by Nir Rotenberg - University of Toronto T-Space · Nir Rotenberg Doctor of Philosophy Graduate...

Ultrafast Active Plasmonics on Gold Films

by

Nir Rotenberg

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright c© 2011 by Nir Rotenberg

Abstract

Ultrafast Active Plasmonics on Gold Films

Nir Rotenberg

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2011

Active plasmonics combines the manipulation of light on both sub-wavelength length

and ultrashort time scales, a unique meld that holds promise for developments in many

scientific fields. This thesis reports on a novel approach to ultrafast, all-optical control of

grating-assisted excitation of surface plasmon polaritons based on opto-thermally modi-

fying the optical properties of gold. In contrast to prior works, this approach results in

plasmonic modulation on picosecond and even sub-picosecond time scales, and is com-

patible with modern, multi-GHz information processing technology. Finally, an analytic

model is developed that allows for the rapid and accurate calculation of the coupling

efficiency of beams with arbitrary spatial profile.

First, the ultrafast dynamics of existing plasmonic coupling resonances, on gold films

with grating overlayers, are studied with spectrally resolved pump-probe measurements.

Irradiation of the metal by 700 fs, 775 nm laser pulses results in modulations of the

plasmonic coupling efficiency of ∼ 20% near the center, or ∼ 60% off-center, of resonances

centered between 540 nm and 700 nm. The modulations decay with a time constant of

770 ± 70 fs. The experimental results are consistent with simulations based on the

thermal-dynamics of the electron-lattice gold system, coupled with numerical modeling

of light-grating interactions.

Next, two 150 fs, 810 nm laser beams are interfered on the surface of a planar gold

film, leading to an absorption/refraction grating in the metal. Optical pump-probe spec-

troscopy measurements of the first (-1) diffracted order in transmission identify plasmonic

ii

coupling resonances between 520 nm and 570 nm. The observed coupling efficiency is

∼ 10−5, and the launch window decays with a time constant of 620± 100 fs.

Lastly, a Green function-based analytic model is developed to describe grating assisted

plasmonic coupling, culminating in a first-order differential equation with coefficients

that have both clear physical significance as well as analytic forms. Comparison of this

technique with standard numerical modeling methods shows that plasmonic coupling

efficiencies in excess of 0.8 are predicted within an error of 15%. This model is used

to study plasmonic excitation by finite-size beams, showing the spatial evolution of the

intensity of both the surface plasmon polariton and the reflected beam.

iii

Acknowledgements

I’m not sure how others find their doctoral studies, but for me it was mainly one crazy

soap opera; there was plenty of tension, gossip, pressure, success, failure, friendship,

confusion, and above all, learning. I certainly would not have made it through this

process without the support of many people, most of which briefly overlapped with me,

and are too numerous to name here. Those whom I thank below are broadly split into

two categories: those with whom I interacted in an academic setting, and, well, the rest.

So, to all the people without whom I would have been lost, lonely, and, mostly, confused

I thank you from my heart, whether or not your name is found below.

First and foremost I would like to thank my supervisor, Prof. Henry van Driel for

making this thesis possible, and for his unbelievable patience and perseverance during

the past few years. You’ve never given up trying to improve me as a scientist, whether

it’s teaching me some physics, trying to get me to say ‘Z-Scan’ correctly, or getting me

to think precisely and analytically about my work; you never got angry and always let

me make my own mistakes, life lessons that I will always remember.

I would also like to thank Prof. John Sipe for teaching an experimentalist to (some-

times) think like a theorist and for showing me that not all Green functions are scary.

I also thank my committee, Prof. Daniel James and Prof. Young-June Kim for making

sure that I stayed on the right course, and of course, Prof. Arthur Smirl, my external

examiner for coming from Iowa.

This work would not have been possible without the other members of our research

group who contributed in a different manner, both scientifically and socially. Markus

Betz was the first to believe that my ideas might be interesting, and he always seemed to

be able to explain everything; I thank him even though he didn’t really wait for me while

we were hiking in the Alps. I thank Niklas Caspers for all the time he spent working

with me on these projects, for learning Labview instead of me; mostly, I would thank

Niklas for making me believe that I can teach, a skill-set that I began developing with

Adam Mattacchione (whom I would also like to thank for attempting all-nighters with

me, even if he only made it to midnight). Ryan Newson, thank you for the countless

discussions and all the help with the math. I also enjoyed my talks with Jesse Dean who

never let me get away with anything, and who forced me to really understand what I

was talking about. I thank Jean-Michel Menard for his interesting perspectives on pretty

much anything, as well as his enthusiasm for all things physics. Lastly, I would like to

iv

acknowledge the latest arrivals to our group, Dr. Mohamed Swillam and Dr. Christoph

Lange, who in a short time have come up with some very interesting ideas.

The counterbalance to all the people who helped me develop scientifically are those

who helped me stay sane while doing so. I need to thank Elie Sibony for all of the hours

he spent listening to me either talk enthusiastically, or complain bitterly, about physics;

there’s a good chance that I would have gone crazy otherwise. I want to also thank

John Elias and Dave Ratcliffe for reminding me that some things hurt more than failed

experiments, unbelievable generosity, and an introduction to an amazing sport where I

can work out my frustrations.

A special thanks goes to my girlfriend Jessica Fiebig who, for reasons I can’t un-

derstand, has stayed with me throughout my doctoral studies. Thank you for sharing

this adventure with me, for the countless hours spent talking, for taking care of me

when I was to busy to, and for driving me (a little) crazy when you needed to. I can

not imagine going through this experience without you, and want you to know that my

accomplishments are also yours.

I goes without saying that nothing would have been possible without the support

of my family; words can not express what I owe you, and what you mean to me. To

my aunt and uncle, Sabina and Yehuda Rotenberg, thank you for the quiet support,

encouragement, and understanding. To my sister, Nitzan, for the belief, love, and care.

And, mostly, to my parents, Esti and Shlomo for 30 years of love and unconditional

support leading to this. I love you, and thank you, for everything.

v

Contents

Abstract ii

Acknowledgements iv

Table of Contents vi

List of Tables ix

List of Figures x

1 Introduction 1

1.1 Surface plasmon polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Modeling plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Research objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Theoretical Background 9

2.1 Optical properties of Gold . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Surface plasmon polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Formalism and notation . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Dispersion relation and properties of SPPs . . . . . . . . . . . . . 14

2.2.3 Coupling of light to SPPs . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Numerical model of grating diffraction: the C-Method . . . . . . . . . . . 19

2.4 Ultrafast modulation of the optical properties of gold . . . . . . . . . . . 23

2.4.1 Optically induced thermal dynamics of electrons in metal films . . 24

2.4.2 Thermal changes to εm . . . . . . . . . . . . . . . . . . . . . . . . 28

vi

3 Active Plasmonics on Gold Gratings 32

3.1 Proof of concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Gold gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Optical sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 Tuneable wavelength pump-probe experiments . . . . . . . . . . . 34

3.1.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Dielectric gratings on gold . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2 Optical sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Broadband continuum pump-probe experiments . . . . . . . . . . 47

3.2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Design considerations and limitations . . . . . . . . . . . . . . . . . . . . 60

4 Active Plasmonics on Planar Gold Films 62

4.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Planar gold film . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Two pump - continuum probe experiments . . . . . . . . . . . . . 63

4.2 Optically induced transient thermal gratings . . . . . . . . . . . . . . . . 66

4.3 Plasmonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Improvements and limitations . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Analytic Model: Grating Assisted Plasmonic Coupling 74

5.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Grating assisted plasmonic coupling . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.2 Approximation and simplifications . . . . . . . . . . . . . . . . . 79

5.2.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.4 Fields at the SPP wavevector . . . . . . . . . . . . . . . . . . . . 80

5.2.5 Fields at the incident wavevector . . . . . . . . . . . . . . . . . . 82

5.2.6 Self-consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.7 Intensities and plane wave excitation . . . . . . . . . . . . . . . . 84

5.3 Performance of the analytic model . . . . . . . . . . . . . . . . . . . . . . 86

5.3.1 Sinusoidal grating . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

vii

5.3.2 Rectangular gratings . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Example: spatial evolution of the fields . . . . . . . . . . . . . . . . . . . 92

5.5 Improvements and Limitations . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions 97

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1.1 Active control of plasmonic coupling . . . . . . . . . . . . . . . . 97

6.1.2 Model: grating-mediated plasmonic coupling . . . . . . . . . . . . 98

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A Green Function Formalism 101

A.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 Uniform media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.3 Structured media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Bibliography 107

viii

List of Tables

2.1 Two-temperature model parameters . . . . . . . . . . . . . . . . . . . . . 25

2.2 The parameters of the band structure of gold near the L point . . . . . . 30

3.1 Gold grating parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ix

List of Figures

1.1 The original reflection spectra showing plasmonic resonances . . . . . . . 2

1.2 Surface plasmon and surface plasmon polariton . . . . . . . . . . . . . . 3

1.3 Typical dispersion relation for a surface plasmon polariton . . . . . . . . 4

2.1 The complex dielectric function and index of refraction of gold . . . . . . 10

2.2 Conceptual representation of a SPP on a metal-dielectric interface . . . . 12

2.3 Plane-wave incidence on a planar interfac including notation . . . . . . . 13

2.4 Penetration depth of the SPP field into the dielectric and the metal . . . 15

2.5 The propagation lengths of SPPs on different gold-dielectric interfaces . . 16

2.6 The dispersion relation for SPPs for different gold-dielectric interfaces . . 17

2.7 A surface relief and a planar grating . . . . . . . . . . . . . . . . . . . . 19

2.8 Typical C-Method reflectivity calculations for a sinusoidal gold grating . 22

2.9 Transient temperatures and energy density from calculation based on the

Two-Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 The thermal and non-thermal changes to the electronic occupancy as func-

tions of wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 The band structure of gold near the L point . . . . . . . . . . . . . . . . 29

2.12 Changes to the complex dielectric function of gold and the corresponding

differential reflectivity and transmissivity due to elevated electron temper-

atures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Typical broadband continuum . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Schematic of a pump-probe setup, including a pulse-shaper that selects

short spectral ranges from a broadband continuum probe . . . . . . . . . 35

3.3 Spectra of different probe pulses selected from the continuum by the pulse-

shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 P- and s-polarized reflectivities from the commercial gold gratings . . . . 37

x

3.5 P- and s-polarized transient reflectivities near the plasmonic resonance at

570 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 P- and s-polarized differential reflectivity spectra for different time delays,

for the plasmonic resonance at 570 nm . . . . . . . . . . . . . . . . . . . 39

3.7 Schematic: A spectral shift to the plasmonic resonance and the resultant

changes to the reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8 The transient spectral shift of the center of the plasmonic resonance . . . 40

3.9 Explanation of the spectral shift of the plasmonic resonance from disper-

sion relation considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 P- and s-polarized differential reflectivity spectra for different time delays,

for the plasmonic resonance at 600 nm . . . . . . . . . . . . . . . . . . . 42

3.11 Schematic: A spectral broadening of the plasmonic resonance and the

resultant changes to the reflectivity . . . . . . . . . . . . . . . . . . . . . 43

3.12 P-polarized transient differential reflectivity at 590 nm . . . . . . . . . . 44

3.13 SEM image of a custom grating . . . . . . . . . . . . . . . . . . . . . . . 46

3.14 Schematic of a pump-probe setup with a broadband continuum probe . . 47

3.15 P-polarized reflectivities from different angles of incidence, showing differ-

ent plasmonic resonance for the gold film with a dielectric grating overlayer 48

3.16 P-polarized zero-order reflectivity spectra as a function of the time delay

between the pump and the probe . . . . . . . . . . . . . . . . . . . . . . 49

3.17 An explanation of the plasmonic coupling efficiency . . . . . . . . . . . . 50

3.18 Change to the plasmonic coupling efficiency as a function of both the

wavelength and the time delay . . . . . . . . . . . . . . . . . . . . . . . . 50

3.19 Spectra and transients of the change to the plasmonic coupling efficiency 51

3.20 Characterization of the all-optical modulation of the plasmonic coupling . 53

3.21 The simulated change in plasmonic coupling efficiency as a function of

both wavelength and time delay . . . . . . . . . . . . . . . . . . . . . . . 56

3.22 Simulation results: The transient spectral shift of the center of plasmonic

resonance. The transient of the change to the plasmonic coupling effi-

ciency. The change to the plasmonic coupling efficiency as a function of

wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.23 A power study of the simulated changes to the plasmonic coupling . . . . 59

xi

4.1 Schematic of a pump-probe setup, including two pump beams and a broad-

band continuum probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Schematic of the sample and incident geometry of the beams . . . . . . . 65

4.3 Proof of the presence of the transient grating in the gold film . . . . . . . 67

4.4 Transient electron temperature difference in the gold film . . . . . . . . . 68

4.5 Theoretical and experimental transient T (−1) spectra . . . . . . . . . . . 69

4.6 The T (−1) spectra for both probe polarizations at the time of peak grating

contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7 ∆T (−1) as a function of time delay at 552 nm . . . . . . . . . . . . . . . 71

4.8 The ∆T (−1) spectra for different incident angles, showing different plas-

monic coupling resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 The electric fields and polarizations of the plasmonic coupling problem, in

k -space, to be calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Typical results for the Green function based simulations . . . . . . . . . 87

5.3 Comparison of the results of simulations using both the current method,

as well as the C-Method, for a 5 nm sinusoidal gold grating . . . . . . . . 88

5.4 Comparison of the results of simulations of a sinusoidal grating using both

the current method, as well as the C-Method, for a range of grating am-

plitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.5 Comparison of the results of simulations of a square grating using both

the current method, as well as RCWA, for a range of grating amplitudes 91

5.6 The spatial evolution of the incident, reflected, and SPP intensities for

gaussian excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.7 The plasmon excitation efficiency as a function of the incident spot size . 95

xii

Chapter 1

Introduction

I have been fortunate to work on a variety of projects during the tenure of my graduate

studies, exploring subjects ranging from nonlinear effects in metals and semiconductors,

to ultrafast active control of plasmonic coupling, and most recently, theoretical modeling

of grating assisted plasmonic coupling. Though these subjects may seem disconnected,

they were all approached from the view of ultrafast spectroscopy, and, in fact, the expe-

rience and knowledge gained from one often led to the next.

A common theme that emerged, as this work progressed, was my interest in the

ultrafast dynamics of surface plasmon polaritons, and in particular in the ability to

control their properties on ultrashort time-scales. In and of themselves, these modes

are unique: a non-radiating nature, sub-wavelength mode confinement, and propaga-

tion along a metal-dielectric interface are all properties that have led to high profile

plasmonic research across several disciplines. Moreover, these features, coupled with an

ability to control the properties and dynamics of these modes on picosecond time-scales

open new and intriguing avenues of research: from all-optical transistors, to plasmonic

switches and modulators, the possibilities are only limited by creativity, ingenuity, and

an understanding of the underlaying physical processes.

This work, then, is a summary of my venture into this exciting field. After building

the required theoretical framework, my studies on ultrafast control of grating assisted

plasmonic coupling (Opt. Lett. 33, 2137, and Phys. Rev. B 80, 245420) are detailed.

Here, a plasmonic resonance is spectrally tuned by irradiating gold gratings with ul-

trashort laser pulses. I then propose and demonstrate an all-optical method to couple

light to plasmons on unstructured gold films (Phys. Rev. Lett. 105, 017402), before

concluding with the development of a new, analytic method to describe plasmonic cou-

1

Chapter 1. Introduction 2

pling (Phys. Rev. B submitted). These works would not have been possible without the

contributions of many quality scientists, whose names appear in the related publications,

and as such this thesis is a testament that true progress is due to a healthy mixture of

luck, team-work, stubbornness, and creativity, though not necessarily in that order.

1.1 Surface plasmon polaritons

The discovery of surface plasmon polaritons (SPPs) dates back to 1902 [1]. While study-

ing p-polarized reflection spectra from metallic gratings, Robert W. Wood found that

certain spectral components were missing: for a given angle, a narrow range of wave-

lengths were not reflected, as shown in Fig. 1.1. This figure shows the reflection spectra

for p-polarized light between 500 and 630 nm (see top axis). An example of a plasmonic

coupling resonance is shown on the first line, for an incident angle of 4.2◦, as a dark

spot near 520 nm where energy is coupled to a SPP. However, at this time these spectral

features were neither understood, nor attributed much significance, and they became

known, simply, as Wood’s anomalies.

Figure 1.1: Robert Wood’s original measurements from Ref. 1 showing plasmonic reso-

nances. The incident angle is given on the left axis, while the wavelength of the spectra

is given on the top axis in units of 10 nm. The first plasmonic resonance is near 520 nm,

on the top row.


Half a century later, in the late 1950’s, three studies on light-metal interactions laid

the theoretical foundation for the understanding of SPPs. First, David Pines attributed

the losses experienced by electrons propagating through a metal to collective electronic

oscillations [2], introducing these oscillations as plasmons. Rufus Ritchie then showed

that these oscillations can exist at the surface of metals [3], theorizing the existence of

surface plasmons. In an unrelated study, John Hopfield coined the term polariton to

describe the coupling of the oscillation of electrons in a medium to an external field [4].

Metal

electricfield

Dielectric

surface plasmon surface plasmon polariton

electroncloud

Figure 1.2: The electromagnetic and surface charge characteristics of both a surface

plasmon on a metallic particle and a surface plasmon polariton at a metal-dielectric

interface. The electric fields are represented by purple arrows, while the free-electrons in

the metal are shown in red.

These modes are shown in Fig. 1.2. The surface plasmon is a localized mode, shown

here for a round metallic nanoparticle, where an electric field induces oscillations of

the free-electrons at the particle’s surface. In contrast, the surface plasmon polariton

is a propagating mode, where the collective free-electron oscillations at the surface of

the metal couple to electromagnetic fields in both media and travel along the metal-

dielectric interface. As these modes are dependent on the presence of the free-electrons

they are bound to the surface of the metal and require an external mechanism, such as

grating-mediated momentum transfer, to couple to light.

In fact, another decade passed before, in 1968, this was understood when Ritchie et

al. offered a theoretical treatment of Wood’s experiments, explaining Wood’s anomalies


in terms of the coupling of light to SPPs [5], though the term itself was not coined until

1974 [6]. Shortly thereafter, several studies also showed that SPPs could be excited by

taking advantage of the frustrated total internal reflection in a prism [7, 8]. Both the

coupling of the light to SPPs with gratings and prisms, as well as a variety of plas-

monic properties, can be understood in terms of the SPP wavevector, as is shown in

chapter 2. For example, the propagation length is calculated from the imaginary part of

the wavevector while the dispersion relation derives from the real part. A diagram of a

typical SPP dispersion relation is shown in Fig. 1.3 where the mismatch in wavevectors

is shown with blue arrows, and is what must be overcome to couple light to SPPs. Thus,

almost 70 years after their discovery, the childhood of plasmonic based research was at

an end; some fundamental properties of SPPs were known [9], as were several simple

plasmonic coupling techniques.

An

gu

lar

fre

qu

en

cy

Wavevector

light line

SPP dispersion

Figure 1.3: A typical dispersion relation for a surface plasmon polariton, including the

light line. The wavevector mismatch between the light and the SPP is shown with the

blue arrows.

Yet, the technological capabilities of the times lagged behind the theoretical progress

and consequently plasmonic-based research stalled. At the time, it was understood that

plasmonic modes localized fields to sub-wavelength dimensions, but it was only in the


1990’s that advances to nanofabrication techniques allowed for the efficient and reli-

able creation of suitably scaled structures. Concurrently, the development of solid-state

lasers [10] provided a coherent, high intensity, and stable light source for plasmonic ex-

periments.

The ability to consistently fabricate nanoscale structures coupled with the availability

of lasers have generated considerable scientific excitement, leading to numerous applica-

tions and publications across a variety of fields [11–16]. The high field confinement, as

well as the sensitivity of the plasmon to the optical properties of the material adjacent

to the metal, have led to the development of plasmonic based biosensors [11, 12]. Here,

the propagation or excitation properties of a SPP are affected by the presence of small

molecules on top of the metal, differentiating, for example, between healthy and diseased

cells. The high mode confinement has also opened a new avenue towards sub-wavelength

imaging [13], while the accompanying high field strengths can greatly enhance nonlinear

effects [14]. Lastly, the nanometer length-scales of plasmonics, which are the same as

those used for electronic-based information processing, coupled with the ultrashort time-

scales of modern pulsed lasers, which are compatible with multi-GHz applications, have

led to the development of plasmonic-based nanophotonic devices [13–16].

One of the major challenges in the actualization of a SPP-based nanophotnic device

is the design of an active plasmonic element: the SPP must be modulated or switched

on the time-scale at which the device performs. For modern information processing tech-

nology, the relevant time scale is sub-nanosecond. Such active control is achieved by

modulating the optical properties of one of the materials in the plasmonic device, which

then alters the properties of the SPP, such as the conditions in which light couples to

the plasmon or the plasmon’s propagation length. Early plasmonic switches suffered

from slow switching times, often on the order of milliseconds, due to their underlaying

physical mechanisms; examples of such devices include switching due to the injection of

free carriers in a semiconductor [17, 18] and phase transitions in semiconductor-metal

alloys [19] or organic films [20]. Only recently, and only in a small number of studies,

has true ultrafast active plasmonics been achieved [21–26]. Studies not presented in this

thesis include plasmonic switching due to free-carrier injection in a semiconductor super-

lattice [21], or through nonlinear transient grating techniques, both degenerate [22] and

non-degenerate [23]. In contrast, the experiments described in this thesis demonstrate

ultrafast active plasmonic switching due to a modulation of linear optical properties,

both in periodically structured [24, 25] and planar [26] gold films.


1.2 Modeling plasmonics

A common theme of the plasmonic experiments described above is the use of gratings to

couple light to SPPs [27]. Currently, three approaches that model the interaction between

light and grating are most prevalent in scientific works. These methods are all numerical

in nature, in that they operate like a black-box: given a set of input parameters, including

the grating and incident geometry as well as the material properties, they output a set of

numbers for the resultant electromagnetic fields or reflection and transmission coefficients.

The main reason for this lack of physical insight into the actual interaction is that there

exist no rigorous analytic solutions to Maxwell’s equations across a periodic boundary.

Regardless, these methods are arbitrarily accurate, though this performance is dependent

on ever-increasing computational times.

The first approach is the Finite-Difference Time-Domain (FDTD) method [28] where

the problem space is divided into a large number of small, square elements. As long as

the size of each element is much smaller than the characteristic length scale associated

with the problem – for example, the period of the grating, or the skin-depth of the

material used – Maxwell’s equations have exact solutions across its boundaries. At each

time step all the fields associated with the problem must be solved in each element,

using the stored fields in both the element and its neighbors. Hence, the accuracy of

this approach is determined by the size of the elements, and the grating and incident

radiation can be arbitrarily defined. Yet, while this is the only approach which allows for

pulsed excitations and which temporally resolves the response of the system, it is also

the most computationally intensive of the methods.

Another popular model for light-grating interactions is the Rigorous Coupled Wave

Analysis (RCWA) [29–31], where the grating profile is divided into a series of planar

interfaces, each with a rectangular profile. This approach is also known as the Fourier

Modal Method because the fields are expressed in the Bloch basis and the dielectric in

terms of its Fourier harmonics. The local modes of the fields for each slab are found in

this basis, and the subsequent application of the boundary matching conditions leads to

the solution for the grating system. The accuracy of this approach is dependent on the

number of Fourier harmonics retained and on the thickness of each slice of the surface

profile. While this technique can model arbitrary grating profiles, it is limited to plane-

wave excitations, and in general performs much better for s-polarized waves than for

p-polarized sources.


Lastly, a coordinate transformation method (C-Method) [32, 33] also models gratings

by moving to a coordinate system where the grating profile is flat. Since the electro-

magnetic fields also transform, there exists no analytic solution. This approach does,

however, reduce the problem in dimensions, leading to a wave equation in only one di-

mension. Again, this equation is solved in terms of the Bloch modes of this system, and

consequently the accuracy of this approach is solely dependent on the number of modes

retained. Consequently, this is the most efficient solution to the grating problem, though

the surface profile is limited to functions which are continuous and single valued.

While these methods can all model grating assisted plasmonic coupling, they do not

offer much physical insight into this process. Rather, to determine how a grating parame-

ter affects the coupling, the entire calculation must be repeated, sweeping over the desired

range of values of the parameter, and the behavior of the system is derived posteriori. As

a result, there are still fundamental aspects of grating assisted plasmonic coupling that

are not fully understood, and which must be studied for the field of nanoplasmonics to

evolve to its full potential.

1.3 Research objective

The first aim of this dissertation is to demonstrate that it is possible to actively control

the coupling of light to surface plasmon polaritons on gold films with grating overlayers,

on ultrashort time-scales. This sort of proof-of-concept experiment is typically used to

expand the tool kit of researchers in an emerging field, in this case the field of plasmon-

ics. As is usual, this preliminary experiment is followed with a full characterization of

the new method. Here, time- and wavelength-resolved pump-probe measurements reveal

a spectral shift of a plasmonic resonance that is due to the pump-induced thermal dy-

namics of the electrons in the gold [34, 35]. In order to study plasmonic resonances at

different wavelengths, the incident geometry of the probe is changed. This phenomenon

is investigated first on deep gold gratings, then on shallower dielectric gratings on top of

bulk gold. These results quantify both the temporal dynamics, as well as the magnitude

of the modulation of the plasmonic coupling.

This work continues with a demonstration that it is possible to achieve plasmonic

coupling to unstructured gold films, the logical extension of the previous experiment.

Here, a plasmonic coupling resonance is actively introduced to the film, while in the

previous case an existing coupling resonance was actively controlled. By interfering two


810 nm beams on the surface of a planar gold film, a periodic modulation of the electron

temperature is induced. Since the optical properties of the gold are dependent on the

electron temperature, this results in an absorption and index of refraction grating, which

decays on the same, picosecond, time-scale over which the electrons cool. With the

proper incident geometry [27], this transient grating [36] couples a spectral component

of a broad-band probe to a SPP. To differentiate between the thermally induced changes

to the linear optical properties of the film, and the induced grating effects, the diffracted

transmission order is detected. Analysis of the Wood’s anomaly in the diffracted order

reveals both the temporal dynamics as well as the efficiency of this plasmonic coupling

technique.

The last objective of this work is to develop an analytic model to describe grating

assisted plasmonic coupling to metal surfaces. Using a Green function formalism [37, 38]

leads to a simple, first order differential equation for the excitation and evolution of the

SPP field. Unlike previous numerical methods, the constants of the plasmonic coupling

equation are required to have both a clear physical significance, as well as be analytic

functions of the problem parameters. Finally, this analytic method must agree with the

currently accepted numerical calculations.

1.4 Thesis outline

This thesis presents both the experimental work which demonstrates that ultrafast active

control of plasmonic coupling to gold surfaces is possible, as well as a theoretical model

that can be used to describe aspects of this process. In Chapter 2, the theoretical basis for

the rest of the thesis is developed. Chapter 3 contains a demonstration of how irradiation

by ultrashort laser pulses can actively control grating-mediated plasmonic coupling on

gold films on ultrafast time-scales, culminating in a characterization of this approach.

This technique is then extended, in chapter 6, to all-optically couple light to SPP modes

on unstructured gold films, again on ultrafast time-scales. In chapter 5 the focus shifts

from experimental to theoretical considerations, and an analytic model that describes

surface-relief grating-based plasmonic coupling is developed based on a Green function

formalism. Lastly, chapter 6 contains both the conclusion, as well as the author’s outlook

for extensions to this work.

Chapter 2

Theoretical Background

This chapter contains the theoretical formalism necessary to understand and model ultra-

fast active control of grating assisted plasmonic coupling. First, the fundamental optical

properties of gold (Sec. 2.1) are presented, followed by a review of surface plasmon polari-

tons (Sec. 2.2), including both their characteristics as well as the methods by which they

are generated. Section 2.3 then presents a numerical method that models light-grating

interactions, showing a clear signature of plasmonic resonances for metal gratings. Lastly,

Sec. 2.4 demonstrates how irradiation of metallic surfaces by ultrashort laser pulses leads

to ultrafast modulation of the optical properties of the gold, which can then be used to

control the plasmonic coupling.

2.1 Optical properties of Gold

At its heart, the study of light-solid interactions is the investigation of the response of

a material’s charge carriers to incident photons. The electromagnetic field of a narrow

bandwidth pulse can be written as,

E (r, t) = E0 (r, t) e−iωt + c.c., (2.1)

where E0 (r, t) is the envelope function of the electric field and ω is the central frequency

of the pulse. For gold, the interaction with pulses with frequencies in the visible is

dominated by linear effects [39]. Consequently, an electric field oscillating at ω induces

a polarization density (in units of dipole moment per unit volume per unit frequency) at

the same frequency,

P (ω) = ε0χ1 (−ω; ω) : E (ω) , (2.2)

9

Chapter 2. Theoretical Background 10

1 2 3 4 5

0.0

0.5

1.0

1.5

Rea

l par

t of t

he in

dex

of re

fract

ion

Angular frequency (1015 rad/s)

(b)

2000 1000 Wavelength (nm)

500

n

0

5

10

15Im

aginary part of the index of refraction

1 2 3 4 5

-200

-150

-100

-50

0

Rea

l par

t of t

he d

iele

ctric

func

tion

Angular frequency (1015 rad/s)

(a)

i

2000 1000 Wavelength (nm)

500

r

0

5

10

15

20

25

Imaginary part of the dielectric function

Figure 2.1: The complex (a) dielectric function and (b) index of refraction of gold, as

a function of both angular frequency and wavelength. The dots are the experimentally

determined values (Ref. [40]), while the lines are guides for the eye.

where ε0 is the vacuum permittivity and χ1 is the first order electric susceptibility tensor

which quantifies the material response to the electric field.

It is often more convenient to describe the material response in terms of the relative

dielectric function, instead of the susceptibility, as it also accounts for the vacuum in

which the material is present. The relative dielectric function,

ε (ω) = 1 + χ1 (−ω; ω) , (2.3)

is in general complex, and can be decomposed into the real and imaginary components

as follows:

ε = εr + iεi, (2.4)

where the explicit frequency dependence is suppressed. However, it is sometimes more

useful to express the optical properties of the material in terms of the complex index of

refraction,

n = nr + ini =√

ε. (2.5)

The complex dielectric function and index of refraction of gold [40], for visible and near-

infrared radiation are shown in Fig. 2.1 as a function of both angular frequency and

wavelength. These are the values that will be used throughout this thesis.

The physical significance of these quantities, and in particular of the components

of n, is readily understood. The real part of the index of refraction, nr, describes the

evolution of the phase of the electromagnetic radiation as it propagates through the


medium; consequently, it determines the phase velocity of the radiation: vg = c/nr,

where c is the speed of light in a vacuum. Conversely, the imaginary component of

the index of refraction, ni, quantifies the absorption that occurs as the electromagnetic

wave propagates through the medium; thus, it is related to the extinction coefficient:

α = 4πni/λ, where λ is the vacuum wavelength of the radiation.

For gold, in the spectral region of interest to this work (Fig. 2.1) nr is generally

small, while ni is relatively large; this is determined by the band-structure. The Fermi-

energy of gold, and indeed of all metals, is located inside the conduction band. Thus,

there are always electrons in the conduction band for which intra-band transitions are

available. Consequently, these free electrons are easily accelerated by electric fields and

for interactions with light in the visible and near-infrared, this typically results in a high

absorption (large ni). Conversely, for semi-conductors, the Fermi-energy is located in

between the valence band and conduction band. Consequently, for photon energies less

than the band-gap only inter-band transitions are available and there is no absorption.

2.2 Surface plasmon polaritons

In the previous section, the optical properties of bulk gold were discussed. However, met-

als also have surfaces and these can support optical modes. One such mode is the surface

plasmon polariton (SPP), a composite entity comprised of the collective oscillations of

the free electrons near the surface of a metal which are coupled to an electromagnetic

field. A schematic of this mode is shown in Fig. 2.2: Essentially, an oscillating electric

field accelerates the free-electrons at the metal’s surface, which then generate an electric

field, which then accelerate electrons further along on the surface of the metal, and so

on. In this manner the SPP propagates along the metal-dielectric interface. Because

the electrons oscillate in the direction of propagation, the field that accelerates them

must have a component in this direction. Consequently, only a p-polarized component

of incident radiation can excite SPPs.

This section introduces the formalism that describes surface plasmon polaritons. The

wavevector and dispersion relation of the SPP is found by considering the modes of

a metal-dielectric interface that are resonantly excited. The various properties of the

plasmonic mode, such as the propagation length or the field penetration into the metal

or dielectric, are then found by ensuring that momentum is conserved as the SPP is

generated from incident radiation; this leads to a discussion of the mechanisms by which


0 y

zDielectric

Metal

+++++ +++++-----

Figure 2.2: A conceptual representation of a SPP on a metal-dielectric interface: the

oscillating free-electrons (+’s and -’s show regions of positive and negative charge, re-

spectively) near the metals surface couple to the electric field (lines).

free-space radiation may couple to a SPP. Particular attention is given to grating assisted

plasmonic coupling, which is integral to the work contained in this thesis.

2.2.1 Formalism and notation

The formalism that is used to describe surface plasmons throughout this work is developed

by considering a plane-wave that is incident on a metal-dielectric interface (Fig. 2.3).

As discussed above, only p-polarized waves can excite plasmons, and consequently the

discussion is restricted to this polarization.

When a plane wave is incident on an interface, a part of it is reflected back within

the original media, and a part is transmitted though to the underlaying substance. A

plane-wave with wavelength λ is characterized by a vacuum wavevector,

ω =2π

λ. (2.6)

When the wave travels through a different medium, with a relative dielectric constant

εj, its wavevector becomes

νj = ωnj, (2.7)

where nj can be complex, and is related to εj by Eq. (2.5). For example, in Fig. 2.3 νd is

the total wavevector of the light in the dielectric, and νm is the total wavevector of the

light in the metal.


0 y

zDielectric

Metal

nd

nm

k

wd

wm

pd-

^ pd+^

pm^

-

Figure 2.3: The situation that results when a plane-wave is incident on a planar interface.

Labels include the relevant wavevectors, as well as the polarization unit-vectors.

The total wavevector is decomposed into a component that is parallel to the surface,

κ, and a component that is normal to the surface,

wj =√

ω2εj − κ2. (2.8)

Here the square root is defined so that Im {wi} ≥ 0, and Re {wi} ≥ 0 if Im {wi} = 0. This

form of wj ensures that momentum is conserved when light, traveling in the ±z direction,

passes through the interface at z = 0. Consequently, the perpendicular components

wavevector of the incident and reflected waves, wd, are not equal to that of the transmitted

wave, wm. Conversely, there is no interface in the ±y directions, and all three waves share

the same κ.

Knowledge that the polarization of the wave is normal to the direction of propagation,

and the restriction of this discussion to p-polarized radiation, leads to the following unit-

vectors [37]:

pj± =κz ∓ wjκ

νj

, (2.9)

where pj+ corresponds to a wave traveling through media j in the upwards direction,

and pj− represents a downward propagating wave.

The interaction of the incident wave with the interface, leads to a reflected wave and

a transmitted wave, whose amplitudes and phases are determined by the solution to

Maxwell’s equations. For a planar interface, the ratios of the amplitude of the reflected


and transmitted fields to that of the incident field are known as the Fresnel coefficients,

rij =wiεj − wjεi

wiεj + wjεi

, (2.10a)

tij =2ninjwi

wiεj + wjεi

, (2.10b)

for p-polarized radiation.

With the above notation, and for a given incident field amplitude, Einc, the incident,

reflected, and transmitted fields in Fig. 2.3 are

Einc = pd−Eince−iνd·r, (2.11a)

Eref = pd+rdmEinceiνd·r, (2.11b)

Etra = pm−tdmEince−iνm·r, (2.11c)

where the frequency dependence has been suppressed for convenience, and the intensity

is given by the square of the modulus of the corresponding field.

2.2.2 Dispersion relation and properties of SPPs

Surface plasmon polaritons are modes of a metal-dielectric structure that can be res-

onantly excited by a p-polarized electromagnetic wave. As such, a signature of their

presence is a pole in the Fresnel coefficients. Setting the denominator of Eq. (2.10) to

zero and solving for κ results in an equation for the SPP wavevector,

κsp = ω

(εdεm

εd + εm

) 12

. (2.12)

This wavevector describes the propagation of the SPP along the interface and, in the

notation of the previous section, is used to represent the electric field of the plasmon as

Esp = Esp (z) eiκspy,

for a plasmon traveling in the positive y direction.

The behavior of the SPP field in the z direction is determined by conservation of

momentum, and hence the component of the SPP wavevector in this direction is given

by Eq. (2.8). As expected, this component has a different value in the dielectric than in

the metal,

wOd =

√ω2εd − κ2

O, (2.13a)

wOm =

√ω2εm − κ2

O, (2.13b)


500 1000 15000

1000

2000

10

20

30

40

Pene

tratio

n de

pth

(nm

)

Wavelength (nm)

(a) Metal

Silicon

Glass

Air

(b) Dielectric

Figure 2.4: The penetration depth of the SPP field into (a) the gold and (b) the dielectric,

as a function of the vacuum wavelength, for air (εd = 1), glass (εd = 2.40), and silicon

(εd from Ref. [41]). The kinks in the curves are due to use of the experimental values for

the dielectric function of gold which are discrete.

where κO is the real part of κsp:

κsp = κO + iγ. (2.14)

From this relation, as well as Eq. (2.12), it is evident that κO > ω√

εd(m), and hence both

wOd and wO

m are complex. Thus, the SPP mode is purely evanescent in both directions

normal to the interface, and a SPP propagating along the surface of a metal does not

experience radiation losses. Fig. 2.4 shows the penetration depths both in the gold,

δm = 1/ Im wmO, as well as in the dielectric, δd = 1/ Im wd

O, as a function of vacuum

wavelength. The penetration depth is seen to depend inversely on the dielectric function

of the overlayer, decreasing from air (εd = 1), to glass (εd = 2.40), to silicon, which has

a wavelength dependent εd [41]. It is interesting that δm ∼ 20 nm, regardless of the

wavelength or the dielectric overlayer, while δd ranges over three orders of magnitude.


500 1000 1500

1

10

100

Prop

agat

ion

leng

th (

m)

Wavelength (nm)

Glass

Air

Silicon

Figure 2.5: The propagation length of plasmons on gold-air, gold-glass, and gold-silicon

interfaces, as a function of the vacuum wavelength.

Before proceeding to other properties of SPPs, it is worth rewriting Esp,

Esp =[pO

d+EdeiwO

d zθ (z) + pOm−Eme−iwO

mzθ (−z)]eiκspy, (2.15)

where θ (z) is the Heaviside function, and Ed(m) is the field amplitude in the dielectric

(metal). This, then, is a bound mode, with exponentially decaying fields, that propagates

along the surface of the metal.

As the SPP propagates its energy is converted into heat by absorption in the metal.

This leads to a characteristic propagation length for the SPP, which is given by the

imaginary part of κsp

δsp =1

2γ, (2.16)

where the factor of 2 occurs because the energy, and not the field, of the SPP is considered.

Figure 2.5 shows δsp for the same materials as in Fig. 2.4 and again, δsp decreases as higher

dielectric function materials are used. Physically, this is understood as follows: as the

dielectric function of the overlayer increases the plasmonic field concentration in the

material decreases. However, to conserve energy, there must be a corresponding increase

of the field contained in the metal, and since most (if not all) of the absorption occurs

inside the metal, the propagation length decreases.

While the discussion above focuses on the properties of SPPs once they are excited,

there is also important information to be gleaned from the differences between the mo-


mentum of the incident light and that of the SPP, which derives from the real part of

their respective wavevectors: ωnd and κO. Figure 2.6 shows the dispersion relations, as

well as the light lines, for air, glass, and silicon. It is evident that the wavevector of the

light in the dielectric is smaller than that of the corresponding SPP. This is also evident

from Eq. (2.12). In fact, as the light-frequency increases, so too does the difference be-

tween the momenta of the light and the SPP. This statement only breaks down in the

shaded region of Fig. 2.6, where the imaginary part of the dielectric function of gold is

comparable to the real component (Fig. 2.1).

0.0 0.5 1.0 1.5 2.00

1

2

3

4

Angu

lar f

requ

ency

(1015

rad/

s)

Wavevector (107 m-1)

GlassAir

Silicon

Figure 2.6: Dispersion relations for SPP (solid) on gold-air, gold-glass, and gold-silicon

interfaces, as well as the light line (dashed) for the same dielectrics. The shaded region

shows the spectral region where the imaginary component of the dielectric function is

comparable to, or even greater, than its real component.

The momentum mismatch between the light and the SPP is a restatement of the

bound nature of these surface modes. For a SPP to decouple and propagate away from

the surface of the metal as free-space radiation, a mechanism by which it can lose some

momentum must be present. Conversely, the coupling of free-space radiation to a SPP

requires that additional momentum be added to that of the incident light. The coupling

(and decoupling) of radiation to and from SPPs is discussed in the next section.


2.2.3 Coupling of light to SPPs

Consider light with a vacuum wavevector ω propagating through a medium with index of

refraction nj, at an angle θ to the normal, and impinging on a planar metal surface. The

in-plane component of the wavevector is κI = ωnj sin θ; the corresponding wavevector of

a SPP on this interface, κsp, is greater than κI (Eq. (2.12)). Consequently, momentum

must be added to the light so that it can couple to the SPP. Schematically, this is evident

in Fig. 2.6 where, for any horizontal line, the light-line has a smaller wavevector than does

the SPP dispersion curve; the difference between the two is the momentum mismatch

that must be overcome.

There are several techniques that are employed to couple light to SPPs, the most

popular of which involve prism [7, 8] and grating [27] couplers. As this thesis deals

chiefly with grating assisted plasmonic couplers, a detailed discussion of this method

follows a brief description of the prism-based coupler.

Since a SPP propagates along a metallic interface, prism-based plasmonic couplers

are, in essence, dielectric prisms that are coated with a thin film of metal. Consequently, a

beam propagating through the prism has a wavevector component parallel to the surface

of the metal, κI = ωnj sin θ. As a result of the total internal reflection at the interface

an evanescent field is present in the metal. If the metal is sufficiently thin such that this

near field reaches the far side of the film, and if in the dielectric adjacent to the far side

κsp = κI , then plasmonic excitation occurs.

It is well known that a grating with period Λ can increase the component of the light

wavevector along the surface by an integer multiple of G, where

G =2π

Λ. (2.17)

This increase can bridge the momentum-gap between the incident beam and the SPP.

Consequently, for plasmonic coupling to occur the following equation must be satisfied,

κO = ωnj sin θ + mG, (2.18)

where κO is given by Eqs. (2.12) and (2.14), and m is an integer.

There are many forms of gratings, from surface relief gratings where the periodicity

is in the shape of the interface between the two media (Fig. 2.7(a)), to planar gratings

where the interface is flat but the properties of the material are periodically modulated

(Fig. 2.7(b)). Some of the gratings are permanent and some are transient, existing for


0 y

zDielectric

Metal

h(y)

0 y

zDielectric

Metal

L L

e e+ eD

(a) (b)

Figure 2.7: Different types of gratings: (a) A surface relief grating and (b) a planar

grating. Although a square grating profile is shown for both the surface relief as well as

the planar grating, any periodic function, h (y), can be used.

a brief moment in time [36]. Regardless, if the periodicity of the grating as well as the

material properties and the incident geometry are such that Eq. (2.18) is valid, plasmonic

coupling occurs.

2.3 Numerical model of grating diffraction: the C-

Method

Most of the numerical modeling of grating assisted plasmonic coupling contained in this

work has been done with the C-Method [32, 33]. This section contains a basic review of

this method and is based on Ref. 33, which offers a detailed discussion of this approach.

For simplicity, the analysis that follows is given for the case of p-polarized excitation.

The interaction between a plane-wave of wavelength λ and a grating with an arbitrary

surface profile h (y), as shown in Fig. 2.7(a), is described by Maxwell’s equations. These

can be reduced to the wave equation,

(∂2

∂y2+

∂2

∂z2+ ω2µε

)F = 0, (2.19)

where µ is the relative magnetic permeability of the material (here taken to equal 1),

and for p-polarized radiation F = Hx, the x-component of the magnetic field. However,

there exists no exact analytic solution across the periodic interface h (y).

This numerical model relies on two main steps: First, the fields in the regions above

the grating, where only the dielectric is present, and below the grating, where only the


substrate is present are decomposed into the Bloch modes of this system,

F (j) =∑

s

A(j)s e

i(αsy±β

(j)s z

), (2.20)

where j = d, m determines whether this is the field in the dielectric or in the metal,

respectively, and a positive z component is used in the prior case, while a negative z

component is used for the latter. Here, the Bloch wavevector components are the usual

αs = nd ω sin θ + sG,

β(j)s =

(n2

j ω2 − α2s

)1/2,

where θ is the angle of incidence. From these, it is evident that if β(j)s is real the field will

propagate away from the grating region, while a complex value represents an evanescent

field; consequently, the values of A(d)s that correspond to real β

(d)s represent the amplitudes

of the reflected fields, and similarly, the A(m)s that correspond to real β

(m)s represent the

amplitudes of the transmitted fields. As expected, a metal substrate with a complex nm

has no propagating transmitted modes.

However, the problem of matching the fields across h (y) remains, and the second

step of this approach is taken. The coordinate system is transformed to remove the

y-dependence of the surface profile:

v = y,

u = z − h (y) .

In this new coordinate system the grating surface is planar. However, the wave equation

(Eq. (2.19)) must also be transformed:

(∂2

∂v2− 2h

∂2

∂v∂u− h

∂

∂u+

(1 + h2

) ∂2

∂u2+ ω2µεj

)F = 0,

where h is the first derivative, and h is the second derivative, of the surface profile with

respect to u. This, then, is the wave equation for a semi-infinite space either above or

below the grating surface, where the properties of this region are determined by εj.

Next, a Fourier transformation of the above equation, and of the surface profile, leads

to an eigenvalue equation BΨ = 1/ρΨ (Eq. 12 in Ref. 33) for each half-space, where,

Ψ =

(F

F ′

),


and the derivative is taken with respect to u. If the qth eigenvalue ρq has a corresponding

eigenvector Fq, then Eq. (2.20) can be rewritten to include the grating interface as,

F+ = ei(α0y−β

(d)0 z

)+

∑o

ei(αoy+β

(d)o z

)A(d)

o +∑

s

eiαsy∑

q

F+sqe

iρ+q uC+

q , (2.21)

for the half-space about the interface, and

F− =∑

k

ei(αky−β

(m)k z

)A

(m)k +

∑s

eiαsy∑

r

F−sre

iρ−r uC−r , (2.22)

for the half space below the interface. Here, each term represents a type of field: In

Eq. (2.21) the first term is simply the incident field, to which the rest of the amplitudes

will later be normalized. The first sum in this equation represents the reflected fields

which are given by the Bloch modes, though the corresponding eigenvalues are real;

consequently, A(d)o are the amplitudes of the reflected fields. The last summation in

this equation, gives the reflected evanescent modes of the grating, with amplitudes C+q .

Likewise, in Eq. (2.22) the first summation represents the transmitted propagating modes,

with amplitudes A(m)k , and the second represents the transmitted evanescent modes, with

amplitudes, C−q .

To find the field amplitudes, the substitution z = u + h (y) is made in Eqs. (2.21)

and (2.22), and F+ is matched to F− at u = 0. However, if the truncation to N modes

is made, then this boundary condition results in N equations, but 2N unknowns, N for

the fields in the dielectric and N for the fields in the metal.

The other N equations are derived from the second boundary condition, which is that

the tangential component of the electric field must also be continuous across u = 0. For

a grating profile h (y), this tangential component is,

G = Ey + h (y) Ez,

where Ey and Ez are related to F = Hx through Maxwell’s equations. This, then, pro-

vides a set of 2N equations for the 2N unknowns, whose solutions are the field amplitudes.

These amplitudes are used to calculate the diffraction efficiencies,

ηro =

β(d)o

β(d)0

∣∣A(d)o

∣∣2 , (2.23a)

ηtk =

εmβ(m)k

εdβ(d)0

∣∣A(m)n

∣∣2 . (2.23b)


10 20 30

500

600

700

800

900 (d)

Cen

tral w

avel

engt

h (n

m)

Angle (degrees)500 600 700 800

0.0

0.5

1.0(c)

32O

27O

23O

19O

Zero

-ord

er re

flect

ivity

Wavelength (nm)

13O

500 550 600 650 7000.02

0.03

0.04

0.05 (b)

Firs

t-ord

er re

flect

ivity

Wavelength (nm)

p-polarized

s-polarized

500 550 600 650 700

0.4

0.6

0.8

1.0

p-polarizedZe

ro-o

rder

refle

ctiv

ity

Wavelength (nm)

s-polarized(a)

Figure 2.8: Typical C-Method reflectivity calculations for a sinusoidal gold grating with

a period of 1000 nm, and a 40 nm amplitude: The (a) zero-order and (b) negative first-

order reflectivities for both p- and s-polarizations, at an incident angle θ = 27◦. (c) The

p-polarized reflectivities for incident angles ranging from 13◦ to 32◦, showing a clear shift

of the plasmonic resonance. (d) The central wavelength of the plasmonic resonance as

a function of the angle of incidence, both from the C-Method (points) and well as from

Eq. (2.18) (line) for m = 1.

Here, o represent the diffraction order in reflection, and k the diffracted order in trans-

mission. The corresponding calculations for s-polarized light are done by making the

following substitutions: E ↔ H, ε0 ↔ −µ0, and ε ↔ −µ.

Since results from C-Method calculations are solutions to Maxwell’s equations they

model all the modes of the grating system, including SPPs. Figure 2.8 shows the results

of reflectivity calculations for a sinusoidal gold grating with a 1000 nm period and 40 nm

height are shown, using the truncation order N = 35. Indeed, the plasmonic resonance is


clearly seen in the both the ηr0 and ηr

−1 spectra for the p-polarized excitation, as compared

to the smooth s-polarized spectra [part (a) and (b)].

As expected, the plasmonic resonance shifts when the angle of incidence changes. In

Fig. 2.8(c) this shift is clearly evident in the different ηr0 curves, each of which corresponds

to a different θ, ranging from 13◦ to 32◦. In part (d) the central wavelength of the

plasmonic resonance is shown for the different incident angles (points), and is in good

agreement with theoretical predictions from Eq. (2.18) (curve). It is interesting to note

that the magnitude of the resonance decreases, and the width increases, as it shifts

to shorter wavelengths. This is attributed to the increased absorption in this region

(see Fig. 2.1) that both decreases the amount of energy that is available to the SPP,

decreasing the amplitude of the resonance, as well as shortening the propagation length

of any plasmon that is excited, leading to a broadening of the resonance.

Overall, it is clear that the C-Method reproduces the expected behavior of grating

assisted plasmonic coupling, and consequently will be used for this purpose throughout

this work.

2.4 Ultrafast modulation of the optical properties of

gold

In this section a mechanism for the ultrafast modulation of the optical properties of gold

is presented. This effect is used subsequently both to control the plasmonic coupling

on a gold grating, as well as to induce a grating in a planar gold film. In short, when

a femtosecond laser pulse impinges on a metal surface some of the light is absorbed by

the free electrons. Their excess energy then manifests as a change to the occupancy

of the electrons. The electrons rapidly thermalize due to electron-electron collisions,

which further changes the occupancy distribution. The system then slowly relaxes as

the electrons cool, losing energy to the lattice through electron-phonon collisions. This

non-equilibrium thermal dynamics of the electrons is discussed in Sec. 2.4.1.

The change in occupancy of the conduction band electrons alters the optical properties

of the gold. Due to the irradiating laser pulse, in the region below the Fermi-level

there will be more vacant states, and subsequent transitions into these states become

more likely; this is quantified by an increase in the imaginary part of the dielectric

function in this region, leading to a corresponding change to the real part of the dielectric


function through the Kramers-Kronig relation. In the region above the Fermi-energy the

irradiating laser pulse induces the opposite effect. The details of this effect are given in

Sec. 2.4.2.

The transient changes to the dielectric function of gold are used to model optically

induced shifts of plasmonic resonances on gratings throughout this work. As the dielectric

constant of gold changes, so too does the wavevector of the SPP supported by the gold-

dielectric interface (Eq. (2.12)); consequently, the wavelength at which the momentum

of the SPP matches that of the free-space radiation also shifts (Eq. (2.18)). Thus, this

process leads to ultrafast modulation of grating assisted plasmonic coupling.

2.4.1 Optically induced thermal dynamics of electrons in metal

films

When an ultrashort laser pulse shines on a metal surface the absorption of its photons

changes the energy density, ρ, and temperature of the free-electrons, Te. Because the

laser pulse leads to an elevated Te, relative to the lattice temperature, T`, the model that

describes this process and the subsequent equilibration of the temperatures is known as

the Two-Temperature Model (TTM) [34, 35, 42]. The nonequilibrium thermal dynamics

induced by the interaction of the laser pulse with the metal are modeled by the following

set of coupled differential equations:

∂ρ

∂t= − ρ

τ1

− ρ

τ2

+ P (z, t) , (2.24a)

Ce∂Te

∂t=

∂

∂z

(Ke

∂Te

∂z

)− g (Te − T`) +

ρ

τ1

, (2.24b)

C`∂T`

∂t= g (Te − T`) +

ρ

τ2

, (2.24c)

where

P (z, t) = (1−R0) αe−αzI (t) , (2.25)

is the energy density absorbed by the sample and I (t) = I0 exp[−4 (ln 2) t2/τ 2

p

]is the

temporal intensity profile of the beam. The beam is characterized by the following: I0

is the peak intensity, R0 is the coefficient of reflection, α is the extinction coefficient,

and τp is the full-width at half of the maximum (FWHM) pulse length. These three

parameters depend on the optical properties of the material as well as the wavelength

of the beam. However, α−1 is set to 100 nm to account for the ballistic nature of the


Symbol Name Value Reference

τ1 Electron thermalization time 0.5 ps [35]

τ2 Lattice thermalization time 1.0 ps [35]

Ce = γTe Electron heat capacity (γ) 62.7 J m−3 K−2 [42]

C` Lattice heat capacity 2.5× 106 J m−3 K−1 [42]

Ke Electron thermal conductivity 310 W m−1 K−1 [42]

g Electron-phonon coupling constant 2.2× 1016 W m−3 K−1 [42]

Table 2.1: The parameters for the two-temperature model, for Te < 3000 K.

excited electrons [43], as well as the absorption coefficient. The rest of the parameters of

Eqs. (2.24) are given in Tab. 2.1, and are taken to be temperature independent, except

for Ce which is linear in Te, for Te < 3000 K [42].

The equations of the TTM have clear physical significance. Equation (2.24a) de-

scribes the initial perturbation of the metal: In this model, energy is assumed to be

instantaneously transferred from the laser-pulse to the electrons of the metal through

linear absorption. Since the power profile of the pulse, P (z, t), is dependent on both

time and depth into the sample, so too will N depend on t and z. This non-thermal

energy density then decays through electron-electron and electron-phonon collisions, el-

evating the temperature of the electrons and the lattice, respectively. For gold these are

both ultrafast processes which occur in 500 fs in the case of the electrons, and 1.0 ps in

the case of the lattice [35].

The temporal dynamics of the electron temperature are encapsulated by Eq. (2.24b).

The first term on the right side describes the diffusion of the heat throughout the sample.

Because, in general, the spot size of a laser-pulse is much larger than the thickness of the

metal, the temperature gradient in z will be much larger than that in y, and consequently

only diffusion into the sample is considered. The second term describes the equilibration

between the electron and the lattice temperatures. Since the heat capacity of the electrons

is much smaller than that of the lattice, and since the electrons thermalize faster than

the lattice (τ1 < τ2), Te > T` and consequently this term will always lead to a cooling of

the electrons. Finally, there is the heating term which represent energy transferred from

the non-thermal energy density to the thermalized electrons.

Lastly, Eq. (2.24c) deals with the temporal evolution of the lattice temperature. The

terms on the right side of this equation are analogous to the final two terms of the


previous equation, and since heat flows from the electrons to the lattice, both terms lead

to heating of the lattice. Due to the characteristic time-scales in these equations the

lattice thermalizes with the electrons after several picoseconds and in general, the large

value of C` ensures that the gold experiences only a small change in temperature; indeed,

changes of hundreds of degrees to Te lead to changes of only tens of degrees to T`.

Although the cooling of the lattice is not contained in Eqs. (2.24), it can be estimated

from the speed at which phonons travel through the metal. For gold, the speed of sound

is vs = 3240 m/s. Consequently, a 100 nm film will cool in ∼ 30 ps.

To obtain the thermal dynamics of the electrons in the metal film, Eqs. (2.24) are

solved numerically. For example, when a 50 mW average power, 700 fs long laser pulse

from a 1 kHz Ti:Sapphire laser, which operates at 775 nm, is focused down to a 500 µm

(FWHM) gaussian spot size on a gold surface, it results in I0 = 24 GW cm−2 (a peak

fluence of 18 mJ cm−2); the subsequent transient thermal and non-thermal dynamics are

shown in Fig. 2.9. From (a), which is calculated for a 50 nm thick gold film, it is evident

that most of the energy that is absorbed from the laser pulse is initially contained in the

non-thermal energy density. Over ∼ 1 ps, most of this energy transfers to the thermal

electrons, resulting in a peak Te = 1515 K. The electrons then cool, with a time constant

of ∼ 5 ps, as heat is transferred to the lattice, resulting in thermal equilibrium after ∼ 10

ps at an elevated temperature of 360 K.

Figure 2.9(b) shows the transient behavior of the electron temperature for gold films

of different thicknesses, ranging from 50 nm to 200 nm. As expected the peak electron

temperature decreases for increasingly thick films, since thermal diffusion spreads the

total absorbed energy over increasingly large volumes. Since this process is nonlinear

doubling the film thickness from 50 nm to 100 nm results in a change of ∆Te = Tmaxe −300

from 1215 K to 525 K.

Both the electron temperature as well as the non-thermal energy density contribute

to a change in the distribution of the electrons in the conduction band. The thermalized

carriers are described by the Fermi-distribution

f (E, Te) =1

1 + exp (E/kBTe), (2.26)

where E is the energy of the electron state relative to the Fermi-energy (i.e. EF = 0 eV)

and kB is the Boltzmann constant. Thus, if the electron temperature is elevated from its

original temperature, T 0e , the occupancy changes as follows:

∆fTh (E) = f (E, Te)− f(E, T 0

e

). (2.27)


0 2 4 6

500

1000

1500

500

1000

1500

(b)50 nm75 nm

Thickness

Elec

tron

tem

pera

ture

(K)

Time delay (ps)

200 nm100 nm

(a)

Lattice

Tem

pera

ture

(K)

Electron

0

30

60 Energy density (J cm-3)

Figure 2.9: Typical results from TTM calculations. (a) The transient electron and lat-

tice temperatures (left axis) as well as the transient non-thermal energy density of the

electrons (right axis) for a 50 nm gold film. (b) The transient electron temperatures for

gold films with thicknesses ranging from 50 nm to 200 nm. Time t = 0 is the instant

when the peak intensity impinges on the film.

Here, ~ω is the energy of the electron state being considered.

The non-thermal energy density induces a very different change to the occupancy. In

essence, an electron with energy E that absorbs a photon with energy ~ωp will move to

a final state with energy E + ~ωp, and consequently the non-thermal component to the

change in the distribution is written as [35]

∆fNT

(E) = ∆ρ0NT

{f

(E − ~ωp, T

0e

) [1− f

(E, T 0

e

)]

−f(E, T 0

e

) [1− f

(E + ~ωp, T

0e

)]},

(2.28)

where ∆ρ0NT

is determined by ensuring that the total energy in ∆ρNT

is equal to that in

∆ρTh

for Tmaxe .


As an example, Fig. 2.10 shows both ∆ρTh

and ∆ρNT

, as functions of wavelength,

for Te = 945 K. The spectral region investigated in this work is shaded. The initial

non-thermal distribution is broadband in nature, ranging from 300 nm to beyond 1500

nm for an excitation wavelength of 775 nm. In contrast, the thermal distribution, with a

width that is dependant on kbTe, extends from 400 nm to 650 nm. However, as shown in

this figure, the magnitude of the non-thermal distribution is 50× magnified and conse-

quently in the region that concerns this work the thermal effects dominate. Thus, further

discussion will focus on thermally induced changes to the dielectric constant, though it

is worth noting that the non-thermal changes allow for the methods presented herein to

be extended to longer-wavelength regions.

200 400 600 800

-0.2

-0.1

0.0

0.1

0.2

Cha

nge

in o

ccup

ancy

Wavelength (nm)

Resonances

x 50

Figure 2.10: Change in occupancy of electronic states in gold for Te = 945 K. The initial

non-thermal (blue, magnification) and the subsequent thermal (red) distributions, as

well as the spectral region investigated in this work (shaded) are shown.

2.4.2 Thermal changes to εm

When an electron absorbs a photon of energy ~ωp it gains this energy, vacating an initial

state of energy E and filling an excited state with energy E + ~ωp. The likelihood that

such an absorption event occurs depends on the number of electrons that can absorb


the photon, as well as the availability of empty excited states for the electron to occupy.

Quantitatively, this part of the material’s optical response is given by the imaginary part

of the dielectric constant, εi (Eq. (2.4)).

Consequently, when a laser pulse induces a change to the occupancy of the electronic

states it will change the optical properties of the material; since the thermal dynamics

described in Sec. 2.4.1 lead to ultrafast transient changes to the occupancy, they result

in modulations of the optical properties on the same time-scales.

For light with a wavelength in the visible, the optical properties of gold are primar-

ily determined by the d -band to conduction-band transitions, located at the L point

of the band-structure (Fig. 2.11) [44, 45]. The separation between the d -band and the

Fermi-energy is 2.38 eV, which corresponds to a wavelength of 520 nm. This is also the

wavelength about which the change to the occupancy is centered (Fig. 2.10); consequen-

tially, the process by which a change of the electron temperature leads to a change in the

dielectric function of the material is known as fermi-smearing.

W L G

EF

ÿwF

ÿw0

mu

mu

mdmd

Figure 2.11: The band structure of gold near the L point, including the Fermi-level, EF .

The upper band is the conduction-band, and the lower is the d -band. The parameters

characterizing this region are given in Tab. 2.2

Rosei et al. [44] relate the change in the occupancy to the change of the imaginary

part of the dielectric function by integrating over all the available transitions, taking into


account the effective mass (and hence the curvature) of the bands:

∆εi (~ω) =κ′

(~ω)2

∫∆fo (E) d (E)√

~ω − E − ~ (ωF + ω0)− mu⊥m`⊥

(E + ~ωF )

, (2.29)

where κ′ = 135.85 is determined numerically by fitting the data of Scouler [46] and

ensuring that (∆εi)max = 1.5× 10−2 for a gold film heated from 120 K to 121 K.

Symbol Name Value

~ωF Conduction-band to Fermi-level energy 0.39 eV

~ω0 d -band to Fermi-level energy 1.99 eV

mu⊥ Conduction-band: effective mass from L−W 0.220m0

mu‖ Conduction-band: effective mass from L− Γ 0.251m0

m`⊥ d -band: effective mass from L−W 0.862m0

m`‖ d -band: effective mass from L− Γ 0.804m0

Table 2.2: The parameters of the band structure of gold near the L point [45]. Note that

⊥ and ‖ are relative to the [111] plane, and m0 = 9.11× 10−31 kg.

Once ∆εi (~ω) is known, ∆εr (~ω) is calculated with the Kramers-Kronig relation

∆εr (~ω) =2

πP

∫ ∞

0

ω′∆εi (~ω′)ω′2 − ω2

dω′, (2.30)

where P denotes that the Principal Cauchy Value of the integral is taken.

Examples of Fermi-smearing induced changes to the complex dielectric function are

shown in Fig. 2.12(a) for Te = 500 K, 750 K, and 1000 K. In the spectral region considered

in this work (shaded), ∆εi relative to εi is larger than ∆εr relative to εr (see Fig. 2.1

for ε). For example at 600 nm, and for an electron temperature of 1000 K, ∆εi = 0.348

while ∆εr = 0.884. However, since ε (600 nm) = −9.42 + i1.50, εi changes by 23.1%

while εr changes by only 9.4%. As is seen in Fig. 2.12(a), the changes to the dielectric

function, and hence these percentages, grow towards the shorter-wavelength part of the

region considered.

For completeness, the corresponding differential reflectivity and transmissivity for the

same temperature range are shown in Fig. 2.12(b), for p-polarized light that is incident at

25◦ on a 50 nm gold film (from air) that is on a glass substrate (n = 1.55). Note that even


400 500 600 700 800-0.6

-0.4

-0.2

0.0

0.2

-0.2

0.0

0.2

T/T 0

Wavelength (nm)

1000 K750 K500 K

R/R

0400 500 600 700 800

-2

-1

0

1

0

1

2

Cha

nge

to

r

Wavelength (nm)

1000 K750 K500 K

Cha

nge

to

i

ba

Figure 2.12: (a) The changes to both the imaginary and real parts of the dielectric

function of gold, for electron temperatures of 500 K, 750 K, and 1000 K. (b) The cor-

responding differential reflectivity and transmissivity for a 50 nm gold film with a glass

substrate and a 25◦ angle of incidence, for the same elevated electron temperatures as in

(a). The region considered in this work is shaded.

for a relatively modest electron temperature of 500 K the reflectivity and transmissivity

at 600 nm change by -0.6% and 3.2%, respectively; for an electron temperature of 1000

K these changes become -5.7% and 16.1%, respectively. These relatively large opto-

thermally induced changes to the dielectric function of gold, and hence its response, are

therefore attractive as the basis of an active plasmonic coupler; this is particularly true

since electron temperatures in excess of 10,000 K have been reported [47].

Chapter 3

Active Plasmonics on Gold Gratings

In this chapter a technique to optically control the coupling of light to surface plasmon

polaritons on gold gratings is introduced and demonstrated. As explained in Chapter 2,

irradiating a gold surface with an ultrashort laser pulse leads to an ultrafast modulation

of its optical properties due to a smearing of the electron occupancy distribution near

the Fermi-energy. If the surface under irradiation is a gold grating, which can resonantly

couple light to SPPs, the change to the optical properties leads to a change to the

plasmonic coupling conditions. Essentially, as the dielectric function of the metal changes,

so too does the dispersion relation of the SPP [Eq. (2.12)], and hence the wavelength of

light at which coupling occurs also shifts [Eq. (2.18)]. For gold, this plasmonic modulation

occurs in picosecond time-scales.

In section 3.1 all-optical ultrafast control of plasmonic resonances of gold gratings

is demonstrated. Subsequently, section 3.2 contains the results of both a power study,

and an investigation into the spectra extent to which this technique can be utilized. A

procedure that models the active control technique is developed in section 3.3, and the

results of the simulations are contrasted with those of the experiments. Lastly, section 3.4

contains a discussion of both the design considerations, as well as the limitations of this

approach to active plasmonic control.

3.1 Proof of concept

This section contains a demonstration of the viability of all-optical, ultrafast control

of grating mediated plasmonic coupling. It begins with a brief description of the sam-

ples used and continues with a detailed discussion of the experimental apparatus and

32

Chapter 3. Active Plasmonics on Gold Gratings 33

procedure. Lastly, typical results of these measurements are presented and discussed.

3.1.1 Gold gratings

The ultrafast dynamics of plasmonic resonances are investigated on two gold gratings.

The parameters of these gratings are given in Tab. 3.1. Both gratings are coated with

∼ 1.2 µm of unprotected gold, and consequently are completely opaque.

Grating Period Amplitude Profile

A 830 nm 400 nm Sawtooth: blazed at 29.9◦

B 500 nm 400 nm Sinusoidal

Table 3.1: The surface profile parameters of the gold gratings used to demonstrate ul-

trafast control of plasmonic coupling.

3.1.2 Optical sources

The ultrashort laser pulses required for this study are generated by a Coherent Mira 900

Ti:sapphire oscillator that is pumped by a 5 W Verdi laser (also from Coherent Inc.).

This laser operates with a repetition rate of 80 MHz, producing pulses that are centered

at λ = 810 nm, and have ∼ 1 nJ of energy. The pulse-length is initially measured to be

100 fs full-width at half-maximum (FWHM) long, by auto-correlator techniques.

Since the thermal mechanism that changes the optical properties of the gold gratings

is dependent on the pulse energy a Coherent RegA, which is pumped by a 10 W Verdi

laser (also from Coherent Inc.), is used to amplify the Mira pulses, resulting in a 250

kHz pulse train. The resultant pulses have 4 µJ of energy, while their pulse-length and

central wavelength are unchanged.

The broadband continuum that is used to excite the SPPs is generated by passing

∼ 30% of the laser light through a sapphire window. A typical continuum is shown

in Fig. 3.1. In general, the continuum is not uniform but has spectral features, and

in particular there is usually a strong, and unstable peak near the laser fundamental

at 810 nm. This is seen spectrum shown above, though here the peak appears to be

shifted to ∼ 750 nm as a result of a 750 nm short-pass filter that is used to attenuate

the fundamental.


500 600 700 8000.0

0.5

1.0

Inte

nsity

(a.u

.)

Wavelength (nm)

Figure 3.1: The spectrum of a typical broadband continuum generated in the sapphire.

Both the onset of the continuum, as well as which regions are spectrally flat are

controlled by changing the average power and spot size of the beam as it passes through

the sapphire. For example, Fig. 3.1 shows a continuum that is suitable for the experiments

for wavelengths ranging from ∼ 550 nm to 700 nm. Generally, it is possible to achieve a

stable continuum that spans at least 100 nm in between 450 nm, and 700 nm.

3.1.3 Tuneable wavelength pump-probe experiments

Pump-probe experiments are performed on the gratings to demonstrate active control of

plasmonic coupling. A schematic of these experiments is presented in Fig. 3.2. In general,

a pump pulse interacts with the sample, changing its optical properties. A probe pulse

then samples these changes at a fixed time delay from the pump, and is subsequently

detected. By changing the time delay between the pump and the probe and repeating

the measurement, a transient of the changes is constructed. Consequently, these results

provide information, such as the associated decay-time or strength, of the underlaying

physical mechanism by which the pump changes the optical properties of the sample.

As shown in Fig. 3.2, a portion of the beam from the laser is used to pump the sample.

The pump beam passes through a variable delay path, where by moving mirrors on a

motorized translation stage, the distance, and hence the light propagation time, between

the optical source and the grating is changed. The beam is then focused, at near normal


Ti:Sapphire

Sapphire Sample

Slit

Pump

ProbeContinuum

Delay

4f-PulseShaper

HWP

Detector

Figure 3.2: Schematic of a pump-probe setup, including a pulse-shaper that selects short

spectral ranges from a broadband continuum probe. The half-wave plate (HWP) controls

the polarization of the probe.

incidence, to a 200 µm (FWHM) spot on the grating. Consequently, the peak on-sample

intensity of the pump beam is 20 GW cm−2 (a peak fluence of 2.1 mJ cm−2).

The remaining laser light is used to excite SPPs on the gold grating, and hence to

probe the transient changes to the plasmonic coupling caused by the pump. First, the

probe pulse is passed through a sapphire window, generating a broadband continuum

(Sec. 3.1.2). A prism based 4f-pulse shaper [48] then selects a spectrally narrow probe

pulse from the continuum, as follows: First, the broadband continuum is diffracted by a

prism, and then it is collimated by a cylindrical lens that is one focal length away from

the prism. A further focal length away from the lens, a mirror reflects the diffracted light

back through the lens and into the prism, where the spectral components are recombined

into a temporally-short probe pulse. A slit is placed just in front of the mirror such that it

blocks most of the diffracted light. Consequently, the bandwidth and central wavelength

of the probe pulse are determined by the width and position of the slit, respectively

(Fig. 3.3). The resultant probe pulses have a spectral bandwidth ranging from 5 to

10 nm and a pulse length of ∼ 400 fs (FWHM) across the spectrum, as measured by

auto-correlation techniques.

The probe pulse then passes through a half-wave plate which rotates its polarization,


520 560 600 640 680

0.0

0.5

1.0

Inte

nsity

(a.u

.)

Wavelength (nm)

Figure 3.3: Spectra of different probe pulses selected from the continuum by the pulse-

shaper.

and hence allows for comparison between p-polarized light which is expected to excite

plasmons, and s-polarized light which is not. Finally, the probe pulses are focused to a

100 µm (FWHM) spot on the surface of the grating, where they are both spatially and

temporally overlapped with the pump pulses. This spot size is confirmed by knife-edge

measurements. The incident angle, θ, of the probe is controlled by rotating the grating,

and is chosen such that a Eq. (2.18) is satisfied and plasmonic coupling occurs. For

example, for θ ∼ −18◦ Eq. (2.18) holds for m = 2, and radiation at λ = 568nm excites

a plasmon which then propagates in the opposite direction in the plane of the grating.

The probe pulse reflects from the grating and is detected either by a silicon photodi-

ode (FDS100) that produced a current that is proportional to the intensity of the incident

signal, or by a fiber-coupled spectrometer (Ocean Optics, USB4000 VIS-NIR) that cap-

tures the entire visible spectrum simultaneously and with a high spectral resolution of

∼ 0.5 nm. In the latter case, the spectrometer is used with the slit of the pulse-shaper

wide-open and the pump beam blocked to capture a background spectrum of both polar-

izations and show the presence of the SPP coupling resonance. The reflectivities for two

cases, one for each grating, are shown in Fig. 3.4; in either case, the plasmonic resonance

is clearly seen as a sharp dip in the p-polarized reflectivity as compared to the s-polarized

reflectivity.

Once such a plasmon resonance is located, the slit of the pulse-shaper is narrowed to


500 550 600 650

Wavelength (nm)

R010O

Grating: BSPP

480 520 560 600 6400.0

0.5

1.0R018O

s-polarized

Zero

-ord

er re

flect

ivity

(a.u

.)

Wavelength (nm)

p-polarized

Grating: ASPP

Figure 3.4: The p- and s-polarized zero-order reflectivities for the commercial gold grat-

ings. Left: reflectivity on grating A for θ = −18◦ showing a dip near 570 nm, for a

counter-propagating SPP. Right: reflectivity on grating B for θ = 10◦ showing a dip near

600 nm, for a forward propagating SPP.

produce probe pulses with a bandwidth of ∼ 5 nm, and the reflection from the grating

is focused onto the silicon photodiode. Lock-in techniques are then used to detect the

transient changes in the reflection with high sensitivity, and by repeating these scans for

the different wavelengths that span the plasmonic resonance a picture of the temporal

dynamics of the plasmonic coupling is created.

3.1.4 Results and discussion

A. Plasmonic resonance at 570 nm: A spectral shift

Pump-probe experiments on the plasmonic resonance near 570 nm (left side of Fig. 3.4)

reveal the pump-induced temporal dynamics of the light-plasmon coupling. For example,

Fig. 3.5 shows the transient differential reflectivities, ∆R/R0, for both the p- and s-

polarized radiation at λ = 568 nm. In the case of the s-polarized radiation, ∆R/R0 < 0

which is the expected transient behavior of a gold surface, for λ = 568 nm, when its

optical properties are changed due to Fermi-smearing [35]. In contrast, for p-polarized

radiation ∆R/R0 > 0 due to the influence of the pump. As the optical response of

gold for both p- and s-polarized radiation should follow the same trend, the observed

difference is attributed to the plasmonic coupling resonance that is present only for the

p-polarized radiation.


0 1 2 3

0

2

4

6

R/R

0 (%

)

Time delay (ps)

p-polarized

s-polarized

Figure 3.5: The p- and s-polarized transient reflectivities, at λ = 568 nm, for a plasmonic

resonance at 570 nm.

The transients of both the polarizations decay with a time constant of ∼ 1 ps. This

is consistent with both theoretical simulations, as described in Sec. 2.4, as well as ex-

perimental observations [49]. This ultrafast thermalization is mainly attributed to the

diffusion of heat throughout the gold, which, in this case, is optically thick.

These transient pump-probe experiments are repeated for different wavelengths and

then joined to form ∆R/R0 spectra as shown in Fig. 3.6. Here, both s- and p-polarized

spectra are shown for different delay times: for 400 fs corresponding to the peak of the

pump-induced changes, for 1000 fs by which time the electrons are mostly thermalized,

and for a longer time delay at 4000 fs. As expected, the s-polarized spectra shows rela-

tively small changes that are spectrally smooth, consistent with theoretical expectation

for differential changes to the reflectivity of gold, due to Fermi-smearing [35].

In contrast, the p-polarized spectra have a distinct dispersive feature that is located

near the plasmonic resonance, at 570 nm. As was noted earlier, this feature peaks for

a time delay of ∼ 400 fs, after the energy transfer from the pump pulse occurs and the

electrons thermalize to a peak temperature [50]. The shape of this feature is characteristic

of a spectral shift of the plasmonic resonance, as is shown schematically in Fig. 3.7. If

a plasmon resonance is shifted by +∆λ from its original spectral location, as the red

curve is shifted from the black curve, then to the right of the intersection of the two

curves ∆R/R0 < 0, while to the left ∆R/R0 > 0, as is shown by the purple arrows.


540 570 600 630

Wavelength (nm)

p-polarized

540 570 600 630

-4

-2

0

2

4

6

R/R

0 (%

)

Wavelength (nm)

400 fs 1000 fs 4000 fs

s-polarized

Figure 3.6: The p- and s-polarized differential reflectivity spectra for different time delays,

for the plasmonic resonance at 570 nm.

-200 -100 0 100 200

1.0

0.5

R/R0 > 0

Ref

lect

ivity

(a.u

.)

Wavelength (nm)

R/R0 < 0

0.0

Figure 3.7: A schematic showing a shift of the plasmonic resonance (black curve to red

curve) and the resultant changes to the reflectivity. The resonance shifts by ∆λ, and

the resulting changes to the reflectivity are shown by purple arrows, with the regions of

positive and negative ∆R/R0 labeled on the top axis.


0 2 40.0

0.4

0.8

(nm

)

Time delay (ps)

550 570 590-6

0

6

R/R

0 (%

)

Wavelength (nm)

Figure 3.8: The transient spectral shift of the center of the plasmonic resonance. The

inset shows both ∆R/R0 (red curve with squares) as well as the best-fit to the difference

in reflectivities for a shift to the original spectra (blue curve); the time delay is 400 fs.

Consequently, the spectral shape of ∆R is the difference between the two curves, and is

the same as that of ∆R/R0 for the p-polarized radiation in Fig. 3.6.

For every time delay the initial p-polarized spectra (Fig. 3.4, left) is shifted and

the difference is fitted to the ∆R/R0 spectra to determine the shift of the plasmonic

resonance. Figure 3.8 shows the results of this procedure, which extracts a maximum

shift of ∼ 0.75 nm to the center of the plasmonic resonance, as well as a recovery time-

constant of ∼ 1 ps. The inset shows the experimentally measured ∆R/R0 spectra (red

curve with squares) for a delay time of 400 fs, along with the best fit (blue curve). As

explained above, this fit is the difference between the original plasmonic resonance, and

one that has been shifted by ∆λ = 0.75 nm.

As seen in Fig. 3.6 and the inset to Fig. 3.8, the differential reflectivity is asymmetric

about the wavelength at which ∆R/R0= 0. However, for a pure shift of the plasmonic

resonance these curves are expected to be symmetric about this axis. Consequently, this

asymmetry is attributed to a broadening of the resonance which occurs at the same time

as the spectral shift. This is expected since both the real and imaginary components of

the dielectric function of gold are changed due to the optical pumping. Since the real

part of the dielectric function of gold in large part determines κO [Eqs. (2.12) and (2.14)]


-3 0 3 6 9

2.0

2.5

3.0

3.5

(1015

rad/

s)

O (107 m-1)

c sin~

1.16 1.20 1.24

300 K

O (107 m-1)

2G

1000 K

600 K

300 K

580

570

560

(nm)

Figure 3.9: An explanation of the spectral shift of the plasmonic resonance. Main figure:

The in-plane component of the light line, ωc sin θ for θ = −18◦ is shifted by 2G. At the

point where it intersects the SPP line plasmonic coupling occurs for the unheated gold

(Te = 300 K). Inset: The point of intersection of the light-line with the SPP line changes

as the plasmonic dispersion relation shifts when Te increases.

which in turn determines the geometry for which plasmonic coupling occurs [Eq. (2.18)],

a change to εr leads to a shift of the plasmonic resonance. Likewise, since the imaginary

component of the dielectric function of metal in large part determines the imaginary part

of the SPP wavevector, γ, increasing εi reduces the propagation length [Eq. (2.16)] of the

SPP, and consequently broadens the plasmonic resonance.

The shift to the plasmonic resonance is also explained from considerations of the

dispersion relation of the SPP [Eqs. (2.12)], as is shown in Fig. 3.9. The main part

of this figure shows the momentum matching equation [Eq. (2.18)] for the unheated

gold grating. Plasmonic coupling occurs were the shifted light-line (grey) intersects the

SPP-line (blue). The inset of this figure shows the SPP dispersion relations for gold

with different electron temperatures. A peak electron temperature of Te = 600 K is

calculated using the Two-Temperature Model (Sec. 2.4.1), due to the optical pumping.

The expected spectral shift of the plasmonic resonance is determined by the intersection

of the light-line with the SPP-lines for Te = 300 K and Te = 600 K, respectively. As

shown in this figure, the intersection shifts by 0.9 nm, which is in good agreement with

the experimentally measured value of 0.75 nm considering both the experimental errors


570 600 630-1.2

-0.8

-0.4

0.0

0.4

400 fs 1000 fs 5000 fs

R/R

0 (%

)

Wavelength (nm)

Figure 3.10: The p- and s-polarized differential reflectivity spectra for different time

delays, for the plasmonic resonance at 600 nm.

(such as errors in the determination of the spot size or average power), as well as those

of the optical constants of gold in the literature.

B. Plasmonic resonance at 600 nm: A spectral broadening

Similar pump-probe measurements on the plasmonic resonance centered near 600 nm

(right-hand side, Fig. 3.4) for grating B ensure that this is not a sample-dependent

phenomenon. Figure. 3.10 shows the p-polarized ∆R/R0 spectra for this plasmonic

resonance for different time delays, by analogy to those shown for the plasmonic resonance

centered at 570 nm (Fig. 3.6). The s-polarized spectra (not shown) for the plasmonic

resonance at 600 nm is again featureless.

There are three main differences between these ∆R/R0 spectra, and those of the 570

nm resonance. First, for this resonance the magnitude of ∆R/R0 peaks at ∼ 1%, while

for the 570 nm resonance ∆R/R0 peaks at ∼ 6%. However, this in not unexpected. As

discussed in Sec. 2.4.2, the strength of the changes to the dielectric function of gold, due

to the heating of the electrons, decrease as one moves spectrally further from the d -band

transition resonance (520 nm). At 600 nm, the changes to the dielectric function are

smaller by ∼ 40% than at 570 nm, and consequently stronger pumping is required to

achieve the same peak ∆R/R0.


-200 -100 0 100 200

0.0

0.5

1.0

R/R0 > 0R/R0 < 0 R/R0 < 0

Ref

lect

ivity

(a.u

.)

Wavelength (nm)

Figure 3.11: A schematic showing a broadening of the plasmonic resonance (black curve

to red curve) and the resultant changes to the reflectivity (purple arrows). The regions

of positive and negative ∆R/R0 are labeled on the top axis.

The second difference between Fig. 3.10 and Fig. 3.6 is in the shape of the ∆R/R0

spectra. Unlike the 570 nm resonance, here the magnitude of ∆R/R0 at the center

of the resonance is smaller then at the outlaying spectral components, and sometimes

even has the opposite sign. Further, the outlaying spectral components of this ∆R/R0

spectra both have the same sign. These spectral features are signatures of a broadening

of the resonance, as is sketched in Fig. 3.11. Both the spectral asymmetry and the

fact that for certain delay times (such as 400 fs) ∆R/R0 is entirely negative show that

the resonance shifts as well as broadens, in much the same way that both effects were

present in the experiments on the 570 nm plasmonic resonance. However, unlike the 570

nm resonance, where the shift of the central wavelength dominate, here the effect of the

spectral broadening on ∆R/R0 is larger than that of the shift. Further, for the 600 nm

resonance, the relative magnitudes of these effects are closer than is the case at 570 nm.

Lastly, from Fig. 3.10, and in particular the data near 590 nm, the effect of the non-

thermal energy density (c.f. Sec. 2.4.1) is seen. For these wavelengths, the contribution to

the change to the dielectric function of gold due to the non-thermal electrons is opposite

in sign to that of the thermal electrons [35]. This is seen in Fig. 3.10 where for select

spectral components ∆R/R0 is initially negative, but then flips signs for longer delay


0 1 2 3 4

-0.4

-0.2

0.0

0.2

Thermal

R/R

0 (%

)

Time delay (ps)

Non-thermal

Figure 3.12: The p-polarized transient differential reflectivity at λ = 590 nm, showing

the effects of both the non-thermal, as well as the thermal electrons on the plasmonic

resonance.

times. Figure 3.12 shows ∆R/R0 as a function of time for λ = 590 nm. This curve clearly

shows that the change to the reflectivity, and hence to the plasmonic resonance, is affected

by two competing mechanisms that act on two different time-scales: for short time delays,

when the most of the absorbed pump energy is in a non-thermal energy distribution,

∆R/R0 < 0, while for longer time delays, after the electrons have thermalized, ∆R/R0

> 0.

Observations

Although, in this section, all-optical and ultrafast modulation of the plasmonic coupling

is demonstrated, only a small effect is observed. The plasmonic resonance shifts by

∼ 0.75 nm, corresponding to a peak ∆R/R0 of ∼ 6%. The magnitude of the observed

changes is mainly limited by two factors. First, the change to the optical properties of

the gold is dependent on the fluence of the pump pulses (Sec. 2.4.1) and consequently in

these experiments the relatively large spot size of the pump limits the modulation of the

plasmonic coupling.

Second, the amplitude of the gold gratings used is very large (400 µm) compared to

the skin depth of the light (∼ 15 nm) and the penetration depth of the SPP into the

metal (∼ 25 nm). From the perspective of the coupling mechanism, the large grating


amplitude means that higher order scattering events are no longer negligible. Likewise,

once the SPP has been excited it no longer propagates along a flat interface, but rather it

follows the grating profile. Due to the disparity in both these length scales the plasmonic

resonance broadens, and hence is less dependent on the optical parameters of the gold;

this is confirmed by numerical calculations (not shown).

3.2 Characterization

Section 3.1 shows that it is possible to achieve ultrafast control of plasmonic coupling

by optically pumping gold gratings. This section contains a characterization of this

technique, studying the changes of the plasmonic coupling as a function of both the

central wavelength of the resonance, as well as of the pump fluence.

This section begins with a description of the gratings used to couple light to the SPPs

and then continues with a discussion of the experimental setup. Lastly, the characteri-

zation of this plasmonic modulation technique is presented.

3.2.1 Dielectric gratings on gold

This plasmonic modulation technique is characterized on custom dielectric gratings that

overlay thick gold films. The gratings are prepared in the following manner. First, 200±2

nm of gold are evaporated onto a silicon (100) wafer, with a 20±1 nm intermediate layer of

chromium for adhesion. Gold of this thickness is opaque to light in the visible spectrum.

The gold is then spin-coated with 220 ± 5 nm of polymethyl methacrylate (PMMA)

into which a 820 nm period grating is exposed via e-beam lithography. The sample is

then developed, and the grating profile is confirmed with scanning-electron microscope

measurements (Fig. 3.13).

The use of the dielectric for the grating, as opposed to gold, results in a plasmonic

resonance that is more sensitive to changes of the optical properties of the underlaying

metal. In particular, the penetration depth of a SPP into the dielectric is generally on

the order of 100 nm, as compared to the penetration depth into the metal which is on

the order of 10 nm (Fig. 2.4). Consequently, the SPP propagates along the interface of

the metal and the dielectric grating, rather than along the surface of the grating, and

hence its properties are accurately described by the theory presented in Sec. 2.2.

Finally, PMMA is used because it is transparent for wavelengths in the visible and


1 �m

Distance (�m)0 1 2 3 4 5

0

100

200

Hei

ght (

nm)

Figure 3.13: Top: A SEM image of a custom dielectric grating that overlays an opticlly

thick gold film. Bottom: The surface profile of the grating; one period is highlighted.

near infrared [51]. Consequently, neither the probe nor the pump are absorbed by the

grating. In particular, this allows the maximum amount of power to reach, and be

absorbed by, the gold film. Since the changes to the dielectric function are related to the

amount of pump-power absorbed by the gold, and since the modulation of the plasmonic

coupling is dependent on the change to the dielectric function, this arrangement results

in the maximum change to the plasmonic coupling.

3.2.2 Optical sources

The optical source for this experiment is a CLARK-2001 1 kHz Ti:sapphire amplifier that

produces 250 fs (FWHM) pulses that are centered at λ = 775 nm, and have ∼ 900 µJ of

energy. To avoid nonlinear effects in the PMMA [51] the grating compressor in the laser


is detuned to produce 700 fs pulses.

The high pulse energy is used to reach higher peak electron temperatures, as com-

pared to those reached in the previous experiment (Sec. 3.1), and consequently leads to

greater changes of the dielectric function of gold. This allows for the investigation of the

limitations of this optical technique for the active control of plasmonic coupling.

A continuum is generated in a sapphire window in the same manner as in the previous

experiments (c.f. Sec. 3.1.2). Due to the higher pulse power in these experiments only

5% of the laser light generates the continuum, as opposed to the 30% used previously.

3.2.3 Broadband continuum pump-probe experiments

The schematic of the pump-probe experiments that characterize the optical control of

plasmonic coupling is shown in Fig. 3.14. The basic features of this setup are unchanged

Ti:Sapphire

Sapphire Sample

Detector

Pump

ProbeContinuum

Delay

HWP

Figure 3.14: Schematic of a pump-probe setup with a broadband continuum probe. The

half-wave plate (HWP) controls the polarization of the probe.

from those used in the earlier experiments, and consequently the reader is referred to

Sec. 3.1.3 for a detailed discussion of pump-probe techniques. Here, the new aspects of

this experiment are highlighted.

Like the experiment described in Sec. 3.1.3, in these experiments a pump pulse is

used to optically heat the electrons in the gold, changing its dielectric function and

subsequently modulation of the coupling of a probe pulse to a SPP. However, in these

experiments the pulse energy is much higher, and therefore the induced changes to the

plasmonic coupling are also expected to be much larger. Consequently, the detection


scheme is altered to one that is less sensitive, but has a greater spectral resolution.

Instead of selecting probe wavelengths before the sample, as was done earlier, here the

entire continuum is used as a probe and is detected by a fiber-coupled spectrometer

(Ocean Optics, USB4000 VIS-NIR). This arrangement allows the entire spectrum to be

simultaneously detected at every time step, with a spectral resolution of ∼ 0.5 nm.

3.2.4 Results and discussion

A. Typical results

Once a plasmonic resonance has been identified, by finding a spectral dip in the reflectivity

that is only present for the p-polarized probe and that shifts according to Eq. (2.18) when

the grating is rotated (Fig. 3.15), the influence of the pump on the coupling efficiency

is characterized. A transient of the behavior of the plasmonic coupling is constructed

500 550 600 650 7000.0

0.2

0.4

0.6

0.8 26.5o29.5o34.1o39.3o

Zero

-ord

er re

flect

ion

Wavelength (nm)

48.3o

Figure 3.15: The p-polarized zero-order reflectivities for the custom dielectric grating

overlaying a gold film. The angle of incidence at which each resonance is found is labeled

on the top axis.

by varying the delay time between the pump and the probe pulses and recording the

zero-order reflectivity for each step, as shown in Fig. 3.16. The plasmonic resonance is

seen as a large dip in the reflectivity near 600 nm.

It is easier to interpret the results of these experiments if the coupling efficiency, η,

and not the reflectivity, R, is considered. Schematically, the relation between R and η


Tim

e de

lay

(ps)

Wavelength (nm)

550 600 650−0.5

0

0.5

1

1.5

2

2.5

3

Zero−

order reflection

0

0.2

0.4

0.6

0.8

1

Figure 3.16: P-polarized zero-order reflectivity spectra as a function of the time delay

between the pump and the probe. The region of interest, containing the ultrafast mod-

ulation of the plasmonic coupling, is bounded.

is shown in Fig. 3.17; essentially, the plasmonic coupling efficiency is simply the energy

that, instead of being reflected, is coupled to the SPP. Here, η0 is the plasmonic coupling

efficiency of the sample before the pump arrives, while η (t) is the plasmonic coupling

efficiency for time delay t. Hence, η (−∞) = η0.

Figure 3.18 shows the change to the plasmonic coupling efficiency, ∆η (t) = η (t)−η0,

as a function of both the wavelength and delay time. There are two clear regions where

the plasmonic coupling is changed: On the longer-wavelength side ∆η > 0, while for

the shorter wavelength region ∆η < 0. As was shown in the discussion of the previous

experiment (Sec. 3.1.4), this spectral feature is a signature of a shift of central wavelength

of the plasmonic resonance.

To facilitate the comparison of these results with those derived in the first set of

experiments the modulation of the plasmonic coupling efficiency is defined to be

M (t) =∆η (t)

η0

. (3.1)

Further, for clarity, ∆η is given as a function of wavelength for certain time delays

[Fig. 3.19(a)] and at fixed wavelengths – along with M (t) – as a function time delay

[Fig. 3.19(b)]. The similarities between both the spectral (part a) and temporal (part b)

behavior of the current resonance is similar to that of the resonance observed in the first

experiment, shown in Fig. 3.6 and Fig. 3.5, respectively. However, here the peak value


500 550 6000.0

0.5

1.0

1.5

0Ze

ro-o

rder

refle

ctiv

ity

Wavelength (nm)

t = -

1t = t

1t

Figure 3.17: An explanation of the plasmonic coupling efficiency. Two zero-order re-

flectivity spectra, vertically shifted for convenience, represent the initial, t = −∞, and

pumped, t = t1, situations along with their respective coupling efficiencies at 550 nm, η0

and η (t1).

Tim

e de

lay

(ps)

Wavelength (nm)

570 590 610 630 650

0

0.5

1

1.5

2

2.5

Change in coupling efficiency (%

)−10

−5

0

5

10

Figure 3.18: The change to the plasmonic coupling efficiency as a function of both the

wavelength and the time delay, for the region shown in Fig. 3.16. Both t = 0.8 ps and

t = 2.0 ps when ∆η is at its peak and when ∆η has mostly recovered, respectively, are

marked by dashed lines.

of M (t) ≈ 60%, which is an order of magnitude larger than that observed in the first

experiment. This large increase in modulation efficiency is due to the higher pulse fluence


-1 0 1 2 3 4

-10

-5

0

5

10

(%)

Time delay (ps)

-60

-40

-20

0

20

40

60

599 nm

M (%

)

623 nm

560 580 600 620 640 660

-10

-5

0

5

10

15

20

25

30

0.8 ps

(%)

Wavelength (nm)

2.0 ps

-20

-16

-12

-8

-4

0

4

(b)

(a)

Figure 3.19: (a) Wavelength-dependent changes to theSPP coupling at peak (0.8 ps) and

long (2.0 ps) delay times. (b) Transient changes (and modulation – dotted line, right

axis) of the SPP coupling efficiency at 599 and 623 nm.

used in the current experiments.

However, the peak modulation of ≈ 60% occurs at λ = 623 nm, which is well away

from the center of the plasmonic resonance, at λ = 603 nm. At λ = 623 nm, η0 = 16% and

at the peak of the changes, ηmax = 26% corresponding to the quoted value of Mmax = 61%.

However, at the center of the plasmonic resonance (λ = 603 nm) η0 = 70%, while at the

time of peak modulation ηmax = 60%, corresponding to Mmax = −14%. Thus, there

exists a balance between the maximum achievable ∆η and Mmax; as one probes further

from the center of the plasmonic resonance the modulation to the coupling efficiency


grows, however the absolute coupling efficiency concurrently decreases.

Further, the sign of the modulation of any given spectral component depends on

whether it is located on the long- or shortwave side of the SPP resonance. As is shown

below, the ∆η spectra of a resonance centered above 560 nm redshifts, resulting in positive

modulations on the long-wave side and negative for shorter wavelengths. Conversely,

below 560 nm the SPP resonance blueshifts and the regions of positive and negative

modulations reverse. This is attributed to the different slope of the induced change

in the real component of the dielectric constant at different wavelengths (Fig. 2.12; at

wavelengths shorter than 560 nm it is negative, while for longer wavelengths it is positive,

and, recalling Eq. (2.18), this leads to resonances in the two spectral regions shifting in

opposite directions. Consequently, with careful consideration of the excitation geometry,

a positive or negative modulation can be induced at any wavelength. Also, as discussed

in Sec. 2.4.1, the different times and processes by which the electrons and the lattice reach

thermal equilibrium [35] have an intriguing effect. It is often possible to first modulate

the SPP coupling with one sign and then, within a few hundred femtoseconds, to reverse

the sign [Fig. 3.19(b)].

From these results and in particular from Fig. 3.19(b), the time scales associated with

this switching are extracted. The peak modulation to the coupling efficiency occurs about

800fs after temporal overlap, when most of the pump-pulse energy has been transferred

to the electrons in the metal and before it can be dissipated to the lattice or spread due

to diffusion. Given the pulse lengths, this peak time is consistent with accepted thermal

models [34, 35]. By fitting an exponential decay to these curves, taking account pump

and probe convolution effects, a time constant of 770 ± 70 fs is extracted. This time

constant is consistent with earlier observations [39, 49].

B. Fluence and wavelength dependencies

The modulation of the plasmonic coupling by optically induced Fermi-smearing of the

gold electrons is characterized by repeating the experiments described above for different

pump fluences [Fig. 3.20(a)], holding the initial resonance at 603 nm, and by repeating

these measurements for different plasmonic resonances [Fig. 3.20(b)]. The peak modula-

tion, Mmax, increases in a nonlinear fashion as a function of the pump fluence, to a peak

value of ∼ 60% for pump fluences of 60 mJ cm−2 [Fig. 3.20(a)]. This is both expected,

as the thermal dynamics of the carriers are highly nonlinear [35], as well as confirmed by


0 20 40 60-20

0

20

40

60

Mmax

(%)

Pump fluence (mJ cm-2)

(a)

540 600 660 720

-6

-4

-2

0

2

Shift

of r

eson

ance

(nm

)

Central wavelength (nm)

(b)

550 600 650 700-20

0

20

40

60

Mmax

(%)

Central wavelength (nm)

Figure 3.20: Characterization of modulation and shift of SPP resonances. (a) Peak

modulations as a function of pump fluence for a SPP resonance at 603 nm. Shown are

the peak positive (red circles) and negative (green triangles) modulations, as well as the

modulation at the center of the plasmonic resonance (blue squares). The guide lines are

exponential fits. (b) The peak shifts of the SPP resonance, as function of the original

center of the resonance. The gray curve is proportional to the slope of the induced change

to the real part of the dielectric function. The inset shows the peak positive and negative

modulations achievable for different resonances. Pump fluences are: 45 mJ cm−2 at 540

nm, 50 mJ cm−2 between 600 and 650 nm, and 55 mJ cm−2 at 700 nm.


earlier studies [39].

The sample degrades at fluences >75 mJ cm−2 due to multiphoton absorption induced

damage to the PMMA grating [51]. At this threshold, Mmax = 100% is observed, for

wavelengths between 540 and 560 nm, though the PMMA melts over the course of several

scans. This peak modulation can be recovered, and the damage to the grating can be

avoided in one of two ways: First, a metal grating, rather than a dielectric grating, can

be used. It is evident that the gold film is not damaged by the pump since, once the

PMMA has melted, the underlaying gold reflects in the normal manner. Second, the

pulse length can be stretched, since the two-photon absorption in the PMMA depends

on the peak intensity, while the change to the dielectric function of gold depends on the

pulse fluence.

The modulation of SPP coupling can be achieved across a broad range of wavelengths;

this is shown by studying the shifts of plasmonic resonances that are excited at different

wavelengths while the pump fluence is held constant at ∼50 mJ cm−2 [Fig. 3.20(b)]. As

expected [Eqs. 2.12 and 2.18], the peak spectral shifts to the center of the plasmonic

resonances follow the relative changes to the real part of the dielectric constant. More

interesting, we observe high modulations, in excess of 20%, for all visible wavelengths

above 540 nm and only slightly smaller changes for wavelengths as long as 700 nm. As

noted earlier, the peak modulation of the plasmonic coupling depends nonlinearly on the

pump fluence, suggesting that modulations in excess of 50% are possible for this entire

region, if the pump fluence is increased.

3.3 Simulations

The previous sections of this chapter detail experimental investigations demonstrating

an all-optical technique to modulate the plasmonic coupling on gold films with grating

overlayers. In this section, a procedure that fully models this technique is developed. An

example of these simulation is presented, and is compared with the experimental results.

Modeling the transient spectral shifts of the plasmonic resonances, and the changes

to the plasmonic coupling efficiencies, is achieved with a combination of the models and

numerical techniques presented in chapter 2, as follows:

1. For a given set of pump-pulse parameters, such as the fluence and central wave-

length of the pulse, as well as the corresponding material properties, such as the


dielectric function data and film thickness of the gold, the transient changes to the

electron temperature is calculated using the Two-Temperature Model (Sec. 2.4.1).

2. Once the time-dependent change to the electron temperature is known, Rosei’s

model [44] is used to calculate the transient changes to the imaginary part of

the dielectric function of the gold (Sec. 2.4.2). With this information, and using

the Kramers-Kronig relation, the time-dependent changes to the real part of the

dielectric function is then calculated.

3. Given the grating and incident radiation parameters, the reflectivities are calcu-

lated using the numerical C-Method outlined in Sec. 2.3. These reflectivities are

recalculated at each delay time, using the corresponding altered complex dielectric

function of the gold.

4. The transient plasmonic coupling efficiency spectra are then extracted from the

reflectivity data following the procedure developed in Sec. 3.2.

As an example, this procedure is used to model the ultrafast plasmonic coupling

dynamics for a sinusoidal gold grating with a period, Λ = 820 nm and an amplitude,

h = 80 nm. Since the underlaying physical mechanism of the plasmonic coupling is

independent of the shape of the grating, it is expected that these simulations will closely

match the experimental results.

The angle of incidence is fixed at θ = 19◦, and is expected to allow the incident radi-

ation to couple to a SPP at λ = 603 nm, according to Eq. (2.18). Indeed, the resonance

(not shown) is found at this wavelength, though its spectral bandwidth (∼ 25 nm) is nar-

rower than that of the similar plasmonic resonance (∼ 35 nm) that was observed in the

experiments (Fig. 3.16). The difference in the bandwidths of the plasmonic resonances

is mainly attributed to the following: The simulations implicitly assume a continuous

plane wave excitation, whereas in the experiments tightly focused gaussian pulses ex-

cite the plasmons. Consequently, the limited spatial extent of the gaussian pulse can be

thought of as introducing additional Fourier components to the grating profile, leading

to a spectral broadening. Further, the gaussian pulse can be thought of as being com-

posed of a superposition of plane waves, each with a different phase and incident angle.

Consequently, the distribution of the incident energy over a range of angles leads to a

broadening of the plasmonic resonance that is not reproduced by the simulations.


The procedure outlined above is followed for a 700 fs (FWHM) pump pulse, with

a central wavelength λ = 775 nm and a pulse fluence of 80 mJ cm−2. This fluence

corresponds to the largest experimental fluence that did not damage the gold film, and

results in a peak electron temperature of 950 K. Figure 3.21 shows the simulated transient

∆η spectra, for comparison with the experimental data shown in Fig. 3.18.

Wavelength (nm)

Tim

e D

elay

(ps

)

580 590 600 610 620 630−1

0

1

2

3

4

5

∆η (%

)

−30

−20

−10

0

10

20

30

Figure 3.21: The simulated change in plasmonic coupling efficiency as a function of both

wavelength and time delay.

To facilitate the interpretations of these results, various cross-sections of the simula-

tions are presented in Fig. 3.22. The effects of the two different pump-induced changes

to the optical properties are most clearly evident in the change in the spectral position

of the SPP resonance as a function of the delay time between the pump and probe pulses

[Fig. 3.22(a)]. The initial nonthermal effects pull the resonance toward shorter wave-

lengths until the thermal effects can compensate and, eventually, dominate and shift the

resonance toward the longer wavelengths.

From transient changes to the coupling efficiency [Fig. 3.22(b)], it is evident that the

time scales in the simulations match those in the experiments: the peak changes occur 0.8

ps after overlap, and the system recovers in 1.4± 0.1 ps. The marginally longer recovery

time is due to the simplicity of the model, which assumes a perfect sample and limits

the electronic energy loss to heat transfer with the lattice. It is interesting that, for all

delay times, the modulation to the coupling efficiency is negative, in contrast with the

shift of the resonance center [Fig. 3.22(a)]. This hints at the complexity of the dynamics

of the system: even as the resonance shifts, it also broadens, so that when it is passes


580 600 620-40

-20

0

20

(%)

Wavelength (nm)

0

20

40

60

80

54%

-24%

-44%

(%

)-37%

-1 0 1 2 3 4 5

604

606

608

610

Cen

ter o

f SPP

reso

nanc

e (n

m)

Delay Time (ps)

-1 0 1 2 3 4 5

-30

-20

-10

0

(%)

Time delay (ps)

-40

-20

0

M (%

)

(c)

(b)

(a)

Figure 3.22: Results extracted from the full time- and wavelength-dependent plasmonic

coupling efficiency simulations for a pump fluence of 80 mJ cm−2. (a) The location of

the center of the SPP resonance as a function of the time delay between the pump and

probe pulses. (b) The time-dependent changes, ∆η (left axis), and modulation, M , (right

axis) of the coupling efficiency at the center of the SPP resonance (603 nm). (c) The

initial, η0 (right axis), and change, ∆η (left axis), of the coupling efficiency as a function

of wavelength at a delay time of 0.8 ps. At selected wavelengths, the modulation, M , is

shown (circles).


through its original position the coupling efficiency is lower at the center, but higher for

some outlying regions.

Figure 3.22(c) shows the ∆η spectra (red curve) of the changes to the coupling effi-

ciency at a delay time of 0.8 ps, corresponding to the greatest pump-induced changes at

the center of the resonance. To facilitate the understanding of the significance of these

results, the spectra of the plasmonic coupling efficiency for the unheated grating, η0, is

included. This figure shows good agreement with the experimental results [Fig. 3.19(a)],

in that the short wavelength side of the resonance experiences a negative change, while

to the long-wave side additional light is coupled to the SPP due to pumping. However,

the simulations predict that the changes occur in a narrower range than experimentally

observed; this is expected, since the simulations also predict a narrower light-plasmon

coupling resonance than is observed experimentally.

Furthermore, unlike the experimental changes, the magnitude of the negative changes

predicted by the simulations is larger than that of the positive changes. As discussed

above, this difference is due to the assumptions inherent in the numerical modeling

method, the result of which is that the shift of the resonance dominates over the broad-

ening. This nicely demonstrates the advantages of a narrower resonance and shows that,

for the practical applications of this method, the gratings used should be optimized to

provide the narrowest possible SPP resonance.

Another consequence of this feature is that while the changes at the center of the

resonance are greatly increased, those at the outlying regions are only slightly decreased.

As a reference, we provide the corresponding modulations at several points along the

∆η spectra [Fig. 3.22(c), circles]. At the center of the resonance at 603 nm, where

initially 77% of the light is coupled to the SPP, there is a peak-induced change of −34%,

corresponding to a coupling efficiency of 43% or a peak modulation of −44%. At the same

time, near 618 nm, there is a peak change to the coupling efficiency of 11%, corresponding

to a modulation of 54%. As a comparison, recall the peak experimental modulations of

−14% and 61% at the center and outlaying regions, respectively.

There are two more reasons that help account for the increase in the coupling effi-

ciency at the center of the resonance. (1) The simulations employ a pump fluence of

∼ 80 mJ cm−2 while the peak experimental values are for a fluence of ∼ 60 mJ cm−2, as

a pure gold grating has a higher damage threshold than does one made out of PMMA.

(2) An assumption inherent to the calculations of the changes to the optical properties of

the gold (c.f. Sec. 2.4.1 and 2.4.2) is that the entire probe spot is pumped by the fluence


given. Experimentally, however, the probe size is ∼ 140 µm (FWHM), and though this is

much smaller than the spot size of the pump (∼ 570 µm, FWHM) this assumption still

introduces some error.

0 20 40 60 80

-40

-20

0

20

40

Mmax

(%)

Pump Fluence (mJ cm-2)

Figure 3.23: Power study of the simulations. the peak achievable positive (circles) and

negative (triangles) modulations, as well as the greatest modulation at the center of the

SPP resonance (squares), are shown. The curves are guides for the eyes.

As both the relationship of the pump fluence to the change to dielectric function of the

gold, as well as the value of the dielectric function to the response of the grating are highly

nonlinear, they cannot be accounted for before the simulations. Consequently, ∆η and M

are calculated for different pump fluences, and the results are summarized in Fig. 3.23 for

comparison with the experimental power study [Fig. 3.20(a)]. By interpolating between

the points, an expected peak modulation of −31% is calculated, at the center of the

resonance at a pump fluence of 60 mJ cm−2. This value is corrected to account for the

temporal gaussian profile of the experimental pulses by taking a weighted average of

M for the different pump fluences probed and arrive at a peak modulation of −28%, a

factor of 2 greater than experimentally observed. For a pump fluence of 80 mJ cm−2,

the maximum fluence, the corrected modulation at the center of the resonance is −41%;

as discussed above, this increase in M is mainly due to the narrow spectral width of the

SPP resonance in the simulations, as compared to the width observed in the experiments.


3.4 Design considerations and limitations

In this chapter ultrafast control of grating-assisted light-plasmon coupling on gold films

is both demonstrated and characterized. Essentially, energy transfers from a pump pulse

to the free electrons in the gold, leading to an elevation of the electron temperature. This

change in electron temperature manifests as a change to the dielectric function of the

gold, and since the plasmonic coupling is dependent on this parameter, it also results in

a modulation of the coupling efficiency. Since both the heating and the cooling of the

electrons occur on ultrashort time-scales, so too does the plasmonic modulation.

However, there are some limitations to this approach to active plasmonic control that,

in general, hint at an underlaying balance in the parameters of the grating coupler. First,

the modulation of the light-plasmon coupling depends on the peak achievable electron

temperature. Here, the maximum electron temperature is 950 K, and this value mainly

depends on two parameters: the fluence of the pump pulses, and the thickness of the

gold film. In these experiments, the fluence dependence is explored and it is seen that

∼ 80 mJ cm−2 damage the gold film. Further, in these experiments the gold films either

200 nm or 1.2 µm thick. Reducing the thickness will limit the volume that the heat

diffuses into, and consequently increase the peak electron temperature. For example,

Two-Temperature Model [35] based calculations show that reducing the film thickness

from 200 nm to 50 nm doubles the peak electron temperature, while employing a 20 nm

film results in a peak electron temperature that is more than 7 times larger than that

of the 200 nm film. However, as the film thickness approaches the penetration depth

of the SPP (∼ 25 nm), the coupling between the SPP modes on the two metal surfaces

increases and the properties of the resultant modes are different from those of SPP on

thick films [52]. While the presence of these modes is not necessarily undesirable, they

must be accounted for in any calculations.

These experiments also investigate the wavelength dependence of this approach, and

it is in the spectral distance from the d -band resonance (∼ 520 nm) to the center of

the plasmonic resonance that a delicate balance is struck. The changes to the dielectric

function of gold due to the Fermi-smearing of the conduction electrons increase as this

separation decreases (Fig. 2.12), and hence one expects that so too will the modulation to

the plasmonic coupling. However, near a wavelength of 520 nm the resonant coupling to

the SPP competes with the d -band absorption resonance, and less light is coupled to the

SPP; consequently, the plasmonic resonance both widens and decreases in amplitude.


This is most easily seen in Fig. 3.15, where the plasmonic resonance near 550 nm is

noticeably shallower and broader than the resonances centered at longer wavelengths.

Thus, the excitation wavelength of the SPP must be chosen with care.

Chapter 4

Active Plasmonics on Planar Gold

Films

In this chapter the focus shifts from the excitation of surface plasmon polaritons with

periodically structured gold surfaces, to the excitation of SPPs on unstructured gold films.

While planar gold surfaces are known to support plasmonic modes, they traditionally

require an extra structure, such as a prism or grating, to overcome the momentum-

mismatch between the light and the SPP; hence, unstructured metal surfaces typically

cannot couple light to SPPs.

Here, a new, all-optical technique to couple free-space radiation to SPPs on planar

gold films is presented. Essentially, two pump-beams interfere on the surface of the

gold film, and the subsequent absorption of light with a periodic intensity leads to a

grating of the free electrons’ temperature. As discussed in chapter 2, the elevated electron

temperature manifests itself as a change to the dielectric function of the gold, and since

the electrons cool on an ultrafast time-scale, this pumping ultimately results in a transient

absorption and refraction grating in the gold film. This grating is then used to couple

light to a SPP.

This chapter begins with a discussion of the experiments that demonstrate plasmonic

coupling on periodically irradiated planar films. First, the experimental set-up, including

a description of the gold film, is provided in Sec. 4.1; here, the focus is on the experimental

conditions necessary to generate a transient grating in the gold. Section 4.2 contains a

demonstration of the optically-induced transient grating in the gold through observations

of the diffraction of the transmitted light. Pump-probe type experiments show that the

grating decays in ∼ 1 ps. In Sec. 4.3 the focus shifts to the identification of plasmonic

62

Chapter 4. Active Plasmonics on Planar Gold Films 63

coupling; this is accomplished by identifying a polarization-dependent feature, in the

diffracted transmissivity, that shifts when the angle of incidence changes, as predicted by

Eq. (2.18). Lastly, in section 4.4 the limitations and possible extensions of this approach

to plasmonic coupling on planar films are discussed.

4.1 Experimental procedure

This section begins with a brief description of the planar gold film used in the experiment.

It then continues with a detailed discussion of the experimental setup that was used to

detect the SPPs that couple to the gold film.

4.1.1 Planar gold film

For these experiments, a film of 99.9% pure gold is evaporated at a rate of 1 nm/s onto

a microscope slide, to a thickness of 50± 2 nm. This thickness is chosen for two reasons:

(1) 50 nm of gold is sufficiently thin that the effects of heat diffusion (c.f. Sec. 2.4.1)

are limited, and electron temperature changes of hundreds of degrees can be achieved

with pump fluences of the order of 10 mJ cm−2. (2) Conversely, 50 nm of gold are thick

enough to effectively decouple the two surfaces; that is, as discussed in Sec. 2.2, the

SPP penetration depth into the metal is ∼ 15 nm and consequently there is very little

interaction between the plasmonic modes of the two surfaces [52]. In fact, for a 50 nm

gold film, for λ = 560 nm, the real component of the SPP wavevector (which determines

the coupling conditions) is changed by less than 1% from that of an SPP mode on a bulk

gold interface.

4.1.2 Two pump - continuum probe experiments

To show plasmonic coupling on the planar gold films, pump-probe experiments are con-

ducted with two pump beams and a broadband continuum probe, as shown in Fig 4.1.

Because both the laser and the broadband continuum that are used in these experiments

are the same as those that were used in the initial investigation of gold gratings, their

details are provided in Sec. 3.1.2 and will not be repeated here save to remind the reader

that the pump pulses are 150 fs (FWHM) long and centered at 810 nm; the continuum

is stable between 450 and 750 nm.


Ti:Sapphire

Sapphire

Sample Detector

Pump

ProbeContinuum

Delay

HWP

T(0)

T(-1)

Figure 4.1: Schematic of a pump-probe setup, including two pump beams and a broad-

band continuum probe. The half-wave plate (HWP) controls the polarization of the

probe, of which the first order diffraction signal in transmission is detected.

As Fig. 4.1 shows, and unlike in the previous pump-probe experiments, two pump

beams are used. Figure 4.2 shows a close-up of the sample and both the incident, as well

as the transmitted beams; the pump beams are focused to a spot size of 200 µm and

are both spatially and temporally overlapped on the surface of the gold. The first pump

beam makes an angle of θ1 ∼ −2◦ with the normal of the gold and has an on-sample peak

intensity of I1 = 7.5 GW cm−2, while the second pump beam makes an angle of θ2 ∼ 43◦

and has a peak on-sample intensity of I2 = 5.3 GW cm−2. The interference of the two

co-polarized pump beams results in an intensity profile on the surface of the sample

I (y, t) = I1 (t) + I2 (t) + 2√

I1 (t) I2 (t) cos

(2π

Λy

), (4.1)

where I1(2) (t) is the time-dependent intensity of the first (second) pump beam, y is the

displacement along the surface of the film, and Λ, the period of the modulation, is

Λ =λ

− sin θ1 + sin θ2

. (4.2)

For the angles given above, the period of the intensity pattern is ∼ 1200 nm. As is

shown in the following section, the time-dependent, periodic intensity profile ultimately

results in a transient planar absorption/refraction grating (c.f. Fig. 2.7). The pump

beams are chopped, allowing the use of a lock-in amplifier (Stanford Research SR830),

and increasing the sensitivity of the detection system.


A broadband continuum probe passes through a half-wave plate for polarization con-

trol, and is then overlapped with the pump beams on the gold sample such that it

impinges on the transient grating. Consequently, a portion of the energy of a p-polarized

probe pulse that fulfills the phase-matching conditions [Eq. (2.18)] couples to a SPP,

shown in Fig. 4.2(b).

Most of the energy of the probe beam is transmitted through the gold film, and is

labeled T (0) in both Fig. 4.1 and Fig. 4.2(a). This transmitted beam is used to facilitate

the overlap between the pump and the probe beams, but is not used to identify the

plasmonic coupling due to the thermal modulation of the transmission as was described

in Sec. 2.4.2. In fact, in the spectral region in which plasmonic coupling is expected,

the elevated electron temperature decreases the transmission by several percent [35].

Consequently, this effect masks the plasmonic coupling, which is difficult to extract from

T (0).

qq1

2q

q-1

pump 1

pump 2probe

T(-1)

T(0)

kI G

kO

kI

G

Gk-1

a b

c

Figure 4.2: Schematic of the sample and incident geometry of the beams. (a) The gold

film and incident pump and probe beams, as well as the zero- and first-order diffracted

beams in transmission. (b) The phase-matching diagram for the plasmonic coupling. (c)

The phase-matching diagram for the first-order diffracted beam.

However, a small portion – no greater than 10−3 – of the transmitted probe light

is diffracted into the first (-1) order; this light is labeled T (−1) in both Fig. 4.1 and

Fig. 4.2(a). The angle at which this beam diffracts, θ−1, is calculated from phase-


matching conditions as is shown in Fig. 4.2(c); this angle is

sin θ−1 = sin θ − λ

Λ. (4.3)

Consequently, the diffracted beam is imaged onto the entrance slit of a monochromator

(Triax-320) and detected at the exit slit with a photomultiplier tube, with a spectral

resolution of 2 nm. The resultant p- and s-polarized transmissivity spectra are then

compared to isolate the energy coupled to the SPP.

By varying the time delay between the probe and the pump pulses and taking spec-

tra for both probe polarizations, the dynamics of this plasmonic coupling process are

explored.

Finally, by rotating the sample the angles of the two pump beams, as well as of

the probe beam, are changed. The sample is rotated over a range of 9◦, resulting in

grating periods ranging from 1200 nm to 1350 nm according to Eq. (4.2), and angles

of incidence for the probe between 36◦ and 45◦. From the phase-matching conditions,

given by Eq. (2.18), these conditions are expected to yield plasmonic coupling resonances

between 520 nm and 575 nm.

4.2 Optically induced transient thermal gratings

Through linear absorption and due to the ultrashort nature of the pump pulses, the

interference of the two pump beams leads to a transient periodic modulation of the

electron temperature in the gold (c.f. Sec. 2.4.1); as was shown in Sec. 2.4.2, an elevation

of the electron temperature leads to a change in the complex index of refraction of the

gold and hence, in the case of the interfering pump beams, results in a transient grating

in the gold.

To confirm the presence of a transient grating, the intensity at the angle of the first-

order diffraction signal in transmission, θ−1 – as predicted by Eq. (4.3) and shown in

Fig. 4.1 – is detected for the following cases: (1) When only pump beam 1 is incident on

the gold; (2) when only pump beam 2 is incident on the gold; and (3) when both pump

beams are incident on the gold. These are shown in Fig. 4.3. When only one of the pump

beams is overlapped with the probe beam on the sample, only a small, negative signal is

detected for T (−1). Since this is a lock-in detection scheme, this signal is interpreted as

light scattered from the T (0) beam that arrives where one expects to measure T (−1); due


0 2 4 6

0

2

4

6

8

10

both pumpsT (-

1) (a

.u.)

Time delay (ps)

pump 2pump 1

Figure 4.3: The pump-probe signal for the first-order diffracted signal in transmission

with: only the first pump, only the second pump, or both pumps incident on the sample.

to the thermal effects, for the spectra region considered in this experiment, the magnitude

of the transmission is reduced [35].

Conversely, when both pump beams are overlapped with the probe beam, a large,

positive signal is observed. This corresponds to the presence of the diffracted order,

which is only present for the duration of the transient grating. The signal peaks at a

time delay of ∼ 0.5 ps, once the electrons have thermalized and corresponding to the

peak temperature contrast in the grating, and decays with a time constant of 1.6 ± 0.1

ps.

Both the heating and cooling times observed experimentally agree with theoretical

prediction. Figure 4.4 shows the electron temperature difference, ∆Te, between the hot

and cold regions of the optically induced thermal grating (shown in the inset) as a function

of time delay between probe and pump pulses. This calculation uses the two-temperature

model, as introduced in Sec. 2.4.1, and the experimental pump parameters (Sec. 4.1.2).

As is evident by a direct comparison between Fig. 4.4 and Fig. 4.3, the results of the

calculation and the experiments are in good agreement.

The periodic temperature profile leads to a transient absorption and refraction grating

in the gold, and the resulting T (−1) is calculated according to [36]

T (−1) =

(πL

λ

)2 [(∆n)2 + (∆κ)2] e−αL (4.4)


0 2 4 60

100

200

300

400

T e (K

)

Time delay (ps)

50 n

m

710 K

1200 nm

340 K

Figure 4.4: The transient electron temperature difference in the gold film. The inset

shows the depth profile of the thermal grating on the sample at the time of peak changes,

∼ 550 fs after overlap.

where L is the sample thickness. The spectrum of T (−1) is measured by repeating the

pump-probe measurements for different wavelengths. Typical theoretical and experi-

mental results are summarized in Fig. 4.5, showing T (−1) as a function of both the probe

wavelength and the pump-probe time delay. Again, the experimental results are in good

agreement with the theoretical calculations.

As expected, T (−1) is strongly wavelength dependent, increasing rapidly for shorter

wavelengths; this follows the same trend as the change to the optical properties of gold

due to Fermi-smearing (c.f. Fig. 2.12), which increase as one approaches the d -band

resonance at 520 nm. The absolute diffraction efficiency, measured at 520 nm where the

diffraction is the strongest, is ∼ 5×10−4, becoming ∼ 1×10−4 at 540 nm and ∼ 1×10−5

at 570 nm; the absolute efficiencies predicted by Eq. (4.4) are 6.8 × 10−4, 2.4 × 10−4,

and 4.7× 10−5 at the same wavelengths, in very good agreement with the experimental

results. For both the experimental results, as well as the theoretical predictions, the

grating decays with a time constant of ∼ 1 ps, due to thermal diffusion and the loss of

the electrons’ energy to the gold lattice through electron-phonon scattering [35]; this,

then, is the ultrashort window during which SPPs can be launched. Furthermore, this

ultrafast mechanism helps prevent re-radiation of the SPP, as both the coupling and

decoupling are dependent on the presence of a grating.


540 550 560 570

0

1

2

3

T(-1) (a.u.)

Wavelength (nm)

Tim

e de

lay

(ps)

0

0.5

1.0

540 550 560 570

0

1

2

3

T(-1)p

(a.u.)

Wavelength (nm)

Tim

e de

lay

(ps)

0

0.5

1.0

(a)

(b)

Figure 4.5: Transient T (−1) spectra from both (a) theoretical calculations, and (b) exper-

imental measurements. Note that only the p-polarized data is shown, as the s-polarized

data is similar when viewed in this manner.

4.3 Plasmonic coupling

To extract information about the plasmonic coupling the transient diffraction spectra for

p- and s-polarized probe pulses, T(−1)p and T

(−1)s respectively, are compared; recall that

this is done as only the p-polarized probe is expected to couple to SPPs. Figure 4.6 shows

T(−1)p and T

(−1)s as a function of wavelength, at the delay time corresponding to the peak

electron temperature contrast in the gold. Near the center of the plot, at ∼555 nm, the

difference between T(−1)p and T

(−1)s is large since energy that would otherwise have been

transmitted from the p-polarized probe is coupled to a SPP mode. This is more evident


540 550 560 570

0.0

0.5

1.0

p-polarized

T (-1)

(a.u

.)

Wavelength (nm)

s-polarized

545 555 5650

1

2

-0.08

-0.01

0.06

T (-1) (a.u.)

Wavelength (nm)

Tim

e de

lay

(ps)

Figure 4.6: The T (−1) spectra for both probe polarizations at the time of peak grating

contrast. The inset shows the difference in T (−1) between the two probe polarizations as

a function of both time delay and wavelength.

in the inset, which shows the transient behavior of ∆T (−1) = T(−1)p − T

(−1)s as a function

of wavelength.

In order to verify the dynamics of the plasmonic coupling, ∆T (−1) curves for individual

wavelengths are plotted as a function of time delay, as shown in Fig. 4.7 for λ = 552

nm. Exponentially decaying curves are fitted to this data and a decay time constant

of 620 ± 100 fs is extracted. That this decay time is faster than that observed for the

thermal effects is not surprising, since heat diffusion washes away the grating and, since

the coupling of light to SPP is not linear in the magnitude of the grating. This figure

contains two interesting features: First, note that initially ∆T (−1) > 0. This feature is

present for all wavelengths, and as such is not attributed to a plasmonic effect. Rather,

this is likely due to the non-thermal energy density (c.f. Sec. 2.4.1) whose effects dominate

over those of the elevated electron temperature, at early delay times, and which change

the specular transmission in opposite ways at 550 nm [35]. Second, for long time delays

∆T (−1) does not recover completely. Again, this is attributed to energy scattered from

the specular transmission which is detected where the diffracted beam is expected. Since,

for these delay times, the thermal effects dominate, the residual ∆T (−1) < 0.

At the temporal peak of ∆T (−1), 8± 1% of the light that would have been diffracted

into the (-1) order is coupled to a SPP. Taken together with the absolute diffraction


0 1 2 3

-0.06

-0.04

-0.02

0.00

0.02

T(-1) (a

.u.)

Time delay (ps)

Figure 4.7: ∆T (−1) as a function of time delay at 552 nm The data is shown as a thick

grey line, while the thinner purple line is an exponential decay curve.

efficiencies given above, this yields a plasmonic coupling efficiency of 10−5 which, given

the relative mode confinements, results in a ratio of peak plasmonic to incident radiation

intensities of 10−3. However, given that most of the light is contained in the zero-order

transmission and reflection, it is reasonable to assume that the total coupling efficiency

is, in fact, much greater.

To ensure that the energy missing from the ∆T (−1) couples to the SPP, this transient

differential spectra is retaken for different incident geometries, as described in the exper-

imental section (Sec. 4.1). The different ∆T (−1) spectra, at the time of peak temperature

contrast, are shown in Fig. 4.8; here, both the plasmonic resonances themselves, as well

as their spectral shift due to the change of the incident angles, are evident. As expected,

the plasmonic feature is a sharp dip and not a slowly varying change as a function of the

wavelength.

The inset of Fig. 4.8 shows the spectral shifts of the plasmonic coupling resonance

both as determined experimentally (dots), and as calculated from the phase-matching

conditions given in Eq. (2.18) (line); The central three points correspond to the resonances

shown in the main part of this figure. The experiment and theory are in good agreement.

Spectrally tuning the plasmonic resonance also reveals the limitations of the optically

induced transient free-electron temperature grating coupling. While the three resonances

near 550 nm have dips that are in the same order of magnitude, the dips of the other


540 550 560 570

-1.0

-0.5

0.0

T(-1)

p-T

(-1)

s (a

.u.)

Wavelength (nm)

520 550 580

-7

0

7

(deg

rees

)

(nm)

Figure 4.8: The ∆T (−1) spectra at the time of peak grating diffraction for different

excitation geometries show different plasmonic resonances. The data is shown as discrete

points while the curves are guides to the eye. The inset shows the measured spectral

displacement of the resonances versus the change in incident angle (points), as well as

the theoretically expected behavior (line).

two resonances shown in the inset of Fig. 4.8 are much smaller. As we expect, the longer

wavelength resonance near 570 nm is smaller in amplitude than those near 550 nm by a

factor of ∼40; this is due to the large spectral separation from the d -band resonance at

520 nm and the consequent small changes to the density of states at these wavelengths,

and hence to a small induced ∆n and ∆κ. More surprising, the resonance near 520 nm

is also ∼10 times smaller in amplitude than those near 550 nm even though it occurs

near the peak of the induced changes to the optical properties of the gold; this occurs

because of the proximity of the d -band resonance: another absorption resonance, other

than plasmonic coupling, exists and consequently energy that would otherwise have been

coupled to a plasmon is lost to this pathway.

4.4 Improvements and limitations

In this chapter it was shown that, through the interference of two ultrafast laser pulses,

a transient grating can be induced in a planar gold film, creating an ultrashort launch

window for SPP excitation; this, truly, is an ultrafast technique: the magnitude of the


SPP launched decays with a time constant of 1.0 ps, relative to the temporal overlap with

the pump beams. However, there also exists a spectral window in which this approach

is effective. One must be sufficiently far from the d -band resonance at 520 nm to avoid

energy loss to this absorption pathway, yet not too far removed to ensure that the changes

to the dielectric function of the gold are sufficiently large to create a high contrast grating.

Near 550 nm, ∼ 8% of the energy from the T(−1)p diffracted beam is seen to couple

to a SPP, corresponding to a coupling efficiency of ∼ 10−5. This is achieved for a peak

electron temperature contrast of only 360 K. Clearly, increasing this contrast will enhance

the grating effects, leading to larger SPP coupling. The simplest approach to raising the

electron temperature is to use a thinner gold film; for example, it has been shown that

electron temperatures in excess of 5000 K are attainable for 20 nm thick gold films [39].

Care must be taken with the thinner films that coupling effects between the two surfaces

are included [52]; however, this need not be a disadvantage as, for example, an intriguing

extension of this experiment would be to attempt all-optical coupling into a long-range

SPP mode, which can be excited when the dielectric at both interfaces is the same [53, 54].

A further improvement to the method might be to increase the length of the pump

pulses relative to the probe pulse. First, longer pump pulses would allow for a higher peak

fluence before the peak intensity becomes sufficient to ablate the gold. Second, increasing

the length of the pump pulses, to several picoseconds for example, will lengthen the launch

window for the plasmons, leading to a higher coupling efficiency.

Due to the high mode confinement of the SPP, as compared to that of the original

free-space radiation, a coupling efficiency of 10−5 means that the peak intensity of the

SPP is only a factor of 10−3 smaller than that of the incident light. Further, this value

only considers energy coupled to the SPP from T (−1), which itself accounts for only

∼ 10−4 of the total energy. Preliminary FDTD simulations performed by Mohamed

Swillam indicate that the total coupling efficiency can be as high as 10−2, as calculated

from absorption considerations using all of the diffracted orders, both in transmission and

reflection; in this case the peak intensity of the SPP would comparable or even greater

than that of the incident free-space radiation, and would clearly be large enough for

plasmonic-based applications. However, experiments detecting this energy loss from the

zero-order transmission and reflection need to be performed to verify these calculations.

Chapter 5

Analytic Model: Grating Assisted

Plasmonic Coupling

In this chapter an analytic model is developed to describe the grating-mediated coupling

of light to surface plasmon polaritons. As explained in the discussion of the theoretical

background of SPPs (Sec. 2.2), gratings can provide the momentum necessary to satisfy

the phase-matching conditions between the free-space radiation and the SPPs, and hence

can resonantly couple light to plasmonic modes.

However, while the process of plasmonic coupling is qualitatively understood, prior to

this work there had been no quantitative description of the spatial dynamics of grating

assisted coupling. Rather, researchers rely on numerical methods [28–33] (c.f. Secs. 1.2

and 2.3) to model the interaction between light and metals in the presence of a grating;

plasmonic properties such as coupling efficiencies are subsequently extracted from the

quantities – such as the reflectivities or the numerical values for the electric fields – that

are extracted from these calculations. Frequently, there is no way to separate plasmonic

contributions, such as reradiation of the SPP into propagating orders, from grating effects

such as diffraction.

In this chapter, a Green function based [37] analytic model is developed that de-

scribes the process of surface relief grating assisted plasmonic coupling by a one dimen-

sional ordinary differential equation; furthermore, analytic expressions are provided for

the constants of this equations. In section 5.1 the problem of plasmonic coupling is

clearly defined and an outline of the approach taken is presented. Next, section 5.2 con-

tains the analytic model that describes surface-relief grating assisted plasmonic coupling,

culminating in expressions for the intensity of the SPP and reflected fields in the case of

74

Chapter 5. Analytic Model: Grating Assisted Plasmonic Coupling 75

plane-wave excitation. Section 5.3 then presents a summary of the performance of this

model, as compared to accepted numerical models. Section 5.4 provides an example of

the application of this model when the excitation beam is not plane-wave but rather has

a gaussian profile, allowing for a discussion of the spatial dynamics of plasmonic cou-

pling. Lastly, in section 5.5 the limitations of this method are summarized, and possible

improvements are suggested.

5.1 Statement of the problem

The purpose of this chapter is to calculate both the SPP and the reflected fields that can

result when light is incident on a metallic grating, such as the one shown in Fig. 2.7(a).

The notation used in this work is presented in Sec. 2.2, and in particular in Sec. 2.2.1, and

is shown schematically in Fig. 2.3. Recall that the metallic grating and substrate have

a frequency dependent dielectric constant, εm (ω), and are below a transparent medium

with a real dielectric constant, εd (ω). The grating has amplitude h0 in the z-direction

and period Λ in the y-direction, with a surface profile h (y) that can be expanded in a

Fourier series

h (y) =∑

p

hpGeipGy, (5.1)

where the reader is reminded that G = 2π/Λ is the grating wave vector, and p is an

integer.

This system can be thought of as three distinct regions: (1) The region above the

grating, z > h0, where only the dielectric is present; (2) the region of the grating,

h0 ≥ z ≥ 0, where both the dielectric and the metal are present; (3) the region below the

grating, z < 0, where only the metal is present. To solve for the electric fields in these

regions, given an incident field, the presence of the grating is treated as a perturbation.

For a planar interface between the metal and the dielectric, the electromagnetic fields

are determined by the Fresnel coefficients [see Eq. (2.10)]; the presence of the grating in

region (2) results in a small correction. That is, the material in region (2) is initially

treated as only a dielectric, and then the presence of the metal grating is considered to

induce a polarization that then acts like a source term, driving the electric fields.

It is easier to develop this model in terms of the Fourier components of the fields (k -

space) and obtain the field dependence in real space by a Fourier transform. A schematic

of the plasmonic coupling process, in k -space, is shown in Fig. 5.1. The electric fields, and


0 k

I

kykOw

~

D

G

E EO

IP PO

I

Figure 5.1: The electric fields and polarizations, in k -space, that need to be calculated.

Superscript I and O show that the relevant quantities are evaluated at the incident and

SPP wave vectors, respectively.

the induced polarizations, at the incident wave vector, κI , are considered to be separate

from those at the SPP wave vector, κO. That is, the electric fields and the polarizations

are written as

E = EI + EO,

P = P I + P O,

and as shown in the schematic, it is assumed that the components do not overlap in

k -space.

By allowing for a small detuning that is defined as

∆ = κI − κO + G, (5.2)

the effect of the grating can be described as follows: (1) An electric field is incident at

κI , and through grating assisted momentum transfer induces a polarization at κO. (2)

This polarization drives the SPP field at κO. (3) Through reverse scattering, the field at

κO sets up a polarization at κI . (4) The polarization at κI drives an electric field at this

wave vector that contributes to the reflected electric field. In the following sections, the

mathematical formulation of this scenario is presented.

5.2 Grating assisted plasmonic coupling

In this section the model that describes grating assisted plasmonic coupling is developed.

First the exact equations for the electric fields associated with this problem are presented


(Sec. 5.2.1); these equations are written employing a Green function formalism, as de-

scribed in Appendix A. The approximations and simplifications that lead to solutions for

the electric fields are explained in Sec. 5.2.2, followed by a description of the polarization

induced by the presence of the grating (Sec. 5.2.3). Next, in Sec. 5.2.4 an expression

for the SPP electric field envelope is developed, followed by an expression for the fields

at the incident wavevector, κI , in Sec. 5.2.5. These are then combined, in Sec. 5.2.6, to

derive a one-dimensional ordinary differential equation for the spatial evolution of the

SPP field. Finally, in Sec. 5.2.7 the intensities of the excited surface plasmon polariton

and the reflection, both for the case of a plane-wave excitation and in the general case,

are derived.

5.2.1 Basic equations

Recall that monochromatic electric fields can be written in the same manner as a narrow

bandwidth pulse [Eq. (2.1)]:

E (r, t) = E (r) e−iωt + c.c.,

where

E (r) =

∫dκy

2πeiκyyE (κy, z).

Here the spatial components have been explicitly separated, and the y dependence of

the field is Fourier transformed; this allows for the use of the Fourier coefficients of the

grating profile [Eq. 5.1] instead of its profile.

First, general expressions for the electric fields in the three regions of the problem are

given (see Appendix A for the derivation). Using a Green function formalism [37], the

field above the grating, for z > h0, is

E (κy, z)

= pd−Einc (κy) e−iwdz + pd+rdmEinc (κy) eiwdz

+iω2

2ε0wd

pd+pd+eiwdz

∫ h0

0

e−iwdz′P (κy, z′) dz′ (5.3)

+iω2

2ε0wd

pd+rdmpd−eiwdz

∫ h0

0

eiwdz′P (κy, z′) dz′,

where Einc is the incident electric field, ε0 is the permittivity of free-space, and rdm is

the κy-dependent Fresnel coefficient of reflection [Eq. (2.10a)]. The polarization vector,

P (κy, z′), should be understood to be taken as a scalar product with the preceding unit


vector (e.g. pd+). Each of the four terms in the above equation has a clear physical

significance: The first term represents the electric field that is incident on the system;

the second term is the reflection from the planar metal dielectric interface at z = 0 that

occurs as though there was no grating present; the last two terms are the corrections

due to the presence of the grating, where the first corresponds to the field generated by

the polarization in the grating that propagates up, away from this region, while the last

is due to this induced field which initially propagates down and is then reflected by the

metal.

In the region of the grating, for h0 ≥ z ≥ 0, the electric field is,

E (κy, z)

= pd−Einc (κy) e−iwdz + pd+rdmEinc (κy) eiwdz

+iω2

2ε0wd

pd+pd+

∫ h0

0

θ (z − z′) eiwd(z−z′)P (κy, z′) dz′

+iω2

2ε0wd

pd−pd−

∫ h0

0

θ (z′ − z) e−iwd(z−z′)P (κy, z′) dz′

− zz

ε0εd

P (κy, z) (5.4)

+iω2

2ε0wd

pd+rdmpd−

∫ h0

0

eiwd(z+z′)P (κy, z′) dz′,

where the Heaviside function used is

θ (z − z′) =

{0 for z < z′,

1 for z > z′.

While the first two terms, as well as the last, have the same physical significance as

those in Eq. (5.3), the remaining three terms must be interpreted separately. The first

represents the electric field, generated in this region, that propagates upwards; for this

region, we need only consider the field generated below the point of interest (i.e. z > z′)

which will then propagate to z. Conversely, the second term represents the generated

electric field that then propagates downwards, but has not been reflected yet. Lastly, the

zz term describes the part of the field not associated with the aforementioned propagating

waves.


Finally, below the grating and in the metal, for z < 0, the electric field is

E (κy, z)

= pm−tdmEinc (κy) e−iwmz (5.5)

+iω2

2ε0wd

pm−tdmpd−e−iwmz

∫ h0

0

eiwdz′P (κy, z′) dz′.

Here the first term represents the part of the incident field that is transmitted across the

metal-dielectric interface, while the second term provides the correction to this field due

to the grating.

At this point the equations are exact. If the polarization, P (κy, z), is known then

Eqs. (5.3)-(5.5) would give the full electric field.

5.2.2 Approximation and simplifications

The following strategy is adopted: The electric fields at the incident and SPP wavevectors,

κI and κO, respectively, are approximated based on the supposition of a small grating

height (h0wd ¿ 1). These fields are coupled by the polarization that exists in the region

of the grating. The polarization itself is expressed in terms of the generating fields, as

well as the Fourier coefficients of the surface profile. This self-consistent approach leads

to a simple dynamic equation for the evolution of the SPP envelope function with respect

to the displacement along the surface of the grating, as well as to an equation for the

envelope function of the reflected field.

The calculations that follow are simplified by defining the following:

C (κy, z) ≡ E (κy, z) +zz

εoεd

P (κy, z) , (5.6)

Q (κy) ≡∫ h0

0

P (κy, z) dz, (5.7)

where using C (κy, z) the electric fields in the two regions above the metal are combined,

and Q (κy) simplifies the equation for the grating induced polarization in k -space.

5.2.3 Polarization

First, the dependence of the polarization densities on the electric fields is found, as these

polarizations will drive the SPP field, as well as couple back and alter the reflected field

for energy lost to the SPP. Using the susceptibility of the metal, χm = εm − 1, the


polarization induced by an electric field is

P (y, z) = ε0 (χm − χd) θ (h (y)− z) θ (z) E (y, z) ,

where this form insures that the polarization is only present in the grating region that

contains the metal. This is rewritten by using Eq. (5.6) and by grouping by directional

unit vectors:

P (y, z) = ε0 θ (h (y)− z) θ (z) L ·C (y, z) , (5.8)

where

L = (εm − εd) (xx + yy) +εd (εm − εd)

εm

zz. (5.9)

In k -space, the electric field requires a single grating scattering event, gaining or losing

G, to set up the polarization. This is reflected by the Fourier transform of Eq. (5.8) that,

by taking Eq. (5.7) into account and by using the form of the surface profile given in

Eq. (5.1), is

QO (κy) = ε0hGL ·CI (κy −G, 0) , (5.10)

QI (κy) = ε0h−GL ·CO (κy + G, 0) , (5.11)

for the polarization at the SPP and incident wavevectors, respectively.

Here one of the fundamental assumptions of this model is made: because the grating

amplitude is assumed to be small, the electric field and the polarization in the region of

the grating are explicitly taken to be uniform in z; that is, C (κ, z) ≈ C (κ, 0). A more

sophisticated assumption can be made, and the functional form of the field expanded and

used in Eq. (5.8); if some knowledge of the form of the field is known a priori, for exam-

ple from FDTD simulations, then this could lead to a more accurate, yet complicated,

solution.

5.2.4 Fields at the SPP wavevector

The electric field at κO, EO, is calculated assuming that the grating perturbation is

small, and hence wOd h0 ¿ 1. By recalling that Einc (κO) = 0 and making use of Eq. (5.6)

and Eq. (5.7), Eqs. (5.3)-(5.5) are rewritten as

CO (κy, z) =iω2

2ε0wOd

pOd+pO

d+eiwOd zQO (κy)

+iω2

2ε0wOd

pOd+rdmpO

d−eiwOd zQO (κy) ,


for z > h0. For h0 ≥ z ≥ 0,

CO (κy, z)

=iω2

2ε0wOd

pOd+pO

d+eiwOd z

∫ h0

0

θ (z − z′) P O (κy, z′) dz′

+iω2

2ε0wOd

pOd−pO

d−e−iwOd z

∫ h0

0

θ (z′ − z) P O (κy, z′) dz′

+iω2

2ε0wOd

pOd+rdmpO

d−eiwOd zQO (κy) ,

and for z < 0,

CO (κy, z) =iω2

2ε0wOd

pOm−tdmpO

d−e−iwOmzQO (κy) .

Since SPPs are resonant modes, they are signaled by poles in the Fresnel coefficients

[see Eq. (2.10)] at κsp. Consequently, if the above equations are multiplied by (κy − κSP )

only the terms containing Fresnel coefficients do not vanish, and they are combined to

yield

(κy − κSP ) CO (κy, z) =iω2ρdm

2ε0wOd

eO (z) pOd−QO (κy) , (5.12)

for all z, where here a new field is defined:

eO (z) ≡ pOd+eiwO

d zθ (z) + pOm−e−iwO

mz τdm

ρdm

θ (−z) . (5.13)

This field decays exponentially both into the metal, as well as into the dielectric; it is

the SPP electric field. In fact, this field was introduced in the discussion of SPPs [c.f.

Eq. (2.15)]. Here, the relative amplitudes of this field are explicitly shown:

Ed = 1,

Em =τdm

ρdm

,

for the field in the dielectric and metal, respectively.

Next, an envelope function for the SPP field, fO, is introduced, allowing the total

field at κO to be written as

CO (κy, z) = eO (z) fO (κy − κO) ,

and, by inserting this into Eq. (5.12), the equation for the SPP envelope function is

derived,

(κy − iγ) fO (κy) = F (κy) , (5.14)


where

F (κy) =iω2ρdm

2ε0wOd

pOd− ·QO (κy + κO) . (5.15)

That is, once F (κy) is known, the SPP envelope function can be calculated. The Fourier

transform of this equation is then used to find the differential equation for the SPP

envelope function in real-space,

dfO (y)

dy= −γfO (y) + iF (y) . (5.16)

To solve this equation, the polarization at κO is required; from Eq. (5.10) it is evident

that this polarization is induced by the field at κI that is scattered by the grating.

Consequently, to proceed the electric field at κI must be found.

5.2.5 Fields at the incident wavevector

In order to correctly determine the SPP field, the electric field at the incident wave vector,

κI , must be known, as this is the field that induces P O [Eq. (5.10)]. Once energy passes

into the metal (i.e. for z < 0) it is absorbed and subsequently lost to heat, and therefore

only the field above the metal surface needs be determined. However, the incident field

does not vanish here, and consequently all terms need to be considered.

In the region of the grating, at κI , using Eq. (2.9) and noting that θ (z − z′) +

θ (z′ − z) = 1, Eq. (5.4) is written as

CI (κy, z)

= pId−Einc (κy) + pI

d+rIdmEinc (κy)

+i

2ε0εd

(κ2

I

wId

zz + wIdκκ +

ω2εd

wId

pId+rI

dmpId−

)QI (κy)

− iκI

2ε0εd

(zκ + κz)

∫ h0

0

θ (z − z′) P I (κy, z′) dz′

+iκI

2ε0εd

(zκ + κz)

∫ h0

0

θ (z′ − z) P I (κy, z′) dz′.

The sum of the final two terms in this expression is identically zero if the polarization is

assumed to be uniform in z. Even for arbitrary polarizations, the sum of these terms at

z = 0 is the negative value of the sum of the terms at z = h0. Hence, these terms are

neglected and, consequently, the field in the grating region simplifies to

CI (κy, z) (5.17)

= pId−Einc (κy) + pI

d+rIdmEinc (κy) +

1

ε0εd

N ·QI (κy) ,


where

N =i

2

(κ2

I

wId

zz + wIdκκ +

ω2εd

wId

pId+rI

dmpId−

). (5.18)

To calculate the field above the grating, the envelope function for the incident field

is introduced:

Einc (κy) = finc (κy − κI) ,

which is taken to be strongly peaked at κy = κI . to determine the field above the grating

this envelope function is inserted into Eq. (5.3), all other quantities on the right-hand

side are evaluated at κI and Eqs. (5.11) and (5.2) are used:

EI (κy + κI , z) = pId−finc (κy) e−iwI

dz

+pId+eiwI

dz[rIdmfinc (κy) + sIfO (κy + ∆)

], (5.19)

where

sI =iω2h−G

2wId

(pI

d+ + rIdmpI

d−) · L · pO

d+. (5.20)

Hence, the field above the grating, near κI , has two components: the first, propagating

downwards, is the incident field; the second, propagating upwards, is the reflected field.

This reflection is further split into two terms: the first, with coefficient rIdm, is the reflec-

tion that would occur from a planar interface, while the second, with coefficient sI , gives

the correction due to the presence of the grating. In essence, this second term represents

the feedback from the SPP field to the field at κI .

5.2.6 Self-consistency

Now that the electric fields at κI are known, they are used to find the closed dynamical

equation for the SPP field. As it is the field in the region of the grating that creates the

polarization that, in turn, drives the SPP field, Eqs. (5.10), (5.11), (5.17), and (5.2) are

inserted into Eq. (5.15), resulting in

F (κy) = ΓXfinc (κy −∆) + ΓSfO (κy) ,

where

ΓX =iω2ρdmhG

2wOd

pOd− · L ·

(pI

d− + pId+rI

dm

), (5.21)

ΓS =iω2ρdmhGh−G

2εdwOd

pOd− · L · N · L · pO

d+, (5.22)


This equation is Fourier transformed back into real space,

F (y) = ΓXei∆yfinc (y) + ΓSfO (y) ,

and is inserted into Eq. (5.16) to arrive at the final, differential equation for the SPP

envelope function,

dfO (y)

dy=

[−γ + iΓS]fO (y) + iΓXei∆yfinc (y) , (5.23)

a one dimensional ordinary differential equation. Thus, given a grating and excitation

geometry, as well as the incident field, finc, the SPP field, fO, can be calculated by solving

Eq. (5.23); this can be done analytically for a large number of cases and, failing that,

numerically. Note that this equation can be rearranged to solve for fO as an integral of

finc:

fO (y) = iΓXe(γ−iΓS)y

∫ y

−∞e(i∆−γ+iΓS)y′finc (y′) dy′. (5.24)

The physical significance of the three terms in Eq. (5.23) is as follows: the two terms

preceding fO relate to the losses of the SPP fields. First are the ohmic losses intrinsic

to the propagation of the SPP along a planar metal surface, which are determined by

γ. Second, ΓS is a correction that is composed of both the reradiation of the SPP

into a propagating mode, and a correction due to the redistribution of the field in the

presence of a grating. The latter may be either a further loss, or even a gain if the

electric field distribution is such that the field amplitude in the metal is less than would

be the case without a grating; consequently, the net effect of ΓS can be to either increase,

or decrease the losses. The coefficient of the final term in the equation, ΓX , gives the

strength of the grating assisted coupling of free-space radiation to the SPP mode. Often,

a comparison of the relative magnitude of these three terms will suffice to determine the

relative efficiency of the grating coupler, without having to completely solve the system.

An example follows in Sec. 5.4.

5.2.7 Intensities and plane wave excitation

Previously, a theory was developed to calculate the electric fields of both the reflected

light and the SPP, associated with a metallic grating coupler. However, it is the relevant

intensities, not fields, which are most often measured, or even calculated numerically.

Below, are the expressions for the intensities associated with this problem. Also shown

is the form of the solution for the intensities under excitation by a plane wave. This


is useful as numerical models, such as the C-Method or RCWA, assume a plane wave

excitation; consequently, this allows for the testing of the accuracy of this model.

Given an incident field envelope, finc (y), the incident intensity is

Iinc (y) = |finc (y)|2 .

The reflected intensity can be read directly from the latter part of Eq. (5.19)

Iref (y) =∣∣rI

dmfinc (y) + sIe−i∆yfO (y)∣∣2 , (5.25)

and the SPP intensity is

ISP (y) = |fO (y)|2 . (5.26)

It is the coefficient of reflection,

R =Iref

Iinc

,

predicted by this method that is compared with the numerical models. Plane wave

excitation requires that finc (y) = finc, and with this used in Eq. (5.23) it is evident the

the SPP envelope must have the form

fO (y) = FOei∆y.

Solving the differential equation yields the following SPP envelope function:

fO (y) =iΓXei∆y

γ − iΓS + i∆finc, (5.27)

and consequently, from Eq. (5.25) the coefficient of reflection for the grating coupler, due

to plane-wave excitation is:

R =

∣∣∣∣rIdm +

iΓXsI

γ − iΓS + i∆

∣∣∣∣2

. (5.28)

Note that for a planar interface the coefficient of reflection would be simply

Rplanar =∣∣rI

dm

∣∣2 ,

and it is the second term in R that accounts for the effect of the grating and the presence

of the SPP. Thus, a comparison of R and Rplanar yields information on the magnitude of

the SPP coupling.


5.3 Performance of the analytic model

To ensure the validity of this model the results of these calculations are compared with

those obtained via C-Method calculations [32, 33] and RCWA calculations [29–31]. First,

simulations of a sinusoidal grating using this method and the C-Method are compared.

Then, results of similar simulations of a square grating with both this model and RCWA

are contrasted.

5.3.1 Sinusoidal grating

Whereas in this model a grating is treated as a small perturbation to an otherwise flat

surface and an approximate analytic expression for the generated SPP field is extracted,

the C-Method (c.f. Sec. 2.3) relies on a co-ordinate transformation that maps the peri-

odic grating boundary to a flat interface. Unfortunately, the electromagnetic fields are

also transformed, and an analytic solution is still impossible. However, by expanding

the incident, reflected, and transmitted modes into a finite number Bloch-modes of the

system, a solution can be calculated. Consequently, the C-Method yields very accurate

reflection and transmission coefficients, although it is computationally intensive.

To contrast the results of this method with those of the C-Method plane waves ex-

citation of the SPP is required. Here, a sinusoidal gold grating is simulated and it is

assumed that the adjacent dielectric is air. The grating period is 1200 nm, and the am-

plitude varies between 0.1 and 60 nm to determine when this model breaks down. An

incident wavelength λ = 1000 nm is used, at which gold has a relative dielectric constant

εm = −41.8 + 2.95i [40]. For this wavelength the SPP resonance occurs at an angle of

incidence of 10.3◦; consequently, the simulations are performed for angles ranging from

8 to 12◦ to ensure that the entire resonance is captured. Typical results are shown in

Fig. 5.2. Part (a) displays the p-polarized coefficient of reflection for a range of incident

angles; both Rplanar (dashed line) and R [solid line, Eq. (5.28)] are shown. As expected,

near 10.3◦, there is a dip in R relative to Rplanar. This is because R includes the grating

effects and consequently for this angle of incidence the momentum matching conditions

κI + G = κO are met and some of the incident radiation is coupled to the SPP. Thus,

some energy that would otherwise be present in the reflected beam - as in Rplanar, which

does not include grating effects - is lost.

Figure. 5.2(b) shows ∆R = R − Rplanar for several different grating amplitudes; in

essence, what is displayed here is solely the grating induced adjustment to the reflected


9.5 10.0 10.5 11.0-0.12

-0.08

-0.04

0.00(b)

R

(degrees)

Height0.5 nm1.0 nm2.0 nm5.0 nm10.0 nm

9.5 10.0 10.5 11.0

0.95

0.96

0.97

0.98

R

(degrees)

(a)

Figure 5.2: Typical results for the Green function based simulations. (a) Reflection

spectra as a function of incidence angle for a gold film with a 5 nm sinusoidal grating

(solid line), and for a planar gold film (dashed line) for λ = 1000 nm. (b) Different ∆R

curves, as functions of the incident angle, for different amplitude gratings. The arrow

points in the direction of decreasing heights, following the inset.

intensity, and as such can be directly interpreted as the intensity transferred to the SPP

by the grating. As expected, the higher the grating, the stronger its effect on the incident

radiation, and the more light that is coupled to the SPP.

The accuracy of this method is quantified by repeating the R and ∆R calculations,

employing the C-Method, for the same structure and incident conditions. For the case

of the 5 nm grating, the results from the two methods are compared in Fig. 5.3. Clearly,

the magnitudes of the peaks are in excellent agreements: the current method results in


9.5 10.0 10.5 11.0

-0.03

-0.02

-0.01

0.00C-Method

Current Method

R

(degrees)

9.5 10.0 10.5 11.0

0.95

0.96

0.97

0.98

C-Method

R

(degrees)

Current Method

(b)

(a)

Figure 5.3: Comparison of (a) R and (b) ∆R curves calculated using both the current

method and the C-Method, for a 5 nm sinusoidal grating.

(∆R)max = 0.03039 while C-Method calculations yield (∆R)max = 0.03007, an error of

only 1.0%. There are some differences between the curves produced by the two methods,

which offer insight into the nature of the error. First, as is evident from the R curve

[Fig. 5.3(a)], there is a positive offset between the C-Method curve and ours. That is,

the current model predicts that more energy be present in the reflection, as compared

to the C-Method. This is easily explained as the C-Method solves for all the propagat-

ing modes of the system, including the diffracted modes that the current calculations

omit. Consequently, in the current work energy from the diffracted modes is attributed

to the reflected mode. Since some energy from these diffracted modes also couples to the


SPP, the error in the plasmonic coupling efficiency calculated with the current method is

expected to be smaller than the error in the amount of reflected intensity. Not surpris-

ingly, the offset, and hence the error, becomes more significant as the grating amplitude

increases and more light is coupled to the diffracted modes.

Secondly, as is more noticeable in the ∆R plot [Fig. 5.3(b)], the C-Method curve has

features that the calculations based on the current model do not. Specifically, for smaller

incident angles, there is a bump near 9.7◦ that this work does not reproduce. Again,

this is explained by the full modal nature of the C-Method calculation. This bump

corresponds to a Wood-Rayleigh anomaly, where a scattered mode propagates along the

surface of the grating. Since the current model includes only the reflected and SPP mode,

this feature is not reproduced.

To determine the range of heights over which this method remains accurate, the above

calculations are repeated for grating heights reaching 60 nm, and the amplitude of the

dip as calculated by both methods are compared (Fig. 5.4). From this figure, it is evident

0 20 40 601E-3

0.01

0.1

1

(R

) MA

X

Height (nm)

0 20 40 60-20

0

20

Erro

r (%

)

Height (nm)

Figure 5.4: The magnitude of the dip in the reflection spectra calculated both with the

current model (line) as well as the C-Method (circles) for excitation by a 1000 nm plane-

wave and a sinusoidal grating with a 1200 nm period and amplitudes ranging from 0.1

to 60 nm. The inset shows the error between this method and the C-Method.

that the two methods agree very well up to an amplitude of ∼ 50 nm, which, for λ = 1000

nm, is about four times longer than the skin-depth of gold. Further, (∆R)max > 0.5 are

predicted to within 10%, while (∆R)max > 0.8 are predicted to within 15%, which is


very good considering the simplicity of the current model, and that these calculations

are over 250 times faster than those of the C-Method. Further, by taking account of the

energy coupled from the higher diffracted orders results in a peak error in the plasmonic

coupling efficiency of ∼ 13%.

For grating amplitudes in excess of 50 nm the current model begins to break down.

This breakdown occurs due to the linearity in h of the grating scattering [Eq. (5.11)

and (5.10)]. Consequently, as h increases so do parameters such as ΓS or ΓX ; where in

reality these parameters must have a maximum value, in the current model they can grow

indefinitely, leading to breakdown. This manifests itself in one of two ways, depending

on the sign of Im {Γs} [Eq. (5.22)]:

1. Im {Γs} > 0: This corresponds to grating induced losses in the system. Since

this constant grows as h2, given the assumptions inherent in the current work, at

some point it will be comparable in value to the natural losses of the propagating

plasmon, which are related to γ. Since the product sIΓX also grows as h2, near the

resonant angle the ratio of this product to ΓS will dominate the second term in R,

[Eq. (5.28)], which will then approach or even exceed unity.

2. Im {Γs} < 0: This corresponds to a decrease of losses in the system. While this

might seem unreasonable, corresponding to gain in the system, this is not the case.

In this model it is assumed that both the electric field and the polarization in the

region of the grating are uniform. A consequence of this is that less field can be in

the metal than would otherwise be the case for a planar film, and the lessening of

the ohmic losses can dominate over the reradiation losses, resulting in what appears

to be a gain term. As the grating amplitude increases, the assumption of electric

field uniformity becomes increasingly erroneous, until |Im {Γs}| > γ resulting in a

net gain, and a catastrophic failure of the theory.

5.3.2 Rectangular gratings

The previous section shows that the current approach to plasmonic coupling calculations

agrees with numerical methods for the case of a sinusoidal grating. However, this might

not seem surprising, given that the current method only uses the first Fourier coefficient

of the grating profile, and that a sinusoidal grating only has one component. This sug-

gests changing the profile of the grating to a more complicated structure, one that has


higher order Fourier components, and repeating the above analysis to see if the first-order

coefficient still dominates. An obvious choice for a new surface profile is the rectangular

grating.

Since the only difference in these calculations will be in the Fourier components of

the grating profile, the entire set of calculation does not need to be repeated. For a

sinusoidal grating h±G = h0/4 while for a square grating h±G = h0/π, and consequently

the results of a calculation for a sinusoidal grating of height h(sin)0 will be identical to

those of a square grating of height, h(sq)0 = π/4 · h(sin)

0 . This is a direct consequence of

only including the first order Fourier coefficient, and can introduce additional errors for

grating profiles with higher harmonics.

This sharp profile presents difficulties for simulations using the C-Method and conse-

quently the RCWA [29–31] approach is employed. Results for a square grating, with the

same parameters as the sinusoidal grating, are shown in Fig. 5.5. Clearly, the two theo-

0 10 20 30 40 501E-3

0.01

0.1

1

(R

) MA

X

Height (nm)

0 10 20 30 40 50-20

0

20

Erro

r (%

)

Height (nm)

Figure 5.5: The magnitude of the dip in the reflection spectra calculated both with the

current method (line) as well as the RCWA (circles) for excitation by a 1000 nm plane-

wave and a square grating with a 1200 nm period and amplitudes ranging from 0.1 to 50

nm. The inset shows the error between this method and the RCWA.

ries are in good agreement, even for large coupling efficiencies. That the theory breaks

down earlier than for the sinusoidal grating is expected, since the Fourier component for

a given grating amplitude is large for the square grating. Further, the sharp edges of this

profile tend to result in hot spots, areas where the field is concentrated, and consequently


it is expected that the assumption of a uniform field (and polarization) is worse than for

a smoothly varying grating. Regardless, the agreement is still very good to upwards of

30 nm amplitude square gratings, where the coupling efficiency is in excess of 0.8.

Even with the improved algorithm, the RCWA technique does not converge quickly

for all grating heights, and as such Fig. 5.5 contains data calculated with both truncation

number N = 101 and N = 151. Even for the prior case, the calculation time for the

current technique is ∼ 3.5× 103 times faster than with RCWA, while for the latter case

the ratio is ∼ 104. For comparison, a calculation with N = 41, using the RCWA is still

500 times slower than calculations performed with the current, Green function based

model.

5.4 Example: spatial evolution of the fields

The previous calculations assumed plane-wave excitation, which allowed for direct com-

parison with numerical methods. This section presents a more complex example which

demonstrates the physical insight that can be gleaned from the current approach. It uses

the same grating as in the previous sections, a sinusoidal gold grating with Λ = 1200

nm and h0 = 20 nm adjacent to air, and considers illumination with λ = 1000 nm radi-

ation. However, instead of plane-wave excitation, a gaussian beam with a full-width at

half-maximum spot size of 5 µm is now used to excite the plasmons. This allows for the

study of the spatial evolution of both the plasmon and reflected field; fO (y) is calculated

using Eq. (5.23), and fref (y) from Eq. (5.19).

At λ = 1000 nm the surface plasmon wavevector is κsp = (6.36 + 5.46× 10−3i) µm−1.

Consequently, the angle of incidence which excites a SPP, and which is used in this ex-

ample, is 10.3◦. Furthermore, the imaginary part of κsp is used to calculate a propagation

length of 92 µm, while the real part results in the plasmonic wavelength, λSP = 988 nm.

Before calculating fO (y) and fref (y), the coefficients from the differential equation

[Eq. (5.23)] must be found. The solution to Eq. (5.23) in a region with no incident field

is

fO (y) ∼ e−(γ−iΓS)y,

and therefore the SPP intensity is

ISP (y) ∼ e−2(γ+Im{ΓS})y.


Consequently, the effect of the grating on the decay of the SPP is given by ΓS. For this

situation ΓS = (5.37× 10−3 + 1.83× 10−4i) µm−1, with the positive sign of the imaginary

component corresponding to net losses due to reradiation and the field distribution in

the grating region. The imaginary component of ΓS leads to a propagation length of

2.7 mm, which is ≈ 100 times the intrinsic length of 92 µm.

From Eq. (5.23), it is evident that the coupling between the incident field and the

SPP field is described by iΓX . Further, since the coupling from the incident wave to the

SPP is linear, the coupling length is defined as δ+SP = 1/ Im

{ΓX

}; this is the distance

over which the SPP envelope intensity will build to the same value as a constant incident

intensity. Here, ΓX = (−5.05× 10−2 − 6.64× 10−3i) µm−1 which results in δ+SP = 151

µm. It is the interplay between ΓX and ΓS that determines the total plasmonic intensity

that can be generated, as the incident beam must be large enough to allow sufficient

coupling, but short enough to avoid unnecessary losses.

Finally, Eq. (5.19) predicts the reradiation of SPPs back into propagating modes.

For this example, the reradiation constant is, sI = 0.0030 − 0.0265i and, consequently,∣∣sI∣∣2 = 7.1× 10−4 of the plasmonic intensity will be reradiated.

Equation (5.23) is solved to determine fO (y) and Eq. (5.19) is used to find fref (y);

the corresponding intensities are shown in Fig. 5.6. Part (a) of this figure shows both

the incident, gaussian intensity, as well as the generated plasmonic intensity. The peak

intensity of the SPP is 0.13 of the incident intensity and, as expected, it decays with a

constant of 92 µm (see inset). Part (b) shows both the incident intensity (solid curve) and

the reflected intensity (dashed curve). In the bottom, the tail of the reflected intensity is

clearly seen, and is due to the reradiation of the SPP; the dip in this curve shows that,

as expected, the plasmon is radiating out of phase with the incident field. The top inset

displays the difference between the incident and reflected intensities at their peaks.

The reflectance for the gaussian excitation is found according to

∆I =Iref − Iinc

Iinc

, (5.29)

where

Iref(inc) =

∫Iref(inc) (y) dy,

and Iref(inc) are the reflected and incident curves given in Fig. 5.6, respectively; for this

example ∆I = 0.034. It is interesting to note that this value is much smaller than the

peak intensity of the SPP. This disparity is readily understood, since the plasmonic mode

is more tightly confinement than the incident and reflected modes.


0 20 40

0.0

0.2

0.4

0.6

0.8

1.0

Incident

PlasmonIn

tens

ity (a

.u.)

y ( m)

0 100 200 3000.00

0.05

0.10

Inte

nsity

(a.u

.)

y ( m)

-10 0 10 20 30

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

y ( m)

(b)

(a)

Figure 5.6: Spatial evolution of the intensities of this problem. (a) The SPP intensity

built up from the original incident, gaussian intensity. The inset shows the long range

decay of the SPP intensity. (b) The incident (solid) and reflected (dashed) intensities.

The top inset zooms in on the peak of the intensities, while the inset on the bottom

zooms on the tail end of the intensities, showing the reradiation of the SPP.

These results do suggest another important consideration that is often neglected: the

spatial extent of the coupling beam. In the literature it is often implicitly assumed that

the size of the incident beam does not matter, possibly since most of the simulations are

performed assuming plane-waves. However, for the conditions in this example RPW =

0.263, which is an order of magnitude larger than that calculate for a 5 µm gaussian

beam. Indeed, if a quarter of the energy of the incident beam is transferred to the SPP,

which has a skin-depth on the order of 500 nm, one would expect that the peak plasmonic


intensity be larger than the peak incident intensity.

The disparity between the plane-wave and gaussian results is bridged by recalling

that the plasmon intensity is not created instantly, but rather that it builds up as the

plasmon propagated along the surface of the grating [see Eq. (5.23)] as long as the

incident field is present. In essence, the beam size in this example is too small for a high

plasmonic intensity to build up. To verify this, the above calculations are repeated for

beam sizes (FWHM) ranging from 1 to 400 µm (Fig. 5.7). This figure shows both (a)

1 10 1000.0

0.1

0.2

0.30

20

40

(b)

(a)

20 nm

10 nm

10 nm

20 nm

5 nm

5 nm

I

Beam size ( m)

(I sp)

max

Figure 5.7: (a) The peak plasmonic intensity (assuming a peak incident intensity of unity)

as a function of the incident beam size (FWHM) for gratings amplitudes: 5 µm, 10 µm,

and 20 µm. (b) ∆I, as defined in Eq. (5.29) (solid), and for plane-wave excitation [unity

minus Eq. (5.28), dashed], as functions of the incident beam size for the same gratings

as in (a).

the peak plasmonic intensity, keeping the peak incident intensity at unity, and (b) R

curves, for different amplitude gratings. Clearly, as the beam size grows, so too does the

peak plasmonic intensity, increasing to over 40 times the incident intensity for a grating


amplitude of 20 nm. Correspondingly, ∆I for a gaussian beam also increases with the

beam size, eventually surpassing that of a plane-wave excitation. This should not be

too surprising since if the coupling and decoupling constants are sufficiently large, then

non-negligible amounts of light can be reradiated in spatial regions where the incident

beam is not present. As expected, smaller amplitude gratings do not show this effect.

5.5 Improvements and Limitations

The model introduced in this chapter describes surface-relief grating assisted plasmonic

coupling with good accuracy for grating amplitudes of a few tens of nanometers; for

this range of amplitudes plasmonic coupling efficiencies can exceed 0.8. Furthermore,

calculations based on this approach are hundreds of times faster, at worse, than standard

numerical modeling techniques.

However, this model has several limitations: First, this model breaks down for larger

amplitude gratings, as discussed in Sec. 5.3.1. Second, the current approach only models

two modes of the system – the SPP and the reflected field – consequently, higher order

plasmonic modes or diffracted modes, which are present in the numerical calculations,

are not reproduced here. Third, while it was demonstrated that this model is accurate

for grating profiles where the first order Fourier harmonic dominates, it is quite likely

that as the higher orders become increasingly significant the predictions of this approach

will become increasingly erroneous.

The improvement which allows this approach to overcome the second and third lim-

itations is obvious: more modes must be included in the model. While, in principle,

this can be done by increasing the complexity of the polarization equations [Eqs. (5.10)

and (5.11)], there is no guarantee that the elegant nature of this model be preserved;

eventually, increasing the intensity will render analytic calculations impossible, and this

approach will also become numerical in nature.

To overcome the first of the limitations, and increase the range of amplitudes over

which this model remains valid, a different approximation can be used. For example, a

local mode theory (see, for example, [55]) where it is the ratio between the amplitude

and period of the grating that is important will allow accurate predictions for greater

amplitudes. However, in this case the model will fail for short period gratings, or for

gratings with a sharp profile such as the square grating that was investigated herein.

Chapter 6

Conclusions

6.1 Summary

The experiments contained in this work demonstrate new techniques for active control

of plasmonic coupling on both structured and unstructured gold films. This is achieved

by altering the optical properties of the gold on ultrafast time-scales due to irradiation

by ultrashort, near-infrared laser pulses: the energy absorbed by the free-electrons in the

metal raises their temperature and hence changes their distribution about the Fermi-level.

This mechanism is then used to both modulate grating-mediated plasmonic coupling to

gold films, as well as to optically induce a grating which then couples light to SPPs on

planar gold films. Lastly, a theoretical model is developed culminating in an ordinary

differential equation, whose constants have analytic forms, that describes the spatial

dynamics of surface-relief grating mediated coupling of light to SPPs.

6.1.1 Active control of plasmonic coupling

Ultrafast active control of plasmonic coupling is demonstrated using both gold and di-

electric surface-relief gratings, where the latter overlays a gold film. For these, a peak

modulation of ∼ 60% is demonstrated on the spectral outskirts of a plasmonic resonance,

where the plasmonic coupling is increased from 16% to 26%; at the center of the res-

onance (λ = 603 nm), a modulation of −14% is observed, as the plasmonic coupling

decreases from 70% to 60% due to the optical pumping. Further, the recovery time for

this system is measured to be 770± 70 fs, as the electron temperature equilibrates with

the lattice.

97

Chapter 6. Conclusions 98

Moreover, this technique is demonstrated over a range of wavelengths spanning 150

nm, from 550 to 700 nm. Over this spectral range peak modulations to the plasmonic

coupling of 20% are observed, and a fluence study suggests that modulations of about

50% are possible. Further, the plasmonic coupling efficiency is either decreased, as is

seen for wavelengths near the center of the resonance, or increases as is more common for

wavelengths on the outskirts of the resonance; consequently, through careful consideration

of the excitation geometry, the coupling of light of any wavelength to a SPP may be either

negatively or positively changed.

All-optical control of the coupling of light to SPPs is also demonstrated on plain gold

films. Here, the first (-1) diffracted order in transmission is detected, and a polarization

dependent dip in the spectra that shifts in the expected manner, as a function of the

incident angle, is taken as a signature of plasmonic coupling. The magnitude of the

spectral dip determines that about 8% of the light from the diffracted beam is coupled to

a SPP, corresponding to a total plasmonic coupling efficiency of 10−5; given the differences

in the mode confinements, this corresponds to a ratio of the peak intensity of the SPP to

that of the incident light of 10−2. Further, pump-probe type experiments reveal that the

launch window for plasmonic excitation is shorter than a picosecond. These all-optical

and ultrafast excitations of SPPs are identified for wavelengths ranging from 520 nm to

570 nm.

6.1.2 Model: grating-mediated plasmonic coupling

A model that describes the interaction of light with a metallic surface-relief grating,

and the subsequent possible excitation of SPPs is developed based on a Green function

formalism. This model is shown to predict the magnitude of the plasmonic coupling

to within 15% for both sinusoidal and square gratings for coupling efficiencies in excess

of 0.8, in the case of plane-wave excitations. In addition, the constants in the model

have clear physical significance – for example, the coupling and and grating induced

loss terms are readily identifiable – and the computational time is over two orders of

magnitude faster than for current numerical methods.

Further, the model described herein also allows for the excitation of SPPs by beams

with a finite spatial extent. An example is given for the excitation of SPP by a gaussian

spot, showing the spatial nature of the generation of the SPP field. The magnitude of

the SPP is then shown to have a strong dependence on the spot size of the incident


beam. Thus, this technique allows for the theoretical exploration of the spatial dynamics

of plasmonic coupling, something that was not previously possible for such a large array

of cases, or with such efficiency.

6.2 Outlook

While the development of active plasmonic elements, such as the ultrafast active couplers

presented in chapters 3 and 4, represents an important step towards the realization of

an integrated nanophotonic information processing device, it is obvious that much work

remains to be done; this is true even within the select subfield of active plasmonics. In

the context of the work presented within this thesis, future endeavors in this field are

separated into two categories: fundamental physics research and application oriented

development, where this latter case involves the integration of these active elements in

nanophotonic devices.

The fundamental research that can be done to develop and extend the ideas pre-

sented herein is multifaceted. Most importantly, different physical mechanisms than the

heating of the free-electrons must be utilized to switch the plasmonics in a more effec-

tive (higher on-off contrast) and efficient (lower pump fluences) manner. Two different

classes of materials show promise in this regard: organic films and semiconductors. In-

deed, it was previously noted that these have been used to actively control plasmonic

properties [17, 18, 20] however the switching timescales tend towards milliseconds, rather

than the picosecond times that are required for modern multi-GHz information process-

ing technology. To this end, ongoing experiments utilizing free-carrier dispersion effects

in sub-micron silicon films, to switch plasmonic coupling with adjacent gold films, show

great promise.

A further avenue of improvement utilizing opto-thermally induced changes to the

optical properties of gold, as is done in this thesis, to control plasmonic coupling is to

use spectrally sharper resonances. Researchers have recently shown that through careful

design and nanofabrication of an array of metal nanoparticles it is possible to reduce

the spectral width of a plasmonic resonance, for local plasmon modes, from tens of

nanometers to a few nanometers [56]. If similarly sharp coupling resonances were to be

found for SPPs, they could be employed in the scheme presented in this work and would

greatly increase the effectiveness of the modulator as the shift of the coupling resonance

would be on the same order as its FWHM.


Another logical extension to the all-optical coupling of light to SPPs on planar sur-

faces is to demonstrate this effect in the specular transmission (or reflection) and not in a

diffracted beam, as was done in this work. This experiment is needed to explore the total

magnitude of the plasmonic coupling that is achievable in this manner. Two possible ap-

proaches are suggested: First, one can repeat the experiments as described in this thesis,

attempting to reduce the noise and measuring the specular transmission. If a sufficiently

large pool of data can be gathered for, for example, different angles of incidence (and

hence different coupling resonances) and pump fluences, it might be possible to separate

the thermal contributions to the differential transmission from those due to plasmonic

coupling. A second more elegant approach would be to fabricate a small grating on top

of a planar gold film. The transient techniques introduced herein could be used to couple

light to a SPP on the planar surface, and the SPP would then propagate to the region of

the grating, which will decouple a portion of the energy; measurements of the resultant

radiation will quantify the total coupling efficiency.

The model presented in chapter 5 will hopefully assist researcher in both simulating

and understanding the process of grating mediated plasmonic coupling. Possible exten-

sions to this model are numerous and, broadly, fall into two categories. First, different

approximations may be used to arrive at analytic constants for the plasmonic coupling

equation. In this work it is assumed that the amplitude of the grating is small compared

to the wavevector components that are normal to the surface of the grating. However,

other approaches such as a local mode theory (for example see Ref. 55) show promise in

extending this model to larger amplitude gratings.

Secondly, while the model presented herein is very specific, in that it deals only with

a continuous-wave excitation of a SPP with a surface-relief grating, the approach to

the problem is very general and is easily adapted and extended to different situations.

Current work includes an extension to planar grating mediated plasmonic coupling, such

as was used experimentally in chapter 4. Further, the model can be adapted to include

pulsed excitation, or even coupling between waveguide and plasmonic modes. The goal,

of course, is to allow researchers to both understand and design new, and improved,

plasmonic elements.

Appendix A

Green Function Formalism

Here, the Green function that is used in the analytic model of the grating assisted plas-

monic coupling (Chapter 5) is derived; this is done using the approach developed by

Sipe [37]. The Green function is derived for a uniform media and the extension to a

multilayered structure is explained. The discussion is limited to p-polarized fields and

only gives the explicit form of the electric fields, as these are used in the model; the

magnetic fields are only shown in a general form. The notation introduced in Chapter 2

is used.

This derivation starts with a brief summary of Maxwell’s equations as they apply to

this problem (Sec. A.1), and proceeds with a solution for the case of a uniform media

(Sec. A.2). Lastly, the Green function solution is determined for a structured media,

where the structure is treated like an extra polarization term that then acts like a source

term for the electric fields (Sec. A.3).

A.1 Basic equations

Maxwell’s equations can be written in the following form:

∂B

∂t= −∇×E

∂D

∂t= ∇×H , (A.1a)

∇ ·B = 0 ∇ ·D = 0, (A.1b)

where

D = ε0E + P , (A.2a)

H =B

µ0

−M . (A.2b)

101

Appendix A. Green Function Formalism 102

Here, P is the polarization density (with units of electric dipole moment per unit volume)

and M is the magnetization (with units of magnetic dipole moment per unit volume)

that determine the material response to an electric or magnetic field, respectively. Note

that if Eqs. (A.1b) are the initial conditions (i.e. are true at t = 0), then it is sufficient

to solve Eqs. (A.1a) for all other times.

The electric and magnetic fields are decomposed in terms of their Fourier components

f (r, t) =

∫dω

2πf (r, ω)eiωt, (A.3)

and hence Eqs. (A.1a) become

−iωB (r, ω) = −∇×E (r, ω) , (A.4a)

−iωD (r, ω) = ∇×H (r, ω) . (A.4b)

For the purpose of this work it is sufficient to limit the discussion to non-magnetic

materials and hence M = 0. Further, the polarization density (now written in units of

electric dipole moment per unit volume per unit frequency) is split into two components:

P (r, ω) = P (ω) + P (r, ω) , (A.5)

where P (ω) represents the polarization for a uniform media and P (r, ω) is the polar-

ization due to any structure in the material. Inserting Eq. (A.5) into Eq. (A.2a) results

in

D (r, ω) = ε0ε (ω) E (r, ω) + P (r, ω) . (A.6)

In the case of an uniform media only the first term on the right side of the equation is

present, while a structured material response contains both terms. The following sections

solve Eqs. (A.4) for these two cases.

A.2 Uniform media

The dielectric constant of a uniform media has no spatial dependence and consequently

Eqs. A.4 become

−iωB (r) = −∇×E (r) , (A.7a)

−iωε

c2E (r) = ∇×B (r) , (A.7b)


where the explicit ω dependence of the fields and the dielectric function have been sup-

pressed. Taking the curl of the first equation and using the second leads to the normal

wave equation (∇2 + ω2ε)E (r) = 0. (A.8)

The solution to this equation is discussed in detail in any text on electromagnetic theory

(see, for example, [57]). Here, it is sufficient to note that if the waves propagate in the

y − z plane (Fig. 2.3), the solution to this equation is,

E (r) = E+eiκyeiwz + E−eiκye−iwz, (A.9)

where the first term corresponds to an upward propagating wave and the second to a

downward propagating wave. Since only p-polarized radiation is considered

E± = p±E±, (A.10)

where E± represent the amplitude of the upward or downward propagating fields.

The total electric field is then given by

E (r, t) = E (r) e−iωt + c.c., (A.11)

in analogy with Eq. (2.1).

A.3 Structured media

In the case of the structured media considered in this thesis the dielectric has both a

spatially invariant and a spatially dependent component. As such, Eqs. A.4 become

iωB (r) = ∇×E (r) , (A.12a)

−iωε

c2E (r) = ∇×B (r) + iωµ0P (r) , (A.12b)

where the explicit ω dependence has been suppressed for convenience.

In general, the spatial dependencies of the polarization can be considered separately,

and the y component Fourier transformed such that

P (r) =

∫dκ

2πP (κ; z) eiκy. (A.13)

It is easy to imagine that the polarization might be different at different z, which is made

explicit by writing,

P (r) =

∫dκ

2πdz0

[δ (z − z0) P (κ; z0) eiκy

],


Likewise, due to the linearity of Eqs. (A.12), the fields can be written as

E (r) =

∫dκ

2πdz0E (r; κ, z0), (A.14a)

B (r) =

∫dκ

2πdz0B (r; κ, z0), (A.14b)

where

E (r; κ, z0) = δ (z − z0) E (κ; z0) eiκy,

and the expression for B is identical to the above, with the substitution of B for E.

Consequently, Eqs. (A.12) are rewritten as

iωB (r; κ, z0) = ∇× E (r; κ, z0) , (A.15a)

−iωε

c2E (r; κ, z0) = ∇× B (r; κ, z0) + iωµ0δ (z − z0) P (κ; z0) eiκy, (A.15b)

Since the solution to these equations requires both an upward and a downward prop-

agating wave, in the same manner as that for the uniform media [Eq. (A.9)], an educated

guess of the solution to Eqs. (A.15) is

E (r; κ, z0) (A.16a)

= θ (z − z0) e−iwz0 E+ (r; κ, z0) + θ (z0 − z) eiwz0 E− (r; κ, z0) + Eδ (z − z0) eiκy,

B (r; κ, z0) (A.16b)

= θ (z − z0) e−iwz0B+ (r; κ, z0) + θ (z0 − z) eiwz0B− (r; κ, z0) ,

where the delta-function in Eq. (A.16a) is included because it is present in Eq. (A.15b).

Further, the Heaviside functions ensure that, for a source at z0, only points above the

source (i.e. z > z0) have an upward propagating field, and that only points below the

source, for z < z0, have a downward propagating field.

Consequently, three unknowns in Eqs. (A.16) – the vector E , and the two scalars E±

as in Eq. (A.10) – must be found such that Eqs. (A.15) are satisfied. A substitution of

Eqs. (A.16) into Eq. (A.15a) and simplification of the result, shows that the following

must hold:

0 = δ (z − z0) e−iwz0 z × E+ (r; κ, z0)− δ (z − z0) eiwz0 z × E− (r; κ, z0)

+δ′ (z − z0) z × Eeiκy + δ (z − z0)(iκ× Eeiκy

).


Clearly, the terms containing δ′ (z − z0) and δ (z − z0) must independently vanish; with

the use of the Eq. (A.10), these conditions lead to

0 = z × Eeiκy, (A.17a)

0 =(z × Ep

+p+eiκy)e−iwz − (

z × Ep−p−eiκy

)eiwz +

(iκ× Eeiκy

), (A.17b)

where the superscript p shows that these are the amplitude of the p-polarized waves.

The first of these equations insures that of the three components of the E vector (s, κ,

and z – where s = κ× z) the only non-vanishing term is

E = Ezz.

Consequently, Eq. (A.17b) leads to

0 = Ep+ (z × p+) e−iwz − Ep

− (z × p−) eiwz + iEz (κ× z) ,

which leads to

Ep+e−iwz + Ep

−eiwz = −iκν

wEz. (A.18)

Similarly, inserting Eqs. (A.16) into Eq. (A.15b) results in

−iωε

c2zEzδ (z − z0) eiκy =

n

cδ (z − z0) Ep

+ (z × s) e−iwz0eiκy (A.19)

−n

cδ (z − z0) Ep

− (z × s) eiwz0eiκy + iωµ0δ (z − z0) P (κ; z0) eiκy.

This equation is separated into the z and κ components that must be individually satis-

fied. From the z component:

−iωε

c2Ez = iωµ0P

z (κ; z0) ,

and hence the first constant is

Ez = − 1

ε0εP z (κ; z0) = − 1

ε0εz · P (κ; z0) . (A.20)

The condition resulting from the κ component of Eq. (A.19) is

0 =n

cEp

+e−iwz − n

cEp−eiwz + iωµ0P

κ (κ; z0) ,

which is rewritten as

Ep+e−iwz − Ep

−eiwz = − iω2

ε0νP κ (κ; z0) . (A.21)


The combination of this equation, Eq. (A.18), and Eq. (A.20) results in solutions for the

last two constants:

Ep+ =

iω2

2ε0weiwz

(κ

νP z (κ; z0)− w

νP κ (κ; z0)

)=

iω2

2ε0weiwzp+ · P (κ; z0) , (A.22)

Ep− =

iω2

2ε0we−iwz

(κ

νP z (κ; z0) +

w

νP κ (κ; z0)

)=

iω2

2ε0we−iwzp− · P (κ; z0) .(A.23)

Finally, the electric field associated with this problem is found by inserting Eqs. (A.20),

(A.22), and (A.23) into Eq. (A.16a)

E (r; κ, z0) =iω2

2ε0weiw(z−z0)θ (z − z0) p+

(p+ · P (κ; z0)

)eiκy

+iω2

2ε0we−iw(z−z0)θ (z0 − z) p−

(p− · P (κ; z0)

)eiκy (A.24)

− 1

ε0εδ (z − z0) z

(z · P (κ; z0)

)eiκy.

This field is formally written as

E (r; κ, z0) = GE (κ; z − z0) · P (κ; z0) eiκy,

where

GE (κ; z − z0) =iω2

2ε0weiw(z−z0)θ (z − z0) p+p+

+iω2

2ε0we−iw(z−z0)θ (z0 − z) p−p− (A.25)

− 1

ε0εδ (z − z0) zz,

is the Green function for this problem.

Recall, however, that it is not E (r; κ, z0), but E (r) that is the desired electric field

[Eq. (A.14)]; by changing the variable z0 to z′ this field is written as

E (r) =

∫dκ

2πE (κ, z) eiκy, (A.26)

where

E (κ, z) =

∫GE (κ; z − z′) · P (κ; z′) dz′. (A.27)

Here, P (κ; z′) is given in Eq. (A.13). This is the solution that is used in Sec. 5.2.1,

though for a multilayer structure; the increased complexity of the structure is dealt with

by accounting for the reflections and transmissions with the Fresnel coefficients.

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