusions and Directions . . . . . . . . . . . . . . . . . . . . . . . . 134

10 Silicon Coupled Plasmonic Nanocircuits: 4-way Power-Splitters and
Resonant Networks

137

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10.2 Plasmonic Modes of the V-Groove Configuration . . . . . . . . . . . . 140
10.3 Mode Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.4 CPP Waveguide Mode Properties . . . . . . . . . . . . . . . . . . . . 143
10.5 Ultracompact 4-Way Power-Splitters . . . . . . . . . . . . . . . . . . 146
10.6 2×2 Plasmonic Logical Device . . . . . . . . . . . . . . . . . . . . . . 148
10.7 Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Bibliography

155

List of Figures
2.1

Real and Imaginary parts of the permittivity data for Ag at visible
frequencies. Solid line plots are obtained by fitting the Drude model to
tabulated Palik data. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Surface plasmon geometry composed of permittivity 1 for y > 0 and
permittivity 2 for y < 0. The corresponding propagating wave vector
and associated field vectors are displayed by black arrows inside the
material, along with a schematic of the resulting field intensity, shown
in red. The ’+’ signs represent a charge depletion area, which is periodic
along the propagation wave vector direction of the surface plasmon. . .

2.3

Dispersion relation properties (Eq. 2.15) for the case where 1 = Si and
2 = Ag . Panel (a) plots the real part of the Ag/Si plasmon propagation
wave vector <[kz ] (red), together with that of bulk Si (green), bulk Ag
(blue), and air (black). Panel (b) plots the propagation length, Lz =
1/2= [kz ], defined as the propagation distance at which the field intensity
reaches a fractional value of 1/e. This quantity is plotted for the Ag/Si
surface plasmon (red), bulk Si (green), and bulk Ag (blue). Panel (c)
plots the surface plasmon penetration depth, Ly(1,2) = 1/2= ky(1,2) ,
into both the Si (green) and Ag (blue), similarly defined as perpendicular
distance away from the interface at which the field intensity reaches a
fractional value of 1/e from its value right at the interface. . . . . . . .

xvi
3.1

SEM images of the cross-sectional profile of coaxial plasmon waveguides
with lengths of 485 nm, dielectric channel widths of ∼100 nm (a) and
∼50 nm (b), and outer radii of ∼175 nm. The insets show top-view
SEM images of the waveguides before cross sectioning. Scale bars are
100 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Transmission measurement (red dotted spectrum, I multiplied by 100) of
a 485-nm-long coaxial waveguide with a ∼100-nm-wide dielectric channel (see Fig. 3.1b) and a reference spectrum (blue dashed line, I0 ). The
transmittance defined as the waveguide transmission spectrum divided
by the reference spectrum is depicted by the green curve (green drawn
line, T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Transmittance spectra of coaxial waveguides with varying lengths. The
outer radius and channel width were ∼175 and ∼100 nm, respectively,
while the waveguide length was decreased from 485 nm (top curve) to 265
nm (bottom curve) in increments of 20 nm. Data are shifted vertically
for clarity. The black dashed lines are guides for the eye and connect
the resonance peaks m = 1 − 3). . . . . . . . . . . . . . . . . . . . . .

3.4

Measured dispersion data and calculated index-averaged dispersion relations (red drawn lines) for coaxial plasmon waveguides with (a) ∼50nm-wide air channel and (b) ∼50-nm-wide spin-on-glass (SOG) filled
channel. Light lines for air (a) and SOG (b) are also shown (dotted
green lines), along with the plasmon dispersion (dashed orange curves)
at a flat Ag/air interface (a) and Ag/SOG interface (b). Symbols in
the figure correspond to different Fabry-Perot mode numbers, m, where
= 1, ∗ = 2, 4 = 3, and ◦ = 4. The inset in (a) shows the calculated
electric field distribution of the mode of azimuthal order n = 1, in an
SOG-filled coaxial waveguide with outer radius of 200 nm and channel
width of 100 nm at ω = 3.5 × 1015 rad/s. . . . . . . . . . . . . . . . . .

xvii
3.5

Measured dispersion data and calculated index-averaged dispersion relations (red drawn lines) for coaxial plasmon waveguides with a 100nm-wide air channel. In (a) the dispersion data are plotted, assuming
a zero net phase shift as a result of reflections at the end facets of the
cavity. In (b) the data are plotted for an overall reflection phase shift
of 0.77π, which was determined using simulations. Light lines for air
are also shown (dotted green lines), along with the plasmon dispersion
(dashed orange curves) at a flat Ag/air interface. Symbols in the figure
correspond to different Fabry-Perot mode numbers, m, where = 1,
∗ = 2, 4 = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

Steady-state simulation results of the electric field intensity profile inside
a 485-nm-long coaxial cavity with an air-filled ∼100-nm-wide dielectric
channel excited at an angular frequency of ω = 4.2×1015 rad/s (freespace
wavelength of 450 nm). (a) Electric field intensity distribution inside
the coaxial cavity for light incident from the left. (b, c) Intensity as a
function of position integrated along the lateral direction (dotted lines)
for light incident from the left (b) and right (c). The red drawn curves
are fits of the intensity profile that were used to find the reflectance and
reflection phase shifts at the distal end facets. The reflection phase shift
is 1.0π at the substrate side and −0.12π at the air side of the cavity,
while the values for the reflectance are 75% and 20%, respectively. . . .

xviii
3.7

Reflection phase (∆φ) and reflectance (|r|) of the end facets of an untapered coaxial waveguide with 75-nm-wide dielectric channel and outer
radius of 175 nm at a wavelength λ0 = 800 nm, derived from finite
difference time domain simulations. (a) ∆φ and |r| are plotted as a
function of the surrounding dielectric index for waveguides with fixed
dielectric channel index, nin ) 1.5. The dotted blue line gives the result obtained from Fresnel equations using the (single) mode index of
the coaxial cavity. (b) Plot of ∆φ and |r| as a function of the refractive index of the dielectric channel of the coaxial waveguide, while the
surrounding dielectric index is kept fixed, nout = 1.0. . . . . . . . . . .

4.1

Coaxial plasmon waveguide geometry and numerical mode solving method.
(a) Schematic cross-section of a coaxial waveguide with the definition of
the cylindrical polar coordinates, r, φ and z. The metallic inner core and
outer cladding separate a dielectric channel. A schematic wave propagating in the waveguide in the direction of positive z is also indicated.
(b) Argument θ of the determinant, det[M (k)], plotted in the complex
k-plane for a Ag/Si/Ag waveguide with 75 nm inner core diameter and
10-nm-wide dielectric channel at ω = 3 × 1015 rad/s . By cycling around
the closed loop, indicated by the dashed square, the net number of discontinuities in θ is determined. Zero positions are indicated by the white
circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix
4.2

Dispersion relations of the three lowest-order modes of a coaxial waveguide with 75-nm-diameter Ag core, 25-nm-wide Si channel, and infinite outer Ag cladding. Radial frequency is plotted versus propagation constant k 0 (a), attenuation constant k 00 (b), and figure-of-merit
k 0 /k 00 (c).

The Ag/Si surface plasmon resonance frequency ωSP =

3.15 × 1015 rad/s (λ0 = 598 nm) is indicated by the horizontal line.
Panel (a) shows two modes with positive index (blue dashed curve and
green dotted curve) and one mode with a negative index below a frequency of ∼ 3.8 × 1015 rad/s (red drawn curve). The insets in (a) show
the Hy field distribution in the transverse plane of the waveguide at
2.8 × 1015 rad/s for the positive-index mode (blue dashed dispersion
curve) and at 3.6 × 1015 rad/s for the negative-index mode. . . . . . .
4.3

Dispersion relations for negative-index coaxial waveguides with Ag core
and cladding and Si dielectric channel, (a): ω(k 0 ); (b): ω(k 00 ). The inner
core diameter is fixed at 75 nm, and the Si-channel thickness w is 10 nm,
30 nm, and 70 nm. Positive-index modes [as shown in Fig. 4.2(a)] are not
shown in the figure. The bold sections of the dispersion curves indicate
the spectral range over which the negative-index mode is dominant, i.e.,
has lower loss than the positive index modes. The frequency where the
red and green dispersion curves cross k 0 = 0 is indicated by the starsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Dispersion relations for coaxial waveguides with Ag core and cladding
and 70-nm-wide Si dielectric channel, (a): ω(k 0 ); (b): ω(k 00 ). The inner
core diameter, dcore , is 45 nm (blue curves), 75 nm (green curves) and
100 nm (red curves). Only modes with negative index are plotted. Bold
lines indicate the spectral range where the mode is dominant over the
positive-index mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xx
4.5

Dispersion relations for coaxial waveguides with 75-nm-diameter metal
core, 25-nm-wide dielectric channel and infinite metal cladding, (a):
ω(k 0 ); (b): FOM k 0 /k 00 . The type of metal in the core and cladding,
as well as the dielectric material, was varied. The frequency axes are
normalized to the corresponding surface plasmon resonance frequency
ωSP . Blue curves are for a Si channel surrounded by Ag (ωSP =
3.15×1015 rad/s), green curves for GaP in Ag (ωSP = 3.49×1015 rad/s),
red curves for SiO2 in Ag (ωSP = 5.24 × 1015 rad/s) and purple curves
for Si in Au (ωSP = 2.77 × 1015 rad/s). Bold curves indicate the spectral
range where the negative-index mode is dominant. . . . . . . . . . . .

4.6

Magnetic field images of a coaxial waveguide with 75 nm Ag core diameter, 25-nm-wide GaP-filled ring and infinite Ag cladding at a frequency
of 3.75 × 1015 rad/s . In (a) we plot the Hy field distribution on the
boundary of the Ag cladding with the dielectric channel. Note that the
phase-fronts in the waveguide are in general not perpendicular to the
optical axis (z-axis). In (b-d) we plot the polar magnetic field components in the transverse plane. The amplitude of the fields plotted in the
figure has the same order of magnitude in all of the four panels. . . . .

5.1

Negative-index metamaterial geometry. (a) Single-layer NIM slab consisting of a hexagonal array of subwavelength coaxial waveguide structures. The inner radius r1 , outer radius r2 and array pitch p are defined
in the image. (b) Unit cell of the periodic structure. The angle-ofincidence θ is shown, as well as the in-plane (p-) and out-of-plane (s-)
polarization directions associated with the incident wavevector k. . . .

xxi
5.2

Coaxial waveguide dispersion relations. The coaxial waveguide consists
of an infinitely long 25-nm GaP annular channel with a 75-nm inner
diameter embedded in Ag. Plotted are the two lowest-order linearly polarized modes that most strongly couple to free space radiation. (ac),
Energy is plotted versus β 0 (a), β 00 (b), and mode index nmode (c). (d)
The figure-of-merit FOM = |β 0 /β 00 |. The Ag/GaP planar surface plasmon energy at ~ωSP = 2.3 eV (λ0 = 540 nm) is indicated by the black
dashed horizontal line. All panels show one mode with positive index
(red curve) and one mode with a negative index (blue curve) below an
energy of 2.7 eV (λ0 = 460 nm). The insets in (a) show the Re(Hy )
(out-of-page) field distribution in the waveguide at a wavelength of λ0
= 650 nm for the positive-index mode and at λ0 = 483 nm for the
negative-index mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Coaxial waveguide negative-index mode. Lateral cross-section of a coaxial waveguide consisting of an infinitely long 25-nm GaP annular channel
with a 75-nm inner diameter embedded in Ag. The dielectric channel
is schematically indicated. Plotted is the real part of the H-field distribution of the n = 1 negative index mode at λ0 = 483 nm, where n
refers to the azimuthal dependence of the fields. The in-plane Re(Hxz )
field distribution is depicted with arrows, while the out-of-plane Re(Hy )
fields are plotted using a color scale. . . . . . . . . . . . . . . . . . . .

5.4

Metamaterial index. (a) Light at λ0 = 483 nm is incident on a semiinfinite slab of single-layer negative index metamaterial at an angle of
30◦ from air. Shown is a time-snapshot of the magnetic field distribution
Re(Hy ), taken along the polarization plane. Arrows denote the direction
of energy flow S and phase velocity k. The coax center-to-center pitch
is schematically indicated. (b) Constant-frequency surface plot at λ0
= 483 nm, showing the relation between kx and kz for a semi-infinite
metamaterial slab over a 50◦ range of incidence angles. The wavevector
k and Poynting vector S data are derived from FDTD simulations. . .

xxii
5.5

Summary of effective refractive index for varying angle of incidence. The
metamaterial effective refractive index neff is plotted for λ0 = 483 nm
s- and p-polarized light incident at angles ranging from 0 − 50◦ , derived
from slab wave vector angles as in Fig. 5.4a, as well as from refraction
angle measurements in wedge-shaped samples as in Fig. 5. The dashed
line indicates the calculated mode index of a single coaxial waveguide.

5.6

Modal reconstruction. A semi-infinite coaxial waveguide consisting of
a 25-nm GaP annular channel with a 75-nm inner diameter embedded
in Ag is illuminated from air with λ0 = 483 nm light at a 30◦ angle-ofincidence. Plotted are the real and imaginary parts of Hy . The two
panels on the left (a, c) show the mode excited inside the waveguide,
and the two right-side panels (b, d) show the mode reconstructed from
a superposition of 87% n=1 mode and 13% n=0 mode, where n refers
to the azimuthal dependence of the fields. . . . . . . . . . . . . . . . .

5.7

Coaxial waveguide mode dispersion relations. The coaxial waveguide
consists of an infinitely long 25-nm GaP annular channel with a 75nm inner diameter embedded in Ag. Plotted are the n = 0, 1, and 2
dispersion relations, where n refers to the azimuthal dependence of the
fields in the waveguide, described by the harmonic function einψ of order
n. Energy is plotted versus β 0 in (a), β 0 in (b). The Ag/GaP planar
surface plasmon energy at ~ωSP = 2.3 eV (λ0 = 540 nm) and the target
negative-index operation wavelength (λ0 = 483 nm) are indicated by
black dashed horizontal lines. . . . . . . . . . . . . . . . . . . . . . . .

5.8

Coaxial waveguide eigenmodes. The coaxial waveguide consists of an
infinitely long 25-nm GaP annular channel with a 75-nm inner diameter
embedded in Ag. Plotted are the real (a, b, c) and imaginary (d, e, f)
parts of the Hy field components of the n = 0 (a, d), 1 (b, e), and 2 (c,
f) modes at λ0 = 483 nm, where n refers to the azimuthal dependence
of the fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii
5.9

Effective refractive index as a function of pitch. The effective refractive
index neff derived from wave vector angles is plotted as a function of
pitch for λ0 = 483 nm p-polarized light incident at 30◦ on a variable pitch
waveguide array slab similar to that shown in Fig. 5.4a. The dashed line
indicates the calculated mode index of a single coaxial waveguide. . . .

5.10

NIM effective parameters. Effective parameters are calculated for a 100nm-thick NIM slab excited at normal incidence over the 400–500 nm
spectral range. (ad) The real (0 ) and imaginary (00 ) parts of the retrieved
effective relative impedance zef f (a), index nef f (b), relative permittivity
ef f (c) and relative permeability µef f (d). . . . . . . . . . . . . . . . .

5.11

Wedge refraction. (af) A ∼300-nm-thick metamaterial slab is illuminated from the left at normal (a,b,e,f) and 30◦ off-normal incidence
(c,d). The right side of the slab is cut at a 3◦ angle to allow refraction (black dashed line indicates the surface normal). The wavelength
of incident light is 483 nm (ad) and 650 nm (e,f). The three panels
on the left (a,c,e) depict the calculated power flow (green arrows), and
the three corresponding right-side panels (b,d,f) show the steady-state
electric field intensity in a polar plot, monitored at a distance of 10 µm
behind the exit side of the slab. The output plane surface normal is indicated on the polar plots by a grey dot. In (a) we also plot the Re(Hx )
field distribution along the plane of refraction. . . . . . . . . . . . . . .

xxiv
6.1

(a) Back illuminated microscope images of the fabricated hole array
filters. Each filter consists of 16 × 16 hexagonally packed hole arrays.
The vertical axis corresponds to hole diameter, ranging from 80 to 280
nm in 20-nm steps, and the horizontal axis corresponds to hole period,
ranging from 220 to 500 nm in 40-nm steps. The white bar on the lower
part of the images corresponds to a 10-µm scale. Inset (b) shows a
SEM image of a representative hole array filter consisting of hexagonally
aligned 16 × 16 holes with p = 420 nm and d = 240 nm. Measured hole
array spectra for filters optimized to transmit (c) red (p = 420 nm, d
= 160280 nm), (d) green (p = 340 nm, d = 120240 nm), and (e) blue
light (p = 260 nm, d = 100180 nm) are plotted in dotted lines, and
the simulated spectra are plotted in solid lines. Each plot of three color
spectra is in steps of 40 nm in hole diameter. . . . . . . . . . . . . . .

6.2

The simulated spectra of the hole array filter with p = 420 nm and d =
240 nm, which is optimized to transmit red color. The top four panels
plot the electric field distribution at the wavelengths of interest along
the diameter of the holes, parallel to the polarization of the plane wave
used to excite the structure. . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Transmission spectra of the hole array filters optimized to red (p = 420
nm, d = 240 nm), green (p = 340 nm, d = 180 nm), and blue (p = 260
nm, d = 140 nm) of different filter sizes of (a) 10 µm-, (b) 5 µm-, (c)
2.4 µm-, and (d) 1.2 µm-squared size filters. The insets of (a-d) panels
show the back illuminated microscope images of the filter with the field of
views corresponding to 13-µm-, 6.5-µm-, 5.0-µm-, and 4.0-µm-squared.
(e)-(f) SEM images are the 1.2-µm-size filters, RGB, respectively. . . .

xxv
6.4

Sliced transmission spectra of color filter pairs with zero separation. (a)
A representative SEM image of boundary between two such filters. (b-d)
Back-illuminated microscope images of the color filter pairs taken with
a color CCD camera. The white lines correspond to 20-µm scale bars.
The spectra of each color filter pair are taken over 1-µm-wide areas
centered at the positions indicated by the yellow ticks in (b). Sliced
spectra are shown for (e) blue/red, (g) green/blue, and (i) red/green
filter pairs. The panels next to the sliced spectra in (f), (h), and (j) plot
the correlation of each measured spectra with respect to the averaged
spectrum of colors in the filter pair. . . . . . . . . . . . . . . . . . . . .

6.5

(a) Transmission spectra of green hole array filter consisting of 32 ×
32 holes with p = 340 nm and d = 180 nm for different random defect
(missing hole) density. (b) Normalized transmission spectra from (a).
(c) Plot of the relative peak efficiency versus defect rate. The data plotted by blue dots corresponds to the transmission efficiency of a green
filter with defect density ranging from 0 to 50%, and the red line is the
analytically estimated degradation curve. (d) A SEM image of green filter with 50% defect density. (e) An analytically-calculated transmission
intensity map of the filter from (d). (f) A back-illuminated microscope
image corresponding to the filter from (d). . . . . . . . . . . . . . . . .

xxvi
7.1

Integrated CMOS image sensor with hole array filter. a) Schematic of
integrated front-side illumination CMOS image sensor with RGB plasmonic hole array filters in Bayer layout. b) Scanning electron micrograph
of RGB hole array filters in Bayer layout. c) Scanning electron micrograph of 11.2µm alignment grid lines separating 40×40 filter blocks. d)
Image of full 360×320 pixel (2016×1792 µm2 ) plasmonic hole array filter array on quartz. Each square on the image corresponds to a 40×40
filter block (224m×224µm2 ) separated by 11.2µm alignment grid lines.
e) Image of integrated CMOS image sensor with plasmonic hole array
filter. The white on the far edges of the filter corresponds to electronic
grade putty used to hold the filter in place after alignment. f) Image of
CMOS image sensor before integration. . . . . . . . . . . . . . . . . . .

7.2

xxvii
7.3

Alignment of plasmonic hole array filter with CMOS image sensor. The
lined grid represents the pixel array of the CMOS image sensor, and the
transparent RGB box grid represents the plasmonic hole array color filter
array in Bayer layout. The pixels are labeled using matrix convention
(i,j) with i coming from the horizontal number axis and j coming from
the vertical number axis, and the double letters inside of the grid refer
to the parity of the pixel label, with E for even and O for odd. Pixel and
filter array are shown with a) perfect alignment, b) translation offset,
and c) rotational offset. d) Images of the difference parity set of pixel
readouts, even-even (EE), even-odd (EO), odd-even (OE), and odd-odd
(OO), after demosaicing the gray wall image taken with the integrated
CMOS image sensor with aligned filter . . . . . . . . . . . . . . . . . .

7.4

Integrated CMOS image sensor response. a) Measured and c) simulated
spectral response of unmounted RGB plasmonic hole array filters. b)
Measured and d) simulated relative efficiency of integrated CMOS image
sensor with RGB plasmonic hole array filters. The horizontal error bars
correspond to the spectral width of the band-bass filter used for the
measurements, and the vertical error bars correspond to the averaged
data. Simulation field map cross-sections taken at the center of the pixel
and at the center transmission wavelengths for the e) blue (λ0 = 450nm),
f) green (λ0 = 550nm), and g) red (λ0 = 650nm) hole array color filters
integrated with the CMOS image sensor. . . . . . . . . . . . . . . . . .

xxviii
7.5

Image reconstruction process. a) Raw image of 24-color Macbeth color
chart positioned in a scene taken with integrated CMOS image sensor
with RGB plasmonic hole array filter. Image after b) demosaicing, c)
white balancing, d) linear matrix correcting, and e) gamma correcting
the image. The yellow dashed line in a) shows the pixel signal that is
being plotted in f-h), corresponding to the images above them. i) Linear
matrix used on image after applying white balance to remove color cross
talk. j) Gamma transformation used to convert the image sensors linear
response to brightness to the logarithmic response of the human eye. .

7.6

Focal length dependence and outdoor lighting conditions. Images of 24color Macbeth chart positioned in a scene taken with integrated CMOS
image sensor with RGB plasmonic hole array filter with a 5.6 f-number
and a a) 6mm, b) 9mm, c) 12.5mm, and d) 50mm lens. Images taken
with outdoor lighting conditions of e) Watson Patio, Caltech, f) Beckman Auditorium, Caltech, g) Atwater Group, Aug 2012, Caltech, and
h) Red Door Cafe, Caltech. . . . . . . . . . . . . . . . . . . . . . . . .

7.7

Green filter angular response and integrated CMOS IS f-number dependence. Simulated spectral transmission response of green hole array
filter as a function of incident angle for a) TM and b) TE polarizations.
c) Simulated spectral response of green filter operating with a maximum half-angle aperture of 15 degrees, corresponding to an f/number
of about 1.8, obtained by averaging the spectral response for incident
angles ranging from 0-15 degrees over both polarizations. Images of
24-color Macbeth chart taken with the integrated CMOS image sensor
with RGB plasmonic hole array filter with a 50mm lens and with an
f-number (maximum half-aperture angle) of d) 1.8 (15.5 degrees), e) 2.8
(10.1 degrees), f) 4 (7.1 degrees), g) 5.6 (5.1 degrees), h) 8 (3.6 degrees),
and i) 11 (2.6 degrees). . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxix
8.1

(a) Schematic of the second nearest-neighbor (n.n.) scattering-absorption
model used to reconstruct the transmission spectra of a square-shaped
triangular-lattice hole array (b). The black circles represent holes in a
triangular lattice, connected by black lines for reference. The first and
second nearest neighbors surrounding the central lattice point are shown
in red and blue, respectively. The scattering intensity of a contributing
lattice site is depicted by green ovals, with the enclosed arrows corresponding to the spatial scattering efficiency amplitudes (β). The absorption efficiencies (β 0 ) of the central lattice are depicted by dashed arrows.
(b) Transmission efficiency curves extracted from the scattering model
(a) as a function of array size for the square-shaped triangular-lattice
hole array shown on the inset, consisting of 180-nm-diameter holes set
at a pitch of 430 nm in a 150-nm-thick Al film embedded in SiO2 . The
red curve corresponds to a 40×40 µm array, which we call ‘∞-array’,
due to its asymptotic behavior. We normalize to the peak transmission
efficiency of this curve for reference. The other curves correspond to
normalized transmission efficiencies for different size hole arrays ranging
from ∼ 4 × 4 – 10 × 10 µm2 in size. The horizontal dashed curve at 0.4
corresponds to the normalized transmission efficiency of a single isolated
hole.

8.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

(b) Transmission efficiency as a function of pitch for a symmetric 4hole triangular-lattice unit cell (a), consisting of 180-nm-diameter holes
set at a pitch of 430 nm in a 150-nm-thick Al film embedded in SiO2 .
The red dotted spectrum is extracted from FDTD simulations, and the
blue dashed spectrum corresponds to the fitted scattering-absorption
model. The horizontal dashed curve at 1 corresponds to the normalized
transmission efficiency of a single isolated hole. (c) Spectrally resolved
scattering-absorption parameters obtained by varying the wavelength
from 400 – 800 nm and fitting as done in (b). . . . . . . . . . . . . . . 108

xxx
8.3

Normalized transmission efficiency curves extracted from FDTD as a
function of array size for the square-shaped triangular-lattice hole array
shown on the inset, consisting of 180-nm-diameter holes set at a pitch
of 430 nm in a 150-nm-thick Al film embedded in SiO2 . The red curve
corresponds to an infinite array, to which we normalize for reference.
The other curves correspond to normalized transmission efficiencies for
different size hole arrays ranging from ∼ 4×4 – 10×10 µm2 in size. The
horizontal dashed curve at 0.35 corresponds to the normalized transmission efficiency of a single isolated hole. . . . . . . . . . . . . . . . . . . 109

8.4

Transmission efficiency for different hole array configurations as a function of number of contributing nearest neighbors (n.n.). Data is shown
for spectra calculated with FDTD, as well as with the nearest neighbor
scattering model with second and third n.n. contributions. The hole
array configurations (see insets) consist of 180-nm-diameter holes set at
a pitch of 430 nm in a 150-nm-thick Al film embedded in SiO2 . . . . . 110

8.5

Absolute transmission efficiency curves extracted from the scattering
model as a function of array pitch for a ∼ 10 × 10 µm2 square-shaped
triangular-lattice hole array, consisting of 180-nm-diameter holes set at
a pitch of 430 nm in a 150-nm-thick Al film embedded in SiO2 . . . . . 111

9.1

Schematic illustration of (a) a 4-terminal equal power-splitting element
and (b) a local resonance in a 2x2 RGWN. . . . . . . . . . . . . . . . . 115

9.2

Power-splitting properties of the emerging pulses in an X-junction: (a)
intensity relative to the exciting pulse, and (b) phase difference at λ0 =1.5
µm [48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.3

Resonance build-up in a 2×2 RGWN. (a) Two in/out-of phase input
pulses result in destructive/constructive interference inside the network.
(b) Steady-state of waves resonating in a 2×2 network where each pair
of pulses excites the X-junctions out of phase [48]. . . . . . . . . . . . . 119

xxxi
9.4

Time snapshots of Hz (normalized to the instantaneous maximum value)
in a 2×2 plasmonic RGWN recorded at the third to the seventh powersplitting events for a 2D-FDTD simulation. The MDM waveguides are
0.25 µm thick and 6 µm long [48]. . . . . . . . . . . . . . . . . . . . . . 121

9.5

Q-factor of 22 RGWN resonator from simulation results compared with
those resulting from incoherent power-splitting [48]. . . . . . . . . . . . 122

9.6

Photonic band structure of infinitely large periodic RGWNs [48]. . . . 123

9.7

Mathematical representation scheme of (a) a 2×2 RGWN system and
its components, (b) a waveguide component, and (c) an X-junction component [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9.8

2×2 RGWN programmed to function as a dichroic router: (a) schematic
drawing, and (b,c) time snapshots of the H-field at the two operation
frequencies [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.9

Flow chart of the RGWN S-matrix optimization procedure [49]. . . . . 129

9.10

3×3 RGWN programmed to function as a trichroic router. Time snapshots of the steady state H-field at the three operation frequencies [49]:
a) λ1 , b)λ2 , c) λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9.11

3D RGWN: (a) rendering of a 3D RGWN building block (6-arm junction). (b, c) Optical DOS of an infinite 3D network spaced periodically
with cubic periodic unit cell with different spacing [48]. . . . . . . . . . 134

xxxii
10.1

Schematic of Si-photonic/v-groove plasmonic hybrid device and
experiment. (a) Scanning electron micrograph of hybrid device overlaid with schematic of fabricated device. Dashed lines represent geometry underneath the sample surface. Illumination condition used is
schematically drawn in addition to the definition of positive and negative
excitation angles for the grating coupler. Inset shows CPP mode profile calculated with an eigenmode solver (Lumerical FDTD v8.0) along
with an experimental NSOM scan of the modal cross-section obtained
at λ0 = 1520 nm. (b) Close-up of Si-ridge/v-groove hybrid device fabricated with electron beam lithography and focused ion beam milling. (c)
Resulting experimental NSOM scan of hybrid structure shown on (b)
taken at λ0 = 1520 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10.2

Field distributions and near-field profiles of v-groove waveguide
modes. (a,b) Dominant field distributions obtained from eigenmode
solver (Lumerical v8.0) calculations of CPP (Ex ) and SPP (Ey ) modes
supported inside and near the surface of the v-groove configuration.
(d,e) Corresponding field intensity distributions of CPP (d) and SPP
(e) modes taken at a distance of 75 nm above the surface of the vgroove structure. Black line scans correspond to FDTD calculations, red
line scans correspond to NSOM measurements. (c) Field distribution
obtained from eigenmode solver calculations of TE mode in Si-ridge
waveguide used to excite the CPP mode of the v-groove structure. . . . 141

xxxiii
10.3

NSOM images of CPP v-groove waveguide mode. Experimental
NSOM image taken at λ0 = 1520 nm of (a) 30 µm and (d) 10 µm long vgroove waveguides. (c) Propagation length of v-groove waveguide mode
obtained by FEM calculations (black line) and by fitting the decaying
NSOM intensity of long v-groove waveguides (blue dotted data). (d)
Effective index of v-grove waveguide mode extracted from eigenmode
solver (Lumerical v8.0) calculations (black line) and the standing wave
pattern observed in NSOM measurements of short v-groove waveguides
(blue dotted points).

10.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Equal power-splitting x-junction plasmonic structure. (a) Scanning electron micrograph of two v-groove waveguides crossed at 90◦ ,
forming an x-junction coupled from one of its ports by a Si-ridge waveguide. (b) NSOM image taken at λ0 = 1520 nm of x-junction shown in
(a). (c) Splitting parameters extracted from fitting the intensity pattern of the NSOM image shown in (b). (d) Simulated optical response
of v-groove waveguide crossing excited through one of its ports with the
CPP mode at λ0 = 1520 nm. . . . . . . . . . . . . . . . . . . . . . . . 147

10.5

2×2 RGWN optical logic device. (a) Scanning electron micrograph of fabricated device consisting of four 15-µm-long waveguides in
an evenly spaced 2×2 configuration, coupled from one of the arms with a
Si-ridge waveguide. (d) Measured near field response of device shown in
(a), when exciting it with λ0 = 1520 nm TE polarized light from the Siridge waveguide. The output ports of interest are labeled for reference.
(c) Simulated optical response of device shown in (a) when exciting one
of its arms with the CPP mode at λ0 = 1520 nm. . . . . . . . . . . . . 149

xxxiv
10.6

2×2 RGWN logic device operation. (a-c) Simulated near field intensity of plasmonic logic device consisting of four 15-µm-long v-groove
waveguides in a 2×2 configuration, excited with the CPP mode from
the bottom left port at (a) λ0 = 1470 nm, (b) λ0 = 1570 nm, and (c)
λ0 = 1670 nm. The top ports of interest are labeled along with their
on/off state configuration based on the excitation wavelength. (d) Intensity cross-sections taken at top output ports of interest at the excitation
wavelengths shown in (a-c). (e) Measured and simulated intensity response at the output ports of interest for λ0 = 1505 nm. . . . . . . . . 151

10.7

Coupling efficiency as a function of waveguide position at λ0 =
1520 nm. Horizontal axis corresponds to vertical offset between Si-ridge
and v-groove waveguides relative to their surface tops. The three dotted
curves correspond to three different waveguide separations, with the blue
corresponding to zero separation, black to 500 nm separation, and red
to 1000 nm separation. The green dotted data corresponds to coupling
efficiencies extracted from NSOM measurements for wavelengths λ0 =
1500, 1510, and 1520 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 152

xxxv

List of Tables
2.1

Plasma ωp , damping Γ, and interband ωib frequency values of some metals used in plasmonics. The plasma and damping values are obtained
by fitting to the Drude model, and the interband frequency values are
obtained by fitting to a Drude-Lorentz model. . . . . . . . . . . . . . .

7.1

F-numbers and their respective ∆E values . . . . . . . . . . . . . . . . 100

9.1

Set of optimized parameters for 2×2 RGWN dichroic router operating
at λ1 =2 µm and λ2 =1.26 µm. . . . . . . . . . . . . . . . . . . . . . . . 130

9.2

Set of optimized parameters for 3×3 RGWN trichroic router operating
at λ1 =1.59 µmm, λ2 =1.97 µm, and λ3 =1.23 µm. . . . . . . . . . . . . 133

Chapter 1
Introduction
1.1

Perspective on Metal Optics

The interaction of naturally occurring materials (whether insulating, semiconducting,
or metallic) with electromagnetic radiation is characterized by the material’s permittivity () and permeability (µ) – the two quantities that tell us how susceptible the
material’s electronic composition is to the two driving fields of electromagnetic radiation. For visible frequencies, it turns out that most materials have a very weak
magnetic response, which has to do with the strength of their atomic magnetic dipole
density, thus making their relative permeabilities very close to that of freespace µ0 .
Fortunately, however, the story for permittivity is a lot more interesting. The permittivity of a material relates to the strength of its polarizability, which is appreciable
since, electrons, whether bound or free, can be driven to oscillate in response to an
electric field. For insulators, the permittivity is determined by the oscillation of bound
electrons, and as a result we get dielectrics with relative permittivities > 1. Combining this index platform with Maxwell’s equations, we get the field of photonics,
which hinges on our ability to confine and manipulate light through the engineering
of interesting dielectric index environments such as photonic crystals and fiber optics.
The field of photonics has given rise to such critical technology as fiber optic
telecommunication and lasers, but it also suffers from the limitation that it can not
be easily miniaturized due to the diffraction limit of light, ∼ λ/2n. Among other
driving factors, we find that as the density of transistors keeps increasing and the size

of technology keeps being miniaturized, we find ourselves looking for ways to engineer
new smaller photonic technologies in order to keep up with current technology trends.
To circumvent this limitation, we find that we can shrink light past the diffraction
limit if we are willing to trade off loss for confinement by using metals. Loss can in
general be considered bad for designing efficient systems, but as long as the system
functionality is within the propagation length of the light inside the system, metals
in optics could still be beneficial.
Metals, unlike dielectrics, are characterized by negative permittivities that originate from the restoring force their free electrons experience as they scatter off of
the metal’s lattice in response to a driving electric field. This sloshing of electrons
transfers some of the field energy to the lattice and, as a result, we get Ohmic loss
in the material. However, at optical frequencies, these electrons also form some very
useful and tunable charge density oscillations both in the material bulk as well as
the surface, and, as a result, we get bulk and surface plasmon polaritons [108], with
which we can manipulate the properties of light. Similar to the field of photonics, we
can confine and manipulate light by designing interesting high-order metal/dielectric
systems, such as metal/dielectric/metal (MDM) slab [33] and coaxial waveguides [5],
and coupling them to create still higher-order systems such as coupled waveguide networks [135] and hole arrays [55, 40]. However, unlike photonics, plasmonics has the
advantage of being able to confine light to sub-wavelength dimensions and enhancing
the field as its starting point.
Although plasmonics and photonic crystals both existed in some form before the
1980s, both fields were kicked off at around the same time ∼ 30 years ago (at the
time of this thesis, 2013), and within that time the number of publications per year
of plasmonics have exceeded that of photonic crystals [1]. Part of this increase is
due to technological advances that have occurred within that time frame in nanofabrication, -characterization, and -imaging, as well as advances in computation power
and techniques [namely, the finite-difference time-domain (FDTD) method and the
finite-element method (FEM)] – allowing us to probe deeper into the field of plasmonics. However, another factor is, of course, the many advantages brought about

by integrating metals into optics. For example, the strong field enhancements in
plasmonics allow us to study applications in nano-lasers [9], sensing [16], non-linear
behavior [98, 20, 24], ultra-fast electron processes [77], and photovoltaics [4], to name
a few. Other applications include using the Ohmic loss induced by plasmonics for
useful heating applications, as in heat assisted magnetic recording (HAMR) [100].
And still, there are applications that are just beginning to arise, such as the use of
acoustics [106], magnetic field [26], and superconductivity [41] to modulate the plasmon response. At the time of this thesis, quantum- and graphene-plasmonics were
at the forefront of the field, making us reflect how far the field has come in only ∼30
years, and wonder where it will be in the next 10.

1.2

Scope of this Thesis

This thesis is divided into three parts, describing the theory and application of plasmonics as it pertains to I) coaxial plasmonic waveguides and metamaterials, II) hole
array color filters and CMOS image sensing, and III) resonant guided wave networks
and hybrid plasmo-photonic circuitry.

1.2.1

Coaxial Plasmonic Waveguides and Metamaterials

As mentioned in §1.1, naturally occurring materials are characterized either by decaying negative refractive indices (metals) or by propagating positive indices (dielectrics) – plasmonics allows us to bridge the gap between these two regimes through
the use of propagating negative index modes, which brings us into another recently
developed field in optics called metamaterials [122] (∼14 years old at the time of this
thesis, 2013). Within this field, artificial photonic materials are designed to have almost arbitrarily selected effective refractive indices via the design of their constituent
meta-atoms. Metamaterials started off at microwave frequencies in the late 1990s
where mm-size split-ring resonators were printed on circuit boards to generate the
desired diamagnetic response that gives rise to negative index. Since then, aided by
advances in nano-fabrication, researchers incrementally miniaturized the size of the

resonant meta-atoms to bring their operation into the telecommunication and visible
part of the electromagnetic spectrum, where they could be used for imaging [46] and
communication applications [91]. However, due to the small scattering cross-section
of these constituent meta-atoms as they are miniaturized, the operation of resonantbased metamaterials saturates at near-infrared frequencies, making it difficult to bring
down their operation into the visible [75].
Plasmonics, via the metal/dielectric/metal (MDM) waveguide configuration, is
able to access H-field anti-symmetric negative index modes near the surface plasmon
resonance of the metal/dielectric material combination [33]. As discussed in §1.1,
this resonance can be tuned into the visible by setting the dielectric permittivity
to some high value. As a result, we can bring the negative index waveguide mode
out of the ultraviolet into the visible – working from the opposite direction than
conventional metamaterials. However, due to the transverse-magnetic nature of SPs,
the planar MDM configuration is polarization and angle-of-incidence sensitive [34],
but this limitation can be circumvented by wrapping the planar MDM geometry to
form coaxial MDM waveguides, which support similar types of modes [30].
Because the negative index mode exists near the SP resonance, the mode is characterized by high loss, i.e.,unlike the H-field symmetric positive index mode, its modal
volume is heavily delocalized into the cladding metal layer [29]. As such, in making
an array of negative index coaxial waveguides, they can easily couple to each other
through the connecting metal layer, forming a type of metallic photonic crystal in the
in-plane direction [131]. What we find is that the in-plane Bloch mode created by the
coupled propagating negative index axial mode can be tuned to have the same negative index behavior as that of an isolated negative index waveguide mode. Through
this method, we are able to design a negative index plasmonic metamaterial that
operates at visible frequencies, all with a single thin layer of coupled negative index
coaxial waveguides [17].
The basis for coaxial MIM waveguides is discussed in Chapter 3, consisting of
transmission measurements done in combination with an analytic and numeric study
to experimentally determine the dispersion relation of their positive (H-field symmet-

ric) index modes. This study is followed by a theoretical investigation in Chapter 4, in
which we study the nature of the negative index modes in MDM coaxial waveguides,
along with a study of the effect of geometry and materials on the configuration. Having studied single waveguide geometries, we move to coupled waveguide geometries
in Chapter 5, in which we present the negative index metamaterial design consisting
of coupled negative index waveguides.

1.2.2

Hole Array Color Filters and Plasmonic CMOS Image
Sensing

Outside of metamaterials, plasmonics also has tremendous potential in imaging applications. In contemporary Si-based image sensor technologies such as charge-couple
devices (CCDs) and complementary metal-oxide semiconductor (CMOS) image sensors, color sensitivity is added to photo detective pixels by equipping them with
on-chip color filters, composed of organic dye-based absorption filters [88]. However,
organic dye filters are not durable at high temperatures or under long exposure to
ultraviolet (UV) radiation [61] and cannot be made much thinner than a few hundred
nanometers due to the low absorption coefficient of the dye material. Furthermore,
on-chip color filter implementation using organic dye filters requires carefully aligned
lithography steps for each type of color filter over the entire photodiode array, thus
making their fabrication costly and highly impractical for multi-color and hyperspectral imaging devices composed of more than the three primary or complementary
colors.
It is well known that plasmonic hole arrays in thin metal films can be engineered as
optical band-pass filters, owing to the interference of surface plasmons (SPs) between
adjacent holes. Unlike current on-chip organic color filters, plasmonic filters have the
advantage of being highly tunable across the visible spectrum and require only a single
perforated metal layer to fabricate many colors. Plasmonic hole array color filters
have recently been integrated with a CMOS image sensor [21, 23], demonstrating
filter viability in the visible, but full color imaging using the plasmonic color filter

technology platform still remains to be reported. Furthermore, hole array research
has mostly been focused on the transmission properties of large size filters, with little
emphasis given to other important filter performance aspects necessary for state-ofthe art image sensor applications, such as the filter transmission dependence on array
size, spatial color-crosstalk, and robustness against defects.
In Chapter 6, we report on such optical properties as they pertain to various configurations of hexagonal arrays of subwavelength holes fabricated in 150-nm-thick Al
films suitable for image sensor integration. In Chapter 7, we investigate the transmission and imaging characteristics of a 360 × 320 pixel plasmonic color filter array
composed on 5.6×5.6 µm2 size RGB color filters integrated onto a commercial black
and white 1/2.8 inch CMOS image sensor, the first demonstration of high resolution full color plasmonic imaging. And, finally, in Chapter 8 we demonstrate that
the transmission spectra of hole array filters can be accurately described by the 2nd
nearest neighbor scattering-absorption interactions of hole pairs – thus making hole
arrays appealing for close packed hole array filters for imaging applications.

1.2.3

Resonant Guided Wave Networks and Hybrid PlasmoPhotonic Circuitry

As mentioned in §1.1, one of the most compelling aspects of plasmonics is the ability to confine electromagnetic radiation in subwavelength modes at metal/dielectric
interfaces – a promising characteristic for miniaturizing photonic communications
technology at the scale and density of electronics. However, in order to simultaneously achieve low waveguide propagation loss and high mode confinement, we require
chip-based hybrid photonic/plasmonic circuits that feature (1) low-loss silicon photonic waveguides, (2) high-confinement plasmonic waveguide building blocks, and (3)
methods for efficient mode coupling between them. The channel plasmon polariton (CPP) configuration supports and confines slot plasmon polaritons in a highly
confined channel (similar to the H-field symmetric MDM plasmonic mode) [13].
Incidentally, these same CPP channels can also serve as plasmonic building blocks

for another type of artificial material design called resonant guided wave networks
(RGWNs) [48] consisting of isolated plasmonic waveguides in a network configuration. RGWNs, through the use of waveguides to accumulate phase and waveguide
crossings to split power, can serve as artificial materials for engineering dispersion
through the interaction of closed-loop wave resonances that arise throughout the
waveguide network as a result of the isolated waveguides that couple only through
their intersections.
In Chapter 9 we give a theoretical premise for RGWNs, describing their building blocks and demonstrating how they can be engineered to form high-Q plasmonic
resonators. We then move on to describe a more comprehensive example of how the
RGWN resonances can be engineered in anisotropic layout to function as di- and
tri-chroic routers operating at telecommunication wavelengths. In Chapter 10 we
provide an experimental realization of a plasmonic RGWN coupled to Si photonics, demonstrating that light can be efficiently coupled from silicon-on-insulator ridge
waveguides to channel plasmon polariton waveguides. By proper control of mode
polarization in the silicon-on-insulator waveguide, we demonstrate that parasitic excitation of surface plasmon polaritons (SPPs) on the metal surface is suppressed,
only exciting the desired CPP mode of the RGWN structure. We substantiate these
findings with near-field scanning optical measurements (NSOM) to demonstrate efficient coupling into the channel plasmon polariton mode from Si ridge waveguides
at λ0 = 1520 nm with light incoupled via grating couplers. Using NSOM, we also
demonstrate an ultracompact 4-way equal power splitter, the fundamental element
for ultracompact resonators and plasmonic circuits. We highlight their functionality
by fabricating and measuring a 2×2 plasmonic resonator operating as logic device,
similar to that described in Chapter 9.

Chapter 2
Introduction to Metal Optics
2.1

Optical Properties of Metals

As mentioned in §1.1, the optical response of metals is characterized by their complex
permittivitity, which, in general, depends on frequency, due to the different types of
processes that occur within the material at different parts of the electromagnetic spectrum. For example, at near-infrared frequencies, a metal’s permittivity is determined
by its free electron response to a driving electric field (an intraband process), while at
visible frequencies, there is an additional contribution that comes from the interband
promotion of electrons to higher energy bands. The free electron dynamics can be
accurately described by the Drude model, whereas the interband transitions can be
described by the Lorentz (damped oscillator) model.
In the Drude model, electrons are considered as charged (but non-interacting)
particles, each with effective mass me and charge q, which can be accelerated in the
presence of a driving electric field, and decelerated as they scatter off of the metal’s
atomic lattice. Classically considering the free electrons to be at rest in the absence
of an external force, they undergo a displacement r from their rest positions, creating
a density of dipole moments p = qr within the material. These dipole moments
manifest themselves as a macroscopic polarization per unit volume of the material
P = np, where n is the number of electrons per unit volume.
As mentioned in section §1.1, the polarization P of a material relates to how
the charges respond to an incident driving electric field. These two quantities are

proportional to each other, connected via the electric susceptibility of the material, χe ,
whose value depends on its electronic response to the incident electric field (Eq. 2.1).
P = 0 χe E

(2.1)

Now, in the absence of the material, light would simply propagate according to the
permittivity of freespace, 0 E, so in the precense of a material, we are adding a
term to this expression (see Eq. 2.2). We designate this quantity the name electric
displacement vector, D.
D = 0 E + P
= 0 E + 0 χe E
= 0 (1 + χe )E
= E

(2.2)

From Eqs. 2.1 and 2.2 and we get that the permittivity of a material is defined
according to Eq. 2.3.
= 0 (1 + χe )E
= 0 1 +

(2.3)

Since the polarization P of a material is related to the dipole moments p within
the material, all that remains to be known is the average displacement vector of its
electrons, r. We do this by applying the equations of motion to the free electron gas
as they are accelerated by the driving field and are decelerated by collisions with the
atomic lattice.

2.2

The Drude Model

Starting with electrons at rest, an applied electric field E with an eiωt temporal
dependence will have the effect of applying a accelerating force of FE = qEeiωt to
the electrons. The electrons are accelerated until they encounter a scattering lattice
site, causing them to decelerate. This damping term is similar to that of a drag force
in that it opposes the initial direction of the electron’s momentum, with a magnitude
that is inversely proportional to the average collision time, Γ = 1/τ . Putting it all
together, we get Eq. 2.4.
me

∂ 2r
∂r
− me Γ
= qEeiωt
∂ t
∂t

(2.4)

Assuming an eiωt response of the displacement vector, which is sensible since the
electrons are being driven at this frequency, we get that the material polarization is
given by Eq. 2.5.
P = qr
−q 2 /m
ω 2 + iΓω

(2.5)

Combining this result with Eq. 2.3, we arrive at the expression for the frequency
dependent permittivity function of a metal (Eq. 2.6), where ωp = q 2 n/me 0 is the
characteristic bulk plasma frequency of the metal.
r (ω) = (ω)/0
ωp2
= 1− 2
ω + iΓω

(2.6)

The plasma ωp and damping Γ frequency values can be obtained by fitting Eq. 2.6 to
empirical permittivity measurements, obtained, for example, by ellipsometry. Values
obtained by fitting to Palik data are shown in Table 2.1. Using the conversion 1 eV
= 1240 nm, we see that the plasma frequency for these metals is in the ultraviolet, at
around λ0 = 200 nm. The permittivity data along with the fitted curves are shown
in Fig. 2.1. A couple of things stand out from Fig. 2.1 and Eq. 2.6 that are worth

11
Metal

ωp (eV)

Γ (meV)

ωib (eV)

Silver (Ag)
Gold (Au)
Copper (Cu)
Aluminum (Al)

7.65
7.96
12.94
6.78

75.9
90.3
177.8
127.0

4.13
2.48
1.55
2.25

Table 2.1: Plasma ωp , damping Γ, and interband ωib frequency values of some metals
used in plasmonics. The plasma and damping values are obtained by fitting to the
Drude model, and the interband frequency values are obtained by fitting to a DrudeLorentz model.

Permittivity

Angular Frequency (Hz)
ωib 4.7
1.9x1015
9.4
3.1
2.4
10
imaginary
−10
−20
real

−30
−40
−50
200

Ag (Palik)
400
600
800
Freespace Wavelength (nm)

1000

Figure 2.1: Real and Imaginary parts of the permittivity data for Ag at visible
frequencies. Solid line plots are obtained by fitting the Drude model to tabulated
Palik data.

mentioning. In the high frequency regime, ω >> Γ, the Drude permittivity can be
further approximated as the purely real quantity shown in Eq. 2.7.
ωp2
r (ω) = 1 − 2

(2.7)

In looking at Eq. 2.7 together with Fig. 2.1, we see that the real part of the permittivity crosses from being negative (ω < ωp ) to positive (ω > ωp ) at the plasma frequency
ωp . Thus, given a metallic permittivity function, one only needs to see where the
real part of the permittivity crosses zero to approximate the plasma frequency of the
material. Similarly, a negative permittivity corresponds to exponentially decaying

12
solutions, i.e., imaginary wave vectors, whereas a positive permittivity corresponds
to real valued propagating wave vectors. Thus, in this high frequency limit, we see
that light will propagate inside the metal for frequency values larger than ωp and
decay exponentially for frequency values smaller than ωp . Physically, this has to do
with the ability of the bulk electron plasma to oscillate at the driving frequency of
light for low values of frequency (ω < ωp ), but unable to keep up at higher frequencies
(ω > ωp ).
Furthermore, we see from Fig. 2.1 that the Drude model is good at predicting the
premittivity of metals at near infrared wavelengths, but fails to capture the interband
transitions, which happen at different parts of the visible spectrum for different metals
(see Table 2.1). These can be modeled classically as damped harmonic oscillators via
a Drude-Lorentz model, but will not be discussed further here for brevity.

2.3

Maxwell’s Equations

In order to understand how light will propagate in the presence of a metal, we need to
examine how light propagates in a material with a general complex permittivity ˜(ω).
As with every electromagnetic problem, we begin by writing down the set of four
electromagnetic equations, which are individually named after the famous physicists
that empirically discovered their form, but collectively take the name of Maxwell,
since it was he who unified them via the addition of the displacement current term
D. Maxwell’s equations are summarized in Eq. 2.8.
∂B
∂t
∂D
∇×H=J+
∂t
∇×E=−

(2.8a)
(2.8b)

∇·D=ρ

(2.8c)

∇·B=0

(2.8d)

13
We can connect these equations by taking the curl of Eq. 2.8a and relating it to
Eq. 2.8b via the vector identity ∇ (∇ × A) = ∇ (∇ · A) − ∇2 A and Eq. 2.8c. Note
how this connection would not be possible without Maxwell’s displacement current
term. Then, using the relations D = E, B = µH, and J = σE, we arrive at the
general form of the wave equation (Eq. 2.9).
∇2 E = µσ

∂E
∂ 2E
− ∇ (∇ · E) + µ 2
∂t
∂t

(2.9)

In the absence of sources, σ = 0 and ∇ · E = 0, and assuming a harmonic response of the form ei(k·r−ωt) , we arrive at the expression known as the dispersion of
light in freespace, k2 = µω 2 . This expression gives us the propagation wave vector
|k| = 2π/λ of light, given its frequency and the material in which it propagates,
characterized by (, µ). Since c = 1/ µ, we get that the propagation wave vector
is proportional to the frequency of the light and inversely proportional to its phase
velocity.

2.4

Surface Plasmons

In order to understand how electron oscillations couple to light at a metal/dielectric
interface, we need to think about what solutions are admissible by the problem.
Assuming a system with permittivity 1 for y > 0 and permittivity 2 for y < 0 (see
Fig. 2.2), we look for solutions that are bound at and propagating along the material
interface. More specifically, we want oscillatory solutions along the z-direction (could
have equally chosen the x-axis) and exponentially decaying solutions along the ydirection. Mathematically, this means we are looking for solutions of the form shown
in Eq. 2.10, where the number subscripts refer to which permittivity medium (1 or
2) the equation applies. Note that I have conveniently selected the TM form of the
solution in anticipation of surface plasmons.

Figure 2.2: Surface plasmon geometry composed of permittivity 1 for y > 0 and
permittivity 2 for y < 0. The corresponding propagating wave vector and associated
field vectors are displayed by black arrows inside the material, along with a schematic
of the resulting field intensity, shown in red. The ’+’ signs represent a charge depletion
area, which is periodic along the propagation wave vector direction of the surface
plasmon.

E(1,2) = Ey(1,2)  ei(kz z−ωt) eiky(1,2) y
Ez(1,2)

(2.10)

But in order for Eq. 2.10 to be a solution we must (1) impose restrictions on the decomposed wave vectors based on the total wave vector as it applies to each material,
(2) make sure that it satisfies the wave equation (Eq. 2.9), and (3) apply the appropriate boundary conditions at the material interface so that our solution is consistent
with Maxwell’s equations.
(1) To restrict the wave vectors, we simply note that in the absence of an interface,
light would simply propagate with wave vector k(1,2)
= (1,2) k02 in each medium. Thus,

in decomposing the field, as done in Eq. 2.10, we need to make sure that in each
medium, the magnitude of the decomposed wave vectors equals magnitude of the

15
medium wave vector (Eq. 2.11).
(1,2) k02 = ky(1,2)
+ kz2

(2.11)

(2) To impose that Eq. 2.10 satisfies the wave equation (Eq. 2.9), we impose the
condition that ∇ · E = 0, since there are no static sources generating electric field.
(Note that in this case σ 6= 0, since we do have a material conductivity that arises
from the oscillating free electrons.) Applying this restriction to Eq. 2.10, we obtain
a relation between the amplitude of the fields in each medium (see Eq. 2.12).
ky(1,2) Ey + kz Ez(1,2) = 0

(2.12)

(3) Lastly, we apply the boundary conditions for an electric field at an interface
with no static sources, E(1)k = E(2)k and 1 E(1)⊥ = 1 E(2)⊥ , which yields the set of
equations shown in Eq. 2.13.
Ez(1) = Ez(2)
1 Ey(1) = 2 Ey(2)

(2.13)

Equations 2.12 and 2.13 form a homogeneous set of equations (Eq. 2.14a), whose
solution is determined by finding the nontrivial solution to det[M] = 0 (Eq. 2.14b),
which yields the relation 1 ky(2) = 2 ky(1) (Eq. 2.14c).

ky(1) kz

 0
M=
 0
1

ky(2)

−2

kz 
−1

(2.14a)

det[M] = 0

(2.14b)

1 ky(2) = 2 ky(1)

(2.14c)

Equation 2.14c allows us to combine the two (up to now independent) dispersion

16
relations of Eq. 2.11 to obtain the combined dispersion relation for the surface wave
supported at the interface between the two materials (Eq. 2.15).

1 2
k0
1 + 2
1,2
ky(1,2) = √
k0
1 + 2
kz =

(2.15a)
(2.15b)

Requiring exponentially decaying solutions away from the interface translates into
having imaginary ky(1,2) wave vectors, which, in looking at Eq. 2.15b, is only possible
if the sum (1 + 2 ) < 0. Similarly, requiring that we get propagating solutions
parallel to the material interface means that we need to have real kz wave vectors.
Combining the form Eq. 2.15a with the restriction imposed by Eq. 2.15b indicates
that real propagating wave vectors will exist when the additional condition (1 2 < 0)
is satisfied. Thus, we get that in order to have solutions with these desired properties,
we require one of the permittivities to be negative, thus requiring one of the materials
to be metallic.
To illustrate some of the properties of these surface waves, some properties of the
dispersion relation (Eq. 2.15) are plotted for the case where 1 = Si and 2 = Ag
(Fig. 2.3). Panel (a) plots the real part of the Ag/Si plasmon propagation wave vector
<[kz ] (red), together with that of bulk Si (green), bulk Ag (blue), and air (black).
Panel (b) plots the propagation length, Lz = 1/2= [kz ], defined as the propagation
distance at which the field intensity reaches a fractional value of 1/e. This quantity
is plotted for the Ag/Si surface plasmon (red), bulk Si (green), and bulk Ag (blue).
Panel (c) plots the surface plasmon penetration depth, Ly(1,2) = 1/2= ky(1,2) , into
both the Si (green) and Ag (blue) – similarly defined as perpendicular distance away
from the interface at which the field intensity reaches a fractional value of 1/e from
its value right at the interface.
Starting at low frequencies (long freespace wavelengths), we see from Fig. 2.3a,
that the surface plasmon propagation wave vector (green) equals that of Si (red), i.e.,
light behaves as if there is no metal present in the system. The only way this could
happen is if the light had a negligible penetration depth into metal. Indeed, we see

200

9.42 x1015

400

4.71

600

ωSP

3.14

800

2.36

1000

1.89

1200

0.05 0.1 0.15 100 101
102 1030
10
-1
L z = 1 / 2 [ kz] (nm) L y = 1/ 2 [ ky] (nm)
[ kz] (nm )

Angular Frequency (Hz)

Freespace Wavelength (nm)

1.57

Figure 2.3: Dispersion relation properties (Eq. 2.15) for the case where 1 = Si and
2 = Ag . Panel (a) plots the real part of the Ag/Si plasmon propagation wave vector
<[kz ] (red), together with that of bulk Si (green), bulk Ag (blue), and air (black).
Panel (b) plots the propagation length, Lz = 1/2= [kz ], defined as the propagation
distance at which the field intensity reaches a fractional value of 1/e. This quantity
is plotted for the Ag/Si surface plasmon (red), bulk Si (green), and bulk
Ag (blue).
Panel (c) plots the surface plasmon penetration depth, Ly(1,2) = 1/2= ky(1,2) , into
both the Si (green) and Ag (blue), similarly defined as perpendicular distance away
from the interface at which the field intensity reaches a fractional value of 1/e from
its value right at the interface.

18
from Fig. 2.3c, that the surface plasmon mode is almost completely expelled from the
metal, having most of its modal overlap with the Si. This has to do with the fact that
the Ag permittivity is large and negative (see Fig. 2.1), as the surface electrons in the
metal are able to keep up with the low driving frequency of the light, thus expelling
its penetration. For these frequencies, we thus get a surface plasmon mode that
resembles a mode just propagating in the Si. Indeed, we see from Fig. 2.3b that for
long wavelengths, we get a surface plasmon propagation length (red) that approaches
that of bulk Si (green). In this regime, the surface plasmon mode is characterized as
having a weak field confinement, which leads to it having little dispersion and long
propagation lengths (limited only by the absorption in Si in this example).
However, as the frequency of light is increased to visible frequencies (going towards
shorter wavelengths), we see from Fig. 2.3c that the penetration depth into the metal
increases, which has to do with the fact that the real part of the Ag permittivity
is becoming less negative (see Fig. 2.1) as the surface electrons in the metal start
becoming less efficient at keeping up with the increasing driving frequency of the
light. As a result, light begins to slow down as it couples with the plasma oscillations
of the metal at its surface. This is reflected by the fact that the surface plasmon
propagation wave vector deviates away from the Si light line towards higher values.
This means that the mode effective wavelength is shortened, giving rise to dispersion.
But, as mentioned in §1.1, this interaction of light with the metal’s free electrons
comes at the cost of Ohmic losses in the metal, as reflected in Fig. 2.3b, where we see
that the surface plasmon propagation length decreases as the penetration depth and
dispersion increase.
This trend continues until we reach the frequency point where the permittivities
satisfy the relation (1 + 2 ) ∼ 0 (see Eq. 2.15b), a frequency known as the surface
plasmon resonance of the system. (I say approximate because in this case both the
Ag and Si permittivities are complex, making it impossible to get both real and
imaginary parts to cancel at the same frequency point. But to good approximation,
this frequency point can be found by satisfying the relation with the real parts of
the permittivities). At this point, the electrons are in resonance with the driving

19
frequency of light, forming a coupled photo-electron system. This point is reflected
in Fig. 2.3a as the highest wave vector point in the dispersion. But again, this comes
at the cost of Ohmic loss in the system as is evident in the low propagation lengths
which arise from the high penetration depth into the metal.
As the frequency of light is increased further toward shorter freespace wavelengths,
larger in frequency than the surface plasmon resonance but lower than the bulk plasmon resonance, we pass a transition regime in which we go from a surface plasmon
mode into a bulk plasmon mode, which is characterized by a forbidden band in the
dispersion described by small wave vectors (smaller than that of Si) and high loss
(Fig. 2.3b). In this regime, as the frequency is increased, the surface electrons are
increasingly inefficient at oscillating at the driving frequency of the light, leading to
us lose the confinement of the SP mode. Meanwhile, the bulk electrons also lose their
ability to oscillate at the frequency of the driving electric field, causing them to be
less efficient at screening its penetration. Finally, as we increase the frequency past
the plasmon resonance of the material, the bulk electrons are completely unable to
keep up with the driving frequency of light, and we get propagation modes inside
of the metal as discussed in §2.2. This is evident from Fig. 2.3a, where we see the
propagation wave vector increase again past the light line.

Part I
Coaxial Plasmonic Waveguides and
Metamaterials

Chapter 3
Plasmon Dispersion in Coaxial
Waveguides from Single-Cavity
Optical Transmission
Measurements
Abstract: This chapter introduces plasmonic coaxial waveguides, describing how
we experimentally determine their dispersion relation via optical transmission measurements performed on isolated coaxial nanoapertures fabricated on a Ag film using
focused ion-beam lithography. The dispersion depends strongly on the dielectric material and layer thickness. Our experimental results agree well with an analytical
model for plasmon dispersion in coaxial waveguides. We observe large phase shifts at
reflection from the end facets of the coaxial cavity, which strongly affect the waveguide resonances and can be tuned by changing the coax geometry, composition, and
surrounding dielectric index, enabling coaxial cavities with ultrasmall mode volumes.

3.1

Introduction

In only a few years’ time the rapidly growing field of plasmonics has generated a
large array of new nanophotonic concepts and applications. The ability of metal
nanostructures to localize light to subwavelength volumes [86, 32, 132] has provided
opportunities for sensing applications[89] and nanoscale optical lithography [65, 60].
Plasmonics also provides a way to finely tailor the dispersion relation of light, giving

22
the ability to shrink the wavelength of light down to only a few tens of nanometers
at optical frequencies,[83] or create materials with negative index of refraction [103].
Possible applications where precise control of the optical dispersion is essential range
from true-nanoscale optical components for integration on semiconductor chips to
lenses for subwavelength imaging [103, 45] and invisibility cloak [105, 2].
Recently, coaxial plasmon waveguides with a circular dielectric channel separating a metallic core and cladding have received a great deal of attention in connection
to observed enhanced transmission from two-dimensional arrays of coaxial nanoapertures at infrared wavelengths [44, 111, 94]. The transmission enhancements have been
ascribed to standing optical waves along the axis of the coaxial apertures [5, 58, 59].
Past studies have been limited to short (<200 nm) coaxial channels, allowing observation only of the lowest order resonances [44, 111, 94]. Furthermore, optical transmission has only been measured for arrays of coaxial waveguides, making it hard to
separate transmission enhancements owing to channel resonances from enhancements
related to collective resonances of the array [44, 111, 94, 130]. To investigate the
optical resonances of coaxial waveguides in detail, measurements on single coaxial
nanostructures are necessary. Furthermore, a systematic study is needed of the phase
shifts upon reflection from the waveguide ends, as they strongly affect the resonances
of a short waveguide.
Here, we report optical transmission measurements of isolated coaxial plasmon
waveguides in Ag with systematically varied lengths in the range 265 – 485 nm. By
variation of the channel length, the dispersion relation for these structures was determined for the first time. The experimental results agree well with an analytical model
for plasmon dispersion in coaxial waveguides. We observe a significant enhancement
of the wave vector of light when coupled to coaxial waveguides from freespace, even
at frequencies well below the surface plasmon resonance. It is found that the phase
shift upon reflection off the waveguide ends can be tuned by changing the waveguide
geometry. We anticipate that the combination of strong optical confinement and relatively low propagation loss make coaxial waveguides very promising as nanoscale
optical components.

Figure 3.1: SEM images of the cross-sectional profile of coaxial plasmon waveguides
with lengths of 485 nm, dielectric channel widths of ∼100 nm (a) and ∼50 nm (b),
and outer radii of ∼175 nm. The insets show top-view SEM images of the waveguides
before cross sectioning. Scale bars are 100 nm.

3.2

Measuring the Dispersion of a Coaxial Plasmonic Waveguide

The plasmon dispersion in coaxial waveguides was determined from transmission measurements of isolated coaxial channels prepared using focused ion beam (FIB) milling
on films of Ag. Ag was deposited by thermal evaporation on quartz substrates. The
Ag thickness was varied between 265 and 485 nm in 20 nm increments, using a shutter.
We fabricated coaxial waveguides by FIB milling 20–100-nm-wide circular channels
through the Ag layer. Figure 3.1 shows scanning electron microscope (SEM) images
of 485-nm-long coaxial channels with an outer radius of ∼175 nm and channel widths
of ∼100 nm (a) and ∼50 nm (b). The main panels in Fig. 3.1 were taken at a 55◦
angle with respect to the sample normal, after a cross section was made using FIB
milling. The images display a small degree of tapering of the channels (∼7◦ taper
angle), which is mainly caused by redeposition of Ag in the waveguide during FIB
milling, an effect that becomes more pronounced deeper in the Ag layer. The insets
in Fig. 3.1 are top-view SEM images of the coaxial channels that show excellent uniformity of the channel radii and width. Coaxial channels were separated by 50 µm to
avoid coupling between waveguides.

Figure 3.2: Transmission measurement (red dotted spectrum, I multiplied by 100) of a
485-nm-long coaxial waveguide with a ∼100-nm-wide dielectric channel (see Fig. 3.1b)
and a reference spectrum (blue dashed line, I0 ). The transmittance defined as the
waveguide transmission spectrum divided by the reference spectrum is depicted by
the green curve (green drawn line, T ).

To investigate the influence of channel length and width on the optical response
of coaxial waveguides, we performed optical transmission measurements. Radiation
from a supercontinuum white light source was focused onto individual coax structures using a 0.95 NA objective. The transmitted light was collected by a 0.7 NA
objective and directed into a spectrometer equipped with a CCD detector to measure optical spectra. For reference, we measured transmission spectra of 10 × 10 µm
open squares, in which the Ag layer had been completely removed by FIB milling.
Figure 3.2 shows transmission spectra of a 100-nm-wide, 485-nm-long coax channel
(I) and of a reference area close to the waveguide (I0 ). The transmittance spectrum
(T = I/I0 ) is obtained by normalizing the waveguide transmission to the reference
spectrum and is also shown in the figure. Three distinct maxima can be resolved in
the transmittance spectrum, at ∼450, ∼600, and >820 nm, which we attribute to
Fabry-Perot cavity resonances of the coaxial waveguide. To corroborate this hypothesis, we have measured the transmittance of a series of coaxial waveguides of equal
channel width but varying channel lengths.

Figure 3.3: Transmittance spectra of coaxial waveguides with varying lengths. The
outer radius and channel width were ∼175 and ∼100 nm, respectively, while the
waveguide length was decreased from 485 nm (top curve) to 265 nm (bottom curve)
in increments of 20 nm. Data are shifted vertically for clarity. The black dashed lines
are guides for the eye and connect the resonance peaks m = 1 − 3).

26
Figure 3.3 shows transmittance spectra of 100-nm-wide coaxial channels with
length decreasing from 485 nm (same data as in Fig. 3.2) to 265 nm in 20-nm increments. Several features are observed. First, the resonance at 600 nm for the longest
waveguide gradually blue shifts to 450 nm for the shortest waveguide. Also, the
resonance at 450 nm, observed for the largest channel length, blue shifts to a wavelength below 400 nm for waveguides shorter than 400 nm. The broad peak in the
long-wavelength range of the spectra also blue shifts as the length of the cavity is decreased, showing a main peak at a wavelength of 650 nm for the shortest waveguides.
Furthermore, the lowest-order resonances observed in the three shortest waveguides
appear to be much broader than other resonances. This is possibly the result of a
longer-wavelength resonance that arises when all propagating modes in the resonator
are in cutoff [130].
Fabry-Perot resonances result from interference between forward and backward
propagating plasmon waves in the cavity. On resonance, the condition shown in
Eq. 3.1 must be fulfilled, with L the length of the waveguide, kspp (ω) the wave vector
of the plasmon at frequency ω, ∆φ1,2 the phase shifts as a result of plasmon reflection
at either end of the waveguide, and m the mode number.

2Lkspp (ω) + ∆φ1 + ∆φ2 = 2πm

(3.1)

Before we can determine the plasmon dispersion in coaxial waveguides, it is necessary to assign mode numbers to the measured Fabry-Perot resonances. With this
in mind, we performed exact calculations of kspp (ω) by solving Maxwell’s equations
for a cylindrical structure of infinite length [5, 58, 93, 79, 128]. The azimuthal dependence of the electric and magnetic fields in the waveguide is described by the
harmonic function einψ of order n. Note that we expect to only excite modes of odd
azimuthal order in the experiment as the incident electric field has even parity about
the center of the waveguide aperture. The radial dependence of the fields in all three
domains (Ag-dielectric-Ag) is described by solutions to the second-order Bessel differential equation. We apply a Bessel function of the first kind, Jn , to the Ag core

27
(1)

and a Hankel function of the first kind, Hn , to the Ag cladding. Inside the dielectric
channel the radial field dependence is described by a linear combination of Bessel and
Hankel functions of the first kind. On each domain boundary we formulate four continuity conditions for the tangential components of the electric and magnetic fields.
The optical eigenmodes of the coaxial waveguide are found when the determinant of
the resulting homogeneous system of eight equations with eight unknown coefficients
vanishes, similar to that described in §2.4.

3.3

Comparing Experiment with Theory

In this way dispersion relations were determined for waveguides of any chosen channel
width. To account for the tapered profile of the resonators in the experiment, we
calculated the effective dispersion relation by index-averaging kspp (ω) over a series of
dispersion curves corresponding to the varying lateral dimensions of the waveguide
determined from SEM images (Fig. 3.1). By inserting this index-averaged dispersion
into Eq. 3.1, we obtain the resonance frequency as function of cavity length and
mode number m. We compared these results to experimental values of the resonance
frequency, obtained by fitting the transmittance spectra with Lorentzian line shapes,
as a function of waveguide length. This comparison makes it possible to assign mode
numbers to the measured resonances and construct the plasmon dispersion relation
for the coaxial waveguides. In the following first analysis, we assume that the plasmon
phase shift upon reflection at the cavity ends, which will be discussed further on, is
zero.
Figure 3.4a shows the dispersion data for coaxial channels with an average outer
diameter of ∼175 nm and a ∼50-nm-wide air channel (see Fig. 3.1b) along with
the calculated dispersion curve for azimuthal order n = 1, taking into account the
tapering in the structures as determined from SEM data (Fig. 3.1). Good agreement
between experiment and calculations is observed. Different symbols indicate measured
resonances characterized by mode numbers, m = 1 and 2. Dispersion in air and at
a Ag/air interface are plotted for reference. The figure shows up to ∼ 60% larger

Figure 3.4: Measured dispersion data and calculated index-averaged dispersion relations (red drawn lines) for coaxial plasmon waveguides with (a) ∼50-nm-wide air
channel and (b) ∼50-nm-wide spin-on-glass (SOG) filled channel. Light lines for air
(a) and SOG (b) are also shown (dotted green lines), along with the plasmon dispersion (dashed orange curves) at a flat Ag/air interface (a) and Ag/SOG interface (b).
Symbols in the figure correspond to different Fabry-Perot mode numbers, m, where
= 1, ∗ = 2, 4 = 3, and ◦ = 4. The inset in (a) shows the calculated electric field
distribution of the mode of azimuthal order n = 1, in an SOG-filled coaxial waveguide
with outer radius of 200 nm and channel width of 100 nm at ω = 3.5 × 1015 rad/s.

29
wave vectors in coaxial waveguides compared to free space. At lower frequencies
ω < 2.5×1015 rad/s, as the mode approaches cutoff, the calculated dispersion relation
flattens slightly and crosses the air light line, in agreement with experimental data
in that frequency range. This behavior is not observed in studies of cylindrical metal
waveguides when the excited plasmon mode has azimuthal symmetry (n = 0),[35, 113]
as that mode does not experience cutoff.
To further increase the dispersion, we infilled the coaxial nanostructures by spincoating the sample with a ∼200-nm-thick layer of spin-on-glass (SOG, n = 1.46).
SEM of FIB-milled cross sections confirmed that SOG infilled the structures entirely.
Figure 3.4b shows the dispersion data for infilled coaxial waveguides of the same
dimensions as in (a). In this case, resonances with mode numbers m = 2 − 4 were
observed. We further note a shift of the dispersion data to higher wavenumbers
compared to the air case of Fig. 3.4a, as well as a clear increase in the curvature of
the dispersion relation. In this case we observe up to ∼2.2 times larger wave vectors
in coaxial waveguides compared to free space. Figure 3.4b also shows the calculated
plasmon dispersion (red drawn curve).

3.4

The Phase Shift at Reflection

Thus far we have demonstrated cases where the calculated dispersion relations match
the experimental data quite well. Figure 3.5a compares the dispersion data (assuming
zero phase change on reflection) and calculated dispersion relation (red drawn curve)
for air-filled waveguides with a dielectric channel width of ∼100 nm. Although the
data follow the same general trend as the calculated curve, the two show a clear offset
in wavenumber with respect to each other. In the final part of this chapter we will
show that the observed discrepancy between the data and theory results from a net
phase shift (∆φ1 + ∆φ2 ) that the plasmons gain when they reflect off the cavity ends,
which can be tuned by changing the cavity geometry.
As Eq. 3.1 shows, a nonzero net phase shift causes the resonance wavelengths
to shift. To study the phase shift, we have performed finite difference time domain

Figure 3.5: Measured dispersion data and calculated index-averaged dispersion relations (red drawn lines) for coaxial plasmon waveguides with a 100-nm-wide air
channel. In (a) the dispersion data are plotted, assuming a zero net phase shift as a
result of reflections at the end facets of the cavity. In (b) the data are plotted for an
overall reflection phase shift of 0.77π, which was determined using simulations. Light
lines for air are also shown (dotted green lines), along with the plasmon dispersion
(dashed orange curves) at a flat Ag/air interface. Symbols in the figure correspond
to different Fabry-Perot mode numbers, m, where = 1, ∗ = 2, 4 = 3.

31
(FDTD) simulations to obtain the field profile in structures similar to those used in
the experiment. In the simulations we used a broad band optical pulse with Gaussian
beam profile to excite the structures. By applying a discrete Fourier transform of
the time-dependent fields, we obtain the spatial field intensity profiles at any given
optical frequency. In Fig. 3.1 we show the simulation results for an air-filled coaxial
waveguide of the same dimensions as the structure shown in Fig. 3.1a, excited at
an angular frequency of 4.2 × 1015 rad/s (freespace wavelength of 450 nm). Figure
3.6a shows the steady-state intensity distribution in the plane of polarization for the
electric field component that is parallel to the polarization direction of the incident
light. As in the experiment, the waveguide is excited at the air-side (left-side in the
figure).
In the steady state, the field profile in the cavity is a superposition of plasmon
waves propagating in forward and backward direction after any number of reflections
at the input or distal end of the cavity. The analytical expression of the resulting
electric field in the cavity as a function of position in the direction parallel to the
waveguide axis, is given by Eq. 3.2, where k is the plasmon wave vector (which
depends on z as a result of waveguide tapering),hki is the index-averaged wave vector,
and |r1 | and ∆φ1 are the reflectance and reflection phase shift at the distal end of the
cavity, respectively.

E(z) ∝ eikz + |r1 |ei(k(L−z)+hkiL+∆φ1 )

(3.2)

Note that the field profile inside the cavity is not affected by the reflectivity of the
input end of the waveguide. In fact, the field profile is simply proportional to the
original plasmon wave and the plasmon wave after one reflection, added together.
Figure 3.6b plots the intensity distribution in the cavity as a function of position
along the waveguide axis (blue dotted curve), obtained by vertically summing the intensity values in Fig. 3.6a. To obtain the phase shift at the distal end of the waveguide
(right end in Fig. 3.6a), we fit the intensity distribution with |E(z)|2 (Eq. 3.2). As
the plasmon wave vector is calculated analytically, the only fit parameters, besides an

Figure 3.6: Steady-state simulation results of the electric field intensity profile inside a
485-nm-long coaxial cavity with an air-filled ∼100-nm-wide dielectric channel excited
at an angular frequency of ω = 4.2 × 1015 rad/s (freespace wavelength of 450 nm). (a)
Electric field intensity distribution inside the coaxial cavity for light incident from the
left. (b, c) Intensity as a function of position integrated along the lateral direction
(dotted lines) for light incident from the left (b) and right (c). The red drawn curves
are fits of the intensity profile that were used to find the reflectance and reflection
phase shifts at the distal end facets. The reflection phase shift is 1.0π at the substrate
side and −0.12π at the air side of the cavity, while the values for the reflectance are
75% and 20%, respectively.

33
amplitude constant, are ∆φ1 and |r1 |. The result of the fit is plotted in Fig. 3.6b (red
drawn curve). From the fit it follows that the phase shift as a result of the reflection
at the substrate end of the waveguide is close to π. The visibility of the oscillation
depends on the reflectance |r1 | at the distal cavity end. On the basis of the fit we
find that the reflectance of the substrate-side end facet is equal to 75%.
To obtain the reflection phase shift at the input end facet, ∆φ2 , the intensity
distribution in the waveguide was simulated for light impinging on the nanostructure from the substrate side of the cavity. We show the result of this simulation in
Fig. 3.6c. Owing to a lower reflectance of the air-side end facet of 20%, the visibility
of the intensity distribution is smaller. Furthermore, we find that the phase shift
upon reflection at the air-side cavity end is close to zero. Using the reflectance values
as determined from the fits and the calculated waveguide losses, we obtain a cavity
quality factor of only ∼4, which explains the broad spectral width of resonances in
measured spectra. We note, however, that the cavity losses are almost entirely due to
the rather poorly reflecting end facets of the cavity. By improving the end face reflectivity, it should be possible to attain quality factors of more than 80. Furthermore,
on the basis of comparisons with simulations of untapered waveguides, it is important
to note that waveguide tapering does not significantly affect the end facet reflectance
and, as a result, does not add to the resonance line width. However, tapering does
result in a larger dielectric channel width at the input side of the cavity, which, as
shown, may give rise to lower end facet reflectance, resulting in a reduction of the
quality factor of the cavity.

3.5

A More Comprehensive Study

The analysis in Fig. 3.6 was done at a frequency of 4.2 × 1015 rad/s (freespace wavelength of 450 nm). At lower frequencies, we find that the overall phase shift tends to
decrease (data not shown). The average net phase shift we find for frequencies within
the experimental bandwidth equals ∼ 0.77π. In Fig. 3.5b we plot the dispersion
relation taking into account this average phase shift and observe a close agreement

34
between theory and experiment. Note that in our analysis of waveguides with a ∼50nm-wide dielectric channel (Fig. 3.4) or waveguides filled with SOG, best agreement
between experiment and theory was found for phase shifts close to zero, demonstrating that the phase shifts on reflection can be tuned by changing the geometry.
To further investigate the tunability of the reflection coefficients of the coaxial
end facets, we simulate the response of waveguides as we vary the refractive index of
the dielectric channel and surrounding medium. The simulations are performed using
untapered waveguides of 485 nm length and 175 nm outer radius, that are composed
of a 75-nm-wide dielectric channel separating a Ag core and cladding.
Figure 3.7 shows the reflectance |r1 | and reflection phase ∆φ1 obtained by fitting
the longitudinal field intensity profile using Eq. 3.1 for a waveguide with Ag core
and cladding and a dielectric channel at λ0 = 800 nm. In Fig. 3.7a the refractive
index of the dielectric channel is kept fixed at nin = 1.5, while the refractive index
of the surrounding medium is varied from 1.0 to 3.5. The data demonstrate that the
reflection phase depends strongly on the surrounding index and can be tuned to any
value between 0 and π. It is worthwhile to note that the observed trend qualitatively
agrees with the trend given by a calculation using the Fresnel equations (blue dotted
line) using the mode index of the cavity. Quantitatively, however, the trend observed
for coaxes is very different, and may only be obtained analytically if we consider mode
overlap between waveguide modes, surface waves, and freespace modes. Interestingly,
a change in the refractive index of the surrounding medium hardly affects the end
facet reflectance of ∼ 70%.
Figure 3.7b shows the influence of a change in the dielectric channel index on
the reflection coefficients of the coaxial waveguide when the refractive index of the
surrounding medium is fixed at nout = 1.0. As the refractive index of the dielectric
in the coaxial waveguide is increased from 1.0 to 2.5, we find that the end facet
reflectance increases from ∼ 55% to ∼ 85%, while the phase shift on reflection off the
cavity ends remains at a value of ∼ 0.1π. In general, we find that an increase of the
effective mode index of the waveguide either by a change in the refractive index of the
dielectric or by a change in the geometry leads to an improved cavity end reflectance.

Figure 3.7: Reflection phase (∆φ) and reflectance (|r|) of the end facets of an untapered coaxial waveguide with 75-nm-wide dielectric channel and outer radius of
175 nm at a wavelength λ0 = 800 nm, derived from finite difference time domain
simulations. (a) ∆φ and |r| are plotted as a function of the surrounding dielectric
index for waveguides with fixed dielectric channel index, nin ) 1.5. The dotted blue
line gives the result obtained from Fresnel equations using the (single) mode index of
the coaxial cavity. (b) Plot of ∆φ and |r| as a function of the refractive index of the
dielectric channel of the coaxial waveguide, while the surrounding dielectric index is
kept fixed, nout = 1.0.

36
Figure 3.7 demonstrates that the reflection phase and reflectance of the cavity end
facets can be independently tuned. This opens the way to realization of plasmonic
cavities with ultrasmall mode volumes, where the reflection phase can effectively cancel the phase accumulated during propagation in the coaxial waveguide [52], enabling
cavities with a length considerably shorter than λ/2. The quality factor of the cavity
is mainly dependent on the reflectance of the cavity mirrors, which, as we have shown,
can be improved by increasing the effective index inside the coax, for instance, by
reducing the dielectric channel width in the cavity. Counterintuitively, a reduction
in channel width may thus give rise to a smaller mode volume as well as a greater
quality factor.

3.6

Conclusion

In conclusion, we have shown that the plasmon dispersion in coaxial waveguides with
subwavelength dimensions can be determined from single-cavity transmission measurements. Our dispersion data agree well with an analytical model for dispersion in
coaxial waveguides of infinite length and demonstrate the large degree of tunability
by varying the coaxial cavity dimensions and dielectric medium. A plasmon phase
shift up to π occurs upon reflection off the cavity ends and strongly affects the cavity
resonance. The phase shift depends greatly on the waveguide geometry and dielectric medium inside and outside the cavity, providing further tunability of the coaxial
cavity resonances and enabling cavities with ultrasmall mode volumes. The fundamental insights obtained in this chapter are important in further studies of nanoscale
waveguiding, field enhancement, and imaging with coaxial cavities, as well as their
use in negative-index metamaterials as will be discussed in Chapters 4 and 5.

Chapter 4
Negative Refractive Index in
Coaxial Plasmon Waveguides
Abstract: Having seen in Chapter 3 how we can experimentally measure the positive
index mode dispersion of plasmonic coaxial waveguides, in this chapter we focus on a
theoretical study showing that coaxial waveguides also exhibit negative refractive index
modes over a broad spectral range in the visible. For narrow dielectric gaps (10 nm
GaP embedded in Ag) a figure-of-merit of 18 can be achieved at λ0 = 460 nm. For
larger dielectric gaps the negative index spectral range extends well below the surface
plasmon resonance frequency. By fine-tuning the coaxial geometry, the special case of
n = 1 at a figure-of-merit of 5, or n = 0 for a decay length of 500 nm can be achieved.

4.1

Introduction

Controlling the propagation of light at the nanoscale is one of the challenges in photonics. Surface plasmons, electromagnetic modes that propagate at a metal/dielectric
interface, provide a key opportunity to achieve this goal, due to their relatively small
evanescent fields [6, 95]. Moreover, as their dispersion can be strongly controlled by
geometry, their effective wavelength can be shrunk well below the freespace wavelength, enabling further miniaturization of optical components. Initial experiments
on plasmon optics were carried out at planar metal/dielectric interfaces, demonstrating basic control of plasmons. Plasmonic components such as mirrors [141, 126] and
waveguides [8, 112] were realized, however, still of relatively large size due to the

38
> 100-nm evanescent tails, and with limited control over dispersion. Subsequently,
dielectric-metal-dielectric structures were investigated, and have demonstrated confinement of light to < 100 nm length scales in taper geometries [127, 132], though at
high loss. The reverse, metal-dielectric-metal (MDM) geometries, have demonstrated
lower loss, higher dispersion [33, 83], and, recently, the attainment of a negative index
of refraction [70, 119, 34].
A disadvantage of planar MDM structures is that they only confine light in one
transverse direction. Recently, coaxial MDM waveguides, composed of a metal core
surrounded by a dielectric cylinder clad by a metal outer layer have been introduced,
that confine light in all transverse directions [44, 5]. We have recently reported
optical transmission measurements through single coaxial waveguides, from which
the dispersion diagram for these nanoscale waveguides was determined [31].
Inspired by the earlier work on MDM waveguides, a natural question arises:
whether coaxial waveguides would posses a negative refractive index, and, if so,
for what geometry and over what spectral range.

Since the coaxial waveguides

are essentially 3-dimensional objects, the observation of negative index in individual coaxes also inspires the design of 3-dimensional negative-index metamaterials
[103, 122, 36, 114, 129] composed of arrays of coaxial waveguides [17].
Here, we theoretically study the dispersion of coaxial Ag/Si/Ag plasmon waveguides and demonstrate that, for well-chosen geometries, modes with a negative refractive index are observed. These modes are dominant over other waveguide modes for a
wide range of frequencies above the surface plasmon resonance frequency. We discuss
the influence of waveguide geometry and material on the mode index and demonstrate that the figure-of-merit (FOM), defined as the magnitude of the real part of
the propagation constant in the waveguide divided by the imaginary part [34, ?], can
be as high as 18.

Figure 4.1: Coaxial plasmon waveguide geometry and numerical mode solving
method. (a) Schematic cross-section of a coaxial waveguide with the definition of the
cylindrical polar coordinates, r, φ and z. The metallic inner core and outer cladding
separate a dielectric channel. A schematic wave propagating in the waveguide in
the direction of positive z is also indicated. (b) Argument θ of the determinant,
det[M (k)], plotted in the complex k-plane for a Ag/Si/Ag waveguide with 75 nm
inner core diameter and 10-nm-wide dielectric channel at ω = 3 × 1015 rad/s . By
cycling around the closed loop, indicated by the dashed square, the net number of
discontinuities in θ is determined. Zero positions are indicated by the white circles.

4.2

Anlaytic Methods

The azimuthal dependence of the fields is described by the harmonic function einφ of
order n. In the remainder we only consider modes with n = 1, since these are the
lowest order modes that couple to freespace radiation. As seen in §3.2, the radial
dependence of the fields in all three domains (metal-dielectric-metal) is described by
solutions to the 2nd order Bessel differential equation. We apply a Bessel function of
the first kind, Jn , to the metal core, as that function remains finite at the waveguide
(1)

axis. A Hankel function of the first kind, Hn , is applied to the metal cladding. Inside
the dielectric channel the radial field is described by two linearly independent cylinder
functions. The arguments of the cylinder functions in each of the three domains is
κi r, where κi is the radial wave number in medium i, defined via Eq. 4.1 where i is
the complex dielectric constant in domain i. To satisfy the condition that fields decay
to zero at radial infinity, we take the square root of Eq. 4.1, such that the radial wave
number has a positive imaginary part.

40
ω2
κ2i = i 2 − k 2

(4.1)

On each domain boundary we formulate four continuity conditions for the tangential components of the electric and magnetic fields. The optical eigenmodes of
the coaxial waveguide are found when the determinant of the resulting homogeneous
system of eight equations with eight unknown coefficients vanishes (Eq. 4.2), where
M is the matrix of the system of equations.

det[M (k)] = 0

(4.2)

We have used two independent methods for determining the optical modes, k(ω), of
the structure. One involved a numerical procedure developed to detect local minima
of the determinant of the system in the complex k-plane. The other method relies on
the fact that the argument, θ, given by 4.3 is undefined when det[M (k)] = 0.

det[M (k)] = |det[M (k)]| eiθ

(4.3)

This can be visualized in a plot of θ in the complex k-plane. An example is shown in
Fig. 4.1(b), where θ is plotted for a coaxial waveguide with a Ag core and cladding
and a 10 nm silicon spacer layer. We used empirically determined optical constants
for the metal [64] and dielectric [97]. Contour lines in the figure appear to close in on
each other at each of the zeros, which are indicated by the white dots in the figure.
By counting each discontinuity −π → π and π → −π about a closed loop in the figure
(for an example, see the dashed square loop in Fig. 4.1b) we are able to determine the
number of zeros in the enclosed area. In case we find that one or more zeros reside
in the area, we split the area up in smaller pieces and repeat the procedure until the
location of the zero(s) is determined with double computer precision.
Using this method, solutions for k were found for real frequency, so that dispersion
relations, ω(k), could be constructed. Calculations were performed in the optical
angular frequency regime 1 × 1015 rad/s < ω < 5 × 1015 rad/s (free-space wavelength,

Figure 4.2: Dispersion relations of the three lowest-order modes of a coaxial waveguide
with 75-nm-diameter Ag core, 25-nm-wide Si channel, and infinite outer Ag cladding.
Radial frequency is plotted versus propagation constant k 0 (a), attenuation constant
k 00 (b), and figure-of-merit k 0 /k 00 (c). The Ag/Si surface plasmon resonance frequency
ωSP = 3.15 × 1015 rad/s (λ0 = 598 nm) is indicated by the horizontal line. Panel (a)
shows two modes with positive index (blue dashed curve and green dotted curve) and
one mode with a negative index below a frequency of ∼ 3.8 × 1015 rad/s (red drawn
curve). The insets in (a) show the Hy field distribution in the transverse plane of the
waveguide at 2.8 × 1015 rad/s for the positive-index mode (blue dashed dispersion
curve) and at 3.6 × 1015 rad/s for the negative-index mode.

λ0 = 377–1884 nm). We only consider modes with positive energy velocity, ve , or
equivalently, positive attenuation constant k 00 [34]. Therefore, to achieve antiparallel
energy and phase velocity, which is the unique requirement for a negative mode index,
the propagation constant k needs to be negative.

4.3

Coaxial Waveguide Dispersion

Figure 4.2 shows the dispersion relation, ω(k), for the three lowest-order modes in a
coaxial waveguide consisting of a 75-nm-diameter Ag core, surrounded by a 25-nmthick Si layer and infinite Ag cladding. In (a) the angular frequency is plotted against

42
k 0 , while (b) shows the frequency as function of k 00 , which determines the propagation
length of light in the waveguide via Eq. 4.4. The surface plasmon resonance frequency
ωSP = 3.15 × 1015 rad/s (λ0 = 598 nm) is indicated by the horizontal line.

2k 00

(4.4)

Figure 4.2a shows two coaxial modes (blue dashed line and green dotted line)
with positive propagation constants over the entire spectral range. Both dispersion
curves closely resemble the dispersion of a surface plasmon polariton propagating
along a planar Si/Ag interface. However, the corresponding propagation constants
(Fig. 4.2a), are nearly three times as large as for the planar single interface plasmon.
This is due to the fact that confinement of the plasmon in the coaxial waveguide
geometry leads to larger mode overlap with the metal. Figure 4.2a also shows the
existence of a third mode (red drawn curve) that has a negative propagation constant
k for frequencies below 3.8 × 1015 rad/s (λ0 = 496 nm). The effective index n =
ck 0 /ω ranges from 9 < n < 5 in the frequency range of Fig. 4.2. The insets in (a)
show the Hy field in the transverse plane for the negative-index mode (calculated
at ω = 3.6 × 1015 rad/s), as well as for the most dispersive positive mode (blue
dashed curve in Fig. 4.2a, ω = 2.8 × 1015 rad/s). From the images it is clear that
the mode with positive effective index has a symmetric field distribution with respect
to the two centers of the dielectric channel on the x-axis. The Hy field is primarily
concentrated at the boundary between the metal core and dielectric channel. The
negative-index mode, in contrast, has its field primarily concentrated at the outermost
channel boundary and has an Hy field distribution that is anti-symmetric about the
center of the dielectric channel, similar to modes with negative index in planar metaldielectric-metal waveguides [34].
Figure 4.2(b) shows that for frequencies below the surface plasmon resonance
frequency ωSP , the lowest-order positive-index mode (blue dashed line) has lowest loss
and will therefore be dominant over other modes. Interestingly, above SP the negativeindex mode (red curve) becomes the dominant mode, as its losses are significantly

Figure 4.3: Dispersion relations for negative-index coaxial waveguides with Ag core
and cladding and Si dielectric channel, (a): ω(k 0 ); (b): ω(k 00 ). The inner core diameter
is fixed at 75 nm, and the Si-channel thickness w is 10 nm, 30 nm, and 70 nm. Positiveindex modes [as shown in Fig. 4.2(a)] are not shown in the figure. The bold sections of
the dispersion curves indicate the spectral range over which the negative-index mode
is dominant, i.e., has lower loss than the positive index modes. The frequency where
the red and green dispersion curves cross k 0 = 0 is indicated by the star-symbols.

lower than those for the positive-index modes. Figure 4.2(c) shows the figure-of-merit
(FOM), k 0 /k 00 , of the modes. As can be seen, the negative-index mode has a FOM that
approaches 10 for a narrow frequency interval around 3.4 × 1015 rad/s (λ0 = 554 nm).
The data in Fig. 4.2 clearly demonstrate that dominant modes of negative index
indeed exist in coaxial plasmon waveguides.

4.4

Conditions for Achieving a Negative Mode Index

Next, we investigate the conditions that are required to achieve a negative index by
varying the geometry and materials of the waveguide. Figure 4.3 shows the effect of
changing the dielectric layer thickness on the dispersion of the negative index mode.
Calculations were performed for a Ag/Si/Ag coaxial waveguide with a core diameter
of 75 nm for a dielectric layer thickness of 10 nm, 30 nm, and 70 nm. Figure 4.3a

Figure 4.4: Dispersion relations for coaxial waveguides with Ag core and cladding and
70-nm-wide Si dielectric channel, (a): ω(k 0 ); (b): ω(k 00 ). The inner core diameter,
dcore , is 45 nm (blue curves), 75 nm (green curves) and 100 nm (red curves). Only
modes with negative index are plotted. Bold lines indicate the spectral range where
the mode is dominant over the positive-index mode.

shows that the variation in dielectric layer thickness has a very dramatic effect on
the dispersion of the negative-index mode. First of all, the largest negative index is
observed for the thinnest dielectric. Second, while for the 10-nm and 30-nm dielectric
gaps the frequency of the resonance associated with the negative index mode appears
close to the surface plasmon resonance at 3.2 × 1015 rad/s, for the 70-nm gap this
resonance is significantly red-shifted to 2.5 × 1015 rad/s. The spectral range over
which the mode is dominant, indicated by the bold curves in Fig. 4.3a, also extends
to lower frequencies when increasing the channel width. For the 70-nm gaps a narrow
frequency range is found near 2.4 × 1015 rad/s (λ0 = 785 nm), where the index is
negative and the figure-of-merit is 5.
As the dispersion branches cross the k 00 = 0 line, the effective refractive index
of the mode vanishes [3]. Coaxial waveguides with a narrow dielectric gap suffer
high loss at this frequency. The green star in Fig. 4.3(b) indicates the frequency at
which the dispersion curve crosses the k 0 = 0 line for the 30-nm gap; a high value of
k 00 = 107 m−1 is found. In contrast, for the 70-nm-wide dielectric channel waveguides,
the losses at the k 0 = 0 crossing (red star in Fig. 4.3b) are much lower (k 00 < 106 m−1 ),

45
corresponding to a decay length of 500 nm. Note that in the spectral range where
the phase velocity ω/k 0 goes to zero, the group velocity, vg = dω/dk, is much larger
than zero.
The two striking effects observed here are: a) waveguide resonances that shift
with geometry, and b) increased propagation length for k 0 = 0 modes for increasing dielectric thickness are in strong contrast to what is observed in planar metaldielectric-metal waveguides [34]. This suggests that coupling of plasmon fields across
the nanoscale diameter of the metal core strongly influences the dispersion of the
negative index modes. To investigate this, we studied the influence of the metal core
diameter on the dispersion of the mode, while keeping the channel width fixed to
70 nm. Figure 4.4 shows the results for waveguides with inner core diameters of 45,
75 and 100 nm. The figure shows that the spectral range where the mode is both
dominant and characterized by a negative index becomes smaller, going from a 45nm-diameter core to a 75 nm core, and vanishes when the core size is increased to
100 nm. This behavior coincides with a red-shift of the resonance in k 00 when the core
diameter is increased (Fig. 4.4b). We attribute the resonance red-shift for increasing
core diameter to a depolarization effect similar to what is known for bulk metallic particles, which show a plasmon resonance red-shift for increasing diameter [81]. Based
on this insight, we predict that a large degree of control over dispersion and resonance
red-shift may also be attained in planar structures composed of a multi-layered stack
of metal and dielectric. In fact, negative index materials based on metal-dielectric
multi-layers have been reported in literature [63]. A final observation that can be
made in Fig. 4.4 regards the special case of n = −1; for a coaxial waveguide with a
core diameter of 50 nm and a 70-nm-wide Si channel, a mode with n = −1 is observed
with a FOM=5 at ω = 2.61 × 1015 rad/s (λ0 = 720 nm).

4.5

Effect of Materials and Geometry

Thus far we have studied the influence of the coax geometry on the frequency dispersion of k. Next, we will investigate the effect of changing the type of metal in core

Figure 4.5: Dispersion relations for coaxial waveguides with 75-nm-diameter metal
core, 25-nm-wide dielectric channel and infinite metal cladding, (a): ω(k 0 ); (b): FOM
k 0 /k 00 . The type of metal in the core and cladding, as well as the dielectric material,
was varied. The frequency axes are normalized to the corresponding surface plasmon
resonance frequency ωSP . Blue curves are for a Si channel surrounded by Ag (ωSP =
3.15 × 1015 rad/s), green curves for GaP in Ag (ωSP = 3.49 × 1015 rad/s), red curves
for SiO2 in Ag (ωSP = 5.24 × 1015 rad/s) and purple curves for Si in Au (ωSP =
2.77 × 1015 rad/s). Bold curves indicate the spectral range where the negative-index
mode is dominant.

47
and cladding, as well as the dielectric in the cylindrical channel. Figure 4.5 shows
the frequency dispersion for a coaxial waveguide with inner metal core diameter of 75
nm and dielectric channel width of 25 nm surrounded by an infinite metal cladding.
We compare the results for Si (blue curves), GaP (green curves), and silica (red
curves) channels in Ag, and for a Si channel in Au (purple curves). The curves are
normalized to the surface plasmon resonance frequency for the corresponding planar
metal/dielectric geometry. Interestingly, coaxial plasmon waveguides support dominant negative index modes regardless of the investigated choice of materials. The
propagation vector k 0 is most strongly negative for waveguides filled with GaP and
least negative for silica, which indicates that, to obtain a strong effect, the dielectric
constant needs to be high (nGaP = 3.5 at the GaP/Ag surface plasmon resonance).
The figure-of-merit for the modes in Fig. 4.5a is plotted in Fig. 4.5b. Due to the low
loss and high index of GaP, the highest figure-of-merit is found in these waveguides.
The highest number we found (FOM = 18) was for a 10-nm-wide GaP channel in Ag
at ω = 4.1 × 1015 rad/s (λ0 = 460 nm, data not shown). Waveguides composed of
Au show a lower FOM than those with Ag, which is attributed to the higher losses
in Au.

4.6

Visualizing the Negative Index Mode

Finally, in Fig. 4.6 we present the field distribution in a coaxial waveguide with
negative index. Figure 4.6a shows the Hy -field on the outer metal-dielectric interface
of a coaxial waveguide with 75-nm-diameter Ag core and 25-nm-wide GaP channel
for a frequency ω = 3.75 × 1015 rad/s (λ0 = 460 nm, ω/ωSP = 1.08 text in Fig. 4.5).
Clearly, the phase fronts in the waveguide are not planar. As a result, a non-zero zcomponent of the electromagnetic field is observed. Figure 4.6b shows the distribution
of Hr in the transverse plane. Its magnitude is similar to that of the Hy -field. For
completeness, Figs. 4.6c and 4.6d show the Hφ and Hz components of the field. Note
that these fields are antisymmetric with respect to the center of the dielectric channel
and are located mostly inside the metal.

Figure 4.6: Magnetic field images of a coaxial waveguide with 75 nm Ag core
diameter, 25-nm-wide GaP-filled ring and infinite Ag cladding at a frequency of
3.75 × 1015 rad/s . In (a) we plot the Hy field distribution on the boundary of the
Ag cladding with the dielectric channel. Note that the phase-fronts in the waveguide
are in general not perpendicular to the optical axis (z-axis). In (b-d) we plot the
polar magnetic field components in the transverse plane. The amplitude of the fields
plotted in the figure has the same order of magnitude in all of the four panels.

4.7

Conclusion

We have theoretically demonstrated that coaxial plasmon waveguides sustain modes
with negative refractive index at optical frequencies. The negative-index modes have a
larger prop- agation length than the positive-index modes over a large spectral range,
depending on the dielectric thickness. For a 10-nm-wide GaP dielectric and a 75-nmwide Ag core, a figure-of-merit k 0 /k 00 = 18 is found at λ0 = 460 nm. For Ag/Si/Ag
coaxial waveguides with increasing Si-channel thickness, the dominant negative-index
mode shifts well below the surface plasmon resonance frequency: for a 70-nm Sichannel it is found at λ0 = 750 nm. The mode index can be fine-tuned to a value
of -1 with a figure-of-merit as high as 5 at λ0 = 720 nm. At slightly higher frequencies, the same mode has an effective index n = 0 with positive group velocity, and
a decay length of 500 nm. Overall, higher tunability and figure-of-merit are found
for coaxial waveguides of Ag rather than Au, and filled with a dielectric of highest
optical constant. Based on the large degree of dispersion control that can be achieved
with coaxial plasmon waveguides, we anticipate that these structures will find use in
new designs for nanoscale photonic integrated circuits (waveguides, splitters, multiplexers), in invisibility cloaks, and three-dimensional negative-index metamaterials,
as will be discussed in the next chapter.

Chapter 5
A Single-Layer Wide-Angle
Negative-Index Metamaterial
at Visible Frequencies
Abstract: In the last two chapters, we have seen how we can measure the dispersion of plasmonic coaxial waveguides and access their negative index modes. In
this chapter, we demonstrate how we can couple an array of negative index coaxial
waveguides to serve as a single-layer wide-angle negative-index metamaterial at visible frequencies. Metamaterials are materials with artificial electromagnetic properties defined by their sub-wavelength structure rather than their chemical composition.
Negative-index materials (NIMs) are a special class of metamaterials characterized
by an effective negative index that gives rise to such unusual wave behaviour as backwards phase propagation and negative refraction. These extraordinary properties lead
to many interesting functions, such as sub-diffraction imaging [103, 45] and invisibility cloaking [2, 18, 73, 105]. So far, NIMs have been realized through layering
of resonant structures, such as split-ring resonators, and have been demonstrated at
microwave[101, 117] to infrared [36, 37, 115, 144] frequencies over a narrow range of
angles-of-incidence and polarization. However, resonant-element NIM designs suffer
from the limitations of not being scalable to operate at visible frequencies because of intrinsic fabrication limitations[38], require multiple functional layers to achieve strong
scattering[144, 38], and have refractive indices that are highly dependent on angle of
incidence and polarization. Here we report a metamaterial composed of a single layer

51
of coupled plasmonic coaxial waveguides that exhibits an effective refractive index of
−2 in the blue spectral region with a figure-of-merit larger than 8. The resulting NIM
refractive index is insensitive to both polarization and angle-of-incidence over a ±50◦
angular range, yielding a wide-angle NIM at visible frequencies.

5.1

Introduction

Negative-index materials were first predicted theoretically by Veselago [136] in 1968,
but it was only in the late 1990s that Pendry [104] defined NIM designs suitable for
experimental realization. In these resonant-element based NIMs, the unusual ‘left
handed’ behaviour of light originates from subwavelength resonant elements that behave like ‘artificial atoms’ with engineered diamagnetic resonances that are the source
of the materials’ negative-index response. As such, NIMs were first demonstrated experimentally with arrays of millimetre-size copper strips and split-ring resonators
operating at microwave frequencies [104, 121]. This discovery sparked a considerable
effort to scale down the size of the constituent resonant components to enable operation at higher frequencies. As a result, micrometre-size structures have been successfully fabricated to produce negative refractive indices at terahertz frequencies. More
recently, NIMs have been fabricated to operate in the near-infrared spectral region.
However, for operation at optical frequencies, the required size of sub wavelength
scatterers is very close to practical fabrication limits. So far, the highest reported
operational frequency of NIMs has been demonstrated at the deep-red side of the visible spectrum (λ0 = 780 nm), using fishnet structures with features as small as 8 nm
[38]. Moreover, to achieve strong scattering, the material was built up from a stack of
multiple physical layers, thus complicating the fabrication of resonant-element based
NIMs for operation at visible frequencies.
Recently, using waveguides, a conceptually different approach was taken to achieve
a negative refractive index in the optical spectral range. Investigation of the mode
structure of two-dimensional metal/dielectric/metal (MDM) plasmonic slab waveguides [34, 70] reveals that certain MDM waveguide geometries support negative- index

Figure 5.1: Negative-index metamaterial geometry. (a) Single-layer NIM slab consisting of a hexagonal array of subwavelength coaxial waveguide structures. The inner
radius r1 , outer radius r2 and array pitch p are defined in the image. (b) Unit cell of
the periodic structure. The angle-of-incidence θ is shown, as well as the in-plane (p-)
and out-of-plane (s-) polarization directions associated with the incident wavevector
k.

modes at visible frequencies. Arrays of such negative-index MDM slab waveguides can
serve as a quasi three-dimensional metamaterial [119]. However, the negative-index
mode in MDM waveguides [34] can only be excited from free space with the perpendicular polarization and off-normal angles of incidence because of the polarization
and symmetry of the mode, respectively.
These practical limitations of planar MDM geometries can be circumvented in a
coaxial MDM geometry in which the planar MDM waveguide is wrapped onto itself
(Fig. 5.1). Similar to the modes supported by planar MDM plasmonic waveguides,
the coaxial waveguide geometry is found to also support field symmetric and antisymmetric modes that correspond to positive- and negative-index modes, respectively.
However, unlike planar MDM waveguides, calculations of individual MDM plasmonic
coaxial waveguides show a negative-index mode that, owing to the cylindrical symmetry of the structure, is accessible from free space independent of both incidence
angle and polarization. Here, we demonstrate that a two-dimensional array of vertically oriented MDM coaxial waveguides, arranged in a dense hexagonal configuration,
functions as a single-layer wide-angle negative index material down to the blue part
of the visible spectrum. Through parameter retrieval analysis, we verify the NIM to

53
have a double-negative [25] index band in the 450 – 500 nm spectral range. Furthermore, we find that the effective refractive index of this geometry is insensitive to both
polarization and angle of incidence up to ±50◦ . Unlike the wire arrays of Liu et al.,
which exhibit negative refraction but not a negative index [74], the coupled coaxial
waveguide array exhibits a true negative refractive index characterized by negative
refraction and backwards phase propagation.
Figure 5.1a schematically depicts the NIM, consisting of a hexagonal close-packed
array of Ag/GaP/Ag MDM coaxial waveguides composed of 25-nm GaP annular
channels with a 75-nm inner diameter set at a pitch of p=165nm in a Ag layer.
We study the metamaterial response both by analytic waveguide modal analysis for
single coaxial structures, and using finite-difference time-domain (FDTD) simulations
for the array of coupled coaxial waveguides.

5.2

Isolated Coaxial Waveguide Dispersion

To estimate the effective refractive index of the material, we first calculate the mode
index dispersion of the constituent coaxial elements by solving Maxwells equations in
cylindrical coordinates [5, 93] for a single MDM coaxial waveguide of infinite length.
As discussed in §4.2, the dispersion relation of the constituent coaxial elements
is calculated by solving Maxwell’s equations in cylindrical coordinates for a single
MDM coaxial waveguide of infinite length. The azimuthal dependence of the electric
and magnetic fields in the waveguide is described by the harmonic function einψ of
order n. We consider only the modes with n = 1, as these are the lowest order
linearly polarized modes that most strongly couple to freespace radiation. The radial
dependence of the fields in all three domains (metal-dielectric-metal) is described by
solutions to the second order Bessel differential equation. We apply a Bessel function
of the first kind Jn to the Ag core; a Hankel function of the first kind Hn to the
Ag cladding; and a linear combination of both Jn and Hn functions to the dielectric
channel. On each domain boundary we formulate four continuity conditions for the
tangential components of the electric and magnetic fields. The optical eigenmodes of

Figure 5.2: Coaxial waveguide dispersion relations. The coaxial waveguide consists of
an infinitely long 25-nm GaP annular channel with a 75-nm inner diameter embedded
in Ag. Plotted are the two lowest-order linearly polarized modes that most strongly
couple to free space radiation. (ac), Energy is plotted versus β 0 (a), β 00 (b), and mode
index nmode (c). (d) The figure-of-merit FOM = |β 0 /β 00 |. The Ag/GaP planar surface
plasmon energy at ~ωSP = 2.3 eV (λ0 = 540 nm) is indicated by the black dashed
horizontal line. All panels show one mode with positive index (red curve) and one
mode with a negative index (blue curve) below an energy of 2.7 eV (λ0 = 460 nm).
The insets in (a) show the Re(Hy ) (out-of-page) field distribution in the waveguide
at a wavelength of λ0 = 650 nm for the positive-index mode and at λ0 = 483 nm for
the negative-index mode.

the coaxial waveguide are found when the determinant of the resulting homogeneous
system of eight equations with eight unknown coefficients vanishes.
The waveguide eigenmodes are characterized by a complex propagation constant
along the z axis β(ω) = β 0 + iβ 00 ,where β 0 and β 00 are the real and imaginary parts
of the propagation constant, respectively. Complex optical constants for Ag [64] and
GaP [97] are taken from tabulated literature data.
Figure 5.2a and b show the calculated dispersion relations ω(β 0 ) and ω(β 00 ) of a
single Ag/GaP/Ag coaxial waveguide. The mode index nmode = cβ 0 /ω is plotted in
Fig. 5.2c. Similar to what is reported for planar MDM structures [34], we find one
mode with a positive index over the entire spectral range (red curve), and a second
mode with a negative index for energies below 2.7 eV (blue curve). The index of
the second mode ranges from −9 < nmode < 1 in the energy range of Fig. 5.2. The
insets in Fig. 5.2a show the Re(Hy ) field profiles corresponding to the positive-index
mode at λ0 = 650 nm (nmode = 8.5) and the negative-index mode at λ0 = 483 nm

Figure 5.3: Coaxial waveguide negative-index mode. Lateral cross-section of a coaxial
waveguide consisting of an infinitely long 25-nm GaP annular channel with a 75-nm
inner diameter embedded in Ag. The dielectric channel is schematically indicated.
Plotted is the real part of the H-field distribution of the n = 1 negative index mode at
λ0 = 483 nm, where n refers to the azimuthal dependence of the fields. The in-plane
Re(Hxz ) field distribution is depicted with arrows, while the out-of-plane Re(Hy )
fields are plotted using a color scale.

(nmode = −2.0). A full field map of the negative-index mode can be found in Fig. 5.3.
Figure 5.2b shows that for energies below ~ωSP = 2.3 eV the positive-index mode (red
curve) has the lowest attenuation and will therefore be dominant, whereas for energies
above ~ωSP the negative-index mode (blue curve) is dominant. Figure 5.2d shows the
figure-of-merit, FOM = |β/β 00 |, of the two modes. The lowest attenuation constant
ω(β 00 ) for the negative-index mode is found at λ0 = 483 nm, with a corresponding
FOM of 8.3.

5.3

NIM Slab Refraction

Next, we analyze the collective response of the coupled coaxial waveguide array using
the FDTD method. For the semi-infinite metamaterial slab calculations, the NIM is
modelled in FDTD (Lumerical FDTD Solutions 6.0) as a single unit cell (Fig. 5.1b)
embedded in air with Bloch boundary conditions along the in-plane direction. The

Figure 5.4: Metamaterial index. (a) Light at λ0 = 483 nm is incident on a semiinfinite slab of single-layer negative index metamaterial at an angle of 30◦ from air.
Shown is a time-snapshot of the magnetic field distribution Re(Hy ), taken along the
polarization plane. Arrows denote the direction of energy flow S and phase velocity k.
The coax center-to-center pitch is schematically indicated. (b) Constant-frequency
surface plot at λ0 = 483 nm, showing the relation between kx and kz for a semiinfinite metamaterial slab over a 50◦ range of incidence angles. The wavevector k and
Poynting vector S data are derived from FDTD simulations.

57
structure is excited with a continuous plane wave source incident at an angle θ. The
appropriate electromagnetic fields are recorded to reconstruct the refraction of phase
(Fig. 5.4a) along the plane of incidence. The refraction of power is obtained by
spatially averaging the steady state Poynting vector components inside the material
along the plane of incidence. The steady state electromagnetic fields are obtained
by calculating the systems impulse response to a plane wave source with a Gaussian
frequency spectrum centred at the frequency of interest.
Figure 5.4a shows a time-snapshot of Re(Hy ) inside a 165-nm-pitch coaxial waveguide array illuminated by a λ0 = 483 nm p-polarized plane wave incident at 30◦ . At
this pitch the waveguides are separated by 40 nm, corresponding to twice the radial
skin depth (δ ∼ 20 nm) of an isolated coaxial waveguide mode into the Ag cladding.
Phase fronts are observed to clearly refract in the negative direction, that is, to the
same side of the interface normal. By following the phase fronts in time we observe backward phase propagation at an angle of −12.5◦ with respect to the interface
normal, as indicated in Fig. 5.4a by the blue arrow labelled k.
Using Snell’s law and the wavevector refraction angle inside the material, we find
that the metamaterial has an effective refractive index of nef f = –2.3, close to the
mode index found for an individual coaxial waveguide at this wavelength (nmode =
–2.0). Furthermore, the direction of energy flow S inside the NIM layer, depicted
by the green arrow in Fig. 5.4a, is found to be antiparallel to the phase velocity
– a signature of a true negative index material. Thus, at a separation of 40 nm,
the waveguides are coupled just enough to allow both power and phase to refract
negatively across the waveguide structures with antiparallel directions, while only
perturbing the metamaterials effective index from that of a single coax by ∆n = –
0.3. From the wavelength in the metamaterial and the exponential energy decay in
the waveguides we find a metamaterial FOM of 8, equal to the FOM calculated for
isolated coaxial structures.
By repeating this analysis for angles ranging from 10◦ to 50◦ , for both s- and
p-polarized light at λ0 = 483 nm, we find very similar results for the materials response, with nef f varying between –2.1 and –2.4. The data for p-polarized light are

Figure 5.5: Summary of effective refractive index for varying angle of incidence. The
metamaterial effective refractive index neff is plotted for λ0 = 483 nm s- and ppolarized light incident at angles ranging from 0 − 50◦ , derived from slab wave vector
angles as in Fig. 5.4a, as well as from refraction angle measurements in wedge-shaped
samples as in Fig. 5. The dashed line indicates the calculated mode index of a single
coaxial waveguide.

summarized in Fig. 5.5. Such a small dependence of the index on polarization and
angle-of-incidence has not been demonstrated in any other NIM reported so far. For
example, Valentine et al. investigated only normal incidence excitation for a NIM operational in the near-infrared spectral region [129], whereas the negative-index mode
in planar MDM structures can only be excited at off-normal incidence angles and
at a specific polarization [34]. We note that simulations as in Fig. 5.4a show the
semi-infinite NIM slab to reflect ∼ 35% of the incident light, depending on angle,
indicating that a significant fraction of light is coupled into the NIM layer. Notably,
for oblique incidence, in addition to exciting the lowest order linearly polarized coaxial waveguide modes (n = 1), we also excite a minor contribution from the radially
polarized mode (n = 0). However, this small modal overlap does not significantly
change the material index response, as both modes have similar dispersion relations
around the operation wavelength of λ0 = 483 nm.
To demonstrate this, we show that in conducting a modal decomposition on a
single coaxial waveguide structure excited at the maximum incidence angle of 50◦ with

Figure 5.6: Modal reconstruction. A semi-infinite coaxial waveguide consisting of a
25-nm GaP annular channel with a 75-nm inner diameter embedded in Ag is illuminated from air with λ0 = 483 nm light at a 30◦ angle-of-incidence. Plotted are the
real and imaginary parts of Hy . The two panels on the left (a, c) show the mode
excited inside the waveguide, and the two right-side panels (b, d) show the mode
reconstructed from a superposition of 87% n=1 mode and 13% n=0 mode, where n
refers to the azimuthal dependence of the fields.

Figure 5.7: Coaxial waveguide mode dispersion relations. The coaxial waveguide
consists of an infinitely long 25-nm GaP annular channel with a 75-nm inner diameter
embedded in Ag. Plotted are the n = 0, 1, and 2 dispersion relations, where n
refers to the azimuthal dependence of the fields in the waveguide, described by the
harmonic function einψ of order n. Energy is plotted versus β 0 in (a), β 0 in (b). The
Ag/GaP planar surface plasmon energy at ~ωSP = 2.3 eV (λ0 = 540 nm) and the
target negative-index operation wavelength (λ0 = 483 nm) are indicated by black
dashed horizontal lines.

λ0 =483 nm light, we find that the resulting excited waveguide mode is composed of
87% n=1 mode and only 13% n=0 mode, where n refers to the azimuthal dependence
of the fields in the waveguide described by the harmonic function einψ of order n. The
accuracy of the modal decomposition can be seen in Fig. 5.6, where the measured and
reconstructed waveguide modes are plotted with 99.97% modal overlap. Furthermore,
in looking at the dispersion relations of these two modes, we find that their complex
indices are similar around the operation wavelength λ0 = 483 nm (Fig. 5.7), thus
explaining why the minor contribution from the n = 0 mode does not significantly
affect the overall functionality of the predominantly n = 1 mode material at offnormal angles of incidence. For reference, Hy field cross-sections of the three lowest
order modes (n = 0, 1, 2) are plotted in Fig. 5.8.

Figure 5.8: Coaxial waveguide eigenmodes. The coaxial waveguide consists of an
infinitely long 25-nm GaP annular channel with a 75-nm inner diameter embedded
in Ag. Plotted are the real (a, b, c) and imaginary (d, e, f) parts of the Hy field
components of the n = 0 (a, d), 1 (b, e), and 2 (c, f) modes at λ0 = 483 nm, where
n refers to the azimuthal dependence of the fields.

Figure 5.9: Effective refractive index as a function of pitch. The effective refractive
index neff derived from wave vector angles is plotted as a function of pitch for λ0
= 483 nm p-polarized light incident at 30◦ on a variable pitch waveguide array slab
similar to that shown in Fig. 5.4a. The dashed line indicates the calculated mode
index of a single coaxial waveguide.

5.4

Pitch Dependence

To further investigate the effect of coupling between coaxial waveguides, we also performed calculations for which the pitch is set to p = 330 nm, corresponding to a
waveguide separation of ∼10 skin depths. As expected, we find that for the same
excitation conditions of λ0 = 483 nm light incident at 30◦ , the waveguides are effectively decoupled, with power flowing straight down the coaxial waveguides. Indeed,
by calculating the average Poynting vector inside the layer, we establish that no net
horizontal power flow occurs for this configuration. Thus, in the limiting case of completely decoupled waveguides, we obtain a uniaxial anisotropic medium with power
flowing straight down the one-dimensional waveguides, irrespective of angle of incidence. In that case, we cannot assign an effective materials index in terms of either
power or phase, but rather assign a local effective mode index that is characteristic
of an isolated waveguide mode.

63
However, as the array pitch is decreased, the waveguides begin to couple in such a
way that both power and phase are able to propagate across the waveguide array. We
find that coupling is easily achieved for the negative-index mode, because its strongly
delocalized field distribution, which resides primarily in the metal, allows neighbouring structures to easily couple (see Fig. 5.2a insets and Fig. 5.8). At a waveguide
separation of roughly twice the mode skin depth of a single coaxial waveguide mode
into the surrounding Ag, we find not only that power and phase are antiparallel, but
also that they refract with an effective index close to the mode index of the constituent waveguides (Fig. 5.9). By varying the incidence angle, the array response is
found to be isotropic within a ±50◦ angular range (Fig. 5.5).
This level of isotropy can be observed in Fig. 5.4b, which shows the constantfrequency surface formed by the wavevector k and Poynting vector S data derived
from FDTD simulations. The figure clearly illustrates that both phase and power are
antiparallel within a ±50◦ range of incidence angles (corresponding to a ±20◦ angular
range inside the material).

5.5

Parameter Retrieval

To confirm the validity of assigning an effective index to the coupled coaxial waveguide
structure, we used FDTD to perform a parameter retrieval procedure on a 100-nmthick NIM slab (Fig. 5.1a) over the 400 – 500 nm spectral range.
The parameter retrieval is calculated using FDTD by exciting a 100-nm-thick
NIM slab embedded in air with a broadband plane wave source ranging from λ0 =
400 to 500 nm at normal incidence. The steady state field distributions are recorded,
and the complex reflection r and transmission t coefficients are calculated by taking
the ratios r = Er /E0 and t = Et /E0 , where E0 is the electric field amplitude
of the incident wave and Er and Et are the reflected and transmitted electric field
amplitudes, respectively. Standard inverted reflection and transmission parameter
equations found in literature[123, 82] are used to relate r and t to the layers effective
impedance zef f and index nef f . The effective permittivity ef f and permeability µef f

Figure 5.10: NIM effective parameters. Effective parameters are calculated for a 100nm-thick NIM slab excited at normal incidence over the 400–500 nm spectral range.
(ad) The real (0 ) and imaginary (00 ) parts of the retrieved effective relative impedance
zef f (a), index nef f (b), relative permittivity ef f (c) and relative permeability µef f
(d).

are calculated through the relations = n/z and µ = nz.
Figure 5.10 shows the resulting curves corresponding to the effective relative
impedance zef f , index nef f , relative permittivity ef f , and relative permeability µef f
of the coupled coaxial NIM structure, with (0 ) and (00 ) denoting the parameters’ real
and imaginary parts, respectively. The extracted effective index curve (Fig. 5.10b)
is found to closely resemble the mode index dispersion of a single coaxial waveguide
structure (Fig. 5.2c) with n0ef f going from positive to negative values at λ0 ∼420 nm,
with increasing wavelength. At λ0 = 483 nm, we obtain a retrieved effective index
of nef f = 2.1 + i0.2, corresponding to a FOM∼10 that is consistent with the single coaxial waveguide FOM∼8. For the retrieved effective relative permittivity ef f
(Fig. 5.10c), we observe a material with 0ef f < 0 over the entire simulated spectral
region, whereas for the effective relative permeability µef f (Fig. 5.10d), we get a material that is diamagnetic withµ0ef f < 0 over the 450–500nm spectral range. Thus,

65
we confirm that the coupled coaxial waveguide NIM structure has a double-negative
index [25] composed of simultaneously negative real parts of the permittivity and
permeability over the 450–500nm spectral range.

5.6

Wedge Refraction

To further corroborate our results, we have simulated the Snell-Descartes refraction of
a ∼300-nm-thick wedged-shaped metamaterial slab cut at a 3◦ angle. The NIM wedge
is modelled in FDTD as a ∼300-nm-thick metamaterial-slab cut at a 3◦ angle. The
structure is excited with a continuous wave source centered at the desired excitation
frequency using a ∼1-µm-wide hollow metallic waveguide oriented perpendicular to
the metamaterial’s input plane. The appropriate electromagnetic fields are recorded
to reconstruct the refraction of phase (Fig. 5.11a) along the plane of incidence. The
structure is also excited from free space with a ∼1.5-µm spot size Gaussian beam.
Using the steady state field distribution at the output side of the wedge, a nearto-far-field transformation is performed to determine the refracted-beam profile at a
distance of 10 µm behind the exit side of the structure.
Figure 5.11a depicts a time snapshot of the steady state Re(Hx ) field distribution
along the plane of refraction for λ0 =483nm s-polarized light at normal incidence.
Figure 5.11a shows that light refracts negatively at the angled side of the prism,
exiting the structure at an angle of −6◦ with respect to the surface normal. Figure
5.11b shows a polar plot of the refracted light projected into the far field. Using Snells
law and the observed negative refraction angle, we derive a refractive index for the
metamaterial slab of nef f =–1.8, in agreement with the effective index derived from
the observed wavevector inside the semi-infinite slab.
To demonstrate the insensitivity of the metamaterial-index to incidence angle,
we also simulated the refraction of off-normal incident light through the 3◦ wedge.
Figure 5.11c and d show the simulation results for λ0 =483nm radiation incident at
30◦ . The green arrows in Fig. 5.11c indicate the direction of the Poynting vector for
the incident and refracted beams. In this case the beam is refracted at an angle of

Figure 5.11: Wedge refraction. (af) A ∼300-nm-thick metamaterial slab is illuminated
from the left at normal (a,b,e,f) and 30◦ off-normal incidence (c,d). The right side
of the slab is cut at a 3◦ angle to allow refraction (black dashed line indicates the
surface normal). The wavelength of incident light is 483 nm (ad) and 650 nm (e,f).
The three panels on the left (a,c,e) depict the calculated power flow (green arrows),
and the three corresponding right-side panels (b,d,f) show the steady-state electric
field intensity in a polar plot, monitored at a distance of 10 µm behind the exit side
of the slab. The output plane surface normal is indicated on the polar plots by a grey
dot. In (a) we also plot the Re(Hx ) field distribution along the plane of refraction.

67
−38◦ with respect to the surface normal, corresponding to an effective index nef f =–
2.2, again consistent with the effective index found for the wedge excited at normal
incidence. By varying the angle-of-incidence from normal incidence up to 50◦ for
both s- and p-polarized light, we find consistent refractive indices ranging from –1.8
to –2.4. These data are summarized in Supplementary Fig. 5.5.
To illustrate the metamaterial’s tunability with wavelength, we study the refraction of λ0 =650nm radiation, for which the mode index dispersion of a single waveguide
element (Fig. 5.2a) shows a positive index (nmode =8.5). Indeed, as Fig. 5.11e and f
show, the beam is now refracted to the opposite side of the interface normal, at an
angle of 23◦ , corresponding to a positive effective index of nef f =7.4. We attribute
the difference between the metamaterial effective index and isolated waveguide mode
index to the possible excitation of higher-order waveguide modes. Thus, by illuminating the structure with frequencies either above or below the Ag/GaP surface plasmon
resonance, we can excite both positive and negative refractive indices within the same
metamaterial.

5.7

Conclusion

Realization of the metamaterial structure reported here involves the challenge of
fabricating high-aspect-ratio nanoscale Ag channels. However, we have shown that
50-nm-wide coaxial apertures with aspect ratios >10 can readily be fabricated using
focused ion beam milling [31]. For large-scale fabrication, a more tractable approach
would be to use electron beam lithography in combination with high-aspect-ratio
reactive ion etching. We also note that observation of the negative-index response
requires only a modest total material thickness, that is, thick enough for a single
waveguide to support the negative index waveguide mode. This can, for instance, be
seen in our wedge simulations (for example, Fig. 5.11a) where the wedge thickness
is only ∼300nm, and the material’s negative index response is clearly observed. At
λ0 =483nm for an index of n = −2, a minimum thickness of ∼120nm is required,
corresponding to a modest coaxial channel aspect ratio of 4.

Part II
Hole Array Color Filters

Chapter 6
Plasmonic Color Filters for CMOS
Image Sensor Applications
Abstract: Having seen how coaxial plasmonic waveguides can serve as building
blocks for metamaterials, in this section we explore how plasmonic hole arrays can
serve as efficient color filtering elements for imaging application. This chapter discusses the optical properties of plasmonic hole arrays as they apply to requirements
for plasmonic color filters designed for state-of-the-art Si CMOS image sensors. The
hole arrays are composed of hexagonally packed subwavelength sized holes on a 150-nm
Al film designed to operate at the primary colors of red, green, and blue. Hole array
plasmonic filters show peak transmission in the 40−50% range for large (> 5×5µm2 )
size filters and maintain their filtering function for pixel sizes as small as ∼ 1×1µm2 ,
albeit at a cost in transmission efficiency. Hole array filters are found to be robust
with respect to spatial crosstalk between pixels within our detection limit, and they
preserve their filtering function in arrays containing random defects. Analysis of hole
array filter transmittance and crosstalk suggests that nearest neighbor hole-hole interactions rather than long-range interactions play the dominant role in the transmission
properties of plasmonic hole array filters. We verify this via a simple nearest neighbor
model that correctly predicts the hole array transmission efficiency as a function of
the number of holes.

6.1

Introduction

Metal films with subwavelength-size periodic hole arrays are known to act as optical
filters, owing to the interference of surface plasmon polaritons (SPPs) between adjacent holes. Unlike current on-chip organic color filters, plasmonic filters have the
advantage of high color-tunability with only a single perforated metal layer and do
not suffer from performance degradation after ultraviolet (UV) radiation. However,
for successful on-chip implementation, the plasmonic filter must also be compatible
with contemporary image sensors featuring small (∼ 1 × 1µm2 ) pixel size, and have
a large functional array size, spatial color-crosstalk effects, and robustness against
random defects.
Extensive studies of enhanced transmission through optically thick metal films
perforated with arrays of subwavelength-size holes have been performed by numerous
groups [39, 56, 6, 7, 71, 96, 102]. The enhanced transmission observed in hole arrays
is explained by excitation of surface plasmon polaritons (SPPs) at the metal surface
that are launched at each hole and interfered among adjacent holes. Transmission
enhancements are found to occur at central frequencies determined by the physical
size of the holes and the thickness of the metal film, as well as the optical properties of
metal and dielectric medium [71, 96]. With reported peak transmission efficiencies of
more than 30% at visible wavelengths, hole array films have received much attention
for their potential to serve as spectral filters for imaging applications [69, 22, 62].
In contemporary image sensor technologies such as CCDs and CMOS image sensors (CISs), color sensitivity is added to photodetective pixels by equipping them with
on-chip color filters (OCCFs), typically composed of organic dyes corresponding to
the three primary colors. However, organic dye filters are not durable at high temperatures or under long-duration ultraviolet irradiation exposure and cannot be made
much thinner than several hundred nanometers, due to the low absorption coefficient
of the dye material. Furthermore, fabrication of each of the three organic dye filters
for a red/green/blue or cyan/magenta/yellow color scheme requires carefully aligned
lithography of each type of color filter over the entire photodiode array, thus mak-

71
ing impractical the fabrication of multicolor imaging devices with both large array
formats and very small pixels.
On the other hand, the transmission properties of plasmonic metal filters composed of periodic hole arrays are mainly defined by their physical structure. This
means that by simply changing the hole size, shape, and separation, the transmission
spectra of the hole array can easily be controlled with only a single thin metal layer.
Owing to this feature, plasmonic color filters are very cost competitive, especially for
multicolor imaging applications. Furthermore, plasmonic filters have many other advantages over conventional filters, such as higher reliability under high temperature,
humidity, and long-term UV radiation exposure conditions.
More recently, a plasmonic hole array color filter was integrated with a CMOS
image sensor, demonstrating filtering functionality in the visible [21, 23]. However,
works such as this have mostly focused on the transmission properties of large size
filters, with little emphasis given to other important filter performance aspects necessary for state-of-the art image sensor applications, such as the filter transmission
dependence on array size, spatial color-crosstalk, and robustness against defects. In
this work, we report on such optical properties as they pertain to various configurations of hexagonal arrays of subwavelength holes fabricated in 150-nm-thick Al films
suitable for image sensor integration.

6.2

Experimental

In our experiments, hole array filter fabrication was done using a 30 kV, 10 pA
focused ion beam (FIB) to mill the desired hole array patterns in a 150-nm-thick
Al film evaporated over 1 in. square quartz substrates. In order to characterize hole
array transmission as a function of hole diameter and period in the visible wavelength
range, we fabricated isolated square filters composed of 16 × 16 hexagonally aligned
holes with period (p) ranging from 220 to 500 nm in 40-nm steps and diameter (d)
ranging from 80 to 280 nm in 20-nm steps (see Figure 6.2).
To analyze the transmission dependence on filter size, we designed filters that were

Figure 6.1: (a) Back illuminated microscope images of the fabricated hole array filters.
Each filter consists of 16 × 16 hexagonally packed hole arrays. The vertical axis
corresponds to hole diameter, ranging from 80 to 280 nm in 20-nm steps, and the
horizontal axis corresponds to hole period, ranging from 220 to 500 nm in 40-nm steps.
The white bar on the lower part of the images corresponds to a 10-µm scale. Inset
(b) shows a SEM image of a representative hole array filter consisting of hexagonally
aligned 16 × 16 holes with p = 420 nm and d = 240 nm. Measured hole array spectra
for filters optimized to transmit (c) red (p = 420 nm, d = 160280 nm), (d) green (p
= 340 nm, d = 120240 nm), and (e) blue light (p = 260 nm, d = 100180 nm) are
plotted in dotted lines, and the simulated spectra are plotted in solid lines. Each plot
of three color spectra is in steps of 40 nm in hole diameter.

73
optimized to operate at the three primary colors of (R) red (p = 420 nm, d = 240
nm), (G) green (p = 340 nm, d = 180 nm), and (B) blue (p = 260 nm,d = 140 nm),
and fabricated four different hole array filter sizes of each corresponding to 1.2×1.2
µm2 -, 2.4×2.4 µm2 -, 5×5 µm2 -, and 10×10 µm2 -size filters. To measure the effect
of spatial color crosstalk between different color filters, we fabricated adjacent color
filters adjoined with zero separation, and to determine the effect of random defects
we fabricated 10×10 µm2 -size green filters with a density of intentionally designed
random defects ranging in area fraction from 1 to 50%.
Open windows with no metal film with the same physical aperture as the square
hole array filters were fabricated in order to normalize filter absolute transmission
efficiency. After milling the desired hole array configurations, the sample was covered
with 200 nm of spin-coated Honeywell 312B spin-on-glass (SOG).
Spectral transmission measurements were performed on a Zeiss Axio Observer
inverted microscope coupled to a grating spectrometer and nitrogen-cooled CCD system. The sample was illuminated with a halogen lamp filtered by a temperature
conversion filter that gave a sunlike blackbody emission (color temperature of 5500
K) incident light. The microscope field diaphragm and aperture stop were both closed
in order to have collimated incident light. All filter spectra were measured with a
spectrometer utilizing a 100×1340-pixel liquid nitrogen cooled CCD detector with
sensitivity in the 300–800-nm wavelength range. The transmission spectrum of each
hole array filter was divided by the spectrum of the corresponding-size open-window
in order to measure the absolute transmission of each filter. To check measurement
system stability during characterization, the open window spectra were measured
every hour.
Figure 6.2a shows a back-illuminated microscope image of the fabricated hole
array filters consisting of arrays of 16 × 16 holes for each, with hole period ranging
from 220 to 500 nm from left to right and hole diameter ranging from 80 to 280 nm
from bottom to top. The smallest 200 nm period filters correspond to ∼3.5×3.0µm2 size filters, and the largest 500-nm period filters to ∼8.0×6.9µm2 . The inset image
(Fig. 6.2b) shows a scanning electron micrograph (SEM) image of a representative

74
filter with p = 420 nm and d = 240 nm.
The measured transmission spectra for the hole array periodicities corresponding
to red (p = 420 nm), green (p = 340 nm), and blue (p = 260 nm) are plotted in Fig.
6.2c,d, respectively, with each panel illustrating the spectra for different diameters.
From the measured spectra, indicated as the color dotted lines, we see that the peak
transmission position shifts to longer wavelength with increasing hole period and
that the peak transmission efficiency increases with increasing hole diameter. The
maximum filter transmission is found to be in the range of 40–50% in the visible
spectrum with a spectral full-width half-maximum (FWHM) of ∼150 nm.

6.3

Comparing with FDTD

To validate the measured spectra, we simulated the fabricated filter transmission response using three-dimensional full-field electromagnetic simulations. The simulation
model consists of a SiO2 matrix with an embedded 150-nm-thick Al film perforated
with hexagonally aligned holes of the same pitch and diameter as those fabricated.
The optical response of the resulting hole array film is excited with a broadband
planewave source in the 300–900-nm wavelength range launched at normal incidence.
The red solid lines in Fig. 6.2c,d plot the resulting simulated transmission spectra.
In general, the simulated profiles show good agreement with the measured spectra,
with the only notable differences being that the measured spectra have relatively
broader profiles, which can be attributed to fabrication imperfections (e.g., tapered
hole profiles) or surface roughness of the Al film.
Although the fabricated color filters reveal vivid colors and have fairly good peak
transmission efficiency over the visible spectrum, there is some undesirable transmission in the blue spectral range for the red color filter designs. As evident from Fig. 6.2,
the red color filters (p = 420 – 500 nm, d ≥ 200 nm), optimized for transmission in
the 600 – 800 nm wavelength range, demonstrate some spectral crosstalk with the
blue part of the spectrum (λ0 < 500 nm wavelengths). Owing to this undesirable
transmission, the red color filters appear slightly pink, magenta, and violet as illus-

Figure 6.2: The simulated spectra of the hole array filter with p = 420 nm and d
= 240 nm, which is optimized to transmit red color. The top four panels plot the
electric field distribution at the wavelengths of interest along the diameter of the
holes, parallel to the polarization of the plane wave used to excite the structure.

trated in Fig. 6.2a. It is worth noting that although having perfect RGB filters would
be most ideal for imaging applications, spectral crosstalk such as this can easily be
corrected using signal processing.
Nevertheless, to better understand the origin of the spectral crosstalk, we used fullfield simulations to spectrally resolved electric field distribution along the diameter
of the holes, parallel to the polarization of the incident planewave. Figure 6.3 shows
the simulated transmission profile of the red (R) filter along with the cross-sectional
electric field distributions at various wavelengths of interest. Consistent with our
measurements, the simulated spectrum shows two broad transmission bands separated
by a null at 540 nm that corresponds to the reciprocal lattice vector of the hexagonally

76
aligned hole array structure [96].
From the electric field intensity maps, we see that the transmission peaks at
582 and 670 nm of the longer wavelength transmission band correspond to strongly
localized electric field distributions at the top and bottom of the hole, whereas for the
transmission null at 540 nm, the localized electric field exists only at the top of the
hole. This is consistent with the interpretation that there is an intrinsic π/2 phase
shift resulting from the SPP emission and recapture process at the exit side of the
film, leading to a minimum in the transmission at wavelengths corresponding to the
reciprocal lattice vector of the array [96].
On the other hand, whereas the electric field distribution at 582 and 670 nm
show strongly localized field distributions at the top and bottom of the holes, the
field distribution at 462 nm shows a high electric field intensity distribution inside
the hole itself, suggesting that the source of transmission at shorter wavelengths is
dominated by the photonic modes supported by the structure. We note that one way
to reduce the excess blue transmission for the red filters may be to suppress these
photonic modes via the use of coaxial structures rather than holes, formed by adding
concentric metallic cylinders inside the hole. However, the absolute transmission
efficiency of coaxial structures has been reported to be only a few percent [17, 120].

6.4

Size Dependence

Next, we investigate the effect of filter size on the transmission properties of hole array
filters, an important feature in determining the smallest pixel size that can be used for
imaging applications. The heavy lines in Fig. 6.4 indicate the transmission spectra of
the three primary color (RGB) hole array filters fabricated at four different filter sizes,
corresponding to squares of side-lengths equal to approximately 10 µm (Fig. 6.4a), 5
µm (Fig. 6.4b), 2.4 µm (Fig. 6.4c), and 1.2 µm (Fig. 6.4d). It is interesting to note
that regardless of size, all the filters maintain their filtering functionality down to the
1 µm-size filters, although at a cost in transmission efficiency. Both the 10-µm and
5-µm-size filters have similar peak transmission efficiencies, with blue peaking at 28%

Figure 6.3: Transmission spectra of the hole array filters optimized to red (p = 420
nm, d = 240 nm), green (p = 340 nm, d = 180 nm), and blue (p = 260 nm, d = 140 nm)
of different filter sizes of (a) 10 µm-, (b) 5 µm-, (c) 2.4 µm-, and (d) 1.2 µm-squared
size filters. The insets of (a-d) panels show the back illuminated microscope images
of the filter with the field of views corresponding to 13-µm-, 6.5-µm-, 5.0-µm-, and
4.0-µm-squared. (e)-(f) SEM images are the 1.2-µm-size filters, RGB, respectively.

and green and red both at 38%. Transmission drops by ∼ 25% for the 2.4-µm-size
filters relative to 10-µm-size filters and by ∼ 40% for the 1.2-µm-size filters.

6.5

Spatial Crosstalk

To investigate the effects of spatial crosstalk of colors between adjacent pixels, we
measure the spectra of three different sets of color filter pairs with zero separation,
consisting of B/R, G/B, and R/G color combinations, with each color filter consisting
of a 10 × 10 µm2 array. Figure 6.5 a shows a representative SEM image of the
boundary between two such hole array filters, and Fig. 6.5b-d show back illuminated
images of the filter pairs. Figure 6.5e illustrates the measured sliced spectra for the
B/R-filter combination, with each spectrum corresponding to a 1-µm-wide integrated
cross-sectional area, taken perpendicular to the filter pair boundary at the positions
indicated by the yellow ticks in Fig. 6.5b. Figure 6.5g,i show the corresponding spectra

Figure 6.4: Sliced transmission spectra of color filter pairs with zero separation.
(a) A representative SEM image of boundary between two such filters. (b-d) Backilluminated microscope images of the color filter pairs taken with a color CCD camera.
The white lines correspond to 20-µm scale bars. The spectra of each color filter pair
are taken over 1-µm-wide areas centered at the positions indicated by the yellow ticks
in (b). Sliced spectra are shown for (e) blue/red, (g) green/blue, and (i) red/green
filter pairs. The panels next to the sliced spectra in (f), (h), and (j) plot the correlation
of each measured spectra with respect to the averaged spectrum of colors in the filter
pair.

for the other two filter pair color combinations.
The panel next to each sliced spectra, for example, Fig. 6.5f, plots the correlation
function ∆i , which measures the difference between the measured spectrum, Ti (λ),
where i runs from 1 to 20 over the slice number, and the averaged spectrum of each
filter color T̄R,G,B (λ), obtained by averaging the middle eight spectra of each color
filter. The correlation function is then defined by Eq. 6.1, where αi is a normalizing
constant for each measured spectrum, allowing ∆i to take on values from 0 to 1,
depending on the correlation of the measured spectra, with respect to the averaged
spectrum.

79
Z λmin =400 nm
(Ti (λ) − T̄R,G,B (λ))dλ

∆i = αi

(6.1)

λmax =700 nm

In Figs. 6.5f,h,j, the correlation functions undergo an inversion in going from one color
filter to another. From these plots, we can estimate that the spatial color crosstalk
between filters is as small as 1 µm. This length contains not only the contribution of
the diffraction limit but also some defocus from the measurement settings, and hence
it is possible that the crosstalk between filters is almost comparable to our detection
limit. It is worth pointing out that although there is zero separation between the
filters, Fig. 6.5b-d clearly show dark lines between filter pairs, evidence of the small
spatial color crosstalk between filters.

6.6

Random Defects

Because fabrication imperfections are inevitable, it is important to characterize the
sensitivity of filter transmission and spectral shape to random defects. To evaluate
the robustness of the hole array filter design, we measured the transmission of 32 ×
32 hole green filters with randomly positioned missing holes at a density of 0, 1, 2, 5,
10, 20, and 50%. The resulting spectra are indicated in Fig. 6.6a, from which we can
see that the filter transmission efficiency and shape is unaffected for random defect
densities ⩽ 5%, but filter transmission monotonically decreases for random defect
densities ⩾ 10%. However, if we normalize each spectrum by its peak transmission,
as shown in Fig. 6.6b, we find that all the spectra have nearly identical line shapes,
independent of filter defect density, suggesting that the random defects only affect
the filter transmission efficiency and not the spectral shape.
To better understand the transmission efficiency degradation with increased defect
density, we show the relative peak transmission efficiency versus random defect density
in Fig. 6.6c, from which it is clear that the dependence is not simply linear, indicating
that it is not simply a geometrical effect. To gain insight into the optical processes
that govern this trend, we introduce a simple analytic model to explain hole array

Figure 6.5: (a) Transmission spectra of green hole array filter consisting of 32 ×
32 holes with p = 340 nm and d = 180 nm for different random defect (missing
hole) density. (b) Normalized transmission spectra from (a). (c) Plot of the relative
peak efficiency versus defect rate. The data plotted by blue dots corresponds to the
transmission efficiency of a green filter with defect density ranging from 0 to 50%, and
the red line is the analytically estimated degradation curve. (d) A SEM image of green
filter with 50% defect density. (e) An analytically-calculated transmission intensity
map of the filter from (d). (f) A back-illuminated microscope image corresponding
to the filter from (d).

81
transmission efficiency based on nearest neighbor interactions.

6.7

Nearest Neighbor Model

Because the hole array layout is three-fold symmetric, the filter transmission efficiency
should not depend on the polarization of the incident light [62]. Thus, any given hole
in the array, characterized by coordinates x = i,y = j, will have six equally distanced
and thus equally contributing nearest neighbor sites, labeled A−F . Using this model,
the transmission efficiency ηi,j of each site in the array can be calculated by simply
averaging the contribution from its nearest neighbors, as given by Eq. 6.2, where δk
represents delta functions which will take on its value only if there is a hole at the
nearest neighbor site, k.

1X
ηi,j =
δk
6 k=A

(6.2)

Using this model, the transmission efficiency of a given hole in the array will vary from
0 to 1, depending on the number of holes surrounding it. The relative transmission
efficiency of a given filter, η, can then be calculated by summing over the efficiency
of each hole in the array and normalizing it by the efficiency of a defect-free array of
the same size, η0 = 6 (number of holes in array); see Eq. 6.3.
η=

i,j∈array ηi,j

6 (number of holes in array)

(6.3)

To check the accuracy of the model, we plot the predicted relative transmission
efficiency of a 32 × 32 hole green filter as a function of random defect density in
Fig. 6.6c. From the correlation between the measured data and predicted curve, we
see that the nearest-neighbor model correctly captures the degradation in efficiency
with the increasing number in random defects, suggesting that nearest neighbors
indeed play the most dominant role in the transmission efficiency of hole arrays.
This nearest neighbor transmission model can also be used to reconstruct the
spatial transmission distribution of a hole array filter with random defects. Figure

82
5d shows a SEM image corresponding to 32 × 32 hole green filter with 50% random
defects. To predict the spatial transmission efficiency of this filter, we take the SEM
image and apply the nearest neighbor transmission model to each site in the array.
The resulting modeled transmission images are plotted in Fig. 6.6e, which shows
good qualitative agreement with the measured back illuminated image of the filter,
Fig. 6.6f.

6.8

Size-Corrected Transmission Efficiencies

Using this model, we can also explain the effect of filter size on the transmission
properties of finite-size defect-free hole array filters. For example, for a 3×3 filter (corresponding SEM image shown in Fig. 6.4e), owing to the limited number of
holes in the filter, only the center hole has six nearest neighbors and thus the maximum transmission. The other eight holes at the filter edge will have fewer than six
neighbors, bringing down the average of the entire filter to 60%, as given by Eq. 6.2.
However, for larger > 24 × 24 hole filters, the transmission efficiencies go up to > 95%
because almost all the holes in the array have six nearest neighbors.
Using these relative transmission efficiencies, we can calculate the size-corrected
transmission efficiency between two size filters of the same color by simply dividing
their relative transmission efficiencies. To check this, we calculate and plot the sizecorrected transmission profiles of the measured finite-size filters in Fig. 3. The light
lines in Fig. 3b-d show the size-corrected spectrum of the 5-, 2.4-, and 1.2-µm size
filters as compared to the 10-µm-size filter. After the correction, we see that all
four filters have fairly constant transmission efficiency, suggesting the model is also
accurate in explaining finite size effects.

6.9

Conclusion

In conclusion, plasmonic hole array color filters are found to be a great alternative to
conventional on-chip dielectric filters due to their tunability across the visible spec-

83
trum with only a single layer of perforated metal, especially for multicolor imaging
applications. In this work, we show that because their scattering dynamic is only determined by nearest neighbor interactions, the transmission properties of hole array
filters are extremely robust with respect to array size, random defects, and spatial
crosstalk from neighboring filters of different color. The hole array’s ability to filter
light with only a small number of holes, combined with their small spatial crosstalk,
are attractive features for 2D image sensor applications requiring densely packed filtering elements. Furthermore, their robustness at filtering functionality against random
defects makes them especially tractable for industrial mass-production. However, for
further developments of plasmonic filter technology to industrial application, additional studies, especially an actual filter implementation to a CMOS image sensor, are
required to demonstrate their feasibility, reliability, and advantages over conventional
technologies.

Chapter 7
Color Imaging via Integrated
Plasmonic Color Filters on a
CMOS Image Sensor
Abstract: In Chapter 6 we explored several filter performance aspects necessary
for state-of-the art image sensor applications, such as filter transmission dependence
on array size, spatial color-crosstalk, and robustness against defects. In this chapter
we use these findings to integrate a plasmonic hole array color filter onto a CMOS
IS, demonstrating the first realization of high-resolution full-color plasmonic imaging.
Plasmonic hole arrays have been the subject of enormous scientific interest over the
last 15 years, since the first observation of extraordinary light transmission [39, 56, 6].
Since then, the physics of hole array spectral filtering has been intensely debated [71,
96, 7, 80, 107], motivated by color imaging applications [19, 68]. Recently, advances
in plasmonic hole array filter design and performance, as well as CMOS integration,
have been described [22, 143, 14, 23, 21], but to date, no functioning camera has been
reported. In this chapter, I demonstrate the imaging characteristics of a 360×320
pixel color camera by integrating a plasmonic color filter array with a commercial
black and white 1/2.8 inch CMOS image sensor, and taking high resolution full color
images with the integrated image sensor. The color filters, consisting of 5.6×5.6µm2 size color pixels in a 150-nm Al film, were chosen to correspond to the RGB
primary colors and arranged in a Bayer mosaic layout. The color images are taken
with C-mount lenses coupled to the image sensor with focal lengths ranging from 6-

85
50 mm, all showing good color fidelity with a 6-color-averaged (Red, Green, Blue,
Yellow, Magenta, Cyan) CIE ∆E 2000 = 16.6–19.3, after a white balance and color
matrix correction is applied to the raw image over the wide range of f-numbers ranging
from 1.8 – 16. The integrated peak filter transmission efficiencies are measured to
be in the 50% range, with a FWHM of 200 nm for all three RGB filters in good
agreement with the spectral response of isolated unmounted color filters [22, 143, 69].
We also investigate light coupling from hole array filters to CMOS pixels using full
wave electromagnetic simulations.

7.1

Introduction

In contemporary Si-based image sensor technologies such as charge-couple devices
(CCDs) and complementary metal-oxide semiconductor (CMOS) image sensors, color
sensitivity is added to photo detective pixels by equipping them with on-chip color
filters, composed of organic dye-based absorption filters [88]. However, organic dye
filters are not durable at high temperatures or under long exposure to ultraviolet (UV)
radiation [61] and cannot be made much thinner than a few hundred nanometers due
to the low absorption coefficient of the dye material. Furthermore, on-chip color
filter implementation using organic dye filters requires carefully aligned lithography
steps for each type of color filter over the entire photodiode array, thus making their
fabrication costly and highly impractical for multi-color and hyperspectral imaging
devices composed of more than the three primary or complementary colors.
It is well known that plasmonic hole arrays in thin metal films can be engineered as
optical band-pass filters, owing to the interference of surface plasmons (SPs) between
adjacent holes. Unlike current on-chip organic color filters, plasmonic filters have
the advantage of being highly tunable across the visible spectrum and require only
a single perforated metal layer to fabricate many colors. Plasmonic hole array color
filters have recently been integrated with a CMOS image sensor, demonstrating filter
viability in the visible [143], but full color imaging using the plasmonic color filter
technology platform has not be reported. In Chapter 6, we investigated several filter

86
performance aspects necessary for state-of-the art image sensor applications, such as
filter transmission dependence on array size, spatial color-crosstalk, and robustness
against defects [23, 21].
In this chapter, we investigate the transmission and imaging characteristics of
CMOS plasmonic color imaging. A commercial black and white 1/2.8-inch CMOS
image sensor is integrated with a 360×320 pixel plasmonic color filter array composed
of 5.6×5.6 µm2 -size RGB color filters in a Bayer mosaic layout (Fig. 7.1a). Full color
images are taken by coupling the image sensor to C-mount lenses with focal lengths
ranging from 6-50 mm, all showing good color fidelity with the 6-color-averaged (Red,
Green, Blue, Yellow, Magenta, Cyan) CIE ∆E 2000 = 16.6–19.3 after, a white balance
and color matrix correction is applied to the raw image over the wide range of fnumbers ranging from 1.8 – 16.

7.2

Filter Design

Following well-known principles for spectral transmission of hole array filters [22, 143,
62], we used a 150-nm-thick Al film perforated with hexagonally packed holes with
period (p) and diameter (d) equal to p=420nm, d=240nm for red (R), p=340nm,
d=180nm for green (G) and p=260nm, d=140nm for blue (B). For the color filter
layout, we used the Bayer-pattern color filter array layout, which consists of a 2×2
color unit cell with two green filters (Gb, Gr) in the diagonal positions (the lower case
letter following the G corresponds to color that is next to the green filter), and blue
(B) and red (R) in the off-diagonal positions (Fig. 7.1b). Each individual color filter
was designed with the size of 5.6 × 5.6 µm2 , which is exactly twice the pixel size of the
CMOS image sensor, in order to account for alignment errors in the integration (see
Fig. 7.3). E-beam lithography and lift-off were used to stitch 180×160 Bayer unit cells
onto a quartz substrate (see Fig. 7.2), resulting in a 360×320 pixel plasmonic color
filter array of dimensions 2016×1792 µm2 (Figs. 7.1c-d). Here we see 11.2-µm-width
grid lines separating 40×40 filter blocks (corresponding to 224×224 µm2 ), which were
created during the fabrication to both prevent overexposure of the peripheral region of

PMMA

Al
SiO2

SiO2

Gr 2 μm

Si3N4

20 μm

Figure 7.1: Integrated CMOS image sensor with hole array filter. a) Schematic
of integrated front-side illumination CMOS image sensor with RGB plasmonic hole
array filters in Bayer layout. b) Scanning electron micrograph of RGB hole array
filters in Bayer layout. c) Scanning electron micrograph of 11.2µm alignment grid
lines separating 40×40 filter blocks. d) Image of full 360×320 pixel (2016×1792 µm2 )
plasmonic hole array filter array on quartz. Each square on the image corresponds to
a 40×40 filter block (224m×224µm2 ) separated by 11.2µm alignment grid lines. e)
Image of integrated CMOS image sensor with plasmonic hole array filter. The white
on the far edges of the filter corresponds to electronic grade putty used to hold the
filter in place after alignment. f) Image of CMOS image sensor before integration.

88
a) Thin Cr layer

b) PMMA

c) EPBG

f) Lift-off

e) Al layer

d) Develop

g) Spin-on-glass (SOG)

h) RGB filter array
k) Contact

i) CMOS IS

Quartz
Cr

j) PMMA

PMMA
Al

SOG
Si

Si3N4
Cu

Figure 7.2: Integrated CMOS image sensor with hole array filter. a) Schematic
of integrated front-side illumination CMOS image sensor with RGB plasmonic hole
array filters in Bayer layout. b) Scanning electron micrograph of RGB hole array
filters in Bayer layout. c) Scanning electron micrograph of 11.2µm alignment grid
lines separating 40×40 filter blocks. d) Image of full 360×320 pixel (2016×1792 µm2 )
plasmonic hole array filter array on quartz. Each square on the image corresponds to
a 40×40 filter block (224m×224µm2 ) separated by 11.2µm alignment grid lines. e)
Image of integrated CMOS image sensor with plasmonic hole array filter. The white
on the far edges of the filter corresponds to electronic grade putty used to hold the
filter in place after alignment. f) Image of CMOS image sensor before integration.

each block during electron beam lithography and to serve as guide lines for alignment
with the CMOS image sensor pixel array.

7.3

Fabrication and Integration

Fabrication of the plasmonic hole array filter was done using electron beam lithography and lift-off (see Fig. 7.2). A thick Cr layer was used to reduce charging of the
quartz substrate during patterning (Fig. 7.2a) and a 150 nm Al layer was used for

89
the final hole array filter metal layer (Fig. 7.2e). After developing, a 300-nm-thick
layer of spin-on-glass (SOG) was used to infill the hole array filters (Fig. 7.2g). Once
fabricated, the integration was done by directly contacting the color filter array onto
the CMOS image sensor (Fig. 7.2k), using PMMA to first planarize the microlenses
of the image sensor (Fig. 7.2j).
For the integration, we used a front-side-illumination black and white CMOS
image sensor composed of 1920×1200 2.8×2.8 µm2 size pixels, corresponding to a
5376×3360 µm2 effective pixel area (Fig. 7.1). The integration was done by directly
contacting the color filter array onto the CMOS image sensor (see Fig. 7.2). As shown
in Fig. 7.1e, due to the difference in size between the plasmonic color filter and the
CMOS chip, only the center 1/3 of the horizontal area and 3/5 of the vertical area of
the CMOS image sensor was equipped with plasmonic color filter functionality.

7.4

Alignment

Alignment of the hole array filter array with the CMOS image sensor was done by
hand under a microscope. Since each individual color in the 360×320 RGB filter
array was designed to be exactly twice the pixel size of the CMOS image sensor
(see Fig. 7.3a), small translational misalignments were easily corrected by selecting
either the even or odd pixels in each direction of the CMOS image sensor array (see
Fig. 7.3b). Note that rotational misalignments (Fig. 7.3c) cannot be corrected using
this scheme, since the rotational alignment error is not constant throughout the chip.
After aligning the filter array as best as possible, electronic grade putty was used to
hold the filter in place.
To check the alignment, an image was taken of a gray wall, and the color uniformity
was checked for the different parity set of pixels after demosaicing the image but before
applying white balance, i.e., before equalizing the RGB value readouts. Thus, the gray
color of the wall will show up as green since the green pixels have a larger signal than
the red and blue pixels before they are equalized by the white balance transformation
(R,G,B = 1.7, 1.0, 1.3). The images for the different parity set of pixels are show

b) Translation offset

1 2 3 4 ...

1 OO OE
2 EO EE

...

c) Rotation offset
1 2 3 4 ...
EO
EO
2 EO
EO
EO
4 EO

...

a) Perfect alignment

d) Alignment check
EE

Properly aligned pixels

Figure 7.3: Alignment of plasmonic hole array filter with CMOS image sensor. The
lined grid represents the pixel array of the CMOS image sensor, and the transparent
RGB box grid represents the plasmonic hole array color filter array in Bayer layout.
The pixels are labeled using matrix convention (i,j) with i coming from the horizontal
number axis and j coming from the vertical number axis, and the double letters inside
of the grid refer to the parity of the pixel label, with E for even and O for odd. Pixel
and filter array are shown with a) perfect alignment, b) translation offset, and c)
rotational offset. d) Images of the difference parity set of pixel readouts, even-even
(EE), even-odd (EO), odd-even (OE), and odd-odd (OO), after demosaicing the gray
wall image taken with the integrated CMOS image sensor with aligned filter

91
in Fig. 7.3d, with the even-even (EE), even-odd (EO), odd-even (OE), and odd-odd
(OO) pixel readouts showing a spatial color map which varies from green to magenta.
The green areas correspond to pixels that are entirely covered by a single color filter
in the RGB filter array (see e.g., Fig. 7.3b), and the magenta areas correspond to
pixels that must be sitting predominantly at the boundaries between different color
filters (see e.g., Fig. 7.3c).

7.5

Transmission Measurements

The optical efficiency of the color filter design was assessed with spectral transmission
measurements of isolated 22.4×22.4-µm2 -size RGB plasmonic hole array filters. The
spectral transmission measurements of isolated unmounted 22.4×22.4-µm2 -size RGB
plasmonic hole array filters were performed on an inverted microscope coupled to
a grating spectrometer and liquid nitrogen cooled CCD system. The sample was
illuminated with a collimated halogen lamp filtered by a temperature conversion filter
that gave a sun-like blackbody emission (color temperature of 5500 K). All filter
spectra were measured with a spectrometer utilizing a 100×1340 pixel liquid nitrogen
cooled CCD detector with sensitivity in the 300-800 nm wavelength range. The
transmission spectrum of each hole array filter was divided by the spectrum of an
open window of the corresponding size in order to measure the absolute transmission
of each filter.
The resulting spectra, plotted in Fig. 7.4a, show all three RGB filter designs having
peak efficiencies in the 50–60% range with FWHM in the 150–200 nm range, consistent with the transmission data extracted from full field electromagnetic simulations
using finite-difference time-domain calculations for unmounted plasmonic color filters
of corresponding dimensions, embedded in a quartz matrix illuminated at normal
incidence (Fig. 2c).
Direct comparison of the transmission efficiencies between the unmounted (Fig. 7.4a)
and CMOS integrated (Fig. 7.4b) plasmonic color filters gives a quantitative measurement of how well the plasmonic color filters are integrated onto the CMOS image sen-

0.8

Blue

0.6

Green

Relative Efficiency

0.8

Measured Hole Array Transmision

Red

0.4
0.2

400

Absolute Transmission

0.8

450

700

Blue

Green

Blue (λ0 = 450nm)

SiO2

Blue

Red

PMMA

Red (λ0 = 650nm)

Green (λ0 = 550nm)

SiO2

Al
Green

SiO2

0.6

Al
PMMA

log|E|2

PMMA

0.2

0.8
Red

Measured Image Sensor Response

0.4

SiO2
400 450 500 550 600 650 700 750
Wavelength (nm)

Simulated Image Sensor Response

Si3N4

SiO2

Si3N4

SiO2

Si3N4

d)
Blue

Green

Red

-1

0.4
0.2

0.2
400

0.6

750

Simulated Hole Array Transmision

0.6
0.4

500 550 600 650
Wavelength (nm)

Relative Efficiency

Absolute Transmission

450

500 550 600 650
Wavelength (nm)

700

750

1μm
400 450 500 550 600 650 700 750
Wavelength (nm)

1μm

-2

Figure 7.4: Integrated CMOS image sensor response. a) Measured and c) simulated
spectral response of unmounted RGB plasmonic hole array filters. b) Measured and
d) simulated relative efficiency of integrated CMOS image sensor with RGB plasmonic
hole array filters. The horizontal error bars correspond to the spectral width of the
band-bass filter used for the measurements, and the vertical error bars correspond
to the averaged data. Simulation field map cross-sections taken at the center of the
pixel and at the center transmission wavelengths for the e) blue (λ0 = 450nm), f)
green (λ0 = 550nm), and g) red (λ0 = 650nm) hole array color filters integrated with
the CMOS image sensor.

93
sor. In comparing these two data sets, we see that the integrated peak transmission
efficiencies are in the 50% range, which is only slightly lower than those of the unmounted color filters, indicating that although the direct contact integration scheme
is not optimal, it does not degrade the efficiency significantly. Because the color filter
was integrated by simply by pressing the filter and image sensor into intimate contact, we expected that there could be a low refractive gap between the bottom of the
plasmonic filter and the top surface of the image sensor that could reduce the light
coupling efficiency. However we find from full field electromagnetic simulations of the
integrated device that the experimentally measured absolute light coupling efficiencies shown in Fig. 7.4c are in the same range as those simulated under perfect light
coupling conditions in intimate mechanical contact (Fig. 7.4d), indicating that the
integration is nearly ideal.

7.6

Transmission Simulations

The transmission of isolated RGB filters was simulated in FDTD (Lumerical Solutions 7.0) with a single unit cell of the hexagonal Al hole array lattice embedded in
quartz with Bloch boundary conditions in the in-plane directions and PML boundary
conditions in the excitation direction. Complex optical constants for Al and SiO2
are taken from tabulated data [97]. For normal incidence simulations, the structure
was excited with a broadband planewave source, and the steady-state electromagnetic
fields were recorded on the opposite side of the structure. The transmission of the
structure was calculated as the time average of the Poynting vector normalized by
the source. For angled illumination, it was done similarly to the normal incidence
simulations, expect that care was given to the angles imposed by the Bloch boundary
condition on all the wavelengths.
The steady state field response of the integrated structure (Fig. 7.1a) was simulated using FDTD. Complex optical constants for Al, SiO2, Si3 N4 , and Si were taken
from tabulated data[97], PMMA was modeled as SiO2 , and Cu as a perfect electric
conductor (PEC) in order to reduce the simulation time. The structure was illumi-

94
nated with a broadband planewave, and the steady-state electromagnetic fields were
recorded at the center cross-section of the structure (Figs. 7.1e-g) to observe the field
distribution response.
The resulting simulated steady state intensity field distributions of the three filters
at their center wavelengths are shown in Figs. 7.4e-g, showing that a significant factor
in the high light coupling efficiency comes from the nitride waveguide, which directs
the light into the active region of the Si image sensor pixel. We note also that, as
expected, the simulations indicate that blue light absorption occurs in the Si pixel
near-surface region, while green light absorption occurs at a depth of approximately
1µm, and red light absorption occurs at a depth of approximately 3µm below the
Si surface. Importantly, the simulations also indicate that the integrated plasmonic
filter and image sensor pixel exhibits negligible crosstalk in the form of light scattering
into adjacent pixels after light is coupled through the plasmonic filter.

7.7

Color Imaging

Having verified the plasmonic color filter transmission efficiency as integrated on the
CMOS image sensor, we demonstrate color camera performance by taking full color
images using a 12.5 mm lens and a 5.6 f-number, corresponding to a half-aperture
angle of 5◦ . In Fig. 7.5a, we display a representative raw image taken under these
conditions of a 24-color Macbeth chart positioned in a scene. In Fig. 7.5f we plot
the signal of the pixels indicated by the dashed yellow line in Fig. 7.5a, which spans
across the gray color patches on the Macbeth color chart. At this point, each pixel
has 10 bit data, ranging from 0-1023, and we subtract the optical black (OPB) signal
from the image, which corresponds to a dark current signal offset, as indicated in
Fig. 7.5f.

95
Demosaicing

300

800

R0
G0
B0

250

OPB

300

250

100

200

OPB

100
150
pixel number

200

R'
G'
B'

100
150
pixel number

200

300

1.7 R
= 1.0 G
1.3 B

R"
G"
B"

150

400

200

600

Gamma Correction
e)

counts’

counts0

1000

Linear Matrix

counts

1200

White Balance

250

1.8 -0.2 -0.6
1.0
0 -2.4 3.4

R'
G'
B'

y=Ax1/ γ;γ=2.2

y (pixel counts''')

Raw Image
a)

200
150
100
50

100
150
pixel number

200

100 150 200
x (pixel counts")

250

300

Figure 7.5: Image reconstruction process. a) Raw image of 24-color Macbeth color
chart positioned in a scene taken with integrated CMOS image sensor with RGB
plasmonic hole array filter. Image after b) demosaicing, c) white balancing, d) linear
matrix correcting, and e) gamma correcting the image. The yellow dashed line in
a) shows the pixel signal that is being plotted in f-h), corresponding to the images
above them. i) Linear matrix used on image after applying white balance to remove
color cross talk. j) Gamma transformation used to convert the image sensors linear
response to brightness to the logarithmic response of the human eye.

7.8

Image Reconstruction

Since the active pixels on the CMOS image sensor are filtered by a Bayer color
filter array, each pixel only has intensity information of one of the three primary
colors. To reconstruct the full-color image, a bilinear demosaicing algorithm is used
to interpolate the set RGB values for each pixel from its neighbors. For example, the
red value of a non-red pixel is computed as the average of the two or four adjacent red
pixels, and similarly for blue and green. The resulting bitmap image after demosaicing
is shown in Fig. 7.5b, with corresponding RGB signal levels (ranging from 0-255)
shown in Fig. 7.5g.
After demosaicing, the image is white balanced by equalizing the RGB signal
levels of the gray patches on the Macbeth color chart (Fig. 7.5c,h). Next, we apply a
linear matrix correction to correct for the filter’s spectral color crosstalk (Fig. 7.5d,i).
Here we see that the Red pixels are applied a transformation that subtracts Green
and Blue values from it, R00 = 1.8R0 − 0.2G0 − 0.6B 0 , as expected, since the Red

96
pixels experience the most color crosstalk as seen from the spectra plotted in Fig. 2.
Similarly, the Blue pixels are subtracted a nearly equal amount of the Green spectra
from itself, B 00 = −2.4G0 + 3.4B 0 , in order to shift the original Blue filter’s spectral
response further into the blue. However, we leave the Green filter spectra untouched,
G00 = 1.0G0 , since subtracting any amount of Red or Blue from the Green would
significantly reduce its counts due to their large spectral overlaps (see Fig. 2). In
order to realize a pure Green spectra with a much narrower FWHM and smaller
linear matrix cross-terms, we would need to incorporate slightly smaller filter hole
diameters or apply novel filtering structures having higher transmission efficiency
with narrower FWHM than those of hole array filters[120].
After applying the linear matrix pixel transformation, a gamma correction (Fig. 7.5j)
is applied to convert the image sensor’s linear response to brightness into the logarithmic response of the human eye. The result is the natural color image shown in
Fig. 7.5e. Here we see the 11.2-µm alignment grid lines, which contain 40 color pixels
in between them along the two normal directions.

7.9

Measuring the Color Quality

The color quality of the image can be measured using the CIE ∆E 2000 metric [116],
which measures the color difference between the reference color on a Macbeth color
chart and what the actual color is in the signal-processed image (Fig. 7.5e). Via this
metric, the higher the ∆E, the more noticeable the color difference is to the human
eye, and ∆E = 1 is regarded as the just noticeable difference. Using established color
code standards [116], the measured CIE ∆E of the integrated CMOS image sensor
increases from 3.8 for Blue, to 10.2 for Green, and 28.6 for Red, reflecting the fact
that the hole array filter design is efficient at filtering short (blue) wavelengths, but
has increasing color crosstalk from the shorter wavelengths as the hole array pitch
and diameter is increased to filter the longer Green and Red wavelengths [143]. Note
that the ∆E (color difference) being measured here is after signal processing, meaning
that filter’s original spectral response (see Fig. 7.4d) has already been modified, in

97
a)

6mm, f/5.6

9mm, f/5.6

Watson Patio, Caltech

12.5mm, f/5.6

50mm, f/5.6

Beckman Auditorium, Caltech

Atwater Group, Aug 2012

Red Door Cafe, Caltech

Figure 7.6: Focal length dependence and outdoor lighting conditions. Images of 24color Macbeth chart positioned in a scene taken with integrated CMOS image sensor
with RGB plasmonic hole array filter with a 5.6 f-number and a a) 6mm, b) 9mm,
c) 12.5mm, and d) 50mm lens. Images taken with outdoor lighting conditions of e)
Watson Patio, Caltech, f) Beckman Auditorium, Caltech, g) Atwater Group, Aug
2012, Caltech, and h) Red Door Cafe, Caltech.

particular with the linear matrix correction (see Fig. 7.5d), which corrects for the
filter’s original color crosstalk.
Similarly, for the complementary colors, which can be considered as linear superpositions of the primary colors, the measured ∆E is 5.7 for Yellow (Green + Red),
24.7 for Magenta (Red + Blue), and 33.3 for Cyan (Blue + Green), which is consistent with the ∆Es of the primary colors that compose them. Averaging the ∆E
values of the 3 primary and 3 complementary colors, we get an averaged ∆E of 17.7
for the integrated CMOS image sensor, a value which, although not ideal, is capable
of capturing vivid full color images as that shown in Fig. 7.5e. Following this image
reconstruction process, we show in Fig. 7.6 several additional images taken with the
integrated CMOS image sensor, other demonstrations of the technology’s versatility
with respect to focal length and outdoor lighting conditions.

7.10

Angle Dependence

These representative images are taken with a 5.6 f-number, corresponding to a nearly
normal incident half-aperture angle of 5◦ ; however, for proper filter performance, it is
important to check that the RGB hole array filters retain their designed filter functionality for larger aperture angles. To investigate this, we simulated the transmission
of the unmounted plasmonic color filter arrays for incidence angles ranging from 030◦ . The resulting spectra for the green color filter is plotted in Figs. 7.7a and b. As
can be seen from these figures, the filter does have an angular dependence [56, 96, 87];
however, in order to see if this deviation will play a significant role in the filter functionality, we need to consider what range of incidence angles we are collecting for
a given f-number (F ). From the definition of f-number, which is the ratio of focal
length (f ) to aperture diameter (D), the half-aperture angle for a given f-number is
equal to θ1/2 = atan(1/2F ). This means that for an f-number of f/16, the maximum
half-aperture collection angle will be 1.8◦ , meaning that the image sensor is collecting
nearly collimated incident light. As a result, the filter function should be consistent
with the normal incidence filter design for images taken with large f-numbers.
However, for f/1.8, the half-aperture angle is 15.5◦ , which, referring to Figs. 7.7a,b,
should have some effect on the image color quality. The extent to which the image
will be affected by off-normal incident light is determined by pixel signal integrated
over all the incidence angles accepted by its f-number. Thus, we need to consider
the transmission through a given filter color averaged over incidence angle (within
its aperture angle) and polarization (to account for unpolarized light). For the green
filter operating at f/1.8, we estimate the filter spectral response by averaging the filter
angular transmission response for incident angles within its 15 degree acceptance angle
and over both TE and TM polarizations. The resulting averaged spectrum is plotted
in Fig. 7.7c, from which we see that it is similar to the normal incidence spectrum.
Thus, we should expect to see admissible difference in the color of images taken with
large f-number (corresponding to a small acceptance angle) and a small f-number
(corresponding to relatively large acceptance angles).

400

500

600
Wavelength (nm)

700

0.3

TE data?

0.2

0.1

0.4

0.2

800

0.5
0.5

0.3

0.4

Absolute Transmission (TE)

0.5

0.1

Averaged AbsoluteTransmission

T(θ = 0 deg)

T(θ = 0-15 deg)
σT

0.4
Absolute Transmission

Absolute Transmission (TM)

Incident Angle θ (degrees)

0.3
0.2
0.1

400

500

600
Wavelength (nm)

700

800

400

500

600
Wavelength (nm)

700

d) 50mm, f/1.8 (θ1/2=15.5 deg)

e) 50mm, f/2.8 (θ1/2=10.1 deg)

f) 50mm, f/4 (θ1/2=7.1 deg)

g) 50mm, f/5.6 (θ1/2=5.1 deg)

h) 50mm, f/8 (θ1/2=3.6 deg)

i) 50mm, f/11 (θ1/2=2.6 deg)

800

Figure 7.7: Green filter angular response and integrated CMOS IS f-number dependence. Simulated spectral transmission response of green hole array filter as a function
of incident angle for a) TM and b) TE polarizations. c) Simulated spectral response of
green filter operating with a maximum half-angle aperture of 15 degrees, corresponding to an f/number of about 1.8, obtained by averaging the spectral response for
incident angles ranging from 0-15 degrees over both polarizations. Images of 24-color
Macbeth chart taken with the integrated CMOS image sensor with RGB plasmonic
hole array filter with a 50mm lens and with an f-number (maximum half-aperture
angle) of d) 1.8 (15.5 degrees), e) 2.8 (10.1 degrees), f) 4 (7.1 degrees), g) 5.6 (5.1
degrees), h) 8 (3.6 degrees), and i) 11 (2.6 degrees).

100
Table 7.1: F-numbers and their respective ∆E values
F-number
1.8 2.8 4.0 5.6 8.0
11
16
θ1/2 (degs) 15.5 10.1 7.1 5.1 3.6 2.6 1.8
Blue
4.0
Green
15.1
Red
34.4
Yellow
5.9
Magenta
34.6
Cyan
20.7
6-color ave. 19.1

5.8
14.5
31.3
5.5
27.5
31.3
19.3

2.9
11.0
32.1
7.0
31.7
30.1
19.1

3.8
10.2
28.6
5.7
24.7
33.3
17.7

3.7 7.0 3.7
10.7 8.3 10.1
25.0 21.6 27.5
5.1 5.6 5.8
23.9 23.7 24.6
31.3 33.6 32.6
16.6 16.6 17.4

To confirm the f-number dependence, we took several images of a 24-color Macbeth
chart using a 50 mm lens and changing the f-number from 1.8-16 (see Figs. 7.7d-i).
As expected, the color looks fairly constant among all the f-number images, showing slightly dimmer color for smaller f-numbers due to the larger acceptance angles.
The color is analyzed using the standard color error index, CIE ∆E 200021, which,
when averaged over the 3 primary (Red, Green, Blue) and 3 complementary (Yellow,
Magenta, Cyan) colors, shows a small variation of only 2.8 in going from f/16 to
f/1.8 (see Table 7.1). For the primary colors, note that the CIE ∆E 2000 values for
Blue are smaller than those of Green and Red across all f-numbers, demonstrating
that, independent of incident angle, the plasmonic color filter design retains its good
color fidelity for shorter wavelengths owing, to the suppression of the photonic modes,
which remain accessible for the longer wavelength filter hole diameters [143]. Further
improvement of the color fidelity between Red and Blue filters is an important direction for future research into imaging device applications of plasmonic color filters
[120, 17, 78].

7.11

Conclusion

In this chapter I have demonstrated the imaging characteristics of a CMOS image
sensor with an integrated primary color RGB plasmonic filter that is competitive
with conventional dye-filter technology in terms of transmission and insensitivity to

101
f-number (incident angle), all with a single lithographic step. The ability to add
color sensitivity to an image with a single perforated metal layer not only reduces the
complexity and cost of fabricating tri-color filters, but also allows for new technology,
such as hyperspectral imaging devices for sensing that consist of any number of color
arrays. In this work the integration was done by directly contacting the plasmonic
color filter array onto the CMOS image sensor, which already showed almost optimal integrated transmission efficiencies; however, future filter-chip integration will
require the fabrication of the plasmonic color filter directly on the CMOS image sensor. Although the hole array filter dimensions are subwavelength, with the smallest
inter-hole spacing being only 120nm for the blue (B) color filter, these dimensions
are compatible with current patterning methods using conventional deep ultraviolet
lithography an attractive feature for large scale manufacturing.

102

Chapter 8
Scattering-Absorption
Nearest-Neighbor Model
Description of Hole Arrays
Abstract: Having seen in the last two chapters how we can use finite size hole array
filters in a closed packed configuration for imaging applications with little to no spatial
crosstalk, it is interesting to investigate the transmission properties of finite-size hole
arrays as a function of nearest neighbor contributions. The analysis is similar to what
was done in Chapter 6, except that here we use a fully analytic model, in which we
account for the scattering, absorption, and transmission contributions of the holes in
the array. We can, of course, account for the scattering contributions of all the sites
in the hole array lattice, but given our findings in the last two chapters, there is an
indication that only nearest neighbors play a dominant role in setting the transmission
properties of hole arrays, especially for large finite-size filters. Following this analysis method, we demonstrate that the transmission spectra of hole array filters can
be accurately described by second nearest neighbor scattering-absorption interactions
of hole pairs, thus explaining our observations in the last two chapters and making
hole arrays especially appealing for close-packed hole array filters for imaging applications. Furthermore, using this model, we find that the peak transmission efficiency
of a square-shaped triangular-lattice hole array reaches ∼ 90% of that of an infinite
array at ∼ 6 × 6 µm2 ; the smallest size array showing near-infinite array transmission properties, making them compatible with the current size of CMOS image sensor

103
technology.

8.1

Nearest Neighbor Scattering-Absorption
Interactions

By investigating the scattering-absorption efficiencies of surface plasmons between
hole pairs in a thin-film hole-array, we find that we can reconstruct the transmission
efficiencies of large-size hole arrays using only second nearest neighbor interactions,
which can, in general, be described as a superposition of scattering and absorbing
events within the array [96, 109]. This model is based on considering each hole as
a dipole scatterer, launching cylindrical surface plasmons with efficiency β and a
Lambert’s cosine law spatial distribution when illuminated, and absorbing surface
plasmons that are incident upon it with efficiency β 0 , and subsequently re-emitting
them with efficiency β back into the array (Fig. 8.1). This interaction is modeled as
occurring on goth sides of the thin metal film, coupled by the transmission, t, of light
from the illumination-side to the transmission-side of the array.
Thus, the amount of field intensity at the mouth a given hole, at position m in
the array, will be that of the incident field, H0 , plus the scattering contributions from
the holes surrounding it, located at positions j 6= m. The fields scattered from these
sites have amplitudes H0 × βjm , where βjm is the efficiency with which the holes
scatter the incident light into circular surface plasmons. These plasmons spread with
a cosine squared circular distribution along the direction of the incident polarization,
and a 1/ r radial dependence decay to ensure constant integrated power as the wave
spreads. As usual, these plasmons also decay and accumulate phase according to the
propagation wavevector of the ’regular’ linear SP, exp[ikspp r].
In general, the mth site has scattering contributions from all sites in the lattice, at
positions j, such that j 6= m; however, here we separate the contributions in a more
suggestive way by separating them into contributions coming from the first, second,

104
etc. nearest neighbors in the array relative to the mth site of interest (Eq.8.1),
Hm,top
(8.1)
H0
cos2 (θjm − θp )
= 1+
βjm
exp[i(kspp ajm )]
ajm
j6=m
cos2 (θjm − θp )
exp[i(kspp ajm )]
= 1+
+ ... βjm
ajm
st
nd

hm,top =

j∈1 n.n.

j∈2

n.n.

where θp is the polarization direction, θjm and ajm are the angle and distance from
site j to m, respectively, and kspp is the surface plasmon propagation wavevector.
Once these interactions occur at the top side of the array, they are transmitted
across the thin film metal layer with transmission t, where the whole scatteringabsorption process takes place all over again. The field from the mth site is simply
transmitted and is contributed to by the fields transmitted and scattered by the holes
surrounding it (Eq. 8.2),
hm,bot =

Hm,bot

(8.2)

cos2 (θjm − θp )
exp[i(kspp ajm )]
jm
j6=m
cos2 (θjm − θp )
exp[i(kspp ajm )]
= hm,top +
+ ... hj,top βjm
ajm
st
nd

= hm,top +

hj,top βjm

j∈1 n.n.

j∈2

n.n.

These field amplitudes then radiate through spherical waves emanating from the
exit side of the array, summing in the far-field to form the transmission spectra of the
array. The efficiency of the array can then be defined as the field amplitude squared
of the sum of all the scattering contributions from the holes at the exit side of the
array, normalized to the amplitude squared of N times a single hole, tH0 (Eq. 8.3),

PN
ηN =

m=1 hm,bot

(8.3)

In this set of equations, Eq. 8.1–8.3,for a given wavelength, kspp is set by the

105
materials used, θp by the illumination polarization, θjm and ajm by the array configuration, and t divides out in Eq. 8.3, thus leaving only βjm , the scattering-absorption
efficiency, as the only unknown variable.

8.2

Extracting the Hole-Pair Scattering-Absorption
Coefficients

The scattering-absorption efficiencies are found by fitting finite-difference time-domain
(FDTD) transmission data to a scattering-absorption model described in Eq. 8.1–
8.3, essentially consisting of a truncated Fraby-Perot resonance between hole pairs
[96, 109]. The fit is done using FDTD to simulate the transmission spectra of a
symmetric triangular lattice unit cell consisting of a 4-hole diamond-shaped configuration (Fig. 8.2a) as a function of pitch and wavelength. The holes are modeled
as 180-nm-diameter SiO2 cylinders in a 150-nm Al film cladded form top and bottom by SiO2 , illuminated with visible frequency pulse polarized along the 1-3 hole
orientation (Fig. 8.2a). Since the scattering-absorption efficiencies are expected to
be wavelength dependent, we fit the spectra by fixing the wavelength and fitting the
hole-pair scattering-absorption model as a function of pitch [96]. Note that this configuration only consists of first nearest neighbor contributions, and thus, we account
for all the contributions of the array for the model fit.
The FDTD data, along with the fitted curve, are shown in Fig. 8.2b for λ0 =
400 nm, from which we can see excellent agreement between the simulation and
model results. We note that although subwavelength holes are known to scatter both
spherically expanding SPs [96, 54] as well as regular linear SPs [110], in this model we
only account for the scattering contribution of spherical SPs, which we find dominate
the transmission spectra of the array. Thus, fitting to the spherical surface plasmon
alone yields fits that are almost in perfect agreement with the FDTD data, but perfect
agreement is obtained only by including the linear SP contributions, thus highlighting
their lower weight in determining the overall hole array transmission spectra. By

106
varying the wavelength and repeating the fitting procedure, we extract the spectrally
resolved complex-valued scattering-absorption parameters for the 180-nm-diameter
hole (Fig. 8.2b).
By spectrally resolving this scattering-absorbing coefficient, we find that it has
a amplitude maximum at a surface plasmon wavelength that is roughly twice the
diameter of the hole (see Fig. 8.2c), i.e., d ∼ λSP /2 [28]. Since the transmission of
the array is expected to be dominated by the amplitude of the scattering-absorption
efficiency coefficient, in selecting the best suited hole diameter for filtering a given
color, we need only to look for the maximum of this curve. Thus, in looking at
Fig. 8.2c, we see that a 180-nm hole is best suited for filtering ∼550 nm (green) light.
By following a similar fitting process for different size holes, we find that the best
suited holes for filtering red (∼650 nm) and blue (∼450 nm) wavelengths are d = 240
nm and d = 140 nm, respectively, in agreement with known values [143].

8.3

Number of Contributing Nearest Neighbors

Given the combined scattering-absorption efficiencies, in order to build a nearest
neighbor model, we need to determine the spatial extent to which these efficiencies
play a role in setting the transmission spectra of finite-size hole arrays. Since we are
treating the transmission of a hole array as a superposition of scattering-absorption
events between hole pairs, in considering the contributing scattering events surrounding a given hole site, it is sensible to only consider those scattering events that occur
at hole positions that are in the line-of-sight of that hole of interest. Thus, to first order, we approximate the transmission of an array by only considering second nearest
neighbor interactions, since scattering events occurring at positions farther than this
are screened by the first and second nearest neighbors of a hole site (see Fig. 8.1a). To
validate the accuracy of the second nearest-neighbor scattering model, we compare
the transmission spectra of several large-size hole array configurations obtained by
both FDTD (Fig. 8.3) and the model (Fig. 8.1b), with both spectra in good agreement
of each other. We note that the second nearest neighbor approximation is only valid

107

Transmission Efficiency as a Function of Array Size

2nd n.n.
1st n.n.

normalized efficiency (T/T0/max[T'])

(Lx x Ly)

6.5x6.5 +m2

0.8

'-array

9.2x9.4 +m2

3.7x3.5 +m2

0.6

`’

0.4
0.2
400

500

600
wavelength(nm)

700

800

Figure 8.1: (a) Schematic of the second nearest-neighbor (n.n.) scattering-absorption
model used to reconstruct the transmission spectra of a square-shaped triangularlattice hole array (b). The black circles represent holes in a triangular lattice, connected by black lines for reference. The first and second nearest neighbors surrounding
the central lattice point are shown in red and blue, respectively. The scattering intensity of a contributing lattice site is depicted by green ovals, with the enclosed arrows
corresponding to the spatial scattering efficiency amplitudes (β). The absorption efficiencies (β 0 ) of the central lattice are depicted by dashed arrows. (b) Transmission
efficiency curves extracted from the scattering model (a) as a function of array size
for the square-shaped triangular-lattice hole array shown on the inset, consisting of
180-nm-diameter holes set at a pitch of 430 nm in a 150-nm-thick Al film embedded
in SiO2 . The red curve corresponds to a 40×40 µm array, which we call ‘∞-array’,
due to its asymptotic behavior. We normalize to the peak transmission efficiency of
this curve for reference. The other curves correspond to normalized transmission efficiencies for different size hole arrays ranging from ∼ 4 × 4 – 10 × 10 µm2 in size. The
horizontal dashed curve at 0.4 corresponds to the normalized transmission efficiency
of a single isolated hole.

108
1.5

FDTD
Model

|``'|

`24

`34

`14

x 10

q ``'

`13

efficiency (T/T )

`23

`12

b)
h = 400 nm

Transmisison Efficiency of a 4-Hole Configuration

0.5
200

400

600
800
pitch (nm)

1000

1200

400

500

600
700
wavelength(nm)

800

Figure 8.2: (b) Transmission efficiency as a function of pitch for a symmetric 4-hole
triangular-lattice unit cell (a), consisting of 180-nm-diameter holes set at a pitch of
430 nm in a 150-nm-thick Al film embedded in SiO2 . The red dotted spectrum is
extracted from FDTD simulations, and the blue dashed spectrum corresponds to the
fitted scattering-absorption model. The horizontal dashed curve at 1 corresponds to
the normalized transmission efficiency of a single isolated hole. (c) Spectrally resolved
scattering-absorption parameters obtained by varying the wavelength from 400 – 800
nm and fitting as done in (b).

for hole arrays in which the array extent is much larger than the second nearest neighbor distance, with smaller arrays requiring third and higher order nearest neighbor
contributions to properly set the in-plane phase that determines their transmission
(see Fig. 8.4).

8.4

Setting the Periodicity of the Array

Given a hole array configuration, say, composed of 29×17 180nm-diameter holes, we
note that the position of the transmission peak can be tuned by varying the period of
the array (see Fig 8.5). However, the amplitude of the transmission peak is maximum
for periodicities that set the transmission peak to have maximum overlap with the
amplitude of the SP scattering-absorption coefficient (see Figs. 8.2c and 8.5). Thus,
we find that the best suited pitch for filtering ∼550 nm (green) light with 180-nmdiameter holes is p ∼ 340 nm. By following a similar period variation for the 240-nmand 140-nm-diameter holes, we find that their optimal pitch for filtering red (∼650
nm) and blue (∼450 nm) light are 420 nm and 260 nm, respectively. These (p, d)

109

normalized efficiency (T/T0/max[T∞])

FDTD − Transmission Efficiency as a Function of Array Size
∞-array

0.8

(Lx x Ly)

9.2x8.8 μm2

6.5x6.5 μm2
3.7x4.1 μm2

0.6
0.4
0.2
400

500

600
wavelength(nm)

700

800

Figure 8.3: Normalized transmission efficiency curves extracted from FDTD as a
function of array size for the square-shaped triangular-lattice hole array shown on the
inset, consisting of 180-nm-diameter holes set at a pitch of 430 nm in a 150-nm-thick
Al film embedded in SiO2 . The red curve corresponds to an infinite array, to which
we normalize for reference. The other curves correspond to normalized transmission
efficiencies for different size hole arrays ranging from ∼ 4×4 – 10×10 µm2 in size. The
horizontal dashed curve at 0.35 corresponds to the normalized transmission efficiency
of a single isolated hole.

110

FDTD
3rd n.n.

efficiency(T/T )

2nd n.n.

400

500

600
wavelength(nm)

700

800

Figure 8.4: Transmission efficiency for different hole array configurations as a function
of number of contributing nearest neighbors (n.n.). Data is shown for spectra calculated with FDTD, as well as with the nearest neighbor scattering model with second
and third n.n. contributions. The hole array configurations (see insets) consist of
180-nm-diameter holes set at a pitch of 430 nm in a 150-nm-thick Al film embedded
in SiO2 .

111
Transmission Efficiency as a Function of Pitch

efficiency(T/T )

2.5

360 nm
340 nm
320 nm
300 nm

380 nm
400 nm

1.5
0.5
400

500

600
wavelength(nm)

700

800

Figure 8.5: Absolute transmission efficiency curves extracted from the scattering
model as a function of array pitch for a ∼ 10 × 10 µm2 square-shaped triangularlattice hole array, consisting of 180-nm-diameter holes set at a pitch of 430 nm in a
150-nm-thick Al film embedded in SiO2 .

values for filtering RGB light are verified with FDTD by doing small variations around
the predicted values.
Having found the optimal (p,d) values for filtering RGB light, we use the model
to determine the transmission properties of finite-size hole arrays as they pertain
to imaging applications. More specifically, we are looking for the smallest size hole
arrays showing near-infinite transmission properties. To do this, we calculate the
transmission spectra of a square-shaped triangular-lattice array as a function of array
size (Fig. 8.1b). Starting with a ∼ 3 × 3-µm2 array, which is on the order of a CMOS
IS pixel size, we systematically add rows and columns to the array and monitor the
evolution of the transmission efficiency spectra, normalizing to an ‘infinite’ 40×40µm2 array for reference. In looking at the asymptotic behavior of the spectra, we find
that the finite array spectra is already at ∼ 80% the peak efficiency with respect to the
infinite array for sizes as small as ∼ 4 × 4 µm2 , consisting of only ∼ 10 × 6 = 60 holes.
This value increases to ∼ 90% for array sizes ∼ 6 × 6 µm2 , with minimal incremental

112
increase after ∼ 10 × 10 µm2 . Thus, we find that ∼ 6 × 6 µm2 is the smallest array
showing near-infinite array transmission properties, with any additional increase in
size only bringing marginal benefit in transmission. These findings are substantiated
by our findings in Chapter 6, where we found a spatial crosstalk of only ∼1 µm
between different color hole filters, and ∼ 5 × 5 and ∼ 10 × 10 µm2 filters showing
similar peak transmissions in the 80 − 90% range.

8.5

Conclusion

In this chapter, we have seen how the transmission of hole arrays, once believed to
be dominated by the long range grating vector condition, are actually dominated
by nearest neighbor scattering contributions. Furthermore, we have shown that the
scattering-absorption inefficiencies of hole pairs peak at a plasmon wavelength that
is roughly twice the size of the hole diameter, i.e., d ∼ λSP /2, allowing us to easily
select the best suited hole for filtering a given wavelength of light. Similarly, as
previously reported by other groups, we find that the transmission peak position
of a hole array is tunable with the hole array pitch, but here we show that the
transmission peak amplitude is maximum when we select a pitch that has maximum
overlap with the hole-pair scattering-absorption efficiency. Then, by analyzing the
contributions of nearest neighbors, we find that second contributions are sufficient to
describe the transmission properties of large finite size hole arrays, but requiring third
contributions for smaller-size configurations. In this manner, we have shown that the
peak transmission efficiency of a hole array reaches ∼ 90% that of an infinite array at
∼ 6 × 6 µm2 , thus demonstrating their potential for full scale CMOS IS integration,
as demonstrated in Chapter 7. But the ability to add color sensitivity to an image
with a single perforated metal layer dominated by nearest neighbor interactions not
only reduces the complexity and cost of fabricating tri-color filters, it also allows for
new technology such as hyperspectral imaging devices for sensing that consist of any
number of color arrays.

113

Part III
Resonant Guided Wave Networks

114

Chapter 9
Resonant Guided Wave Networks
Abstract: In the last two sections, we have seen how plasmonic coaxial and hole
arrays can serve as efficient platforms for designing negative index metamaterials and
color filters for imaging applications. However, as mentioned in Chapter 1, one of the
most appealing aspects of plasmonics is its subwavelength confinement, which serves as
the ideal platform for designing photonic circuitry. In this section, we investigate the
properties of another new type of artificial photonic material called Resonant Guided
Wave Networks (RGWNs), in which isolated waveguides are assembled in a network
layout to form closed loop resonances, with which we can engineer material dispersion
and circuit functionality. Furthermore, we present an experimental realization of this
concept, in which we integrate it with conventional Si technology, bringing plasmonic
circuitry closer toward chip implementation.

9.1

Introduction

In the last two decades, the development of new photonic material design paradigms
has opened up new avenues for designing photonic properties based on different underlying physics. For example, photonic crystals are based on dispersive Bloch wave
modes that arise in periodic index structures. Different in operation than photonic
crystals, metamaterials [122, 114] are based on subwavelength resonant elements (or
‘meta-atoms’) that interact with incident radiation to give rise to complex refractive
indices. In this chapter, we introduce a new approach to optical dispersion control

115

Figure 9.1: Schematic illustration of (a) a 4-terminal equal power-splitting element
and (b) a local resonance in a 2x2 RGWN.

based on resonant guided wave networks (RGWNs) in which power-splitting elements
are arranged in two- and three-dimensional waveguide networks.
A possible framework for comparing and classifying photonic design paradigms
is according to their basic resonating elements, with which light interacts to give
the desired artificial dispersion. Under this classification scheme, we can think of
materials that operate based on the local interaction of waves with sub-wavelength
resonating elements (i.e. metamaterials), structures based on the nonlocal interference of Bragg periodic waves (i.e. photonic crystals), and arrays of coupled resonator
optical waveguides (CROWs), where adjacent resonators are evanescently coupled
[142]. Different from these existing concepts, the dispersion that arises in RGWNs is
a result of the multiple closed-path loops that localized guided waves form as they
propagate through a network of waveguides connected by wave-splitting elements.
The resulting multiple resonances within the network give rise to wave dispersion
that is tunable according to the network layout. These distinctive properties, which
will be described here, allow us to formulate a new method for designing photonic
components and artificial photonic materials.
A RGWN is comprised of power-splitting elements connected by isolated waveguides. The function of the splitting element is to distribute a wave entering any of
its terminals between all of its terminals, as illustrated in Fig. 9.1a. The waves
are then propagated in isolated waveguides between the splitting elements, where

116
the local waves from different waveguides are coupled together. For example, four
splitting elements arranged in a rectangular network layout form a 2×2 RGWN (see
Fig. 9.1b). When one of the terminals is excited, the multiple splitting occurrences
of the incident wave within the network form closed path resonances that reshape
the dispersion of the emerging waves according to the network layout and is different
from the dispersion of the individual waveguides. Properly designing this network
layout reshapes the interference pattern and the optical function of the RGWN, as
will be exemplified later in this chapter. The 2×2 RGWN consists of one closed loop
resonance; however, larger two- or three-dimensional networks can support multiple
resonances, which give rise to more design possibilities.
Although the concept of RGWNs is quite general, we will first illustrate the underlying physics of this paradigm using plasmonics, since it allows for a simple topological
implementation. After introducing this implementation, in the following sections we
will demonstrate how the local wave interference can be designed to engineer small
(2×2) energy storage RGWN resonators, and also how we can program the optical
transmission function of inhomogeneous RGWNs using transfer matrix formalism.
We will also address how the same design principles can be utilized to control the
optical dispersion properties of infinitely large RGWNs that behave like artificial optical materials. After addressing other possible implementation and practical issues,
we will conclude with possible future directions and a more detailed comparison to
other optical design paradigms.

9.2

Plasmonic RGWN Components

The operation of RGWNs is based on two basic components: power-splitting elements
and isolated waveguides. While the waveguides could easily be implemented using
dielectric waveguides [47], the power-splitting elements at the intersection of two such
waveguides could not be achieved using dielectrics alone. Nevertheless, this splitting
operation, which is the key enabler of this technology, is native to the intersection
of two plasmonic waveguides. Consequently, a possible implementation of a RGWN

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is by using plasmonics via a mesh of intersecting sub-wavelength air gaps in a metal
matrix.
Surface plasmon polaritons (referred here to as plasmons for brevity) are slow
surface waves that propagate at metal-dielectric interfaces. Adding another metaldielectric interface to this system results in a metal-dielectric-metal (MDM) waveguide, which supports a highly confined plasmon wave (the lowest order transverse
magnetic mode - TM0 ) that does not get structural cut-off as the dielectric gap between the metal layers becomes vanishingly small. The existence of this lowest-order
plasmonic mode in MDM waveguides allows for such plasmonic components as power
splitters [51] and high transmission sharp waveguide bends (for a review of MDM
waveguides and their possible applications, see [50]). However, the existence of metal
in the MDM waveguide configuration does add a source of a modal attenuation to the
system as a result of the usual loss mechanisms present in any real metal-containing
system. This results in a trade-off between the compression of the modal cross-section
and the modal attenuation as the air gap size is decreased. Since the loss in metals is
strongly frequency- and material-dependent, the focus here will be on RGWNs composed of Au-air-Au MDM waveguides operating at telecommunication frequencies,
where the modal propagation lengths are on the order of tens of microns, which are
substantially larger than the propagation lengths at visible frequencies. The optical
properties of the materials throughout this chapter are based on tabulated data [97].
In this implementation, the intersection of two sub-wavelength MDM waveguides
forms an X-junction that functions as the power-splitting elements in the network [51],
and the MDM segments between the intersections serve as the isolated waveguides
connecting the X-junctions. Through this implementation, X-junctions can be tuned
to split power equally at infrared wavelengths both for continuous waves and for
short pulse waves consisting of only a few optical cycles while conserving the shape
of the input signal. The observed equal-power split is a result of the subwavelength
modal cross-section of the input plasmonic waveguide that excites the junction with
a broad spectrum of plane waves. As such, equal four-way optical power-splitting is
enabled for transmission lines (e.g., MDM and coaxial configurations), but cannot be

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Figure 9.2: Power-splitting properties of the emerging pulses in an X-junction: (a)
intensity relative to the exciting pulse, and (b) phase difference at λ0 =1.5 µm [48].

easily achieved using purely dielectric waveguides, due to their half-wavelength modal
cross-section limit. Thus, through a plasmonic implementation, the strong coupling
to all four neighboring X-junctions gives the plasmonic RGWN structure an optical
response different from a cross-coupled network of purely dielectric waveguides, where
most of the power would be transmitted in the forward direction, with only weak
coupling to perpendicular waveguides.
As the MDM waveguide air gap thickness is varied, the power-split between the
X-junction terminals can be tuned both in terms of amplitude and phase [48]. This,
in addition to determining the phase accumulation in the waveguide segments, sets
independent controls in designing the interference pattern that governs the operation of a RGWN. The power-splitting in the Au/air X-junction was investigated
using the 2D finite-difference time-domain (FDTD) method with short pulse excitation and two equal-thickness intersecting MDM waveguides. Through this study, it
was found that for small (0.25-µm) MDM gaps, these plasmonic X-junctions exhibit
equal power-splitting with the reflected pulse being out-of-phase (i.e., approximately
π-phase shifted) with respect to the sideways and forward transmitted pulses. As
illustrated in Fig. 9.2a, as the MDM gap size is increased, the optical power flow
deviates from equal power-splitting between the terminals toward dominant power
transmission directly across the X-junction, which resembles the wavelength-scale
photonic mode limit. Furthermore, in these calculations, the phase shift between the

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Figure 9.3: Resonance build-up in a 2×2 RGWN. (a) Two in/out-of phase input
pulses result in destructive/constructive interference inside the network. (b) Steadystate of waves resonating in a 2×2 network where each pair of pulses excites the
X-junctions out of phase [48].

sideways (S) and the forward (F) transmitted pulses is consistent with the geometrical
difference in their pulse propagation trajectories (see Fig. 9.2b).

9.3

Resonators

After characterizing the properties of the RGWN building blocks, we illustrate the
working principles of RGWNs by investigating the dynamics of a compact 2×2 RGWN
resonator. In order to form a resonance, the network is designed such that when an
X-junction is excited from the internal ports, the exciting waves are out-of-phase,
resulting in constructive interference inside the network, as illustrated in Fig. 9.3. For
such out-of-phase excitation the fields in the external terminals interfere destructively,
and the power is coupled back into the resonator, enhancing the energy storage quality
factor (Q-factor).
When the 2×2 RGWN is excited from the lower-left arm (see Fig. 9.4), after a
transient that includes the first five splitting events, the resonant state is reached as
pairs of pulses resonate between junctions 1 and 3 (exemplified by snapshot t6 ) and
junctions 2 and 4 (exemplified by snapshot t7 ). However, before this steady state
is reached, it is instructive to follow the dynamics that lead up to this resonance.
Starting with the third power-split, this event occurs as junctions 1 and 3 are both

120
simultaneously excited by two waveguides. The incoming pulses arrive at both junctions in-phase, which would result in destructive interference inside the network if
the R and S split components of each pulse were exactly π-phase shifted. However,
the interference is not completely destructive, due to the finite size of the waveguides, which causes the phase difference to deviate from a perfect π-phase shift (in
accordance with Fig. 9.2). This power-splitting event determines how much power
couples into the network. For all future power-splitting events after the third one,
the two pulses arriving simultaneously at each junction are out-of-phase and therefore
interfere constructively inside the resonator. The trade-off between coupling power
into the resonator and maintaining it inside suggests that MDM gap sizes that are
subwavelength, but not arbitrarily small, will maximize the network resonance. To
interpret the FDTD observations and arrive at the conclusion described above, a simplified analytical description of pulse propagation in the network is derived, in which
only a few parameters are tracked: phase, amplitude, position and direction. The
pulses are assumed to travel in the waveguides and split into four new pulses upon
arrival at an X-junction. This model also illustrates the compactness of the possible
mathematical representation of RGWNs, and the importance of this advantage becomes more substantial when considering the dynamics of larger 2D and 3D network
topologies.
Calculating the Q-factor of such 2×2 RGWN resonators (Fig. 9.5) illustrates the
role of interference in generating a strong network resonance, which causes the network Q-factor to be an order of magnitude larger than what would be expected if
optical power-splitting in the X-junctions operated incoherently, i.e., we lost half the
power in each splitting event. Increasing the MDM gap size causes the phase of the
interfering waves to deviate from being π-phase shifted, resulting in a degradation
of the constructive interference inside the resonator and a decrease in the overall
network Q-factor. On the other hand, as the gap size is decreased, the plasmonic
mode attenuation increases, due to metallic losses in the waveguides. Between these
two competing effects, the maximal Q-factor value is obtained for a gap size of 250
nm. These RGWN Q-factor values are considerable for plasmonic resonators and

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Figure 9.4: Time snapshots of Hz (normalized to the instantaneous maximum value)
in a 2×2 plasmonic RGWN recorded at the third to the seventh power-splitting events
for a 2D-FDTD simulation. The MDM waveguides are 0.25 µm thick and 6 µm long
[48].

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Figure 9.5: Q-factor of 22 RGWN resonator from simulation results compared with
those resulting from incoherent power-splitting [48].

even comparable to typical values of wavelength-size dielectric resonators that are
dominated by radiation loss (e.g., a cylindrical dielectric cavity of radius 1.3λ with a
purely real refractive index of n = 2.5 surrounded with air has a Q 100). If we were to
artificially decrease the Au loss at 1.5 µm (or alternatively go to longer wavelengths),
the Q-factor of the resonator would increase appreciably (e.g., Q ∼ 750 for a 200 nm
gap width), indicating that the resonator Q-factor is primarily limited by the material
loss.

9.4

Tailoring the Optical Properties of Artificial
Materials

After studying the resonance effects in a small RGWN, we now investigate the dispersion characteristics of infinitely large 2D periodic RGWNs by modelling the structure
unit cell in FDTD with Bloch boundary conditions. Through this analysis, we find
that RGWNs exhibit wave dispersion and photonic bandgaps due to interference
effects, and that their band structure can be controlled by modifying the network
structural parameters. Two different length-scales control the network dispersion:

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Figure 9.6: Photonic band structure of infinitely large periodic RGWNs [48].

the subwavelength width of the MDM gaps determines the phase shift at each Xjunction, and the wavelength-order distance between the nodes, along with network
topology, determine the interference scheme.
The same interference dynamics that govern the energy storage in finite-size 2×2
RGWN resonators also control the optical properties of artificially-designed RGWN
materials of infinite size. If the network parameters are chosen such that a planewave
excitation at a given incidence angle results in a resonance effect similar to the one
demonstrated for the 2×2 network, then this would correspond to a forbidden state
of propagation in the photonic band diagram. Examining the optical density of states
(DOS) for different wave vectors over frequencies in the near infrared range, where the
material (Au in this case) dispersion is small, we observe a photonic band structure,
which is only due to dispersion resulting from the network topology, as shown in
Fig. 9.6a. The functionality of the infinitely large RGWN is not hindered by loss,
since its dispersion depends on the waveguide decay length being much larger than
the size of the largest resonant feedback loop that has dominant contribution to the
RGWN dispersion. Further possibilities for achieving band dispersion control are
illustrated in Fig. 9.6b, showing a flat band over a wide range of wavevectors at 130
and 170 THz, as well as the formation of a photonic bandgap between 140–160 THz,
for appropriately chosen network parameters.
The infinitely large RGWN is illustrated in Fig. 9.6c, along with a few schematic
resonance orders that represent the resonances that could arise within the network.

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The operating mechanism of the RGWN is very different from that of photonic crystals composed of metal/dielectric alternating materials. Although the schematic layout might look similar, the difference between the two classes of artificially designed
optical materials becomes clear when considering the difference in the length scales
of their composite elements. Whereas photonic crystals operate based on non-local
interaction of Bloch waves with the entire array, RGWNs rely on the interference of
local waves. Therefore, RGWNs are not sensitive to the actual topology of waveguides between junctions but only to its trajectory length, whereas the properties of
photonic crystals would greatly depend on the shape of the periodic metallic islands.
Additionally, RGWNs do not necessarily have to be periodic to operate as resonant
guided wave networks, and, for the same reason, planting a defect in a RGWN would
not have the same effect as it would in a photonic crystal.

9.5

Programming the Optical Properties of a Network

Because the underlying physics of RGWNs is based on the interference of local waves,
it allows for layouts that are inhomogeneous and non-periodic across the network. Unlike photonic crystals, which are restricted to Bragg wave effects in periodic structures,
the flexibility of RGWNs opens up design possibilities, where the wave properties are
varied across the structure. With respect to metamaterials, which could inherently
be nonhomogeneous due to the local nature of the interaction between light and
the meta-atoms, RGWNs have the advantage of having interference effects within the
network, which allows for frequency spectrum reshaping designs through these effects.
An additional unique feature of RGWNs relates to the constraints on wave propagation within the structure. Unlike other photonic designs, RGWNs have a limited
number of modes that are allowed to propagate within the structure (e.g., only the
TM0 mode for the case of the plasmonic implementation described previously). Furthermore, the waves can propagate only inside the waveguides connecting the splitting

125
elements. The different waveguides are coupled only by X-junctions, which each have
only a limited number of terminals. This level of control is beneficial for several reasons. First, the interference pattern in the network can be controlled more directly.
Second, it allows for a comprehensive mathematical representation of the RGWN
by scattering matrix (S-matrix) formalism that greatly reduces the computational
complexity of programming the network. Third, since the waveguides are isolated
from each other, their only contribution to the network is to serve as phase retardation elements between the splitting elements. As a result, the waveguide length is
the only effective parameter in its contour, as long as the bending is not too severe.
This waveguide feature allows for the network to maintain its engineered function
even when distorted. Additionally, the ability to utilize curved or bent waveguides
to accommodate long contours is useful when designing the interference pattern of
RGWNs.
These distinctive RGWN characteristics open up new opportunities for designing
photonic devices by programming the entire network rather than by assembling interconnected discrete components with traceable functions. The usual way of designing
photonic devices is to target the desired subsystem functions, map them logically into
sub-functions, and then assembling components that carry out these sub-functions
in the desired system. For example, a wavelength router could be designed using
add/drop ports where the input and output waveguides are coupled by wavelength
sensitive ring resonators [72] or by defects in a photonic crystal [43]. Similarly, in
free space optics, this function could be achieved through the use of collinear beam
splitters, each designed to deflect a desired wavelength band. In these schemes, the
couplers and waveguides are discrete components that are associated with a specific
function, and are combined in a logical way to carry out the overall system function.
An alternative approach is to use a network of components that carries out the desired function but, unlike traditional designs, there is no specific logical sub-function
associated with any individual component. While the inner connectivity of the device
will be less intuitive, it has the potential to result in more efficient designs of complex
and compact devices.

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Figure 9.7: Mathematical representation scheme of (a) a 2×2 RGWN system and its
components, (b) a waveguide component, and (c) an X-junction component [49].

One possible way of representing a system function in a RGWN is through the
use of a scattering operator that maps the set of local waves entering the device
terminals to the set of the waves exiting from the same terminals [49]. Since a
RGWN is composed of a discrete set of components (waveguides and X-junctions)
and terminals, the system function is represented by a scattering matrix (S-matrix)
connecting the vectors of the waves entering and emerging from the RGWN via the
external ports (see Fig. 9.7). Designing the system function of the RGWN is then
mathematically equivalent to designing the S-matrix to yield a desired output, given
a set of inputs.
Programming an optical function onto a network, according to the design principle
described above, will first be demonstrated for a plasmonic 2×2 RGWN, in which the
constituent MDM waveguides are allowed to differ in width, length, and contour. The
device has eight terminals, numbered from 1 to 8, as illustrated in Fig. 9.7a. The
input vector lists the complex amplitudes of the magnetic fields (H-fields) entering
the network in the eight terminals, and, similarly, the output vector describes the

127
complex H-field amplitudes of the waves exiting the network through these same
terminals.
The network S-matrix is assembled from the mathematical representation of its
components according to the network layout. As a first step, a function library
of mathematical representations is generated for all the possible network components (i.e., waveguides and X-junctions) using finite difference time domain (FDTD)
full-wave electromagnetic simulations. Once this library is established, the RGWN
S-matrix can be assembled according to the network layout. It is worth pointing
out that the S-matrix calculation scheme is almost always found to be much faster
than resolving the RGWN behavior from full-wave electromagnetic simulations, yet
reproduces the same information about the network. This becomes significant for
optimization tasks and especially as the network size increases.
To carry out this formalism, the two basic RGWN components (waveguides and
X-junctions) need to first be represented mathematically. The waveguides are mathematically represented by their complex phase retardation, determined by the complex
propagation constant of the wave and the waveguide length. The propagation constants are extracted from FDTD simulations for waveguides with various widths at
different frequencies. The X-junctions, which are comprised of two intersecting waveguides with four terminals, are mathematically represented by a (4×4) S-matrix. For a
given set of waveguide widths, the complex transmission coefficients of the X-junction
ports are extracted from FDTD simulations by measuring the amplitude and phase
of the wave transmitted to the different ports when excited from one of the terminals
at a given wavelength.
The S-matrix of the 2×2 RGWN is then assembled from the mathematical representation of its constituent components according to the network layout [49]. When
validating the field amplitude predictions of the S-matrix representation with FDTD
simulations, less than 5% difference is found for various test cases. The two major
contributions to this small deviation result from the interpolation between the parameter space points, where the library components were calculated, and from the error
added when the waveguides are bent. For cases where no interpolation or waveguide

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bending occurs, the FDTD results differ by only 1% from the S-matrix predictions.
The ability to accurately predict the RGWN interference using S-matrix representation reduces the complicated task of programming a desired optical function of a
RGWN into an efficient optimization of its S-matrix.
For example, the RGWN can be programmed by minimizing the difference between the actual network output and the desired one (for a given input), as the
network parameter space is swept across the various waveguide widths and lengths.
The optimization process then results in a set of network parameters that can be
translated to a network layout and then validated with FDTD simulations.

9.6

Multi-Chroic Filters using RGWNs

The S-matrix programming method can be exemplified by designing a 2×2 RGWN
to function as a dichroic router (Fig. 9.8a). Although simple in concept, the exercise
of setting a passive device to have different functions at different wavelengths is quite
instructive. Explicitly, the required function is to route two different wavelengths
(λ1 and λ2 ) to a different set of ports (1 and 6 for λ1 and 2 and 5 for λ2 ) when the
two bottom ports (7 and 8) are simultaneously excited with equal power. Mathematically, we can represent the device as an 8×8 S-matrix S(λ1 , λ1 ) connecting the
input and the output vectors. For both wavelengths, the input vector is nonzero for
the bottom ports (i.e. In=(0,0,0,0,0,0,1,1)) and the desired output vectors would be
Out(λ1 )=(1,0,0,0,0,1,0,0) for λ1 and Out(λ2 )=(0,1,0,0,1,0,0,0) for λ2 . Because we
do not have enough degrees of freedom in this small 2×2 network to exactly attain
the desired outputs, we instead optimize the ratio of power going to the two sets of
ports at the different wavelengths.
The optimization procedure is implemented in Matlab using the pre-calculated
mathematical representation data set of the RGWN components obtained from fullfield electromagnetic FDTD simulations excited with continuous wave sources (see
illustration in Fig. 9).
The dichroic router network is defined by eight parameters: the length and width

129

Figure 9.8: 2×2 RGWN programmed to function as a dichroic router: (a) schematic
drawing, and (b,c) time snapshots of the H-field at the two operation frequencies [49].

Figure 9.9: Flow chart of the RGWN S-matrix optimization procedure [49].

130
Waveguides

Width (µm)

Length (µm)

Lower
Side
Upper

0.47
0.31
0.38

5.40
1.34
6.60

Table 9.1: Set of optimized parameters for 2×2 RGWN dichroic router operating at
λ1 =2 µm and λ2 =1.26 µm.
of the upper, lower, and side waveguides and the two wavelengths. The waveguide
widths determine the effective index in the waveguides as well as the transmission
coefficients of the X-junctions. The optimization procedure is conducted in Matlab
by optimizing the ratio of power of the top and bottom set of sideways port for the
two different wavelengths.
After defining the optimization function, we constrain the parameter space based
on practical considerations. The parameter space includes the width and length of
the upper, lower, and side waveguides, as well as the two wavelengths of operation
(λ1 and λ2 ). We decrease the number of parameters to optimize by restricting the
device to have left-right symmetry based on the desired operation. We restrain the
design to operate in the infrared frequency range (λ0 = 1.2–2.0 µm), where the material dispersion and loss are less pronounced than in the visible. Furthermore, the
waveguide thickness is constrained to be small enough to only support the lowest
order plasmonic mode (air gap widths 100-500 nm).
The optimization procedure yields the network parameters given in Table 9.1,
which reveal that the required RGWN for color routing is distributed inhomogeneously.
When translating the optimized network parameters into the network layout, we
learn that the upper waveguide is longer than the lower one, and therefore needs
to be bent. Importing the resulting layout into FDTD, we obtain the steady state
H-field distribution shown in Fig. 9.8b and c, which show time snap shots at the two
operation wavelengths. The FDTD simulation results validate the S-matrix design,
with λ1 and λ2 clearly routed to a different set of sideways ports as illustrated in
Fig. 9.8b and c, respectively. From these FDTD results, it is also possible to observe

131
the build-up of local resonance inside the network, which results in the filtering out
of the desired output ports. We note that the transmission (‘3’ and ‘4’) and reflection
(‘7’ and ‘8’) ports from the device are not identically zero, since the device does not
have enough degrees of freedom and were therefore not included in the optimization
function.
The matrix representation can also be used to understand the interference conditions through which the RGWN accomplishes its desired function. From the known
input vector and the network S-matrix, the wave complex amplitudes can be identified
at any point in the network. For each wavelength, we resolve the excitation conditions
of the X-junctions that have the ports that are to be filtered out. For example, for
λ1 to be filtered out from terminals ‘2’ and ‘5’, we examine the excitation conditions
in X-junction ‘3’, which has four terminals. Two of the terminals are external device
ports (‘4’ and ‘5’) and the other two are internal network terminals. There is no
input signal incident on the two external ports, so it is the excitation conditions of
the remaining two junction terminals that null the output in terminal ‘5’. Indeed, the
excitation amplitudes of junction ‘3’ obtained from the S-matrix representation are
0.23e−i0.21π and 0.34ei0.64π , which are close in amplitude and ∼ π phase-shifted. This
is consistent with the results from section 3, which show that when an X-junction
is simultaneously excited π phase-shifted from two adjacent terminals, the two other
terminals will be filtered out (Fig. 9.3a). The fact that the excitations are not exactly
the same in amplitude and π phase-shifted is attributed to the additional constraints
the design has on the other wavelength as well as the limitations imposed on the
parameter space.
Similarly, the excitation conditions necessary for filtering out terminals ‘1’ and ‘6’
at λ2 (Fig. 9.8b) are examined by focusing on the S-matrix amplitudes of X-junction
‘4’. In this case, there are three terminals being excited: the lower terminal of the
X-junction (port ‘7’) is given by the network excitation, so the excitation of the other
two internal ports will determine the filtering out of port ‘6’. Intuitively, the condition
to filter out terminal ‘6’ will be simply a π phase-shifted excitation of the upper and
lower terminals of junction ‘4’, with zero excitation from the side port. From the case

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Figure 9.10: 3×3 RGWN programmed to function as a trichroic router. Time snapshots of the steady state H-field at the three operation frequencies [49]: a) λ1 , b)λ2 ,
c) λ3 .

of λ1 , we also know that additional constraints might cause a residual wave emerging
from terminal ‘6’, which could be compensated by a small amplitude excitation at
the other side terminal of junction ‘4’. Indeed, the excitation amplitudes of junction
‘4’ in the S-matrix representation are 1 in the lower terminal, 0.9ei0.82π in the upper
terminal, and 0.3e−i0.32π in the side terminal.
To further exemplify the programmability of RGWNs via S-matrix formalism, we
consider a 3×3 RGWN programmed to function as a trichroic router. In order to
implement the more complex task of routing three wavelengths, we allow for more
degrees of freedom in the network by increasing the number of components, effectively
increasing the amount of data contained. The function is defined as an extension of the
dichroic router, but here when the three bottom terminals are simultaneously excited
at three different frequencies, the frequencies are filtered out to three different sets of
side terminals, as illustrated in Fig. 9.10. The analysis results in the optimal RGWN
parameters shown in Table 9.2.
It is interesting to note that the wavelengths are not mapped monotonically to the
output terminals (i.e. from bottom/top ports as the wavelength increases/decreases),
which would be the usual case for devices relying on material dispersion, such as a
glass prism.

133
Waveguides
Vertical

Horizontal

center-bottom
Side-Bottom
Center-Top
Side-Top
Top
Middle
Bottom

Width (µm)

Length (µm)

0.45
0.10
0.45
0.10
0.29
0.26
0.30

13.25
8.15
3.55
4.00
12.80
7.30
11.95

Table 9.2: Set of optimized parameters for 3×3 RGWN trichroic router operating at
λ1 =1.59 µmm, λ2 =1.97 µm, and λ3 =1.23 µm.

9.7

Possible Implementations

The underlying physics and the working principles of the RGWNs were demonstrated
in the previous sections with an idealized 2D implementation using MDM waveguides.
However, for the same 2D network topology, as shown in Fig. 9.4, but implemented
with 3D high aspect ratio Au-air channel plasmon waveguides [13], the observed wave
dynamics are found to closely resemble that of the 2D MDM waveguide network, as
studied with 3D full-field simulations. If the aspect ratio of the channel plasmon
waveguide is high enough, the propagating mode within the channels strongly resembles the MDM gap plasmonic mode. This can, for instance, be seen in the measured
quality factors of RGWNs comprised of channel plasmon waveguides (3D simulations)
and MDM slot waveguides (2D simulations), which have Q-factor values of 82 and
83, respectively, at a wavelength of 1.5 µm. Furthermore, the two power splitting
events that define the RGWN resonant state are similar for both the channel and
MDM waveguides (Fig. 9.4).
The dispersion design in a volume can be addressed by 3D-RGWN topologies, for
example, constructing an array of orthogonally intersecting 3D networks of coaxial
Au-air waveguides aligned in a Cartesian grid (Fig. 9.11a). In this case, the four-arm
X-junction element of the 2D network is replaced by a six-arm 3D junction element.
Using 3D FDTD, we have verified that six-way equal power splitting occurs for pulsed
excitation in a coaxial Au-air waveguide junction. Like for the 2D-RGWN, the dispersion of the infinitely large periodic 3D-RGWN is predominantly determined by

134

Figure 9.11: 3D RGWN: (a) rendering of a 3D RGWN building block (6-arm junction). (b, c) Optical DOS of an infinite 3D network spaced periodically with cubic
periodic unit cell with different spacing [48].

the network parameters rather than the waveguide dispersion. This is demonstrated
by the noticeable difference in the band diagrams (Fig. 9.5b and c) obtained for two
networks comprised of the same waveguides but with different inter-node spacing.

9.8

Conclusions and Directions

RGWNs offer a different approach for designing dispersive photonic materials. Whereas
photonic crystals rely on the formation of Bloch wave states by interference of waves
diffracted from an array of periodic elements, a truly non-local phenomenon, RGWNs rely on the coherent superposition of power flowing along isolated waveguides
and splitting at X-junctions. Furthermore, in photonic crystals, the interference pattern of the diffracted waves depends on the non-local periodic spatial arrangement of
the diffracting elements, and in RGWNs it is the local network topology that determines the dispersion and resonance features. For example, in a RGWN, the coherent
wave propagation through the network is determined only by the total path length
along the waveguide and the phase shift added at a power splitting event, having
no restriction on whether the waveguides are straight or curved. Metamaterials also
feature a design approach based on the attributes of localized resonances, but their

135
dispersive properties do not depend on any length scale between resonant elements,
thus differing substantially from RGWNs. Arrays of coupled resonator optical waveguides (CROWs) feature discrete identifiable resonators that act as the energy storage
elements, and dispersion occurs as modes of adjacent resonators are evanescently coupled. By contrast, in RGWNs energy is not stored resonantly in discrete resonators,
but rather in the network of waveguides that are designed to exhibit a collective
resonant behavior.
The operation of RGWNs was demonstrated in this chapter using plasmonics,
which allowed for a simple layout and broadband range of operation; however, this
implementation also brought about substantial attenuation due to the fundamental
loss of plasmonic modes. As indicated above, the plasmonic MDM modes used here
have typical propagation lengths of about 50 microns due to metal loss. Since the
RGWN scope is broader than the field of plasmonics, it calls for an all-dielectric
implementation to mitigate the losses brought on by plasmonics. Implementing RGWNs using photonic circuitry would also address the coupling loss associated with
the difference in the modal overlaps between the plasmonic modes in the RGWN and
the interfacing dielectric optics.
This new design paradigm is based on different underlying physics and thus opens
up new directions for the design of artificial optical materials and devices. Since the
RGWN design relies on the interference of local waves, we can use these accessible
design parameters to program optical functions directly onto the network. Furthermore, the constraints on the propagation and coupling of the local waves in RGWNs
allow for the device operation to reduce to a simple mathematical representation using S-matrix formalism. This allows for the network programming to take the form
of an optimization procedure over a relatively small parameter space. The RGWN
S-matrix representation was demonstrated here where the inputs were given and the
S-matrix of the device was designed to give a desired output (e.g., routing, mode converting). However, this formalism could be extended to different type of functions,
such as sensing, in which the inputs are given and the output changes are monitored.
In this chapter, dichroic and trichroic RGWN color routing was demonstrated as

136
a proof of concept; however, incorporating more components into the RGWN, and
therefore increasing the possible degrees of freedom, could allow for more complex
devices or, alternatively, for devices with enhanced performance. Furthermore, we
exemplified the RGWN design paradigm using plasmonics, nesting a split element
simply by intersecting waveguides; still the concept is broad, and implementing the
concept using a photonic component could open new opportunities in the design of
photonic circuitry devices.

137

Chapter 10
Silicon Coupled Plasmonic
Nanocircuits: 4-way
Power-Splitters and Resonant
Networks
Abstract: One of the most compelling aspects of plasmonics is the ability to confine electromagnetic radiation in subwavelength modes at metal/dielectric interfaces
[108, 6] – a promising characteristic for miniaturizing photonic communications technology at the scale and density of electronics.

However, in order to simultane-

ously achieve low waveguide propagation loss and high mode confinement, we require chip-based hybrid photonic/plasmonic circuits [67, 124] that feature (1) lowloss silicon photonic waveguides, (2) high-confinement plasmonic waveguide building
blocks [10, 11, 50, 84, 57], and (3) methods for efficient mode coupling between them
[134, 42, 50]. The v-groove waveguide configuration supports and confines channel
slot plasmon polaritons (CPP) [84, 92, 50, 10] in a highly confined channel (similar
to the TM metal/dielectric/metal plasmonic mode) [50, 33], which have been experimentally investigated by several groups [13, 12, 137, 138, 139]. However, care must be
given to properly distinguish between CPP and the presence of the longer propagation
length corner-bound surface plasmon polariton (SPP) mode that is also supported by
the structure. In this work, we use near field scanning optical measurements (NSOM)
to demonstrate that λ0 = 1520 nm light can be coupled from conventional silicon-oninsulator ridge waveguides to subwavelength channel plasmon polariton waveguides

138
with an efficiency of 10%, consistent with FDTD calculations. By proper control
of mode polarization in the silicon-on-insulator waveguide [15], we demonstrate that
parasitic excitation of surface plasmon polaritons (SPPs) on the metal surface is suppressed, only exciting the desired CPP mode of the v-groove structure – serving as
the perfect platform for designing truly subwavelength plasmonic nano-circuit devices
such, as resonant guided wave networks (RGWNs) [48, 49]. RGWNs, through the use
of isolated waveguides to accumulate phase and waveguide crossings (x-junction) to
split power [51, 133], have been theoretically demonstrated to be programmable to serve
as plasmonic resonators [48] and color routers [49]; however, they still remain to be
experimentally demonstrated. In this work, using the Si-ridge coupled CPP waveguides
as a platform, we also demonstrate that a 90-degree CPP waveguide crossing serves as
an ultra-compact 4-way power-splitter. Furthermore, we demonstrate how the layout
of these two elements, subwavelength isolated CPP waveguides and their crossings,
can be designed to operate as a compact optical logic device [27, 53, 99] operating
at telecommunication wavelengths. The work presented here not only demonstrates
the integration of Si-photonics with truly subwavelength plasmonic waveguides [140]
[15, 85, 118, 125], but also illustrates how the crossing of these plasmonic waveguides
serve as ultra-compact 4-way power-splitters for plasmonic networks – forming the
platform for next-generation truly subwavelength integrated plasmonic circuits.

10.1

Introduction

With the size of transistors approaching the sub-nanometer scale and Si-based photonics pinned at the micrometer scale, due to the diffraction limit of light, we are
unable to easily integrate the high transfer speeds of this comparably bulky technology with the increasingly smaller architecture of state-of-the-art chip technology.
However, we find that we can bridge the gap between these two technologies by
directly coupling electrons to photons through the use of dispersive metals in plasmonics [108, 6]. Doing so allows us to access surface electromagnetic wave excitations
at metal/dielectric interfaces, a feature which both confines and enhances light in

(a)

intensity [a.u.]

139
3.0

FWHM
450 nm

2.0

-2 -1 0 1 2
position [μm]

200nm

Si ridge waveguide

Air

+35 o

(c)

(b)

1.0

900nm

plasmonic channel

grating

Si
Au
50μm

10μm

(c)

Figure 10.1: Schematic of Si-photonic/v-groove plasmonic hybrid device
and experiment. (a) Scanning electron micrograph of hybrid device overlaid with
schematic of fabricated device. Dashed lines represent geometry underneath the sample surface. Illumination condition used is schematically drawn in addition to the
definition of positive and negative excitation angles for the grating coupler. Inset
shows CPP mode profile calculated with an eigenmode solver (Lumerical FDTD v8.0)
along with an experimental NSOM scan of the modal cross-section obtained at λ0 =
1520 nm. (b) Close-up of Si-ridge/v-groove hybrid device fabricated with electron
beam lithography and focused ion beam milling. (c) Resulting experimental NSOM
scan of hybrid structure shown on (b) taken at λ0 = 1520 nm.

subwavelength dimensions – two promising characteristics for the development of integrated chip technology. However, the confinement of light in plasmonics comes
at the cost of loss in the metal, thus making plasmonics appealing for use as optical nanocircuits, but impractical for propagating light across long distances. Thus,
we turn to hybrid photonic/plasmonic systems [67, 124] as a way of achieving both
low loss propagation for inter-chip communication and high mode confinement for
chip-size compatible processing. However, this configuration also requires efficient
coupling between the silicon-based photonic and the metal-based plasmonic waveguides [134, 42, 50]. The photonic waveguide of choice is the Si-ridge waveguide, due
to its lossless propagation lengths at telecommunication wavelengths, while that of
the plasmonic waveguide is the v-groove configuration due to its supported plasmonic
modes [84, 92, 50, 10].
In this work we bridge the gap between these two technologies by demonstrat-

140
ing efficient end-coupling of the diffraction limited Si-ridge waveguide platform with
the subwavelength architecture of plasmonic v-groove waveguides at telecommunication frequencies (Fig. 10.1). The CPP mode of the v-groove waveguide is selectively coupled by proper control of the mode polarization of the silicon-on-insulator
waveguide, suppressing the SPP mode that is also supported by the v-groove structure, with efficiencies upwards of ∼ 40% for well optimized v-groove taper couplers.
Furthermore, we demonstrate how a 90-degree CPP waveguide crossing operates as
an ultra-compact equal-power-splitting element for developing truly subwavelength
plasmonic nano-circuit devices [48]. Having these two basic plasmonic network elements, waveguides and splitters, we demonstrate the possible type of useful circuitries
that subwavelength waveguides and splitters can be applied towards by studying a
plasmonic network composed of four v-groove waveguides in an evenly spaced 2×2
configuration, which is shown to function as a compact optical logic device [27, 53, 99]
at telecommunication wavelengths, routing different wavelengths in different on/off
combinations to the same set of transmission ports.

10.2

Plasmonic Modes of the V-Groove Configuration

The v-groove channel plasmon waveguide configuration [84, 92, 50, 10] supports
two fundamental plasmonic modes, the confined slot plasmon polariton (CPP) (Fig.
10.2a), residing in between the v-groove sidewalls, and the surface plasmon mode
(SPP), which is a delocalized surface wave weakly bound at the top of the v-groove
geometry by its corners (Fig. 10.2b). As evident from Figs. 10.2a,b, the CPP mode
is characterized by E-field perpendicular to the v-groove sidewalls (similar to the TM
metal/dielectric/metal plasmonic mode), while the SPP mode is characterized by Efield perpendicular to the top of the v-groove surface (similar to that of a regular
SPP mode). Thus, we get that in addition to the difference in modal volume and
confinement between these two modes, they are also characterized by orthogonal po-

141
(a) Channel plasmon polariton (CPP)

(b) Surface plasmon polariton (SPP)

L ~10μm

L ~100μm

200nm

offset

Air

Si
SiO 2

Air

900nm

740nm

1.0

(d)

measured
simulated

Normalized intensity

0.8

(e)

measured
simulated

0.6

0.4

0.2

0.0
-3000 -2000 -1000

1000

position [nm]

2000

3000

-2000 -1000

1000

2000

3000

position [nm]

Figure 10.2: Field distributions and near-field profiles of v-groove waveguide
modes. (a,b) Dominant field distributions obtained from eigenmode solver (Lumerical v8.0) calculations of CPP (Ex ) and SPP (Ey ) modes supported inside and near the
surface of the v-groove configuration. (d,e) Corresponding field intensity distributions
of CPP (d) and SPP (e) modes taken at a distance of 75 nm above the surface of the
v-groove structure. Black line scans correspond to FDTD calculations, red line scans
correspond to NSOM measurements. (c) Field distribution obtained from eigenmode
solver calculations of TE mode in Si-ridge waveguide used to excite the CPP mode
of the v-groove structure.

larizations, making for a convenient polarization-based selection rule between the two
modes.
However, the confinement also manifests itself in the propagation length of the
two modes, with the CPP mode having a propagation length of L ∼ 10µm, an order
of magnitude shorter than that of the SPP with L > 100µm. This propagation
length difference means that if we were to simply try to end-couple to the v-groove
waveguide configuration from freespace and measure the output down the v-groove
waveguide past the scattering-dominated area, we would only be able detect the SPP
mode, since it alone would have a propagation length long enough to survive past

142
the scattering dominated area. Thus, an alternative coupling scheme is required for
selectively coupling into the CPP mode, preferably one which takes advantage of the
polarization difference of the two supported v-groove modes and is compatible with
current Si-photonics.

10.3

Mode Selectivity

To address this, we fabricated a hybrid Si-photonic/v-groove plasmonic chip consisting of 300-µm-long Si-ridge waveguides end-coupled to v-groove channel waveguides
of variable length and layout (see Figs. 10.1a,b). The hybrid device was fabricated on
an SOI chip with a 220 nm Si device layer on a 2 µm buried oxide (BOX) layer, using
aligned-write electron beam lithography, reactive ion etching, electron beam evaporation, and focused ion beam milling. Arrays of 100×100-µm2 -size Au pads were
defined by electron-beam lithography using PMMA resist (MicroChem), where the
polymer was used both as a mask for SF6 -based plasma etching of the top Si layer
and as a liftoff layer for metallization. To define the required metal depth for the
v-groove waveguide configuration, the chips were successively etched using buffered
hydrofluoric acid to remove approximately 1 µm of SiO2 from the BOX layer prior to
metallization. A 900-nm Au layer was deposited into the etched regions by electronbeam evaporation. The SOI waveguides were patterned with a ridge width of 740
nm using negative-tone electron-beam resist (Micro Resist Technology ma-N 2403),
and the exposed Si was partially etched to a depth of 30 nm with a C4 F8 /O2 plasma
etching process. The v-groove channel waveguides were fabricated using multi-pass
focused ion beam (FIB) milling on the Au pads at positions corresponding to that
of the ridge waveguide (Fig. 10.1). An interface FIB ‘cleaning’ step was finally used
to ensure maximum coupling from the Si-ridge waveguide to the v-groove waveguide,
although this also added more separation between the waveguides. The resulting integrated structure then consisted of Si-ridge waveguide end-coupled to the v-groove
Au waveguide with a waveguide separation of ∼500 nm, depending on the device,
and a vertical offset of ∼ -50 nm.

143
Telecommunication wavelength light from a tunable diode laser was coupled into
the Si-ridge waveguide through a grating coupler fabricated at the distal end from
the Si-ridge/v-groove interface (see Fig. 10.1a,b), thus separating the incoupling scattering event from the excitation of the v-groove mode. With this configuration, we
were able to access the either TE (E-field parallel to the substrate, see Fig. 10.2c) or
TM (E-field perpendicular to the substrate) modes of the Si-ridge waveguide simply
by controlling the angle of excitation, due to the different phase velocities of these
two modes. By illuminating with λ0 = 1520 nm light at an incident angle of +35◦
(see Fig. 10.1a) and adjusting the polarization of the incident light, we were able to
access the Si-ridge TE mode, which matches the dominant E-field distribution of the
CPP mode (see Fig. 10.2c). The optical response of the system was measured using
a Nanonics MV2000 NSOM, where a near field optical probe was scanned over the
area of interest, collecting the near-field intensity information at the surface of the
structure, and sending it to an InGaAs APD to be recorded.
A representative NSOM measurement of a Si-ridge waveguide coupled to a vgroove waveguide at 1520 nm is shown in Fig. 10.1c, demonstrating a very clear
coupling intensity profile in going from the Si-ridge TE waveguide mode to the vgroove waveguide mode. The ridge waveguide shows a standing wave pattern due
to the reflection, which happens at its facet end. Also, there is a clear scattering
intensity field profile in the intermediate region between the waveguides. However,
this scattering is predominantly out of plane, not affecting the field intensity of the
mode excited in the v-groove waveguide.

10.4

CPP Waveguide Mode Properties

To determine the nature of the light coupled into the v-groove waveguide, we calculate
its modal properties, namely its propagation length, effective index, and intensity
full-width half-max (FWHM). The propagation length was calculated by coupling
the Si-ridge waveguides into 30-µm-long v-groove waveguides and measuring the field
decay of the light down their length (see Fig 10.3a). By fitting a decaying exponential

144
to the measured intensity decay, we measured a propagation length of L ∼10 µm for
λ0 = 1520 nm, consistent with the FDTD calculated propagation length of the CPP
mode (see Figs. 10.2a,b). The resulting data as as a function of wavelength is plotted
in Fig. 10.3c, demonstrating good agreement with the calculated CPP eigenmode of
the v-groove sructure.
Having found a coupled light propagation length of L ∼10 µm for the v-groove
waveguide, we proceeded to measure the effective mode index of the structure by
observing the standing wave pattern of short 10-µm-long waveguides, which are on
the order length of the propagation length (see Fig. 10.3a). The resulting interference
patterns for different excitation wavelengths were fitted to a sinusoid, which results
in the structure due to the reflection at the waveguide end. From this fit, we extract
an effective index of n = 1.05 at 1520 nm, consistent with the index of a bound mode.
The resulting data as a function of wavelength is plotted in Fig. 10.3d, demonstrating
good agreement with the eigenmode of the CPP mode dispersion.
Furthermore, in addition to the differences in propagation lengths, the SPP and
CPP modes also differ in their modal cross-sections, with the CPP having a single
peak FWHM of only 300 nm, and the SPP having a double peak with a total FWHM
of > 1 µm (see Figs. 10.2a,b). Thus, to definitively determine that we have selectively
coupled into the CPP mode of the v-groove channel, we also measured the FWHM of
the NSOM intensity obtained at cross-sections of the v-groove waveguide excited with
TE polarized light from the Si waveguide. A typical line scan at 1520 nm is shown in
Fig 10.2d (red curve), showing a single peak FWHM of ∼400nm, in agreement with
the CPP mode of the structure.
Lastly, we demonstrate that we can also access the SPP mode of the structure
by changing the excitation angle on the Si-ridge grating. By rotating the excitation
angle to −35◦ (see Fig. 10.1a), we were able to access the TM mode of the Si-ridge
waveguide, which matches the E-field profile of the SPP mode of the v-groove waveguide (Fig. 10.2b). By coupling in this manner, and measuring the resulting near field
distribution along the v-groove waveguide structure, we obtain the line scan shown
in Fig 10.2b, demonstrating that the measured mode is indeed double-peaked with a

145

(a)

(b)
long waveguide

short waveguide

L=10μm

L=30μm

(c)

1.20

14
13
12

simulated
measured

1.15

11
index

propagation length [μm]

(d)

simulated
measured

1.10

1.05

1450

1475

1500
wavelength [nm]

1525

1550

1450

1475

1500

1525

1550

1575

wavelength [nm]

Figure 10.3: NSOM images of CPP v-groove waveguide mode. Experimental
NSOM image taken at λ0 = 1520 nm of (a) 30 µm and (d) 10 µm long v-groove waveguides. (c) Propagation length of v-groove waveguide mode obtained by FEM calculations (black line) and by fitting the decaying NSOM intensity of long v-groove waveguides (blue dotted data). (d) Effective index of v-grove waveguide mode extracted
from eigenmode solver (Lumerical v8.0) calculations (black line) and the standing
wave pattern observed in NSOM measurements of short v-groove waveguides (blue
dotted points).

146
FWHM > 1 µm, consistent with the simulated field distribution of the SPP mode.
Thus, we have demonstrated that by proper control of the polarization in the Siridge structure, we can access either the CPP or SPP mode of the v-groove structure;
however, only the CPP mode is truly subwavelength, lending itself as the perfect infrastructure for designing confined plasmonic circuitry such as ultra-compact resonant
guided wave networks (RGWNs) [48, 51].

10.5

Ultracompact 4-Way Power-Splitters

Having verified that we can selectively couple into the subwavelength CPP mode of the
v-groove structure, we use the v-groove platform to demonstrate their functionality
as power splitting elements. Deriving from the concept of RGWNs, a power-splitting
element in plasmonic representation can be engineered by the crossing of two subwavelength mode waveguides, termed x-junctions. In Fig. 10.4a, we show a scanning
electron micrograph of such a structure, consisting of two 15-µm-long v-groove waveguides crossed at their centers at a 90◦ angle. The x-junction is excited from one of its
ports with the λ0 = 1520 nm TE mode of the Si waveguide, thus coupling into the
CPP mode as demonstrated from the single waveguide measurements.
The resulting NSOM image of the x-junction is shown in Fig. 10.4b, from which
we can clearly see that power is split amongst the four ports of the x-junction. In
addition, we also observe a standing wave pattern in each of the x-junction arms,
with that of the forward and sideways ports (relative to the excitation port) arising
from the waveguide-end reflections, and that of the excitation port coming from the
reflection at the x-junction. To extract the amount of power coupled into each port,
we fit the intensity amplitudes of each arm at the onset of the x-junction, getting
approximately equal power-splitting into each port at λ0 = 1520 nm (see Fig. 10.4c).
We note that although the x-junction scattering coefficients are complex in nature,
we are only able to extract their amplitudes from intensity measurements, due to
the non-interferometric nature of our measurements. Nevertheless, we determine the
dispersion of the splitting coefficient amplitudes by varying the incident wavelength

147

(a)

SOI

P in

(b)

Si ridge waveguide

x-junction

5μ

5 μm

splitting coefficient [%]

(c)

(d)

forward
side
reflected

P in

26
25

simulated

5 μm

22
1470

1480

1490

1500

1510

1520

1530

wavelength [nm]

Figure 10.4: Equal power-splitting x-junction plasmonic structure. (a) Scanning electron micrograph of two v-groove waveguides crossed at 90◦ , forming an xjunction coupled from one of its ports by a Si-ridge waveguide. (b) NSOM image
taken at λ0 = 1520 nm of x-junction shown in (a). (c) Splitting parameters extracted
from fitting the intensity pattern of the NSOM image shown in (b). (d) Simulated
optical response of v-groove waveguide crossing excited through one of its ports with
the CPP mode at λ0 = 1520 nm.

148
and repeating the amplitude fitting routine described above. The resulting data as
a function of wavelength is plotted in Fig. 10.4c, demonstrating that the splitting
amplitude is fairly insensitive to wavelength over the wavelength range of 60 nm.
To substantiate our findings, we investigate the expected splitting amplitudes
using FDTD, superimposed onto the measured data in Fig. 10.4c, with both sets in
good agreement with each other. For visual comparison, in Fig. 10.4d, we also plot
the simulated field intensity profile of a 200×900nm v-groove waveguide x-junction. A
CPP mode is launched from one of the x-junction arms and the resulting steady state
field is recorded at the surface of the structure. We observe from from Figs. 10.4b,d
that both near field spectra are in good agreement of each other.

10.6

2×2 Plasmonic Logical Device

As a demonstration of the possible type of useful circuitries that subwavelength waveguides and splitters can be applied towards, we fabricated a plasmonic network composed of four 15-µm v-groove waveguides in a 2×2 configuration (see Fig. 10.5a),
which is shown to operate as a compact optical logic device at telecommunication
wavelengths, routing different wavelengths in different on/off combinations to the
same set of transmission ports.
First, we investigate the device properties using FDTD, where we excite the bottom left port with a broadband CPP mode and monitor the output at the two top
ports of the structure. From Figs. 10.6a-c, we see that 1670 nm light selectively routs
to the left port (on/off configuration), 1570 nm routes to both ports (on/on configuration), and 1470 nm routes to neither (off/off configuration) – forming a wavelength
selection logical device. The optical response of the ports of interest at the operation
wavelengths is shown in Fig. 10.6d. The observed behavior is similar to that described
in our previous work [49] where we design 2×2 and 3×3 resonators to function as color
routers based on the complex scattering coefficients of the junctions and the phase
accumulated by the isolated waveguides. There we found that the network parameters can be engineered to have a different set of resonances within the network for

149

(a)

Si ridge waveguide
2x2
resonator
SOI
Au

5 μm

(b)

P in

port 2
port 1

exp.

(c)

P in

port 2

sim.

port 1

Figure 10.5: 2×2 RGWN optical logic device. (a) Scanning electron micrograph
of fabricated device consisting of four 15-µm-long waveguides in an evenly spaced 2×2
configuration, coupled from one of the arms with a Si-ridge waveguide. (d) Measured
near field response of device shown in (a), when exciting it with λ0 = 1520 nm TE
polarized light from the Si-ridge waveguide. The output ports of interest are labeled
for reference. (c) Simulated optical response of device shown in (a) when exciting one
of its arms with the CPP mode at λ0 = 1520 nm.

150
different wavelengths, causing light to route to different waveguide ports for different
wavelengths.
The fabricated device is shown in Fig. 10.5a, coupled from one of its ports by
the Si-ridge waveguide mode, which excites the CPP mode of the v-groove structure.
The resulting NSOM intensity pattern for λ0 = 1505 nm is shown in Fig. 10.5b, from
which we can see that the network resonances result in selectively coupling into port
1 of the structure (on/off configuration). For comparison, we also plot in Fig. 10.5c
the corresponding FDTD calculated field intensity at the surface of the structure,
which shows to be in good agreement with the measurement. In Fig. 10.6e we plot
the optical response of the ports of interest at the operation wavelength of λ0 = 1505
nm.

10.7

Coupling Efficiency

Having demonstrated that hybrid Si-ridge coupled v-groove structures can serve as a
useful platform for selectively coupling into the subwavelength CPP mode, which we
have further demonstrated to serve as ideal elements for formulating power-splitters
and logical devices, we proceed to investigate the coupling efficiencies associated in
going from the photonic Si-ridge waveguide mode to the subwavelength plasmonic
v-groove mode [134, 42, 50]. Previous work has been focused on the coupling to
the SPP mode of various structures, including DLSPP [66, 15, 90], v-groove mode
structures [13, 12, 137, 138, 139], and hybrid plasmo-photonic modes [76]. Here we
investigate the coupling properties in going from the large modal volume of a Si-ridge
waveguide to a subwavelength CPP waveguide mode. The modal volume of the Siridge TE mode is plotted in Fig. 10.2c, alongside that of the v-groove CPP mode
10.2d. From these images, we can see that the CPP mode is ∼1/5 the transverse size
of the Si-ridge TE mode, so that we should expect to have coupling efficiencies on
this order.
However, because the separation between the v-groove and Si-ridge waveguides
is nonzero due to fabrication limitations, this coupling efficiency is expected to be

151

(a) Port1

Port2
(off)

2μm

1470nm

(off)

P in

(b) Port1

Port2
(on)

(on)

P in

(c)

Port1
(on)

P in

1570nm

Port2
(off)

1670nm

1.0
1470nm
1570nm
1670nm

(d)

intensity [a.u.]

0.8

(e)

1670nm
measured
simulated

0.6

0.4

0.2

0.0
-6

-4

-2

position [μm]

-6

-4

-2

position [μm]

Figure 10.6: 2×2 RGWN logic device operation. (a-c) Simulated near field
intensity of plasmonic logic device consisting of four 15-µm-long v-groove waveguides
in a 2×2 configuration, excited with the CPP mode from the bottom left port at
(a) λ0 = 1470 nm, (b) λ0 = 1570 nm, and (c) λ0 = 1670 nm. The top ports of
interest are labeled along with their on/off state configuration based on the excitation
wavelength. (d) Intensity cross-sections taken at top output ports of interest at the
excitation wavelengths shown in (a-c). (e) Measured and simulated intensity response
at the output ports of interest for λ0 = 1505 nm.

152

coupling efficiency [%]

14
s=0

12
10

s = 500nm

measured

s = 1000nm

-400

-200

200

400

vertical offset [nm]

Figure 10.7: Coupling efficiency as a function of waveguide position at λ0
= 1520 nm. Horizontal axis corresponds to vertical offset between Si-ridge and
v-groove waveguides relative to their surface tops. The three dotted curves correspond to three different waveguide separations, with the blue corresponding to zero
separation, black to 500 nm separation, and red to 1000 nm separation. The green
dotted data corresponds to coupling efficiencies extracted from NSOM measurements
for wavelengths λ0 = 1500, 1510, and 1520 nm.

153
smaller than their modal volume ratios. The exact efficiency will, of course, depend
on the amount of light radiated at the interface, along with the number of accessible
modes that couple light from the Si-ridge to the CPP waveguide. However, since we
are coupling through a volume of freespace, there are an infinite number of freespace
modes which contribute to the coupling, making this calculation difficult to do analytically. Thus, we reduce the complexity of the calculation by resorting to FDTD
simulations, in which we launch light from an aligned Si-ridge waveguide to a v-groove
waveguide separated by some distance and having different offsets relative to their
top surfaces.
Thus, we calculate the coupling efficiency simply by monitoring the amount of
power transmitted into the v-groove waveguide when excited by the TE Si-ridge
mode at 1520 nm as a function of the waveguide separation and vertical offset. The
resulting transmission data is shown in Fig. 10.7, from which we can see that, indeed,
for a zero separation, we get a maximum coupling efficiency of ∼14% at an offset of
∼ −100 nm, which is only slightly lower than expected based on their modal volumes
alone. However, we see that this efficiency quickly drops as the separation is increased,
going down to < 10% for 500 nm separation, and < 5% for 1000 nm separation. From
FIB cross-sections of the fabricated device, we get that the waveguides are separated
by ∼500nm and offset by ∼ -50nm, thus placing our devices in the 10% theoretical
range for 1520 nm light.
From NSOM measurements of the Si-ridge/v-groove interface at 1520 nm, the
amount of light coupled from the Si-ridge waveguide into the v-groove CPP waveguide
can be calculated by measuring the near field intensity distribution at the Si-ridge/vgroove waveguide junction and using FDTD to relate the intensity amplitudes to the
power in the waveguides. By comparing the intensity amplitudes before the junction,
which includes contributions from the incoming and reflected photonic modes of the
structure, and after the junction, which includes the amount coupled into the CPP
waveguide mode, we calculate a coupling efficiency of ∼8%, consistent with FDTD
calculated values for similar geometries (see Fig. 10.7, green dotted data). We measure
the wavelength-dependent coupling efficiencies by varying the excitation wavelength

154
between 1490-1520 nm and repeating the power coupling calculation for the resulting
NSOM images. The resulting data is shown in Fig 10.7, showing coupling efficiencies
in the 7-8% range.

10.8

Conclusion

In this work, we have demonstrated how a hybrid Si-photonic/v-groove plasmonic
platform can serve as an efficient platform for designing integrated optical circuits
exhibiting both the low loss propagation of Si photonics for long distance data transfer, and the compact subwavelength advantages of the confined plasmonic v-groove
modes for designing compact optical devices. Furthermore, we have demonstrated
how we can use this platform to selectively couple into the CPP v-groove waveguide
mode, discarding the SPP mode based on the polarization of the incident light. The
coupling efficiencies demonstrated here, although not high, are in good agreement
with the theoretical values for direct coupling – demonstrating that for the geometry
parameters of the reported configuration, our coupling in near ideal. Lastly, we have
shown how, once coupled into the CPP mode, it can be used as a subwavelength platform for designing ultracompact power-splitters and logic devices, just two examples
of what is possible with the design of waveguides and power-splitters in integrated
subwavelength plasmonic circuitry.

155

Bibliography
[1] Surface plasmon resurrection. Nat Photon, 6(11):707–707, 11 2012.
[2] A. Alu and N. Engheta. Multifrequency optical invisibility cloak with layered
plasmonic shells. Physical Review Letters, 100(11):113901, 2008.
[3] A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta. Epsilon-near-zero
metamaterials and electromagnetic sources: Tailoring the radiation phase pattern. Physical Review B, 75(15):155410, 2007.
[4] H. A. Atwater and A. Polman. Plasmonics for improved photovoltaic devices.
Nature materials, 9(3):205–213, 2010.
[5] F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous. Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes. Physical Review B, 74(20):205419, 2006.
[6] W. L. Barnes, A. Dereux, and T. W. Ebbesen. Surface plasmon subwavelength
optics. Nature, 424(6950):824–830, 2003.
[7] W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen.
Surface plasmon polaritons and their role in the enhanced transmission of light
through periodic arrays of subwavelength holes in a metal film. Physical Review
Letters, 92(10):107401, 2004.
[8] P. Berini. Plasmon-polariton modes guided by a metal film of finite width.
Optics Letters, 24(15):1011–1013, 1999.

156
[9] P. Berini and I. De Leon. Surface plasmon-polariton amplifiers and lasers.
Nature Photonics, 6(1):16–24, 2011.
[10] S. I. Bozhevolnyi. Effective-index modeling of channel plasmon polaritons. Opt.
Express, 14(20):9467–9476, Oct 2006.
[11] S. I. Bozhevolnyi and J. Jung. Scaling for gap plasmon based waveguides. Opt.
Express, 16(4):2676–2684, Feb 2008.
[12] S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen. Channel
plasmon-polariton guiding by subwavelength metal grooves. Phys. Rev. Lett.,
95:046802, Jul 2005.
[13] S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen.
Channel plasmon subwavelength waveguide components including interferometers and ring resonators. Nature, 440(7083):508–511, 03 2006.
[14] J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. Garcia-Vidal, L. MartinMoreno, and T. Ebbesen. How light emerges from an illuminated array of
subwavelength holes. Nature Physics, 2(2):120–123, 2006.
[15] R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater. Efficient coupling between dielectric-loaded plasmonic and silicon photonic
waveguides. Nano Letters, 10(12):4851–4857, 2010.
[16] A. G. Brolo, S. C. Kwok, M. D. Cooper, M. G. Moffitt, C.-W. Wang, R. Gordon,
J. Riordon, and K. L. Kavanagh. Surface plasmon-quantum dot coupling from
arrays of nanoholes. The Journal of Physical Chemistry B, 110(16):8307–8313,
2006.
[17] S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater. A single-layer wideangle negative-index metamaterial at visible frequencies. Nature Materials,
9(5):407–412, 2010.

157
[18] W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev. Optical cloaking
with metamaterials. Nature Photonics, 1(4):224–227, 2007.
[19] P. Catrysse and B. Wandell. Integrated color pixels in 0.18-mu m complementary metal oxide semiconductor technology. Journal of the Optical Society of
America a-Optics Image Science and Vision, 20(12):2293–2306, 2003.
[20] C. Chen, A. De Castro, and Y. Shen. Surface-enhanced second-harmonic generation. Physical Review Letters, 46(2):145–148, 1981.
[21] Q. Chen, D. Chitnis, K. Walls, T. D. Drysdale, S. Collins, and D. R. S. Cumming. CMOS photodetectors integrated with plasmonic color filters. Ieee Photonics Technology Letters, 24(3):197–199, 2012.
[22] Q. Chen and D. R. S. Cumming. High transmission and low color cross-talk
plasmonic color filters using triangular-lattice hole arrays in aluminum films.
Optics Express, 18(13):14056–14062, 2010.
[23] Q. Chen, D. Das, D. Chitnis, K. Walls, T. D. Drysdale, S. Collins, and D. R. S.
Cumming. A CMOS image sensor integrated with plasmonic colour filters.
Plasmonics, 7(4):695–699, 2012.
[24] Y. J. Chen and G. M. Carter. Measurement of third order nonlinear susceptibilities by surface plasmons. Applied Physics Letters, 41(4):307–309, 1982.
[25] U. K. Chettiar, A. V. Kildishev, H.-K. Yuan, W. Cai, S. Xiao, V. P. Drachev,
and V. M. Shalaev. Dual-band negative index metamaterial: double negative at
813nm and single negative at 772nm. Optics Letters, 32(12):1671–1673, 2007.
[26] C. Clavero, K. Yang, J. Skuza, and R. Lukaszew. Magnetic field modulation of
intense surface plasmon polaritons. Optics Express, 18(8):7743–7752, 2010.
[27] M. Cohen, Z. Zalevsky, and R. Shavit. Towards integrated nanoplasmonic logic
circuitry. Nanoscale, pages –, 2013.

158
[28] F. de León-Pérez, F. J. Garcı́a-Vidal, and L. Martı́n-Moreno. Role of surface
plasmon polaritons in the optical response of a hole pair. Physical Review B,
84(12):125414, 2011.
[29] R. de Waele, S. P. Burgos, H. A. Atwater, and A. Polman. Negative refractive
index in coaxial plasmon waveguides. Optics Express, 18(12):12770–12778, 2010.
[30] R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater. Plasmon dispersion
in coaxial waveguides from single-cavity optical transmission measurements.
Nano letters, 9(8):2832–2837, 2009.
[31] R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater. Plasmon dispersion
in coaxial waveguides from single-cavity optical transmission measurements.
Nano Letters, 9(8):2832–2837, 2009.
[32] R. de Waele, A. F. Koenderink, and A. Polman. Tunable nanoscale localization
of energy on plasmon particle arrays. Nano Letters, 7(7):2004–2008, 2007.
[33] J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman. Plasmon slot
waveguides: Towards chip-scale propagation with subwavelength-scale localization. Physical Review B, 73(3):035407, 2006.
[34] J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater. Are negative index
materials achievable with surface plasmon waveguides? A case study of three
plasmonic geometries. Opt. Express, 16(23):19001–19017, Nov 2008.
[35] H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R.
Aussenegg, and J. R. Krenn. Silver nanowires as surface plasmon resonators.
Physical Review Letters, 95(25):257403, 2005.
[36] G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden. Simultaneous negative phase and group velocity of light in a metamaterial. Science,
312(5775):892–894, 2006.

159
[37] G. Dolling, M. Wegener, and S. Linden. Realization of a three-functional-layer
negative-index photonic metamaterial. Optics Letters, 32(5):551–553, 2007.
[38] G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden. Negative-index metamaterial at 780 nm wavelength. Optics Letters, 32(1):53–55, 2007.
[39] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff. Extraordinary optical transmission through sub-wavelength hole arrays. Nature,
391(6668):667–669, 1998.
[40] F. Eftekhari, C. Escobedo, J. Ferreira, X. Duan, E. M. Girotto, A. G. Brolo,
R. Gordon, and D. Sinton. Nanoholes as nanochannels: flow-through plasmonic
sensing. Analytical chemistry, 81(11):4308–4311, 2009.
[41] A. Eftekharian, H. Atikian, and A. H. Majedi. Plasmonic superconducting
nanowire single photon detector. Optics Express, 21(3):3043–3054, 2013.
[42] A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. M. Fedeli, B. de Salvo, H. A. Atwater, and R. E. de Lamaestre. Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides. Applied Physics Letters, 101(25):251117, 2012.
[43] S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus. Channel drop filters in
photonic crystals. Optics Express, 3(1):4–11, 1998.
[44] W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck. Enhanced
infrared transmission through subwavelength coaxial metallic arrays. Physical
Review Letters, 94(3):033902, 2005.
[45] N. Fang, H. Lee, C. Sun, and X. Zhang. Sub-diffraction-limited optical imaging
with a silver superlens. Science, 308(5721):534–537, 2005.
[46] N. Fang and X. Zhang. Imaging properties of a metamaterial superlens. Applied
Physics Letters, 82(2):161–163, 2003.

160
[47] E. Feigenbaum and H. Atwater. Dielectric based resonant guided wave networks.
Optics Express, 20(10):10674–10683, 2012.
[48] E. Feigenbaum and H. A. Atwater. Resonant guided wave networks. Physical
review letters, 104(14):147402, 2010.
[49] E. Feigenbaum, S. Burgos, and H. Atwater. Programming of inhomogeneous
resonant guided wave networks. Optics Express, 18(25):25584–25595, 2010.
[50] E. Feigenbaum and M. Orenstein. Modeling of complementary (void) plasmon
waveguiding. Journal of Lightwave Technology, 25(9):2547–2562, 2007.
[51] E. Feigenbaum and M. Orenstein. Perfect 4-way splitting in nano plasmonic
x-junctions. Optics Express, 15(26):17948–17953, 2007.
[52] E. Feigenbaum and M. Orenstein. Ultrasmall volume plasmons, yet with complete retardation effects. Physical Review Letters, 101(16):163902, 2008.
[53] Y. Fu, X. Hu, C. Lu, S. Yue, H. Yang, and Q. Gong. All-optical logic gates
based on nanoscale plasmonic slot waveguides. Nano Letters, 12(11):5784–5790,
2013/05/14 2012.
[54] H. Gao, J. Henzie, and T. W. Odom. Direct evidence for surface plasmonmediated enhanced light transmission through metallic nanohole arrays. Nano
letters, 6(9):2104–2108, 2006.
[55] F. Garcia-Vidal, L. Martin-Moreno, and J. Pendry. Surfaces with holes in
them: new plasmonic metamaterials. Journal of optics A: Pure and applied
optics, 7(2):S97, 2005.
[56] H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec. Surface
plasmons enhance optical transmission through subwavelength holes. Physical
Review B, 58(11):6779–6782, 1998.
[57] D. K. Gramotnev and S. I. Bozhevolnyi. Plasmonics beyond the diffraction
limit. Nat Photon, 4(2):83–91, 02 2010.

161
[58] M. I. Haftel, C. Schlockermann, and G. Blumberg. Enhanced transmission with
coaxial nanoapertures: Role of cylindrical surface plasmons. Physical Review
B, 74(23):235405, 2006.
[59] M. I. Haftel, C. Schlockermann, and G. Blumberg. Role of cylindrical surface
plasmons in enhanced transmission. Applied Physics Letters, 88(19):193104,
2006.
[60] C. Hubert, A. Rumyantseva, G. Lerondel, J. Grand, S. Kostcheev, L. Billot,
A. Vial, R. Bachelot, P. Royer, S. H. Chang, S. K. Gray, G. P. Wiederrecht, and
G. C. Schatz. Near-field photochemical imaging of noble metal nanostructures.
Nano Letters, 5(4):615–619, 2005.
[61] Y. Inaba, M. Kasano, K. Tanaka, and T. Yamaguchi. Degradation-free MOS
image sensor with photonic crystal color filter. Ieee Electron Device Letters,
27(6):457–459, 2006.
[62] D. Inoue, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya,
Y. Sugimoto, and Y. Koide. Polarization independent visible color filter comprising an aluminum film with surface-plasmon enhanced transmission through
a subwavelength array of holes. Applied Physics Letters, 98(9):093113, 2011.
[63] Z. Jacob, L. V. Alekseyev, and E. Narimanov. Optical hyperlens: Far-field
imaging beyond the diffraction limit. Optics Express, 14(18):8247–8256, 2006.
[64] P. B. Johnson and R. W. Christy. Optical constants of noble metals. Physical
Review B, 6(12):4370–4379, 1972.
[65] A. F. Koenderink, J. V. Hernandez, F. Robicheaux, L. D. Noordam, and A. Polman. Programmable nanolithography with plasmon nanoparticle arrays. Nano
Letters, 7(3):745–749, 2007.
[66] A. V. Krasavin and A. V. Zayats.

Passive photonic elements based on

dielectric-loaded surface plasmon polariton waveguides. Applied Physics Letters, 90(21):211101, 2007.

162
[67] B. Lau, M. A. Swillam, and A. S. Helmy. Hybrid orthogonal junctions: wideband plasmonic slot-silicon waveguide couplers. Opt. Express, 18(26):27048–
27059, Dec 2010.
[68] E. Laux, C. Genet, T. Skauli, and T. Ebbesen. Plasmonic photon sorters for
spectral and polarimetric imaging. Nature Photonics, 2(3):161–164, 2008.
[69] H.-S. Lee, Y.-T. Yoon, S.-S. Lee, S.-H. Kim, and K.-D. Lee. Color filter based on
a subwavelength patterned metal grating. Optics Express, 15(23):15457–15463,
2007.
[70] H. J. Lezec, J. A. Dionne, and H. A. Atwater. Negative refraction at visible
frequencies. Science, 316(5823):430–432, 2007.
[71] H. J. Lezec and T. Thio. Diffracted evanescent wave model for enhanced and
suppressed optical transmission through subwavelength hole arrays. Optics Express, 12(16):3629–3651, 2004.
[72] B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine. Microring resonator channel
dropping filters. Journal of Lightwave Technology, 15(6):998–1005, 1997.
[73] R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith. Broadband
ground-plane cloak. Science, 323(5912):366–369, 2009.
[74] Y. Liu, G. Bartal, and X. Zhang. All-angle negative refraction and imaging in a
bulk medium made of metallic nanowires in the visible region. Optics Express,
16(20):15439–15448, 2008.
[75] V. Lomakin, Y. Fainman, Y. Urzhumov, G. Shvets, et al. Doubly negative metamaterials in the near infrared and visible regimes based on thin film nanocomposites. Opt. Express, 14(23):11164–11177, 2006.
[76] F. Lou, Z. Wang, D. Dai, L. Thylen, and L. Wosinski. Experimental demonstration of ultra-compact directional couplers based on silicon hybrid plasmonic
waveguides. Applied Physics Letters, 100(24):241105, 2012.

163
[77] K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev. Ultrafast
active plasmonics. Nature Photonics, 3(1):55–58, 2008.
[78] R. Marani, A. D’Orazio, V. Petruzzelli, S. Rodrigo, L. Martin-Moreno,
F. Garcia-Vidal, and J. Bravo-Abad. Gain-assisted extraordinary optical transmission through periodic arrays of subwavelength apertures. New Journal of
Physics, 14:013020, 2012.
[79] D. Marcuse. Light transmission optics. Van Nostrand Reinhold, New York,
1972.
[80] L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry,
and T. Ebbesen. Theory of extraordinary optical transmission through subwavelength hole arrays. Physical Review Letters, 86(6):1114–1117, 2001.
[81] M. Meier and A. Wokaun. Enhanced fields on large metal particles - dynamic
depolarization. Optics Letters, 8(11):581–583, 1983.
[82] Y. Minowa, T. Fujii, M. Nagai, T. Ochiai, K. Sakoda, K. Hirao, and K. Tanaka.
Evaluation of effective electric permittivity and magnetic permeability in
metamaterial slabs by terahertz time-domain spectroscopy. Optics Express,
16(7):4785–4796, 2008.
[83] H. T. Miyazaki and Y. Kurokawa. Squeezing visible light waves into a 3-nmthick and 55-nm-long plasmon cavity. Physical Review Letters, 96(9):097401,
2006.
[84] E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I.
Bozhevolnyi. Channel plasmon-polaritons: modal shape, dispersion, and losses.
Opt. Lett., 31(23):3447–3449, Dec 2006.
[85] R. G. Mote, H.-S. Chu, P. Bai, and E.-P. Li. Compact and efficient coupler
to interface hybrid dielectric-loaded plasmonic waveguide with silicon photonic
slab waveguide. Optics Communications, 285(18):3709 – 3713, 2012.

164
[86] P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl.
Resonant optical antennas. Science, 308(5728):1607–1609, 2005.
[87] W. Murray, S. Astilean, and W. Barnes. Transition from localized surface plasmon resonance to extended surface plasmon-polariton as metallic nanoparticles
merge to form a periodic hole array. Physical Review B, 69(16):165407, 2004.
[88] J. Nakamura. Image sensors and signal processing for digital still cameras. CRC
Press, Boca Raton, 2005.
[89] S. M. Nie and S. R. Emery. Probing single molecules and single nanoparticles
by surface-enhanced raman scattering. Science, 275(5303):1102–1106, 1997.
[90] T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi. Polymerbased surface-plasmon-polariton stripe waveguides at telecommunication wavelengths. Applied Physics Letters, 82(5):668–670, 2003.
[91] M. Noginov, L. Gu, J. Livenere, G. Zhu, A. Pradhan, R. Mundle, M. Bahoura,
Y. A. Barnakov, and V. Podolskiy. Transparent conductive oxides: Plasmonic
materials for telecom wavelengths.

Applied Physics Letters, 99(2):021101–

021101, 2011.
[92] I. V. Novikov and A. A. Maradudin.

Channel polaritons.

Phys. Rev. B,

66:035403, Jun 2002.
[93] L. Novotny and C. Hafner. Light-propagation in a cylindrical waveguide with
a complex, metallic, dielectric function. Physical Review E, 50(5):4094–4106,
1994.
[94] S. M. Orbons, A. Roberts, D. N. Jamieson, M. I. Haftel, C. Schlockermann,
D. Freeman, and B. Luther-Davies. Extraordinary optical transmission with
coaxial apertures. Applied Physics Letters, 90(25):10896–10904, 2007.
[95] E. Ozbay. Plasmonics: Merging photonics and electronics at nanoscale dimensions. Science, 311(5758):189–193, 2006.

165
[96] D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A. Atwater.
Universal optical transmission features in periodic and quasiperiodic hole arrays.
Optics Express, 16(12):9222–9238, 2008.
[97] E. D. Palik and G. Ghosh. Handbook of optical constants of solids. Academic
Press, Orlando, 1985.
[98] S. Palomba and L. Novotny. Nonlinear excitation of surface plasmon polaritons
by four-wave mixing. Physical review letters, 101(5):056802, 2008.
[99] D. Pan, H. Wei, and H. Xu. Optical interferometric logic gates based on metal
slot waveguide network realizing whole fundamental logic operations. Opt. Express, 21(8):9556–9562, Apr 2013.
[100] L. Pan and D. B. Bogy. Data storage: Heat-assisted magnetic recording. Nature
Photonics, 3(4):189–190, 2009.
[101] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian.
Experimental verification and simulation of negative index of refraction using
snell’s law. Physical Review Letters, 90(10):107401, 2003.
[102] J. Parsons, E. Hendry, C. P. Burrows, B. Auguie, J. R. Sambles, and W. L.
Barnes. Localized surface-plasmon resonances in periodic nondiffracting metallic nanoparticle and nanohole arrays. Physical Review B, 79(7):073412, 2009.
[103] J. B. Pendry. Negative refraction makes a perfect lens. Physical Review Letters,
85(18):3966–3969, 2000.
[104] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Magnetism
from conductors and enhanced nonlinear phenomena. Ieee Transactions on
Microwave Theory and Techniques, 47(11):2075–2084, 1999.
[105] J. B. Pendry, D. Schurig, and D. R. Smith. Controlling electromagnetic fields.
Science, 312(5781):1780–1782, 2006.

166
[106] J. Pitarke, V. Silkin, E. Chulkov, and P. Echenique. Theory of surface plasmons
and surface-plasmon polaritons. Reports on progress in physics, 70(1):1, 2007.
[107] E. Popov, M. Neviere, S. Enoch, and R. Reinisch. Theory of light transmission
through subwavelength periodic hole arrays. Physical Review B, 62(23):16100–
16108, 2000.
[108] H. Raether. Surface plasmons on smooth and rough surfaces and on gratings.
Springer tracts in modern physics, 111, 1988.
[109] S. Ravets, J.-C. Rodier, B. Ea Kim, J.-P. Hugonin, L. Jacubowiez, and
P. Lalanne. Surface plasmons in the young slit doublet experiment. JOSA
B, 26(12):B28–B33, 2009.
[110] N. Rotenberg, M. Spasenović, T. Krijger, B. Le Feber, F. Garcı́a de Abajo,
and L. Kuipers. Plasmon scattering from single subwavelength holes. Physical
Review Letters, 108(12):127402, 2012.
[111] J. Salvi, M. Roussey, F. I. Baida, M. P. Bernal, A. Mussot, T. Sylvestre, H. Maillotte, D. Van Labeke, A. Perentes, I. Utke, C. Sandu, P. Hoffmann, and B. Dwir.
Annular aperture arrays: study in the visible region of the electromagnetic spectrum. Optics Letters, 30(13):1611–1613, 2005.
[112] M. Sandtke and L. Kuipers. Slow guided surface plasmons at telecom frequencies. Nature Photonics, 1(10):573–576, 2007.
[113] G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R.
Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman. Plasmon
dispersion relation of Au and Ag nanowires. Physical Review B, 68(15):155427,
2003.
[114] V. M. Shalaev.
1(1):41–48, 2007.

Optical negative-index metamaterials.

Nature Photonics,

167
[115] V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P.
Drachev, and A. V. Kildishev. Negative index of refraction in optical metamaterials. Optics Letters, 30(24):3356–3358, 2005.
[116] G. Sharma, W. Wu, and E. Daa. The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations.
Color Research and Application, 30(1):21–30, 2005.
[117] R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a
negative index of refraction. Science, 292(5514):77–79, 2001.
[118] P. Shi, G. Zhou, and F. S. Chau. Enhanced coupling efficiency between dielectric
and hybrid plasmonic waveguides. J. Opt. Soc. Am. B, 30(6):1426–1431, Jun
2013.
[119] G. Shvets. Photonic approach to making a material with a negative index of
refraction. Physical Review B, 67(3):035109, 2003.
[120] G. Si, Y. Zhao, H. Liu, S. Teo, M. Zhang, T. J. Huang, A. J. Danner, and
J. Teng. Annular aperture array based color filter. Applied Physics Letters,
99(3):033105, 2011.
[121] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz.
Composite medium with simultaneously negative permeability and permittivity.
Physical Review Letters, 84(18):4184–4187, 2000.
[122] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire. Metamaterials and negative
refractive index. Science, 305(5685):788–792, 2004.
[123] D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis. Electromagnetic
parameter retrieval from inhomogeneous metamaterials. Physical Review E,
71(3):036617, 2005. Part 2.

168
[124] Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu. Broadband coupler between
silicon waveguide and hybrid plasmonic waveguide. Opt. Express, 18(12):13173–
13179, Jun 2010.
[125] Y. Song, J. Wang, M. Yan, and M. Qiu. Efficient coupling between dielectric and
hybrid plasmonic waveguides by multimode interference power splitter. Journal
of Optics, 13(7):075002, 2011.
[126] B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R.
Aussenegg, A. Leitner, and J. R. Krenn. Dielectric stripes on gold as surface
plasmon waveguides. Applied Physics Letters, 88(9):094104, 2006.
[127] M. I. Stockman. Nanofocusing of optical energy in tapered plasmonic waveguides. Physical Review Letters, 93(13):137404, 2004.
[128] J. A. Stratton. Electromagnetic Theory. McGraw-Hill, New York, 1st edition,
1941.
[129] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and
X. Zhang. Three-dimensional optical metamaterial with a negative refractive
index. Nature, 455(7211):376–379, 2008.
[130] D. Van Labeke, D. Gerard, B. Guizal, F. I. Baida, and L. Li. An angleindependent frequency selective surface in the optical range. Optics Express,
14(25):11945–11951, 2006.
[131] E. Verhagen, R. de Waele, L. Kuipers, and A. Polman. Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides.
Physical review letters, 105(22):223901, 2010.
[132] E. Verhagen, A. Polman, and L. Kuipers. Nanofocusing in laterally tapered
plasmonic waveguides. Optics Express, 16(1):45–57, 2008.
[133] G. Veronis and S. Fan. Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides. Applied Physics Letters, 87(13):131102, 2005.

169
[134] G. Veronis and S. Fan. Theoretical investigation of compact couplers between
dielectric slabwaveguides and two-dimensional metal-dielectric-metal plasmonicwaveguides. Opt. Express, 15(3):1211–1221, Feb 2007.
[135] G. Veronis, S. E. Kocabas, D. A. Miller, and S. Fan. Modeling of plasmonic
waveguide components and networks. J. Comput. Theor. Nanosci, 6(8):1808–
1826, 2009.
[136] V. G. Veselago. Electrodynamics of substances with simultaneously negative
values of epsilon and mu. Soviet Physics Uspekhi-Ussr, 10(4):509, 1968.
[137] V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen. Bend loss for
channel plasmon polaritons. Applied Physics Letters, 89(14):143108, 2006.
[138] V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen. Compact
gradual bends for channel plasmon polaritons. Opt. Express, 14(10):4494–4503,
May 2006.
[139] V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martı́n-Moreno, F. J. Garcı́aVidal, E. Devaux, and T. W. Ebbesen. Nanofocusing with channel plasmon
polaritons. Nano Letters, 9(3):1278–1282, 2013/05/12 2009.
[140] J. Wang, X. Guan, Y. He, Y. Shi, Z. Wang, S. He, P. Holmström, L. Wosinski, L. Thylen, and D. Dai. Sub-µm2 power splitters by using silicon hybrid
plasmonic waveguides. Opt. Express, 19(2):838–847, Jan 2011.
[141] J. C. Weeber, M. U. Gonzalez, A. L. Baudrion, and A. Dereux. Surface plasmon
routing along right angle bent metal strips. Applied Physics Letters, 87(22),
2005.
[142] A. Yariv, Y. Xu, R. Lee, and A. Scherer. Coupled-resonator optical waveguide:
a proposal and analysis. Optics Letters, 24(11):711–713, 1999.
[143] S. Yokogawa, S. Burgos, and H. Atwater. Plasmonic color filters for CMOS
image sensor applications. Nano Letters, 12(8):4349–4354, 2012.

170
[144] S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J.
Brueck. Experimental demonstration of near-infrared negative-index metamaterials. Physical Review Letters, 95(13):137404, 2005.