Electromagnetic effects of metamaterials with negative parameters

Filipa Isabel Rodrigues Prudêncio

Dissertation submitted for obtaining the degree of Master in Electrical and Computer Engineering

Jury

President: Professor José Bioucas Dias Supervisor: Professor Carlos Manuel dos Reis Paiva Co-Supervisor: Professor António Luís Campos da Silva Topa Member: Professor Maria Emília da Costa Manso

November, 2009 Abstract

This thesis has as central goal: the analysis of the electromagnetic wave propagation in the double negative media (DNG). The interest on this topic has been increasing worldwide while potential applications of complex media are being proposed in different areas, such as optical devices, radar and mobile communications, microwaves and invisibility cloaks.

The most exciting property of the DNG medium, the negative refraction, is also carefully investigated in this thesis. Unusual propagation characteristics of these metamaterials are studied, namely the Poynting and wave vectors are antiparallel, thereby creating backward waves and the anomalous dispersion in DNG metamaterial interfaces.

The guided electromagnetic wave propagation in planar structures containing both double positive media (DPS) and DNG materials is approached. The DNG interface will logically require a dispersive model, such as the Lorentz’s model, which is also discussed. Neglecting losses in the dispersive model may lead to unphysical solutions. The numerical results for negative refractive index media show that the dispersive model for the permeability and the permittivity must include losses. The DNG dielectric slab shows the existence of super slow modes and mode bifurcation.

The electromagnetic wave propagation in DNG waveguiding structures based on the H-guide is also introduced. Hybrid longitudinal-section electric (LSE) and longitudinal- section magnetic (LSM) modes are studied. Finally, the double-slab DNG/DPS H-guide and the H-guide directional coupler are addressed: it is shown that both co-directional and contra- directional coupling effects may take place in these structures.

Keywords

Double negative media, Complex media, Metamaterials, Negative refraction, Planar waveguides, Backward waves, Material dispersion and losses, H-guides, Directional couplers, Microwaves, Photonics.

i ii Sumário

Esta dissertação tem como estudo central a propagação de ondas electromagnéticas em meios complexos, nomeadamente metamateriais DNG. Tem surgido, por todo o mundo, um grande interesse neste tema por parte de investigadores e cientistas. Este estudo de meios complexos inovadores tem sido aplicado em diversas áreas, tais como, antenas, mantos de invisibilidade, super lentes e também aplicações no domínio das microondas e da óptica.

A existência de um índice de refracção negativo neste tipo de materiais é considerado por muitos cientistas como a propriedade mais importante dos metamateriais DNG. Algumas características de propagação destes metamateriais foram referidas, sendo estas fisicamente diferentes dos resultados obtidos em materiais que encontramos na natureza. Pode salientar-se o aparecimento de ondas regressivas no caso em que o vector de Poynting é anti-paralelo ao vector de onda, assim como a existência de dispersão material nas interfaces dos metamateriais DNG.

O estudo da propagação electromagnética guiada em estruturas planares, contendo simultaneamente meios DNG e DPS está subjacente em todos os capítulos de desenvolvimento desta dissertação. Na análise da interface DNG-DPS é necessário incluir-se o modelo dispersivo de Lorentz, com perdas, para que haja significado físico durante a evolução do índice de refracção efectivo em função da frequência. No estudo da placa dieléctrica DNG-DPS surge o aparecimento de modos super lentos, assim como o conceito da bifurcação de modos.

Por último, e igualmente interessante, a análise da propagação electromagnética guiada em estruturas derivadas do guia H é introduzida. Os modos híbridos LSE e LSM propagam-se nesta estrutura sendo também objecto de estudo. Seguidamente, analisam-se estruturas como a dupla placa dieléctrica DNG/DPS com guia H e o acoplamento direccional DNG/DPS com guia H que permitem a identificação de extraordinários efeitos físicos. Estas estruturas exibem nos seus diagramas de dispersão regiões de acoplamento co-direccional e contra-direccional.

iii Palavras-chave

Meios duplamente negativos, Meios complexos, Metamaterials, Refracção negativa, Guias planares, Ondas regressivas, Material dispersivo, guias-H, Acopladores direccionais, Microondas, Fotónica.

iv Acknowledgements

First, I would like to thank my supervisor Professor Carlos Paiva for supporting me to on this thesis and helped me with suggestions and comments.

Also, I would like to express my gratitude to my co-supervisor Professor António Topa for his attention and for all the invaluable discussions about some of the results presented in this work.

Finally, my thoughts go to my family and to my friends for their sensible advises and optimism.

v

vi Table of Contents

ABSTRACT ...... I

KEYWORDS ...... I

SUMÁRIO...... III

PALAVRAS-CHAVE...... IV

ACKNOWLEDGEMENTS ...... V

TABLE OF CONTENTS ...... VII

LIST OF FIGURES...... IX

LIST OF ACRONYMS ...... XI

LIST OF SYMBOLS ...... XIII

CHAPTER 1...... 1

INTRODUCTION ...... 1 1. 1 State of the art...... 2 1. 2 Motivations and objectives...... 5 1. 3 Structure of dissertation...... 7 1. 4 Main Contributions...... 9 References ...... 10

CHAPTER 2...... 13

PLANE WAVE PROPAGATION IN DNG MEDIA ...... 13 2.1. Introduction ...... 14 2.2. Classification of material media...... 15 2.3 Metamaterial characterization ...... 18 2.3.1 Dispersion...... 26 2.3.1.1 Group velocity and velocity of energy transport...... 27 2.3.1.2 Lorentz’s model...... 29 2.3.2 Transmission and Reflectivity...... 31 2.3.3 Negative refraction ...... 32 2.3.3.1 Snell’s Law...... 34 2.4 Conclusion...... 35

CHAPTER 3...... 37

PROPAGATION OF ELECTROMAGNETIC WAVES IN DNG GUIDES ...... 37

vii 3.1. Introduction ...... 38 3.2 DPS-DNG Interface...... 38 3.2.1 Energy on the DPS-DNG interface ...... 49 3.3 DNG dielectric slab ...... 50 3.3.1 Surface Modes...... 53 3.4 Conclusion...... 58

CHAPTER 4...... 61

PROPAGATION OF ELECTROMAGNETIC WAVES IN H-GUIDES AND H-GUIDE COUPLERS ...... 61 4.1 Introduction ...... 62 4.2 The DNG H-Guide ...... 63 4.3 Double-slab DNG/DPS H-guide ...... 69 4.4 DNG/DPS H-guide directional coupler...... 71 4.5 Conclusion...... 74

CHAPTER 5...... 75

CONCLUSION ...... 75 5.1 Summary...... 76 5.2 Future work ...... 79 5.2.1 Indefinite media for H-guide ...... 79 5.2.2 Geometric algebra applied to indefinite media ...... 80

APPENDIX A...... 81

CONVERGENT INTEGRAL ...... 81 Convergent integral ...... 82

APPENDIX B...... 83

THE HILBERT TRANSFORM...... 83 The Hilbert transform ...... 84

BIBLIOGRAPHY...... 85

viii List of Figures

CHAPTER 2

FIGURE 2. 1 MATERIAL CLASSIFICATION...... 15

FIGURE 2. 2 TRIPLET VECTORS AND TO THE DPS AND DNG MEDIA...... 19

FIGURE 2. 3 ELECTRIC PERMITTIVITY REPRESENTED BY SPHERICAL COORDINATES...... 20

FIGURE 2. 4 REPRESENTATION OF, , USING SPHERICAL COORDINATES...... 21

FIGURE 2. 5 REFRACTIVE INDEX REPRESENTED BY SPHERICAL COORDINATES...... 25

FIGURE 2. 6 DIELECTRIC SLAB ...... 31

FIGURE 2. 7 GRAPHICAL INTERPRETATION OF THE NEGATIVE REFRACTION...... 33

FIGURE 2. 8 INTERFACE Z=0...... 34

CHAPTER 3

FIGURE 3. 1 PLANAR INTERFACE BETWEEN A DPS MEDIUM AND A DNG MEDIUM...... 40 FIGURE 3. 2 LOSSLESS DISPERSIVE MODEL FOR ε AND µ ...... 42

FIGURE 3. 3 LOSSLESS DISPERSIVE MODEL FOR n ...... 42

FIGURE 3. 4 DISPERSION DIAGRAM FOR TE MODE IN THE LOSSLESS CASE...... 43 € € € FIGURE 3. 5 VARIATION OF α1 AND α2 AS A FUNCTION OF FREQUENCY, FOR TE MODE...... 44 FIGURE 3. 6 DISPERSIVE DIAGRAM WITHOUT LOSSES FOR THE TM MODE...... 45 € € FIGURE 3. 7 VARIATION OF α1 AND α2 AS A FUNCTION OF FREQUENCY, FOR THE TM MODE...... 45 FIGURE 3. 8 LOSSY DISPERSIVE€ € MODEL FOR ε AND µ ...... 46

FIGURE 3. 9 LOSSY DISPERSIVE MODEL FOR n...... 46

FIGURE 3. 10 DISPERSION DIAGRAM FOR THE TE MODE IN THE LOSSY CASE...... 47 € € FIGURE 3. 11 DISPERSION DIAGRAM FO€ R THE TM MODE IN THE LOSSY CASE...... 47

FIGURE 3. 12 VARIATION OF α1 AND α2 AS A FUNCTION OF FREQUENCY, FOR TE MODE. .... 48

FIGURE 3. 13 VARIATION OF α1 AND α2 AS A FUNCTION OF FREQUENCY, FOR TM MODE. ... 48 € € FIGURE 3. 14 VARIATION OF THE ELECTRIC FIELD Ey (t = 0, x,z), AS FUNCTION OF, x k0 , ON € € THE DPS-DNG INTERFACE...... 49

FIGURE 3. 15 DNG DIELECTRIC SLAB...... 51 € FIGURE 3. 16 THE MODAL SOLUTIONS FOR€ A CONVENTIONAL DPS DIELECTRIC SLAB, WHERE

ε1 = 1,µ1 = 1...... 54

FIGURE 3. 17 THE MODAL SOLUTIONS, WHERE ε1 =1,µ1 =1,ε2 = −2,µ2 = −2 ...... 55

€ € ix FIGURE 3. 18 DISPERSION DIAGRAM FOR THE TE MODES OF A DNG DIELECTRIC SLAB

CHARACTERIZED ...... 56

FIGURE 3. 19 DNG DIELECTRIC SLAB DISPERSION DIAGRAM CHARACTERIZED

BY ε1 = 2,µ1 =1,ε2 = −0.8,µ2 = −1.3, FOR TE MODE...... 57 FIGURE 3. 20 DNG DIELECTRIC SLAB DISPERSION DIAGRAM CHARACTERIZED € BY ε1 = 2,µ1 = 2,ε2 = −1.5,µ2 = −1.5, FOR TE MODE...... 58

€ CHAPTER 4

FIGURE 4. 1 DNG H-GUIDE...... 63

FIGURE 4. 2 OPERATIONAL DIAGRAM FOR A H-GUIDE, CONTAINING BOTH DNG AND DPS

MATERIALS...... 65

b FIGURE 4. 3 DISPERSION DIAGRAM FOR = 0.4 ...... 66 λ b FIGURE 4. 4 DISPERSION DIAGRAM FOR = 0.6 ...... 67 λ FIGURE 4. 5 DISPERSION DIAGRAM€ FOR ζ =1.25...... 68 FIGURE 4. 6 DISPERSION DIAGRAM€ FOR ζ =1.75...... 68

FIGURE 4. 7 DOUBLE-SLAB DNG/DPS€ H-GUIDE ...... 69 FIGURE 4. 8 OPERATIONAL DIAGRAM€ FOR ξ = 0.25 AND ξ = 4...... 70

FIGURE 4. 9 DISPERSION DIAGRAM FOR ξ = 0.15 AND ξ = 4...... 71 € € FIGURE 4. 10 DNG/DPS H-GUIDE DIRECTIONAL COUPLER...... 71

FIGURE 4. 11 OPERATIONAL € DIAGRAM FOR€ THE DNG/DPS H-GUIDE DIRECTIONAL

COUPLER...... 72 b FIGURE 4. 12 DISPERSION DIAGRAM FOR = 0.5...... 73 λ b FIGURE 4. 13 DISPERSION DIAGRAM FOR =1...... 74 l €

€

x List of Acronyms

BW Back-ward wave

DNG Double negative media

DPS Double positive media

ENG Epsilon-negative media

IR Infrared radiation

LHM Left-handed media

LSE Longitudinal-section magnetic

LSM Longitudinal-section electric

MNG Mu-negative media

NIM Negative-index metamaterial

SRR Split ring resonator

TE Transverse electric mode

TM Transverse magnetic mode

xi xii List of Symbols

A Amplitude of electric field and magnetic field for x > 0

α € Transverse attenuation constant

α1 Transverse attenuation constant in the DPS € medium €

α2 Transverse attenuation constant in the DNG medium € αz Attenuation constant

B Amplitude of electric field and magnetic field € for x < 0

B € Magnetic flux density (vector)

Plat separation of DNG H-guide € b

β Propagation constant €

β Real part of longitudinal wavenumber € z

c Velocity of light € Thickness of dielectric slab € d

χe Electric susceptibility €

χm Magnetic susceptibility €

Electric flux density (vector) € D

Electric field intensity (vector) € E

E0 Electric field for the referential where € ˆ ˆ ikz E0 = (Exx + Eyy )e . €

€

xiii Ex Electric field (x-axis)

Ey Transverse electric field €

Ez Electric field (z-axis) €

Electric permittivity € ε

Real part of permittivity € ε'

ε' ' Imaginary part of permittivity

€ ε0 Vacuum permittivity

ε1 Electric permittivity of medium 1 € €

ε2 Electric permittivity of medium 2 €

η Wave impedance €

η0 Vacuum wave impedance €

Magnetic field intensity (vector) € H

H0 Magnetic field for the referential where € H  H xˆ +H yˆ e ikz 0 =  x y  €

* Complex conjugated vector of the magnetic H € field € * H0 Complex conjugated vector of the magnetic field for the referential where

€ H  H xˆ +H yˆ e ikz 0 =  x y 

H x Magnetic field (x-axis) €

H y Magnetic field (y-axis) €

Hz Magnetic field (z-axis) €

J Electric current density (vector) € ext

h Transverse wavenumber €

€

xiv h1 Transverse wavenumber in DPS medium

h2 Transverse wavenumber in DNG medium €

ˆ Wave normal € k

k Wave vector

k Wavenumber € €

k0 Vacuum wavenumber €

Real part of the wavenumber € k'

kx Transverse wavenumber

€ k' ' Imaginary part of the wavenumber € ky Wavenumber (axis-y)

kz Longitudinal wavenumber € €

k2 Transverse wavenumber in DNG media €

Slab thickness of DNG H-guide € l

leff Imaginary part of effective refractive index €

l Thickness of DPS medium € 1

l2 Thickness of DNG medium €

Poynting vector € S

S Time-average of the Poynting vector (DNG- € total DPS interface)

€ Sω Time-average of the Poynting vector

s Half distance between two different slabs €

€ U U =−iu

µ Magnetic permeability € €

µ0 Vacuum magnetic permeability €

€ xv µ' Real part of permeability

µ' ' Imaginary part of permeability

€ µ1 Magnetic permeability of medium 1

µ2 Magnetic permeability of medium 2 € €

W Time-averaged energy density €

We Time-averaged electric energy density €

Wm Time-average magnetic energy density €

ξ Quotient between two different thickness €

ζ Aspect ratio (chapter 4) €

ζ Normalized wave impedance (chapter 2) €

ζ Vacuum normalized wave impedance € vacuum

ζDNG DNG normalized wave impedance € n Refractive index € Real part of refractive index € n'

n' ' Imaginary part of refractive index

€ neff Real part of effective refractive index

€ Γ Collision frequency € Γe Collision frequency of electric permittivity €

Γm Collision frequency of magnetic permeability €

Transmission coefficient € τ

θ Angle between the real part and the € ε imaginary part of electric permittivity € θµ Angle between the real part and the imaginary part of magnetic permeability

€

xvi θn Angle between the real part and the imaginary part of refractive index € θtrans Angle of transmission on the interface

θinci Angle of incidence on the interface € u Wavenumber € Wavenumber € v

vp Phase velocity €

v Group velocity € g

v Velocity of energy transport E € R Reflectivity coefficient € ρ Electric charge density (scalar) €

ρε Absolute value of electric permittivity € ρµ Absolute value of magnetic permeability €

w Wavenumber €

ω Angular frequency

ω0 Resonance frequency €

ω0e Resonance frequency of electric permittivity €

ω0m Resonance frequency of magnetic € permeability € ω1 Beginning of the interval intersection

between [ωε,ωb] and [ωa,ωµ]. €

ω1e,m Beginning of the interval where ℜ ε < 0 and € € ( ) ℜ(µ) < 0 € €

ω2 Ending of the interval intersection between €

€

xvii [ωε,ωb] and [ωa,ωµ].

ω2e,m Ending of the interval where ℜ ε < 0 and € € ( ) ℜ(µ) < 0 € €

ωa Beginning the interval where ℜ µ < 0 € ( )

ωb Ending the interval where ℜ ε < 0 € € ( )

ωp Plasma frequency € €

ωpe Plasma frequency of electric permittivity €

ωpm Plasma frequency of magnetic permeability €

ωε Beginning the interval where ℜ ε < 0 € ( )

ωµ Ending the interval where ℜ µ < 0 € € ( )

Wavelength € λ €

ϕ(x, y,z) Field component function €

ϕ(x) Field component function as a function of x €

€ €

xviii

Chapter 1

Introduction

This chapter contains a brief review about complex media since the mid XIX century to actuality. The objectives and the motivations of this dissertation are addressed. In addition, the main applications of these materials are described to illustrate the potential interest of their properties. The structure of dissertation is also presented allowing global view of each chapter.

1 1. 1 State of the art

In the mid-nineteenth century, several experiments and science demonstrations of the fundamental nature of electricity were intensively investigated. Michael Faraday is best known for his work with electricity and magnetism. Andre Marie Ampere developed a mathematical and a physical theory to understand the relationship between electricity and magnetism.

Several theories of electromagnetism, which unified all these knowledge, were proposed, putting in evidence Weber and Maxwell [1] theory. The last one prevailed because it used a simple and a practical method. Actually, the electromagnetism is universally known as the Maxwell’s electromagnetic theory. It relates the optical phenomenon with electric and magnetic concept. Maxwell’s equations describe the properties of the electric and magnetic fields, as well as, their relation with charge and current densities.

The classical electromagnetic theory received a mathematical structure for the first time, in 1873 with the Maxwell’s treaty. It proved that the velocity of light is defined by the vacuum permittivity and permeability, concluding that, light is an electromagnetic effect. The physicist Einstein has proposed the special theory of relativity [2] to show that the electromagnetic theory, based on Maxwell’s equations, had not succeeded to explain all physical properties. However, he assumed that the velocity of light remained constant in all frames of reference, as required by classical Maxwellian theory. Associated with these equations system, the relation between the constitutive relations and the classical electromagnetic theory is defined in a recent review [3]. In 1968, the concept of a bianisotropic medium [4] was coined by Cheng and Kong [5]- [6] defining a medium with the most general linear constitutive relations. In microwaves, bianisotropic media [7]-[8] are conceived as artificial structures.

Recently, many researchers all over the world have suggested future electromagnetic applications, based on the interaction between electromagnetic fields and media.

Nowadays, artificial electromagnetic materials with effective negative permeability and permittivity have attracted the attention of the electromagnetic community. This new class of composite materials with extraordinary electromagnetic properties cannot be found in nature but can be artificially achieved.

On a first approach, the unconventional response functions of these metamaterials [9]

2 are generated by artificial fabricated inclusions in a host medium or in a host surface. Manipulating and tailoring the electromagnetic wave properties, several impacts of metamaterials may envisage, such as the significant decrease in the size and weight of components and devices.

The history of complex media, with negative permittivity and permeability, starts with the concept of “artificial” materials in 1898, when Lagidis Chunder Bose developed the first microwave experiment on twisted structures. Currently, these elements immersed in a host medium are denominated by artificial chiral medium.

Karl Ferdinand Lindman, in 1914, had studied the wave interaction with collections of randomly - oriented small wire helices, in order to create an artificial chiral media. Later in 1948, Kock, made lightweight microwave lenses by combination of conducting spheres, strips periodically and disks. These metamaterials, built for lower frequencies, can be designed for higher frequencies by length scaling.

Since then, many research groups worldwide have studied various types of electromagnetic composite media, such as chiral materials, wire media, omega media, bianisotropic media, linear and nonlinear media. New techniques have allowed the building of structures and composite materials that mimic known material responses. The corresponding physically realizable response functions do not occur or may not be readily available in nature.

The electromagnetic waves in composite media may induce both electric and magnetic moments, which influence the macroscopic effective permittivity and permeability. Metamaterials can also be synthesized by embedding artificially components in a specific host medium. The design of these materials, such as the size, the shape, and composition of the inclusions, the density, arrangement and the alignment of these inclusions works in order to material engineering. These properties and the electromagnetic response function, not found in each of the individual constituent, provide new possibilities for metamaterial applications.

During the 1960s, the idea of negative refraction first arose when a physicist, Veselago, considered the optical properties of an imaginary material. In 1967, he investigated the plane wave propagation in a material which permittivity and permeability were simultaneously negative [10]. He demonstrated that for a monochromatic uniform plane wave in such a medium, contrary to the conventional simple media, the direction of the Poynting vector is antiparallel to the direction of phase velocity. In the 1980s and 1990s, interesting artificial dielectrics were investigated extensively for microwave radar.

3 Later, during the late 1990s, Pendry and his colleagues at Imperial College began to produce structures with the right kind of properties. Pendry was interested in developing materials with negative permeability. Also, he created an array of closely spaced, thin, conducting elements, such as metal hoops [11]. In 1999, he described how he adjusted the array’s properties and he developed an array with negative permeability. This structure consisted of periodic array of split-ring resonators (SRRs) [12] that expressed negative effective permeability over a narrow frequency band. This is possible if the magnetic field of incident wave was normal to the plane of the structure. Veselago medium is probably the most famous class of metamaterials in the present wave in complex electromagnetic media. Veselago medium has been known by several names, as negative-index media, negative- refraction media, backward wave media, double-negative media, media with simultaneously negative permittivity and permeability and even left-handed media (LHM).

Following up on this work, Smith created a material with negative refractive index [13]-[16], composed by interlocking units of thin fibreglass sheets imprinted with copper rings and wires. The material exhibited negative effective permittivity bellow a cutoff frequency. Metamaterials are able to work in the GHz and optical frequency ranges [17]-[29]. Therefore, he studied and measured the characteristics of negative index materials (NIMs).

For metamaterials with negative permittivity and permeability, some terminologies have been proposed, such as “left-handed” media, “back-ward wave media”(BW media) and “double negative (DNG)”, just to name a few.

In April 2001, Smith and his colleagues constructed a composite medium for the microwave regime, and announced the experimental evidence of an unusual form of refraction.

Many research groups all over the world are now studying various aspects of this class of metamaterials and suggesting future applications. Perfect lenses, or the creation of acoustic metamaterials are just some examples. The latter one causes a bizarre reverse effect in the Doppler concept, which was recently investigated by a group of researchers from Korea and

China.

Causality in DNG media is one of the most important concepts related to metamaterials. If one ignores the temporal dispersion in a DNG medium, one will immediately state a causality paradox in the time domain, that is, a nondispersive DNG medium will be necessarily noncausal. In fact, the causality of waves propagating in a dispersive DNG metamaterial was investigated using the one-dimensional electromagnetic

4 plane-wave radiation from a current sheet source in this medium.

Presently, metamaterials are synthesized by combining an array of thin metallic wires with an array of split-ring resonators. This structure possesses both negative permittivity and permeability in a certain frequency region. However, its electromagnetic properties are only exhibited for one particular direction of propagation of the electromagnetic waves. Therefore, the magnetic field must be oriented perpendicularly to the plane of split rings until the electric field is oriented parallel to the metallic wires. The negative refraction of electromagnetic waves is being investigated in this structure and it is being prepared in laboratories. Recently, there has been a growing interest in the theoretical and experimental study of metamaterials.

1. 2 Motivations and objectives

Artificial electromagnetic materials with effective negative permittivity and permeability, at least at certain frequency band, form a new electromagnetic concept. The effective negative refractive index is an interesting electromagnetic property for a medium and provides new electromagnetic effects.

Actually, metamaterial researchers have not only demonstrated new interesting physical phenomena but have also lead to the development of new design procedures. The realization of promising new types of microwave devices and their application to mobile antennas has attracted widespread interest. In fact, metamaterials may significantly improve the performance of several devices and antennas. Specially, a lot attention has been drawn to using periodic structures in the design of antennas and of microwave components.

With negative refractive index availability, one can dramatically improve the performance of antennas by reducing interference. Materials with this property are also able to reverse the Doppler effect. In fact, new-type of metamaterials would open up a new field for automotive electronics applications, such as scanning-beam antennas for radar and mobile communications. Surface periodic structures, with thickness much smaller than the wavelength, that contain new electromagnetic properties provide the development of high impedance surfaces, known as artificial magnetic conductors. These engineered surfaces can have interesting applications in antenna design, because a magnetic conductor ground plane enhances the radiation of an antenna with horizontal profile. A new generation of low profile antennas for wireless communication systems is being developed, based on low profile

5 microstrip patch antennas, whose lead to reduced mobile terminals. In a near future, personal mobile communications will incorporate multi-functions devices, which means that the same terminal will have to operate in several frequencies. Reconfigurable antenna, capable of switching the operating frequency and having a low profile, is a promising scenario.

Researchers are working in the design of a radiating element above a textured surface that can be reconfigured and adapted to operate at different frequencies.

John Pendry and his co-researchers, that first suggested how materials with negative permeability could be artificially built, have shown as a negative refractive index material could be used to make perfect lens [20]. The concept is related to focus an image with resolution not restricted by the wavelength of light, which does not happens with the conventional lens. The perfect lens would also support the evanescent waves, which result in a perfect image of object and develop a high resolution of lens.

A recent interest in the design of metamaterials is the creation of an object invisible to radar to be experimentally demonstrated in the near-term. Physicists and engineers know that the ability to control the properties of metamaterials can be exploited to develop the refractive index profile needed to make an object invisible by bending the electromagnetic radiation. The design of cloaking devices that turn an object macroscopically invisible, due to the flexibility of manipulating electromagnetic waves, may lead to new interesting functionalities. Several research teams are working in order to create new devices that could render objects invisible to the human eye. This will be possible only because metamaterials have extraordinary capabilities to bend electromagnetic wave. In the case of invisibility cloaks, the material would need to curve light waves completely around the object.

Pendry thought to make things invisible [21]-[22] as a rubber sheet around the object to conceal it. The permittivity and permeability, in tangential directions, along the surface of the cloak, remain finite, then, the electromagnetic waves have no problem passing around the object [23]-[24]. The light reflected of an object makes it visible for the observers. However, if the invisibility shield wraps light around the object, it is cloaked. Using any metamaterial that absorbs a bit of light and casts a slight shadow, many researchers say that the cloaking is almost perfect. A cloak could bend radiation of a just single frequency, so it could only hide an object of one color. Therefore, invisibility is a staple for science fiction.

Nowadays, it is already possible, under very low frequency electromagnetic waves, to force a zero magnetic field inside of a body, but not altering the exterior field. In this case, this object becomes completely undetectable to these waves. A very recently important stage is now being reached, that is, building a prototype in the laboratory and applying this device

6 to improving magnetic field detection technology. The researchers are using metamaterials with specific magnetic properties to create invisible areas to the magnetic field at very low frequencies. This discovery can be applied to medical purposes, such as magnetoencephalographic or magnetocardiographic techniques, which are used to measure the magnetic fields created by the brain or in the heart.

Also, these studies can be used in other areas, where magnetic field detection is important, such as in sensors, or to prevent the magnetic detection of ships or submarines. On the other hand, additional layers of invisibility cloak can adjust some types of systematic errors of telescope optics, or after designing a certain configuration for permittivity and permeability, it is possible to make objects look smaller, larger, shifted or deformed in rather general. Finally strong investigation is being developed to create invisibility, not only for radar but potentially for optical wavelengths as well.

Recently, the researchers have been realized nanostructured materials with a custom- designed refractive index at optical wavelengths. Several aspects of metamaterials have been published [25]-[30] in the literature and new suggestions are being studied. This means that the march of scientific progress will lead to further advances.

1. 3 Structure of dissertation

Five chapters compose this dissertation, each one possessing several sections and subsections. The first chapter includes the introduction. It contains a brief history of complex media since the mid XIX century to actuality. Several key scientists are put in evidence and their experimental demonstrations of complex media are explained. Also, one shows the objectives and the motivations of this dissertation. Moreover, in the introduction, the main applications of these materials are explained to illustrate the potential interest of their properties.

In the second chapter, the main properties of metamaterials DNG [31]-[32] are studied. As it is well-known, these media are characterized by both negative real part permittivity ε and permeability µ. Several concepts and mathematical expressions are displayed. Applying Maxwell’s equations, a plane wave propagating in an isotropic unlimited medium is studied. According to distinct vector triplets in a DNG medium, the electromagnetic wave and the electromagnetic energy have opposite sides, which creates a backward wave. When considering a DNG metamaterial, conventional electric and magnetic

7 energy density relations have to be replaced by more general others that include the dispersion section. In addition, one considers the air-DNG interface to analyze the reflection and transmission properties of these materials. One of the most important properties of DNG metamaterials, providing interesting potential applications, is the negative refractive index, which is addressed in the last section.

Numerical simulations and respective interpretation for the DNG-DPS interface and for the dielectric DNG slab are presented in the third chapter. As it is well-known, the DNG medium is necessarily dispersive, so, a dispersive model must to be used in the analysis. Therefore, the Lorentz dispersive model is applied to the DPS-DNG interface. Important differences between the dispersive model and the nondispersive model are put in evidence. The TE and the TM modes are studied for both models. The complex values assumed by the permeability and the permittivity, as well as the refractive index, are depicted for the case of a dispersive model.

Applying the boundary conditions to the tangential field components, as well as the relations between normalized wavenumbers, leads to the modal equation for the TE odd and TE even modes. While a DPS medium only allows modal solutions with real transverse wavenumbers providing slow modes, when using a dielectric DNG slab, solutions with imaginary transverse wavenumbers may also be found, providing super-slow modes. In the super-slow modes the phase velocity is lower than the speed of light in the medium, while for conventional slow modes, the phase velocity is limited by a lower bound.

The simulations include the dispersion diagrams for both TE odd modes and TE even modes. Several results and solutions for different values of the permeability and permittivity, and are presented.

In the fourth chapter, the electromagnetic wave propagation in DNG waveguiding structures based on the H-guide is considered. These structures may contain both DNG and DPS materials. The operation of the DNG H-guide is, as usually, divided into the closed- waveguide regime and the open-waveguide regime. The modes propagating in this waveguide are hybrid longitudinal-section electric (LSE) modes and longitudinal -section magnetic (LSM) modes. Only the last are analyzed and the modal equations for both even and odd modes are derived. The operational diagram of a DNG H-guide is compared with that of a conventional DPS H-guide. Different types of results are depicted for different values geometrical and constitutive parameters. Several modes exhibit mode bifurcation above cutoff. Moreover the super-slow modes are depicted in the dispersion diagrams.

8 In the following section, the double-slab DNG/DPS H-guide is analyzed. The modes propagating in this structure are of the same type as before. Two juxtaposed dielectric slabs (DNG slab and conventional DPS slab) form the structure.

Using several sets of values for the geometrical and constitutive parameters, the operational diagram for the DNG/DPS H-guide [33]-[35] is analyzed.

This chapter ends with the analysis of a H-guide directional coupler having a DPS and a DNG slab. As the previous sections, the modal equation for LSM modes is studied and the operational and dispersion diagrams are obtained. Mode coupling regions are shown in the operational diagram of the DNG/DPS H-guide directional coupler. Co-directional and contra- directional coupling regions occur in the same coupler.

Finally, the main conclusions are explained and several physical contents with potential interest that can be developed in future are addressed.

1. 4 Main Contributions

The topic of metamaterials is receiving special attention among the electromagnetics research community. The main contribution of this work is the systematic analysis of a set of canonical electromagnetic problems involving metamaterials, which were spread over the published literature. These problems are now collected and presented in an original way that can be useful for future researchers in the field.

Other important contribution is the analysis of electromagnetic wave propagation in waveguiding structures based on the H-guide, containing both DPS and DNG materials. Some unusual features, such as anomalous waveguide dispersion, mode bifurcation, and the existence of super-slow modes, are investigated in this thesis. These features providing physical insight into the properties of waveguides filled with such metamaterials have potential interest in the design of novel devices and components.

9 References

[1] J. C. Maxwell, A Treatise on Electricity & Magnetism. Dover Publications, New York, 1954.

[2] H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity. Dover Publications, New York, 1952.

[3] F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric. Boston, Birkhäuser, 2003.

[4] V. G. Veselago, “The electrodynamics of substances with simultaneously negatives values of ε and µ,” Sov. Phys. Uspekhi, vol. 10, no. 4, pp. 509-514, 1968. [Usp. Fiz. Nauk, vol. 92, pp. 517-526, 1967.]

[5] J. B. Pendry, A. J. Holden, W. J. Robbins, and J. Stewart, “ Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter, vol. 10, pp. 4785-4809, 1998.

[6] A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bianisotropic Materials. Theory and Applications, Amsterdam, 2001.

[7] Cheng, D. K. and J. A. Kong, “On guided waves in moving anisotropic media,” IEEE Trans. Microwave Theory and Tech., Vol. MT-16, pp. 99-103, 1968..

[8] J. A. Kong, Electromagnetic Wave Theory. New York: Wiley, 1986.

[9] W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics. Bellingham, Washington, 2003.

[10] V. Dmitriev, “Plane wave solutions for homogeneous bianisotropic media described by the mgantic group D_h(C_v),”Microwave and Optical Technology Letters.

[11] S. Zouhdi, A. Sihvola, and M. Arsalane, Advances in Electromagnetics of Complex Media and Metamaterials. Kluwer Academic Publishers, Dordrecht, 2002.

[12] J. B. Pendry, A. J. Holden, D. J. Robbins, and W J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech., MTT-47, pp. 195, 1999.

10 [13] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S.Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184-4187, May 2000.

[14] D. R. Smith and N. Kroll, “Negative refractive index in the leaft-handed materials,” Phys. Rev. Lett., vol. 85, no. 14, pp. 2933-2936, Oct. 2000.

[15] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett., vol. 78, no. 4, pp. 489-491, Jan. 2001.

[16] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, no. 5514, pp. 77-79, April 2001.

[17] A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left- handed material that obeys Snell’s law,” Phys. Rev. Lett., vol. 90, 137401, 2003.

[18] A. Grbic, G. V. Eleftheriades, “Overcoming the Diffraction Limit with a Planar Left- Handed Transmission-Line Lens,” Phys. Rev. Lett., vol. 92, 117403, 2004.

[19] V. M. Shalaev et al, “Negative index of refraction in optical metamaterials,” Opt. Lett., vol. 30, pp. 3356-3358, 2005.

[20] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966-3969, 2000.

[21] D. Mackenzie, “What’s Happening in the Mathematical Sciences,” AMS, vol. 7, pp. 62-68, 1993.

[22] M. Born and E. Wolf, “Principles of Optics,” Cambridge Univ. Press, Cambridge, 1999.

[23] U. Leonhartd, J. B. Pendry, D. Schuring, and D. R. Smith, “Optical Conformal Mapping,” Science, 1126493, 2006.

[24] U. Leonhartd, J. B. Pendry, D. Schuring, and D. R. Smith, “Controlling Electromagnetic Fields,” Science, 1125907, 2006.

[25] C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configuration,” J Appl Phys 90, p.p 5483-5486, 2001.

11 [26] L. Liu, C. Caloz, C.-C. Chang, and T. Itoh, “Forward coupling phenomena artificial left-handed transmission lines,” J Appl Phys 90, pp. 5560-5565, 2002.

[27] D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves”, Appl Phys Lett 81, pp. 2713-2715, 2002.

[28] S. Zouhdi, A. H. Sihvola, and M. Arsalane, “Ideas for potential applications of metamaterials with negative permittivity and permeability, in Advances in Electromagnetics of Complex Media and Metamaterials,” NATO Science, pp. 19-37, 2001.

[29] A. Alù, and N. Engheta, “Paring an epsilon-negative slab with a mu-negative slab: Resonance, tunnelling and transparency,” IEEE Trans Antennas and Propagation 51, pp. 2558-2571, 2003.

[30] B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J Appl Phys 93, pp. 2558-2571, 2003.

[31] R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys Rev E 64, 2001.

[32] R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys Rev E 63, 2001.

[33] F. J. Tisher, “Properties of the H guide at microwaves and millimmiter waves,” Proc IEE 106B, suppl. 13, pp. 47-53, 1959.

[34] I. S. Nefedov and S.A. Tretyakov, “Theoretical study of waveguiding strucutures containing back-ward materials,” In Proc. Of XXVIIth URSI General Assembly, 2002.

[35] S. Hrabar, J. Bartolic, and Z. Sipus, ”Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans Antennas Propagate 53, pp. 110- 119, 2005.

12

Chapter 2

Plane wave propagation in DNG media

In this chapter, the propagation of plane waves in complex media is characterized and several unexpected analytic results are put in evidence. Important properties like the negative refraction and the anomalous dispersion in double negative (DNG) metamaterials are studied.

13 2.1. Introduction

In this chapter, the propagation properties of double negative (DNG) metamaterials are addressed. The DNG metamaterial is characterized by two complex constants, permittivity ε, and permeability µ, both with negative real part. Passive DNG metamaterials have to be inherently dispersive. Using Maxwell’s equations, plane wave propagation in an isotropic and unlimited DNG medium is studied and several important results are shown, like the case of negative refraction, anti-parallel Poynting vector and anomalous refraction.

In the third section, the two distinct vector triplets related with DPS and DNG medium are compared. In a DNG medium, the wave vector and the Poynting vector have opposite orientations, which corresponds to a backward wave.

In the fourth section, the material dispersion in DNG metamaterials is addressed and the average densities of electric and magnetic energies are obtained.

Another interesting topic is the planar interface between a DNG material and the air. Considering normal incidence from the air at the interface the reflection and transmission coefficients at this interface are derived in the fifth section.

Finally, one of the most discussed properties of this type of DNG metamaterials is the negative sign of the real part of the refraction index. Graphical and analytical explanation is used to prove that.

14 2.2. Classification of material media

In this chapter we consider a DNG metamaterial characterized by two complex constitutive parameters, permittivity ε, and permeability µ, which are described by the following relations

ε = ε′ + iε′′ (2. 1) and,

€ µ = µ′ + iµ′′ (2. 2) with .

According to the value€ of the real part of permeability and permittivity, these materials can be classified as shown in Figure 2.1.

µ'

ENG material DPS material

Plasmas € Dielectrics

ε'

DNG material MNG material Not found in nature, but Gyrotropic magnetic physically realizable materials €

Figure 2. 1 Material classification.

In certain frequency ranges, a plasma medium exhibits the same characteristics as an ENG medium, as it is the case of some metals, gold and silver at very high frequencies. In fact, these materials can be considered as ENG materials in the infrared or in the visible spectra. Moreover, the gyrotropic magnetic materials can be classified as MNG medium in a specific frequency band.

15 The constitutive relations of the medium may be written as a function of frequency, according to

, (2. 3) where and are the electric and magnetic susceptibilities.

Assuming that the electric field polarization is linear and along the axis-x and considering z as the wave propagation direction, the electric and magnetic fields can be written by

. (2. 4)

The vacuum wavenumber , , can be expressed as

, (2. 5) such that, using the refractive index, , the complex wave vector, , is defined by

. (2. 6)

The wave impedance is given by the equation

. (2. 7) where and are the electric and the magnetic field intensities, respectively.

Applying Maxwell’s equations to the plane waves propagating in the isotropic and unlimited medium, one gets

, (2. 8) where E and H are the electric and the magnetic fields vectors, respectively.

16 Expressing,

, (2. 9) where , the refractive index can be written as

, (2. 10) or

, (2. 11)

where and,

, (2. 12) with representing the vacuum wave impedance. Finally, the normalized wave impedance can be written as

. (2. 13)

From the previous relation, one may write

. (2. 14)

The time-average of the Poynting vector is given by

(2. 15)

A passive medium must verify the condition, , for an electromagnetic wave that propagates the positive z-axis.

17 In general, a complex refractive index must be considered

, (2. 16) where and .

Therefore, the electric field can be described by the following relation

. (2. 17)

Again, a passive medium must impose , otherwise, one would not have electromagnetic waves whose energy propagates in positive z-axis.

Finally, the phase velocity of the electromagnetic wave is defined as

. (2. 18)

2.3 Metamaterial characterization

The left-hand rule can be easily understood from the Maxwell’s equations in the differential form,

, (2. 19)

and the two constitutive relations, which describe the response of the medium to the applied fields,

, (2. 20)

In the time-harmonic regime, the following relations between operators can be derived:

and .

18 The sign of the Poynting vector, , and of the wave vector, , are computed using the following relations which govern the spatial orientation of the electric and magnetic field vectors, E and H, as it is shown in the Figure 2.2,

k × E = ωµ0µH . (2. 21) k × H = −ωε0εE

€

Figure 2. 2 Triplet vectors and to the DPS and DNG media.

According to the triplet vectors in a DNG medium, vectors and have opposite orientations, which creates a backward wave. It means that the electromagnetic wave and the electromagnetic energy have opposite directions. In the same way, it easily to see that, in a DPS medium, the vectors and have the same orientation, providing a forward wave. On the DNG medium the triplet vector is left, while, in the DPS medium, the same triplet vector is right. On the other hand, the triplet vector is right in both media.

A physical solution can be obtained from equations (2.11) and (2.12). Note that, for both DPS and DNG medium, one has

(2. 22)

19 Considering a DNG medium, where and , and assuming a passive material, and , the following condition must be verified

n′µ′ + n′′µ′′ ˆ 2 2 > 0 ⇒ z ⋅ Sω > 0 (2. 23) (µ′) + (µ′′)

On the other hand, for a DPS medium, with and , and assuming a passive material, and , one shows the same result € n′µ′ + n′′µ′′ ˆ 2 2 > 0 ⇒ z ⋅ Sω > 0 (2. 24) (µ′) + (µ′′)

In the following, one will use spherical coordinates to represent the complex permeability and the permittivity, as shown in Fig. 2.3 €

ε′

€

Figure 2. 3 Electric permittivity represented by spherical coordinates.

Writing,

, (2. 25)

20 and using the trigonometric relation

, (2. 26) the following system can be derived

(2. 27)

Again in spherical coordinates, is depicted in Figure 2.4

Figure 2. 4 Representation of, , using spherical coordinates.

Since for the DNG medium, with <0, it means that permittivity angle is π π θ π confined by π ≥ θ ≥ and the ≥ ε ≥ , where ε 2 2 2 4

€ €

21 1 θ 1 i ε   ′ ′′ 2 2 2  θε   θε  nε = nε + inε = ε = ρε e = iρε sin   − icos  . (2. 28)   2   2 

€

According to the following relations

. (2. 29)

if one replaces the equation (2.27) in (2.29), it results

(2.30)

Replacing the equations (2.30) and (2.27) by (2.28), it yields to

, (2.31)

and,

. (2. 32)

One should note that , due to the DNG medium condition .

22 In the limit case of the absence of losses ( ), the system can be described by

, (2. 33)

where

. (2. 34)

In the same way,

, (2. 35)

and,

. (2. 36)

Similarly to the permittivity case, the positive sign is chosen for the squares roots, therefore leading to , and also with .

Again, in the limit case of no losses ( ),

(2. 37)

and also,

. (2.38)

According the relations (2.34) and (2.38), one has

23 (2. 39)

One can easily verify that this last relation is always negative.

In general case, the refractive index can be written as

(2. 40) that can be split in

. (2.41)

Considering,

(2. 42) where,

. (2. 43) it results

(2.44)

This condition shows that the real part of the vector and the Poynting vector have an opposite sign, therefore corresponding to backward waves. This result leads to the same conclusion obtained from the triplet vectors.

Considering now, a lossless DNG medium, with, , where the refractive index is given by

(2.45) and,

. (2.46) it is clear that the conditions (2.43) and (2.44) also apply in this case, which means that it will also propagate backward waves.

24 Note that, in the DPS medium, with positive parameters, and . it is obvious that,

, (2. 47) and, consequently,

. (2. 48)

This result provides the same conclusion as the DPS triplet vector has suggested in the last section. The equation (2.48) is verified for both lossy and lossless DPS media due to the positive real part of refractive index. This fact suggests forward electromagnetic waves.

In the Figure 2.5, the refractive index is depicted in spherical coordinates for a DNG medium.

+n n′′

θ n € €

n′

€

€ −n

ρn

Figure 2. 5 Refractive index represented€ by spherical coordinates. € Therefore, the complex wave impedance is expressed by

. (2. 49)

By neglecting losses, one has

25 . (2. 50)

These results have been obtained from the condition .

However, another possible solution, with , can be written as

. (2. 51)

There is a positive direction to the Poynting vector in the first solution and a negative direction in the second solution. Also, both solutions verify the condition (2.44); it means that, for a passive media, the Poynting vector has always an opposite direction to the increasing electric field.

2.3.1 Dispersion

Electromagnetic theory shows that, for a certain isotropic medium, characterized by permittivity ε and permeability µ, the time-averaged electric and magnetic energy densities, and are, respectively, given by

, (2. 52)

. (2. 53)

Since ε and µ are both negative in the case of DNG metamaterials, it is easily proven that the previous expressions should not be used, because the DNG medium is necessarily dispersive. Therefore, the previous time-averaged electric and magnetic energy densities, (2.52) and (2.53), become invalid and new expressions have been proposed for lossless dispersive media,

26 t ∂D W = ∫ Edt′ , (2. 54) e t −∞ ∂ ′ and, € t ∂B W = ∫ Hdt′ , (2. 55) m t −∞ ∂ ′ where the electric and magnetic fields are almost monochromatic, such that, € and , with a slowly variation in the period , which means . The previous expressions, (2.54) and (2.55), show that and are positive.

2.3.1.1 Group velocity and velocity of energy transport

Let one consider a lossless medium, where the time-averaged energy density is defined by

. (2. 56)

Moreover, the time-averaged of the Poynting vector for a uniform plane wave in this lossless medium is given by

(2. 57) or

. (2. 58)

Monochromatic plane waves propagates with a group velocity such that

, (2. 59) which is defined as the velocity with which the whole wave packet moves without distortion.

27 In an isotropic medium, is a function of the magnitude of the wave vector, , or with .

Hence,

, (2. 60)

This means that the group velocity is parallel to the wave vector.

The ratio between the Poynting vector and the energy density expresses a quantity whioch has velocity dimensions. This ratio is denoted as energy transportation velocity,

, (2. 61) which never exceeds the speed of light.

Using the relation between, and , (2.6), the group velocity can be derived as

. (2. 62)

Replacing (2.54) and (2.52) in (2.57), the energy transportation velocity is given by

. (2. 63)

The energy transportation velocity coincides with the group velocity in a lossless, nondispersive isotropic medium.

28 2.3.1.2 Lorentz’s model

In this section, one considers a Lorentz’s model for the frequency dispersion

, (2.64)

where expresses the resonance frequencies, represents the collision frequencies and express the plasma frequencies.

Therefore, within the Lorentz’s model, is possible to find a certain frequency range , where the parameters and exhibit negative real part, such that

. (2. 65)

To simplify the previous expressions, one removes subscripts ‘e’ and ‘m’ of the electric permittivity and the magnetic permeability, respectively, getting

. (2. 66)

If one neglects the losses along the electromagnetic wave propagation, , these equations (2.54) reduce to

(2. 67)

In the case of a DNG medium, the frequencies and must be confined by

(2. 68)

29 Assuming the existence of this interval and considering the condition , then and are obtained. Therefore, the interval of DNG medium is .

On the other hand, considering the Drude’s ( ), and so, and

, the following conclusions are derived

(2. 69) which corresponds for DNG medium,

(2. 70)

Moreover, when using a causal model, losses and the dispersion are two concepts of the same physical reality. According to causal principle, the Kramers-Kroning relations are expressed by the following set of equations

. (2. 71)

which have necessarily to be satisfied

The Kramers-Kroning relations prove the mutual dependency of the real and the imaginary parts of the refractive index. This means that one only have a real part, , if the imaginary part, , also exists and vice-versa. This is especially important when dealing with metamaterials and proves how dispersive they are.

The symbol in the previous integrals stands for the Cauchy main value. In addition, using the Appendix A and B, the sum rules can be described as

. (2. 72)

30 2.3.2 Transmission and Reflectivity

Let one now consider an interface between a DNG medium and the air. Assuming normal incidence at the interface from airside, the reflectivity can be derived from

(2.73)

The parameter , represents the normalized wave impedance as given by the equation (2.13). From the energy conservation principle, the reflectivity must be confined to .

One will now consider a dielectric slab, characterized by permittivity ε and permeability µ, with thickness d and immersed in the vacuum, as depicted in Figure 2.6.

τ ε

µ Γ €

€ Figure 2. 6 Dielectric slab

The reflection coefficient at each of the dielectric slab interfaces satisfies the following equation

, (2. 74)

while the transmission coefficient is given by

, (2.75)

where and .

31 Assuming, now, that the dielectric slab is adapted to the vacuum, one must have , or and .

If the thickness , the dielectric slab reduces to a simple interface between two semi- infinite media. Therefore, using one can infer

, (2. 76) which implies

, (2. 77)

Therefore recovering equation (2.73), as expected. In this case, there is no transmission to the third medium, as it does not exist, and so, .

2.3.3 Negative refraction

As already seen, metamaterials must exhibit complex permittivity and permeability, and these constitutive parameters may be written as

, (2. 78) and

, (2. 79) where the permittivity and the permeability phases are confined to and .

The refractive index, n = εµ has also a complex form

(2. 80) € with .

As one well knows, in the case of DNG metamaterials, the real parts of ε and µ are both negative. In the passive materials the imaginary part of the refractive index , permittivity ε′′and permeability µ′′, are positive, as depicted in the Figure 2.7.

32€ €

µ

ε n

z

Figure 2. 7 Graphical interpretation of the negative refraction.

Combining the last three equations, the refractive index phase assumes the following expression

, (2.81) where can be zero (solid line), or one (dashed line) – see Figure 2.7. Only a negative satisfies a positive , in order to obtain a physical solution. The negative real part of refractive index is one of the most unusual properties of the DNG metamaterials. Moreover, the real part of the wave impedance, , must be positive to lead to a physical condition.

From another point of view, and using the equations, (2.1), (2.2) and (2.10), the negative sign of the real part of the refractive index can be proved as follows.

. (2. 82)

On the other hand,

. (2. 83)

Comparing the last two equations, one can obtain

. (2.84)

Now assuming negative real parts ε′ and , and the positive imaginary parts ε′′, µ′′ and , then the sign of the real part of refractive index, n′, must be negative, as shown before.

€ € € €

33 2.3.3.1 Snell’s law

In this section one will rewrite the Snell’s law applied to DNG metamaterial. One will start by considering an interface placed at z = 0, that splits two different materials, as shown in Figure 2.8.

€

x

DPS DPS € z

DNG

€

Figure 2. 8 Interface z=0.

One will assume that region z < 0 represents a DPS medium with refraction index , and region z > 0 and x > 0 another DPS medium , but with refraction index > 0, while the DNG medium, with refraction n 0, is confined by, and x 0 . Both materials are € 2 < < considered to be lossless, or at least, that losses can be neglected. € €

Applying Snell’s€ law at the interface, , one has€

, (2. 85)

proving that there exists negative refraction in , since .

It is easily verify in the Figure2.8, that in the DNG medium, confined by z > 0 and , the transmitted wave propagates along a negative angle. In this case, the vector and the vector have opposite orientations due to the previously choice of a negative refractive index. € However, in the case where, n′′, is negative, the angle would be positive in the DNG medium and the vector and the vector would have the same orientation. Then, the

34 € Snell’s law would be the same as in the conventional DPS medium. However, in the DPS medium the vector would have an opposite orientation then before. With this interpretation, it is possible to conclude that, n′′, is positive is the correct solution for the DNG medium with negative refraction index.

€

2.4 Conclusion

This chapter has presented a comprehensive study about DNG metamaterial. It was shown that the real parts of the permeability, the permittivity and the refraction index are all negative which represents the major result to be used in the next chapters. These new characteristics generate a new diversified phenomenology and new electromagnetic effects.

In the case of a DNG medium, one has shown the existence of backward waves propagating with a wave vector and an energy flow with opposite directions.

The dispersion in the DNG metamaterial has been also analyzed and two different phenomenology interpretations, based on the Lorentz and Drude models, were presented. Transmission and the reflectivity at the interface DNG, which are important features of metamaterials, were also studied.

35

36

Chapter 3

Propagation of electromagnetic waves in DNG guides

In this chapter, the propagation of complex structures that containing DNG and DPS media such as DNG interface and DNG dielectric slab is studied. The modal characterization of these planar structures is analysed and several effects are addressed.

37 3.1. Introduction

The study of unlimited isotropic DNG media suggests new interesting electromagnetic properties. The analysis of a simple DPS-DNG interface provides non standard electromagnetic effects, such as the reflection and refraction of backward waves, which do not exist in DPS media. The propagation of electromagnetic waves in complex structures that contain both DNG and DPS media, leads to new physical phenomena, which can be used in the development of new devices.

In this chapter, the modal characterization of some planar structures involving DNG metamaterials is addressed, such as the DPS-DNG interface and the DNG dielectric slab. In Section 3.2, using the Lorentz dispersive model, a general lossless DPS-DNG interface is analyzed. Then, applying the same model but now considering losses, several important results are obtained. The inclusion of losses eliminates the existence of unphysical solutions, and provides new effects, which do not show on a conventional DPS interface.

Finally, in Section 3.3, the DPS-DNG slab is analyzed. The frequency dispersion of the surface modes is studied with the inclusion of the Drude-Lorentz model, providing a set of new physical effects that are put in evidence.

3.2 DPS-DNG Interface

Considering the Maxwell equations in differential forms for derivate the electric and magnetic fields

 ∂B ∇ × E = −  ∂t  ∂D ext  ∇ × H = + J  ∂t . (3. 1)  ∇⋅ B = 0   ∇⋅ D = ρ 

In this section, a planar interface between a DPS medium and a DNG medium, as shown in € Fig. 3.1, is analyzed. The z-axis represents the direction of the propagation, x-axis the

38 transverse direction and y-axis the transversal infinite direction, where the electric and the ∂ magnetic fields verify the condition, = 0 . ∂y

To transform Maxwell equations of a monochromatic field from time to frequency

domain, we choose the time€- dependent factor e−iωt . Therefore, the electric and magnetic field components can be written as,

€ ik z β z −α z i t ψ(x, y,z) = ψ(x)e z = ψ(x)e z e z e− ω , (3. 2)

where, kz = βz + iαz. € The constant kz is the longitudinal wavenumber, βz is the real part of this wavenumber € and αz the attenuation constant. Also, the operator ∇ has the following form

€ ∂ € ∇ = xˆ + ikzzˆ . (3. 3) € ∂€x

€ From Maxwell equations applied to an isotropic medium and using the planar wave formalism, the following equation systems are obtained

 kz  kz  H x = − Ey  Ex = Hy  ωµ0µ  ωε0ε  1  ∂Ez   1  ∂Hz   H y =  kz Ex + i   Ey = −  kz H x + i  . (3. 4)  ωµ0µ  ∂x   ωε0ε  ∂x   1 ∂Ey  1 ∂H y  Hz = −i  Ez = i  ωµ0µ ∂x  ωε0ε ∂x

€

39 The interface between the DPS medium and the DNG medium is shown in Figure 3.1.

x DPS

ε1 > 0, µ1 > 0

€ z

€ ε < 0, µ < 0 2 2

y DNG €

Figure 3. 1 Planar€ interface between a DPS medium and a DNG medium. € For the TE modes, the transverse electric field Ey can be written as

 Aexp(−α1x)exp(ikzz), x > 0 Ey =  (3. 5) Bexp(α x)exp(€ik z), x < 0  2 z ,

where α1 and α2 represent, respectively, the transverse attenuation constant in the DPS € medium and in the DNG medium, which are given by the following relations

€ € α 2 = n2 −ε µ 1 eff 1 1, (3. 6)

and, € α 2 = n2 −ε µ 2 eff 2 2 . (3. 7)

By using (3.4), the following equation is obtained €

 iAα1  exp(−α1x)exp(ikzz), x > 0  ωµ0µ1 Hz =  (3. 8)  −iBα2 exp(α2x)exp(ikzz), x < 0  ωµ µ  0 2 .

Using now, the boundary conditions € E (x = 0+ ) = E (x = 0− ) y y , (3. 9)

and, €

40 + − Hz (x = 0 ) = Hz (x = 0 ), (3. 10)

the modal equation for the TE modes can be expressed as € α2 µ2 = − (3. 11) α1 µ1 .

Using a similar procedure for the TM modes, it results €

α2 ε2 = − (3. 12) α1 ε1 .

Therefore, valid propagating solutions are obtained from both modal equations, which does not happen in a DPS/DPS €interface. One should note that, when replacing the DNG medium by a plasma or a magnetoplasma, only TE or TM modes exist, respectively.

Moreover, when using the Lorentz’s model as a frequency dispersive model, the modal equation exhibits more than one single solution. In this model, the variation of the permittivity and permeability as a function of frequency, assume the following relations

ω2 ( ) 1 pe ε ω = + 2 2 (3. 13) ω0e − iΓeω −ω ,

and, € ω2 ( ) 1 pm µ ω = + 2 2 (3. 14) ω0m − iΓmω −ω .

9 In this section, we choose the following values ω0e = 2π × 2.6 ×10 rad /s, € 9 9 9 ωom = 2π × 2.2 ×10 rad /s, ω pm = 2π × 5 ×10 rad /s and ω pe = 2π × 6 ×10 rad /s. € Moreover, we assume µ1(ω) = ε1(ω) =1 to the DPS medium and using the Lorentz’s € model to represent€ the DNG medium to construct€ the Figure 3.2. It shows the parameters of

the dispersive model, when neglecting losses: the magnetic permeability variation µ (ω) and € 2 the electric permittivity variation ε2 (ω). Also, the variation of the refraction index with frequency is depicted in Figure 3.3. € € It is clear that n(ω) is only real when the magnetic permeability µ2 (ω) and the electric permittivity ( ) have negative or positive values, simultaneously. In the second ε2 ω chapter, one has shown the occurrence of negative refraction in the DNG medium frequency € € € 41 range of Figure 3.3, i.e., there is negative refraction in the same frequency range, where the

parameters ε2 (ω) and µ2 (ω) are both negative. In the same way, there is a positive

refraction for a DPS medium, which corresponds to a positive parameters of ε2 (ω) and

µ2 (ω). There is a ENG region in the lossless dispersive model, where the permeability is € € positive and the permittivity is negative. The refractive index becomes purely imaginary. € €

DNG ENG DPS

Figure 3. 2 Lossless dispersive model for ε and µ .

€ €

DNG ENG DPS

Figure 3. 3 Lossless dispersive model for n.

€

42 Using equations (3.6), (3.7) as well as (3.13), (3.14), the following equation is obtained for the effective refraction index of the TE mode

  2 µ2 (ω) ε2 (ω)µ2 (ω) −  ε1(ω)µ1(ω)  µ1(ω)  neff (ω) = (3. 15)   2 µ2 (ω) 1−  µ ω  1( )  .

In Figure 3.4 the real and imaginary parts of effective refractive index, , for TE € modes is depicted.

In the lossless case, one can easily see that the effective refractive index, , is

purely real. Considering µ1(ω) =1, the effective refractive index goes to infinity in the same

frequency where µ2 (ω) = −1, since it that corresponds to a null for the denominator of (3.15).

€

€

Figure 3. 4 Dispersion diagram for TE mode in the lossless case.

From equation (3.5), on can see that the transverse attenuation constants α1 and α2 ,

must be always positive, otherwise, the electric field Ey, tend to infinite with an increasing

distance to the interface, which is unphysical. € €

From the modal equation (3.11), one€ can see that α1 and α2 , change from real to imaginary exactly at the same point, as shown in Figure 3.5. In fact, from (3.5), the constant α is real when the effective refraction index is greater than one. In the same way, α is 1 € € 1

43 € € purely imaginary when the effective refraction index is lower then one.

Figure 3. 5 Variation of α1 and α2 as a function of frequency, for TE mode.

Following the same steps for the TM mode, the effective refraction index is € € now given by

 2 ε2 (ω) ε2 (ω)µ2 (ω) −  ε1(ω)µ1(ω)  ε1(ω)  neff (ω) = 2 (3. 16)   ε2 (ω) 1−   ε1(ω)  and its variation with frequency is shown in Fig. 3.6 The attenuation constants and for the TM€ mode are shown in Figure 3.7.

44

Figure 3. 6 Dispersive diagram without losses for the TM mode.

Figure 3. 7 Variation of α1 and α2 as a function of frequency, for the TM mode.

ε ω From the modal equations€ (3.11)€ and (3.12), one can see that, when 2 ( ) = −1 or ε1(ω)

µ2 (ω) = −1, for the TE and TM modes, respectively, the effective refractive index, n (ω) eff µ1(ω) € goes to infinite at a frequency which is not the resonance frequency. This is an unphysical behavior. € €

Considering now, a lossy dispersive model with Γe = Γm = 0.04 ×ω0e . Figure 3.8 shows

the magnetic permeability µ2 (ω), and the electric permittivity ε2 (ω) as a function of the

€ € € 45 frequency, while the refraction index, n(ω), is depicted in Figure 3.9.

The imaginary parts of µ2 and ε2 must be positive, when considering a dispersive medium. The frequency range,€ in the lossless dispersive model, where the real parts of the permeability and permittivity are negative (DNG), corresponds approximately to the same € € frequency range in the lossy dispersive model. This just happens, because one uses very low values to the losses.

DNG ENG DPS

Figure 3. 8 Lossy dispersive model for ε and µ .

€ €

DNG ENG DPS

Figure 3. 9 Lossy dispersive model for n .

€

46 In this case, for both TE and TM mode, the refraction index does not tend to infinite, and so, there are physical results, as shown in the Figure 3.10 and the in the Figure

3.11. It is important to note that the imaginary part of neff exhibits a considerable increase in the frequency range where, before, there was an unphysical solution. In the same way, leff have negligible values in the frequency range where, previously, there was a physical € solution. €

Figure 3. 10 Dispersion diagram for the TE mode in the lossy case.

Figure 3. 11 Dispersion diagram for the TM mode in the lossy case.

Results for the attenuation constants and , for both TE and TM modes, are depicted in Figure 3.12 and Figure 3.13, respectively. In the lossy case, the real part of both

47 attenuation constants must be positive to attenuate exponentially the transverse electric field when one moves away of the interface. However, the imaginary part of both attenuation constants is related with electromagnetic propagation. Therefore, they must be negative for the electric field going in the positive axis-z.

Figure 3. 12 Variation of α1 and α2 as a function of frequency, for TE mode.

€ €

Figure 3. 13 Variation of α1 and α2 as a function of frequency, for TM mode.

The electric field, E (t = 0, x,z), as function of, x k , for TE modes, is shown in the € y € 0 Figure 3.14. Having different values for the permeability of the inner and outer medium, such

€ € 48 that µ2 > µ1 and using the modal equation (3.11), one easily verify α2 > α1 . Then, it

provides a grater slope for the curve of the negative x k0-axis. Having, µ2 = µ1 , the slope

€ are the same for both curves. €

€ €

Ey

€

x k0 Figure 3. 14 Variation of the electric field Ey (t = 0, x,z), as function of, x k0, on the DPS-DNG interface. € € €

3.2.1 Energy on the DPS-DNG interface

Considering an interface between a DNG medium and the vacuum (DPS medium). As one well know, the parameters for the first medium are both negative ε′ < 0 and µ′ < 0 and both positive for the second medium ε′ > 0 and µ′ > 0 .

The Poynting vector applied to the interface with,€z = 0, can €be written as € € 2 E  1 S(z 0) zˆ 0 . (3. 17) = = ℜ€  2η0  ζ 

Using the conditions (2.24) and (2.25), it is clear that the Poynting vector has a positive orientation that for the€ DNG medium and a negative orientation for the vacuum.

In the DPS-DNG interface, the total energy is defined by the sum of the two Poynting vectors,

49 2 E0   1   1  Stotal (z = 0) = zˆ  ℜ  + ℜ  . (3. 18) η0   ζDNG   ζvacuum 

Note that, using a greater value for ζDNG than ζvacuum , the total vector Poynting has a positive € direction in axis-z, which provides backward waves. Otherwise, the total vector Poynting has a negative direction in axis-z if one considers smaller values for ζ than € € DNG ζvacuum , which creates forward waves.

€ € 3.3 DNG dielectric slab

In this section, the DNG dielectric slab, as shown in the Figure 3.12, is addressed. The modal

solutions of this structure can be divided into TE and TM modes. From the geometrical symmetry of the structure, these modes can be termed as odd or even modes.

For the even TE modes, the transverse electric field can be expressed as

 Aexp[−α1(x − d)], x ≥ d Ey =  . (3. 19)  Bcos(k2 x), 0 ≤ x ≤ d

while the same expression for the odd TE modes would be €  Aexp[−α1(x − d)], x ≥ d Ey =  , (3. 20)  Bsen(k2 x), 0 ≤ x ≤ d

where α1 is the attenuation constant in the outer medium and k 2 is the transverse wavenumber €in the inner medium.

€ €

50

x DPS µ1 > 0,ε1 > 0

€ € z 2d µ2 < 0,ε2 < 0

DNG

€ € y

€ Figure 3. 15 DNG dielectric slab. € Applying the boundary conditions to the tangential field components, the following relation can be derived for the even TE mode

µ w = 1 u tan(u), (3. 21) µ2 while for TE odd mode, € µ w = − 1 u cot(u), (3. 22) µ2 where € w = α1d , (3. 23) and € u = k2d . (3. 24)

On the other and, the relation between the normalized wavenumbers is given by € u2 + w2 = v2 , (3. 25) where € v = k0d ε2µ2 −ε1µ1 (3. 26)

€

51

The same procedure can be applied to the TM modes, leading to the following expression for TM even modes

ε w = 1 u tan(u), (3. 27) ε2

while for the TM odd modes € ε w = − 1 u cot(u). (3. 28) ε2

Using (3.19) and (3.20) in the magnetic and electric wave equations, one obtains € ∂ 2 H y + (ω2εµ − β 2 )H = 0, (3. 29) ∂x2 y

and, € ∂ 2E y + (ω2εµ − β 2 )E = 0 , (3. 30) ∂x2 y

where β is the propagation constant. € Also, the following relations for the outer and inner medium, respectively, are obtained

€ 2 2 2 α1 = β −ω ε1µ1, (3. 31)

and,

€ 2 2 2 k2 = ω ε2µ2 − β . (3. 32)

According to the previous relations,

€ 2 2 2 2 2 (k2 d) + (α1 d) − (ωd) (ε2µ2 −ε1µ1 ). (3. 33)

The transverse wavenumber k2 , is real whenever € β < ε2µ2 k0 , (3. 34) € and imaginary for, €

52 β > ε2µ2 k0 . (3. 35)

€ 3.3.1 Surface Modes

In this section, we study the surface modes propagating in a DNG dielectric slab. When the phase velocity, defined by

ω v = (3. 36) p β is smaller than the speed of light in the outer unlimited medium, the modes are termed as super slow modes. In this€ case, assuming that U = −iu, with u ∈ ℜ, and replacing in equation (3.25), yields to

w2 =U 2 + v2. (3. 37) € € Also, rewriting the modal equations (3.21) and (3.22), one gets € µ w = − 1 U tanh(U), (3. 38) µ2 and € µ w = − 1 U coth(U), (3. 39) µ2 since , € tan(ix) = i tanh(x) (3. 40) and € cot(ix) = −icoth(x) (3. 41)

The numerical solutions of the modal equations can be found, in plane (u,w) or (U,w) through the intersection€ of the corresponding modal curves and the curve representing equations (3.25) or (3.37). In Figure 3.17, the curves for the modal equations of the TE modes for a DNG dielectric slab and the curve of equation (3.25) and (3.37) are depicted. Also, the conventional slab, a DPS dielectric slab is depicted in Figure 3.16.

53 The positive abscissa semi-axis expresses the transversal wavenumber, u, as real, while in the negative direction the imaginary part of, u, is represented. It is well known that, for the DPS slab, only modal solutions with real u are allowed. €

However, in the case of a DNG slab, solutions€ with imaginary u are also possible. This case corresponds to the super slow modes,€ for which the phase velocity is lower than the c speed of light in the outer medium, vp < . On the€ other hand, with u as real, the ε1µ1 superficial modes are denominated by slow modes, and one easily verify c c € v . The same analysis can be done for the TM modes by duality. < p < € ε2µ2 ε1µ1

One should note that, due to the DNG medium, the sign of the right hand side of the TE

€ modal equations changes due to µ2 < 0. This causes a change in the slope of the branches of the tangent and cotangent function. Also, this inversion inserts two solutions for a slow mode

with the same k0d value in some regions. It develops the limited modes that are confined by € the limited interval of k0 values.

€ € w

€

U u

€ € Modal equation for TE even mode

Modal equation for TE odd mode v = 2

Figure 3. 16 The €modal solutions for a conventional DPS dielectric slab, where ε1 = 1,µ1 = 1.

€

54

w

€

U u

€ Modal equation for TE even mode € Modal equation for TE odd mode

k0d = 2.6

k0d = 0.2

€ €

Figure 3. 17 The modal solutions, where ε1 =1,µ1 =1,ε2 = −2,µ2 = −2 .

The respective dispersion diagram for the TE modes for the DNG dielectric slab is shown in € Figure 3.18. The graphic expresses k0d as a function of βd variation.

The electric fields Ey of the even slow modes vary according to cosine, while those of the odd slow modes € vary according to € a sine function. However, if the transverse wavenumber is imaginary, these trigonometric functions become hyperbolic functions, € respectively.

Note that, when if the outer medium is less dense that inner medium there are slow modes and, given a DNG metamaterial inner medium also arise super slow modes.

55

k0d

€

Odd slow mode

Even slow mode 1

Even slow mode 2

Odd slow mode Odd super slow mode

βd

Figure 3. 18 Dispersion diagram for the TE modes of a DNG dielectric slab characterized

byε1 =1,µ1 =1,ε2 = −2,µ2 = −2 , for TE mode. €

βd € The straight line, with a lower slope, corresponding to k0d = expresses the transition ε2µ2 between the super slow modes and the slow modes. On the other hand, w = 0, it corresponds βd to the cutoff, or k d . The odd super slow mode is the fundamental mode, which is 0 = € ε1µ1 € µ excited from null frequency. This mode becomes a slow mode when u = 0, that is, v = 1 µ2 € π and it propagates until v = . The other modes with two distinct solutions, one goes to 2 € € second straight line, and the other goes to the cutoff. The transitions of the double solutions for the same mode, for the even and the odd modes, can be described as €

2 2 2 cos (x)+ µx (sin (x)+ x tan(x)) = 0 (3. 42) 2 2 2 sin (x)+ µx (cos (x)− x cot(x)) = 0 €

where, µx expresses the quotient between µ1 and µ2. €

€ € € 56 It is clear that two degenerated modes are excited in the dielectric slab, a conventional mode and a limited conventional mode, which can be verified in the Figure 3.18.

Moreover, for this structure, quite different dispersive diagrams can be obtained,

depending on the relation between the constitutive parameters of the media. In the case when

ε2µ2 < ε1µ1 there are several important consequences to be reported.

Using the equation (3.31) and ensuring w ≥ 0 for the dielectric slab where the inner 2 2 € medium is less dense than the outer medium, the following inequality is applied, U + v ≥ 0. Also, according to the equation (3.26), one obtains v2 < 0 . Although, the equation € U 2 + v2 ≥ 0, is only verified when the super slow modes exist, otherwise the inequalities € u2 v2 0 are expressed, which is always false. Therefore, from (3.25), using u2 w2 0 − + ≥ € + < € for the less dense medium (interior medium). The propagation, with w ≥ 0, is only possible if 2 the following condition u < 0 until is verified. The previous condition is inserted in the € € super slow modes, it means that the propagation of the less dense medium in only satisfied € using a DNG dielectric slab. € In the case of a less dense inner medium, the dispersion diagrams can be divided in two different cases:

i) µ2 > µ1 , in Figure 3.19; ii) µ2 < µ1 , in Figure 3.20.

k0d € €

€

β d

Figure 3. 19 DNG dielectric slab dispersion diagram characterized by ε1 = 2,µ1 =1,ε2 = −0.8,µ2 = −1.3, for TE mode. €

€

57 k0d

€

βd

Figure 3. 20 DNG Dielectric slab dispersion diagram characterized by ε1 = 2,µ1 = 2,ε2 = −1.5,µ2 = −1.5, € for TE mode.

€ The first figure shows an even and an odd super slow mode, with a null cutoff frequency. The longitudinal wavenumbers for both these modes, tend to the same value with an increasing frequency. In fact, the even and odd super slow modes become degenerate modes.

In second case, only the even super slow mode propagates, which is limited by frequency. In this frequency range, there are always two solutions that tend to same point.

Also, there are propagation conditions of super slow modes, for all values of k0 .

€ 3.4 Conclusion

The use of DNG metamaterials to replace the common isotropic DPS media in waveguiding structures, creates a set of new physical effects to be explored. A simple DPS-DNG interface may support two surface modes: a TE and a TM mode both propagating in the interface.

The frequency range in the lossless dispersive model where the real part of permeability and permittivity are both negative corresponds to the same frequency range region in the lossy dispersive model. It just happens because one uses low values to the losses.

The electric field, Ey (t = 0, x,z), as function of, x k0 , for TE modes, is shown. Using different values for the permeability of inter and outer medium, it provides a variation slope

€ € 58 for the curves.

The understanding of this new type of propagation is essential to analyze more complex phenomena observed in other structures. The DNG dielectric slab is a waveguiding structure exhibiting new electromagnetic features.

This structure provides the existence of the super slow modes, where. The phase velocity is lower than the phase velocity in the outer medium. On the other hand, the surface slow modes exhibit a double solution in certain limited frequency ranges.

When analyzing the DNG dielectric slab dispersion diagrams, several important effects can be reported. In the case where the slab is less dense then the outer medium, µ2 < µ1, there are only super slow modes propagating in the structure, odd and even, in an unlimited frequency band. In the reverse case, µ2 > µ1, when the slab is more dense then the outer € medium, common slow mode solutions start to exist. Moreover, there may be single super slow mode solutions and double solutions. The results obtained in this chapter may help to € envisage the application of DNG metamaterials in propagation structures.

59 60

Chapter 4

Propagation of electromagnetic waves in H- guides and H-guide Couplers

In this chapter the electromagnetic wave propagating in waveguiding structures H-guide is studied. Several effects such as co-directional and contra-directional coupling regions are verified in the DNG/DPS H-guide directional.

61 4.1 Introduction

In this chapter, the electromagnetic wave propagating in waveguiding structures H-guide is addressed, containing both DPS and DNG materials. Assuming that these materials are homogeneous, isotropic and exhibit losses.

Some unusual features, such as anomalous waveguide dispersion, mode bifurcation, and the existence of super-slow modes, are put in evidence. These features providing physical insight into the properties of waveguides filled with such metamaterials have potential interest in the design of novel devices and components.

A DNG slab in the DNG H-guide replaces the common DPS slab. From the operational diagram, which has two distinct regions, the closed-waveguide regime and the open-waveguide regime, one observes several important results. The corresponding dispersion diagrams are also addressed.

In the following section, two juxtaposed dielectric slabs (DNG and DPS) form the double-slab DNG/DPS H-guide. Also, one addresses the operational and dispersion diagrams.

Finally, the DNG/DPS H-guide directional coupler is studied. It is possible to verify interesting mode coupling regions and co-directional and contra-directional coupling regions.

62 4.2 The DNG H-Guide

In this section, the propagation characteristics of an H-guide are analyzed. The common DPS slab is replaced by a DNG slab with ε < 0,µ < 0, as shown in the Figure 4.1.

x €

€

ε < 0, µ < 0 b

y € € 2l €

Figure 4. 1 DNG H-guide. € Considering the constant b as the parallel plat separation while 2l is the slab thickness and b ζ = is the aspect ratio. The operation of the H-guide is divided in two regimes: the closed- l b b waveguide regime, when < 0.5 and the open-waveguide regime, when > 0.5. λ λ € The modes propagating are hybrid and have five field components to the DNG or DPS media. Classifying€ these modes in two families: longitudinal€ -section electric (LSE)

modes and longitudinal-section magnetic (LSM) modes. The LSM modes have Ex = 0 and

the LSM modes have H x = 0. Both modes can be separated into even and odd modes, due to spatial symmetry. € €

63 Assuming monochromatic time-harmonic variation of the form e jωt and planewave

− jkz propagation along z of the form e . The modal equations for the even and odd LSM mn modes, respectively, can be written as €

€ € ε α + h tan(hl) = 0 , € (4. 1)

and € ε α − hcot(hl) = 0 , (4. 2)

€ 2 π where α 2 = k 2 + k 2 − k 2, h2 = εµk − k 2 − k 2 and also k = ω ε µ , k = n , with n y 0 0 y 0 0 0 y b being an integer. The mode cutoff in the open-waveguide regime is obtained by α = 0, while in the closed-waveguide, it is obtained by β = 0. € € € € € The modal equations for the LSE modes can be easily determined€ by duality directly

from the equations (4.1) and (4.2),€ that is, ε being replaced by µ . The LSM 01 mode is the most interesting mode when operating in the closed-waveguide regime, due to its lower attenuation losses. Therefore, the LSM mode is analyzed without losses. € € € In the Figure 4.2 is depicted the operational diagram of a DNG H-guide, where ε = µ = −2, and simultaneously is shown the operational diagram of a DPS H-guide, where ε = µ = 2. The curves of the DNG H-guide and DPS H-guide do not have significant b € differences, when > 0.5, that is, in the open-waveguide regime. λ €

€

64 LSM21

LSM21

LSM11

LSM11

LSM01 LSM01

Figure 4. 2 Operational diagram for a H-guide, containing both DNG and DPS materials.

However, in the closed-waveguide regime, one finds multiple points of operation for the same mode, for the DNG H-guide. The corresponding dispersion diagrams are shown in Figure 4.3 and Figure 4.4.

b The case = 0.4 , that includes the closed-waveguide regime, is depicted in Figure λ b 4.3 and the second case = 0.6, that uses the open-waveguide regime, is shown in Figure λ 4.4. €

€

65 LSM11

LSM01 LSM21

b Figure 4. 3 Dispersion diagram for = 0.4 . λ

€ k In the Figure 4.4 the normalized longitudinal wavenumber β = is represented as a k0 l function of . There are dashed lines for h = 0 (upper line) and α = 0 (lower line). λ € It is easy to verify, in the dispersion diagrams, how many modes are propagating € € l above€ cutoff for certain value of . It is possible to consider the waveguide miniaturization λ because there is no cutoff for the fundamental mode, which does not happen in a conventional H-guide. Some dispersion curves are no longer monotonically increasing curves with € frequency. The group of velocity of certain modes, in some regions, becomes negative for the l H-guide, which is different in a conventional H-guide. Also, when the value of is λ increased above some critical point, where the derivate of β is infinity, two solutions arise. Also, the mode bifurcation is verified. The electromagnetic field inside the DNG slab and the € field outside are such that the total power flowing along the waveguide is null. €

Also, there is a super slow mode to LSM 01 , when β is greater than εµ . At this point, the variable h becomes purely imaginary, although the real solutions are still

66 € € € € permitted. The surface waves whose fields decay exponentially, inside and outside the DNG slab, can propagate.

LSM11 LSM01

LSM21

b Figure 4. 4 Dispersion diagram for = 0.6 . λ

Fixing two values for the aspect ratio ζ , ζ =1.25 andζ =1.75. The first case, € without super-slow mode is shown in Figure 4.5 and the second case, exhibiting super-slow mode is depicted in Figure 4.6. These conditions form a new type of dispersion diagrams. € € € Again, there are dashed lines for h = 0 (upper line) and α = 0(lower line). There is possible to find critical points where the derivate of β is infinity and the total net power flowing in the

waveguide is null. The fundamental mode LSM 01 can go below cutoff for higher values of € € b . € λ €

€

67 LSM11

LSM01

Figure 4. 5 Dispersion diagram for ζ =1.25.

€

LSM11 LSM21

LSM01

Figure 4. 6 Dispersion diagram for ζ =1.75.

€

68 4.3 Double-slab DNG/DPS H-guide

In this section two juxtaposed dielectric slabs, a conventional DPS slab and a DNG slab form the double-slab DNG/DPS H-guide, as it shown in Figure 4.7. The DPS slab is characterized

by ε1,µ1 > 0 and the DNG slab is characterized by ε2,µ2 < 0 .

The total thickness of this structure, l = l1 + l2, is formed by the DPS slab thickness € € l2 l1 , and the DNG slab thickness l2 , while ξ = . € l1

€ € € x

€

DPS DNG b y

€ l1 l2 € Figure 4. 7 Double-slab DNG/DPS H-guide

From spatial symmetry€ of this structure,€ the modes propagating are hybrid LSE and LSM modes. The modal equation for LSM modes can be expressed by

ε1h2 [h1 +ε1α tan(h1l1)][ε2 α + h2 tan(h2l2 )]+ ε2 h1[ε1α − h1 tan(h1l1 )] , (4. 3) [h2 − ε2 α tan(h2l2 )] = 0

2 2 2 2 2 2 2 2 where h1 = ε1µ1k0 − k − ky and h2 = ε2µ2k0 − k − ky . € Considering, DPS and DNG media with ε1 = −ε2 = ε and µ1 = −µ2 = µ, which provide DNG

€ and DPS isorefractive €media. Having h1 = h2 = h, the modal equation can be factorized as

 €  hl   €  hl  h +εα tan   εα − h tan  = 0. (4. 4)  €  2    2 

Note that, there are no even or odd modes propagating. €

69 Then, in the open-waveguide regime, α = 0 , one has to replace the equation (4.4) by

tan(hl) = 0. (4. 5) € That leading to € l m 1+ξ = , (4. 6) λ 4 εµ −1 1−ξ where m is an integer. € € In Figure 4.8, the operational diagram is shown for two different values of ξ . b Verifying that there are no differences between them in the open-waveguide regime, > 0.5. λ € The double-slab DNG/DPS H-guide has ε1 = µ1 = 2 to the DPS slab andε1 = µ1 = 2 to the DNG slab. The last one, in the closed-waveguide regime, is the main influence in the € structure if the value of ξ is grater than one. Otherwise, is ξ is lower than one, the main € € influence is the DPS slab.

€ €

LSM11

LSM01

Figure 4. 8 Operational diagram for ξ = 0.25 and ξ = 4.

b Having, = 0.5, in the Figure 4.9 is€ constructed€ the dispersion diagram for two λ different values of ξ . There is no cutoff for the fundamental mode, when ξ = 4. Also, there are super-slow modes propagating which have no upper bound for the value of β . € € € 70 €

Figure 4. 9 Dispersion diagram for ξ = 0.15 and ξ = 4.

4.4 DNG/DPS H-guide directional coupler€ €

In this section, the DNG/DPS H-guide directional coupler is addressed. The DPS slab and the DNG slab separated by a distance 2s form that structure, as shown in Figure 4.10.

€

x

DPS€ DNG b y

l2 € l1 2s € Figure 4. 10 DNG/DPS H-guide directional coupler. € € In fact, the power fluxes€ in each slab have opposite directions.

71 The LSM modes have the following modal equation

[h1 +ε1α tan(h1l1)]{ε1αh2 cosh(2αs)[ε2 α + h2 tan(h2l2 )] 2 +2ε1 ε2 α sinh(αs)cosh(αs)[h2 −ε2 αtan(h2l2 )]}+ [ε1α (4. 7) −h1 tan(h1l1)]{2h1h2 sinh(αs)cosh(αs)[ε2 α + h2 tan(h2l2 )]

−ε2αh1 cosh(2αs)[h2 −ε2 αtan(h2l2 )]} = 0

For the LSE modes, the modal equation is derived by duality. In Figure 4.11, the operational€ diagram is depicted for a DNG/DPS H-guide directional coupler, having s ε = µ = 2 and ε = µ = −2, for ξ = 2 and = 0.4 . Again, the distance l, is defined by 1 1 2 2 l

l = l1 + l2. € € € € Verifying the coupling regions€ in the crossing points between the dispersion curves of € the isolated DPS and DNG H-guides.

Figure 4. 11 Operational diagram for the DNG/DPS H-guide directional coupler.

l In the Figure 4.12 the dispersion diagram is depicted as a function of with λ b = 0.5 . In the same coupler, there are co-directional and contra-directional coupling regions. λ The last one tends to frequency stop-band. €

€ 72

b Figure 4. 12 Dispersion diagram for = 0.5. λ

b In the same way, the dispersion diagram is addressed€ in the Figure 4.13, as a function of , λ b with a fixed value, =1. Using an appropriate model for the dispersion, the coupler l reflectivity can be adjusted as a function of material parameters. €

€

73

b Figure 4. 13 Dispersion diagram for =1. l

4.5 Conclusion €

In this chapter, the electromagnetic-wave propagation in waveguiding structures H-guide is studied, using both DNG and DPS materials. Several effects, where the common DPS slab is replaced by a DNG slab, were put in evidence, as well, the mode bifurcation, anomalous dispersion and the existence of super-slow modes.

In the double-slab DNG/DPS H-guide, one concludes that, in the closed waveguide regime, the thicker slab dictates the dispersion characteristic.

Also, the co-directional and contra-directional mode coupling in DNG/DPS H-guide directional couplers is verified. It can be applied in the design of new devices.

74

Chapter 5

Conclusion

This is the final chapter where the most important results and conclusions are addressed. Summarising conclusions and put in evidence some contents that can be developed in future work.

75 5.1 Summary

In this section, a global overview about all the contents in this dissertation and its most important results are addressed. The main objective of this thesis was to characterize the modal propagation of electromagnetic waves in waveguiding planar structures containing metamaterials. The possibility of using complex media, such as DNG metamaterials, in these structures may generate new electromagnetic properties that can be applied in the design of new microwave and millimeter-wave devices.

In the first chapter, the state of the art in this field is addressed and several important contributions of known investigators are referred. The brief history of complex media investigation, as well as, the first experimental demonstrations of the electromagnetic properties of DNG metamaterials are described. The main objectives and motivations of this dissertation are also addressed, putting in evidence the potential interest of these contents. Furthermore, the structure of this dissertation is presented.

The DNG medium as a complex medium is addressed in the second chapter. In fact, the consequences of having negative values for the real part of the permittivity ε, and the permeability µ in DNG metamaterials, provides new properties such as the negative real part of refractive index. The constitutive relations for this medium are presented in this chapter € and the expressions for the polarization due to the electric field, the wave impedance, the refractive index, the average of the Poynting vector and the phase velocity are addressed. The Maxwell’s equations are applied to the plane waves propagating in the isotropic and unlimited DNG medium and several results are shown.

Several other concepts about the medium characterization are addressed, such as the velocity of energy transport, which coincides with the group velocity in a lossless, nondispersive isotropic medium. In a DNG medium, the electromagnetic wave phase and energy have opposite direction, which creates a backward wave. This concept cannot be verified in any conventional DPS media. In this case, the electromagnetic wave phase and energy have the same orientations and forward waves are presented. The anomalous dispersion in DNG medium is also studied. Using the general electromagnetic theory, the expressions for the mean values of the electric and magnetic energy densities must be corrected for a DNG medium. The conventional energy relations must be replaced because the DNG medium is necessarily dispersive, and the electromagnetic energy densities would be negative. The Lorentz’s model for the dispersion is used to find a range of frequencies

76 where the medium behaves like a DNG complex medium. When applied to a DNG medium, the Kramers-Kroning relations lead to a causal dispersion model. The DNG-DPS dielectric slab is also considered, and the transmission and the reflectivity coefficients are derived. One of most interesting properties of the DNG metamaterials is the negative refraction. The geometrical interpretation and the analytic explanation of negative refraction are addressed. The Snell’s law is applied to the interface between two different media, DNG medium with negative refraction and DPS medium. The opposite direction of the wave vector k and the Poynting vector S are shown to be due to the choice of the negative sign for the refraction index, as one must always choose a positive imaginary part for this index, for the permittivity and the permeability. The reason for this is because we are considering passive media, where the electromagnetic wave must be attenuated.

In the third chapter, the study of the DPS-DNG interface reveals a new form of propagation: an interface mode. In general, the refractive index in only real when the magnetic permeability and the electric permittivity have simultaneously negative or positive values. Neglecting the losses in the dispersive model, the refractive index has only real part. Negative refraction in the DNG medium, will occur in the same frequency range, where the parameters ε2 (ω) and µ2 (ω) are both negative. On the other hand, there will be positive refraction (DPS medium) whenε2 (ω) and µ2 (ω) are both positive. Finally, there will be an ENG region in the lossless dispersive model, when the permeability is positive and the € € permittivity is negative. In this case, the refraction index becomes purely imaginary. € € The frequency range, in the lossless dispersive model, where the real parts of the permeability and permittivity are both negative corresponds approximately to the same frequency range in the lossy dispersive model, if one considers very low values for the loss coefficients.

In the second chapter, we show that in the case of a lossy dispersive model the imaginary part of the permittivity, the permeability and the refractive index must be all positive. Moreover, the real part of both attenuation constants must be positive to assure transverse exponential attenuation for the fields away from the interface. In fact, the fields would go to infinite if the real part of the transverse wavenumbers α1 and α2 was negative. However, the imaginary part of both attenuation constants, which is related with the electromagnetic wave propagation direction, must be negative.€ These€ two last variables change characteristic at the same point, both real and imaginary parts. In the lossless case, the first wavenumber is real when the effective refractive index neff is grater than one, and imaginary when it is lower than one. €

77 Applying the boundary conditions to the tangential field components, an important relation between the wavenumbers is obtained for both TE and TM modes. A special result is

obtained from the modal equations. In the lossless case, when ε2 =−1 and µ2 =−1, for TE ε1 µ1 mode and TM mode, respectively, there is no physical solution because the effective € € refractive index neff tends to infinite. However, when considering a dispersive lossy model, physical solutions arise. €

Also in this chapter, the electric field, Ey (t = 0, x,z), as function of, x k0 , for TE modes, is shown. Using different values for the permeability of the inner and outer media, a great variety of curves are obtained. € € The DNG dielectric slab proves to be a structure with new electromagnetic features. Applying the boundary conditions on the tangential fields components, an important relations between wavenumbers are shown for the TE mode and TM mode.

This structure provides the propagation of super slow modes, which are characterized by a phase velocity lower than the light velocity in the DPS medium. On the other hand, one has shown that the surface slow modes lead to double solutions in a certain limited frequency range.

In the dispersion diagrams of the DNG dielectric slab several important aspects are

analyzed. When the slab is less dense then the outer medium, µ2 < µ1, there are only super slow modes, odd and even, propagating in an unlimited frequency range. On the contrary, when µ > µ , that is, when the slab is denser then the outer medium, conventional single 2 1 € and double slow mode solutions are possible.

€ The electric fields Ey of the even slow modes vary according to the cosine function, while those of the odd slow modes vary according to a sine function. However, if the transverse wavenumber is imaginary, this trigonometric dependence becomes hyperbolic. € In the fourth chapter, the electromagnetic-wave propagation in H-guides using both DNG and DPS materials is analyzed, and its application to the design of new devices is addressed. In the DNG H-guide, the common DPS slab is replaced by a DNG slab and several effects were put in evidence. The operational and dispersion diagrams are obtained, showing mode bifurcation, anomalous dispersion and super-slow mode propagation. In the double-slab DNG/DPS H-guide, the thicker slab will dictate the dispersion characteristics of the waveguide. Finally, co-directional and contra-directional coupling in DNG/DPS H-guide directional couplers is shown.

78 5.2 Future work

Real experiences applied to the study electromagnetic effects of metamaterials, show that the anisotropic properties of the media should be considered, therefore replacing the isotropic model.

Indefinite media, as a kind of artificial materials, have been recently considered in the literature. These media provide a variety of unusual effects. For example, the structure of split-ring resonators is a uniaxial indefinite medium for some particular polarized waves within a certain frequency range. An indefinite medium slab has been demonstrated to have focusing effects. Moreover, this kind of medium has shown to have the possibility of a several number of refraction channels.

5.2.1 Indefinite media for H-guide

The basic properties of DNG H-guides involving metamaterials, like the double-slab DNG/DPS H-guide and the DNG/DPS H-guide directional coupler, were analyzed in this thesis. The analysis of the surface waves propagating along the boundary between DNG and DPS media is very useful to understand the physical properties of metamaterials.

However, some characteristics of the H-guide related with other artificially structured materials can also be investigated. The use of indefinite anisotropic media in H-guides could be considered in a future work.

In indefinite media, the permittivity , and the permeability , tensors can be expressed using the following uniaxial form

and

,

79 where is negative, and are positive and, in the same way, is negative, and are positive. This media can provide new electromagnetic effects for H-guides. This topic can be highly relevant and a continued research about H-guides with indefinite media is an interesting study.

5.2.2 Geometric algebra applied to indefinite media

Several electromagnetic properties of anisotropic media were found by an innovator mathematical method that uses a direct geometric interpretation based on Clifford algebras. The problem of interfaces has been addressed in several publications worldwide.

More recently, the development of metamaterial technology based on anisotropic media has suggested new applications, such as invisibility cloaking devices.

The geometric algebra is a new mathematical framework and provides a complete understanding of anisotropic media. This could be an interesting future topic of research if one considers the interface between two indefinite media, with different values for the electric and magnetic parameters in each direction of the medium. Important electromagnetic effects could be shown and new technology devices will be suggested.

80

Appendix A

Convergent integral

81 Convergent integral

Having an integral containing an integrand that presents singularities in certain integration

b points, the principal value must to be considered. Considering the integral f x dx , where ∫ a ( )

f (x) has a singularity in the point x0, such that a < x0 < b. The convergence of this integral depends the existence of following limits, €

x0 −e b € I €lim f x€dx lim f x dx. (A.1) = + ( ) + + ( ) e→0 ∫ a δ→0 ∫ x0 +δ

b Supposing that this limit exists and it is finite, then, the integral f x dx is convergent, ∫ a ( )

€ b independently the limit lim f x is finite or divergent. As example, considering the + ( ) δ→0 ∫ x0 +δ following cases: €

€ 2 dx 1−e dx 2 dx I1 = 1 = lim 1 + lim 1 ∫ 0 e→0+ ∫ 0 δ→0+ ∫ 1+δ (A.2) (x −1) 3 (x −1) 3 (x −1) 3

and, € 2 dx 1−e dx 2 dx I = = lim + lim . 2 ∫ 2 + ∫ 2 + ∫ 2 (A.3) 0 (x −1) e→0 0 (x −1) δ→0 1+δ (x −1)

Note that, one have I1 = 2.2500 −1.2990i while I2 is divergent. Using e = δ in the previous equation,€ so, the principal value is considered by Cauchy and it can be written as

b x0 −e b € € P f x dx = lim € f x dx + f x dx . (A.4) ∫ ( ) + ∫ ( ) ∫ ( ) a e→0 ( a x0 +e )

2 dx As example, in spite of = lim ln e − lim lnδ + ln(2) does not exist, one can ∫1 + [ ] + [ ] € x e→0 δ→0 affirms that the Cauchy principal value exists,

2 dx P = lim ln e − ln e + ln(2) = ln(2) ≈ 0.6931. + [ ] ∫1 x e→0 €

€

82

Appendix B

The Hilbert transform

83 The Hilbert transform

The Hilbert transform was originally defined for periodic functions as

1 ∞ f (ζ) H f x = P dx. (B.1) [ ( )] ∫ −∞ π ζ − x

As well, € x n′′(x) ∞ x n′′(x) ∞ f1(x) f (x) = → = dx 1 x +ω ∫ 0 x2 −ω2 ∫ 0 x −ω , (B.2)

n′(x) −1 ∞ n′(x) −1 ∞ f2 (x) f (x) = → = dx 2 x +ω ∫ 0 x2 −ω2 ∫ 0 x −ω the Kramers-Kroning relations can be written using the transform Hilbert. € Then, the following system expressed the Kramers-Kroning relations

  x n′′(x)  n′(ω) −1= 2H  x    +ω   (B.3)   n′(x)−1  n′′(ω) = −2ωH     x +ω 

€

84 Bibliography

[1] C. R. Paiva, “Introdução aos metamateriais,” IST, 2008.

[2] A. L. Topa, C. R. Paiva and A. M. Barbosa, “Novel propagation features of double negative H-guides and H-guide couplers,” Microwave and Optical Technology Letters, vol. 47, pp. 185-190, October 2005.

[3] M. W. McCall, A. Lakhtakia and W. S. Weiglhofer, “The negative index of refraction demystified,” European Journal of Physics, vol. 23, pp. 353-359, May 2002.

[4] R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Physical Review, vol. 64, 06625, October 2001.

[5] J. Pendry, “ Invisibility cloaks come closer to reality,” Thaindian News, July 2009

[6] A. Boyle, “Here’s how to make an invisibility cloak,” Technology & science, May 2006.

85