phdthesis_sunilsudhakaran

Negative Refraction from

Electromagnetic Periodic Structures and

Its Applications

A Dissertation Submitted for the Degree of Doctor of

Philosophy at the University of London

Sunil Sudhakaran

Department of Electronic Engineering

Queen Mary

University of London

June 2006

i

Abstract

Left Handed Metamaterials (LHMs) are materials with negative permittivity

and permeability. Negative refraction occurs when electromagnetic waves pass through

LHMs at an oblique angle. It was reported that such a phenomenon can be found in

Electromagnetic Bandgap structures (EBGs) at the bandgap edges. However, the

effects of spatial dispersion, which is inherent in all periodic structures, have not fully

been probed yet. These effects are negligible only when the period is much smaller than

wavelength. Also, the conventional approaches for analysing EBGs ignore incident

field coupling and evanescent fields.

In this thesis, spatial dispersion effects in finite EBGs on negative refraction and

the possibility of obtaining analogous LHMs are investigated. It is observed that high

order Floquet’s harmonics in finite periodic structures affect the overall dispersion of

the EBGs, and hence a ‘super-cell’ model analysis is proposed and validated through

numerical simulation and experiments.

Verification of negative refraction from an EBG prism consisting of thick

metallic wires at multiple frequency bands is presented. Enhancement of negative

refraction at certain frequencies by selection of negatively refracted harmonic is

demonstrated. High resolution focusing at low microwave frequencies in the passband

from a slab structure consisting of thick metallic wires is detailed. Numerical

simulation and experimental results are presented in good agreement. Finally, the sub-

wavelength imaging using the canalization regime in a thin metallic wire medium is

experimentally investigated. The source and wire medium interaction in near field

imaging is studied and reduction in bandwidth is noticed.

ii

Acknowledgements

I express my deep sense of gratitude for my supervisor Dr. Yang Hao. This dissertation

is truly an outcome of his constant support, valuable ideas and suggestions. The

discussions with him always provided me great motivation.

I also thank Prof. Clive G. Parini for his suggestions and support.

I am thankful to the Department of Electronic Engineering, Queen Mary, University of

London for awarding me the scholarship and The Royal Academy of Engineering, UK

for the financial support for the research and travel grants.

I sincerely thank Mr. John Dupuy for his constant help in measurements.

I would like to thank Dr. Pavel Belov, working with him always provided me an

opportunity to learn more.

There are no words to express my thanks towards my colleagues at the Department of

Electronic Engineering. I express my sincere thanks to all of them.

A special line of thanks is due to my colleagues Mr. Akram Alomainy and Miss

Mariana Setta for their suggestions.

I am deeply indebted to my wife and parents; their love and support were the pillars for

the success of my research work.

Above all I thank Almighty God for showering his infinite bounties and grace upon me,

which has seen me successfully complete my research work.

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Publications

JOURNAL PUBLICATIONS

1. Pavel Belov, Yang Hao, Sunil Sudhakaran, ‘Subwavelength microwave

imaging using an array of parallel conducting wires as a lens’, Physical Review

B, Vol. 73, , pp. 033108-0033112, 2006

2. S. Sudhakaran, Y. Hao, C. G Parini, ‘Focusing of waves at low microwave

frequencies using metallic wire media’, Microwave and Optical Technology

Letters, , Vol. 48, pp.133-138, 2006

3. S. Sudhakaran, Y. Hao, C. G Parini, 'Negative refraction phenomenon at

multiple frequency bands from electromagnetic crystals’, Microwave and

Optical Technology Letters, Vol.45, pp. 465-469, 2005

4. S. Sudhakaran, Y. Hao, C. G Parini, 'An enhanced prediction of negative

refraction from EBG-like structures', Microwave and Optical Technology

Letters, Vol.41, Issue 4, pp. 258-261, 2004

CONFERENCE PUBLICATIONS

1. Pavel Belov, Yang Hao and Sunil Sudhakaran, ‘Sub wavelength imaging

using a lens formed by an array of parallel conducting wires’, Proc.

Loughborough Antennas and Propagation Conference 2006 (LAPC 2006)

Loughborough, UK, April 2006.

2. S. Sudhakaran, Y. Hao, D. Nyberg, C.G Parini, 'Negative refraction from thick

metallic wire medium', Proc. URSI General Assembly, October, New Delhi,

India, October 2005

3. Yang Hao, S. Sudhakaran, D. Nyberg, C. G Parini, H. Cory, 'Metamaterial

realisation and its applications to antennas', Proc. Progress in Electromagnetics

(PIERS) 2005, Hangzhou, China, August 2005.

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4. S. Sudhakaran, Y. Hao, C. G Parini, 'Experimental verification of wave

focussing from metallic wire media', Proc. Loughborough Antennas and

Propagation Conference (LAPC) 2005, Loughborough, UK, April 2005

5. Y. Hao, S. Sudhakaran, C.G Parini, 'Spatial harmonics selection in

metamaterials for antenna applications', Proc. of Bianisotropics 2004 (BIAN'

04)-10th Conference on Complex Media and Metamaterials, Ghent, Belgium,

September 2004

6. S. Sudhakaran, Y. Hao, C. G Parini, 'An experimental verification of spatial

harmonics effects on negative refraction', Proc. IEEE AP-S International

Symposium on Antennas and Propagation and USNC/URSI National Radio

Science, Monterey, California, USA on June 2004.

7. S. Sudhakaran, Y. Hao, C. G Parini, J. Dupuy, ' Negative Refraction from

Generalised Periodic Structures - Measurement Verification', 27th ESA

Antenna Technology Workshop on Innovative Periodic Antennas, Spain, March

2004

8. Y. Hao, L. Lu, S. Sudhakaran and C. G. Parini, 'Numerical Techniques in Left-

handed Materials Modelling', Proc. Progress in Electromagnetics Research

Symposium (PIERS 2004), Pisa, Italy, March 2004

9. Y. Hao, L Lu, S Sudhakaran and C G Parini, 'From electromagnetic bandgap

to left handed metamaterials: modelling and applications', Proc. Meta materials

for Microwave and (Sub) millimetre Wave Applications: Photonic Bandgap and

Double Negative Designs, Components and Experiments, Seminar, London,

UK, November 2003

10. Y. Hao, S. Sudhakaran, C. G Parini, ' Spatial harmonic effects on

characterisation of left handed materials’, Proc. Asia Pacific Microwave

Conference (APMC '03), Seoul, South Korea., November 2003

v

11. S. Sudhakaran, Y. Hao, C. G Parini, ' Characterisation of negative meta

materials based on Generalised periodic structure concept', Proc. 8th IEEE High

Frequency Post Graduate Student Colloquium, Belfast, Ireland, September 2003

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Table of Contents

Abstract i

Acknowledgements ii

Publications iii

Table of contents vi

List of figures xi

List of tables xxiii

List of abbreviations xxiv

Chapter 1: Introduction

1.1 Introduction 1

1.2 Left Handed Metamaterials (LHMs): A new class of materials 1

1.3 Photonic or electromagnetic Bandgap Structures (PBG/EBG) 4

1.4 Negative refraction from EBGs 6

1.5 Motivation for the research 8

1.6 Organisation of the thesis 10

References 12

Chapter 2: A Review of Negative Refraction Phenomenon

2.1 Introduction 16

2.1.2 Negative Permittivity and Permeability of materials 17

2.2 Dispersion in materials 18

2.3 Phase and group velocities 19

2.4 A brief history of negative refraction phenomenon 20

2.5 Maxwell equations and left handed phenomenon 21

2.6 Backward radiation in LHMs 23

2.7 Inverse Doppler Effect in LHMs 24

2.8 Terminology on negative refraction phenomenon 25

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2.9 Recent importance of LHMs studies 26

2.10 Realisation of LHMs 27

2.10.1 The Resonance Model- Split Ring Resonator (SRR) and Wire 28

2.10.1.1Inverse Snell’s Law verification 32

2.10.2 A planar equivalent high pass C-L medium 35

2.10.3 Composite right/ left handed (CRLH) transmission lines 39

2.10.4 Negative refraction from EBGs- The analogous LHMs 42

2.10.4.1 Analogy between EBGs and electronic crystals 42

2.10.4.2 All Angle Negative Refraction (AANR) 48

2.10.5 Drawbacks of the realisations 49

2.11 Applications of LHMs 53

2.11.1 Perfect lens 53

2.11.2 Scattering reduction using LHMs 56

2.11.3 Microwave components using LHM 58

2.11.5 Optical receivers using EBG based analogous LHM 62

2.12 Conclusions 64

References 66

Chapter3: Periodic Structures: Modelling and Negative Refraction Phenomenon

3.1 Introduction 73

3.2 Analysis of periodic structures 75

3.2.1 Circuit or network analysis 75

3.2.2 Wave analysis 75

3.3 Floquet’s theorem and spatial harmonics 78

3.4 ω- β Diagram (Brillouin diagram) 81

3.5 Unit cell and ‘Super cell’ analysis 83

3.6 Computational Electromagnetics (CEM) and simulation packages 87

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3.6.1 Finite Element Method (FEM) 89

3.6.1.1 Finite element method for electroamgentic problems 90

3.6.2 Finite Integration Technique (FIT) 91

3.7 EBG modelling in commercial packages 97

3.8 Numerical simulations of LHMs and analogous LHMs 95

3.9. Lumped element circuit simulations 103

3.9.1 C-L High pass filter circuit 104

3.9.2 L-C low pass filter circuit 106

3.10 Spatial harmonics suppression in periodic structures 108

3.11 Conclusions 111

References 112

Chapter 4: Negative Refraction Phenomenon from Metallic Wire Medium

4.1 Introduction 117

4.2 Bragg’s Law and bandgaps EBGs 118

4.3 Metallic electromagnetic bandgap structures (MEBGs) 123 4.3.1 Design of the MEBG Slab 120

4.3.2 Numerical simulation and discussion 125

4.3.3 Prism structure for Inverse Snell’s law verification 128

4.3.3.1 Design of the MEBG prism 128

4.3.3. 2. Measurement verification of refraction phenomenon 136

4.3.3.3 Inclusion of defects for enhancing negative refraction 142

4.4 Conclusions 148

References 149

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Chapter 5: Wave Focusing at Low Microwave Frequencies Using Metallic Wire

Medium


5.2 Design and numerical simulation 157

5.3. Measurement of focusing from MEBG slab 163

5.4 Discussion of results 166

5.5 Conclusions 170

References 172

Chapter 6: Experimental Investigation of Canalization and Imaging Using

Metallic Thin Wire Medium


6.2 The canalization regime for near field imaging 174

6.3. Design of Wire Medium Lens (WML) 178

6.4. Experimental verification 179

6.5. Interaction between source and lens 184

6.6 Conclusions 193

References 195

Chapter 7: Conclusions and Future work

7.1 Summary 197

7.2 Conclusions 197

7.3 Future work 198

7.3.1 Novel applications 199

7.3.1.1. Dispersive EBG for demultiplexing and beam scanning

applications 199

7.3.1.2 Sub-wavelength antennas 200

7.3.2 Theoretical extensions 200

x

7.3.2.1 Transmission enhancement 200

7.3.2.2. All angle negative refraction 200

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List of figures

Fig.1.1.A diagram showing the electric, magnetic, Poynting and wave vector directions

in a conventional right handed material (left) and in a left handed material (right). It can

be seen that the wave vector and Poynting vector are in opposite directions in

LHM…………………………………………………………………………………… 2

Fig.1.2.A ray diagram showing the wave propagation direction in a slab of RHM (left)

and LHM (right) when an electromagnetic wave obliquely incidents on it …..………..3

Fig.1.3.Diagram illustrating the application of EBG as a mirror and its comparison with

a metal reflector The EBG reflector allows propagation for waves at passband

frequencies and the metal inhibits waves at all frequencies……………………….…....6

Fig.2.1. A diagram showing the possible domains of electromagnetic materials and

wave refraction or reflection directions based on the signs of permittivity and

permeability. The arrows represent wave vector directions in each medium. There is

wave transmission only when both parameters are of same sign. Wave refracted

positively in conventional materials and negatively in LHMs

(Reproduced from [9])…………………………………………………………………18

Fig.2.2. Energy flow and wave vector diagrams in a slab having negative refractive

index n = -1. The energy flow vectors are in same direction (shown in red) and flow

wave vectors are in opposite directions (shown in green). Since n = -1 the angles of

incidence and refraction are having same magnitude. (Reproduced from [21])……...22

Fig.2.3.Doppler effect in a right handed medium (top) and in left handed medium

(bottom). A is the stationary source and B is the detector which moves with a velocity

v. The reference point is indicated in green……………………….…………………...25

Fig.2.4: Unit cell of split ring contributing negative permeability (left) and wire (right).

The wires contribute negative permittivity. SRR can be characterised by the parameters

xii

w, d, g and c. Wire array can be characterised by radius a and the spacing between

each wires. The EM wave propagation direction is along the z

direction…….………………………………………………………………………….29

Fig.2.5.A split ring and wire array having negative refractive index. The split rings

contribute negative permeability and wires contribute negative permittivity.

(Reproduced from [29])……………………………………………………………….31

Fig.2.6. Transmission through a medium consisting of SRRs only, wires only and

composite structure consisting of SRRs and wires showing increased transmission at

the frequency band 8.4-9.3GHz where the composite material posses negative

permittivity and permeability. The plasma frequency of the wire medium in this case is

around 20GHz. (Reproduced from [32)]………………………………………………31

Fig.2.7.A ray diagram showing refraction from a prism of RHM and LHM. For an

RHM prism the refracted ray bends towards the base making a positive angle with the

normal and in LHM prism the refracted ray bends away from the base making a

positive angle with the normal. θ1 is the angle of incidence and θ2 is the angle of

refraction……………………………………………………………………………….33

Fig.2.8.An experimental set-up used for Inverse Snell’s Law verification from a prism

composite medium consisting of SRRs and wires. (Reproduced from [24])…………34

Fig.2.9.A plot of refractive index vs. frequency for SRR-Wire medium. The red curve

is the real component of theoretical LHM data and the red dotted curve is the imaginary

part. The black curve gives the measured value in which the dotted line represents the

non reliable region of measurement due to experimental limitations. (Reproduced from

[24])……………………………………………………………………………………34

Fig.2.10. Measured transmitted power at 10.5GHz as a function of refracted angle from

an LHM (SRR–Wire) prism and a Teflon prism. The Teflon prism shows an angle of

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refraction of +27º (dashed curve) and for the LHM prism the angle of refraction is -61º

(solid curve) (Reproduced from [24])………………………………………………….35

Fig.2.11.A unit cell of 2D loaded C-L high pass transmission line consisting of

capacitor and inductor. Each capacitor is having impedance per unit length ZC and each

inductor has admittance per unit length YL. The unit cell length (∆d) is equal in both x

and y directions. (Reproduced from [35])…………………………………………….36

Fig.2.12. A backward antenna based on highpass C-L circuit (above) and the enlarged

view of the unit cell shown in the inset (below). (Reproduced from [36])……………38

Fig.2.13.Radiation pattern of a backward radiating CL structure. The backward

radiation is strong by 12dB at an angle of -40˚. (Reproduced from [36])…………….39

Fig.2.14.The CR-LH bandpass structure consisting of right handed transmission line

(RH-TL- high pass) and left handed transmission line (LH-TL- low pass)…………..40

Fig.2.15. Propagation (β) and attenuation (α) diagrams for a dominant mode leaky

wave antenna consisting of four regions. The LH and RH guided wave regions exist

when | β | > k0 (region I-LH and IV-RH), and leaky wave region exists when | β | < k0.

When β < 0 the structure supports LH backfire (region II) and when β > 0 the structure

supports RH endfire (region III). (Reproduced from [40])……………………………41

Fig.2.16.EFS diagram showing wave propagation through a dielectric material. The

incident wave direction shown by red arrow and the refracted wave direction is denoted

by blue arrow. The propagation direction is determined using the normal to the EFS at

the k point……………………………………………………………………………. .43

Fig.2.17. EFS in an EBG in the long wave length limit and the medium can be

considered homogeneous………………………………………………………………44

Fig.2.18. Dispersion surface in a photonic crystal at high frequency where the medium

becomes inhomogeneous. The two EFS circles overlaps and form a bandgap. Around

the region Lo Bragg’s law is satisfied and this region the bandgap exists……………..45

xiv

Fig.2.19.Schematic diagram of wave propagation through a weakly modulated

hexagonal photonic crystal (left) and for a strongly modulated photonic crystal at

different frequencies. The bandgap frequency is at ω = 0.48. ( Reproduced from

[45])……………………………………………………………………………………46

Fig.2.20.A diagram showing analogy between EBGs and electronic crystals. The

effective mass in electronic crystal (right) is analogous to refractive index in EBG

(left)……………………………………………………………………………………47

Fig. 2.21.Several iso frequency surfaces for the dielectric photonic crystal. The

frequency is written in units of 2πc/a. It can be seen that the surfaces are convex in the

vicinity of M point. (Reproduced from [44])………………………………………….49

Fig.2.22.Dispersion diagram for the photonic crystal. The all angle negative refraction

region is highlighted in red. (Reproduced from [44])………………………………….49

Fig.2.23.An illustration of the perfect lens formation from an LHM slab. The first focus

point is inside the slab and the second focus point is outside the slab at the right side.

(Reproduced from [57])……………………………………………………………….54

Fig.2.24. Experimental result showing focusing of electric field from an EBG due to

negative refraction [46]. The yellow point at the left side is the source point and yellow

point at right side is the image point (Reproduced from [42])………………………...55

Fig.2.25. Experimental result showing focusing of electric field from an EBG due to

negative refraction when the source is moved 4cm upward. It can be seen that the

image at the left position also moved by the same distance (Reproduced from

[42]……………………………………………………………………………………..56

Fig.2.26. Scattering pattern for a circular cylinder with coating. Curve l corresponds to

scattering when surface of the cylinder coated with parameters µ = -1-i*0.6 and ε = -1-

i*1, curve 2 represents the formal sign reversal for the real parts µ = 1-i*0.6 and ε = 1-

i*1. Curve 3 is the scattering when the material parameters are that of absorbing foam

xv

glass (µ = 1 and ε = 1- i*0.6). Curve 4 shows scattering from an uncoated cylinder

(Reproduced from [64])………………………………………………………………..57

Fig. 2.27.A two layer structure consisting of positive and negative refractive index

materials. The left layer is a dielectric with ε > 0, µ >0 and the right layer is a lossless

material with negative permittivity and permeability ( ε < 0, µ < 0). In the left layer the

Poyting vector (S) and wave vector (k) are parallel. In the second layer they are anti

parallel………………………………………………………………………………....58

Fig.2.28.The two layer resonating structure consisting of RHM (green) and LHM

(yellow) slabs backed by conductors at both ends (shown in red)…………………… 60

Fig.2.29. Focusing from a crystal at negative refraction frequency. All the rays being

emitted from point A pass through point B and gather again at point C. (Reproduced

from [50])………………………………………………………………………………63

Fig.2.30. Schematic drawing of a field of view expander. The boundaries have the

curvatures r1 and r2 with the same centre of origin. The inset shows a magnified part of

the crystal surface. (Reproduced from [50])…………………………………………..63

Fig.2.31. Schematic drawing showing the reduction of illumination area obtained by

applying a photonic crystal with negative refraction characteristics (Reproduced from

[50])……………………………………………………………………………………64

Fig.3.1. Diagram showing wave amplitudes in periodic structure. cn+, c+

n+1 are forward

propagating waves at the nth and (n+1)th terminals respectively and cn-, c-

n+1 are the

corresponding backward waves………………………………………………………..76

Fig.3.2.ω-β diagram of a C-L circuit showing phase velocity and group velocity at a

point x on the curve. The ω/β at the point x gives phase velocity at that point and the

slope of the curve at the point x gives group velocity…………………………………82

Fig.3.3. A Diagram illustrating the case of an infinite photonic crystal. The entire space

is occupied by the photonic crystal……………………………………………………85

xvi

Fig.3.4.A Diagram illustrating the case of a finite periodic structure. The surrounding

extends to infinity and there exist Bloch modes and evanescent modes………………86

Fig.3.5.A Diagram illustrating the numerical approach for a finite periodic structure.

The boundaries are terminated by absorbing boundary conditions……………………86

Fig.3.6. Categories within computational electromagnetics and the domain of the

packages CST Microwave studio and Ansoft HFSS………………………………….88

Fig.3.7.A cell Vi,j,k-1 of the cell complex G with the allocation of the electric grid

voltages e on the edges of A and the magnitude facet flux b through this surface……8

Fig.3.8. The allocation of the six magnetic facet fluxes which have to be considered in

the evaluation of the closed surface integral for non-existence of magnetic charges

within the cell volume…………………………………………………………………91

Fig.3.9. The spatial allocation of of a cell and a dual cell of the grid doublet G, G1

Fig.3.10. Near field intensity through multilayer LHM structure with refractive index n

= -1. (Reproduced from [34])………………………………………………………..100

Fig.3.11: Simulated field showing refraction from a wedge with positive refractive

index (left) and negative refractive index (right). Refracted rays are shown in dotted

arrows. (Reproduced from [42])……………………………………………………..101

Fig.3.12. Simulated electric field intensity showing positive refraction at 9GHz and

positive refraction at 7GHz from a metallic EBG (Reproduced from [43])………….103

Fig.3.13. A unit cell of a high pass C-L filter with C= 0.5pF and L= 1.0nH………...104

Fig.3.14. Response of the C-L filter showing pass band from about 5 GHz…………104

Fig.3.15. Dispersion diagram for the high pass C-L filter with unit element………...105

Fig.3.16.Dispersion diagram for the high pass C-L filter with three elements………105

Fig.3.17. A unit cell of a low pass L-C filter with C= 1.5pF and L= 1.5nH…………106

Fig.3.18 Response of the L-C filter showing pass band up to about 4.8 GHz……….107

Fig.3.19. Dispersion diagram for the high pass L-C filter with unit element………..107

xvii

Fig.3.20.Dispersion diagram for the high pass L-C filter with three elements………108

Fig.3.21.A Pseudo Periodic Waveguide: A comb type structure (Reproduced from

[46])…………………………………………………………………………………..109

Fig.4.1.A diagram showing passband and bandgap behaviour of an EBG…………..118

Fig.4.2.A diagram illustrating Bragg’s law…………………………………………..119

Fig.4.3.Complete bandgaps for TE modes as functions of r/a for an EBG with square

lattice consisting of metallic wires (Reproduced from [11])…………………………120

Fig.4.3.Schematic diagram of square lattice of period a consisting of metallic wires

of radius r……………………………………………………………………………..121

Fig.4.4. A diagram showing a metallic wire array which acts like a metallic surface at

low frequencies (since the wavelength is large in this case the spacing between the

wires are very small compared to the wavelength) and when frequency increases

(wavelength decreases) the spacing between the elements becomes comparable to the

wavelength……………………………………………………………………………122

Fig.4.5. A schematic diagram of the MEBG structure used for numerical simulation

studies and vector directions of electric field E, magnetic field H and propagation

vector k……………………………………………………………………………….124

Fig.4.6: Transmission response of the wire periodic structure showing bandgap around

8.5 GHz……………………………………………………………………………….126

Fig.4.7: Dispersion diagram for the wire periodic structure with period 3.15 for a

direction of propagation along x axis……………………………………………...….126

Fig.4.8: A plot indicating magnetic field propagation directions outside and inside the

periodic structure at 7.5 GHz(left) and 6.5 GHz (right). The propagation directions in

each media are indicated by arrows. Surface normal is shown in dotted lines………127

Fig.4.9. Diagram of the prism consisting of metallic wires with period 1.75

cm…………………………………………………………………………………….129

xviii

Fig.4.10. Unit cell assignment in CST simulation model for the simulation of bandgap

diagram for EBGs consisting of square lattice and wave propagation directions Γ, X

and M…………………………………………………………………………………130

Fig.4.11.CST Simulation model for bandgap determination. Four side walls are defined

as periodic boundaries and bottom and top walls are defined as magnetic

boundaries……………………………………………………………………………131

Fig.4.12.Bandgap diagram for the metallic wire structure with r/a = 0.36 (TE

polarisation). Green region indicates complete bandgap…………………………….132

Fig. 4.13. Measured transmission response for the prism structure showing bandgap

around 5.7GHz……………………………………………………………………….133

Fig.4.14: Dispersion Diagram for the prism periodic structure with metallic wires

having a period of 1.75 cm. The bandgap region is marked in blue…………………134

Fig.4.15. Electric field plot showing positive refraction at 6.8 GHz. Units are

in V/m………………………………………………………………………………..135

Fig.4.16. Electric field plot showing negative refraction at 7.4 GHz. Units are

in V/m………………………………………………………………………………..135

Fig.4.17. A diagram showing the setup used for experimental studies………………136

Fig.4.18. Two prism structures used for measurement: Prism structure consists of

metallic wires (left) and Teflon prism (right)……………………………………...…137

Fig.4.19. Simulated electric field showing positive refraction from Teflon prism at

4.7GHz………………………………………………………………………………..138

Fig.4.20. Simulated electric field showing positive refraction from Teflon prism at

4.7GHz………………………………………………………………………………..139

Fig.4.21. Measured refracted field for Teflon prism at 4.7 GHz. The angle of refraction

is +23º……………………………………………………………………………….139

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Fig.4.22: Measured refracted field for wire prism showing positive refraction at 4.7

GHz and negative refraction at 5.6 GHz, 7.2 GHz and 7.4 GHz……………………..140

Fig.4.23. Diagram of radiating dipole inside an EBG with defects for changing the

radiation pattern and corresponding radiation patterns. The blue dots indicate the defect

positions. (Reproduced from [32])…………………………………………………...144

Fig.4.24. Simulated electric field at 4.3 GHz on prism with defects showing negative

refraction……………………………………………………………………………..145

Fig.4.25. Measured refracted field from wire prism structure showing positive

refraction at 4.3 GHz, 4.7 GHz and negative refraction at 4.3 GHz after the introduction

of the defects into the structure………………………………………………………146

Fig.4.26. Dispersion diagrams for the wire structure without the defects and with the

defects has been introduced………………………………………………………….146

Fig.5.1. Wave focusing in a slab with negative refractive index. n = -1. A is the source

point, B and C are first and second image points respectively……………………….154

Fig.5.2. Simulated electric field plot showing focusing from a slab of refractive index n

= -1……………………………………………………………………………………155

Fig.5.3. A graphical representation simulated and measured average power intensity

distribution at the image plane with the EBG and the intensity without the EBG

structure. The red curve shows the FDTD simulation result and the blue line represents

the experimental result. The green dotted curve shows the field intensity without the

EBG. It can be seen that there is strong field intensity at the centre corresponding to the

image point. (Reproduced from [3])…………………………………………………156

Fig.5.4. Schematic diagrams of the structure used for focusing studies (left) and the

experimental setup used for the experiment. There are four elements in z direction and

8 elements in y direction in the wire structure……………………………………….158

xx

Fig.5.5.Dispersion diagram obtained for incident field perpendicular to the wire axis.

The negative refraction regions and bandgap regions for the incident field direction

along the z axis are marked in the frequency region starting from 5.5GHz………….159

Fig.5.7.Simulated electric field intensity along the yz plane at 2.1GHz showing

focusing. The focus point is marked in the rectangular box………………………….160

Fig.5.8.Simulated electric field intensity along the yz plane at 3.4GHz showing no

focusing……………………………………………………………………………….161

Fig.5.9.Simulated electric field intensity along xz plane at a distance of 2.4cm from the

source…………………………………………………………………………………161

Fig.5.10. Simulated electric field intensity along xz plane at a distance of 3.4cm from

the source…………………………………………………………………………….162

Fig.5.11. Simulated total electric field intensity along xz plane at a distance of 7cm

from the source……………………………………………………………………….162

Fig.5.12.A schematic diagram of the probe used (left) and a picture of the coaxial fed

probe used in the experiment for transmission and reception (right) ………………..163

Fig.5.13. Measured electric field intensity along xy plane from the slab consisting of

metallic wires at 2.1GHz showing focusing………………………………………….164


metallic wires at 3.4GHz showing no focusing………………………………………165


metallic wires at 4.2GHz showing focusing………………………………………….166

Fig.5.16. Simulated electric field intensity along yz plane at 2.1 GHz with one layer of

wires…………………………………………………………………………………..168

Fig.5.17. Simulated electric field intensity along the yz plane at 2.1GHz without any

slab structure showing no wave focusing ……………………………………………168

xxi

Fig.5.18. Simulated electric field intensity along the yz plane at 2.1GHz with a slab of

permittivity 2.0 showing no wave focusing…………………………………………..169

Fig.5.19. Simulated electric field intensity long the yz plane at 2.1GHz with a slab of

permittivity 80.0 showing no wave focusing…………………………………………170

Fig.6.1. Geometry of a slab lens consisting of metallic wires. The length of the wires is

an integer multiple of λ/2…………………………………………….………………175

Fig.6.2. A photograph of the wire medium lens with the P loop source (left) and a

diagram showing the dimensions of the P loop source……………………………….179

Fig.6.3. Distribution of electric field and its absolute value: (A), (C) in the vicinity of

the source (at 2.5mm distance from the front interface; (B), (D) at 2.5mm distance from

the back interface; (E) the transverse plane and (F) the intensity along the transverse

plane (reproduced from [9])…………………………………………………………180

Fig.6.4. A photograph of the two probes used for the measurement studies…………181

Fig.6.5. A diagram of the straight probe and the bend probe used for the measurements

and corresponding electric field directions…………………………………………...181

Fig.6.6. Measured near field intensity distribution along the yz plane: Absolute values

of x, y and z components of the electric fields (A, C, E, G) in source plane in image

plane (B, D, F, H)…………………………………………………………………….183

Fig.6.7.A photograph of the Meander shaped radiating source used for the experiment

and a diagram showing dimensions of the loop………………………………………185

Fig.6.8.Measured electric field intensity distribution in the yz plane at 0.85GHz (A) and

0.86GHz (B). Source fields with and without lens (left and right respectively) and the

image field (middle) are given………………………………………………………..187

Fig.6.9.Measured electric field intensity distribution in the yz planes at 0.87GHz (A),

0.88GHz (B), 0.89GHz (C) and 0.9GHz (D). Source fields with and without lens (left

and right respectively) and the image field (middle) are given………………………188

xxii

Fig. 6.10. Measured electric field intensity distribution in the yz plane at 0.91GHz (A)

and 0.92GHz (B), 0.93GHz (C) and 0.94GHz (D). Source fields with and without lens

(left and right respectively) and the image field (middle) are given………………..189

Fig.6.11.Measured electric field intensity distribution in the yz plane at 0.95 GHz (A),

0.96 GHz (B), 0.99 GHz (C) and at 1 GHz (D). Source fields with and without lens

(left and right respectively) and the image field (middle) are

given………………………………………………………………………………...190

Fig.6.12.Measured electric field distribution at 155 mm away in front of the source

showing no imaging………………………………………………………………....192

Fig.6.13.Measured electric field distribution at 3 mm away from the foam sheet at the

backside of the source (12 mm from the source) and at 3 mm away from the foam sheet

at the front side at 0.88 GHz. In both cases the near field distribution is the same and

there is no considerable transmission loss……………………………………………193

Fig.7.1. A schematic diagram illustrating the application of MEBG prism as a spatial

frequency demultiplexer. The red arrow represents the polychromatic wave and the out

coming beams of different frequencies are shown in different colours and the

corresponding angles from the normal are marked…………..………………………199

xxiii

List of tables

Table.3.1. A comparison of FIT, FEM methods…………………………………….97

Table.4.1. Three steps for obtaining the complete bandgap diagram and

their corresponding phase shifts for the parameter sweep………………………….131

Table: 4.2. Comparison of measurement results at 7.2GHz with similar works for

verification of inverse Snell’s law using different structures………………………142

xxiv

List of abbreviations

AANR All Angle Negative Refraction

CR-LH Composite Right Handed- Left Handed

DNG Double Negative

EBG Electromagnetic Bandgap

FEM Finite Element Method

FIT Finite Integration technique

LHM Left Handed Metamaterial

RHM Right Handed Material

RH-TL Right Handed Transmission Line

LH-TL Left Handed Transmission Line

PC Photonic Crystal

MEBG Metallic Electromagnetic Bandgap

SRR Split Ring Resonator

TL Transmission Line

WML Wire Medium Lens

1

Chapter 1

Introduction

1.1 Introduction

Electromagnetics has received great attention among researchers all over the

world because of its immense civilian and defence applications. During the Second

World War, the use of radar and thereafter the wide use of microwave communication

systems facilitated the transformation from radio to microwave frequency. This

dramatic change demanded more advanced materials for high frequency performance

and opened up new dimensions in the field of electromagnetic materials. Artificial

materials were developed with desired dielectric and magnetic characteristics. Now

modern fabrication facilities can enable more advanced materials with superior

characteristics which cannot be obtained in nature. Such artificial materials with the

properties which cannot be found in ubiquitous materials are called Metamaterials.

Nano-composites and electromagnetic bandgap structures are examples of

metamaterials.

1.2 Left Handed Metamaterials (LHMs): A new class of materials

In 1968 Russian physicist Vaselago proposed the possibility of materials with

both dielectric parameter permittivity (ε) and magnetic parameter permeability (µ)

negative [1]. He referred these materials with negative parameters as Left Handed


2

Metamaterials (LHMs). When an electromagnetic wave passes through these materials,

the electric field vector ( Ev

), magnetic field vector ( Hv

) and the wave vector ( kv

) obey

a left handed rule whereas in natural materials these vectors obey a right handed rule

and they have positive material parameters. Such materials with positive parameters are

termed as Right Handed Materials (RHMs). Fig.1.1 shows the wave vector directions in

RHMs and in LHMs. It can be seen that the pointing vector Sv

and wave vector kv

are

in opposite directions in an LHM.

Fig.1.1. A diagram showing the electric, magnetic, Poynting and wave vector directions in a

conventional right handed material (left) and in a left handed metamaterial (right). It can be

seen that the wave vector and Poynting vector are in opposite directions in LHM

LHMs show interesting characteristics such as Inverse Snell’s law, backward

radiation and Inverse Doppler effect [1]. Fig.1.2 shows the wave propagation directions

using ray diagrams in a slab of conventional RHM and LHM, when an electromagnetic

wave obliquely incident on it. It can be seen that the direction of wave refraction at the

first interface is upward (inside the slab) with respect to the normal in the RHM slab

and downward in the LHM slab. Recently, a composite structure consisting of split ring

resonators (SRRs) and conducting wires was identified as a medium with negative

refractive index [2]. In this composite medium, an array of split rings gives rise to

Hv

Ev

kv

Sv

Ev

kv

Sv

Hv

RHM LHM


3

effective negative permeability and an array of wires exhibit negative effective

permittivity.

Fig.1.2.A ray diagram showing the wave propagation direction in a slab of RHM (left) and

LHM (right) when an electromagnetic wave obliquely incidents on it.

When these structures are combined, an effective negative refractive index at the

overlapping frequency bands can be enabled. This was experimentally verified by

Shelby et al. [3] in demonstrating the Inverse Snell’s Law from a structure consisting of

SRRs and wires. Later, a transmission line model was proposed having LHM

characteristics. It was demonstrated in [4] that a periodic array of capacitor-inductor

(C-L) high pass filter network showed equivalent effective negative parameters at the

regimes of effective medium theory (EMT).

In this medium, the capacitors contribute an effective negative permittivity and

inductors give effective negative permeability. Lately, some interesting applications of

LHMs such as a perfect lens which is free from all aberrations, sub-wavelength

resonators, zero phase delay lines etc. were put forward [5, 6]. A detailed analysis of

LHMs realisations and its applications will be presented in chapter 2 of this thesis.

RHM

LHM


4

1.3 Photonic or Electromagnetic Bandgap Structures (PBG/EBG)

PBGs are periodic arrangements of dielectric or metallic elements in one, two or

three dimensional manners. PBGs inhibit the passage of electromagnetic wave at

certain angles of incidence at some frequencies. These frequencies are called partial

bandgaps. At a specific frequency band, a PBG does not allow the propagation of wave

in all directions and this frequency region is called the complete bandgap or global

bandgap [7-9]. The original idea of PBGs was put forward by physicists and some

recent studies revealed the interesting fact that PBGs exist in living organisms [10-12].

The well known examples are the butterfly scales [11] and eyes of some insects [10]. In

biological systems, there is no metal. In this case, a metallic like reflection effect is

obtained by using refractive index differences. A multilayer thin film with different

refractive indices in animals is a good example for this. Recently, these ideas were

undergone a preliminary study for its commercialisation such as paints for certain

applications [10].

PBGs can be realised in one, two and three dimensional forms. The

dimensionality depends on the periodicity directions. Three dimensional PBGs are

more appropriate for getting a complete bandgap because they can inhibit waves for all

incident angles. The bandgap in PBGs is analogous to a forbidden energy gap in

electronic crystals. Hence PBGs are also termed as photonic crystals (PCs). The first

attempts towards three dimensional structures were realised in the form of face centred

cubic (fcc) lattice structures [13]. At the initial stages of PBG research, due to the lack

of theoretical predictions, a ‘cut and try’ approach was adopted in experimentally

predicting the bandgap. At the beginning, the investigations of PBGs were mainly on

wave interactions of these structures at optical frequencies and hence PBGs emerged

with the name of photonic bandgap structures. Now, vast extensions of PBGs at

microwave [14], millimetre [15] and sub-millimetre [16] wave frequencies are


5

common. At microwave frequencies, such structures are popularly called as

Electromagnetic Bandgap (EBG) structures [17]. In the next sections of this thesis,

these materials are addressed as EBGs.

A periodic structure can give rise to multiple bandgaps. However, it should be

noted that the bandgap in EBG is not only due to the periodicity of the structure but

also due to the individual resonance of one element. A study by S. John et al. [13]

revealed the mechanisms to form a bandgap in an EBG. He proposed that the bandgap

formation in EBGs is due to the interplay between macroscopic and microscopic

resonances of a periodic structure. The periodicity governs the macroscopic resonance

or the Bragg resonance. It is also called the lattice resonance. Microscopic resonance is

due to the element characteristics and it is called the Mie resonance [18]. When the two

resonances coincide, the structure possesses a bandgap having maximum width.

Depending on the structural characteristics and polarisation of the wave, one resonance

mechanism (i.e. either the multiple scattering resonance or the single element scatterer

resonances) can dominate over the other [18]. The characteristic property of stop bands

at certain frequencies enables many applications using EBGs. At this stop band, all

electromagnetic wave will be reflected back and the structure will act like a mirror [18].

The advantage over a metal reflector is that for an EBG, reflection takes place only at

stopband frequency. At other frequencies it will act as transparent medium. This

concept is illustrated in fig.1.3.

EBGs found their applications in improving the performance of microstrip

antennas. Microstrip antennas mounted on a substrate can radiate only a small amount

of its power into free space because of the power leak through the dielectric substrate

[20]. In order to increase the efficiency of the antenna, the propagation through the

substrate must be prohibited. In this case, the antenna can radiate more towards the

main beam direction and hence increase its efficiency.


6

Fig.1.3.Diagram illustrating the application of EBG as a mirror and its comparison with a metal

reflector The EBG reflector allows propagation for waves at passband frequencies and the

metal inhibits waves at all frequencies.

An antenna with an EBG substrate can radiate more efficiently at frequencies

corresponding to the bandgap region [17, 20-22]. At this region, the substrate will not

allow radiated power to pass through it and the efficiency of the antenna will be

increased. In [20] such a design was detailed for a patch antenna operating at 14GHz.

The gain was increased by 1.8 dB corresponding to a radiated efficiency increment of

2%. EBGs found interesting wave-guiding applications by applying line defects which

can be obtained by removing a line of elements from the EBG. This produces defect

propagation modes inside the bandgap. Using this approach, a 90 degree bend [23, 24]

can be obtained which is not possible with conventional waveguides. The

characteristics of EBGs are completely not obtainable from naturally occurring

materials and hence EBGs belong to the class of Metamaterials.

1.4 Negative refraction from EBGs

It was revealed that negative refraction phenomenon can be observed in EBGs

due to the strong dispersion characteristics at certain frequencies [25, 26]. In [27], it

was shown that such a phenomenon was possible in the regimes of negative group


7

velocity and negative effective index at higher bands. It was also identified that at

certain frequencies, due to the negative effective mass of the crystal which is analogous

to a negative effective refractive index, negative refraction can be obtained [28,29]. The

negative refraction phenomenon from a dielectric EBG was investigated in [30] and it

was observed that the EBG supports the phenomenon for all incident angles at a narrow

band frequency. This was due to the negative ‘photonic effective mass’ of the crystal.

Wave focusing from a dielectric slab with periodic air holes was demonstrated using

numerical simulation in [31]. It was made clear that neither a negative group velocity

nor a negative effective index is a prerequisite for negative refraction [31, 32].

Negative refraction phenomenon in EBGs is different from that in LHMs [29].

The advantage in obtaining negative refraction from EBGs is that it can explore the

phenomenon at optical frequencies at which the resonance models [2] for LHMs will

show limitations due to effective medium theory approximations. It appears that such

approximations are possible in very narrow band frequencies in the microwave regime

[30]. An EBG at negative refraction frequencies may not have negative effective

parameters [29]. This narrow frequency band at which the EBG exhibits negative

refraction characteristics were identified near the bandgap edges [29, 30]. The wave

vector diagrams for infinite EBGs at a constant frequency, called the equi-frequency

surfaces [25-29] were used for all these investigations. However, in an EBG, the

characteristics can not be completely governed by a single wave vector but by a

combination of many [30]. Also, in a finite EBG the dispersion characteristics are not

only governed by Bloch modes inside the EBG but also the evanescent modes and

coupling factors between the incident field and the finite structure [30], which were

completely neglected in the infinite EBGs analysis. Since the mechanism of negative

refraction from EBGs is different from that in LHMs, it is more accurate to regard such

EBGs as analogous LHMs rather than LHMs.


8

1.5 Motivation for the research

In the last few years, the field of metamaterials (LHMs and EBGs) received

much interest. Novel realisations of LHMs and their applications are in great demand.

It is noted that the LHMs or the analogous LHMs realised so far belong to the general

class of EBGs. This leads to great enthusiasm on the roles of periodic structures in

negative refraction phenomenon and the factors that need to be addressed in finite

structures especially the spatial dispersion which is inherent in all EBGs. Even in the

larger wavelength limit, the spatial dispersion can be strong in some EBGs, for

example, a metallic wire medium [25].

The research carried out in this thesis investigates the possibilities of obtaining

the negative refraction phenomenon from EBGs. Based on the EMT approach, the

period of the structure should be very small compared to the operating wavelength

(typically 1/100th of the wavelength) [32]. This limits its applicability for most of the

practically realisable structures at high frequencies and even at microwave frequencies.

When the period is comparable to the wavelength, the spatial dispersion in the structure

will be very high [25]. Also, in this case the effects of high order Floquet’s harmonics

are not negligible [34]. So the medium can not be characterised by assigning a single

permittivity and permeability value for throughout the finite structure. In other words,

the parameters depend on the spatial coordinates and this causes spatial dispersion. As

a result of this, the medium will not be homogenous.

The conventional analysis by using a unit cell of the infinite model with

periodic boundaries has taken only the characteristics of Bloch modes into account. In

fact, in a finite EBG there exist evanescent modes and a coupling between the incident

field and the structure. The unit cell analysis will give only fundamental Floquet’s

harmonic interaction which is not adequate for the complete dispersion nature of finite


9

crystals. This indicates that finite EBGs need to be addressed separately for negative

refraction studies.

In this work, the possibility of negative refraction due to the spatial dispersion

and from Floquet’s harmonics interactions at frequencies in the passband is

investigated. Imaging applications using the metamaterial nature of the EBGs are also

presented. An approach to enhance the dispersion prediction of finite EBGs is

proposed. It has been identified that at certain narrow frequency bands the structure

gives rise to negative refraction. Introduction of defects into the structure for enhancing

negative refraction at certain frequencies is verified. Negative refraction at narrow

frequency bands in passband are identified and confirmed by numerical simulations and

experiments.

In this work, wave focusing from metallic EBGs is demonstrated at low

microwave frequencies. Earlier works [34, 35] in this subject considered dielectric

EBGs and the measurement was at higher frequencies in one dimensional plane. This

does not guarantee the exact resolution of the image. The proposed idea of negative

refraction using spatial dispersion enables wave focusing at low frequencies in the

passband. In order to investigate the resolution of the image formed by a finite EBG

consisting of thick metallic wires, focusing measurement was performed on a two

dimensional scanning plane and good resolution of an image is obtained. The

phenomenon is observed from 2.1 GHz at multiple bands. In this work, dispersion and

negative refraction studies for realisable, finite EBGs is carried out and identification of

frequency regions where the structure shows negative refraction is carried out.

The metamaterial behaviour of a medium consisting of periodic array of thin

metallic wires is demonstrated for sub-wavelength imaging. Experimental observations

illustrated sub-wavelength imaging over a broad bandwidth. This imaging regime is


10

called canalisation. A study on the interaction of source and the proposed wire medium

lens was studied experimentally.

Simulation results were obtained using commercial packages High Frequency

Structure Simulator (HFSS™) from Ansoft Inc., Advanced Design System™ (ADS)

from Agilent Technologies and CST Microwave Studio™. Measurements were carried

out and the results obtained were found to be in good matching with numerical

simulations.

1.6 Organisation of the thesis

The thesis is divided into seven chapters. The organisation of the chapters can be

outlined as follows.

Chapter 1: In this chapter, an introduction to metameterials, LHMs and EBGs are

presented with the motivation factors that lead to the work presented in the next

chapters of the thesis.

Chapter 2: Literature review on LHMs and their characteristics, realisations of LHMs

including resonance model, EBG model, high pass C-L model, the composite right

handed- left handed media (CR-LH) and negative refraction from infinite EBGs are

presented in this chapter. Interesting applications of LHMs such as perfect lens

focusing, backward radiating antennas, sub-wavelength resonators and phase shifters

are also detailed.

Chapter 3: This chapter describes general approaches on the analysis of EBGs; wave

amplitude analysis and unit cell approach for infinite structures. Spatial dispersion in

finite models and its effects on negative refraction in finite EBGs are also discussed.

An overview on computational approaches used in this thesis and a brief review on

numerical simulations on LHMs and analogous LHMs is also presented. The proposed

super cell analysis to include higher order spatial harmonics and dispersion has been


11

verified with numerical simulations of lumped circuit models using Agilent ADS™. It

is observed that when more elements are included for analysis the backward radiation

frequency regions start deviating from that of the unit cell model.

Chapter 4: In this chapter, metallic electromagnetic periodic structures (MEBGs) are

studied in detail for negative refraction phenomenon. It is observed that such a

phenomenon can be obtained at narrow frequencies in passband due to the strong

spatial dispersion from the finite structures. Negative refraction is verified using

numerical simulations and measurements. It is shown that introduction of defects into

the structure can enhance the negative refraction at certain frequencies. Measurement

results are in good agreement with simulation results.

Chapter 5: Wave focusing from MEBGs is detailed in this chapter. High resolution

focusing at low microwave frequencies in the passband is presented. A resolution of

λ/7 is demonstrated. Two dimensional scanning is performed to verify the focusing

phenomenon from the structure. The measurement results are matching with that

obtained from numerical simulations. The metamaterial behaviour of the slab under

study has been further verified with numerical simulations using more cases. It is

illustrated that the observed phenomenon is due to the metamaterial nature of the

MEBG slab.

Chapter 6: In this chapter, experimental investigation on canalization, a near field sub-

wavelength imaging regime using metamaterials consisting of metallic wires is

presented. High resolution images were obtained using metallic wire media at a

resolution of λ/15. It is proved using experimental results that the source - wire medium

interactions in near field imaging systems reduce the bandwidth of the imaging regime

but the image resolution is maintained.

Chapter 7: This chapter presents the conclusions of the research carried out and the

proposed future work directions.


12

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13

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746-754, 2003


15

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16

Chapter 2

A Review of Negative Refraction

Phenomenon

2.1 Introduction

In mechanical, electrical and thermal scenarios, materials exhibit different

characteristics. The electrical and magnetic properties are used to describe the

characteristics of a material when it is exposed to an electromagnetic field. Permittivity

(ε) and permeability (µ) are two parameters to govern electromagnetic performance of

all materials. The permittivity is a measure of how much a medium changes to absorb

energy when subject to an electric field [1-3]. It is defined as the ratio EDv

v

where Dv

is

the electric displacement by the medium and Ev

is the electric field strength. The

common term dielectric constant is the ratio of permittivity of the material to that of

free space (ε0 = 8.85 * 10-12 F/m). It is also termed as the relative permittivity [2, 3].

Permeability is a constant of proportionality that exists between magnetic induction and

magnetic field intensity. Free space permeability (µ0) is approximately 1.257 x 10-6

H/m. In other materials, it is usually greater than the free-space value. Due to the recent

advances in material fabrication techniques, highly engineered materials that exhibit

superior properties which can not be observed in naturally occurring materials can be

made. These structured composite materials are known as Metamaterials [4, 5].

Chapter 2: A Review of Negative Refraction Phenomenon_

17

Left handed metamaterials (LHMs), EBGs and artificial magneto-dielectric materials

[6] are examples of metamaterials. Among them, LHMs received great research

interests in the past few years. In the first part of this chapter, an overview of LHMs,

their properties and realisations based on resonance, high pass C-L and dispersive EBG

models are presented. This is followed by some exciting applications of LHMs and

their analogous.

2.1.2 Negative Permittivity and Permeability of materials

Negative permittivity can be obtained from some naturally available materials.

It can be achieved from metals at frequencies near their plasmonic resonance [7, 8].

Free electron gas with permittivity ε = 1- ω2p /ω2 is negative when the frequency

ω < ωp, where ωp is the plasma frequency. Some crystals such as SiC, LiTaO3, LiF and

ZnSe can have negative permittivity at certain frequencies governed by their dispersion

relations [8]. Materials with negative permeability do not occur in nature. They have to

be constructed artificially.

Fig.2.1 illustrates a graphical representation of different material possibilities

for electromagnetic applications based on the signs of their permittivity and

permeability values and refraction and reflection at an interface between air and each

medium [9]. There are four regions in the diagram. Plasma belongs to the region with

negative permittivity and positive permeability. Split rings belong to the region of

negative permeability and positive permittivity. It can be seen that when the two

parameters are of opposite signs, there is no wave transmission. This is because when

the parameters are of opposite signs, the wave vector becomes imaginary. When both

parameters are positive, refraction occurs positively and when they are negative,

refraction occurs negatively.


18

Fig.2.1.A diagram showing the possible domains of electromagnetic materials and

wave refraction or reflection directions based on the signs of permittivity and

permeability. The arrows represent wave vector directions in each medium. There is

wave transmission only when both parameters are of same sign. Wave refracted

positively in conventional materials and negatively in LHMs (Reproduced from [9]).

2.2 Dispersion in materials

Dispersion is a phenomenon that causes the separation of a wave into

components with different frequencies, due to a dependence of the velocity of wave on

its frequency [3, 10]. A well familiar example of dispersion is the splitting-up of white

light into its components when it passes through a glass prism. There are generally two

sources of dispersion. First is the material dispersion, which results from a frequency

dependent response of a material when a wave passes through it. The second source is

the waveguide dispersion, which arises when the transverse mode solutions for waves

confined within a finite waveguide. This depends upon the frequency of operation.

LHMs


19

There is also modal dispersion, which comes when a signal consists of a superposition

of multiple modes at single frequency. In this case, different modes generally travel at

different speeds resulting in dispersion of temporal features.

2.3 Phase and group velocities

The phase velocity of a wave is the rate at which the phase of the wave

propagates in a medium [11]. This is the velocity at which the phase of any one

frequency component of the wave will propagate. One particular phase of the wave (for

example the crest) would appear to travel at the phase velocity. The phase velocity is

given by

k

v p vv ω

= (2.1)

where ω is the frequency and kv

is the wave vector.

Group velocity is the velocity with which the envelope of the wave propagates [3, 11].

The group velocity gvv is given by

1−

−=

λλ

ddn

ncvgv

Where n is the refractive index of the medium, c is the velocity of light and λ is the

wavelength. In most cases, this is true and the group velocity is same as the signal

velocity of the waveform. But in certain cases, where the wavelength of light is close to

an absorption resonance of the medium, it is possible for the group velocity to exceed

the speed of light (vg > c), leading to the conclusion that superluminal (faster than light)

communication is possible [3]. In such situations, the value of group velocity becomes

meaningless, and does not represent the true signal velocity of the wave. The group

(2.2)


20

velocity itself is a function of frequency. This results in group velocity dispersion

(GVD), which can be quantified as the group delay dispersion parameter D.

= 2

2

λλ

dnd

cD

Due to GVD, a short pulse of light spread in time as different frequency components

can travel at different velocities. If D is less than zero, the medium is having positive

dispersion. If D is greater than zero, the medium has anomalous dispersion. If light

propagates through a normally dispersive medium, the result is that the higher

frequency components travel slower than the lower frequency components. If a pulse

travels through an anomalously dispersive medium, high frequency components travel

faster than the lower ones. So frequency decreases when the time increases. Dispersion

management and control is very important in communication systems especially at

optical frequencies [9].

2.4 A brief history of negative refraction phenomenon

Even though negative refraction received great attention recently, its possibility

has been suggested in some other ways long time ago. The existence of backward

waves in mechanical systems was suggested by H. Lamb [12, 13]. He identified that for

these waves the phase moves in the direction opposite to the energy flow indicating

opposite signs for phase and group velocities. However, Lamb has not given insight

into electromagnetic systems. Backward waves in electromagnetic systems were first

investigated by A. Schuster [14]. He detailed an optical refraction phenomenon in such

a media, if a material with such properties was ever to be found. So the possibility of

negative refraction was first appeared in this work. Schuster has not detailed what

could be the characteristic properties of such a medium. Pocklington [15] illustrated

(2.3)


21

that in a specific backward-wave medium, a source produces a wave whose group

velocity is directed away from the source, while its phase velocity moves toward the

source. In 1968 V. G Veselago [16] predicted the electrodynamics of materials with

both negative parameters [16] and he showed that such materials can support reversed

optical phenomenon such as Inverse Doppler Effect, backward radiation etc. This was

a major milestone in the history of LHMs.

2.5 Maxwell equations and left handed phenomenon

If a material has negative parameters, it must hold the energy propagation

relations given by Maxwell Equations [20]. The four Maxwell equations are given by

0=•∇

=•∇

+∂∂

=×∇

∂∂

−=×∇

B

D

jtDH

tBE

v

v

vv

v

vv

ρ

In order to see the effect of sign in material parameters, the relations in which ε, µ

appear separately rather than in the form of their product need to be taken. These

equations are the two Maxwell equations 2.4 and 2.5 given above and the constitutive

relations.

ED

HB

vv

vv

ε

µ

=

=

(2.4) (2.5)

(2.6)

(2.8) (2.9)

(2.7)


22

Applying these equations for a plane monochromatic wave, it can be seen that if µ < 0

and ε < 0, then Ev

, Hv

and kv

form a left handed set of vectors [16]. If αi, βi and γi are the

direction cosines for Ev

, Hv

and kv

vectors respectively, then a wave propagating in a

given medium will be characterised by the matrix

=

21

21

1

3

3

γγγβββααα 32

G

The determinant of this matrix p equals +1 if Ev

, Hv

and kv

vectors are right handed set.

It is -1 if this set is left handed [16]. If the material parameters are negative, the

determinant p is -1. Hence these types of materials are called Left Handed

Metamaterials (LHMs).

Fig.2.2. Energy flow and wave vector diagrams in a slab having negative refractive index n = -

1. The energy flow vectors are in same direction (shown in red) and flow wave vectors are in

opposite directions (shown in green). Since n = -1 the angles of incidence and refraction are

having same magnitude. (Reproduced from [21])

(2.10)


23

2.6 Backward radiation in LHMs

Consider an electric field Ev

polarized along the unit vector direction pv . The

electric field Ev

and magnetic field Hr

can then be expressed as [17, 18]

rkiepEvvvv

•=

rkiepkHvv

vvv •×= )(1ωµ

From Maxwell’s equation, the permeability and permittivity are related with electric

and magnetic fields as follows [17]

From Poynting Theorem, the power flow direction can be obtained by the time

averaged Poynting vector and is given by [18]

)(1

*)(

pkp

HERS

vvv

vvv

××=

×=⟩⟨

ωµ

where R is a real operator. On expanding the triple product in eqn.2.15, we get

[ ]pkpkppS vvvvvvv).().(1 −=⟩⟨

ωµ

and 0).( =kp

vv due to Gauss’s Law. Hence eqn.2.16 becomes

(2.16)

(2.11)

(2.12)

(2.14)

(2.13)

(2.15)

)()(- )(

)()( )(

rErHk

rHrEk

vvvvv

vvvvv

ωωε

ωωµ

=×

=×


24

Considering the fact that for wave propagation, both ε, µ should have the same sign (to

obtain a real wave vector), eqn.2.17 is valid only when both parameters have the same

sign. Hence it is clear from eqn. 2.17 that when both ε and µ are negative, then time

averaged Poynting vector Sv

will be in opposite direction of the phase propagation

vector. This leads to backward radiation characteristic of LHM.

2.7 Inverse Doppler Effect in LHMs

Doppler Effect is the noticeable change in frequency of a wave when an

observer moving away or towards the source of the wave. Let a detector of radiation

move relative to a source of radiation with a frequency ω0. During its motion, the

detector will observe points of the wave which correspond to some definite phase. For

example, crest of the wave. The formula for Doppler shift can be written as [16]

+=

uvp10ωω

where v is observer velocity and u is the energy flux velocity. In this equation p denotes

that the medium is either RHM or LHM. If p = +1 the medium is RHM and if p = -1 the

medium is LHM. If the detector is in a medium of positive refractive index, the

frequency received by the detector will be larger than ω0 since the argument in the

bracket in eqn. 2.18 is greater than 1 ( p = +1 in this case). From fig.2.3, it can be seen

that in an RHM, the detector B and k vector are moving in opposite directions. In this

case, the detector will pick up the reference points (shown in green) fast and this leads

µkS ω

vv =⟩⟨ (2.17)

(2.18)


25

to increase in frequency. If the detector is in a medium with negative parameters then p

= -1 giving the argument less than 1. So the frequency received by the detector will be

decreasing in an LHM when it travels towards the source.

Fig.2.3.Doppler Effect in a right handed medium (top) and in left handed medium (bottom). A

is the stationary source and B is the detector which moves with a velocity v. The reference

point is indicated in green.

In other words, the wave vector k and the detector velocity v are in the same direction

and hence the detector has to travel for more time to pick up the reference point. This

leads to a decrease in frequency. Hence it can be concluded that waves will observe a

reversed Doppler Effect in an LHM [16].

2.8 Terminology on negative refraction phenomenon

Vaselago [16] termed the materials with negative permittivity and permeability

as Left Handed Metamaterials (LHMs) or double negative materials (DNGs). Inverse

Doppler Effect and backward radiation are characteristic properties observed in these

materials. These are also referred as backward radiating metamaterials [19]. However,

one could find much literature using these terms frequently irrespective of the

uniqueness of LHMs. It is worth making these terms more clear. Metamaterials are


26

those materials whose properties can not be obtained in naturally occurring materials. It

is not necessarily to be an LHM. A medium with negative permittivity and permeability

can be accurately termed as LHM or double negative (DNG) metamaterial. In EBGs,

negative refraction phenomenon arises due to the dispersion nature of the medium at

some frequencies. Such materials can be accurately termed as negative refraction

medium or backward metamaterials or analogous LHMs rather than DNGs or LHMs.

Another possibility is a case where the material offers negative group delay (hence

negative group velocity) and negative refractive index as proposed by [20] and these

materials can be referred as negative anomalous refraction materials.

2.9 Recent importance of LHMs studies

Since Vaselago’s prediction in 1968, it has taken more than three decades to

realise a composite material with LHM parameters. Following the ideas that an array of

split ring resonators (SRR) can posses negative permeability and an array of conducting

wires have negative permittivity [21,22], Smith et al. put forward a composite medium

which consists of a periodic array of interspaced conducting non-magnetic split ring

resonators (SRRs) and continuous wires [22,23]. Shelby et al. first experimentally

verified the negative refraction phenomenon and the existence of negative permittivity

and permeability from a composite medium consisting of two dimensional arrays of

split rings and wires [24].

The realisation of negative refraction and its experimental verification has been

questioned by some researchers. Valanju et al. [25] believed that the transmission of

energy or a signal is possible when the wave comes in a range of frequencies, which

combine to form energy packets. The phase velocity is given by the speed of each

frequency component and the group velocity is the speed of the energy packet. A wave

with just one frequency component could be bent in the wrong way, but this is


27

irrelevant because real light never has just one frequency. It was argued that if more

than one frequency were combined together to form a packet, the resulting packet

would bend and travel in the same direction. But this argument have been set away by

considering the fact that in [26], it has to be taken into consideration that the wave

packet does not extend to infinity in both directions. Even if the wave fronts of such a

finite pack will bend as suggested in [25], they still propagate in the negative

directions. This verified the existence of negative refraction phenomenon.

Since Shelby’s first experimental verification of negative refraction, there has

been a lot of attention in the measurement verification of the characteristics of LHMs

for the last few years. Transmission measurements of LHMs constructed by combining

split ring resonators and wires were detailed in [27]. The advantage of this method was

that both split rings and wires were on the same substrate board. Transmission and

reflection measurements for a double negative medium consisting of SRRs and

discontinuous wires were investigated by Ozbay [28].

In free space measurements, it is difficult to determine the material parameters

accurately because of the complexity in proper calibration to avoid extraneous

contributions. The transmission performance in a T junction waveguide filled with

negative media was presented in [29]. The negative refraction verification by numerical

simulations and measurements from an EBG prism structure consisting of metallic

wires at frequencies near bandgap edges was done by Parimi et al. [30]. The negative

refraction from a prism structure consisting split ring resonators and wires was studied

experimentally and the near field refraction measurements at different distances from

the structure were reported in [31]. It will be interesting to have a close view on the

major milestones in the realisation of LHMs or their analogous obtained in the past few

years, which provided great motivation to perform the works detailed in forthcoming

sections in this thesis.


28

2.10 Realisation of LHMs

Materials with LHM performance have been realised in many ways. Various

theories have been put forward for the LHM performance from composite media. Some

of these realisations are not really an LHM but an analogous LHM medium which can

give rise to the same properties of an LHM. However, in some literature, for

convenience it was preferred to address these analogous materials as LHMs or negative

index media. The main strategies in obtaining LHMs can be categorised as follows.

a) Resonance model

b) High pass filter model

c) Negative Refraction from EBGs- the analogous LHMs

2.10.1 The Resonance Model- Split Ring Resonator (SRR) and Wire

In 1999 Smith et al. found that periodic structure containing SRRs and thin

wires can perform as a medium having negative refractive index [22, 23]. The SRR

array can perform as a medium with negative permeability (µ) and the wire array can

perform as a medium of negative permittivity (ε) near their corresponding resonant

frequencies. The split ring resonator (SRR) exhibits resonant magnetic response to

electromagnetic wave, with the magnetic field vector Hv

parallel to the axis of the SRR.

A periodic array of SRRs having an effective magnetic permeability given by the

following equation [22].

)(

1)( 20

2

2

ωγωωωωµ

iF

eff +−−= (2.19)

where ω0 is the resonant frequency given by


29

3

2

0 2ln

3)2(b

dtcL x

=π

πω

F is the filling fraction of the SRR and γ is the damping factor, 2πγ = 2Lxρ /bµ0, where

ρ is the resistivity of the metal and Lx is the unit cell size along the x axis. c is the

velocity of light and t, d and b are the parameters of the split ring (fig.2.4) and ω is the

operating frequency.

Fig.2.4.Unit cell of split ring contributing negative permeability (left) and wire contributing

negative permittivity (right). The wires contribute negative permittivity. SRR can be

characterised by the parameters w, d, g and b. Wire array can be characterised by radius a and

the spacing between each wires. The EM wave propagation direction is along the z direction.

From eqn. (2.19) it can be seen that real part of µeff is negative at an interval ∆ω around

the resonant frequency. If this negative permeability medium is combined with a

medium of negative real part of permittivity, the resulting structure would possess

negative refraction in a frequency band. An array of thin metallic wires, which acts as a

high pass filter for electromagnetic wave polarized with electric field parallel to wires,

can perform as a medium of negative permittivity. This can be considered as plasma

w

g

z

b 2r

y

x

td

(2.20)


30

medium. The plasma frequency ωp of this medium is determined by the dimensions of

the wire lattice. It is given by [22, 23, 28]

=

raa

cp

ln2 2

22

πω

where a is the lattice period and r is the radius of the wire.

In this array the effective permittivity is given by [23]

2

2

1)(ω

ωωε p

eff −=

From eqn.2.22, it can be seen that the permittivity is negative when ω < ωp. By

combining the above two structures, a medium having negative permittivity and

negative permeability at a narrow frequency band can be realised. When both material

parameters are negative, the refractive index becomes negative. It is given by the

equation

εµµε −=×−−= ||||n

Fig.2.5 shows the photograph of the composite medium consisting of split rings and

wires. Fig.2.6 shows the transmission through the left handed medium consisting of

SRRs and wires [32]. From the figure, it can be observed that the transmission through

an array of SRRs shows a bandgap at the region 8.5-11GHz. This is due to the negative

(2.23)

(2.22)

(2.21)


31

effective permeability at this frequency region. The transmission of wire array is very

low below the plasma frequency due to its effective negative permittivity.

Fig.2.5. A split ring and wire array having negative refractive index. The split rings

contribute negative permeability and wires contribute negative permittivity.

(Reproduced from [29]).

Fig.2.6. Measured transmission through a medium consisting of SRRs only, wires only and

composite structure consisting of SRRs and wires showing increased transmission at the

frequency band 8.4-9.3GHz where the composite material posses negative permittivity and

permeability. The plasma frequency of the wire medium in this case is 20GHz. (Reproduced

from [32)

0

-40 -80 -120 -160 -200 -240 -280

T

rans

mis

sion

dB

Frequency GHz


32

The plasma frequency of the wire array under study was 20 GHz. Thus the transmission

response will be very low below 20 GHz. For the structure obtained by combining

SRRs and wires, the transmission is increased at 8.4-9.3GHz band. This is due to the

fact that at this interval the material has both negative effective permeability and

permittivity. The limitation of this methodology was that it failed in probing the

effective index corresponding to the extremes of the left handed frequency band. At

these frequencies, the wavelength in the material was very large and at the extreme

frequencies it was larger than the dimensions of the sample. So ray optics will fail to

characterise the sample. In this case, a scattering cross section analysis [33] is needed

to characterise the phenomenon accurately. It is a method to characterise

inhomogeneous materials by considering the effective area of collision wave.

2.10.1.1 Inverse Snell’s Law verification

According to the Snell’s law, refractive index can be expressed as

2

1sinsin

θθ

=n

where θ1 is the angle of incidence and θ2 is the angle of refraction.

When both material parameters are negative, the refractive index becomes negative. It

is interesting to see the validity of Snell’s law in such a media. When the refractive

index n is negative, the electromagnetic wave observes a negative angle with the

normal. In this case, the parallel component of the wave vector is preserved in the

transmission but the energy flow and the wave vector directions are opposite. As a

consequence, in an LHM prism the wave observes refraction in exactly opposite

direction to that from an RHM prism.

(2.24)


33

Fig.2.7 illustrates positive and negative refraction from a prism. From the

figure, it can be seen that in the case of an LHM prism, the refracted ray is going in a

direction away from the base of the prism which is opposite to the direction of

refraction from an ordinary (positive refractive index) RHM prism. Shelby et al. [24]

performed the Inverse Snell’s Law measurement using the SRR and wire structure.

Fig.2.8 shows the experimental setup used in Shelby’s experiment. Fig.2.9 shows

variation of negative refractive index with frequency [24]. Fig.2.10 is the measured

refracted field with a Teflon prism and an LHM prism showing positive refraction for

Teflon prism and negative refraction for LHM prism. The main disadvantages of the

split ring-wire model were large dissipation [34] and anisotropy in the structure.

Fig.2.7: A ray diagram showing refraction from a prism of RHM and LHM. For an RHM

prism, the refracted ray bends towards the base making a positive angle with the normal. In an

LHM prism, the refracted ray bends away from the base making a negative angle with the

normal. θ1 is the angle of incidence and θ2 is the angle of refraction.

RHM

LHM

θ2 θ1

θ2 θ2


34

Fig.2.8.An experimental setup used for Inverse Snell’s Law verification from a prism

composite medium consisting of SRRs and wires. (Reproduced from [24])

Fig.2.9. A plot of refractive index vs. frequency for SRR-Wire medium. The red curve is the

real component of theoretical LHM data and the red dotted curve is the imaginary part. The

black curve gives the measured value in which the dotted line represents the non reliable region

of measurement due to experimental limitations. (Reproduced from [24])

Microwave absorber

Detector

Sample

8 9 10 11 12 Frequency GHz

In

dex

of r

efra

ctio

n

3 2 1 0 -1 -2 -3


35

Fig.2.10.Measured transmitted power at 10.5GHz as a function of refracted angle from an

LHM (SRR–Wire) prism and a Teflon prism. The Teflon prism shows an angle of refraction of

+27º (dashed curve) and for the LHM prism the angle of refraction is -61º (solid curve).

(Reproduced from [24])

2.10.2 A planar equivalent high pass C-L medium

It was illustrated that transmission line structures comprising of capacitors in

series and inductors in parallel exhibit high pass filter characteristics and they

demonstrate negative refraction as equivalent to a material having both negative

parameters [35-37]. This was based on the fact that the fundamental component of

propagation constant of this C-L network was negative (indicating negative phase

velocity) and the group velocity positive. So the structure supports backward radiation

with an effective negative refractive index. In the EMT limit, the capacitors can be

expressed as an analogous uniform dielectric and the inductors with analogous uniform

magnetic media. The unit cell of the high pass C-L filter is given in figure 2.11. Let C

and L are the capacitance and inductance per unit length respectively.


36

Fig.2.11: A unit cell of 2D loaded C-L high pass transmission line consisting of capacitor and

inductor. Each capacitor is having impedance per unit length ZC and each inductor has

admittance per unit length YL. The unit cell length (∆d) is equal in both x and y directions.


The analogous expressions for permittivity and permeability can be written as

dC

dL

∆−=

∆−=

2

2

1

1

ωµ

ωε

Let i be the current and v be the voltage of which the directions are denoted by suffixes

x and y. From the figure one can write down the equations for each node [35],

∆

−=∂

∂

∆

−=∂

∂

dCji

xv

dCji

zv

xy

zy

ωω1, 1

∆

−=∂∂

+∂∂

dLjv

xi

zi

yxz

ω1

(2.27)

(2.28)

(2.25)

(2.26)


37

Combining equations 2.27 and 2.28 gives

dLCv

z

v

x

vy

yy

∆−==+

∂

∂+

∂

∂

ωββ 1 ,02

2

2

2

2

where β is the propagation constant. The phase and group velocities are given by

dLCv p ∆−== 2ωβω

dLCvg ∆+=

∂∂

=−

21

ωωβ

From equations 2.30 and 2.31 it can be seen that the phase and group velocities are in

opposite directions. Hence the structure supports backward radiation. The refractive

index is give by [35]

dLCvcnp ∆

−==00

21

εµω

and n is found to be negative. So this structure has a negative refractive index and

supports backward radiation. A coplanar waveguide implementation of the above

circuit has been presented at 15 GHz [36]. Fig.2.12 shows the backward radiating high

pass periodic structure studied in [36]. The capacitor and inductor were designed by

controlling the slot and conductor dimensions. Fig.2.13 gives the radiation pattern from

the structure [36] indicating very strong backward radiation at an angle of -40 degrees.

(2.29)

(2.30)

(2.32)

(2.31)


38

The radiation is 12 dB stronger than the forward radiated component. These facts are

valid only in the regime of which the Effective Medium Theory (EMT) is satisfied, that

is the spacing of the EBG is very small compared to the operating wavelength so that

the material can have an effective material parameters and refractive index.

Fig.2.12. A backward antenna based on high pass C-L circuit (above) and the enlarged view of

the unit cell shown in the inset (below). (Reproduced from [36])

The backward radiation, negative refraction and focusing effect were verified by

simulation and measurements. But the wide band negative refraction from this structure

still remains as a question. In theory, the above structure is a wide band negative

refractive planar transmission line structure. However, realising a planar high pass filter

is extremely difficult because of the deviations of transmission line approximations at

high frequencies and also because of the loss and material parameter deviation at high

frequencies.


39

Fig.2.13.Radiation pattern of a backward radiating CL structure. The backward radiation is

strong by 12dB at an angle of -40˚. (Reproduced from [36])

2.10.3 Composite right/ left handed (CRLH) transmission lines

Recently, Caloz et al. [38, 39] studied the fundamentals of ideal transmission

line metamaterials with their physical characteristics and the synthesis of practical

artificial transmission line metamaterials (TL-MM). He put forward the idea of

microstrip implementation of composite right/left handed (CRLH) structures. Novel

applications such as backfire to end fire leaky wave antenna, bandwidth enhanced

hybrid ring, a dual band harmonic branch line coupler and a negative

reflection/refraction phase conjugation interface were also analysed in [39] using the

CRLH approach.

The CRLH structure is a combination of right handed TL (RH-TL, a low pass

filter) and its dual the left-handed TL (LH-TL, a high pass filter) forming a band pass

TL. A diagram of the circuit is shown in fig.2.14. Let L and C be inductance and

capacitance respectively. In order to indicate its position in the high pass (left handed)

and low pass (right handed), suffixes 'L and 'R are used with L and C. Let βc

represents the propagation constant for the CRLH medium.


40

At low frequencies, the CRLH structure is dominantly LH with a hyperbolic

dispersion relation [39]

Fig.2.14. The CR-LH bandpass structure consisting of right handed transmission line (RH-TL-

high pass) and left handed transmission line (LH-TL- low pass)

''1

LLL CLω

ββ −=≈

while at high frequencies it is dominantly RH with a linear non dispersive relation

'' RRR CLωββ =≈

LRc βββ =

The transition frequency from LH to RH is given by

(2.33)

(2.34)


41

||/2 where,)(

0)(.

c0c

04 ''''

0 ''

c

cRRLL

and

CLCL

βπλωλ

ωβω

=∞=

==

ω0 is the centre frequency of the bandpass filter. Matching (Z0=ZL=ZR) can be obtained

by properly adjusting the L and C values. A microstrip equivalent model of the circuit

in fig.2.14 has been studied for its radiation properties [40]. The dispersion behaviour

of the structure is shown in fig.2.15 in which there are four regions. These are LH-

guided, LH-leaky, RH-leaky and RH-guided. Based on these principles, an antenna

with backward to forward scanning capability was introduced in [40]. This antenna

does not have the drawback of conventional uniform transmission line leaky wave

antennas, which can be used only for scanning the half space from broadside to end

fire.

Fig.2.15. Propagation (β) and attenuation (α) diagrams for a dominant mode leaky wave

antenna consisting of four regions. The LH and RH guided wave regions exist when | β | > k0

(region I- LH and IV- RH), and leaky wave region exists when | β | < k0. When β < 0 the

structure supports LH backfire (region II) and when β > 0 the structure supports RH endfire

(region III). (Reproduced from [40])

(2.35)


42

In fig.2.14, there are four frequency regions: region I (f <3.1 GHz) and region

IV (f > 6.3 GHz) are having | β | > k0 and represents left handed (LH) and right handed

guided (RH) wave regions respectively. The region between 3.1 GHz to 6.3 GHz

includes an LH backfire region with β < 0 (region II) and an RH end fire region with

β > 0 (region III). The transition from LH backfires to RH end fire occurs at β = 0 at 3.9

GHz. The verification of backward radiation was detailed in [40]. A more practical

realisation of an LH-TL structure was done by ladder network as an ideal LH- TL does

not exist in nature. It can be noticed that lumped element implementation of such

structures are not wide band as it includes numerous stop bands.

2.10.4 Negative refraction from EBGs- The analogous LHMs

In the previous sections of this chapter, various realisations of LHMs have been

detailed. In this section, negative refraction phenomenon in EBGs is detailed. These

EBG based realisations can be termed as analogous LHMs, indicating that these EBGs

will show certain characteristics same as that of an LHM.

2.10.4.1 Analogy between EBG and electronic crystal

For the last few years, there has been a growing research interest on the

negative refraction phenomenon in EBGs [41-50, 53, 54]. The propagation of wave in

an EBG is different from that in conventional materials [51, 52]. Due to the periodic

nature in the structure, the wave vector k is not conserved in the EBG. Due to this

reason, it is too complicated to define a specific refractive index in weakly modulated

EBGs (Modulation in an EBG is the periodic disturbance on transmission due to

periodicity of the medium). Strong modulation takes place when the period is much

smaller than the wavelength and weak modulation occurs when the period is

comparable to the wavelength.


43

Most of the theoretical analysis of negative refraction in EBGs is based on

Equi- Frequency Surface (EFS) analysis. EFSs are contours in the wave vector space at

a specific frequency. The direction of normal to the tangent on the EFS gives the

Poynting vector. Since Poynting vector indicates the direction of energy flow, the

normal to the tangent gives the direction of energy propagation through the crystal. In

vacuum, the dispersion relation in k space is given by a sphere of radius k= 2π/λ, where

λ is the wavelength in the medium. Fig.2.16 shows schematic diagram of light

propagation through a dielectric media using EFS. The smaller circle represents EFS in

vacuum and the circle with larger radius represents EFS in the dielectric medium. The

incident field direction is shown by the red arrow. When the wave enters into the

dielectric media, it undergoes refraction and hence the direction of propagation changes

as shown by the blue arrow.

Fig.2.16. EFS diagram showing wave propagation through a dielectric material. The incident

wave direction shown by red arrow and the refracted wave direction is denoted by blue arrow.

The propagation direction is determined using the normal to the EFS at the k point.

The EFS are drawn in k space i.e. in coordinates of kx and ky where kx and ky are the x

and y components of k vector. A vertical line is drawn to study the k vector

Incident wave kx

ky

ky

Excited wave kx

EFS for vacuum

EFS for dielectric


44

conservation. The normal to the tangent on the EFS at any k point gives the wave

propagation direction at that point. This schematic diagram (fig.2.16) illustrates the

Snell’s law in a dielectric medium.

The above case is for a homogenous dielectric medium. The EFS of an EBG is

different from that of a uniform dielectric. For an EBG in the long wavelength limit,

the medium can be considered as homogeneous. Let λav be the average wavelength in

the medium. In this case, dispersion surfaces are spheres of radius k = 2π/λav. Fig.2.17

shows the EFS in an EBG in the long wavelength limit. When frequency increases, λav

decreases and the medium start losing its homogeneity. Then λav becomes small and the

radius of the circles increase. They start partially overlapping each other as shown in

fig.2.18. Fig. 2.18 shows the dispersion surfaces in a crystal in the long wavelength

(when the medium can be homogenous) limit. This is the point at which the Bragg’s

law is satisfied. The overlapping of the spheres leads to complex dispersion surfaces

and the bandgap appears.

Fig.2.17. EFS in an EBG in the long wave length limit and the medium can be considered

homogeneous.

The circles are with radius k = 2π/λav. hv

is a vector, the modulus of which gives the

average wave number in the crystal. Bragg’s diffraction law is satisfied around point L0

and this results a bandgap.

O Hhv

2π/λav


45

Fig.2.18. Dispersion surface in a photonic crystal at high frequency where the medium becomes

inhomogeneous. The two EFS circles overlaps and form a bandgap. Around the region Lo

Bragg’s law is satisfied and this region the bandgap exists.

In [48], it was shown that the effective mass in electronic crystals is analogous

to the effective refractive index of an EBG at narrow frequency band near the bandgap.

In strongly modulated EBGs, an effective refractive index can be defined near the

bandgap frequency [48]. For strong modulation in EBG, the period must be much

smaller than the wavelength. In such cases, the effective refractive index of photonic

crystal is analogous to the effective mass approximation in electron-band theory. In

solid state physics, a particle's effective mass is the mass it carries during its transport

in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal

respond to electric and magnetic fields almost as if they were free particles in a

vacuum, but with a different mass. In this approximation regime, one could observe

EFS at certain frequencies near bandgap as circles and hence at this region the medium

is similar to a homogeneous dielectric. In this sense, the wave propagation inside the

crystal can be defined in the narrow frequency region. However, in the other frequency

2π/λav

O H

L0 Xh

δ

Xo Bandgap

Dispersion surface


46

regions, for a strongly modulated crystal, the excited waves inside consist of a mixture

of many diffracted wave components. This indicates that the EFSs can be decomposed

into circles and due to this reason the situation can be characterised as chaotic [48]. For

such crystals, the EFS shape becomes rounded as the frequency approaches the

bandgap.

Fig.2.19 shows the EFSs for a weakly modulated and strongly modulated

photonic crystal [48]. In both cases, two dimensional EBGs with hexagonal lattice

consisting of air holes in a dielectric (GaAs with dielectric constant 3.6) were

considered. From fig. 2.19, it can be seen that the EFS of the strongly modulated PC

has circular shape at frequencies near bandgap (at ω = 0.45). Frequencies are

normalised (ωa/2πc) in the plot. The bandgap was at ω = 0.48.

Fig.2.19.Schematic diagram of wave propagation through a weakly modulated hexagonal

photonic crystal (left) and for a strongly modulated photonic crystal (right) at different

frequencies. The bandgap frequency is at ω = 0.48. (Reproduced from [45]).

It can be seen that the EFS becomes circular in shape at ω = 0.45. This suggests that at

this frequency region, very close to the bandgap there can be an effective refractive

index.


47

When the modulation is not strong, EFS generally has a star-like shape

consisting of arcs belonging to diffracted waves as shown in fig.2.19. When the

modulation becomes strong, EFS becomes circular near the bandgap. This is similar to

the EFS of a dielectric and at this region one could define an effective refractive index.

The direction of propagation inside the EBG can be determined from the tangent to the

EFS [48, 54]. For a conventional dielectric, the wave propagation direction is outward

tangent direction but in an EBG, it is inward, indicating a negative propagation

direction. At this point, the EBG is having a negative effective refractive index [48].

The bandgap phenomenon in a strongly modulated EBG is analogous to

electronic bandgap in semiconductors. Likewise, there is a promising analogy between

the effective mass state in electronic crystal and effective refractive index in an EBG

[48]. In a semiconductor, the negative effective mass state appears below the energy

gap and a positive effective mass state appears above the gap.

Fig.2.20.A diagram showing analogy between EBGs and electronic crystals. The effective mass

in electronic crystal (right) is analogous to refractive index in EBG (left).

Positive index state Positive mass state (Electron band)

Photonic bandgap Energy bandgap

Negative index state Negative mass state (Hole band)

ω E

Effective index approx. Effective mass approx.

EBG Electronic crystal


48

The negative mass state is the hole band and positive mass state is the electron band.

This is similar to the effective index states in EBGs. The effective mass state

approximation is valid near the gap whereas in EBGs, the effective refractive index

state is valid near bandgap edges. Fig. 2.20 illustrates the analogy between EBGs and

electronic crystals. It can be seen that at band gap edges, there exists a negative index

state for EBG same as negative mass state in electronic crystals.

2.10.4.2 All Angle Negative Refraction (AANR)

Inspired by the ideas of Notomi [48], Luo et al. [43-45] demonstrated all angle

negative refraction from photonic crystals without having an effective negative index of

refraction. It is observed that in the lowest photonic band near Brillouin-zone, where

positive group velocity and positive refractive index can be found and with a negative

effective mass. In this band, for all angles of incidence, a negatively refracted beam

will result.

In [44], a two dimensional array of dielectric photonic crystal consisting of air

holes in dielectric εr = 12.0 with lattice constant a and hole radius r = 0.35a was

studied. The EFSs were obtained at different frequencies as shown in fig.2.21. Fig.2.22

shows the bandgap diagram for the photonic crystal under study in [44] and the all

angle negative refraction region is marked in red. The photonic effective mass is

ji kk ∂∂∂ ω2 which is negative at the M point. The frequency at which the EFS is convex

has got negative refraction. However, the medium will still have a positive refractive

index [44].


49

Fig.2.21.Several EFSs for the dielectric photonic crystal. The frequency is written in units of

2πc/a. It can be seen that the surfaces are convex in the vicinity of M point. (Reproduced from

[44])

Fig.2.22.Dispersion diagram for the photonic crystal. The all angle negative refraction region is

highlighted in red. (Reproduced from [44])

The theoretical analysis of negative refraction in photonic crystals makes use of EFS. It

has to be mentioned that out of the many wave vectors that coexist, it is not accurate to

analyse the phenomenon by selecting one among the different possible wave vectors in

Nor

mal

ised

freq

uenc

y ω

a/2π

c


50

the crystal [54]. The difference between these wave vectors governs the positively and

negatively refracted power at the exit surface of the crystal. If one follows the EFS

approach for EBGs for negative refraction prediction, it is needed to incorporate case of

incident wave on the crystal. That means one has to consider the dispersion surfaces in

the two half spaces. In this case the thickness of the EBG plays a role in determining

the dispersion [54].

2.10.5 Drawbacks of LHM realisations

In the previous sections, various realisations of LHMs or analogous LHMs are

presented. It was noticed that SRR-wire medium works entirely on resonance and high

pass C-L network is based on the backward radiation characteristics of the C-L unit

cell. In the former work, the effective permeability depends on the filling fraction of the

SRR and hence periodicity plays an important role. In order to obtain a truly quasi-

continuous medium, unit element sizes must be at least one hundredth of the operating

wavelength [55]. It is very difficult to achieve in practice and therefore it is interesting

to investigate the effects of periodicity towards negative refraction. The photonic

crystal model for negative refraction has taken only infinite structures into account

which have the drawbacks which will be discussed in detail in chapter 3. This model

does not take into consideration some of the dispersion sources that exist in finite

structures. The need of finite model analysis for negative refraction studies are to be

presented in chapter 3.

In a periodic structure, the field of a Bloch wave repeats at every terminal plane

having a propagation factor e-γd where d is the length of the unit cell and γ is the

propagation constant. When the loss in the structure is negligible, γ equals propagation

constant β. By Floquet’s theorem, a periodic structure can give rise to spatial

harmonics (Floquet’s harmonics).


51

Each harmonic has a propagation phase constant βn [56].

dn

nπββ 2

+= (2.36)

Where n = ....-2,-1, 0,1,2,3…

Depending on the sign and value of n, these waves can have positive and negative

phase velocities as shown in equations below.

nddv

nnp πβ

ωβω

20, +

== (2.37)

Equation (2.37) can be rewritten into

dn

dd

ndcc

vc n

np

000

, 22 λ

πβ

ωπβ

ωβ

+=

+⋅==

(2.38)

β0 is a large value and when n is not a very large value, from eqn.2.38, it can be seen

that the term pnvc is independent from the value of n when the value is

d0λ high, then

the sign of pnvc only relies on the fundamental propagation constant β0. Omission of

higher order spatial harmonics can be done when the ratio d

0λ >> 1, where λ0 is the

resonance wavelength [56].

In contradiction, when d

0λ ≈ 1, spatial harmonics contribution is considerable

and overall response of the structure can be considered as an aggregation of individual


52

responses from fundamental component (n = 0 ) and other spatial harmonics (here n

can be positive or negative) [72]. This leads to change in dispersion curve at some

frequency bands. Indeed, according to [35-37], one can design a high pass type

transmission line to achieve principal propagation constant β0 < 0 subject to the

condition that d/λ0 is infinitesimally small. However, it is worthy mentioning that

negative dispersion can also be achieved through contribution from high-order

harmonics in any EBGs. As a common practice, higher order harmonics contribution

has been neglected and EBGs are analysed only in terms of ‘unit cell’ concept. The unit

cell analysis is sufficient for studying the pass band behaviour of a periodic system but

inadequate to predict negative refraction frequencies.

As discussed before in section 2.10.1, LHM performance in a narrow frequency

band can be obtained from numerical simulation and experiments using the analysis of

a periodic structure consisting of SRRs and wires using resonance theory and EMT. In

a periodic structure consisting of SRR alone or wire alone, it can also have a negative

refraction frequency region because of the dispersion of the entire structure and the

frequency regions may differ from the prediction using unit cell approach. When the

period increases, the structure starts deviating from EMT approximations. However, the

wire medium and split ring medium are EBGs. Based on the previous discussion,

negative refraction can be obtained from these models without combining them.

Strictly speaking, in this case the structure does not posses a negative effective

refractive index but instead acts as an analogous LHM.

The approach to obtain an LHM medium based on high pass C-L structure was

based on the unit cell dispersion behaviour of the high pass periodic structure. In the

work detailed in this thesis, using numerical simulations of lumped circuit models it is

demonstrated that a high pass filter structure can demonstrate negative refraction only

at some frequencies in the pass band and hence the LHM characteristics are not wide


53

band as claimed in [35-37]. This is more pronounced when the spacing of the structure

is increased. The unit cell analysis can effectively predict the stop or pass band

behaviour but it is not accurate to predict the effects of spatial harmonics in the

dispersion diagram. In this case, we need to consider high order harmonic interactions.

In this way, one can obtain frequencies other than lower edges of band gap in an EBG

structure, at which it can give negative refraction.

2.11 Applications of LHMs

While the research of LHMs is focused on the realisation of such materials

using different approaches, a lot of effort has also been made in exploring the

applications of LHMs or their analogous. There have been many works done towards

this direction [18, 50, 57, 63, 64]. Applications can be found in the construction of

novel devices such as perfect lens [57], sub-wavelength resonating cavities and phase

shifters [63, 65], LHMs as radar absorbing materials [18, 64], LHMs in optical

receivers [50] etc. Some of these applications were experimentally verified. In the

previous section, it was shown that planar LHM media have been used as backward

radiating and scanning antennas. In the next sections, some of the interesting

applications of LHMs are introduced.

2.11.1 Perfect lens

Pendry proposed that a slab of LHM can be used as a lens which is free from all

aberrations observed in a lens made with positive refractive index [57]. In contradiction

to the conventional lens, the perfect lens formed by LHMs will be flat. Evanescent

waves in a conventional lens lead to diffraction limitation and aberrations on the quality

of an image. Due to this, very small details of the object in the image can not be

obtained. Pendry found that these decaying waves would actually amplify as they pass


54

through a left-handed material, making a perfect lens possible. Fig. 2.23 shows the ray

diagram of the focusing mechanism using a slab of LHM. The proposed lens is free

from all aberrations. So it is called a perfect lens.

Fig.2.23.An illustration of the perfect lens formation from an LHM slab. The first focus point is

inside the slab and the second focus point is outside the slab at the right side. (Reproduced from

[57])

Some researchers believed that the perfect lens seems to be in contradiction

with fundamental physics laws [58-62]. The argument by Valanju et al. [26] was that

the negative refraction phenomenon is not possible due to the reasons stated in section

2.9. However, these arguments were shown to be wrong by Pendry [26]. The second

objection was on the effect of loss of materials on the ‘perfection’ of the image

obtained by the lens. The limitation on perfect lens formation due to the material loss

has been detailed [59]. For a perfect lens formation, the absorption loss should be zero.

So the resolution depends on the loss but still one can obtain a ‘super lens’ (if not

perfect) that can focus features of an object with a resolution well below the

wavelength leading to sub-wavelength imaging.


55

Taking into consideration all these facts it can be concluded that the negative

refraction phenomenon and the perfect lens formation can fundamentally exist. This

application can lead to very small integrated circuits and more digitally ‘enhanced’

CDs and DVDs [6] by making use of high resolution focusing of lasers.

As discussed before, an LHM or analogous LHM can be obtained in many

ways. Focusing using LHM based on a C-L high pass filter model is demonstrated in

[35]. Wave focusing using a composite medium consisting of metallic wires and

spirals, which was similar to the LHM medium proposed by Smith et al. [22, 23] was

presented in [67]. On the other hand, focusing can be obtained from analogous LHM

using EBGs. Such an approach was presented in [42] using dielectric EBG. Fig.2.24

shows measured results for the perfect lens focusing from a PBG panel of 37.5 cm X 30

cm, at a frequency of 9.3 GHz [42]. Now the source position is displaced by a distance

of 4 cm upward and it can be noticed that the focus point also moved by the same

distance [42] as shown in Fig.2.25.

Fig.2.24.Experimental result showing focusing of electric field from an EBG due to negative

refraction [46]. The yellow point at the left side is the source point and yellow point at right

side is the image point (Reproduced from [42])

Image Source


56

Fig.2.25.Experimental result showing focusing of electric field from an EBG due to

negative refraction when the source is moved 4cm upward. It can be seen that the

image at the left position also moved by the same distance (Reproduced from [42])

2.11.2 Scattering reduction using LHMs

Negative refractive index materials can be used for the reduction of

electromagnetic wave scattering [18, 64, 72]. In [64], the scattering from a metallic

cylinder coated with RHM / LHM with same material parameters in magnitude but

opposite in signs was presented. Fig.2.26 illustrates the effective scattering surface

patterns of a cylinder with different coatings (the angle = 180 degrees corresponds to

the backscattering direction). From the figure, it can be seen that the use of a material

with a negative refractive index -n is much more advantageous than that of an RHM

with refractive index +n for reducing the back scattering from the metallic cylinder.

Image Source


57

Fig.2.26.Scattering pattern for a circular cylinder with coating. Curve l corresponds to

scattering when surface of the cylinder coated with parameters µ = -1-i*0.6 and ε = -1-

i*1, curve 2 represents the formal sign reversal for the real parts µ = 1-i*0.6 and ε = 1-

i*1. Curve 3 is the scattering when the material parameters are that of absorbing foam

glass (µ = 1 and ε = 1- i*0.6). Curve 4 shows scattering from an uncoated cylinder


In [18], the reflection from a metallic surface with coatings of RHM and LHM

was presented and it was argued that LHM is superior from an application point of

view due to its scalability and reduced weight. The limitation of this comparison was

that in this case the comparison between RHM and LHM performance was done for

materials with parameters of different magnitudes and opposite signs. This was

comparing the performance of two materials that have refractive indices +nr and –n1,

where nr and n1 are different values. The subscript r represents RHM and l represents

LHM. Based on the comparison, it was shown that LHM was superior for absorber


58

applications. Recently a theoretical analysis based on Mie scattering was presented in

[72] which indicated that when a metal is coated with an LHM the scattering

coefficient will be very small.

2.11.3 Microwave components using LHMs

LHM materials can be employed as sub-wavelength resonators and zero phase

delay lines. The advantage over that of RHM materials is that the dimension of the

resonator will be very small [63, 65]. If there are two layers of materials for the phase

compensator, with thickness d1 and d2, in the case where one layer is with LHM then

the resonator size will be proportional to d1/d2 on the other hand the if they both are of

RHM, then the resonator size will be proportional to d1+d2. The schematic diagram of

such a two layer structure is shown in figure 2.27.

Fig.2.27. A two layer structure consisting of positive and negative refractive index

materials. The left layer is a dielectric with ε > 0, µ >0 and the right layer is a lossless

material with negative permittivity and permeability (ε < 0, µ < 0). In the left layer the

Poynting vector (S) and wave vector (k) are parallel. In the second layer they are anti

parallel.

ε1, µ1 > 0 ε2, µ2 < 0

d2 d1

k0 k1 k2 k0

S0 S1 S2 S2

I II


59

Consider a uniform plane wave incident on layer I and if the impedance of layer I is

same as that of air (i.e. η0 = η1) there will not be any reflection. When the wave reaches

the end of slab I, there will be a phase change by an amount 1011 dkn=ϕ where

000 εµω=k . An LHM with refractive index n2 < 0 was placed next to slab I (i.e. slab

II in the fig. 2.27). In the second slab, the directions of Poynting vector and phase

velocity are anti-parallel. Hence the wave vector k1 and k2 in slabs I and slab II are in

opposite directions and the phase change occurred from the beginning and end of slab

II is given by 2022 dkn−=ϕ . This shows that the phase change that occurred during

the travelling of wave through slab I, can be decreased or can be cancelled by

traversing the slab II. When the thickness of the slabs are taken in such a way that they

satisfy the following equation [63] then the total phase difference becomes zero.

1

2

2

1

nn

dd

=

In this way, the structure can be used for phase compensation / conjugate applications.

The phase change can be controlled by the ratio of d1 and d2. In principle, the

thicknesses can be any value as long as the ratio d1/d2 satisfies the above relation.

The same structure can be converted into a one 1D sub-wavelength cavity

resonator by employing conductors at both ends (as shown in fig.2.28, conductor layer

is shown in red). From fig.2.28, it can be seen that the first conductor is at z = 0 and

the other is at z = d1+d2. For a 1D cavity resonator all quantities are independent of x

and y coordinates. In the slab I (i.e. in region ≤the electric and magnetic fields can be

written as follows.

(2.39)


60

)cos(

)sin(

01011

011

01011

zknEi

knH

zknEE

y

x

ωµ=

=

Fig.2.28. The two layer resonating structure consisting of RHM (green) and LHM (yellow)

slabs backed by conductors at both ends (shown in red).

In slab II the fields can be expressed as

)](cos[

)](sin[

2102022

022

2102022

zddknEi

knH

zddknEE

y

x

−+=

−+=

ωµ

where subscripts1 and 2 represent regions I and II respectively. To satisfy the boundary

conditions at the interface between regions I and II

I ε1, µ1 > 0

II ε2, µ2 < 0

d2 d1

x

z

(2.42) (2.43)

(2.40) (2.41)


61

11

11

||

||

21

21

dzydzy

dzxdzx

HH

EE

==

==

=

=

This leads to

0)sin()sin( 2020210101 =− dknEdknE

0)cos()cos( 202022

0210101

1

01 =+ dknEkndknEknµµ

In order to have E01≠0 and E02≠0, the determinant in of the above equations must

vanish. i.e.

0)cos()sin(

)cos()sin(

1012021

1

2021012

2

=

+

dkndknn

dkndknn

µ

µ

This can be written as

0)tan()tan( 2021

1101

2

2 =+ dknndknnµµ

The permeabilities µ1 and µ2 can be written as µ1 = | µ1 | and µ2 = -| µ2 | and substituting

into eqn. (2.49)

(2.44) (2.45)

(2.48) (2.49)

(2.46)

(2.47)


62

0)tan(||

)tan( 2021

1101

2

2 =+− dknndknnµµ

This indicates that at a frequency ω, if ε1 > 0, µ1 >0 and ε2 < 0, µ2 <0, a non trivial

solution for a one dimensional cavity can be obtained as

||||

)tan()tan(

12

21

202

101µµ

nn

dkndkn

=

Hence in principle d1 and d2 can be as thin or as thick as long as the above ratio is

satisfied. This proves the possibility of sub-wavelength resonators using LHMs. Some

applications such as backward wave couplers with high coupling coefficient, broad

band antenna feed network etc. [65] have also been reported.

2.11.5 Optical receivers using EBG based analogous LHM

The spreading of the illumination area in the focal plane of a lens can be

suppressed by using negative refraction behaviour of photonic crystals. In [50], a field

of view expander (FOVE) technique based on this approach for optical receivers was

proposed. In optical detectors, the light acceptance area is limited so that a reduction of

focal spot’s illumination area at reception of aberrated beam will take place. The

propagation angles of the received beam in front of optical devices need to be taken

into account. The proposed method [50] can significantly reduce the focal spot size

without expanding the propagation angles.

When a point source A is located on a slab of LHM, light entering will focus at

point B inside the crystal and then again gathers at point C outside the crystal as shown

in fig.2.29. The line connecting the points A, B and C are perpendicular to the

(2.50)

(2.51)


63

boundaries and each ray in front of the slab is parallel to the corresponding ray behind

the slab. If the boundaries are slightly bend, then the location of the point B where the

rays gather will be shifted. So boundaries properly designed by connecting many short

bent lines may contribute to a reduction of beam spreading due to wave front

aberrations after the light passed through the crystal. If the radius of curvature is large

enough, the propagation angles of the rays in front of and behind the crystal will not

change very much. Making use of the aforementioned approach a FOVE technique for

optical receivers can be developed. The following figures 2.29-2.31 illustrate this

concept clearly. In this case, illumination range behind the crystal is reduced to 31% of

that in front of the crystal [50].

Fig.2.29. Focusing from a crystal at negative refraction frequency. All the rays being

emitted from point A pass through point B and gather again at point C. (Reproduced

from [50])

EBG

Y

X

A

B

C


64

Fig.2.30. Schematic drawing of a field of view expander. The boundaries have the

curvatures r1 and r2 with the same centre of origin. The inset shows a magnified part of

the crystal surface (Reproduced from [50])

Fig.2.31. Schematic drawing showing the reduction of illumination area obtained by applying a

photonic crystal with negative refraction characteristics. Reduction in illumination area is

marked by W. (Reproduced from [50])


65

2.12 Conclusions

In this chapter, a background survey of negative refraction phenomenon has

been presented. The characteristic properties of LHMs such as backward radiation,

inverse Snell’s Law and inverse Doppler Effect have been identified. Recent

realisations of LHMs using resonance model, high pass C-L model and analogous

LHMs using EBGs, numerical simulations of the performance of LHMs, their

analogous media and applications have been reviewed. It has been observed that LHMs

are capable of potential electromagnetic applications such as in the construction of

perfect lens, beam scanning antenna, microwave and optical circuit components. The

resonance models are very narrow band and they work on EMT approximations. The

limitation of EMT is that in order to get a pure homogeneous medium, the period

should be very small compared to wavelength. This makes the design difficult at both

microwave and optical frequencies. Also, the resonance models have the disadvantages

of strong dissipation and anisotropy in the metamaterial. However, the anisotropy in

such LHMs has not been addressed in-depth. It has been noted that negative refraction

can be obtained from EBGs. The negative refraction from EBGs does not guarantee a

negative index of refraction and this phenomenon arises due to the dispersion nature of

EBGs. All aforementioned studies were proposed for infinite models, which have not

taken into consideration the higher order Floquet’s harmonics interactions and the

incident field coupling. These factors can affect the dispersion of the model especially

when the period is getting comparable to the wavelength. This has indicated the need of

considering Floquet’s harmonics interactions and finiteness of EBGs for negative

refraction studies.


66

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73

Chapter 3

Periodic Structures: Modelling and Negative

Refraction Phenomenon

3.1 Introduction

Collin [1] defined periodic structures as waveguides and transmission lines

loaded at periodic intervals with identical obstacles. Periodic structures have many

applications at microwave and optical frequencies due to their inherent properties of

forming pass/stop bands and supporting propagation of waves with phase velocities

much less than velocity of light (slow wave). The first characteristic ensures their role

in frequency selective surfaces (FSS) or filters and EBGs. The second property leads to

the realisation of slow wave structures or travelling wave tubes (TWTs).

At present, periodic structures are realised in various forms in one, two and

three dimensions using various materials including dielectrics or metals. Therefore it

opens up more innovations such as photonic/electromagnetic bandgap (PBG/EBG)

structures and novel applications in microwave and antenna engineering. For example,

radiation pattern and gain enhancement of microstrip antennas can be achieved by

using EBGs as ground planes [2, 3] or above the radiating patch [4] to suppress

radiation in certain directions. There have been many approaches for the analysis of

EBGs both analytically and numerically [4-7]. Analytical models can be obtained only

for a limited class of electromagnetic periodic structures with certain approximations.

So majority of the analysis were based on numerical modelling.

Chapter 3: Periodic Structures: Modelling and Negative Refraction Phenomenon

74

Recently, it has been observed that negative refraction can be obtained from

EBGs at frequencies at the edges of bandgap [8-15]. However, all such numerical

analysis considered infinite photonic crystals which extended in the entire space and

there was no incident field excitation used in this approach. In this case, the properties

of the crystal were determined from wave vectors inside the photonic crystal [16]. The

analysis with unit cell alone will only take into consideration fundamental spatial /

Floquet’s harmonic. An analytical study on the excitation of a crystal consisting of

periodic dipole scatterers was performed in [7]. In comparison with the initial approach

[17] which considered only fundamental Floquet’s harmonic interaction, in [7] the

interaction between higher order Floquet’s harmonics was presented. In fact, the

Floquet’s harmonics interaction, the coupling between the finite structure and incident

field and evanescent modes which exist in finite EBGs can contribute towards the

dispersion behaviour. This dispersion can be classified as spatial dispersion in which

the effective parameters change with spatial coordinates inside the structure and also

with the finiteness of the structure.

In this chapter, the methodology based on the unit cell approach, dispersion

characteristics of infinite and finite periodic medium with special emphasis to negative

refraction phenomenon and Floquet’s harmonics effects and spatial dispersion in finite

EBGs are detailed. It is observed that negative refraction phenomenon exists at narrow

frequency bands due to the spatial dispersion in the structure. This dispersion arises

due to the Floquet’s harmonics interactions. Lumped element circuit simulations are

carried out for the initial study. The role of Floquet’s harmonics and backward

radiation from periodic structures will be presented.


75

3.2 Analysis of periodic structures

Periodic structures can be analysed by either lumped circuit/ network analysis

or wave analysis method. The selection of analysis method depends on nature of the

problem. For simple periodic structures, circuit approximations can be obtained without

much complexity. Circuit network analysis is preferred in this case. For complex

structures, wave analysis is much easier.

3.2.1 Circuit or network analysis

This method of analysis starts with constructing an equivalent network for a

single basic section or unit cell of the structure. It is followed by a process to determine

the voltage and current in the structure connecting an infinite number of basic networks

[1] by applying periodic boundary conditions. This method is complicated in periodic

structures where it is difficult to get an equivalent analytical model for the structure and

also the approximations to obtain an equivalent model limit the applicability. The

approximations mainly depend on the size of the unit cell and the frequency of

operation.

3.2.2 Wave analysis

Periodic structures can be analysed in terms of forward and backward

propagating waves which can exist in each unit cell of the structure. This approach is

based on wave amplitude transmission matrix. This transmission matrix relates the

incident and reflected wave amplitudes on the input side of the junction to that on the

output side of the junction. This matrix can hold the same relationship to the scattering

matrix as the voltage-current transmission matrix related with the impedance matrix.

Fig. 3.1 shows the diagram of a periodic structure consisting of identical elements.


76

Fig.3.1. Diagram showing wave amplitudes in periodic structure. cn+, c+

n+1 are forward

propagating waves at the nth and (n+1)th terminals respectively and cn-, c-

n+1 are the

corresponding backward waves.

Let the amplitudes of forward and backward propagating waves at the nth and (n+1) th

terminal plane be cn+,cn

-,cn+1+,cn+1

-. They are related by the wave amplitude

transmission matrix by the relation [1]

=

−+

++

−

+

1

1

2212

1211

n

n

n

n

cc

AAAA

cc

(3.1)

where A is wave amplitude matrix.

In eqn.3.1 amplitude matrix A describes the circuit. In terms of scattering matrix S for a

single junction we have [1]

=

=

+

+

+

−

−

−

2

1

2212

1211

2

1

2

1

V

VSSSS

c

c

V

V

This gives [1]

=

−

+

−

−

2

1

2212

1211

2

1

c

cSSSS

V

V (3.2)

On solving these equations for c1+ and c1

- give [1]

n

n+1

n+1

Unit cell

cn+ c+

n+1

cn- c-

n+1


77

−

−

=

−

+

−

+

2

2

12

22112

12

12

11

12

22

12

1

1

1)( c

c

SSSS

SS

SS

S

c

c

For a Bloch wave solution it requires that

+−++ = n

dn cec γ

1

−−−+ = n

dn cec γ

1

Hence eqn. (3.1) becomes in the general form for nth and (n+1 )th element

01

1

2221

1211 =

−

−++

−+

n

nd

d

c

c

eAA

AeAγ

γ

A nontrivial solution for c+n+1 and c-

n+1 can be obtained only when the determinant

becomes zero. i.e

0)( 22112

21122211 =+−+− AAeeAAAA dd γγ

or

2cos 2211 AAd +

=γ

For a two port network, a conversion from scattering matrix S to amplitude matrix A

can be obtained using the following equations.

(3.3)

(3.5)

(3.4)


78

12

22112

1222

12

1121

12

2212

1211

1

SSSS

A

SS

A

SS

A

SA

−=

=

−=

=

Using equations 3.5 and 3.6-3.9 the dispersion characteristics which relate propagation

constant and frequency can be obtained.

3.3 Floquet’s theorem and spatial harmonics

In an infinite periodic structure, the field of a Bloch wave repeats at every

terminal plane having a propagation factor e-γd , where d is the length of the unit cell

and γ is the propagation constant when the loss is negligible [1]. Thus, if the field in the

unit cell between 0 < z < d is ),,(),,,( zyxHzyxE .The field in the unit cell located in

the region d< z < 2d is

),,( ),,,( dzyxHedzyxEe dd −− −− γγ

)

Then the field in the periodic structure is described by a solution of the form

),,(),,( zyxEezyxE pdγ−=

),,(),,( zyxHezyxH pdγ−=

(3.6)

(3.7) (3.8) (3.9)

(3.10)

(3.11)


79

Where Ep and Hp are periodic functions of z with period d.

),,(),,( zyxEndzyxE pp =+

This is called Floquet’s theorem. The field at z1+d is related to the field at z1 as

),,(),,( 1)(

11 dzyxEedzyxE p

dzp +=+ +−γ

),,(),,( 1)(

11 zyxEedzyxE p

dz +−=+ γ

which gives ),,(),,( 11 zyxEedzyxE dγ−=+ (3.12)

This is the property of Bloch wave.

Using the property of Fourier series, any periodic function can be expanded into an

infinite number of Fourier series. Thus

dπnzjpnp (x, y) eE(x, y, z) E /2−=

where Epn are wave functions of x and y.

dz e (x, y, z) E d

(x, y) E πmz/d-jpd

pn 2

0

1∫=

Exponential functions from a complete orthogonal set, i.e.


80

=≠

=∫ nm nm 0

0

22d

dze e (x, y, z)Ed

πmz/djπnz/d-jp

The field in a periodic structure can now be represented as

πnz/dπnz-j-jpnα

-αn (x,y) E E(x, y, z) 22∑

==

z/d-jβpn

α

-αn

n ( x, y) eE E(x,y,z) ∑=

=

where γ = jβ and

dn

nπββ 2

0 ±=

Each term in the expansion is called spatial harmonic and has a propagation phase

constant βn. The phase velocity of the nth harmonic is given by

ndv

npn πβ

ωβω

20 ±==

The group velocity of the nth harmonic is

βω

ddv gn =

(3.13)

(3.14)


81

and is the same for all harmonics. From the relations, we can see that some of the

spatial harmonics have phase and group velocities in opposite directions. This leads to

backward radiations from the structure.

3.4 ω- β Diagram (Brillouin diagram)

The frequency bands of operation of a periodic structure can easily be

determined by plotting the curves of β versus k0 where β is the propagation constant

and k0 is given by

εµω=0k

It will give the frequency bands for propagation and attenuation for the structure. Such

k0d- βd diagram is also called Brillouin diagram. It can also be represented in terms of

ω and β. So they are also termed as ω-β diagrams. The phase velocity and group

velocity at a frequency point can be determined from the Brillouin Diagram. The β/ω

value gives the phase velocity and the slope of the curve dω /d β gives group velocity.

Brillouin diagram is a good tool for determining frequency regions where backward

radiation or negative refraction can be obtained from an electromagnetic periodic

structure. Brillouin diagrams are also known as dispersion diagrams [18-20]. This also

can be plotted in kod- βd axes. Figure 3.2 shows the dispersion diagram of a C-L

periodic structure. This diagram shows positive phase velocity and negative group

velocity. Since they are in opposite directions it will lead to backward radiations.

(3.15)


82

Fig.3.2. ω-β diagram of a C-L circuit showing phase velocity and group velocity at a point x on

the curve. The ω/β at the point x gives phase velocity at that point and the slope of the curve at

the point x gives group velocity.

The dispersion diagram can be obtained from the unit cell analysis of the

periodic structure. This represents the variation of fundamental spatial harmonic with

frequency. As discussed before, in a periodic structure a number of spatial harmonics

can exist and the interaction can change the behaviour of the structure at some

frequency points. Depending on the interaction among harmonics, different composite

dispersion characteristics will be obtained [21]. For example, in a band gap structure,

the fundamental propagation constant is not adequate to represent the dispersion nature,

especially at band gap edges. Single dominant mode (n = 0 spatial harmonic) is

sufficient to describe the field adequately for a transmission problem. However, near

the bandgap edges a single spatial harmonic does not satisfy the prescribed boundary

conditions on the structure and does not constitute the normal mode by itself. In this

case the field can be represented by the aggregate of all spatial harmonics [21]. For a

minimum accuracy, at least two higher order harmonics should be considered [21]. In

this case, the dispersion is determined by the aggregate of all the harmonics.

Depending on the periodic structure, there can be a coupling between the spatial

harmonic modes and this can change the ω-β diagram at some frequencies. This is

β

ω

x

Slope= vg < 0


83

called mode coupling [21]. In this study, it was observed that phase velocity and group

velocity can be different from the frequencies predicted by unit cell analysis at some

frequency bands. This gives the indication that negative refraction phenomenon can be

found in frequency points where phase velocity is negative and group velocity is

positive at these frequencies there may not be any backward radiation or negative

refraction when analysed with unit cell alone.

In the past, defects in EBGs were analysed by considering a finite number of

cells which contain the defect, as a repeating element and this method is called super

cell analysis [22]. In this research, a similar analysis is carried out with special

emphasis to negative refraction phenomenon from EBGs. When the spacing is very

small compared to the operating wavelength the spatial harmonics effects can be

negligible. This is not the case when the spacing is comparable to the operating

wavelength; the case where most of the Electromagnetic Band Gap structures (EBGs)

are realised. So in practical structures, the spatial harmonics effects are important in

determining the dispersion characteristics. It is interesting to see these effects and their

contributions towards negative refraction so that one can make use of negative

refraction phenomenon in the passband of EBGs which can lead to novel applications

in antenna and electromagnetic engineering.

3.5 Unit cell and ‘Super cell’ analysis

This section will show how the dispersion behaviour of a periodic structure

changes from that of a unit cell in practically realisable structures. When periodic

structures are analysed using unit cell, it will not give spatial harmonics effects but only

the fundamental harmonic. Even though unit cell analysis can predict transmission

characteristics of the structure, it is not accurate to predict negative refraction

phenomenon completely. This is because at some frequencies the spatial harmonics


84

effects are considerable and thereby the dispersion behaviour changes at narrow

frequency bands.

Most of the EBGs realised are finite in the microwave frequency region and in

this case, there is a coupling between the incident field and Bloch waves inside the

structure. This coupling depends on numerous parameters and the field in a finite EBG

can never be reduced to a combination of Bloch waves [16]. Bloch theory can predict

the properties of the crystal to a great extent (e.g. transmission characteristics). It does

not take into consideration the fact that all realisable EBGs are of finite extent, which is

generally excited by an incident field. So it neglects evanescent waves and coupling

that exist in all real situations. This suggests that an analysis is needed which include

more realistic situations.

By considering many elements of the periodic structure excited by an incident

field, one could incorporate the interaction of higher order spatial harmonics and

evanescent wave appears in finite structures. Figures 3.3 and 3.4 illustrate the

difference between the infinite and finite periodic structures and the numerical domain

for finite structures. In the case of infinite crystal, it can be seen that the entire space is

filled with the crystal itself and there exist only Bloch modes inside the crystal and

there is no incident field. For a finite crystal which is limited in extend, there exist an

incident field excitation and this leads to evanescent modes in addition to the Bloch

modes. In a computational procedure for the finite crystal, the infinite extending

surroundings have to be terminated with absorbing boundaries (fig.3.5).


85

Fig.3.3. A Diagram illustrating the case of an infinite photonic crystal. The entire space is

occupied by the photonic crystal.

In the subsequent studies of this work, various periodic structures are

considered. As a first step, structures consisting of lumped elements will be discussed

in this chapter. In the proposed method, more elements are considered for the analysis.

It is observed that there can be narrow frequency regions at which one could obtain

backward radiation phenomenon. In the next chapter, metallic periodic structures are

analysed using plane wave illumination in certain directions and negative refractions at

frequencies other than lower edges of bandgap are identified. In this case, the

interaction of higher order Floquet’s harmonics and the coupling between the incident

field and the structure are taken into consideration as it appears in the calculated S

parameters.

Photonic Crystal Propagating Bloch modes No incident field

-∞ +∞ Problem space

- ∞

+∞

P

robl

em sp

ace


86

Fig.3.4. A Diagram illustrating the case of a finite periodic structure. The surrounding extends

to infinity and there exist Bloch modes and evanescent modes.

Fig.3.5. A Diagram illustrating the numerical approach for a finite periodic structure. The

boundaries are terminated by absorbing boundary conditions.

Air

Periodic structure Propagating Bloch modes + Evanescent modes Incident Field

Absorbing boundary

Abs

orbi

ng b

ound

ary

Absorbing boundary

Inci

dent

fiel

d Air

Periodic structure Propagating Bloch modes + Evanescent modes Incident Field

Inci

dent

fiel

d

Problem space-∞ +∞

- ∞

Prob

lem

spac

e

+∞


87

3.6 Computational Electromagnetics (CEM) and simulation packages

Numerical simulation tools are vastly used these days in all branches of both

engineering and biological sciences. Computational Electromagnetics (CEM) is the

discipline that intrinsically and routinely involves the use of digital computer to obtain

numerical results for electromagnetic problems [23]. It is not uncommon to employ

analysis and /or CEM to understand experimental results, nor is it uncommon to verify

analysis results and CEM results with experimental results. In this work, commercial

electromagnetic simulations tools Ansoft High Frequency Structure Simulator

(HFSS™), CST Microwave Studio™ were used for various studies. Agilent Advanced

Design System (ADS™) was used for lumped circuit simulations.

HFSS™ is the industry-standard software for S-parameter and full-wave SPICE

extraction and for the electromagnetic simulation of high-frequency and high-speed

components [24]. It can be used for the design of on-chip embedded passives, PCB

interconnects, antennas, RF/microwave components, and high-frequency IC packages,

scattering/radiation problems etc. It solves problems in frequency domain using Finite

Element Method (FEM). The recent versions of HFSS™ can solve coupled thermal and

stress analysis by incorporating ePhysics™, a software from Ansoft which can take into

account of mechanical and thermal parameter variations [24]. Ansoft HFSS™ uses

FEM in frequency domain with some advanced numerical methods for matrix solving.

Another commonly used high frequency simulation tool is CST Microwave

Studio™. This software uses the Finite Integral Technique (FIT) for solving

electromagnetic problems with combination of some advanced techniques to increase

the computational performance. These techniques are mainly the Perfect Boundary

Approximation technique (PBA®), Thin Sheet Technique (TST™) and Multilevel Sub

Griding scheme (MSS™) [25]. This is based on time domain techniques.


88

The package can give frequency domain information by performing Fourier transform

of the time domain data.

All the electromagnetic computational tools fundamentally deal with solving

Maxwell’s equations at different problem scenarios. Since the Maxwell equations can

be written either in integral or differential forms, different computational methods can

be deployed. Fig.3.6 shows the categories in CEM and the domains. The domains

where CST Microwave Studio™ and Ansoft HFSS™ are also shown.

Fig.3.6. Categories within computational electromagnetics and the domain of the packages CST

Microwave studio and Ansoft HFSS

In order to avoid any misinterpretation of the results obtained from commercial

packages, it is advantageous to know about the computational schemes used in each

package. In the next sections, the basis of the main computational schemes; Finite

Element Method (FEM) and Finite Integration Technique (FIT) will be explained.


89

3.6.1 Finite Element Method (FEM)

Many popular electromagnetic simulation packages (e.g. HFSS™ and

EMSolve™) employ the finite element method and its hybrid versions for robust and

adaptable modelling. The basis of FEM is to take a complex problem whose solution

may be difficult to obtain, and decompose it into pieces upon each of which a simple

approximation of the solution may be constructed, and then put the local approximate

solutions together to obtain a global approximate solution [27]. In order to find the

local solutions, the problem space is divided into small elements (meshes).

FEM is used for solving partial differential equations (PDE) approximately.

Solutions are approximated by either eliminating the differential equation completely

(steady state problems), or converting the PDE into an equivalent ordinary differential

equation (without any partial differentiation involved), which is then solved using

techniques such as finite differences. In solving partial differential equations, the

primary challenge is to create an equation which approximates the equation to be

studied. This equation should be numerically stable, meaning that errors in the input

data and intermediate calculations do not accumulate and cause the resulting output to

be of no use.

A PDE will involve a function u(x) defined for all x in the domain with respect

to some given boundary condition. The purpose of the method is to determine an

approximation to the function u(x). The method requires the discretisation of the

domain into sub regions or cells. For example, a two-dimensional domain can be

divided and approximated by a set of triangles (the cells). On each cell, the function is

approximated by a characteristic form. Here u(x) can be approximated by a linear

function on each triangle [27].


90

3.6.1.1 Finite element method for electromagnetic problems

In electromagnetic problems, Maxwell’s equations can be solved using FEM. In

Ansoft HFSS™ the first step is to mesh the problem space into tetrahedral elements. In

the second step, it solves the field equation derived from Maxwell’s equation. Consider

the electric field equation [48]

Obtain a basis function Wn which define conditions between nodal locations in the

overall mesh of tetrahedra, based on problem inputs. Third step is to multiply the basis

functions with equation (3.16) and then integrate the result over the volume V. This

gives

Where n = 1, 2, 3... N and N is the total number of unknowns.

Rewriting equation (3.17) using Green’s and Divergence theorems gives

Writing E field as a summation of unknowns using the same basis function used before

gives

01 2 =−

×∇×∇ EkE ro

r

vvε

µ (3.16)

∫ =

−

×∇×∇⋅

V nror

n dVEWkEW 01 2 vvε

µ (3.17)

( )∫ ∫=

−

×∇•×∇

VS

nror

n dStermboundarydVEWkEW )(1 2 vε

µ (3.18)


91

Substituting equation (3.19) in (3.18) yields

Now equation (3.20) is in the form of a matrix equation

AX = B

where A is the basis functions and field equation, in a known N x N matrix, X is the

unknowns to be solved for and B is the excitation. Equation (3.20) can be solved by

using matrix solution methods. HFSS uses an iterative algorithm for the matrix

solution.

3.6.2 Finite Integration Technique (FIT)

The FIT technique was developed by Weiland [29, 30] in 1977 which provided

volume type discrete reformulations of Maxwell’s equations in an integral form. The

first step of the finite integral technique is to restrict the electromagnetic problem to a

bounded space region Ω Є R3, which contains the space region of interest. Next step is

to decompose the computational domain Ω into a finite number of cells Vi in a way that

all cells have to fit exactly to each other. This decomposition gives finite cells complex

G, which serves as computational grid. Fig. 3.7 shows the diagram of such a grid in the

∑=

=N

mmm WxE

1

v

( ) ∫∫∑ =

−

×∇•×∇⋅

= SVmnrom

rn

N

mm dStermboundarydVWWkWWx )(1 2

1ε

µ

(3.19)

(3.20) (3.20) (3.20)


92

cell complex G.

Fig. 3.7. A cell Vi,j,k-1 of the cell complex G with the allocation of the electric grid voltages e on

the edges of A and the magnitude facet flux b through this surface.

Assume a rectangular shaped computational domain Ω. Also, each edge of the cells

includes a direction. Let i, j and k be the number of mesh points along x, y and z

directions. After the definition of grid cell G, the further introduction of the finite

integral theory can be restricted to a single cell volume Vn. Starting with Faraday’s law

in integral form [30]

).,().,( AdtrBt

sdtrEA

A

vvvvvv∫∫∫ ∂

∂=

∂

can be rewritten for a facet A z( i, j ,k) of Vn as the ordinary differential equation

),,(),,1(),1,(),,1(),,( kjib

tkjiekjiekjiekjie zyxyx ∂

∂−=+−+−++

as shown in fig. 3.7.

The scalar values

),1,( kjiex +

),,( kjiey

),,( kjibz

),,( kjiex

),,1( kjiey +

(3.21)

(3.22)


93

sdEkjie kii

kii

zyxzyxx

vv.),,( ),,(

),,(1∫ +=

is the voltage along one edge of the surface Az ( i ,j ,k), representing the exact value of

the integral over of the electric field along this edge. The scalar value

AdBkjibkjiAz

z

vv.),,(

),,,(∫=

represents the magnetic flux.

Assuming a lexicographical ordering of the electric voltages e(i,j,k) and the

magnetic fluxes b(i,j,k) over the whole cell complex G and their assembly into column

vectors in such a way, that we compose the degrees of freedom first in the x, y and z

directions. The eqn.3.21 of all grid cell surfaces of the complex G can be collected in a

matrix form using finite integral approximation similar to eqn. 3.22 for the three facets.

∂∂

−=

M

M

M

M

n

n

n

n

bt

e

e

e

3

2

1

... ... ... 1- ... 1- ... 1 ... 1

... ... ...

C . e = b

The matrix C represents a discrete curl operator on the grid G. The second discrete

operator to be considered is the divergence.

(3.25)

(3.23)

(3.24)


94

Fig.3.8.The allocation of the six magnetic facet fluxes which have to be considered in the

evaluation of the closed surface integral for non-existence of magnetic charges within the cell

volume.

Maxwell’s equations describing the non existence of magnetic charges is [30]

0.),( =∫∫∂ AdtrBv

vvv

which is considered for a volume cell Vi,j,k as shown in fig.3.8.

The evaluation of the integral in eqn.3.26 for the cell shown above gives

0)1,,(),,(

),1,(),,(),,1(),,(

=++−

++−++−

kjibkjib

kjibkjibkjibkjib

xz

xyxx

This relation for the single cell can be expanded to the whole cell G and this yields a

discrete divergence matrix

),1,( kjiby +

),,( kjibz

),,1( kjibx + ),,( kjib x

)1,,( +kjibz

),,( kjib y

(3.27)

(3.26)


95

0 ... ...

. 1 1- 1 1- 1 1- .... ...

6

5

4

3

2

1

=

M

M

m

m

m

m

m

m

bbbbbb

S . b = 0

In the above discussion, it has been shown that how to perform the discretisation of two

of the four Maxwell’s equations. For the discretisation of these two equations using

finite integration technique, another cell complex G1 which is the dual to the primary

cell G is considered. For the Cartesian coordinates, G1 is obtained by taking the foci of

the cells of G as grid points for the mesh cells of G1. Fig.3.9 shows the spatial

allocation of a cell G and its dual G1.

The complete integral of the charge density within a dual cell V1 can be related

to a discrete charge onto the single grid point of the primary grid G placed inside V1.

The discretisation of Ampere’s law in integral form

( ) AdtrJtrDt

sdtrHA

A

vvvvvvv.),(),(.),(

11

+∂∂

= ∫∫∫∂

(3.28)


96

Fig. 3.9. The spatial allocation of a cell and a dual cell of the grid doublet G, G1

can be performed for a facet A1 of dual grid cell V1 in the same way as Faraday’s law.

Finally, Gauss Law in integral form can be discretised for dual grid cells. These will

result in matrix equations. These equations are called Maxwell grid equations. Let C1

be the grid operator for the dual discrete curl and S1 for the dual discrete divergence.

For the cell complex pair G, G1, the complete set of discrete matrix equations are

given by [30, 31]

qdSbS

jddtdhCb

dtdeC

==

+=−=

vvv

vvvv

0

1

This system of matrix equations can be solved using numerical techniques. This is the

basis of FIT. Table 3.1 presents a comparison of FEM and FIT.

Dual Grid G1

Grid G


97

Method Basis Capabilities and limitations

Finite Element

Method (FEM)

Frequency domain

differential

equations

Can handle complex geometries,

Dispersive modelling will result in longer

simulation time, Gives approximate

solution

Finite Integration

Technique (FIT)

Time domain

integral equations

Can handle complex geometries and

dispersive materials, Having numerical

dispersion

Table. 3.1. A comparison of FIT, FEM methods

3.7 EBG modelling in commercial packages

Most of the commercial packages are capable of modelling periodic structures

using periodic boundary conditions. Periodic boundary conditions can be implemented

using the symmetry in the structure and this way the computational time can be reduced

for determining the transmission response of the system. Periodic boundary conditions

are the computational implementation of Bloch theorem as given in eqn. 3.12. In CST

Microwave Studio™ and Ansoft HFSS™ these problems can be solved by considering

it as an eigen mode problem. This gives the result for source free solution under given

boundary conditions. In this case, the periodic structure in the computational model is

similar to a cavity problem and the solutions are calculated in a source free

environment for electric or magnetic fields satisfying the given boundary conditions.

For a given structure, there can be multiple solutions and these solutions are termed as

the eigen solutions of the problem. For example, the gap between the first and second

eigen modes in the wave vector space can be identified as the first bandgap of an EBG.


98

3.8 Numerical simulations of LHMs and analogous LHMs

Following the theoretical and experimental verification of the LHMs, analogous

LHMs and their characteristics, there have been lots of interests in numerically

studying the effects on wave propagation through in LHMs. These investigations

provided deep insight into the phenomenon especially from an application point of

view. The main complexities in numerical studies of LHMs are the dispersive

modelling and computational resources. Both time and frequency domain methods

were employed in various studies.

Investigations of LHMs characteristics began with the numerical simulation of

Inverse Snell’s law, focusing and applications of LHMs such as phase shifters. Some of

these studies considered the dispersive material characteristics of LHMs by applying

the Drude model [40]. Dispersive material modelling in frequency domain needs large

computational resources. So time domain modelling is usually preferred. For an

accurate numerical study of LHMs, the simulation model has to take care of the

material dispersion. Using this dispersion, the existence of negative material parameters

can be explained [40]. Let W be the energy density, E, H be the electric and magnetic

fields respectively. Then

22

21

21 HEW µε +=

In a case where there is no dispersion and if both material parameters are negative, the

total energy becomes negative which is a meaningless concept. If one takes the fact that

there is dispersion eqn. 3.29 becomes [34, 39]

(3.29)


99

∂∂

+∂

∂= 22 ])([])([

21 HEW

ωωωµ

ωωωε

From the equation (3.30), it can be seen that for the energy density W to be positive, the

material parameters must obey the following conditions.

0)])([>

ωωωε

dd

0)])([>

ωωωµ

dd

Following this criteria ε, µ can be defined as a plasma medium model [40]

])(

1[)(2

0e

pe

iνωωω

εωε−

−=

])(

1[)(2

0m

pm

iνωωω

µωµ−

−=

Where peω is the electronic plasma frequency, pmω is the magnetic plasma frequency.

eν is electric collision frequency and mν is the magnetic collision frequency. The above

relations (eqns.3.33 and 3.34) represent the Drude model for material parameters.

FDTD simulations of LHMs were widely employed by many researchers [32-39].

Wave focusing using an LHM slab is demonstrated using dispersive FDTD in [33, 37]

considering the material parameter variations with frequency. Wave propagation in a

multilayer stack which consisted of thin alternating layers of conventional materials

and lossy LHMs were demonstrated using dispersive FDTD in [34, 39]. This structure

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)


100

can act as a wave transportation system for evanescent waves over a large distance. The

simulated near field intensity through a multilayer LHM structure with refractive index

n = -1 is presented in fig. 3.10. It can be noticed that the image is transferred through

four layers of LHM which are equally spaced apart. It was also shown in [34] that the

LHM slab

450 500 550 600 650

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ImagePlane

ObjectPlane

LHMs slab: 3 wavelength long 1/10 wavelength wide cell=wavelength/220

Ez

Cell

Fig.3.10. Diagram showing the simulated near field intensity through multilayer LHM structure

with refractive index n = -1. (Reproduced from [34])

compensates for the phase delay of the wave as it passes through the vacuum layers in

between.

Numerical studies of a composite medium consisting of split rings and wires

using Finite Integration Technique (FIT) were detailed in [41]. In this work, a unit cell

consisting of a conducting wire and a split ring resonator with periodic boundary

conditions was selected for modelling. The scattering parameters were calculated and

from this an effective refractive index was obtained. Numerical simulations were

performed to study the losses in the split ring wire model. The Ohmic and dielectric


101

losses in the structure were studied in detail and it was observed that in addition to the

two lossy mechanisms, the transmission will also be affected by the thickness of the

conducting layer. Low dielectric substrates were found to be causing conductive losses

high.

Frequency domain solvers can also be applied for LHMs problems to some

extend. The latest versions of HFSS™ can solve these problems to some extent [42]. In

[42], Inverse Snell’s Law verification and slab lens focusing were verified using HFSS.

However, due to the difficulties in implementing a non linear dispersive material

properties and the high computational resource requirement, this package has not

received much interest in LHMs studies. Fig.3.11 shows the Ansoft HFSS™ simulation

results demonstrating positive refraction from a wedge of permittivity ε = 2.2 and µ = 1

and negative refractions from a wedge of ε = -2.2 and µ = -1 [42]. It can be seen that

the refraction directions are opposite directions in each case.

Fig.3.11: Simulated field showing refraction from a wedge with positive refractive index (left)

and negative refractive index (right). Refracted rays are shown in dotted arrows. (Reproduced

from [42])


102

The above mentioned works were on the characteristics of an LHM media which is

homogeneous (if such a medium exists). The wave propagation in analogous LHMs

obtained from EBGs was also studied using numerical simulations. As discussed in

chapter 2, all these analysis considered infinite EBGs, showing negative refraction

phenomenon near the bandgap edges.

Inverse Snell’s Law verification using FDTD from a periodic structure

consisting of triangular lattice of metallic wires for TM polarization was presented in

[43]. The radius to period ratio was 0.2. The simulated electric field intensity plots for

positive and negative refractions are given in fig. 3.12. It can be observed that at 7

GHz, the EBG exhibits positive refraction. At 9 GHz, the EBG shows negative

refraction. However it has to be noted that at 9 GHz there exists a positively refracted

components, even though it is low in intensity compared to the negatively refracted

component. This establishes the fact that in an EBG there exist both positively and

negatively refracted components. There have been many works one could find in

literature on the numerical simulation of wave propagation at negative refraction

frequencies in EBGs and some of them were briefed in chapter2. The intuitive

observation is that it considered the possibility of getting negative refraction from

infinite EBGs.


103

Fig.3.12. Simulated electric field intensity showing positive refraction at 7 GHz and negative

refraction at 9 GHz from a metallic EBG. (Reproduced from [43])

3.9 Lumped element circuit simulations

In order to verify the above discussed facts, numerical simulations of lumped

element circuits using ADS were carried out. The aim of the simulations was to

establish that the ‘super cell’ analysis can predict backward radiation frequencies due to

the spatial dispersion in the structure. In these simulations, the finite periodic structure

itself acts as a repeating element of an infinite periodic structure. The dispersion

diagram obtained revealed that backward radiation exists at narrow frequency bands

due to spatial dispersion in the structure when more elements were included in the

model. The dispersion behaviour was obtained from S parameters as detailed in section

3.2.2. Bandgap regions are those regions with group velocity zero (slope of the

dispersion diagram is zero) and phase constant is either zero or π .


104

3.9.1 C-L High pass filter circuit

Consider a high pass filer consisting of unit cells of capacitors and inductors (C-

L) as shown in fig.3.13. As discussed in chapter 2 (sections 2.10.2), this structure was

identified as an equivalent LHM medium. Let C = 0.5 pF and L= 1 nH. The

transmission response of the unit cell with periodic boundary condition is given in

fig.3.14 which shows a high pass filter response with a cut-off frequency around 5.0

GHz. The dispersion diagram obtained using unit cell analysis is given in fig. 3.15.

From fig.3.15, it can be noticed that this structure is a backward radiating structure

throughout the passband [44] when unit cell analysis is used for the analysis.

Fig.3.13. A unit cell of a high pass C-L filter with C= 0.5pF and L= 1.0nH

Fig.3.14. Response of the C-L filter showing pass band from about 5 GHz


105

Fig.3.15. Dispersion diagram for the high pass C-L filter with unit element

From this diagram, it can be seen that from 5 GHz onwards there is a passband, which

agreed well with frequency response in fig. 3.14. From the unit-cell analysis, planar

structure analogy to such a lumped-element circuit is a backward radiating structure

throughout the pass band starting from 5 GHz, which can be observed from the

dispersion diagram. Next, more elements were included in the simulation model to see

the effect of spatial harmonics in the model. When more elements were considered as

repeating elements in numerical analysis, the high order spatial harmonics were taken

Fig.3.16.Dispersion diagram for the high pass C-L filter with three elements

P

hase

con

stan

t

Phas

e co

nsta

nt


106

into account. Fig.3.16 shows the dispersion diagram when three elements were

considered. From the figure, it can be seen that the backward radiation regions were

narrowed to a frequency region of 5.0 GHz-5.7 GHz. When high-order spatial

harmonics are taken into account by considering many elements, the dispersion curve

has changed and negative dispersion disappeared at some frequency bands. This

indicates that when high order harmonics are considered, the dispersion behaviour

changes from that of a unit cell analysis. Hence the high pass model can not satisfy the

claim [44] that the medium can act as an ultra wide band backward radiating antenna

throughout the passband.

3.9.2 L-C low pass filter circuit

A similar analysis was carried out for a low pass type L-C filter. The unit cell is

shown in fig. 3.17, in which C=1.5 pF and L = 1.5 nH. The transmission response of

the filter shows a passband up to 4.3 GHz as in figure 3.18. Dispersion diagram

obtained from a unit cell is given in figure.3.19. From fig.3.19, it can be seen that there

is no backward radiation throughout the passband for the unit cell dispersion diagram.

Fig. 3.20 shows the dispersion diagram with three elements. It can be seen that the

dispersion behaviour is changed in the three elements case. There is a backward

radiation frequency region starting at 2.5 GHz, which was absent in the unit cell

analysis case.

Fig.3.17. A unit cell of a low pass L-C filter with C= 1.5 pF and L= 1.5 nH


107

Fig.3.18 Response of the low pass L-C filter showing pass band up to 4.3 GHz

Fig.3.19. Dispersion diagram for the low pass L-C filter with unit element

2 3 4 5 6 7 8 91 10

-2

0

2

-4

4

freq, GHz

beta

*dPh

ase

cons

tant

S

(2,1

) dB


108

Fig.3.20.Dispersion diagram for the low pass L-C filter with three elements

3.10 Spatial harmonics suppression in periodic structures

The suppression of one or more spatial harmonics in a periodic structure can

lead to novel applications in electromagnetics. Solntsev [46, 47] has recently studied

the spatial harmonics selection in pseudo periodic waveguides depending on frequency

[46,47]. This is based on the coordinated variation of the period and the phase

distribution of elements along the waveguide and thereby giving constant phase

velocity to one spatial harmonic and suppressing other harmonics. The principle of

selection based in using electrodynamic systems with non periodic spacing of its

elements and a specified relation between the step Lq and the field phase θq of the

elements, which makes it possible to select one spatial harmonic and suppress the

others. In this method, the phase distribution is determined by the shape and

dimensions of the system elements which are to be chosen from the conditions of

spatial harmonics selection. This is analogous to a filter whose frequency

characteristics also determined by the choice of elements and the period.

2 3 4 5 6 7 8 91 10

-2

0

2

-4

4

freq, GHz

beta

*dPh

ase

cons

tant


109

Fig.3.21. A Pseudo Periodic Waveguide: A comb type structure (Reproduced from [46])

The longitudinal electric field distribution along the system with Q steps and different

length Lq (q=1, 2, 3,…Q) as shown in fig.3.21 is given by [46]

)],(exp[),(),( 0 ωψωω zizfEzEz = (3.35)

The type of the periodic system (uniform periodic or non uniform) governs the

distribution of real amplitude f (z, ω) and phase θ (z, ω). Applying Fourier transform,

the amplitudes E(h, ω) of spatial harmonics is given by [46].

dzihzzEl

hEl

z )exp(),(1),(0

−= ∫ ωω

In general, the amplitudes E (h, ω) are continuous functions of the wave number h and

frequency ω and differ in spectral density by a factor l, where l is the length of the

system. Now, expressing E (h, ω) as a sum over Q steps of the system gives

(3.36)

0 Z1 Z2 …………………………… ZQ= l Z

q=1 2---------------------------Q

dq

Lq


110

)]])((exp[1),(1

qqq

Q

qq hziMU

lhE −= ∑

=

ωψω

Where θq(ω) = θ(zq, ω) is the average field phase at the qth step. Mq(h) is the local

electron interaction coefficient. Uq is the RF voltage at the qth step, zq dq are the mean

coordinate and effective width of the qth gap. If the field is constant in the gap f(z) = fq,

we have the expression [46]

2

2sin

q

q

q hd

hd

M

=

From eqn. (3.37), we can get the maximal values E (h, ω) for the wave numbers h = hm

that satisfy Q conditions

...3,2,1,2 =+= qqmzh qqm πψ

m is an integer m = …..-2,-1, 0, 1, 2…determines the number of the field spatial

harmonic with maximal amplitude. Now, introducing a field phase shift φq = ψq+1 – ψq

at the qth step taking into account that Lq = zq+1 -zq. This yields the equivalent

conditions of synchronism for every step as

QqmLh qqm ,...2,1,2)()( =+= πωϕω

In a periodic waveguide, Lq = L, φq = φ and φq = qφ, so the conditions for

synchronism will meet for an infinite number of spatial harmonics m’ = m. In a non

uniform waveguide with different steps Lq, the synchronism condition can only be

(3.37)

(3.38)

(3.39)


111

satisfied by one harmonic by appropriately choosing the phase φq. In this way,

selection of spatial harmonics takes place [35]. In other words, the spatial harmonics

selection can be done by making the structure pseudo periodic or by introducing a

phase changing mechanism. This idea is applied to metallic periodic structures for

negative refraction enhancement and is presented in chapter 4 of this thesis.

3.11 Conclusions

In this chapter, a brief discussion on periodic structure analysis is presented. It

is shown that conventional unit-cell approach for infinite periodic structures is not

adequate to predict negative refraction phenomenon at certain frequency bands. It is

seen that there are forward and backward radiating spatial harmonics components in a

periodic structure. These spatial harmonics can interact with each other and change the

dispersion characteristics. It has been identified that in a realisable periodic structure

which is always finite, there exist dispersion sources which are not included in the unit

cell model analysis. The method of considering finite structure as a super cell with

incident field excitation will take into account the higher order spatial harmonic

interaction, incident field coupling with the structure and evanescent wave interactions

towards dispersion characteristics. Numerical simulations of structures consisting of

lumped capacitor and inductor models were performed to verify the finite structure

analysis and spatial harmonics effects on dispersion characteristics. The variation of

backward radiating frequency regions from the unit cell was observed when more

elements were included in the analysis.


112

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117

Chapter 4

Negative Refraction Phenomenon from

Metallic Wire Medium

4.1 Introduction

In the previous chapters, it was shown that periodic structures are capable of

backward radiation and negative refraction at certain frequencies in the passband due to

the strong spatial dispersion effects. In this chapter, further verification of the idea

using EBGs consisting of metallic wires is considered in depth. Such structures are

generally called Metallic Electromagnetic Bandgap Structures (MEBGs) or metallic

photonic crystals (MPCs). The extensive studies on electromagnetic bandgap (EBG)

structures began with dielectric EBGs [1-6] and most recently a growing interest in

metallic EBGs [7-12] has emerged. Negative refraction phenomenon at the lower edges

of bandgap frequencies in dielectric EBGs was presented in [13-17] and the same

phenomenon in MEBGs [18, 19]. The limitation of this work was that the observed

negative refraction phenomenon was at frequencies in the bandgap edges and this limits

the transmitted power. Fig.4.1 depicts the frequency region for bandgaps and bandgap

edges. Also, the observed phenomenon was not reported for multiple frequency bands

using an EBG.


118

Fig. 4.1. A diagram showing passband and bandgap behaviour of an EBG

All the aforementioned works were based on the unit cell analysis which is not

adequate for refraction studies. In this chapter, it is demonstrated that negative

refraction can be obtained at frequencies in passband in addition to the lower edges of

the bandgap. This enables higher wave transmission. However, as in these structures

there are no negative effective parameters, the observed phenomenon should be more

accurately termed as refraction like phenomenon. For simplicity, the term refraction is

used in the following sections of this chapter.

MEBG structures are studied for negative refraction verification. Numerical

simulations were carried out for a slab model and a prism structure. Experimental

results in good agreement with simulations are presented for the prism model and an

approach for enhancing the negative refraction at certain frequencies is proposed and

verified.

4.2 Bragg’s Law and bandgaps in EBGs

As discussed before, EBGs consist of periodic arrangement dielectric/metallic

shapes. The basic principle of an EBG is the Bragg’s reflection. This occurs in a crystal

when the period of the crystal obeys Bragg’s law [20]. It is given by the equation

T

rans

mis

sion

Frequency

B

andg

ap

Passband

Passband

Bandgap edges

-15dB

0dB


119

θλ sin2dn =

where n is an integer

λ is the wavelength

d is the period

θ is the angle between the scattering plane and the incident wave

Figure 4.2 illustrates the Bragg reflection from a crystal. The red lines indicate atomic

layers in the crystal. The law expressing the condition under which a crystal reflects

electromagnetic wave with the maximum amplitude and, at the same time, denoting the

angle at which the maximum reflection occurs. This reflection causes partial and

complete bandgaps in EBGs.

Fig.4.2. A diagram illustrating Bragg’s law

Equation 4.1 determines the centre frequency of the bandgap. The bandgap properties

are governed by the symmetry of the crystal, periodicity and the thickness of the EBG

slab. For example, the complete bandgap of an EBG consisting of metallic wires with

radius r and period a depends on both r and a. Fig. 4.3 shows the bandgap variation

d

θ θ

(4.1)


120

with respect to r/a ratio for an infinite metallic wire EBG having square lattice for TE

polarisation.

Fig.4.3. Complete bandgaps for TE modes as functions of r/a for an EBG with square lattice

consisting of metallic wires (Reproduced from [11])

4.3 Metallic electromagnetic bandgap structures (MEBGs)

MEBGs are of great interest in the electromagnetics community due to its ease

of fabrication and clear cut-off at bandgap frequencies compared to dielectric bandgap

structures. MEBGs consisting of square lattices of metallic wires are selected for this

study. Throughout the following discussions, the radius of wires is denoted by r and

the period by a. A schematic diagram of the square lattice is given in figure 4.3.The

filling ratio of the EBG is the ratio of radius to period (r/a), which is the wire radius

normalised by the period.


121

Fig.4.3. Schematic diagram of square lattice of period a consisting of metallic wires of radius r

For an MEBG consisting of infinitely long thin wires, the bandgap depends mainly on

the filling ratio (r/a). For transverse magnetic(TM) polarisation, MEBGs will act as a

high pass filter having a low frequency cut-off [11,21] and for transverse electric (TE)

polarisation it does not posses a bandgap at lower modes of frequencies unless the

filling ratio r/a >0.2 [11]. This can be observed from fig.4.2. This difference in

performance for the two polarisations can be explained by considering the polarisation

of the cylinders induced by an external field. If the electric field is directed along the

wire axis i.e. TM polarisation, the cylinder will get easily polarised and this leads to

strong interaction of the cylinders in the structure. This interaction makes the array to

act like a metal surface. When an incident field with electric field vector ( Ev

) is parallel

to the wire axis incident on, it will set up currents along the wires just like on a metal

surface. This current cancels the incident field. This leads to complete reflection [22] of

the wave. When frequency increases the period is getting comparable to the wavelength

and the metal surface approximation starts deviating. Fig.4.4 illustrates this concept

clearly.

2r

a

a


122

Fig.4.4. A diagram showing a metallic wire array which acts like a metallic surface at low

frequencies (since the wavelength is large in this case, the spacing between the wires are very

small compared to the wavelength) and when frequency increases (wavelength decreases) the

spacing between the elements becomes comparable to the wavelength.

When the electric field is perpendicular to the wire axis, there will not be any

current flow since the interaction between the wires is small. This is because, for thin

wires the wire diameter is small (so that current along the radial direction is negligible)

hence there will not be any bandgap for TE polarisation, if the elements are not too

close (i.e. when r/a is small). The bandgap starts appearing when r/a becomes large. For

a given radius, this can be achieved by making the period ‘a’ smaller and thereby

making the wires more closely. Thus the interaction between the wires will increase.

When the filling ratio greater than 0.2 there exists a bandgap at higher modes. When r/a

>0.3 there exists a narrow bandgap in the first few modes also and the width of the

bandgap increases with further increase in r/a ratio [11].

The bandgap in finite wire MEBG for TE polarisation can arise not only due to

the lattice resonance but also due to the self resonance of each wire [19, 21]. Recently,

it was revealed that the bandgap due to self resonance have the same characteristics as

λ λ Metallic wires

Incident field ↑Ev


123

that due to periodicity [19]. Negative refraction can be observed at frequencies near

these bandgaps. The negative refraction frequency shifts in wires of finite length,

loaded with active elements is given in [21]. Let l be the length of each wire in a finite

length wire MEBG. Then the first bandgap will occur at the first resonance frequency

of the wire. The resonating wavelength λr is given by lr =2λ

.

Negative refraction in a structure can be observed by verifying the inverse

Snell’s law. Two structures were considered for numerical studies. One was a slab and

the other was a prism structure consisting of metallic wires. Negative refraction at

multiple frequency bands including the passband is demonstrated in the subsequent

sections and experimental verification for a prism structure is presented.

4.3.1 Design of the MEBG slab

Firstly, a periodic structure consisting of metallic wires of radius of 0.63 cm and

height 1.26 cm was considered. A diagram of the structure used for the numerical

simulation along with the field vector directions are given in fig.4.5. The spacing

between the centres of wires was 3.15 cm. The filling ratio (r/a) was 0.2. There were

three wires in the x direction and six in the y direction. TM polarisation (electric field

vector parallel to the axis of the wires) was employed. The wave propagation was along

the x direction as shown in fig.4.5.

The lower cut-off plasmonic frequency of the MEBG when the wires are

infinitely long is given by the following equation [9].


124

Fig.4.5. A schematic diagram of the MEBG structure used for numerical simulation studies and

vector directions of electric field E, magnetic field H and propagation vector k

−≈

ar

a2120 π

λ

where a is the period, r is the radius of the wire and λ0 is the wavelength corresponding

to the lower cut-off frequency. Substituting the values equation 4.1 gives the λ0 as

1.225cm corresponding to plasmonic frequency of 24.4 GHz. The period is 3.15cm and

therefore the lattice resonance is at λ = 2*3.15cm= 6.3cm corresponding to a frequency

of 4.76 GHz. Since the fundamental lattice resonance is lower than the plasmonic

resonance, the structure exhibits a pass band from 4.76 GHz. The second lattice

resonance will occur around 9.4 GHz. In the present model, the wires are finite with a

length of 1.26 cm. Hence, there arises a bandgap due to the self resonance of the wire

[19]. This self resonance corresponds to a wavelength of 1.26*2 = 2.52cm, which gives

a frequency of 11.9GHz. The passband of the MEBG starts from the lowest possible

resonance frequency. In the present case, it is due to the lattice resonance at 4.7 GHz.

So the MEBG is having a passband from 4.7 GHz.

z

y

Ev

kv

Hv

x

1.26cm

7.56cm

17.01cm

(4.1)


125

4.3.2 Numerical simulation and discussion

Numerical simulations were carried out using Ansoft HFSS™. In the simulation

model, the metallic wire structure was enclosed with an air box having absorbing

boundaries except the faces at which the ports or incident field excitations were

assigned. For S parameter simulation, wave ports were used as the source. For refracted

field study, incident field excitations had been employed. The angle of incidence was

assigned by providing direction vector values for Ev

, Hv

and kv

and vectors in the

HFSS™ incident field options. The S parameters were obtained for the wave

propagation direction perpendicular to the wire axis. i.e. along the x axis. The

refraction occurred when the wave incidents obliquely at the interface and leaves the

EBG was studied. If the refracted field inside the structure is propagating towards the

negative z direction, it is considered as positive refraction.

The S parameters were obtained from frequency sweep and the dispersion

diagram was calculated from the S parameters. The simulated S(2, 1) response is given

in Fig.4.6. From the S(2,1) it can be seen that the bandgap is from 7.7-8.8GHz. This

bandgap is the second lattice resonance bandgap. The shift in frequency towards the

lower side (from 9.4 GHz to 8.5 GHz) can be explained by the fact that the effective

wavelength increases in the MEBG.

The dispersion diagram obtained is presented in fig.4.7. From the dispersion

diagram, the frequency regions where the structure supports backward radiation and

negative refraction (regions with phase and group velocities in opposite directions)

were determined based on the criteria presented in the previous chapter. In the band gap

region, the dispersion diagram shows some propagation. This can be explained by the

fact that the structure used was a small structure (with less elements or with less

number of periodicity), and hence the band gap is not deep. In theory, deep band gaps

can be obtained with more elements because when the number of elements increases, a


126

greater amount of reflection of wave occurs at the bandgap frequencies [18].

Fig.4.6: Transmission response of the wire periodic structure showing bandgap around 8.5 GHz

-4

-3

-2

-1

0

1

2

3

4

6 7 8 9 10 11 12 13

Frequency (GHz)

Phas

e co

nsta

nt

Fig.4.7: Dispersion diagram for the wire periodic structure with period 3.15 for a direction of

propagation along x axis

Band Gap


127

From the dispersion diagram, there is negative phase velocity and positive

group velocity at 6.5 GHz. This indicates that there is negative refraction at this

frequency. At 7.5 GHz there is a positive phase velocity and a positive group velocity;

hence the structure supports positive refraction. In order to study the negative

refraction, an oblique incident field at one face of the model with an angle of incidence

45˚ was used. The magnetic field propagation in the structure was observed at 6.5GHz

and 7.5GHz. The simulated field plots at these frequencies are given in Fig.4.8. The

propagation directions of the field are denoted by arrows. From this field view, it can

be noticed that at 6.5 GHz, there is bending of the magnetic field propagation direction

towards the negative side. This wave bending is in the opposite direction i.e. towards

the positive side at 7.5 GHz.

Fig.4.8: A plot indicating magnetic field propagation directions inside and outside the periodic

structure at 7.5 GHz(left) and 6.5 GHz (right). The propagation directions in each media are

indicated by arrows. Surface normal is shown in dotted lines.

From fig.4.8 it can be seen that the incident field at an angle 45º reach upon the

structure and it gets refracted at the air and wire medium interface. When it comes out


128

of the structure again refracted at the wire medium and air interface. At 7.5GHz, it

refracted in such a way that the structure supports positive refraction. At 6.5GHz, the

structure supports negative refraction with an angle of -41º as observed in fig.4.8. From

the transmission response in fig. 4.6, it can be seen that 6.5 GHz is in the passband, 1.2

GHz below the bandgap. In terms of normalised frequency (ωa / 2πc), the bandgap is at

0.81 and the observed negative refraction is at 0.68. If this occurred due to the all angle

negative refraction, it should be very close to 0.81. The above results clearly show the

existence of negative refraction from the MEBG in the passband.

4.3.3 Prism structure for Inverse Snell’s Law verification

The studies in the previous section showed negative refraction from a slab

shaped structure for oblique incidence. In the following section, the Inverse Snell’s

Law verification at frequencies in the passband, in addition to the bandgap edges from

a prism shaped EBG formed by periodic array of metallic wires is detailed.

4.3.3.1 Design of the MEBG prism

A prism structure made with square lattices consisting of metallic cylinders with

length 12 cm and the radius 0.63 cm was selected. A diagram of the prism is shown in

fig. 4.9. The period was 1.75cm and the filling ratio was 0.36. In this prism structure

there were 10 elements both on base and vertical sides which give minimum side

dimension of 17.01 cm corresponding to 2.68 λ at 4.3GHz.


129

Fig.4.9.Diagram of the prism consisting of metallic wires with period 1.75cm.

The plasmonic cut-off frequency for the EBG structure for TM polarisation is given by

eqn.4.1 which gives a frequency of around 13GHz. Since the lattice resonance

frequency (corresponds to wavelength λ = 2a = 3.5cm) is at 9GHz which is far below

the plasma resonance frequency, the EBG acts as a high pass filter for TM polarisation

having cut-off frequency of 9GHz.

For TE polarisation the structure acts as a dielectric EBG and since r/a > 0.3 the

first complete bandgap occurs within the first and second modes [11]. The simulation

model for the structure used for the calculation of bandgap and wave propagation

directions Γ, X and M is shown in fig 4.10. An eigen value solution model was selected

with periodic boundary conditions. TM and TE polarisations can be achieved by

changing the top and bottom walls as Electric (i.e. Etan = 0) and Magnetic (i.e. Htan =0)

respectively. The bandgap diagram was obtained from numerical simulation using CST

Microwave Studio. Fig.4.11 illustrates the CST simulation model used for obtaining the

bandgap diagram.


130

Fig.4.10. Unit cell assignment in CST simulation model for the simulation of bandgap diagram

for EBGs consisting of square lattice and wave propagation directions Γ, X and M

In a bandgap diagram eigen frequencies are plotted against wave number which

is equivalent to eigen solutions against boundary condition phases [9]. From fig. 4.10 it

can be seen that there are three directional sweeps for the model. They are Г to X, X to

M and Г to M. So the dispersion diagram consists of these three regions. The dispersion

curve for each region was produced separately with two phase shifts. The phase shift

assignments for the parameter sweep function through the three steps are given in

table.4.1. The three sweeps were combined to get the complete bandgap diagram. In

step 1, phase shift 1 is varied from 0-180, keeping phase shift 2 as zero. This is

equivalent to the wave vector sweep from 0 to 180 degrees in the Г to X domain. In

step 2, phase shift 1 is kept at 180 and phase shift 2 is varied from 0-180. In step 3, both

phases 1 and 2 are swept from 180-0. Then step 3 is inverted because the required

phase shift is 0˚-180˚. The dispersion diagram is obtained by combining the three steps.

The bandgap diagram of the structure for TE polarisation obtained from the CST

simulation is shown in fig.4.12.

Γ X

M

Phase shift 1

Phas

e sh

ift 2


131

From this figure, it can be seen that the complete bandgap is from 9.3- 10.2 GHz as

shaded in green.

Fig.4.11.CST Simulation model for bandgap determination. Four side walls are defined as

periodic boundaries. The top and bottom walls are defined as magnetic/electric boundaries

depending on the polarisation.

Table.4.1. Three steps for obtaining the complete bandgap diagram and their corresponding phase

shifts for the parameter sweep.

Step 1 : Г X

Step 2: X M

Step 3: Г M

Phase Shift 1: 0-180

Phase Shift 1 : 180


Phase Shift 2: 0



Magnetic/Electric Periodic

Periodic

Periodic

Periodic


132

Fig.4.12. Bandgap diagram for the metallic wire structure with r/a = 0.36 (TE polarisation).

Green region indicates complete bandgap

The prism structure used was with less number of elements and the number of

elements was not same throughout the wave propagation domain because the number

of elements is decreasing from base to top. Unsurprisingly, a clear and wide bandgap

can not exist except for an infinite number wire prism. The S parameters were obtained

in the incident field direction perpendicular to one side of the prism. The transmitted

fields were measured at different frequencies using two WR-187 waveguides; one for

transmission and one for reception. The waveguides have a lower cut-off frequency of

3.5 GHz. The measured transmission response S (2, 1)) obtained for the prism structure

is shown in fig.4.13. From the figure it can be seen that the bandgap is from 5.6 to 6.1

GHz.

0

2

4

6

8

10

12

14

16

Freq

uenc

y G

Hz

Г X M Г


133

Fig.4.13.Measured transmission response for the prism structure showing bandgap around

5.7GHz. The bandgap region is marked in blue.

The dispersion diagram was obtained for incident field direction perpendicular

to the wire axis, from the simulated S parameters. Dispersion diagram in fig.4.14

indicates the bandgap region around 5.54 to 6.5 GHz, which agrees with the measured

transmission response. The measured transmission shows bandgap from 5.6 - 6.1 GHz.

The difference between the dispersion diagram in fig.4.12 and 4.14 is that the first one

predicts the complete bandgap (bandgap for all incident field directions) and the second

figure gives the bandgap for the incident field direction perpendicular to the wire axis.

It can be noticed that the bandgap width is narrowed in the finite prism. The backward

radiation frequency regions were identified from the dispersion diagram as in the

previous case. For studying the refraction properties of the structure, using numerical

simulation, a rectangular dielectric slab waveguide with opening dimensions of 5 mm

was used for the excitation. The dielectric constant of the slab was 40. The design

equation for the cut-off frequency less than 10 GHz this slab waveguide is given in by

the equation [35]


134

141010 9

−××

=r

c tf

ε

Where t is the side dimension of the slab in centimetres. This gives the lower cut-off

frequency of 0.8 GHz.

-4

-3

-2

-1

0

1

2

3

4

4 5 6 7 8

Frequency GHz

βd

Fig.4.14. Dispersion Diagram for the prism periodic structure with metallic wires having a

period of 1.75 cm. The bandgap region is marked in blue.

From the dispersion diagram it was observed that there are multiple regions that

can yield negative refraction. These regions were 4.0 - 4.5 GHz and 7.2 - 8.0 GHz. It

can be seen from the dispersion diagram that positive refraction occurs at 4.7 GHz and

negative refraction at 7.4 GHz. Simulated refracted field obtained at these frequencies

verified this. The simulated field plots are given in figs.4.15 and 4.16. It is clear from

the field plots that the wave refracted at an angle of +37˚ at 4.7 GHz and -45˚ at 7.4

GHz. The wave propagation directions are indicated in dark blue arrows and the

surface normal is denoted by the dotted line. Different scales were used at the two

frequencies for better view of the refraction phenomenon.

Bandgap

Phas

e co

nsta

nt

(4.2)


135

The animation of phase obtained from simulation also supported the refraction at

positive and negative angles at the two frequencies.

Fig.4.15. Electric field plot showing positive refraction at 4.7 GHz. Units are in V/m

Fig.4.16. Electric field plot showing negative refraction at 7.4 GHz. Units are in V/m


136

4.3.3.2 Measurement verification of refraction phenomenon

The effects of Floquet’s harmonics and spatial dispersion on refraction

properties of EBG like structures and negative refraction phenomenon from prism

periodic structures were studied in the previous sections and the theoretical predictions

were verified using numerical simulations. In this section, the experimental work

carried out for the verification of simulation results for the prism structure with period

1.75cm is detailed. The main goal of the experiments was to verify the existence of

positive and negative refraction from prism structure at various frequencies.

Measurement results have shown good agreement with simulation results. Based on the

available literature, so far there has not been any extensive experimental verification on

the role of Floquet’s harmonics and spatial dispersion in the study of LHMs or

analogous LHMs.

From the numerical simulations discussed in the previous sections, it has been

demonstrated that negative refraction exists at narrow bands of frequencies in the

passband. A diagram of the experimental setup used is shown in fig.4.17.

Fig.4.17. A diagram showing the setup used for experimental studies.

Rx Tx

Rotation path

30cm 30cm 30cm

Waveguide Waveguide

Absorber Absorber


137

Two models were selected for the experimental study. One was a Teflon prism with

known parameters (εr = 2.1 and µ = 1.0), having base dimensions of 7.4 cm, 10.5cm

and height of 30 cm and the other was the wire prism under study. For the excitation, a

WR-187 waveguide with 30cm length and opening dimensions of 4.68cm and 2.1 cm

was used. A waveguide of same dimensions as that at the transmitting side was used for

reception. A photograph of the two prism models used for the experimental studies is

given in fig.4.18. The waveguides and the equipments were surrounded by absorbers to

prevent any reflected wave reaching into the structure under study or to the receiving

waveguide.

Fig.4.18. Two prism structures used for measurement: Prism structure consists of metallic

wires (left) and Teflon prism (right)

In order to substantiate the experimental setup, the refracted field from a Teflon

prism was measured and positive refraction was identified. Figure 4.19 is the simulated

electric field plot for the Teflon prism showing positive refraction at an angle of +20º.

For the measurement of refraction, the structure under study was placed in front of the

transmitting waveguide and then rotated the transmitting setup (equivalent to receiving

refracted power at different angles). First, the transmitted field was measured with the


138

waveguides at a distance of 30cm. The receiver was rotated from +90 degrees to -90

degrees and the electric field was measured at around 1200 steps (positions) at each

frequency. The received field thus obtained without any structure in between the

waveguides is given in fig.4.20. The figure shows a peak when the receiver is at the

boresight position i.e. at an angle of zero degrees. The negative angle side corresponds

to negative refraction and positive angle side corresponds to positive refraction. The

measured field with the Teflon prism is given in fig.4.21 which shows an angle of

refraction of +23º.

Fig.4.19. Simulated electric field showing positive refraction from Teflon prism at 4.7GHz.

The angle of refraction is +20º.


139

0

0.2

0.4

0.6

0.8

1

1.2

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

Angle (Deg.)

Norm

alis

ed P

ower

Fig.4.20.Measured pattern with flanges only at 7.4 GHz showing the peak amplitude at

boresight. The half power beam width is 34º.

Fig.4.21.Measured refracted field for Teflon prism at 4.7 GHz. The angle of refraction is +23º

Next, the angle of refraction was measured by replacing the Teflon prism with

the EBG. The metallic wires were fixed on two foam sheets of 1cm thickness as shown

in fig.4.22. There were 8 wires in base and vertical sides of the prism. This gives a side

dimension of 2.19 λ at 4.3GHz. The waveguide opening was kept very close (4mm


140

from the first air wire interface) to the structure to avoid field spreading at the incident

face. The two waveguides were at a distance of 30cm. At frequencies corresponding to

positive refraction in the dispersion diagram, the refracted wave will bend towards the

negative side. At positive refraction frequencies, it will bend towards positive side. The

refracted field was measured at different frequencies corresponding to positive and

negative refraction. The measured refracted field intensity at different frequencies is

presented in fig.4.22. It was found that at 4.7 GHz, the structure exhibited positive

refraction with a refraction angle of +34˚. From simulated field plot, it can be seen that

there is a positive refraction at +37˚. At 7.4 GHz the measured field showed negative

refraction with an angle of -44˚, where -45˚ was observed in numerical simulation.

0

0.2

0.4

0.6

0.8

1

1.2

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

Angle (Degree)

Nor

mal

ised

Pow

er

7.4 GHz

5.6 GHz

4.7 GHz

7.2 GHz

Fig.4.22: Measured refracted field for wire prism showing positive refraction at 4.7 GHz and

negative refraction at 5.6 GHz, 7.2 GHz and 7.4 GHz

The experiment was repeated for other frequency points where the dispersion diagram

predicted negative refraction (a potential 2% bandwidth around 7GHz) and found

dominant negative refraction phenomena. The measured field at 7.2GHz also gave

negative refraction with an angle of -64˚ (fig.4.22). Such a phenomenon has also


141

measured at 5.6 GHz which was at the lower edge of bandgap, shown negative

refraction with an angle of -19˚ as depicted in fig.4.22. The measurement results

indicated that there was a beam narrowing at negative refraction frequencies.

Results obtained from measurement at 7.2 GHz were compared with similar

results previously reported on Inverse Snell’s Law verification [6-8] using different

prism structures. Table.4.2 presents the comparison of measurement result at 7.2 GHz

with similar works using prisms consisting of metallic wires [18], split rings and wires

[28.29]. The parameters compared were the main beam refraction angle, 3dB beam

width in each case at negative refraction frequency and the strength of negatively

refracted beam by the positively refracted one. It was noticed from previously reported

results, that even though there was dominant negative refraction there associated a

positive refracted component. This can be explained by the fact that there always some

Floquet’s harmonics radiating towards the positive side [24-26]. In order to quantify the

amplitude of negative refracted field compared to that of positive refracted field, a

parameter Negative Refraction Excess (NRE) was introduced. NRE is defined as

%100×−

=n

pn

AAA

NRE

Where An and Ap are the maximum amplitudes of negative and positive refracted beams

respectively.

(4.3)


142

Structure

under study

Negative refraction

angle (Deg.)

3dB Beam width at NR

frequency(Deg.)

NRE

λ/d

Metallic wires

[18]

-35

15

70

1.05

SRR+ Wire [28]

-61

33

81

6

SRR+Wire [29]

-34

17

72

7.2

Metallic wires [This work]

-64

17

85

2.3

Table: 4.2.Comparison of measurement results at 7.2GHz with similar work for verification of

inverse Snell’s law using different structures.

In [18], NRE was reported 70% from an EBG consisting of metallic wires,

while in [27], [28], it was 81% and 72 % for a structure of Split Ring Resonators

(SRRs) and wires respectively. In this measurement, an NRE of more than 80% (85%

at 7.2 GHz and 81% at 7.4 GHz) is achieved. This can be explained by the fact that the

measurement was performed in the pass band with less wave attenuation compared

with those achieved at the lower edge of bandgap. At 5.6 GHz NRE was 69%. The fifth

column in table.1 indicating the effective medium applicability on the basis of

operating wavelength (λ) and the period (d) of the model used in each case. In the

present study, a λ/d ratio of 2.3 was used. This is comparable to the models which were

used previously for negative refraction measurement.

4.3.3.3 Inclusion of defects for enhancing negative refraction

According to the dispersion diagram in fig.4.14 there can be negative refraction

from 4.0 GHz to 4.5 GHz. However, from the measurement we observed that the

negative refraction phenomenon is not significant, for example at 4.3 GHz. In this case


143

the positive refracted component was strong by 42% then the negatively refracted

component. This can be accommodated by the fact that this frequency point is very

close to the positive refraction frequency band.

In [30] a motivating idea was presented to enhance ‘anomalous refractions’ at

its bandgap of a periodic structure by introducing some defects into the structure. In

[31] the defect states in a dielectric periodic structure have been detailed. From the

dispersion diagram presented for a square array of GaAs dielectric rods with defects, it

was observed that the introduction of defects changes the refraction properties of the

band dispersion diagram even in the passband. In the conventional treatment of EBGs

using EFS diagrams, when we introduce anisotropy in the crystal the EFS shape can be

changed and one can obtain a case where the EFS circles become ellipsoids [32]. The

reason for this change is that the introduction of anisotropy causes a phase

displacement. The radiation patterns of a dipole surrounded by an EBG having different

line defects introduced on it to enhance the radiation pattern in certain directions is

experimentally studied in [33]. Fig.4.23 shows the two switching mechanisms used and

the corresponding radiation patterns. From the figure, it can be seen that the two

switches are mechanically controlled to turn it on or off. On or off position indicates

the removal or introduction of the rods. For example, ON-ON indicates that both top

and bottom defect lines are active. The radiation pattern is changed due to the removal

or introduction of the defect lines.


144

Fig.4.23. Diagram of radiating dipole inside an EBG with defects for changing the radiation

pattern and corresponding radiation patterns. The blue dots indicate the defect positions.

(Reproduced from [32] )

From the discussion in chapter 3 (section. 3.10), it was found that a phase

altering mechanism by introducing pseudo periodicity in the structure one could

suppress some of the Floquet’s harmonic components. In a prism structure like the one

under study this suppression can lead enhanced refracted field in one direction. A

similar approach was used to restore the negative refraction phenomena at the pass

bands at which both negative and positive refractions are equally significant. Four

wires from the prism structure were removed to enhance the negatively refracted beam.

Fig.4.24 shows that four metallic rods are missing from the prism, creating defects in

the prism structure. It is verified from the HFSS™ simulation (fig. 4.24) that the

introduction of defects indeed enhances the negative refraction phenomena in the way

of suppressing unwanted spatial harmonics.


145

Fig.4.24. Simulated electric field at 4.3 GHz on prism with defects showing negative refraction

This has been verified by measurement. Fig.4.25 shows that introduction of the defects

can lead to an increase of 27% in the negative refraction. This can be explained by the

frequency shift in the dispersion diagram of the prism structure with the defects. The

dispersion diagrams for both cases for an incident field direction normal to the wire

axis for TE polarisation is given in fig.4.26. From the diagrams it can be seen that

there is a frequency shift and that the frequency point of interest (4.3GHz) has shifted

more towards the negative phase velocity side.


146

Fig.4.25. Measured refracted field from wire prism structure showing positive refraction at 4.3

GHz and negative refraction at 4.3 GHz after the introduction of the defects into the structure.

-4

-3

-2

-1

0

1

2

3

4

4 5 6 7 8

Frequency (GHz)

βd

with defectswithout defects

Fig.4.26. Dispersion diagrams for the wire structure without the defects and with the defects

has been introduced

Phas

e co

nsta

nt


147

In [30], negative refraction at bandgap frequencies was obtained by the

introduction of defects. In the present work this was obtained in the passband. This is

the first realisation of negative refraction in the passband using defects. The change in

the dispersion diagram supports the enhancement of negative refraction by introducing

defects. In a periodic structure, the removal of the wires can be done in many ways. In

order to get an optimised performance, one needs to use optimisation routines in which

the removal of various wires with different degrees of freedom can be studied to predict

the phenomenon for better performance.

For a normal prism of a natural material the frequency dispersion is very small.

The transmitted power in the previously reported works based on split ring and wire

medium was found to be very low (less than 20%) [28, 16]. In the present work, the

measured transmission from the prism structure was 54% at 7.2GHz. This has shown

that the transmitted power obtained from the EBG prism is more than twice the

transmission from split ring and wire medium.

A comparison was carried out on the frequency regions of interest with the

negative refraction reported in [15] (fig. 2.22, chapter.2). In terms of normalised

frequency (ωa/2πc), the negative refraction observed in this work was at 0.25 (at 4.3

GHz), 0.32 (at 5.6GHz) and 0.42 (at 7.2GHz). The complete bandgap for the EBG was

at 0.548. In [15] the bandgap was at 0.25 and the all angle negative refraction was

obtained at 0.19 (i.e. 0.06 below the bandgap). This indicated that the observed

negative refraction phenomenon in the present study was far below the complete

bandgap. If the observed phenomenon was due to all angle negative refraction proposed

in [15], it should be very close to the complete bandgap at 0.548.


148

4.4 Conclusions

In this chapter, negative refraction phenomenon from metallic EBGs has been

presented. It was found that negative refraction can be obtained from the EBG at

multiple frequency bands even though it does not posses a negative effective refractive

index. It has been identified that strong spatial dispersion in the structure due to high

order Floquet’s harmonic interactions, incident field coupling and evanescent modes

can give rise to negative refraction at certain frequencies. These frequencies are not

very close to complete bandgap. In conventional analysis of infinite EBGs, such a

phenomenon was obtained near the complete bandgap edges only. This limited the

transmitted power. In this work, transmitted power of 54% was obtained which was

much higher than that from a split rings and wires medium. A negatively refracted

beam which was 85% stronger than the positively refracted beam at 7.2 GHz was

obtained. This was higher than that of the previously reported works. To enhance the

negative refraction, a method is proposed by applying defects into the finite structure

and negative refraction enhancement at certain frequencies is verified. An enhancement

of 27% for the negatively refracted beam is obtained. Numerical simulations results

were verified with measurement results and both were in very good agreement. The

dispersion obtained from the MEBG can lead to novel applications in beam scanning

and spatial frequency demultiplexing. It is very difficult to obtain such type of

dispersion from natural materials at microwave frequencies.


149

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153

Chapter 5

Wave Focusing at Low Microwave

Frequencies Using Metallic Wire Medium

5.1 Introduction

Pendry [1] proposed the possibility of obtaining a perfect lens using a slab of

LHM. It was claimed that such a lens can lead to novel applications in medical imaging

and digital storage media with extremely high capacity due to its unique characteristic

of aberration-free image. Ordinary lenses are of curved shape, which suffer from

spherical aberrations, and hence low image resolution. A perfect lens formed by an

LHM enables focusing of electromagnetic waves to a size much smaller than the

wavelength [2-5]. This lens is free from all aberrations which limit the image quality

from conventional lens. Conventional lenses can not focus waves to a size significantly

sharper than λ/2 [2]. Fig.5.1 shows the focusing mechanism in a slab with negative

refractive index n = -1. Let the distance of the point source from the slab be l and the

thickness of the slab be 2l. In fig.5.1, A is a point source, B is the first focus formed

inside the slab and C is second focus formed outside the slab. Since the refractive index

is -1 the first focus B is formed at the centre of the slab.

Chapter 5: Wave Focusing at Low Microwave Frequencies Using Metallic Wire Medium

154

Fig.5.1.A Ray diagram showing wave focusing in a slab with negative refractive index. n = -1.

A is the source point, B and C are first and second focus points respectively.

After the pioneering proposal by Pendry [1], there has been growing curiosity in

realising such a lens. Due to the loss of all materials, obtaining a ‘perfect’ lens is a

difficult task. However, getting a ‘super lens’ is a more practical solution. There has

been a lot of work in studying focusing using an LHM slab by numerical simulations

and experiments. Fig.5.2 shows the numerical simulation result showing wave focusing

from a slab of refractive index -1. The simulation was carried out using HFSS®. From

the figure, it can be seen that the source point at the right side is getting focused in the

middle of the slab and the second focus point is obtained outside of the slab on the left

side.

Recently, a medium consisting of resonant inclusions of conducting wires and

spirals was selected for LHM performance and focusing studies [4]. In this medium, the

spiral arrays contribute negative permeability and the wires contribute negative

permittivity [4]. One dimensional measurement of electric field was carried out using

two identical dipoles for transmission and reception. Since this model is based on EMT

approximations, the bandwidth is very narrow.

n = -1

A B C

l l2l


155

Fig.5.2. Simulated electric field plot showing focusing from a slab of refractive index n = -1

Wave focusing using negative refraction properties of EBGs was widely studied

in [3, 5-7]. Sub-wavelength focusing at microwave frequencies using an EBG

consisting of a square array of dielectric rods of dielectric constant ε = 9.61, the length

of rods l = 15cm and the radius r = 11.75mm was presented in [3]. The period of the

EBG was 4.79mm. TM mode polarisation was employed for the study and the wave

focusing was observed at a very narrow band with a centre frequency 14.27 GHz.

There were 21 layers in lateral direction and 15 layers in the propagation direction. The

source was at 0.7mm away from the surface of the EBG and an image was obtained at

0.7 mm on the other side. Figure 5.3 shows the results for focusing studies detailed in

[3]. It represents one dimensional power distribution showing focusing from the slab.

In [7], wave focusing from a two dimensional EBG slab consisting of cylindrical

alumina rods was detailed. The results are presented in chapter 2 (figs. 2.23 and 2.24).

The frequency at which the focusing observed was 9.3GHz.


156

Fig.5.3. A graphical representation simulated and measured average power intensity

distribution at the image plane with and without the EBG. The red curve shows the FDTD

simulation result and the blue line represents the experimental result. The green dotted curve

shows the field intensity without the EBG. It can be seen that there is strong field intensity at

the centre corresponding to the image point. (Reproduced from [3])

From the ray diagram in fig. 5.1 and the numerical simulation result of electric

field intensity in fig. 5.2, it can be seen that in addition to the focus point outside the

left handed slab, the refracted ray converges to another point inside the slab. This

indicates that objects in the left hand space produce a real image in the right hand space

[8]. Imaging using EBGs is close to the imaging by a mirror. However, in a mirror the

image is virtual one. The resolution of image obtained using an EBG depends on the

period. The maximum resolution obtained can not exceed the lattice spacing [8].

The experimental work detailed above considers only one dimensional

distribution of the field. This will not fully guarantee the exact resolution of the image

that is important in finite EBGs. There is also an issue that of the frequency at which

imaging was obtained was at higher frequencies. In [3], the frequency of focusing is

around 14GHz and in [7] it was at 9.3 GHz.


157

Next sections of this chapter will address the focusing from a finite MEBG at

low microwave frequencies. The design details of the EBG slab will be followed by

numerical simulations using HFSS® and experimental verification. It is shown that the

phenomenon is due to the metamaterial nature of the slab other than any diffraction or

array antenna effect. This measurement was the first extensive measurement of wave

focusing from an MEBG along two directions; lateral and longitudinal to determine the

exact size and resolution of the focus, obtained from a finite EBG. This work is the first

realisation of focusing using metallic EBGs.

5.2 Design and numerical simulation

From the previous chapter, the complete bandgap frequency region was found

to be 9.3 GHz-10.2 GHz, for the MEBG consisting of metallic wires with square

lattice. The wires were having radius 0.63cm and length 12cm. The radius to period

ratio (r/a) was 0.36. The proposed slab was having a height of 13.4cm, thickness

6.51cm and width 12 cm (length of wire). It was formed with 4 wires in z direction and

8 in y direction (a total of 32 wires). Fig.5.4 shows the schematic diagram of the

structure and the setup used for experimental studies.


158

Fig.5.4. Schematic diagrams of the structure used for focusing studies (left) and the setup used

for the experiment. There are four elements in z direction and 8 elements in y direction in the

wire structure.

The dispersion diagram of the proposed slab obtained for a direction of incidence

perpendicular to the wire axis (the wave propagation is along z axis) with electric field

along the y axis is given in fig.5.5. This diagram gives the phase velocity versus

frequencies for a certain incident angle (perpendicular to the axis of the wire). In both

figs. 5.4 and 5.5, the incident wave propagation is along the z axis.

The difference between the figure 4.16 in chapter 4 and figure 5.5 is that the

first represents the eigen frequencies for different wave vector directions inside the

infinite EBG and the latter gives the phase constant over the structure for different

frequencies for a specific angle of incidence for the finite EBG. The latter diagram

shows a bandgap at 5.5 GHz and this bandgap exists for the incident wave direction

perpendicular to wires. The dispersion diagram of the structure gives negative phase

velocity and positive group velocity at some frequency regions. Phase velocity has the


159

same sign as phase constant and the slope of the dispersion curve gives the group

velocity.

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9

Frequency GHz

Pha

se c

onst

ant

NR NRBandgap

Fig.5.5.Dispersion diagram of the MEBG slab obtained for incident field perpendicular to the

wire axis (along z axis). The negative refraction regions and bandgap regions are marked. The

bandgap starts from 5.5GHz.

From the dispersion diagram in fig.5.5 it can be observed that there can be multiple

frequency bands with negative phase velocity and positive group velocity (e.g. 2.0-2.6

GHz and 4.0-5.0 GHz). Since at these frequencies the medium can act as an analogous

LHM, it can give rise to wave focusing.

Numerical simulation was performed using Ansoft HFSS™ to verify the

focusing phenomenon from the proposed slab given in fig.5.4. A wave port with a

small opening of 5mm at the plane containing the origin of coordinates at and a

distance of 2.4 cm away from the air-wire interface was used as the source with all

other boundaries as absorbing walls. Simulated electric field intensity along the yz

plane at 2.1 GHz is given in fig.5.7. The focus point is marked in the rectangular box.

At 3.4GHz was a positive refraction frequency point at which wave focusing was not

observed. Fig. 5.8 shows the simulated electric field at 3.4 GHz. It can be seen that


160

there is no focusing of waves and there is a low intensity point at the right hand side of

the slab.

It is shown that at 3.4 GHz the wave is spreading at the right side of the slab and

at the position corresponding to focus point there is no wave focusing while at negative

refraction frequency 2.1 GHz there is a high intensity point marked in a small

rectangular box in the field plot in fig.5.7. This high intensity point is at a distance of

11.31 cm from the source (i.e. from the origin of coordinates). The electric field

intensity along the xy plane at various positions along the z axis was taken. This

gradual movement of electric field plotting plane shows the wave is getting focused at

the middle of the slab, when it travels through the slab and then gets diverged. The

electric field intensity at 2.4 cm (at the left air slab interface), 3.4cm and 7.0cm from

the source are depicted in fig.5.9, 5.10 and 5.11 respectively.

Fig.5.7.Simulated electric field intensity along the yz plane at 2.1GHz showing focusing. The

focus point is marked in the rectangular box


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Fig.5.8. Simulated electric field intensity along the yz plane at 3.4GHz showing no focusing.

This shows the progressive movement of the field along the wire slab. From these field

plots, it can be noticed that the wave propagates inside the model in a way shown in the

ray diagram in fig.5.1. So the focus obtained at outside the slab is the second focus

point.

Fig.5.9. Simulated electric field intensity along xy plane at a distance of 2.4cm from the source


162

Fig.5.10. Simulated electric field intensity along xy plane at a distance of 3.4cm from the

source

Fig.5.11.Simulated electric field intensity along xy plane at a distance of 7cm from the source


163

5.3 Measurement of focusing from an MEBG slab

The experiment was performed with a slab structure consisting of metallic wires as

used in the simulation. A monopole with resonant length 1cm was used for the

excitation. The antenna radiates almost uniformly in almost all directions except the

backfire. A probe with the same dimensions was used for receiving the transmitted

power. Figure 5.12 shows the schematic diagram and a picture of the monopole probe

used for the measurement.

Fig.5.12.A schematic diagram of the probe used (left) and a picture of the coaxial fed probe

used in the experiment for transmission and reception (right)

The source was placed at the origin which is 2.4 cm away from the air wire

interface and the field intensity was measured in the xy plane using a near field NSI™

scanner. The receiving probe was fitted to the arm of the scanner which can move in x

and y directions. Fig.5.4 shows the schematic diagram of the measurement setup used

for the experiment. Calibration was done to eliminate the effects of surroundings. The

four sides of the slab were covered by absorbers. Electric field was scanned at points

with steps of 5 mm along the x direction and 2mm along y direction. The criteria

adopted for determining the focusing phenomenon was when there was a high field

intensity compared to the surrounding region at the position corresponding to the

source monopole in the image plane, it indicated a focus point. The measured field


164

intensity along xy plane at 2.1 GHz at a distance of 11.31cm from the source is shown

in fig.5.13. In the figure, the field values are given in dB. The reference was taken as

the maximum value measured in each case. It shows focusing at the right hand side of

the slab. The focus point is indicated in rectangular box. It can be seen that the

difference between the values at the focus point and the surrounding area was 6dB.

This indicated clear wave focusing. The focus point is marked in a rectangular box.

Fig.5.13. Measured electric field intensity along xy plane from the slab consisting of metallic

wires at 2.1GHz showing focusing

Measurement results in the region 2.7-3.8 GHz did not show any focusing since

it belonged to the positive refraction region in the dispersion diagram. The measured

field intensity along the xy plane at 3.4GHz which is a frequency point at which

positive refraction (hence no focusing) occurs is shown in fig.5.14. From the figure, it


165

can be seen that there is no wave focusing at 3.4 GHz. Measurements were carried out

for different frequencies and focusing was also observed at 4.2-5.0 GHz band. This

band corresponds to a negative refraction frequency region in the dispersion diagram.

The measured electric field intensity at 4.2 GHz is shown fig.5.15.


wires at 3.4 GHz showing no focusing


166


wires at 4.2GHz showing focusing

5.4 Discussion of results

The focusing in the MEBG at lower frequencies was due to strong spatial

dispersion in the structure. Unlike the results reported earlier in [6] and [8], this study

considers the scanning of field intensity in two dimensional planes. From simulation

and measurement results at 2.1 GHz, it can be seen that at 2.4 cm away from the right

edge of the slab, the electric field intensity at the centre is higher than that in the area

surrounding. In the measurement, an additional high intensity point at the edge was

observed and this can be explained by the edge effects from the slab and air interface.

The simulated and measured results clearly verified wave focusing from an EBG

consisting of metallic wires.


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There may be an argument that the observed focusing was arisen by the

diffraction from the edges. The simulated electric field intensity at various xy planes

along the z axis indicated that wave is getting focused at the middle of the slab and then

diverged when it travels through the slab (fig.5.9-5.11). If the phenomenon observed is

due to diffraction of two sources separated by a distance, then there will be focusing

points at the centre of all xy planes at different z positions. In other words, if the

phenomenon is due to diffraction or due to an array antenna effect (where multiple

number of beams can be obtained depending on the number of elements used ), there

will be focusing with one layer of wires and this will repeat at multiple points as the

wave passes through the slab. This was not the case observed from simulation results.

In fig.5.9, the field plots at 3.4 cm away from the source (which is inside the wire

structure by 1cm from the left air- slab interface) indicating no focusing. Also the

simulated field distribution with one layer of wires is given in fig.5.16. No focusing

was observed in this case.

In order to verify that the focusing phenomenon was due to the metamaterial

behaviour of the MEBG slab, further numerical studies were carried out. In this aspect,

the MEBG slab was replaced with slabs of different materials, each having same

dimensions as that of the MEBG slab. The electric field distributions obtained from

numerical simulations was studied in three cases. In the first case, the MEBG was

removed (air box only). In the second case, a dielectric slab of permittivity 2.0 was

placed instead of the MEBG. In third case, another slab of permittivity 80.0 was

employed. In all of the above cases, high resolution focusing phenomenon was not

observed. Fig.5.17 shows the electric field distribution in the yz plane for the air box

with radiation boundaries only (i.e. no slab in between) and fig. 5.18 shows the electric

intensity distribution with a dielectric slab with low permittivity εr = 2.0. Simulation

was performed with a dielectric slab with high dielectric constant (εr = 80.0). Simulated


168

electric field at 2.1 GHz is presented in fig.5.19. From figures 5.16-5.19, it can be

observed that there is no wave focusing in all the cases. One may argue that the peaks

at the right side of the slab can be a focus but it is easily seen that the physical size of

the peak intensity area is three times larger than that in the case of metallic slab. In

other words, waves are not focused in this case.

Fig.5.16. Simulated electric field intensity along yz plane at 2.1 GHz with one layer of wires

indicating no wave focusing.

Fig.5.17.Simulated electric field intensity along the yz plane at 2.1GHz without any slab

structure showing no wave focusing


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This confirmed that the observed wave focusing from the MEBG is due to the

metamaterial behaviour of the slab and can not be obtained with ubiquitous materials.

Compared to the field intensity distribution at 2.1GHz, the resolution of the focusing at

4.2 GHz was reduced. At 2.1GHz the image size was observed as 2cm corresponding to

a resolution of λ/7 and at 4.2 GHz the image size was 2.7cm yielding a resolution of

λ/2.64. In both cases, an image size of 6mm was noticed in the y direction. From the

above discussions, it was shown that high resolution focusing can be obtained from an

MEBG at low microwave frequencies in the passband of the EBG. The observed

focusing was far below the complete bandgap. The complete bandgap of the EBG starts

only at 9.2 GHz.

Fig.5.18.Simulated electric field intensity along the yz plane at 2.1GHz with a slab of

permittivity 2.0 showing no wave focusing


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Fig.5.19. Simulated electric field intensity along the yz plane at 2.1GHz with a slab of

permittivity 80.0 showing no wave focusing

5.5 Conclusions

Focusing at low microwave frequencies from a slab structure consisting of thick

metallic wires is studied in this chapter. The structure behaves like a slab having

negative refractive index or an LHM slab. So it can be considered as an analogous

LHM. Compared to the infinite model analysis which predicts negative refraction at

bandgap edges, in this study wave focusing at low frequencies in the passband is

demonstrated. At 2.1 GHz the phenomenon is observed with good resolution and this is

the lowest frequency reported for focusing using EBGs. This is far below from the

frequencies where focusing was obtained in previous work. Numerical simulation and

measurement results were presented with good agreement. In order to establish further,

that the observed phenomenon is due to the metamaterial nature of the MEBG, various

cases have been studied numerically. For this, the MEBG slab was replaced with

dielectric slabs with low and high dielectric constants. It has been demonstrated using

simulation results that the observed phenomenon is not due to array antenna or


171

diffraction effects. The two dimensional focusing phenomenon observed at lower

microwave frequencies can lead to applications in high resolution imaging.


172

References

[1] J. B. Pendry, “Negative refraction makes a perfect lens’’, Physical Review

Letters, Vol. 85, pp. 3966–3969, 2000

[2] Gennady Shvets, “Photonic approach to making a material with a negative index

of refraction”, Physical Review B, Vol. 67, pp. 035109-1-8, 2003

[3] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinpolou and C. M Soukoulis, “Sub-

wavelength resolution in a two-dimensional photonic-crystal-based super lens”,

Physical Review Letters, Vol.91, pp. 207401-1-4, 2003

[4] A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system

based on left-handed material plate”, Physical Review Letters, Vol. 92, pp.

077401, 2004

[5] P. A. Belov, “Flat lenses formed by capacitively loaded wire media”, Proc.

Progress in Electromagentics Research, 2005, China


refraction without negative effective index”, Physical Review B, Vol. 65, 201104,

2003


by flat lens using negative refraction”, Nature, Vol. 426, pp. 404, 2003

[8] Mayasa Notomi, “Negative refraction in photonic crystals”, Optical and Quantum

Electronics, Vol. 34, pp. 133-143, 2002


properties of photonic crystals”, Journal of Optical Society of America, Vol. 17,

pp. 1012-1020, 2000

173

Chapter 6

Experimental Investigation of Canalization

and Imaging Using Metallic Thin Wire

Medium

6.1 Introduction

In the previous chapter, wave focusing at low microwave frequencies using

thick metallic wire medium was demonstrated. It was shown that the wire medium act

as a metamaterial at certain frequency bands and the phenomenon observed can not be

obtained using natural materials. A sub-wavelength imaging regime, recently proposed

by Belov et al. [1] using periodic structures which acts as a metamaterial with superior

imaging performance is investigated in detail. In this study, periodic structures

consisting of thin metallic wires were used. In the canalization imaging regime, the

source should be very close to the periodic medium so that the near field information

can be carried out to the image side without distortion. From the extensive

measurement studies, it was remarkably noted that some factors needed to be

considered for a precise study of the imaging systems. The first is related to the lens

performance due to the interaction of the source with the lens and the second is the

imaging resolution and bandwidth of the lens. However, these two issues are

interrelated.

Chapter 6: Experimental Investigation of Canalization and Imaging Using Metallic Thin Wire

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174

In this chapter, the high resolution imaging regime proposed in [1, 2] is briefly

discussed with its merits over other near field imaging approaches. This is followed by

experimental studies of the imaging phenomenon. A wire medium slab lens was

constructed and used for the imaging studies. Imaging measurements were carried out

in two steps. In the first case, a source in the shape of letter ‘P’ was used for the study.

The results obtained from the P source imaging measurements verified the applicability

of the wire medium lens for sub-wavelength imaging. In the second case, the

measurement was carried out with a meander like periodic radiating source. Compared

to the P loop source, the meander-like source has more complex near field distribution.

From this experiment, the source-lens interaction and bandwidth issues were studied

experimentally. Very high resolution with broad bandwidth was obtained in both cases

and it was found that there will be a reduction in bandwidth due to the interaction of

source and lens.

6.2 The canalization regime for near field imaging

Pendry [3] proposed imaging using a slab of left handed material. However, the

major challenge is still to obtain the homogenous materials possessing negative

magnetic properties at very high frequencies (optical and terahertz) and to overcome

the issues related to losses. The other option to reach the sub-wavelength resolution

was suggested by Wiltshire et al. [4]. The idea was based on the use of an array of

magnetic wires which were called as Swiss rolls [5-7] to transfer sub-wavelength

information directly from the source to the image plane (pixel-to-pixel imaging

principle). The lens formed by the Swiss rolls has to be placed in the near field of the

source since it is capable to transport rather than amplify evanescent harmonics. This

realization of sub-wavelength imaging systems experiences similar problems as those


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in the development of left-handed medium. It is essential to obtain a metamaterial with

magnetic properties with negligible losses. The metamaterials consisting of Swiss rolls,

and the composite medium consisting of split rings and wires with negative parameters

were found to be very lossy at high frequencies.

To overcome these limitations of the existing models, Pavel et al. [1] proposed

the alternative approach to construct the sub-wavelength lens, which does not require

magnetic properties. The imaging device is formed by an array of parallel conducting

wires, so-called wire medium, a typical geometry of which is shown in Fig.6.1. At the

first sight it seems that this structure is an electrical analogue of Wiltshire's system [4-

7]. An array of Swiss rolls, being similar to magnetic wires, is capable to transmit s-

polarized (transverse electric, TE) spatial harmonics of the source spectrum. An array

of wires operates in the same manner, but for p-polarized (transverse magnetic, TM)

waves. In the other words, an array of Swiss rolls restores at the back interface normal

components of magnetic field produced by the source.

Fig.6.1. Geometry of a slab lens consisting of metallic wires. The length of the wires is an

integer multiple of λ/2.

nλ/2


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An array of wires restores normal components of electric field. At the same

time, there is a considerable difference between the Wiltshire's system and the wire

medium slab. The Swiss rolls are artificial resonant structures which behave as

magnetic wires only at the frequencies in vicinity of the resonance. This fact restricts

the Swiss rolls to be narrow-band and very lossy. However, the conducting wires in this

sense are natural electrical wires. It means that they are wide-band and practically

lossless. The absence of strong losses (inherent in Swiss rolls) in ordinary wires

removes restriction on the lens thickness. It allows the creation of sub-wavelength

lenses of nearly arbitrary thickness that deliver images with sub-wavelength resolution

into an image plane in the far-field region of the source and beyond. The imaging

system effectively works as a telegraph formed by a multi-conductor transmission line.

Different spatial harmonics incident to the front interface of the lens formed by

wire medium experience different reflection/transmission properties. It happens due to

impedance mismatch between air and wire medium. The wire medium has surface

impedance for p-polarization which is independent of incidence angles in contrast to

the air which the surface impedance varies for different angles of incidence. The

reflections from the thin slabs are negligibly small, but become significant for thick

layers. This problem can be solved by choosing an appropriate thickness of the slab in

order to fulfil condition for the Fabry-Perot resonance and reduce reflections. Actually,

the reflections can be eliminated completely in the present case in contrast of those in

the classical Fabry-Perot resonator where nonzero reflections are inevitable for oblique

incidences.

For any incidence angle, the wire medium supports propagating modes, so-

called transmission line modes, which travel across the slab with the same phase

velocity equal to the speed of light. If the slab thickness is chosen to be integer number


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of half-wavelengths then the Fabry-Perot condition holds for any incidence angle and

hence such a slab experiences total wave transmission. This phenomenon of collective

reduction of reflections for all angles of incidence together with the fact that the waves

are allowed to transfer energy only across the slab (along the wires) with a fixed phase

velocity is called as canalization regime [1]. This regime can be observed in various

electromagnetic crystals which possess flat iso frequency contours or the equi

frequency surfaces (EFS) at certain frequencies [7, 8].

The wire medium is a unique example of electromagnetic crystals with such

properties observed at very long wavelengths as compared to the period of the crystal,

which opens up a possibility to obtain nearly unlimited resolution of sub-wavelength

imaging. The transmission line modes exist in the slab have TEM polarisation and

travel along the wires with speed of light [1, 2]. Let the longitudinal component of the

wave vector is ql and transverse wave vector be qt. Then for the TL modes ql is equal to

k. This means that the medium has a flat equi frequency contour for all values of

transverse component of wave vector [1]. When the thickness of the slab equals half

wavelength, it acts as a Fabry-Perot resonator with complete transmission for p

polarised (TM) wave.

The resolution of a lens formed by a wire medium is restricted only by its

period which can be made as small as necessary for certain applications. In the present

case, the resolution is equal to the double period of the lattice [1]. This means that two

different objects can be distinguished if they are located close to two different wires,

but their location within one elementary cell cannot be determined. The sub-

wavelength imaging regime using canalization can be applied in optical microscopy

and near field scanning applications.


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6.3 Design of Wire Medium Lens (WML)

A wire medium lens (WML) consisting of copper wires with radius 1mm was

selected for the studies. The array consists of 21 elements both in horizontal and

vertical directions. The period of the WML was 1cm. The frequency of interest was

1GHz (which corresponds to a wavelength of 30cm in free space). Based on the

discussion in the previous section (section 6.2), the length of the wires should be a

minimum of 15 cm (i.e. half the wave length at 1GHz). The wires were fixed on two

thin foam sheets. The foam sheets offer transmission without any considerable loss

around 1GHz. The source used was a loop in the shape of the letter P, formed from a

copper wire of radius 1mm. One arm of the loop was connected to the inner conductor

of a 50 Ohm coaxial cable and the other arm to the outer conductor. The P loop was

placed firmly at the surface of the foam (approximately 3 mm away from the

terminations of wires). A picture of the WML with the P shaped source used for

experimental study and the dimensions of the P loop are given in fig.6.2. The source

was expected to give a good near field distribution at the desired frequencies. The

measured return loss was -5 dB which confirmed that the source would serve its

purpose. The WML transports the sub-wavelength distribution of electric field from the

front interface (source plane) to the back interface (image plane).


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Fig.6.2. A photograph of the wire medium lens with the P loop source (left) and a diagram

showing the dimensions of the P loop source.

6.4 Experimental verification

Numerical simulation results obtained from CST Microwave Studio verified the

imaging using WML. Fig.6.3 shows the simulated electric field intensity at the source

and at the image planes. From fig.6.3, it can be noticed that the energy at the source

plane was transported to the image plane (back interface). The details were not lost

during the transmission. The corners of the loop produced local maxima as shown in

fig.6.3.D. The p-polarized harmonics are guided by wires and canalized from the front

interface to the back one. The trace of their propagation is visible inside the slab. These

waves form an image at the back interface (fig.6.3.D). The resolution of the image

obtained was noted as 2cm which was λ/15 at 1 GHz. Very good agreement between

theoretical predictions and results from numerical simulations proves the validity of the

canalization regime. This indicates the canalization of images and the high resolution

imaging using WML.

8cm 10cm

6cm

1cm


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Fig.6.3. Distribution of electric field and its absolute value: (A), (C) in the vicinity of the source

(at 2.5mm distance from the front interface; (B), (D) at 2.5mm distance from the back interface;

(E) the transverse plane and (F) the intensity along the transverse plane (reproduced from [9])

Further verification of the simulated results was carried out by measurements.

The main aims of the experiments were to demonstrate the canalization regime using

WML and to determine the imaging efficiency in comparison with simulation results.

In order to verify the canalization regime, the electric field components coming out of

the WML at the image plane needs to be considered separately; Ex, Ey and Ez where Ex,


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Ey and Ez are the electric field components along x,y and z respectively. The WML

predicted to transport the Ex component effectively, which is TM polarised and the Ey,

Ez components will suffer much loss since they are TE polarised. The source field

distribution was measured at a distance of 5mm from the source at the front interface

and the image was measured at 5 mm from the image plane at the back interface. To

measure various field components accurately, two different electric field probes were

used (Fig.6.4). Fig.6.5 shows the direction of electric field in each probe.

Fig.6.4. A photograph of the two probes used for the measurement studies

Fig.6.5. A diagram of the straight probe and the bent probe used for the measurements and

corresponding electric field directions

Ev

Ev


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To scan the x component of the electric field, a coaxial probe of length 1cm was used.

This probe was held in parallel to the wire axis. Field intensity was measured at various

points in a rectangular area of 400cm2 (70 points along y and z direction with scanning

centre at the centre of the air – WML interface). To measure the y component of the

electric field, a bent probe which was similar to the previous one but with a bend as

shown in fig.6.4 was employed. Fig.6.5 shows the electric field directions in each

probe. This bend was introduced due to two reasons. One reason was the ease of fixing

the probe to the scanner without having mechanical distortions during the scanning

process. The second reason was that the reduction of source or image field interaction

with the nearby conducting cable. The probes were tested for good transmission in the

required frequency band. The measured electric field intensity distribution along the yz

plane at 0.98 GHz is given in fig.6.6. The wire medium was designed at 1 GHz.

However, in measurement the best quality of image was obtained at 0.98 GHz. This can

be explained by the fact that there can be a small shift in the Fabry- Perot resonance

due to the fabrication tolerance of the WML.

From the measurement results given in fig.6.6, it can be seen that the x

component of electric field is completely recovered at the back interface. The other two

components suffered image degradation since they are not in the TE polarisation. The

plots of the absolute values of source and image fields (fig.6.6.G, H) were in very good

agreement with the simulation results (fig.6.3.C, D). The probes used for the

measurements only give an average value over a very small volume (approximately

1cm3 surrounding the probe) and due to this reason some of the local maxima

corresponding to the termination of the wires that appeared in simulation results, got

diminished in measurement.


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Fig.6.6. Measured near field intensity distribution along the yz plane: Absolute values of x, y

and z components of the electric fields (A, C, E, G) in source plane in image plane (B, D, F, H).


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From the measurement, it was clear that there was a resolution of 2 cm at 0.98GHz.

This corresponds to resolution of about λ/15. From the extensive measurements, it was

made clear that sub-wavelength imaging occurred from 0.92- 1.1 GHz. The bandwidth

can be calculated using the formula

c

lu

fffBW −

= (6.1)

where fc is the centre frequency, fu and fl are upper and lower frequencies respectively.

Substituting the values equation 6.1 gives a bandwidth of 17.8%. This is higher than the

bandwidth of imaging obtained using Swiss rolls in [4].

6.5 Interaction between source and lens

In the previous discussions, the canalization and imaging using WML was

studied experimentally and good agreement with numerical simulation results were

observed. However, the interaction of the source and the lens was not adequately

studied in the P loop imaging work. The basic principle of mutual induction explains

that there exist an induced current on a conductor when it is placed near a current

carrying element. This leads to close by near fields to interact and this can reduce the

bandwidth of the imaging regime. It depends on the source and the period of the lens.

When the source contains more complex elements, there will be much more interaction

compared the case of a P loop source. In the case of a P loop, the source was very

simple and it was not showing much interaction with the WML. In the following

discussions, the experimental observations of the resolution of images and bandwidth

of imaging from the WML, when the P loop source was replaced with a meander-like


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radiating loop are detailed. The loop was constructed on a foam sheet using a thin wire

of 0.22 mm radius, by shaping it in a Meander like pattern. A photo of the Meander like

radiating source and a diagram showing its dimensions is given in fig.6.7. The spacing

between each folding of the wires was 2 cm. The corners were fixed using metallic

pins.

Fig.6.7.A photograph of the Meander shaped radiating source used for the experiment and a

diagram showing dimensions of the loop.

These fixing pins may affect the near field distribution. However, the main interest was

to create a complex near field pattern which can interact with WML effectively and to

study the canalization efficiency and bandwidth. So the presence of the pins was found

not affecting the aims of the studies. The source was fed using a coaxial cable as in the

previous case.

The WML proposed in the previous experiment was used in this study as well.

Near fields were measured at source and image planes in similar manner as in the

previous experiment. Also, fields were measured at the source plane with and without

the lens. In the previous experiment, it was noted that the x component of the electric


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field was canalised very effectively. The source plane was at 3 mm from the first

interface and the image plane was 3mm distance from the back interface. As mentioned

earlier, the aim of the experiment was to accurately determine the imaging bandwidth

and resolution and to study the effect of the source – WML interaction. The

measurement observations revealed the practical applicability of the WML. As it will

serve the purpose of the experiment, only the x component of the electric field was

measured in this case.

The design frequency of the WML was same as before i.e. at 1 GHz. The

electric fields were measured using the horizontal probe employed in the previous

experiment. Fields were measured in the frequency range 0.85- 1.1 GHz with a step of

0.01GHz. For each frequency, there were three field measurements; the source

distribution at the first interface at a distance of 3 mm from the backside of the foam

sheet without lens, the source distribution at the same point with the WML in front of

the source and the field distribution at the image plane at a distance of 3 mm from the

back interface (image plane). The source was placed very close (3 mm) to the WML.

The first set of measurements yields the near field distribution which is the

source for the WML. The second set of measurements gives the near field distribution

after the source field gets interacted with the WML. The last set of measurements

yields the image obtained from the WML. The measured results at 0.85-0.96 GHz and

at 0.99 and 1GHz at frequencies are given in figs.6.8 - 6.11. In the figures, the field plot

at the left is the source with lens and the plot at right is the source distribution without

the WML. The image field distribution is given in the middle.

Fig.6.8 shows that source distribution got distorted due to the insertion of WML

in front of the meander-like radiator. Also, from the image field distribution it can be

seen that there is no imaging takes place till 0.86 GHz. Fig. 6.9 A-D shows the field


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distribution from 0.87- 0.9 GHz. The field distributions at different frequencies are

given in figs. 6.10- 6.11. From fig.6.8 it can be noticed that no imaging takes at 0.85

GHz and 0.86 GHz. It can be seen from fig.6.9 that the source with and without lens

(left and right in the figures) are different and this indicates that source has a strong

interaction with the WML at 0.87 -0.88 GHz. When the frequency is increased in steps

of 0.01GHz, the gradual change in the field distributions at typical frequencies can be

observed (Figs.6.9-6.11).

Fig.6.8.Measured electric field intensity distribution in the yz plane at 0.85GHz (A) and

0.86GHz (B). Source fields with and without lens (left and right respectively) and the image

field (middle) are given.

A

B


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Fig.6.9.Measured electric field intensity distributions in the yz plane at 0.87GHz (A),

0.88GHz (B), 0.89GHz (C) and 0.9 GHz (D). Source fields with and without lens (left and

right respectively) and the image field (middle) are given.

A

B

C

D


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Fig. 6.10. Measured electric field intensity distribution in the yz plane at 0.91GHz (A) and

0.92GHz (B), 0.93GHz (C) and 0.94GHz (D). Source fields with and without lens (left and

right respectively) and the image field (middle) are given.

A

B

C

D


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Fig.6.11.Measured electric field intensity distribution in the yz plane at 0.95 GHz (A), 0.96

GHz (B), 0.99 GHz (C) and at 1 GHz (D). Source fields with and without lens (left and right

respectively) and the image field (middle) are given.

A

B

C

D


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From the results it can be noticed that the canalization regime is from 0.89-0.94

GHz. At 0.92-0.93 GHz (Figs.6.10.B, C), the best quality image was obtained.

Resolution in this case was 2.1cm. This gives rise to a resolution of about λ/15 at

0.92GHz. At frequencies lower than 0.89 GHz, the source field plot with lens is

different from that without the lens. This indicates that the source field was reflected

back and distorted due to the introduction of the WML. It has to be noted that the

canalization regime is active till 0.87 GHz because whatever be the field at the source

plane got transported to the image plane. This disappears below 0.87 GHz. A

frequency above 0.94 GHz, the source is distorted due to the interaction of the WML

(fig.6.11 A, B). However, the canalization regime exists till 0.96 GHz. This indicated

that the actual canalization bandwidth was reduced from the actual one due to the

interaction between the source and WML. Here an imaging bandwidth of 5.3 % was

noted whereas in the case of P loop source, a bandwidth of 17.8% was obtained. This

reduction in bandwidth due to the source field alteration and can be explained by the

effect of induced field interaction.

The measurement results revealed that the actual bandwidth of the imaging

system was influenced by the WML and the type of source. However, in both cases the

resolution was maintained as λ/15. The resolution and bandwidth was found to be in

good agreement with the theoretical predictions given in [10]. It is proved in [10] that

the minimum bandwidth of imaging for the WML is 5%.

Next, the electric field was measured at the image plane without the WML in

between the source and scanning probe. This was to further establish the fact that the

observed imaging can not be obtained without the WML. The WML acts similar to a

transmission line medium supporting TL modes (for the x component of the field). The

measured field had not given a distribution similar to the one at the source, but instead


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almost a uniform distribution was obtained. Measured electric field distribution at a

distance of 155mm from the source at 0.92 (which was at the centre of the canalization

regime) is given in fig.6.12, indicating no imaging.

Finally, the effect of the foam sheet used for the support was verified. Fig.6.13

shows the measured x component of the electric field in front of the source and at the

back, very close to the foam sheet (3 mm away from the foam sheet). In both cases, the

near field distribution is same (with very low transmission loss with the foam sheet).

This confirmed that the effect of the foam sheet was negligible. It was not reducing the

transmitted power considerably and it was not distorting the field distribution.

Fig.6.12. Measured electric field distribution at 0.92 GHz at 155 mm away in front of the

source showing no imaging.

z (cm)

y (c

m)


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Fig.6.13. Measured electric field distribution at 3 mm away from the foam sheet at the backside

of the source (12 mm from the source) and at 3 mm away from the foam sheet at the front side

at 0.88 GHz. In both cases the near field distribution is the same and there is no considerable

transmission loss.

The imaging can be observed at the same frequencies with longer WML as long

as the wire length is an integer multiple of half wavelength. The same phenomenon can

be obtained at higher frequencies when the frequency of operation is multiples of

Fabry-Perot resonance frequency.

6.6 Conclusions

In this chapter Canalization, a regime in which the wire medium acts as a

transparent slab for a near field source was studied experimentally. Images with sub-

wavelength resolution were obtained. The imaging of a P letter shaped source was

demonstrated with experimental results. The bandwidth of the observed phenomenon

was 17.8%. The observed phenomenon can not be obtained from slabs of naturally

available materials. In the present case, the wire medium is a metamaterial or it behaves

like a left handed slab (without any field amplification). Near field interaction of the

source with the proposed lens was an issue, which was not considered in previous

studies. For this purpose, a complex source compared to the P loop source was used.

The results obtained from the second experiment revealed that for complex sources the


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bandwidth of the imaging regime was reduced but the imaging was obtained with the

same resolution as in the case of a simple source. For the meander-like source, the

bandwidth obtained was 5.3%. Extensive experimental results confirmed the existence

of canalisation regime. The WML has the best possible imaging resolution (λ/15)

among other near field imaging approaches based on periodic structures.


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References

[1] Pavel A. Belov, Constantin R. Simovski, Pekka Ikonen, “Canalization of sub-

wavelength images by electromagnetic crystals”, Physical review B, Vol. 71, pp.

193105-193108, 2005

[2] Pavel A. Belov, “Flat lens formed by a capacitively loaded wire media”, Proc. of

the Progress in Electromagnetics Research (PIERS), China, August, 2005

[3] J. B Pendry, “Negative refraction makes a perfect lens”, Physical Review Letters,

Vol. 85, pp.3966-3969, 2000

[4] M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J.

Gilderdale, J. V. Hajnal, “Microstructured magnetic materials for RF flux guides

in magnetic resonance imaging”, Science, pp. 849 – 851, 2002

[5] M. C. K. Wiltshire, E. Shamonia, L. Solymar, “Experimental and theoretical

study of magneto- inductive waves supported by one dimensional arrays of swiss

rolls”, Journal of Applied Physics, Vol. 95, pp. 4488-8893, 2004

[6] S. Anantha Ramakrishna, J. B. Pendry, M C K Wiltshire, W. J. Stewart,

“Imaging the near field” , Journal of Modern Optics, Vol. 50, pp.1419 – 1430,

2003

[7] C. Luo, S.G Johnson, J.D Joannopolous and J.B Pendry, “All-angle negative

refraction without negative effective index”, Physical Review B, Vol. 65, pp.

201104, 2003

[8] C. Luo, S. G Johnson, J. D Joannopolous and J. B Pendry, “Sub-wavelength

imaging in photonic crystals”, Physical Review B, Vol. 68, pp. 045115, 2003

[9] Pavel. A Belov, Yang Hao, Sunil Sudhakaran, “Sub-wavelength microwave

imaging using an array of parallel conducting wires as a lens”, Physical review B,

Vol. 73, pp. 033108 - 033111, 2006


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[10] Pavel A. Belov, “Resolution of sub-wavelength transmission devices formed by a

wire-medium”, ArXiv:Physics/0511139v2, March 2006 (Online resource)

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Chapter 7

Conclusions and Future work

7.1 Summary

Study of Metamaterials in particular LHMs is an emerging and interesting topic

of research so an in-depth understanding of LHMs and analogous LHMs and their

characteristics is very challenging. In this thesis, substantial effort has been made to

understand the concepts and realisations of negative refraction or refraction like

phenomenon.

The main objectives of the research were to investigate the possibilities of

obtaining negative refraction-like phenomenon at frequencies other than the complete

bandgap edges of EBGs and to extend the concepts of EBG based metamaterial into

high resolution wave focusing and imaging. These goals were achieved in this work.

7. 2 Conclusions

The following conclusions can be drawn from this study:

• Numerical simulation results confirmed the spatial dispersion effects due to the

finiteness of the periodic structure. Change in the backward radiation frequency

regions was noticed between unit cell analysis for infinite models and the

proposed ‘super cell’ model.

• Negative refraction from slab and prism models were presented with numerical

simulation results and the occurrence was further established with measurement

Chapter7: Conclusions and Future Work

198

results using a prism model. Extensive measurement results indicated that there

were multiple frequency regions in the pass band at which the structure gave

negative refraction. A transmission of 54% was observed which was higher

than that reported from the split ring and wire medium.

• It was observed that the obtained result had higher negative refraction excess

rate (NRE) of 85%. This was higher than that from previously reported values.

• Enhancement of negatively refracted beam in the passband by introducing local

defects into the structure was demonstrated. An enhancement of 27% for the

negatively refracted beam was obtained.

• Wave focusing from an MEBG slab was demonstrated at low frequencies in the

passband with numerical simulation and experiments. A resolution of λ/7 was

obtained at 2.1GHz. Numerical simulation results confirmed that such a high

resolution focusing cannot be obtained from ubiquitous materials. The

phenomenon observed was at a lower frequency compared to similar works

reported before.

• Sub-wavelength imaging using canalization was experimentally verified using a

P Loop source and a resolution of λ/15 with a bandwidth of 17.8% was

obtained. It was observed that the interaction between the source and WML

degrades the bandwidth of the system. The bandwidth is reduced to 5.3% with a

complex source.

7.3 Future work

The study presented in this thesis opens up a lot of possibilities in future

research. The outcomes obtained in the study can be extended to novel applications of

metamaterials. Also, novel approaches can be developed for transmission enhancement

and all angle negative refraction from EBGs.


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7.3.1 Novel applications

7. 3.1.1. Dispersive EBG for demultiplexing and beam scanning applications

The MEBG used in the study shown capability of forming beams at different

angles at different frequencies. For beam scanning application or in a frequency

demultiplexer, only the properties of the exiting wave are important. So the structure

can find its applications as a frequency spatial Demultiplexer. Fig.7.1 shows the

diagram illustrating the mechanism of the spatial demultiplexer using the MEBG prism.

Fig.7.1. A schematic diagram illustrating the application of MEBG prism as a spatial frequency

demultiplexer. The red arrow represents the polychromatic wave and the out coming beams of

different frequencies are shown in different colours and the corresponding angles from the

normal are marked.

The refracted fields were obtained from measurements discussed in chapter 4. This also

can be used as a beam scanner at microwave frequencies. Another possibility is to

consider hemi-circular shaped EBGs for multiple beam applications. This can lead to

novel EBG based antennas. The advantage is the EBG can act as a filter and antenna

with the same hardware leading to joint filter/antenna applications.

EBG Prism

4.5GHz, +32˚ 5.6GHz, -21˚ 7.4GHz, -45˚ 7.2GHz, -58˚

Spa

tial d

etec

tor

Normal

Polychromatic wave


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7.3.1.2 Sub-wavelength antennas

It was noticed that spatial dispersion exists in all periodic structures and this can

cause backward radiation at certain frequencies. This concept can be applied to planar

periodic structures and novel antennas with beam scanning capability can be obtained.

In conventional backward wave antennas, backward radiation can be obtained only

when the period is comparable to the wavelength of operation. This makes the antenna

electrically large. The spatial dispersion in EBGs leads backward radiation at low

frequencies. This reduces the size of the antenna resulting sub-wavelength antennas.

Due to its reduced size, these antennas can be promising candidates for future on body

and indoor communication antennas.

7.3.2 Theoretical extensions

7.3.2.1 Transmission enhancement

From the studies performed, it was noted that the transmitted power obtained

from the MEBG was 54%. The main cause of the reduced transmitted power is the

impedance mismatch at the air and the MEBG interface. Methods can be developed for

impedance matching at the interface and transmission increase. However, it has to be

mentioned that the available matching techniques will cause reduction in bandwidth as

the reported techniques such as Fabry Perot resonator model for transmission

enhancement are for narrow band applications. So a broadband matching technique can

be developed.

7.3.2.2 All angle negative refraction

The study presented in this thesis has not considered all angle negative

refraction. All angle negative refraction is not a mandatory for beam scanning

applications. However, this could lead to improved performance in focusing and


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imaging applications. As a possible extension of the work, all angle dispersion analysis

can be obtained by considering different angles of incidence.

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