Thesis

METAMATERIALS: AN OVERVIEW FROM THEORY TO PRACTICE

by

Richard Allen Dudley

A thesis submitted to the faculty ofUniversity of North Carolina at Charlottein partial fulfillment of the requirements

for the degree of Master of Science inApplied Physics

Charlotte

2012

Dr. Michael A. Fiddy

Dr. Donald Jacobs

Dr. Greg Gbur

http://uncc.edu

ii

c© 2012Richard Allen Dudley

ALL RIGHTS RESERVED

iii

ABSTRACT

RICHARD ALLEN DUDLEY. Metamaterials: An overview from theory to practice(Under the direction of DR. MICHAEL A. FIDDY)

A theoretical study of the optical properties of the relatively newly formed genre

of metamaterials is presented. A focus on the limitations of current theoretical expla-

nations for such a materials macroscopic properties as well as the limits such materials

are confronted with using off the shelf modeling software, specifically COMSOL R©

Multiphysics. Attention is paid to two specific class of metamaterials, namely form

birefringent structures as well as metallic resonators, which both provide mechanism

for unique material properties not found readily in nature. We also study the prop-

erties of homogeneous composite consisting of metallic flakes embedded in a dielectric

substrate, which is shown to have an unusually high index of refraction due to its ex-

tremely high permittivity. Materials with prescribed real parts of the permittivity are

highly valuable for imaging, cloaking and detection technologies. We demonstrate a

theoretical explanation of this unusually high index material using and expansion of the

Lorentz-Drude model. We also demonstrate tunability using several core parameters,

dominated by metal flake size, the dielectric gap thickness and metal flake conduc-

tivity, using COMSOL R© Multiphysics 4.2s RF module. COMSOL R© Multiphysics was

also used in verification of our measurement technique for the real material, which was

tested using free space and S-parameter measurements with a HP 8510a vector network

analyzer.

iv

DEDICATION

To my spouse and partner Desiree Tara Self, without whom this process and

this life would be less vibrant, joyful, intriguing and meaningfull. I would not be here

without your challenges and support or your constant love and friendship. Thank you

for never giving up either.

v

ACKNOWLEDGMENTS

First, I will have to thank my Advisor Professor M.A. Fiddy. This process would

not have been completed in the time frame it was without his constant support and

guiding. Throughout the years I have been at UNC-Charlotte he has been nothing but

the most wonderful influence on my decision making and has provided much insightful

into the many matters of Optics and Physics, which I have spent my undergraduate and

graduate career striving to better understand. His investment in time and resources,

into my development have afforded me the ability to continue to study and accomplish

the work presented within this body. I would also like to acknowledge the many scien-

tific, political and philosophical discussions at the university and elsewhere that have

made my time at UNC-Charlotte a time of enlightenment and growth.

Secondly, would also like to thank Dr. Robert Ingel, whose persistence, guidance

and assistance in the modeling and experimentation proved to be a practical guide to

all scientific endeavorers. This guide has allowed for the research to never loose focus on

attaining and disseminating useful knowledge to the scientific community as a whole.

Furthermore, I would also like to thank Professor Donald Jacobs and Professor

Greg Gbur whose challenging coursework, thorough and enthusiastic lectures and work

ethic, has inspired me to become a better physicist and mathematician and a more

motivated researcher and teacher.

Additionally I would like to thank my fellow students Jason Case who has assisted

in my development as a student and has refreshed my thirst for academic knowledge

and consistently challenged my understanding.

Finally, I would like to thank my friends and family, who have supported and

allowed me to continue this pursuit. I would also like to acknowledge the thousands

of individuals who have coded for the LaTeX project for free. It is due to their efforts

that we can generate professionally typeset PDFs now.

vi

TABLE OF CONTENTS

LIST OF FIGURES ix

LIST OF TABLES xv

LIST OF ABBREVIATIONS

CHAPTER 1 :INTRODUCTION 1

1.1 The Forerunners 1

1.1.1 Maxwell’s Equations 3

1.2 Material Properties Overview 5

1.2.1 Permittivity 6

1.2.2 Permeability 7

1.2.3 Metals 9

1.2.4 Drude model 9

1.2.5 Dielectrics 11

1.2.6 Lorentz oscillator model 11

1.2.7 Semi-conductors 15

CHAPTER 2 :THE OPTICAL PROPERTIES OF METAMATERIALS 17

2.1 History of Negative Index 17

2.1.1 Principle of least action 18

2.1.2 Re-radiation from a negative index material 19

2.1.3 Double negative index material (DNIM) possibilities 21

2.1.4 The perfect lens 24

2.2 The Six Velocities of Light 25

2.2.1 Free space velocities 25

vii

2.2.2 Waves in a medium 28

2.2.3 Super-luminal speeds 30

2.3 Pendry’s Perfect Lens Revisited 32

CHAPTER 3 :META-ATOMIC CONSTITUENTS 35

3.1 Overview of Possible Metamaterial Constituents 35

3.2 Metal Resonators 35

3.2.1 Size limitations 36

3.2.2 Geometrical scaling 37

3.2.3 Scaling down 40

3.2.4 Continued scaling possibilities 45

CHAPTER 4 :FORM BIREFRINGENT STRUCTURES 47

4.1 Form-Birefringent Materials 47

4.1.1 Gigantic Anisotropies 50

4.1.2 The Search for Higher Index Materials 53

4.1.3 Tunable Negative Group Delay 57

4.2 FBS Discussion 57

4.3 Measurement Techniques 59

4.4 Experimental Setup 61

4.5 VNA phase change measurement setup 61

4.6 Measurement calculations 61

4.7 Stationary Horns Measurement 62

4.8 Teflon R© Rod Polarization Test 68

4.9 Additional Considerations for Measurement Error 71

4.9.1 Cabling Sensitivity 71

CHAPTER 5 :MODELING TECHNIQUES 76

5.1 Modeling Overview 76

5.1.1 Drude or Drude-Lorentz models 76

viii

5.2 Negative Index Properties and Computational Restrictions 84

5.2.1 What is meant by ’exactly’ n = −1 84

5.2.2 Mesh restrictions 88

5.2.3 Restrictions based on conductivity 101

5.2.4 Meta-atomic approximation 102

CHAPTER 6 :NOVEL THEORETICAL TREATMENTS 118

6.1 Return to the Foundations of Material Properties 118

6.1.1 High index materials 118

6.2 Uncovering the Material Properties 121

6.2.1 Understanding why it Works 121

6.3 Magnetic Meta-atoms 129

CHAPTER 7 :DISCUSSIONS AND CONCLUSIONS 132

7.1 Summary 132

7.2 Conclusions 132

REFERENCES 134

APPENDIX A 137

ix

LIST OF FIGURES

FIGURE 1.1 : Sphere of Lorentz local field. 12

FIGURE 2.1 : Huygens re-radiation 19

FIGURE 2.2 : Ray diagram for simple lens. 22

FIGURE 2.3 : Airy disc 23

FIGURE 2.4 : Demonstration of a double negative index slab lens 25

FIGURE 2.5 : Demonstration of a simple lens using a DNIM 26

FIGURE 2.6 : Wave packet showing a combination of waves 27

FIGURE 2.7 : Brillouin’s diagram of integration paths 29

FIGURE 2.8 : Brillouin’s differentiation of signal and front velocities. 30

FIGURE 2.9 : Brillouin’s diagram of propagation speed in a medium. 31

FIGURE 2.10 : Negative index lens with plane wave incident. 32

FIGURE 2.11 : Negative index lens with cylindrical scatter 33

FIGURE 2.12 : Cylindrical scatterer 34

FIGURE 3.1 : General schematic for the SRR as proposed by Tretyakov(40) 38

FIGURE 3.2 : This graphs shows a response from the initial reading (40) 41

FIGURE 3.3 : Graph of split ring resonator resonant response 42

FIGURE 3.4 : Plot of the resonant frequency 43

FIGURE 3.5 : Scaling continued 44

FIGURE 3.6 : Scaling factor further reduced 45

FIGURE 3.7 : Backward wave 46

FIGURE 4.1 : Simple Mandatori structure 48

FIGURE 4.2 : Schematic of incoming wave onto birefringent layer 49

x

FIGURE 4.3 : Graph of bandwidth vs splitting ratio 51

FIGURE 4.4 : Form birefringent ABS plastic 52

FIGURE 4.5 : Graph of index value for TE, nx and TM ny 53

FIGURE 4.6 : SrTiO3Epoxy 54

FIGURE 4.7 : TiO2Nylon 55

FIGURE 4.8 : Graph of index value for TE, nx and TM ny 56

FIGURE 4.9 : Phase relations of ABS 57

FIGURE 4.10 : Phase relations of Gold Coated ABS 58

FIGURE 4.11 : Phase relations of Ti02 doped polyamides plastic 58

FIGURE 4.12 : Transmission for ABS stack 59

FIGURE 4.13 : Transmission for gold coated ABS stack 60

FIGURE 4.14 : Transmission for Ti02 doped polyamides plastic Stack 60

FIGURE 4.15 : Horn Measurement Reference 62

FIGURE 4.16 : Experimental diagram of volume filled with air 63

FIGURE 4.17 : Reference for two horns initially touching measurement type 65

FIGURE 4.18 : Air as either reference or demonstration 65

FIGURE 4.19 : Measurement for DUT 1 65

FIGURE 4.20 : Air as a reference measurement 66



FIGURE 4.23 : Air as a reference measurement 67

FIGURE 4.24 : Measurement for a DUT 4 67

FIGURE 4.25 : Measurement for a DUT 4 67

FIGURE 4.26 : Teflon R© rod initial 68

FIGURE 4.27 : Teflon R© rod polarization graph 69

FIGURE 4.28 : Teflon R© rod angle measurements 70

FIGURE 4.29 : Metal Horn Graph 70

xi

FIGURE 4.30 : Metal Horn Angle 71

FIGURE 4.31 : Bend Angle Response 1 72

FIGURE 4.32 : Bend Angle Response 2 73

FIGURE 4.33 : Bend Angle 3 74

FIGURE 4.34 : Bend Angle 4 75

FIGURE 5.1 : Lorentz-Drude Refractive Index of Copper 78

FIGURE 5.2 : Loreontz-Drude Refractive Index of Silver 79

FIGURE 5.3 : Cu Lorentz-Drude and Drude Permittivity Comparisons 79

FIGURE 5.4 : Plasma Frequency Responses 80

FIGURE 5.5 : Ag Lorentz-Drude and Drude Permittivity Comparisons 80

FIGURE 5.6 : Plasma frequency Responses 81

FIGURE 5.7 : Conductivity only Response 82

FIGURE 5.8 : Frequency Dependent Model 82

FIGURE 5.9 : Figure text for Fig 5.7 and Fig 5.8 82

FIGURE 5.10 : Frequency Dependent Permittivity SRR 83

FIGURE 5.11 : Non-Frequency Dependent Permittivity SRR 83

FIGURE 5.12 : This image is a simple sweep from 0.801 ≤ n ≤ −1.201 85

FIGURE 5.13 : The image is a sweep from 0.98001 ≤ n ≤ −1.20001 86

FIGURE 5.14 : The image is a sweep from 0.999 ≤ n ≤ −1.002 87

FIGURE 5.15 : Normal Mesh 88

FIGURE 5.16 : Normal Mesh Results 88

FIGURE 5.17 : Fine Mesh 89

FIGURE 5.18 : Fine Mesh Results 89

FIGURE 5.19 : Finer Mesh 90

FIGURE 5.20 : Finer Mesh Results 91

FIGURE 5.21 : Extra Fine Mesh 91

FIGURE 5.22 : Extra Fine Mesh Results 92

xii

FIGURE 5.23 : Extremely Fine Mesh 92

FIGURE 5.24 : Extremely Fine Mesh Results 93

FIGURE 5.25 : Fine Custom Mesh 93

FIGURE 5.26 : Fine Custom Mesh results 94

FIGURE 5.27 : Finer Custom Mesh 94

FIGURE 5.28 : Fine Custom Mesh results 95

FIGURE 5.29 : Finest Custom Mesh 95

FIGURE 5.30 : Finest Custom Mesh results 96



FIGURE 5.33 : Combination Results 1 97

FIGURE 5.34 : Combination Results 2 98

FIGURE 5.35 : Evanescent Amplification 98

FIGURE 5.36 : Evanescent Amplification 99

FIGURE 5.37 : Mesh Refinement 1 99




FIGURE 5.41 : Tall Slab1 103

FIGURE 5.42 : Tall Slab 2 103


FIGURE 5.44 : Finer Mesh 104

FIGURE 5.45 : Extremely Fine Mesh 104


FIGURE 5.47 : Extremely Fine Mesh Revisited 105

FIGURE 5.48 : Tall Slab Extremely Fine Mesh Revisited 105


xiii

FIGURE 5.50 : Image Revealed 106

FIGURE 5.51 : Image Revealed 2 106


FIGURE 5.53 : Large Scaled Conductivity Sweep 107

FIGURE 5.54 : Narrow Scaled Conductivity Sweep 108

FIGURE 5.55 : Refined Narrow Scaled Conductivity Sweep 109

FIGURE 5.56 : Broken1 110

FIGURE 5.57 : Best Case 1 111

FIGURE 5.58 : Best Case 2 111


FIGURE 5.60 : Conductivity restrictions 112

FIGURE 5.61 : Conductivity restrictions 2 112


FIGURE 5.63 : Further Conductivity Restrictions 113

FIGURE 5.64 : Further Conductivity Restrictions 2 113


FIGURE 5.66 : Broken Slab type 2 114

FIGURE 5.67 : Mesh Size for Following Set of Simulations 115

FIGURE 5.68 : Low Conductivity Comparisons 116

FIGURE 5.69 : Low Conductivity Comparisons 116

FIGURE 5.70 : Super meta-atomic structure 117

FIGURE 6.1 : Gold color coating ABS disc 120

FIGURE 6.2 : SEM Image of metallic flake 121

FIGURE 6.3 : SEM Image of cleaved metallic flake 121


FIGURE 6.5 : EDAX for Copper(Magenta) 122

FIGURE 6.6 : EDAX for Carbon(Red) 122

xiv


FIGURE 6.8 : Plane wave incident on periodic structure 123

FIGURE 6.9 : Incident wave on periodic structure 124




FIGURE 6.13 : This is the general schematic for our periodic structure. 125


FIGURE 6.15 : Full Dipole Sheet 126

FIGURE 6.16 : Flake Geometry 128

FIGURE 6.17 : Flake PEC Field 129

FIGURE 6.18 : SEM of magnetic particle inclusions in a dielectric substrate 130

FIGURE 6.19 : Diagram of magnetic response. 130

FIGURE 6.20 : Data of magnetic response of FBS disc 131

xv

LIST OF TABLES

TABLE 4.1 : Tabulated data for gold color coated ABS disc. 55

TABLE 4.2 : Teflon R© rod used 64

TABLE 6.1 : Bulk measured gold colored coating 120

xvi

LIST OF ABBREVIATIONS

ABS Acrylonitrile butadiene styrene plastic

DBE Degenerate Band Edge

DNIM Double Negative Index Material

FBS Form Birefringent Structures

LRC, RLC Inductor, Capacitor and Resistor

NIM Negative Index Material

OPL Optical Path Length

PBC Periodic Boundary Condition

PEC Perfect Electric Conductor

SBE Split Band Edge

SRR Split Ring Resonator

SNIM Single Negative Index Material

TM Magnetic Field Aligned

TE Electric Field Aligned

VNA Vector Network Analyzer

CHAPTER 1 : INTRODUCTION

1.1 The Forerunners

The history of electromagnetic radiation should be said to start at the exact

moment humans became curious about the world around them. Though the

historical record goes only so far, even the very first of the known presocratics

philosophers Hesiod acknowledged the differences between darkness and light.

From Chaos there came into being Erebos(Darkness) and black night

From Night, Aither(bright upper air) and Hemera(Day)

which she conceived and bore after uniting in love with Erebos. (24)

For much of the history that followed, western civilizations knowledge of

electromagnetic radiation was coupled to the ideas presented by Hesiod. Though

studies of reflection or refraction of some kind followed with every major

philosopher or scientist since Aristotle, there was a common premise that held back

the theoretical understanding of light. The physical world was made of all the same

kinds of stuff, atoms, substance, or matter. This paradox frustrated scientist and

philosophers for millennia, as debates about the origins of the universe collided with

human experience. Even with this barrier much of the knowledge about the nature

of light was discovered. Although possibly suggested first by Thales of Miletos,

when comparing the attractive nature of amber when rubbed with fur to that of a

lodestone, it was not until over 2000 years later that the links between electricity

and magnetism were formalized. Though we might find progress slow, we should

remind ourselves that it was not until the ’voltaic pile’ around 1800, that a

2

consistent source of current was available. Not long after, in 1820 while using this

type of battery, Hans Christian Oersted noticed that a nearby compass needle

deflected by a parallel wire. Almost simultaneously, Ampere presented a detailed

study of this phenomena which relates the closed path integral of the magnetic field

to the electric current density, which can be formalized to a form similar to(32)

∮C

B · dl = µ0

∫ ∫J · ndS (1.1)

This lead to the obvious hypothesis that magnetism could, in some way, also create

and electric response. This proved more difficult as source of magnetic current are

limited to say the least, even if there were magnetic monopoles. Faraday was

performing such an experiment when he recognized that by moving a magnet within

a wire loop he was able to create a current. Thus, the changing magnetic flux was

causing an induced electrical current, which can be expressed by

Emf =

∣∣∣∣∂ΦB

∂t

∣∣∣∣ (1.2)

Of course these formalisms are missing a few key components, even Faraday realized

that there was a more complicated law of induction at work. The idea of flux was

carried further by Gauss to relate the electric flux through a closed surface due to

electric charge.

ΦE =Qenc

ε0(1.3)

which, as far as we know, can be extended to magnetism using the same principle of

superposition with the assumption that there are not any magnetic monopoles

ΦB = 0 (1.4)

3

It was a considerable feat to realize that the relationship among these equations can

be combined to formulate the fundamentals of electromagnetics. Though we have to

note that Maxwell initially started with over 20 equations to relate these, and it was

in fact Oliver Heaviside that reduced them further to four equations now known as

Maxwell’s equations. (28)

1.1.1 Maxwell’s Equations

We now state the current differential formulation of Maxwell’s equations

which we intend to use for the remaining derivations within this text.

∇ · E(r, t) =ρ(r, t)

ε0(1.5a)

∇ · B(r, t) = 0 (1.5b)

∇× H(r, t) =∂

∂tD(r, t) + Je(r, t) (1.5c)

∇× E(r, t) = − ∂

∂tB(r, t) (1.5d)

4

where

E(r, t) = electric field strength (volts/meter)

B(r, t) = magnetic flux density (webers/meter)2

D(r, t) = electric flux density (amperes/meter)

H(r, t) = magnetic field strength (coulombs/meter)2

Je(r, t) = electric current density (amperes/meter)2

p(r, t) = electric field strength (coulombs/meter)3

We should also note that this requires the addition of two more equations to

formally complete fundamental derivations.(42)

D(r, t) = ε0E(r, t) + P(r, t) = ε0εrE(r, t) = εE(r, t) (1.6)

B(r, t) = µ0H(r, t) + M(r, t) = µ0µrH(r, t) = µH(r, t) (1.7)

5

where

P(r, t) = electric polarization vector (coulombs/meter)

M(r, t) = magnetic polarization vector (webers/meter)2

ε = permittivity(amperes× seconds)

(volts×meters)

ε0 = permittivity of free space 8.8541878× 10−6 (amperes× seconds)

(volts×meters)

εr = relative permittivity dimensionless

µ = permeability(amperes× seconds)

(volts×meters)

µ0 = permeability 1.25663706× 10−12 (amperes× seconds)

(volts×meters)

µr = relative permeability dimensionless

(13) The current formulation and understanding of these physical quantities

requires the duality of the wave and particle nature of light, though many of the

derivations are still based on classical approximations. It was Lorentz who removed

the barrier in his 1892 theory of electrons by separating mechanical qualities from

that of light, thus effectively paving the way for the separation of electromagnetic

radiation and its interaction with normal matter (16).

1.2 Material Properties Overview

Originally material interaction with light was studied only in the visible

spectrum because of the human detector. Once the relationships between the

electromotive force and fields were settled, pioneers like Heinrich Hertz quickly

expanded other frequency ranges into the realm of electromagnetic radiation, until

it became clear that both visible light and lower frequencies waves were all part of a

continuum. Traditionally, the index of refraction has served as the figure of merit in

6

which to evaluate the optical properties of a material. However; with our focus on

the comparison and relationship between the magnetic and electric fields in

metamaterials, we will need to consider the fundamental interaction of light with

materials by separating their magnetic and electric responses, namely permittivity

and permeability.

1.2.1 Permittivity

The importance of permittivity to the electromagnetic properties has been the

primary focus of optics for well over a century, since at optical frequencies one

typically assumed there to be little magnetic response. It was re-termed recently

from dielectric constant to relative permittivity (17). It is generally assumed to be

the dominant, if not the only contribution to the index of refraction of a bulk

material at optical frequencies. We will see later how this assumption is

considerably inaccurate for many metamaterials. We can begin by using Ampere’s

law in differential form

∇× H(r, t) =∂

∂tD(r, t) + Je(r, t) (1.5c)

In order to make an association at a specific frequency or frequency ranges we will

consider a steady state case. The general form for a harmonic field with frequency

given as ω is,

<D(r)e−iωt

= D(r, t) (1.8)

(21) This follows for E(r, t), H(r, t), B(r, t), Je(r, t), p(r, t), Jm(r, t) as well. Using

EQ. 1.5c we can easily see,

<[∇× H(r)− ∂

∂tD(r)− Je(r)

]e−iωt

= 0 (1.9)

7

We can now substitute ∂∂tD(r)e−iωt = −iωDe−iωt and

∇× H(r) = −iωD(r) + Je(r) (1.10)

By definition in a source free region the free current density Je(r) = σeE(r), where

σe is the electrical conductivity. We can also define D as

D(ω) = ε0E(ω) + ε0χeE(ω), (1.11)

with ε0 equal to the permittivity of free space and χe as the electrical susceptibility.

We can now fully define the permittivity through this relationship.

ε(ω) = ε0 (ε∞ + χe(ω)) +iσeω

(1.12)

The complicated nature of ε(ω) can be seen from EQ. 1.12. We can generally note

that the imaginary part of the electric susceptibility will overlap with the

contributions due to the bulk conductivity of the material. This is primarily

because of this fact that permittivity for metals, semiconductors and dielectrics are

treated from separate theoretical standpoints.

1.2.2 Permeability

As mentioned above, a material’s permeability is often ignored in optics. This

is the case often due to many materials having a permeability near unity and thus

are governed solely by their electrical response. We will see that the permeability or

its approximate equivalent circuit counterpart, the inductance, can play a huge role

in finding unique metamaterial properties not readily available in nature. The

permeability is derived in a similar means as the permittivity, however for most

cases the idea of a magnetic conductivity is ignored. We will include it here as a

theoretical possibility, which can be explored for real materials later on. Using

8

Faraday’s law, with the addition of a magnetic current density term, we have

∇× E(r, t) =∂

∂tB(r, t) + Jm(r, t) (1.5d)

As we have stated previously in EQ. 1.8, we would like to move to the frequency

domain, which leads us to some frequency ω,

<B(r)e−iωt

= B(r, t) (1.13)

We are again lead to the equation for the real part, this time for Faraday’s law

<[∇× E(r)− ∂

∂tB(r)− Jm(r)

]e−iωt

= 0 (1.14)

The simple substitution of ∂∂tB(r)e−iωt = −iωBe−iωt leads to

∇× E(r) = −iωB(r) + Jm(r) (1.15)

By definition in a source free region the free current Jm(r) = σmH(r), where σm is

now the magnetic conductivity. We can also define B as

B(ω) = µ0H(ω) + µ0χmH(ω), (1.16)

This completes the expression for the permeability as a function of µ0, equal to the

permeability of free space, and χm which is the magnetic susceptibility.

µ(ω) = µ0 (µ∞ + χm(ω)) +iσmω

(1.17)

Though we may not have an easy identifier for what may constitute a magnetic

conductance, we can still see how this is exactly as complicated as the permittivity.

9

We know know the general forms of the permittivity and permeability, relating this

to real materials turns out to be a bit more complex.

1.2.3 Metals

There are a few derivations for the bulk properties of a metal which accurately

describe its interaction with incoming electromagnetic radiation. Almost by

definition the traditional classification of materials ignores permeability. Thus we

will only take a closer look at the permeability when specific materials necessitate

this. We will also intentionally ignore deeper derivations such as quantum

polarization models as they do not apply broadly, but do provide an avenue for

exploration for smaller nanoparticles and are required for several interesting

properties found later. As we have already shown the permittivity is a function of

the susceptibility χe and the conductivity σe. We already have some clues from

these definitions that there is a significant physical variation between these two,

which is determined by whether the charges are bound or free. Remembering that

the relative permittivity is defined through the relationship ε = εrε0, we can

separate the relative permittivity into free and bound charges (4).

εr(ω) = ε(f)r (ω) + ε(b)r (ω) (1.18)

In doing so we can now analyze some physical models to further refine the

permittivity of free and bound charges.

1.2.4 Drude model

The free electron effects or intraband effects are summarized using the Drude

model, which is based on a simple free electron gas model. (35) We will simply start

with a model based on the classical driven damped spring, as it serves a much more

10

general purpose.

F = Fexternal + Fspring + Fdamping

We will assume the external electric field will be the only external force,

me∂2x

∂t2= qE(t) +meΩ

2x+meΓ∂x

∂t(1.19)

where meΓ is the damping coefficient and meΩ2 is the associated spring constant.

As we are first modeling a free electron gas, we are only considering conduction

band electrons. Thus we can allow the spring constant, meΩ2 → 0. We also can

preform the same operation for x(t)→ x(ω)e−iω, as we did previously, in EQ. 1.9,

under the assumption that the motion will follow the driving force in some manner

in a steady state case.

me(−iω)2x(ω) = qE(ω) +meΓ(−iω)x(ω) (1.20)

We will now define the electric polarization vector in terms of the dipole moment

created by n dipoles per unit volume P(ω) = nqx(ω). Plugging this in for x(ω).

P(ω) = − nq2

meω2 −meiΓωE(ω) (1.21)

Using the plasma frequency at low temperatures, ωp =(nq2

ε0

) 12

(38). We make a

further substitution for the electric susceptibility χ(ω), which is related to the

electric field and polarization through the relationship P(ω) = ε0χeE(ω). A full

derivation for the plasma frequency can be found in Appendix A EQ. 7 . We have

χe = −ε0nq2

ε0me

ω2 − iΓω= −ε0

ω2p

ω2 − iΓω(1.22)

11

which gives a complete solution for the relative permeability from free charges as

ε(f)r (ω) = εr(∞)−

ω2p

ω2 − iΓ0ω(1.23)

where εr(∞) is generally considered to be unity. There are more accurate

derivations as we can add the bound charge or interband contributions to the

relative permittivity, however this provides a good general description of the

permittivity of a metal and will suffice for most microwave applications.

1.2.5 Dielectrics

Much like the original driven damp spring example model, a dielectric has a

similar polarization response which follows this example except it has no free

charge. Thus the restoring force or spring force will be relatively large and play a

considerable role. We will also have to take into account the local field due to the

shift in polarization of nearby molecules, since its magnitude is large enough, on the

order of the individual response of the electric field, to contribute to nearest

neighbors. Typically this is classifed as the Lorentz oscillator or an adaptation of

the Lorentz-Drude model,

1.2.6 Lorentz oscillator model

Using the original equation for our electron oscillator in an electric field,

me∂2x

∂t2= qE(t) +meΩ

2x+meΓ∂x

∂t(1.19)

then as before we wish to move to the frequency domain, which gives us

x(t)→ x(ω)e−iω

me(−iω)2x(ω) = qE(ω) +meΩ2x(ω) +meΓ(−iω)x(ω) (1.24)

12

We will use the same identity for the polarization vector P(ω) = nqx(ω).

P(ω) = − nq2

meω2 +meΩ2 −meiΓωE(ω) (1.25)

Now we need to look at the polarization vector a little more closely, since previously

we assumed that only external field contributions were a factor in the case where

charges are free to move. We will redefine E(ω) = EExternal(ω) + ELocal(ω), where

the local field contributions are simply due to the nearby electrons displaced from

equilibrium. In this case we need to consider a single molecule in a spherical volume

element, which is surrounded by similar atoms. It is important to note that the

volume is considered large when compared to the volume element of the molecule,

however this volume is still small on the order of the wavelength since we are

assuming a uniformly polarized material. Although we are assuming a uniformly

polarized material, inside this sphere we can assume the field contributions will add

to zero. (6)

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

Figure 1.1: Sphere of Lorentz local field.Illustration demonstrates the effect of a background field both in and outside of thevolume we are considering the local field. We will return to this diagram later as we

discuss some novel metamaterial constituents.

This assumption is based on the fact that P is uniform and therefore is not a

13

function of the position. However; if we assume zero field inside the sphere and

some mean field exterior, this creates a discontinuity which is difficult to resolve.

Instead we can look at the complimentary situation concurrently. A situation with

zero field to the exterior of the sphere and a uniform polarization inside similar to

Fig 1.1 allows us to talk about the potential at the surface of the sphere

→ φOuter + φInner = 0. The potential can be found from the integral.

φOuter = −φInner = −P ·∫∇ 1

RdV (1.26)

We also know that ∇× E = 0, therefore the electric field is path independent.

Where

∫∇ 1

RdV → ∇

∫1

RdV , and the value of

∫1

RdV = φ0, which may be

interpreted as potential of a uniformly charged sphere with charge density ρ = −1.

The components of the field strength can then be given as

−∂φ∂x

=∂

∂x

[Px∂φ0

∂x+ Py

∂φ0

∂y+ Pz

∂φ0

∂z

]=

[Px∂2φ0

∂x2+ Py

∂2φ0

∂x∂y+ Pz

∂2φ0

∂x∂z

](1.27)

(6)

−∂φ∂y

=∂

∂y

[Px∂φ0

∂x+ Py

∂φ0

∂y+ Pz

∂φ0

∂z

]=

[Px

∂2φ0

∂y∂x+ Py

∂2φ0

∂y2+ Pz

∂2φ0

∂y∂z

]

−∂φ∂z

=∂

∂z

[Px∂φ0

∂x+ Py

∂φ0

∂y+ Pz

∂φ0

∂z

]=

[Px

∂2φ0

∂z∂x+ Py

∂2φ0

∂z∂y+ Pz

∂2φ0

∂z2

]

By symmetry we can see that all cross terms are zero.

∂2φ0

∂z∂x=∂2φ0

∂z∂y=

∂2φ0

∂y∂x= 0 (1.28)

14

This leaves only the squared terms which again by symmetry gives

∂2φ0

∂x2=∂2φ0

∂y2=∂2φ0

∂z2(1.29)

which simply results in −∇φ =1

3ε0P. We now have our field in terms of the

polarization vector.

E(ω) = EExternal(ω) +1

3ε0P(ω) (1.30)

We can now insert the new value for the electric field for EQ. 1.19 and we have

me(−iω)2x(ω) = q

[E(ω) +

1

3ε0P(ω)

]+meΩ

2x(ω) +meΓ(−iω)x(ω) (1.31)

⇒ me(−iω)2x(ω)− q

3ε0P(ω) +meΩ

2x(ω) +meΓ(−iω)x(ω) = qE(ω) (1.32)

Previously we have used P(ω) = nqx(ω), which will come in handy here and leads

to,

P(ω)

nq

[me(−iω)2 − nq2

3ε0+meΩ

2 +meΓ(−iω)

]= qE(ω) (1.33)

Substitution for the polarization vector leads to,

χe(ω) =nq2

ε0

[−meω2 − nq2

3ε0+meΩ2 +meΓ(−iω)

] (1.34)

and rearranging for future simplification gives,

χe(ω) =3nq2

3ε0me [−ω2 + Ω2 + Γ(−iω)]− nq2(1.35)

15

Solving for the permittivity gives us the full relationship.

εr(ω) =2nq2 − 3ε0m (Ω2 − ω2 + iωΓ)

3ε0m (Ω2 − ω2 + iωΓ)− nq2(1.36)

This is a similar result for bound charges and we can see that the only difference

here is that there is an additional factor for the local field and there is no necessity

to derive the oscillator restoration force, only to determine the bound charges per

unit.

1.2.7 Semi-conductors

Our discussion of semi-conductors will be brief as we predominately have

found the classification of metals and dielectrics to be the foundations of most of the

materials we investigated in the microwave regime, however, semiconductors offer a

host of unique possibilities especially when entering the near infrared region or more

so as we move towards optical regimes. Initially, one might think that a doped

semi-conductor could simply be modeled as a quasi-metal, having a larger damping

rate Γ. A quick study of conduction in semiconductors shows that this is almost the

case, since electrons have a very long carrier collision time, which leads to a very low

effective damping term much like a metal, albeit 108 smaller. However it also turns

out the carrier density is significantly reduced and thus the combination of the

larger damping term and the smaller density of charge carriers allows this to be

incorporated into the local field, Lorentz like calculation. Thus we can generally

ignore the bulk fee charges found in the original Drude model. It is also generally

difficult to classify all semiconductors into the same group when it comes to

conduction, as lattice structure and doping play a large role in the conduction bands

and possible carrier densities. It turns out that this can be encapsulated in a

modified Debye relaxation model, which is similar to the Lorentz oscillator model.

The name of this particular modification is called the GavrilyakNegami (30)

16

relaxation curve. It allows for both asymmetrical, α, dispersion curves as well as

broadened, β, dispersion curves, which range from 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1.

εr(ω) = ε(∞) +∆ε

(1 + (iωτD)α)β(1.37)

This, of course, reduces to the Debye relaxation model for α = β = 1. Here τd,

dubbed the relaxation dispersion relation in this case, is found using the exponential

increase in polarization given an external field, or exponential decay from a

polarized state, which is the reciprocal of Γ. We can calculate these for metals

semiconductors and dielectrics alike using

τD =εr(∞)

4πσ(1.38)

The values found for the relaxation tend to be near DC and then this is used as the

relaxation constant (8).

CHAPTER 2 : THE OPTICAL PROPERTIES OF METAMATERIALS

2.1 History of Negative Index

Thus far, we have explored the possible material parameters that influence

bulk electromagnetic properties which lead to effective parameters for homogenized

media. This description is limited by the initial assumptions that bulk properties

are restricted to a macroscopic effect of the linear addition of atomic properties that

make up the material constituents. There is little attention to resonant features, as

these are usually also areas of high absorption, nonlinear effects, which are inherent

in all materials, as well as no geometrical treatment except the dipole classification

of individual atoms. This changed with the review of a paper published in 1968 by

Victor Vesalago (15), where he described what would happen if a material had both

simultaneously a negative permeability as well as a negative permittivity. In the 2-D

case if we choose an arbitrary propagation direction, since k = ωc

√εµ, a negative ε

leads to an imaginary k value. We have known√εµ = n, thus using the properties

of =(ε) > 0 and =(µ) > 0, then =√ε > 0 and =√µ > 0, thus we can argue that

√εµ

is negative if both ε and µ are negative(19)(15). In this case a ray would reflect on

the same side of the normal as the incident ray(15).

Sir John Pendry, upon reviewing this, first proposed a design for a

magnetically negative metamaterial, which lead the way for a practical material

having negative permeability and permittivity. (19) Pendry later theorized that

there was a way to make these types of materials using metal wires embedded in a

dielectric. This idea turned into the first ’swiss-roll’ design, which had almost

immediate practical applications to enhance MRI resolution. Pendry realized that

18

typical diffraction resolutions limits which constrain imaging in every regime, could

be overcome with ease if metamaterials were possible, given an index of n = −1

(23). We have studied the requirements a material that would have both a negative

permeability and negative permittivity and will address these particular material

possibilities in the following sections.

2.1.1 Principle of least action

First we should determine whether if a solution with negative refraction is in

fact a possible path of action. Originally, Fermat’s principle of least time was

considered a formal approach, but with a negative index the idea of ’least’ time was

not applicable. This was because the direction of the path was not considered with

a negative index material. Further consideration lead to the fact that both

Feynman’s derivation of ’least action’ for quantum mechanical situations and the

more applicable formulation of Format’s principle of least time are suggested as

equivalent mechanical descriptions that both allow negative index as a possibility.

This observation paved the way for a theoretical description of a negative index

dipole sheet to be made, along the same lines of argument in which a positive index

dipole sheet was described.

The intent to show that spherically re-radiated waves interfere and

approximate a new wave front with a retarded velocity when compared to the

incoming field, based on the rate of re-radiation has to be looked at more closely to

understand physically when a negative index might occur. It is because this picture

does not provide an intuitive idea for a phase advance, which we would expect from

a material with a negative index, that we need to look closer at what occurs near or

at resonances in a material. When each of the point sources near a resonance

re-radiate, the scattered field can be phase advanced if the re-radiation is phase

advanced as well.

19

Figure 2.1: Huygens re-radiation - Illustration of refractive index due to superposi-tion of smaller re-radiating point sources or Huygens Principle. (31).

2.1.2 Re-radiation from a negative index material

As a simplistic approach the used by David W. Ward is to consider a plane

wave from a source incident on a material and then again on a detector. (12) With

a negative index material the detector would detect a phase advance when

compared to that of no material being present. We can look at a polarized

monochromatic plane wave of the form in vacuum incident on a plane of material of

thickness d. We can then describe the wave at the detector as,

Edz = E0e

− ωc0d(n−1)

eiω(t− x

c0)

(2.1)

This is an approximation assuming that the transport mean free path, or the length

at which the direction of propagation becomes completely randomized is large when

compared to the mean free path, thus the scattered amplitude is neglected.

Feynman rewrote this by expanding e− ωc0d(n−1)

, showing that the equation can be

20

broken into the incident field without the material and a response term (34).

Edz w E0e

iω(t− xc0

) − i ωc0

E0d(n− 1)eiω(t− x

c0)

(2.2)

This is also reminiscent of the Born approximation applied to the

Lippmann-Schwinger equation(29), which we know is similarly the linear

superposition of the Greens function and the perturbation caused by the medium.

We again follow Ward’s suggestion to imagine both a sheet of electric and magnetic

dipoles, which can simultaneously exhibit a Lorentz electric dipole resonance,

discussed in Section 1.2.6. We can also imagine a similar equation for the magnetic

susceptibility. Much like the electric susceptibility term the magnetic susceptibility

is given in terms of a Bloch resonance in a static background magnetic field. Since a

resonance is induce during a locally static magnetic field, the magnetic susceptibility

has the same features as the electric susceptibility.

χµ =

χµ02

Ω0T2

1 + (ω − Ω0)2 T 22

[(Ω0 − ω)T2 + i] (2.3)

With the knowledge that n = 1 + 12

(χe + χm) this will lead us to the equation for

the detected wave from the sheet of dipoles as

Esz = −E0i

ωχε2c

d(χε + χµ)eiω(t− x

c0)

(2.4)

The resulting re-radiated field from the superposition of a double dipole sheet is

Esz = −iωd

2c(χε + χµ)e

iω(t− xc0

)(2.5)

Ward(12) further shows that this can be expanded to a layered double dipole sheet,

where the phase propagation due to the microscopic susceptibilities allow

21

propagation for simultaneous values of negative permittivities and permeabilities. A

consideration of negative phase due to the condition that χε+χµ2

< −1 demonstrates

that energy conservation, or power conservation holds in this case because the group

velocity is this region is actually positive, even while the phase velocity is

negative(12). As we are comfortable that such material properties might be

possible, we should investigate the advantages of such a material in the first place.

2.1.3 Double negative index material (DNIM) possibilities

As we used in the Ward demonstration of a dipole sheet o show the possibility

of a DNIM, we can look at the far field of a single scattering object. Using far field

contributions, the optical resolution limit is set by the maximum value of θ for a

simple lens diagram as shown below in Fig 2.2. As a most simplistic model, we can

begin by looking for solutions to the equation of a directional point source, pointing

towards a lens of arbitrary height.

Ese(ikzcosθ+ikxsinθ−iωt) (2.6)

In this case we can see that in order to obtain the highest spacial resolution we can

simplify the dispersion through k → ωc

= 2πλ

. Then as θ → 90o, we expect that

∆x ≈ 2πk

= 2πcω

= λ in order for any noticeable contribution to the image plane from

and addition ∆x . A much more accurate derivation follows from Fraunhofer

diffraction, which is considered in the far field as well. Using a circular aperture of

radius r, we can describe the diffraction from a source which lies displaced from the

origin, along the line of propagation Ps. Using a formulation devised from Sharma

(36), we can express the spatial frequencies u and v as their polar equivalents.

u =r′′cosθ′′

λR′′0= ρcosθ′′ (2.7)

22

-

6

?

Image

6

Object

x

z?

@@

@@

@@@

@@@@@@@

ZZ

ZZ

ZZZ

ZZZZZZZ

HHHHH

HH

HHHHHHH

PPPP

PPP

PPPPPPP

(((((((

hhhhhhh

hhhhhhh(((((((

θ

Figure 2.2: Ray diagram for simple lens.This serves to illustrate the resolution limits set by the converging rays at the image

plane.

v =r′′sinθ′′

λR′′0= ρsinθ′′ (2.8)

The solutions to the Fourier transform for

F (ρ, θ′′) =

∫ ∞0

∫ 2π

0

t(r, θ)e−i2π(urcosθ+vrsinθ)rdrdθ are not based on θ which reduces

to an integral that can be categorized as a Bessel function.

F (ρ) =

∫ a

0

rdr

(∫ 2π

0

t(r, θ)e−i2πρrcosθdθ

)(2.9)

Bessel functions have a recursion relation that allows us to integrate for a first order

(5)

∫ [∂

∂x[xpJp(x)] = xpJp−1(x)

]= x′J1(x′) =

∫ x′

0

xJ0(x)dx (2.10)

This leads to our solution for F (ρ) = F (0)J1(2πρa)2πρa

. Plugging ρ = r′′

λR′′0back into the

equation allows us to find the first minimum at the first zero of

J1(2πρa) = J1(1.22π). If two different sources were created by two points, they

would be indistinguishable because of the principle of linear superposition. Thus

23

this directly relates to the diffraction limit of optics

2r′′ = 1.22λ

aR′′0 (2.11)

We can imagine several possible techniques to decompose the superposition of two

Figure 2.3: Airy disc - Fraunhofer diffraction from a circular aperture of radius5.0cm at 10[GHz].

such simple functions, however this limit has remained significant because of several

other limiting factors due to refraction based optics. We must note however, that

recent work has shown that in special cases, specifically biological ones, quantum

dots can be injected or attached to attain resolution limits of ≈ λ30

(20). This seems

to beg the question how does a DNIM allow for resolution beyond a normal material

lens.

24

2.1.4 The perfect lens

We can first see a very intriguing difference in the near field of a DNIM.

Looking at the wave vector k of a plane wave much like the one described Fig 2.2,

we know kz =√k2x −

(ωc

)2, thus if k2

x <(ωc

)2, kz ∈ = This implies that these near

field waves, also called evanescent waves, exponentially decay. We also notice that

because of the exponential decay our source must be relatively close in order to

capture these high spatial frequency waves. We can quickly notice that if we intend

to amplify these waves in the medium, instead of allowing them to be exponentially

suppressed it requires an ε < 0 and a µ < 0(33). We already know that we can

model a set of successive dipole sheets as a DNIM according to the work done by

Ward(12). We know that for ε and µ of the same sign we have a real k value, thus

with k = ωc

√µε = nω

c, we are allowed by Maxwell’s equations to have a negative

value for n. There seems to be no principle restricting us from a DNIM, except the

real material! Snell’s law states that n = sin(θ1)sin(θ2)

and is no different because of the

negative index material except the aforementioned refraction on the same side of the

normal. This leads to new simplistic imaging designs like that of Fig 2.4.

This fact alone is not enough to improve the resolution limit of a lens. The

amplification of evanescent waves is the key to the increased resolution limit of a

DNIM. We can look a a low loss dielectric with ε = −1 + iδ. This leads to an

approximated transmission

Tp ≈e−kxd

δ2 + e−2kxd(2.12)

(11). We can look at the limit as losses approach zero,

limδ→0

Tp = limδ→0

e−kxd

δ2 + e−2kxd= ekxd (2.13)

25

NegativeSpace

@@@@@R

>

ZZZZZ~

*

HHHHHj

:

XXXXXz

@

@@@@@@@@R

>Z

ZZZZZZZZ~

*HHHHHH

HHHj

:XXXXXXXXXz

@@@@@R

>

ZZZZZ~

*

HHHHHj

:

XXXXXz~ ~Object Image

Airn=1

NIMn=-1

Figure 2.4: Demonstration of a double negative index slab lenswith index of n = −1This is the Perfect Lens as stated by Pendry, which creates a

space(equal to the volume of the DNIM lens and and equal volume of air orvacuum) that appears to the detector not to exist, at least spatially.

However, amplification does not occur if kx >lnδd⇒ ∆pl = 2π

kmax≈ 2πd|lnδ| . We can

compare this to the limit of our simplistic normal lens ∆x ≈ λ and our new

∆pl ≈ 2πd|lnδ| . It is quite interesting to note that now the resolution limit is no longer a

function of the wavelength, but instead a function of the distance from the object to

the lens face. Of course we will hope for a near lossless case, and in this limit we

expect our resolution limit ⇒∞.

2.2 The Six Velocities of Light

Since metamaterials are specifically designed with the manipulation of both

the group velocity vg and the phase velocity vp, a closer look into these phenomena

and others surrounding electromagnetic radiation is fundamental to insuring no non

physical results appear from our theoretical explanations.

2.2.1 Free space velocities

Following the notations used by Brillouin (7) we know,

c =√µ0ε0 (2.14)

26

EvanescentMagnitude

~ ~Object Image

Airn=1

NIMn=-1

Figure 2.5: Demonstration of a simple lens using a DNIMwith index of n = −1. In this case the bottom curve(red) represents the amplitude

of the evanescent component of the wave.

where ε0 and µ0 are the permittivity and permeability of free space as defined

immediately following EQ. 1.5 . It is important to note that, though it may seem

possible initially through metamaterial designs, based on special relativity,

information in a wave cannot travel faster through space than this value. The speed

at which the phase of a specific frequency ω0 travels through space is defined as the

phase velocity.

W = vp =ω

k(2.15)

In free space this is simply vp = c. This velocity can be thought of as the motion of

elementary wavelets in the carrier. The group velocity vg is defined as the speed at

which a packet or collection of larger disturbances travel through space. In free

space the group velocity is not dependent upon the collection of frequencies that

make up the wave packet, since

U = vg =∂ω

∂k(2.16)

27

where ω is and individual frequency with k = 2πλ

, with k called the wave number

and λ is the wavelength of the individual frequency. The group can be thought of as

the modulation impressed on the carrier and is the result of a building up of some of

the groups of individual wavelets into a large amplitude which moves along with

velocity vg.

-Group

XXXXXXXXz?

Modulation

Figure 2.6: Wave packet showing a combination of wavesresulting in an envelope modulation. These individual waves, each moving at theirown vp, of a carrier are causing the larger modulation, which move at vg. This wastaken from Brillouin (7). We note the change in terminology from, the combinationof individual wavelets resulting in a larger carrier modulation used by Brillouin, to

the above.

We can now define the signal velocity, which governs the transfer of

information through a medium. Signal velocity, S, can be defined as the moment

when forced oscillations of the characteristic frequency of the incoming wave become

detectable in a medium. Generally speaking the signal velocity refers to the moment

when oscillations are on the order of magnitude of the input signal, but this does

not always need to be the case, as in the case of a lossy material. In free space, once

again

S = c = vp (2.17)

Energy velocity can be defined depending on the medium. In a non-dispersive

medium, or a medium which has absorption far from the from the frequency of the

28

incoming signal, it can be defined as the group velocity. However, in a dispersive

medium the energy velocity has to be defined as the ratio of the Poynting vector to

the energy density.

We can now define the last primary velocity which is the front velocity. The

front velocity, or the velocity of the wavefront, is defined as the speed at which the

smallest and most minute disturbance of the field propagates through a medium.

Since the energies at the wavefront are so small this makes them undetectable, thus

representing a rather theoretical space where the wave exist. In Brillouin’s (7)

derivation the wave requires a finite beginning and end, otherwise most of the

previous definitions become meaningless since the wave would otherwise exist

everywhere throughout space. In this case the front velocity also has a counterpart

called the end velocity, which is created by the beginning of a new wave of opposite

phase but equal shape and amplitude.Through destructive interference, these

amplitudes cancel, thus equivalently destroying all oscillations in the carrier. Both

of these are physically needed and they set limits on the other velocities.

2.2.2 Waves in a medium

The value of c was defined in vacuum and it does not change in a medium.It is

vp that is a function of ω or k and which is dependent upon the type of medium.

Lord Rayleigh used ocean waves as an example, using ∂2y∂t2

+K2b2 ∂4y∂x4

= 0 as the wave

equation for surface waves in deep ocean water.(7). With a phase velocity defined as

W = 2πλKb = Kbk = Bλn = B′k−n → U = W (1− n) then from EQ. 2.16 we can

now show that vg 6= c. With electromagnetic waves phase velocity is constrained by

the reaction of local oscillators. In typical metals we consider this to be made up of

electron, which usually have a characteristic plasma frequency in the UV range. In

the case that the phase velocity is near a local oscillators characteristic frequency,

the phase velocity is now a function of the local oscillators in the medium thus

vp → vp(k, ρ, t) with k = 2πλ

as the wave number, ρ as the density of oscillators and

29

T , which is temperature. We can now also look at the group velocity in a medium.

vg =∂ω

∂k=∂kvp∂k

= vp + k∂vp∂k

(2.18)

Demonstration of differences in signal, group and phase velocities can be shown

through the different paths of integration of two wave of opposite phase but equal

amplitude one beginning at time t = 0 the other at time t = T = nτ with τ equal to

a single period

f(t) =1

τ

∫ ∞−∞

dn

n2 −(

2πτ

)2

(ein(t−T ) − eint

)(2.19)

The most complicated velocity to predict in a dispersive medium is the signal

Figure 2.7: Brillouin’s diagram of integration paths - This picture is an extractionfrom” Wave Propagation and Group Velocity” (7) and is a representation of thedifferent results of taking different paths of integration from EQ. 2.19. — representsc/W where W = phase velocity , 2em represents c/U where U = Group velocity ,· · · represents c/S where S = signal velocity , ·-·- represents c/U1 whereU1 = energyvelocity

30

velocity. This requires rigorous work to define the time and space between the

wavefront and the onset of the first forced oscillations after the forerunners. Since

the forerunners exist before the forced oscillations they do not have the

characteristic frequency of the signal and in a dispersive medium actually follow two

different paths.

Figure 2.8: Brillouin’s differentiation of signal and front velocities. - This picture istaken from” Wave Propagation and Group Velocity” (7). This is a schematic of wavepropagation in a medium, where the signal velocity can be differentiated by the firsttwo groups of forerunners through its characteristic frequency and magnitude.

2.2.3 Super-luminal speeds

The initial interest in meta-materials was to create phase velocities or group

velocities not found in nature. There is no restriction on vp or vg in a dispersive

medium, but this still does not imply that the signal velocity is greater than c. This

important consequence also shows some of the theoretical limits of our previous

derivations as we have, for the most part, neglected the time evolution of the wave

and considered only steady state solutions.

31

Figure 2.9: Brillouin’s diagram of propagation speed in a medium. - This pictureis taken from” Wave Propagation and Group Velocity” (7). This is a demonstrationof the propagation speeds of the phase and wavefront, or c as a function of distancein a medium.

32

Figure 2.10: Negative index lens with plane wave incident.Uses COMSOL R©Multiphysics 4.1a RF module

2.3 Pendry’s Perfect Lens Revisited

33

Figure 2.11: Negative index lens with cylindrical scatterand plane wave incident using COMSOL R©Multiphysics 4.1a RF module

34

Figure 2.12: Cylindrical scattererand plane wave incident using COMSOL R©Multiphysics 4.1a RF module

CHAPTER 3 : Meta-Atomic Constituents

3.1 Overview of Possible Metamaterial Constituents

There has been much focus on metallic like resonators or metal lattices as a

primary way to induce both electrical and magnetic resonances. This has been the

predominate focus of authors such as Pendry (19) and Kong (2) for almost a

decade. The strengths of these types of resonators also lead to their inherent

difficulties, predominately conductivity, which inherently contains a mechanism for

resonance as well as losses. Despite this fact, the ease of design due to modern

circuit theory as well as the ability to produce inductance, or a magnetic response,

has lead these materials to be some of the most useful metamaterials to date.

3.2 Metal Resonators

Split Ring Resonators (SRR), U-shaped or S-shaped resonators are the most

popular choices for early metamaterial designs due to their known LRC Circuit

equivalences. These structures typically are sized on the order of or preferably

smaller than the wavelength at which they resonate. Most groups work with these

materials because of the induced magnetic response near or at resonances which in

most cases can become the dominant feature in the effect of the SRR on wave

propagation in the bulk material. Current SRR’s are focused on a bulk material

response and are designed to push the barrier for metamaterials farther into the

optical regime.

36

3.2.1 Size limitations

There are fundamental limits to this simple scaling down of SRR’s. As these

get physically smaller and smaller and so does their effective operational

wavelength. The first issue is based on the frequency dependence of metallic

conductivity. or the frequency dependence of the electrical susceptibility. We will

show in section 5.1 that the standard, in the microwave regime, of approximating a

metal as a simple electron gas with conductivity as its governing factor does not

apply in the regime where dimensions become small and where plasmonics effects

can dominate. There is another limiting factor that was not normally considered.

Once we scale down this effective LRC circuit, the magnetic energy used to draw a

current through the circuit also decreases. Initially we have a simple relationship

where U∝12LI2. As this approaches smaller and smaller sizes, it begins to infringe

upon the metals natural ’kinetic inductance’ (43), where

1

2Nmv2 =

1

2LkinI

2 (3.1)

but this does not depend on the size of the SRR. We know with a LRC equivalent

circuit the resonant frequency based on the inductive and capacitive properties is

ωLC =1√LtotC

(3.2)

where Ltot = L+ Lkin, thus the resonance is dominated by the kinetic term at small

frequencies.

Additional limitations arise when thinking of practical applications, as we can

envision trying to create the perfect lens Fig 2.5, and we would need this to be a

omni-directional negative index material. We could imagine using a biaxial or

uniaxial metamaterial for many purposes, but this would not display the

37

characteristics of the perfect lens that was first suggested. There is also a question

of how we expect evanescent waves to couple into the material. If the predominate

effect is a resonance created in the SRR, and the SRR are on the order of the

wavelength this seems to indicate that the highest spatial response will also be on

the order of the wavelength. We have been looking for practical solutions in the

microwave regime currently, thus we can take these limitations and redirect our

study towards reducing the size of the SRR’s in order to address both of these

concerns.

Realization of subwavelength, λ50

, SRRs will allow for bulk material

properties to resemble those of the individual resonators or have less predictable

effective index values, depending on the effective coupling with their neighbors.

Given sufficiently small resonators we can randomize the orientations and embed

these in a substrate of our choice and create this type of omni-directional

metamaterial and this has been shown with frequency selective surfaces (18).

3.2.2 Geometrical scaling

We will assume we are generally outside the region in which kenetic

inductance plays a role as we intend this structure for the microwave regime, nearly

105 orders larger. We can begin with a simple model for a standard type of SRR.

Using the Drude model of a material we previously defined however in this case we

would like the addition of the intraband transition, which after some simplification

takes the form (40)

ε = ε0

(εr −

ω2p

ω2

)(3.3)

Where εr is the relative permittivity due to the intraband transitions. Making the

general assumption that the current density is uniform across the cross section of

38

the SRR, we can then relate this to the external field

Je = jωε0

(εr −

ω2p

ω2

)E (3.4)

The relationship between the total current passing through the cross section and the

voltage along the SRR perimeter can be found using the current amplitude

I = jewh (3.5)

where je = |J| and a voltage

V = E(leff ) (3.6)

where leff =(π2

)l, due to simple circular resonator approximation, with length l

equal to the physical length of the SRR.

⊗

⊗

⊗

⊗ ⊗

⊗ ⊗

⊗

⊗

⊗

⊗

⊗ -δ

6

?

l2

- w

Figure 3.1: General schematic for the SRR as proposed by Tretyakov(40)It should be noted that this δ is of no relation to the previous notation and simply

signifies the gap distance between the arms of the split ring resonator.

39

Simple substitution leads to

I =

(jωε0εrwh

leff+

ε0wh

jωleff

)V (3.7)

Simply identifying units we can pull out an additional capacitance

Cadd =ε0εrwh

leff(3.8)

and an inductance

Ladd =leff

ε0whω2p

(3.9)

which leads to the equation for the current as a function of the LRC equivalents

I =

(jωCadd +

1

jωLadd

)V (3.10)

This leads to a resonant frequency ω0 = 1√(L+Ladd)(C+Cadd)

(40) from the equivalent

circuit model. Simplifying the geometric model of an SRR, the inductance can be

approximated as

L ≈ µ0l

4lv

8l

w + h(3.11)

while the capacitance is written as

C ≈ ε0wh

δ(3.12)

A further approximation could be made for a round loop inductor L = µ0aln8ar0

,

where the loop radius is a = l4

and the equivalent wire radius r ≈ h+w4

(39). This

40

leads to a resonant frequency,

ω0 =1√[

1 +(π2

) (lδ

)] [ε0µ0

(wh2π

)ln(

8lw+h

)+ 1

ω2p

] (3.13)

Typically the goal has been to reduce the denominator in order to increase the

resonant frequency in order to make SRR’s effective in the optical regime.With this

is mind the typical graph has shown something on the order of terahertz as a

limiting factor. This is due to the fact that even if all geometric terms go to zero,

(40)

ω0 →1√[

1 +(π2

) (lδ

)] [1ω2p

] (3.14)

then we can also say that(lδ

)max≈ 1, since the largest values for δ will always be

less than l is our simple model. This limiting case indicates, without the

introduction of losses in the metal, at the resonant frequency approaches the order

of the plasma frequency as the SRR size approaches zero.

ω0 →[

2

1 + π

] 12

(ωp) (3.15)

This general trend can be seen in Fig 3.2, as the plasma frequency used here

was ωp = 2.17× 1015Hz.

3.2.3 Scaling down

Ideally we would like to shrink the size of the SRR while maintaining the

resonant frequency. With the advantages of shrinking the size of the SRR discussed

above, we move towards modeling the initial SRR in order to investigate what effect

changes to the shape or material composition might have to the resonate frequency.

Using the LRC equivalent circuit model we examined we will look at the trends for

41

varying several variables, such as permittivity of the metal, the scaling factor,

defined as β, and the gap distance δ.

We need to guage the contributions of the individual parameters mentioned.

Fist we will take l = βli, h = βhi, w = βwi and δ = βδi. There should be no reason

this model can not be scaled from the initial terahertz region to the microwave,

[GHz] region, since this is our region of interest. Figure 3.3 shows the resonant

frequency f0 as a function of 1β.

The attempt to show that scaling can bring the resonant frequency to the

−1

Figure 3.2: This graphs shows a response from the initial reading (40), where the resonant frequency can be seen to converge to near the plasma

frequency, even at size near the wavelength

42

y = 0.4389× 1014x+ 797.36

−1

Figure 3.3: Graph of split ring resonator resonant responseFrequency f0 versus the inverse of the scaling factor 1

βis shown.

microwave range shows an approximate value of β = 6.39× 104 for the scaling

factor. Figure 3.3 also shows that the geometric scaling can easily be applied in the

GHz frequency range. The cost of testing equipment, availability of materials and

military applicability are driving factors to first show proof of concept of a

metamaterial’s properties in the microwave regime before developing them for higher

frequencies. In this case we are looking for a resonant frequency near or at 10[GHz].

We can now solve our equation for the given resonant frequencyf0 = 1010Hz to

arrive at scaling factor of β = 6.39× 104. We can now move to trying to reduce the

scaling factor while manipulating other variables in order to reduce the overall size

43

of the SRR. We will first look at the gap between the SRR ends δ, from figure 3.1.

First we will take a look at δ at the current scale factorβ = 6.39× 104. We will now

define δ = κ (βδi) as an intent to scale the gap distance, δ, while keeping the

remaining dimensions constant. The figure 3.4 demonstrates that it is possible to

Scale Factor= 6.39× 104

Figure 3.4: Plot of the resonant frequency,when compared to the independent scaling of the gap distance δ at scale factor

β = 6.39× 104

move the resonant frequency by simply adjusting the gap between then ends of the

SRR. We can now begin to change the scale factorβ and see if we can compensate

with the scaling of the gap length δ. Figure 3.5, is our first guess at rescaling both

the scale factor and the gap size. The figure above is a first step in proof of concept,

44


Figure 3.5: Scaling continuedBy decreasing the initial scale factor β to β = 6.39× 103

since we have now scaled our original SRR by a factor of 10−1. We have shown that

it is possible to compensate for the offsets in scaling by rescaling the gap distance δ

by a scale factor of κ ≈ 10−2. Unfortunately this does not come near our initial goal

of a size reduction to λ50

. Thus we can try once again to see if further rescaling can

be done by simply adjusting the gap size.

45

FIG. 3.6 is the final attempt we present here to use the separation between

the ends as a means of rescaling the SRR. We further reduce the scaling factor of

the SRR. With a scaling factor of β = 6.39× 102 we are nearing our goal of λ50

. This

however pushes the scaling factor of the gap another two orders of magnitude to

κ = 10−4. With such a small scaling factor the gap distance is approaching normal

limits in manufacturing δ ≈ 10−8m = 10nm.


Figure 3.6: Scaling factor further reducedto β = 6.39× 102 the gap size has to decrease by a factor of κ = 10−4.

3.2.4 Continued scaling possibilities

Since the entire SRR model was introduced as a metal conductor in air, it is

possible to immerse this into a dielectric and through modification of the dielectric

46

Figure 3.7: Backward waveWhen a negative index metamaterial is realized, then because the phase and group

velocites have opposite sign we observe a so-called backward wave. In one of themetamaterial structures we have studied, based on a meandering line or S-structure,

we can see this phenomenae

properties, expect that new resonances might be found. An initial attempt was

made to simply increase the permittivity of the surrounding medium from εr = 1 to

εr = 4. The intention being that this would possibly quadruple the resonant

frequency, since the resonator would essentially see a new shorter effective

wavelength λnew = λ/4 equal to 1/4 the original wavelength. This, in retrospect,

was naive attempt since the wave, upon exiting the dielectric, would simply return

to the original dimensions, thus removing any effect we might hope to have.

CHAPTER 4 : FORM BIREFRINGENT STRUCTURES

4.1 Form-Birefringent Materials

Unlike resonators where much of the focus lies in the possibility of creating a

negative phase velocity from simultaneously having both negative permittivity and

negative permeability, form-birefringent material structures can attempt to exploit

the group velocity to create a negative group velocity as well as other interesting

unique properties. It is due to this group velocity manipulation that birefringent

materials have the possibility of being narrow band filters, producing large field

enhancements and even being negative group index materials.

In 2003 a group headed by A. Mandatori published a paper looking at a 1D

system consisting of alternating dielectric layers. Each layer consisted of an

anisotropic material. By alternating specific anisotropy the group showed that it

was possible to create a negative group index through an anomalous phase shift

these structures. The simplest version of the Mandatori structure is shown in Fig

4.1 (1) (26)

Although it was shown that a birefringent material could produce a negative

group index, this was using structures with high anisotropies. With normal

materials this index contrast is relatively low. Naturally occurring anisotropies

occur in materials such as quartz with ∆nn

= 0.6% and LiNbO3 with ∆nn

= 3.8% ,

which are not high enough to achieve noticeable results with the anomalous phase

behavior. (1).

The published Mandatori stack experiment showed instead of a large stack, a

simplistic single layer stack like that in Fig 4.1 could be modeled more easily. Given

48

Figure 4.1: Simple Mandatori structureThis consisting of a single anisotropic layer, which combined with a transmitted

linearly polarized wave and a received linearly polarized wave, ccontrols one periodof variable path length(26).

the initial direction of the Poynting vector, or the direction of travel of a polarized

plane wave as z, the birefringent layers are aligned such that for nx and ny and have

nxny> 1. With a given thickness d along the direction of propagation for the

birefringent layer, this then implies a splitting ratio

α =|Ex||Ey|

(4.1)

which is determined by the angle between the electric field alignment of the

polarizer or horn and the x direction of the birefringent material. This then leads to

a center operating frequency of

ωm = mcπ

d∆n(4.2)

where m is and integer greater than zero, c is the speed of light and ∆n = nx − ny.

49

Figure 4.2: Schematic of incoming wave onto birefringent layerAlso Shows with exit polarization change.(1)

This leads to the group index of refraction at the center frequency ωm as

ng = ny

(1− α∆n

(1− α)ny

)(4.3)

We can also define the frequency range, or bandwidth, of the anomalous behavior as,

∆ω =2c arccosα

d∆n(4.4)

where the transmission can also be shown to be

T = (1− α)2 (4.5)

Using the equations above it is easy to see that standard materials, which

have a birefringence ∆n ≈ 0.1%, would require a splitting ratio of almost 1 in order

to have a negative index, which leads to transmission loss of the order 40dB.

Though a few materials exhibit birefringence in the range of .6% ≤ ∆n ≤ 4% (13),

50

we intend on attaining a much higher value, as a ∆n = n = 100%, with a

transmission throughput of around 25%, would be useful in many applications.

4.1.1 Gigantic Anisotropies

The large birefringence needed for the above applications for negative group

delays or effective negative index can be attained using form birefringence in

appropriately doped host polymeric materials. Low cost materials such as plastics

or other easily produced materials can be formed with slits, columns or other

patterned structures to induce form birefringence. The birefringence due to simply

lines or slits can easily be calculated using the effective index neff of the original

material and the filling ratio r, where r ≤ 1. We can calculate the index along the

polarized plane wave electric field oscillations(TE),

nx =√rn2

eff + 1− r (4.6)

Likewise, we can calculate the index along the polarized plane wave magnetic

field oscillations(TM)

ny =

(r

n2eff

+ 1− r

)− 12

(4.7)

A quick guess might lead one to speculate that a maximum contrast would be

found at a fill percent of r = .5 and this is the case. For r = .5,

(∆n

nx

)=

1

4

(n− 1

n

)2

(4.8)

With materials that are readily available index contrast of greater than 20%

are possible. Initial attempts to create such a material were done with a Stratasys

FDM Titan rapid prototyping machine at Western Carolina. The effective index of

ABS plastic is 1.57 at 10GHz (25).

51

Figure 4.3: Graph of bandwidth vs splitting ratioThis demonstrates the large variability dependent upon the rotation angle of the

form birefringent disc

Initially discs were manufactured with a ∆n ≈ 0.14 which leads to an index

contrast of ∆nnx

= 0.141.31

= 11%. However in order to move the center frequency ωm2π

given in EQ. 4.2 to 10[GHz] the thickness of the disc has to be d ≈ 9× 10−2m or

9.0cm.

Although not initially problematic, this stack of ABS discs is only for the

simplistic Mandatori structure, which requires only one period. The initial

conception of the Mandatori stack was motivated by exploiting the large field

enhancement that occurs near band edges, spcifically at regions the degenerate band

edge or split band edge regions. The regions have field enhancements proportional

to the fourth power of the number of periods, which consist of alternating

birefringent discs. In order to realize these field enhancements in a real material, a

52

Figure 4.4: Form birefringent ABS plasticMade at Western Carolina University by a Stratasys FDM Titan rapid prototyping

machine, with .5mm and larger slits.

higher ∆nnx

needed to be realized. Since the losses of the material would increase in a

real experiment or material considerably over a greater length, lower losses were also

required. This is not due to the intrinsic losses in the material, but rather the

stringent requirements for each period to be nearly geometrically identical.

Estimates suggest that anything above 0.1% disorder will make drastic changes to

the field enhancement and concerns are known (14), but exact numbers are not yet

published.

53

Figure 4.5: Graph of index value for TE, nx and TM nyThis is for a form birefringent disc of ABS plastics with neff = 1.57 as seen in

4.4(26).

4.1.2 The Search for Higher Index Materials

At microwave frequencies, specifically at X-band, materials with an index

above 2 are not difficult to find, but are usually expensive and hard to machine,

thus the search was limited to readily available inexpensive materials. The form of

the ABS disc was still perfectly useful and thus was used as the test subjects for all

kinds of ABS dopants and coatings.

With one of the highest permittivities of easily obtainable materials, SrT iO3

was high on the list of dopants to test as a suitable rapidly manufacturable form

54

birefringent metamaterial. However dopant particle sizes were limited and this lead

to the use of an epoxy based host, which allowed for a very low mass fraction of

SrT iO3 to be successfully mixed. Because of the fast drying of the epoxy that was

chosen inhomogeneity of dopant particles was an issue as well. Particle

dispersement in the host polymer became a larger and larger problem as the mass

fraction increased. This is the cause of the noticeable dip in the increasing index for

the highest mass fraction sample, as surface defects and air pockets in the host

lowered the effective index.

Figure 4.6: SrTiO3Epoxy - Graph for SrT iO3 doped epoxy formed disc

An investigation into easily accessible dopants lead to a few very widely used

materials. One example is rutile TiO2, which is available in almost any size due to

its widespread use and its listed permittivity. Its permittivity is listed as

ε11 = ε22 = 86(13), though this is only listed for 10[kHz]. In the microwave regime

we know it has a εr ≈ 20. Mixing the TiO2 into host materials was successful on

many fronts, several photopolymers were used as well as a polyamides plastic host.

We were able to place a higher mass fraction in the polyamides plastic host εr = 2,

because of the technique used to form the disc, injection molding, which allowed a

much more viscus material to form and did not cause the anticipated issue problems

55

found with the photopolymers.

Figure 4.7: TiO2Nylon - Graph for TiO2 doped polyamides plastic

Though these discs definitely provided a high ∆N , it was easy to see they had

reached their limit and we needed an approach which would lead us to a much

larger index.

Through some intuition and sheer luck a specific coating was found that

increased the index significantly with additional layers. These gold colored ABS

discs were initially tested to have an increased ∆nnx≈ 35%.

After ≈ 30 coats, TE index values approached nx ≈ 4.0− 5.0

Gold ABS Index Comparison

Disc Index Index ∆nNumber TM(ny) TE(nx)1 1.406 4.508 3.1022 1.378 4.479 3.1013 1.476 4.750 3.2754 1.453 4.812 3.3595 1.508 4.830 3.3226 1.448 4.039 2.591

.

Table 4.1: Tabulated data for gold color coated ABS disc.

With such a high index testing became difficult since we were using only an

56

Figure 4.8: Graph of index value for TE, nx and TM nyThis is for a form birefringent disc of 6 coats of gold colored coating ABS plastic.

approximate plane wave and Fresnel reflection at the boundary became an issue.

Since the ratio of the transmitted wave along nx compared to ny was affected by

this as well, readings became even more difficult at angles with lower transmission

coefficients. Nevertheless, the benefit of such discs is still apparent since with a

∆n = 3.3 to find a center frequency near 10GHz a stack of thickness of only

5.0× 10−3m or 5.0mm thick was needed to observe a phase advance. Though

transmission is still theoretically possibly near a resonance, this was largely ignored

for our initial experiments.

57

4.1.3 Tunable Negative Group Delay

The next phase in using these stacks was to show a tunable negative group

delay around the calculated center frequency. This was done using the effective

delay detected by a VNA as the disc was rotated to change the splitting ratio and

thus the effective index given by equation EQ. 4.3. The first attempt was made

using the original ABS disc with low ∆n, shown in Fig. 4.12 and Fig. 4.9. Further

experimental data are provided with other types of form birefringent disc. Gold

painted discs are shown with a stack thickness of d = 2.5cm and a ∆n = 0.6 in Fig

4.10 and Fig 4.13. Investigation led to the production of TiO2 doped polymer discs

with .5mm slits water machined into them. Advantages of these disc were a higher

anisotropy, ∆n = 1.2 and low losses compared to other materials with a similar ∆n.

A stack of these discs was even shorter with a d = 1.29cm, but it turns out there

were still issues with measurements due to losses Fig 4.11 and Fig 4.14.

Figure 4.9: Phase relations of ABS - Graph of phase response at differing anglesversus frequency of the d = 9.0cm stack of ABS disc.

4.2 FBS Discussion

Fig 4.9 to Fig4.10 are a demonstration of the ability to control the relative

phase delay of a form birefringent structure through mechanical rotation. The plots

also verify the relations stated in EQ. 4.2 through EQ. 4.6. The last plots in each

58

Figure 4.10: Phase relations of Gold Coated ABS - Graph of phase response atdiffering angles versus frequency of the d = 2.5cm stack of gold painted ABS disc.

Figure 4.11: Phase relations of Ti02 doped polyamides plastic - Graph of phaseresponse at differing angles versus frequency of a d = 1.29cm stack of TiO2 dopedpolyamides plastic disc.

59

group Fig 4.11 and Fig 4.14, showed some additional unexplained data points, that

needed to be looked at. It turns out this was not the first case of irregularities such

as this tended to be the case for very high loss materials, or those with higher

reflection coefficients. Additional restraints on test equipment, such as reduction is

cross sectional area for testing caused additional transmission drops and this began

to effect the reliability of our test set. This general line of questioning has recently

caused an complete revision of our test setup in order to better prepare for

materials with higher losses that would normally be expected.

4.3 Measurement Techniques

relative index measurements

Figure 4.12: Transmission for ABS stack - Graph of transmission versus frequencyat varying angles of the d = 9.0cm stack of ABS disc.

60

Figure 4.13: Transmission for gold coated ABS stack - Graph of transmission versusfrequency at varying angles of the d = 2.5cm stack of gold painted ABS disc.

Figure 4.14: Transmission for Ti02 doped polyamides plastic Stack - Graph oftransmission versus frequency at varying angles of a d = 1.29cm stack of TiO2 dopedpolymer disc. The odd data spike is noted and will be discussed later in the measure-ment techniques section.

61

4.4 Experimental Setup

Typically our experimental results are verified by a time domain technique as

well as a S-parameter technique. As we attempt to investigate what could be

materials with and index n < 1 we need to be clear what kinds of results we expect

and also acknowledge any flaws inherent to the types of data we are collecting.

4.5 VNA phase change measurement setup

The current VNA test setup uses the incident wave amplitude to derive the

phase of the received wave and compares this to the transmitted wave received with

a material in place in order to derive the phase change. An HP 8150A Test Set is

used in conjunction with a set of free space rectangular horn antennae. Both cabling

and horns are optimized for performance from 8.2-12.4[GHz]. Once the equipment

is set up, the initial phase can be changed by an electronic delay that merely offsets

the output driving frequency, thus with the current setup the initial phase can be

set to anything we choose −180o ≤ θ ≤ 180o. With careful observation this allows

any material that has a relative phase shift of less than ±360o to be calculated with

confidence. Relative phase shift is an important distinction as we are typically

comparing this shift to that of air or approximately vacuum.

4.6 Measurement calculations

The calculation of phase shift can be done with two different methods. With a

gantry system, a transmitter and detector/receiver can be placed against each other

as in Fig 4.17. This will define the reference phase for this system. Then they can

be separated to the exact thickness (d) of the DUT as in Fig 4.22 in order for a

second measurement to be taken. In this case the calculation of the index of the

DUT is n = cv, with c as the speed of the EM wave in vacuum, v as the speed of the

EM wave in the medium. Using v = fλ ⇒ cfλ

This leads to λ = dN

, where λ is the

wavelength is the medium, d is the thickness of the medium and N is the number of

oscillations. ⇒ cf dN

. The number of oscillation can be calculated from the phase

62

Figure 4.15: Horn Measurement Reference - Reference for two horns initially touch-ing measurement

change where N = ∆θ360o

The final calculation for the index becomes

n =∆θ × c

f × d× 360o(4.9)

4.7 Stationary Horns Measurement

In this case the horns will already be separated by some distance d. As in Fis

4.23, the reference already has a phase delay associated with the volume of air with

thickness d. This will define our reference phase for this system. In this case, we

take into account the phase delay created by the existing volume of air. Using EQ.

4.9 we already know the associated phase delay

1 =∆θair × c

f × d× 360o(4.10)

Replacing the entire volume of air with a DUT with measured phase delay of

∆θDUT we end up with

n = (∆θair+∆θDUT )×cf×d×360o

We can then manipulate this equation to

63

Figure 4.16: Experimental diagram of volume filled with air - Reference measure-ment for DUT with stationary horns

n = ∆θair×cf×d×360o

+ ∆θDUT×cf×d×360o

,

since the first contribution is EQ. 4.10, this leads to

n = 1 +∆θDUT × cf × d× 360o

(4.11)

This is the equation we have used consistently since October 2008 for

calculations to determine the effective refractive index from the VNA. If d is found

in cm this equation is simplified further as

n = 1 + ∆θ×3.0×108

10×109×d(cm)×10−2

⇒ n = 1 +∆θ × 3.0

360od(cm)(4.12)

64

The following pages contain a set of illustrations meant to clarify the

measurement system and explore the possibilities for various phase changes and

example possible negative index material measurements.

This is used as an example because of a continuing debate on the ability of

scattering parameters to accurately describe the phase response near a resonant

peak. Since the structure specifically is engineered to behave this way near the

measured frequency, a more through investigation into the physical interpretation of

all constituent variables in the scattering parameter matrix and approximations

made internally with out VNA must be looked at. We also have some limitations

with the free space horns which required modification in this particular experiment.

The cross sectional area of the horns was much greater than that of the stack of S’s

thus the predominant signal detected was that which simply propagated by the

material. Since the VNA processes the total received signal, the magnitude of the

unaffected wave alters the outcome quite significantly and it approximates a black

box with a mix of air and material as a uniform 1-D scatterer. This problem is what

led to the motivation to use a dielectric wave guide. Though losses were greater in

the dielectric waveguide, the transmitted wave was passed solely through the

intended material thus measurement accuracy was increased greatly.

Results

Phase ThicknessMaterial Advance Index (cm)2-S 243.06o 0.138 2.3514-S 590.47o −0.045 4.7076-S 962.25o −0.121 7.154

Table 4.2: Teflon R© rod usedComplete phase wraps were inserted 4-S( 1 wrap) and 6-S(2-wraps) structure.

Because of high losses, the results for the 6-S structure have up to ±50o of variation.

65

Figure 4.17: Reference for two horns initially touching measurement type

Figure 4.18: Air as either reference or demonstrationThis displayes the separation of horns from Fig 4.17

Figure 4.19: Measurement for DUT 1This is to help demonstrate different possibilities for phase delay of normal material

66

Figure 4.20: Air as a reference measurement

Figure 4.21: Measurement for DUT 2This is to help demonstrate different possibilities for phase de-lay of normal material. In this case the measured phase delaywould be X, as the reference would already have a delay of 2X.

Figure 4.22: Measurement for DUT 3This is to help demonstrate different possibilities for phase delay of normal material.Again this situation would have a measured delay of 2X as the reference value alreadycontained the additional 2X phase delay.

67

Figure 4.23: Air as a reference measurement

Figure 4.24: Measurement for a DUT 4This shows a phase advance. In this case the measured phasewould be -2X as the reference already contained 2X phase delay

Figure 4.25: Measurement for a DUT 4Again, this shows a phase advance. In this case the measurement would be -4X asthe reference contained a +2X

68

Figure 4.26: Teflon R© rod initial - A solid Teflon R© rod was used to test the polar-ization of the exit wave

4.8 Teflon R© Rod Polarization Test

We have had several questions about how well the Teflon R© rod maintains the

wave’s polarization. Below is comparison data between our initial rectangular horns

and the Teflon R© rod, both of which are rotated through 180o to test transmission

polarization. The setup in Figure 4.27 has the waveguide adapters at a 20o angle

and has no transmission (implies transmission magnitude is equal to the power with

no waveguides present) without the Teflon R© rod.

69

Figure 4.27: Teflon R© rod polarization graph - Graph of Transmission loss as thereceiving waveguide adapter is rotated 180 o

Although initial transmission power is greater with the rectangular horns,

both show the same general profile. The profile of the horns also demonstrate a

slight hump, which is a know effect of this shape, which is avoided with the use of

the dielectric waveguide.

70

Figure 4.28: Teflon R© rod angle measurements - Illustration of how the angle wasmeasured, which results in approx ±0.5o error

Figure 4.29: Metal Horn Graph - Graph of transmission loss as the receiving hornand waveguide adapter are rotated 180 o

71

Figure 4.30: Metal Horn Angle - Illustration of how the angle was measured, whichresults in approx ±0.5o error

4.9 Additional Considerations for Measurement Error

4.9.1 Cabling Sensitivity

Variability in measurements was noticed when preforming the initial Teflon R©

rod test. Rotating the horn 1800 did not have the desired and expected result for

either the horns or the rod. The peak Transmission power was not realized at 0o

and 180o. This was because of a slight bend in the cable. For this particular set of

data this was not very important, but for scattering parameter measurements it

would ruin the measurements system. Provided are some images to show how

sensitive the test set is and why calibrations must be performed regularly.

We have also recorded how simple orientation of the experimenters body

makes a significant , though < 2o change as well. All of the variations can be

eliminated if the system becomes automated. This suggested setting up Labview R©

and using the 8512 test set with external calibration provided free through NIST

and a possible automated mechanism for placing the DUT in and out of the field.

72

Figure 4.31: Bend Angle Response 1 - This image illustrates the greater bend anglethat caused ≈ −1.5dB change in transmission

73

Figure 4.32: Bend Angle Response 2 - This image is used as the reference for theimage above the change in angle

74

For further discussions on s-parameter measurements. (10)

Figure 4.33: Bend Angle 3 - Demonstration of difficulty detecting the cable benddifference, as this is the greater bend

75

Figure 4.34: Bend Angle 4 - This image is used as the reference, again, to showdifficulties if we are constantly moving cabling.

CHAPTER 5 : Modeling Techniques

5.1 Modeling Overview

Over the course of several years we have conducted a large number of

numerical experiments using several different off-the-shelf softwares as well as

creating code on our own. Currently, and for most of the finite element simulations

shown below, COMSOL R©4.1a was used on the Mosaic Linux Redhat provided

through the William States Lee College of Engineering here at UNC-Charlotte. In

conjunction with Matlab R©Comsol R©4.1a has provided valuable insight into the

physical expectations of not only a negative index lens, but a plethora of insight

into FBS structure as well as scattering parameter calculations and more. We will

attempt to demonstrate some of the various limitations to the theories, assumptions

and even the software itself in this section.

5.1.1 Drude or Drude-Lorentz models

By default, COMSOL R©uses the bulk conductivity as an approximation for

metallic response when using the RF package(9). We need to take a look at the

limits of this assumption as we simulate a wide range of devices and materials for

optical to microwave regimes. As before we broke up the bound and free charges,

however we never looked at the resultant combination for metals. Making a slight

notation change when compared to 1.23 and a dimensionless weight factor fi, the

free electron effects or intraband effects are summarized using the Drude model,

ε(f)r (ω) = ε(∞)−

foω2p

ω (ω − iΓ0)(5.1)

77

The effects of the interband charges, can be summed up exactly the same as the

Lorentz oscillator model but with a dimensionless weight factor to represent the

density of electrons with this lifetime plasma frequency etc...

ε(b)r (ω) =k∑j=1

fjω2p(

Ω2j − ω2

)− iωΓj

(5.2)

We know at long wavelengths that the contributions from interband transition

states will be small, but where does the simplistic Drude model break down and

require this more complicated structure? We already have an idea as the plasma

frequency of the free electron model gives us a regime in which we expect a resonant

feature to occur. Thus at least in this area any other contributions, no matter how

small can have significant impact on the response of the metal. We can take a quick

look at this if we simply compare the values for a frequency range of given

metals.Using a previously developed Matlab code ”Drude-Lorentz and

Debye-Lorentz models for the dielectric constant of metals and water” (41), which

contains experimental values for several noble metals for fj, ωj and Ωj from the

work of Aleksander D. Rakic (4).

78

Figure 5.1: Lorentz-Drude Refractive Index of Copper - . Demonstrates the dras-tic fluctuation near the plasma frequencies associated with both the intraband andinterband free electron charges.

79

Figure 5.2: Loreontz-Drude Refractive Index of Silver - . Demonstrates the dras-tic fluctuation near the plasma frequencies associated with both the intraband andinterband free electron charges.

Figure 5.3: Cu Lorentz-Drude and Drude Permittivity Comparisons - . There is littledifference to be seen at this scale, which highlights why it is of little consequence formost cases to include interband transition effects

80

Figure 5.4: Plasma Frequency Responses - Loretz-Drude and Drude Comparisons.A close up view around the plasma frequencies illustrates the major difference ifworking within this regime.

Figure 5.5: Ag Lorentz-Drude and Drude Permittivity Comparisons - . Little dif-ference can be seen at this scale, which highlights why it is of little consequence formost cases to include interband transition effects

81

Figure 5.6: Plasma frequency Responses - Lorentz-Drude and Drude Comparisons.A close up view around the plasma frequencies illustrates the major difference ifworking within this regime.

82

Much of our modeling has been to assist or illustrate principles, but modeling

can only solve the equations we provide. We can see that for simple metallic

scatterers there is hardly and detectable difference at all even when approaching the

optical regime.

Figure 5.7: Conductivity only Response Figure 5.8: Frequency Dependent Model

Figure 5.9: Figure text for Fig 5.7 and Fig 5.8Comparisons between figures in order to demonstrate the slight differences createdthrough a frequency dependent model which includes interband contributions to theconductivity.

The two illustrations of the SRR’s one with a frequency dependent model the

other using bulk conductivity shows large difference due to the interband

contributions compared to the Drude model. We highlight this difference specifically

to demonstrate the difficulty when simulating, in this case, not only near a resonant

geometrical feature, but a material resonance as well. It turns out we can see clearly

that this difference will even affect the far field scattering pattern significantly.

83

Figure 5.10: Frequency Dependent Permittivity SRR - Frequency dependent splitring resonator with scattered field shown. When compared to Fig 5.11, we can clearlysee a significant change in the field distribution.

Figure 5.11: Non-Frequency Dependent Permittivity SRR - As we compare thisimage to the previous 5.10 we can see that when working in or near a resonance, eventhe smallest difference are magnified.

84

5.2 Negative Index Properties and Computational Restrictions

We have looked at the plasmon resonances but we have never discussed where

these effects really play a large role, namely on the surface of several types of

materials. The set up to excite surface plasmons does not require that we have and

specific structure, only a boundary with an alternating ±ε, or possibly ±µ.

k(ω) =ω

c

√ε1ε2µ1µ2

ε1µ1 + ε2µ2

(5.3)

Given the correct conditions, which will easily be satisfied in any orientation

of the metal sphere, a surface plasmon resonance can occur and cause a greatly

enhanced scattered field. However the focus of this section is to point out that if by

chance we have ε1µ1 = −ε2µ2 our k vector will tend to ∞, which usually causes

problems for any solver, as even a high spatial cutoff will then arbitrarily be loosing

information, where the real material may not. We will see that this applies rather

well to our next section on reviewing the limits of Pendry’s Perfect lens

5.2.1 What is meant by ’exactly’ n = −1

We begin by examining again what is meant by the perfect lens. We should

already expect that attaining Pendry’s exact n = −1 will be nearly impossible, but

what if we inch ever so close, will this lead us to anything significant? In all of the

following simulations a pair of in phase dipoles emit 10[GHz]

85

Figure 5.12: This image is a simple sweep from 0.801 ≤ n ≤ −1.201This contains point sources at 0.2m apart which is 2/3λ. Initially the intent was to

simply show the effect of a varying Index, where index=Ep in this case. Itimmediately became apparent that though the far field image was reproduced at theexpected image planes, the image was not super-resolved except for the case of the

index almost exactly atn− 1. This first section was done with no losses orconductivity of 0. It was also noted that the evanescent build up due to couplingbetween faces was also absent from most of the simulations, which corresponded

with a result from Pendry (23) which stated that the modes of the evanescent fieldare also dependent on this thickness and for perfect imaging the lens needs to be

some m |n|λ thick.

86

Figure 5.13: The image is a sweep from 0.98001 ≤ n ≤ −1.20001The sweep was further narrowed in order to verify how close to the perfect lens with

regard to index values one needs to be to have some super-resolution. This sweepillustrates that there is a build up of evanescent waves close to and index of -1, butthat it is non-uniform, which brings light to the complication of coupling evanescent

modes that occur within the lens faces. For the majority of the sweep themagnitude of the evanescent waves is greater on the incident face, which indicates

that the exit face has some fundamental limits to the coupling across the slab.According to Pendry this has to do with coupling across the face, where |nd| = λ

allows for perfect coupling.

87

Figure 5.14: The image is a sweep from 0.999 ≤ n ≤ −1.002Though it may seem unlikely, it was impossible not to try a further refinement of a

’how close to negative 1’ sweep. The results of this emphasized how difficult itwould be to ever make a material with these near perfect conditions. This

simulation was limited by mesh and a finite size slab and because of the boundaryconditions all information above this height was lost. The mesh size was restrictedto the finest mesh COMSOL default allowed which in this case allowed for mesh

elements around 0.024m in size. This leads to the two following sets of simulations,which are regarding the restrictions of the accuracy of the simulations due to mesh

quality and the slab height..

88

5.2.2 Mesh restrictions

Figure 5.15: Normal Mesh - Mesh for the Following Simulation

The images here the illustrate the resolution limits placed on the simulation

by using a versus coarse mesh, or ’normal’ by COMSOL’s standards. The artifacts

of the mesh are easily seen as even the general field has what appear to the eye as

discontinuities. The image resolutions were set to the finest COMSOL has available

to ensure that no artifacts of image regeneration were responsible for the granulated

looking form of the solution.

Figure 5.16: Normal Mesh Results - Index of n = −1.0001

89

Figure 5.17: Fine Mesh - Mesh for the Following Simulation

The images above and below illustrate the resolution limits placed on the

simulation by using a mesh called ’fine’. This mesh still has similarities to the

pixelated image from the previous mesh, but we can at least begin to see a uniformly

expanding wave from the point sources. The strange fields at the boundaries of the

slab illustrate the abrupt changes the evanescent fields are experiencing. It is

obvious that we are still seeing artifacts of the mesh, which most likely is partly

causing our inability to resolve the two point sources in the image plane.

Figure 5.18: Fine Mesh Results - Index of n = −1.0001

90

Figure 5.19: Finer Mesh - Mesh for the Following Simulation

It appears that we are getting cleaner far fields but so far this is doing nothing

for our resolution of the two point sources. In this case the mesh is set to ’finer’

which results in a mesh of about 0.088m, which is only about the distance from our

point sources. It appears as if there is a slight difference in the image plane, and

this is the first sign of a possible subwavelength resolution. Although this might

seem intuitive, we can specify that the equations across a domain, from mesh

boundaries are higher order equations. Ina situation such as this, where we have

infinitely small point sources and care about resolving them to that limit, a higher

order quadratic might be a better choice vs the default, but we will not explore this

here. We are content to note that once near the subwavlength limit the information

appears to be transmitted proportional to the mesh size.

91

Figure 5.20: Finer Mesh Results - Index of n = −1.0001

Figure 5.21: Extra Fine Mesh - Mesh for the Following Simulation

The images here are beginning to demonstrate the coupling of evanescent

waves that we would expect from a near perfect DNIM lens. We can definitely see

two specific areas of field concentration where the image plane would be, but we do

not have a image nearly as strong or well defined as the original sources. We

continued the mesh refinement further beyond ’extra fine’ to see how much further

the image can be improved with the mesh or if other limitations are causing

restricting our resolution.

92

Figure 5.22: Extra Fine Mesh Results - Index of n = −1.0001

Figure 5.23: Extremely Fine Mesh - Mesh for the Following Simulation

It appears as though the mesh is no longer the restriction to the resolution of

the image at this mesh density. As before we can see two clearly localized points in

the image plane, but they have increased in intensity relative to the surrounding

field. Changes in the evanescent fields can be seen and are most likely a result of

high order effects from coupling from the faces.

93

Figure 5.24: Extremely Fine Mesh Results - Index of n = −1.0001

Figure 5.25: Fine Custom Mesh - Mesh for the Following Simulation

Decreasing the mesh size further to a restriction of 0.02m seems to make little

difference once again on the image plane. There does seem to be some localization

of the field, but cut line plots show this is not any different from the previous plot.

We repeated a few more refinements just to be sure!

94

Figure 5.26: Fine Custom Mesh results - Index of n = −1.0001

Figure 5.27: Finer Custom Mesh - Mesh for the Following Simulation

The mesh is now set to .01m and can barely be seen from the image. Yet

again, the image plane sees no improvements, but the evanescent field has change

structure yet again. We might begin to consider that since an infinite number of

modes might be available that this will be the case for an up to an infinite mesh.

Further results will show that only a limited number of modes are coupling into the

structure because of several different factors, thus the evanescent field pattern is still

changing due to information loss from the near field of the point sources.

95

Figure 5.28: Fine Custom Mesh results - Index of n = −1.0001

Figure 5.29: Finest Custom Mesh - Mesh for the Following Simulation

At a mesh size of .005m it seems as though our solution has converged. Both

the evanescent field and the image plane have stabilized and we know that the

restrictions to the image are due to other factors at this point. We next have to

check where our restriction to the index has actually restricted our mesh

consideration

96

Figure 5.30: Finest Custom Mesh results - Index of n = −1.0001

Figure 5.31: Extremely Fine Mesh Results - n = −1.0001

Here is a comparison of the ’extremely fine’ mesh with our original n = −1.001

(top) and a mush less perfect index of n = −1.01(bottom). It is apparent that the

change in index has completely removed any added benefits of the finer mesh.

97

Figure 5.32: Extremely Fine Mesh Results - n = −1.01

Figure 5.33: Combination Results 1 - Combination Results 1

A change towards the perfect negative 1 index to −1.000001 makes quite an

enormous difference as the image plane is now practically drowned out by the

evanescent field. It is tough to discern whether subwavelegth resolution is possible

in this case, but we could always use a hyperlens at the image plane to capture and

propagate the total fields and hopefully calculate the evanescent fields due to our

sources.

98

Figure 5.34: Combination Results 2 - Combination Results 2

Figure 5.35: Evanescent Amplification - n = −1.001

These images are for a mesh of 0.02m. It was not noted in the images above

but the scale change can easily be seen here. The image with an index of

n = −1.001(top) has a field enhancement of about 1/100 that of the field shown on

the index of −1.000001(below). Again the evanescent field strength is so high that it

overshadows the image plane. One would think this would be a valid way to test for

the perfect lens, and also a good problem to have since it implies that you at least

have a larger information in the image plane.

99

Figure 5.36: Evanescent Amplification - n = −1.000001

Figure 5.37: Mesh Refinement 1 - n = −1.001

Decreasing the mesh further reiterates the fact that the accuracy of the

perfect lens has to be beyond .01 in order to see the subwavelength transfer or

information. Mesh size here equals 0.01m

100



Decreasing the mesh further again still shows no effect in the simulation,

which at least helps forward the idea that at a mesh of around 1/10th the point

source separation distance we have reached a point where we are comfortable with

the results. So we have successfully decided on a mesh density that provides

accuracy for a point source to provide enough information to the matrix to transfer

this to an image plane, which is on the order of 1/10th the feature size we wish to

resolve. Now we move onto the question of slab height and conductivity

101


5.2.3 Restrictions based on conductivity

102

5.2.4 Meta-atomic approximation

Simulations on broken(=segmented) structures become difficult as evanescent

coupling between partitions becomes more complicated. Due to processing limits

and restrictions on the size we have already noticed from our height considerations,

two slabs are used as a demonstration of the effect of parameters imposed by

different periodic arrays of a negative index material. The first slab is a periodic

array of 19cm blocks with 1 cm gaps between 40 tall and 3 wide. The initial mesh

density did not show any results when compared to the original slab, thus

refinements were needed. The difference here is that the initial mesh outside of the

block was on the order of the original 8m tall slab, but the mesh size within the slab

itself was on the order of 20 times smaller. Solutions for this block were relatively

easy to find as long as the mesh density was relatively low on the outside of the

slab. Once the mesh was increased on the outside solutions began to stop

converging because of the many boundaries with -1 to 1 index values. To

compensate for this conductivity was added, but as we know this reduced the

possibility for the transfer of high spatial frequency information to the image plane.

Note: this restriction lead to sweeps to having to be done from higher values of

conductivity similar to those sweeps previously showed down to relatively low values

around 10−25[S/m]. These solutions only converged because the initial condition

was set as the result from the previous solution, thus any interruption in simulations

or any attempt to overstep the progression from high values of conductivity to lower

conductivity cause the simulation to have to be restarted.

103

Figure 5.41: Tall Slab1 Figure 5.42: Tall Slab 2

Figure 5.43: Figure text for Fig 5.41 and Fig 5.42Using a taller slab should capture more evanescent waves and in this simulation, sincethe scattered field off the boundaries was pointed outward from the domain of interest,this can be done by simple increasing the total domain height. We can already seean improvement even with the ’finer’ mesh used on the left. We must also not that’finer’ is a relative term to COMSOL, in such that because of the height increaseby a factor of 2, the mesh size is also increased by a factor of 2. Thus we have anincrease image plane with a decrease in mesh size, which was well below our previousobserved threshold. The image to the right was done with a ’extremely fine’ mesh,which, again, was twice as large as the previous extremely fine mesh used before. Theresults demonstrate the great effect the height has on capturing the evanescent field.

104

Figure 5.44: Finer Mesh Figure 5.45: Extremely Fine Mesh

Figure 5.46: Figure text for Fig 5.42 and Fig 5.45Here we have encountered once again, a mesh artifact with the ’finer’ mesh used onthe left. This mesh is now a factor of 4 times larger than our original slab, thusan image is barely even transferred and the objects are barely rendered either. Theimage on the right shows the power of the height of the slab, as the ’extremely fine’mesh has been used, and even though it has 4 times the mesh size has resolved the twopoints and has enhanced evanescent fields almost to the point of the more ’perfect’negative slab simulations. We will increase the mesh more to see how much of animprovement can be gained by the height.

105

Figure 5.47: Extremely Fine Mesh Revis-ited

Figure 5.48: Tall Slab Extremely FineMesh Revisited

Figure 5.49: Figure text for Fig 5.47 and Fig 5.48As the mesh is decreased to the original ’extremely fine’ mesh size, or 0.025m, we canfurther see the symmetry and even greater resolution in the image plane. With sucha long distance over which to absorb the energy the evanescent field doesn’t build upquite as high and we can still see the object plane clearly. It should be noted thatthere is a slight shift in the object plane, but this is to be expected, as we have chosenan index slightly below that of -1.

106

Figure 5.50: Image Revealed Figure 5.51: Image Revealed 2

Figure 5.52: Figure text for Fig 5.50 and Fig 5.51In order to illustrate how well our point sources are imaged, the color range wasrestricted. We can still see that the magnitude of the image is less that the magnitudeof the objects, but only by 20%. We now know what cases allow for maximum transferto the image plane and can now look at a less perfect lens to see what limits it holdsas well.

107

Figure 5.53: Large Scaled Conductivity SweepWe have considered only the loss-less case of a negative lens so far. We considerlosses and where they apply restrictions on our attempt to attain sub-wavelength

imaging. The broad sweep below is a demonstration of losses from a badsemi-conductor to a regular dielectric. We can see that at even small values of

conductivity the losses quickly drown out any transfer of evanescent waves, even inthe case of a perfect -1 Index slab. We observe that around 10−-5 S/m a thresholdoccurs where propagation is viable. We will look closer near this threshold to see ifanything noteworthy occurs The image below is a simple sweep from 1 to 10−30[S/m]

108

Figure 5.54: Narrow Scaled Conductivity SweepIn order to understand where the threshold for the transfer of evanescent waves lies

a smaller sweep was done to demonstrate this. The image shows that at around10−3 we can already see an image, though not super-resolved at all. The huge buildup at 10−4 calls for further investigation as it drops off after this value, though thiscould simply be related so some sort of symmetry in the evanescent fields due to acutoff of higher order modes. This image is a simple sweep from 1 to 10−5[S/m]

109

Figure 5.55: Refined Narrow Scaled Conductivity SweepA more refined image was used in this animation to get a better idea of the

threshold for subwavelength resolution. Somewhere near 10−-4 we can see that twopeaks start to appear on the image plane. We now have some guidance for the

necessary conductivity to at least transfer enough information to the image planefor subwavelength resolution. We can now look at a periodic lens structure itself.

The image below is a simple sweep from 10−3.5[S/m] to 10−5[S/m]

110

Figure 5.56: Broken1 - 1st Broken slab type

111

Figure 5.57: Best Case 1 Figure 5.58: Best Case 2

Figure 5.59: Figure text for Fig 5.57 and Fig 5.58The images here represent the best resolution possible at the moment, due to re-strictions on solutions since further solutions with a more refined mesh would notconverge without the addition of conductivity sweeps. These images were taken withmesh sizes of 0.02m(left) and 0.01m(right). With the previous continuous slabs theseshowed very good resolutions of two distinct points, but do not in the case shownhere. We require smaller mesh sizes in order to see if the resolution limit is due tothe slabs or the mesh

112

Figure 5.60: Conductivity restrictions Figure 5.61: Conductivity restrictions 2

Figure 5.62: Figure text for Fig 5.60 and Fig 5.61We can see that the addition of conductivity has not resulted in a further decreasein our solution’s resolution. With the increase in mesh density we have overcomethe limits placed on the image by the conductivity increase almost immediately. Theimages below represent the maximum mesh density with no conductivity(left) and amesh density of 5 times this(left) with a conductivity of 10−6

113

Figure 5.63: Further Conductivity Re-strictions

Figure 5.64: Further Conductivity Re-strictions 2

Figure 5.65: Figure text for Fig 5.63 and Fig 5.64We can begin to see an image with a conductivity of 10−10(left). Further attemptslead to results, but the file size causes an error which has taken more than a fewweeks to resolve, thus we were unable to improve at this point further on the imagequality. The mesh to the right shows how dense the mesh is to obtain this image, asthis is a single corner of a segment of the structure, and is also a reflection on thedifficulty we may have at obtaining three dimensional images.

114

Figure 5.66: Broken Slab type 2 - Second Type of Broken slab to investigate possiblemeta-atomic restrictions to information transfer across the DNIM slab

The set up here ?? represents a perfect negative one periodic slab with 28 cm

with a 2 cm gap periods of 2 X 26. The same limits from the previous slides showed

up in this setup, with the gap distance being the restricting element in both since

the mesh is generated based on the smallest features. Ironically, with a smaller

mesh compared to the size of the building blocks to this lens solutions did not

converge for even larger meshes than the previous. This seems to follow the logic

that the finer mesh allowed for higher order modes to be produced on the edges of

the blocks, which is what is the restricting feature of the solution. The same

method was employed to obtain solution with larger meshes.

115

Figure 5.67: Mesh Size for Following Set of Simulations -

The image on top was a result at 10−7[S/m] and the image on bottom with

10−15[S/m]. T illustrate how little effect this had on the overall image plane. We

can barely begin to make out some non-uniformity in the image plane that may be

the beginning of a subwavelength image. The mesh shown below is the mesh used

for these simulations.

116

Figure 5.68: Low Conductivity Comparisons - 10−7[S/m]

Figure 5.69: Low Conductivity Comparisons - 10−15[S/m]

117

The obvious next step is to further reduce the size of the constituent elements

and check and see whether this can again increase the ease with which a

subwavelength image can be formed. The image below is a 4 cm block with 1 cm

gap, which is 160× 12 periods. Initial results are that this structure does in fact

have the capabilities to resolve the subwavlength scale features into an image, but

the computational time was > 3 days for even the worst mesh possible! We seem to

have reach a computational limit, and so we sought another simplification in order

to better understand possible meta-atomic constituents

Figure 5.70: Super meta-atomic structure - Structure consist of 12 rows and 160columns

CHAPTER 6 : NOVEL THEORETICAL TREATMENTS

6.1 Return to the Foundations of Material Properties

We have summarized as an overview the treatment of several metamaterial

designs that focus on expanding the group or phase velocities to new ranges namely

vg < 0 or vp < 0. These materials are based on creating structures that we are

calling meta-atoms, which are small enough to allow for a homogeneous bulk

material response at a specified wavelength. As the highly sought after perfect lens

is laid out before us, we should reflect to make sure our new perspectives on

manipulating what we expect from traditional material properties cannot benefit us

in some other way. We can take a look at the four generalize quadrants of materials

and notice a unusual fact, almost all real materials lie relatively close to the lines

where either µ = 1 or ε = 1

6.1.1 High index materials

It seems as though we might be missing opportunities when it comes to

determining what could be a good candidate for a high index material. Though as

we have stated, we might have come across such a material when attempting to find

off the shelf products to dope our ABS disc with in order to increase their ∆n, we

take a closer look at the possible index values of this type of metamaterial. We

already have the formulation from Section 4.1. We will modify this since we are

generally concerned with only the permittivity of this particular material as it

showed no magnetic response. We also know that the permittivity of the ABS

plastic is extremely small when compared to our new gold colored coating. We will

119

6

?

-

εr > 0µr > 0

εr < 0µr > 0

εr > 0µr < 0

εr < 0µr < 0

εr

µr

1

1

Normal Dielectrics

Metals

?Metamaterials?Metmaterials

?Metmaterials?

n < 1

assume the ABS is acting merely as a scaffold and only provides structure, thus the

material is the only FBS involved (25)

ε0TE = fεs + (1− f)ε0 (6.1)

Using this equation and generalizing our coat thickness, then after 30 coats our

thickness is 0.089cm, where we have an average delay of 50× 10−12s or equivalently

a phase delay of ≈ 170o. This gives an effective index of n ≈ 17 or εTE0 = 289. This

120

is huge compared to normal materials in the microwave regime, though some such

materials do exist. If we insert this value in to exploit form birefringence, assuming

a 50% fill factor and the filling material as air, ε0 = 1, this leads to an expected

permittivity for the base coating of εs = 577 or an index of n ≈ 24. . A quick

Figure 6.1: Gold color coating ABS disc - This is our gold colored coated ABS discbeing measure between two rectangular transmission and receiving horns

Measured Index Values for≈ 30 Layers

Index Minimum Index maximum Average Value0.56mm 0.38mm15.364 22.642 19.108

Table 6.1: Bulk measured gold colored coatingusing time domain methods. Significant but linear variations from a 6cm× 6cm

square.

literature search will enlighten us into the potential similarities with Cube and H or

I shaped high permittivity or high index metallic metamaterial constituent

structures with an intention to achieve exactly what we have found off the

shelf.(3)(37)

6.2 Uncovering the Material Properties

121

While we might be enthusiastic about the revelation, our primary goal

remains to understand the mechanism by which this readily available coating

material exhibits a normally difficult to create high index value. We can take a look

under the SEM and EDAX to gain a better understanding of what size particles are

involved and exactly what their chemical makeup is. We already have some insight

as to the mechanism given the work of authors like Fan (37) who have focused on

the coupling ability of very small gaps between nearest neighbor capacitances.

Figure 6.2: SEM Image of metallic flakeStructure embedded in some dielectric

substrate.

Figure 6.3: SEM Image of cleavedmetallic flake

Structure embedded in somedielectric substrate.

Figure 6.4: Figure text for Fig 6.3 and Fig 6.3Though the material is not yet known, we can tell by the size, ≈ 30µm that it is wellbelow what we would consider a diffraction effect of any kind.

6.2.1 Understanding why it Works

Since our material that exhibits some unique properties, we should return to

the basics to fully understand what other possibilities might be responsible for our

measured observations. All of the models we have looked at involved an assumption

of a homogeneous structure, thus we can look at size considerations first as we know

much about size requirements for a homogeneous medium. In order to move away

from diffraction effects and towards real bulk material properties we discuss a

relatively simple technique used to discover the limits of the size of meta-atoms by

122

Figure 6.5: EDAX for Cop-per(Magenta)

Signature on metallic flake structure.Figure 6.6: EDAX for Carbon(Red)Signature on metallic flake structure.

Figure 6.7: Figure text for Fig 6.6 and Fig 6.6Flake EDAX copper and carbon signatures. We can now see the material is exactlywhat other have been working to create as a copper flake embedded in a dielectricsubstrate.

way of periodic structures.

As a first approximation we can look at the effects of a plane wave incident

onto a periodic structure. This simplification allows for both a qualitative and

quantitative look at some of the relationship of the size of a periodic structure and

its effect on an incoming plane wave. We will first assume a lossless medium in

order to better grasp the effects of the periodicity on the incoming wave. A simple

qualitative assessment of the wave amplitude inside the periodic structure will

illuminate whether the wave is experiencing a homogeneous material since reflections

at the boundaries will indicate whether or not the material is seen as homogeneous.

We can estimate the maximum thickness of a layer, to provide a homogeneous

response through a simple Matlab R© code dubbed PlanewaveGUI,(22) which solves

an incident plane wave electromagnetic field onto a stack of periodic elements.

123

]

εr2 εr2εr1 εr1

-6

E

Bk . . . . . .. . . . . .. . . . . .

Period

d1 d2

Period

Figure 6.8: Plane wave incident on periodic structureThis is the general schematic for our periodic structure. With an arbitrary choice of

the permittivities, εr1 or εr2 , we adjust the resulting path length that the incidentwave experiences, thus further constraining out physical dimensions for each period

As a quick check we can see that the optical path length for the structure in

both cases, given by OPL = n1d1 + n2d2, where the periods are d = 0.04m is

OPL = 1.1.5(.04) + 3.4.5(.04) ≈ 0.1m,. At 1[GHz]→ λ = 0.3m and with

1[GHz]→ λ0.5m . This essential leads us to the approximation that for OPL ≤ λ4

we can assume a homogeneous response for a periodic structure. This quickly gets

us into trouble if we consider a object scattering into our periodic structure, much

like if we were attempting to use it as a lens of some sort. As shown in Fig. 6.13 We

now have a even more restrictive case for the thickness of our periods if we ever

intend to use a material off axis at all, since OPLn = OPLicosθ

, which implies as

θ → 90o then OPL→∞. Unfortunately this doesn’t help much because this would

be the case for everything except an infinite slab of material. We can look to some

other clues to see what we might expect from the dimensionality of a meta atomic

constituent. Since we are leaning towards including metals, as we already know the

124

Figure 6.9: Incident wave on periodicstructureWhich consisting of alternate layers of

ε1 = 1.1 and ε2 = 3.4


ε1 = 1.1 and ε2 = 3.4

Figure 6.11: Figure text for Fig 6.9 and Fig 6.10The change from a an inhomogeneous response to a homogeneous one is due to thefact that the OPL ≤ λ

4for a period.

constituents of our high index coating, we can use the skin depth to limit the

thickness of the metal based meta-atom we would use.

δ =

√2

ωµσ(6.2)

which results in a skin depth of 0.6510−6m at 10[GHz] or ≈ λ104

. This restriction

may put some perspective on why we don’t find metallic constituents very often, as

fabrication techniques are quite capable to design something on this scale, but the

cost associated with trying to create a large enough sample is considerable. Once

again this fits with our newly discovered high index coating, as the copper flake

inclusions are < 35µm in diameter, and only about 1 micron in thickness. Since no

objections come to mind, we will now look at a more formal way of defining the

additional effective index seen in this coating.

We should return first to the idea of the difference between metals and

dielectrics, where bound and free charges play the divisive role in deciding which

125


ε1 = 1.1 and ε2 = 3.4

]

εr2 εr2εr1 εr1

@@I @@I

AAUAAU

E

B

k

θ

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

Period

d1 d2

Figure 6.13: This is the generalschematic for our periodic structure.

If we now consider directionality, wewill adjust the resulting path length

that the incident wave experiences, thusfurther constraining out physical

dimensions for each period

Figure 6.14: Figure text for Fig 6.12 and Fig 6.13As we attempt to collect more information from the scatterer our new OPLn = OPLi

cosθ

belongs to which category. What separates the free charge from the bound is simply

where the boundary lies for the charge to stop. We can see that a metal does not

have to be an infinite sheet in order to have ’free charge’ but the electron transport

does have to be of some order of the wavelength in order to ’flow’ as if it were a free

charge. It seems then very clear that our metallic flakes, which from the SEM do

not touch at all, and though we have tested metallic inclusions or flakes that do

make contact with one another , they exhibit no enhanced response. Thus we can

model our metal flake as a dipole response to the electric field, exactly as we did for

a dielectric with the Lorentz local field

126

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

Figure 6.15: Full Dipole SheetWhich allows for coupling between adjacent meta-Atoms an exact analog to the

Lorentz local field for atomic dipoles

We will start with the result of the contributions of the background field of a

volume surrounding the ’particle’ or our meta-atom. In this case me must take a

closer look at the polarization vector as it holds the fundamental physical insight

127

into what is happening with these flakes.

E(ω) = EExternal(ω) +1

3ε0P(ω) (1.30)

Since the polarization vector is in fact merely the polarization density, we can

compare what this means to standard atoms, versus our new Meta atom. With a

standard atom we expect that the dipole shift will stay within the inter-atomic

distance of the lattice of the solid. This means for every atom there is a dipole

response which is contained within its own inter-atomic distance. If we were to ask

the question what is the magnitude of the polarization for an individual dipole, we

have,

Pi(ω) = nqxi(ω) (6.3)

We can then add up all of the polarization vectors of all the dipoles that would take

up the same linear space as our meta-atom and we can maximize this for a static

background field or at some ω0 while every dipole is extended exactly to its

maximum contribution, the inter-atomic distance defined as a.

N∑1

Pi(ω0) = nqxi(ω0)|xi=a = Nnqa (6.4)

This will represent the polarization per meta-atomic linear volume and we can

compare this to the polarization of a maximally poled meta-atom of length Na− a,

which will also contain Nn free charge which would simultaneously be displaced at

this maximum. We will assume N large for this case thus our length Na− a→ Na,

thus

Pi(ω0) = Nnq(Na) (6.5)

128

Comparing these two quantities w can see that the magnitude of polarization vector

for the meta-atom PMeta−Atom = NPIndividual. As our flakes are a few microns in

size, N ∝ 1026, thus we expect a huge polarization. Of course we recognize that this

is an extreme simplification, as the spacing between adjacent meta-atoms will be

large on the order of the spacing of actual atoms, this will reduce the effective

coupling between elements and will reduce the total polarizability of the medium.

Work has already been done to show the proportionality of this effect based on the

gap distance between adjacent elements.(3)(37).

We can now take a look at some proposed simulations to parameterize this

material with the intent to improve its response.

Figure 6.16: Flake Geometry - Proposed Flake Geometry for COMSOL R© Multi-physics simulations. Flakes vary in size from 5µm to 30µm

129

Figure 6.17: Flake PEC Field - Proposed Flake Geometry for COMSOL R© Multi-physics simulations. Flakes vary in size from 5µm to 30µm

6.3 Magnetic Meta-atoms

Along the same like we can envision a magnetic meta-atom, in this case we do

not expect magnetic charges to flow, but as most magnetic materials are metals, the

previous formulation provides for a highly reactive magnetic response, without the

bulk conductivity usually associated with a metal. Again we are able to find such a

material readily available as shown in Fig 6.18. We can now imagine that with such

a coating we can tune the response of the material by use of a background magnetic

field to induce a response in the Fe3O4 inclusions. Though initial test were done

with only a 0.04Tesla background field we were able to completely reorient the

ordinary axis of the FBS coated disc!

130

Figure 6.18: SEM of magnetic particle inclusions in a dielectric substrate - 10-20micron Fe3O4 inclusions which are listed as large volume fraction of this material andEDAX verifies iron signature.

Figure 6.19: Diagram of magnetic response. - of FBS disc with magneticallyresponsive coating. With a background magnetic field, the permeability increasechanges the dominant axis to which an incoming electromagnetic wave responds.

131

Figure 6.20: Data of magnetic response of FBS disc - . The data shown here notonly shows a large material index, and in general a tunable index through the use ofa relatively low background magnetic field, it demonstrates that for this particularapplication we have been able to reorient the ordinary axis of the material to alignwith the direction of the incident B-field

CHAPTER 7 : DISCUSSIONS AND CONCLUSIONS

7.1 Summary

We have looked at a wide range of materials which all fall into the category of

metamaterials. We have looked at their possible applications for imaging with

subwavelength resolution, field enhancements, large index materials and tunable

index structures. It is easy to extrapolate some more exotic uses of materials with

negative index values, using them as broad band visible cloaks, which may one day

be possible, or event cloaks, which may already have been demonstrated (27). We

have outlined the material necessities for constituent materials to create such

materials. At the moment restrictions to losses and limits to our models and

knowledge of the parameters needed to manipulate index values successfully

continue to be outstanding challenges. While we have demonstrated that both form

birefringent structures and SRRs are fairly well understood, their practical

applications have not yet been developed to their potential. The future of

metamaterials will be focused on provided higher transmission throughput while

maintaining negative index values, and finding way around the current directionality

of form birefringent materials and SRRs.

7.2 Conclusions

We have presented many of the applicable theoretical models for normal

materials and shown how these can be applied to metamaterials meta-atomic

constituents. Through this understanding we have shown how these subwavelength

sized structures can be used to realize properties not found in nature. These

structures can be engineered to exhibit resonant or average properties nor normally

133

found in nature because because impurities and random phase effects eliminate

coherent scattered field effects that can combine to give a larger positive or negative

constituent parameter. We also have demonstrated theoretically and experimentally

a specific exploitation of these new material properties through the results of our

high index and magnetically addressable coatings. We also looked at specific

restrictions and challenges to engineering and implementing materials to be used

with a super-resolving or ’perfect’ lens. in doing so this work has shown that like

Lorentz, Veselago overcame a fundamental barrier when simply removing the

assumptions or an index greater than 1, paving the way for a new vision on how

materials could be engineered from the ’atom’ up.

134

REFERENCES

[1] M. Bertolotti S. Zhukovsky J. W. Haus A. Mandatori, C. Sibilia and M. Scalora.Anomalous phase in one dimensional, multilayer, periodic structures with bire-fringent materials. Phys. Rev. B, 70(16), 2004. 47, 49

[2] J. A. Kong Et Al. Magnetic properties of s-shaped split-ring resonators. Progressin Electromagnetics research, 51:231–247, 2005. 35

[3] Muhan Choi Et Al. A terahertz metamaterial with unnaturally high refractiveindex. Nature, 470:369–474, Feb 2011. 120, 128

[4] Jovan M. Elazar Aleksandar D. Rakic, Aleksandra B. Djurisic and Marian L.Majewski. Optical properties of metallic films for vertical-cavity optoelectronicdevices. APPLIED OPTICS, (5):5271–5283, 1998. 9, 77

[5] Nakhle Asmar. Partial Differential Equations with Fourier Series and BoundaryValue Problems. Printice Hall, 2nd edition, 2005. 22

[6] Max Born and Emil Wolf. Principles of Optics. Pergamon Press, 1980. 12, 13

[7] L. Brillouin. Wave Propagation and Group Velocity. Academic Press, 1960. 25,27, 28, 29, 30, 31

[8] Richard H. Bube. Electrons in solids: an introductory survey. Academic Press,3rd edition, 1992. 16

[9] COMSOL AB., Los Angeles, CA 90024. COMSOL Multiphysics Users Guide,version 3.4a edition, 2007. 76

[10] V. V. Varadan D. K. Ghodgaonkar and V. K. Varadan. Free-space measure-ment of complex permittivity and complex permeability of magnetic materialsat microwave frequencies. TRANSACTIONS ON INSTRUMENTATlON ANDMEASUREMENT., (2):387–394, 1990. 74

[11] et al D. R. Smith. Limitations on subdiffraction imaging with a negative refrac-tive index slab. Appl. Phys. Lett., 82, 2003. 24

[12] Keith A. Nelson David W. Ward and Kevin J. Webb. On the physical origins ofthe negative index of refraction. New journal of Physics, 7:1–8, 2005. 19, 20, 21,24

[13] Online Publication NIST EPQLibrary. Nist physical constants. Web Document,March 2012. 5, 49, 54

135

[14] Alex Figotin and Ilya Vitebskiy. Frozen light in photonic crystals with degenerateband edge. PHYSICAL REVIEW E, 74:066613–1–066613–14, 2006. 52

[15] V. G.Veselago. The electrodynamics of substances with simultaneously negativevalues of and . Physics-Uspekhi, (10):509–514, 1968. 17

[16] Tetu Hirosige. Origins of lorentz’ theory of electrons and the concept of theelectromagnetic field. Tetu Horosige, 1:151–209, 1969. 5

[17] K. Homann N. Kallay K. Kuchitsu I. Mills, T. Cvitas. Quantities, Units andSymbols in Physical Chemistry. Blackwell Science, 2rd edition, 1993. 6

[18] D. J. Shelton M. B. Sinclair I. Brener J. A. DArchangel, G. D. Boreman. Re-leasable infrared materials. Journal of Vaccuum Technology B, 1:51806–1–51806–5, 2011. 37

[19] D. J. Robbins J. B. Pendry, A.J. Holden and W. J. Stewart. Magnetism fromconductors and enhanced nonlinear phenomena. IEEE Trans Microwave Theoryand Techniques, 47(1):2075–2084, 1999. 17, 35

[20] Thomas M. Jovin Keith A. Lidke, Bernd Rieger and Rainer Heintzmann. Su-perresolution by localization of quantum dots using blinking statistics. OPTICSEXPRESS, (1):7025–7062, 2005. 23

[21] Jin Au Kong. Electromagnetic Wave Theory. EMW Publishing, 2008. 6

[22] Lorenzo Luini. Gui for te tm electromagnetic plane waves propagation throughmultilayered structures. BSD Licenset, May 2010. 122

[23] J.V. Hajnal D.J. Edwards M.C.K. Wiltshire, J.B. Pendry and C.J. Stevens. Nearfield imaging with magnetic wires. Optics Express, 11:709–715, 2003. 18, 85

[24] Richard D. Mckirahan. Philosophy Before Socrates. Hackett Publishing CompanyInc, 1994. 1

[25] Weiguo Yang et al Michael A. Fiddy. Internal field distribution measurementin 1-d strongly anisotropicsub-wavelength periodic structures of finite length.Optics Express, 19(1), December 2010. 50, 119

[26] Weiguo Yang et al Michael A. Fiddy. Tunable negative group index in metama-terial structures with large form birefringence. Optics Express, 18(2), January2010. 47, 48, 53

[27] Yoshitomo Okawachi Moti Fridman, Alessandro Farsi and Alexander L. Gaeta.Demonstration of temporal cloaking. Nature, (1):61–64, 2012. 132

[28] Paul J. Nahin. Oliver Heavyside The Life, Work, and Times of an ElectricalGenius of the Victorian Age. The John Hopkins University Press, 1987. 3

136

[29] J. J. Sakurai Jim Napolitano. Modern Quantum Mechanics, volume 1. Addison-Wesley, 2nd edition, 1994. 20

[30] R. R. Nigmatullin and Ya. E. Ryabov. Coledavidson dielectric relaxation as aself-similar relaxation process. Physics of Solid State, (1):87–92, 1997. 15

[31] Arne Nordmann. Refraction in the manner of huygens. Web Document Image,April 2007. 19

[32] George E. Owen. Introduction to Electromagnetic Theory. Allyn and Bacon,1963. 2

[33] J. B. Pendry. Negative index makes a perfect lens. Phys. Rev. Lett., (85):3966–3969, 2000. 24

[34] Matthew Sands Richard Phillips Feynman, Robert B. Leighton. The FeynmanLectures on Physics, volume 1. Basic Books, 2011. 20

[35] John B. Schneider. Understanding the FDTD Method. Washington State Uni-versity Online Publication, 2010. 9

[36] K. K. Sharma. Optics Principles and Applications. Academic Press, 1st edition,2006. 21

[37] Jonghwa Shin Jung-Tsung Shen and Shanhui Fan. Three-dimensional metamate-rials with an ultrahigh effective refractive indexover a broad bandwidth. Physicalreview Letters, 102(1), December 2009. 120, 121, 128

[38] Carlo G. Someda. Electromagnetic waves. EMW Publishing, 2008. 10

[39] Sergei Tretyakov. Analytical Modeling in Applied Electromagnetics. ArtechHouse, 1st edition, 2003. 39

[40] Sergei Tretyakov. On geometrical scaling of split-ring and double-bar resonatorsat optical frequencies. Metamaterials, 1(1):40–43, March 2007. ix, 37, 38, 39, 40,41

[41] B. Ung and Y. Sheng. Interference of surface waves in a metallic nanoslit. OpticsExpress, 15:1182–1187, 2001. 77

[42] Waser. On the notation of maxwells field equations. Web Document AndreWaser, 1:1–14, 2000. 4

[43] Martin Wegener and Stefan Linden. Shaping optical space with metamaterials.Physics today, 1:32–36, 2010. 36

137

APPENDIX A

Derivation of plasma frequency,

We can look at Maxwell’s equations in differential form to find that

∇2E =1

c2

∂2

∂t2E + µ0

∂

∂tJe (1)

We will assume a free electron in 1D, though not necessary, under and plane wave

electric field given by

E = E0ei(kx−ωt)x (2)

where the force on exact electron is

mex = qE (3)

where me is the electron mass and q is its charge. The current density is defined as

J = nqx, where n is the number density of electrons. tis allows us to place the current

density in terms of the electric field, if we consider[µ0

∂∂tJe

]µ0∂

∂t[nqx] = µ0[nqx] = µ0

[nq2

me

E

](4)

Inserting this as well as ∇2E = −k2E and 1c2

∂2

∂t2E = −ω2

c2E into the original wave

equations leads to

−k2 = −ω2

c2+ µ0

[nq2

me

](5)

138

Using 1c2

= ε0µ0 and rearranging we get the dispersion relation

c2k2 = ω2 −[nq2

ε0me

](6)

where we define the last term as the plasma frequency

ωp =

[nq2

ε0me

](7)

Documents

Thesis