Citation
Transmission properties of sub-wavelength hole arrays in metal films

Material Information

Title:
Transmission properties of sub-wavelength hole arrays in metal films
Creator:
Woo, Kwangje ( Dissertant )
Tanner, David B. ( Thesis advisor )
Hebard, Arthur ( Reviewer )
Hill, Stephen O. ( Reviewer )
Hershfield, Selman P. ( Reviewer )
Holloway, Paul H. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
2006
Language:
English

Subjects

Subjects / Keywords:
Brewsters angle ( jstor )
Dielectric materials ( jstor )
Geometric angles ( jstor )
Photons ( jstor )
Plasmons ( jstor )
Spectrometers ( jstor )
Squares ( jstor )
Transmittance ( jstor )
Wave diffraction ( jstor )
Wavelengths ( jstor )
Physics thesis, Ph.D
Dissertations, Academic -- UF -- Physics

Notes

Abstract:
We have measured the optical transmittance of sub-wavelength hole arrays in metal films. We investigated the spectral behavior of transmittance (the peak positions, intensities, line-widths, and the dip positions) as a function of the geometrical parameters of the hole arrays, the angle of incidence, the polarization angle and the refractive indices of the substrates. We calculated the positions of transmittance peaks and dips with equations from the surface plasmon theory and the diffraction theory, and compared the calculated positions of peaks and dips with measured transmittance data. We found that there is a discrepancy of 3 ~ 5% between the peak positions calculated with the surface plasmon equation and the peak positions in the measured transmittance data. We explain this discrepancy as possibly due to the approximations of the surface plasmon equation. However, the positions of the dips in the spectra, as calculated with the diffraction grating equation, were well matched to the measured data. We also observed splittings and shifts of the peaks and dips when changing the angle of incidence and the polarization of the light. We confirmed this spectral behavior qualitatively with calculation of momentum conservation equations for oblique incidence and showed that the diffraction modes are degenerate for s-polarization, while the modes are not degenerate for p-polarization. We studied the dependence of hole size and shape on the transmittance while also changing the in-plane polarization angle. We observed that the transmittance peak is strongly dependent on the length of the hole edge perpendicular to the polarization direction. In addition, we investigated the dependence on film thickness and the refractive index of dielectric substrate.
Abstract:
array, diffraction, hole, plasmon, subwavelength, surface, transmission
General Note:
Title from title page of source document.
General Note:
Document formatted into pages; contains 159 pages.
General Note:
Includes vita.
Thesis:
Thesis (Ph.D.)--University of Florida, 2006.
Bibliography:
Includes bibliographical references.
General Note:
Text (Electronic thesis) in PDF format.

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Woo, Kwangje. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
8/31/2006
Resource Identifier:
649814512 ( OCLC )

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Full Text












TRANSMISSION PROPERTIES OF SUB-WAVELENGTH HOLE ARRAYS IN
METAL FILMS

















By

KWANGJE WOO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Kwangj e Woo















ACKNOWLEDGMENTS

For last 5 years for my Ph.D. work, there are many people whom I have to thank

for their support, advice and encouragement.

First, I would like to thank my advisor, Professor David B. Tanner. Since I have

become his research assistant, I have received so much valuable advice, encouragement

and support.

Also, I would like to thank Professor Arthur F. Hebard, Professor Stephen O. Hill,

Professor Selman P. Hershfield and Professor Paul H. Holloway for serving on my

supervisory committee.

It was a great time for me to work in Prof Tanner's lab for last four years because I

had good colleagues in this lab: Dr. Andrew Wint, Dr. Hedenori Tashiro, Dr. Maria

Nikolou, Dr. Minghan Chen, Haidong Zhang, Naveen Margankunte, Nathan Heston,

Daniel Arenas and Layla Booshehri. I would like to thank these people.

Especially, I would like to thank my collaborator Sinan Selcuk for supplying

samples, scientific discussions and a truthful friendship.

I would like to thank my parents. They have supported me throughout my life.

Finally, my wife and children, Ohsoon, Jisoo and Jiwon, have supported me with their

love and patience. I would like to express my deepest thanks to them.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES ....................................................... ............ .............. .. vii

L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii

ABSTRACT ........ .............. ............. .. ...... .......... .......... xii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Background and M otivation ...................................................... .......................
O rg animation .................................................. .......................... 2

2 RIVIEW OF SURFACE PLASMON AND DIFFRACTION THEORY ..................4

Bethe's Theory for Transmittance of a Single Sub-Wavelength Hole.........................5
Surface P lasm on ................................................... ...................... 7
D definition of Surface Plasm on ........................................ ......................... 7
Dispersion Relation of Surface Plasmon............... ..............................................7
Dispersion relation for the p-polarization ............................................... 8
Dispersion relation for the s-polarization...............................................10
D ispersion curves ......... .... ........ ........................ .. ....... .................12
Propagation Length of the Surface Plasmon ...................................................14
Surface Plasmon Excitation................. ...... ........ .................14
Mechanism of Transmission via Surface Plasmon Coupling in Periodic Hole
A rra y ....................................... ............................................ ............... 1 7
CDEW (Composite Diffractive Evanescent Wave) .................. ............... 19
Basic Picture of the CDEW ................ ..... ....................... .........19
CDEW for an Aperture with Periodic Corrugation ................ ...............22
CDEW for a Periodic Sub-Wavelength Hole Array............... ............... 23
F ano P profile A naly sis .............................. .... ...................... .. ...... .... ...... ...... 25

3 IN STR U M E N TA TIO N ................................................................... .....................29

Perkin-Elmer 16U Monochromatic Spectrometer.................................................29
Light Sources and D etectors.......................................... ........... ............... 29









Grating Monochromator ........... ..... .............. ............... 30
M onochrom ator configuration .......................................... ............... 31
Resolution of monochromator............ ......... .............................32
The D iffraction G rating .............................................. ..... ....................... 33
Grating equation and diffraction orders ....................................... .......... 33
Blaze angle of the grating................................ ........................ ......... 34
R solving pow er of grating ...................... ......... ...................................... 34
Bruker 113v Fourier Transform Infrared (FTIR) Spectrometer..............................35
Interferom eter ............... .. ......................................... .. .. ......... 35
Description of FTIR Spectrometer System ............... ............................... 38

4 SAM PLE AND M EASUREM ENT ........................................ ........ ............... 41

Sample Preparation..................... .......................... .......... 41
S u b stra te s .............................................................................................................. 4 1
M easurem ent Setup ............................................................... .... .. ....43

5 EXPERIMENTAL RESULTS ............................................................................46

Enhanced Optical Transmission of Sub-wavelength Periodic Hole Array ...............46
Comparison of Enhanced Transmission with Classical Electromagnetic Theory......47
Dependence of Period, Film Thickness and Substrate on Transmission....................48
Dependence on Period of Hole Array .................................................. ... ........... 49
Dependence on the Thickness of Metal Film.............. .... .................50
Dependence on the Substrate M material ..................................... ............... 52
Dependence on the Angle of Incidence.............................. ...............54
Dependence on Hole Shape ......... ..... ......... .......................... 58
Squ are H ole A rray s ........................... ......................... .. ...... .. ...... ............58
Rectangular Hole Array ......... .. .. ......... ............... ............... 59
Slit A rray s ............... ................. ... ............. .............................. 6 1
Transmission of Square Hole Array on Rectangular Grid ................................ 62
Refractive Index Symmetry of Dielectric Materials Interfaced with Hole Array ......63

6 ANALYSIS AND DISCU SSION ........................................ ......................... 68

Prediction of Positions of Transmission Peaks......... ................ .... ..................... 68
Comparison of Calculated and Measured Positions of Transmittance Peaks and
D ip s .......................... ....... ...... .................................... ............... 7 0
Dependence of the Angle of Incidence on Transmission.................................74
Drawbacks of Surface Plasmon and CDEW ................................... ............... 82
Dependence of Hole Shape, Size and Polarization Angle on Transmission ..............84
CDEW and Trapped Modes for Transmission Dependence on Hole Size.................87

7 C O N C L U SIO N ......... ......................................................................... ........ .. ..... .. 9 1

APPENDIX

A TRANSMITTANCE DATA OF DOUBLE LAYER SLIT ARRAYS ....................95









B POINT SPREAD FUNCTIONS AND FOCUSING IMAGES OF PHOTON
S IE V E S ...................................... .................................................. 10 7

C TRANSMITTANCE DATA OF BULL'S EYE STRUCTURE............................136

L IST O F R E F E R E N C E S ...................... .. .. ......... .. .................................................. 142

BIOGRAPHICAL SKETCH ................ ............................................ ............... 146
















LIST OF TABLES


Table page

4-1 List of the periodic sub-wavelength hole arrays .................................................43

6-1 Calculated positions of surface plasmon resonant transmittance peaks for three
interfaces of 2000 nm period hole arrays at normal incidence (Ed of air, fused
silica and ZnSe are 1.0, 2.0 and 6.0, respectively)................................ ...........69

6-2 Calculated positions of transmittance dips for three interfaces of 2 atm period
hole arrays at normal incidence (Ed of air, fused silica and ZnSe are 1.0, 2.0 and
6.0, respectively) ......................................................................72















LIST OF FIGURES


Figure p

2-1 Schematic diagram for p-polarized (TM) light incident on a dielectric/metal
interface ............................................................... .... ..... ......... 12

2-2 Schematic diagram for s-polarized (TE) light incident on a dielectric/metal
interface ............................................................... .... ..... ......... 12

2-3 Dispersion curves of surface plasmon at air/metal interface and at quartz/metal
interface, light lines in air and fused silica.................................. ............... 13

2-4 Schematic diagrams of (a) the excitation of the surface plasmon by the incident
photon on a metallic grating surface and (b) the dispersion curves of the incident
photon, the scattered photon and the surface plasmon...........................................15

2-5 Schematic diagrams of the excitation of the surface plasmon by the incident
photon on a two dimensional metallic grating surface ...........................................18

2-6 Schematic diagram of transmission mechanism in a sub-wavelength hole array.... 18

2-7 Geometry of optical scattering by a hole in a real screen in (a) real space and (b)
k-space for a range that kx is close to zero.................................... ............... 21

2-8 CDEW lateral field profile at z = 0 boundary, a plot of Eq. (2-44) ..... ......... 22

2-9 CDEW picture for an aperture with periodic corrugations on the input and
output surfaces. Red arrows indicate the CDEWs generated on the input and
ou tpu t su races ................................................. ................ 2 4

2-10 A CDEW picture for a periodic sub-wavelength hole array. Red arrows indicate
the CDEWs generated on the input and output surfaces .......................................24

2-11 Schematic diagrams for Fano profile analysis. .............................. ......... ...... .27

2-12 A schematic diagram of the non-resonant transmission (Bethe's contribution)
and the resonant transmission (surface plasmon contribution) .............................27

2-13 Schematic diagram of the interference between the resonant and non-resonant
diffraction in transmission of sub-wavelength hole array ................... ..............28

3-1 Schematic diagram ofPerkin-Elmer 16U monochromatic spectrometer.................30









3-2 Schematic diagram of the Littrow configuration in the monochromator of
Perkin-Elmer 16U spectrometer...................... ..... ............................ 31

3-3 Schematic diagram of a reflection grating. ................................... ............... 33

3-4 Schematic diagram of a blazed grating ........................................ ............... 34

3-5 Schematic diagram of Michelson interferometer................. ............... .............36

3-6 Schematic diagram of the Bruker 113v FTIR spectrometer.................................39

4-1 SEM images of periodic hole arrays samples ..............................42

4-2 Picture of the sample holder used to measure transmittance with changing the
angle of incidence and the in-plane azimuthal angle ............................................44

5-1 Transmittance of the square hole array (A14-1) and a silver film ...........................47

5-2 Comparison between Bethe's calculation and the transmittance measured with
the square hole array (A 14-1)........................................................ ............... 48

5-3 Transmittance of square hole arrays with periods of 1 tm (A15) and 2 tm (A18-
1).............. .................... ...................................... ........ ...... 4 9

5-4 Transmittance vs. scaling variable, ks = k/(nd x period), for the square hole
arrays of 1 tm period (A15) and 2 tm period (A18-1) made on fused silica
su b states (n d = 1.4 ) ............... ............................................. ................ 5 1

5-5 (a) Transmittance vs. wavelength (b) transmittance vs. scaling variable, ks =
X/(nd x period), for the square hole arrays of 2 tm period (A14-1) made on a
fused silica substrate (nd = 1.4) and a ZnSe substrate (nd = 2.4)...........................53

5-6 Transmittance of a square hole array (A14-1) with three different polarizations
at norm al incidence ............................................ ................. ........ 54

5-7 Measurement of transmittance with s-polarized incident light as a function of the
incident angle. ........................................................................56

5-8 Measurement of transmittance with p-polarized incident light as a function of
the incident angle. .....................................................................57

5-9 Transmittance of square hole array (A18-1) as a function of polarization angle.
The inset shows a SEM image of the square hole array.................................59

5-10 Transmittance of a rectangular hole array (A18-2) for in-plane polarization
angles of 0 0 and 90 The inset shows a SEM image of the rectangular hole
array ............... .... ... ......... .. ............................................60









5-11 Transmittance of a slit array (A18-3) for in-plane polarization angles of 0 and
90 The inset shows a SEM image of the slit array. ............. ...............61

5-12 Transmittance of a square hole array on a rectangular grid (A18-4) for
polarization angles of 0 45 o and 90 The inset shows a SEM image of the
square hole array in a rectangular grid. .......................................... ............... 62

5-13 Schematic diagram of sample preparation .................................... ............... 65

5-14 Transmittance of a square hole array (A14-1) on fused silica substrate with and
without PR coated on the top ............................................................................65

5-15 Transmittance of a square hole array (A14-1) on ZnSe substrate with and
without PR coated on the top of hole array ....... ........ ......................... ........ 66

5-16 Transmittance of a square hole array (A14-1) on fused silica substrate with and
without PMMA coated on the top of hole array with the second fused silica
substrate attached on the top of PM M A....................................................... 66

6-1 Comparison of calculated peak positions with measured transmittance data.
Transmittance measured with a square hole array (A18-1) is shown. P1, P2 and
P3 are the calculated positions of three transmittance peaks ................................70

6-2 Comparison of the calculated transmittance peaks and dips with the
transmittance measured with a square hole array (A18-1) made on a fused silica
substrate. P1, P2 and P3 are the calculated positions of the first three peaks and
D1, D2 and D3 are the calculated positions of the first three dips...........................73

6-3 Comparison of the calculated transmittance peaks and dips with the
transmittance measured with a square hole array (A14-1) made on a ZnSe
substrate. P4 and P5 are the calculated peak positions and D4 and D5 are the
calculated dip positions for the ZnSe-metal interface. P2, P3, D2 and D3 are the
positions of the peaks and the dips for the air-metal interface.............................74

6-4 Transmittance of a square hole array (A14-1) measured using unpolarized light
at norm al incidence. ........................................... .. .... ......... .. ..... .. 75

6-5 Schematic diagram of an excitation of surface plasmon by the incident light on
two dimensional metallic grating surface.............. ............................. ..............76

6-6 Transmittance with s-polarized incident light...................... ........ ........................ 77

6-7 Transmittance with s-polarized incident light...................... ........ ........................ 78

6-8 Peak and dip position vs. incident angle for s-polarization..................................79

6-9 Peak and dip position vs. incident angle for p-polarization. .............. ...............80









6-10 Transmittance of square, rectangular and slit arrays with polarization angle of
00.. ....................................................89

6-11 Transmittance of square, rectangular and slit arrays with polarization angle of
900. ...................................... .................. .. 90















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

TRANSMISSION PROPERTIES OF SUB-WAVELENGTH HOLE ARRAYS IN
METAL FILMS

By

Kwangj e Woo

August 2006

Chair: David B. Tanner
Major Department: Physics

We have measured the optical transmittance of sub-wavelength hole arrays in metal

films. We investigated the spectral behavior of transmittance (the peak positions,

intensities, line-widths, and the dip positions) as a function of the geometrical parameters

of the hole arrays, the angle of incidence, the polarization angle and the refractive indices

of the substrates. We calculated the positions of transmittance peaks and dips with

equations from the surface plasmon theory and the diffraction theory, and compared the

calculated positions of peaks and dips with measured transmittance data. We found that

there is a discrepancy of 3 5% between the peak positions calculated with the surface

plasmon equation and the peak positions in the measured transmittance data. We explain

this discrepancy as possibly due to the approximations of the surface plasmon equation.

However, the positions of the dips in the spectra, as calculated with the diffraction grating

equation, were well matched to the measured data. We also observed splitting and shifts

of the peaks and dips when changing the angle of incidence and the polarization of the









light. We confirmed this spectral behavior qualitatively with calculation of momentum

conservation equations for oblique incidence and showed that the diffraction modes are

degenerate for s-polarization, while the modes are not degenerate for p-polarization. We

studied the dependence of hole size and shape on the transmittance while also changing

the in-plane polarization angle. We observed that the transmittance peak is strongly

dependent on the length of the hole edge perpendicular to the polarization direction. In

addition, we investigated the dependence on film thickness and the refractive index of

dielectric substrate.














CHAPTER 1
INTRODUCTION

Background and Motivation

Recently, many workers in the area of optics have reported very interesting results

in a new regime of optics called nano optics, sub-wavelength optics, or plasmonic optics

[1]. In this area of optics, the physical dimension of objects for optical measurements is

on a sub-wavelength scale. Interestingly, the optical properties of sub-wavelength

structures are different from what we predict from classical electromagnetic theory [2]. In

addition, this new field of optics makes it possible to manipulate light via sub-wavelength

structures. This capability of controlling light attracts a lot of applications in various

fields of science and technology, for instance, Raman spectroscopy, photonic circuits, the

display devices, nanolithography and biosensors [3-6].

Since the first research on enhanced optical transmission of an array of sub-

wavelength holes was reported in 1998 by Ebbesen et al. [7], no theory has explained this

phenomenon, even though a lot of work has been carried out. But theoretical studies are

still actively going on, with the most prominent one being the surface plasmon polariton

(SPP) theory [8]. In addition to the surface plasmon polariton, the diffraction theory is

also a very strong candidate as an explanation of the enhanced transmission of sub-

wavelength hole array [9-11]. Another model [12-14] proposed to explain this enhanced

transmission phenomenon is the superposition of a resonant process and a non resonant

process which shows the Fano profile [15]. Since the surface plasmon polariton model

has some drawbacks [9] and shows a discrepancy between calculated and measured data









[16], other models are considered as strong explanations of this enhanced transmission

phenomenon.

Many experiments also have been done for a wide spectral range. The enhanced

transmission of periodic hole arrays for the optical region, the near-infrared region [17],

and the terahertz (THz) region [18-24] was reported.

Other scientific and technological interest is focused on the enhanced transmission

of a single sub-wavelength aperture. The enhanced transmission of a single plain

rectangular aperture, which depends on the polarization direction, was reported [25, 26].

And an aperture with corrugations on the input side showed an enhanced transmission as

well as a beaming of the transmitted light with corrugations on the output side [27-29].

In this dissertation, we present experimental transmission data for sub-wavelength

hole arrays as a function of their geometrical parameters, the angle of incidence, the

polarization of the light, and for two values of the refractive index of the dielectric

substrate material. For the theoretical models, we will discuss surface plasmon,

composite diffractive evanescent wave, Fano profile analysis and trapped mode.

Organization

This dissertation consists of seven chapters, including this introduction chapter.

The details of each chapter are as follows:

In Chapter 2, we review the basic theories of surface plasmon and diffraction. The

surface plasmon theory includes surface plasmon excitation by incident light, the

plasmon dispersion relation, and an introduction of the transmission mechanism via

surface plasmon coupling. The diffraction theory includes the CDEW (composite

diffractive evanescent wave) model and Fano profile analysis. In Chapter 3, we describe

our experimental setup for transmission measurement. Two spectrometers (a grating









monochromatic spectrometer and a FTIR spectrometer) are introduced. In Chapter 4, the

sample preparation and the measurement technique with the specifications of samples are

presented.

In Chapter 5, the measured transmittance data are presented. The transmittance data

are shown as a function of the geometrical parameters of hole array, the polarization and

the incident angle of light, and the refractive indices of the substrate material.

In Chapter 6, we analyze and discuss the experimental results based on the surface

plasmon and diffraction theories. We discuss the positions of peaks and dips, spectral

changes with variation of the incident angle and polarization, and the dependence on hole

shape and size. Finally, Chapter 7 has the conclusions of this dissertation and briefly

introduces some additional studies which are necessary for a future study.














CHAPTER 2
REVIEW OF SURFACE PLASMON AND DIFFRACTION THEORY

There are two independent theories which explain the transmission enhancement by

periodic arrays of sub-wavelength holes: the surface plasmon polariton and the

diffraction theory. When the enhanced transmission was reported by Ebbesen and his co-

workers, they interpreted their results with the surface plasmon [7]. The surface plasmon

is still the most generally accepted explanation of the enhanced phenomenon [30-33].

With dispersion relation of the surface plasmon and momentum conservation equation of

periodic grating, one can predict the positions of the enhanced transmission peak pretty

accurately. But the prediction still shows some differences with the experimental results

[16]. For this difference, there might be two reasons. First, the surface plasmon theory is

based on the long-wavelength approximation (A >> d), which means that it does not

depend on the hole size of the structures. Second, the surface plasmon theory, which is

currently used in most papers, is still limited to the dispersion relation for a single

interface between a dielectric and a metal (in which both are infinitely thick) while the

experiments deal with structures containing double interfaces with a finite thickness for

the metal film [34]

As we know, the classical diffraction theory for an electromagnetic wave impinging

on a sub-wavelength aperture in an optically opaque conducting plane predicts an

extremely low transmittance [2]. In this paper, Bethe showed the transmittance intensity

of a sub-wavelength aperture proportional to (d/k)4. But many calculations for diffraction









by periodic hole arrays show an enhanced transmission which is very similar to

experimental data [35-38].

The composite diffractive evanescent wave (CDEW) [9] is one of the diffraction

models explaining the enhanced transmission by periodic structures. The CDEW means a

constructive interference of electromagnetic waves diffracted by periodic sub-wavelength

structure and it is another strong candidate responsible for the enhanced transmission

phenomenon. This diffraction model (CDEW) can explain the enhanced transmission of

hole array in a perfect conductor or in non-metallic materials which the surface plasmon

model cannot explain.

Another transmission model explaining the enhanced transmission is a unifying one

of both the surface plasmon and the diffraction model [12, 13]. This unifying model

proposes an analysis with Fano profile in transmission spectra which is attributed to a

superposition of the resonant process and non-resonant process.

Recently, A. G. Borisov et al. [39] proposed another diffraction model for the

enhanced transmission of sub-wavelength structures. They suggested that the enhanced

transmission of sub-wavelength hole arrays is due to the interference of diffractive and

resonant scattering. The contribution of the resonant scattering comes from the

electromagnetic modes trapped in the vicinity of structures. This trapped electromagnetic

mode is a long-lived quasistationary mode and gives an explanation of extraordinary

resonant transmission.

Bethe's Theory for Transmittance of a Single Sub-Wavelength Hole

Bethe [2] reported that the transmittance of electromagnetic waves through a single

hole in an infinite plane conducting screen, which is very thin but optically opaque, is

very small when wavelength of the incident light is much larger than the hole size. With









this long-wavelength condition, d/k << 1, where d is the diameter of hole and X is

wavelength of the incident light, Bethe has calculated "diffraction cross section" of the

hole for the s- and p-polarization:


A, k4 Cos0 (2-1)
S27r 2

64 d4 1 2 + I
A = 27k4 \ l+Sin 20 (2-2)
P 27r 7 2) 4

The s-polarized (TE mode) wave has an electric field perpendicular to the plane of

incidence whereas the p-polarized (TM mode) wave has a magnetic field perpendicular to

the plane of incidence. These polarizations are schematically shown in Figures 2-1 and 2-

2.

In Eqs. (2-1) and (2-2), one can recognize that the diffraction cross sections for two

polarizations are the same for normal incidence, 0= 0. If the diffraction cross section is

normalized to hole area, the normalized diffraction cross section becomes

A 64 (kd-4 (d 4
=~ 2 23 = T (2-3)
(d 2 27r2 2"A
2\-

2j
where k = k and 2 are wave number and wavelength of the incident wave,

respectively, and d is diameter of hole. This normalized diffraction cross section can be

considered as transmission normalized to hole area, T.

Eq. (2-3) is actually an expression for a circular aperture. If we change the circular

aperture to a rectangular aperture which has a dimension ofD x D, Eq. (2-3) can be

changed as









A 64 (kD)4 D) T (2-4)
2 18 = T (2-4)
D 2 27fr 26 l

Surface Plasmon

The presence of a surface or an interface between materials with different dielectric

constants leads to specific surface-related excitations. One example of this phenomenon

is the surface plasmon. The interface between a medium with a positive dielectric

constant and a medium with negative dielectric constant, such as a metal, can give rise to

special propagating electromagnetic waves called surface plasmons, which stays confined

near the interface.

Definition of Surface Plasmon

Sometimes the surface plasmon is also called the surface plasmon polariton. To

understand this surface plasmon polariton, we need to define some terms: plasmon,

polariton and surface plasmon. First, a plasmon is the quasiparticle resulting from the

quantization of plasma oscillations. They are collective oscillations of the free electron

gas. If this collective oscillation happens at the surface of metal, it is called a surface

plasmon. Therefore, we define the surface plasmon as a collective oscillation of free

electrons at the interface of metal and insulator [8]. The surface plasmon is also called the

surface plasmon polariton. A polariton is the quasiparticle resulting from strong coupling

of electromagnetic waves with an electric or magnetic dipole-carrying excitation.

Therefore, if an electromagnetic wave excites the surface plasmons on a metal surface

and is coupled with the surface plasmon, it is called the surface plasmon polariton.

Dispersion Relation of Surface Plasmon

To get the dispersion relation for surface plasmons [8, 34, 40], we need to consider

an interface between two semi-infinite isotropic media with dielectric functions, P1 and 82.









The x and y axes are on a plane of the interface and the z axis is perpendicular to the

interface. Medium 1 dielectricc function 1e) and medium 2 dielectricc function 82) occupy

each half of the space, z > 0 and z < 0, respectively. The electromagnetic fields for the

surface wave which propagate in the x direction and are confined in the z direction on this

interface are of the form:

E, = E>e )-t) e-a1 z > 0 (2-5)

E, = Eo
where E> and E< are electromagnetic fields in each half space, Eo> and Eo< are amplitudes,

co is angular frequency, t is time, kx is the wave vector of surface wave propagating along

the x-axis and al, a2 are positive real quantities.

Dispersion relation for the p-polarization

For p-polarized electromagnetic wave (TM wave), the magnetic field is

perpendicular to the plane of incidence and the electric field is in the plane of incidence.

In Figure 2-1, the H-field is along the y-axis and the E-field is in the x-z plane. Thus, the

E and H fields in each region can be expressed as

E= (A, 0, B)ey(kx-t)e z > 0 (2-7)

H =(0, C, O)e'(k-x-t)e- z > 0 (2-8)

E2 (D, 0, E)e'(kx )t)e'a2 z <0 (2-9)

H2 = (0, F, 0)e' -'t)e"2 z <0 (2-10)

The boundary condition that needs to be considered is that the components of E

and H parallel to the surface are continuous at the interface, z = 0, that is

El = E2xlz (2-11)










Hix =0 H 2xz0 (2-12)

Substituting Eq. (2-7) through Eq. (2-10) into Eq. (2-11) and Eq. (2-12), the boundary

conditions give A = D and C = F. One of the Maxwell's equations for continuous media

is

E aE
Vx H (2-13)
c 8t

For region 1 and 2, the x components in Eq. (2-13) give


aC= io A z>0 (2-14)
c


aF = -icoi D z <0 (2-15)
c

With A = D and C = F, division of Eq. (2-14) by Eq. (2-15) gives

= 1 (2-16)
a'2 2

This equation is a condition for the surface plasmon mode and demonstrates that one of

the two dielectric functions must be negative, so that, for example, the interface of

metal/vacuum or metal/dielectric supports the surface plasmon mode.

To get the dispersion relation of the surface plasmon, we use two Maxwell's

equations:

E dE
VxH = (2-13)
c 8t

1 8H
VxE=- (2-17)
c 8t

Operating V x on both sides of Eq. (2-17) and substituting for V x H from Eq. (2-14)

gives










I a E 2 2E
Vx(Vx E) = (V x H) = (2-18)
c at C2 Qt2

Using V x (V x E) = V(V. E)- V E and V E = 0 for a transverse wave, we get the

transverse wave equation:


V2E = (2-19)
C at2

In the region ofz > 0, the x and z components of the solution of Eq. (2-19) are


x-component: az2A+acikB= 2 2A (2-20)



z-component: azikxA- kZB = c--2B (2-21)
c

Combining Eq. (2-20) and (2-21), we get

x 1 1 2
-kx+a 2 z >0 (2-22)
c

Similarly, in the region ofz < 0:


81 2
-k2+a =--c- z<0 (2-23)


Combining Eqs. (2-16), (2-22), and (2-23) we obtain the dispersion relation of the surface

plasmon:


kx= Dispersion relation of surface plasmon (2-24)
C 8 +82

Dispersion relation for the s-polarization

As shown in Figure 2-2, the s-polarization has the E field perpendicular to the plane

of incidence and the H field in the plane of incidence. Then, we have a set of E and H

fields:









E1 =(O,A, O)e' -c)e- z > 0 (2-25)

Hi = (B, 0, C)e -c)e- z >0 (2-26)

E2 = (0, D, 0)e "-t)e"2 z <0 (2-27)

H2 =(E, 0,F)e' -o)eC2 z <0 (2-28)

As in the p-polarization case, we apply the boundary conditions Eqs. (2-11) and (2-

12) and get A = D and B = E. Then we use the Maxwell's equation:

1 dH
Vx E (2-29)
c at

Solving Eq. (2-29) with Eq. (2-25) through Eq. (2-28) for both regions ofz > 0 and z < 0

give solutions with x and z components for each region:


x-component: B= A > 0 (2-30)
io)

ke
z-component: C = kc A z > 0 (2-31)
ca


x-component: E= 2 D z <0 (2-32)
io)


z-component: F = kx D z <0 (2-33)
0)

With the results from the boundary conditions, A = D and B = E, Eqs. (2-30) through (2-

33) can be combined and simplified


c (al +a2)A= 0 (2-34)
i')

Since we defined a1 and a2 positive, thus A = 0 and all other constants (B, C, D, E, and F)

also become zero. Therefore, the surface plasmon mode does not exist for the s-

polarization.






12


Z




E H
| ko
0
E Ez
Dielectric E1



Figure 2-1. Schematic diagram for p-polarized (TM) light incident on a dielectric/metal
interface

Z




H E
| ko
H Hz
Dielectric E1



Figure 2-2. Schematic diagram for s-polarized (TE) light incident on a dielectric/metal
interface
Dispersion curves
Figure 2-3 shows the dispersion curves of surface plasmons at the interface of

metal/air, metal/quartz and the light lines in vacuum and fused silica glass, respectively.

The momentum k is calculated by Eq. (2-24). The dielectric constant of metal, E2, in the

Eq. (2-24) is described by the Drude dielectric function [41]:











2 2 2
02 A2
P


(2-35)


where Ap is the bulk plasma wavelength of the metal (cop is the bulk plasma frequency). Ap

is 324 nm for the silver film used in this experiment. The dielectric constants of air and

fused silica substrate are 1.0 and 2.0, respectively.

In Figure 2-3, the thickness of the metal film is considered to be infinite; thus, the

interaction of the surface plasmons on both interfaces is ignored. But if the thickness is

finite, then there will be an interaction between two surface plasmons which will distort

the dispersion curves of surface plasmons [34]

light in air

5x10' 1 I I II
S--- light in quartz

4x10 -
4x 1" -- SP at air/metal



3x10o SP at quartz/meta[



32x10



1x10' -




0.0 5 0x10 1.0x10l1 1.5x10' 2 0x10'" 2.5x10'1 3.0x101

ck(s-')
Figure 2-3. Dispersion curves of surface plasmon at air/metal interface and at
quartz/metal interface, light lines in air and fused silica









Propagation Length of the Surface Plasmon

The propagation length of the surface plasmon can be defined by the imaginary part

of the wave vector k, in Eq. (2-24) as follows [8, 34, 40]


L (2-36)
2k,

The dielectric function E2 is a function of co. At each co, it is a complex number,

E2 = ,2r + iE2 where e- and E2, are the real and the imaginary parts of the dielectric

function. The wave vector kx is also a complex number, kx = k.r + ik .


kx, = -I2r2 1 (2-37)



k = 2 2 (2-38)
C E + E 2Er

From Eqs. (2-30) and (2-32), we can get the propagation length of the surface plasmon:


Lc 2r (2-39)
0 1 +2rJ E2

Using parameters for silver [42], we can evaluate the propagation lengths at air/silver

interface are about 20 ptm and 500 pm for X = 500 nm and X = 1 pm, respectively.

Surface Plasmon Excitation

As seen above, light does not couple to the surface plasmon on metal surface due to

no crossing point between the dispersion curves of the incident light and the surface

plasmon except for k = 0. There are two ways to excite the surface plasmon optically on

an interface of a dielectric and a metal. First, one can use a dielectric prism to make

coupling between the incident photons and the surface plasmon on an interface between

the prism and the metal [8]. But this is not a case which is studying in this dissertation, so









I am going to skip this part. Second, one can use periodic structures on the metal surface.

When light is incident on the grating surface, the incident light is scattered by the grating

structure. The surface component of the scattered light gets an additional "momentum"

from the periodic grating structure. This additional momentum enables the surface

component of the scattered light to excite the surface plasmon on metal surface.

Z


Photon, ko
On



Air E,


Metal E n ao


Surface plasmon, ksp

LZZI


t photon Scattered photon
o)=ck I


./1


I / Surface plasmon
co s p ......................./ ......... ....................................






I
ko ksp kx
(b)

Figure 2-4. Schematic diagrams of (a) the excitation of the surface plasmon by the
incident photon on a metallic grating surface and (b) the dispersion curves of
the incident photon, the scattered photon and the surface plasmon


Sx









Let us consider this case for one dimensional grating, as shown in Figure 2-4 (a).

When light with a wave number ko is incident on a periodic gating on a metal surface

with an incident angle 00, the incident light excites the surface plasmon on the metal

surface. The momentum conservation equation allows this surface plasmon to have a

wave vector, kp, equal to a sum of the x-component of the incident wave vector and an

additional wave vector which is the Bragg vector associated with the period of the

structure:


k = ko sin00 +m ko= (2-40)
ao c

where ko is the wave number of the incident light, and ao is the period of the grating

structure, and m is an integer.

As shown in Figure 2-4 (b), this additional wave vector shifts the dispersion line of

the incident light to the dispersion line of the diffracted photon. This light line crosses the

dispersion curve of the surface plasmon. This crossing means that the incident light

couples with the surface plasmon on the metal grating surface.

If we consider a two dimensional grating on the metal surface, as shown in Figure

2-5, the momentum conservation equation becomes


kp =k +k +igx +jg, g g =2" (2-41)
ao

where kx and ky are surface components of the incident wave vector, gx and gy are the

Bragg vectors, ao is a period of the grating, i andj are intergers. From Eqs. (2-24) and (2-

41), we get an equation which predicts the resonant coupling wavelengths of the incident

light and the surface plasmon on metallic grating surface. Putting k- = k in Eq. (2-24),


we get an equation:










S= 2- i sino, +(i2+j2) d dm _J2 sin20 (2-42)
S+j Ed + Em

From this equation, one can predict the wavelength where the incident light excites the

surface plasmon on the metallic grating surface.

The surface plasmon excitation wavelength is used to explain the enhanced

transmission phenomenon of the sub-wavelength hole array because the excitation

wavelengths are close to the wavelength of the enhanced transmission [7]. But the surface

plasmon excitation wavelength shows a 15 % difference between theoretical calculation

and experimental measurement [9].

Mechanism of Transmission via Surface Plasmon Coupling in Periodic Hole Array

As we mentioned, the surface plasmon is a collective excitation of the electrons at

the interface between metal and insulator. This surface plasmon can couple to photons

incident on the interface of metal and insulator if there exists a periodic grating structure

on the metal surface. So, the coupling between photon and surface plasmon forms the

surface plasmon modes on the interface. If both sides of metal film have the same

periodic structure, such as an array of holes, the surface plasmon modes on the input and

exit sides couple and transfer energy from the input side to the exit side. The surface

plasmon modes on the exit side decouple the photons for re-emission. In this optical

transmission process, the energy transfer by the resonant coupling of surface plasmon on

the two sides is a tunneling process through the sub-wavelength apertures. Thus, the

intensity of transmitted light decays with a film thickness exponentially.

To compensate this decay, a localized surface plasmon (LSP) [43-46] plays a role

in this process. The LSP is a dipole moment formed on the edges of a single aperture due





















Figure 2-5 Schematic diagrams of the excitation of the surface plasmon by the incident
photon on a two dimensional metallic grating surface
Photon


Metal


SSPin
(1)... .


101


SPout
Photon


Figure 2-6. Schematic diagram of transmission mechanism in a sub-wavelength hole
array. (1) excitation of surface plasmon by the incident photon on the front
surface (2) resonant coupling of surface plasmons of the front and back
surfaces (3) re-emission of photon from surface plasmon on the back surface









to an electromagnetic field near the aperture and it depends mainly on the geometrical

parameters of each hole. The LSP makes a very high electromagnetic field in the aperture

and increases the probability of transmission of the incident light.

CDEW (Composite Diffractive Evanescent Wave)

A recently proposed theory competing with the surface plasmon theory is the

CDEW [9, 47-49]. The CDEW is a second model explaining the enhanced transmission

phenomenon of sub-wavelength periodic structures.

Basic Picture of the CDEW

The CDEW model originates from the scalar near-field diffraction. Kowarz [50]

has explained that an electromagnetic wave diffracted by a two dimensional structure can

be separated into two contributions: a radiative (homogeneous) and an evanescent

inhomogeneouss) contributions. The diffracted wave equation for the 2-D structure is

based on the solution to the 2-D Helmoltz equation:

(V2 + k2)E(x, z) = 0 (2-43)

a2 a2 2r
where V2 + ,k = and E(x, z) = Eoe kz),the amplitude of the wave
2x -y A

propagating in the x, z directions. As mentioned, the diffracted wave is a sum of the

radiative (homogeneous) and the evanescent inhomogeneouss) contributions:

E(x, z) = Er (x, z) + E, (x, z) (2-44)

We note that the homogeneous and the evanescent components separately satisfy the

Helmoltz equation.

If we consider that the incident plane wave with a wave vector ko impinges on a

single slit of width d in an opaque screen, as shown in Figure 2-7 (a), the momentum

conservation of the incident wave and the diffracted wave should satisfy








k = 2- k2 (2-45)

where kx and k, are the wave vectors of the diffracted wave in the x and z directions. If kx

is real and if kx > ko, then

k = i k2 (2-46)

This result means that the diffractive wave propagates in the x direction while being

confined and evanescent in z direction. This evanescent mode of the diffracted wave

emerging from the aperture grows as d/ becomes smaller. In contrast, for kx < k, kz

remains a real quantity and the light is diffracted into a continuum of the radiative,

homogeneous mode. In Figure 2-7, the diffraction by an aperture is described in real

space (a) and k-space (b). The blue lines represent the radiative modes (kx < k0), whereas

the red lines represent the evanescent mode (kx > k ). The surface plasmon mode in this

picture is the green line which is one of the evanescent modes diffracted by the aperture.

Now, in order to find the specific solutions for the radiative and evanescent modes,

we need to solve Eq. (2-43). The solution for Eev at the z = 0 is

Ee, (x,0)= E Si ko x+ Si k0 x -- for x >- (2-47)


E (x, 0)= E Si kox+d +Sil kx x- for x < (2-48)

8 sin2 t
where Eo is the amplitude of the incident plane wave and Si(/Y) ol tdt.

If we consider the surface wave on the metal, Eq. (2-47) can be simplified with a

good approximation as [9]









E, d i
Eev -cos(k x+-) (2-49)
Zx 2

From the expression of CDEW in Eq. (2-49), we notice that the amplitude of the

CDEW decreases as 1/x with the lateral distance, x, and its phase is shifted by 7r/2 from

the propagating wave at the center of the slit. These results are different from the surface

plasmon. The phase of the surface plasmon is equal to that of the incident wave and its

amplitude is constant if absorption is not considered [9] Figure 2-8 shows the lateral field

profile of CDEW.



(a) z

k < Iko[ jacadiave modes
Ikol ...........






kx > Ikol evanescent modes

=i A2 2

(b) ", o =ck ,,SP



kx< Ikol k > jk










Figure 2-7 Geometry of optical scattering by a hole in a real screen in (a) real space and
(b) k-space for a range that k, is close to zero [9].










2

1.5



0.1


-4 -2 2 4
-0.5 xld
-d/2 d/2
-1

-1.5

-2

Figure 2-8. CDEW lateral field profile at z = 0 boundary, a plot of Eq. (2-44) [47]

CDEW for an Aperture with Periodic Corrugation

So far we have been discussing the diffraction by a single aperture. Now we are

going to extend our discussion to the periodic corrugation around a single aperture as

shown in Figure 2-9. The corrugations are on both input and output surfaces and actually

play a role as CDEW generating points. The individual corrugation also becomes a

radiating source.

As shown in Figure 2-9, when a plane wave impinges on the periodically

corrugated input surface with an aperture at the center, only a small part of the incident

light is directly transmitted through the aperture. Of the rest, part of incident light is

directly reflected by the metal surface and part of the incident light is scattered by the

corrugations. This scattering produces CDEWs on the input surface (red arrows). The

CDEWs propagate on the input surface and are scattered by the corrugations. The









corrugations on the input surface act as point sources for the scattered light which is

radiating back to the space. Part of the CDEWs propagating on the input surface is

scattered at the aperture and transmitted to the output surface along with the light directly

transmitted through the aperture. When the transmitted light (directly transmitted light

and CDEWs) arrives at the output surface, a small part of the light radiates directly into

space and the rest of the light is scattered again by the aperture and corrugations on the

output surface. The output surface CDEWs are now produced by the scattering of the

transmitted light and it propagates on the output surface between the aperture and the

corrugations. These propagating CDEWs on the output surface are scattered again by the

corrugations and radiated into the front space. This means that the each corrugation on

the output surface also becomes a radiation source. Thus, the transmitted light can be

observed from all over the corrugation structure at the near field. At the far field, the

radiation from the corrugations and the transmitted light from the aperture are superposed

and interfere with each other. As discussed before, the CDEW has 7t/2-phase difference

from the transmitted light. Therefore, the CDEWs and the directly transmitted light make

an interference pattern. The interference pattern of these two waves at the far field has

been observed experimentally. [49]

CDEW for a Periodic Sub-Wavelength Hole Array

Now we are going to develop the CDEW model for a periodic array of sub-

wavelength holes. The CDEW model for the periodic hole array is similar to that of an

aperture with periodic corrugations, except there are many holes rather than one.

As shown in Figure 2-10, a plane wave is incident on the input surface of a periodic

hole array. The incident wave is partially reflected, diffracted, and transmitted. The












I


I


I


I


I


Transmitted wave
Scattered wave
>- CDEWs
Scattered wave
Reflected wave
Incident plane wave


Figure 2-9.CDEW picture for an aperture with periodic corrugations on the input and
output surfaces. Red arrows indicate the CDEWs generated on the input and
output surfaces.

Transmitted wave


S:: Scattered wa
CDEWs
Scattered wE
, Reflected we


I


I


I


I


I


ive


ive
ive


Incident plane wave


Figure 2-10. A CDEW picture for a periodic sub-wavelength hole array. Red arrows
indicate the CDEWs generated on the input and output surfaces.









reflected wave consists of a direct reflection by the metal surface and the back scattering

from the hole, similar to the case of the hole with corrugations in the previous section.

Like the corrugations in Figure 2-9, each hole acts as a point for scattering and radiation

of the CDEWs on the input surface. The CDEWs on the input surface are partially

scattered back to space and partially transmitted along with the directly transmitted wave

through the holes to the output surface. Thus, the transmitted wave is a superposition of

the CDEW and the wave directly transmitted through the holes. When the transmitted

light arrives at the output surface, it is partially scattered (generates CDEWs on the

output surface) and partially radiated into space. The CDEWs generated on the output

surface propagate on the surface, and are partially scattered and radiated into space. In the

front space, the directly transmitted wave from the holes and the radiation from the

CDEWs are superposed to be the total transmission of the hole array for detection at the

far field observation point.

Fano Profile Analysis

Genet et al. [13] proposed that the Fano line shape in transmittance of periodic sub-

wavelength hole arrays is a strong evidence of an interference between a resonant and a

non-resonant processes. Figure 2-11 shows schematic diagrams for the coupling of the

resonant and non-resonant processes in a hole array. In Figure 2-11, the period of hole

array is ao, the thickness is h and the hole radius is r. As shown in this figure, there are

two different scattering channels: one open channel q1 corresponding to the continuum of

states and one closed channel q2 with a resonant state which is coupled to the open

channel with is called "direct" or "non-resonant" scattering process. The other possible

transition is that the input state transits to the resonant state (sometimes called

quasibound state) of the closed channel and then couples to the open channel via the









coupling term V. This is called "resonant" scattering process meaning opposed to the first

one. The "non-resonant" scattering process simply means the direct scattering of the

input wave by the sub-wavelength hole array. This scattering can be called Bethe's

contribution. Bethe's contribution is the direct transmission through the holes in the array

which is proportional to (d/ )4 and will be detected as a background in transmttance. In

contrast, the "resonant" scattering process is a contribution from the surface plasmon

excitation. This resonant scattering process basically consists of two steps: (1) the

excitation of the surface plasmon on the periodic structure of metal surface by the input

wave and (2) the scattering of the surface plasmon wave by the periodic structure. The

surface plasmon wave can be scattered into free space (reflection) or into the holes in the

array (transmission). A simple transmission diagram of this model can be described via

Figure 2-12. The total transmission amplitude is decided with the interference of the non-

resonant contribution (Bethe's contribution) and the resonant contribution (surface

plasmon contribution).

A paper published by Sarrazin et al. [12] has also discussed the Fano profile

analysis and the interference of resonant and non-resonant processes. In Figure 2-13, the

homogeneous input wave (i) incident on the diffraction element A is diffracted and

generates a non-homogeneous resonant diffraction wave (e) which is characterized by the

resonance wavelength, X,. This resonant wave (e) is diffracted by the diffraction element

B and makes a contribution to the homogeneous zero diffraction order. On the other hand,

the other input wave is incident on the diffraction element B and generates a non-resonant

homogeneous zero diffraction order. This non-resonant scattered wave from B interferes

with the resonant wave of X, from A. The Fano profile in transmittance of the sub-








wavelength hole array results from a superposition of the resonant and the non-resonant
scattering processes.


rcflcction
t


transmission


-X


Y


Figure 2-11. Schematic diagrams for Fano profile analysis. (a) Formal representation of
the Fano model for coupled channels and (b) physical picture of the scattering
process through the hole array directly (straight arrows) or via SP excitation
[13]


~I


ma


rJ
MI


Figure 2-12. A schematic diagram of the non-resonant transmission (Bethe's
contribution) and the resonant transmission (surface plasmon contribution)
[14]


a V2


C--


k-,

D]


t aSat
Ldwri W













Interference between
resonant and nonresonant
processes tn.


Resonant profile


Figure 2-13. Schematic diagram of the interference between the resonant and non-
resonant diffraction in transmission of sub-wavelength hole array [12]














CHAPTER 3
INSTRUMENTATION

Optical transmittance measurements have been taken using two spectrometers: a

Perkin-Elmer 16U monochromatic spectrometer and a Bruker 113v fourier transform

infrared (FTIR) spectrometer. The Perkin-Elmer 16U monochromatic spectrometer was

used for the wavelength range from ultraviolet (UV), throughout visible (VIS) and to

near-infrared (NIR), i.e., between 0.25 [m and 3.3 [m. Measurement for longer

wavelengths (> 2.5 [am) employed the Bruker 113v FTIR spectrometer. The FTIR

spectrometer is able to measure up to 500 [m, but in this experiment it was used for a

range between 2.5 [m and 25 [m, i.e., near-infrared (NIR) and mid-infrared (MIR).

Perkin-Elmer 16U Monochromatic Spectrometer

A spectrometer is an apparatus designed to measure the distribution of radiation in

a particular wavelength region. The Perkin-Elmer 16U monochromatic spectrometer

consists of three principal parts; light source, monochromator and detector. Figure 3-1

shows a schematic diagram of the Perkin-Elmer 16U monochromatic spectrometer. Here,

the spectrometer has three light sources, two detectors and a gating monochromator.

Light Sources and Detectors

This spectrometer has three different light sources installed: a tungsten lamp, a

deuterium lamp and a glowbar. The tungsten lamp is for VIS and NIR (0.5 [am 3.3 [am),

and the deuterium lamp is for VIS and UV (0.2 [am 0.6 am). This spectrometer has the

glowbar for MIR region, but it was not used because the matching detector for MIR

region has not been installed. This monochromatic spectrometer has two detectors: a lead









sulfide (PbS) detector for VIS and NIR range (0.5 C[m 3.3 km) and a Si photo

conductive detector (Hamamatsu 576) for UV and VIS range (0.2 [m 0.6 km).


Figure 3-1. Schematic diagram of Perkin-Elmer 16U monochromatic spectrometer

Grating Monochromator

A monochromator is an optical device that transmits a selectable narrow band of

wavelengths of light chosen from a broad range of wavelengths of input light.









Monochromators usually use a prism or a grating as a dispersive element. In prism

monochromators, the optical dispersion phenomenon of a prism is used to separate

spatially the wavelengths of light, whereas the optical diffraction phenomenon of grating

is used in the grating monochromators for the same purpose. In this section, only the

grating monochromator will be discussed.

Monochromator configuration

There are several kinds of monochromator configurations. The configuration of

monochromator which is used in Perkin-Elmer 16U spectrometer is the Littrow

configuration. A schematic diagram of the Littrow configuration is shown in Figure 3-2.



Slit B


MirrorA

SlitA Mirror B



Grating

Figure 3-2. Schematic diagram of the Littrow configuration in the monochromator of
Perkin-Elmer 16U spectrometer

In this configuration, the broad-band light enters the monochromator through slit A,

which is the entrance slit. This entrance slit controls the amount of light which is

available for measurement and the width of the source image. The light that enters

through the entrance slit (slit A) is collimated by mirror A, which is a parabolic mirror.

The collimated light is such that all of the rays are parallel and focused at infinity. The

collimated light is diffracted from the grating and collected again by the parabolic mirror









(mirror A) to be refocused. The light is then reflected by the plane mirror (mirror B), and

sent to the exit slit (slit B). At the exit slit, the wavelengths of light are spread out and

focus their own images of the entrance slit at different positions on the plane of exit slit.

The light passing through the exit slit contains an image of the entrance slit with the

selected wavelength and the part of the image with the nearby wavelengths. Rotation of

the grating controls the wavelength of light which can pass through the exit slit. The

widths of the entrance and exit slits can be simultaneously controlled to adjust the

illumination strength. When the illumination strength of the input light becomes stronger,

the signal to noise (S/N) ratio becomes higher but, at the same time, the resolution of

measurement becomes lower because the exit slit opens wider and passes a broader band

of the light.

Resolution of monochromator

One of the important optical quantities of monochromator is its resolution. The

resolution of monochromator in the Littrow configuration (a = = 0) can be expressed as

[51]

R -- (3-1)
AA (1 R) + (1 Rg)


R, = (3-2)
2f


Rg (3-3)
h(a)

where R, is the resolving power contributed from optical quantities of all components

except for the grating, Rg is the ultimate resolving power of the grating, S is the slit width,

0 is the angle of incidence and diffraction,f is the focal length of collimating mirror, h(a)









is an error function, and Ro is the resolving power of the grating. Thus, the resolution of

monochromator is dependent not only on the grating but also on other optical and

geometrical quantities of the monochromator.

The Diffraction Grating

A diffraction grating is one of the dispersing elements which are used to spread out

the broad band of light and spatially separate the wavelengths.

Grating equation and diffraction orders

Figure 3-3 shows the conventional diagram for a reflection grating. In this Figure,

the general equation of grating can be expressed as [52]

Path difference = PQ + QR

d sin a + d sin =mA (3-4)

where m is diffraction order which is 0, +1, +2 ....

diffracted
incident zero order




P

R

d Q

Figure 3-3. Schematic diagram of a reflection grating.

Iff/ = -a, m becomes zero, the zero order diffraction. When the diffraction angle fl

is on the left-side of the zero order angle, the diffraction orders are all positive, m > 0,

whereas if the angle fl crosses over the zero order and is on the right side of the zero order,

the diffraction order m becomes negative, m < 0.









Blaze angle of the grating

Most modern gratings have a saw-tooth profile with one side longer than the other

as shown in Figure 3-4. The angle made by a groove's longer side and the plane of the

grating is the blaze angle. The purpose of this blaze angle is so that, by controlling the

blaze angle, the diffracted light is concentrated to a specific region of the spectrum,

increasing the efficiency of the grating.

grating normal
facet normal diffracted
incident zero order







d
4 blaze angle


Figure 3-4. Schematic diagram of a blazed grating

Resolving power of grating

As mentioned before, the resolving power of a grating is one of the important

optical quantities contributing in the resolution of monochromator. If we use the Rayleigh

criterion, the resolving power of grating becomes


R= = mN = (sina + sin ) (3-5)
AA A

where Nis the total number of grooves on the grating, Wis the physical width of the

grating, 2 is the central wavelength of the spectral line to be resolved, a and /f are the

angles of incidence and diffraction, respectively. Consequently, the resolving power of









grating is dependent on the width of grating, the center wavelength to be resolved, and

the geometry of the optical setup.

Bruker 113v Fourier Transform Infrared (FTIR) Spectrometer

As mentioned before, the Bruker 113v FTIR spectrometer was used to measure

transmission in the range of MIR (2.5 am 25 r[m). Basically, this FTIR spectrometer

can cover up to the range of far-infrared (FIR) which is up to 500 am. The entire system

is evacuated to avoid absorption of H20 and CO2 for all of the measurements.

Interferometer

The interferometer is the most important part in FTIR spectrometer. The

interferometer in a FTIR spectrometer is a Michelson interferometer with a movable

mirror. The Michelson interferometer is shown in Figure 3-6.

The electric field from the source can be expressed as

E() = Eoek-x (3-6)

where 2 is a position vector, k is a wave vector and Eo is an amplitude of the electric

field. As shown in Figure 3-6, 11, 12, 1s and 12+x/2 are the distances between the source

and the beam splitter, the beam splitter and the fixed mirror, the beam splitter and the

detector, and the beam splitter and the movable mirror, respectively. The reflection and

transmission coefficients of the beam splitter are rb and tb, and the reflection coefficients

and the phases of the fixed mirror and the movable mirror are rf, pf and rm, P9m,

respectively.

The electric field Ed which arrives at the detector consists of two electric field

components: one from the fixed mirror, Ef, and the other from the movable mirror, Em.






36

fixed mirror



I movable mirror
source
1 12,+X/2
I r tb

beam splitter"





Detector


Figure 3-5. Schematic diagram of Michelson interferometer

Thus, Ed, Ef, and Em are

Ed = E + Em (3-7)


Ef = Eoe'kl rbk rfek2 e'ki2eftb ekl3 (3-8)

Em = Eoe'kl tb ek(12+x/2) +x/2) e rme'2 e rbe 'kl3 (3-9)

To simplify, consider the mirrors as perfect mirrors, so rfand rm are 1. Also, we define

the frequency v as follows

2xyv 27c
k= = -o) (3-10)
c A

With c = 1, Eq. (3-10) becomes 2irv = c) and we measure x in cm and v in cm-1. If we let

(o()) = (P ,(f q =( k( + 22 +/3)+(~

Ed = Eo rbtb e' (1 +(+))) (3-11)

The light intensity at the detector is









Sd =EdE, = 2SRbTb[1 + cos( oix + P(o(o))] (3-12)

where So = EoEo, Rb and Tb are the reflectance and the transmittance of the beam splitter.

Sd is the intensity of light at the detector for a given frequency co. Then the total

intensity for all frequencies is

Id(x) = J Sd (o)do = 2J SORbb [1 + cos(cx + ((w))]doe (3-13)

For an ideal beam splitter, Tb = 1 Rb and RbTb with Rb = 1/2 is


RbTb =Rb(1-Rb) = (3-14)
4

Here we define the beam splitter efficiency, Eb, as follows

eb -4RbTb = 4Rb(1-Rb) (3-15)

Then, Eq. (3-13) becomes


Id (x)=- = SO (C)) (C)[1 + cos(Cx + p(*))])dt (3-16)

Here we have two special cases, x oo and x = 0. For x oo, Id in Eq. (3-17)

becomes Id (oo) called the average value:


Id (0o)= f SO (C)b(m)da (3-17)
20

With x = 0 and p (co) = 0 (zero path difference or ZPD), Id becomes Id (0) called the white

light value:

Id(0) = J So (o)Eb (c)do = 2Id (o) (3-18)

Now we need to define another quantity which is the difference between the intensity at

each point and the average value called the interferogram:

(x) Id () = S() cos(cx + (pw(o))do) (3-19)









where S(o) So(wc)Eb(co) and y(x) is the cosine Fourier Transform of S(c). If we assume

that S(c) is hermitian, then y(x) is


y(x) = 4 JS(c4)eO(w)e"tdc (3-20)
4-

and


S(O)eO(O=) j2 (x)e ,-dx (3-21)


From the measurement with the interferometer, we get the interferogram, y(x) and

compute the Fourier transform to get the spectrum, S(o).

The resolution of a Fourier spectrometer consists of two terms: one contributed

from the source and the collimation mirror and the other decided by the maximum path

difference.

1 1 1
-+ (3-22)
R R, R2


8f2
R, (3-23)


R2 =lv (3-24)

wherefis the focal length of the collimating mirror, h is the diameter of the circular

source, / is the maximum path difference or the scan length and v is the wave number in
-1
cm.

Description of FTIR Spectrometer System

A simple description of interferometer of the FTIR spectrometer is as follows. The

light from a source is focused on a beam splitter after reflected by a collimation mirror.

This beam splitter divides the input light into two beams: one reflected and the other

transmitted. Both beams are collimated by two identical spherical mirrors to be sent to a










two-sided moving mirror. The moving mirror reflects both beams back to the beam

splitter to be recombined and the recombined beam is sent to the sample chamber for

measurement.




















I Source Chamber III Sample Chamber
a Near- mkd- or far- IR sourwsw i Trwaritatne focum
b Automated Aperture I Rertlance focus
II Irterferometer Chamber IV Detecor Chamber
c Opbcal filter k Near-. rd-. or ar-IR
d Automabc beamsplier changer detector
e Two-side movable mror
f ContrWo i fromater
g Reference aser
h Remole control algnment rmnor

Figure 3-6. Schematic diagram of the Bruker 113v FTIR spectrometer

As shown in Figure 3-6, the Bruker 113v FTIR spectrometer consists of 4 main

chambers: a source chamber, an interferometer chamber, a sample chamber and a

detector chamber. In the source chamber, there are two light sources: a mercury arc lamp

for FIR (500 [m 15 [m) and a glowbar source for MIR (25 [m 2 am). The

interferometer chamber has actually two interferometers for a white light source and a

helium-neon (He-Ne) laser. As we know the exact wavelength of the laser, the small

interferometer with the He-Ne laser is used as a reference to mark the zero-crossings of

its interference pattern which defines the positions where the interferogram is sampled.

This is the process of digitization of interferogram.






40


White light transmission and reflection are measured in the sample chamber. The

transmission is measured in the front side of the sample chamber and the reflection is

measured in the back. There are two detectors installed in the detector chamber: a liquid

helium cooled silicon bolometer and a room temperature pyroelectric deuterated

triglycine sulfate (DTGS) detector. The bolometer detects light signals in the FIR range

(2 am 20 km) and the DTGS detector is for MIR (2 km 25 km).














CHAPTER 4
SAMPLE AND MEASUREMENT

Sample Preparation

The sub-wavelength periodic array samples were prepared using electron-beam

lithography and dry etching. The sample fabrication process is simply described as

follows. Silver films with thickness between 50 nm and 100 nm were deposited on

substrates using thermal evaporation. Fused silica and ZnSe were used for the substrates.

Before the E-beam writing process, a PMMA film is coated on the silver film. The

PMMA coated samples were baked on a hot plate at 180 OC for a minute. The baked

PMMA film was exposed by the electron beam to make a periodic pattern on it. After the

E-beam writing, the sample developed with the area of the PMMA film, which was not

exposed by the electron beam, removed by the developing solution. The patterned

PMMA film is going to be used to mask the silver film from the dry etching. During the

dry etching process, Ar-ions strike the surface of the sample to make holes on the silver

film. Finally, the remaining PMMA mask was removed with the stripping solution.

With this fabrication process, a variety of samples have been prepared for this

research, listed in Table 4-1. SEM images of the selected samples are also shown in

Figure 4-1.

Substrates

Fused silica and ZnSe were used for the substrates. When the enhanced

transmittance is expected to occur at wavelengths shorter than 5000 nm, fused silica

substrate is used. If the transmission peaks are supposed to occur at wavelengths longer







42
















tal 1 mn BIT 4W SidA *IrAm Oft 10 Fb 2004 IRit 1C t EIN. 4C W S4d A I.nL DOS AFe Nf
Magaot x K WD llrn UrN, reTRAINING T7_ 7It 2 ag5n 1m x [- I n 11 r U flrm TRMaING TirmN 11,50

(a) (b)

















RbiC Iu n ET' LwW SI Gi M4 ht. Oftfl n7 N RNu C lmC i, x MtB00W3 OW A*an IN-7J0W
M6 KX ----- IN-- 11In- Ur0 N-l rTRAINING -TiT rmK 11-.U-7 MN4- 20 uKX 3 3llm -rN =TRAINING TN n 13", 25

(c) (d)
















R I-, 5HW5 SW AhLO -C RfN Do,? J. 2M5 11AMW gMA-N Dal. J.2004
M.. 102 -li I.M 11., WWN-C RAn*,T0716 M t-94715 La-- ----- AM-11- NwN* R.CAINw TI- 10M21

(e) (f)

Figure 4-1. SEM images of periodic hole arrays samples. (a) A14-1, (b)A14-3, (c) A18-1,
(d) A18-2, (e) A18-3, and (f) A18-4









than 5000 nm, ZnSe substrate is used. It is because fused silica is transparent between

300 nm and 5000 nm, while ZnSe is transparent between 500 nm and 15000 nm [53].

Measurement Setup

We have used the Perkin-Elmer 16U monochromatic spectrometer and Bruker 113v

FTIR spectrometer for transmittance measurement. Transmittance of an open aperture

and of the sample has been measured. We first measured an open aperture as a reference

and then measured the sample. We used the same diameter aperture when measuring the

sample to keep the measurement area the same. Then we calculated the ratio of the

transmission of the sample to that of the open aperture to get the transmittance of the

sample.

Table 4-1. List of the periodic sub-wavelength hole arrays

film thickness
sample hole shape hole size (nm) period (nm) fm thkn
(nm)
A14-1 square 900 x 900 2000 70
A14-2 square 900 x 900 3000 70
900 x 900(out)
A14-3 square donut 00 x 900(out) 2000 70
500 x 500(in)
A15 square 500 x 500 1000 50
A18-1 square 840 x 840 2000 100
A18-2 rectangular 900 x 1300 2000 100
A18-3 slit 1000 (width) 2000 100
8-4 square on 2000 (x-axis) 100
rectangular grid 1500 (y-axis)

We measured transmittance as a function of the angle of incidence. The samples

were mounted on a transmission sample holder that allows changes in the angle of

incidence. A picture of the transmission sample holder is shown in Figure 4-2. By rotating

about an axis perpendicular to the direction of the incident light, the angle of incidence is

changed. We measured transmittance at every 2 degrees between 0 degrees and 20

degrees. Also, we have varied the in-plane azimuthal angle. The azimuthal angle can be









varied from 0 degrees to 360 degrees. We used this measurement to study the effect of

polarization direction. This angle can be controlled using the same transmission sample

holder by rotating the sample mounting plate shown in Figure 4-2.




Incident angle rotator








Azimuthal angle rotator


Sample mounting plate
(An open aperture at center)


Figure 4-2. Picture of the sample holder used to measure transmittance with changing the
angle of incidence and the in-plane azimuthal angle

After the exit slit of the monochromator of Perkin-Elmer 16U spectrometer, we

could installed one of three different polarizers. A wire grid polarizer that is made of gold

wires deposited on a silver bromide substrate is used for the MIR region, and two

dichroic polarizers are used for NIR, VIS and UV regions. We can get the either s-

polarized or p-polarized incident light by using these polarizers. Another wire grid

polarizer has been installed on the exit aperture of the interferometer chamber of the

Bruker 113v FTIR spectrometer to get polarized light in the MIR region. In the FTIR

spectrometer, the polarizer is rotated instead of the sample.






45


In the Perkin-Elmer spectrometer, an optical solid half angle of the incident light on

samples is adjustable with an iris aperture installed on the spherical mirror before the

transmission sample holder (see Figure 3-1), but for most of measurement, we set the iris

aperture to make this angle 1 o, to minimize the incident angle effect. The optical solid

half angle of the Bruker 113v spectrometer is about 8.5 and was not adjusted.

Once we measured the samples with both spectrometers, the two transmittance data

have been merged into one transmittance data by our own data merging program.














CHAPTER 5
EXPERIMENTAL RESULTS

In this chapter, we present our experimental results. These experimental results will

be shown as follows. First, we present experimental data for the transmittance of the

arrays of square holes. We discuss the dependence on the period of the hole arrays, and

also on the thickness of the metal films. Second, transmittance of the square hole array as

a function of the angle of incidence using polarized light is presented. Third,

transmittance with different hole shapes, hole sizes and in-plane polarization angles are

shown. Finally, transmittance with different dielectric materials interfaced to the metal

film is presented.

Enhanced Optical Transmission of Sub-wavelength Periodic Hole Array

Figure 5-1 shows the transmittance of square hole array (A14-1) between 300 nm

and 5000 nm. As shown in this figure, the transmittance maximum occurs at 3070 nm

which shows an intensity of 60 %. This is about 3 times greater than the fraction of open

area. This means that the light which is impinging not only on the hole area but also on

the metal surface transmits into the output surface of the hole array via a certain

transmission mechanism. This enhanced transmission of sub-wavelength hole array was

first reported by Ebessen et al. in 1998. [7] The reason why it is called "enhanced" is that

the transmittance intensity is not only greater than the fraction of open area but also much

greater than a prediction from the classical electromagnetic theory for transmission of an

isolated aperture proposed by Bethe in 1944 [2]. Other spectral features we see from this

transmittance are the second highest peak at 2450 nm and another sharp peak at 323 nm.









The sharp peak at 323 nm is the bulk plasmon peak of silver and this is an intrinsic

property of the metal which is silver.


1000 2000 3000 4000
Wavelength(nm)


5000


Figure 5-1. Transmittance of the square hole array (A14-1) and a silver film

Comparison of Enhanced Transmission with Classical Electromagnetic Theory

For comparison we need to recall the Bethe's transmittance for a single sub-

wavelength hole, Eq. (2-4):

A 64 (kD)4 T (2-4)
2- 2 18 =T (2-4)
D 2 27_ r 26 z

In Figure 5-2, we show the transmittance calculated with Eq. (2-4) for wavelengths

up to 5000 nm and compare with the transmittance measured with the square hole array.

As shown in Figure 5-2, Bethe's calculation is reasonable for wavelengths longer than

2000 nm which is 2 times greater than the dimension of hole. For wavelengths shorter










than 2000 nm, the calculated transmittance increases very rapidly and is not compatible

with the measured transmittance.

At 3070 nm the intensity of the transmittance maximum is 2.93, while the

transmission amplitude of Bethe's calculation at the same wavelength is 0.19. Thus, the

measured transmittance is 15 times greater than the calculated one at the wavelength of

the transmittance maximum.




Bethe
o measurement
CU


0


-N

0

S\ Open fraction






0 1000 2000 3000 4000 5000
Wavelength(nm)

Figure 5-2. Comparison between Bethe's calculation and the transmittance measured
with the square hole array (A14-1)

Dependence of Period, Film Thickness and Substrate on Transmission

The experimental data shows that the enhanced transmission of sub-wavelength

hole array depends on materials and geometrical parameters of sample. In this section, we

discuss the dependence on the period of the hole array, the film thickness and the

substrate material on transmission.









Dependence on Period of Hole Array

For this experiment we prepared two different hole array samples which have

different periods of 1 pm and 2 pm, respectively. These samples are fabricated on silver

film. The thicknesses are 50 nm for the 1 [pm period sample and 100 nm for the 2 tpm

period sample. The hole size for the 2 pm period sample is 1 pm x 1 pm and that of the 1

jpm period sample is 0.5 pm x 0.5 jpm. Both samples are prepared on fused silica

substrates.


1000 2000 3000 4000
Wavelength(nm)


5000


Figure 5-3. Transmittance of square hole arrays with periods of 1 pm (A15) and 2 pm
(A18-1)

Figure 5-3 shows the transmittance of both samples. The transmittance maxima for

1 pm and 2 pm period samples appear at 1560 nm and 2940 nm, respectively. The ratio

of the two peak positions is about 1.88. This is very close to 2 which is the ratio of the

periods of both samples. The second highest peaks are located at 1170 nm and 2180 nm.









The ratio of the second highest peak positions of both samples is 1.86 and it is almost the

same with that of the maximum peak positions. The transmittance minimum or the dip

more closely follows the ratio of the periods. The dip located between two highest peaks

occurs at k = 1410 nm for the 1000 nm period sample and at k = 2800 nm for the 2000

nm period sample. The ratio of dip positions is 1.98, almost same as the ratio of the

periods of the two samples. From this simple consideration we are able to predict that the

positions of peaks and dips in transmittance of sub-wavelength periodic hole arrays are

closely associated with the periods of hole arrays.

Dependence on the Thickness of Metal Film

Another feature in Figure 5-3 is the dependence of the transmittance on the

thickness of the metal film. As indicated in this figure, the thickness of the metal film in

the 1 [tm period array sample is 50 nm and that of the 2 [tm period array sample is 100

nm. Two transmittances from these hole arrays show different spectral behaviors. The

transmittance of the hole array with 50 nm thickness shows a stronger maximum peak, a

higher background, and a broader line-width compared to the transmittance of the hole

array with 100 nm thickness [55].

For a direct comparison between these hole array samples, we rescaled the x-axis to

wavelength divided by the period of each array. These rescaled transmittances are shown

in Figure 5-4. The background in the transmittance for the hole array with 1 [tm period is

higher than that of the hole array with 2 [tm period, due to the difference of thickness in

the metal film. For a thinner metal film, transmission through leakage paths in the film or

direct transmission through metal film increase. These kinds of contribution decrease

when the thickness of film increases. Thus, the background for the hole array with 2 tm







51


period decreases. The difference between the backgrounds of the 1 [m period hole array

and the 2 am period hole array is about 10 %.


1.0

0.9

0.8

0.7

0.6 -

0.5

0.4 -

0.3

0.2-

0.1 -

0.0 -
0.0


0.5 1.0 1.5 2.0


Figure 5-4. Transmittance vs. scaling variable, ks = k/(nd x period), for the square hole
arrays of 1 [m period (A15) and 2 [m period (A18-1) made on fused silica
substrates (nd = 1.4)

From Figure 5-4, we can see a shift of the transmittance maximum even though

these hole arrays are supposed to have the maximum at the same position in the rescaled

x-axis. And also the positions of the dips in the transmittance of 1 and 2 [m period hole

arrays do not coincide but are slightly different. This difference in position of peak or dip

might be attributed to an imperfection in the geometrical structure of the hole arrays. But,

if we take a closer look in the figure, the peak of the 1 am period array has a little broader

line-width than that of the 2 am period array. The broadness of transmission peak is

basically coming from factors such as a larger hole size and a thinner film which increase









the coupling strength between front and back surfaces. This coupling also probably

causes the shift in peak position.

Dependence on the Substrate Material

The transmittance of the hole arrays depends on the dielectric materials interfaced

with the hole array. In particularly, the positions of peaks and dips are strongly dependent

on the dielectric material. In order to see the effect of the dielectric material in

transmittance, we used two different substrates: fused silica and ZnSe. The dielectric

constants of fused silica and ZnSe are 2.0 and 6.0, and the transmittances of bare

substrates are 90 % and 70 %, respectively [53].

Figure 5-5 shows the transmittance of a 2 [m period square hole array (A14-1) on

different substrates: one on a fused silica substrate and the other on a ZnSe substrate.

Even though those samples are on different substrates, the film thickness of films was 70

nm for both transmittances. In Figure 5-5 (a), the hole array on fused silica has its

transmittance maximum at 3070 nm while the maximum for the array on ZnSe substrate

is at 5180 nm. The ratio of the peak positions of the two samples is about 1.69. We know

the refractive indices of fused silica and ZnSe which are 1.4 and 2.4, respectively, so that

nznse / nsio2 = 1.7, close to the ratio of the peak wavelengths. This result indicates that the

most dominant factor for this big red shift in the peak positions of these two hole arrays is

the refractive index of the substrate material. In Figure 5-5 (b), the x-axis is rescaled with

wavelength divided by a product of the refractive index and the period, ks = k / (nd x

period). Even though the effect of the period and the refractive index is eliminated by the

rescaling, the dip positions are still different between the two spectra. This is probably

due to imperfections of the samples such as a difference in the thickness or the period.


































0 1000 2000 3000 4000 5000 6000 7000 8000
Wavelength(nm)


0.0 L-
0_0


0-5 1-0 1.5
s


Figure 5-5. (a) Transmittance vs. wavelength (b) transmittance vs. scaling variable, ks =
X/(nd x period), for the square hole arrays of 2 pm period (A14-1) made on a
fused silica substrate (nd = 1.4) and a ZnSe substrate (nd = 2.4)










Dependence on the Angle of Incidence

In this section, we will discuss the effect of the incident angle on the transmittance.

For this measurement we used the square hole array with 2 pm period (A14). As

mentioned in chapter 4, the incident angle is changed by rotating about an axis

perpendicular to the incident light and the plane of incidence. For this measurement we

used polarizers to get the s- and p-polarized incident light. We also measured with nearly

unpolarized light. The transmittance was measured every 2 from 0 to 20




0.9
unpolarized
0.8 00 polarization

0.7 90 polarization

S0.6

S0.5 ,

04
I--
03

0.2

0.1

0.0 I I
2200 2400 2600 2800 3000 3200 3400
Wavelength(nm)


Figure 5-6. Transmittance of a square hole array (A14-1) with three different
polarizations at normal incidence

From this experiment, we found a very strong dependence of the transmittance on

the incident angle. In addition, a significant polarization dependence of the transmittance

at non-normal angle of incidence is also observed. The spectral behavior of transmittance

of s and p-polarized light differ when the incident angle is changed [14, 56, 57].









Figure 5-6 shows the normal incidence transmittance of a square hole array (A14-1)

for three different polarizations. These spectra are almost the same except for the second

highest peak. The intensity of the second peak for the case of unpolarization is a little

higher than the peaks of others. A reason of this similarity in transmittance at normal

incidence is that the sample (A14-1) used in this experiment has a geometrical symmetry

for the two orthogonal polarizations.

Figure 5-7 (a) shows schematically the s-polarized light incident on a hole array

sample. The lower panel, Figure 5-7 (b) shows the transmittance of a square hole array

(A14-1) with s-polarized incident light as a function of the incident angle. The s-

polarization (TE mode) has a transverse electric field which is perpendicular to the plane

of incidence. The magnetic field is in the plane of incidence. Figure 2-2 in Chapter 2

shows a schematic diagram for s-polarization.

In Figure 5-7 (b), we can see some dependence on the transmittance on the angle of

incidence. The intensity of the maximum transmission peak decreases and the line-width

of the peak increases when the incident angle increases. The locations of both the

maximum peak and of the dip shift to shorter wavelengths with increasing incident angle,

while the second highest peak shifts the longer wavelengths.

Figure 5-8 (a) shows a schematic diagram of p-polarized light incident on a hole

array. Figure 5-8 (b) shows the transmittance of the same square hole array using p-

polarized incident light as a function of the incident angle. The p-polarization (TM mode)

has a transverse magnetic field, perpendicular to the plane of incidence. The electric field

is in the plane of incidence. Figure 2-1 in Chapter 2 shows schematically the case of p-

polarization. For p-polarization, the transmittance is quite different from that of the s-







56


polarization as the incident angle changes. The maximum peak at 3070 nm at normal

incidence splits into two peaks. One peak shifts to the longer wavelengths while the other

peak shifts to the shorter wavelengths with increasing incident angle.


1.0

0.9

0.8

0.7

0.6

0.5
E
- 0.4

S0.3

0.2

0.1


0.0 1
220


0


k
/~


2400 2600 2800 3000 3200


3400


Wavelength(nm)
(b)

Figure 5-7. Measurement of transmittance with s-polarized incident light as a function of
the incident angle. (a) Schematic diagram of s-polarized light incident on a
hole array and (b) transmittance of a square hole array (A14-1)


6''' ~ '


--- 0
-- 2
-- 4
-- 6
80
8
--100
120
--140
---16
---180
- 200


I I I I *


00


00











00


1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

no-


k

J-7


A


2200 2400 2600 2800 3000 3200 3400
Wavelength(nm)
(b)

Figure 5-8. Measurement of transmittance with p-polarized incident light as a function of
the incident angle. (a) Schematic diagram of p-polarized light incident on a
hole array and (b) transmittance of a square hole array (A14-1)

The dip at 2860 nm also shows the same spectral behavior when the incident angle

increases, splitting into two dips, one of which shifts to shorter wavelengths and the other

dip shifts to longer wavelengths with increasing incident angle.


--02
-204
- 60

80
--6
8



140
- 160
- 180









We cannot easily distinguish how the second highest peak at 2450 nm changes. It

is a very interesting feature that the transmittance of the s- and p-polarizations behave

very differently as a function of the angle of incidence.

Dependence on Hole Shape

In this section, we discuss dependence of the transmittance on the hole shape and

the in-plane azimuthal angle of polarization. For this measurement, we prepared four hole

array samples which have different shapes and sizes of holes. Those arrays are shown in

Figure 4-1. The four samples are: 1) an array of square holes with 1000 nm x 1000 nm

hole size and 2000 nm period (A18-1), 2) an array of rectangular holes with 1000 nm x

1500 nm hole size and 2000 nm period (A18-2), 3) an array of slits with 1000 nm width

and 2000 nm period (A18-3) and 4) an array of square holes on rectangular grid with

1000 nm x 1000 nm hole size and 1500 nm period for x-axis direction and 2000 nm

period fory-axis direction (A18-4).

Square Hole Arrays

Figure 5-9 shows the transmittance of the square hole array as a function of

polarization angle. The spectra at all polarization angles (0 45 90 ) are the same.

The transmittance maximum occurs at 2940 nm with an intensity of 60% for all three

polarization angles. The behaviors at 0 0 and 90 0 polarization angles are due to

geometrical symmetry of the square hole array. For 45 polarization angle, the electric

field has decomposed into 0 0 and 90 0 components, making the spectra at 0 0 and 90

polarization angles to be the same.

The transmittance peak at 2940 nm shows Fano line-shape which we discussed in

Chapter 2. This Fano line-shape is a typical feature of the enhanced transmission of sub-






59


wavelength hole arrays even though it is still not clear if it is due to the superposition of

contributions from the resonant and non-resonant scattering processes in transmission

mechanism.


1000 2000 3000 4000
Wavelength(i(m)


Figure 5-9. Transmittance of square hole array (A18-1) as a function of polarization angle.
The inset shows a SEM image of the square hole array.

Rectangular Hole Array

The transmittance of the rectangular hole array for polarization angles of 0 and 90

0 are shown in Figure 5-10. As shown in the figure, it is evident that the transmission of

the 0 0 polarization angle is very different from that of the 90 0 polarization angle.

For the 90 0 polarization angle, the transmittance maximum has an intensity of

83 % at 3300 nm. This peak disappears for 0 0 polarization angle while another peak

appears at 2900 nm which shows an intensity of 43 %. This difference between the









transmittance of 0 0 and 90 0 polarization angles shows that the position of maximum

transmittance strongly depends on polarization angle due to the asymmetry of rectangular

holes.


0 1000 2000 3000 4000
Wavelength(pm)


Figure 5-10. Transmittance of a rectangular hole array (A18-2) for in-plane polarization
angles of 0 0 and 90 The inset shows a SEM image of the rectangular hole
array.

Another interesting difference between the transmittance of 0 0 and 90

polarization angles is the line-width of the maximum peak. Figure 5-10 shows that the

line-width of the maximum peak in the transmittance of the 90 0 polarization angle is

much broader that that of the 0 0 polarization angle.

There is the second highest peak around 2300 nm in the transmittance spectra of 0 0

and 90 0 polarization angles. These peaks are located at the same position with a similar










line-width. This is a different spectral behavior compared to the large peaks at 2900 nm

and 3300 nm.


1.0 I i I i

0.9 900
Slit array 90
0-8 -00 00
07 90 0
007

a 0.6 [

._ 0.5
(U
= 0.4
I-
0.3

0.2

0.1

0 0
0 1000 2000 3000 4000
Wavelength(jpm)


Figure 5-11. Transmittance of a slit array (A18-3) for in-plane polarization angles of 0 o
and 90 0. The inset shows a SEM image of the slit array.

Slit Arrays

The transmittances of the slit array for 0 0 and 90 0 polarization angles are shown in

Figure 5-11, along with a SEM picture of the array (inset). The 0 0 polarization direction

is parallel to the slit direction and the 90 0 polarization is perpendicular to the slit

direction. The transmittance at 90 0 polarization angle shows a very broad transmittance

peak around 4000 nm with an intensity of 73 %. This peak disappears for 0 0 polarization

angle. This transmittance behavior of slit array is expected as slit arrays are used as a

wire grid polarizer [52].









The transmittance of the slit array also shows a second maximum peak for both 0

and 90 polarizations around 2300 nm which is the same position as the square and the

rectangular hole arrays. But, in the transmittance of the 0 polarization angle, we hardly

recognize the dips which exist in the transmittance of the 90 polarization angle at 2000

nm and 2800 nm. This is probably due to an absence of periodic grating structure in the

direction of 0 0 polarization angle.


1000 2000 3000 4000
Wavelength(nm)


Figure 5-12. Transmittance of a square hole array on a rectangular grid (A18-4) for
polarization angles of 0 45 and 90 The inset shows a SEM image of the
square hole array in a rectangular grid.

Transmission of Square Hole Array on Rectangular Grid

In order to see the effect of different periods in two orthogonal polarization angles,

we prepared a square hole array on a rectangular grid (A18-4). As mentioned previously,

the periods in the 0 0 and 90 0 polarization angles are 1500 [m and 2000 [m, respectively.









The hole size is 1000 nm x 1000 nm which is the same as that of the square hole array

(A18-1).

Figure 5-12 shows the transmittance of the square hole array on a rectangular grid

for 0 o, 45 o and 90 o polarization angles. The transmittance at the 90 0 polarization shows

a sharp maximum peak at 3020 nm and a second maximum at 2270 nm. The peak at 3020

nm disappears for the transmittance of the 0 0 polarization angle. But the peak at 2270 nm

remains at the same position with a little higher intensity for the 0 0 polarization angle.

There is a small peak at 3000 nm in the spectrum of the 0 0 polarization angle and this

might be due to a misalignment of polarization at the angle of 0 .

Refractive Index Symmetry of Dielectric Materials Interfaced with Hole Array

Most of the samples that we have prepared are asymmetric structures with a fused

silica substrate (or ZnSe substrate)/a periodic array on sliver film/air, as shown in Figure

5-13 (a). But there were some reports proposed an increase of the transmittance when

sample has refractive index symmetry of dielectric materials on both sides of hole array

[58] In order to test an effect from this refractive index symmetry, we used photo resist

(Microposit S1800, Shipley) and PMMA (NanoPMMA, MicroChem) as a dielectric

material to make the refractive index symmetry with fused silica substrate. The refractive

indices of PR and PMMA are approximately 1.6 and 1.5, respectively [59, 60], and the

refractive index of fused silica is about 1.4 [42].

First, we measured transmittance of an original sample which is the square hole

array (A14-1). Then, we coated PR or PMMA with a thickness of 150 nm on the top of

hole array and measured the transmittance. Figure 5-13 shows schematic diagrams of

each step of the sample preparation for measurement.









Figure 5-14 and Figure 5-15 show the transmittance of square hole arrays on fused

silica substrate and ZnSe substrate, and the same hole arrays with PR coated on the top.

When the PR (n 1.6) is coated on the hole arrays, the transmittance maximum of the

hole array on fused silica substrate shifts more than 600 nm to longer wavelengths while

the peak of the hole array on ZnSe substrate shifts only 60 nm which is small compared

to that of the hole array on fused silica substrate. There is a small increase in the peak

intensity for the hole array on ZnSe substrate but there is almost no increase for the hole

array on fused silica substrate. The dip at 2800 nm also shifts about 100 nm to longer

wavelengths in the hole array on fused silica substrate but the same dip of ZnSe substrate

sample shifts to longer wavelengths slightly.

In addition, we used PMMA (n z 1.5) for this index symmetry experiment. As we

know, the refractive index of PMMA is almost same as the refractive index of fused silica.

Figure 5-16 shows transmission spectra of the square hole array (A14-1) with and

without PMMA on top of the hole array. The transmittance of PMMA coated hole array

shows the maximum transmittance at 3210 nm. This peak is shifted about 200 nm to

longer wavelengths from 3010 nm where the maximum transmittance of the hole array

without PMMA coating occurs. Another transmittance in Figure 5-16 is measured with

the same hole array but with another fused silica substrate attached on the top of PMMA.

The transmittance with the second fused silica substrate shows no shift in the positions of

peak and dip but a small decrease in transmittance intensity compared to the spectrum of

the PMMA coated hole array. The transmittance decrease is probably due to reflection

and absorption by the additional fused silica substrate attached on the top of PMMA.










Ag pattern (100 nm)


(a)
Fused silica (1mm)

Photo resist or PMMA (150nm)


Fused silica


Fused silica


Fused silica


Figure 5-13. Schematic diagram of sample preparation (a) an original square hole array
(b) a PR (or PMMA) coated square hole array (c) another fused silica
substrate attached on top of PR (or PMMA)


1.0



0.8



O
) 0.6
C-,


E
u,
c 0-4
I-


0-2



0.0


1000 2000 3000 4000
Wavelength(nm)


5000


Figure 5-14. Transmittance of a square hole array (A14-1)
and without PR coated on the top


on fused silica substrate with











1.0 -i


Square hole array on Ag/ZnSe
0.8 -- uncoated
-- PR coated


S0.6


E
o
c 0.4



0.2



0.0
0 2000 4000 6000 8000 10000
Wavelength(nm)


Figure 5-15. Transmittance of a square hole array (A14-1) on ZnSe substrate with and
without PR coated on the top of hole array


00 1
2200


2400 2600 2800 3000 3200
Wavelength(nm)


Figure 5-16. Transmittance of a square hole array (A14-1) on fused silica substrate with
and without PMMA coated on the top of hole array with the second fused
silica substrate attached on the top of PMMA.


Square hole array on Ag/fused silica glass (quartz)
uartz / Ag film / Air
Quartz I Ag film / PMMA
-Quartz / Ag film / PMMA / Quartz






67


Even though we expected a remarkable increase of the transmittance in the case of

the fused silica substrate samples, it is hard to observe an increase in the measured

transmittance. But this result shows that the peak and the dip of the hole array on fused

silica substrate shift a lot more than the hole array on ZnSe substrate. It means that the

spectral shifts of peak and dip by an addition of the index symmetry layer depend on the

substrate material of the hole array.














CHAPTER 6
ANALYSIS AND DISCUSSION

In Chapter 5, we have shown the transmittance of various structures of hole arrays,

which have different geometrical parameters (period, film thickness, incident angle and

hole size) and the refractive indices of dielectric material. In this chapter, we will analyze

and discuss a few important features. First, we compute the theoretical predictions for the

positions of peaks and dips, and compare them with experimental data. Second, we

discuss the transmittance dependence on incident angle for s- and p-polarized light. Third,

we discuss the dependence on hole shape and size.

Prediction of Positions of Transmission Peaks

We need to recall one of surface plasmon equations which predicts the position of

resonant transmittance peaks in two dimensional hole array.

a = isin0, +(i2+j2) dm -j2 sin2 20
l i2 +j2 d + Em


for non-normal incidence (Oo # 0) (2-42)

a
A+ao dm 2 for normal incidence (Oo = 0) (6-1)


With this equation, we can calculate wavelengths of the surface plamon resonant

transmission peaks of a two dimensional hole array. For this calculation, we need the

dielectric constants of air, substrate materials and metal which is silver in this work. First,

we know that the dielectric constant of air is 1. The substrate we mostly used is fused

silica glass substrate. The dielectric constant of fused silica glass is 2.0 for a wavelength









range between 2000 nm and 3000 nm. We also need to calculate the dielectric constant of

silver. Generally, the dielectric constant of a metal is a strong function of frequency (or

wavelength) and has a complex form:

,m = ,mr + im, (6-2)

where ,mr and m are real and imaginary parts of es. em, is mainly associated with

absorption of metal. em in Eqs. (2-42) and (6-1) is usually considered as the real part of

dielectric constant of metal, Emr.

For calculation of em in Eq. (6-1), we consider silver as an ideal metal and use the

Drude model for free electrons. Eq. (2-35) gives the dielectric function of a Drude metal:



0) V
,1 = 1-- 1 (2-35)


where Ap is the bulk plasma wavelength of the metal (cop is the bulk plasma frequency).

We use 324 nm for the bulk plasma wavelength, as measured in this experiment.

From calculation of the dielectric constant of silver, we found that EAg for = 3000

nm is about -84.75 (and eAg = -49.71 for X = 2000 nm). With these numbers, we get the

wavelengths of the resonant transmittance peaks for hole arrays with a period of 2 [tm

using Eq. (6-1). The result of calculation is shown in Table 6-1.

Table 6-1. Calculated positions of surface plasmon resonant transmittance peaks for three
interfaces of 2000 nm period hole arrays at normal incidence (Ed of air, fused
silica and ZnSe are 1.0, 2.0 and 6.0, respectively)

/ fused silica / metal ZnSe / metal
( i, j ) air / metal interface interface interface
interface interface
(0, 1l) and (l, 0) 2020 nm (P2) 2860 nm (P1) 5080 nm (P4)

(+1, +1) 1450 nm (P3) 2040 nm (P2) 3590 nm (P5)










Comparison of Calculated and Measured Positions of Transmittance Peaks and
Dips

Figure 6-1 indicates the calculated positions of the transmittance peaks in the

measured transmittance of a square hole array made on a fused silica substrate (A18-1).

As shown in this figure, the calculated positions of the peaks do not match accurately

with the peak positions in the measured transmittance. The difference between P1 and the

maximum peak position in the measured transmittance is about 80 nm. The spectral

difference for the second highest peaks is 140 nm. Even though many people still believe

in the role of surface plasmon in the enhanced transmission of sub-wavelength hole

arrays, the discrepancy between the peak positions calculated with Eq. (6-1) and the

measured peak positions still remains as an unsolved problem.


1.0 I I I I



0.8

P1
S0.6 P2
a P3

E
c 0.4
I--

0.2



0.0
0 1000 2000 3000 4000
Wavelength(nm)


Figure 6-1. Comparison of calculated peak positions with measured transmittance data.
Transmittance measured with a square hole array (A18-1) is shown. P1, P2
and P3 are the calculated positions of three transmittance peaks.









Actually, the surface plasmon equation, Eq. (2-42) (or, Eq. (6-1) for normal

incidence), has some approximations that are not applicable to real systems. First, the

dispersion relation of surface plasmon which is used to derive Eq. (2-42) is not for a

system of periodic hole array structure but for a plane interface of metal and dielectric

those are infinitely thick. This will give a difference in the dielectric constant of the

system. Second, the surface plasmon equation is based on the long wavelength

approximation. Thus, it does not depend on the shapes and the sizes of holes, but it

depends only on the periods of hole arrays. Third, as we mentioned in Chapter 2, the

surface plasmon equation is derived for a system with an infinitely thick metal which is

not possible in a real system. As the metal film is infinitely thick, it does not consider the

effect from an interaction between two interfaces. But, in a real system, the thickness of

metal film is finite, so there must be the interaction between two interfaces. Furthermore,

if there are holes in the metal film, the interaction will be stronger. These approximations

could be a reason for the difference between the calculated and the measured peak

positions.

Another interesting feature is the dips in the transmittance. It is known that the

transmittance minima of sub-wavelength hole arrays are due to Wood's anomaly.

According to Wood's anomaly, the minima (dips) appear at wavelengths where the

incident light is diffracted into the surface direction by periodic grating structures, and the

transmittance becomes a minimum. Eq. (6-2) is the diffraction equation of one

dimensional grating for normal incidence [52].


An = d sin 0 (6-3)
n









where d is the groove spacing, n is an integer, Ed is the dielectric constant of the dielectric

material and 0is the diffraction angle. As Wood's anomaly happens at the diffraction

angle 0= 90 so there is no transmitted light at the wavelength:


Al =d (6-4)
n

If we consider two dimensional grating structure such as a hole array, n in Eq. (6-4)

is replaced by ,j2 + j2 and the equation becomes


A= ra (6-5)


where i andj are integers and a is the period of two dimensional hole array. Eq. (6-5) is

very similar with the surface plasmon equation, Eq. (6-1), except for the dielectric

constant. Because the dielectric constant of the metal is much bigger than that of

dielectric material, the peak positions predicted by Eq. (6-1) is very close to the dip

positions predicted by Eq. (6-5).

Table 6-2. Calculated positions of transmittance dips for three interfaces of 2 [tm period
hole arrays at normal incidence (Ed of air, fused silica and ZnSe are 1.0, 2.0
and 6.0, respectively)

fused silica / metal ZnSe / metal
( i, j ) air / metal interface interface interface
interface interface
(0, 1l) and (l, 0) 2000 nm (D2) 2800 nm (D1) 4900 nm (D4)

(+1, +1) 1430 nm (D3) 2000 nm (D2) 3460 nm (D5)

Table 6-2 shows the calculated positions of dips. Figure 6-2 shows the same

transmittance shown in Figure 6-1 with the positions of the dips indicated. As we can see

in Figure 6-2, the calculated positions of the dips coincide well with the positions of the

dips in the measured transmittance. This is different from the discrepancy of the peak









positions. The reasons why the positions of transmittance minima are matched better than

the transmittance maxima are: 1) the diffraction grating equation is derived for a periodic

structure, not for a plane surface as the surface plasmon equation, 2) the diffraction

grating equation is not dependent on the refractive index of the grating material (metal),

but only depends on the refractive index of the dielectric material. Figure 6-3 shows the

positions of the peaks and the dips for the ZnSe-metal interface with the measured

transmittance of a square hole array (A14-1) made on a ZnSe substrate. This comparison

between the calculation and the measurement for a hole array on a ZnSe substrate also

shows a discrepancy in the peak positions and a good coincidence in the dip positions.

1.0 I



0.8

P1
S0.6 P2
a P3

E D1
S0.4 D3 D2
t--

0.2



0.0
0 1000 2000 3000 4000
Wavelength(nm)

Figure 6-2. Comparison of the calculated transmittance peaks and dips with the
transmittance measured with a square hole array (A18-1) made on a fused
silica substrate. P1, P2 and P3 are the calculated positions of the first three
peaks and D1, D2 and D3 are the calculated positions of the first three dips.










1 -0 *--------- -- i -- ---------
1.0



0.8
P4
P5
S0.6 P2
c P3
D4
E D5
o, D3 D2
C 0.4



0.2
I- '






0.0
0 2000 4000 6000 8000
Wavelength(n rm)


Figure 6-3. Comparison of the calculated transmittance peaks and dips with the
transmittance measured with a square hole array (A14-1) made on a ZnSe
substrate. P4 and P5 are the calculated peak positions and D4 and D5 are the
calculated dip positions for the ZnSe-metal interface. P2, P3, D2 and D3 are
the positions of the peaks and the dips for the air-metal interface.

Dependence of the Angle of Incidence on Transmission

Fig. 6-4 shows the transmittance of an array of square holes (A14-1) on a silver

film. This transmittance was measured using unpolarized light at normal incidence. As

discussed before, the peak A and the dip B are attributed to (i, j) = (+1, 0) or (0, 1)

modes on the fused silica-metal interface, and they don't vary with changing the

polarization direction of the incident light at normal incidence.

In the previous chapter, we have seen that the transmittance varies with the angle of

incidence and also strongly depends on the polarization of the incident light.










In order to explain the spectral behavior of transmittance maximum on the angle of

incidence, we need to recall the surface plasmon equation, Eq. (2-42). Even though the

surface plasmon equation has some drawbacks in its approximation, it is still useful to

explain the spectral behavior on the angle of incidence qualitatively. The surface plasmon

equation for oblique incidence was already introduced in Eq. (2-42) of Chapter 2, and

here we derive Eq. (2-42) using Eqs. (2-24) and (2-41):


k = -d-" Dispersion relation of surface plasmon (2-18)
C ed +


k = k +ky +ig +jg,


1.0-

0.9

08 -

0.7

0.6

0.5

0.4

03

0.2

0.1

0.0
2200


g, = g = 2a,
a,


2400 2600 2800 3000
Wavelength(nm)


3200 3400


Figure 6-4. Transmittance of a square hole array (A14-1) measured using unpolarized
light at normal incidence.


(2-34)


\/B (+1,0)Q, (0,+1)Q
| I I













Plane of incidence


Photon o0


Figure 6-5. Schematic diagram of an excitation of surface plasmon by the incident light
on two dimensional metallic grating surface. An azimuthal angle of the
incident light is 0 so that the wave vector of the incident light is always on
the plane of incidence and on the x-axis.

As we did in Chapter 2, we set the in-plane azimuthal angle to be 0 so that the

incident light is on the x-z plane which is the plane of incidence. This is shown in Figure

6-4. The magnitude of k in oblique incidence with 0o is ko sin0o and k = 0 Therefore,

the magnitude of ks is


,2 2 1n 1/2+
aI ) a,


(6-6)


From Eq. (6-6) and Eq. (2-24), we get an equation as


c red m


k, sin8, +i- + ]-
S2) (


(6-7)


ksp














Phot


0,2

0.1

0,0
2200


:on


Wavelenglh(nmr

(b)


Figure 6-6. Transmittance with s-polarized incident light. (a) Schematic diagram of (0, 1)
and (0, -1) modes excited on a square hole array for s-polarization and (b)
transmittance of a square hole array (A14-1) as a function of incident angle
for s-polarization. The peak A and the dip B are attributed to (0, 1) and (0, -1)
modes on the fused silica-metal interface that are degenerated in the s-
polarization case.


I i I I
--0
---2



14
--6---- -
8 A





120 /
--- 20 f ..... .



: -"i l... B(O, 1)Q


mX




V








































4-(1, 0)Q
(-1. O)Q
/ N-


t'-
---18,-
-- 20' ^-- *""*



"-, ., < ---^-"


-.


1, o0)QB(:1, 0)q
i00 2800 300C
Wavelenglh(nmr


N


(b)

Figure 6-7. Transmittance with s-polarized incident light. (a) Schematic diagram of(1, 0)
and (-1, 0) modes excited on a square hole array for p-polarization and (b)
transmittance of a square hole array (A14-1) as a function of the incident
angle for s-polarization. The peak A and the dip B are attributed to (1, 0) and
(-1, 0) modes on the fused silica-metal interface that are separated with
changing the angle of incidence in the p-polarization case.


0.2

0.1

0.0
2200


______











3200

3000

2800


I


F


I-- I --mm-Im m--m-m-,-_
Smaximum peak

-m--m---mm---m-,--mmmm

(0,1)S and (0,-1)S-



2nd highest peak



-- m -(0,1)A and (0,-1)A
Sm-m"----m--" m __m.
--I~--
- ---- measurement
-. --- calculation
I I I + I


0 4 8 12 16 20
Incident angle (0)

(a)


3000 -


Incident angle ()

(b)


Figure 6-8. Peak and dip position vs. incident angle for s-polarization. (a) Peak position
and (b) dip position. The red and the blue squares indicate the measured and
the calculated positions, respectively.


E 2600
C

c
2400

0C
. 2200

2000
2000


1800

1600


3200


2800

2600

2400

2200

2000

1800

1600

1400


---1=-- -m--m m
L[m *-U ---,
(0,1)S and (0,-1)S










---- measurement
calculation











3600

3400

3200

S3000

0 2800
Co
- 2600
CD

2400

2200

2000


3600

3400

3200

S3000

o 2800
0
C- 2600

2400

2200

2000


4 8 12 16 20

Incident angle ( )

(a)


4 8 12 16 20

Incident angle ()

(b)


Figure 6-9. Peak and dip position vs. incident angle for p-polarization. (a) Peak position
vs. incident angle and (b) dip positions vs. incident angle for p-polarization.
The red and the blue squares indicate the measured and the calculated
positions, respectively.










With some steps of calculation and k, -, where A is the wavelength of the
c cA

incident light, we get Eq. (2-42) for the position of resonant peak at oblique incidence:


= = a -isin8 + (i2 +j2) EdEm j2 sin 20 (2-42)
p i +j2 e+ JM


For s-polarization case, the electric field of incident light is parallel to the rotating

axis which is y-axis, so that only (0,j) modes are excited. This means that the modes

responsible for the transmittance maximum in the s-polarization case are (0, 1) and (0, -1)

mode on the fused silica-metal interface. This is shown in Figure 6-6.

From Eq. (2-42) we notice that there are onlyj2terms, which means that the (0, 1)

and (0, -1) modes on fused silica-metal interface are degenerate inj2. This is the reason

why there is no splitting in the peak A with changing the angle of incidence in the s-

polarization case.

On the other hand, for the p-polarization case, the electric field of the incident light

has two components which are parallel to the x-axis and the z-axis, but there is noy-axis

component. The x-axis component of electric field allows only (i, 0) modes to be excited

on the metal surface. Therefore, the peak A in the p-polarization case is attributed to (1, 0)

and (-1, 0) modes on the fused silica-metal interface. These modes are governed by a

linear term ofi in the Eq. (2-42) which is i sin 0. By this term, the (1, 0) and (-1, 0)

modes are separated with changing the angle of incidence, which shows a splitting of the

peak A in the transmittance.

In addition, in Figure 6-6 (b) and Figure 6-7 (b), there is the dip B at 2860 gm for

normal incidence. The dip B shows the same spectral behavior as the peak A. As

discussed before, this dip has been known as the Wood's anomaly. Eq. (6-5) is an









equation for the positions of the transmittance dips for normal incidence. If we consider

oblique incidence, the momentum conservation equation is the same with Eq. (2-34). But

the dispersion equation is different from the case of the transmittance peaks. The

dispersion equation for the diffracted (grazing) light is


k=-9 (6-8)
c

Combining with Eq. (2-34) and a few steps of calculation give the positions of

transmission dips:

Adp = a2 {-isinOo + (2+2)d 2 sin2 0 (6-9)


As we see from this equation, the position of the transmittance dip is also

dependent on the angle of incidence, which is same as the transmittance peak. This is the

reason why the dip B also shows the same spectral behavior as the peak A.

Figures (6-8) and (6-9) show the positions of the transmittance peaks and the dips

as a function of the incident angle for the s-polarization and the p-polarization,

respectively. As discussed above, we can see a spatial gap (a discrepancy) between the

calculated peak positions and the measured peak positions. For both polarizations, the

gap of the maximum transmittance peaks is about 200 nm and that of the second highest

peaks is about 400 600 nm. But the positions of the dips between the calculation and

the measurement are well matched. For the p-polarization, Figure 6-9 shows the splitting

of the peak and the dip when the incident angle increases.


Drawbacks of Surface Plasmon and CDEW

Another interesting feature that we observe is that there exists a resonant

transmission in the case of s-polarization. As shown in Chapter 2, the surface plasmon









does not exist for s-polarized incident light. This means that the surface plamon cannot be

the reason for resonant transmission with s-polarization. Moreno et al. [61] reported in

their paper that a resonant transmission is also possible for s-polarization. They proposed

that the resonant transmission is not due to the surface plasmon, but due to a coupling of

the incident light to surface mode. As we noticed, there is no difference between s-

polarization and p-polarization for normal incidence due to the geometrical symmetry.

Therefore, we cannot say that the surface plasmon is only responsible for the resonant

transmission of p-polarization case, while something else is responsible for the

transmission of s-polarization. Therefore, at least, we can say that the resonant

transmission on both s- and p-polarizations is not mainly due to the surface plasmon.

In addition to this inappropriateness of the surface plasmon for the explanation of

the enhanced transmission with s-polarization, in their paper [9], Lezec at al. claimed that

the surface plasmon is not responsible for the enhanced transmission of sub-wavelength

hole arrays because of the following reasons: 1) the difference of the peak positions

between the surface plasmon model and experimental data (we already discussed about

this previously), 2) an observation of the enhanced transmission of the hole arrays in Cr

for NIR region and tungsten for VIS region which do not support the surface plasmon, 3)

the demonstration of the enhanced transmission with numerical simulation for hole arrayz

in a perfect metal that also do not support the surface plasmon.

In contrast, the CDEW cannot explain some parts of experimental features. First,

the CDEW model cannot explain the spectral variations of s-polarization and p-

polarization as a function of the incident angle because the CDEW is based on the scalar

diffraction theory [50], so it does not depend on polarization directions. Second, J.









Gomez Rivas et al. [24] proposed in their paper about the enhanced transmission in

terahertz (THz) region that the enhanced transmission of sub-wavelength hole array in a

doped silicon film depends on temperature, because the mobility of the charge carriers in

the doped silicon film depends on temperature. This means that the enhanced

transmission of hole arrays on the doped silicon is attributed to the charge carriers as the

electrons in a metal film. This could be an evidence of that the surface plasmon is

responsible for the enhanced transmission in the metal.

Dependence of Hole Shape, Size and Polarization Angle on Transmission

In the previous chapter, we showed the transmittance of the different hole array

structures. We have seen that the transmittance of each hole array varied with the in-

plane polarization angle except for the square hole array due to its symmetry in x andy

directions. Now we compare three different hole arrays, square hole array, rectangular

hole array and slit array, with the same polarization angle. Figure 6-10 and Figure 6-11

show transmittance of the three hole arrays with polarization angles of 0 0 and 90 ,

respectively. As each hole array has an open fraction which is different from those of

other hole arrays, we rescaled the x-axis with transmittance divided by open fraction to

compare more directly the data for the different hole array. The open fractions for the

square hole array, rectangular hole array and slit array are 18 %, 29 % and 50 %,

respectively.

For the polarization angle of 0 Figure 6-10 (a) shows schematic diagrams

comparing three different arrays with the polarization angle of 0 0 and the lower panel

shows the transmittance of those arrays with the same polarization angle. The

transmittance of the square hole array (A18-1) shows the maximum peak intensity of 3.3









at 2940 nm. But, the intensity of the maximum peak of the rectangular hole array

decreases to 1.5. Finally, this maximum peak disappears for the slit array. The position of

the maximum peak shifts to shorter wavelengths slightly with increasing length of hole

edge parallel to polarization direction. Thus, the intensity of maximum peak is strongly

dependent on the length of hole edge parallel to polarization direction, whereas the

position of the maximum peak is not affected by changing the length of hole edge parallel

to polarization direction. For the peak positions, there is not enough space in shorter

wavelengths for the peak to be shifted because the shift to shorter wavelengths is stopped

by the dip at 2800 nm.

In addition, we can see a change in the dips at 2800 nm and 2000 nm. The dips in

the transmittance of the square hole array are well established. But, those dips rise up in

the transmittance of the rectangular hole array. These dips finally disappear for the slit

array. As we mentioned in the previous chapter, we understand this disappearance of the

dips for the slit array because there is no grating structure in the 0 0 polarization angle in

the slit array. But, for the rectangular hole array, even though the rectangular hole array

has a grating structure with a period of 2 tm, which is the same as the period of square

hole array, in the 0 0 polarization angle, the transmittance minimum is less well defined.

The increase in the transmittance at the minima is not due to the increase of the open

fraction because we already rescaled the y-axis with transmittance divided by open

fraction. Thus, the effect of larger open fraction is eliminated. The only parameter that we

consider here is the length of hole edge parallel to the polarization angle of 00, which is

different in each array. This indicates that the spectral feature of dips in transmittance

measured with a certain polarization direction is not only dependent on the period of hole









array in the direction parallel to polarization, but also on the length of hole edge parallel

to the polarization direction.

Another interesting feature in these transmittance is the intensity of the second

highest peak. Different from the maximum peak spectra, the second highest peak in each

spectrum shows the intensity which is the same as the open fraction of each array.

Figure 6-11 shows schematic diagrams comparing the three different arrays with

the polarization angle of 90 and the lower panel shows the transmittance of the three

hole arrays with the same polarization angle. Same as the 0 polarization angle, the

transmittance of the square hole array with the polarization angle of 90 shows the

maximum peak at 2940 nm. In the transmittance of the rectangular hole array, the

maximum peak shifts to longer wavelengths and shows a lower intensity with a broader

line-width. The transmission spectrum of slit array shows that the peak shifts even more

to longer wavelengths and has the lowest intensity with the broadest line-width. The y-

axis of these spectra is also rescaled with transmittance divided by open fraction, so the

effect of open fraction in the transmittance is eliminated.

As we mentioned before, we observed the red shift of the maximum peak with

increasing the dimension of hole edge which is perpendicular to polarization direction.

The maximum peaks of the rectangular hole array and the slit array occur at 3300 nm and

4000 nm, which are shifted 350 nm and 1050 nm from the maximum peak position of the

square hole array, respectively. This means that the position of the maximum peak is

strongly dependent on the length of hole edge perpendicular to the polarization direction.

In addition to the red shift of the maximum peak, the transmittance show a lesser

maximum peak intensity and a broader line-width with increasing the length of hole edge









perpendicular to the polarization direction. This observation tells us two different cases:

first, the resonant transmission becomes stronger with a shorter hole edge, which shows

the strong and sharp peak, second, the direct transmission from the front surface to the

back through the bigger holes becomes stronger with longer hole edge, which shows the

low and broad transmittance peak.

The second highest peak is also very interesting in the case of 90 0 polarization

angle. The second highest peak shows almost the same features (the peak position, the

intensity and the line-width) with increasing the length of hole edge perpendicular to the

polarization direction. This is very different from the spectral behavior of the maximum

peak. But, we are still not sure what gives this difference between the maximum peak and

the second highest peak.

The dips appear with a similar intensity at the fixed positions which are 2800 nm

and 2000 nm in all three transmission spectra except the dips of the slit array are a little

higher than others. This is absolutely due to the same periodic grating structures of the

three hole arrays in the polarization direction of 90 .

CDEW and Trapped Modes for Transmission Dependence on Hole Size

The CDEW model predicts the red-shift and the broader line-width for larger holes.

It explains those features with a reduction of the effective number of hole that contributes

to the resonant transmission. When the hole size becomes bigger, the bigger holes act as

leakage channels for the CDEW, so each hole is reached by CDEWs from fewer holes.

This effective reduction in the number of holes contributing to the resonant transmission

causes a weakness of resonant transmission, thus the transmittance shows the red-shift

and the broadening of the peak.




Full Text

PAGE 1

TRANSMISSION PROPERTI ES OF SUB-WAVELENGTH HOLE ARRAYS IN METAL FILMS By KWANGJE WOO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Kwangje Woo

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iii ACKNOWLEDGMENTS For last 5 years for my Ph.D. work, there are many people whom I have to thank for their support, advice and encouragement. First, I would like to thank my advisor, Professor David B. Tanner. Since I have become his research assistant, I have receiv ed so much valuable advice, encouragement and support. Also, I would like to thank Professor Arthur F. Hebard, Professor Stephen O. Hill, Professor Selman P. Hershfield and Prof essor Paul H. Holloway for serving on my supervisory committee. It was a great time for me to work in Prof TannerÂ’s lab for last four years because I had good colleagues in this lab: Dr. Andrew Wint, Dr. Hedenori Tashiro, Dr. Maria Nikolou, Dr. Minghan Chen, Haidong Zhang, Naveen Margankunte, Nathan Heston, Daniel Arenas and Layla Booshehri. I would like to th ank these people. Especially, I would like to thank my collaborator Sinan Selcuk for supplying samples, scientific discussi ons and a truthful friendship. I would like to thank my parents. They have supported me throughout my life. Finally, my wife and children, Ohsoon, Jis oo and Jiwon, have supported me with their love and patience. I would like to ex press my deepest thanks to them.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iiiLIST OF TABLES............................................................................................................viiLIST OF FIGURES.........................................................................................................viiiABSTRACT......................................................................................................................x ii CHAPTER 1 INTRODUCTION........................................................................................................1Background and Motivation.........................................................................................1Organization.................................................................................................................22 RIVIEW OF SURFACE PLASMO N AND DIFFRACTION THEORY....................4BetheÂ’s Theory for Transmittance of a Single Sub-Wavelength Hole.........................5Surface Plasmon...........................................................................................................7Definition of Surface Plasmon..............................................................................7Dispersion Relation of Surface Plasmon...............................................................7Dispersion relation for the p-polarization......................................................8Dispersion relation for the s-polarization.....................................................10Dispersion curves.........................................................................................12Propagation Length of the Surface Plasmon.......................................................14Surface Plasmon Excitation.................................................................................14Mechanism of Transmission via Surface Plasmon Coupling in Periodic Hole Array................................................................................................................17CDEW (Composite Diffractive Evanescent Wave)...................................................19Basic Picture of the CDEW.................................................................................19CDEW for an Aperture with Periodic Corrugation.............................................22CDEW for a Periodic Sub-Wavelength Hole Array............................................23Fano Profile Analysis.................................................................................................253 INSTRUMENTATION..............................................................................................29Perkin-Elmer 16U Monochromatic Spectrometer......................................................29Light Sources and Detectors................................................................................29

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v Grating Monochromator......................................................................................30Monochromator configuration.....................................................................31Resolution of monochromator......................................................................32The Diffraction Grating.......................................................................................33Grating equation and diffraction orders.......................................................33Blaze angle of the grating.............................................................................34Resolving power of grating..........................................................................34Bruker 113v Fourier Transform In frared (FTIR) Spectrometer.................................35Interferometer......................................................................................................35Description of FTIR Spectrometer System.........................................................384 SAMPLE AND MEASUREMENT...........................................................................41Sample Preparation.....................................................................................................41Substrates....................................................................................................................4 1Measurement Setup....................................................................................................435 EXPERIMENTAL RESULTS...................................................................................46Enhanced Optical Transmission of Subwavelength Periodic Hole Array................46Comparison of Enhanced Transmission w ith Classical Electromagnetic Theory......47Dependence of Period, Film Thickness and Substrate on Transmission....................48Dependence on Period of Hole Array.................................................................49Dependence on the Thickness of Metal Film......................................................50Dependence on the Substrate Material................................................................52Dependence on the Angle of Incidence......................................................................54Dependence on Hole Shape........................................................................................58Square Hole Arrays.............................................................................................58Rectangular Hole Array.......................................................................................59Slit Arrays............................................................................................................61Transmission of Square Hole Array on Rectangular Grid..................................62Refractive Index Symmetry of Dielectric Materials Inte rfaced with Hole Array......636 ANALYSIS AND DISCUSSION..............................................................................68Prediction of Positions of Transmission Peaks...........................................................68Comparison of Calculated and Measured Positions of Transmittance Peaks and Dips.........................................................................................................................70Dependence of the Angle of Incidence on Transmission...........................................74Drawbacks of Surface Plasmon and CDEW..............................................................82Dependence of Hole Shape, Size and Polarization Angle on Transmission...............84CDEW and Trapped Modes for Transm ission Dependence on Hole Size.................877 CONCLUSION...........................................................................................................91APPENDIX A TRANSMITTANCE DATA OF DOUBLE LAYER SLIT ARRAYS......................95

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vi B POINT SPREAD FUNCTIONS AND FOCUSING IMAGES OF PHOTON SIEVES.....................................................................................................................107C TRANSMITTANCE DATA OF BULLÂ’S EYE STRUCTURE..............................136LIST OF REFERENCES.................................................................................................142BIOGRAPHICAL SKETCH...........................................................................................146

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vii LIST OF TABLES Table page 4-1 List of the periodic sub-wavelength hole arrays......................................................436-1 Calculated positions of surface plasm on resonant transmittance peaks for three interfaces of 2000 nm period hole arrays at normal incidence ( d of air, fused silica and ZnSe are 1.0, 2.0 and 6.0, respectively)...................................................696-2 Calculated positions of transmittance dips for three interfaces of 2 m period hole arrays at normal incidence ( d of air, fused silica and ZnSe are 1.0, 2.0 and 6.0, respectively)......................................................................................................72

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viii LIST OF FIGURES Figure page 2-1 Schematic diagram for p-polarized (TM) light incident on a dielectric/metal interface....................................................................................................................122-2 Schematic diagram for s-polarized (TE) light incident on a dielectric/metal interface....................................................................................................................122-3 Dispersion curves of surface plasmon at air/metal interface and at quartz/metal interface, light lines in air and fused silica...............................................................132-4 Schematic diagrams of (a) the excitati on of the surface plasmon by the incident photon on a metallic grating surface and (b) th e dispersion curves of the incident photon, the scattered photon and the surface plasmon.............................................152-5 Schematic diagrams of the excitation of the surface plasmon by the incident photon on a two dimensional metallic grating surface.............................................182-6 Schematic diagram of transmission mechanism in a sub-wavelength hole array....182-7 Geometry of optical scattering by a hole in a real screen in (a) real space and (b) k-space for a range that kx is close to zero................................................................212-8 CDEW lateral field profile at z = 0 boundary, a plot of Eq. (2-44) .........................222-9 CDEW picture for an aperture with periodic corrugati ons on the input and output surfaces. Red arrows indicate the CDEWs generated on the input and output surfaces..........................................................................................................242-10 A CDEW picture for a periodic subwavelength hole array. Red arrows indicate the CDEWs generated on the input and output surfaces..........................................242-11 Schematic diagrams for Fano profile analysis.........................................................272-12 A schematic diagram of the non-resona nt transmission (BetheÂ’s contribution) and the resonant transmission (s urface plasmon contribution)................................272-13 Schematic diagram of the interference between the resonant and non-resonant diffraction in transmission of sub-wavelength hole array........................................283-1 Schematic diagram of Perkin-Elmer 16U monochromatic spectrometer.................30

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ix 3-2 Schematic diagram of the Littrow co nfiguration in the monochromator of Perkin-Elmer 16U spectrometer...............................................................................313-3 Schematic diagram of a reflection grating...............................................................333-4 Schematic diagram of a blazed grating....................................................................343-5 Schematic diagram of Michelson interferometer.....................................................363-6 Schematic diagram of the Bruker 113v FTIR spectrometer.....................................394-1 SEM images of periodic hole arrays samples..........................................................424-2 Picture of the sample holder used to measure transmittance with changing the angle of incidence and the in-plane azimuthal angle...............................................445-1 Transmittance of the square hole array (A14-1) and a silver film...........................475-2 Comparison between BetheÂ’s calculati on and the transmittance measured with the square hole array (A14-1)...................................................................................485-3 Transmittance of square hole arrays w ith periods of 1 m (A15) and 2 m (A181)............................................................................................................................. ..495-4 Transmittance vs. scaling variable, s = /( nd period), for the square hole arrays of 1 m period (A15) and 2 m period (A18-1) made on fused silica substrates ( nd = 1.4)..................................................................................................515-5 (a) Transmittance vs. wavelength (b) transmittance vs. scaling variable, s = /( nd period), for the square hole arrays of 2 m period (A14-1) made on a fused silica substrate ( nd = 1.4) and a ZnSe substrate ( nd = 2.4)..............................535-6 Transmittance of a square hole array (A14-1) with three different polarizations at normal incidence..................................................................................................545-7 Measurement of transmittance with s-polar ized incident light as a function of the incident angle...........................................................................................................565-8 Measurement of transmittance with p-pol arized incident light as a function of the incident angle.....................................................................................................575-9 Transmittance of square hole array (A181) as a function of polarization angle. The inset shows a SEM image of the square hole array...........................................595-10 Transmittance of a rectangular hole array (A18-2) for in-plane polarization angles of 0 and 90 The inset shows a SEM imag e of the rectangular hole array.......................................................................................................................... 60

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x 5-11 Transmittance of a slit array (A18-3) for in-plane polarization angles of 0 and 90 The inset shows a SEM im age of the slit array...............................................615-12 Transmittance of a square hole array on a rectangular grid (A18-4) for polarization angles of 0 45 and 90 The inset shows a SEM image of the square hole array in a rectangular grid.....................................................................625-13 Schematic diagram of sample preparation...............................................................655-14 Transmittance of a square hole array (A 14-1) on fused silica substrate with and without PR coated on the top...................................................................................655-15 Transmittance of a square hole arra y (A14-1) on ZnSe substrate with and without PR coated on the top of hole array..............................................................665-16 Transmittance of a square hole array (A 14-1) on fused silica substrate with and without PMMA coated on the top of hol e array with the second fused silica substrate attached on the top of PMMA...................................................................666-1 Comparison of calculated peak positions with measured transmittance data. Transmittance measured with a square hol e array (A18-1) is shown. P1, P2 and P3 are the calculated positions of three transmittance peaks...................................706-2 Comparison of the calculated tran smittance peaks and dips with the transmittance measured with a square hole array (A18-1) made on a fused silica substrate. P1, P2 and P3 are the calculate d positions of the first three peaks and D1, D2 and D3 are the calculated po sitions of the first three dips...........................736-3 Comparison of the calculated tran smittance peaks and dips with the transmittance measured with a square hole array (A14-1) made on a ZnSe substrate. P4 and P5 are the calculated peak positions and D4 and D5 are the calculated dip positions for the ZnSe-metal interface. P2, P3, D2 and D3 are the positions of the peaks and the dips for the air-metal interface.................................746-4 Transmittance of a square hole array (A14-1) measured using unpolarized light at normal incidence..................................................................................................756-5 Schematic diagram of an excitation of surface plasmon by the incident light on two dimensional metallic grating surface.................................................................766-6 Transmittance with s-po larized incident light..........................................................776-7 Transmittance with s-po larized incident light..........................................................786-8 Peak and dip position vs. incident angle for s-polarization......................................796-9 Peak and dip position vs. incident angle for p-polarization.....................................80

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xi 6-10 Transmittance of square, rectangular and slit arrays with polarization angle of 0 ..............................................................................................................................8 96-11 Transmittance of square, rectangular and slit arrays with polarization angle of 90 ............................................................................................................................90

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRANSMISSION PROPERTI ES OF SUB-WAVELENGTH HOLE ARRAYS IN METAL FILMS By Kwangje Woo August 2006 Chair: David B. Tanner Major Department: Physics We have measured the optical transmittance of sub-wavelength hole arrays in metal films. We investigated the spectral behavior of transmittance (the peak positions, intensities, line-widths, and the dip positions) as a function of the geometrical parameters of the hole arrays, the angle of incidence, the polarization angle and the refractive indices of the substrates. We calculated the positions of transmittance peaks and dips with equations from the surface plasmon theory and the diffraction theory, and compared the calculated positions of peaks and dips with measured transmittance data. We found that there is a discrepancy of 3 ~ 5% between the peak positions calculated with the surface plasmon equation and the peak positions in the measured transmittance data. We explain this discrepancy as possibly due to the a pproximations of the surface plasmon equation. However, the positions of the dips in the spec tra, as calculated with the diffraction grating equation, were well matched to the measured da ta. We also observed sp littings and shifts of the peaks and dips when changing the angl e of incidence and th e polarization of the

PAGE 13

xiii light. We confirmed this spectral behavior qualitatively with calc ulation of momentum conservation equations for oblique incidence and showed that the diffraction modes are degenerate for s-polarization, while the mode s are not degenerate for p-polarization. We studied the dependence of hole size and shap e on the transmittance while also changing the in-plane polarization angl e. We observed that the tr ansmittance peak is strongly dependent on the length of the hole edge pe rpendicular to the polar ization direction. In addition, we investigated the dependence on f ilm thickness and the refractive index of dielectric substrate.

PAGE 14

1 CHAPTER 1 INTRODUCTION Background and Motivation Recently, many workers in the area of optics have reported very interesting results in a new regime of optics called nano optics, sub-wavelength optics, or plasmonic optics [1]. In this area of optics, the physical dime nsion of objects for optical measurements is on a sub-wavelength scale. In terestingly, the optical prop erties of sub-wavelength structures are different from what we predict from classi cal electromagnetic theory [2]. In addition, this new field of optics makes it pos sible to manipulate lig ht via sub-wavelength structures. This capability of controlling light attracts a lot of applications in various fields of science and technology, for instan ce, Raman spectroscopy, photonic circuits, the display devices, nanolithography and biosensors [3-6]. Since the first research on enhanced op tical transmission of an array of subwavelength holes was reported in 1998 by Ebbesen et al. [7], no theory has explained this phenomenon, even though a lot of work has b een carried out. But theoretical studies are still actively going on, with the most promin ent one being the surface plasmon polariton (SPP) theory [8]. In addition to the surf ace plasmon polariton, the diffraction theory is also a very strong candidate as an explan ation of the enhanced transmission of subwavelength hole array [9-11]. Another model [ 12-14] proposed to explain this enhanced transmission phenomenon is the superpositi on of a resonant process and a non resonant process which shows the Fano profile [15]. Since the surface plasmon polariton model has some drawbacks [9] and shows a discrepa ncy between calculated and measured data

PAGE 15

2 [16], other models are consider ed as strong explanations of this enhanced transmission phenomenon. Many experiments also have been done for a wide spectral range. The enhanced transmission of periodic hole arrays for the optical region, the near -infrared region [17], and the terahertz (THz) re gion [18-24] was reported. Other scientific and technol ogical interest is focused on the enhanced transmission of a single sub-wavelength aperture. The enhanced transmission of a single plain rectangular aperture, which de pends on the polarization dire ction, was reported [25, 26]. And an aperture with corrugations on the inpu t side showed an enhanced transmission as well as a beaming of the transmitted light wi th corrugations on the output side [27-29]. In this dissertation, we present experime ntal transmission data for sub-wavelength hole arrays as a function of their geometrical parameters, the angle of incidence, the polarization of the light, and for two values of the refract ive index of the dielectric substrate material. For the theoretical models, we will discuss surface plasmon, composite diffractive evanescent wave, Fa no profile analysis and trapped mode. Organization This dissertation consists of seven chapters, including this introduction chapter. The details of each chapter are as follows: In Chapter 2, we review the basic theori es of surface plasmon and diffraction. The surface plasmon theory includes surface plasmon excitation by incident light, the plasmon dispersion relation, and an introduc tion of the transmission mechanism via surface plasmon coupling. The diffraction theory includes th e CDEW (composite diffractive evanescent wave) model and Fano profile analysis. In Chapter 3, we describe our experimental setup for transmission m easurement. Two spectrometers (a grating

PAGE 16

3 monochromatic spectrometer a nd a FTIR spectrometer) are introduced. In Chapter 4, the sample preparation and the measurement techni que with the specifications of samples are presented. In Chapter 5, the measured transmittance da ta are presented. The transmittance data are shown as a function of the geometrical pa rameters of hole arra y, the polarization and the incident angle of light, and the refr active indices of the substrate material. In Chapter 6, we analyze and discuss th e experimental results based on the surface plasmon and diffraction theories. We discuss the positions of peaks and dips, spectral changes with variation of the incident angl e and polarization, a nd the dependence on hole shape and size. Finally, Chapter 7 has the c onclusions of this di ssertation and briefly introduces some additional studies which are necessary for a future study.

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4 CHAPTER 2 RIVIEW OF SURFACE PLASMON AND DIFFRACTION THEORY There are two independent theories whic h explain the transmission enhancement by periodic arrays of sub-wavelength hole s: the surface plas mon polariton and the diffraction theory. When the enhanced transm ission was reported by Ebbesen and his coworkers, they interpreted their results with the surface plasmon [7]. The surface plasmon is still the most generally accepted explan ation of the enhanced phenomenon [30-33]. With dispersion relation of th e surface plasmon and momentum conservation equation of periodic grating, one can predict the positions of the enhanced transmission peak pretty accurately. But the prediction still shows some differences with the experimental results [16]. For this difference, there might be two reasons. First, the surface plasmon theory is based on the long-wavelength approximation ( >> d ), which means that it does not depend on the hole size of the structures. S econd, the surface plasmon theory, which is currently used in most papers, is still lim ited to the dispersion relation for a single interface between a dielectric and a metal (in which both are infinitely thick) while the experiments deal with structur es containing double interfaces with a finite thickness for the metal film [34] As we know, the classical diffraction theory for an electromagnetic wave impinging on a sub-wavelength aperture in an optic ally opaque conducting plane predicts an extremely low transmittance [2]. In this paper, Bethe showed the transmittance intensity of a sub-wavelength aper ture proportional to ( d / )4. But many calculations for diffraction

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5 by periodic hole arrays show an enhanced transmission which is very similar to experimental data [35-38]. The composite diffractive evanescent wave (CDEW) [9] is one of the diffraction models explaining the enhanced transmission by periodic structures The CDEW means a constructive interference of electromagnetic waves diffracted by periodic sub-wavelength structure and it is another st rong candidate responsible for the enhanced transmission phenomenon. This diffraction model (CDEW) can explain the enhanced transmission of hole array in a perfect conductor or in non-me tallic materials which the surface plasmon model cannot explain. Another transmission model e xplaining the enhanced tran smission is a unifying one of both the surface plasmon and the diffrac tion model [12, 13]. This unifying model proposes an analysis with Fano profile in tr ansmission spectra which is attributed to a superposition of the resonant pr ocess and non-resonant process. Recently, A. G. Borisov et al. [39] pr oposed another diffraction model for the enhanced transmission of sub-wavelength stru ctures. They suggested that the enhanced transmission of sub-wavelength hole arrays is due to the interference of diffractive and resonant scattering. The contribution of th e resonant scattering comes from the electromagnetic modes trapped in the vicinity of structures. This trapped electromagnetic mode is a long-lived quasista tionary mode and gives an explanation of extraordinary resonant transmission. BetheÂ’s Theory for Transmittance of a Single Sub-Wavelength Hole Bethe [2] reported that the transmittance of electromagnetic waves through a single hole in an infinite plane c onducting screen, which is very thin but optically opaque, is very small when wavelength of the incident li ght is much larger than the hole size. With

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6 this long-wavelength condition, d / << 1, where d is the diameter of hole and is wavelength of the incident light, Bethe has ca lculated “diffraction cross section” of the hole for the sand p-polarization: cos 2 27 646 4 d k As (2-1) 2 6 4sin 4 1 1 2 27 64 d k Ap (2-2) The s-polarized (TE mode) wave has an elec tric field perpendicu lar to the plane of incidence whereas the p-polarized (TM mode) wave has a magnetic field perpendicular to the plane of incidence. These polarizations are schematically shown in Figures 2-1 and 22. In Eqs. (2-1) and (2-2), one can recognize that the diffraction cross sections for two polarizations are the same for normal incidence, = 0. If the diffraction cross section is normalized to hole area, the normalized diffraction cross section becomes T d kd d A 4 4 2 223 2 27 64 2 (2-3) where 2 k k and are wave number and wavele ngth of the incident wave, respectively, and d is diameter of hole. This normaliz ed diffraction cross section can be considered as transmission normalized to hole area, T Eq. (2-3) is actually an expression for a circ ular aperture. If we change the circular aperture to a rectangular aper ture which has a dimension of D D Eq. (2-3) can be changed as

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7 T D kD D A 4 6 4 218 2 27 64 (2-4) Surface Plasmon The presence of a surface or an interface be tween materials with different dielectric constants leads to specific surface-related excitations. One example of this phenomenon is the surface plasmon. The interface betw een a medium with a positive dielectric constant and a medium with negative dielectric constant, such as a metal, can give rise to special propagating electromagnetic waves cal led surface plasmons, which stays confined near the interface. Definition of Surface Plasmon Sometimes the surface plasmon is also called the surface plasmon polariton. To understand this surface plasmon polariton, we need to define some terms: plasmon, polariton and surface plasmon. First, a plasm on is the quasiparticle resulting from the quantization of plasma oscillations. They ar e collective oscillations of the free electron gas. If this collective oscillation happens at the surface of metal, it is called a surface plasmon. Therefore, we define the surface plasmon as a collective oscillation of free electrons at the interf ace of metal and insulator [8]. The surface plasmon is also called the surface plasmon polariton. A polariton is the quasiparticle resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. Therefore, if an electromagnetic wave ex cites the surface plasmons on a metal surface and is coupled with the surface plasmon, it is called the surface plasmon polariton. Dispersion Relation of Surface Plasmon To get the dispersion relation for surface plas mons [8, 34, 40], we need to consider an interface between two semi -infinite isotropic media w ith dielectric functions, 1 and 2.

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8 The x and y axes are on a plane of the interface and the z axis is perpendicular to the interface. Medium 1 (dielectric function 1) and medium 2 (dielectric function 2) occupy each half of the space, z > 0 and z < 0, respectively. The electromagnetic fields for the surface wave which propagate in the x direction and are confined in the z direction on this interface are of the form: 01) ( 0 z e ez t k ix E E (2-5) 02) ( 0 z e ez t k ix E E (2-6) where E> and E< are electromagnetic fields in each half space, E0> and E0< are amplitudes, is angular frequency, t is time, kx is the wave vector of su rface wave propagating along the x -axis and 1, 2 are positive real quantities. Dispersion relation for the p-polarization For p-polarized electromagnetic wave (TM wave), the magnetic field is perpendicular to the plane of incidence and the electric field is in the plane of incidence. In Figure 2-1, the H-field is along the y -axis and the E-field is in the x-z plane. Thus, the E and H fields in each region can be expressed as 0 01) ( z e e B A,z t x k ix 1E (2-7) 0 0 01) ( z e e C, ,z t x k ix 1H (2-8) 0 02) ( z e e E D,z t x k ix 2E (2-9) 0 0 02) ( z e e F, ,z t x k ix 2H (2-10) The boundary condition that needs to be c onsidered is that the components of E and H parallel to the surface are continuous at the interface, z = 0, that is 0 2 0 1 z x z xE E (2-11)

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9 0 2 0 1 z x z xH H (2-12) Substituting Eq. (2-7) through Eq. (2-10) into Eq. (2-11) and Eq. (2-12), the boundary conditions give A = D and C = F One of the MaxwellÂ’s equations for continuous media is t c E H (2-13) For region 1 and 2, the x components in Eq. (2-13) give 01 1 z A c i C (2-14) 01 1 z D c i F (2-15) With A = D and C = F, division of Eq. (2-14) by Eq. (2-15) gives 2 1 2 1 (2-16) This equation is a condition for the surface plasmon mode and demonstrates that one of the two dielectric functions must be negativ e, so that, for example, the interface of metal/vacuum or metal/dielectric supports the surface plasmon mode. To get the dispersion relation of the surface plasmon, we use two MaxwellÂ’s equations: t c E H (2-13) t c H E1 (2-17) Operating on both sides of Eq. (2-17) and substituting for H from Eq. (2-14) gives

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10 2 2 2) ( 1 ) ( t c t c E H E (2-18) Using E E E2) ( ) ( and 0 E for a transverse wave, we get the transverse wave equation: 2 2 2 2t c E E (2-19) In the region of z > 0, the x and z components of the solution of Eq. (2-19) are x -component: A c ikB A2 2 1 1 1 2 (2-20) z -component: B c B k A ikx x 2 2 1 2 1 (2-21) Combining Eq. (2-20) and (2-21), we get 02 2 1 2 1 z c kx (2-22) Similarly, in the region of z < 0: 02 2 1 2 1 2 z c kx (2-23) Combining Eqs. (2-16), (2-22) and (2-23) we obtain the dispersion relation of the surface plasmon: 2 1 2 1 c kx Dispersion relation of surface plasmon (2-24) Dispersion relation for the s-polarization As shown in Figure 2-2, the s-polarization has the E field perpendicular to the plane of incidence and the H field in the plane of incidence. Then, we have a set of E and H fields:

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11 0 0 01) ( z e e A, ,z t x k ix 1E (2-25) 0 01) ( z e e C B,z t x k ix 1H (2-26) 0 0 02) ( z e e D, ,z t x k ix 2E (2-27) 0 02) ( z e e F E,z t x k ix 2H (2-28) As in the p-polarization case, we apply the boundary conditions Eqs. (2-11) and (212) and get A = D and B = E Then we use the MaxwellÂ’s equation: t c H E1 (2-29) Solving Eq. (2-29) with Eq. (2-25) through Eq. (2-28) for both regions of z > 0 and z < 0 give solutions with x and z components for each region: x-component: 0 z A i c B 1 (2-30) z-component: 0 z A c k Cx (2-31) x-component: 0 z D i c E 2 (2-32) z-component: 0 z D c k Fx (2-33) With the results from the boundary conditions, A = D and B = E Eqs. (2-30) through (233) can be combined and simplified 0 A2 1 i c (2-34) Since we defined 1 and 2 positive, thus A = 0 and all other constants ( B C D E and F ) also become zero. Therefore, the surface plasmon mode does not exist for the spolarization.

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12 Figure 2-1. Schematic diagram for p-polarized (TM) light incident on a dielectric/metal interface Figure 2-2. Schematic diagram for s-polarized (TE) light incident on a dielectric/metal interface Dispersion curves Figure 2-3 shows the dispersion curves of surface plasmons at the interface of metal/air, metal/quartz and the light lines in vacuum and fused silica glass, respectively. The momentum k is calculated by Eq. (2-24). The dielectric constant of metal, 2, in the Eq. (2-24) is described by the Drude dielectric function [41]: Metal 2 E Ex EzH k0z x Dielectric 1 Metal 2 H Hx HzE k0z x Dielectric 1

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13 2 2 2 2 21 1p p (2-35) where p is the bulk plasma wa velength of the metal ( p is the bulk plasma frequency). p is 324 nm for the silver film used in this experiment. The dielectric constants of air and fused silica substrate are 1.0 and 2.0, respectively. In Figure 2-3, the thickness of the metal film is considered to be infinite; thus, the interaction of the surface plasmons on both in terfaces is ignored. But if the thickness is finite, then there will be an interaction between two surface plasmons which will distort the dispersion curves of surface plasmons [34] Figure 2-3. Dispersion curves of surface plasmon at air/metal interface and at quartz/metal interface, light lin es in air and fused silica

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14 Propagation Length of the Surface Plasmon The propagation length of the surface plasmon can be defined by the imaginary part of the wave vector kxi in Eq. (2-24) as follows [8, 34, 40] xi xk L 2 1 (2-36) The dielectric function 2 is a function of At each it is a complex number, i ri2 2 2 where 2r and 2i are the real and the imaginar y parts of the dielectric function. The wave vector kx is also a complex number, xi xr xik k k 2 1 2 1 2 1 r r xrc k (2-37) 2 2 2 2 3 2 1 2 12r i r r xic k (2-38) From Eqs. (2-30) and (2-32), we can get the propagation length of the surface plasmon: i r r r xc L2 2 2 2 3 2 1 2 1 (2-39) Using parameters for silver [42], we can ev aluate the propagation le ngths at air/silver interface are about 20 m and 500 m for = 500 nm and = 1 m, respectively. Surface Plasmon Excitation As seen above, light does not couple to the surface plasmon on metal surface due to no crossing point between the dispersion curves of the in cident light and the surface plasmon except for k = 0. There are two ways to exci te the surface plasmon optically on an interface of a dielectric a nd a metal. First, one can use a dielectric prism to make coupling between the incident photons and the surface plasmon on an interface between the prism and the metal [8]. But this is not a case which is studying in this dissertation, so

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15 I am going to skip this part. Second, one can use periodic structures on the metal surface. When light is incident on the grating surface, the incident light is scattered by the grating structure. The surface component of the scat tered light gets an additional “momentum” from the periodic grating structure. This additional momentum enables the surface component of the scattered light to exc ite the surface plasmon on metal surface. (a) (b) Figure 2-4. Schematic diagrams of (a) th e excitation of the surface plasmon by the incident photon on a metallic grating su rface and (b) the dispersion curves of the incident photon, the scattere d photon and the surface plasmon Incident photon sp k0 ksp =c k kx Scattered photon Surface plasmon z Photon, k0 Surface plasmon, ksp x A ir d a0 Metal m

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16 Let us consider this case for one dimensi onal grating, as shown in Figure 2-4 (a). When light with a wave number k 0 is incident on a periodic gating on a metal surface with an incident angle 0, the incident light excites the surface plasmon on the metal surface. The momentum conservation equation allows this surface plasmon to have a wave vector, ksp, equal to a sum of the x -component of the incident wave vector and an additional wave vector which is the Bragg ve ctor associated with the period of the structure: c k a m k ksp 0 0 0 0, 2 sin (2-40) where k0 is the wave number of the incident light, and a0 is the period of the grating structure, and m is an integer. As shown in Figure 2-4 (b), this additional wave vector shifts the dispersion line of the incident light to the disper sion line of the diffracted photon. This light line crosses the dispersion curve of the surface plasmon. This crossing means that the incident light couples with the surface plasmon on the metal grating surface. If we consider a two dimensional grati ng on the metal surface, as shown in Figure 2-5, the momentum conservation equation becomes 02 a j iy x y x y x sp g g g g k k k (2-41) where kx and ky are surface components of the incident wave vector, gx and gy are the Bragg vectors, a0 is a period of the grating, i and j are intergers. From Eqs. (2-24) and (241), we get an equation which predicts the re sonant coupling wavelengths of the incident light and the surface plasmon on me tallic grating surface. Putting sp xk k in Eq. (2-24), we get an equation:

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17 0 2 2 2 2 0 2 2 0sin ) ( sin j j i i j i am d m d sp (2-42) From this equation, one can predict the wave length where the incident light excites the surface plasmon on the meta llic grating surface. The surface plasmon excitation wavelength is used to explain the enhanced transmission phenomenon of the sub-wave length hole array because the excitation wavelengths are close to the wavelength of the enhanced transmission [7]. But the surface plasmon excitation wavelength shows a 15 % di fference between theoretical calculation and experimental measurement [9]. Mechanism of Transmission via Surface Plasmon Coupling in Periodic Hole Array As we mentioned, the surface plasmon is a collective excitation of the electrons at the interface between metal and insulator. This surface plasmon can couple to photons incident on the interface of meta l and insulator if there exists a periodic grating structure on the metal surface. So, the coupling between photon and surface plasmon forms the surface plasmon modes on the interface. If bot h sides of metal film have the same periodic structure, such as an array of holes, the surface plasmon modes on the input and exit sides couple and transfer energy from the input side to the exit side. The surface plasmon modes on the exit side decouple the photons for re-emission. In this optical transmission process, the energy transfer by the resonant coupling of surface plasmon on the two sides is a tunneling process through the sub-wavelength apertures. Thus, the intensity of transmitted light decays with a film thickness exponentially. To compensate this decay, a localized su rface plasmon (LSP) [43-46] plays a role in this process. The LSP is a dipole moment formed on the edges of a single aperture due

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18 Figure 2-5 Schematic diagrams of the excitati on of the surface plasmon by the incident photon on a two dimensional metallic grating surface Figure 2-6. Schematic diagram of transmis sion mechanism in a sub-wavelength hole array. (1) excitation of surface plasm on by the incident photon on the front surface (2) resonant coupling of surf ace plasmons of the front and back surfaces (3) re-emission of photon from surface plasmon on the back surface Photon SP m d a0 x y z Photon Photon SPinSPout(1) (2) (3) Metal

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19 to an electromagnetic field near the aperture and it depends mainly on the geometrical parameters of each hole. The LSP makes a very high electromagnetic field in the aperture and increases the probability of transmission of the incident light. CDEW (Composite Diffractive Evanescent Wave) A recently proposed theory competing w ith the surface plasmon theory is the CDEW [9, 47-49]. The CDEW is a second mo del explaining the enhanced transmission phenomenon of sub-wavelength periodic structures. Basic Picture of the CDEW The CDEW model originates from the scal ar near-field diffraction. Kowarz [50] has explained that an electr omagnetic wave diffracted by a two dimensional structure can be separated into two cont ributions: a radia tive (homogeneous) and an evanescent (inhomogeneous) contributions. The diffracted wa ve equation for the 2-D structure is based on the solution to the 2-D Helmoltz equation: 0 ) ( ) (2 2 z x E k (2-43) where 2 2 2 2 2y x 2 k and ) ( 0) (z k x k iz xe E z x E, the amplitude of the wave propagating in the x z directions. As mentioned, the diffracted wave is a sum of the radiative (homogeneous) and the evan escent (inhomogeneous) contributions: ) ( ) ( ) ( z x E z x E z x Eev ra (2-44) We note that the homogeneous and the evanes cent components separately satisfy the Helmoltz equation. If we consider that the incident plane wave with a wave vector k0 impinges on a single slit of width d in an opaque screen, as shown in Figure 2-7 (a), the momentum conservation of the incident wave an d the diffracted wave should satisfy

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20 2 2 0 x zk k k (2-45) where kx and kz are the wave vectors of the diffracted wave in the x and z directions. If kx is real and if 0k kx then 2 0 2k k i kx z (2-46) This result means that the diffractive wave propagates in the x direction while being confined and evanescent in z direction. This evanescent mode of the diffracted wave emerging from the aperture grows as d/ becomes smaller. In contrast, for 0k kx kz remains a real quantity and the light is diffr acted into a continuum of the radiative, homogeneous mode. In Figure 27, the diffraction by an apertu re is described in real space (a) and k -space (b). The blue lines re present the radiative modes (0k kx ), whereas the red lines represent the evanescent mode (0k kx ). The surface plasmon mode in this picture is the green line which is one of th e evanescent modes diffracted by the aperture. Now, in order to find the specific solutions for the radiative and evanescent modes, we need to solve Eq. (2-43). The solution for Eev at the z = 0 is 2 for 2 Si 2 Si ) 0 (0 0 0d x d x k d x k E x Eev (2-47) 2 for 2 Si 2 Si ) 0 (0 0 0d x d x k d x k E x Eev (2-48) where E0 is the amplitude of the incident plane wave and 0sin ) ( Si dt t t. If we consider the surface wave on the metal, Eq. (2-47) can be simplified with a good approximation as [9]

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21 ) 2 cos(0 0 x k x d E Eev (2-49) From the expression of CDEW in Eq. (2-49), we notice that the amplitude of the CDEW decreases as 1/ x with the lateral distance, x and its phase is shifted by /2 from the propagating wave at the center of the slit. These results are different from the surface plasmon. The phase of the surface plasmon is eq ual to that of the incident wave and its amplitude is constant if absorption is not c onsidered [9] Figure 2-8 shows the lateral field profile of CDEW. Figure 2-7 Geometry of optical scattering by a ho le in a real screen in (a) real space and (b) k-space for a range that kx is close to zero [9].

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22 Figure 2-8. CDEW lateral field profile at z = 0 boundary, a plot of Eq. (2-44) [47] CDEW for an Aperture with Periodic Corrugation So far we have been discussing the di ffraction by a single aperture. Now we are going to extend our discussion to the period ic corrugation around a single aperture as shown in Figure 2-9. The corrugations are on bo th input and output su rfaces and actually play a role as CDEW generating points. The individual corrugation also becomes a radiating source. As shown in Figure 2-9, when a plan e wave impinges on the periodically corrugated input surface with an aperture at the center, only a small part of the incident light is directly transmitted through the aperture. Of the rest part of incident light is directly reflected by the metal surface and part of the incident light is scattered by the corrugations. This scattering produces CDEWs on the input surface (red arrows). The CDEWs propagate on the input surface and are scattered by the corrugations. The

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23 corrugations on the input surface act as point sources for the scattered light which is radiating back to the space. Part of the CDEWs propagating on the input surface is scattered at the aperture and tr ansmitted to the output surfac e along with the light directly transmitted through the aperture. When the tr ansmitted light (directly transmitted light and CDEWs) arrives at the output surface, a sma ll part of the light ra diates directly into space and the rest of the light is scattered ag ain by the aperture and corrugations on the output surface. The output surface CDEWs ar e now produced by the scattering of the transmitted light and it propagates on the output surface between the aperture and the corrugations. These propagating CDEWs on the output surface are scattered again by the corrugations and radiated into the front space. This means that the each corrugation on the output surface also becomes a radiation source. Thus, the transmitted light can be observed from all over the corrugation structure at the near field. At the far field, the radiation from the corrugations and the transmitted light from the aperture are superposed and interfere with each other. As discussed before, the CDEW has /2-phase difference from the transmitted light. Therefore, the CDEWs and the directly transmitted light make an interference pattern. The interference patt ern of these two waves at the far field has been observed experimentally. [49] CDEW for a Periodic Sub-Wavelength Hole Array Now we are going to develop the CDEW model for a periodic array of subwavelength holes. The CDEW model for the peri odic hole array is similar to that of an aperture with periodic corrugations, excep t there are many holes rather than one. As shown in Figure 2-10, a plane wave is incident on the input surface of a periodic hole array. The incident wave is partiall y reflected, diffracted, and transmitted. The

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24 Figure 2-9.CDEW picture for an aperture with periodic corrugations on the input and output surfaces. Red arrows indicate the CDEWs generated on the input and output surfaces. Figure 2-10. A CDEW picture for a periodic sub-waveleng th hole array. Red arrows indicate the CDEWs generated on the input and output surfaces. Incident plane wave Reflected wave Scattered wave Scattered wave Transmitted wave CDEWs Incident plane wave Reflected wave Scattered wave Scattered wave Transmitted wave CDEWs

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25 reflected wave consists of a direct reflect ion by the metal surface a nd the back scattering from the hole, similar to the case of the hole with corrugations in the previous section. Like the corrugations in Figure 2-9, each hole acts as a point for scattering and radiation of the CDEWs on the input surface. The CDEWs on the input surface are partially scattered back to space and partially transm itted along with the dire ctly transmitted wave through the holes to the output surface. Thus, the transmitted wave is a superposition of the CDEW and the wave directly transm itted through the holes. When the transmitted light arrives at the output surface, it is partially scattered (generates CDEWs on the output surface) and partially radiated into space. The CDEWs generated on the output surface propagate on the surface, a nd are partially scattered and radiated into space. In the front space, the directly transmitted wave fr om the holes and the radiation from the CDEWs are superposed to be the total transm ission of the hole array for detection at the far field observation point. Fano Profile Analysis Genet et al. [13] proposed that the Fano line shape in transmittance of periodic subwavelength hole arrays is a st rong evidence of an interference between a resonant and a non-resonant processes. Figure 2-11 shows sc hematic diagrams for the coupling of the resonant and non-resonant processes in a hol e array. In Figure 2-11, the period of hole array is a0, the thickness is h and the hole radius is r As shown in this figure, there are two different scattering channels: one open channel 1 corresponding to the continuum of states and one closed channel 2 with a resonant state wh ich is coupled to the open channel with is called “direct” or “non-res onant” scattering proce ss. The other possible transition is that the input state transits to the resonant state (sometimes called quasibound state) of the closed channel and then couples to the open channel via the

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26 coupling term V This is called “resonant” scattering process meaning opposed to the first one. The “non-resonant” scattering process si mply means the direct scattering of the input wave by the sub-wavelength hole array. This scattering can be called Bethe’s contribution. Bethe’s contributi on is the direct transmission through the holes in the array which is proportional to ( d/)4 and will be detected as a background in transmttance. In contrast, the “resonant” scattering process is a contribution from the surface plasmon excitation. This resonant scattering process basically consists of two steps: (1) the excitation of the surface plasmon on the peri odic structure of metal surface by the input wave and (2) the scattering of the surface plasmon wave by the periodic structure. The surface plasmon wave can be scattered into free space (reflection) or into the holes in the array (transmission). A simple transmission di agram of this model can be described via Figure 2-12. The total transmission amplitude is decided with the interference of the nonresonant contribution (Bethe’s contributi on) and the resonant contribution (surface plasmon contribution). A paper published by Sarrazin et al. [12] has also discussed the Fano profile analysis and the interference of resonant a nd non-resonant processes. In Figure 2-13, the homogeneous input wave ( i ) incident on the diffraction element A is diffracted and generates a non-homogeneous resonant diffraction wave (e) which is characterized by the resonance wavelength, This resonant wave (e) is di ffracted by the diffraction element B and makes a contribution to the homogeneous zero diffraction order. On the other hand, the other input wave is inci dent on the diffraction element B and generates a non-resonant homogeneous zero diffraction order. This non-re sonant scattered wave from B interferes with the resonant wave of from A. The Fano profile in transmittance of the sub-

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27 wavelength hole array results from a superpos ition of the resonant and the non-resonant scattering processes. Figure 2-11. Schematic diagrams for Fano profil e analysis. (a) Formal representation of the Fano model for coupled channels and (b) physical picture of the scattering process through the hole array directly (straight arrows) or via SP excitation [13] Figure 2-12. A schematic diagram of th e non-resonant transmission (BetheÂ’s contribution) and the resonant transm ission (surface plasmon contribution) [14]

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28 Figure 2-13. Schematic diagram of the in terference between the resonant and nonresonant diffraction in transmission of sub-wavelength hole array [12]

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29 CHAPTER 3 INSTRUMENTATION Optical transmittance measurements have been taken using two spectrometers: a Perkin-Elmer 16U monochromatic spectro meter and a Bruker 113v fourier transform infrared (FTIR) spectrometer. The Perkin -Elmer 16U monochromatic spectrometer was used for the wavelength range from ultrav iolet (UV), throughout visible (VIS) and to near-infrared (NIR), i.e., between 0.25 m and 3.3 m. Measurement for longer wavelengths (> 2.5 m) employed the Br uker 113v FTIR spectrometer. The FTIR spectrometer is able to measure up to 500 m, but in this experiment it was used for a range between 2.5 m and 25 m, i.e., near-i nfrared (NIR) and mid-infrared (MIR). Perkin-Elmer 16U Monoch romatic Spectrometer A spectrometer is an apparatus designed to measure the distribution of radiation in a particular wavelength region. The Perk in-Elmer 16U monochromatic spectrometer consists of three principal parts; light s ource, monochromator a nd detector. Figure 3-1 shows a schematic diagram of the Perkin-Elmer 16U monochromatic spectrometer. Here, the spectrometer has three light sources, two detectors and a gating monochromator. Light Sources and Detectors This spectrometer has three different li ght sources installed: a tungsten lamp, a deuterium lamp and a glowbar. The tungsten lamp is for VIS and NIR (0.5 m ~ 3.3 m), and the deuterium lamp is for VIS and UV (0.2 m ~ 0.6 m). This spectrometer has the glowbar for MIR region, but it was not used because the matching detector for MIR region has not been installed. This monochrom atic spectrometer has two detectors: a lead

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30 sulfide (PbS) detector for VIS and NIR range (0.5 m ~ 3.3 m) and a Si photo conductive detector (Hamamatsu 576) for UV and VIS range (0.2 m ~ 0.6 m). Figure 3-1. Schematic diagram of Perkin-Elmer 16U monochromatic spectrometer Grating Monochromator A monochromator is an optical device that transmits a selectable narrow band of wavelengths of light chosen from a broa d range of wavelengths of input light.

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31 Monochromators usually use a pr ism or a grating as a dispersive element. In prism monochromators, the optical dispersion phenomenon of a pris m is used to separate spatially the wavelengths of light, whereas the optical diffraction phenomenon of grating is used in the grating monochromators for th e same purpose. In this section, only the grating monochromator will be discussed. Monochromator configuration There are several kinds of monochromator configurations. The configuration of monochromator which is used in Perkin -Elmer 16U spectrome ter is the Littrow configuration. A schematic diagram of the L ittrow configuration is shown in Figure 3-2. Figure 3-2. Schematic diagram of the Littro w configuration in the monochromator of Perkin-Elmer 16U spectrometer In this configuration, the broad-band light enters the monochromator through slit A, which is the entrance slit. This entrance sl it controls the amount of light which is available for measurement and the width of the source image. The light that enters through the entrance slit (slit A) is collimated by mirror A, which is a parabolic mirror. The collimated light is such that all of the rays are parallel and focused at infinity. The collimated light is diffracted from the grat ing and collected again by the parabolic mirror

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32 (mirror A) to be refocused. The light is then reflected by the plane mirror (mirror B), and sent to the exit slit (slit B). At the exit slit, the wavelengths of light are spread out and focus their own images of the entrance slit at different positions on the plane of exit slit. The light passing through the exit slit contai ns an image of the entrance slit with the selected wavelength and the part of the imag e with the nearby wavelengths. Rotation of the grating controls the wa velength of light which can pass through the exit slit. The widths of the entrance and exit slits can be simultaneously controlled to adjust the illumination strength. When the illumination st rength of the input light becomes stronger, the signal to noise (S/N) rati o becomes higher but, at the same time, the resolution of measurement becomes lower because the exit slit opens wider and passes a broader band of the light. Resolution of monochromator One of the important optical quantities of monochromator is its resolution. The resolution of monochromator in the Littrow configuration ( = = ) can be expressed as [51] ) 1 ( ) 1 ( 1g sR R R (3-1) f S Rs2 cot (3-2) ) (0h R Rg (3-3) where Rs is the resolving power contributed fr om optical quantities of all components except for the grating, Rg is the ultimate resolving power of the grating, S is the slit width, is the angle of incidence and diffraction, f is the focal length of collimating mirror, h ( )

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33 is an error function, and R0 is the resolving power of the grating. Thus, the resolution of monochromator is dependent not only on th e grating but also on other optical and geometrical quantities of the monochromator. The Diffraction Grating A diffraction grating is one of the dispersi ng elements which are used to spread out the broad band of light and spatia lly separate the wavelengths. Grating equation and diffraction orders Figure 3-3 shows the conventional diagram fo r a reflection grating. In this Figure, the general equation of grati ng can be expressed as [52] QR PQ difference Path m d d sin sin (3-4) where m is diffraction order which is 0, 1, 2 Â…. Figure 3-3. Schematic diagram of a reflection grating. If m becomes zero, the zero order diffr action. When the diffraction angle is on the left-side of the zero order angle, the diffraction orders are all positive, 0 m, whereas if the angle crosses over the zero order and is on the right side of the zero order, the diffraction order m becomes negative, 0 m.

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34 Blaze angle of the grating Most modern gratings have a saw-tooth pr ofile with one side longer than the other as shown in Figure 3-4. The angle made by a grooveÂ’s longer side and the plane of the grating is the blaze angle. The purpose of this blaze angle is so that, by controlling the blaze angle, the diffracted light is concentr ated to a specific region of the spectrum, increasing the efficiency of the grating. Figure 3-4. Schematic diagram of a blazed grating Resolving power of grating As mentioned before, the resolving power of a grating is one of the important optical quantities contributing in the resolution of monochrom ator. If we use the Rayleigh criterion, the resolving power of grating becomes sin sin W mN R (3-5) where N is the total number of grooves on the grating, W is the physical width of the grating, is the central wavelength of th e spectral line to be resolved, and are the angles of incidence and diffraction, respectiv ely. Consequently, th e resolving power of

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35 grating is dependent on the width of grating, the center wavelength to be resolved, and the geometry of the optical setup. Bruker 113v Fourier Transform Infrared (FTIR) Spectrometer As mentioned before, the Bruker 113v FT IR spectrometer was used to measure transmission in the range of MIR (2.5 m ~ 25 m). Basically, this FTIR spectrometer can cover up to the range of far-infrared (FIR) which is up to 500 m. The entire system is evacuated to avoid absorption of H2O and CO2 for all of the measurements. Interferometer The interferometer is th e most important part in FTIR spectrometer. The interferometer in a FTIR spectrometer is a Michelson interferometer with a movable mirror. The Michelson interferometer is shown in Figure 3-6. The electric field from the source can be expressed as x k ie E x E 0) ( (3-6) where x is a position vector, k is a wave vector and E0 is an amplitude of the electric field. As shown in Figure 3-6, l1, l2, l3 and l2+x /2 are the distances between the source and the beam splitter, the beam splitter and the fixed mirror, the beam splitter and the detector, and the beam splitter and the mova ble mirror, respectively. The reflection and transmission coefficients of the beam splitter are rb and tb, and the reflection coefficients and the phases of the fixed mirror and the movable mirror are rf, f and rm, m, respectively. The electric field Ed which arrives at the detector consists of two electric field components: one from the fixed mirror, Ef, and the other from the movable mirror, Em.

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36 Figure 3-5. Schematic diagram of Michelson interferometer Thus, Ed, Ef, and Em are m f dE E E (3-7) 3 2 2 10ikl b i ikl f ikl b ikl fe t e e r e r e E Ef (3-8) 3 2 2 1) 2 ( ) 2 ( 0 ikl b i x l ik m x l ik b ikl me r e e r e t e E Em (3-9) To simplify, consider the mirrors as perfect mirrors, so rf and rm are 1. Also, we define the frequency as follows 2 2c k (3-10) With c = 1, Eq. (3-10) becomes 2 and we measure x in cm and in cm-1. If we let f m ) (, ml l l k ) 2 (3 2 1 ) 1 ()) ( ( 0 x i i b b de e t r E E (3-11) The light intensity at the detector is

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37 ))] ( cos( 1 [ 20 x T R S E E Sb b d d d (3-12) where 0 0 0E E S Rb and Tb are the reflectance and the transmittance of the beam splitter. Sd is the intensity of light at th e detector for a given frequency Then the total intensity for all frequencies is 00 0))] ( cos( 1 [ 2 ) ( ) ( d x T R S d S x Ib b d d (3-13) For an ideal beam splitter, Tb = 1 Rb and RbTb with Rb = 1/2 is 4 1 ) 1 ( b b b bR R T R (3-14) Here we define the beam splitter efficiency, b, as follows ) 1 ( 4 4b b b b bR R T R (3-15) Then, Eq. (3-13) becomes d x S x Ib d 0 0))] ( cos( 1 )[ ( ) ( 2 1 ) ( (3-16) Here we have two special cases, x and x = 0. For x Id in Eq. (3-17) becomes Id ( ) called the average value: 0 0) ( ) ( 2 1 ) ( d S Ib d (3-17) With x = 0 and ( ) = 0 (zero path difference or ZPD), Id becomes Id (0) called the white light value: ) ( 2 ) ( ) ( ) 0 (0 0 d b dI d S I (3-18) Now we need to define another quantity whic h is the difference between the intensity at each point and the average value called the interferogram: d x S I x I xd d)) ( cos( ) ( 2 1 ) ( ) ( ) (0 (3-19)

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38 where S ( ) S0( ) b( ) and ( x ) is the cosine Fourier Transform of S ( ). If we assume that S ( ) is hermitian, then ( x ) is d e e S xx i i ) () ( 4 1 ) ( (3-20) and dx e x e Sx i i ) ( 2 ) () ( (3-21) From the measurement with the interf erometer, we get the interferogram, ( x ) and compute the Fourier transform to get the spectrum, S ( ). The resolution of a Fourier spectrometer consists of two term s: one contributed from the source and the collimation mirror a nd the other decided by the maximum path difference. 2 11 1 1 R R R (3-22) 2 2 18 h f R (3-23) l R2 (3-24) where f is the focal length of the collimating mirror, h is the diameter of the circular source, l is the maximum path differe nce or the scan length and is the wave number in cm-1. Description of FTIR Spectrometer System A simple description of interferometer of the FTIR spectrometer is as follows. The light from a source is focused on a beam splitter after reflected by a collimation mirror. This beam splitter divides th e input light into two beams: one reflected and the other transmitted. Both beams are collimated by two id entical spherical mirrors to be sent to a

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39 two-sided moving mirror. The moving mirror reflects both beams back to the beam splitter to be recombined and the recombined beam is sent to the sample chamber for measurement. Figure 3-6. Schematic diagram of the Bruker 113v FTIR spectrometer As shown in Figure 3-6, the Bruker 113v FTIR spectrometer consists of 4 main chambers: a source chamber, an interferometer chamber, a sample chamber and a detector chamber. In the source chamber, th ere are two light sources: a mercury arc lamp for FIR (500 m ~ 15 m) and a glowbar source for MIR (25 m ~ 2 m). The interferometer chamber has act ually two interferometers fo r a white light source and a helium-neon (He-Ne) laser. As we know the exact wavelength of the laser, the small interferometer with the He-Ne laser is used as a reference to mark the zero-crossings of its interference pattern which defines the positions where the interferogram is sampled. This is the process of digitization of interferogram.

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40 White light transmission and reflection are measured in the sample chamber. The transmission is measured in the front side of the sample chamber and the reflection is measured in the back. There are two detectors installed in the detect or chamber: a liquid helium cooled silicon bolometer and a r oom temperature pyroelectric deuterated triglycine sulfate (DTGS) detector. The bolom eter detects light signals in the FIR range (2 m ~ 20 m) and the DTGS detect or is for MIR (2 m ~ 25 m).

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41 CHAPTER 4 SAMPLE AND MEASUREMENT Sample Preparation The sub-wavelength periodic array sample s were prepared using electron-beam lithography and dry etching. Th e sample fabrication proce ss is simply described as follows. Silver films with thickness be tween 50 nm and 100 nm were deposited on substrates using thermal evaporation. Fused si lica and ZnSe were used for the substrates. Before the E-beam writing process, a PMMA film is coated on the silver film. The PMMA coated samples were baked on a hot plate at 180 C for a minute. The baked PMMA film was exposed by the electron beam to make a periodic pattern on it. After the E-beam writing, the sample developed with the area of the PMMA film, which was not exposed by the electron beam, removed by the developing solution. The patterned PMMA film is going to be used to mask the silver film from the dry etching. During the dry etching process, Ar-ions strike the surface of the sample to make holes on the silver film. Finally, the remaining PMMA mask wa s removed with the stripping solution. With this fabrication process, a variety of samples have been prepared for this research, listed in Table 4-1. SEM images of the selected samples are also shown in Figure 4-1. Substrates Fused silica and ZnSe were used for the substrates. When the enhanced transmittance is expected to occur at wavelengths shorter than 5000 nm, fused silica substrate is used. If the transmission peaks are supposed to occur at wavelengths longer

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42 (a) (b) (c) (d) (e) (f) Figure 4-1. SEM images of peri odic hole arrays samples. (a) A14-1, (b) A14-3, (c) A18-1, (d) A18-2, (e) A18-3, and (f) A18-4

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43 than 5000 nm, ZnSe substrate is used. It is because fused silica is transparent between 300 nm and 5000 nm, while ZnSe is transpar ent between 500 nm and 15000 nm [53]. Measurement Setup We have used the Perkin -Elmer 16U monochromatic spectrometer and Bruker 113v FTIR spectrometer for transmittance measurem ent. Transmittance of an open aperture and of the sample has been measured. We firs t measured an open ap erture as a reference and then measured the sample. We used the same diameter aperture when measuring the sample to keep the measurement area the same. Then we calculated the ratio of the transmission of the sample to that of the open aperture to get the transmittance of the sample. Table 4-1. List of the periodic sub-wavelength hole arrays sample hole shape hole size (nm) period (nm) film thickness (nm) A14-1 square 900 x 900 2000 70 A14-2 square 900 x 900 3000 70 A14-3 square donut 900 x 900(out) 500 x 500(in) 2000 70 A15 square 500 x 500 1000 50 A18-1 square 840 x 840 2000 100 A18-2 rectangular 900 x 1300 2000 100 A18-3 slit 1000 (width) 2000 100 A18-4 square on rectangular grid 900 x 900 2000 (x-axis) 1500 (y-axis) 100 We measured transmittance as a function of the angle of incidence. The samples were mounted on a transmission sample holde r that allows change s in the angle of incidence. A picture of the transmission sample holder is shown in Figure 4-2. By rotating about an axis perpendicular to the direction of the incident li ght, the angle of incidence is changed. We measured transmittance at every 2 degrees between 0 degrees and 20 degrees. Also, we have varied the in-plane azimuthal angle. The azimuthal angle can be

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44 varied from 0 degrees to 360 degrees. We used this measurement to study the effect of polarization direction. This a ngle can be controlled using the same transmission sample holder by rotating the sample mounti ng plate shown in Figure 4-2. Figure 4-2. Picture of the sample holder used to measure transmittance with changing the angle of incidence and the in-plane azimuthal angle After the exit slit of the monochromator of Perkin-Elmer 16U spectrometer, we could installed one of three diff erent polarizers. A wire grid po larizer that is made of gold wires deposited on a silver bromide substr ate is used for the MIR region, and two dichroic polarizers are used for NIR, VIS and UV regions. We can get the either spolarized or p-polarized inci dent light by using these polar izers. Another wire grid polarizer has been installed on the exit apertu re of the interferom eter chamber of the Bruker 113v FTIR spectrometer to get polari zed light in the MIR region. In the FTIR spectrometer, the polarizer is rotated instead of the sample. Incident angle rotator Azimuthal angle rotator Sample mounting plate (An open aperture at center)

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45 In the Perkin-Elmer spectrometer, an optical solid half angle of the incident light on samples is adjustable with an iris aperture installed on the spherical mirror before the transmission sample holder (see Figure 3-1), but for most of measurement, we set the iris aperture to make this angle 1 , to minimi ze the incident angle effect. The optical solid half angle of the Bruker 113v spectrometer is about 8.5 and was not adjusted. Once we measured the samples with both sp ectrometers, the two transmittance data have been merged into one transmittance data by our own data merging program.

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46 CHAPTER 5 EXPERIMENTAL RESULTS In this chapter, we present our experiment al results. These experimental results will be shown as follows. First, we present expe rimental data for the transmittance of the arrays of square holes. We discuss the depe ndence on the period of the hole arrays, and also on the thickness of the metal films. Sec ond, transmittance of the square hole array as a function of the angle of incidence usi ng polarized light is presented. Third, transmittance with different hole shapes, hole sizes and in-plane polarization angles are shown. Finally, transmittance with different di electric materials interfaced to the metal film is presented. Enhanced Optical Transmission of Sub-wavelength Periodic Hole Array Figure 5-1 shows the transmittance of s quare hole array (A14-1) between 300 nm and 5000 nm. As shown in this figure, the transmittance maximum occurs at 3070 nm which shows an intensity of 60 %. This is a bout 3 times greater than the fraction of open area. This means that the light which is impinging not only on the hole area but also on the metal surface transmits into the output surface of the hole array via a certain transmission mechanism. This enhanced tran smission of sub-wavelength hole array was first reported by Ebessen et al in 1998. [7] The reason why it is called “enhanced” is that the transmittance intensity is not only greater than the fraction of open area but also much greater than a prediction from the classical electromagnetic th eory for transmission of an isolated aperture proposed by Bethe in 1944 [2]. Other spectral features we see from this transmittance are the second highest peak at 2450 nm and another sharp peak at 323 nm.

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47 The sharp peak at 323 nm is the bulk plasmon peak of silver and this is an intrinsic property of the metal which is silver. Figure 5-1. Transmittance of the square hole array (A14-1) and a silver film Comparison of Enhanced Transmission with Classical Electromagnetic Theory For comparison we need to recall the BetheÂ’s transmittance for a single subwavelength hole, Eq. (2-4): T D kD D A 4 6 4 218 2 27 64 (2-4) In Figure 5-2, we show the transmittance ca lculated with Eq. (2-4) for wavelengths up to 5000 nm and compare with the transmittanc e measured with the square hole array. As shown in Figure 5-2, BetheÂ’s calculati on is reasonable for wa velengths longer than 2000 nm which is 2 times greater than the di mension of hole. For wavelengths shorter Silver film (thickness 50 nm) A14-1 (thickness 70 nm) Open fraction

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48 than 2000 nm, the calculated tran smittance increases very rapi dly and is not compatible with the measured transmittance. At 3070 nm the intensity of the transmittance maximum is 2.93, while the transmission amplitude of BetheÂ’s calculation at the same wavelength is 0.19. Thus, the measured transmittance is 15 times greater th an the calculated one at the wavelength of the transmittance maximum. Figure 5-2. Comparison between BetheÂ’s ca lculation and the transmittance measured with the square hole array (A14-1) Dependence of Period, Film Thickne ss and Substrate on Transmission The experimental data shows that the enhanced transmission of sub-wavelength hole array depends on materials a nd geometrical parameters of sa mple. In this section, we discuss the dependence on the period of th e hole array, the film thickness and the substrate material on transmission.

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49 Dependence on Period of Hole Array For this experiment we prepared two di fferent hole array samples which have different periods of 1 m and 2 m, respectiv ely. These samples are fabricated on silver film. The thicknesses are 50 nm for the 1 m period sample and 100 nm for the 2 m period sample. The hole size for the 2 m period sample is 1 m 1 m and that of the 1 m period sample is 0.5 m 0.5 m. Both samples are prepared on fused silica substrates. Figure 5-3. Transmittance of square hole arra ys with periods of 1 m (A15) and 2 m (A18-1) Figure 5-3 shows the transmittance of both samples. The transmittance maxima for 1 m and 2 m period samples appear at 1560 nm and 2940 nm, respectively. The ratio of the two peak positions is about 1.88. This is very close to 2 which is the ratio of the periods of both samples. The second highe st peaks are located at 1170 nm and 2180 nm.

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50 The ratio of the second highest peak positions of both samples is 1.86 and it is almost the same with that of the maximum peak positions. The transmittance minimum or the dip more closely follows the ratio of the periods The dip located between two highest peaks occurs at = 1410 nm for the 1000 nm period sample and at = 2800 nm for the 2000 nm period sample. The ratio of dip positions is 1.98, almost same as the ratio of the periods of the two samples. From this simple consideration we are able to predict that the positions of peaks and dips in transmittance of sub-wavelength periodic hole arrays are closely associated with the periods of hole arrays. Dependence on the Thickness of Metal Film Another feature in Figure 5-3 is th e dependence of the transmittance on the thickness of the metal film. As indicated in this figure, the thickness of the metal film in the 1 m period array sample is 50 nm and th at of the 2 m period array sample is 100 nm. Two transmittances from these hole arrays show different spectral behaviors. The transmittance of the hole array with 50 nm thickness shows a stronger maximum peak, a higher background, and a broader line-width co mpared to the transmittance of the hole array with 100 nm thickness [55]. For a direct comparison between these hole array samples, we rescaled the x -axis to wavelength divided by the period of each array. These rescal ed transmittances are shown in Figure 5-4. The background in the transmittance for the hole array with 1 m period is higher than that of the hole array with 2 m period, due to the difference of thickness in the metal film. For a thinner metal film, transm ission through leakage paths in the film or direct transmission through meta l film increase. These kinds of contribution decrease when the thickness of film increases. Thus, the background for the ho le array with 2 m

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51 period decreases. The difference between the backgrounds of the 1 m period hole array and the 2 m period hole array is about 10 %. Figure 5-4. Transmittance vs. scaling variable, s = /( nd period), for the square hole arrays of 1 m period (A15) and 2 m period (A18-1) made on fused silica substrates ( nd = 1.4) From Figure 5-4, we can see a shift of the transmittance maximum even though these hole arrays are supposed to have the maximum at the same position in the rescaled x -axis. And also the positions of the dips in the transmittance of 1 and 2 m period hole arrays do not coincide but are slightly differe nt. This difference in position of peak or dip might be attributed to an imperfection in the geometrical structure of the hole arrays. But, if we take a closer look in th e figure, the peak of the 1 m period array has a little broader line-width than that of the 2 m period a rray. The broadness of transmission peak is basically coming from factors su ch as a larger hole size and a thinner film which increase

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52 the coupling strength between front and b ack surfaces. This coupling also probably causes the shift in peak position. Dependence on the Substrate Material The transmittance of the hole arrays depe nds on the dielectric materials interfaced with the hole array. In particularly, the posit ions of peaks and dips are strongly dependent on the dielectric material. In order to see the effect of the dielectric material in transmittance, we used two different substrates: fused silica and ZnSe. The dielectric constants of fused silica and ZnSe are 2.0 and 6.0, and the transmittances of bare substrates are 90 % and 70 %, respectively [53]. Figure 5-5 shows the transmittance of a 2 m period square hole array (A14-1) on different substrates: one on a fused silica substrate and the other on a ZnSe substrate. Even though those samples are on different s ubstrates, the film thickness of films was 70 nm for both transmittances. In Figure 5-5 (a), the hole array on fused silica has its transmittance maximum at 3070 nm while the maximum for the array on ZnSe substrate is at 5180 nm. The ratio of the peak positions of the two samples is about 1.69. We know the refractive indices of fused silica and ZnSe which are 1.4 and 2.4, respectively, so that nZnSe / nSiO2 = 1.7, close to the ratio of the peak wa velengths. This result indicates that the most dominant factor for this big red shift in the peak positions of these two hole arrays is the refractive index of the substrat e material. In Figure 5-5 (b), the x -axis is rescaled with wavelength divided by a product of th e refractive index and the period, s = / ( nd period). Even though the effect of the period a nd the refractive index is eliminated by the rescaling, the dip posi tions are still different between the two spectra. This is probably due to imperfections of the samples such as a difference in the thickness or the period.

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53 Figure 5-5. (a) Transmittance vs. wavelengt h (b) transmittance vs. scaling variable, s = /( nd period), for the square hole arrays of 2 m period (A14-1) made on a fused silica substrate ( nd = 1.4) and a ZnSe substrate ( nd = 2.4) (a) (b)

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54 Dependence on the Angle of Incidence In this section, we will discuss the effect of the incident angle on the transmittance. For this measurement we used the square hole array with 2 m period (A14). As mentioned in chapter 4, the incident a ngle is changed by rotating about an axis perpendicular to the incident light and the pl ane of incidence. For this measurement we used polarizers to get the sand p-polarized in cident light. We also measured with nearly unpolarized light. The transmittance was measured every 2 from 0 to 20 Figure 5-6. Transmittance of a square hol e array (A14-1) with three different polarizations at normal incidence From this experiment, we found a very strong dependence of the transmittance on the incident angle. In additi on, a significant polarization dependence of the transmittance at non-normal angle of incidence is also obser ved. The spectral behavior of transmittance of s and p-polarized light differ when the incident angle is changed [14, 56, 57].

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55 Figure 5-6 shows the normal incidence transmittance of a square hole array (A14-1) for three different polarizations. These spectra are almost the same except for the second highest peak. The intensity of the second peak for the case of unpolarization is a little higher than the peaks of others. A reason of this similarity in transmittance at normal incidence is that the sample (A14-1) used in this experiment has a geometrical symmetry for the two orthogonal polarizations. Figure 5-7 (a) shows schematically the s-pol arized light incident on a hole array sample. The lower panel, Figure 5-7 (b) shows the transmittance of a square hole array (A14-1) with s-polarized inci dent light as a function of the incident angle. The spolarization (TE mode) has a tran sverse electric field which is perpendicular to the plane of incidence. The magnetic field is in the plane of incidence. Fi gure 2-2 in Chapter 2 shows a schematic diagram for s-polarization. In Figure 5-7 (b), we can see some dependence on the transmittance on the angle of incidence. The intensity of the maximum tran smission peak decrease s and the line-width of the peak increases when the incident angle increases. The locations of both the maximum peak and of the dip shift to shorter wavelengths with increasing incident angle, while the second highest peak sh ifts the longer wavelengths. Figure 5-8 (a) shows a schematic diagram of p-polarized light incident on a hole array. Figure 5-8 (b) shows the transmittance of the same square hole array using ppolarized incident light as a function of the incident angle. The p-polarization (TM mode) has a transverse magnetic field, perpendicular to the plane of inciden ce. The electric field is in the plane of incidence. Figure 2-1 in Chapter 2 show s schematically the case of ppolarization. For p-polarization, the transmitta nce is quite different from that of the s-

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56 polarization as the incident angle changes. The maximum peak at 3070 nm at normal incidence splits into two peaks. One peak shifts to the longer wavelengths while the other peak shifts to the shorter wavelengths with increasing incident angle. (a) (b) Figure 5-7. Measurement of transmittance with s-polarized incident light as a function of the incident angle. (a) Schematic diag ram of s-polarized light incident on a hole array and (b) transmittance of a square hole array (A14-1) 0 k H E

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57 (a) (b) Figure 5-8. Measurement of transmittance with p-polarized incident light as a function of the incident angle. (a) Schematic diagra m of p-polarized light incident on a hole array and (b) transmittance of a square hole array (A14-1) The dip at 2860 nm also shows the same sp ectral behavior when the incident angle increases, splitting into two di ps, one of which shifts to s horter wavelengths and the other dip shifts to longer wavelengths with increasing incident angle. 0 k E H

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58 We cannot easily distinguish how the sec ond highest peak at 2450 nm changes. It is a very interesting feature that the transm ittance of the sand p-polarizations behave very differently as a function of the angle of incidence. Dependence on Hole Shape In this section, we discuss dependence of the transmittance on the hole shape and the in-plane azimuthal angle of polarization. For this measur ement, we prepared four hole array samples which have different shapes a nd sizes of holes. Those arrays are shown in Figure 4-1. The four sample s are: 1) an array of square holes with 1000 nm 1000 nm hole size and 2000 nm period (A18-1), 2) an array of rectangular holes with 1000 nm 1500 nm hole size and 2000 nm period (A18-2), 3) an array of slits with 1000 nm width and 2000 nm period (A18-3) and 4) an array of square holes on rect angular grid with 1000 nm 1000 nm hole size and 1500 nm period for x -axis direction and 2000 nm period for y -axis direction (A18-4). Square Hole Arrays Figure 5-9 shows the transmittance of the square hole array as a function of polarization angle. The spectra at all polarization angles (0 45 90 ) are the same. The transmittance maximum occurs at 2940 nm with an intensity of 60% for all three polarization angles. The behaviors at 0 and 90 polarization angles are due to geometrical symmetry of the square hole array. For 45 polarization angle, the electric field has decomposed into 0 and 90 components, making the spectra at 0 and 90 polarization angles to be the same. The transmittance peak at 2940 nm shows Fano line-shape which we discussed in Chapter 2. This Fano line-shape is a typical feature of the enhanced transmission of sub-

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59 wavelength hole arrays even though it is still not clear if it is due to the superposition of contributions from the resonant and non-re sonant scattering processes in transmission mechanism. Figure 5-9. Transmittance of square hole array (A18-1) as a function of polarization angle. The inset shows a SEM image of the square hole array. Rectangular Hole Array The transmittance of the rectangular hol e array for polarization angles of 0 and 90 are shown in Figure 5-10. As shown in the fi gure, it is evident that the transmission of the 0 polarization angle is very di fferent from that of the 90 polarization angle. For the 90 polarization angle, the transmitta nce maximum has an intensity of 83 % at 3300 nm. This peak disappears for 0 polarization angle while another peak appears at 2900 nm which shows an intens ity of 43 %. This difference between the 0o 45o 90o

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60 transmittance of 0 and 90 polarization angles shows th at the position of maximum transmittance strongly depends on polarization angle due to the asymmetry of rectangular holes. Figure 5-10. Transmittance of a rectangular ho le array (A18-2) for in-plane polarization angles of 0 and 90 The inset shows a SEM imag e of the rectangular hole array. Another interesting difference between the transmittance of 0 and 90 polarization angles is the lin e-width of the maximum peak Figure 5-10 shows that the line-width of the maximum peak in the transmittance of the 90 polarization angle is much broader that that of the 0 polarization angle. There is the second highest peak around 2300 nm in the transmittance spectra of 0 and 90 polarization angles. These peaks are locat ed at the same position with a similar 90o0o

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61 line-width. This is a different spectral beha vior compared to the large peaks at 2900 nm and 3300 nm. Figure 5-11. Transmittance of a slit array (A18-3) for in-plane polarization angles of 0 and 90 The inset shows a SEM image of the slit array. Slit Arrays The transmittances of the slit array for 0 and 90 polarization angles are shown in Figure 5-11, along with a SEM pictur e of the array (inset). The 0 polarization direction is parallel to the slit direction and the 90 polarization is perpendicular to the slit direction. The transmittance at 90 polarization angle shows a very broad transmittance peak around 4000 nm with an intensity of 73 %. This peak disappears for 0 polarization angle. This transmittance behavior of slit arra y is expected as slit arrays are used as a wire grid polarizer [52]. 90o0o

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62 The transmittance of the s lit array also shows a second maximum peak for both 0 and 90 polarizations around 2300 nm which is th e same position as the square and the rectangular hole arrays. But, in the transmittance of the 0 polarization angle, we hardly recognize the dips which exist in the transmittance of the 90 polarization angle at 2000 nm and 2800 nm. This is probably due to an ab sence of periodic grating structure in the direction of 0 polarization angle. Figure 5-12. Transmittance of a square hole array on a rectangular grid (A18-4) for polarization angles of 0 45 and 90 The inset shows a SEM image of the square hole array in a rectangular grid. Transmission of Square Hole Array on Rectangular Grid In order to see the effect of different periods in two orthogona l polarization angles, we prepared a square hole array on a recta ngular grid (A18-4). As mentioned previously, the periods in the 0 and 90 polarization angles are 1500 m and 2000 m, respectively.

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63 The hole size is 1000 nm 1000 nm which is the same as that of the square hole array (A18-1). Figure 5-12 shows the transmittance of the square hole array on a rectangular grid for 0 45 and 90 polarization angles. The transmittance at the 90 polarization shows a sharp maximum peak at 3020 nm and a second maximum at 2270 nm. The peak at 3020 nm disappears for the transmittance of the 0 polarization angle. Bu t the peak at 2270 nm remains at the same position with a little higher intensity for the 0 polarization angle. There is a small peak at 3000 nm in the spectrum of the 0 polarization angle and this might be due to a misalignment of polarization at the angle of 0 Refractive Index Symmetry of Dielectric Materials Interfaced with Hole Array Most of the samples that we have prepared are asymmetric structures with a fused silica substrate (or ZnSe substrate)/a periodic array on sliver film/air, as shown in Figure 5-13 (a). But there were some reports proposed an increase of the transmittance when sample has refractive index symmetry of diel ectric materials on both sides of hole array [58] In order to test an effect from this refractive index symmetry we used photo resist (Microposit S1800, Shipley) and PMMA (N anoPMMA, MicroChem) as a dielectric material to make the refractive index symmetr y with fused silica substrate. The refractive indices of PR and PMMA are approximately 1.6 and 1.5, respectively [59, 60], and the refractive index of fused silica is about 1.4 [42]. First, we measured transmittance of an original sample which is the square hole array (A14-1). Then, we coated PR or PMMA with a thickness of 150 nm on the top of hole array and measured the transmittance. Figure 5-13 shows schematic diagrams of each step of the sample preparation for measurement.

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64 Figure 5-14 and Figure 5-15 show the transmittance of square hole arrays on fused silica substrate and ZnSe substrate, and the sa me hole arrays with PR coated on the top. When the PR ( n 1.6) is coated on the hole arrays, the transmittance maximum of the hole array on fused silica substrate shifts more than 600 nm to longer wavelengths while the peak of the hole array on ZnSe substrate shifts only 60 nm which is small compared to that of the hole array on fused silica subs trate. There is a small increase in the peak intensity for the hole array on ZnSe substrate but there is almost no increase for the hole array on fused silica substrate. The dip at 2800 nm also shifts about 100 nm to longer wavelengths in the hole array on fused silica su bstrate but the same di p of ZnSe substrate sample shifts to longer wavelengths slightly. In addition, we used PMMA ( n 1.5) for this index symmetry experiment. As we know, the refractive index of PMMA is almost same as the refractive index of fused silica. Figure 5-16 shows transmission spectra of the square hole arra y (A14-1) with and without PMMA on top of the hole array. Th e transmittance of PMMA coated hole array shows the maximum transmittance at 3210 nm. This peak is shifted about 200 nm to longer wavelengths from 3010 nm where the maximum transmittance of the hole array without PMMA coating occurs. Another transmittance in Figu re 5-16 is measured with the same hole array but with another fused s ilica substrate attached on the top of PMMA. The transmittance with the second fused silica substrate shows no shift in the positions of peak and dip but a small decrease in transm ittance intensity compared to the spectrum of the PMMA coated hole array. The transmitta nce decrease is probably due to reflection and absorption by the additional fused silica substrate attached on the top of PMMA.

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65 Figure 5-13. Schematic diagram of sample prep aration (a) an origin al square hole array (b) a PR (or PMMA) coated square hole array (c) another fused silica substrate attached on top of PR (or PMMA) Figure 5-14. Transmittance of a square hole a rray (A14-1) on fused silica substrate with and without PR coated on the top Fused silica ( 1mm ) Ag pattern (100 nm) Fused silica Fusedsilica Photo resist or PMMA (150nm) Fused silica (a) (b) (c)

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66 Figure 5-15. Transmittance of a square hole array (A14-1) on ZnSe substrate with and without PR coated on the top of hole array Figure 5-16. Transmittance of a square hole a rray (A14-1) on fused silica substrate with and without PMMA coated on the top of hole array with the second fused silica substrate attached on the top of PMMA. Square hole array on Ag/fused silica glass (quartz)

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67 Even though we expected a remarkable in crease of the transmittance in the case of the fused silica substrate samples, it is hard to observe an increase in the measured transmittance. But this result shows that the peak and the dip of the hole array on fused silica substrate shift a lot more than the hole array on ZnSe substrate. It means that the spectral shifts of peak and dip by an addition of the inde x symmetry layer depend on the substrate material of the hole array.

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68 CHAPTER 6 ANALYSIS AND DISCUSSION In Chapter 5, we have shown the transmittance of various structures of hole arrays, which have different geometrical parameters (period, film thickness, incident angle and hole size) and the refractive indices of dielectric material. In this chapter, we will analyze and discuss a few important feat ures. First, we compute the theoretical predictions for the positions of peaks and dips, and compare them with experimental data. Second, we discuss the transmittance dependence on incident angle for sand p-polarized light. Third, we discuss the dependence on hole shape and size. Prediction of Positions of Transmission Peaks We need to recall one of surface plasmon equations which predicts the position of resonant transmittance peaks in two dimensional hole array. 0 2 2 2 2 0 2 2 0sin ) ( sin j j i i j i am d m d sp for non-normal incidence ( 0 0) (2-42) 2 1 2 2 0 m d m d spj i a for normal incidence ( 0 = 0) (6-1) With this equation, we can calculate wavelengths of the surface plamon resonant transmission peaks of a two dimensional hole array. For this calculation, we need the dielectric constants of air, subs trate materials and metal which is silver in this work. First, we know that the dielectric constant of air is 1. The substrate we mostly used is fused silica glass substrate. The diel ectric constant of fused silica glass is 2.0 for a wavelength

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69 range between 2000 nm and 3000 nm. We also need to calculate the di electric constant of silver. Generally, the dielectri c constant of a metal is a st rong function of frequency (or wavelength) and has a complex form: mi mr mi (6-2) where mr and mi are real and imaginary parts of m. mi is mainly associated with absorption of metal. m in Eqs. (2-42) and (6-1) is usually considered as the real part of dielectric constant of metal, mr. For calculation of m in Eq. (6-1), we consider silv er as an ideal metal and use the Drude model for free electrons. Eq. (2-35) gi ves the dielectric function of a Drude metal: 2 2 2 21 1p p m (2-35) where p is the bulk plasma wa velength of the metal ( p is the bulk plasma frequency). We use 324 nm for the bulk plasma wavelengt h, as measured in this experiment. From calculation of the dielectric constant of silver, we found that Ag for = 3000 nm is about –84.75 (and Ag = –49.71 for = 2000 nm). With these numbers, we get the wavelengths of the resonant transmittance p eaks for hole arrays with a period of 2 m using Eq. (6-1). The result of cal culation is shown in Table 6-1. Table 6-1. Calculated positions of surface pl asmon resonant transmittance peaks for three interfaces of 2000 nm period hole arrays at normal incidence ( d of air, fused silica and ZnSe are 1.0, 2.0 and 6.0, respectively) ( i, j ) air / metal interface fused silica / metal interface ZnSe / metal interface (0, 1) and ( 1, 0) 2020 nm (P2) 2860 nm (P1) 5080 nm (P4) ( 1, 1) 1450 nm (P3) 2040 nm (P2) 3590 nm (P5)

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70Comparison of Calculated and Measured Positions of Transmittance Peaks and Dips Figure 6-1 indicates the calculated positio ns of the transmittance peaks in the measured transmittance of a square hole array made on a fused silica substrate (A18-1). As shown in this figure, the calculated positions of the peaks do not match accurately with the peak positions in the measured transmittance. The difference between P1 and the maximum peak position in the measured transmittance is about 80 nm. The spectral difference for the second highest peaks is 140 nm. Even though many people still believe in the role of surface plasmon in the e nhanced transmission of sub-wavelength hole arrays, the discrepancy between the peak pos itions calculated with Eq. (6-1) and the measured peak positions still remains as an unsolved problem. Figure 6-1. Comparison of calculated peak pos itions with measured transmittance data. Transmittance measured with a square hole array (A18-1) is shown. P1, P2 and P3 are the calculated positions of three transmittance peaks. P1 P2 P3

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71 Actually, the surface plasmon equation, Eq. (2-42) (or, Eq. (6-1) for normal incidence), has some approximati ons that are not applicable to real systems. First, the dispersion relation of surface plasmon which is used to derive Eq. (2-42) is not for a system of periodic hole array structure but fo r a plane interface of metal and dielectric those are infinitely thick. This will give a difference in the dielectric constant of the system. Second, the surface plasmon e quation is based on the long wavelength approximation. Thus, it does not depend on the shapes and the sizes of holes, but it depends only on the periods of hole arrays. Th ird, as we mentioned in Chapter 2, the surface plasmon equation is derived for a system with an infinitely thick metal which is not possible in a real system. As the metal film is infinitely thick, it does not consider the effect from an interaction between two interfaces. But, in a real system, the thickness of metal film is finite, so there must be the interaction between two interfaces. Furthermore, if there are holes in the metal film, the inte raction will be stronger. These approximations could be a reason for the difference between the calculated and the measured peak positions. Another interesting feature is the dips in the transmittance. It is known that the transmittance minima of sub-wavelength hole arrays are due to WoodÂ’s anomaly. According to WoodÂ’s anomaly, the minima (dips) appear at wavelengths where the incident light is diffracted into the surface di rection by periodic gra ting structures, and the transmittance becomes a minimum. Eq. (6-2) is the diffraction equation of one dimensional grating for normal incidence [52]. sind nn d (6-3)

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72 where d is the groove spacing, n is an integer, d is the dielectric cons tant of the dielectric material and is the diffraction angle. As WoodÂ’ s anomaly happens at the diffraction angle = 90 so there is no transmitted light at the wavelength: d nn d (6-4) If we consider two dimensional grati ng structure such as a hole array, n in Eq. (6-4) is replaced by 2 2j i and the equation becomes dj i a 2 2 (6-5) where i and j are integers and a is the period of two dimensional hole array. Eq. (6-5) is very similar with the surface plasmon equati on, Eq. (6-1), except for the dielectric constant. Because the dielectric constant of the metal is much bigger than that of dielectric material, the peak positions predic ted by Eq. (6-1) is very close to the dip positions predicted by Eq. (6-5). Table 6-2. Calculated positions of transmitta nce dips for three interfaces of 2 m period hole arrays at normal incidence (d of air, fused silica and ZnSe are 1.0, 2.0 and 6.0, respectively) ( i, j ) air / metal interface fused silica / metal interface ZnSe / metal interface (0, 1) and (1, 0) 2000 nm (D2) 2800 nm (D1) 4900 nm (D4) (1, 1) 1430 nm (D3) 2000 nm (D2) 3460 nm (D5) Table 6-2 shows the calculated positions of dips. Figure 6-2 shows the same transmittance shown in Figure 6-1 with the positi ons of the dips indicated. As we can see in Figure 6-2, the calculated positions of the dips coincide well with the positions of the dips in the measured transmittance. This is different from the discrepancy of the peak

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73 positions. The reasons why the positions of transmittance minima are matched better than the transmittance maxima are: 1) the diffracti on grating equation is derived for a periodic structure, not for a plane surface as the surface plasmon equation, 2) the diffraction grating equation is not dependent on the refractive index of th e grating material (metal), but only depends on the refractive index of th e dielectric material. Figure 6-3 shows the positions of the peaks and the dips for the ZnSe-metal interface with the measured transmittance of a square hole array (A14-1) made on a ZnSe substrate. This comparison between the calculation and the measurement for a hole array on a ZnSe substrate also shows a discrepancy in the peak positions and a good coincidence in the dip positions. Figure 6-2. Comparison of the calculated transmittance peaks and dips with the transmittance measured with a square hole array (A18-1) made on a fused silica substrate. P1, P2 and P3 are the calculated positions of the first three peaks and D1, D2 and D3 are the calculated positions of the first three dips. P1 P2 D2 D1 P3 D3

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74 Figure 6-3. Comparison of the calculated transmittance peaks and dips with the transmittance measured with a square hole array (A14-1) made on a ZnSe substrate. P4 and P5 are the calculated peak positions and D4 and D5 are the calculated dip positions for the ZnSe-metal interface. P2, P3, D2 and D3 are the positions of the peaks and the dips for the air-metal interface. Dependence of the Angle of Incidence on Transmission Fig. 6-4 shows the transmittance of an array of square holes (A14-1) on a silver film. This transmittance was measured using unpolarized light at normal incidence. As discussed before, the peak A and the dip B are attributed to (i, j) = (1, 0) or (0, 1) modes on the fused silica-metal interface, and they donÂ’t vary with changing the polarization direction of the incide nt light at normal incidence. In the previous chapter, we have seen that the transmittance varies with the angle of incidence and also strongly depends on the polarization of the incident light. P4 P5 D5 D4 P2 D2 P3 D3

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75 In order to explain the spectral behavior of transmittance maximum on the angle of incidence, we need to reca ll the surface plasmon equation, Eq. (2-42). Even though the surface plasmon equation has some drawbacks in its approximation, it is still useful to explain the spectral behavior on the angle of incidence qualitatively. The surface plasmon equation for oblique incidence was already introduced in Eq. (2-42) of Chapter 2, and here we derive Eq. (2-42) us ing Eqs. (2-24) and (2-41): m d m d spc k Dispersion relation of surface plasmon (2-18) 02 ,a j iy x y x y x sp g g g g k k k (2-34) Figure 6-4. Transmittance of a square hole array (A14-1) measured using unpolarized light at normal incidence. A ( 1,0)Q, (0,1)Q B ( 1,0)Q, (0, 1)Q

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76 Figure 6-5. Schematic diagram of an excita tion of surface plasmon by the incident light on two dimensional metallic grating su rface. An azimuthal angle of the incident light is 0 so that the wave vector of the incident light is always on the plane of incidence and on the x-axis. As we did in Chapter 2, we set th e in-plane azimuthal angle to be 0 so that the incident light is on the x-z plane which is the plane of incidence. This is shown in Figure 6-4. The magnitude of kx in oblique incidence with 0 is k0 sin0 and 0 yk Therefore, the magnitude of ksp is 2 1 2 0 2 0 0 02 2 sin a j a i k ksp (6-6) From Eq. (6-6) and Eq. (2-24), we get an equation as 2 1 2 0 2 0 0 02 2 sin a j a i k cm d m d (6-7) m d a0 x y z Photon SP Plane of incidence

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77 (a) (b) Figure 6-6. Transmittance with s-polarized incident light. (a) Schematic diagram of (0, 1) and (0, -1) modes excited on a square hole array for s-polarization and (b) transmittance of a square hole array (A141) as a function of incident angle for s-polarization. The peak A and the dip B are attributed to (0, 1) and (0, -1) modes on the fused silica-metal interface that are degenerated in the spolarization case.

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78 (a) (b) Figure 6-7. Transmittance with s-polarized incident light. (a) Schematic diagram of (1, 0) and (-1, 0) modes excited on a square hole array for p-polarization and (b) transmittance of a square hole array (A14-1) as a function of the incident angle for s-polarization. The peak A and the dip B are attributed to (1, 0) and (-1, 0) modes on the fused silica-meta l interface that are separated with changing the angle of incidence in the p-polarization case.

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79 (a) (b) Figure 6-8. Peak and dip position vs. incident angle for s-polarization. (a) Peak position and (b) dip position. The red and the bl ue squares indicate the measured and the calculated positions, respectively.

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80 (a) (b) Figure 6-9. Peak and dip position vs. incident angle for p-polarization. (a) Peak position vs. incident angle and (b) dip positions vs. incident angle for p-polarization. The red and the blue squares indicate the measured and the calculated positions, respectively.

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81 With some steps of calculation and c c k 20 where is the wavelength of the incident light, we get Eq. (2-42) for the posit ion of resonant peak at oblique incidence: 0 2 2 2 2 0 2 2 0sin ) ( sin j j i i j i am d m d sp (2-42) For s-polarization case, the elect ric field of incident light is parallel to the rotating axis which is y-axis, so that only (0, j ) modes are excited. This means that the modes responsible for the transmittance maximum in th e s-polarization case are (0, 1) and (0, -1) mode on the fused silica-metal interfa ce. This is shown in Figure 6-6. From Eq. (2-42) we not ice that there are only j 2 terms, which means that the (0, 1) and (0, -1) modes on fused silica-met al interface are degenerate in j 2. This is the reason why there is no splitting in the peak A with changing the angle of incidence in the spolarization case. On the other hand, for the p-polarization case, the electric field of the incident light has two components which are parallel to the x -axis and the z -axis, but there is no y -axis component. The x -axis component of electric field allows only ( i 0) modes to be excited on the metal surface. Therefore, the peak A in th e p-polarization case is attributed to (1, 0) and (-1, 0) modes on the fused silica-metal interface. These modes are governed by a linear term of i in the Eq. (2-42) which is 0sin i By this term, the (1, 0) and (-1, 0) modes are separated with changing the angle of incidence, which shows a splitting of the peak A in the transmittance. In addition, in Figure 6-6 (b) and Figur e 6-7 (b), there is the dip B at 2860 m for normal incidence. The dip B shows the same spectral behavior as the peak A. As discussed before, this dip has been known as the WoodÂ’s anomaly. Eq. (6-5) is an

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82 equation for the positions of the transmittance dips for normal incidence. If we consider oblique incidence, the momentum conservation equation is the same with Eq. (2-34). But the dispersion equation is different from the case of the transmittance peaks. The dispersion equation for the diff racted (grazing) light is dc k (6-8) Combining with Eq. (2-34) and a few st eps of calculation give the positions of transmission dips: 0 2 2 2 2 0 2 2 0sin ) ( sin j j i i j i ad dip (6-9) As we see from this equation, the position of the transmittance dip is also dependent on the angle of incidence, which is same as the transmittance peak. This is the reason why the dip B also shows the same spectral behavior as the peak A. Figures (6-8) and (6-9) show the positions of the transmittance peaks and the dips as a function of the incident angle for the s-polarization and the p-polarization, respectively. As discussed above, we can see a spatial gap (a disc repancy) between the calculated peak positions and the measured peak positions. For both polarizations, the gap of the maximum transmittance peaks is a bout 200 nm and that of the second highest peaks is about 400 ~ 600 nm. But the positions of the dips between the calculation and the measurement are well matched. For the ppolarization, Figure 6-9 shows the splitting of the peak and the dip when the incident angle increases. Drawbacks of Surface Plasmon and CDEW Another interesting feature that we observe is that there exists a resonant transmission in the case of s-polarization. As shown in Chapter 2, the surface plasmon

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83 does not exist for s-polarized incident light. This means that the surface plamon cannot be the reason for resonant transmission with s-pol arization. Moreno et al. [61] reported in their paper that a resonant transmission is al so possible for s-polarization. They proposed that the resonant transmission is not due to the surface plasmon, but due to a coupling of the incident light to surface mode. As we noticed, there is no difference between spolarization and p-polarization for normal incidence due to the geometrical symmetry. Therefore, we cannot say that the surface pl asmon is only responsible for the resonant transmission of p-polarization case, while something else is responsible for the transmission of s-polarization. Therefore, at least, we can say that the resonant transmission on both sand ppolarizations is not mainly due to the surface plasmon. In addition to this inappropr iateness of the surf ace plasmon for the explanation of the enhanced transmission with s-polarization, in their paper [9], Lezec at al. claimed that the surface plasmon is not responsible for th e enhanced transmission of sub-wavelength hole arrays because of the following reasons : 1) the difference of the peak positions between the surface plasmon model and experi mental data (we already discussed about this previously), 2) an observation of the e nhanced transmission of th e hole arrays in Cr for NIR region and tungsten for VIS region whic h do not support the surface plasmon, 3) the demonstration of the enhanced transmissi on with numerical simulation for hole arrayz in a perfect metal that also do not support the surface plasmon. In contrast, the CDEW cannot explain some parts of experimental features. First, the CDEW model cannot explain the spectra l variations of spolarization and ppolarization as a function of the incident a ngle because the CDEW is based on the scalar diffraction theory [50], so it does not depe nd on polarization directions. Second, J.

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84 Gomez Rivas et al. [24] proposed in their paper about the enhanced transmission in terahertz (THz) region that th e enhanced transmission of su b-wavelength hole array in a doped silicon film depends on temperature, beca use the mobility of the charge carriers in the doped silicon film depends on temperat ure. This means that the enhanced transmission of hole arrays on the doped silicon is attributed to the charge carriers as the electrons in a metal film. Th is could be an evidence of that the surface plasmon is responsible for the enhanced transmission in the metal. Dependence of Hole Shape, Size and Polarization Angle on Transmission In the previous chapter, we showed th e transmittance of the different hole array structures. We have seen that the transmitta nce of each hole array varied with the inplane polarization angle except for the squa re hole array due to its symmetry in x and y directions. Now we compare th ree different hole arrays, squa re hole array, rectangular hole array and slit array, with the same pol arization angle. Figure 6-10 and Figure 6-11 show transmittance of the three hole arrays with pol arization angles of 0 and 90 respectively. As each hole array has an open fraction which is diffe rent from those of other hole arrays, we rescaled the x -axis with transmittance divided by open fraction to compare more directly the data for the diffe rent hole array. The open fractions for the square hole array, rectangul ar hole array and slit arra y are 18 %, 29 % and 50 %, respectively. For the polarization angle of 0 Figure 6-10 (a) shows schematic diagrams comparing three different arrays with the polarization angle of 0 and the lower panel shows the transmittance of those arrays with the same polarization angle. The transmittance of the square hole array (A18-1) shows the maximum peak intensity of 3.3

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85 at 2940 nm. But, the intensity of the maxi mum peak of the rectangular hole array decreases to 1.5. Finally, this maximum peak disappears for the slit array. The position of the maximum peak shifts to shorter wavelengt hs slightly with increasing length of hole edge parallel to polarization di rection. Thus, the intensity of maximum peak is strongly dependent on the length of hole edge parall el to polarization direction, whereas the position of the maximum peak is not affected by changing the length of hole edge parallel to polarization direction. For the peak pos itions, there is not e nough space in shorter wavelengths for the peak to be shifted because the shift to shorter wavelengths is stopped by the dip at 2800 nm. In addition, we can see a change in the dips at 2800 nm and 2000 nm. The dips in the transmittance of the square hole array are we ll established. But, those dips rise up in the transmittance of the rectangular hole array. These dips finally disappear for the slit array. As we mentioned in the previous chap ter, we understand this disappearance of the dips for the slit array because ther e is no grating structure in the 0 polarization angle in the slit array. But, for the rectangular hole array, even though the rectangular hole array has a grating structure with a period of 2 m, which is the same as the period of square hole array, in the 0 polarization angle, the transmittance minimum is less well defined. The increase in the transmittance at the mini ma is not due to the increase of the open fraction because we already rescaled the y-axis with transmittance divided by open fraction. Thus, the effect of larger open frac tion is eliminated. The only parameter that we consider here is the length of hole edge parallel to the polarization angle of 0 which is different in each array. This indicates that the spectral feature of dips in transmittance measured with a certain polari zation direction is not only de pendent on the period of hole

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86 array in the direction parallel to polarization, but also on the length of hole edge parallel to the polarization direction. Another interesting feature in these transmittance is th e intensity of the second highest peak. Different from the maximum peak spectra, the second highest peak in each spectrum shows the intensity which is the sa me as the open fraction of each array. Figure 6-11 shows schematic diagrams comp aring the three different arrays with the polarization angle of 90 and the lower panel shows the transmittance of the three hole arrays with the same polar ization angle. Same as the 0 polarization angle, the transmittance of the square hole array with the polarization angle of 90 shows the maximum peak at 2940 nm. In the transmittan ce of the rectangular hole array, the maximum peak shifts to longe r wavelengths and shows a lowe r intensity with a broader line-width. The transmission spectrum of slit array shows that the peak shifts even more to longer wavelengths and has the lowest intensity with th e broadest line-width. The y axis of these spectra is also rescaled with transmittance divided by open fraction, so the effect of open fraction in the transmittance is eliminated. As we mentioned before, we observed th e red shift of the maximum peak with increasing the dimension of hole edge which is perpendicular to po larization direction. The maximum peaks of the rectangular hole ar ray and the slit array occur at 3300 nm and 4000 nm, which are shifted 350 nm and 1050 nm from the maximum peak position of the square hole array, respectively. This means that the position of the maximum peak is strongly dependent on the length of hole edge pe rpendicular to the polarization direction. In addition to the red shift of the maximum peak, the transmittance show a lesser maximum peak intensity and a broader line-wid th with increasing the length of hole edge

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87 perpendicular to the polarization direction. This observation tells us two different cases: first, the resonant transmission becomes str onger with a shorter ho le edge, which shows the strong and sharp peak, second, the direct transmission from the front surface to the back through the bigger holes becomes strong er with longer hole edge, which shows the low and broad transmittance peak. The second highest peak is also ve ry interesting in the case of 90 polarization angle. The second highest peak shows almost the same features (the peak position, the intensity and the line-width) with increasing the length of hole edge perpendicular to the polarization direction. This is very different from the spect ral behavior of the maximum peak. But, we are still not sure what gives this difference between the maximum peak and the second highest peak. The dips appear with a similar intensity at the fixed positions which are 2800 nm and 2000 nm in all three transmission spectra ex cept the dips of the sl it array are a little higher than others. This is absolutely due to the same periodic grating structures of the three hole arrays in the polarization direction of 90 CDEW and Trapped Modes for Transmi ssion Dependence on Hole Size The CDEW model predicts the red-shift a nd the broader line-width for larger holes. It explains those features with a reduction of the effective number of hole that contributes to the resonant transmission. When the hole size becomes bigger, the bigger holes act as leakage channels for the CDEW, so each hol e is reached by CDEWs from fewer holes. This effective reduction in the number of holes contributing to the resonant transmission causes a weakness of resonant transmission, thus the transmittance shows the red-shift and the broadening of the peak.

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88 The trapped electromagnetic mode also expl ains the larger hole effect. The trapped mode is a long-lived quasi -stationary state that exists in th e vicinity of structures and is responsible for the resonant transmission. If the hole becomes bigger, the trapped mode becomes short-lived rather than long-lived, then the diffractive scattering dominates in transmission process. Thus, for the larger holes, the transmittance spectra lose their resonant features such as a strong and na rrow peak, but show the red-shift and the broadening of the peak instead.

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89 (a) (b) Figure 6-10. Transmittance of square, rectangular and slit arrays with polarization angle of 0 (a) Schematic diagrams and (b) transmittance of the arrays. The transmittance is normalized with open fraction of each hole array. Square Rectangle E E d 1.5d E Slit

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90 (a) (b) Figure 6-11. Transmittance of square, rectangular and slit arrays with polarization angle of 90 (a) Schematic diagrams and (b) transmittance of the arrays. The transmittance is normalized with open fraction of each hole array. Square Rectangle EE Slit E

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91 CHAPTER 7 CONCLUSION In this dissertation, we measured tr ansmission spectra of sub-wavelength hole arrays as functions of the geometrical parame ters of the hole arrays the incident angle and polarization, for two values of the refrac tive indices of dielectric materials. The subwavelength hole arrays that were measured in cluded square hole arrays with different hole sizes and periods, a recta ngular hole array, a slit array and a square hole array on a rectangular grid. Theoretical models explaining the enhan ced transmission of sub-wavelength hole array; the surface plasmon, CDEW and Fano pr ofile analysis, were discussed and their predictions were compared with the experimental transmittance. What we presented and discussed in this dissertation are as follows: First, we calculated the positions of tr ansmittance peaks and dips with the surface plasmon equation and the diffr action grating equation, and compared these with the experimental data. This comparison showed di screpancies between th e peak positions of the calculation and the measurement. In co ntrast, the positions of dips from the calculation are well matched with the measured data. This discrepanc y in the position of the peaks might be due to the approximati ons of the surface plam on model which are the dispersion relation of plane interface, the long-wavelength approximation and an ignorance of the interaction betw een the front and back surface. Second, we demonstrated transmittance as a function of the angle of incidence. For s-polarized light, the wavelength of the tran smittance maxima and that of the neighboring

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92 dip both shift slightly to shor ter wavelengths, For p-polarizat ion, the same peak and dip split into two peaks (and two dips), and the separated two peaks (and two dips) shift in opposite directions. We explai ned this different spectral behavior between sand ppolarization with the surface plasmon equati on and the diffraction grating equation for oblique incidence as follows. Th e s-polarized light excites (0, j ) and (0, – j ) modes along y -axis which are governed by the j2 term in the equations, so that these two mode are degenerate, thus, they do not show a separa tion of peak or dip. In contrast, the ppolarized light excites ( i 0) and (– i 0) modes along the x -axis and these modes are affected by – i term in the equations. This result means that the two modes are separated by the – i term. Thus, the ( i 0) mode shifts to shorter wavelengths, while the (– i 0) mode shifts to longer wavelengths. In addition, transmittance measured with spolarized incident light, as well as ppolarized incident light, showed enhanced tr ansmittance peaks. This transmission feature conflicts with the surface plasmon theory because no surface plasmon with s-polarization exists due to the boundary conditions. Ther efore, the surface plasmon may not be a critical effect for the enhanced tran smission of sub-wavelength hole arrays. Third, we tested the dependence of hol e shape and size by changing the in-plane polarization angle. For arrays of rectangular holes and slits, the transmittance maxima showed higher intensities and red-shifts when the polarization direction was perpendicular to the longer edge of the hole. Moreover, the maxima show stronger resonant features when the edge of hole perpendicular to the polarization direction becomes shorter. When the edge becomes longer, the transmission peaks show lower intensities, broader line-widths and red-shifts This suggests that th e larger hole size gives

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93 a lesser contribution of res onant transmission but more contribution of direct transmission. The dips and the other peaks in the transmittance are also important. We found that the positions of dips are gove rned by the diffraction grating equation. For example, the dips are very strongly fixed at their positions, if the period of hole array in the direction of polarization was kept the same, then they do not shift with changes in hole size. The second highest peak showed almost the same spectral features (p eak intensity, peak position and line-width) in most of the tran smittance spectra, which is quite different from the spectral behavior of the maximum p eak. We are still not sure what gives this difference. With this work we think that the main contribution of the enhanced transmission of sub-wavelength hole arrays comes from the interference of electromagnetic waves diffracted by hole array. There are two di ffracted waves, a surface wave (CDEW or trapped mode) and a directly transmitted wave. The surface wave is more dominant when the holes are smaller, giving resonant transmi ssion features. In contrast, when the holes are larger, the directly transmitted wave is dominant, giving less resonant transmission features. However, there still exists surface plasmons in the hole array, even though their effect on the transmission is smaller than that of the diffracted waves. For future work, we need to study more systematically the dependences of hole size and film thickness on the transmittance. This additional work will provide a clearer understanding of the enhanced transmission. Also, the measurement of reflectance will be needed for a complete study of optical pr operties of sub-wavelength hole arrays. Reflectance measurements will probably give an evidence of surface wave mode on the

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94 metal surface by observing absorption features. In addition, numerical simulations will be essential to establish an appropriate theo retical background for this optical phenomenon. We need to measure transmittance at different detection angles which means meaning transmittance at different diffraction orders.

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95 APPENDIX A TRANSMITTANCE DATA OF DO UBLE LAYER SLIT ARRAYS

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107 APPENDIX B POINT SPREAD FUNCTIONS AND FOCUSING IMAGES OF PHOTON SIEVES

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108 Focal length = 50.0 mm Focal length = 50.5mm

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109 Focal length = 51.2mm

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110

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111 500 nm wavelength light 600 nm wavelength light

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113 500 nm wavelength light 600 nm wavelength light

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115 500 nm FZP 600 nm FZP

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117 500 nm wavelength light 600 nm wavelength light

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119 500 nm wavelength light 600 nm wavelength light

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121 500 nm wavelength light 600 nm wavelength light

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123 49.74 mm focal length 50.98 mm focal length

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124 51.45 mm focal length 51.92 mm focal length

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125 52.38 mm focal length 53.91 mm focal length

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127 42.19 mm focal length 43.50 mm focal length

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128 44.75 mm focal length 49.23 mm focal length

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129 51.45 mm focal length

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131 53.38 mm focal length 54.54 mm focal length

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132 58.33 mm focal length 59.18 mm focal length

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134 51.45 mm focal length 52.38 mm focal length

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136 APPENDIX C TRANSMITTANCE DATA OF BULLÂ’S EYE STRUCTURE Optical microscopic image of 2 m period bullÂ’s eye structure

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146 BIOGRAPHICAL SKETCH I was born in Seoul, Korea, and grew up with a dream of becoming a famous scientist. It was the time of my gradua tion from Myongji High School when I started thinking about studying physics. In 1987, I entered Chungang University in Seoul and majored in physics. I finished my bachelorÂ’s and masterÂ’s degrees in the same university. In 1996, I graduated with my masterÂ’s degree and I entered Orion El ectric Company as a research engineer to work on flat panel display devices. I worked in Orion Electric Company for five years, and th en I joined in a research group of ETRI which is one of the national labs in Korea. But, I still want ed to study physics more, and decided to go to the University of Florida. I spent my last five years here in Ga inesville to study physics and now I am finishing my Ph.D.