UNIVERSIDADE TÉCNICA DE LISBOA

INSTITUTO SUPERIOR TÉCNICO

Metamaterials with Negative

Permeability and Permittivity:

Analysis and Application

José Manuel Tapadas Alves

Dissertation submitted for obtaining the degree of

Master in Electrical and Computer Engineering

Jury

President: Professor José Manuel Bioucas Dias

Supervisor: Professor Carlos Manuel Dos Reis Paiva

Co-Supervisor: Professor António Luís Campos da Silva Topa

Member: Professor Sérgio de Almeida Matos

October, 2010

Abstract

In this dissertation we study and analyze the electromagnetic phenomena of media with neg-

ative permeability and permittivity, called DNG metamaterials, and how this leads into some

physical phenomena such as the appearance of backward waves and the emergence and impli-

cations of negative refraction.

Two simple DNG waveguiding structures are also studied: the DPS-DNG interface and the

DNG slab. As a DNG medium is necessarily dispersive, the utilization of a known dispersive

model, the Lorentz Dispersive Model, is used in the analysis of the DPS-DNG interface in

order to obtain physical signicant results. The appearance of super-slow modes in the DNG

slab propagation is also a subject of interest.

Finally we address the lens design using DNGmetamaterials. The dependence on the refractive

index of this design process is evidenced. The particular structure of the DNG Veselago's at

lens is also analyzed in order to study a potentially practical application of DNG metamaterials

in optics and the implications of dealing with such materials as this lens structure overruns

some conventional limitations, allowing propagating waves to be brought to a single point

focus producing an image that has sub-wavelength detail.

Keywords

Double Negative Media, Metamaterials, Negative Refraction, Backward Waves, Planar Waveg-

uides, Lens Design, Superlens, Microwaves, Photonics

i

Sumário

Nesta dissertação são estudados e analisados os fenómenos electromagnéticos associados aos

meios com permeabilidade e permitividade negativas, designados por meios duplamente neg-

ativos (DNG), e como esta característica leva ao aparecimento de alguns fenómenos sicos,

como por exemplo o surgimento de ondas regressivas, e o aparecimento, e implicações, de um

índice de refracão negativo.

São também estudadas duas estruturas simples de propagação guiada, mas utilizando meios

DNG: a interface DPS-DNG e a placa DNG. Como ummeio DNG é necessáriamente dispersivo,

a utilização de um modelo dispersivo conhecido, como o modelo de Lorentz, é usado para a

análise da interface DPS-DNG, com vista a obter resultados sicamente signicativos. O

aparecimento de modos super-lentos na propagação na placa DNG é também um assunto em

análise.

Finalmente é focado o estudo do desenho de lentes usando metamateriais DNG. É evidenciada a

dependência deste processo de desenho em relação ao índice de refracção. Estudamos também

a estrutura particular de uma lente DNG chamada lente plana de Veselago com vista a analisar

uma potencial aplicação e implicações deste tipo de materiais, já que este tipo de lente supera

algumas limitações de lentes convencionais permitido que as ondas propagadas sejam focadas

num único ponto produzindo uma imagem com um detalhe ao nível de comprimentos inferiores

ao comprimento de onda.

Palavras-Chave

Meios Duplamente Negativos, Metamateriais, Refracção Negativa, Ondas Regressivas, Guias

de Onda Planares, Desenho de Lentes, Superlentes, Microondas, Fotónica

ii

Acknowledgements

I would like to express my gratitude to both Professor Carlos Paiva and Professor António

Topa for the continuous support on the development of this work. Without the help, the

suggestions, comments and the share of knowledge from these two professors the realization

of this dissertation would not be possible.

I also want to thank my family and friends who have always supported me.

My last acknowledgment goes to my colleagues who are working at the 4th Floor's work-room

of the IST's North Tower for the helpful and cheerful moments that have provided me during

the development of this work.

iii

Contents

Abstract i

Keywords i

Sumário ii

Palavras-Chave ii

Acknowledgements iii

List of Figures ix

List of Tables x

Nomenclature xi

List of Symbols xii

1 Introduction 1

1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

iv

1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Electromagnetics of Double Negative (DNG) Media 11

2.1 Medium Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Negative Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 The Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 A DNG interval using the Lorentz Model . . . . . . . . . . . . . . . . . 31

2.3.2 A DNG interval using the Drude Model . . . . . . . . . . . . . . . . . . 33

2.4 Group Velocity and Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Guided Wave Propagation in DNG Media 39

3.1 Propagation on a Planar DNG-DPS Interface . . . . . . . . . . . . . . . . . . . 39

3.1.1 Modal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.2 Surface Mode Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.2.1 Neglecting Losses in the LDM (ΓL = 0) . . . . . . . . . . . . . 44

3.1.2.2 Considering Losses in the LDM (ΓL = −0.05× ωpe) . . . . . . 49

3.2 Propagation on a DNG Slab Waveguide . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Modal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Surface Mode Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

v

4 Lens Design Using DNG Materials 69

4.1 Optical Path and the Lens Contour . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 The Veselago's Flat Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Conclusions 81

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

References 86

vi

List of Figures

1.1 Photo of a nonlinear tunable metamaterial. The close-up photo square shows

a split-ring resonator with variable-capacity diode. (Source: Ilya, Shadrivov,

Australian National University, Nonlinear Physics Centre, Australia, 2008) . . 4

1.2 Metamaterial at lens consisting of an array of 3 by 20 by 20 unit cells. [?] . . 5

2.1 Material Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The permittivity in the complex plan . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Spatial Representation of the elds, the energy ux and the propagation con-

stant for a DPS and a DNG medium . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Scattering of a wave that incises on a DPS-DNG interface . . . . . . . . . . . . 27

3.1 The planar interface between a DPS and a DNG medium, here represented by

a dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Lorentz lossless dispersive model for εr,L and µr,L . . . . . . . . . . . . . . . . . 45

3.3 Relative refraction index (nr = n√ε0µ0

), using the lossless LDM, on the DPS-

DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Dispersion relation, β(ω), using the lossless LDM, on the DPS-DNG interface . 47

3.5 Attenuation constants α1 and α2 for the TE modes, using the lossless LDM, on

the DPS-DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

vii

3.6 Attenuation constants α1 and α2 for the TM modes, using the lossless LDM,

on the DPS-DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 Lorentz dispersive model for εr,L and µr,L . . . . . . . . . . . . . . . . . . . . . 49

3.8 Relative refraction index (nr = n√ε0µ0

), using the lossy LDM, on the DPS-DNG

interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 Dispersion relation, β(ω), using the lossy LDM, on the DPS-DNG interface. . 51

3.10 Attenuation constants α1 and α2, for the TE modes, using the lossy LDM, on

the DPS-DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.11 Attenuation constants α1, for the TM modes, using the lossy LDM, on the

DPS-DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.12 Variation of the electric eld, Ey(t = 0, x, z), on the DPS-DNG Interface . . . . 53

3.13 A DNG slab waveguide immersed on a DPS media . . . . . . . . . . . . . . . . 54

3.14 The representation of the modal solutions (red dots) given by the intersection

of the curves for a DPS slab with ε1 = µ1 = 1 and ε2 = µ2 = 2. . . . . . . . . . 60

3.15 The representation of the modal solutions (red dots) given by the intersection

of the curves for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −1.5 and V = 0.5 . 61

3.16 The representation of the modal solutions (red dots) given by the intersection

of the curves for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −1.5 and V = 3 . . 62

3.18 Modal solutions (red dots) for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −2,

with (i) V = µ1|µ2| and (ii) V = π

2 . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.17 Dispersion diagram for a DNG slab with ε1 = µ1 = 1 and ε2 = µ2 = −1.5 . . . 63

3.19 Dispersion diagram for a DNG slab with ε1 = 2, µ1 = 1 , ε2 = −1 and µ2 = −1.5 65

3.20 Dispersion diagram for a DNG slab with ε1 = 2, µ1 = 2 , ε2 = −1 and µ2 = −1.5 65

viii

4.1 Lens contour and optical path representation . . . . . . . . . . . . . . . . . . . 69

4.2 The lenses contours for dierent refraction indexes, n = −2.5,−1.5, 100, 1.5, 2.5 71

4.3 Passage of light waves through a Veselago at lens, A: the image source, B:

focused image, i.f.: the internal focus point . . . . . . . . . . . . . . . . . . . . 73

4.4 Evanescent eld variation in the presence of the Veselago's at lens. . . . . . . . 78

ix

List of Tables

3.1 Simulation parameters for the Lorentz Dispersive Model, on the DPS-DNG

interface structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

x

Nomenclature

BW Backward Waves

DNG Double Negative Medium

DPS Double Positive Medium

ENG Epsilon Negative Medium

LDM Lorentz Dispersive Model

MNG Mu Negative Medium

NRI Negative Refraction Index

TE Transverse Electric

TM Transverse Magnetic

xi

List of Symbols

αiTransverse attenuation constant in

medium i

B Magnetic ux density

β Propagation constant

c Velocity of light

d Thickness of dielectric slab

χe Electric susceptibility

χm Magnetic susceptibility

D Electric ux density

E Electric eld intensity

Ex Electric eld (x-axis)

xii

Ey Electric eld (y-axis)

Ez Electric eld (z-axis)

ε Electric permittivity

ε′ Electric permittivity (Real Part)

ε′′ Electric permittivity (Imaginary Part)

ε0 Electric permittivity (Vaccum)

εi Electric permittivity (Medium i)

η Wave impedance

ζ Normalized wave impedance

H Magnetic eld intensity

Hx Magnetic eld (x-axis)

Hy Magnetic eld (y-axis)

Hz Magnetic eld (z-axis)

hi Transverse wavenumber (Medium i)

k Wave vector

xiii

k Wavenumber

k0 Wavenumber (Vaccum)

kx Transverse wavenumber (x-axis)

ky Transverse wavenumber (y-axis)

kz Transverse wavenumber (z-axis)

ki Wavenumber (Medium i)

S Poyting Vector

Sav Time-averaged of the Poynting vector

µ Magnetic permeability

µ′ Magnetic permeability (Real Part)

µ′′Magnetic permeability (Imaginary

Part)

µ0 Magnetic permeability (Vaccum)

µi Magnetic permeability (Medium i)

n Refractive index

xiv

n0 Refractive index (vaccum)

n′ Refractive index (Real Part)

n′′ Refractive index (Imaginary Part)

ni Refractive index (Medium i)

vp Phase Velocity

vg Group Velocity

ω Angular Frequency

ΓL Lorentz damping coecient

χL Lorentz coupling coecient

χe Electric susceptibility

Mi Magnetization eld

Zi Impedance (Medium i)

t Transmission Coecient

r Reection Coecient

Ts Overall Transmission Coecient

xv

Rs Overall Reection Coecient

xvi

Chapter 1

Introduction

1.1 Historical Background

The study of the fundamental theories about the true nature of electricity have been chal-

lenging scientists for centuries. The rst empirical observations and written documents about

electric physical phenomena have their origins in ancient Egypt, from about 3000 B.C.E.,

which referred to the study of electric shocks produced by sh, who are described as the

Thunderers of Nile. These kind of phenomena have also fascinated and inuenced the stud-

ies made by the following civilizations (Greeks, Roman, Arabic, ...) [?]. Ancient writers, such

as Pliny the Elder (23 C.E.) and Scribonius Larges (47 C.E.), wrote about the eect of electric

shocks delivered by shes and concluded about the guiding phenomenon of theses shocks

along conducting objects [?]. Some ancient cultures also observe that some materials, as they

were rubbed against fur, could small attract objects. Based on this observation Thales of

Miletos (600 B.C.E.) wrote some results about the nature of static electricity, where some

amber objects, after being rubbed, rendered magnetic properties in contrast with other ma-

terials that needed no rubbing, such as magnetite [?]. Even thought Thales was incorrect by

believing that the nature of the attraction phenomenon was magnetic, later on science could

1

prove that there was in fact a direct link between magnetism and electricity.

The recognition about a connection between both the electric and magnetic phenomena was

made by André-Marie Ampère and Hans Christian Ørsted in the beginning of the XIX century

[?] This electromagnetic unication theory, rst observed by Michael Faraday but extended by

James Clerk Maxwell, and then partially reformulated by Oliver Heaviside and Heinrich Hert,.

is one of the key accomplishments of XIX century mathematical physics. After Maxwell's pub-

lication of his Treatise of Treatise on Electricity and Magnetism (1873 C.E.), electricity and

magnetism were no longer two separate physical phenomena. He experimentally demonstrated

that: a) Electric charges attract or repel one another with a force inversely proportional to

the square of the distance between them: unlike charges attract, like ones repel; b) Magnetic

poles (or states of polarization at individual points) attract or repel one another in a simi-

lar way and always come in pairs: every north pole is yoked to a south pole; c) An electric

current in a wire creates a circular magnetic eld around the wire, its direction depending

on that of the current; and d) current is induced in a loop of wire when it is moved towards

or away from a magnetic eld, or a magnet is moved towards or away from it, the direction

of current depending on that of the movement. The equations obtained by Maxwell, along

with the Lorentz force law (that was also derived by Maxwell under the name of Equation

for Electromotive Force, fully describe classical electromagnetism. These equations have also

been the starting point for the development of relativity theory by Albert Einstein and are

still fundamental to physics and engineering. These equations show the existence of electro-

magnetic waves, propagating in vacuum and in matter, and seemingly dierent phenomena

like radio waves, visible light, and X-rays are then understood, by interpreting them all as

propagating electromagnetic waves with dierent frequency which is of major scientic and

engineering importance even nowadays [?]

As a consequence of the development of the comprehension of electromagnetism many re-

searchers have explored the interaction between electromagnetic elds and specic media. Ar-

ticial electromagnetic materials, with negative permeability and permittivity, have pr oven to

2

have extraordinary electromagnetic properties. The study of these kind of articial materials

appears in the end of the XIX century, when Bose published his work on the rotation of the

plane of polarization by man-made twisted structures in 1898 [?]. Lindman studied articial

chiral media formed by a collection of randomly oriented small wire helices in in 1914 [?]. After

wards, there were several other investigators in the rst half of the XX century who studied

various man-made materials. In the 1950s and 1960s, articial dielectrics were explored by

Kock, and its application for lightweight microwave antenna lenses [?]. The `bedofnails' wire

grid medium was used in the early 1960s to simulate wave propagation in plasmas [?]. The

research on these kind of articial materials increased as the development of various potential

device and component applications appear [?].

Veselago published a paper in 1967 [?], but it was only translated to English in 1968, where

he considered a homogeneous isotropic electromagnetic material in which the permittivity

and permeability assumed negative real values. He studied the uniform wave propagation

that kind of material, which he named as left-handed (LH) material [?]. He concluded

that, in such medium, the direction of the Poynting vector of the wave is the opposite of

its phase velocity, suggesting that this isotropic medium supports a so called backward-

wave propagation and that its refractive index can be negative. Since such materials were

not available until recently, the interesting concept of negative refraction, and its various

electromagnetic and optical consequences, suggested by Veselago had received little attention.

This was until Smith inspired by the work of Pendry [?] constructed a composite medium

in the microwave regime by arranging periodic arrays of small metallic wires and split-ring

resonators and demonstrated the anomalous refraction at the boundary of this medium, which

is the result of negative refraction in this articial medium [?].

3

Figure 1.1: Photo of a nonlinear tunable metamaterial. The close-up photo square shows asplit-ring resonator with variable-capacity diode. (Source: Ilya, Shadrivov, Australian Na-

tional University, Nonlinear Physics Centre, Australia, 2008)

The negative refractive index propriety of DNG metamaterials could be used to bring radiation

to a focus with a at metamaterial lens, as proposed by Veselago [?] and then expanded by

Pendry [?]. The advantage of a at lens in comparison to a conventional curved lens is that

the focal length could be varied simply by adjusting the distance between the lens and the

electromagnetic wave source. These lens could be constructed using the split-ring resonator

conguration in a periodic array of metallic rings and wires, based on work by researchers at

the University of California at San Diego [?, ?]. A photograph of the at lens array of DNG

metamaterial cells, constructed by NASA [?] is showed in Figure 1.2.

4

Figure 1.2: Metamaterial at lens consisting of an array of 3 by 20 by 20 unit cells. [?]

For microwave radiation at wavelengths about 10 times a cell length, this conguration provides

negative eective values of electric permittivity and magnetic permeability, resulting in a

negative value for the index of refraction. The NASA Glenn Research Center testing have

demonstrated that appears a reversed refraction eect with focusing of the microwave radiation

and nite element models are being developed and an optics ray tracing code in order to create

new lens designs and to develop new congurations that are more amenable for operation

at higher frequencies. These research intends to achieve the applications of a at lens for

biomedical imaging and detection and other applications [?].

5

1.2 Motivation and Objectives

With the introduction of these new physical properties of DNG metamaterials, the study

and interpretation of the associated results is in fact very attractive and challenging. There

are many established physical concepts that must be re-interpreted in order to comply with

this new paradigm and there is also the probability of nding new eects associated with

this kind of materials, since there is a whole new set of resulting physical phenomena. In this

dissertation we have the possibility to associate and consolidate the more conventional and well

known electromagnetic concepts but now, with the introduction of the DNG metamaterials,

in a more generalized perspective, as we study the physical eects found even on simple

guiding structures. As up to today the demonstrations and experiments of the new physical

phenomena associated with DNG metamaterials have lead to the construction of new types

of microwave structures whose applications to mobile communication systems have attracted

a lot of attention from the scientic community. These metamaterials could help improve the

performance of several communication devices, such as antennas, and a lot of eort is being

made on the the design of antennae using this kind of periodic structures.

This kind of material also have implications on lens design. As classical electrodynamics

impose a resolution limit when imaging using conventional lenses, since this fundamental

limit, called the diraction limit, in its ultimate form, is attributed to the nite wavelength

of electromagnetic waves, the introduction of metamaterial lenses is also a subject of great

interest since no longer the resolution is restricted by the wavelength of the propagated light

waves. Conventional lenses focuses only the propagating waves, resulting in an imperfect

image of the object. The ner spatial details (which are smaller than a wavelength) of the

object, carried by the evanescent waves, are lost due to the strong attenuation these waves

experience when traveling from the object to the image. As predicted by Pendry [?], with DNG

metamaterial lenses, the evanescent waves are amplied by just the right amount. These waves

can be brought to a focus at the same position as an object's radiative eld, thereby producing

6

an image that has sub-wavelength detail.

As this kind of materials promise, for optical and microwave, new applications such as, for

example, new types of beam stirrers, modulators, band-pass lters, high resolution lenses,

microwave couplers, and antenna radomes, the study and research on these results is in fact

very encouraging and motivating for anyone who looks into addressing this subject of DNG

metamaterials.

The main objective of this dissertation is the analysis and study of wave propagation in DNG

metamaterial guides, and also the application of this kind of materials in lens design, taking

advantage of its particular electromagnetic properties to achieve results that are not present

in conventional lenses.

We try to understand the new physical phenomena that are associated with double negative

media and the eects when applied to well known propagation guide structures. The study

of lens design using DNG materials is also addressed in order to verify the dierent results

between these kind of lens against the physical limitations of common DPS lenses.

1.3 Structure

The rst chapter of this dissertation has the single purpose of introducing and situating the

reader in the subject that is addressed in this work. In order to do so a brief historical

background analysis is included at rst, where key researchers, publications an results are

mentioned, chronologically, in order to understand the evolution of the research process that

eventually reached to the object of study in this work. In this introductory chapter we also

expose the main motivations and objectives of this dissertation, as well as this explanation of

the work's structure.

In the second chapter we study the electromagnetic phenomena associated with DNG meta-

materials. After formulating the classication of a specic medium as DNG, the implications

7

of having a negative permittivity and permeability leads into studying the characterization

of the medium and the physical phenomena such as the appearance of backward waves and

the emergence and implications of negative refraction. A dispersive analysis is also introduced

in this chapter as we study the Lorentz Dispersive Model and nd a possible frequency in-

terval in which a material can act as DNG. The introduction of dispersion helps us infers

about the nature of behavior of both the phase and group velocities when dealing with DNG

metamaterials.

The third chapter deals with the guided wave propagation with DNG materials. We have

chosen to study two simple structures, the DPS-DNG interface and te DNG slab. A modal

analysis was made for both wave guiding structures and also numerical simulations, with the

respective interpretations. As a DNG medium is necessarily dispersive, the utilization of a

known dispersive model, the Lorentz Dispersive Model, is used in the analysis of the DPS-

DNG interface, and the results with and without the introduction of losses are compared. The

appearance of super-slow modes in the DNG slab propagation is also a subject of analysis on

this subject, consequence of having a phase velocity that is smaller than the outer medium in

which a DNG slab is immersed. The existence of these super-slow modes enables the propa-

gation on the DNG slab even if we use a less dense medium for the slab and this phenomenon

is studied as also. Even though we are using simple wave guiding structures and a somehow

elementary study when addressing the DNG guided propagation it proves to be an ecient

mechanism in order to evidenced this new physical problems and paradigm.

The fourth chapter is dedicated to the study of lens design using DNG metamaterials. First

we address a way to achieve a desired contour for a lens using physical concepts as the optical

path. The dependence of the refractive index on this process evidences the implications that

having a NIR medium as the material for designing lenses. The particular structure of the

Veselago's at lens, that is basically a DNG slab, is also analyzed in this chapter in order to

study a potentially practical application of DNG metamaterials in optics and the implications

of dealing with such materials. The conventional limitations of lens design when dealing with

8

sub-wavelenght detail are overrun by this DNG at lens as with DNG metamaterial lenses,

the evanescent waves are amplied by just the right amount allowing the waves to be brought

to single point focus at the same position as an object's radiative eld on the other side of the

lens and producing an image that has sub-wavelength detail. These results are also studied in

this chapter.

Finally, in the fth chapter, main conclusions are exposed and some developing potential

applications and further investigation hypothesis of the subjects addressed in this dissertation

are introduced.

1.4 Main Contributions

The main contribution of this dissertation is the analysis of known electromagnetic phenomena

but introducing the DNG metamaterials proprieties and concepts into the study of these phys-

ical subjects, hopefully helping further research on this kind of eld. The particular physical

phenomena that are generated by the usage of these materials when dealing with waveguides

or even with the design of lenses can provide a better comprehension of the potential of interest

when designing structures, being it communication devices or other physical components that

take advantage of the DNG media proprieties.

9

10

Chapter 2

Electromagnetics of Double Negative

(DNG) Media

Electromagnetic waves interact with the inclusions of particulate composite materials, inducing

magnetic and electric moments, which aects the macroscopic eective permittivity of the

bulk composite medium. Nowadays, metamaterials can be synthesized by articial fabricated

inclusions in an arbitrary host surface or host medium which provides the designer a wide set

of degrees of freedom, such as the host's size and shape and the composition's density and

alignment of the inclusions, in order to create a specic electromagnetic response that is not

found individually in each of the constituents.

2.1 Medium Characterization

Let us consider a specic material that is characterized by the two electromagnetic macroscopic

constitutive parameters: the electrical permittivity ε and the magnetic permeability µ. As

opposed to the response in vacuum, the response of materials to external elds generally

depends on the frequency of the eld, which reects the fact that a material's polarization

11

does not respond instantaneously to an applied eld. For this reason both the permittivity and

the permeability are often treated as complex functions of the frequency of the applied eld,

since complex numbers allow the specication of magnitude and phase [?]. These parameters

can both be described as follows:

ε = ε′ + ε′′ (2.1)

with ε′, ε′′ ε <. And:

µ = µ′ + µ′′ (2.2)

with µ′, µ′′ ε <

We can now proceed to the classication of the medium by analyzing the value of both ε′ and µ′

(the real parts of the permittivity and permeability). A medium with both the permittivity and

permeability greater than zero (<(ε) > 0 ,<(µ) > 0 ) is called a Double Positive Medium

(DPS), designation in which most naturally occurring media fall into (i.e. dielectrics). A

medium with the permittivity less than zero and the permeability greater than zero (<(ε) <

0 ,<(µ) > 0 ) is called an Epsilon Negative Medium (ENG), characteristic than can be

found, for certain frequency regimes, in many plasmas. A medium with permittivity greater

than zero and the permeability less than zero (<(ε) > 0 ,<(µ) < 0 ) is designated by Mu

Negative Medium (MNG), characteristic which, for certain frequency regimes, is exhibited

by some gyrotropic materials. A medium with both permittivity and permeability less than

zero (<(ε) < 0 ,<(µ) < 0 ) is designated as a Double Negative Medium (DNG), this

characteristic has only been demonstrated, up to this date, in articially constructed materials

[?]. Figure 2.1 shows the location of each medium qualication in a diagram whose axis is

formed by ε′ = <(ε) and µ′ = <(µ).

12

Figure 2.1: Material Classication

Let us now consider a generic media were both the constitutive parameters can be written as

functions of the frequency:

D = ε0ε(ω)E (2.3)

B = µ0µ(ω)H (2.4)

Let us now consider that the electric eld is polarized along the x-axis and the electromagnetic

wave propagates in the z-axis direction. We can write the expressions for both the electric

and magnetic elds in the time and z-axis domain:

E = xE0 exp[i(kz − ωt)] (2.5)

H = yH0 exp[i(kz − ωt)] (2.6)

13

Where the complex wave number k, is given by:

k = kz (2.7)

with k = nk0 (where n is the refraction index).

The vacuum wave-number k0, is given by:

k0 = ω√ε0µ0 =

ω

c(2.8)

where c is the speed of light.

Let us now consider the Maxwell Equations:

∇×E = −∂B∂t

(2.9)

∇×H = J+∂D

∂t(2.10)

∇ ·D = ρ (2.11)

∇ ·E = 0 (2.12)

To allow us to transform both E and H from the time domain to the frequency domain we

use the following Fourier transform pair:

Tω(r, ω) =

ˆ +∞

−∞tω(r, t) exp[iωt]dt (2.13)

14

tω(r, t) =1

2π

ˆ +∞

−∞Tω(r, ω) exp[−iωt]dt (2.14)

where k = xx+ yy + zz.

Applying (2.13) and (2.14) to (2.9)-(2.12) we obtain:

∇×E = iωB(ω) (2.15)

∇×H = J(ω)− iωD(ω) (2.16)

∇ ·D = ρ (2.17)

∇ ·E = 0 (2.18)

In order to express the spatial dependence of the eld quantities in (2.9)-(2.12) in the algebraic

form, we introduce the three-dimensional Fourier transform pair, which allows us to obtain

the Maxwell Equations in the wave number domain (or k-space):

Tk(r, k) =

ˆ +∞

−∞tk(r, t) exp[−ik.r]dr (2.19)

tk(r, ω) = (1

2π)3ˆ +∞

−∞Tk(r, t) exp[−ik.r]dk (2.20)

where k = xx+ yy + zz, dk = dkxdkydkz and k.r = kxx+ kyy + kz z.

Finally we can now work in the (k − ω) space by subjecting all eld quantities to a four-fold

Fourier transform given by the following transform pair:

15

Tk−ω(r, k) =

ˆ +∞

−∞tk(r, t) exp[iωt− ik.r]drdt (2.21)

tk−ω(r, ω) = (1

2π)4ˆ +∞

−∞Tk−ω(r, ω) exp[ik.r− iωt]dkdω (2.22)

where k = xx+ yy + zz, dk = dkxdkydkz and k.r = kxx+ kyy + kz z.

By using (2.21) and (2.22) to transform (2.9)-(2.12) we now obtain:

ik×E = iωB (2.23)

ik×H = J− iωD (2.24)

−ik ·D = ρ (2.25)

−ik ·E = 0 (2.26)

Assuming the inexistence of the conduction current (J=0) we now have from (2.23) and (2.24):

k×E = ωB = ωµ0µH (2.27)

k×H = −ωD = −ωε0εE (2.28)

Now from (2.5) we can write:

16

k×E = yωµ0µH0 exp[i(kz − ωt)] (2.29)

k×H = −xωε0εE0 exp[i(kz − ωt)] (2.30)

As we assumed, the electric eld E is polarized along the x-axis and the magnetic eld is

polarized along the y-axis in a way that the electromagnetic waves propagate trough the

z-axis, in the direction of k. Assuming that the media is isotropic we can state that:

k·E = k ·H = 0 (2.31)

So, from (2.25)-(2.28) now we can say that, as in[?]:

|k|E0 − ωµ0µH0 = 0 (2.32)

−ωε0εE0 + |k|H0 = 0 (2.33)

Or in its matricial form:

|k| −ωµ0µ

−ωε0ε |k|

E0

H0

=

0

0

(2.34)

As we are not trying to nd the solution were there are neither an electric nor a magnetic eld

(E0 = H0 = 0) we will equal the matrix determinant to zero:

|k|2 − ω2µ0µε0ε = 0 (2.35)

17

From (2.7) and with (2.8) we can write:

|k|2 =ω2

c2µε = k20µε (2.36)

This will allow us to dene the wave impedance, the ratio between the transverse components

of the electric and magnetic elds , [?]:

η =E0

H0=ωµµ0|k|

=|k|

ωεε0(2.37)

Now we can dene the frequency defendant refraction index n from (2.36) , (2.7) and (2.8) :

n =√µε (2.38)

And we can also dene the normalized wave impedance, the relation between the intensities

of the electric and the magnetic eld:

ζ =η

η0=

õ

ε=n

ε=µ

n(2.39)

with η0 being the free space intrinsic wave impedance. As we have seen before, the polarization

does not respond instantaneously to an applied eld. This causes dielectric loss, which can

be expressed by a permittivity and permeability that is both complex and frequency depen-

dent. Real materials are not perfect insulators either, i.e. they have non-zero direct current

conductivity [?]. Taking both aspects into consideration, we can dene a complex refraction

index:

n = n′ + in′′ (2.40)

where n′ is the refractive index indicating the phase velocity coecient and n′′is called the

18

extinction coecient, which indicates the amount of absorption loss when the electromag-

netic wave propagates through the material. Both n′ and n′′ are dependent on the frequency

[?]. Let us now dene, based on (2.40) and (2.7), the phase velocity vp, of an electromagnetic

wave:

vp =ω

<(k)=

ω

k0<(n)=

c

n′(2.41)

We can now write the complex amplitude equations for both the electric and the magnetic

eld using (2.40):

E = xE0 exp[ink0z] = xE0 exp[−n′′k0z] exp[in′k0z] (2.42)

H = yE0

ζη0exp[ink0z] = y

E0

ζη0exp[−n′′k0z] exp[in′k0z] (2.43)

The Time-Average Poynting Vector, which can be thought of as a representation of the energy

ux of the electromagnetic eld, is given by:

Sav =1

2<(E×H∗) (2.44)

Using the expressions (2.42) and (2.43) on (2.44) we obtain:

Sav = z|E0|2

η0<[

1

ζ

]exp[−2n′′k0z] (2.45)

In this case the value of n′′ needs to be always positive in order to verify energy extinction

along with the propagation of the wave on the z-axis, as expected since we are dealing with a

passive media where:

19

limz→∞

|E| ≤ E0 (2.46)

Now to take conclusions about the direction of the power ux we need to analyze the sign

of S. As we can see from (2.45) it depends on the sign of the real part of the normalized

impedance's value. From (2.39) we have:

<[

1

ζ

]= <

[n

µ

]= <

[n′ + in′′

µ′ + iµ′′

]=n′µ

′+ n′′µ′′

µ′2 + µ′′2(2.47)

As we have seen above, we are dealing with a passive media, so, as we concluded from (2.46),

we have:

n′′ > 0→ k′′ > 0→ µ′′, ε′′> 0 (2.48)

From (2.48), and by knowing that we are dealing with DNG media (µ′, ε′< 0), we can easily

verify that in (2.47) the divisor is always positive but we really can't conclude, at this moment,

about the sign of S because the sign of (2.47) may depend on the sign of n′(present at its

dividend). In (2.38) we have established a relation between the refraction index and both the

permittivity and permeability, so we will use that in order to infer about the nature of n′.

n = nµnε =√µ√ε (2.49)

We will now study the permittivity ε in the complex plan using polar coordinates. (It is

important to notice that we have chosen to study ε but the analysis is exactly the same for

the permeability).

ε = ρε exp[iθε] (2.50)

20

Graphically represented, as proposed in [?], by Figure 2.2.

Figure 2.2: The permittivity in the complex plan

We can also dene the permittivity dependent part of the refraction index in polar coordinates:

nε =√ε = n

′ε + in

′′ε =√ρε exp

[iθ

2

](2.51)

Using (2.50) we have for ε:

ρε =√ε′2 + ε′′2 (2.52)

cos(θε) =ε′

ρε=

ε′

√ε′2 + ε′′2

(2.53)

sin(θε) =ε′′

ρε=

ε′′

√ε′2 + ε′′2

(2.54)

21

As we are dealing with a passive DNG media (ε′< 0 and ε

′′> 0) we have for θε:

θε =[π

2, π]

(2.55)

We can also dene nε by:

nε =√ε = n

′ε + in

′′ε = i(n

′′ε − in

′ε) = iρε

[sin

(θε2

)− i cos

(θε2

)](2.56)

And from (2.55) we can obtain the argument of nε (by dividing it by 2):

θε2

=[π

4,π

2

](2.57)

If we use the following trigonometric relations:

cos

(θε2

)=

√1 + cos(θε)

2(2.58)

sin

(θε2

)=

√1− cos(θε)

2(2.59)

From (2.52)-(2.54) we can now write the argument of nε depending only on the permittivity:

cos

(θε2

)=

1

2

2√

1 +ε′√

ε′2 + ε′′2(2.60)

sin

(θε2

)=

1

2

2√

1− ε′√ε′2 + ε′′2

(2.61)

From (2.57) we now that for this specic interval both cos(θε2

)and sin

(θε2

)must be greater

than 0 so we must choose the positive root. Knowing this and with (2.51), (2.52)-(2.54) and

22

(2.60)-(2.61) we can now write:

n′ε =

√|ε′ |2

4

√1 +

(ε′′

ε′

)2√√√√√ 1 + sgn(ε′)√

1 +(ε′′

ε′

)2 (2.62)

n′′ε =

√|ε′ |2

4

√1 +

(ε′′

ε′

)2√√√√√ 1− sgn(ε′)√

1 +(ε′′

ε′

)2 (2.63)

We know that we are dealing with DNG media so sgn(ε′) = −1. We can now easily see by

the result in (2.62) and (2.63) , and for the interval that we have dened for θε2 , that n

′′ε > n

′ε

.

Let us now consider the limit case where there are no losses:

ε′′

= 0 (2.64)

From (2.62) and (2.63) we obtain:

n′ε = 0 (2.65)

n′′ε =

√|ε′ | (2.66)

And with these results in (2.65) and (2.66) we can write:

nε = i√|ε′ | (2.67)

As we have mentioned before a similar result can be obtained for the magnetic permeability

by using an analogous process:

23

nµ = i√|µ′ | (2.68)

With these two last results and using the denition in (2.49) we can now easily obtain the

refraction index for a DNG media:

n = nµnε = i√|µ′ |i

√|ε′ | = −

√|µ′ε′ | (2.69)

This proves that for a lossless DNG material the refraction index is negative. Let us now

consider the losses to create a more general solution:

n = nµnε = n′+ in

′′= (n

′ε + in

′′ε )(n

′µ + in

′′µ) = −(n

′′εn′′µ − n

′εn′µ) + i(n

′εn′′µ + n

′′εn′µ) (2.70)

As we have seen from (2.62) and (2.63), n′′ε > n

′ε and the same happens for the permeability

as the demonstration process is analogous so n′′µ > n

′µ so with the result in (2.70):

n′

= −(n′′εn′′µ − n

′εn′µ) < 0 (2.71)

n′′

= (n′εn′′µ + n

′′εn′µ) > 0 (2.72)

The results in (2.71) and (2.72) are indeed very important because they not only corroborate

the result in (2.48) that states that there is an extinction of the eld along the propagation

axis (as n′′> 0) but it also gives us the nal conclusion about the direction of the power ux

since n′< 0 so from (2.45) and (2.47) we can say that:

S · z > 0 (2.73)

24

As n′< 0 we can also state from (2.41) that we are dealing with medium with negative

phase velocity as its direction is the opposite from the energy ow and attenuation, from

(2.73) and from the fact that we are dealing with a passive media. Let us now analyze the

refraction index for the general case (with losses). In polar coordinates we have:

n =√ρn exp(iθn) =

√ρε√ρµ exp

[iθε + θµ

2

](2.74)

We saw that the condition was valid for nε and nµ the argument was in the interval[pi4 ,

pi2

]so it is easy to see from (2.74) that arg(n) is also between those values. The refraction index

on a DNG medium is in fact negative and we can now relate it with the propagation constant:

k = k · z = nk0 · z = z(n′k0 + in

′′k0) (2.75)

As n′k0 < 0 we can see that the direction of propagation is the opposite compared with the

energy ux:

k′ · z < 0 (2.76)

From (2.73) and (2.76) we can create a graphical representation of both the electric and

magnetic elds with the energy ux vector and the propagation constant for a DPS medium

and for a DNG medium and compare the results, represented in Figure 2.3.

25

Figure 2.3: Spatial Representation of the elds, the energy ux and the propagation constantfor a DPS and a DNG medium

Here we can see that, from these two types of medium, both the Poynting vector and the

propagation constant shares the same axis but not the same direction because in the DPS

media there is a right-handed trihedral formed by [E0, H0, k′] . From these results appears the

designated Backward Waves [?] (BW), electromagnetic waves that present a propagation

direction that is the opposite of the associated power ux..

2.2 Negative Refraction

As we have seen on the previous section, the phase velocity for wave propagation in a DNG

media is negative and this has important implications.

Let us consider the scattering of a wave that incises on a DPS-DNG interface as shown in

Figure 2.4.

26

Figure 2.4: Scattering of a wave that incises on a DPS-DNG interface

Now we assume that we have a DNG medium, with a negative refraction index (n2 < 0), in

the area with blue background (x < 0 and z > 0) and a DPS media, with a positive refraction

index (n2 > 0, in x > 0 and z > 0. We also assume that the losses on both the DPS and the

DNG materials can be neglected.

The Snell's law of reection assures us that the angle of reection is equal to the angle of

incidence:

θr = θi (2.77)

If we consider an uniform plane wave incising obliquely on a plane boundary (z=0) between

materials with dierent constitutive parameters (and refraction indexes n1, n2), and enforcing

the boundary conditions at the interface, we can also obtain, from the Snell's law of reection,

the relation between the angle of the transmitted wave and the angle of the incident wave [?],

which is given by:

27

sin(θt)

sin(θi)=n1n2

(2.78)

If we now consider the situation represented by the previous , where there is a DNG mate-

rial with a negative refraction index n2 we see that, for obtaining the correct angle of the

transmitted wave one must write (2.78) in the following form:

θt = sgn(n2) arcsin

[n1|n2|

sin(θi)

](2.79)

We must note that if the refraction index of a medium is negative, according the Snell's Law,

the refracted angle should also become negative and then, as we have seen in the previous

section, the direction of the energy ux, given by S, is the opposite of the wave propagation,

given by k. It's also important to notice that we are considering the solution where n′′> 0,

as we have mentioned in the previous section, because we are dealing with a passive media.

But if we have chosen to use n′′< 0, according to Snell's Law we would not have a negative

refracted angle but a positive one instead, which is the same result as if the transmitted wave

was propagating in a DPS material, with one very important dierence, as we have mentioned

before, that the energy ux was then propagating in the direction of the interface (and the

source) which is the opposite of a causal direction and makes no sense for a passive media.

2.3 The Lorentz Model

The temporal response of a chosen polarization eld component i to the same component of

the electric eld, assuming that the electric charges can move in the same direction as the

electric eld, can be described by a material model called the Lorentz Model [?]. This

model is derived from the description of the electron's motion in terms of a damped harmonic

oscillator:

28

d2

dt2Pi + ΓL

d

dtPi + ω2

0Pi = ε0χLEi (2.80)

Where the rst term describes the acceleration of the electric charges, the second one describes

the reduction of the oscillation's amplitude in terms of the damping coecient ΓL and the

third term describes the restoring forces of the system. On the right hand side of the equation

χL is called the coupling coecient.

The response in the frequency domain, using the operators used on the previous section, is

given by:

−ω2Pi(ω)− iωΓLPi(ω) + ω20Pi(ω) = ε0χLEi(ω) (2.81)

We know that the electric susceptibility χe, a measure of how easily it polarizes in response

to an electric eld, is given by:

χe =P

ε0E(2.82)

With both (2.81) and (2.82) we can obtain the Lorentz frequency dependent electric suscep-

tibility:

χe,Lorentz(ω) =Pi(ω)

ε0Ei(ω)=

χLω20 − iωΓL − ω2

(2.83)

The electric permittivity is given by:

ε = ε0(1 + χe) (2.84)

So with (2.83) and (2.84) we can obtain now the Lorentz electric permittivity:

29

εLorentz(ω) = ε0

(1 +

χLω20 − iωΓL − ω2

)(2.85)

There are also other models which are particular cases of the Lorentz Model when we are

making certain assumptions:

If the term that is related to the charge acceleration is very small when compared with both

the damping and the restoring forces term then we can neglect it, obtaining from (2.81) the

Debye Model:

−iωΓdPi(ω) + ω20Pi(ω) = ε0χdEi(ω) (2.86)

χe,Debye(ω) =χd

ω20 − iωΓL

(2.87)

When we have the case where the restoring forces are neglectful then we obtain from (2.81)

the Drude Model:

−ω2Pi(ω)− iωΓDPi(ω) = ε0χDEi(ω) (2.88)

χe,Drude(ω) =χD

−iωΓD − ω2(2.89)

The couple coecient χL (χd or χD depending on the model that is used) is normally repre-

sented by the plasmas frequency as χL = ω2p.

We have made our analysis of the Lorentz Model in terms of the electric polarization eld, but

the same kind of process can be made in terms of the magnetization eld Mi (instead of the

polarization) and the magnetic susceptibility χm. The magnetic permeability, using similar

analysis, is then given by:

30

µLorentz(ω) = µ0

(1 +

Mi(ω)

Hi(ω)

)= µ0

(1 +

χLω20 − iωΓL − ω2

)(2.90)

2.3.1 A DNG interval using the Lorentz Model

Let us consider the following expressions for the real parts of both the (relative) permittivity

and (relative) permeability obtained using the Lorentz Model [?]:

<(εr,L(ω)) =ω2pe(ω

20e − ω2) + ω2Γ2

Le + (ω20e − ω2)2

(ω20e − ω2)2 + ω2Γ2

Le

(2.91)

<(µr,L(ω)) =ω2pm(ω2

0m − ω2) + ω2Γ2Lm + (ω2

0m − ω2)2

(ω20m − ω2)2 + ω2Γ2

Lm

(2.92)

We now want to obtain a frequency interval, which we will represent as [ω−, ω+] where both

parameters have negative real parts, by denition of a DNG media. This can be formulated

as the following conditions:

[ω−ε , ω+ε ] −→ < (εr,L(ω)) < 0 (2.93)

[ω−µ , ω+µ ] −→ < (µr,L(ω)) < 0 (2.94)

First we will try to nd this interval for the frequencies where the permittivity is negative and

we can assume that for the permeability the computation is analogous.

Initially we must nd the limit in which the permittivity becomes negative by nding where

the real part becomes zero:

< (µr,L(ω)) = 0⇒ ω2pe(ω

20 − ω2) + ω2Γ2

Le + (ω20e − ω2)2 = 0 (2.95)

31

ω4 − ω2(2ω20e + ω2

pe − Γ2Le) + (ω2

peω20e + ω4

0e) = 0 (2.96)

We can nd the zeros by applying the Quadratic Formula to (2.96) from which we obtain the

following result:

2ω2 = 2ω20e + ω4

pe − Γ2Le ±

√Γ4Le + ω4

pe − 4ω20eΓ

2Le − 2ω2

peΓ2Le (2.97)

Admitting that there are no losses, by considering that the oscillation amplitude does not

decrease in time (ΓL = 0), we can simplify expression (2.97):

2ω2 = 2ω20e + ω2

pe ± ω2pe (2.98)

And now we have both the positive and negative solutions:

ω− = ω0e

ω+ =√ω20e + ω2

pe

(2.99)

From this result, and by doing the same kind of computation for the permeability, we can

conclude that there are in fact two frequency intervals, one for the permittivity [ω−ε , ω+ε ] and

one for the permeability [ω−µ , ω+µ ], where they assume negative values:

ω−ε = ω0e

ω+ε =

√ω20e + ω2

pe

,

ω−µ = ω0m

ω+µ =

√ω20m + ω2

pm

(2.100)

So we are in the presence of a DNG medium when the frequencies are in the following interval:

32

[ω−, ω+] = [ω−e , ω+e ] ∩ [ω−µ , ω

+µ ] (2.101)

Admitting that (2.101) it is not an empty set, we can nally write the interval in which, using

the Lorentz Dispersive model for both the permittivity and the permeability, the material acts

as a DNG medium:

[ω−, ω+] =[max

(ω−ε , ω

−µ

),min

(ω+ε , ω

+µ

)](2.102)

2.3.2 A DNG interval using the Drude Model

Let us now consider the Drude Model, a particularization of the Lorentz Model that also allows

negative permeabilities and permittivities but neglects the restoring forces (of the harmonic

model), and apply a similar process as we have done in the previous section. First we separate

the real and imaginary parts of the model's expression. As we have done in the previous

section, we will do the analysis for the permittivity as for the permeability the process is

analogous. The real and imaginary parts of the Drude Model permittivity is given by:

<(εr,D(ω)] =Γ2Deω

2 + ω4 − χDeω2

Γ2Deω

2 + ω4(2.103)

=(εr,D(ω)] =−iχDeΓDeωΓ2Deω

2 + ω4(2.104)

As we have done for the Lorentz Model, the wanted spectral interval can be found when

equaling to zero the real part of the model:

<(εr,D = 0⇒ ω4 + ω2(Γ2De − ω2

pe) = 0 (2.105)

33

Using again the quadratic formula for nding the zeroes on (2.105) we obtain the following

expression:

2ω2 = ω2pe − Γ2

De ± Γ2De − ω2

pe (2.106)

And now we have both the positive and negative solutions for this model:

ω− = 0

ω+ =√ω2pe − Γ2

De

(2.107)

From this result, and by doing the same kind of computation for the permeability, we can

again obtain, as we have done in the previous section, the two frequency intervals, one for

the permittivity [ω−ε , ω+ε ] and one for the permeability [ω−µ , ω

+µ ], where they assume negative

values:

ω−ε = 0

ω+ε =

√ω2pe − Γ2

De

,

ω−µ = 0

ω+µ =

√ω2pm − Γ2

Dm

(2.108)

So the frequency interval in which the media is DNG, when using the Drude Model, is given

by:

[ω−, ω+] =[0,min

(ω+ε , ω

+µ

)](2.109)

2.4 Group Velocity and Phase Velocity

The expression for the time-averaged energy density of a plane wave is [?]:

34

U =1

4

[ε0∂(ωε)

∂ω|E|2 + µ0

∂(ωµ)

∂ω|H|2

](2.110)

From (2.110) we know that:

∂(ωε)

∂ω> 0 (2.111)

∂(ωµ)

∂ω> 0 (2.112)

If we multiply (2.111) by µ and (2.112) by ε and then add them together we obtain the

following expression, that we will call A, which will be useful further in this section:

A = µε+ ωµ∂(ωε)

∂ω+ µε+ ωε

∂(ωµ)

∂ω= 2µε+ ω

[µ∂(ωε)

∂ω+ ε

∂(ωµ)

∂ω

](2.113)

Since we are dealing with a DNG medium, where ε, µ < 0, we can conclude from (2.113) that

A < 0.

We know from (2.35) and (2.39) that for an isotropic medium we have:

k2 = ω2µ0µε0ε (2.114)

Deriving it in order of the frequency ω we obtain:

∂(k2)

∂ω= µ0ε0

∂(ω2µε

)∂ω

= µ0ε0ω

[2εµ+ ωµ

∂(ωε)

∂ω+ ωε

∂(ωµ)

∂ω

]= µ0ε0ωA (2.115)

From (2.45) we know that:

k = nk0 = nω

c(2.116)

35

So we can also write (2.115) as:

∂(k2)

∂ω= 2k

∂(k)

∂ω= 2nk0

∂(k)

∂ω= 2n

ω

c

∂(k)

∂ω(2.117)

The Phase Velocity, vp, and the Group Velocity, vG are given by:

vp =ω

<(k)=

ω

k0<(n)=

c

n′(2.118)

vG =∂ω

∂k(2.119)

Using (2.118) and (2.119) on (2.117) we obtain:

∂(k2)

∂ω= 2ω

1

vp

1

vG(2.120)

Since A < 0 this implies that ∂(k2)∂ω < 0 and from (2.210), for a lossy DNG medium, we can

conclude that, on a dispersive DNG medium, the group velocity and the phase velocity have

dierent signs.

For dispersive media is also easy to prove that the group velocity and phase velocity have

dierent values (since for a non-dispersive media vp = vG). If we derive (2.116) we obtain:

∂(k)

∂ω=∂(ωn)

∂ω

1

c=

1

c

(n+ ω

∂n

∂ω

)(2.121)

From (2.118) and (2.119) we can now write (2.122) as:

1

vG=

1

vp+

1

cω∂n

∂ω(2.122)

36

That shows us that vp = vG is only possible when there is no frequency dependence of the

refraction index.

2.5 Kramers-Kronig Relations

Some general relations were developed to relate the real part of an analytic function to a

integral that contains it's imaginary part, and vice-versa.

Applying this relations to the dielectric function ε(ω) we obtain [?]:

<[ε(ω)] = 1 +1

2π

ˆ +∞

0

Ω=[ε(ω)]

Ω2 − ω2dΩ (2.123)

=[ε(ω)] =−2ω

π

ˆ +∞

0

<[ε(ω)]− 1

Ω2 − ω2dΩ (2.124)

Named after Ralph Kronig and Hendrik Kramer, they are known as the Kramers-Kronig

Relations.

The real part of ε(ω), (2.123), is related with the refraction index and the imaginary part,

(2.124), is related with the eld's extinction (as we have seen in the previous section). In the

computation of these integrals the Cauchy Principal Value method is used.

Equation (2.123) allows us to obtain the refraction index prole and chromatic dispersion, phe-

nomenon where the phase velocity and the group velocity depend on frequency, of a medium

by knowing only it's frequency dependent losses, which can be measured over a large spectral

range. This is a very important result because it demonstrates that there is an interdepen-

dency between losses and dispersion.

Equation (2.124) gives a not so useful result. We can use it to obtain the eld extinction by

knowing the refraction index but it is very dicult to measure this index over a wide frequency

range.

37

38

Chapter 3

Guided Wave Propagation in DNG

Media

3.1 Propagation on a Planar DNG-DPS Interface

In this section we will study the propagation of electromagnetic waves on a planar interface

between a DPS and a DNG medium, which is represented in Figure 3.1.

Figure 3.1: The planar interface between a DPS and a DNG medium, here represented by adashed line.

39

3.1.1 Modal Equations

Let us consider that the propagation direction is given by the z-axis, the transverse direction

by the x-axis and the y-axis as the transverse innite direction, where there is no variation

of both the electric and magnetic elds. Since the surface is homogeneous along the z-axis,

solutions to the wave equation can be taken as:

E(x, t) = Em(x) exp[i(βz − ωt)] (3.1)

H(x, t) = Hm(x) exp[i(βz − ωt)] (3.2)

Or, in the time harmonic form of the elds:

E(x, t) = Em(x) exp[iβz] (3.3)

H(x, t) = Hm(x) exp[iβz] (3.4)

We can now plug the general eld solutions (3.3) and (3.4) into the Homogeneous Wave

Equation, for the Transverse Electric (TE) mode, given by:

∇2E+ ω2εµE = 0 (3.5)

∂2

∂x2E+

∂2

∂z2E+ (k20n

2i )E = 0 (3.6)

∂2

∂x2E+

(k20n

2i + β2

)E = 0 (3.7)

40

Where ni is the refraction index of the medium i (given by ni =√εiµi) and β the propagation

constant.

Since the term between parenthesis in equation (3.7) is constant in x we are dealing with a

constant coecient dierential equation that could have the following solution:

Ey(x) = E0 exp[ihix] + E0 exp[−ihix] (3.8)

Where we can also dene hi as the transverse wave number of the medium i (given by h2i =

k20n2i − β2).

In order to maintain wave guiding on the interface, the elds must be evanescent and decay

with distance away from the separation surface. This requirement causes the propagation

constant to be in the range of k0ni < β, therefore the propagation constant in both the

regions is complex and it is given by:

hi = ±iαi (3.9)

Where αi, the attenuation constant, is given by:

α2i = β2 + k20n

2i (3.10)

The sign of hi is chosen in such way that the eld decays with distance away from the interface

so the resulting elds on both the DPS and DNG regions (that we will call 1 and 2 respectively).

Admitting that the interface is on x = 0, the eld's expressions are given by:

Ey(x) =

E0 exp[−α1x] , x > 0

E0 exp[α2x] , x < 0

(3.11)

41

With α1 , α2 > 0.

We can now use Faraday's Law, from the Maxwell Equations, to compute the magnetic eld:

∇×E = iωµH (3.12)

H =1

iωµ∇×E (3.13)

And for this case of TE propagation mode, expression (3.13) can be written as:

H =1

iωµ

∂Ey∂x

z (3.14)

From Eq. (3.13) we can now obtain the expressions for the magnetic eld on both regions:

Hz(x) =

iE0α1ωµ1

exp[−α1x] , x > 0

−iE0α2ωµ2

exp[α2x] , x < 0

(3.15)

Applying the boundary conditions at the interface (x = 0), and assuring the continuity of the

magnetic eld components (Hz(0)− = Hz(0)+) we can write:

iE0α1

ωµ1exp[−α10] =

−iE0α2

ωµ2exp[α20] (3.16)

α1

µ1= −α2

µ2(3.17)

Now we have obtained the modal equation for the Transverse Magnetic () propagation

mode given by:

42

α2µ1 + α1µ2 = 0 (3.18)

By applying a similar computation process to both the wave equation and eld expressions

for the TM modes, and using the following equation from the Maxwell Equations:

∇×H = −iωεE (3.19)

We can also obtain the modal equation for the TE mode:

α2ε1 + α1ε2 = 0 (3.20)

With these results (3.18) and (3.20) we are now able to infer if there is propagation along the

interface. Since we now that both α1, α2 > 0 and µ1, ε1 > 0, from (3.18) :

α1 = −µ1µ2α2 > 0 =⇒ µ2 < 0 (3.21)

And from (3.20):

α1 = −ε1ε2α2 > 0 =⇒ ε2 < 0 (3.22)

So, from the implications on (3.21) and (3.22), we can conclude that it is in fact possible to

have propagation on an interface between a DPS medium and a DNG medium (ε2, µ2 < 0).

3.1.2 Surface Mode Propagation

We will now use the Lorentz Dispersive Model (LDM), which was introduced on the previous

chapter, to study the solutions of the modal equations. The model frequency dependent

permittivity and permeability are given, as seen before, by:

43

εr,L(ω) = 1 +ω2pe

ω20e − iωΓL − ω2

(3.23)

µr,L(ω) = 1 +ω2pm

ω20m − iωΓL − ω2

(3.24)

These models will be used to describe the frequency dependence of the parameters on the

DNG medium (region 2), and have chosen the following values for the plasma's frequencies

ωpm, ωpe, central frequencies ω0e, ω0m, damping coecient ΓL, as well as the parameters of the

DPS medium ε1,r, µ1,r. The simulation parameters are represented at Table 2.1.

Parameter Value

ωpe 2π × 7× 109rad.s−1

ωpm 2π × 6× 109rad.s−1

ω0e 2π × 2.5× 109rad.s−1

ω0m 2π × 2.3× 109rad.s−1

ΓL 0.05× ωpeε1,r 1µ1,r 1

Table 3.1: Simulation parameters for the Lorentz Dispersive Model, on the DPS-DNG interfacestructure

3.1.2.1 Neglecting Losses in the LDM (ΓL = 0)

First lets analyze the variation of the parameters εr,L(ω) and µr,L(ω). The representation is

presented in Figure 3.2.

From Figure 3.1 we can easily identify three regions:

• region (1) where: ε < 0 and µ < 0 (DNG),

• region (2) where: ε < 0 and µ > 0 (ENG) ,

• region (3) where: ε > 0 and µ < 0 (DPS).

44

Figure 3.2: Lorentz lossless dispersive model for εr,L and µr,L

With relation (3.27), and using the Lorentz dispersive model, we can now analyze the variation

of the refraction index with frequency, represented on Figure 3.3.

We can nd in Figure 3.3 that we have the three regions, as mentioned before. As expected, on

the DNG region, we have a negative refraction index and on the DPS region we have a positive

refraction index. On the ENG region, as the permittivity is negative and the permeability is

positive, we have a purely imaginary refraction index. The eect that n varies with frequency

(except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion.

In regions of the spectrum where the material does not absorb, the real part of the refractive

index tends to increase with frequency, as seen in Figure 3.3. Near absorption peaks, the

curve of the refractive index is a complex form given by the KramersKronig relations,

and can decrease with frequency. The real and imaginary parts of the complex refractive index

are related through use of the KramersKronig relations (one can determine a material's full

complex refractive index as a function of wavelength from an absorption spectrum of the

45

Figure 3.3: Relative refraction index (nr = n√ε0µ0

), using the lossless LDM, on the DPS-DNGinterface

material).

Using equations (3.18), (3.20) and (3.11) we can now establish a relation that expresses the

variation of the propagation constant β with frequency, called the dispersion relation. The

dispersion relation describe the interrelations of wave properties such as wavelength, frequency,

velocities, refraction index, attenuation coecient. For the TE mode, the relation is given by:

β(ω) =

√√√√√µ2(ω)ε2(ω)− µ1ε1(µ2(ω)2

µ21

)1−

(µ2(ω)2

µ21

) k0 (3.25)

And for the TM mode:

β(ω) =

√√√√√µ2(ω)ε2(ω)− µ1ε1(ε2(ω)2

ε21

)1−

(ε2(ω)2

ε21

) k0 (3.26)

46

Using the Lorentz Dispersive Model, and the expressions in (3.26) and (3.25), we can now

obtain the dispersion relation graphical representation for both the TE and the TM modes,

presented in Figure 3.4.

Figure 3.4: Dispersion relation, β(ω), using the lossless LDM, on the DPS-DNG interface

From both Figures 2.4 and 2.5 we can see that when∣∣∣µ2(ω)2µ21

∣∣∣ = 1 (or∣∣∣ ε2(ω)2ε21

∣∣∣ = 1) the value

of β(ω) goes to innity which represents an unphysical solution, as an innite propagation

constant, at a given frequency, is not a valid electromagnetic phenomenon.

The graphical representation of the attenuation constants, for both the TE and TM modes, is

represented in Figure 3.5 and 3.6.

47

Figure 3.5: Attenuation constants α1 and α2 for the TE modes, using the lossless LDM, onthe DPS-DNG interface

Figure 3.6: Attenuation constants α1 and α2 for the TM modes, using the lossless LDM, onthe DPS-DNG interface

48

The attenuation constants have, for this frequency interval, a positive real part, as a condition

to have propagation along the interface and exponential attenuation as we move away from

it, as stated in (3.21) and (3.22). Since we are not dealing with losses, the imaginary parts of

both α1 and α2 are both zero, in the intervals where we have propagation.

3.1.2.2 Considering Losses in the LDM (ΓL = −0.05× ωpe)

Le us now consider a lossy structure using the Lorentz Dispersive mode. The constitutive

parameters are graphically represented in Figure 3.7.

Figure 3.7: Lorentz dispersive model for εr,L and µr,L

From Figure 3.7 we can identify that the three regions are approximately the same from the

previous structure when we were neglecting losses, and this happens since we are dealing with

a small value for ΓL . The positive imaginary parts are the result of a negative damping

present on the Lorentz's Model.

If we consider a relative refraction index given by:

49

nr =n

√ε0µ0

(3.27)

For the lossy situation, a representation of the refraction index can be obtained and is shown

in Figure 3.8.

Figure 3.8: Relative refraction index (nr = n√ε0µ0

), using the lossy LDM, on the DPS-DNGinterface.

In Figure 3.8 we also have the representation of the same three regions, as mentioned before.

As we expected on the DNG region we have a negative refraction index and on the DPS

region we have a positive refraction index. From the variation n on the ENG region, where

we also have a negative real component of the refraction index, we can take an important

conclusion. The existence of a negative real refraction index on this ENG region proves that a

DNG medium has always the designation of (Negative Refraction Index) but a NRI medium

does not have to be DNG, as we can see when considering losses and dispersion.

The representation of the dispersion relation, β(ω), using the Lorentz Dispersive Model, and

50

considering losses, is showed in Figure 3.9.

Figure 3.9: Dispersion relation, β(ω), using the lossy LDM, on the DPS-DNG interface.

From the graphical representation of the dispersion relation in Figure 3.9 we can see that

the value of β(ω) no longer goes to innity (as it does when neglecting losses, Figure 3.4).

Both the real and the imaginary parts of β(ω) experience a signicant increase in the range

of frequencies where there were asymptotes (∣∣∣µ2(ω)2µ21

∣∣∣ = 1 (or∣∣∣ ε2(ω)2ε21

∣∣∣ = 1) but they can now

represent valid physical solutions as the propagation constant is no longer innite.

The representation of the attenuation constants for both the TE and TM modes are depicted

in Figure 3.10 and Figure 3.11.

51

Figure 3.10: Attenuation constants α1 and α2, for the TE modes, using the lossy LDM, onthe DPS-DNG interface

Figure 3.11: Attenuation constants α1, for the TM modes, using the lossy LDM, on theDPS-DNG interface

52

The attenuation constants have also, for this frequency interval, a positive real part, as a

condition to have propagation along the interface and exponential attenuation as we move

away from it, as stated in (3.21) and (3.22) but now the imaginary parts of both α1 and α2

are always negative, condition that is needed in order to have propagation along the z-axis.

From the expressions in (3.11) we can also have a graphical representation of the electric eld's

variation along the x-axis dimension. This is shown in Figure 3.12. The variation of the eld

shows us that the eld intensity increases as we approach x = 0, as we expected, because

this is a representation of the eld in the interface (which is at x = 0) and the attenuation as

we get further from it. From modal equations (3.18) and (3.20) we can also verify that the

slope of the eld branches is also inuenced by the values of both the permittivities and the

permeabilities of the DPS and DNG media.

Figure 3.12: Variation of the electric eld, Ey(t = 0, x, z), on the DPS-DNG Interface

53

3.2 Propagation on a DNG Slab Waveguide

3.2.1 Modal Equations

In this section we will study the propagation of electromagnetic waves on a DNG slab waveg-

uide represented by Figure 3.13.

Figure 3.13: A DNG slab waveguide immersed on a DPS media

For the TE modes, and as we have done in the previous chapter, we have the following wave

equation:

∂2

∂x2E+

(k20n

2i + β2

)E = 0 (3.28)

The solutions form this equation, considering that −d < x < d, can take the form of:

Ey = A cos(h1x) +B sin(h1x) (3.29)

with h1 = ω2µ1ε1 + β2 .

For this structure we have presented for the slab, we want that the electric eld decays with

distance as we get away from the slab, so the evanescence of the electric eld can be represented

by:

54

Ey(x) =

C exp(ih2x) , x ≥ d

D exp(ih2x) , x ≤ −d

(3.30)

Where the transverse wave number is dened by:

h2 = ±jα2 (3.31)

With the attenuation constant, α2, given by:

α22 = β2 − k22 = β2 − ω2ε2µ2 (3.32)

Placing this attenuation constant in (3.30) we can now establish for the evanescent elds the

following expressions:

Ey(x) =

C exp(−α2x) , x ≥ d

D exp(α2x) , x ≤ −d

(3.33)

From the result on (3.30) we can see that there are two kinds of solutions:

• one even solution, given by the cos(h1x) term,

• one odd solution, given by the sin(h1x) term.

We will show the manipulation only for the odd mode since the procedure is the same for the

even mode.

We can now represent the electric eld, inside and outside the slab, by the following relations:

55

Ey(x) =

B sin(h1x) exp(iβz) , |x| ≤ d

C exp(−α2x) exp(iβz) , x ≥ d

D exp(α2x) exp(iβz) , x ≤ −d

(3.34)

Obtaining the magnetic eld expression can be done by using Faraday's Law, from the

Maxwell's Equations:

∇×E = iωµH (3.35)

H =1

iωµ

(−∂Ey∂z

x+∂Ey∂x

z

)(3.36)

Applying this equation on the resultant eld expression on (3.34) we obtain the magnetic eld

for this structure:

Hy(x) =

(− Bβωµ1

sin(h1x)x+ ih1Bωµ1

cos(h1x)z)

exp(iβz) , |x| ≤ d

(− Cβωµ2

exp(−α2x)x+ iCα2ωµ2

exp(−α2x)z)

exp(iβz) , x ≥ d

(− Dβωµ2

exp(α2x)x− iCα2ωµ2

exp(α2x)z)

exp(iβz) , x ≤ d

(3.37)

Applying the boundary conditions at the interface (x = d), assuring the continuity of the

magnetic eld components z, and assuming that B = A = E1 and C = D = E2, we can

obtain, from (3.37):

B sin(h1d)− C exp(−α2d) = 0 (3.38)

56

−h1µ2µ1

cot(h1d) = α2 (3.39)

We call to this result in (3.39) the asymmetric or odd TE modal equation, as we have used

the odd solution of the wave equation.

Repeating the same kind of algebraic manipulation procedure to the even solution of the wave

equation we obtain the even or symmetric TE modal equation:

h1µ2µ1

tan(h1d) = α2 (3.40)

Achieving the results for the both the odd and even TE modal equations for the slab structure:

−h1dµ2µ1 cot(h1d) = α2d (Odd Modes)

h1dµ2µ1

tan(h1d) = α2d (Even Modes)

(3.41)

Using the same procedure to obtain the TM modes we get:

−h1d ε2ε1 cot(h1d) = α2d (Odd Modes)

h1dε2ε1

tan(h1d) = α2d (Even Modes)

(3.42)

We can now simplify the modal equations by making the following substitutions:

a = α2d (3.43)

b = h1d (3.44)

57

Obtaining for the TE modes he following relations:

a = −µ2

µ1b cot(b) (assymetric mode)

a = µ2µ1b tan(b) (symmetric mode)

(3.45)

The relation between the normalized propagation's constants is given by:

a2 + b2 = V 2 (3.46)

Where V , the normalized frequency, is given by:

V = k0d√ε2µ2 − ε1µ1 (3.47)

The intersection of the curve(3.46) with the modal equations will represent the modal solutions

for these modes in the slab.

3.2.2 Surface Mode Propagation

We will now study the surface modes on the DNG slab. From (3.29) we can easily nd that the

transverse propagation constant h1 can take real values if β < ω√ε1µ1 and imaginary values if

β > ω√ε1µ1 and, for the analysis of the slab, we know that assuming either imaginary or real

values for h1 we will maintain the surface mode conditions where we have the wave diminish

with distance from the slab. Let us now assume that B = −ib, if we consider the following

relations:

tan(ix) = i tanh(x)

cot(ix) = −i coth(x)

(3.48)

58

We can now rewrite equations (3.45) and (3.46):

a = −µ2µ1B coth(B) (3.49)

a = −µ2µ1B tanh(B) (3.50)

a2 = B2 + V 2 (3.51)

We can now nd the numerical solutions for the modes graphically. These solutions can be

found as the result of the intersection of the curves obtained from the modal equations (3.50)

and (3.51) and the curve from (3.46) or (3.51), as we have said before.

At rst we will consider the DPS situation where ε1 = µ1 = 1 and ε2 = µ2 = 2, the graphical

solution is shown on Figure 3.14, where the horizontal positive semi-axis represents the trans-

verse propagation constant b and the negative semi-axis represent its imaginary value, that

we have previously called B.

On Figure 3.15 we have the modal solution's representation, but now considering a DNG slab

with ε1 = µ1 = 1 and ε2 = µ2 = −1.5.

As we can see from the Figures 3.14 and Figure 3.15, for the DNG slab there are also solutions

with imaginary values of b, that we dened as B. These modes are called super-slow modes,

since the phase velocity, given by vp = ωβ , assumes such values that:

vp <c

√ε2µ2

(3.52)

The graphical solution for a dierent value of V is shown in Figure 3.16.

Form Figure 3.16 we can also see positive modal solutions, with b being real, as we have

59

Figure 3.14: The representation of the modal solutions (red dots) given by the intersection ofthe curves for a DPS slab with ε1 = µ1 = 1 and ε2 = µ2 = 2.

seen on the DPS slab, represented on Figure 3.14. These positive-b surface modes are called

slow-modes since the value of the phase velocity assumes values on the interval:

c√ε2µ2

< vp <c

√ε1µ1

(3.53)

Since we are now dealing with a DNG medium for the slab, ε2, µ2 < 0, this inverses the

signal of the modal equations in such way that the slopes of the tangents and cotangents are

changed, and we also have some slow-modes that, for a given range of frequencies, can have

more than one solution for the same h1d value, as we can see on Figure 3.16. The slow/super-

slow transitions for multiple solutions of the same mode can be described by the next relations.

For the even modes:

cos2(b) +

(µ1|µ2|

)2 [sin2(b) + b tan(b)

]= 0 (3.54)

60

Figure 3.15: The representation of the modal solutions (red dots) given by the intersection ofthe curves for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −1.5 and V = 0.5

And for the odd modes:

sin2(b) +

(µ1|µ2|

)2 [cos2(b) + b cot(b)

]= 0 (3.55)

The representation of the dispersion diagram for the DNG dielectric slab, shown on Figure

3.17.

Here we can see the two dashed lines that represent transition limits dened by functions of

βd(k0d). The rst one, given by k0d = βd√µ1ε1

, represents the cuto condition of the surface

modes on the slab, where h1d = 0. The second limit line, from the relation k0d = βd√µ2ε2

, gives

us the transition border from a slow-mode to a super-slow surface mode, as we can see from

Figure 3.17 where the fundamental mode is a super-slow odd mode represented by a red curve.

This super-slow mode, from Figure 3.17, becomes a slow-mode when V = µ1|µ2| and propagates

61

Figure 3.16: The representation of the modal solutions (red dots) given by the intersection ofthe curves for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −1.5 and V = 3

until V = π2 as we can see from Figure 3.18.

i) ii)

Figure 3.18: Modal solutions (red dots) for a DNG slab with ε1 = µ1 = 1 , ε2 = µ2 = −2,with (i) V = µ1

|µ2| and (ii) V = π2

On this DNG slab structure, from the results on Figure 3.18, we can conclude that there is a

direct relation between the constitutive parameters and the resultant dispersive diagram, as

the point from which the fundamental mode transitions from a super-slow mode to a slow-

mode depends on the value of both µ1 and µ2. On the previous situation, that we have used to

62

Figure 3.17: Dispersion diagram for a DNG slab with ε1 = µ1 = 1 and ε2 = µ2 = −1.5

generate the results on both Figure 3.16 and Figure 3.17, we have assumed that µ1ε1 < µ2ε2

and that µ1 < |µ2|, however, if we consider a case where the slab's inner medium is less dense

than the outer medium, µ1ε1 > µ2ε2, we obtain dierent and important results.

From the expression (3.50), where we dened the normalized frequency, we can easily nd that

if we consider µ1ε1 > µ2ε2 we obtain:

V 2 < 0 (3.56)

From this result, and still considering the situation where the outer medium is more dense

than the slab's inner medium, we have from (3.49) :

b2 + a2 < 0 (3.57)

Considering that, in order to have propagation one must satisfy the condition:

63

a2 ≥ 0 (3.58)

So now we can conclude, from equations (3.57) and (3.58), that the following relation must be

veried in order to have propagation on the slab:

b2 < 0 (3.59)

These conditions can only be true if we are in the presence of super-slow modes, as we can

see from Figure 3.19, which verify (3.58) , (3.59) and B2 + V 2 ≥ 0. From this result we can

say that the propagation on less dense interior medium, as stated by the inequality (3.56), is

only possible if we are in the presence of super-slow modes and this is a phenomenon that is

veried when using a DNG slab.

We will now analyze the dispersion diagrams for this situations but considering the inuence

of the constitutive parameters, as we have mentioned before. As we have done for the denser

inner medium, we will rst consider the case where |µ2| > µ1. The dispersion diagram is

shown in Figure 3.19.

Here we can see that there is propagation of two super-slow modes where, as we increase in

frequency, or k0 = ωc , both transverse propagation constants, β tend for the same value. Both

these modes have a null cuto frequency, one being a conventional mode, the even one, and a

limited odd mode.

The dispersion diagram where |µ2| < µ1 is shown on Figure 3.20 in order to compare with the

results obtained in Figure 3.19.

64

Figure 3.19: Dispersion diagram for a DNG slab with ε1 = 2, µ1 = 1 , ε2 = −1 and µ2 = −1.5

Figure 3.20: Dispersion diagram for a DNG slab with ε1 = 2, µ1 = 2 , ε2 = −1 and µ2 = −1.5

65

On this situation we can see that only one even super-slow mode propagates and on a limited

frequency band. We can also notice that, for all the frequency band in which the mode

propagates, there are always two modal solutions and these tend to the same value as we

increase the frequency. The point where the double-solutions intersect represents the limit

from which there is no surface mode propagation on the DNG slab.

3.3 Conclusions

In this chapter we have studied and presented the propagation in a DNG metamaterial medium

by analyzing the physical phenomena and implications of having both a negative magnetic

permeability and electric permittivity. We have shown that in this kind of media the existence

of waves that propagate in a antiparticle direction of the power ux is noticed, called the

Backward Waves, and that we are in the presence of a Negative Index of Refraction material,

which implies some modications in the interpretation of Snell's Law. A dispersive analysis

is also made using the Lorentz Dispersive Model, and for a particularization called the Drude

Model, and using these models we have also shown that it is possible to nd a DNG interval

even when considering dispersion. From this introduction of losses we have also concluded that

both the group and phase velocities have dierent values and, for this kind of DNG media,

they even have opposite directions.

We have also presented, in this chapter the study of two wave guiding structures using DNG

metamaterials: the DPS-DNG interface and the DNG slab. This is important because these

kind of waveguides present particular physical eects that could be used in wave propagation

structures and even replace some most common DPS guides. The dispersive models mentioned

before where used in the study of theses structures.

First we have showed that it is possible to have both TE and TM surface mode wave prop-

agation on a DPS-DNG interface. This kind of propagation mode is new and does not exist

in other more conventional DPS wave guiding structures. We have also found that, when

66

not neglecting losses, the feature of being a NIR medium can be applied to all DNG media

but the NIR designation is not exclusive of DNG materials since there are other non DNG

frequency bands where the medium also acts as a NIR. When dealing with the propagation of

this surface waves we have also seen that it permits large attenuation outside the interface.

Finally we have analyzed the guided propagation on a DNG slab, whose electromagnetic

proprieties can be of large interest for the practical application of DNG materials to the

construction of waveguides. In this structure there's also the possibility of having surface

wave mode propagation but the most important result is the propagation of super-slow modes

that are a consequence of having a phase velocity that is smaller than the outer medium in

which a DNG slab is immersed (there is also slow-mode propagation that exhibits a double

modal solution for some frequency bands). The existence of these super-slow modes enables the

propagation on the DNG slab even if we use a less dense medium for the slab (i.e., medium

2) when compared to the outer medium (i.e. medium 2), ε1µ1 > ε2µ2. This phenomenon

is veried, fullling the propagation conditions, only on double negative materials. When

analyzing the dispersion relation diagrams for the ε1µ1 > ε2µ2 structure we could see that

for a medium with |µ2| > µ1 the are two super-slow modes and for a medium with µ1 > |µ2|

only one super-slow mode propagates on a limited frequency band, but that there are, in this

band, always two possible modal solutions.

The study of DNG media characterization and the application of DNG materials in the pre-

sented wave guiding structures could help the understanding of implications and capabilities

of using this kind of material on propagation structures.

67

68

Chapter 4

Lens Design Using DNG Materials

4.1 Optical Path and the Lens Contour

Let us consider that there are light rays emanating from a source at point O and that they

are being transmitted in the θ direction. In order to convert these light rays to plane waves

we must use a lens to assure that the optical paths for the dierent directions are equal as

they reach a plane wavefront. This can be represented by Figure 4.1.

Figure 4.1: Lens contour and optical path representation

69

Considering, in this 2D representation, a plane wavefront dened by the line formed with

points P1 and P2, we can state that the two optical paths must be equal, one from O to P1

(where we have free space propagation), and one from O to P2 (where there is free space

propagation from O to c, and the propagation in a medium with a refraction index of n from

c to P2). This equality can be represented by the following expression:

OP1 = R = OC + nCP2 (4.1)

Or in polar coordinates:

R = d+ n[R cos(θ − d)] (4.2)

R =d(1− n)

[1− n cos(θ)](4.3)

Where n is the refraction index of the material of the lens.

Also from Figure 4.1 we can establish the Cartesian coordinates:

x = R

d cos(θ)

y = Rd sin(θ)

(4.4)

Where,

R

d=√x2 + y2 =

1− n1− n cos(θ)

(4.5)

From expression (4.5) we can also establish a direct relation between the coordinates and only

the refraction index:

70

√x2 + y2 =

1− n1− n x√

x2+y2

(4.6)

And after some algebraic manipulation:

(x− n

n+ 1

)2

− y2

n2 − 1=

1

(n+ 1)2(4.7)

From this expression we can achieve the lens contour in order to verify the equality we have

shown in (4.1). We can also see that expression (4.7) is in fact an elliptical formula, which

degenerates on a circumference as the refractive index approaches n = 0. In a matter of fact

the 2D lens contours are also commonly called as circles [19]. Dierent lens' contours, for

dierent values of n, are represented in Figure 4.2.

Figure 4.2: The lenses contours for dierent refraction indexes, n = −2.5,−1.5, 100, 1.5, 2.5

From (4.2) we can see that there is an asymptote in n = 1cos(θ) , making the contours hyperbolic.

We can also see from Figure 4.2 that the curvature is the opposite depending on n being either

positive or negative.

After we have calculated the optical path we can now analyze the design in terms of focal

71

length. The focal length can be seen as a measure of how strongly the lens converges or

diverges light, or in geometric terms, the distance over which initially transmitted rays are

brought to a focus. It can be given, as f , by the following expression [?]:

f =

∣∣∣∣ Rc1− n

∣∣∣∣ (4.8)

From this expression we can see that this length depends on the refractive index, n, and

the radius-of-curvature of the lens surface, Rc. We can note, as an example, that a concave

cylindrical lens with n = −1 has the same focusing properties as a convex lens with n = +3

so, if we consider n to be negative, a lens with that properties can alter the trajectory of

transmitted waves as if the material possessed a much larger index.

4.2 The Veselago's Flat Lens

As we have seen from the expression (4.7) and in Figure 4.2, as the refraction index tends

to large values (or even innity), the contour tends to a straight line, which can be called

as a at lens [?]. Knowing that such a large refractive index does not have any important

practical application [?], a functional at lens was proposed by Victor Georgievich Veselago

in 1968 [?]. In Veselago's paper [?] he proposed that a planar slab, composed by a material

with the refractive index n = −n0, with n0 being the refractive index of the medium in which

the slab was immersed, would focus the light waves emitted by a source to a single point. This

can be showed by a simple application of Snell's law, using a structure with two consecutive

boundaries. This structure is called the Veselago's at lens [?] and a graphical representation

is shown in Figure 4.3.

72

Figure 4.3: Passage of light waves through a Veselago at lens, A: the image source, B: focusedimage, i.f.: the internal focus point

This lens geometry and structure, which converts a diverging beam to a converging one, and

vice-versa, creates the existence of a particular point called the internal focus, represented in

Figure 4.3. Knowing that the optical path from the external focus point to the internal focus

point must be zero, we can also processed to the computation of the lens contour, as we have

done in the previous section. From Figure 4.3 we can state that, in order to have an equality

of optical paths, one must have:

r1 + nr2 = d1 + nd2 (4.9)

r22 = r21 + (d1 + d2)2 − 2r1(d1 + d2) cos(θ) (4.10)

where, n is the refractive index of the lens. Assuming that the optical path from focus to focus

is zero one must have:

d1 + nd2 = 0 (4.11)

73

In polar coordinates, after some manipulation:

(n+ 1)

(r1d1

)2

− 2n cos(θ)

(r1d1

)+ (n− 1) = 0 (4.12)

As done in (4.4), using the Cartesian co-ordinates we obtain the following equation for the

lens contour:

(x− n

n+ 1

)2

+ y2 =1

(n+ 1)2(4.13)

Which is the expression of a circumference centered at(

nn+1 , 0

), and when n = −1 we also

obtain a at lens. The equality imposed by expression (4.4) clearly implies that one must have

a NIR medium, which include all DNG media.

Let us now consider the general expression for the impedance of a specic medium:

Zi =

√µ0µiε0εi

(4.14)

If we consider that the slab's material has, for the relative permittivity and permeability,

both εr = µr = −1, we can state that this DNG medium is a perfect match to free space

(ε0r = µ0r = 1). From this result, one of the conclusions is that there will not be reections

at the interfaces between the lens and freespace and even at the far boundary interface there

is again an impedance match, and the light is again perfectly transmitted to vaccum.

If the propagation is done in the z axis, in order to have all the energy transmitted through

the slab it is required that we have a propagation constant:

k′z = −√ω

c2− k2x − k2y (4.15)

With the overall Transmission coecient being:

74

T = tt′ = exp(ik′zd) = exp

[−i(√

ω

c2− k2x − k2y

)d

](4.16)

where d is the thickness of the slab. The choice of the propagation constant is done in order to

maintain causality and this phase correction is what grants the lens the capability of refocusing

the image by canceling the phase of the transmitted wave as it propagates from its source [?].

Let us consider a TE wave propagating in the vaccum, medium 1, with the following eld

expression:

E1 = exp(ikzz + ikxx− iωt) (4.17)

where the propagation constant is:

kz = i

√k2x + k2y −

ω2

c2(4.18)

with k2x + k2y >ωc2. From this eld expression in (4.17) we can easily identify that we are

dealing with an exponentially evanescent eld. At the interface, between media 1 and 2, the

waves experience both transmission (into medium 2) and the reection (back to medium 1).

It is also important to notice that in order to maintain causality the elds must decay as the

get away from the interface, so the eld expression for the transmitted wave can be:

Et2 = t exp(ik′zz + ikxx− iωt) (4.19)

And for the reected wave, the following expression:

Er1 = r exp(−ikzz + ikxx− iωt) (4.20)

where the propagation constant is given by:

75

k′z = i

√k2x + k2y − ε2µ2

ω

c2(4.21)

with ε2 and µ2 being the permittivity and permeability of the slab, and also having k2x + k2y >

ε2µ2ωc2.

When matching the wave elds at the interface from medium 1 to medium 2 we obtain the

reection and transmission coecients, t and r:

t =2µkz

µkz + k′z(4.22)

r =µkz − k′zµkz + k′z

(4.23)

And for the transmission and reection coecients of the transition from inside medium 2 to

medium 1:

t′ =2k′z

µkz + k′z(4.24)

r′ =k′z − µkzk′z + µkz

(4.25)

Now in order to obtain the expression for the transmission of light through both the interfaces

one must sum the multiple scattering events, from [?]:

Ts = tt′ exp(ik′zd) + tt′r′2 exp(3ik′zd) + tt′r′3 exp(5ik′zd) + (...) (4.26)

Ts =tt′ exp(ik′zd)

1− r′2 exp(2ik′zd)(4.27)

76

Considering the DNG situation (with ε = µ = −1), and using (4.22)-(4.27), we can compute

the limit to this values of permittivity and permeability in order to nd the overall transmission

coecient. The solution for this special kind of structure is calculated asymptotically as n

approaches −1:

limµ→−1,ε→−1(Ts) =

= limµ→−1,ε→−1

(tt′ exp(ik′zd)

1−r′2 exp(2ik′zd)

)=

= limµ→−1,ε→−1

(2µkzµkz+k′z

2k′zµkz+k′z

exp(ik′zd)

1−(k′z−µkzk′z+µkz

)2exp(2ik′zd)

)=

= exp(−ik′zd) = exp(−ikzd)

(4.28)

This result in (4.29) is very important. As another consequence of having a negative index of

refraction, we have waves of the form exp(−kz), outside the lens, that couple to waves of the

form exp(kz) inside the lens. So, even if the waves decay outside the lens, they are amplied

on the inside of it, recovering an image on the opposite side of the lens, from the source, and

all done by the transmission process. On Figure 4.4 we can see the evolution of the evanescent

eld variation in the presence of the Veselago's at lens.

Since the waves decay in amplitude and not in phase, as they get further from the source, the

lens focus the image by amplifying these waves rather than correcting the phase. This is a

proof that this medium does in fact amplify the evanescent waves, and so, with this kind of

lens, both the propagating and evanescent waves contribute to the resolution of the resulting

image [?].

As we have stated before, as the result of the perfect matched impedance, there will be no

reected wave on the interface as we can also see by the asymptotic analysis of the overall

reection coecient:

77

Figure 4.4: Evanescent eld variation in the presence of the Veselago's at lens.

limµ→−1,ε→−1(Rs) =

= limµ→−1,ε→−1

(r + tt′ exp(ik′zd)

1−r′2 exp(2ik′zd)

)= 0

(4.29)

Which conrms that all the energy is transmitted between the media transitions.

4.3 Conclusions

In this chapter we have presented the optical lens design using the concept of optical path,

and we have particularized the design process with the usage of DNG materials. In order to

do so we have studied the situation where a DNG slab is used in order to produce an high

resolution lens, which is called the Veselago's at lens.

From the concept of optical path we have obtained an expression that enables us to infer

about the geometrical form of the lens contour and about its dependence on the value of the

refractive index of the lens material. The curvature of the lens contour can be concave if we are

dealing with positive refraction indexes and convex if we are dealing with negative refraction

78

index materials. The at lens contour is obtained when using large values of n, which is really

unpractical, or if n→ −1. The concept of focal lenght is also introduced into the lens design

and we have seen that for a lens made of a DNG material, we can obtain the same focal lenght

as we would if a DPS material was used, but with the implication of having a much smaller

refraction index.

Then we introduce the Veselago DNG at lens. This lens structure consists of a planar DNG

slab, composed by a material with the refractive index n = −n0, with n0 being the refractive

index of the medium in which the slab was immersed, that can focus the light waves emitted by

a source to a single point. This phenomenon is achieved by a simple ray tracing problem using

Snell's Law. When considering that the slab's material has, for the relative permittivity and

permeability, both εr = µr = −1 we could see that there was a perfect impedance match for

both interfaces between the DNG slab and the medium in which it was immersed. From this

result, one of the conclusions is that there will not be reections at the interfaces between the

lens and free space and even at the far boundary interface there is again an impedance match,

and the light is again perfectly transmitted to vaccum to a single point. The DNG material

properties creates a physical phenomenon where we have waves of the form exp(−kz), outside

the lens, that couple to waves of the form exp(kz) inside the lens. So, even if the waves decay

outside the lens, they amplied inside of it, recovering an image on the opposite side of the

lens, from the source. These two results are the responsible of granting the lens a capability

of refocusing the image by canceling the phase of the transmitted wave as it propagates from

its source.

This introduction into the lens design using DNG materials is important as its particular

physical properties could enable the creation of high resolution lenses and it is a proof of

practical application of DNG media in optics and engineering.

79

80

Chapter 5

Conclusions

In this chapter main conclusions are exposed as well as some developing potential applica-

tions and further investigation hypothesis of the subjects addressed in this dissertation are

introduced.

5.1 Summary

In the second chapter we study the electromagnetic phenomena associated with DNG meta-

materials. After formulating the classication of a specic medium as DNG, the implications

of having a negative permittivity and permeability lead into studying the characterization of

the medium and the physical phenomena. We have shown that in this kind of media the exis-

tence of waves that propagate in a antiparalell direction of the power ux is noticed, called the

Backward Waves, and that we are in the presence of a Negative Index of Refraction material,

which implies some modications in the interpretation of Snell's Law. A dispersive analysis

is also made using the Lorentz Dispersive Model, and for a particularization called the Drude

Model, and using these models we have also shown that it is possible to nd a DNG interval

even when considering dispersion. From this introduction of losses we have also concluded that

81

both the group and phase velocities have dierent values and, for this kind of DNG media,

they even have opposite directions.

The third chapter deals with the guided wave propagation with DNG materials. We have

chosen to study two simple structures, the DPS-DNG interface and te DNG slab. A modal

analysis was made for both waveguiding structures and also numerical simulations, with the

respective interpretations. First we have showed that it is possible to have both TE and TM

surface mode wave propagation on a DPS-DNG interface. This kind of propagation mode is

new and does not exist in other more conventional DPS wave guiding structures. We have

also found that, when not neglecting losses, the feature of being a NIR medium can be applied

to all DNG media but the NIR designation is not exclusive of DNG materials since there

are other non DNG frequency bands where the medium also acts as a NIR. When dealing

with the propagation of this surface waves we have also seen that it permits large attenuation

outside the interface. Finally we have analyzed the guided propagation on a DNG slab,

whose electromagnetic proprieties can be of large interest for the practical application of DNG

materials to the construction of waveguides. In this structure there's also the possibility of

having surface wave mode propagation but the most important result is the propagation of

super-slow modes that are a consequence of having a phase velocity that is smaller than the

outer medium in which a DNG slab is immersed (there is also slow-mode propagation that

exhibits a double modal solution for some frequency bands). The existence of these super-slow

modes enables the propagation on the DNG slab even if we use a less dense medium for the

slab (i.e., medium 2) when compared to the outer medium (i.e. medium 2), ε1µ1 > ε2µ2.

This phenomenon is veried, fullling the propagation conditions, only on double negative

materials. When analyzing the dispersion relation diagrams for the ε1µ1 > ε2µ2 structure we

could see that for a medium with |µ2| > µ1 the are two super-slow modes and for a medium

with µ1 > |µ2| only one super-slow mode propagates on a limited frequency band, but that

there are, in this band, always two possible modal solutions.

The fourth chapter is dedicated to the study of lens design using DNG metamaterials. We have

82

presented the optical lens design using the concept of optical path, and we have particularized

the design process with the usage of DNG materials. In order to do so we have studied the

situation where a DNG slab is used in order to produce an high resolution lens, which is called

the Veselago's at lens. From the concept of optical path we have obtained an expression that

enables us to infer about the geometrical form of the lens contour and about its dependence

on the value of the refractive index of the lens material. The curvature of the lens contour can

be concave if we are dealing with positive refraction indexes and convex if we are dealing with

negative refraction index materials. The concept of focal lenght is also introduced into the lens

design and we have seen that for a lens made of a DNG material, we can obtain the same focal

lenght as we would if a DPS material was used, but with the implication of having a much

smaller refraction index. Then we introduced the Veselago DNG at lens. This lens structure

consists of a planar DNG slab, composed by a material with the refractive index n = −n0,

with n0 being the refractive index of the medium in which the slab was immersed, that can

focus the light waves emitted by a source to a single point. This phenomenon is achieved by a

simple ray tracing problem using Snell's Law. When considering that the slab's material has,

for the relative permittivity and permeability, both εr = µr = −1 we could see that there was a

perfect impedance match for both interfaces between the DNG slab and the medium in which

it was immersed. From this result, one of the conclusions is that there will not be reections at

the interfaces between the lens and free space and even at the far boundary interface there is

again an impedance match, and the light is again perfectly transmitted to vaccum to a single

point. The DNG material properties creates a physical phenomenon where we have waves of

the form exp(−kz), outside the lens, that couple to waves of the form exp(kz) inside the lens.

So, even if the waves decay outside the lens, they amplied inside of it, recovering an image on

the opposite side of the lens, from the source. These two results are the responsible of granting

the lens a capability of refocusing the image by canceling the phase of the transmitted wave

as it propagates from its source. This is indeed a very important result since no longer the

resolution is restricted by the wavelength of the propagated light waves, as we can found in

83

conventional lens structures.

5.2 Future Work

When dealing with DNG media propagation the physical implications are more profound than

one could nd at rst sight. One could study the physical aspects of DNG media in motion,

the non linear eects of a DNG medium and the study of the anisotropic properties of media,

since we have dealt with the commonly used isotropic model.

We have also addressed two simple DNG waveguiding structures but there is a large set of

known DPS guides in which one could replace or add, one or several, DNG media, leading to

new results and guiding structures understanding.

Since no longer the resolution is restricted by the wavelength of the propagated light waves,

as we can found in conventional lens structures, the design of DNG metamaterial lens can also

lead to the development of the so called Superlens structures.

84

85

86

References

[1] K. B. Moller, Peter, Review: Electric sh, BioScience (American Institute of Biological

Sciences), 1991.

[2] T. H. Bullock, Electroreception. Springer, 2005.

[3] J. Stewart, Intermediate Electromagnetic Theory. World Scientic, 2001.

[4] R. S. Kirby, Engineering in History. Courier Dover Publications, 1990.

[5] Electromagnetism, maxwells equations, and microwaves, IEEE Virtual Museum., 2008.

[6] J. C. Bose, On the rotation of plane of polarization of electric waves by a twisted struc-

ture, Proc. R. Soc., 1988.

[7] A. H. S. I. V. Lindell and J. Kurkijarvi, Karl f. lindman: The last hertzian, and a

harbinger of electromagnetic chirality, IEEE Antennas Propag. Mag vol. 34, no. 3, 1992.

[8] W. E. Kock, Metallic delay lenses, Bell Syst. Tech. J., vol. 27, 1948.

[9] W. Rotman, Plasma simulation by articial dielectrics and parallelplate media, IRE

Trans. Antennas Propag. , vol. 10, no. 1, 1962.

[10] R. Z. N. Engheta, Metamaterials: Physics and Engeneering Explorations. Addison-

Wesley, 206.

86

[11] V. G. Veselago, The electrodynamics of substances with simultaneously negative values

of ε and µ, Soviet Physics USP, vol. 47, 1968.

[12] J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., vol. 85, 2000.

[13] R. Z. N. Engheta, A positive future for double-negative metamaterials, IEEE Transac-

tions on Microwave Theory and Techniques NO. 4,, vol. 53, 2005.

[14] D. Smith, Composite medium with simultaneously negative permeability and permittiv-

ity, Phys. Rev. Lett., vol. 84, 2000.

[15] R. Shelby, Microwave transmission through a two-dimensional, isotropic, left-handed

metamaterial, Phys. Rev. Lett., vol. 78, 2001.

[16] C. T. C. A. N. D. Dr. Jerey D. Wilson, Zachary D. Schwartz and K. R. Vaden, Flat lens

focusing demonstrated with left-handed metamaterial, NASA Glenn Research Center,

2005.

[17] D. K. Cheng, Field and Wave Electromagnetics. Addison-Wesley, 1989.

[18] C. R. Paiva, Introdução aos metamateriais. DEEC-IST, 2008.

[19] E. S. Laszlo Solymar, Waves in Metamaterials. Oxford University Press, USA, 2009.

[20] D. R. S. P. Kolinko, Numerical study of electromagnetic waves interacting with negative

index materials, OPTICS EXPRESS 647, 2003.

87