In the Name of ALLAH, The Most Beneficent, The Most Merciful

THEORETICAL ANALYSIS OF THE EFFECTIVE

PARAMETERS OF ROD METAMATERIALS

A PH.D. THESIS

BY

MUHAMMAD RAZA

CENTRE FOR ADVANCED STUDIES

IN PURE AND APPLIED MATHEMATICS

BAHAUDDIN ZAKARIYA UNIVERSITY

MULTAN, PAKISTAN.

JANUARY 2011

THEORETICAL ANALYSIS OF THE EFFECTIVE

PARAMETERS OF ROD METAMATERIALS BY

MUHAMMAD RAZA

SUPERVISED BY

DR. OLEG N. RYBIN

PROF. DR. TAHIR ABBAS

CENTRE FOR ADVANCED STUDIES

IN PURE AND APPLIED MATHEMATICS

BAHAUDDIN ZAKARIYA UNIVERSITY

MULTAN, PAKISTAN.

JANUARY 2011

THEORETICAL ANALYSIS OF THE EFFECTIVE

PARAMETERS OF ROD METAMATERIALS

A DISSERTATION SUBMITTED IN PARTIAL

FULFILLMENT OF THE REQUIREMENT

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN MATHEMATICS

BY

MUHAMMAD RAZA

CENTRE FOR ADVANCED STUDIES

IN PURE AND APPLIED MATHEMATICS

BAHAUDDIN ZAKARIYA UNIVERSITY

MULTAN, PAKISTAN.

JANUARY 2011

Dedicated to Abu Turab Hazrat Ali Ibn-e-Abitalib a.s.

CERTIFICATE

It is certified that the work contained in this thesis is original and carried out by Mr. Muhammad Raza under our supervision.

Dr. Oleg Rybin (Supervisor) Associate Professor Head of the Department of Information Technology Kharkov State University of Food Technology and Trade 333, Klochkovskaya St. Kharkov, 61051, Ukraine

Prof. Dr. Tahir Abbas (Supervisor) Department of Physics, B. Z. University, Multan, Pakistan.

CERTIFICATE

It is certified that this work has not already been submitted for any degree and shall not,

in future, be submitted for obtaining Ph. D degree of any other university.

Candidate _______________________________ Muhammad Raza

1

Abstract

This thesis is dedicated to theoretical characterizing of two component metamaterials as

arrays of metal rods/wires periodically immersed in a dielectric isotropic matrix. The rods

material is supposed to be non-magnetic. Only a circular cross section case of the rods is

considered in the thesis while the rods do not touch each other. At the same time, there is no

restriction to the radius of cross section of the rods. The microwave frequency range (from 0 to 5

GHz) has only been considered in this study. The metamaterial media/structures are being

considered in the thesis as artificial semiconductors with their own effective complex dielectric

and magnetic constants. So, the presented metamaterial media/structures in the thesis are

considered as perfect crystals with their own dispersive properties studied in the microwave

frequency range. The lattice constant of the crystal is equal to the constant of unit cell of

metamaterial under consideration.

The characterizations of considered metamaterial structures in the thesis are identified with

the study of properties of the effective dielectric and magnetic constants as functions of the

frequency (in the GHz frequencies) of incident electromagnetic wave and the volume fraction of

metal rods in the unit cell. The above characterizing is the key for defining unusual material

properties unavailable in real nature: enhancement of the effective parameters; a possibility to

get negative values of the effective parameters; ultra-low values of the refractive index.

Throughout the thesis, we consider the initial plane electromagnetic wave that is normally

incident to the flat boundaries of the chain. The wave has the magnetic induction vector parallel

to the axes of rods while the electric intensity vector is perpendicular to the ones.

The above effective complex dielectric and magnetic constants have been obtained for the

first time by author of the thesis on base of extension of the case of 2-D infinite metamaterial

medium to a slab metamaterial scatterer under consideration through the implementation of the

Effective Medium Theory (EMT) in appropriate frequency range. The expressions of the

2 appropriate effective constants for the infinite medium are obtained by other authors. These

expressions of the effective constants obtained in this thesis takes in account multipole effects

for the case of composite with a very small value of the rods volume fraction while dipole effects

are taken in account for the case of large volume fraction values.

The accuracy of obtained mathematical models was always benchmarked through a

comparison with numerical calculations obtained via the implementations of Finite-Domain

Time-Difference (FDTD) method for calculating S-parameters of a metamaterial structure under

considerations. S-parameters were used to calculate the effective constants by means of the using

Nicolson-Ross approach.

All of the numerical experiments presented in the thesis have been carried out with the

help of the free Meep FDTD software package while analytical modeling has been done using

MatLab software.

In this thesis, an improved broadband method for determining complex effective refractive

index, dielectric and magnetic constants of an arbitrary passive metamaterial has been proposed.

Evaluation of the effective parameters is realized using the reflection-transmission S-parameters

obtained by simulation or experimental measurements and analytically evaluated interface

reflection coefficient of the slab.

In consideration of practical party of this thesis, the obtained qualitative and quantitative

results in this thesis have allowed to formulating some properties of two component slab

metamaterial structures as arrays of metal rods/wires periodically immersed in a dielectric

isotropic matrix:

1. The effective electromagnetic properties of infinite 2-D array of copper cylinders

immersed in metal-dielectric matrix in the GHz frequencies shows the existence of the

enhancement of effective dielectric constant and low absorption in the microwave

frequencies.

3

2. The obtained analytical models of the composite in the thesis gives a good qualitative but

a weak quantitative correlation with results of numerical simulations in the case if

cylinders touch each other.

3. The above analytical models of infinite metamaterial medium quantitatively describes

well the slabs embedded with the above metamaterial medium if there is some relation

between the width of slabs and the dimension of unit cell of the metamaterial medium for

appropriate frequency range.

4. The considered artificial material medium can be used to increase the directivity of patch

antenna and to obtain ULI structures in the GHz frequency range and to design a new

type of waveguides.

5. The obtained mathematical models cannot reveal negative values of the effective

dielectric and/or magnetic constants (their real parts) in the GHz frequency range.

The main theoretical results of this thesis can be presented by two theoretical methods of

characterizing of any 2-D slab metamaterial structures in the microwave frequencies via EMT

approach:

1. Non-destructive broadband method for the evaluation of the effective complex

dielectric and magnetic parameters of 2-D slab metamaterials.

2. The analytical and numerical optimization method for separating a slab metamaterial

into its elementary sub-slabs of the order of the unit cell dimension of the slab.

It is important to mention that the above methods have been designed irrespective to the

shape of inclusion in the unit cell. Moreover these methods allow to easy evaluating the optical

and transport properties of slab metamaterial structures including magnetic ones trough using the

relation for the total reflection and transmit ion coefficients and the above constants of single

layer.

The obtained results in this thesis are in a good quantitative and qualitative agreement with

the results of experimental research carried out earlier by one of the supervisors. Moreover these

4 results can be used for creating the course of laboratory works with the using of personal

computers for students of Engineer and Sciences directions (Industrial Mathematics, Theoretical

Physics, Electrical & Electronic Engineering, Material Science) to study the optical and transport

properties of slab metamaterial structures in the microwave frequency range.

ii

Contents

List of Figures iv

List of Tables vi

1 METAMATERIALS AND THEIR UNUSUAL PROPERTIES 11.1 E¤ective Medium Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Classi�cation of Metamaterials by Physical Phenomena . . . . . . . . . . . 3

1.2.1 Left-Handed Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Metamaterials with ultra low index . . . . . . . . . . . . . . . . . . . 81.2.3 Enhanced Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 Nihility Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Metamaterial Quadrants Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 The Rod Metamaterials Structures . . . . . . . . . . . . . . . . . . . . . . . 141.5 Planning of the research and method of investigation . . . . . . . . . . . . . 15

1.5.1 Proposed Work and Possible Practical Issues of Planned Research . 151.5.2 Objective of Method of Investigations . . . . . . . . . . . . . . . . . 16

2 MATHEMATICAL MODELS 182.1 The Rod Metamaterials Structures . . . . . . . . . . . . . . . . . . . . . . . 192.2 The Mathematical Models of Rod Metamaterials Structures . . . . . . . . . 212.3 Elements of S- and T-Matrices Algebra . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 Scattering parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 Transfer parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 S- and T- Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.1 T-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.2 S- scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 THE EVALUATION OF EFFECTIVE PARAMETERS OF SLABMETA-MATERIALS 443.1 An Improved Broadband Method for the Evaluation of E¤ective Parameters

of Slab Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iii

3.1.2 Evaluation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1.3 Precision and Sensitivity Discussion . . . . . . . . . . . . . . . . . . 56

3.2 Evaluation of Layer Properties of E¤ective Parameters of Metallic Rod Meta-materials in GHz Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 LONG WAVE LAYER-SPECIFIC REPRESENTATION OF THE OPTI-CAL PROPERTIES OF SLAB METAMATERIALS 674.1 Cascaded Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 The Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 EFFECTIVE ELECTROMAGNETIC RESPONSE OF THE INFINITECHAIN OF CIRCULAR METALLIC CYLINDERS 895.1 Homogenization of Slab Metal-Rod Periodic Media . . . . . . . . . . . . . . 905.2 The Case of Simple Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Main Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2.2 Modeling and simulation results . . . . . . . . . . . . . . . . . . . . 95

5.3 The Case of Embedded Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.1 Main relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.2 Modeling and simulation results of the case . . . . . . . . . . . . . . 104

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Conclusion 109

Bibliography 113

iv

List of Figures

1.1 To the problem of interpretation of left handed materials . . . . . . . . . . 51.2 Ray comparison between wave propagation in doubly negative medium and

doubly positive one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 To the question of materials classi�cation . . . . . . . . . . . . . . . . . . . 13

2.1 Periodic array of metallic cylinders of diameter d arranged in a square arraywith lattice constant a. The cylinders are in�nitely long in the y direction . 22

2.2 Unit cell of the composite with!k = kz �

!z0;�!E = Ex � �!x0;

�!H = Hy � �!y0 : . . . . 26

2.3 Model of the composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 The two-port network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 For T-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.6 For S-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Change of real part of e¤ective refractive index of metamaterial versus thefrequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Change of imaginary part of e¤ective refractive index of metamaterial versusthe frequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Change of real part of e¤ective dielectric constant of metamaterial versus thefrequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Change of imaginary part of e¤ective dielectric constant of metamaterialversus the frequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Change of real part of e¤ective magnetic constant of metamaterial versus thefrequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 Change of imaginary part of e¤ective magnetic constant of metamaterialsversus the frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7 The Basic Layer in 2-D Image . . . . . . . . . . . . . . . . . . . . . . . . . . 603.8 Change of real part of e¤ective refractive index of metamaterial versus metal

volume fraction F at 1 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Representation of a slab obstacle as a cascaded network of sub-slabs . . . . 704.2 Representation in the complex plane of the calculated and simulated S-

parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

v

4.3 Change of real part of the e¤ective dielectric constant of the elementary slabversus metal volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . 86

4.4 Change of real part of the e¤ective magnetic constant of the elementary slabversus metal volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . 87

5.1 The chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2 Change of real part of e¤ective dielectric constant of the chain versus metal

volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Change of real part of e¤ective magnetic constant of the chain versus metal

volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.4 The chain immersed in the imbedded matrix . . . . . . . . . . . . . . . . . 995.5 Representation of the metamaterial slab as a four-terminal network of sub-slabs1015.6 Change of real part of the e¤ective dielectric constant of the chain versus

metal volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . . . . . 1055.7 Change of real part of the e¤ective magnetic constant of the chain versus

metal volume fraction F at 1GHz . . . . . . . . . . . . . . . . . . . . . . . . 106

vi

List of Tables

3.1 Value of the real part of the e¤ective Dielectric Constant at 1 GHz . . . . . 613.2 Value of the real part of the e¤ective Magnetic Constant at 1 GHz . . . . . 61

vii

Acknowledgments

Praise and glory is to Allah, The Cherisher and The Sustainer of the worlds, without His

will, and faith in HIS support, I would not have been able to accomplish this laborious task.

I bow to Him in reverence and gratitude. Peace and blessing to Hazrat Muhammad (Peace

Be Upon Him) and His Ahlalbait (a.s.) who own all knowledges and wisdom supreme.

I �nd myself very lucky to have sincere and e¤ectionate supervisor like Dr. Oleg N.

Rybin, for his consistent encouragement, friendly behavior and kind supervision enabled me

to overcome the hinderances that came in my way during the tenure of research. In short

he is a perfect model of professionalism, understanding and having a wealth of knowledge.

I am also thankful to the family of Dr Oleg Rybin especially mother for providing moral

support.

I am highly obliged and very grateful to my, eminent supervisor, Prof. Dr. Tahir

Abbas, Chairman, Department of Physics, B. Z. University, Multan, whom I always found

helpful with smiling face. I am very much obliged for his keen supervision and friendly

a¤ection. He has always been the most cordial and cooperative to me throughout the

research work and the compilation of this thesis.

I am highly obliged and very greatful to Director CASPAM, Dr. Muhammad

Anwar Chaudhary for providing me every type of latest research facility and for creating

congenial research environment in the centre.

I owe a great debt of gratitude to Prof. Dr. Nazir Ahmed Mir, Ex-Director,

CASPAM, B. Z. University, Multan, as he always helped and encouraged me in problems

during research with his valuable suggestions.

viii

Thanks also to all my teachers at CASPAM. They have always been a ray of hope

and light in the darkness of di¢ cult moments. It is my pleasant duty to pay special thanks

to sta¤ of the library especially Mr. Almas (librarian) of CASPAM.

Thanks are also due for hardworking members of the o¢ ce of CASPAM especially

Mr Abdul Jabbar, for their full and friendly cooperation, during my studies.

I am thankful especially to my batch fellows Ms Tahira Nawaz, Ms Fiza Zafar,

Ms Nikhat Bano, Mr Farooq Ahmed, Mr Sifat Husain, Mr. Amjad Islam, Mr Muhammad

Ishaq and all other PhD Scholars at CASPAM, B. Z. University Multan whom I always

found helpful and ready to guide me.

Very special thanks to Higher Education Commission (HEC), Islamabad, Pakistan

as without its �nancial help (scholarship) I could not have been able to reach this desti-

nation. This is HEC which supported our project �nancially and we were able to perform

high level research and produced publications.

Very special thanks to my parents, brothers Ali Imran and Hadi Raza, wife and

lovely children as they helped and supported me and also they su¤ered much during my

research tenure.

Muhammad Raza

1

Chapter 1

METAMATERIALS AND THEIR

UNUSUAL PROPERTIES

The engineered response of metamaterials has a dramatic impact on the physics,

applied mathematics, optics and engineering communities. These can o¤er electromagnetic

properties that are di¢ cult or impossible to achieve with conventional and naturally oc-

curring materials. The advent of metamaterials has yielded new opportunities to realize

physical phenomena which were previously only theoretical exercises. Metamaterials have

been of particular interest to scientists and engineers for the last two decades.

The unusual properties of metamaterials which have been de�ned in the following

sections, de�ne the possible practical application of metamaterials in the modern indus-

try. Particularly, these properties promise distinct advantages to the microwave range in

electromagnetics. Indeed, the rapid development of the microwave technology suggests

elaboration of methods for e¢ cient manipulations with microwaves. Metamaterials being

2

arti�cial structures, arranged as a lattice of sets of identical components, usually represent

arti�cial crystals designed for microwaves and, therefore, seems to be especially suitable

for this purpose. That is why, a huge numbers of scientists intend to join the stream of

metamaterials research regarding microwave frequencies. Thus, throughout of this work we

carry out the characterization of metamaterials in the microwave frequency range.

1.1 E¤ective Medium Theory

Metamaterials are composite materials consisting of two or more natural ingre-

dients with di¤erent values of such parameters as permittivity, permeability and conduc-

tivity. Electromagnetic waves interact with the inclusions, inducing electric and magnetic

moments, which in turn a¤ect the macroscopic e¤ective dielectric "r and magnetic �r con-

stants of the bulk composite medium. Taking in account the fact that metamaterials have

unit cell dimensions much smaller than the wavelength of interest, they can be considered

as e¤ective media in some frequency range when describing electromagnetic phenomena

are considered at a macroscopic level. The scatteres (inclusions) emulate the atoms or

molecules in a dielectric and produce a net average polarization and magnetization that

give rise to an e¤ective permittivity and permeability, correspondingly1. In general, for

some frequency range, the electromagnetic properties of such structures can be expressed in

terms of an e¤ective electric permittivity and e¤ective magnetic permeability following the

e¤ective refractive index [7,8]. This approach is called the E¤ective Medium Theory [9,10]

or Homogenization Approach [11,12]. For example, a structure composed of a periodic ar-

1Sometimes, people make mistake considering unit cell of a metamaterial as an e¤ective atom or molecule.In reality the direct analogy between the e¤ective atom or molecule and the unit cell does not exist.

3

ray of metallic cylinders or metallic strips with a small spatial period with respect to the

wavelength, behaves as a homogenous material with an e¤ective dielectric permittivity and

magnetic permeability. This homogenous material exhibits a plasma-like behavior [13-16].

1.2 Classi�cation of Metamaterials by Physical Phenomena

The electromagnetic (EM) properties of a dielectric material can be described by

its corresponding electric permittivity " and magnetic permeability �. These two parame-

ters macroscopically describe the e¤ects of induced electric and magnetic polarization. Also

permittivity and permeability are the keys to calculate the refractive index and character-

istic impedance. The refractive index is a key parameter which describes the interaction of

light with matter. So, having full knowledge of permittivity and permeability of a dielectric

material, we can learn its optical and transport properties.

Metamaterials enable us to get the permittivity and permeability that are di¤erent

from that of its naturally existing parental materials. Thus, it is tenable to consider the

dispersive properties of materials with a viewpoint of complex permittivity (" = "0 + i"00)

and permeability (� = �0 + i�00) of materials because of analogous consideration of the com-

plex refractive index (n = n0 + in00). Considering the real parts of the above parameters,

we can make some conclusions about optical and transport properties of materials in some

frequency range while considering the imaginary parts of these parameters can bring us to

some conclusions about absorption properties of materials in some frequency range.

It is possible to make a quite convenient classi�cation of metamaterials on the basis

of the range of values of the real parts of permittivity and permeability or the real part of

4

the refractive index, because of a number of phenomena classi�ed by di¤ractive properties

of materials: left-handed materials, materials with ultra low index, and enhanced materials.

1.2.1 Left-Handed Materials

The real part of refractive index is usually considered to be positive, which is in

fact true for all naturally existing materials. However, a refractive index with a negative

real part does not violate any physical law. Instead some years ago, Soviet scientist Veselago

[17] made a theoretical study of materials in which the sign of the electrical permittivity

(its real part), and magnetic permeability (its real part), were simultaneously negative. He

concluded that although these materials should support electromagnetic waves having a

well de�ned wave vector!k

�����!k ���� = "�!2

c2

�, there was something decidedly peculiar about

these waves: theory predicted that the energy �ow as dictated by Poynting vector would be

in the opposite direction to the wave vector, the implication being that rays travel in the

opposite direction to waves. Since Veselago has shown in his paper that in these material

vectors!E;

!H and

!k , form the left-hand triplet, Fig.1.1. For this reason, he referred to

his new materials as being �left handed�(LH)2. Veselago also showed that at an interface

between doubly positive and doubly negative materials light would be bent in the wrong

way relative to the normal, Fig.1.2. Since such materials do not exist in nature they have

to be arti�cially fabricated.

2Throughout this work, the electromagnetic waves with the right-hand triplet of vectors, are called �righthanded�(RH) ones

5

Figure 1.1: To the problem of interpretation of left handed materials

6

The left-handed materials have many strange properties [18, 19]: a reversed Doppler

shift, and reversed Cherenkov radiation that emerges in the opposite direction to the con-

ventional direction of radiation. The strangest property follows directly from negative

refraction and is the ability of the material to focus light. The left-handed materials based

on lenses have the very important advantage as compared to ordinary optical lenses: they

capture the near �eld but ordinary ones do not capture the near �eld; hence no limitation

of the resolution. Moreover, numerical studies indicate that nearly-perfect imaging should

be expected even under realistic conditions when both dispersion and losses are taken into

account [20]. This extraordinary property of the Veselago lens was experimentally pointed

out in ref. [21] and [22]. Lenses with this sub wavelength focusing are referred to as �per-

fect lenses�.Recently a new property of left-handed materials has been predicted [23]; the

left-handed material structures can demonstrate a giant lateral Goos-Hanchen shift of the

scattered beam which can have a number of practical applications.

The recent implementation of a microstructured material with negative magnetic

permeability in the gigahertz range has quickly led to the development of a composite

medium with simultaneously negative values of permeability and permittivity over a �nite

frequency range [24]. The reported metamaterial combines an array of metallic wires (to

attain negative e¤ective permittivity) with an array of split-ring resonators (to achieve

negative permeability) [25].

7

Figure 1.2: Ray comparison between wave propagation in doubly negative medium anddoubly positive one

8

Pendry has shown that left-handed materials can be consisted of split ring res-

onators (SRRs) and continuous wires [26, 27], and suggested that they can also act as

perfect lenses [20]. Various samples were prepared [28, 29] according to the Pendry�s the-

ory of split ring resonators (SRRs) and long wires all of which have been shown to exhibit

a passband in which it was assumed that " (its real part) and � (its real part) are both

negative.

Thus it can be assumed that the most practical application of left-handed meta-

materials is a design of lenses without to have principal limitations to their resolution for a

near �eld image that is di¢ cult to be realized well by the ordinary optical lenses.

1.2.2 Metamaterials with ultra low index

Enoch et al., have proposed a novel idea of generating directive emission with a

metamaterial composed of slices of foam of copper grids [30]. Operating at the frequencies

just over the plasma frequency, the slab has a positive refractive index, but itsvalue is close

to zero (it is called ultra low index). This approach has been incarnated experimentally in

[31] for patch antenna system with high directivity.

We often encounter with a necessity to consider the media with ultra low index

(ULI). In optical micro-resonators [32], the e¤ective cavity length and the enhancement of

spontaneous emission is directly dependent on the index contrast. In photonic crystals [33],

the photonic bandgap width is directly related to the index contrast. In semiconductor op-

toelectronics, dielectric material with a very low refractive index is the key component for

the optical parts in the devices [34, 35]. This motivates the development of new �air-like�

optical materials with a refractive index smaller than 1.0 optical micro and nanostructured

9

materials are intriguing because of their ability to control light in unconventional ways.

Among these are photonic crystals and metamaterials, which have potential applications

in integrated photonics [36]. Very important advantage of ULI materials with regard to

electromagnetic theory is the presence of intriguing optical properties including total exter-

nal re�ection. Moreover, in the reference [37], ultra low refractive-index metamaterials has

been proposed as two-dimensional square array of cylindrical silver wires embedded in an

air host medium for purposes of a building block for photonic applications. E¤ect of total

external re�ection in ULI metamaterials has also been used in [7] to design and analyze

slab waveguide structures that guide visible light in an air core.

ULI materials enable us to design band gaps in a wide range of frequencies up to

the visible. So, the paper [38], theoretically discusses the possibility of having materials

with zero e¤ective permittivity. The physical realization of these materials is also discussed

in terms of embedding metallic nanoparticles and nanowires in a dielectric medium. In the

limit of long wavelengths, these composites will behave like a homogeneous medium with

zero permittivity that will completely re�ect electromagnetic waves.

Thus ULI materials have three important properties. 1) A beam incident on a

planar interface between air (n = 1) and a ULI material would be refracted away from

the normal, instead of refracted toward the normal like most of optical materials. If one

were to build a plano-concave lens with a ULIM, the result would be a converging device.

2) Ultralow-refractive-index metamaterials exhibit an interesting new optical property at

visible wavelengths: total external re�ection.

10

1.2.3 Enhanced Metamaterials

A possibility to produce the doubly negative media and media with ULI has made

the metamaterial subject a very popular and perspective at present times. But metama-

terials can exhibit one more property which cannot be observed in ordinary materials �

enhancement of the e¤ective permittivity and permeability. For the �rst time, the e¤ective

permittivity enhancement has been predicted by Tretyakov [39]. Unfortunately, theoretical

approach designed by Tretyakov did not allow to obtain the essential enhancement of the

e¤ective permittivity. In 2004, for the �rst time, above phenomenon has been obtained

theoretically [40], and in 2005, experimentally [41], by Metamaterials Group of Zouganelis.

Aforementioned result is very promising for many future applications of metama-

terials in communications and new devices. For instance, the use of electrically enhanced

substrates can be exploited for the miniaturization of antennas [42, 43].

Detached enhancement phenomenon with a viewpoint of practical applications is

enhancement of the e¤ective permeability that can be of interest for building the theory of

metaferrites of new generation, [44].

1.2.4 Nihility Materials

A quite new and popular subject of metamaterial investigations is the nihility

property.

Nihility is the electromagnetic nilpotent expressed in terms of zero dielectric and

magnetic constants [45]. The directionality of the phase velocity relative to the wave-vector

in nihility is thus a non-issue. So, the direct practical application of the nihility phenomenon

11

is Stealth-technology.

The realization of nihility is predicted as a composite either by mixing anti-vacuum

("r = �1 , �r = �1) and vacuum ("r = 1 , �r = 1), or by mixing an LHM with an

isoimpedant commonplace material (such as te�on). This is because the polarizability and

the magnetizability (per unit volume) of an isotropic dielectric-magnetic sphere embedded

in nihility do not depend on the constitutive parameters of the medium that the sphere is

made of.

It is interesting to mention that the Maxwell-Garnett formalism [46] predicts that

a homogeneous dispersion of electrically small, anti-vacuum spheres in vacuum is e¤ectively

equivalent to nihility, provided the volume fraction of anti-vacuum is 0:25. Conversely, if a

matter was scooped out of anti-vacuum in the form of electrically small spheres, the resulting

Swiss-cheese composite would also mimic nihility, if the volume fraction of anti-vacuum were

0:75.

1.3 Metamaterial Quadrants Set

So, as we have seen above, metamaterials are arti�cial materials constructed with

arrays of elements having a huge potential to provide various exotic properties which are not

found in the nature. Such properties have a great promise in applications in microwave and

radio engineering, optics and spectroscopy, covering the frequency region from microwaves

to visible range. This is because slight changes to a repeated unit cell of the above arrays

can be used to tune the e¤ective bulk material properties of a metamaterial, replacing the

need to discover suitable materials for an application with the ability to design a structure

12

for a desired e¤ect. However, the range of metamaterials applications is not con�ned to the

demonstration and use of the negative refraction phenomenon regardless of the popularity

of this phenomenon for study at the moment. This is because any couple of values of

permittivity and permeability is of interest to fundamental research. Indeed, while a medium

with both permittivity less than zero (" < 0 and � < 0) designated a doubly negative (DNG)

medium, a medium with both permittivity greater than zero (" > 0 and � > 0) designated a

doubly positive (DPS) medium. Most naturally occurring media (e.g., dielectrics) fall under

this designation. A medium with permittivity less than zero and permeability greater than

zero (" < 0 and � > 0) will be designated as an epsilon-negative (ENG) medium. In

certain frequency regimes, many plasmas exhibit this characteristic. For example, noble

metals (e.g., silver, gold) behave in this manner in the infrared (IR) and visible frequency

domains. A medium with the permittivity greater than zero and permeability less than

zero (" > 0 and � < 0) will be designated as a munegative (MNG) medium. In certain

frequency regimes, some gyrotropic materials exhibit this characteristic. Arti�cial materials

have been constructed in such a way that they also have DPS, ENG, and MNG properties.

This medium classi�cation can be graphically illustrated as shown in Fig.3, [47].The above

mentioned material classi�cation is, of course, related to exactly real part of both the

permittivity (or the dielectric constant) and permeability (or the magnetic constant). But

in reality, we have to consider the complex e¤ective parameters where the imaginary parts

play a very important role with a viewpoint of practical applications. Study of both real

and imaginary parts of the e¤ective constants of metamaterials as functions of frequency is

integral part of this work.

13

Figure 1.3: To the question of materials classi�cation

14

1.4 The Rod Metamaterials Structures

In the recent years, the periodic rod lattices have found many applications both in

optical and microwave ranges. However, some fundamental problems have not been resolved

yet, even for typical metallic electromagnetic crystals. One of them is the problem of low-

frequency spatial dispersion in wire media (WM). The low-frequency spatial dispersion of

a simple wire medium (a doubly periodic regular array of parallel wires) has been studied

only recently in [40], using so called tensor-predictable approach.

The operational principle of a medium consisting of periodically arranged wires

has been known for a long time [7, 40]. This kind of wire medium (also called rod medium)

is known to operate as an arti�cial dielectric with negative e¤ective permittivity at low fre-

quencies. Owing to the periodic nature of the wire medium, the medium can be considered

as an electromagnetic band-gap (EBG) structure introducing frequency bands in which

electromagnetic waves cannot propagate [42]. A detailed analysis of the electromagnetic

band structures of the wire medium can be found in references [43, 44]. The transmission

and absorption properties of a two-dimensional lattice of conducting cylinders have been

presented [45], and the e¤ective electronic response of a system of metallic cylinders has

been introduced [46].

The nature of wave propagation inside the wire medium and its dispersion proper-

ties have also been a subject of comprehensive studies. In [47] the wave propagation inside

the wire medium was studied in detail. Comprehensive analytical study of the dispersion

and re�ection properties of the wire medium can be found, for example, in [40, 48-50].

Authors of [49] considered also a wire medium periodically loaded with bulk reactance and

15

presented the dispersion in relation to this kind of loaded wire medium. A quasi-static

model for the loaded wire medium can be found in [14].

1.5 Planning of the research and method of investigation

1.5.1 Proposed Work and Possible Practical Issues of Planned Research

As we can see from aforementioned review of the literature of last years, recently,

extensive e¤orts have been devoted to the utilization of the rod/wire medium in microwave

applications. It is in connection with that of a periodic array of conducting or magnetic

elements. They can behave as an e¤ective medium for electromagnetic scattering when the

wavelength is much longer than both the element dimension and lattice spacing. What

makes the resulting media special is that the e¤ective permittivity "eff and permeability

�eff can have values not observed in ordinary materials [19]. Moreover, there is an essen-

tial interest for metamaterials modeling for �low�frequency microwave range because the

desirable properties of conventional materials are seriously degraded for frequency above 1

GHz [51]. Thus it is of the interest to design the new analytical approaches by modeling of

the e¤ective parameters of the wire metamaterial structures consisting of metallic rod/wire

lattice embedded in dielectric matrix.

We plan to carry out the frequency behavior of e¤ective permittivity and per-

meability of the metallic wire metamaterial structures with an arbitrary �nite number of

lattice layers in long length wave/microwave approximation. This kind of metamaterial

structures can be of interest for the design of a new kind of tunable perfect lenses, because

of unusual properties of such kind of structures [52], a new kind of miniaturized tunable

16

beam antenna with arti�cial wire medium lens [53], and a new kind of the low band pass

spatial �lters [54]. This type of work is very important in modern theoretical physics and

applied mathematics.

1.5.2 Objective of Method of Investigations

Analytical approaches for the study of frequency dependence of e¤ective permit-

tivity and permeability are being developed still. There are two basic analytical approaches

for calculation of the e¤ective parameters of metamaterial structures. First approach is

based on Bregman�s method [55]. This method has been developed as theory of e¤ective

parameters of composites. One is very convenient to treat two component composites with

a spherical or elliptic at shape component as one of them. But this approach has a number

of disadvantages: 1) Theoretical calculations become as complicate analytical and numeri-

cal ones for case of the inclusion shapes which di¤er from spherical or elliptical ones [56].

2) This method is rather convenient for a calculation of one e¤ective parameter (permit-

tivity or permeability) than for a calculation of both ones. 3) This method becomes quite

complicated for a calculation of magnetic composites.

Second approach for a calculation of the e¤ective permittivity and permeability

of metamaterials is based on solving the corresponding di¤raction problems under some

assumption regarding the kind of permittivity tensor, so called tensor-predictable approach

[10-11, 40, 48-50]. Using this approach Tretyakov has explained a basic mechanism of such

phenomena as ULI, NRI, and enhancement of e¤ective parameters with a viewpoint of elec-

tromagnetic theory but regardless of that, above approach has a number of disadvantages:

1) This approach has been developed for the ideal metamaterial structures formed

17

by the conductor wires. That is why the approach cannot be used for a study of the e¤ective

magnetic properties of metal-dielectric metamaterial structures.

2) This approach has been developed for thin wires and rare wire lattices. That

is why the approach cannot be used for a study of e¤ective permittivity enhancement since

one can be found only on conditions of either non-thin wires or dense lattices [57].

3) The approach does not allow to study the e¤ective parameters of structures

with an arbitrary �nite number of lattice-layers.

We will propose a new analytical approach for calculation of the e¤ective para-

meters of the multi-lattice metallic wire metamaterial structures. This approach is based

on the combined use of Average Boundary Conditions Method that has been developed for

the electromagnetic di¤raction problems by Soviet scientist Kontorovich [58] and Astrakhan

[59], and S- and T-parameters approaches of electromagnetic analysis for macroscopic elec-

trodynamics media.

18

Chapter 2

MATHEMATICAL MODELS

Since metamaterials can be synthesized by embedding arti�cially fabricated in-

clusions in a speci�ed host medium (or on a host surface), this provides the designer to

play with a large number of independent parameters (or degrees of freedom) �such as the

properties of the host materials; the size, shape, and composition of the inclusions; and the

density, arrangement, and alignment of these inclusions �to work with in order to engineer

a metamaterial with speci�c electromagnetic response function not found in its individual

constituents.

All of the above mentioned design parameters can play a key role in the �nal

outcome of the synthesis process. Among these, the geometry (or shape) of the inclusions

is one that can provide a variety of new possibilities for metamaterials processing.

The simplest physical model of metamaterial is 2-D model that can be represented

as a speci�ed host medium periodically embedded with some type of inclusions subject to

their unit cells. Throughout this work we will generally consider this type of metamaterial

19

media.

2.1 The Rod Metamaterials Structures

In the recent years, the periodic rod lattices have found many applications in both

optical and microwave frequency ranges. This is because the useful properties of metal grid

structures are quite well known: they can act as cages for electromagnetic waves, as polarized

retarding/accelerating �eld structures, as stop/pass band �lters and etc. Consideration of

such structures as metamaterial structures creates new possibilities for carrying out the

investigation of such structures for practical application [31, 52-54].

However, some fundamental problems of mathematical physics are still not resolved

for the cases of typical metallic electromagnetic crystals. One of them is the problem of

low-frequency spatial dispersion in wire media. The low-frequency spatial dispersion of a

simple wire medium (a doubly periodic regular array of parallel wires) has been studied

only recently in [49], using so called tensor-predictable approach.

The operational principle of a medium consisting of periodically arranged wires

has been known for a long time [50-51]. This kind of wire medium (also called rod medium)

is known to operate as an arti�cial dielectric with negative e¤ective permittivity at low

frequencies. Due to periodic nature of the wire medium, it can be considered as an electro-

magnetic band-gap (EBG) structure introducing frequency bands in which electromagnetic

waves cannot propagate [42]. A detailed analysis of the electromagnetic band structures of

the wire medium can be found, for example, in [43, 44]. The transmission and absorption

properties of a two-dimensional lattice of conducting cylinders have been presented in [45],

20

and the e¤ective electronic response of a system of metallic cylinders has been introduced

in [46].

The nature of wave propagation inside the wire medium and its dispersion prop-

erties have also been subjects of comprehensive studies. In [47] the wave propagation inside

the wire medium was studied in detail. Comprehensive analytical study of the dispersion

and re�ection properties of the wire medium can be found, for example, in [40, 48-50].

Authors of [49] considered a wire medium periodically loaded with bulk reactance and pre-

sented the dispersion relation for this kind of loaded wire medium. A quasi-static model for

the loaded wire medium can be found in [14].

This thesis is dedicated to the study of properties of the e¤ective electromagnetic

response from two-component metamaterial structures consisted of non-magnetic metal

cylinders of circular cross section periodically embedded in a dielectric matrix. An air

matrix is mostly considered in this work in order to prevent losses. However, as it will

be shown in this study, the above metamaterial structures possess quite low losses. The

next paragraph is related to a short review of metamaterial structures similar to the above

mentioned ones with a view point of mathematical models of the e¤ective dielectric and

magnetic constants in order to substantiate the choice of the models used throughout this

thesis.

21

2.2 The Mathematical Models of Rod Metamaterials Struc-

tures

There are two approaches to model the e¤ective parameters of metamaterials.

According to the �rst approach, which can be called as the circuit approach, some RLC-

circuit can be assigned in compliance with any metamaterial structure under consideration.

In this way, the complex e¤ective dielectric and magnetic constants are to be expressed

in terms of the e¤ective resistivity, inductance, and capacitance which can be evaluated

through some way, for example as mentioned in references [57-59]. According to the second

one, which can be called as the wave approach, the complex e¤ective dielectric and magnetic

constants can be expressed in terms of equations of mathematical physics, for example [23,

60 and 61]. Throughout this thesis, author will consider metamaterials drawing upon the

wave approach.

The problem of electromagnetic wave propagation and scattering on metal-wire

periodic media (Fig.2.1) is a longstanding problem in electromagnetics [49, 51, 55 and

56]. But the problem of electromagnetic wave propagation and scattering on metal-wire

periodic media is related to Contemporary Electromagnetics. One of the �rst successful

attempts for obtaining the e¤ective parameters of metal-wire periodic media is related to

the work [14]. The wave long approximations of the complex propagation constant and

complex dielectric constant as functions of the frequency of initial monochromatic wave

were obtained experimentally in this work, where the metal wire medium was considered

as an arti�cial dielectric.

22

Figure 2.1: Periodic array of metallic cylinders of diameter d arranged in a square arraywith lattice constant a. The cylinders are in�nitely long in the y direction

23

This work has, in fact, created a new era in characterizing arti�cial media. How-

ever, a surge of interest in characterizing metal-wire periodic media via their expressions

for the e¤ective complex parameters is related to subsequent years. This is possibly caused

by the idea of the principal possibility for designing an arti�cial medium with negative

permittivity and permeability. This possibility has presented and substantiated �rst time

by Soviet scientist Veselago [17].Last three decades can be characterized by a huge surge

of interest in characterizing metal-wire metamaterials. But research was mostly performed

on two idealized type of such structures: metamaterials with perfectly conducting rods [48,

62-64] or metamaterials with thin rods [65-68]. However, some relative successful attempts

regarding obtaining of expressions of the e¤ective parameters of metal-wire metamaterial

structures have been made. Let us shortly most interesting of them pointing out their

disadvantages.

In work [14], for purposes of simulations of the metal-wire grid considered as a

lossy arti�cial isotropic dielectric, is presented as plasma in absence of DC magnetic �eld

with the complex dielectric "eff constant and propagation constant eff respectively given

by:

"eff = "0

"1�

!2pv2 + !2

+ i!2p � v=!v2 + !2

#; (2.1)

eff = i!p�0"eff

where !p is the plasma frequency, v is the collision frequency, and "0 and �0 are the dielectric

constant and magnetic permeability of free space respectively.

As it is mentioned in the work [14], the proposed model does not take into account

24

nonlinear interaction with electromagnetic radiation and self-radiation but two more disad-

vantages of that model are too obvious: 1) any metal-wire grid is an anisotropic medium

[39, 53]; 2) the presented model is the model of non-magnetic media while the dielectric

constant of a real metal-wire medium can essentially be di¤erent from 1, [44,69,70].

The most famous and popular approach to get the expressions for the e¤ective

dielectric constants of in�nite medium as a host dielectric matrix periodically embedded

with in�nitely long metal cylinders (rods) is the Bergman-Milton theory [71,72].

It was shown by Bergman and Milton that, in the long wavelength limit, the

e¤ective dielectric function of any macroscopically homogeneous two-component system of

dielectric constants "p and "m and volume fractions F and 1 � F , respectively, can be

expressed as a sum of simple poles that depend only on the microgeometry of the composite

material and not on the dielectric functions of the components:

"eff = "r

"1� F

Xs

Bsu�ms

#(2.2)

where u is the spectral variable

u =1

1� "p="r(2.3)

are the depolarization factors, are the strengths of the corresponding normal modes, that

all add up to unity:

Xs

Bs = 1 (2.4)

The problem of de�ning values of the depolarization factors and the strengths of the corre-

sponding normal modes in (2.2-4) is not solvable analytically. So, only some approximate

25

values for ms and Bs can be obtained for some de�nite (�xed) upper limit of s. The type of

approximation depends on the value of upper limit of s. Over the years, the interaction be-

tween the particles in a composite has been considered within so called Maxwell-Garnet ap-

proximation [73]. According to that approximation, there is only one mode with a strength

di¤erent from zero, i.e.:

"eff = "r

�1� F B1

u�m1

�where B1 t 1.

The basic assumption of the Maxwell-Garnet approximation is that the average

electric �eld within a particle located in a system of identical particles is related to the av-

erage �eld in the medium outside as in the case of a single isolated particle, thus only dipole

interactions are taken into account. So, as it was mentioned above, the main disadvantage

of Maxwell-Garnet approximation is that it takes into account only dipole interactions be-

tween particles under an incident electromagnetic �eld. Moreover, it is possible to show

that Maxwell-Garnet approximation works well for the case of small volume fraction of

inclusions [71].

Some attempts to improve the Maxwell-Garnet approximation have been made.

For example, in the paper [61], the Bergman spectrum of a two-dimensional composite with

parallel cylinders embedded in the matrix was calculated and analyzed. The author also

calculated the e¤ective dielectric constant. However, the above results were exactly obtained

for the above mentioned medium placed in an in�nite capacitor. So, in the reference [61],

the partial cases of the metamaterial that is of interest to us, has been considered.

26

Figure 2.2: Unit cell of the composite with!k = kz �

!z0;�!E = Ex � �!x0;

�!H = Hy � �!y0 :

27

In this study, we utilize the results obtained in the work [74]. According to that,

the e¤ective dielectric constant of 2-D component composite medium presented in Fig. 2.1

is de�ned by:

"eff = 1�2F

(1+")(1�") + F

; (2.6)

where F = �r2=a2 is the volume fraction of rod in the unit cell (see Fig.2.2), r is the radius

of rods, a is the constant of the unit cell, " = "p="r and "p is the dielectric constant of rod

material, "r is the dielectric constant of the host medium. Taking into account that the

rods are metallic, the dielectric constant of rod material de�ned by Drude�s model as

"r (!) = 1�!2p

! (! + i )(2.7)

where !2p =ne2

m"0is the plasma frequency, is the inverse relaxation time, e is the charge

on an electron, n is the concentration of electrons, m is the mass of electron and "0 is the

permittivity of free space. Due to (2.7), the expression (2.6) takes in account a dispersive

nature of the composite medium under consideration. It means that the dielectric constant

is a function of frequency of an incident electromagnetic wave.

It is important to mention that throughout this work the author considers only

incident electromagnetic waves with the electric intensity vector parallel to the rods.

Equation (2.2) is related to the dipole approximation valid for any value of the

cylinders volume fraction. However, this formula also takes into account multipole e¤ects

for the case of composite with a very small value of the cylinders volume fraction, [74].

In order to �nd out the expression for the e¤ective magnetic constant of the com-

28

posite with non magnetic inclusions (rods), we can use the approximation similar to that

presented by (2.5), [71], for example, in the form of [74]:

�eff � �r�eff + 2�r

= F�p � �r�p + 2�r

(2.8)

where �m is the magnetic constant of host medium (matrix), �p is the magnetic constant

of cylinders. But it is logical to conclude that the approximation (2.8) possesses all of the

disadvantages inherent in the approximation (2.5).

In this study, the author uses the results obtained in the work [27]. According to

this work, the expression of frequency dependent magnetic constant of the medium shown

on the Fig.2.1 for the case of air matrix is given by:

�eff (!) = 1��r2

a2

�1 + i

2�

!r�0

��1(2.9)

where � is the resistance of the cylinder surface per unit area. The expression (2.9) is for

the case of air matrix. Let us extend Eq. 2.9 to the case of arbitrary isotropic dielectric

matrix with the dielectric constant �r. In order to do that we will use the same approach

used in the work [27]. Let us consider an in�nite square array of metal circular cylinders

immersed in a dielectric isotropic matrix with the dielectric constant �r, Fig.2.3. Here j is

a current per unit length �ow. Let us apply an external �eld H0, which we will take to be

parallel to the cylinders. The magnetic �eld inside the cylinders is given by [27]:

H = H0 + j ��r2

a2j: (2.10)

We consider very long or in�nite cylinders in order the depolarizing �eld to be uniformly

spread over the unit cell.

29

Figure 2.3: Model of the composite

30

The total electromotive force (EMF) around the circumference of a cylinder is to

be de�ned by:

EMF = ��r2�r�0@

@t

�H0 + j �

�r2

a2j

�� 2�j�: (2.11)

Taking into account that EMF must be balanced (EMF = 0) and that � = 1=2�r� where

� is the conductivity of rod material, we can write the identity for the balanced EMF in

the form:

i!�r2�r�0

�H0 + j �

�r2

a2j

�� j

r�= 0: (2.12)

The factor i! has been obtained in (2.12) from (2.11) due to the harmonic time dependence�� ei!t

�for the functions H0 and j. It means: @

@t ! �i! in the case of a monochromatic

incident wave.

Solving (2.12) with respect to j gives:

j = � H0

1� �r2

a2+ i

!�r2�r��

(2.13)

The average of the B-�eld over the entire unit cell is given by:

Bave = �r�0H0 (2.14)

The average of the H-�eld over a line lying entirely outside the cylinders is given by [27]:

Have = H0 ��r2

a2j (2.15)

So, we de�ne

�eff =Bave�0Have

(2.16)

Making simple manipulations in (2.16) gives:

�eff = �r

"1� F 1

1 + i(��r�0r

2�!)

#(2.17)

31

where F = �r2=a2.

It is important to note that the expression (2.17) takes in account the multipole

e¤ects at least for the case of composite with a very small value of the rods (cylinders) volume

fraction. The main disadvantage of the last formula is the singularity at the frequency ! = 0.

This is related to the de�nition of resistance of the cylinder surface per unit area �. At the

high frequencies, the skin layer depth is close to zero. In this case, the conductivity of rod

material � can be easily expressed in terms of the cylinder surface per unit area. Anyway

(2.10) works well in the microwave frequency range.

Thus, through out this thesis, the formulas (2.6-7) and (2.17) are used for the

characterization of 2-D metal-dielectric metamaterial media/objects as a host dielectric

matrix periodically embedded with in�nitely long metal cylinders (rods).

2.3 Elements of S- and T-Matrices Algebra

Current work is dedicated to the characterizing of slab metamaterials. The charac-

terizing is carried out by means of the study of behavior of the e¤ective dielectric and mag-

netic constants of metamaterial scatterer under consideration as functions of the frequency1

of initial electromagnetic wave that is always normally incident to the �at boundary of the

scatterer throughout this study. The above e¤ective parameters are calculated through

S-parameters of the scatterer using Nicolson-Ross approach [76-77]. These parameters are

also functions of the frequency of initial electromagnetic wave.

Let us introduce the elements of and -parameters algebra according to [78].

1Throughout this thesis, only microwave frequency range considered

32

2.3.1 Scattering parameters

Scattering parameters (S-parameters) are the properties which are used in the �eld

of electrical engineering, electronics engineering and communication system engineering.

These properties describe the electrical behavior of linear networks when undergoing various

steady state stimuli by small signals. These parameters do not use open or short circuits

conditions to characterize a linear electrical network. Quantities are measured in terms of

power.

Although S-parameters are applicable at any frequency, yet they are mostly used

for networks operating at radio frequency (RF) and microwave frequencies. S-parameters

change with the measurement frequency and are represented in matrix form and obey the

rules of matrix algebra.

An electrical network to be described by S-parameter may have any number of

parts. Parts are the points at which electrical current either enter or exit the network.

The S-parameter matrix describing an N -port network will be square of dimension N and

will therefore contain N2 elements. At the test frequency each element or S-parameter is

represented by unit less complex number, thus representing magnitude and angle. The S-

parameter magnitude may be expressed in linear or logarithmic form. In logarithmic form

magnitude has the dimensionless unit of decibels. The S-parameter angle is most frequently

expressed in degrees but occasionally in radians. Any S-parameter may be displayed on a

polar diagram by a dot for one frequency or a locus for a range of frequencies. If it applies

to one port only (being of the form ), it may be displayed on an impedance or admittance

Smith Chart normalized to the system impedance.

33

Figure 2.4: The two-port network

34

For the purposes of simplicity, let us consider the simplest case of two-port network,

Fig.2.3.The S-parameter matrix for the 2-port network is probably the most common and

it serves as the basic building block for generating the higher order matrices for larger

networks. In this case the relationship between the re�ected, incident power waves and the

S-parameter matrix is given by:

�b1b2

�=

0BB@ S11 S12

S21 S22

1CCA�a1a2�

(2.18)

Expanding the matrices in (2.11) into equations gives:

b1 = S11a1 + S12a2;

b2 = S21a1 + S22a2

9>>=>>; (2.19)

As we can see from (2.19), each of above equations gives a relationship between the re�ected

and incident power waves at each of the network ports, 1 and 2, in terms of the network�s

individual S-parameters S11, S12, S21 and S22. If one considers an incident power wave at

port 1 (a1) there may result from it waves exiting from either port 1 (b1) itself or port 2

(b2). However if, according to the de�nition of S-parameters, port 2 is terminated in a load

identical to the system impedance (Z0) then, by the maximum power transfer theorem (b2),

will be totally absorbed making (a2) equal to zero. Therefore:

S11 =b1a1

S21 =b2a1

9>>=>>; (2.20)

35

Similarly, if port 1 is terminated in the system impedance then a1 becomes zero, giving

S12 =b1a2

S22 =b2a2

9>>=>>; (2.21)

Each 2-port S-parameter has the following generic descriptions:

S11 and is the input port voltage re�ection coe¢ cient, S12 is the reverse voltage

gain, S21 is the forward voltage gain, S22 is the output port voltage re�ection coe¢ cient.

2.3.2 Transfer parameters

The Scattering Transfer parameters or S-parameters of a 2-port network are ex-

pressed by the T-parameter matrix and are closely related to the corresponding S-parameter

matrix. The T-parameter matrix is related to the incident and re�ected normalized waves

at each of the ports as follows:

�b1a1

�=

0BB@ T11 T12

T21 T22

1CCA�a2b2�

(2.22)

The advantage of T-parameters compared to S-parameters is that they may be used to read-

ily determine the e¤ect of cascading two or more than 2-port networks by simply multiplying

the associated individual T-parameter matrices. If the T-parameters of say three di¤erent

2-port networks 1, 2 and 3 are , T1; T2; and T3 and respectively then the T-parameter

matrix for the cascade of all three networks (TT ) in serial order is given by:

(TT ) = (T1) � (T2) � (T3) (2.23)

36

As with S-parameters, T-parameters are complex and there is a direct conversion between

the two types. Although the cascaded T-parameters is a simple matrix multiplication of the

individual T-parameter, the conversion for each network�s S-parameters to the correspond-

ing T-parameters and the conversion of the cascaded T-parameters back to the equivalent

cascaded S-parameters, which are usually required, is not trivial. However once the oper-

ation is completed, the complex full wave interaction between all ports in both directions

will be taken into account. The following equations will provide conversion between - and

- parameters for -port network. From S- to T-parameters we have:

T11 = �det(S)S21

;

T12 =S11S21;

T21 = �S22S21;

T22 =1S21:

9>>>>>>>>>>=>>>>>>>>>>;(2.24)

From T- to S- parameters we have:

S11 =T12T22;

S12 =det(T )T22

;

S21 =1T22;

S22 =�T21T22

;

9>>>>>>>>>>=>>>>>>>>>>;(2.25)

where det(S) indicates the determinant of the matrix S.

The presented de�nitions of S- and T- parameters for 2-port network will be very

useful for characterizing and analysis of the optical and transport properties of slab meta-

material structures on microwave in the next chapters. But the representation of S- and T-

parameters will be normally used in terms of S- and T- schemes designed in [79]. Let us

shortly present these schemes.

37

2.4 S- and T- Schemes

2.4.1 T-scheme

Transfer matrices relate the forward and backward modes on the far side of a

structure to the forward and backward modes on the near side of that structure (Fig. 2.4):2664 FII

BII

3775 = T2664 FI

BI

3775 : (2.26)

The attractiveness of transverse matrices lies in the fact that they are easy to cascade:

the T-matrix of a structure composed from substructures is simply the product of the

T-matrices of each substructures.

To determine the transfer matrices of an interface from its re�ection and trans-

mission matrix, we start from (see Fig.2.4 for the meaning of the subscripts):

F2 = T1;2 � F1 +R2;1 �B2 (2.27)

B1 = R1;2 � F1 + T2;1 �B2 (2.28)

These equations express that e.g. the forward �eld in medium 2 is the superposition of the

transmitted forward �eld from medium 1 and the re�ected backward �eld from medium

2. Rearranging eq. 2.27 and 2.28 into the form of eq. 2.26, yields after some matrix

manipulations the matrix I12:2664 F2

B2

3775 =2664 T1;2 � R21R12

T21R21T21

�R12T21

1T21

37752664 F1

B1

3775 : (2.29)

38

Figure 2.5: For T-scheme

39

The transfer matrix P23 of z�invariant layer is easy to write down using indepen-

dent of the amplitude of electromagnetic �eld component of z�coordinate (for example, for

the electric intensity vector, we write:��!E (�!r ) =

Pk

Ak�!E (x; y) e�i�kz

�:

2664 F3

B3

3775 =2664 diag

�e�j�id23

�0

0 diag�ej�id23

�37752664 F2

B2

3775 (2.30)

where the submatrices are diagonal matrices containing the propagation factors of each

eigenmodes and where d23 is the thickness of that layer. After calculating the I and P

matrices for the interfaces and layers, the transfer matrix for the whole stack is

T = In�1;n ::::: P45 � I34 � P23 � I12: (2.31)

With the knowledge of the transfer matrix of the stack, we can calculate the transmission

and re�ection matrices from eq. 2.27 and 2.28. First we write the transfer matrix in block

form:

T =

2664 A B

C D

3775 (2.32)

R12 is then simply �D�1 � C, the other matrices follow in the similar way. The big dis-

advantage of the T-scheme is that for evanescent modes Eq. 2.30 contains a mixture of

increasing and decreasing exponentials, which is very detrimental for numerical stability.

Indeed, during the calculation very small numbers will be added to very large numbers,

which will obviously result in a loss of precision. In the next section, we will discuss a

di¤erent scheme, where this problem will not occur.

40

2.4.2 S- scheme

The scattering matrix relates the outward-propagating �elds to the inward prop-

agating �elds. From eq. 2.27 and 2.28, we know that for an interface this is (see Fig. 2.5

for the slightly modi�ed sub-scripts)2664 F1+B1

3775 =2664 T1;1+ R1+;1

R1;1+ T1+;1

37752664 F1

B1+

3775 (2.33)

In the S-scheme, we divide the stack into chunks consisting of an interface and propagating

over a given distance in the exit medium of the interface. For e.g. the �rst chunk, we write

the scattering matrix as 2664 F2

B1

3775 =2664 t1;2 r2;1

r1;2 t2;1

37752664 F1

B2

3775 : (2.34)

The chunk matrices r and t follow easily by combining eq. 2.33 and 2.30:

t1;2 = diag�e�j�id12

�� T1;1+;

r2;1 = diag�e�j�id12

��R1+;1 � diag

�e�j�id12

�;

r1;2 = R1;1+;

t2;1 = T1+;1 � diag�e�j�id12

�:

9>>>>>>>>>>=>>>>>>>>>>;(2.35)

41

Figure 2.6: For S-scheme

42

We can write such expression for any chunk in the stack. Note that only expo-

nentials with the same sign occur in these equations, which makes them numerically more

stable than the T-scheme.

Now we only need to determine how to calculate the scattering matrix of the entire

stack, given the scattering matrices of the individual chunks. More speci�cally, we will do

this by calculating S for the �rst p chunks. Given that S for the �rst (p� 1) chunks is

already known. So, the following quantities are given:2664 Fp

B1

3775 =2664 T1;p Rp;1

R1;p Tp;1

37752664 F1

Bp

3775 (2.36)

We want to calculate the matrices2664 Fp+1B1

3775 =2664 T1;p+1 Rp+1;1

R1;p+1 Tp+1;1

37752664 F1

Bp+1

3775 (2.37)

We do this by combining eq. 2.28 with the expressions for the chunk matrices in chunk p.

After some straightforward but rather tedious algebra, we eventually arrive at (I is the unit

matrix)

T1;p+1 = tp;p+1:(I �Rp;1 � rp;p+1)�1 � T1;p

Rp+1;1 = tp;p+1:(I �Rp;1 � rp;p+1)�1 �Rp;1 � tp+1;p + rp+1;p

R1;p+1 = Tp;1:(I � rp;p+1 �Rp;1)�1 � rp;p+1 � T1;p +R1;p

Tp+1;1 = Tp;1:(I � rp;p+1 �Rp;1)�1 � tp+1;p:

9>>>>>>>>>>=>>>>>>>>>>;(2.38)

Using these formulas, we can calculate the scattering matrix of the entire stack, starting

from the �rs chunk (eq. 2.33) and working our way through the rest of the chunks. We

already mention that only decreasing exponentials occur in these formulas. More rigorous

studies show that the numerical stability of the S-scheme is indeed for better than that of

43

the T-scheme. However, this stability comes at a price, because more matrix multiplications

have to be performed in the S-scheme.

44

Chapter 3

THE EVALUATION OF

EFFECTIVE PARAMETERS OF

SLAB METAMATERIALS

In this chapter, the characterization of slab metamaterial structures as slab arti-

�cial dielectric scatterers is proposed to be carried out by means of the study of dispersive

behavior of the e¤ective complex dielectric and magnetic constants of the scatterers in ap-

propriate (microwave) frequency range. The above characterization is the key for de�ning

unusual material properties (see Chapter 1) unavailable in the nature. The method to eval-

uate the complex e¤ective parameters of slab metamaterial objects is the main subject of

the current chapter. As well as the chapter is dedicated to the evaluation of e¤ective electric

and magnetic properties of slab metamaterial structures in GHz frequencies through the

use of the above evaluation of parameters as designed in the next paragraph.

45

3.1 An Improved Broadband Method for the Evaluation of

E¤ective Parameters of Slab Metamaterials

3.1.1 Introduction

Retrieval techniques are nondestructive and contactless methods, to evaluate the

e¤ective dispersive parameters of an inhomogeneous material. They have a big practical

interest due to fast development of metamaterials in the last decade. Majority of these

techniques are modi�ed retrieval of so-called Nicolson-Ross method [76-77], very well devel-

oped for parameter characterizations of materials at microwave frequencies. Development

of such techniques is especially important for microwave frequency range because the desir-

able properties of conventional materials are seriously degraded for frequency above 1 GHz

[80]. Such techniques are, as usually, based on E¤ective Medium Theory approach [9, 80].

Knowledge of microwave dispersive parameters of metamaterials is very important in order

to design a variety of microwave circuits such as circulators, phase shifters and �lters.

Retrieval parameter techniques give a possibility to evaluate e¤ective dielectric

constant "eff and magnetic constant �eff of metamaterials implying that e¤ective refrac-

tive index should be evaluated as p"eff�eff but the techniques do not mention how to

de�ne correct branch of square root with mathematical viewpoint. Moreover the calcula-

tion procedure of techniques requires for choosing the complex function branch of square

root inclusive function of S-parameters like

�(S11; S21) = K(S11; S21)�pK2(S11; S21)� 1:

The procedural choice implies that j�(S11; S21)j < 1: Sometimes this way to choose the

46

complex branch gives mistaken result at some test frequencies since both possible branches

satisfy the above conditions. In this paper, we propose a method that does not require

making a choice of square root branch of complex functions of evaluated S-parameters in

order to evaluate any of the refractive index, permittivity or permeability. Instead we obtain

a formula for evaluation of complex refractive index which is simple.

It is well known that Nicolson-Ross like techniques have been designed for macro-

scopically homogenous materials but an evaluation of e¤ective parameters of metamaterials

sometimes requires introducing the boundaries of e¤ective �at material. This problem is

being discussed in this paragraph.

Assessment and precision of the presented method is made using a comparison of

the simulation results obtained by our method and the Nicolson-Ross method.

3.1.2 Evaluation Method

Consider normal incidence of an electromagnetic wave on a slab. We consider an

inhomogeneous slab of thickness which is inhomogeneous in the direction of the incident

wave. We can, in principle, replace this inhomogeneous slab with a homogeneous slab of the

same thickness. This approach is called �the thin sample principle�i.e. we imply that the

thickness of the metamaterial sample is much less then the other dimensions of the same.

Regardless the �eld produced by the induced currents is not uniform, we can consider a

�eld beyond the slab as a plane wave like �eld for the incident wave. This approach is

quite acceptable for frequencies up to 10 GHz [40, 44, 76-77, 81-83] and is in the domain

of the E¤ective Medium Theory valid for the case of a slab like inhomogeneous samples.

S-parameters of an inhomogeneous slab can be described closely equal to [76, 84]:

47

S11 =

�1�e�2ik0neff (!)d

�R(!)

1�R2(!)e�2ik0neff (!)d

S21 =(1�R2(!))e�ik0neff (!)d

1�R2(!)e�2ik0neff (!)d

9>>=>>; (3.1)

where ! is a frequency of the incident electromagnetic wave, R(!) is the interface re�ection

coe¢ cient of the slab, k0 is the wavenumber of the incident wave outside the slab, neff is

e¤ective refractive index of the slab under investigation.

Leaving out the term e�2ik0neff (!)d; expression (1) can be rewritten as:

�i!cneff (!)d = ln

�S21(!)

1� S11(!)R(!)

�(3.2)

where c� the velocity of light in vacuum.

Since an argument of logarithm in equation (2) is complex there are multiple values

for neff (!). If we de�ne the argument of logarithm function on the right side of (2) as:

F (!) =S21(!)

1� S11(!)R(!)

= jF (!)j ei�(!); (3.3)

then neff (!) is given by:

neff (!) = �c

!dfi [�(!) + 2�n]� ln(jF (!)j)g (3.4)

where n = 0;�1;�2; :::.

We consider the wavelength range �m > d where �m is the wavelength in sample

material. In this case, n = 0; [77].

48

Within the region of sample there is an e¤ective characteristic impedance Zeff (!) =q�eff (!)="eff (!)Z0;where Z0 is the characteristic impedance of the air. Re�ection coef-

�cients for normal incidence of wave on the interface from air-�lled outer can be given

by:

R?(!) = �Rk(!) =

q�eff (!)="eff (!)� 1q�eff (!)="eff (!) + 1

(3.5)

where R?(!) is the Fresnel coe¢ cient of vertical polarization and Rk(!) is the Fresnel

Re�ection coe¢ cient of parallel polarization for the case of normal incident. Using sim-

ple manipulation we can obtain formulas to evaluate the e¤ective constant for both the

polarizations

"eff (!) =1�R?(!)1+R?(!)

neff (!)

�eff (!) =1+R?(!)1�R?(!)neff (!)

9>>=>>; (3.6)

"eff (!) =1+Rq(!)1�Rq(!)neff (!)

�eff (!) =1�Rq(!)1+Rq(!)

neff (!)

9>>=>>; (3.7)

As we see from formulas (3:4) and (3:6� 7) in the case of a symmetrical metamaterial

slab, we do not have to care about the structure of metamaterial. We care only about the

interface re�ection coe¢ cient of metamaterial slab and S-parameters obtained theoretically

or by means of experimental measurements. It is due to this fact that an impedance of

homogenous slab does not depend on its thickness.

49

Figure 3.1: Change of real part of e¤ective refractive index of metamaterial versus thefrequence.

50

Figure 3.2: Change of imaginary part of e¤ective refractive index of metamaterial versusthe frequence.

51

Figure 3.3: Change of real part of e¤ective dielectric constant of metamaterial versus thefrequence.

52

Expressions (3:4) and (3:6� 7) are very convenient for evaluating the e¤ective

parameters for the case of homogenous dielectric spacing between the inhomogeneous ma-

terial in�ll and �at boundaries of sample since the interface re�ection coe¢ cient is de�ned

by a simple formula. The result of calculations for a �at metamaterial slab with a unit cell

2 � 2 � 2 mm with the rod radius 0.2 mm as shown in Fig:2:2, is plotted in Figs:3:1 � 6.

Extension of the unit cell has been done in x� and y � axes. Unit cell is presented by

symmetrically spaced copper wires and some hypothetical material matrix ("r = 1; �r = 6).

Calculation of S-parameters has been performed by FDTD method for the case of vertical

polarization of normally incident monochromatic wave.

The plots for complex e¤ective refractive index evaluated by both Nicolson-Ross

approach and proposed method have been shown in Fig.3.1-2 respectively while the plots

for complex e¤ective dielectric and magnetic constants evaluated by both Nicolson-Ross

approach and proposed method have been shown in Fig.3.3-6 respectively. Fig.3.5 and

Fig.3.6 contains the plots of complex e¤ective magnetic constants evaluated using analytical

approximation (2.10) where ("r = 1; �r = 6).

Fig.3.1-6 show a remarkable coincidence between the calculated results on the basis

of Nicolson-Ross approach, proposed method and the theoretical approximation. Numerous

numerical experiments for a di¤erent kind of �at metamaterial structures have shown a

remarkable coincidence between the results obtained by Nicolson-Ross approach and the

proposed method.

It is important to note another advantage of presented method: it does not require,

in principle, determining of the �rst e¤ective boundary as it has been done in [85].

53

Figure 3.4: Change of imaginary part of e¤ective dielectric constant of metamaterial versusthe frequence.

54

Figure 3.5: Change of real part of e¤ective magnetic constant of metamaterial versus thefrequence.

55

Figure 3.6: Change of imaginary part of e¤ective magnetic constant of metamaterials versusthe frequency.

56

The plots for complex e¤ective refractive index evaluated by both Nicolson-Ross

approach and proposed method have been shown in Fig.3.1-2 respectively while the plots

for complex e¤ective dielectric and magnetic constants evaluated by both Nicolson-Ross

approach and proposed method have been shown in Fig.3.3-6 respectively. Fig.3.5 and

Fig.3.6 contains the plots of complex e¤ective magnetic constants evaluated using analytical

approximation (2.10) where , .

Fig.3.1-6 show a remarkable coincidence between the calculated results on the basis

of Nicolson-Ross approach, proposed method and the theoretical approximation. Numerous

numerical experiments for a di¤erent kind of �at metamaterial structures have shown a

remarkable coincidence between the results obtained by Nicolson-Ross approach and the

proposed method.

It is important to note another advantage of presented method: it does not require,

in principle, determining of the �rst e¤ective boundary as it has been done in [85].

3.1.3 Precision and Sensitivity Discussion

Proposed method like any other existing methods, should be used under some

restrictions. First of all clear choice of the complex function branch in (4) should be made

if �m > d. It can be resolved in checking the phase of evaluated dielectric constant or

magnetic constant for any concrete metamaterial structure. Secondly, the method is under

the restrictions of E¤ective Medium Theory. It means that a slab metamaterial must

have unit cell dimensions much smaller than the wavelength of interest (�m > d). This

restriction bounds the observed frequencies for the use of proposed method relative to

the concrete type of slab metamaterial. If the frequency restriction is imposed by the

57

above mentioned requirement then the precision is quite acceptable, for instance, calculation

results for metamaterials with di¤erent numbers of unit cells match well with what sometime

is not observed using the other methods [85].

The use of proposed method for a slab metamaterial with elements embedded in

the matrix that touch with the �at boundaries of the slab provides lower precision than for

the case of slab with elements embedded in the matrix not touching with the �at boundaries.

It is easier to satisfy the second case for the condition: �m > d:

The very important matter is the sensitivity of the proposed method regarding an

error in the calculation/measurement of S-parameters of the slab. In the frequency region,

where the transmission is small, especially for thicker metamaterials, as we can see from

(4), an error of calculation/measurement of S21 can produce an essential error of calculation

of e¤ective parameters. It can be seen more clearly from the formula for di¤erential of the

neff (!) as a function of S11- and S21-parameters:

dneff (S11; S21) =ic

!d

�1

S21dS21 �

R

1�RS11dS11

�(3.8)

This type of error can create �extra�peaks in the frequency dependency of e¤ective parame-

ters which can be mistakenly treated as peaks due to a plasma frequency of metamaterial.

We can also see from Eqs. (3:4) and (3:8) that an error in S11�parameter does

not essentially introduce an error in the calculation except for the case when RS11 � 1 but

practically realizability of this negligibly zero.

In the next paragraph, analytical modeling and numerical simulation of the com-

plex e¤ective dielectric, magnetic constants and refractive index of a metallic rod metama-

terial in microwave frequency range is presented. This modeling will enables us to �nd out

58

the general properties of slab metamaterial structures that can be virtually separated into

a set of same subslabs having dimensions equal to the unit cell dimension. It is important

to go inside a metamaterial to the level of �the e¤ective atom�.

3.2 Evaluation of Layer Properties of E¤ective Parameters

of Metallic Rod Metamaterials in GHz Frequencies

There is a lot of work devoted to analytical evaluation of the complex e¤ective di-

electric and magnetic constants under the E¤ective Medium Theory [52, 86-87]. However,

mostly, those works are devoted to thin rods of hypothetical perfect conducting material.

Modifying results for complex expressions of the dielectric constants obtained by di¤erent

authors, we avoid the above-mentioned disadvantages by: 1) Considering the metamate-

rial structure for di¤erent volume fractions of the rods including cases of thick ones; 2)

Considering a real metal (copper) rod metamaterial structure.

Let us focus our attention on a 2-D rod metamaterial to be de�ned as follows: the

periodic grid of in�nite circular cylinders made from real metal (Copper) with the same

period d in two dimensions embedded in a host medium, which we will call as the �matrix�

throughout the chapter. For convenience, we take dielectric constant "r and magnetic

constant �r of the matrix to be unity.

Let the vector�!k to be wave vector of incident monochromatic plane wave prop-

agating in the direction perpendicular to the rod axes. In addition, the electric intensity

vector�!E and magnetic intensity vector

�!H are parallel and perpendicular to the rod axes

respectively. Then the dielectric constant of the metamaterial medium is to be described

59

by the Eq. 2.10.

Considering Cu under the room temperature gives

"r (!) = 1�i�

!"0(3.9)

Formulas (2:6);�(2:10) and (3:9) can be used to analyze optical properties of the considered

metamaterial medium de�ned by its refractive index neff (!) =q"eff (!)�eff (!), and

absorption properties of the medium is de�ned by imaginary parts of both "eff (!) and

�eff (!). However, this is not the matter of our primary interest. The target of the work

is to �nd the relation between the in�nite medium model and the model of �nite thickness

because any real scatterer has �nite dimensions. For this reason, we pick out one layer of

the in�nite composite medium, which can be considered as 2-D grid of cylinders of radius

r embedded in magnitodielectric matrix of thickness d = a with dielectric constant "r and

magnetic constant �r, we name this layer as �the basic layer�. If we extend the basic

layer in both remaining directions then we will get our periodically in�nite medium. The

question is, how to extend the basic layer in order to get a remarkable coincidence between

the e¤ective constants of extended �nite medium and the e¤ective constants of our in�nite

medium for microwave frequency range from a point of view of e¤ective medium theory

(EMT)? In order to answer this question, we performed FDTD simulations of S-parameters

of the above-mentioned �nite layer-structure with unit cell shown in Fig.2.2. Nicolson-Ross

approach has been used for evaluating e¤ective constants from S-parameters.

In order to be sure in the authenticity of our further conclusions, we have performed

all of numerical calculations and analytical modeling for the in�nite metal for di¤erent values

of volume fraction F of the rods.

60

Figure 3.7: The Basic Layer in 2-D Image

61

Table 3.1: Value of the real part of the e¤ective Dielectric Constant at 1 GHz

F 1layer 3layer 5layer 7layers 14layers 24layers Infinite

0:03 1:0873 1:07935 1:0752 1:0721 1:0711 1:0720 1:0649

0:10 1:2487 1:23150 1:2340 1:2300 1:2266 1:2230 1:2129

0:20 1:5903 1:53970 1:5301 1:5240 1:5100 1:5070 1:4886

0:30 1:9149 1:90680 1:8997 1:8996 1:8906 1:8713 1:8649

0:44 2:8492 2:83630 2:7654 2:7341 2:7153 2:6950 2:5829

0:53 3:8179 3:65110 3:6362 3:6046 3:4879 3:4816 3:2882

0:64 5:7374 5:64450 5:6333 5:5087 5:2702 5:2678 4:4971

Table 3.2: Value of the real part of the e¤ective Magnetic Constant at 1 GHz

F 1layer 3layer 5layer 7layers 14layers 24layers Infinite

0:03 0:98660 0:97991 0:97880 0:98259 0:97831 0:96804 0:96875

0:10 0:94829 0:93461 0:93437 0:92615 0:92508 0:92477 0:90384

0:20 0:88562 0:85550 0:85140 0:84389 0:83876 0:83505 0:80368

0:30 0:77101 0:75176 0:74823 0:74515 0:73528 0:73098 0:69811

0:44 0:62663 0:62469 0:62025 0:60720 0:60711 0:59889 0:55823

0:53 0:57781 0:55449 0:54766 0:54525 0:52917 0:52800 0:47191

0:64 0:48341 0:43818 0:43653 0:43576 0:42811 0:41470 0:36384

62

Results of such calculations and modeling are shown in Table 3.1 for real part of

e¤ective dielectric constant and in Table 3.2 for real part of magnetic constant. The last

column of these tables is corresponding to the result of analytical modeling and the rest of

the columns are for the numerical simulations for di¤erent numbers of the basic layer.

As we can see from the Tables 3.1-2, roughly, the result of simulation for any

number of layers are very closely related to the result of analytical modeling for the in�nite

metamaterial medium. This generalized result can be explained by the fact that wavelength

of the initial wave in the metamaterial medium in the GHz frequency range is much smaller

than the dimension of the heterogeneity (radius of the rods in this case). Therefore, hy-

pothesis of the applicability of EMT for considered radii of rods is real for GHz frequencies.

However, to be exact, we have to mention that the best coincidence between the results of

analytical modeling and numerical simulation starts from 15th basic layer.

63

Figure 3.8: Change of real part of e¤ective refractive index of metamaterial versus metalvolume fraction F at 1 GHz.

64

Let us observe �the layer evolution�of the e¤ective parameters of the metamaterial

in Fig.3.8 where the frequency dependence of real part of the e¤ective refractive index of the

metamaterial has been plotted for the considered unit cell. Thus, the expected �e¤ective

parameters saturation�occurs from 15th basic layer. This number of the layers, we will call

as the �saturation number�N .

We also conclude that the saturation number logically, is a function of parameters

of the basic layer and frequency of initial wave i.e.

N = N ("m; �m; �; !; d; r) : (3.10)

Moreover, numerical simulation has shown that the dependence of N on parameters "r; �r

and �; is weaker whereas the dependence on parameters !; d and r is stronger.

In this study, we have not considered the imaginary parts of the e¤ective constants.

It is because both analytical modeling and numerical simulation have shown that values of

the above-mentioned parameters are not increasing in the order of 10�4 excluding the case

when the rods touch each other. Last case is not of practical interest because of no wave

propagation in the material. Small values of imaginary part of the e¤ective constants

indicate low losses of considered metamaterial structure that, in turn, indicates low level of

damping of electromagnetic waves in the material. FDTD simulations of S-parameters also

con�rm this fact.

The obtained results provide us with theoretical con�rmation of the enhancement

of e¤ective dielectric constant (its real part) takes place in metal rod metamaterial structures

similar to the considered ones. Indeed, Table 3.1 reveals the enhancement of dielectric

constant for considered metamaterial due to the spatial dispersion [39]. It is interesting to

65

mention that the enhancement of e¤ective dielectric constant has been found in the same

kind of metamaterials but with di¤erent orientation of the electric �eld vector relative to

the cylinder axes [41]. It has also been found that for considered orientations of the electric

�eld and cylinder axes, that in case of cylinders, touching each other, the e¤ective dielectric

constant increases exponentially. We do not show this result due to strong quantitative

divergence between the results obtained by numerical simulation and analytical modeling

regardless of good qualitative coincidence between them.

With a viewpoint of possible practical applications, observed enhancement of the

properties the considered metamaterial can be used for designing highly directive patch

antennas. Indeed, as it has been shown in [31], a patch antenna with the metamaterial

under consideration placed at the top of the antenna increases the directivity due to an

ultra low refractive index (ULI) of the system �patch antenna-metasubstrate�. In turn, the

phenomenon of ULI may have a number of other practical applications [36].

3.3 Conclusion

In this chapter, a simple way has been described to calculate the complex e¤ective

dielectric and magnetic constants of 2-D slab metamaterial structures without any ambiguity

inherent in the existing similar methods due to the necessity for choosing branches of the

square root of complex functions. The method has been designed irrespective to the shape

of unit cell.

In this chapter, the analysis has also been carried out of the e¤ective electromag-

netic properties of 2-D array of copper cylinders embedded in metal-dielectric matrix in the

66

GHz frequencies that has shown existence of the enhancement of e¤ective dielectric constant

and low absorption in the microwave frequencies. The considered analytical models of the

composite gives a good qualitative but a weak quantitative correlation with results of nu-

merical simulation in case, if cylinders touch each other. The considered arti�cial material

can be used to increase the directivity of patch antenna and to obtain ULI structures in the

GHz frequency range.

67

Chapter 4

LONG WAVE LAYER-SPECIFIC

REPRESENTATION OF THE

OPTICAL PROPERTIES OF

SLAB METAMATERIALS

During the last decade, a number of approaches based on the E¤ective Medium

Theory [10, 63, 75, 89-90] were developed for evaluating the complex e¤ective dielectric and

magnetic constants of the metamaterial media. These techniques were based on calculating

the e¤ective parameters of unbounded media without taking into account the di¤raction

from the boundaries.

Certain e¤orts to take in account �nite dimensions of metamaterials where per-

formed, for example, in [61,91]. In [61], the author addresses to the problem of determining

68

the quasi-static dielectric response of a two-component composite consisting of either a �-

nite or an in�nite chain of cylinders of one material embedded in a matrix made of a second

material. The results are limited to certain spectral properties of the response. Thus, re-

sults can be considered as a next step in the way of characterizing metamaterials on �the

e¤ective interatomic�level.

In reference [91], the Bergman spectrum and e¤ective dielectric constant of a two-

component composite with dielectric cylinders immersed, parallel, in a dielectric medium,

with the whole system placed in an in�nite parallel-plate capacitor. The results do not

allow to characterizing the material with the dimensions of the power of the unit cell of

metamaterial presented as an arti�cial crystal.

In this chapter, utilizing the separation of the arti�cial material onto its elementary

slabs [66], we represent a 2-D slab metamaterial as a set of in�nitely long elementary layers

(slabs) whose thickness is equal to the dimension of the unit cell of the metamaterial. Using

the E¤ective Medium Theory and the of S-parameters de�nition, we produce matrix equa-

tions that include the complex e¤ective dielectric and magnetic constants of the elementary

slab and the complex e¤ective dielectric and magnetic constants of the bulk metamaterial

slab. We calculate the complex e¤ective dielectric and magnetic constants of the elemen-

tary slab utilizing the optimization procedure. The obtained results are benchmarked using

standard FDTD.

69

4.1 Cascaded Network Theory

Consider the in�nite slab as a cascaded network of sub-slabs of thickness di (i =

1; 2; :::) where i is the number of the sub-slab in the network. It is always assumed that

the object is located in the free air. Geometry of the problem is given in Fig.4.1 where

F and B are �eld functions of electromagnetic �eld outside of the object. The incident

electromagnetic �eld contains only the component perpendicular to the interface of the

object and the normal incident is considered. Here F and B are the scalar components of

the electric intensity vector.According to [79], the matrix expression for the �eld functions

is de�ned by Eq. 2.19

For the �rst sub-slab of the obstacle, the matrix expression for and near the left

boundary is expressed in the terms of S-scheme expressed by Eq. 2.26 as follows:

2664 F1+B1

3775 =2664 T1;1+ R1+;1

R1;1+ T1+;1

37752664 F1

B1+

3775 (4.1)

where T1;1+ = 1+R1; R1+;1 = �R1; R1;1+ = R1; T1+;1 = 1 � R1, and R1 is the Fresnel

re�ection coe¢ cients on the interface of left boundary of the �rst sub-slab.

Thus, equation (4.1) we can rewrite as follows:2664 F1+

B1+

3775 =2664 T1;1+ �R1;1+T�11+;1R1+;1 R1+;1T

�11+;1

�R1;1+T�11+;1 T�11+;1

37752664 F1

B1

3775 (4.2)

70

Figure 4.1: Representation of a slab obstacle as a cascaded network of sub-slabs

71

The �eld functions at the end of the �rst sub-slab can be evaluated using T-scheme

[79]: 2664 F2

B2

3775 =

2664 e�i d 0

0 ei d

3775 �2664 F1+

B1+

3775

=

2664 e�i idi�T1;1+ �

R1+;1R1;1+

T1+;1

�e�i idi

R1+;1T1+;1

�ei idi R1;1+T1+;1ei idi 1

T1+;1

3775 �2664 F1

B1

3775 ; (4.3)

where i = ni!=c = k0ni is the complex propagation constant of the i-th sub-slab, ni =

p"i�i is the complex refractive index of the i-th sub-slab, "i and �i are the complex dielectric

and magnetic constants of the i-th sub-slab, respectively and c is the velocity of light in

vacuum.

Further implementation of the T-scheme yields [79]:2664 F2+

B2+

3775 =

2664 T2;2+ �R2+;2R2;2+

T2+;2

R2+;2T2+;2

�R2;2+

T2+;2

1T2+;2

3775 �2664 F2

B2

3775

=

2664 T2;2+ �R2+;2R2;2+

T2+;2

R2+;2T2+;2

�R2;2+

T2+;2

1T2+;2

37752664 e�i d

�T1;1+ �

R1+;1R1;1+

T1+;1

�e�i d

R1+;1T1+;1

�ei d R1;1+T1+;1ei d 1

T1+;1

3775 �2664 F1

B1

3775

=

2664 a11 a12

a21 a22

3775 �2664 F1

B1

3775 (4.4)

where

72

a11 = (e�i dT2;2+T2+;2T1;1+T1+;1 � e�i dT2;2+T2+;2R1;1+R1+;1

�e�i dR2;2+R2+;2T1;1+T1+;1 + e�i dR2;2+R2+;2R1;1+R1+;1

�R2+;2R1;1+ei d)=(T2+;2T1+;1)

a12 =e�i dT2;2+T2+;2R1+;1 � e�i dR2;2+R2+;2R1+;1 + ei dR2+;2

T2+;2T1+;1

a21 = �e�i dR2;2+T1+;1T1;1+ � e�i dR2;2+R1+;1R1;1+ + ei dR1;1+

T2+;2T1+;1

a22 =�e�i dR2;2+R1+;1 + ei d

T2+;2T1+;1:

Recalling S-scheme yields [79]:2664 F2+B1

3775 =

2664 a11 � a12a21a22

a12a22

�a21a22

1a22

3775 �2664 F1

B2+

3775

=

2664 S11 S12

S21 S22

37752664 F1

B2+

3775 ; (4.5)

where

S11 = � T2;2+T1;1+e�i 1d1R2;2+R1+;1 � ei 1d1

(4.6)

S21 = �e�i 1d1R2;2+T1+;1T1;1+ � e�i 1d1R2;2+R1+;1R1;1+ + ei 1d1R1;1+

e�i 1d1R2;2+R1+;1 � ei 1d1(4.7)

For the case of a one sub-slab object, R1;1+ = R =n�"n+" ; R2;2+ = �R; R1+;1 = �R;R2+;2 =

R, the expressions (7) and (8) can be presented [76,77] as follows:

S11 =1�e�2ik0nd1�R2e�2ik0ndR;

S21 =1�R2

1�R2e�2ik0nd e�ik0nd;

9>>=>>; (4.8)

73

where R is the Fresnel re�ection coe¢ cient on the interface of left boundary of the obstacle.

The scattering matrix Sn (S-matrix) for the �rst n sub-slabs is, de�ned by the

following matrix equation, [79]:

2664 B1

F(n+1)+

3775 = Sn �

2664 F1

B(n+1)+

3775 (4.9)

=

2664 S11 S12

S21 S22

3775 �2664 F1

B(n+1)+

3775 ;where B1 = BI ; F1 = FI ; B(n+1)+ = BII ; F(n+1)+ = FII according to Fig.4.1.

Assuming that the matrix equation for the �rst (n � 1) sub-slabs is similar to

equation (4.9), the induction gives:

2664 Fn

B1

3775 =2664 T1;n Rn;1

R1;n Tn;1

3775 �2664 F1

Bn

3775 (4.10)

where the coe¢ cients for n = 2 are known [19]:

T1;2 = e�i 1d1T1;1+ ; R2;1 = e

�2i 1d1R1+;1; T1;2 = R1;1+ ; T2;1 = e�i 1d1T1+;1.

Equation (4.10) can be rewritten as:

2664 Fn

B1

3775 =2664 T1;n �R1;nT�1n;1Rn;1 Rn;1T

�1n;1

�T�1n;1R1;n T�1n;1

3775 �2664 F1

Bn

3775 (4.11)

The expression for the �eld functions for the n-th sub-slab is similar to (4.1) gives:

2664 Fn+Bn

3775 =2664 Tn;n+ Rn+;n

Rn;n+ Tn+;n

37752664 Fn

Bn+

3775 (4.12)

74

where

Rn;n+ =

8>><>>:Rn if n is odd

�Rn if n is even(4.13.1)

Rn+;n =

8>><>>:�Rn if n is odd

Rn if n is even

(4.13.2)

Tn;n+ =

8>><>>:1�Rn if n is odd

1 +Rn if n is even

(4.14.1)

Tn+;n =

8>><>>:1 +Rn if n is odd

1�Rn if n is even(4.14.2)

Finally, equation (4.l2) can be presented in the following form:2664 Fn+

Bn+

3775 =2664 Tn;n+ �Rn+;nT�1n+;nRn;n+ Rn+;nT

�1n+;n

�Rn;n+T�1n+;n T�1n+;n

3775 �2664 Fn

Bn

3775 (4.15)

Substituting (4.11) into (4.15) produces the �eld functions near the left boundary of the

n-th sub-slab: 2664 Fn+

Bn+

3775 =

2664 Tn;n+ �Rn+;nT�1n+;nRn;n+ Rn+;nT�1n+;n

�Rn;n+T�1n+;n T�1n+;n

3775

�

2664 T1;n �Rn;1R1;nT�1n;1 Rn;1T�1n;1

�R1;nT�1n;1 T�1n;1

3775 �2664 F1

B1

3775

=

2664 b11 b12

b21 b22

3775 �2664 F1

B1

3775 ; (4.16)

75

where

b11 =

Tn;n+Tn+;nT1;nTn;1 � Tn;n+Tn+;nRn;1R1;n �Rn+;nRn;n+T1;nTn;1

+Rn+;nRn;n+Rn;1R1;n �Rn+;nR1;nTn+;nTn;1

b21 = �Rn;n+T1;nTn;1 �Rn;n+Rn;1R1;n +R1;nTn+;nTn;1

b12 =Rn;1Tn;n+Tn+;n �Rn;1Rn+;nRn;n+ +Rn+;n

Tn+;nTn;1

b22 = �Rn;n+Rn;1 � 1Tn+;nTn;1

The �eld functions at the end of the n-th sub-slab are obtained by applying T-scheme

approach to (4.16):2664 F(n+1)

B(n+1)

3775 =2664 e�i ndn 0

0 ei ndn

3775 �2664 Fn+

Bn+

3775 =2664 c11 c12

c21 c22

37752664 F1

B1

3775 (4.17)

where

c11 =

Tn;n+Tn+;nT1;nTn;1 � Tn;n+Tn+;nRn;1R1;n �Rn+;nRn;n+T1;nTn;1

+Rn+;nRn;n+Rn;1R1;n �Rn+;nR1;nTn+;nTn;1ei ndn

;

c21 = �Rn;n+T1;nTn;1 �Rn;n+Rn;1R1;n +R1;nTn+;nTn;1

;

c12 =Rn;1Tn;n+Tn+;n �R1;nRn;n+Rn;1

Tn+;nTn;1ei ndn;

c22 = �ei ndnRn;n+Rn;1 � 1Tn+;nTn;1

:

After making same tedious algebra manipulations, we rewrite (4.17) to the form:2664 F(n+1)B1

3775 =2664 e�i ndn 0

0 ei ndn

3775 �2664 F1

B(n+1)

3775 (4.18)

where

76

T1;(n+1) = e�i ndn (1�Rn;1Rn;n+)�1 T1;nTn;n+;

R(n+1);1 = e�2i ndn (Rn;1Tn;n+Tn+;n �Rn+;nRn;1 +Rn;n+) (1�Rn;1Rn;n+)�1 ;

R1;(n+1) = (Rn+;nT1;nTn;1 �R1;nRn;1Rn;n+ +R1;n) (1�Rn;1Rn;n+)�1 ;

T(n+1);1 = e�i ndn (1�Rn;1Rn;n+)�1 Tn;1Tn+;n;

I is the unitary matrix.

Repeating the previous step gives the relations between the �eld functions near

the left boundary in the n-th sub-slab and ones near the boundary before the object (see

Appendix A-3):2664 F(n+1)+

B(n+1)+

3775 =

2664T(n+1);(n+1)+�R(n+1)+;(n+1)R(n+1);(n+1)+

T(n+1)+;(n+1)

R(n+1)+;(n+1)T(n+1)+;(n+1)

�R(n+1);(n+1)+T(n+1)+;(n+1)

1T(n+1)+;(n+1)

3775 �2664 F(n+1)

B(n+1)

3775

=

2664 c11 c12

c21 c22

3775 �2664 F1

B1

3775 (4.19)

=

2664 A+B+C+DK

E+FK

G+HK

JK

3775 �2664 F1

B1

3775

77

where

A =T(n+1);(n+1)+T(n+1)+;(n+1) (Tn;n+Tn+;nT1;nTn;1 � Tn;n+Tn+;nRn;1R1;n �Rn+;nRn;n+T1;nTn;1)

ei ndn;

B =

T(n+1);(n+1)+T(n+1)+;(n+1) (Rn+;nRn;n+Rn;1R1;n �Rn+;nR1;n)

�R(n+1);(n+1)+R(n+1)+;(n+1)Tn;n+Tn+;nT1;nTn;1ei ndn

;

C =

R(n+1);(n+1)+R(n+1)+;(n+1)

(Tn;n+Tn+;nRn;1R1;n +Rn+;nRn;n+Tn;1T1;n �Rn+;nRn;n+Rn;1R1;n)

ei ndn

D = ei ndnR(n+1)+;(n+1)�R(n+1);(n+1)+Rn+;nR1;n

e2i ndn�Rn;n+T1;nTn;1 +Rn;n+Rn;1R1;n �R1;n

�

E = e�i ndn

8>><>>:T(n+1);(n+1)+T(n+1)+;(n+1)

hTn;n+Tn+;nRn;1

e�2i ndn�Rn+;nRn;n+Rn;1 +Rn+;n

i�R(n+1);(n+1)+R(n+1)+;(n+1)Tn;n+Tn+;nRn;1

9>>=>>; ;

F = R(n+1);(n+1)+

0BB@ e�i ndnR(n+1);(n+1)+Rn;n+Rn+;nRn;1 � e�i ndnR(n+1);(n+1)Rn+;n

�ei ndnRn;n+Rn;1 + ei ndn

1CCA

G = e�i ndnR(n+1);(n+1)+

0BB@ Tn;n+Tn+;nTn;1T1;n � Tn;n+Tn+;nRn;1R1;n

�Rn;n+Rn+;nTn;1T1;n

1CCAH = e�i ndnR(n+1);(n+1)+ (Tn;n+Tn+;nRn;1R1;n �Rn+;nR1;n)

+ei ndn (Rn;n+Tn;1T1;n �Rn+;nR1;nRn;1 +R1;n) ;

J = e�i ndnR(n+1);(n+1)+ (Tn;n+Tn+;nRn;1 �Rn;n+Rn+;nRn;1 +Rn+;n)

+ei ndn (Rn;n+Rn;1 � 1) ;

K = T(n+1)+;(n+1)Tn+;nTn;1

Rewriting the (4.19) in the form of equation (4.4) gives:

78

2664 F(n+1)+B1

3775 = �2664 L

UMU

NU

WU

3775 �2664 F1

B(n+1)+

3775 (4.20)

where

L = T(n+1);(n+1)+Tn;n+T1;n;

M = e�i ndnT(n+1);(n+1)+T(n+1)+;(n+1) (Rn;1Tn;n+Tn+;n �Rn;1Rn;n+Rn+;n +Rn+;n) +

ei ndn�Rn;1Rn;n+ +R(n+1)+;(n+1) (1�Rn;1Rn;n+)

�

e�i ndnR(n+1);(n+1)+R(n+1)+;(n+1) (Rn;1Tn;n+Tn+;n �Rn;1Rn;n+Rn+;n)

N = e�i ndnR(n+1);(n+1)+ �0BB@ Tn;n+Tn+;nTn;1T1;n � Tn;n+Tn+;nRn;1R1;n �Rn+;nRn;n+T1;nTn;1

+Rn+;nRn;n+R1;nRn;1 �Rn+;nR1;n

1CCA�ei ndnR1;n (1�Rn;n+Rn;1) ;

W = T(n+1)+;(n+1)Tn+;nTn;1;

U = e�i ndnR(n+1);(n+1)+ (Rn;1Tn;n+Tn+;n �Rn;1Rn;n+Rn+;n +Rn+;n)

+ei ndn (Rn;1Rn;n+ � 1)

The S11 and S21 parameters for the case n � 2 can be presented as:

S11 = � Pn;nOn;n

;

S21 = �Hn;nOn;n

9>>=>>; (4.21)

79

where

Pn;n = ei ndnT(n+1);(n+1)+Tn;n+Tn;1

Hn;n = R(n+1);(n+1)+ �

8>><>>:Tn;n+Tn+;n (Tn;1T1;n �Rn;1R1;n)

�Rn+;nRn;n+�T1;nTn;1 �R1;nRn;1 +R�1n+;nR1;n

�9>>=>>;

+e2i ndnR1;n

n1 +Rn;n+

�Tn;1T1;nR

�11;n �Rn;1

�oOn;n = R(n+1);(n+1)+ (Rn;1Tn;n+Tn+;n �Rn;1Rn;n+Rn+;n +Rn+;n)

�e2i ndn (1�Rn;1Rn;n+) :

S11 and S21 parameters are rather �standard�parameters for characterizing both the electric

and magnetic properties of bulk materials.

4.2 Problem Formulation

Here the concept of EMT is applied to the slab object in the microwave frequency

range. The object is substituted by a single slab of bulk material with e¤ective dielectric

and magnetic constants "eff and �eff , respectively, so that the last slab has the same S-

parameters as the initial slab when later is considered as a single slab. It should be noted

that "eff and �eff are functions of frequency !.

The implementation of EMT utilized here is conventionally named as �a single slab

homogenization�(SSH) procedure. Such procedure is valid for any metamaterial medium

that might be considered as a cascaded network of parallel sub-slabs.

Let us consider a metamaterial slab as a cascaded network of identical parallel slabs

(with same e¤ective dielectric and magnetic constants, and thicknesses) with the thickness

equals the constant of the unit cell. In the paper, such slabs are called the �elementary

80

layer�. Assume also that the elementary slabs have a free space gap of zero thickness in

between. Thus, Eq. 4.13-14 can be rewritten:

Rn;n+ =

8>><>>:R if n is odd

�R if n is even(4.22.1)

Rn+;n =

8>><>>:�R if n is odd

R if n is even

(4.22.2)

Tn;n+ =

8>><>>:1�R if n is odd

1 +R if n is even

(4.23.1)

Tn+;n =

8>><>>:1 +R if n is odd

1�R if n is even(4.23.2.)

where is the Fresnel re�ection coe¢ cient on the interface of the left boundary of the each

layer.

Assume that "i = " and �i = � (for any i) are the complex e¤ective dielectric

and magnetic constants of the each layer of the object, respectively; ni = n and Ri = R

(for any integer positive i) are the complex e¤ective refractive index and Fresnel re�ection

coe¢ cient of the each layer, respectively; di = a, where a is the constant of unit cell of the

metamaterial slab under consideration. Then SSH procedure gives the system of the two

non-linear equations:

1�e�2ik0neff d

1�R2eff e�2ik0neff dReff = �

Pn;nOn;n

;

1�R2eff1�R2eff e

�2i !c nde�ik0neffd = �Hn;n

On;n

9>>>=>>>; (4.24)

with respect to the complex unknown variables " and �, or n =p� � " and R

81

subject to the representation of Fresnel re�ection coe¢ cient in the form: R =

p�="�1p�="+1

or

in the form: R = n�"n+" , respectively. neff =

p�eff � "eff is the complex e¤ective refractive

index of the object, d is its length; Reff =p�eff="eff�1p�eff="eff+1

is the complex Fresnel re�ection

coe¢ cient of the object.

It should be noted that elementary slab can be presented as the free space matrix

�lled with ingredient/ingredients layer/layers i.e. can be presented as another metamaterial

slab. In general case,

It is important to mention that the above mentioned constants "; �; �eff ; "eff ; R;Reff

are functions of the frequency !.

4.3 The Optimization Procedure

Since the nonlinear system (4.24) has no solution in closed form, the approxi-

mate solution is found by utilizing the optimization procedure. The procedure is based on

the step-by-step minimization of the di¤erence between the analytically obtained complex

Fresnel re�ection coe¢ cients and empirically evaluated ones.

The idea of the procedure is based on iteration calculation of the complex dielec-

tric and magnetic constants of the elementary slab subject to the corresponding values of

complex S-parameters determined in (4.8) and (4.21). The theoretical complex Fresnel re-

�ection coe¢ cient is evaluated by Nicolson-Ross algorithm using the values of S-parameters

calculated by (4.21) for a current iteration. The empirical complex Fresnel re�ection coef-

�cient is evaluated using the values of complex dielectric and magnetic constants obtained

either by the measurements or the numerical simulations. Thus, introducing the complex

82

dielectric and magnetic constants of the elementary slab on a current step of iteration:

" = "p + i"q

� = �p + i�q

9>>=>>; (4.25)

the parameters "p; �p; "q and �q are given as:

"p = 1 + a: M;

"q = c: M;

�p = 1 + b: M;

�q = d: M

9>>>>>>>>>>=>>>>>>>>>>;(4.26)

where M is the increment of iterations, the parameters a; b; c and d will be de�ned by means

of solving the problem of nonlinear mathematical programming stated as:

min jRth �Rempj ;

1�e�2ik0neff d

1�R2eff e�2ik0neff dReff = �

Pn;nOn;n

;

1�R2eff1�R2eff e

�2i !c nde�ik0neffd = �Hn;n

On;n

9>>>>>>>=>>>>>>>;(4.27)

with respect to the above mentioned parameters. Here Rth and Remp are, the theoretical

and empirical Fresnel re�ection coe¢ cients of a metamaterial �at slab (the object). It is

assumed that the increment M is rather small value limited by computational recourses of

the work station used for the simulations.

A solution of (4.27) can be obtained, for example, by the gradient method [82].

According to the this method, S-parameters of n-th iteration (n � 1) can be presented as a

point on the complex plane, Fig. 4.2. The vectors�!V and

�!W on this plane join zero iteration

to the theoretical evaluated values of S-parameters, and the vectors�!V and

�!W joins the zero

iteration to the previously described ones

83

�!V = a

�!V1 + b

�!V2 + c

�!V3 + d

�!V4 (4.28)

�!W = a

�!W1 + b

�!W2 + c

�!W3 + d

�!W4

where the parameters a; b; c and d are de�ned by (4.26) and are completely de�ned the

parameters "p; �p; "q and �q of these equalities. The increment of iterations M is being

assigned arbitrary small subject to the computational power of computer is used for the

simulations.

The parameters a; b; c and d are to be de�ned from (4.25-26) through the inverse

matrix:266666666664

a

b

c

d

377777777775=

266666666664

Re (V1) Re (V2) Re (V3) Re (V4)

Im (V1) Im (V2) Im (V3) Im (V4)

Re (W1) Re (W2) Re (W3) Re (W4)

Im (W1) Im (W2) Im (W3) Im (W4)

377777777775

�1

�

266666666664

Re (V )

Im (V )

Re (W )

Im (W )

377777777775(4.29)

Four new values for S-parameters (the �rst iteration) are de�ned by iterating "p; �p; "q and

�q successively. Their values can be calculated, for example, from S-parameters of the

in�nite metamaterial medium formed by expanding the elementary layer in the directions

perpendicular to plane of the layer. The appropriate calculation can be done by means of

numerical way using FDTD method.

Recommended initial values of the complex dielectric and magnetic constants (Eq.

4.25-26) of the elementary slab, namely the values of the zero iteration, are recommended

to be: S11 = 0 and S21 = 1.Solving the optimization problem (4.27) through the gradient

method requires a routine programming while some embedded functions of MatLab software

84

Figure 4.2: Representation in the complex plane of the calculated and simulated S-parameters

85

can be used in order to design a quite simple programming realization of solving of the

problem (4.27).

In this thesis, the results of programming realization for the function F = jRth �Rempj

of the optimization problem (4.27) is presented. The programming is made with MatLab

software via using the minimizing functions of several variables fmin.

The proposed algorithm is tested on the following benchmark problem.

Consider a �at slab (an object) of thickness d �lled with the two component meta-

material medium. A periodical array of the parallel copper cylindrical rods of the radius r is

imbedded into the host dielectric material. The medium parameters are the same as those

in the paragraph 3.2 of this thesis, namely, the constant a of the unit cell equals 1 mm. The

expressions for complex e¤ective dielectric and magnetic constants of such medium were

reported by Eq. 2.6-7 and Eq. 2.10. It enables us to approximately consider the above ex-

pressions as the expressions for complex e¤ective parameters of �nite metamaterial medium

which consists of �fteen elementary layers ( in (2.21) and (2.24)). This assumption enables

to evaluate the empirical complex Fresnel re�ection coe¢ cient of the object. At the same

time, the theoretical complex Fresnel re�ection coe¢ cient is obtained as mentioned above.

In this work we evaluate the accuracy of the proposed optimization procedure

by comparing the results obtained using the free Meep XFDTD software package. The S-

parameters of the elementary slab calculated using Meep FDTD are then used to obtain the

e¤ective constants of the elementary slab by Nicolson-Ross algorithm. The results are shown

in Fig.4.2-3, where the dependences on real parts of the e¤ective constants are presented as

functions of the metal volume fraction F = �r2=a2. Here a = d= (n+ 1) is the dimension

86

Figure 4.3: Change of real part of the e¤ective dielectric constant of the elementary slabversus metal volume fraction F at 1GHz

87

Figure 4.4: Change of real part of the e¤ective magnetic constant of the elementary slabversus metal volume fraction F at 1GHz

88

of the unit cell. The plots demonstrate rather good agreement between the ana-

lytical and numerical results when the values of the volume fraction are small (F � 0:25).

The discrepancy becomes lagers with the increase of F . Such result follows the main idea

of EMT.

In our paper, we have not presented the comparison of the graphs for imaginary

parts of the e¤ective constants. This is because the both ways of simulations have shown

that the values of the above mentioned parameters are not increasing in the order of 10�3.

4.4 Conclusion

In this paragraph, the analytical and numerical optimization method for separating

a slab metamaterial into its elementary sub-slabs of the order of the unit cell dimension

of the slab is presented. The proposed method enables us to characterize the electric and

magnetic responses of the elementary slab on the �e¤ective interatomic�dimension level via

joint implementation of S� and T� parameters schemes and the E¤ective Medium Theory.

A good agreement with benchmark solution obtained by Meep FDTD software package is

shown for the small values of the inclusion volume fractions. The method has been designed

irrespective to the shape of unit cell.

89

Chapter 5

EFFECTIVE

ELECTROMAGNETIC

RESPONSE OF THE INFINITE

CHAIN OF CIRCULAR

METALLIC CYLINDERS

In this chapter, we investigate the electric and magnetic responses from the in�nite

chain of in�nite circular metallic cylinders periodically immersed in dielectric matrices of

di¤erent shapes. The long-wavelength approximation for the complex e¤ective dielectric and

magnetic constants is obtained on the basis of the E¤ective Medium Theory. Comparison

of the analytically obtained results with the numerically calculated ones is carried out.

90

Conclusions about applicability range of the proposed analytical model have been made.

5.1 Homogenization of Slab Metal-Rod Periodic Media

The problem of homogenization of metal-wire periodic media for electromagnetic

waves in the long-wavelength limit is a longstanding problem in the contemporary electro-

magnetics [51, 49, 55, 56]. It especially attracts attention of researchers during the last

decade due to the appearance of abundant research on metamaterials [21, 90, 92-95].

In characterizing metamaterials, mostly one considers an unbounded arti�cial di-

electric medium. But real practical applications require the making of mathematical models

of metamaterials with boundaries. There are a few successful attempts where metamaterials

are considered as �at arti�cial media, for example references [61, 66, and 91]. Moreover, in

the work [66], authors have proposed well to separate the arti�cial material into its elemen-

tary planes and then use periodic moment method techniques to individually characterize

each elementary plane.

In this work we present obtaining the long-wavelength approximations for expres-

sions of the complex e¤ective dielectric and magnetic constants of the in�nite chain of

in�nite metallic circular cylinders periodically embedded in a dielectric matrix. Copper

cylinders are considered for the use of visual aids. Correctness of the analytically obtained

solver is assessed in comparison with the numerically obtained with the aid of FDTD. The

proposed solver can be used for developing the idea for separating the arti�cial material

into its elementary planes [66].

91

5.2 The Case of Simple Matrix

5.2.1 Main Relations

Let us consider the in�nite chain of in�nite metallic (copper) circular cylinders

periodically imbedded with dielectric matrix, Fig.5.1 where a is the radius of the cylinder,

d is the spacing between the cylinders. In this work the matrix is just air to prevent losses

due to the dielectric. Let us also consider normal incidence of an electromagnetic wave

of frequency ! and wavenumber k on the chain. Throughout this chapter we consider the

initial electromagnetic wave only with the electric intensity vector perpendicular to the axes

of the cylinders. The long-wave approximations (ka; kd << 1) for the full re�ection and

transmission coe¢ cients is presented by the formulas[56].

R =1

kd

�2b+0 � 4b

�1

�(5.1)

T = 1 +1

kd

�2b+0 + 4b

�1

�(5.2)

where

b+0 =�1

I p0 + it0; I p0 =

2

kd; t0 = N0 +

i

ap0; � = ka;

N0 =�2�log

1:781 � kd4�

� 1:202 � (kd)2

4�3;

ap0 = �i��2

4

24 1� �K + �2

4

n�K

��2

2 + 1�� �2 + 1

2

o1� �2

4

n�2 � 1 + 2 log 2

1:781�� (�K � 1)o35 ;

K = �"

"p; � =

s"p�p

"�; N2 =

�4�3 (kd)2

+1

�� 1:202 � (kd)

2

8�3;

b�1 =�kd

4 + ikdt1; t1 = N0 +N2 +

i

ap1;

92

Figure 5.1: The chain

93

a01 = ��i��2

4

�24 K � � + �2

8

���3 + �2

��K

�1 + 3�2

�K + � + �2

8

n� + 2K � �2 (� + 3K) + 4 log 2

1:781�� (K � �)o35

In the above formulae, "pand �p are the dielectric and magnetic constants, respectively, of

the cylinders embedded in a medium with the dielectric and magnetic constants " and �,

respectively. In our work " = 1 = �; and "p is de�ned by Drude�s model by Eq.2.7.

where !2p =ne2

m"0is the plasma frequency, is the inverse relaxation time, e is the

charge of electron, n is the concentration of electrons, m is the mass of electron and is the

permittivity of free space.

The formulas (5:1 � 2) have been obtained in [56] by means of a Taylor series

expansion of the �elds at one cylinder arising from its neighbours. In making that expansion,

only the terms proportional to �2 were used.

Let us present the considered chain as a homogeneous layer for the frequency range

related to the above mentioned restrictions (ka; kd << 1). In this way, the chain can be

considered as an arti�cial dielectric layer �that is a metamaterial �the layer characterized

by complex e¤ective dielectric "eff and magnetic constants �eff that are, generally speaking,

functions of the frequency. So, optical properties of the considered metamaterial layer are

presented by the complex e¤ective refractive index neff =p"eff�eff .

Let us obtain the long-wave approximation for the complex e¤ective parameters

(dielectric and magnetic constants, refractive index) of the considered metamaterial layer.

According to the general theory of electromagnetic wave propagation through a dielectric

layer [14], the full re�ection and transmission coe¢ cients of the layer are given by:

R = �1� e�2ikl1� �2e�2ikl (5.3)

94

T =1� �2

1� �2e�2ikl e�ikl; (5.4)

where � = Z�1Z+1 ; Z is the normalized characteristic impedance, k = k0n = n!=c is the

wavenumber of the layer, k0 is the wavenumber of air, n is the refractive index of the layer,

l is the thickness of the layer (in our case, l = 2a ), c is the velocity of light in vacuum.

So, the expressions (5.3) and (5.4) can be applied to the considered metamaterial

layer if: .

Resolving (5.3) and (5.4) can be applied to the considered material layer if :

� =Zeff � 1Zeff + 1

; Zeff =

r"eff�eff

; keff = k0neff :

Resolving Eqs (4) and (5) with respect to � gives

� =1

2R

�R2 � T 2 + 1�

q(R2 � T 2 + 1)2 � 4R2

�(5.5)

Resolving Eqs (4) and (5) with respect to e�ikeff l gives

e�ikeff l =R+ T � �1� (R+ T ) � (5.6)

where the expressions of � is de�ned by Eq. (6). The correct sign in Eq. (6) is to be de�ned

by the fact that � is the Fresnel re�ection coe¢ cient for normal incidence of a wave on the

interface from air-�lled outer, the correct solver of Eq. (7) can be chosen such as j�j � 1.

Then �nding the logarithm of left and right parts of Eq. (7) and taking in account that

keff = neff!=c, gives

neff =ic

l!log

�R+ T � �1� (R+ T ) �

�(5.7)

95

or

neff =ic

l![log (jF j) + i (�+ 2�m)] ; m = 0;�1; �2; ::: (5.8)

where

F =R+ T � �1� (R+ T ) � (5.9)

and � is the main arguement of the complex function F and the coe¢ cients R and T are

de�ned by the expressions (5.1) and (5.2)

In this work, we consider such wavelength range that �m > l where �m is the

wavelength in the layer. In this case n = 0, as it was mentioned in paragraph 3.1.

Characterization of the electric and magnetic properties of the layer can be made

via analyzing the complex e¤ective electric and magnetic constants which can be evaluated

by the expressions [96]:

"eff =�1��1+�

�� neff ;

�eff =�1+�1��

�� neff

9>>=>>; (5.10)

where the expression of the e¤ective refractive index neff is de�ned by the formulas (5.8)

and (5.9).

Analyzing the expressions (5.8) and (5.9) is necessary for getting information about

the optical and transport properties of a material medium under consideration.

5.2.2 Modeling and simulation results

Let us asses the accuracy and applicability range of mathematical model presented

by the expressions (5.7) and (5.10). In order to do that, we will compare the results obtained

96

through these formulas with the results obtained numerically. Obtaining the numerical re-

sults is based on evaluating the complex e¤ective parameters from the numerically obtained

S- parameters via Nicolson-Ross approach. The S-parameters are evaluated by FDTD cal-

culations. The example of such comparisons is shown in Fig.5.2-3 where the dependence on

real parts of the e¤ective constants are presented as function of the metal volume fraction

F = �a2=d2. As we see from the graphs, a good agreement between the analytical calcu-

lations and numerical results is observed for low values of the volume fraction (a=d � 0:2)

while a marked disagreement between the analytical calculations and numerical ones is ob-

served as the volume fraction increases. this is in agreement with the results of the work

[56].Furthermore, as we can see from the graphs, better accuracy of the obtained mathemat-

ical model is observed for calculation of the magnetic constant than that of the dielectric

one.

In this study we have not presented the comparison of the imaginary part of the

e¤ective constants. This is because both analytical modeling and numerical simulations have

shown that the values of above-mentioned parameters are not increasing in the order of 10�3

that indicates low losses of considered metamaterial structure. That, in turn, indicates low

level of damping of electromagnetic waves in the material.

5.3 The Case of Embedded Matrix

5.3.1 Main relations

Let us consider the scatterer as 2-D in�nite periodic chain of metal in�nite cylinders

embedded in a dielectric matrix as shown in Fig.5.4.

97

Figure 5.2: Change of real part of e¤ective dielectric constant of the chain versus metalvolume fraction F at 1GHz

98

Figure 5.3: Change of real part of e¤ective magnetic constant of the chain versus metalvolume fraction F at 1GHz

99

Figure 5.4: The chain immersed in the imbedded matrix

100

It is supposed that "0 and �0 are the dielectric and magnetic constants, respectively,

of the cylinders embedded in a medium with the dielectric and magnetic constants " and

�, respectively while�" and

�� are the dielectric and magnetic constants of the matrix,

respectively. As it has been shown in the previous paragraph the long-wave approximation

(ka; kd << 1) of S-parameters for the above �at metamaterial structure for the case�" = ";

�� = �, are de�ned by the formulae:

�S11 =

1

kd

�2b+0 � 4b

+1

�(5.11)

�S21 = 1 +

1

kd

�2b+0 + 4b

+1

�(5.12)

where b+0 ; I00; t0; �;N0; a

00;K; �;N2; b

�1 ; t1; a

01 are de�ned in the sub-paragraph 5.2.1.

Let us consider the presented metamaterial slab as a four-terminal network of sub-

slabs with the �eld functions FI ; BI ; FII ; BII ; Fig.5.5 where "eff and �eff are, respectively,

the complex e¤ective dielectric and magnetic constants of the slab for the case�" = ";

�� = �. The appropriate expressions of the complex e¤ective constants are obtained in the

sub-paragraph 5.2.1 and are:

"eff =ic2a!

�1��1+�

�� log

0@ �S11+

�S21��

1���S11+

�S21��

�1A ;

�eff =ic2a!

�1+�1��

�� log

0@ �S11+

�S21��

1���S11+

�S21��

�1A

9>>>>>>=>>>>>>;(5.13)

where�S11;

�S21 are S-parameters of the presented metamaterial slab

�" = ";

�� = �; � =

Zeff�1Zeff+1

, Zeff =q

�eff"eff

is the normalized complex e¤ective characteristic impedance of the

metamaterial slab for the case�" = ";

�� = �, c is the velocity of light in vacuum.

101

Figure 5.5: Representation of the metamaterial slab as a four-terminal network of sub-slabs

102

If the complex constants "eff and �eff S-parameters of the four-terminal network

are known for the case�" = ";

�� = �, then its S-parameters for the case of arbitrary values of

the constants�";

�� can be obtained through T-matrix of the four-terminal network de�ned

via the matrix equality [10]:2664 FI

BI

3775 =2664 T11 T12

T21 T22

3775 �2664 FII

BII

3775 = T �2664 FII

BII

3775 (5.14)

Let us obtain the elements of T-matrix. The elements can be obtained by multiplying the

�eld functions matrix for each sub-slab region of the four-terminal network of sub-slabs as

follows:

[T ] =

2664 ei�kh 0

0 e�i�kh

3775 �2664 (1�Reff )�1 Reff (1�Reff )�1

Reff (1�Reff )�1 (1�Reff )�1

3775 �2664 e2ikeffa 0

0 e�2ikeffa

3775 �2664 (1 +Reff )

�1 �Reff (1�Reff )�1

�Reff (1 +Reff )�1 � (1 +Reff )�1

3775 �2664 ei

�kh 0

0 e�i�kh

3775 (5.15)

where�k =

q�"��:!=c is the wavenumber in the �rst and third sub-slabs, Reff =

p�eff="eff�1p�eff="eff+1

is

the complex re�ection coe¢ cient at the plane boundaries between the �rst and second sub-

slabs (while �Reff is the complex re�ection coe¢ cient at the plane boundaries between the

second and third sub-slabs), k = p"eff�eff :!=c is the wavenumber in the second sub-slab.

103

Multiplying it, gives:2664 T11 T12

T21 T22

3775=

1

1�R2eff�2664

�e2ikeffa �R2effe�2ikeffa

�e2i

�kh �2iReff sin 2keffa

2iReff sin 2keffa�e�2ikeffa �R2effe2ikeffa

�e�2i

�kh

3775 (5.16)Taking into account (5.16) and the fact that:2664 1

S12�S22S12

S11S12

S212�S11S22S12

3775 =2664 T11 T12

T21 T22

3775gives:

S21 =e2ikeff a(1�R2eff)e�2i

�kh

eikeff a�R2eff e

�ikeff a ;

S11 =2iReff e

�2i�kh sin 2keffa

eikeff a�R2eff e

�ikeff a :

9>>>=>>>; (5.18)

Having the expressions of S-parameters of the considered metamaterial slab we can obtain

the expressions of the e¤ective dielectric and magnetic constants of the slab via Nicolson-

Ross approach:

"eff = neff1�K �

pK2 � 1

1 +K +pK2 � 1

;

�eff = neff1 +K +

pK2 � 1

1�K �pK2 � 1

: (5.19)

where

neff = jneff j � ei� = �ln (jneff j) + i (�� 2�l)

2 (a+ h); l = 0;�1;�2; ::: (5.20)

and

K =S211 � S221 + 1

2S11(5.21)

104

As usually, we consider the case l = 0 for Eq. 5.20. The slab thickness is not less than the

wavelength of initial electromagnetic wave in the slab for this case (see the paragraph 3.1).

5.3.2 Modeling and simulation results of the case

We evaluate the accuracy and applicability range of mathematical model presented

by the expressions (5.18-21) for the case when d = 1mm, " = 1 = � =��,

�" = 2:2 (wax).

The calculations are made at the frequency of 1 GHz.

Let us compare the results obtained through these formulas with the results ob-

tained numerically. Obtaining the numerical results is based on FDTD simulations for

evaluating the complex S-parameters that are used for calculating the complex e¤ective pa-

rameters via the formulas (5.19-21). The example of such comparisons is shown in Fig.5.6-7

where the dependences on real parts of the e¤ective constants are presented as functions

of the cylinders volume fraction F = �a2=4 (a+ d)2. As we see from the graphs, a good

agreement between the analytical and numerical results is observed on the low values of

the volume fraction (a=2 (a+ d) � 0:3) while a worse agreement between the analytical and

numerical results is observed as the volume fraction increases that is in a good agreement

with results of the work [56]. Furthermore as we can see from the graphs, better accuracy

of the obtained mathematical model is observed to calculate the magnetic constant than

that of the dielectric one.

It is also interesting to conclude from the graphs of Fig. 5.6-7 that the enhancement

of real part of the complex e¤ective dielectric constant takes place with respect to the value

of dielectric constant of the matrix. As we can see from the graphs, real part of the e¤ective

dielectric constant increases with the increase in volume fraction.

105

Figure 5.6: Change of real part of the e¤ective dielectric constant of the chain versus metalvolume fraction F at 1GHz

106

Figure 5.7: Change of real part of the e¤ective magnetic constant of the chain versus metalvolume fraction F at 1GHz

107

More calculations have shown that the above enhancement is a function of the

volume fraction.

In these calculations, we do not present the comparison between analytical calcu-

lations and numerical ones for imaginary parts of the e¤ective constants. This is because

both analytical modeling and numerical simulations have shown that the values of the above

mentioned parameters are not increasing in the order of 10�2 for the dielectric constant and

10�3 for magnetic one that indicates low losses of considered metamaterial structure. That,

in turn, indicates low level of damping of electromagnetic waves in the presented material

slab.

Comparing Figure 5.2 and 5.7 gives a very important practical result: immers-

ing the initial chain to a magneto-dielectric matrix increases the enhancement of complex

dielectric constant.

5.4 Conclusion

In this chapter, the product of 2-D electromagnetic response of the in�nite chain of

metallic circular cylinders periodically immersed in two di¤erent kinds of matrix is solved in

the microwave frequency range. The approach used to obtain the response is based on EMT

applied to very well known solution of the 2-D problem of di¤raction of electromagnetic wave

from in�nite chain of circular cylinders placed in air, obtained in the middle of last century.

A good agreement between the analytical calculations and numerical ones was found for

the low values of the volume fraction.

The enhancement of real part of the complex e¤ective dielectric constant with

108

respect to the value of dielectric constant of the matrix was found as a function of the

volume fraction of cylinder inclusions.

The simulations have shown that the considered chain possesses a low level loss.

109

Chapter 6

Conclusion

This thesis is dedicated to theoretically (analytically and numerically) character-

izing the e¤ective electric and magnetic constants of two component slab metamaterial

structures and in�nite metamaterial media as arrays of metal rods/wires periodically im-

mersed in a dielectric isotropic matrix. The rods material is supposed to be non-magnetic.

Only a circular cross section of the rods is considered in this work while the rods do not

touch each other. At the same time, there is no restriction on the radius of the rods. The

microwave frequency range (from 0 to 5 GHz) has only been considered in this study.

The characterizations of considered metamaterial structures in the thesis are iden-

ti�ed with the study of properties of the e¤ective dielectric and magnetic constants as

functions of the frequency (GHz) of initial electromagnetic wave that is normally incident

to the �at boundaries of metamaterial structures under consideration.

The above mentioned e¤ective parameters have been obtained on the basic of

extension of the case of 2-D in�nite metamaterial medium to a slab metamaterial scatterer

110

through the implementation of the E¤ective Medium Theory in appropriate frequency range.

The expressions for the appropriate e¤ective constants for the in�nite medium have been

obtained by other authors. However the expressions of the e¤ective constants obtained in

this thesis take into account multipole e¤ects for the case of composite with a very small

value of the volume fraction of the rod while dipole e¤ects are taken in account for the case

of large volume fraction.

The accuracy of obtained mathematical (analytical) models was always bench-

marked for comparison with numerical calculations obtained via the implementations of

FDTD method for calculating S-parameters of a metamaterial structure under considera-

tions. S-parameters were used to calculate the e¤ective constants by using Nicolson-Ross

approach.

In this thesis, an improved broadband method for determining complex e¤ective

refractive index, dielectric and magnetic constants of an arbitrary passive metamaterial

has been proposed. Evaluation of the e¤ective parameters is realized using the re�ection-

transmission S-parameters obtained by simulation or experimental measurements and ana-

lytically evaluated interface re�ection coe¢ cient of the slab.

In the consideration of practical part of this thesis, the obtained qualitative and

quantitative results have allowed to formulate properties of two component slab metamate-

rial structures as arrays of metal rods/wires periodically immersed in a dielectric isotropic

matrix. Following general conclusions can be drawn:

1. The e¤ective electromagnetic properties of in�nite 2-D array of copper cylinders im-

mersed in metal-dielectric matrix GHz frequencies show the existence of enhancement

111

of e¤ective dielectric constant and low absorption.

2. The analytical models of the composite gives a good qualitative but a weak quanti-

tative correlation with results of numerical simulation in the case if cylinders which

touch each other.

3. The analytical models of in�nite metamaterial medium quantitatively describes well

the slabs imbedded with the metamaterial medium if there is some relation between

the width of the slabs and the dimension of the unit cell of the metamaterial medium

at appropriate frequency range.

4. The arti�cial material medium can be used to increase the directivity of patch antenna

and to obtain ULI structures in the GHz frequency range and to design a new type

of waveguides.

The main theoretical results of this thesis have been presented by two theoret-

ical methods for characterization of a 2-D slab metamaterial structure in the microwave

frequency range:

1. An improved broadband method for the evaluation of the e¤ective complex dielectric

and magnetic parameters of 2-D slab metamaterials.

2. The analytical and numerical optimization method for separating a slab metamaterial

into its elementary sub-slabs of the order of the unit cell dimension of the slab.

It is important to mention that the above method have been designed irrespective

to the shape of inclusion in the unit cell.

112

The obtained results in this thesis are in a good quantitative and qualitative agree-

ment with the results of experimental research carried out earlier by one of the supervisors.

The above results should logically give the start for a new approach in research

on qualitative level. For example, it is planned to carry out an analytical and numerical

characterization of the e¤ective optical and transport properties of 2-D layered slab meta-

material structures with a complicate shape of the unit cell. At the same time, some kind

of experimental research regarding the microwave characterization of bulk arti�cial periodic

structures can be initiated if some facilities are available. So, one of the most important

problem to be resolved in further acquiring some funds (including grants for travel abroad)

will enable to initiate in Pakistan a number of new industry-oriented research projects with

a possibility for involving Pakistani students and international students in such studies to

be carried in Pakistan.

113

Bibliography

[1] Ari Sihvola, "Metamaterials in electromagnetics," Metamaterials, 1(1), pp.2-11, 2007

[2] http://en.wikipedia.org/wiki/Metamaterials

[3] http://www.darpa.mil/dso/thrust/matdev/metamat.htm

[4] T.M. Grzegorczyk and J. A. Kong, �Review of left-handed metamaterials: Evolution

from theoretical and numerical studies to potential applications�, J. of Electromagn.

Waves and Appl., Vol. 20, No. 14, 2053�2064, 2006

[5] G.V. Eleftheriades and K.G. Balmain. Negative-Refraction Metamaterials: Fundamen-

tal Principles and Applications. John Wiley & Sons and IEEE Press, June 2005.

[6] A. K. Sarychev and V. M. Shalaev, Electrodynamics of Metamaterials, World Scienti�c,

Singapore, 2007.

[7] Brian T. Schwartz, and Rafael Piestun, �Waveguiding in air by total external re�ection

from ultra low index metamaterials�, Appl. Phys. Lett., Vol. 85, No. 1 (2004), pp. 1-3.

[8] Anthony Grbic and George V. Eleftheriades, �Experimental veri�cation of backward-

114

wave radiation from a negative refractive index metamaterial�, J. of Appl. Phys., Vol.

92, No. 10 (2002), pp. 5930-5935.

[9] A.N. Lagarkov, A.K. Sarychev, Y.R. Smychkovich, and A.P. Vinogradov, �E¤ective

Medium Theory For Microwave Dielectric Constant and Magnetic Permeability of Con-

ducting Stick Composites�, J. of Electromagnetic Waves and Applications, Vol. 6, Issue

9 (1992), pp. 1159-1176.

[10] T. Koschny and M. Kafesaki, E. N. Economou, C.M. Soukoulis. �E¤ective Medium The-

ory of Left-Handed Materials�, Phys. Rev. Lett., Vol. 93, No 10 (2004), pp. 107402(1-4).

[11] D. Felbacq and G. Bouchitté, �Homogenization of a set of parallel �bers,�Waves in

Random Media, Vol. 7 (1997), pp. 245�256.

[12] Elena Cherkaev, �Inverse homogenization for evaluation of e¤ective properties of a

mixture�, Institute of Physics Publishing, Inverse Problems, Vol. 17 (2001), pp. 1203�

1218.

[13] Mario G. Silveirinha, and Carlos A. Fernandes, �Homogenization of Metamaterial Su-

fases and Slab: The Crossed Wire Mesh Canonical Problem�, IEEE Trans. On Anten-

nas and Propag. Vol.53, Issue 1 (2005), pp. 59-69.

[14] W. Rotman, �Plasma simulation by arti�cial dielectrics and parallelplate media,�IRE

Trans. Antennas Propag., Vol. AP-10, pp. 82�95, Jan. 1962.

[15] P.A. Belov, S.A. Tretyakov, and A.J. Viitanen, �Dispersion and Re�ection Properties

of Arti�cial Media Formed by Regular Lattice of Ideally conducting Wires�, J. of

Electromagnet. Waves and Applicat., Vol. 16, No. 88 (2002), pp. 1153-1170.

115

[16] Tom G. Mackay and Akhlesh Lakhtakia, �Correlation Length Facilitates Voigt Wave

Propagation,�arXiv: physics/0408045 v3 7 Sep 2005.

[17] V. G. Veselago, �The electrodynamics of substances with simultaneously negative val-

ues of " and �,�Sov. Phys. Usp. Vol. 10 (1968), pp. 509-514.

[18] J.B. Pendry and S. A. Ramakrishna, �Focussing Light Using Negative Refraction�, J.

Phys. [Condensed Matter] Vol. 15, (2003), pp. 6345-6367.

[19] J.B. Pendry. �Negative refraction�, Contemporary Physics, Vol. 45, No 3, pp. 191-202,

January-February 2004.

[20] J.B Pendry, �Negative Refraction Makes a Perfect Lens�, Phys. Rev. Lett., Vol. 85,

No. 18, pp. 3966-3969, 30 October 2000.

[21] D.R. Smith, Willie J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, �Com-

posite Medium with Simultaneously Negative Permeability and Permittivity�, Phys.

Rev. Lett., Vol. 84, No. 18 (2000), pp. 4184-4187.

[22] S.A. Cummer and B. Popa, �Wave �elds measured inside a negative refractive index

metamaterial�, Appl Phys Lett, Vol. 85 (2004), pp. 4564�4566.

[23] Ilya V. Shadrivov, Richard W. Ziolkowski, Alexander A. Zharov, and Yuri S. Kivshar,

�Excitation of guided waves in layered structures with negative refraction�, Optics

Express, Vol. 13, No. 2, pp. 481-492, 24 January 2005.

[24] M.C.K. Wiltshire, J.B. Pendry, I.R. Young, D.J. Larkman, D.J. Gilderdale, J.V. Haj-

116

nal, �Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance

Imaging�, Science, Vol. 291, No. 5505 (2001), pp. 849-851.

[25] R.A. Shelby, D.R. Smith, and S. Schultz, �Experimental Veri�cation of a Negative

Index of Refraction�, Science, Vol. 292, No. 5514 (2001), pp. 77-79.

[26] J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, �Extremely Low Frequency

Plasmons in Metallic Mesostructures�, Phys. Rev. Lett., Vol. 76, (1996), pp. 4773-

44777.

[27] B. Pendry, A.J. Holden, D.J, Robbins, and W.J. Stewart, �Magnetism from Conductors

and Enhanced Nonlinear Phenomena�, IEEE Trans. On Microwave Theory And Tech.,

Vol. 47, No. 11, pp. 2075-2084, November 1999.

[28] 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 (2001), pp. 489-491.

[29] K. Li, S.J. McLean, R.B. Gregor, C.G. Parazzoli, and M. Tanielian, �Free-Space

Focused-Beam Characterization of Left-Handed materials�, Appl. Phys. Lett., Vol.

82, No. 15 (2003), pp. 2535-2537.

[30] S. Enoch, G. Taybe, P. Sabouroux, N. Guerin, and P. Vincent, �A Metamaterial for

Directive Emission,�Phys. Rev. Lett., Vol. 89 (2002), pp. 213902.

[31] G.Zouganelis, F.Soma, O.Rybin, H.Ohsato, �Study of Ultra Low Index (0 < " < 1)

metamaterial using a patch antenna. Japan�, Cer. Soc. Conf. 2005 (Fall meeting),

1PD41.

117

[32] S.-T. Ho, S. L. McCall, R. E. Slusher, L. N. Pfei¤er, K. W. West, A. F. J. Levi, G.

E. Blonder, and J. L. Jewell, �High Index Contrast Mirrors for Optical microcavities�,

Appl. Phys. Lett, Vol. 57 (1990), pp. 1387-1390.

[33] B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, �Wavelength

Scalable Optical Fibers for CO2 Laser Transmission�, Nature, Vol. 420 (2000), pp.

650-653.

[34] J.-Q. Xi, M. Ojha, W. Cho, J. L. Plawsky, W. N. Gill, Th. Gessmann, and E. F.

Schubert, �Omnidirectional re�ector using nanoporous SiO2 as a low-refractive-index

material�, Opt. Lett. Vol. 30, No. 12 (2005), pp. 1518-1520.

[35] J.-Q. Xi, M. Ojha, J. L. Plawsky, W. N. Gill, J. K. Kim, and E. F. Schubert, �Internal

high-re�ectivity omni-directional re�ectors�, Appl. Phys. Lett., Vol. 87 (2005), pp.

031111.

[36] T. Schwartz and R. Piestun, "A New Path, �Ultralow-index metamaterials present new

possibilities for controlling light propagation�, OE magazine, Vol. 5 (2005), pp. 30-32.

[37] B. T. Schwartz and R. Piestun, �Total external re�ection from metamaterials with

ultralow refractive index", J. Opt. Soc. Am., B 20, No. 12 (2003), pp. 2448-2453.

[38] N. Garcia, E. V. Ponizovskaya, and John Q. Xiao, �Zero permittivity materials: Band

gaps at the visible�, Appl. Phys. Lett., Vol. 80, No. 7 (2002), pp. 1120-1122.

[39] Sergei Tretyakov, Analytical Modeling in Applied Electromagnetics, Artech House,

2003.

118

[40] G. Zouganelis, T. Tsunooka and M. Andou, �CAD of one-layer frequency selected

surfaces with metamaterials properties�, J. of the European Ceramic Society, Vol. 26,

Issue 10-11 (2006), pp. 2199-2203.

[41] G. Zouganelis, O. Rybin, H. Ogawa, H. Ohsato, �Study of enhancement of internal

magnetic constants of metamaterials made from layers of magnetic wires�, Japan Cer.

Soc. Conf. 2005 (fall meeting), 1PD42.

[42] J. Kula, D. Psychoudakis, C.C. Chen, J.L. Volakis, J.H. Halloran, �Patch antenna

miniaturization using thick truncated textured ceramic substrates�, 20-25 June 2004,

IEEE Antennas and Propagation Society International Symposium 2004, Vol. 4, pp.

3800�3803.

[43] Y-H. Koh, J.W. Halloran, G. Kiziltas, D. Psychoudakis, and J.L. Volakis, �Thermo-

plastic Green Machining for Textured Dielectric Substrate for Broadband Miniature

Antenna,�J. Am. Ceram. Soc., Vol. 88, No. 2 (2005), pp. 297�302.

[44] Georgios Zouganelis, Oleg Rybin. Two layer magnetodielectric metamaterial with en-

hanced dielectric constant as a new ferrite like material. 2006. Japanese Journal of

Applied Physics (Letter). PT.2, Vol. 45, No. 44.

[45] A. Lakhtakia, An electromagnetic trinity from �negative permittivity�and �negative

permeability�, Int J Infrared Millim Waves 22 (2001), 1731�1734; 23 (2002), 813�818.

[46] A. Lakhtakia, On Homogenization of Impedance-Matched Chiral-in-Chiral Composites,

J. Phys. D: Appl. Phys. Vol. 29, 957�962 (1996).

119

[47] N. Engheta and R. W. Ziolkowski, (editors), Metamaterials: Physics and Engineering

Explorations, Wiley-IEEE Press, 414 pages, June 2006.

[48] Belov, P.A., R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silverinha, C.R. Simovski and

S.A. Tretyakov, �Strong spatial dispersion in wire media in the very large wavelength

limit�, Phys. Rev. B, Vol. 67 (2003), 113103-1-7.

[49] Von Ignatowsky, �Zur Theorie der Gitter�, A M. Physik, Vol. 44, p. 369, 1914.

[50] W. Rotman, �Plasma simulation by arti�cial dielectrics and parallelplate media,�IRE

Trans. Antennas Propag., Vol. AP-10, pp. 82�95, Jan. 1962.

[51] M.I. Astrakhan, G.A. Fersman, �Average boundary conditions and coe¢ cients of re-

�ection of a double, dense array of thin, parallel metallic �laments�, Radio Engineering

and Electronic Physics, Vol. 23, p. 24-28, July 1978.

[52] B.-I. Wu, W. Wang, J. Pacheco, X. Chen, T. Grzegorczyk and J. A. Kong, � A Study

of Using Metamaterials as Antenna Substrate to Enhance Gain�. Progress in Electro-

magnetic Research, PIER 51, 2005, pp. 295-328.

[53] G.A. Fersman, �Frequency-Selective Systems Based on Double Metal Grid Structures�,

DrSc thesis, 2002, St. Petersburg State Polytechnic University, St. Petersburg, Russia.

[54] P. Ikonen, M. Kärkkäinen, C. Simovski, P. Belov and S. Tretyakov, �Light-weight

base station antenna with arti�cial wire medium lens�, IEE Proc.-Microw. Antennas

Propag., vol. 153, no. 2, pp. 163-170, 2006.

120

[55] M.I. Kontorovich, N.N. Lebedev, On a Method of Solution of Some Problems of the

Di¤raction theory [in Russian], Zh. Exp. i Teor. Fiz. 8 (1938), 1192-1206.

[56] V. Twersky, On Scattering of Wires by the In�nite Grating of Circular Cylinders, IRE

Transaction on Antennas and Propagation, November 1962, pp. 737-765.

[57] S.A. Tretyakov, Meta-materials with wideband negative permittivity and permeability.

2001, Vol. 31, Issue 3, pp. 163-165.

[58] G.V. Eleftheriades, A.K. Iyer and P.C. Kremer, Planar negative refractive index media

using periodically L-C loaded transmission lines, IEEE Trans. on Microwave Theory

and Techniques, Vol. 50, No. 12, pp. 2702-2712, Dec. 2002.

[59] G. Zouganelis, D. Budimir, Silicon Gap-Loaded Microstrip Slit-Tetragonal Resonator

under Ir-Irradiation, Microwave and Optical Technology Letter. 2007, Vol. 49, No. 3,

pp. 699-702.

[60] Z. Hashin, S. Htrikman, A Variational Approach to the Theory of the E¤ective Mag-

netic Permeability of Multiphase Materials, Journal of Applied Physics. Vol. 33, No.

10, October 1962.

[61] Xiangting Li and H R Ma, The Bergman Spectrum of the E¤ective Dielectric Constant

in Two-Dimensional Composite Media, J. Phys.: Condens. Matter, 11, No 23 (14 June

1999), L241-L246.

[62] S.R. Simovsky and P.A. Belov, Low-Frequency Spatial Dispersion in Wire Media. 2004,

Phys. Review, E 70, pp. 0406616-1-8

121

[63] M. G. Silveirinha, and C. A. Fernandes, Homogenization of metamaterial surfaces and

slab: The crossed wire mesh canonical problem, IEEE Trans. on Antennas and Propag.

vol.53, no.1, 2005, pp. 59-69.

[64] Yan Zhao, Pavel A. Belov and Yang Hao, Spatially dispersive �nite-di¤erence

time-domain analysis of sub-wavelength imaging by the wire medium slabs. 2006,

arXiv:physics/0605025v1

[65] J.B. Pendry, Extremely Low Frequency Plasmons in Metallic Microstructures, Phys.

Rev. Lett. Vol. 76, 4773-6 (1996), pp. 1-12.

[66] C.A. Moses, N. Engheta, Electromagnetic Wave propagation in the Wire Medium: a

Complex Medium with Long Thin Inclusions, Wave Motion, 34 (2001), 301-317.

[67] P. Markos and C. M. Soukoulis, Absorption Losses in Periodic Arrays of Thin Metallic

Wires, Optics Letters. 2003, Vol. 28, No. 4, pp. 846-848.

[68] Pekka M. T. Ikonen, Elena Saenz, Ramon Gonzalo and Sergei A. Tretyakov, Modeling

and Analysis of Composite Antenna Superstrates Consisting on Grids of Loaded Wires.

arXiv:physics/0701035v1, 3 Jan 2007.

[69] O. Rybin, G.Zouganelis, Enhancement of the Dielectric Constant in Magneto-Dielectric

Metamaterial Substrates. 2009, Telecommunications and Radio Engineering, Vol. 68,

No. 9, pp. 835-840.

[70] O. Rybin, Enhancement of Dielectric Constant in Metal-Dielectric Metamaterials. 2009,

The African Physical Review, Vol. 3, No. 1, pp. 49-55.

122

[71] D. Bergman, The Dielectric Constant of a Composite Material �a Problem in Classical

Physics, Phys. Reports, Vol. 43, pp. 377-407, 1978.

[72] G. Milton, Bounds On the Complex Permittivity of a Two-Component Composite

Material, J. Appl. Phys., Vol. 52, pp. 5286-5293, 1981.

[73] J. M. Pitarke, F. J. Garsia-Vidal, J. B. Pendry, E¤ective Electronic Response of a

System of Metallic Cylinders. 1998, Phys. Review B, Vol. 57, No. 24, pp. 15261-15266.

[74] N. A. Nicorovici and R. C. McPhedran, Transport Properties of Arrays of Elliptical

Cylinders, Phys. Rev. E. August 1996, Vol. 54, No. 2, pp. 1945-1957.

[75] V.B. Bregar, M. Pavlin, �E¤ective-susceptibility tensor for a composite with ferromag-

netic inclusions: Enhancement of e¤ective-media theory and alternative ferromagnetic

approach�. J of Applied Phys., Vol.95, No. 11, 1 June 2004, pp. 6289-6293.

[76] A.M. Nicolson and G.F. Ross, Measurement of the Intrinsic Properties of Materials by

Time-Domain Techniques. IEEE Trans. On Instrumentation and Measurement, Sep-

tember. 1970; TM-19 (4), pp. 377�382.

[77] D.K. Ghodgaonkar, V. V. Varadan, Anad V. K. Varadan, �Free-Space Measurement of

Complex Permittivity and Complex Permeability of Magnetic Materials at Microwave

Frequencies�, IEEE Trans. On Instrumentation and Measurement, Vol.39, No2, Apr.

1990, pp.387�394.

[78] http://en.wikipedia.org/wiki/Scattering_parameters

123

[79] P. Bientsman, Rigorous and e¢ cient modeling of wavelength scale photonic compo-

nents, PhD thesis, University of Gent, 2001.

[80] D.J. Kern, D.H. Werner, and M. Lisovich. Metaferrites: Using Electromagnetic

Bandgap Structures to Synthesize Metamaterial Ferrites. IEEE Trans. On Antennas

and Propagation, April 2005; 53 (4): 1382�1389.

[81] N.E. Belhadj-Tahar, A. Fourrier-Lamer. Broad-band simultaneous measurement of

complex permittivity and permeability using a coaxial discontinuity. IEEE Trans. On

Microwave Theory and Techniques, Oct. 1991; 39 (10): 1718-1724.

[82] P. Que¤elec, Ph. Gelin, J. Gieraltowski, J. Loaec. Microstrip Device for the Broad

Band Simultaneous Measurement of Complex Permeability and Permittivity. IEEE

Trans. On Magnetics, March 1994: 30 (2): 224�231.

[83] E. Salahun, P. Que¤elec, M. Le Floc�s and Ph. Gelin. A Broadband Permeameter

for �in situ�Measurements of Rectangular Samples. IEEE Trans. On Magnetics, July

2001; 37 (4): 2743�2745.

[84] Hewlett-Packard Product Note 85 10-3. Measuring dielectric constant with the HP

8510 Network Analyzer, Aug. 1985.

[85] Xudong Chen, Tomasz M. Grzegorczyk, Bae-Ian Wu, Joe Pacheco, Jr., and Jin Au

Kong. Robust method to retrieve the constitutive e¤ective parameters of metamateri-

als. Phys. Rev. E, 2004; 70: 016608 (1-7).

[86] M. G. Silveirinha, and C. A. Fernandes, �Homogenization of metamaterial surfaces

124

and slab: The crossed wire mesh canonical problem�, IEEE Trans. on Antennas and

Propag. vol.53, no.1, 2005, pp. 59-69.

[87] C. R. Simovski, and P. A. Belov, �Low-Frequency Spatial Dispersion in Wire Media�

Phys. Rev. vol. E 70, 046616 1-8, 2004.

[88] R. C. McPhedran and D. R. McKenzie, �Electrostatic and optical resonances of arrays

of cylinders�, J of Appl. Phys., vol. 23, 1980, pp. 223-235.

[89] D.Bariou, P.Que¤elec, Ph. Gelin, and M. Le Floc�h, Extension of the e¤ective medium

approximation for determination of the permeability tensor of unsaturated polycrys-

talline ferrites, IEEE Trans. Magn., 37, No. 6 (November 2001), 3885-3891.

[90] D.R. Smith and S. Schultz, P. Maros and C.M. Soukoulis, Determination of e¤ective

permittivity and permeability of metamaterials from re�ection and transmission coef-

�cients, Phys. Rev. B, 65 (2002), 195104 (1-5).

[91] A.J. Reuben, Spectral decomposition for the optical response of cylinder chains, J.

Phys. A: Math. Gen., 32 (1999), 4299-4310.

[92] D.R. Smith, Willie J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, �Com-

posite Medium with Simultaneously Negative Permeability and Permittivity�, Phys.

Rev. Lett., Vol. 84, No. 18 (2000), pp. 4184-4187.

[93] D.R. Smith, J.B. Pendry, M.C.K. Wilshire, Metamaterials and Negative Refractive

Index, Science, 305 (6 August 2004), 788-792.

125

[94] B.T. Schwartz and R. Piestun, "A New Path, Ultralow-Index Metamaterials Present

New Possibilities for Controlling Light Propagation, OE magazine, 5 (2005), 30-32.

[95] C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J.F. Shou,

Th. Koschny, and C.M. Soukoulis, Magnetic Metamaterials at Telecommunication and

Visible Frequencies, Phys. Rev. Let., 95 (11 November 2005), 203901 1-4.

[96] S.J. Orfanidis, Electromagnetic waves and antennas, Text Book, Rutgers University,

2008.

List of Publications

No. Authors Title Journals/Proceedings/

Conferences Status

HEC Recognized

1. Oleg Rybin, Muhammad

Raza

Long Wave Layer-Specific Representation of the Optical Properties of Slab Metamaterials.

International Journal of Applied Electromagnetics and Mechanics 32 (2010) No.4 pp 207-218

Published yes

2.

Oleg Rybin, Muhammad Raza and S.

Vyalkina

Effective Electromagnetic Response of the Infinite Chain of Metallic Cylinders Immersed in Isotropic Dielectric Matrix.

Telecommunications and Radio Engineering, 69(6):473-480

(2010) Published yes

3. Oleg Rybin, Muhammad

Raza

Effective electric and magnetic properties of the infinite chain of circular metallic cylinders

International Journal of Applied Electromagnetics and Mechanics 31 (2009) 61–66

Published yes

4. O. Rybin,

Muhammad Raza, Tahira Nawaz and T.

Abbas

Evaluation of Layer Properties of Effective Parameters of

Metallic Rod Metamaterials in GHz Frequencies.

International Journal of Electronics and Communications,

63 (2009) 648–652. Published yes

5.

O. Rybin, T. Abbas, M. Raza and

Tahira Nawaz

An Improved Broadband Method for the Evaluation of Effective Parameters of Slab

Metamaterials.

International Journal of Electronics and Communications,

62 (2008) 762 – 767.

Published yes

6.

O. Rybin, T. Abbas,

Muhammad Raza and

Tahira Nawaz

An Improved Broadband Method to Evaluate Effective

Parameters of Slab Metamaterials In Microwave

Frequency Range.

Proceedings Advanced Materials-2007, p57-62. Published -

7. Oleg Rybin, Muhammad

Raza

Long Wave Layer-Specific Representation Of

Metamaterial Slabs In The Microwave Frequency Range

IEEE proceedings of 13th International Conference on Mathematical Methods in Electromagnetic Theory September 6 – 8, 2010, Kyiv, Ukraine

Published -