MODES, EXCITATION AND APPLICATIONS OF PLASMONIC …
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i MODES, EXCITATION AND APPLICATIONS OF PLASMONIC NANO- APERTURES AND NANO-CAVITIES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Feng Wang December, 2012
Transcript of MODES, EXCITATION AND APPLICATIONS OF PLASMONIC …
i
MODES, EXCITATION AND APPLICATIONS OF PLASMONIC NANO-APERTURES AND NANO-CAVITIES
A dissertation submitted
to Kent State University in partial
fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Feng Wang
December, 2012
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UMI Number: 3531366
ii
Dissertation written by
Feng Wang
Ph.D., Kent State University, 2012
M.S., Fudan University, China, 2006
B.S., Xiamen University, China, 2001
Approved by
_________________________________ Chair, Doctoral Dissertation Committee Dr. Qi-Huo Wei
_________________________________ Members, Doctoral Dissertation Committee Dr. Hiroshi Yokoyama
_________________________________ Dr. Deng-Ke Yang
_________________________________ Dr. Mietek Jaroniec
_________________________________ Dr. Elizabeth Mann
Accepted by
________________________________ Chair, Chem. Phys. Interdisciplinary Prog. Dr. Liang-Chy Chien
________________________________ Dean, College of Arts and Sciences Dr. Raymond Craig
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TABLE OF CONTENTS
LIST OF FIGURES AND TABLES .............................................................................. VI
DEDICATION.............................................................................................................. XIV
ACKNOWLEDGEMENTS ..........................................................................................XV
CHAPTER 1 INTRODUCTION ..................................................................................... 1
1.1 Dielectric function of plasma .................................................................................... 4
1.2 Surface Plasmon Wave .............................................................................................. 6
1.3 Surface plasmons in metal-dielectric-metal system ................................................ 15
CHAPTER 2 OPTICAL TRANSMISSION THROUGH CONCENTRIC
CIRCULAR AND SPIRAL NANO-SLITS IN METAL FILMS ................... 23
2.1 Introduction ............................................................................................................. 23
2.2 Experiment---concentric circular nano-slits ............................................................ 25
2.2.1 Sample preparation ........................................................................................ 25
2.2.2 Experimental setup ........................................................................................ 26
2.2.3 Optical microscopy ....................................................................................... 26
2.2.4 Transmission spectra ..................................................................................... 28
2.3 Simulation---concentric circular nano-slits ............................................................. 29
2.4 Discussion---concentric circular nano-slits ............................................................. 32
2.5 Experiment---spiral nano-slits ................................................................................. 37
2.5.1 Sample preparation and experimental setup.................................................. 37
2.5.2 Measured transmission spectra...................................................................... 38
2.6 Discussion---spiral nano-slits .................................................................................. 40
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2.7 Conclusion ............................................................................................................... 43
CHAPTER 3 CAVITY MODES AND THEIR EXCITATIONS IN ELLIPTICAL
PLASMONIC PATCH NANOANTENNAS .................................................... 50
3.1 Introduction ............................................................................................................. 50
3.2 Sample preparation .................................................................................................. 52
3.3 Experimental and simulation results for the cavity modes ..................................... 54
3.4 Analytical expressions for the cavity modes ........................................................... 58
3.4.1 Mathieu equations ......................................................................................... 58
3.4.2 Dispersion relations of gap surface plasmon................................................. 62
3.5 Period effects ........................................................................................................... 64
3.6 Conclusion ............................................................................................................... 67
CHAPTER 4 POLARIZATION CONVERSION WITH ELLIPTICAL
PLASMONIC PATCH NANOANTENNAS .................................................... 72
4.1 Introduction ............................................................................................................. 72
4.2 Sample preparation and experimental measurement ............................................... 74
4.3 Numerically calculated cavity modes ..................................................................... 76
4.4 Polarization conversion at the wavelengths of cavity resonance ............................ 78
4.5 Lorentz oscillator model for the cavity modes ........................................................ 79
4.6 Conclusion ............................................................................................................... 82
CHAPTER 5 PLASMONIC NANOCAVITY NETWORKS ..................................... 86
5.1 Introduction ............................................................................................................. 86
5.2 Sample preparation .................................................................................................. 88
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5.3 Experimental measurements ................................................................................... 90
5.4 Numerical simulations and calculations .................................................................. 91
5.5 Discussion ............................................................................................................... 94
5.6 Conclusion ............................................................................................................. 101
CHAPTER 6 SUMMARY AND CONCLUSIONS .................................................... 106
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LIST OF FIGURES AND TABLES
Figure 1.1. Schematic for surface plasmon polaritons propagation at an interface between
metal and dielectric materials. ..................................................................................... 8
Figure 1.2. Dispersion curve of surface plasmon wave at the metal-dielectric interface
(blue). ......................................................................................................................... 13
Figure 1.3. Schematic representation of P-polarized light incident at the metal-dielectric
interface to excite the propagating surface plasmons along the interface. a) Otto
configuration; b) Kretschmann configuration. .......................................................... 14
Figure 1.4. Excited surface plasmon wave at the surface of a metal grating under normal
illumination. The arrows represent the electric field. ................................................ 15
Figure 1.5. Schematic illustration of the metal-dielectric-metal system. ......................... 16
Figure 2.1. (a) An SEM image of the circular nanoslit sample with 375nm radial period
and 145nm slit width; (b) the experimental setup for imaging and spectral
measurements. ........................................................................................................... 25
Figure 2.2. Polarized optical microscopic images of the circular nanoslits with radial
period at 375 nm (a and d); 320 nm (b and e); and 270 nm (c and f). The polarizer is
oriented all vertically; the analyzer is oriented vertically for a-c and horizontally for
d-f............................................................................................................................... 27
Figure 2.3. Measured transmission spectra for the circular nanoslits with the radial period
at 375nm (a), 320nm (b) and 270nm (c); and simulated transmission spectra for the
circular nanoslits with radial period at 375nm (d), 320nm (e), and 270nm (f)
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respectively. The red dash-dot lines represent the transmission spectra for the cross
polarized microscopy condition; the green dash lines represent the transmission
spectra for the parallel polarized microscopy condition; and the blue solid curves
represent the total transmission (i.e., without an analyzer). ...................................... 28
Figure 2.4. FDTD calculated local field distributions of the transmitted light at a plane
350 nm beneath the Ag film for the circular nanoslits of 375 nm radial period at three
representative free space wavelengths: 850 nm (a, b and c), 550 nm (d, e and f) and
400 nm (g, h and i). The left column (a, d and g) depicts the amplitudes of the
vertical components of the electric field; the middle column (b, e and h) depicts the
amplitudes of the horizontal components of the electric field; and the right column
(c, f and i) depicts the snapshots of the electric field vector distributions. The
incident polarization is vertical for all simulations. The electric field amplitudes have
been normalized by the incident field amplitude. In (c), (f) and (i), the arrow color
represents the electric field amplitude; the arrow orientation represents the electric
field direction. ............................................................................................................ 31
Figure 2.5. Dispersion curves (a) and propagation lengths (b) for the anti-symmetric TM
mode and the anti-symmetric TE mode in a Ag/air/Ag waveguide with the air gap at
145 nm, 120 nm and 90 nm respectively. .................................................................. 34
Figure 2.6. A vector distribution snapshot of the calculated local electrical field for the
transmitted light at a plane 350 nm beneath the Ag film for 270 nm radial period and
400 nm incident wavelength. ..................................................................................... 36
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Figure 2.7. Representative scanning electron microscopic pictures of the spiral
nanotrenches with 2 periods (a), 3 periods (b), and 4 periods (c). The radial
periodicity of these spirals is 370nm. ........................................................................ 37
Figure 2.8. Measured transmission spectra for the spirals of 370nm radial period with the
number of radial period is 2 (a), 3 (b), 4 (c), 5 (d) respectively The black line
represents the transmission spectra for the left circularly polarized incidence; the red
lines represent the transmission spectra for the right circularly polarized incidence. 38
Figure 2.9. Measured transmission difference between left circularly polarized incidence
and right circularly polarized incidence with the number of radial period changing
from 2 to 6. ................................................................................................................ 40
Figure 2.10. (a) The simulated phase of surface wave excited by left circularly polarized
incidence; (b) the simulated phase of surface wave excited by right circularly
polarized incidence. ................................................................................................... 41
Figure 2.11. Calculated near field distribution of Ez (electric field perpendicular to the Ti
film) for the transmitted light at 540 nm wavelength and at a plane 90nm beneath the
Ti film. (a) and (d): Left circularly polarized incidence transmit through 2-periods
and 6-periods spiral trench respectively; (b) and (e) Right circularly polarized
incidence transmit through 2-periods and 6-periods spiral trench; (c) and (f): Ez2
(polar angle averaged) of 2-periods and 6-periods spiral as a function of radius
respectively. ............................................................................................................... 42
Figure 3.1. Schematic illustration of the fabrication procedure of elliptical plasmonic
patch nanoantennas. ................................................................................................... 53
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Figure 3.2. A representative SEM image of a 2D array of plasmonic patch nanoantennas
with a period of 500nm. Two major axis radii of the patches are a=72nm, b=52nm.
The long axis of the patches is tilted about 22º from vertical direction. ................... 54
Figure 3.3. The measured and simulated reflection spectra for two representative patch
sizes: a=93nm, b=74nm (blue) and a=72nm, and b=52nm (red). The polarization of
the incident light is parallel to the long (a, b) and short axes (c, d) of the patches
respectively. The dotted curves in b and d are the local field enhancement spectra
calculated at the cavity edges. The cavity modes are indicated with the mode indices.
The e02 cavity mode is hard to locate for the smaller patch antennas (red curves) and
thus not labeled. ......................................................................................................... 55
Figure 3.4. Excitation configurations and the snapshots of simulated electrical field (Ez)
distributions for different cavity modes for the 93nm×74nm patches: (a) modes
symmetrical to x-axis and anti-symmetric to y-axis; (b) modes symmetrical to y-axis
and anti-symmetric to x-axis; (c) modes symmetric to both x- and y-axes; (d) modes
anti-symmetric to both x- and y-axes. ....................................................................... 57
Figure 3.5. Snapshots of electrical field distributions for different cavity modes.
calculated using the Mathieu functions (Eq. 3) for the 93nm×74nm patches. .......... 60
Figure 3.6. Data points represent the measured cavity resonant frequencies versus the gap
plasmon wave vector calculated using Eq. 3.6 for real patch sizes (a) and for the
effective patch sizes a΄=a+h, b΄=b+h (b). The blue solid curves represent the
dispersion curve for the gap plasmons calculated using Eq. 3.1. .............................. 63
x
Figure 3.7. Snapshots of the z-component Ezd of the electric field calculated at the middle
plane through the dielectric layer at the resonant wavelength 540 nm for two
different patch radii: 72nm×52nm (a) and 93nm×74nm (b). The illumination is
normal to the plane. ................................................................................................... 64
Figure 3.8. Calculated reflection spectra for different periods with the elliptical patch
size: a=70 nm; b=50 nm. a) and c): periods vary between 300nm to 500nm; b) and
d): periods vary between 600nm to 900nm. a) and b): Incident polarization is along
the long axis of elliptical patches for; c) and d): Incident polarization is along the
short axis of elliptical patches. .................................................................................. 67
Figure 4.1. (a) Schematic architecture of plasmonic patch nanoantenna arrays; (b) A
representative SEM picture of fabricated arrays of plasmonic patch nanoantennas
with 300nm period. .................................................................................................... 73
Figure 4.2. Measured (a-c) and calculated (d-f) reflection spectra for the plasmonic patch
nanoantenna arrays. (a) and (d) for sample a with 130×100 nm patch diameters and
300 nm period; (b) and (e) for sample b with 154×120 nm patch diameters and 300
nm period; (c) and (f) for sample c with 130×100 nm patch diameters and 500 nm
period. Red and blue curves represent the reflection spectra for the incident
polarization parallel to the long and short patch axis respectively. Black curves
represent the reflection spectra for un-polarized incident light. ................................ 75
Figure 4.3. (a-f) Snapshots of the simulated electrical field distributions in the middle
plane of the dielectric layer at the resonant frequencies for sample a. Comparing
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them with analytical theories Eq. 1 and 2 indicates that they are e11, o11, e21, e02, e12
and o12 cavity modes.................................................................................................. 76
Figure 4.4. (a-c) Measured reflection spectra at different θ for the sample a, b and c
respectively. Here β is set at 20o; and the results for θ varied only between 0o to 90o
are shown for the sake of clarity. ............................................................................... 78
Figure 4.5. (a) Measured reflection (circles) as the function of θ and fitted reflection (sold
curve) as the function of θ according to Eq. 4.3 for 560 nm wavelength (red), 720 nm
wavelength (green) and 850 nm wavelength (blue). (b) Calculated ellipticity spectra
for sample a (red), b (blue), and c (green). ................................................................ 79
Figure 4.6. (a) Phase of the magnetic field along the long and short axes of the elliptical
patch calculated inside the cavity. Circles represents fitting with the Lorentz
oscillator model. (b) Difference in phase in the magnetic field of the reflected light
calculated along the long and short axes. .................................................................. 80
Figure 4.7. Near field and reflected far field for sample a at 850 nm wavelength with
β=20°. First row: snapshots of calculated local field distributions in the middle plane
of the dielectric gap of plasmonic patch nanoantenna at different time. Second row:
Snapshots of calculated far field reflection at different time. .................................... 82
Figure 5.1. (a) Schematic one unit of the crossbar plasmonic nanocavity network; (b) a
representative SEM picture of the fabricated plasmonic nanocavities with 1.4 µm
periodicity. The bottom and top Au wires are 800nm and 780nm in width
respectively. ............................................................................................................... 90
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Figure 5.2. (a-d): Measured transmission (olive circles) and reflection (blue circles)
spectra for 4 representative sizes of the crossbar plasmonic nanocavity networks. (e-
h): Calculated transmission (olive circles) and reflection spectra (blue circles) for the
crossbar plasmonic nanocavity networks. The dashed lines are the calculated
transmission and reflection spectra for the crossbar structures of the same sizes with
the nanocavities filled with Au. (i-l): Calculated local field enhancement spectra for
normal incidence (olive) and for 20º tilted incidence (blue: s polarization, red: p-
polarization). The 4 representative top and bottom Au wire widths are at 310×330nm
for (a, e, i), 576×560nm for (b, f, j), 800×780nm for (c, g, k) and 942×870nm for (d,
h, l). The red solid lines in a-h were the best fittings with Fano resonances on top of
the Lorentzian profiles. .............................................................................................. 93
Figure 5.3. Snapshots of simulated local electrical field distributions for the cavity modes
with 800nm×780nm wire widths. (a) The first 6 cavity modes excited with a s-
polarized light at 20o tilted incidence; (b) the first 6 cavity modes excited with a p-
polarized light at 20o tilted incidence; (c) the first 3 cavity modes excited with a
normal incidence polarized at 45o to x-and y-axis. ................................................... 95
Figure 5.4. (a) Cavity resonant frequencies versus the averaged nanowire width,
(Lx+Ly)/2. The symbols represent experimental data obtained from Fano fittings; and
the solid curves represent simulation results. (b) Measured cavity resonant
frequencies versus 2 2 2 2x ym L n Lπ + . The solid curve is the dispersion curve for
the gap plasmons based on equation (1). (c) Simulated effective mode volumes
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(104Veff/λ03) for different cavity modes vs. cavity resonance frequency. (d) Measured
quality factors for different cavity modes versus cavity resonance frequency. ......... 98
Figure 5.5. Simulated transmission spectra (a), reflection spectra (b) and local field
enhancement (c) of the crossbar plasmonic nanocavity networks with the top and
bottom wire widths being 576nm and 560nm respectively. The red, green and blue
curves represent the crossbar period of 1.1μm, 1.4μm and 1.7μm respectively. .... 100
Table 5.1. Crossbar cavity excitation condition……………………………………….97
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DEDICATION
To my wife Mengna
and my parents
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ACKNOWLEDGEMENTS
I would like to express my profuse gratitude to my advisor, Dr. Qi-Huo Wei, for
offering me the valuable opportunity to study not only liquid crystals but also
nanotechnology, and providing me the time, training, equipment, guidance and
instructions for my research work. Through these five years’ studies in his group, I have
grown from an outsider to nanotechnology to a researcher knowledgeable and skillful in
nano-fabrications, computer simulations, and device design and testing. Without his great
encouragement and patience, his support and all the advices on how to pursue my career,
I could never come to this point smoothly. Dr. Wei has shared so much knowledge and
experience with me not only in scientific researches but also in all aspects of life.
Special acknowledgement should go to Dr. Thomas R. Nelson, Donald Agresta,
Kevin Leedy and Dennis Walker at Air Force Research Labs in the Wright Patterson Air
Force Base for their support and collaborations in the crossbar nanocavity project. I am
greatly thankful to Dr. Kai Sun at University of Michigan for relentlessly dedicating his
time to help us in fabricating and characterizing various experimental samples and for
many stimulating discussions. Especially for countless cases, Dr. Sun had to work with us
during his weekends and evenings to find time slots of the heavily used tools.
I’m very thankful to Dr. Antal Jakli for his advice and support during my research
rotation. His passion about liquid crystal science has always inspired my interest in soft
matter. I like to thank Dr. Deng-Ke Yang, Dr. Jonathan Selinger, Dr. Liang-Chy Chien,
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Dr. Oleg Lavrentovich, Dr. Peter Palffy-Muhoray and Dr. Philip Bos, for all the
knowledge I have learned from their excellent classes and lectures.
I’d like to thank all the group members in Dr. Wei’s lab, especially, Ayan
Chakrabarty, Dr. Bhuwan Joshi, Jakub Kolacz, Yubing Guo, Fred Minkowski for all their
help, discussions and cooperation on my research, which made my life more enjoyable.
Last but not least, I would like to thank the dissertation committee members Dr.
Hiroshi Yokoyama, Dr. Mietek Jaroniec, Dr. Deng-Ke Yang and Dr. Elizabeth Mann for
their valuable time and advices.
Feng Wang
1
CHAPTER 1
Introduction
Over the last two decades, the demand for integrating photonic devices with nano-
scale electronic devices has been vastly expanded to take advantage of the remarkable
capabilities of photonic devices in information transport and processing [1]. However, it
remains challenging to design functional photonic devices at nano-dimensions by using
conventional dielectric material, due to the physical limit of light diffraction [2]. One
kind of promising “building blocks” for integrated nano-photonics is related to the
plasmonic nanostructures. By converting the free space light wave into surface plasmon
polaritons, plasmonic devices make it possible to miniaturize photonics devices down to
deep subwavelength regime. As a result, the excitation, guiding, focusing and processing
of surface plasmon waves have been extensively studied during the latest decade [3-10].
Surface plasmons are electron density waves that exist at the interface of two
materials whose real parts of the complex permittivity are opposite in signs (e.g. a metal-
dielectric interface). Accompanying the collective electron oscillations is an
electromagnetic wave localized at the metal-dielectric interface with the field strength
decaying exponentially away from the interface. The plasmonic devices are constructed
by employing various metallic nanostructures to couple free space light into surface
plasmons. And the excited surface plasmons can be guided and concentrated at a length
scale much smaller than the wavelength of light in the free space [11, 12]. The
confinement of light into an ultra-small volume in plasmonic nanostructures naturally
2
leads to strong enhancements of local electromagnetic fields, which can promote a series
of linear/nonlinear optical effects and uphold various applications, including
extraordinary optical transmission [13, 14], perfect absorption [15, 16], nano-lasers [17],
single photon light source [18], biosensing [19, 20] and surface enhanced Raman signals
(SERS) [21, 22].
For example, the bull’s eye structure which consists of a sub-wavelength nano-
aperture surrounded by periodic concentric rings on metal surface, can extraordinarily
enhance the light transmission through the sub-wavelength aperture [23]. In bull’s eye
structure, the metallic concentric rings serve as a signal collecting antenna, which couple
the free space incident light into surface plasmons propagating towards the centric sub-
wavelength aperture, resulting in hugely enhanced local field inside the aperture and thus
an extraordinarily enhanced optical transmission.
Other than extraordinary optical transmission, the massively enhanced local field
can be utilized for bio-sensing and SERS. SERS is a surface-sensitive technique that
tremendously enhances Raman scattering by molecules adsorbed on rough metal surfaces
or between two adjacent metal nanoparticles. Once the molecules get adsorbed between
two metal particles, a metal-dielectric-metal (MDM) [24] plasmonic structure is formed.
When the incident light strikes the two adjacent metallic nanoparticles, localized surface
plasmons on each metal particle are excited and they couple with each other to form
various electromagnetic modes called gap plasmons. When the MDM structure is in
resonance with the radiation, the excited gap plasmons in MDM nanostructures can
induce enhancement of local fields by up to 1-3 orders of magnitude, leading to massive
3
increase in intensity of the Raman scattering from the adsorbed molecules. The
enhancement factor of the Raman signal of the adsorbates can be as much as 1010 to 1011,
which means the technique may even be used to detect single molecules.
In recent years, due to the difficulty to control the gap size between two metal
nanoparticles, vertical MDM designs of plasmonic nanoantennas and nanocavities have
been proposed and demonstrated with the great advantage that the nanogap thickness can
be precisely controlled by using the advanced thin film deposition techniques.
The plasmonic nano-patch antennas [15] as a variant of vertical MDM
nanoantennas are composed of flat metal nano-plates and a metal ground spaced by an
ultra-thin dielectric layer. It has been proved that the plasmonic patch antennas can
provide a new avenue towards fabrication of near perfect absorbers which possess huge
potential for greatly improving sun light harvest efficiency in solar cell [25].
In this dissertation, research results of two plasmonics related projects will be
presented. The first project is mainly focused on modeling the circular and spiral metallic
nano-gratings. Their applications in generating azimuthally and radially polarized beam
and focusing near field will be discussed. The second project is to design and fabricate
various MDM plasmonic cavities and to explore the excited resonance modes inside the
cavities. The potential applications of these studied MDM plasmonic structures in
integrated photonics will also be presented. The dissertation will be organized in the
following way: in this chapter, I will present the theoretical basis of the surface plasmons,
and the details of the simulations and the fabrication techniques of the nanoantennas. The
second chapter deals with the design and fabrication of circular and spiral nano-gratings
4
and the novel optical phenomena observed in these nanostructures. The third and the
fourth chapter will discuss on the design and fabrication of the patch nanoantennas and
their optical properties including the excited optical cavity modes. The fifth chapter will
discuss the optical properties of crossbar-plasmonic-nanocavities networks and their
potential applications in electrical addressable optoelectronic devices.
1.1 Dielectric function of plasma
Plasma can be loosely defined as an electrically neutral gas of unbound positive
ions and negative ions/electrons. Most of the highly doped semiconductors and metals
can be seen as plasmas since they comprise equal numbers of free electrons and fixed
positive ions. When these free electrons interact with incident electromagnetic waves,
they will be driven to move with no restoring forces. In contrasts to most dielectric
materials, the electrons are bound and have intrinsic resonant frequencies due to the
restoring forces of the medium. Nowadays, one promising “building block” for novel
nano-photonic devices is the plasmonic metamaterials which utilize noble metals and
highly doped semiconductors. Thus, to know the dielectric function of plasma would
greatly help the designing and devising plasmonic metamaterials.
To study the dielectric function of plasma, we can start by considering the free
electrons’ oscillations induced by the oscillating electric field E(t) of incident
electromagnetic wave [26]. Suppose the polarization direction of the incident
electromagnetic wave is along x-axis. The motion of an electron for the displacement x
can be expressed as:
5
,)( 0tieeEteExmxm ωγ −−=−=+ (1.1)
where m is the mass of electrons, γ is the coefficient of friction damping force of the
medium, ω is the frequency of light, and E0 is its amplitude. This equation means that the
driving force exerted by the light equals to the acceleration force of the electron plus the
friction damping force of the medium.
By substituting tiexx ω−−= 0 into Eq.1.1, we obtain the solution:
( )γωω
ω
imeeEtx
ti
+=
−
20)( (1.2)
The polarization P of the gas is equal to –Nex, where N is the number of electrons per
unit volume. By recalling the definitions of the electric displacement D, we can write:
( ),20
02
0000 γωωεεεε
ωω
imeENeeEPEED
titi
r +−=+==
−−
(1.3)
And thus the dielectric constant at different light frequencies can be written as the
following:
( ),1)( 20
2
γωωεωε
imNe
r +−= (1.4)
Eq. 1.4 is frequently written in the more concise form:
( ),1)( 2
2
γωωω
ωεip
r +−= (1.5)
where 21
0
2
=
mNe
p εω , is known as the plasma frequency.
Real and imaginary parts of Eq. 1.5 are:
6
2 2
' ''2 2 2 2
( ) 1 ( )( ) ( )
p pandω ω γε ω ε ωω γ ω ω γ
= − =+ +
(1.6)
The collision frequency is inverse of relaxation time (τ). At high frequencies near to ωp,
ωτ>>1 real part of the dielectric function
2
2( ) 1 pωε ω
ω= − (1.7)
When ω < ωp dielectric function becomes negative and the field exponentially decays into
metals and they behaves as reflectors. For ω > ωp dielectric function of metals is positive
and the metals become transparent and behave as a dielectrics. The plasma frequency
(ωp) is the characteristic frequency above which metals behaves as dielectrics. Plasma
oscillations are longitudinal in nature therefore light cannot couple with bulk plasmons
above the plasma frequency while below the plasma frequency there is possibility of
coupling with plasmon at the surface.
1.2 Surface Plasmon Wave
Surface Plasmons (SPs) are electromagnetic waves propagate along the interface
of metal and dielectric materials induced by the oscillations of the valence electrons at the
metal (or highly doped semiconductor) surface. SPs resonance in nanometer-sized
structures, such as metallic nanoparticles, is called Localized Surface Plasmon Resonance
(LSPR). Away from the metal-dielectric interfaces, SPs decay exponentially and remain
confined along the interfaces. The physical properties of the surface plasmons can be
understood by solving the Maxwell’s equations with appropriate boundary conditions
[27].
7
In isotropic, linear, homogeneous medium with no free charge and electric
current, the Maxwell’s equations can be expressed in four equations:
0E∇• = (1.8) 0B∇• = (1.9)
BEt
∂∇× = −
∂ (1.10)
DHt
∂∇× =
∂ (1.11)
in which E and H are electric filed and magnetic field respectively while
ED 0εε= (1.12)
HB 0µµ= (1.13)
are the electric displacement and the magnetic flux density respectively.
Combining the Eqs.1.10 and 1.11 leads to wave equation:
2
0 2DEt
µ ∂∇×∇× = −
∂ (1.14)
Eq.1.14 can be simplified as
2
22 2 0EE
c tε ∂
∇ − =∂
(1.15)
We assume the solution of electric field has a harmonic time dependence E(r, t)=E(r)e-iωt.
By inserting the solution into Eq.1.15, this yields the Helmholtz equation
2 20 0E k Eε∇ + = (1.16)
For electromagnetic waves propagating at the interface between two different
materials, we assume for simplicity the propagation is only along x-direction of a
8
Cartesian coordinate system and there is no spatial variation in y-direction. This is a one-
dimensional problem and thus the dielectric constant ε depends only on coordinate z,
ε=ε(z). The plane z=0 coincides with the interface sustaining the propagating surface
waves, which can now be described as E(r, t) = E(x, y, z)e-iωt = E(z)eiβxe-iωt or H(r, t) =
H(x, y, z)e-iωt = H(z)eiβxe-iωt. The complex parameter β=kx is called the propagation
constant of the traveling surface waves and corresponds to the component of the wave
vector in the direction of propagation. Inserting this expression into Eq.1.16 yields the
desired form of the wave equation:
( )2
2 202
( ) 0E z k Ez
ε β∂+ − =
∂ (1.17)
For harmonic time dependence ( it
ω∂= −
∂), propagation along the x-direction ( i
xβ∂
=∂
)
and homogeneity ( 0y∂=
∂) in the y-direction, we can obtain the following set of coupled
equations according to Eqs. 1.10 and 1.11:
Figure 1.1. Schematic for surface plasmon polaritons propagation at an interface
between metal and dielectric materials.
9
0y
xE i Hz
ωµ∂= −
∂ (1.18a)
0x
z yE i E i Hz
β ωµ∂− =
∂ (1.18b)
0y zi E i Hβ ωµ= (1.18c)
0y
xH i Ez
ωε ε∂=
∂ (1.18d)
0x
z yH i H i Ez
β ωε ε∂− = −
∂ (1.18e)
0y zi H i Eβ ωε ε= − (1.18f)
It can easily be shown that the six equations allow two sets of self-consistent solutions
with different polarization properties of the propagating waves. The first set are the
transverse magnetic (TM) modes, where only the field components Ex, Ez and Hy are
nonzero, and the second set is the transverse electric (TE) modes, with only Hx, Hz and Ey
being nonzero. For TM modes, the system of governing eq. 1.18 reduces to
0
1 yx
HE izωε ε
∂= −
∂ (1.19a)
0
z yE Hβωε ε
= − (1.19b)
and the wave equation for TM modes is
( )2
2 202 0y
yH k Hz
ε β∂+ − =
∂ (1.19c)
For TE modes the analogous set is
0
1 yx
EH izωµ
∂=
∂ (1.20a)
10
0
z yH Eβωµ
= (1.20b)
with the TE wave equation
( )2
2 202 0y
yE k Ez
ε β∂+ − =
∂ (1.20c)
Based on Eqs.1.19 and 1.20, we can find the solutions of propagating wave confined to
the z=0 interface, i.e. the solutions of surface plasmon waves. It will be shown that, to
sustain the surface plasmons, the materials of upper half space (i.e. z>0) should be
dielectric and non-absorbing with a positive real dielectric constant εd, while the materials
of the lower half space (z<0) have to be conducting metal which can be described via a
dielectric function εm(ω) as shown in Fig.1.1. The requirement of metallic character
implies that Re(εm)<0. For metals, this condition is fulfilled at frequencies below the bulk
plasmon frequency ωp.
We can first consider about the TM solutions. According to equation set 1.19, in
for upper half space (z>0) we have:
( ) di x k zy dH z A e eβ −= (1.21a)
0
1( ) di x k zx d
dE z iA e eβ
ωε ε−= (1.21b)
0
( ) di x k zz d
dE z A e eββ
ωε ε−= − (1.21c)
while for lower half space (z<0) we have:
( ) mi x k zy mH z A e eβ= (1.22a)
11
10
1( ) mi x k zx m
mE z iA k e eβ
ωε ε= − (1.22b)
0
( ) mi x k zz m
mE z A e eββ
ωε ε= − (1.22c)
Here, km and kd are the components of the wave vectors perpendicular to the interface in
the metal and dielectric media respectively. By applying the boundary conditions at z=0
interface that Hy and εEz should be equal, equation set 1.20, 1.21 and Eq.1.19c will lead
to the conclusions that Am=Ad and
d d
m m
kk
εε
= − (1.23a)
2 2 20m mk kβ ε= − (1.23b)
2 2 20d dk kβ ε= − (1.23c)
It should be noted here that the surface waves exist only at interfaces between materials
with opposite signs of the real part of their dielectric permittivities. Based on the equation
set 1.23, we obtain the dispersion relation of SPPs propagating at the interface between
the two half spaces
0m d
m dk ε εβ
ε ε=
+ (1.24)
For TE mode, the respective expressions for the field components are:
( ) di x k zy dE z A e eβ −= (1.25a)
0
1( ) di x k zx d dH z iA k e eβ
ωµ−= − (1.25b)
12
0
( ) di x k zz dH z A e eββ
ωµ−= (1.25c)
for upper half space (z>0), while for lower half space (z<0) we have:
( ) mi x k zy mE z A e eβ= (1.26a)
10
1( ) mi x k zx mH z iA k e eβ
ωµ= (1.26b)
0
( ) mi x k zz mH z A e eββ
ωµ= (1.26c)
Continuity of Ey and Hx at the interface requires the condition to be satisfied:
Am(km+kd) = 0 (1.27)
Since confinement to the surface requires Re(km)>0 and Re(kd)>0, this condition is only
fulfilled if Am=0, so that Ad=Am=0. Thus, no surface modes exist for TE polarization.
Surface plasmon polaritons only exist for TM polarization.
13
According to Eq. 1.24, the dispersion relation of surface plasmon at metal-
dielectric interface is plotted as in Fig. 1.2. For metal with very small electron collision
losses (Eq.1.7), the dielectric constant is real. When εm=-εd, we have the surface plasma
frequency 1sp p dω ω ε= + . The dispersion curve of surface plasmon, ω(kx), has slope
equal to / dc ε at kx=0 and then monotonically increase with kx. However, for large kx,
the slope of ω(kx) is always smaller than /x dck ε and is asymptotic to ωsp.
Figure 1.2. Dispersion curve of surface plasmon wave at the metal-dielectric interface
(blue).
14
Excitation of surface waves by photons requires the conservation of momentum
(wave vector) parallel to the surface. Surface plasmons supported by the metal-dielectric
interface cannot be directly excited by light beams since ksp=kx> kd, where kd is the
wavevector of light on the dielectric side of the interface. To effectively excite surface
plasmon, the wavevector-matching condition has to be satisfied and can be achieved in a
three-layer system, consisting of a thin insulator film with low dielectric constant
sandwiched between a metal layer and an insulator of high dielectric constant (Otto
configuration), or a thin metal layer sandwiched between two insulators with different
dielectric constants (Kretschmann configuration) as shown in Fig 1.3. For both Otto
configuration and Kretschmann configuration, surface plasmons would only exist at the
interface between the metal and the low refractive index insulators. Besides, surface
plasmons can also be excited at the surface of a metallic grating. At the grating surface,
Figure 1.3. Schematic representation of P-polarized light incident at the metal-dielectric
interface to excite the propagating surface plasmons along the interface. a) Otto
configuration; b) Kretschmann configuration.
15
the conservation of momentum requires that the surface plasmon wave vector ksp and the
component of the incident photon wave vector parallel to the grating surface kd·sinθ
satisfy ksp=kd·sinθ + n·2π/d with n a nonzero integer and d the grating period [28, 29]. For
normally incident light on a metal grating (θ=0), usually the wavelength of the first order
surface plasmon mode (n=1) equals to the period of the grating, as shown in Fig. 1.4.
1.3 Surface plasmons in metal-dielectric-metal system
For a dielectric slab sandwiched between two metal layers (the metal-dielectric-
metal system), surface plasmons can be excited at both metal-dielectric interfaces as
shown in Fig. 1.5. If the slab thickness is larger than the penetration depth in the
dielectric layer (~100 nm), surface plasmons excited at two metal-dielectric interfaces
have little interaction with each other and thus propagate independently. For dielectric
slabs with thickness smaller than 100 nm, surface plasmons at both interfaces strongly
couple with each other, resulting in separation of the degenerate mode into a symmetric
mode and an anti-symmetric mode.
Figure 1.4. Excited surface plasmon wave at the surface of a metal grating under
normal illumination. The arrows represent the electric field.
16
For the metal-dielectric-metal system illuminated by TM polarized light, the field
components at z>a can be expressed as:
( ) mi x k zyH z Ae eβ −= (1.28a)
0
1( ) mi x k zx
mE z iA e eβ
ωε ε−= (1.28b)
0
( ) mi x k zz
mE z A e eββ
ωε ε−= − (1.28c)
while at the region of z<-a, we have:
( ) mi x k zyH z Be eβ=
(1.29a)
0
1( ) mi x k zx
mE z iB e eβ
ωε ε= − (1.29b)
Figure 1.5. Schematic illustration of the metal-dielectric-metal system.
17
0
( ) mi x k zz
mE z B e eββ
ωε ε= − (1.29c)
In the core region –a<z<a, the electromagnetic field at the top and bottom metal-
dielectric interface can interfere with each other and thus the field distribution can be
expressed as:
( ) d di x k z i x k zyH z Ce e De eβ β −= + (1.30a)
0 0
1 1( ) d di x k z i x k zx
d dE z iC e e iC e eβ β
ωε ε ωε ε−= − + (1.30b)
0 0
( ) d di x k z i x k zz
d dE z C e e D e eβ ββ β
ωε ε ωε ε−= + (1.30c)
The requirement of continutity of Hy and Ex leads to
m d dk a k a k aAe Ce De− −= + (1.31a)
m d dk a k a k am d d
m d d
A C Dk e k e k eε ε ε
− −= − + (1.31b)
at z=a and
m d dk a k a k aBe Ce De− −= + (1.31a)
m d dk a k a k am d d
m d d
B C Dk e k e k eε ε ε
− −− = − + (1.31b)
at z=-a. Solving this system of linear equations results in a pair of dispersion relations,
namely:
tanh m dd
d m
kk akεε
= − (1.32a)
18
tanh d md
m d
kk akεε
= −
(1.32b)
As abovementioned, surface plasmon mode at the single interface is split into two
modes because of the strong coupling between the surface plasmon of the two metal-
dielectric interfaces. Eq.1.32a describes the anti-symmetric modes, for which the Ex (z)
component has an odd vector parity, while Eq.1.32b describes symmetric modes, for
which Ex (z) is an even function [24, 30]. The symmetric modes have much higher
energy than the anti-symmetric modes and usually exist at very high frequencies range
[31]. With decreasing the dielectric gap thickness, the energy split between the two
modes will increase due to the enhanced coupling. For anti-symmetric modes, the
amplitude of the electric field inside the dielectric gap is 101 to 103 stronger than incident
electric field from free space, which promises many high impact applications in
photonics.
19
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23
CHAPTER 2
Optical Transmission through Concentric Circular and Spiral Nano-slits in Metal
Films
2.1 Introduction
In this chapter, optical transmission through two types of plasmonic nano-
apertures is studied in both experiments and numerical simulations. The first type of
nano-apertures is composed of metallic concentric circular nano-slits while the second
type is made of Archimedean spiral nano-slits.
The observation of extraordinary optical transmission (EOT) through
subwavelength scale metallic apertures was first reported in 1998 by Ebbesen and his
coworkers [1]. In their experiments, light was normally incident on a metal-film which
was perforated by subwavelength holes, and the transmission exhibited peak intensities
that were much higher than classically predicted [2]. Two different physical mechanisms
have been identified responsible for the enhancements of optical transmission. The
dependence of the enhanced transmission on the grating periodicity is primarily attributed
to the Wood anomalies and the excitation of surface plasmons at the metal surfaces [3-8],
while the effects of aperture shapes are due to the Fabry-Perot type of cavity mode
resonances inside individual subwavelength apertures [9-13]. It has been shown that the
coupling of surface plasmons with other diffraction orders may lead to the suppression of
24
transmission in one-dimensional (1D) slit array [14]. It has also been shown that the
enhanced transmission can be obtained through cavity resonances only [15, 16].
The abovementioned EOT effects are usually observed when light transmit
through either 2D arrays of subwavelength holes or 1D array of subwavelength slits, i.e.
there is a translational repetition of the subwavelength apertures. Actually, enhanced
optical transmission can also be observed in subwavelength apertures with radial
repetition, such as circular gratings and Archimedean spiral nano-slits.
A main topic of this chapter is that a beam of linearly polarized incident light
would become a cylindrical vector beam after transmitting through metallic gratings with
radial repetition (circular or spiral gratings). Cylindrical vector beams, or laser beams
with cylindrical symmetry in polarization have attracted considerable research recently
due to their interesting properties and potential applications [17]. For example, radial
polarization is optimal for laser machining [18, 19]. The laser machining with a radially
polarized beam can be about 2 times more efficient than with a linear polarized beam due
to the polarization dependence of absorption of metal materials [19]. It has also been
shown that a radially polarized light beam can be focused to a much sharper spot than a
linearly polarized light, an effect that might find unique applications in various optical
instruments and devices such as lithography, confocal microscopy, and optical data
storage, as well as in particle trapping [20]. While a myriad of methods have been
developed to generate cylindrical vector polarized light beams, most of these methods
require complex and bulky optical setups [17]; and a miniaturized device for cylindrical
beam generation is highly demanded.
25
2.2 Experiment---concentric circular nano-slits
2.2.1 Sample preparation
The concentric circular nanoslits were perforated in a 100nm Ag film on a fused
silica substrate by using a focused ion beam (FIB) (FEI Nova Nanolab Dual Beam
Workstation). In order to promote the adhesion between the Ag film and the glass
substrate, a 10nm Ti adhesion layer is evaporated on the glass substrate before the Ag
film deposition. Three different radial periods, 270 nm, 320 nm and 375 nm, have been
used in the experiments. The width of the circular slits is designed to be about one third
Figure 2.1. (a) An SEM image of the circular nanoslit sample with 375nm radial
period and 145nm slit width; (b) the experimental setup for imaging and spectral
measurements.
26
of the radial period, which is approximately reproduced in the fabricated samples as can
be seen from the scanning electron microscopic (SEM) pictures (Fig.2.1a).
2.2.2 Experimental setup
For optical measurements, the samples are illuminated from the metal film side
with a tungsten halogen white light source, and the transmitted light is collected by using
a 40× objective (NA=0.6) on an inverted optical microscope. Although the
subwavelength radial periods of the nanoslits exclude the grating effects, the transmitted
light beam is slightly divergent due to the Fraunhofer diffraction caused by the finite size
of the nanoslits. Simple estimation can yield that the objective with 0.6 numerical
aperture can collect more than 90% of the transmitted light for a 3μm diameter aperture
(~the smallest sample in our experiments). A circular pinhole of 300μm in diameter
located at the image plane of a side exit of the microscope is used to select the area of
interest and block the background and stray light. The light transmitted through circular
nanoslits is imaged into this pinhole and then coupled into the entrance slit of an imaging
spectrograph for spectral measurements (Fig.2.1b). Through a switchable mirror in the
optical microscope, the light transmitted through those nanoslits can also be imaged on to
a color CCD camera. With this setup, the measured optical spectra can be correlated with
optical microscopic images of the samples.
2.2.3 Optical microscopy
In the experiments, the incident light is linearly polarized, and a polarization
analyzer is placed after the microscope objective. Under the optical microscope, these
27
circular nanoslits show vivid colors which are spatially inhomogeneous and vary with
their radial periods (Fig. 2.2). When the analyzer is parallel to the polarization of the
incident light (parallel polarized microscopy), the images of these circular slits exhibit
color fan textures (Fig. 2.2a-c). When the analyzer is perpendicular to the incident
polarization (cross polarized microscopy), the optical images exhibit a different type of
fan texture patterns with black crosses (Fig. 2.2d-f). These black crosses indicate that the
polarization of the transmitted light in these black regions is linear and parallel to the
polarization of the incident light. It is important to note that these colors observed with
the CCD camera are slightly different from the measured spectra because the spectrum of
Figure 2.2. Polarized optical microscopic images of the circular nanoslits with radial
period at 375 nm (a and d); 320 nm (b and e); and 270 nm (c and f). The polarizer is
oriented all vertically; the analyzer is oriented vertically for a-c and horizontally for d-f.
28
the illuminating halogen lamp can not be properly normalized with the color CCD
camera.
2.2.4 Transmission spectra
We measured the total transmission spectra without the polarization analyzer and
the transmission spectra under parallel and cross polarized microscopic conditions (Fig.
2.3a-c). The results show that the transmission of the electrical field parallel to the
Figure 2.3. Measured transmission spectra for the circular nanoslits with the radial
period at 375nm (a), 320nm (b) and 270nm (c); and simulated transmission spectra
for the circular nanoslits with radial period at 375nm (d), 320nm (e), and 270nm
(f) respectively. The red dash-dot lines represent the transmission spectra for the
cross polarized microscopy condition; the green dash lines represent the
transmission spectra for the parallel polarized microscopy condition; and the blue
solid curves represent the total transmission (i.e., without an analyzer).
29
incident polarization is much higher than that perpendicular to the incident polarization.
A broad transmission peak can be observed in the total transmission spectra with the peak
wavelength dependent on the radial period of the nanoslits. For the radial period
decreased from 375nm to 270nm, the resonant wavelength decreases from 550nm to
400nm.
2.3 Simulation---concentric circular nano-slits
In order to understand the relationship between the transmission spectra and the
color fan textures in the optical microscopic images, we performed numerical
calculations using finite-difference time domain (FDTD) simulations. The geometrical
parameters of the circular gratings used in the simulations are obtained from SEM
pictures of these samples. The Drude model: ε(ω)=ε∞-ωp2/ω(ω-iωc) is used to describe
the complex dielectric permittivity of Ag, where the parameters, ε∞=3.57, ωp=1.388×1016
rad/s and ωc=1.064×1014Hz, are obtained by fitting to previously measured Ag
permittivity [21].
The transmission spectra were calculated by integrating the Poynting vectors on a
plane placed at 350nm away from the metal surface. Similar calculations for a few
representative cases were also performed for planes placed at 300nm, 350nm, 400nm and
450nm from the metal surface, all calculations yield the same transmission spectra,
excluding near field effects. The calculated transmission spectra (in Fig. 2.3d-f) agree
qualitatively with the experimental results in terms of their overall trend and the
occurrence of the resonance peak. The electric field distributions were calculated at
representative wavelength also at a plane placed 350nm beneath the Ag film. Exemplary
30
results for the 375nm radial period are shown in Fig. 2.4. Clearly, the electric field of the
transmitted light is not linearly polarized. Instead, the transmitted light is in-phase
radially polarized at long wavelengths (e.g., 850nm in Fig. 2.4a-c), while in-phase
azimuthally polarized at short wavelengths (e.g., 400nm in Fig. 2.4g-i). When the
incident wavelength is close to the resonant peak (~550nm), the transmitted electrical
field is oriented primarily perpendicular to the metal surface (Fig. 2.4f), indicating the
propagation of electromagnetic waves along the metal surfaces. Though so, it can be seen
that the transmitted light contains a mixture of azimuthally and radially polarized
components as well (Fig. 2.4d-f).
This wavelength dependence of the polarization of the transmitted light explains
the colored fan textures observed under the optical microscope (Fig. 2.4a-b, d-e and g-h).
For the fan textures observed under the parallel polarization condition, the radially
polarized transmission at long wavelengths is responsible for the two vertical quadrants
(Fig. 2.4a), while the azimuthally polarized transmission is responsible for the two
horizontal quadrants (Fig. 2.4g). In contrast, under cross-polarization condition, both the
radially polarized transmission and the azimuthally polarized transmission contribute to
the transmission in these four quadrants (Fig. 2.4b and 2.4h), since they all have the
electric field components along the crossed polarization.
31
Figure 2.4. FDTD calculated local field distributions of the transmitted light at a plane
350 nm beneath the Ag film for the circular nanoslits of 375 nm radial period at three
representative free space wavelengths: 850 nm (a, b and c), 550 nm (d, e and f) and 400
nm (g, h and i). The left column (a, d and g) depicts the amplitudes of the vertical
components of the electric field; the middle column (b, e and h) depicts the amplitudes
of the horizontal components of the electric field; and the right column (c, f and i)
depicts the snapshots of the electric field vector distributions. The incident polarization
is vertical for all simulations. The electric field amplitudes have been normalized by
the incident field amplitude. In (c), (f) and (i), the arrow color represents the electric
field amplitude; the arrow orientation represents the electric field direction.
32
2.4 Discussion---concentric circular nano-slits
The generation of cylindrical vector polarized light at long and short wavelengths
can be understood as a result of the interplay between the transmission of the TM and TE
waves through the nanoslit arrays. A straight nanoslit can be considered as a planar
metal-dielectric-metal (MDM) waveguide. For a circular nanoslit, the electrical field of
the incident light can be projected into a component tangential to the slit (TE mode) and a
component normal to the slit (TM mode).
In an MDM waveguide, there exist a symmetric and an anti-symmetric mode for
both TE and TM modes. Because the symmetric TM mode has much higher propagation
loss than the anti-symmetric TM mode and the symmetric TE mode has much lower
cutoff wavelength than the anti-symmetric TE mode, only the anti-symmetric TM and
anti-symmetric TE modes contribute to the optical transmission through nanoslits. For
example, for a 250 nm wide nanoslit, the propagation length of light with 600 nm free
space wavelength is around 0.26 μm for the symmetric TM mode, while is 19 μm for the
anti-symmetric TM mode; and the cutoff wavelength is about 600 nm for the anti-
symmetric TE mode, while is about 315 nm for the symmetric TE mode.
As discussed in chapter 1, the dispersion relations can be expressed as:
tan( 2) 0d m m d dk i k k dε ε− = (2.1) for the anti-symmetric TM mode; and expressed as:
tan( 2) 0m d dk ik k d− = (2.2)
33
for the anti-symmetric TE mode [22, 23]. Here, 2 2 2m m wgk c kε ω= − and
2 2 2d d wgk c kε ω= − , and kwg is the wave vector for the MIM waveguide modes. The
propagation length of these waveguide modes can be obtained by 12 Im( )p wgL k
−= .
From these equations, the dispersion curves and propagation lengths can be
calculated for these modes in the vacuum nanoslits of 90 nm, 120 nm and 145 nm widths
in Ag films (Fig. 2.5). It can be seen that the anti-symmetric TE mode has a cutoff
wavelength above 400 nm and that the anti-symmetric TM mode has no cutoff
wavelength. This indicates that the transmission of the anti-symmetric TM mode is much
higher than that of the anti-symmetric TE mode at long wavelengths; and as a result, the
transmitted light should be radially polarized at these wavelengths.
On the contrary, the anti-symmetric TE mode plays an important role at short
wavelengths. The propagation length Lp of the anti-symmetric TE mode increase with
decreasing wavelength and becomes comparable to the Ag film thickness when the
incident wavelength is below 500 nm. As a result, the TE mode has a finite transmission
at short wavelengths [24]. It was previously shown that the transmission enhanced by the
Fabry-Perot type of cavity resonances necessitates 2 ~ 2wgk h nπ with n being an integer
[25]. For the anti-symmetric TM mode at short wavelengths in our experiments,
2 ~wgk h π (for 429 nm wavelength), therefore its transmission is minimal due to the
destructive interference. These imply that the transmitted light at short wavelengths
should be azimuthally polarized.
34
Two possible physical processes, i.e., surface plasmon excitation and Wood
anomaly may be responsible for the transmission peaks and the generation of surface
waves. The excitation of surface plasmons at the metal-dielectric surfaces have been
considered to play a key role in the enhanced transmission through subwavelength
apertures; and its resonant condition is that for a perpendicular incident light, the wave
vector of surface plasmon ( )0sp d m d mk k ε ε ε ε= + should be matched by the reciprocal
vector (2π/d) of the grating. Here k0 is the wave vector of a light wave in vacuum; mε and
dε are the permittivity for the metal (Ag) and the dielectric materials (air or SiO2). For
the grating period d= 375 nm, 320 nm and 270 nm, it can be calculated that the plasmon
resonant wavelength is at 413 nm, 371 nm and 340 nm respectively for the air/metal
interface, and 590 nm, 520 nm and 460 nm respectively for the substrate/metal interface.
Comparisons with the simulated and measured transmission spectra indicate that the
Figure 2.5. Dispersion curves (a) and propagation lengths (b) for the anti-
symmetric TM mode and the anti-symmetric TE mode in a Ag/air/Ag
waveguide with the air gap at 145 nm, 120 nm and 90 nm respectively.
35
resonance wavelengths for plasmons at the glass/Ag interface actually correspond to the
dips in the transmission spectra.
The Wood anomaly occurs when a diffraction order disappears, or 02 dd kπ ε= for
perpendicular illumination [6, 7]. As a result, the resonant wavelength of the Wood
anomaly at the substrate/metal interface should be at 548 nm, 467 nm and 394 nm for
d=375 nm 320 nm and 270 nm. These Wood anomaly resonant wavelengths, though
obtained for 1D gratings, are in good agreement with these transmission peaks in the
measured and simulated spectra. Again, the resonant wavelengths of the Wood anomaly
at the air/metal interface are below 400 nm and outside the wavelength range of the
measurements. For a 1D grating, one signature of the Wood anomaly is a sharp edge in
the transmission spectrum. This distinctive feature of Wood anomaly is smeared out in
our structures, which can be attributed to the curvature and the finite size of the nanoslits.
It can be expected that 1D grating of nanoslits can be used as a good approximation when
the diameter of the circular nanoslit is larger than the incident wavelength. While for
these circular nanoslits of wavelength size or smaller in the middle of the samples, more
rigorous models should be employed [26].
The suppression of transmission at the resonance of surface plasmon excitation,
which seems in contradiction with the common belief of the critical role played by
surface plasmons, actually agrees with a number of recent numerical, experimental and
theoretical studies in 1D array of subwavelength slits [27]. This discrepancy has been
resolved by considering the coupling between surface plasmons and other diffraction
36
orders or a background radiation. One clear physical picture is based on the equipartition
of diffraction orders [28].
Considering the surface plasmons as one diffraction order, a resonant excitation of
surface plasmons suppresses other diffraction orders including the zeroth order
transmission. Similarly, a disappearance of a diffraction order means enhancing other
diffraction orders including the zeroth order transmission, or enhanced transmission at
Wood anomaly frequencies.
When the radial period of the nanoslits is decreased, the resonance of the Wood
anomaly shifts to shorter wavelength. Therefore, it can be expected that a mixture of
surface waves and azimuthally polarized light should exist at short wavelengths. This is
actually the case for the nanoslits with small periods. For example, the resonant peak of
the Wood anomaly for the 270 nm radial period is located around 400 nm; and thus the
Figure 2.6. A vector distribution snapshot of the calculated local electrical
field for the transmitted light at a plane 350 nm beneath the Ag film for 270
nm radial period and 400 nm incident wavelength.
37
calculated electrical field contains both the azimuthally polarized components and surface
wave components (Fig. 2.6).
2.5 Experiment---spiral nano-slits
2.5.1 Sample preparation and experimental setup
For spiral nano-slits preparation, we directly deposited 110nm thick titanium
films on fused quartz substrates by using electron beam evaporation, and milled
nanotrenches of Archimedean spirals into the Ti film using the previously mentioned
focused ion beam (FIB) system (FEI Nova Nanolab Dual Beam Workstation). The
acceleration voltage and current of the Ga ion beam were set at 30keV and 30pA
respectively. Five Archimedean spiral nano-trenches of radial periods ranging from 2 to 6
were milled in the Ti films (see Fig. 2.7 for examples). The radial periodicity of these
Archimedean spirals is 370 nm; and the width of these trenches is about 110nm (Fig.2.7).
Figure 2.7. Representative scanning electron microscopic pictures of the spiral
nanotrenches with 2 periods (a), 3 periods (b), and 4 periods (c). The radial
periodicity of these spirals is 370nm.
38
For the optical transmission measurements, we use the same experimental setup as for the
experiments of the circular nano-slits.
2.5.2 Measured transmission spectra
The measured transmission spectra for spirals with 2 to 5 radial periods are shown
in Fig. 2.8a-d. It can be seen that the transmissions for both right and left circularly
polarized incident light exhibit a peak around the wavelength of 520 nm. Within the
wavelength range of the measurements, the transmission of left circularly polarized
incidence is clearly larger than that of right circularly polarized incidence
Figure 2.8. Measured transmission spectra for the spirals of 370nm radial period with
the number of radial period is 2 (a), 3 (b), 4 (c), 5 (d) respectively The black line
represents the transmission spectra for the left circularly polarized incidence; the red
lines represent the transmission spectra for the right circularly polarized incidence.
39
for the 2-periods spirals. Another interesting observation is that when the radial periods
of the spiral trenches are increased, the transmission difference between left circular
polarization and right circular polarization is gradually decreased, and eventually become
negligible when the radial periods are larger than 5. In another word, the increase of
redial repetition of the spiral trenches leads to the reduction of circular dichroism. This is
the opposite to increasing the translational repetitions of chiral nanostructures in 2D
arrays where the translational repetitions result in enhancements of circular dichroism
[29, 30].
According to the definition of enantiomeric transmission difference [31],
( ) ( )2 LCP RCP LCP RCPT T T T− + , we calculate and plot the transmission difference for spirals
with radial period number changing from 2 to 6. The transmission difference calculated
from the measured transmission spectra was shown in Fig. 2.9. The gray area in Fig. 2.9
indicates the inherent error of our measurement system, which is ±2.5‰. For all the
spirals, the transmission difference has a peak within the wavelength range from 500 nm
to 550 nm. From 2-period spiral to 6-period spiral, the maximum transmission difference
gradually goes down.
40
2.6 Discussion---spiral nano-slits
For the spiral nano-gratings, the excited surface wave (propagating along the
interface of Ti/SiO2 substrate) at the edge of each slit will propagate inwardly (towards
the geometry center of the spiral) and outwardly. The amplitude of outwardly
propagating surface wave is approximately proportional to( )ik R reA
R r
− −
−, which attenuate
very fast and can be ignored. The inwardly propagating surface wave can be expressed as
[32]:
2 cos( ) cos( )2( , )
2i
PR k RiP
swPE R A e e d
φπ θ φ θ φπ φθ φ
π
− − − −− = ∫ (2.3)
for left circular polarization excitation and
Figure 2.9. Measured transmission difference between left circularly polarized
incidence and right circularly polarized incidence with the number of radial period
changing from 2 to 6.
41
2 cos( ) cos( )22( , )
2i
PR k Ri iPsw
PE R A e e e dφπ θ φ θ φ
φπ φθ φπ
− − − −− = ∫ (2.4)
for right circular polarization excitation. Here, A is the amplitude of the excited surface
wave field. R and θ are the radial coordinate and angular coordinate respectively. P
represents the radial pitch of the spirals, ik is the imaginary part of the wavevector of the
excited surface wave, φ represents the angle of the spirals. According to Eq. 2.3 and
2.4, at R=0 (the center of the spirals), the left circular polarization excited swE is always in
phase (Fig. 2.10a) and thus has a huge constructive interference (Fig. 2.11 a and d) while
Figure 2.10. (a) The simulated phase of surface wave excited by left circularly polarized
incidence; (b) the simulated phase of surface wave excited by right circularly polarized
incidence.
42
due to the phase mismatch by 2ie φ (Fig. 2.10b) the resonance amplitude of right circular
polarization is always zero (Fig. 2.11 b and e). According to Eqs. 2.3 and 2.4, the
calculated near-field intensity as the function of radius are in accordance with our
numerical simulation as shown in Fig. 2.11 c and f.
Based on our previous discussion, we know the peak of the transmission spectra is
due to the Wood anomaly effect. At the wavelength of Wood anomaly, surface wave
energy can be re-scattered into far field. Since left circular polarization excited surface
Figure 2.11. Calculated near field distribution of Ez (electric field perpendicular to
the Ti film) for the transmitted light at 540 nm wavelength and at a plane 90nm
beneath the Ti film. (a) and (d): Left circularly polarized incidence transmit through
2-periods and 6-periods spiral trench respectively; (b) and (e) Right circularly
polarized incidence transmit through 2-periods and 6-periods spiral trench; (c) and
(f): Ez2
(polar angle averaged) of 2-periods and 6-periods spiral as a function of
radius respectively.
43
wave have larger electric field intensity at the center of the spirals, more electromagnetic
energy of left circular polarization will be scattered into zero order transmission than
right circular polarization. That’s the reason why transmission of left circularly polarized
incidence is larger than the transmission of right circularly polarized incidence. With
increasing the number of radial periods, the energy at the spiral center takes a smaller
proportion of the total surface wave energy. So the transmission difference between left
circular polarization and right circular polarization will decrease.
2.7 Conclusion
To summarize, for concentric circular nano-slits in Ag films, we have studied the
optical transmission through it. Experimental and numerical results show that the light
transmitted through these apertures is in-phase radially polarized at long wavelengths and
in-phase azimuthally polarized at short wavelengths due to the interplay between the TE
and TM transmission through the nano-slits. Also, the transmission exhibits a peak at the
wavelength of Wood anomaly and a dip at the wavelength of the surface plasmon wave
excitation, and the wavelengths of these peaks and dips vary with the radial period of the
slits.
The spiral nano-slits in Ti film can induce enantiomerically sensitive surface
wave intensities at the geometry center of the spirals, which can lead to small but
observable far field transmission difference between left circularly polarized incidence
and right circularly polarized incidence. However, the transmission difference decrease
very fast when increasing the period number of the spirals. This is attributed to the fact
44
that the difference of surface wave intensity at the geometry center takes up smaller and
smaller proportion of the total excited surface wave energy when increasing the period
number of the spirals.
45
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50
CHAPTER 3
Cavity Modes and Their Excitations in Elliptical Plasmonic Patch Nanoantennas
3.1 Introduction
Plasmonic nanoantennas have attracted significant attention in recent years due to
their ability in coupling free space electromagnetic radiation into sub-diffraction limited
volumes and vice versa based on the electrodynamics reciprocity [1, 2]. Resonantly
enhanced local fields in these nanoantennas facilitate applications in various fields such
as surface enhanced fluorescence and Raman spectroscopy [3, 4] and in nonlinear optics
[5, 6]. For examples, as the optical analogues to their radio frequency counterparts,
Hertzian dipole and Yagi-Uda antennas enable not only focusing light into nanometer
spots but also emitting light unidirectionally by coupling quantum dots with them [7, 8].
Vertical metal-dielectric-metal (MDM) designs of plasmonic nanoantennas and
nanocavities have been proposed and demonstrated with the great advantage that the
nanogap thickness can be precisely controlled by using the advanced thin film deposition
techniques [9-11].
A variant of the MDM nanoantennas is the plasmonic patch nanoantennas where
metal patches are fabricated on a metal film with a dielectric spacer layer [12]. The large
local field enhancements in these patch nanoantennas make them attractive substrates for
surface enhanced Raman spectroscopy (SERS) [12-14]. Recent studies show that the
plasmonic patch antennas provide a new avenue towards various applications such as
near perfect absorbers [15-18], single photon light sources by them with quantum
51
emission systems [19], metamaterials [20], and biosensors [21]. By exciting the cavity
modes between two metal plates (a configuration similar to the patch antennas), a strong
light-induced negative optical pressure can be introduced [22]. To facilitate the design of
plasmonic patch nanoantennas for these various applications, a simple approach to
accurately predict the resonant frequencies of these patch nanoantennas is highly desired.
Previous studies, however, are mostly focused on the first cavity mode for the circular or
square patch shapes, while the high order cavity modes and their resonant/excitation
conditions have rarely been addressed quantitatively.
In this chapter, we present our experimental and theoretical studies of the cavity
modes and their excitation conditions for plasmonic patch nanoantennas in optical
frequencies. Patches with elliptical shapes were used to investigate the effect of the
azimuthal symmetry breaking on the local electrical field distributions for different cavity
modes and on their excitation conditions. Numerical calculations show that breaking the
circular symmetry leads to the presence of both even and odd cavity modes, and that the
excitation configurations for these modes are dictated by their modal symmetries.
Analytical expressions of the modal field distributions agree well with the simulation
results. By using the actual patch radii plus the gap thickness as the effective radii, we
show that the resonant condition based on Neumann boundary conditions show in
excellent agreements with experimental and simulation results. This physical
understanding of the resonant modes and their excitation conditions of the patch
nanoantennas should be extendable to plasmonic patch nanoantennas with other
geometrical shapes.
52
3.2 Sample preparation
To fabricate the plasmonic patch nanoantennas (Fig. 3.1), a 10 nm NiCr adhesion
layer and a 100 nm Ag film were sequentially deposited on a Si wafer using electron
beam evaporation, and then a 15 nm Al2O3 dielectric layer was deposited using atomic
layer deposition (ALD). The 45 nm thick top elliptical Ag patches were fabricated using
the standard electron beam lithography and lift-off processes. ALD is a technique for
depositing thin film one molecular/atomic layer by one molecular/atomic layer using
sequential gas phase chemical process. For Al2O3 ALD, the substrate was sequentially
exposed to trimethyl aluminum (TMA) and H2O vapor. The advantages of using ALD
process include the conformal and uniform film coverage and sub-nm accuracy in
thickness control. An exemplary SEM picture of the fabricated patch nanoantennas is
shown in Fig. 3.2. To measure the reflection spectra, a collimated white light beam was
focused onto the samples using a 40× (0.6NA) objective, and the reflected light was
collected using the same objective for spectral measurements. The reflection spectra are
normalized by the spectra measured from the areas without the patches.
53
Figure 3.1. Schematic illustration of the fabrication procedure of elliptical plasmonic
patch nanoantennas.
54
The major axial radii of the elliptical patches are varied from 40 nm to 96 nm.
Two patch periods, 300 nm and 500 nm, are used for the patch antenna samples. As will
be discussed later, these patch periods are small enough to ensure minimal effect of
surface plasmons excitation outside the cavity on the cavity modes, and large enough to
ensure no overlapping of the fringe fields between neighboring patches.
3.3 Experimental and simulation results for the cavity modes
The measured reflection spectra for two representative patch sizes are presented
in Fig. 3.3a and 3.3c, where several dips with different absorption depths can be
observed. Fig. 3.3a represents a larger patch size with 93 nm long axis radius and 74 nm
Figure 3.2. A representative SEM image of a 2D array of plasmonic patch
nanoantennas with a period of 500nm. Two major axis radii of the patches are a=72nm,
b=52nm. The long axis of the patches is tilted about 22º from vertical direction.
55
short axis radius, while Fig. 3.3c represents a smaller patch size with 72 nm long axis
radius and 52 nm short axis radius. In Fig. 3.3, blue curves and red curves are the
Figure 3.3. The measured and simulated reflection spectra for two representative patch
sizes: a=93nm, b=74nm (blue) and a=72nm, and b=52nm (red). The polarization of the
incident light is parallel to the long (a, b) and short axes (c, d) of the patches respectively.
The dotted curves in b and d are the local field enhancement spectra calculated at the
cavity edges. The cavity modes are indicated with the mode indices. The e02 cavity mode
is hard to locate for the smaller patch antennas (red curves) and thus not labeled.
56
measured reflection spectra for incident light polarized along the long axis and the short
axis respectively. As can be seen, the resonant wavelengths of the dips shift to red with
increasing the patch size. In addition, the resonant wavelengths of these dips shift to blue
when the incident polarization is changed from the direction parallel to the long axis to
the direction parallel to the short axis of the elliptical patches.
To illustrate the physical origins of these resonant dips, we performed numerical
calculations using the finite integration technique (CST-Microwave Studio) as has been
introduced in Chapter 2. The frequency dependent permittivity for silver was described
by the Drude model εAg=ε∞-ω p2/ω(ω+iγ), with ε∞=3.57, ωp=1.388×1016 rad/s, and γ
=1.064×1014 Hz obtained by fitting to the measured bulk permittivity of Ag. The optical
constants from both Ref [23] and [24] have been used in numerical simulations in the
previous studies such as in [25]. Since the imaginary part of the permittivity from Ref
[23] is smaller than that from Palik’s Handbook, simulations normally result in narrower
resonances. While the real part of the permittivity for Ag from these two sources differs
by only a small percentage for wavelength below 1 µm, the difference in the predicted
resonant frequencies is normally very small as verified in also our simulations. The
dielectric permittivity of Al2O3 was fixed at 2.40. To emulate the experimental
conditions, the reflection spectra were obtained by averaging over two incident angles (0º
and 20º) for two polarizations parallel to the major axes of the patches respectively. It can
be observed that the calculated reflection spectra reproduce these resonant features in
experimental data at similar resonant wavelengths (Fig. 3.3a-d). At these resonant
57
wavelengths, the calculated near-field spectra exhibit resonant peaks, indicating cavity
resonances (dotted lines in Fig. 3.3b and d).
We calculated the z-component of the electrical field distributions in the x-y plane
through the middle of the Al2O3 layer at these resonant wavelengths. Fig. 3.4 presents the
results for different cavity modes and their corresponding excitation conditions. Based on
comparing these simulated field distribution patterns with the theoretical results in next
section, these cavity modes and their presences in the far-field spectra have be indexed
(Fig. 3.3 and 3.4). The excitations of these cavity modes are sensitive to both incidence
Figure 3.4. Excitation configurations and the snapshots of simulated electrical field (Ez)
distributions for different cavity modes for the 93nm×74nm patches: (a) modes
symmetrical to x-axis and anti-symmetric to y-axis; (b) modes symmetrical to y-axis
and anti-symmetric to x-axis; (c) modes symmetric to both x- and y-axes; (d) modes
anti-symmetric to both x- and y-axes.
58
angle and polarization directions of the exciting light and will be discussed later in this
chapter.
3.4 Analytical expressions for the cavity modes
3.4.1 Mathieu equations
The cavity modes observed in the elliptical patch nanoantennas are due to
constructive interferences of gap surface plasmons generated and reflected at the patch
boundaries propagating parallel to the interface. To understand these cavity modes, we
consider the electromagnetic waves propagating in the metal-dielectric-metal (MDM)
waveguide structure. The electrical field E can be described by the Helmholtz Eq.
022 =+∇ EE ik where 222 ck ii ωε= are the total wave vectors with the subscript i=m, d
referring to metal and dielectric regions, ω and c are the radial frequency and speed of
light in vacuum. Since the gap surface plasmons need to be evanescent in the direction (z)
perpendicular to the MDM plane, the propagation constant along z should be imaginary,
denoted as ikzi. Therefore, the total wave vector ki is the summation of the propagation
constant ikzi, and the propagation constant along the MDM plane or the gap plasmon
wave vector kgsp, 222zigspi kkk −= .
There exist two gap surface plasmon modes with one symmetric and one anti-
symmetric profile of field distribution [26]. For small dielectric thicknesses, the
symmetric mode exists at high frequencies and can be ignored in our case. For the top
metal layer of the MDM structure with a finite thickness tm, the dispersion relation for the
anti-symmetric mode can be expressed as [27]:
59
,1
1
2121
2121
mzd
dzm
rrrr
rrrrεkεk
+++
+++−= (3.1)
where ,222 ckk mgspzm ωε−= ,222 ckk dgspzd ωε−= ),2exp(1 mzmtkr −= ),2exp(2 dzdtkr −= and td is the
thicknesses of the dielectric layer.
Given the elliptical shape of the nanocavities, it is convenient to consider the
Helmholtz Eqs. in the elliptical cylindrical coordinates (ξ, η, z). The transformation from
the Cartesian coordinates (x, y, z) to the elliptical coordinates is: ,coscosh ηξfx =
,sinsinh ηξfy = z=z where f is the focal length of the ellipse, 2/122 )( baf −= . Since our
primary interest is in the electrical field distributions inside the dielectric layer, we focus
only on the dominant component of the electrical field Ezd (ξ, η, z). After the
transformation, the Helmholtz Eq. for Ezd (ξ,η,z) can be written as:
,0)sin(sinh
1 22
2
2
2
2
2
222 =++
+
+zdd
zdzdzd Ekdz
Edd
Edd
Edf ηξηξ
(3.2)
Assuming Ezd(ξ,η,z)=R(ξ)Φ(η)Z(z), ,)( zkzk zdzd BeAezZ += − separation of variables
decomposes Eq. 3.2 into the radial and angular Mathieu Eqs.:
,02cos2
,02cosh2
2
2
2
2
=−+
=−−
η)Φq(cdη
Φd
)Rq(cdξ
Rd ξ
(3.3)
where c is the separation constant and 422gspkfq = . The fact that Φ should be a periodic
function of η with either a π or 2π period determines the possible eigen values of c [28].
The allowed solutions for Φ are two independent families of even (e) and odd (o) angular
Mathieu functions: ),( qCem η for 0≥m and ),( qSom η for 1≥m respectively. The radial
function R should be a non-periodic, decreasing oscillatory function of ξ in 0≤ξ<∞ with
60
non-singularity at the origin; the standing wave solutions for R are the even and odd
radial Mathieu functions of the first kind: ),( qJem ξ for 0≥m and ),( qJom ξ for 1≥m . For
fixed z, Ezd(ξ,η,z) is proportional to the product of the angular and radial Mathieu
function:
,)(1),,(),()(0),,(),(
oddmqSeqJoevenmqCeqJe
Emm
mmmzd ≥
≥≈
ηξηξ
(3.4)
where the superscript m stands for the m-th order. Since the nanocavities have an open
edge, the electrical field Ezd(ξ,η,z) is approximately at its local maximum at the boundary.
As a result, the cavity resonances are determined by the Neumann boundary condition:
,0),(),(:)(),(),(:)(=
′′
qSeqoJoddqCeqeJeven
mm
mm
ηξηξ
(3.5)
where )/(arcsin0 fbh=ξ defines the cavity boundary. For a given order m, there exists an
infinite number of q values satisfying Eq. 3.5. We use oemnq , to denote the nth zero of
),(' 0 qeJ m ξ or ),( 0 qoJ m ξ′ , and use emn and omn to denote the even and odd cavity modes.
Figure 3.5. Snapshots of electrical field distributions for different cavity modes.
calculated using the Mathieu functions (Eq. 3) for the 93nm×74nm patches.
61
Based on Eq. 3.4 and Eq. 3.5, we calculated the electrical field distributions for
these cavity modes of lowest orders. The results as shown in Fig. 3.5 are in excellent
agreements with the field distributions obtained in simulations (Fig. 3.4). Previous studies
of elliptical patch antennas in microwave frequencies for metamaterials show similar
field distribution patterns though with detailed structures are different from ours [20].
As can be seen in Fig. 3.4, the excitation conditions for these cavity modes are
dictated by their modal symmetries. The cavity modes with odd m can be excited with
both normal and tilted illumination, while the excitation configurations for the even and
odd modes are rotated by 90º from each other. While for the even modes (e21, o02) with
even m which are symmetric to both x- and y-axes, tilted illumination with p-polarization
is required (Fig. 3.4c); and for the odd modes with even m which are anti-symmetric to
both x- and y-axes (o21), tilted illumination with s-polarization is needed. As a result,
these modes excited only by tilted illumination have small resonant signatures in the far-
field reflection spectra (Fig. 3.3).
These cavity excitation rules are set by the phase asymmetry of the excited gap
surface plasmons. For normal incidence, the electric fields Ezd at the opposite cavity sides
are always in opposite phase along the polarization direction while in the same phase
along the direction perpendicular to the polarization (Fig. 3.4a-b). Consequently, in order
to excite cavity modes such as e21, o21 or e02, this asymmetry has to be broken by using
tilted illumination. It is important to note that due to the subwavelength size of the
patches, the tilted illumination cannot completely compensate the π phase difference of
the gap plasmons excited at the opposite edges, and the modal field distributions inside
62
the cavities are always composed of a standing wave superposed on a propagating wave.
This also makes it difficult to identify cavity modes such as e02, because their
appearances in the far-field spectra are expected to be shallow (red curves in Fig 3.3 a, b).
3.4.2 Dispersion relations of gap surface plasmon
For one given oemnq , , the gap plasmons forming the standing waves should satisfy:
,)4( 212, fqk oemngsp = (3.6)
To validate this resonant condition, we calculated the kgsp for these cavity modes by using
Eqs. 3.5 and 3.6 based on their resonant wavelengths and patch sizes observed in the
experiments, and compare it with the kgsp calculated using the dispersion relation (Eq.
3.1). The results for different patch sizes in Fig. 3.6a show reasonable agreements
between the resonant condition and the gap plasmon dispersion curve. To note here, we
only included the data for e11/o11, e12/o12 and e31/o31 modes in Fig. 3.6 because their
features can be easily identified in the measured reflection spectra.
It is also discernible that the resonant frequencies obtained using Eq. 3.6 are
systematically lower than the dispersion curve for the gap surface plasmons obtained
using Eq. 3.1. This systematic discrepancy can be ascribed to the fringing fields, i.e. the
electrical fields do not go to zero beyond the cavity edge. In another word, the antinodes
of the cavity modes are not exactly located at the cavity boundary, and there exists
effectively a phase shift upon excitation/reflection of the gap surface plasmons at the
boundary [9, 29]. This fringing field effect can also be considered as a result of the
capacitance between the patch edges and the bottom metal film [30, 31]. To take this
fringing field into account, one simple while effective empirical approach suggested in
63
microwave frequencies is to use the sum of the actual patch radius and the dielectric layer
thickness as the effective radius [32]. By following this empirical correction, we
recalculated the resonant frequencies using a΄=a+h and b΄=b+h as the two main axial
radii in Eq. 3.6. The results show excellent agreements with the dispersion for gap
surface plasmons (Fig. 3.6b).
One interesting observation is that the resonant frequencies for even mode and
odd modes are noticeably different except for the e31 and o31 modes. This is fortuitous
and because the resonant conditions for these two modes are almost the same. For
example, the value of eq31 and oq31 are 1.9159 and 1.963 for the same patch size with
a=93nm, b=74nm respectively.
Figure 3.6. Data points represent the measured cavity resonant frequencies versus
the gap plasmon wave vector calculated using Eq. 3.6 for real patch sizes (a) and for
the effective patch sizes a΄=a+h, b΄=b+h (b). The blue solid curves represent the
dispersion curve for the gap plasmons calculated using Eq. 3.1.
64
3.5 Period effects
The effects of array periods on the resonant wavelengths are very interesting
while quite complicated. We speculate that there exist three regions: (1) For small periods
where the fringe fields between neighboring patches overlap, the cavity modes are
coupled with each other (in analog to the systems of atoms with overlapping wave
functions), and form a 2D photonic crystal system. In this case, it can be expected that the
gap plasmons will be delocalized at certain frequency bands, while be prohibited to
propagate forming band gaps at the other frequencies. (2) For large periods, the surface
plasmons at the Ag/Al2O3 interface outside the cavities are often excited, causing the
complex interactions between cavity modes and surface plasmon modes. (3) For the
intermediate periods like those in our experiments, the cavity fringe fields are not
overlapping, and the surface plasmon modes are not crowded. In the following, we show
Figure 3.7. Snapshots of the z-component Ezd of the electric field calculated at the
middle plane through the dielectric layer at the resonant wavelength 540 nm for two
different patch radii: 72nm×52nm (a) and 93nm×74nm (b). The illumination is normal
to the plane.
65
numerical studies for the last two situations as a proof that the cavity modes and resonant
conditions in our case are not affected by these array effects.
The 2D periodicity of the metal patches provides a reciprocal vector to
compensate for the momentum mismatch between the incident light and the surface
plasmon waves at the Ag-Al2O3 interface; a natural question is therefore how the surface
plasmon excitation outside the cavity affects the cavity resonances. Although the incident
angles vary in a range determined by the objective N.A., the excitation condition for the
surface plasmons propagating at the direction perpendicular to the incident plane remains
unchanged. For example, for 500 nm array period, the dips at about 540 nm in the
reflection spectra are due to the excitation of the surface plasmons. This can be verified
by looking at the local field distributions at this resonant wavelength, where the
interference patterns of surface plasmons outside the cavities can clearly be seen. For a
small thickness (15nm) of the Al2O3 film, the dispersion for the surface plasmon waves at
the Ag-Al2O3 interface can be approximated as kkk spsp ∆+= 0 where
)1(0 +′′= mmsp ck εεω
is the wave vector for the excited surface plasmons at the Ag-air
interface, λπ
εεεε
εε
εεω∆ 21
111
2
mm
md
m
m
d
dc
k′′−
′−
′−
′−= ,
'mε is the real part of the permittivity
for Ag, and dε is the permittivity for Al2O3 [33]. Simple calculations show that for the
incident light at 540 nm wavelength, kgsp=0.0125 nm-1 matches with the reciprocal vector
of the patch arrays.
66
The coupling effects between surface plasmons and cavity resonances usually
induce Fano type resonances. For a fixed patch period, the excited surface plasmon may
couple with different cavity modes for different patch sizes. As it can be seen in Fig. 3.7,
for 500 nm array period, the surface plasmon excited by 540 nm wavelength light can
couple with either e11 mode or e31 mode depending on different patch sizes. It is also
interesting to note that for the e11 mode not at resonance, the surface plasmon waves
outside and inside the nanocavities are in opposite phase (Fig. 3.7a). While for the e31
mode which is at resonance, the electrical fields due to the surface plasmons outside and
inside the nanocavities are in phase (Fig. 3.7b). It is important to note that the small patch
periods (300 nm and 500 nm) are used in the experiments to minimize the excitations of
surface plasmons outside the cavities and their effects on the cavity resonances. This can
be seen from the simulated reflection spectra for different patch periodicities (Fig. 3.8).
For the patch period below 500nm, the narrow dips due to the excitation of surface
plasmons shift to red with the period increase, while the reflection dips due to the cavity
modes remain unchanged as long as they are not overlapping with the surface plasmon
excitation (Fig. 3.8a). When the patch period is larger than 600 nm, the resonant
wavelengths of these cavity modes start to vary with the changes in the patch period due
to the excitation of multiple surface plasmon modes (Fig. 3.8b).
67
3.6 Conclusion
In this chapter, we have studied the two dimensional periodic arrays of elliptical
plasmonic patch nanoantennas. It is shown that by breaking the azimuthal symmetry with
elliptical patch shapes, even and odd resonant cavity modes can be excited with the
excitation configurations depending on their modal symmetries. An analytical expression
for the cavity modal field distributions based on Mathieu functions has been derived,
yielding excellent agreements with both simulations and experiments.
Figure 3.8. Calculated reflection spectra for different periods with the elliptical patch
size: a=70 nm; b=50 nm. a) and c): periods vary between 300nm to 500nm; b) and d):
periods vary between 600nm to 900nm. a) and b): Incident polarization is along the
long axis of elliptical patches for; c) and d): Incident polarization is along the short axis
of elliptical patches.
68
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72
CHAPTER 4
Polarization Conversion with Elliptical Plasmonic Patch Nanoantennas
4.1 Introduction
Controlling the polarization state of light is critical for a wide range of
applications. Traditionally, wave plates of birefringent materials and optical gratings have
been used for polarization conversion and rotation. Recently, it has been shown that
plasmonic structures and/or metamaterials exhibit extraordinary capabilities in
controlling and manipulating the polarization states of the light [1]. For example,
metamaterials with giant optical activities and circular dichroism can be achieved through
engineering the structural and extrinsic chirality of the system [2-8]. Due to the broken
front-back symmetry, 2D arrays of planar chiral structures of gammadion-shaped
nanoparticles or nanoapertures exhibit large circular dichroism and giant optical activity
[9], similar to conventional three-dimensional materials. For achiral nanostructures,
optical activities and circular dichroism can be enhanced by arranging them in a chiral
maps [7, 8], an effect similar to structural chirality in the cholesteric liquid crystals. In
other approaches H-shaped non-chiral structures [1], elliptical antenna grating [10],
cavity involved structures [11, 12], space-variant grating [13-15], can also be used for
polarization control.
Recently plasmonic patch nanoantennas consisting of metal nanodisks on top of a
metal film with a dielectric spacer layer have attracted considerable attention due to their
capability to enhance local fields for surface enhance Raman spectroscopy [16], gas and
73
chemical sensing [17, 18], and perfect absorption [19-21]. In a previous publication, we
studied the cavity modes and their excitation conditions for elliptical plasmonic patch
nanoantennas, and showed that breaking the circular symmetry leads to the presence of
both even and odd cavity modes and that the excitation configurations for these modes
are dictated by their modal symmetries [22]. Analytical expressions of the modal field
distributions were obtained with resonant conditions in good agreement with
experimental and simulation results [22].
In this work we show that an optical patch antenna array can not only confine
light into ultra-small modal volumes but also provide a polarization conversion in the
reflection. By properly orienting the incident polarization with regards to the patch axis, a
linearly polarized light in resonance with one cavity mode can be converted into a
circular or elliptical polarization after reflection. Simulation studies indicate that the
major cavity modes can be excited at all incident angles, and thus the polarization
conversion can be realized as independent of the incident/reflection angle. It should be
Figure 4.1. (a) Schematic architecture of plasmonic patch nanoantenna arrays; (b) A
representative SEM picture of fabricated arrays of plasmonic patch nanoantennas with
300nm period.
74
pointed out that 2D arrays of metal-dielectric-metal (MDM) nanoantennas with elliptical
shapes should function similarly in polarization conversion, while the smaller aspect ratio
of the top patches in the patch nanoantennas provides a great advantage in the fabrication
processes.
4.2 Sample preparation and experimental measurement
The procedure for fabricating the elliptical patch nanoantenna samples has been
described in Chapter 3. The design for elliptical patch nanoantennas is schematically
shown in Fig. 4.1a. To fabricate the plasmonic patch nanoantennas, a 10nm NiCr
adhesion layer and a 100nm Ag film were sequentially deposited on a polished silicon
substrate; then a 15 nm Al2O3 spacer layer was deposited on top of the Ag film using
atomic layer deposition (ALD). The elliptical Ag disks were then patterned through the
standard procedures of electron beam lithography, electron beam evaporation, and lift-
off. Plasmonic patch nanoantennas with different patch sizes and periodicities (300nm,
500nm) were fabricated. A representative scanning electron microscopic (SEM) picture
of the fabricated sample is shown in Fig.4.1b.
To measure the reflection spectra, the samples were illuminated from the air-
metal side with a focused white light beam (contain both normal and tilted incidence)
using a 40× (0.6 NA) objective. The reflected light was collected by the same objective
and coupled into the entrance slit of an imaging spectrograph for spectral measurements.
A polarizer and an analyzer were inserted into the optical path for controlling the incident
polarization and analyzing the polarization status of the reflected light.
75
The measured reflection spectra for three representative patch nanoantenna
samples are presented in Fig. 4.2a-c. For the sample a with 130nm×100nm patch
diameters and 300nm period, two sets of resonant dips (or cavity modes) are excited by
the incident light with polarization along the long (red curve) and short axis (blue curve)
respectively, and can be observed simultaneously with un-polarized incident light (black
curve) (Fig. 4.2a). For the sample b with the same periodicity (300nm) but increased
patch sizes (154nm×120nm), the measured reflection spectra exhibit similar resonant dips
with red-shifted resonant wavelengths (Fig. 4.2b). For the sample c with a larger period
Figure 4.2. Measured (a-c) and calculated (d-f) reflection spectra for the plasmonic
patch nanoantenna arrays. (a) and (d) for sample a with 130×100 nm patch diameters
and 300 nm period; (b) and (e) for sample b with 154×120 nm patch diameters and 300
nm period; (c) and (f) for sample c with 130×100 nm patch diameters and 500 nm
period. Red and blue curves represent the reflection spectra for the incident polarization
parallel to the long and short patch axis respectively. Black curves represent the
reflection spectra for un-polarized incident light.
76
(500nm) and the patch size of sample a, the resonant dips become shallower with
locations the same as that in sample a (Fig. 4.2c).
4.3 Numerically calculated cavity modes
To elucidate the cavity modes signaled by these dips, we performed numerical
simulations with a commercial software package (CST Microwave Studio), which solves
the Maxwell equations using finite-integration techniques. A hexahedral mesh scheme
was used with mesh sizes much smaller than the surface plasmon wavelength. The
dielectric permittivity of Ag was represented by the Drude model εAg=ε∞-ωp2/ω(ω-iωc)
where ε∞=3.57, ωp=1.388x1016 rad/sec, and ωc=1.064x1014 Hz [23]. The dielectric
permittivity for the aluminum oxide gap was fixed at 2.4. Normal and oblique incidences
with s and p polarizations were used to excite the patch nanoantennas; and the far-field
reflection spectra were obtained by averaging calculated reflections for two (p and s)
Figure 4.3. (a-f) Snapshots of the simulated electrical field distributions in the
middle plane of the dielectric layer at the resonant frequencies for sample a.
Comparing them with analytical theories Eq. 1 and 2 indicates that they are e11, o11,
e21, e02, e12 and o12 cavity modes.
77
polarizations and two incident angles (0° and 20°).
The simulated reflection spectra agree well with the measurements, as seen in Fig.
4.2a-f. To identify the cavity modes, we calculated the electrical field distributions inside
the dielectric layers at these resonant frequencies. As an example, the results for the
sample a are shown in Fig. 4.3.
As demonstrated in chapter 3, the dominant electrical field component Ez for the
cavity modes can be expressed as the product of the even and odd radial and angular
Mathieu functions [24]:
( , ) ( , ), 0 ( )( , ) ( , ), 1 ( )
m mz
m m
Je q Ce q m evenE
Jo q Se q m oddξ ηξ η
≥≈
≥ (4.1)
where 2 2 4gspq f k= is a function of gap plasmon wave vector kgsp and the focal length f of
the elliptical patches. mCe ( ,q )η with 0m ≥ and mSo ( ,q )η with 1m ≥ are two
independent families of even (e) and odd (o) angular Mathieu functions. mJe ( ,q )ξ with
0m ≥ and mJo ( ,q )ξ with 1m ≥ are the even (e) and odd (o) radial Mathieu functions of
the first kind. The edges of the nanocavities can be defined as 0 arcsin ( / )h b fξ = . The
open boundaries at the nanocavity edges imply that the Neumann boundary condition can
be applied to determine the cavity resonance:
( ) : ( , ) ( , )0
( ) : ( , ) ( , )m m
m m
even J e q Ce qodd J o q Se q
ξ ηξ η
′=
′ (4.2)
With Eqs. 4.1 and 4.2, we analytically calculated the field distributions for the even (emn)
and the odd (omn) cavity modes, where m denotes the order of the Mathieu function and n
denotes the nth zero of 0( , )mJ e qξ′ or 0( , )mJ o qξ′ . By comparing the calculated and
78
simulated field distributions, we can determine the cavity modes in Fig. 4.3a-f and for
other samples as marked in Fig. 4.2. As demonstrated in our previous work, the
fundamental modes e11 and o11 can be excited with wide range of illumination angles and
thus present strong absorption in the reflection spectra [22, 25].
4.4 Polarization conversion at the wavelengths of cavity resonance
To analyze the polarization status of the reflected light, we inserted an analyzer in
front of the spectrometer. The angle between the patch long axis and the polarizer is
denoted as β, and that between the polarizer and analyzer is denoted as θ. The reflection
spectra at different θ are shown in Fig 4.4a-c for samples a-c respectively.
Over the whole range of wavelength measurements, we found that the measured
reflection intensity at each wavelength λ can be fitted with:
2 2max min( , ) ( )cos ( )sinI I Iλ θ λ θ λ θ= + (4.3)
This implies that the polarization becomes elliptical after reflection. Take sample a as an
example. The measured reflection as the function of θ at three representative wavelengths
Figure 4.4. (a-c) Measured reflection spectra at different θ for the sample a, b and c
respectively. Here β is set at 20o; and the results for θ varied only between 0o to 90o are
shown for the sake of clarity.
79
(560 nm, 720 nm, 850 nm) has been perfectly fitted according to Eq. 4.3, as shown in Fig.
4.5a. Ellipticity, defined as (Imin/Imax)1/2, is normally employed to characterize polarized
light. Ellipticity equal to 1 and 0 corresponds to the circular and linear polarization status,
respectively. From fitting the reflection spectra at individual wavelength λ with Eq. 4.3,
we obtained the ellipticity spectra. The results, as shown in Fig. 4.5b, clearly indicate that
the ellipticity approaches unity at the resonant wavelength of the e11 mode for samples a
and b, while it is close to zero for wavelengths around 600nm. While for sample c, the
ellipticity is mostly smaller than unity (Fig. 4.5b).
4.5 Lorentz oscillator model for the cavity modes
This phenomenon of polarization conversion can be understood from two physical
aspects. Firstly, the incident and reflected electrical fields can be projected into two
directions parallel to the major axes of the patches; the difference between the resonant
Figure 4.5. (a) Measured reflection (circles) as the function of θ and fitted reflection
(sold curve) as the function of θ according to Eq. 4.3 for 560 nm wavelength (red), 720
nm wavelength (green) and 850 nm wavelength (blue). (b) Calculated ellipticity spectra
for sample a (red), b (blue), and c (green).
80
frequencies of the e11 and o11 modes leads to a phase delay between these two electrical
field components. This can be verified by the numerical simulations. As shown in Fig. 4.6
for sample a, a π phase change from short to long wavelengths can be observed for the
magnetic field inside the cavity for both incident polarizations. This π phase change is
expected if we consider the cavity resonance as a Lorentz oscillator and the phase delay
is given by:
12 20
( ) tan γωδ ωω ω
− = − (4)
where γ and ωo represent the damping constant and the resonant frequency respectively,
which can be obtained by fitting the simulated near field spectra within the cavity with a
Lorenzian shape. As can be seen from Fig 4.6a the Lorentzian oscillator model is in good
agreement with the simulated phase for the magnetic field inside the cavity. At the e11
resonant frequencies, the electrical field parallel to the long axis experiences π/2 phase
Figure 4.6. (a) Phase of the magnetic field along the long and short axes of the elliptical
patch calculated inside the cavity. Circles represents fitting with the Lorentz oscillator
model. (b) Difference in phase in the magnetic field of the reflected light calculated
along the long and short axes.
81
delay from the cavity resonance; while the electrical field perpendicular to the long axis
does not have cavity resonance and thus have a π phase delay. This π/2 difference in
phase delay between two electrical field components provides the foundation for a
circular polarized light at cavity resonant frequencies. The e11 and o11 cavity modes used
for polarization rotation can be excited with both normal and tilted illumination as
observed in our previous study [22]; therefore the polarization control with these
fundamental modes is possible with all incident light.
Secondly, the ellipticity of the reflected light is sensitive to the polarization
direction, i.e., β of the incident light. This is because the angle β determines the
amplitudes of the incident electrical fields along the two axes of the patches. Therefore,
by varying β, the amplitudes of the reflected field components can be tuned to be equal,
leading to circular polarization such as in sample a and b. As shown in Fig. 4.7, when
β=20° the reflected far field has equal amplitudes along the two axes of the patches.
82
4.6 Conclusion
To summarize, we have experimentally and numerically shown that arrays of
elliptical plasmonic nanopatch antennas can convert linear polarized light to circular
polarized light based on resonant excitation of the first cavity mode which is a magnetic
dipole mode; coupling light with these magnetic mode leads to phase retardation which
can be described by a Lorentz oscillator model. The periodicity and the reduced
symmetry of the patch play an important role as well. Although a drawback due to the
light absorption exists, the ability to tune the working wavelength and the little angle
dependence in polarization conversion make the elliptical patch antennas a good
candidate for practical applications.
Figure 4.7. Near field and reflected far field for sample a at 850 nm wavelength with
β=20°. First row: snapshots of calculated local field distributions in the middle plane
of the dielectric gap of plasmonic patch nanoantenna at different time. Second row:
Snapshots of calculated far field reflection at different time.
83
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10. A. Drezet, C. Genet and T. W. Ebbesen, Miniature plasmonic wave plates.
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concentric metallic gratings fabricated on optical fibers. Journal Of Optics, 2011.
13(1): p. 015003.
15. F. Wang, et al., Generation of radially and azimuthally polarized light by optical
transmission through concentric circular nanoslits in Ag films. Optics Express,
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103(36): p. 13300.
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Wavelength Range and Its Application to Hydrogen Sensing. Nano Letters, 2011.
11: p. 4366.
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3557-3563.
85
19. N. Liu, et al., Infrared Perfect Absorber and Its Application As Plasmonic Sensor.
Nano Letters, 2010. 10(7): p. 2342-2348.
20. J. Hao, et al., High performance optical absorber based on a plasmonic
metamaterial. Applied Physics Letters, 2010. 96(25).
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by plasmonic metamaterials. Physical Review B, 2011. 83(16).
22. A. Chakrabarty, et al., Cavity modes and their excitations in elliptical plasmonic
patch nanoantennas. Optics Express, 2012. 20(11): p. 11615-11624.
23. P. B. Johnson and R.W. Christy, Optical-Constants Of Noble-Metals. Physical
Review B, 1972. 6(12): p. 4370-4379.
24. J. C. Gutierrez-Vega, et al., Mathieu functions, a visual approach. American
Journal Of Physics, 2003. 71(3): p. 233-242.
25. B. Zhang, et al., Polarization-independent dual-band infrared perfect absorber
based on a metal-dielectric-metal elliptical nanodisk array. Optics Express, 2011.
19(16): p. 15221-15228.
86
CHAPTER 5
Plasmonic Nanocavity Networks
5.1 Introduction
Nanophotonics, the study of the electromagnetic response of subwavelength-
scaled structures or objects, is leading to new discoveries and insights into the basic
science of these phenomena as well as opening new technologies. One essential pursuit of
nanophotonics is the spatial localization of electromagnetic energy well below the optical
diffraction limit at wavelengths ranging from the visible to infrared. This strongly
confined light energy can take the form of resonant standing-wave modes at specific
positions of the subwavelength-scaled objects.
Optical microcavities, which trap light and enhance light-matter interactions by
recirculating light in various cavity structures, are critical to numerous applications in
light-emitting devices, quantum electrodynamics, enhanced nonlinear optics and
spontaneous emission control [1]. The enhancement of light-matter interactions can be
estimated by the Purcell factor of the optical cavities, which is proportional to the quality
factor Q of the cavities and inversely proportional to their effective mode volume Veff.
Traditional optical microcavities based on dielectric materials can achieve extremely high
quality factors Q (104 to 108), while their effective mode volume Veff is normally above
single cubic wavelength (λ/n)3 due to the interference nature of the confinements [1].
Cavities with high Q values have huge advantages in narrowband applications, especially
87
in lasing. For wideband applications [2], however, the dielectric cavities require a
compromise between the spectral width and Purcell factor. To improve the wideband
response of optical cavities, substantial efforts have been dedicated to developing new
avenues to reducing the mode volumes of the optical microcavities [3-5].
Recent studies have shown that plasmonic nanocavities enable squeezing light
into ultra-small volumes and controlling strong light-matter interactions [6-11], due to the
large wave vectors of the surface plasmons [6-11]. This extreme light confinement
capability of the plasmonic nanocavities naturally break the mode volume limit and have
found various applications, such as semiconductor-metal core-shell nanolasers [12, 13],
cavity-enhanced spontaneous emission [14], single photon light sources [15] and
biosensing [16]. Since the plasmonic materials including Cu, Ag, Au and highly doped
semiconductors are also widely used as the signal carriers in electronic circuits, it has
been envisioned that the electrical control of plasmonic nanocavities and the integration
between these plasmonic nanocavities and nanoscale electronic circuits would be one of
the next chip-scale technology [17]. For example, by electrically controlling the
plasmonic nanocavities filled with nonlinear materials, tunable harmonic generation of
light has been realized [18]. However, for plasmonic nanocavities in current designs, it is
either hard to address them electrically or challenging to fabricate them in large scale.
This is because that the plasmonic nanocavities normally necessitate focused ion beam
(FIB) milling to define the lateral dimensions, making the fabrication process very slow
and inefficient. Therefore, new nanocavity designs with both electrical connectivity and
manufacturability are desired for practical uses.
88
In this chapter, we propose a design of crossbar nanocavity networks with
potential to realize the desired electrical addressability[19]. Experimental and numerical
studies show that myriad sets of cavity modes can be excited with enhanced local fields
and resonant wavelengths tunable from shortwave to longwave infrared (IR) region. We
show that the cavity resonances lead to Fano resonance profiles in the far-field spectra
due to their coupling with a broadband surface wave mode and that the symmetries of the
cavity modes dictate the illumination configurations to excite them. The cavity resonant
conditions based on gap surface plasmon interferences show excellent agreements with
both experiments and simulations. The effective cavity mode volumes can reach ~2×10-
4λ03, or 10-3 times smaller than the microcavities made of semiconductors. This plasmonic
nanocavity design, similar to that of crossbar nanoelectronic devices, potentially allow
for not only electrical control of individual nanocavities, but also large scale
manufacturing with high precision in gap thickness control. The manufacturability and
electrical addressability of the crossbar plasmonic nanocavities may be used to devise
tunable metamaterials and IR detectors with very high sensitivities.
5.2 Sample preparation
Our new design is composed of two layers of perpendicularly oriented Au
nanowires, where the small gaps between the top and bottom nanowires form the
plasmonic nanocavities (Fig. 5.1a). The crossbar samples with different Au nanowire
widths were made in Air Force Research Labs in Wright Patterson Air Force base in
collaboration with Thomas Nelson, Donald Agresta, Kevin Leedy, and Dennis Walker.
Specifically, CaF2 and ZnO were chosen for the substrates and the dielectric gap layers
89
respectively because of their optical transparency from visible to long-wave infrared (~10
µm) wavelengths. The first layer of the one-dimensional (1D) grating of Au nanowires
was fabricated using 100keV electron beam lithography (EBL) on a JEOL JBX-6300FS
and subsequent metal deposition (e-beam evaporation) and lift-off process. The ZnO
dielectric gap for the plasmonic nanocavity was deposited using atomic layer deposition
(ALD) in a Cambridge Nanotech Fiji F200 ALD chamber, with a chuck temperature of
100ºC and a chamber pressure of 0.19 Torr. Diethylzinc was used as the metal-organic
precursor and a 300 W RF O2 plasma was used as the reactant. The second layer of the
1D Au nanowire grating was fabricated through the same procedures as for the first layer.
PMMA was used as the EBL resist, and a conductive polymer (Espacer by Showa Denko
K.K.) were spin-coated on the top of PMMA to eliminate charging effects. No adhesion
material was used for the Au deposition. The thicknesses of the Au layers are 35nm and
40nm for the first and second layers respectively. An exemplary SEM image of the
fabricated samples is shown in Fig. 5.1b.
90
5.3 Experimental measurements
Infrared reflection and transmission spectra were measured using a Fourier
Transform Infrared (FTIR) microscope system (Bruker Hyperion 2000 FTIR
Microscope) with a 15× IR/Vis objective. The samples were illuminated from the
substrate side for transmission measurements and from the sample side for reflection
measurements. The reflection spectra were normalized by the reflection spectrum
measured from a thick gold film; and the transmission spectra were normalized by the
transmission spectrum measured through the pure CaF2 substrate.
In Fig. 5.2a-d, the measured reflection and transmission spectra for four
representative nanowire widths exhibit broad dips (peaks) superposed by narrow resonant
dips/peaks. With the increase of the Au wire width, the position of the broadband peaks
Figure 5.1. (a) Schematic one unit of the crossbar plasmonic nanocavity network; (b)
a representative SEM picture of the fabricated plasmonic nanocavities with 1.4 µm
periodicity. The bottom and top Au wires are 800nm and 780nm in width respectively.
91
(dips) shifts to higher frequencies while the positions of these narrow dips/peaks shift to
lower frequencies.
5.4 Numerical simulations and calculations
To understand these resonance characteristics, numerical simulations were
performed using the commercial software CST Microwave Studio. As for the dielectric
constant of gold in the infrared region, the Drude model εAu=ε∞-ω2p/ω(ω-iγ) was used to
describe its frequency dependence. The parameters, ε∞=3.49, ωp=1.31×1016 rad/s, and γ
=6.01×1013 Hz were obtained by fitting the Drude model to experimentally measured
permittivity for bulk Au material [20]. The Drude model was also used to describe the
frequency dependence of the dielectric constant for ZnO within the frequency range of
our studies, with ε∞=3.5, ωp=2.828×1014 rad/s, and γ =1.52×1014 Hz obtained from fitting
to the measured permittivity of thin ZnO films. The mesh size for simulations is set at ~ 7
nm. The local field enhancement spectra were calculated and averaged over values at the
four corners of the plasmonic nanocavities; the far-field reflection /transmission spectra
were obtained by integrating the Poynting vectors at a plane placed at 500nm above or
below the crossbar samples. Incident angles of 0º and 20º were used. The transmission
and reflection spectra averaged over two polarizations and two incident angles are used to
represent the total spectra of the system; the local field spectra averaged over four
nanocavity corners are used to represent the total local field enhancements. The
agreements between simulated and experimental far-field spectra are excellent (Fig. 5.2a-
h); both the broad and narrow dips/peaks are reproduced at almost the same frequencies.
92
We also calculated the transmission/reflection spectra for the crossbars with the
ZnO gaps replaced by Au, or two-dimensional (2D) hole arrays in thin metal films. The
numerical results suggest that the broad dips/peaks in the reflection/transmission spectra
actually originate from the 2D hole arrays (dashed lines in Fig. 5.2e-h). Extensive prior
studies in enhanced optical transmission through 2D hole arrays suggest that the physical
origin of the resonances is the excitation of coupled surface electromagnetic modes [21].
The broad transmission peak in our system, f~150 THz matches the Wood-Rayleigh
anomalies in the CaF2 side (Fig. 5.2e). As shown in Fig. 5.2e-h, with increasing the width
of the bars, i.e. reducing the size of the holes, the broad transmission peak would shift
towards higher frequency. The reason is attributed to the blue-shift of the cut-off
frequency of the holes. Besides, the thicknesses of the bars (35 nm and 40 nm) are much
smaller compared to the infrared wavelengths. That’s why light can still transmit through
93
Figure 5.2. (a-d): Measured transmission (olive circles) and reflection (blue circles)
spectra for 4 representative sizes of the crossbar plasmonic nanocavity networks. (e-
h): Calculated transmission (olive circles) and reflection spectra (blue circles) for the
crossbar plasmonic nanocavity networks. The dashed lines are the calculated
transmission and reflection spectra for the crossbar structures of the same sizes with
the nanocavities filled with Au. (i-l): Calculated local field enhancement spectra for
normal incidence (olive) and for 20º tilted incidence (blue: s polarization, red: p-
polarization). The 4 representative top and bottom Au wire widths are at 310×330nm
for (a, e, i), 576×560nm for (b, f, j), 800×780nm for (c, g, k) and 942×870nm for (d, h,
l). The red solid lines in a-h were the best fittings with Fano resonances on top of the
Lorentzian profiles.
94
the holes with small attenuation, i.e. a large transmission, even with the cut-off effect of
the holes.
The local field spectra show large field enhancements at those resonant peaks,
achieving ~50 for some modes (Fig. 5.2i-l). A comparison of the near-field with the far-
field spectra indicates that the cavity resonances generate resonant peaks/dips with
typical Fano-profiles in the far-field spectra (Fig. 5.2e-h). Fano resonances are a general
phenomenon occurring when a discrete state interferes with a continuum state [22]. In our
system, the narrow-band plasmonic cavity modes act as the discrete states, while the
broad-band mode due to the 2D hole arrays serves as the continuum state.
5.5 Discussion
To understand these cavity modes, we consider the local field distribution inside
the plasmonic nanocavities at their resonance frequencies. The patterns of the electrical
field distributions are highly dependent on the incident angle and polarization conditions,
as exampled in Fig. 5.3 for the sample with 800×780 nm Au wire width. These patterns
can be ascribed to the constructive interferences of gap surface plasmons excited and
launched from the edges of the nanocavities. The gap plasmons refer to the coupled
surface plasmon modes when two metal-dielectric interfaces are brought into close
proximity. There exist one symmetric and one anti-symmetric gap plasmons, while the
symmetric mode only occurs at high frequencies and thus can be ignored in our case. For
small gap size t, the dispersion relationship for the anti-symmetric mode can be
approximated as [23]:
95
( )d d m m dtanh k t 2 k k= −ε ε (5.1)
2 2 2m gsp mk k c= − ε ω , 2 2 2
d gsp dk k c= − ε ω , where kgsp is the wave vector for the gap
surface plasmon; εm and εd are the permittivity of the metallic and dielectric materials; ω
and c are the radial frequency and speed of light in vacuum.
We assume that the gap plasmons have two wave vector components, kx and ky,
along the two nanowire directions. The open boundary conditions at the cavity edges
imply kxLx=mπ, and kyLy=nπ, where Lx and Ly are the widths of the nanocavities, the
non-negative integers, m and n represent the number of antinodes in x and y directions.
Given 2 2 2x y gspk k k+ = , the cavity resonant condition can be obtained as:
Figure 5.3. Snapshots of simulated local electrical field distributions for the cavity
modes with 800nm×780nm wire widths. (a) The first 6 cavity modes excited with a s-
polarized light at 20o tilted incidence; (b) the first 6 cavity modes excited with a p-
polarized light at 20o tilted incidence; (c) the first 3 cavity modes excited with a
normal incidence polarized at 45o to x-and y-axis.
96
2 22gsp 2 2
x y
m nkL L
= π + , m, n=0, 1, 2, 3… (5.2)
Based on this resonant condition, the cavity modes can be indexed as (m, n).
By examining the cavity modes excited at various illumination conditions, we
extract the cavity mode excitation rules (Table 5.1). These excitation rules can be
understood as a result of the phase asymmetry requirements in the gap plasmons
excitation. For normal incidence, the gap surface plasmons launched at two opposite
cavity edges must be in opposite phase along the polarization direction while must be in
the same phase in the perpendicular direction. Therefore, to excite modes with both even
or both odd indexes, these phase constraints have to be broken by using tilted incidence.
Consequently, the cavity resonances in far-field spectra are distinctive for modes with
one even and one odd index because of their excitations by both normal and oblique
incident light, while less distinctive for the modes with both even or both odd indexes
because of their excitation only by tilted illuminations (Fig. 5.2).
97
We fitted the measured reflection/transmission spectra with a broad Lorentzian
peak/dip multiplied by multiple Fano resonance profiles to extract the cavity resonant
frequencies. To validate this fitting method, we firstly performed this fitting to the
simulation data (red solid curves in Fig. 5.2e-h), and can show that the cavity resonant
frequencies obtained from the Fano fitting agree well with those from the near-field
spectra (Fig. 5.2i-l). The Fano profiles on the broad Lorentzian peak/dip fit the measured
Table 5.1. Crossbar cavity excitation condition
(m, n) mode Incidence plane polarization Examples
both even tilted p (2,0), (0,2), (2,2)
both odd tilted s (1,1), (3,1), (1,3)
odd m, even n
tilted p or s with Ex
component (1,0), (1,2), (3,0)
normal x
even m, odd n
tilted p or s with Ey
component (0,1), (0,3), (2,1)
normal y
98
spectra well, and the acquired resonant frequencies are in close agreements with the
simulation results.
For all cavity modes, the resonant frequencies decrease monotonically with the
increase of cavity width (Fig. 5.4a). By re-plotting the resonant frequencies as a function
of 2 2 2 2x ym L n Lπ + , all data points collapse into a master curve (Fig. 5.4b). A
Figure 5.4. (a) Cavity resonant frequencies versus the averaged nanowire width,
(Lx+Ly)/2. The symbols represent experimental data obtained from Fano fittings; and
the solid curves represent simulation results. (b) Measured cavity resonant frequencies
versus 2 2 2 2x ym L n Lπ + . The solid curve is the dispersion curve for the gap
plasmons based on equation (1). (c) Simulated effective mode volumes (104Veff/λ03)
for different cavity modes vs. cavity resonance frequency. (d) Measured quality
factors for different cavity modes versus cavity resonance frequency.
99
comparison of the data with the gap plasmon dispersion curve calculated using Eq. 5.1
show reasonably good agreements (Fig. 5.4b), indicating the validity of Eq. 5.2.
Fig. 5.4c and 5.4d present the calculated effective mode volumes Veff and quality
factors (both measured and calculated) respectively for four cavity modes as functions of
cavity resonant frequencies. The cavity mode volume is calculated
using [ ]3mV U(r)d r max U(r)= ∫ , where U is total electromagnetic energy stored in the
crossbar nanostructures which should be calculated by the summation of both electric and
magnetic energy, r r2 2 20 0 0 r
1 ( ) 1 1 ( )U(r) E (r) H (r) E (r)2 2 2
∂ ωε ∂ ωε ′= ε + µ = ε ε + ∂ω ∂ω . r′ε and
r′′ε represent the real part and imaginary part of the dielectric permittivity [24]. The
volume integral of the energy density U(r) was performed from 50 nm above the top
layer of the crossbar to 50nm below the bottom layer of the crossbar, including the
regions of air, Au, ZnO and CaF2 substrate. For the Drude model,
rr r
( ) 2∂ ωε ′ ′′= ε + ε γ∂ω
.
The mode volumes are ultra-small and reach a minimal value ~2×10-4λ03 when the
cavity resonant frequencies coincide with the broad band transmission peak frequency
(~150THz). This is about 10-3 smaller than photonic crystal cavities [4]. The quality
factor varies between 4 and 20 and goes up with the increase of the cavity frequency. The
Purcell factor, a measure of the spontaneous emission rate enhancements can reach 2400
in our nanocavity arrays, which is close to previous achievements [7, 8].
100
We have discussed about the cavity resonance condition of our plasmonic
nanocavity networks and reached the conclusion that the k-vectors of the resonant cavity
modes should satisfy Eq. 5.2. According to Eq. 5.2, the positions of the resonant
frequencies are only determined by the widths of the crossbar wires. The period of the
crossbar should have no influence on the resonant frequencies. This is true for the middle
infrared and far infrared frequency range but not for the higher frequencies. In the visible
and near infrared range, some cavity resonances can be easily shifted by changing the
period of crossbar networks.
In this chapter, we focus our discussions on the middle and far infrared frequency
range. In this frequency range, despite changing the repetition period of the nanocavities
can weakly shift the positions of the peak/dip of the transmission/reflection due to the
Fano resonances, the positions of cavity resonance frequencies remain the same with no
apparent shift. We confirm this point in our simulation results. For example, as shown in
Figure 5.5. Simulated transmission spectra (a), reflection spectra (b) and local field
enhancement (c) of the crossbar plasmonic nanocavity networks with the top and
bottom wire widths being 576nm and 560nm respectively. The red, green and blue
curves represent the crossbar period of 1.1μm, 1.4μm and 1.7μm respectively.
101
Fig. 5.5 c, the (1,0) modes always occur around 71THz when varying the period.
However, in Fig.5.5 a, 71THz corresponds to the transmission peak of 1.1 μm period
crossbar while it is close to the transmission dip of 1.7 μm period crossbar. An even
clear case is the (1,1) mode resonance which happens at the frequency of 103THz. This
mode cause a dip in the reflection spectrum of 1.1 μm period crossbar but exactly a peak
in the reflection spectrum of 1.7 μm period crossbar. The physics underlying the
phenomena is that the broad lorentzian resonance induced by the surface wave has been
red shifted with increasing the period. The interaction between the shifted broad
lorentzian resonance and the unshifted local cavity resonance would induce the Fano
resonance. Depending on the positions of the local cavity resonances on the side wall of
the broad lorentzian resonance, the transmission/reflection can appear as either peaks or
dips of the Fano resonances.
5.6 Conclusion
In summary, we demonstrate a new type of crossbar plasmonic nanocavity
network with ultrasmall mode volumes and the potential of electrical addressability. In
the middle and far infrared range, the excitation of cavity resonances strictly obeys the
metal-dielectric-metal dispersion relation. Thus variation of the period would not shift the
cavity resonant frequencies despite the peak/dip of the far field transmission/reflection
could be slightly shifted due to the Fano resonances.
Crossbar design was firstly pioneered as molecular electronic devices [25]; its
advantages in molecular electronics apply for plasmonic cavities, such as its electrical
addressability, tolerance to fabrication errors, ease of multiplexing and 2N electrodes to
102
address N2 nanocavities. Unique applications of this crossbar design of plasmonic
nanocavity networks can be envisioned. For example, integrating plasmonics with the
molecular electronic devices will provide a new platform not only for characterizing the
radiation effects on molecular junctions [26] but also for tunable metamaterials
considering the similarity of crossbars to the fishnets for metamaterials [27]. The large
local field enhancements, ultrasmall mode volume and relative broad band (low Q factor)
provide an optimal solution for the conflicting demands of IR detections for simultaneous
low mode volume and high photo-absorption.
103
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CHAPTER 6
Summary and Conclusions
The development of semiconductor industry has led to the realization of
nanoscale electronic circuits for computation and information storage. However, the
interconnection of semiconductor electronics is limited in speed due to the RC delay
issues [1]. Compared to electronic devices, photonic devices have stronger capabilities
for data transportation and processing. In traditional photonics, dielectric materials are
used to facilitate information transport over long distances due to their low loss and high
transparency. However, photonic devices are limited in size (usually larger than 1 µm)
due to the diffraction laws. Plasmonic devices consisting of deep-subwavelength
structures enable light to be highly concentrated to nanoscale and actively manipulated at
very high speed [2]. Plasmonic structures exactly complement electronics and
conventional photonics [3, 4]. We have discussed the excited electromagnetic modes and
related applications of several representative plasmonic nanostructures, including the
circular and spiral nanogratings [5], patch nanoantennas [6, 7] and crossbar nanocavities.
For the nanoapertures formed by plasmonic circular and spiral nanogratings,
optical transmission through the nanoapertures has been measured and calculated. By
analyzing the transmission spectra, we found that the transmission spectra exhibit peaks
at the wavelengths of Wood anomalies while dips at the wavelengths of surface plasmon
waves. This result is in accordance with other researcher’s conclusion that the excitation
107
of surface plasmons not only can enhance the optical transmission but also in many
situations can induce a suppression of the optical transmission through the nanoapertures
including the 1-dimensional metallic nanogratings [8].
For the spiral nanogratings with 2 periods, we have observed the circular
dichroism between left circularly polarized light and right circularly polarized light
incident on them. We attributed the detected circular dichroism to the in-plane
handedness of the spiral nanogratings. This in-plane handedness always leads to different
surface wave intensity between left circularly polarized incidence and right circularly
polarized incidence. With increasing the number of the radial periods, the near field
difference between left circular polarization and right circular polarization rapidly
decreases so that the circular dichroism can hardly be observed (signal-to-noise ratio<1)
when the number of radial periods is larger than four.
The study of circular and spiral nanogratings will lead to a series of potential
applications including nano-focusing, generation and control of spatially variant
polarization [9] and ultra-compact light handedness detector. For example, only the left
circular polarized incidence can be strongly focused at the center of the spiral
nanogratings with left handedness. By introducing the optoelectronic materials (such as
amorphous silicon) to the spiral center, the optical properties and electrical conductivities
of these materials can be greatly tuned for the left circular polarization while remain the
same for the right circular polarization. This specific property of nano-spirals gives an
optimal solution for detecting the light handedness at nanoscale [10].
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Ultrasmall plasmonic nanoantennas and nanocavities, as the most important
building blocks of plasmonic devices, are currently leading to various promising
applications in nanophotonics. The plasmonic nanocavities are usually composed of
metal-dielectric-metal structures. The dimensions of the structures are much smaller than
the wavelengths of free space light and the thickness of the dielectric layer is only a few
nanometers to tens of nanometers. In this dissertation, we analyzed two types of
plasmonic nanocavities, i.e. the patch nanoantennas and the crossbar nanocavities
network. In spite of their different shapes and configurations, the two types of
nanocavities have common optical features in many aspects since they both are metal-
dielectric-metal structures in essence.
Our experimental studies show that from the middle infrared to near infrared
frequency range, the resonant frequencies of all the plasmonic cavity modes strictly obey
the theoretical dispersion relation of metal-dielectric-metal structures. The periods of
plasmonic nanocavities arrays have little affection on the resonant frequencies of cavity
modes despite the far field transmission/reflection can be changed due to the period
effect. In the visible range, there exists a slight discrepancy between our experimentally
obtained cavity dispersion curve and theoretical dispersion curve. The small discrepancy
comes from the fringe field effect.
The first order plasmonic cavity modes can be simply described by Lorentz
oscillator model. From the frequency lower than the Lorentz oscillation frequency to the
frequency higher than the Lorentz oscillation frequency, a π phase delay would be
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expected. This phase delay can be utilized to convert the linear polarization to
elliptical/circular polarization as discussed in Chapter 4.
All the plasmonic cavity modes have shown ultrasmall mode volumes (~10-4
λ03) and thus a greatly enhanced Purcell factor. These optical properties of plasmonic
cavities can be utilized to tremendously enhance light-matter interaction and have
potentials for various optoelectronic applications [1, 2]. To realize the electrical control
of nanocavities in large scale, we proposed and studied the crossbar nanocavities
network. For the future work, active media including organic conductive polymers and
inorganic semiconductors should be used as the gap materials of the crossbar cavities.
External voltages can be applied to each nanocavities for optoelectronic studies.
110
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