i
Microwave Metamaterials
for Compact Filters and Antennas
vorgelegt von
M.Sc
Merih Palandöken
aus Kahramanmaras
Von der Fakultät IV-Elektrotechnik und Informatik
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
Dr.-Ing
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Klaus Petermann
Berichter: Prof. Dr.-Ing. Heino Henke
Berichter: Prof. Dr. Sergey I. Bozhevolnyi
Tag der wissenschaftliche Aussprache: 06. September 2012
Berlin 2012
D83
ii
Acknowledgements
It is my pleasure to express my deepest gratitude and respect to Prof. Dr.-Ing Heino Henke
for his invaluable guidance, helpful suggestions, endless support and patience. His personal
and academic virtue shaped my academic personality and changed my approach to scientific
study. I am very lucky to have the opportunity to study under his supervision. I would like
to thank to the members of my thesis committee, Prof. Dr. Sergey I. Bozhevolnyi and Prof.
Dr.-Ing. Klaus Petermann for reading the manuscript and commenting on the thesis.
I am very fortunate to have been a member of TET family, where I had a chance to work
together and benefit from intelligent and excited people. I am very thankful to Dr.-Ing
Manfred Filtz for motivating scientific discussions about the electromagnetic field theory,
continuous support, patience and nice chance to be an evidence for his brilliant thinking
approaches. Special and deep thanks to Dr. –Ing Heinrich Büssing for his incredible help,
understanding and irreplaceable friend- and brothership during my research life. I would
like to thank all members of TET family for their help in my doctorate period. I thank also
very much to Dr.-Ing Tolga Tekin in the Department of Nano Interconnect Technology due
to his help during and after my doctorate research.
I am also very thankful to Prof. Dr. Siddik Yarman, the Department Chair of Electrical and
Electronics Engineering at Istanbul University for his continuous motivating advices and
support during my doctorate research.
I would like to express my endless thanks to my mother and father for their love,
encouragement and care.
iii
Zusammenfassung
Werkstoffe mit gleichzeitig negativer Permittivität und Permeabilität in einem bestimmten
Frequenzband wurden zuerst von Veselago im Jahr 1968 theoretisch untersucht. Er
bezeichnet solche Medien als linkshändige Materie (LHM), weil die Vektoren des
elektrischen Feldes, des Magnetfeldes und der Wellenausbreitungsrichtung ein
linkshändiges System aufspannen. Weil solche Materialien in der Natur nicht vorkommen,
konnten die vorhergesagten optischen und elektromagnetischen Konsequenzen aus
Veselago‘s Arbeit erst ca. 30 Jahre nach deren Veröffentlichung beobachtet werden. Im Jahr
2000 wurde an einem künstlich zusammengesetzten Material, das auch als ein Metamaterial
bekannt ist, und aus periodisch angeordneten negativen Permittivitäts- und
Permeabilitätszellen in der Form von metallischen Drähten und Spaltringen aufgebaut ist,
linkshändige Eigenschaft gezeigt. Weil die Abmessung der Drähte und Ringe klein im
Vergleich zu der Wellenlänge sind, ist es möglich, die elektromagnetischen Eigenschaften
dieser Strukturen durch eine effektive Permittivität und Permeabilität zu approximieren. In
dieser Arbeit werden die grundlegenden Eigenschaften negativer Permeabilität, Permittivität
und linkshändiger Materialien beim Entwurf von Antennen und Filtern als potenzielle
Anwendungen in der Mikrowellentechnik erforscht.
Bei der Untersuchung grundlegender Eigenschaften und alternativer Design-Methoden von
Metamaterialien, widmet sich diese Arbeit zunächst den künstlichen magnetischen
Materialien. Diese künstlichen magnetischen Materialien liegen in Form von metallischen
Zylindern mit oder ohne Spalten oder Spaltringresonatoren vor. Approximierte
Formulierungen für die effektive Permeabilität werden abgeleitet. Die Berechnungen der
Transmissions- und Reflexionsparameter von konzentrischen Zylindern mit Spalten werden
auf analytischem Wege und auch numerisch durchgeführt. Die gute Übereinstimmung
zwischen analytischen und numerischen Berechnungen wird bestätigt. Die
Resonanzfrequenz der Spaltringresonatoren ist zusätzlich nach einem quasistationären
Ansatz berechnet worden. Es wird gezeigt, dass die Ergebnisse der numerischen
Berechnungen besser mit den Ergebnissen aus dem quasistationären Ansatz
übereinstimmen. Das 1D-Dispersionsdiagramm der Spaltringresonatoren wird numerisch
untersucht. Effektive Materialparameter werden extrahiert, um die Frequenzabhängigkeit
der effektiven Permeabilität nach dem Lorentz-Modell zu bestätigen. Zur Untersuchung
eines homogeneren Materialdesigns, wird ein fraktaler Spiralresonator als Einheitszelle mit
negativer Permeabilität zum Einsatz gebracht. Es wird bestätigt, dass die effektive
Permeabilität im gesamten Bereich zwischen der magnetischen Plasmafrequenz und der
Resonanzfrequenz negativ ist und einen kapazitiven Wellenwiderstand aufweist. Eine zweite
Art künstliche Materie, bestehend aus periodisch angeordneten metallischen Drähten, wird
durch analytische und numerische Berechnungen untersucht. Es wird gezeigt, dass die
effektive Permittivität einer solchen Drahtanordnung eine Frequenzabhängigkeit nach dem
Drude-Modell aufweist und bei Frequenzen unterhalb der Plasmafrequenz negativ ist. Auch
hier sind die Ergebnisse der numerischen und analytischen Rechnungen in guter
Übereinstimmung. Um stärker homogenisierte Metamaterialien zu entwickeln, wird ein
fraktaler Antispiralresonator (fractal anti-spiral-resonator) als Einheitszelle eines Materials
mit negativer Permittivität untersucht. Es wird gezeigt, dass das Material zwischen der
elektrischen Plasma- und Resonanzfrequenz negative Permittivität bei induktivem
Wellenwiderstand aufweist.
iv
Als letzte Untersuchung der Grundlagen von Metamaterialien, wird das Design
linkshändiger Materialien erforscht. Die Eigenmodegleichung eines rechteckigen
Hohlleiters, der periodisch mit Zellen negativer Permittivität und Permeabilität bestückt
wird, wird analytisch und numerisch berechnet. Es wird gezeigt dass die Ergebnisse der
analytischen und numerischen Rechnungen gut übereinstimmen. Darüber hinaus wird eine
herkömmliche LHM Einheitszelle numerisch untersucht. Es wird bestätigt, dass das
linkshändige Transmissionsband der Überlappungsbereich des negativen Permittivitäts- und
Permeabilitätsbands ist.Eine kompakte Zellgeometrie, die auf Spiralresonatoren basiert,
welche mit Drähten belastet werden, wird als Alternative zu den herkömmlichen Zelldesigns
untersucht. Es wird aus numerischen Berechnungen abgeleitet, dass solche belasteten
Resonatoren bei der Gestaltung von homogener LHM verwendet werden können.
Als nächstes werden zwei mögliche Anwendungsbereiche in der Mikrowellentechnik
untersucht. Der erste Bereich ist Antennendesign. Zwei Meta-Antennen werden untersucht.
Die erste Antenne ist eine breitbandige Dipolantenne, bei der die Dipolelemente durch LHM
Einheitszellen in Form eines Arrays belastet werden. Die Breitbandigkeit dieser Antenne
wird experimentell durch Messung und durch die Ergebnisse der numerischen Rechnung
bestätigt. Es wird mit Hilfe der Oberflächenstromverteilung und Richtcharakteristik gezeigt,
dass LHM Einheitszellen mit einem geringeren Strahlungsgewinn im Vergleich zu der
Dipolantenne abstrahlen. Darüber hinaus ist aus den numerischen Ergebnissen abgeleitet,
dass durch Anpassung der Phasendifferenz von Zelle zu Zelle die Abstrahlungseffizienz
erhöht werden kann. Mögliche Methoden zur Verbesserung des Antennengewinns werden
aufgezeigt, von denen eine im Design der zweiten Antenne verwendet ist. Die zweite
Antenne ist eine Schlitzantenne, welche mit elektrischen und magnetischen Dipolen in Form
von geschlitzten Metamaterialzellen im Abstrahlelement gestaltet wird. Es wird numerisch
bestätigt, dass die Antenne schmalbandig mit hoher Abstrahlungseffizienz und Gewinn ist.
Die zweite Anwendung ist der Filterentwurf. Zwei Meta-Filter sind untersucht. Der erste
Filter ist ein kompakter Bandsperrfilter, welcher durch galvanische Kopplung von vier LHM
Zellen in der Form von λ/4-Resonatoren mit Speiseleitung gestaltet wird. Die Wirkung der
Bandsperre wird durch experimentelle und numerische Ergebnisse bestätigt. Das zweite
Design ist ein kompakter Bandpassfilter, der durch Kopplung von zwei Elementarzellen, die
direkt mit der Speiseleitung verbunden sind, entworfen ist. Es ist numerisch gezeigt, dass
eine niedrige Einfügedämpfung und hohe Selektivität durch die Optimierung der Feld-
Kopplung zwischen den Resonatoren erreicht werden kann. Ein Vorteil der beiden Filter ist,
dass es keine Notwendigkeit für eine Anpassungsschaltung gibt, wodurch die Filtergröße
deutlich reduziert wird.
v
Abstract
Materials with simultaneous negative permittivity and permeability in a certain frequency
band were first studied by Veselago in 1968. He termed such media left-handed media due
to the left-handed triad formed by the electric field, magnetic field and the phase
propagation vector. Because of the inexistence of such materials in nature, the optical and
electromagnetic consequences examined in Veselago’s work could only be observed after
nearly 30 years. In 2000, a composite material, also known as a metamaterial, consisting of
periodic negative permittivity and permeability cells in the form of metallic wires and split
rings was shown to exhibit left-handed properties. Because the dimension of the rods and
rings are small compared to the operation wavelength, it is possible to approximately
describe their bulk electromagnetic properties using an effective permittivity and
permeability. In this thesis, the fundamental properties of negative permeability, permittivity
and left-handed materials are explored in the antenna and filter design as potential
microwave applications.
As an investigation on basic properties and alternative design methods of metamaterials,
artificial magnetic materials are examined at first. Artificial magnetic materials in the form
of metallic cylinders with/without splits and split ring resonators are studied. Approximate
effective permeability formulations are derived. Numerical calculations for the transmission
and reflection parameters of concentric cylinders with splits are carried out in addition to
analytical calculations. The good agreement between analytical and numerical calculations
is confirmed. The resonance frequency of split ring resonators is also approximated with an
alternative formulation derived from quasi-static analysis. It is shown that the numerical
calculations agree better with the derived formulation than the original formulation. 1D
dispersion diagram of split ring resonators is also examined numerically. Effective material
parameters are retrieved to confirm Lorentzian-type frequency dependence of permeability.
As an investigation for more homogenous material design, a fractal spiral resonator is
studied as a unit cell of negative permeability material. It is confirmed that the composite
material has negative permeability between the magnetic plasma and resonance frequency
with capacitive wave impedance. At second, artificial dielectrics composed of periodic
metallic wires are examined with analytical and numerical calculations. It is shown that the
effective permittivity of wire array has Drude type frequency dependence and is negative at
the frequencies smaller than the plasma frequency. The numerical and analytical
calculations are in good agreement. To design more homogeneous metamaterials, a fractal
anti-spiral resonator is examined as a unit cell of negative permittivity material. It is pointed
out that the composite material has negative permittivity between the electric plasma and
resonance frequency with inductive wave impedance. As a last investigation on
metamaterial fundamentals, the design of left-handed materials is explored. The eigenmode
equation of a rectangular waveguide, which is periodically loaded with negative permittivity
and permeability cells is analytically and numerically calculated. Both are in good
agreement. In addition, a conventional LHM unit cell is numerically studied. It is confirmed
that the LH transmission band is the overlapping region of negative permittivity and
permeability bands. A compact cell geometry based on wire loading of spiral resonators is
studied as an alternative to the conventional cell designs. It is deduced from the numerical
calculations that it can be used in the design of more homogeneous LH materials.
vi
Next, two potential microwave applications are explored. The first application is antenna
design. Two meta-antennas are examined. The first antenna is a broadband dipole antenna,
which is designed by loading the dipole element with LHM cells in an array form. The
broadband operation is confirmed by experimental and numerical results. It is shown from
the surface current distribution and radiation patterns that LHM cells are radiating with a
lower gain in comparison to the dipole antenna. In addition, it is deduced from the
numerical results that to adjust the phase difference per unit cell can be one solution to
increase the radiation efficiency and antenna gain. Possible gain enhancement methods are
pointed out, one of which is used in the design of the second antenna. It is a microstrip slot
antenna. It is designed with electric and magnetic dipoles in the form of slotted
metamaterial cells in the radiator. It is confirmed numerically that the antenna is narrowband
with high radiation efficiency and gain.
The second application is the filter design. Two meta-filters are examined. The first filter is
a compact band-stop filter. It is designed by directly connecting four LHM cells in the form
of λ/4 resonators with the feeding line. The bandstop characteristics are confirmed by
experimental and numerical results. The second design is a compact band-pass filter, which
is designed by coupling two unit cells directly connected with the feeding line. It is shown
numerically that low insertion loss and high selectivity can be achieved by optimizing the
field coupling among the resonators. One advantage of both filters is that there is no need of
a matching network, which therefore reduces the filter size significantly.
vii
Contents
1
Introduction................................................................................................1
1.1
Motivation…………………………………………….…………...…1
1.2
Thesis Work…………………………………………….....................3
2
Metamaterials Short Overview………………………………………….5
2.1
Artificial Dielectric Materials……………………………...…….......5
2.2
Artificial Magnetic Materials………………………….......................6
2.3
Left-Handed Materials…………………………………………….....6
3
Negative Permeability Metamaterials…………………...……………...8
3.1
Introduction……………………………………………………..……8
3.2
Theoretical Analysis…………………………………………………8
3.2.1
Periodic Array of Cylindrical Metallic Sheets…...……….......9
3.2.2
Periodic Array of Concentric Cylindrical
Metallic Sheets with Splits …………………………..……..12
3.2.3
Periodic Array of Split- Ring Resonators…………………...15
3.3
Numerical Simulations……………………………….......................18
3.3.1
Resonance Frequency of SRR Periodic Array………………18
3.3.2
Dispersion Relation of SRR Periodic Array………………...21
3.3.3
Effective Material Parameters of
SRR Periodic Array…………………………………………22
3.4
Fractal Spiral Resonator as Magnetic Metamaterial………………..26
3.4.1
Structural Description……………………………………….26
3.4.2
Simulation Results………………………………………….26
3.4.3
Effective Parameters……………………………………...…28
3.5
Chapter Conclusion…………………………………………………29
4
Negative Permittivity Metamaterials……………………………….…31
4.1
Introduction………………………………………………………....31
4.2
Theoretical Analysis………………………………………………..31
4.3
Numerical Simulations……………………………………………...41
4.3.1
Frequency Response of a 1D Wire Array………………...…42
4.3.2
Dispersion Relation of Periodic Wire Array………………...44
4.3.3
Effective Parameters of Periodic Wire Array……………….45
4.4
Fractal Anti-Spiral Resonator as Dielectric Metamaterial………….47
4.4.1
Structural Description……………………………………….47
4.4.2
Simulation Results..................................................................48
4.4.3
Effective Parameters...............................................................50
4.5
Chapter Conclusion…………………………………………………51
viii
5
Left-Handed Metamaterials……………………………………………53
5.1
Introduction………………………………………………………....53
5.2
Theoretical Analysis……………………………………………..…53
5.3
Numerical Simulations……………………………………………...59
5.3.1
Resonance Frequency of LHM Periodic Array……………..59
5.3.2
Dispersion Relation of LHM Periodic Array………………..62
5.3.3
Effective Parameters of LHM Periodic Array………………64
5.4
Wire loaded Spiral Resonator as LHM…………………………..…67
5.4.1
Structural Description…………………………………….…67
5.4.2
Numerical Simulations……………………………………...68
5.4.2.1
Transmission and Reflection Parameters
of Wire loaded Spiral Resonator…………………..68
5.4.2.2
1D Brillouin Diagram
of Wire loaded Spiral Resonator…………………..69
5.4.3
Effective Parameters...............................................................70
5.5
Chapter Conclusion............................................................................73
6
Metamaterial-based Antenna Design………………………………….74
6.1
Introduction………………………………………………………....74
6.2
Fundamental Limits of Small Antennas……………………………74
6.3
LHM-based Broadband Dipole Antenna…………………………...79
6.3.1
Antenna Design……………………………………………...79
6.3.2
Experimental and Numerical Results……………….……….81
6.4
Metamaterial-Inspired Slot Antenna………………………………..86
6.4.1
Metamaterial Slot Radiator Design………………………….86
6.4.2
Antenna Design……………………………………………...87
6.4.3
Simulation Results……………………………………….….88
6.5
Chapter Conclusion............................................................................90
7
Metamaterial-based Filter Design……………………………………...91
7.1
Introduction………………………………………………………....91
7.2
Fundamental Principles of Metamaterial-based Filter Design……...91
7.3
Metamaterial-based Band-Stop Filter………………………………96
7.3.1
Band-Stop Filter Design………………………………….….96
7.3.2
Experimental and Numerical Results
of Band-Stop Filter…………………………………………..97
7.4
Metamaterial-based Band-Pass Filter Design……………………..101
7.4.1
Band-Pass Filter Design…………………………………....101
7.4.2
Numerical Results of Band-Pass Filter…………………….102
7.5
Chapter Conclusion..........................................................................105
8
Conclusion……………………………………………..…………….....106
References……………………………………………………………...108
A
Quasi-static Analysis of Negative Permeability Cells ……………...116
B
List of Author’s Publications …………………………………………120
1
1. Introduction
Electromagnetic waves have important role in daily life, since the electromagnetism is the
main driving phenomenon in many applications, including communication, imaging,
sensing and devices like antennas, light sources, optical fibers, lenses, etc. Extending from
extremely low frequencies up to Gamma rays, the electromagnetic wave phenomenon has
received great interest. Researchers have widely studied on the interaction of
electromagnetic waves with matter. This has led many technological applications and
technical devices to be realized in diverse scientific disciplines. Transmission, reflection,
refraction, diffraction and scattering are among many effects resulting from the interaction
between the electromagnetic fields and materials. This electromagnetic interaction is
determined by electromagnetic material parameters, dielectric permittivity (ε) and magnetic
permeability (μ). Ordinary materials usually have positive values of ε and μ. However, the
electromagnetic response of natural materials can be intentionally extended to the values
that are not readily available by designing artificial materials, so called metamaterials. The
phrase “Meta” is originating from Greek meaning “beyond”. The ordinary materials are
composed of atomic and molecular constituents that are much smaller than the wavelength.
They are therefore termed as homogeneous materials. A similar interpretation could be
analogically established with the artificial materials. In these materials, the basic building
blocks, meta-atoms, are much smaller than the wavelength of interacting electromagnetic
wave. The effective permittivity and permeability can therefore be similarly assigned to the
whole composite medium as in natural materials. The motivating point in the metamaterials-
based designs is the controllable engineering of the electromagnetic material parameters to
obtain any desired value within the theoretical limits, even negative real values.
The progressive interest in metamaterial research started at the beginning of 21th century
after the successful demonstration of artificial material with simultaneously negative
permittivity and permeability [1]. However, it was more than 30 years ago in 1968 that a
Russian physicist, Veselago, examined intensively the electromagnetic consequences of a
conceptual material in which ε and μ had both negative real values [2]. He has termed this
material as left-handed material (LHM) because the electric field, magnetic field, and wave
propagation direction form a left-handed system. In addition, the negative material
parameters result the effective refractive index to be negative. This is not achievable with
known conventional materials [2-4]. Negative electromagnetic parameters result alternative
design methodologies to be implemented in the component design. Novel microwave
components with improved performances are therefore realized. Large number of scientific
papers on the analytical, numerical and experimental investigations of peculiar metamaterial
effects and metamaterial-based components are published [3-6].
1.1 Motivation
Metamaterials offer a wide range of exciting physical phenomena to be observed, which are
not attainable with ubiquitous materials. Artificial magnetism is one of these exotic
electromagnetic properties. Resonant magnetic response can be achieved from the periodic
arrangement of non-magnetic inclusions. These inclusions can be designed as electrically
small resonators with distributed capacitive and inductive elements. A split-ring resonator
(SRR) is an original example of this type [7].
2
The resonant permeability can be tailored to any desired value including negative values
between the magnetic resonance and plasma frequency. On the other hand, an effective
negative permittivity over a desired frequency band can be obtained with the regular lattice
of conducting wires [8,9]. This periodic structure exhibits plasma-like permittivity response
for the wavelength larger than the lattice period. Therefore, the negative permeability and
permittivity materials can be engineered in a controlled manner to obtain any value of
refractive index within a desired frequency range [1], [3-6]. Some possible applications of
metamaterials are shown in Figure 1.1.
(a) (b)
(c)
Figure 1.1. Metamaterial applications (a) magnifiying hyperlens [10] (b) superlens [11] (c) cloaking device [12]
There are some engineering applications derived from the metamaterial concept such as
phase compensation [4,5] and electrically small resonators [6], [13], compact high selective
microwave filters [3],[14], subwavelength waveguides with lateral dimensions below
diffraction limits [15,16], enhanced focusing [17,18], backward wave antennas [5],[19] and
enhanced electrically small antennas [20-23]. Because the metamaterials are geometrically
scalable structures, they offer the same operation and design principles to be conveniently
implemented at higher/lower frequencies. Different metamaterial designs in wide range of
operation frequencies are explained in detail [24-30].
3
1.2 Thesis Work
In this thesis, the fundamental properties of negative permeability, negative permittivity and
LHM at microwave frequencies are analytically and numerically studied. How to utilize
these electrically small cells in the design of compact antennas and filters is experimentally
and numerically investigated with two meta-antenna and two meta-filter designs.
In the second chapter, how to tailor and overcome the restrictions in electromagnetic
material parameters are introduced. Artificial dielectric, magnetic and left-handed materials
are explained shortly. The fundamental electromagnetic properties are studied with the
analytical and numerical calculations in detail in the next chapters.
The third chapter is dedicated to the main principles of non-resonant and resonant artificial
magnetism. The artificial diamagnetism obtained from periodic infinitely long cylinders is
addressed at first. It is important to understand non-resonant magnetism in 2D. Two
concentric cylindrical sheets with opposite splits are then investigated. They are important
to obtain resonant magnetic permeability with negative permeability region. SRR is studied
next. The numerical calculations are done to confirm the negative permeability phenomenon
in SRR-based materials. The effective electromagnetic parameters are retrieved. Dispersion
relation is studied. Negative permeability band is determined. At the end of the chapter, a
novel unit cell based on fractal spiral resonator is proposed. The material parameters, Bloch
impedance and propagation constant are extensively studied. This magnetic cell can be used
to design more homogeneous materials with negative permeability.
In Chapter 4, the fundamental principles of negative permittivity materials are explained in
detail. The analytical calculations on periodic wire array in Pendry’s original work [8,9] are
revised. A more generalized formulation is obtained. The plasma-like permittivity response
is analytically calculated. The effect of periodic wire loading on effective permeability is
also pointed out. More homogeneous alternative designs are referenced. The numerical
calculations are performed for a wire strip model. The effective material parameters are
retrieved. 1D dispersion relation is studied. The plasma frequency and negative permittivity
band are determined. At the end of the chapter, a unit cell based on fractal antispiral
resonator is proposed to design more homogeneous materials with negative permittivity.
LHMs are then explained in Chapter 5. They are basically composed of both negative
permittivity and permeability unit cells in the same host medium. The eigenmode equation
of periodically loaded negative permittivity and permeability materials is analytically and
numerically calculated. This analytical study is important to understand the subwavelength
resonance feature and negative phase velocity in LHMs. The numerical calculations are
done for a conventional LHM cell. The wave propagation in the frequency band of both
negative permeability and permittivity is confirmed. The effective parameters are retrieved.
Dispersion relation is also studied to determine the frequency band of negative refractive
index. At the end of the chapter, an LHM cell based on wire-loaded spiral resonator is
numerically investigated for the design of more homogeneous LHMs.
One potential application of artificial materials in microwave systems is antenna design.
Metamaterials-based antenna design is therefore explained in Chapter 6. Basic concepts in
small antennas are addressed first to understand the fundamental performance limitations.
As a next step, a broadband LHM-loaded dipole antenna is designed. It is studied
numerically and experimentally. The reflection parameter of the fabricated prototype is
measured to confirm the numerical results. This design is a good representative example
how to exploit artificial materials in broadband antenna design.
4
As a second design, a higher profile meta-antenna is designed. In this antenna, electrically
small cells are used in the design of a slot radiator to enhance the antenna gain.
In Chapter 7, the basic design approaches of metamaterials-based filters are explained. They
are addressed with some filter examples in the literature. Compact narrowband / broadband
bandpass filters are illustrated in addition to the alternative wideband filters. As a first meta-
filter example, a compact band-stop filter is designed. The filter performance is studied
numerically and experimentally in detail. As a second example, a compact band-pass filter
is designed with the same resonator geometry. The important geometrical parameters are
pointed out to optimize and tune the filter operation frequencies and performance
parameters.
Chapter 8 concludes the results and implications of various aspects of this research.
5
2. Metamaterials Short Overview
There are some restrictions on the electromagnetic material parameters of naturally
available materials. These materials have lossy frequency dependent material parameters. In
addition, it is sometimes difficult to find a suitable material with desired material parameters
for some applications. However, the electromagnetic material parameters of natural
materials can be tailored artificially in a controlled manner to obtain desired frequency
dispersion. The resulting artificial materials with engineered material parameters are termed
as ‘metamaterials’. Three basic types of metamaterials are introduced shortly in the next
sections.
2.1 Artificial Dielectric Materials
Artificial dielectrics are the first ever known artificial materials. They are composed of
artificially created ‘meta-molecules’ or ‘meta-atoms’ in the form of dielectric or metallic
inclusions of a certain shape. These inclusions can be arranged either periodically or
aperiodically in the host material. The ‘meta-molecules’ size and lattice distances are
assumed to be very small in comparison to the operation wavelength. Therefore, the
macroscopic interaction of the electromagnetic wave with the artificial material can be
described in terms of effective material parameters as in the case of homogeneous materials.
The first artificial dielectric was proposed by W.E.Kock [31] for the design of light weight
microwave lens. The main advantage of artificial dielectrics is to have the design possibility
of high permittivity, low loss dielectric material. The material losses can be intentionally
increased in a controlled manner to design light weight absorbers. Another important
application is to design light weight materials with plasma-like permittivity response at the
microwave frequencies. These materials can be realised from the periodic conducting wires
with smaller radii as compared to the lattice periods as shown in Figure 2.1.
Figure 2.1. Wire medium geometry [9]
6
The wire medium is often called ‘artificial plasma’. They were even used in the plasma
simulations and high-impedance surface design at microwaves [6],[32,33]. Negative
permittivity materials are explained in more detail in Chapter 4.
2.2 Artificial Magnetic Materials
The artificial magnetic materials in the present time are basically in the form of diamagnetic
and resonant magnetic materials in 2D and 3D [7].Their design approach is based on the
periodic arrangement of electrically small resonators to obtain strong magnetic response.
The typical resonant elements are split-ring resonators (SRR) in 3D and Swiss rolls in 2D.
They are shown in Figure 2.2. Magnetic materials made up of periodic SRRs in 3D and
Swiss rolls in 2D have negative permeability within a narrow frequency band between the
magnetic resonance and plasma frequencies [7],[3]. Actually, the design logic of SRR dates
back to the slotted tube resonators in the detection system of nuclear magnetic resonance
spectroscopy [34].
Figure 2.2 Split-ring resonator and Swiss Roll Structure geometry
However, Swiss Roll structures are more effective than SRRs with the design possibility of
compact RF lens and flux guiding elements for MRI systems [35-37]. Negative permeability
materials are explained with alternative homogeneous realizations in Chapter 3.
2.3 Left-Handed Materials
The left-handed concept is first introduced by V.G. Veselago. He examined theoretically the
electromagnetic consequences of plane wave propagating in a material with simultaneous
negative permittivity and permeability in 1968 [2]. He has termed this material as ‘left-
handed material’ because electric field, magnetic field and wave vector form a left-handed
triad rather than right-handed triad in the conventional materials. LH media have
interesting possible applications such as the possibility of perfect lens construction predicted
by Pendry [18], a sub-wavelength cavity resonator invented by Engheta [13], compact high
gain resonant antennas [20], backward wave antennas [19], compact phase shifters [38],
compact high selective filters [14], directional couplers with arbitrary coupling strength [39],
subwavelength waveguides [15,16], cloaking [12]. The uniaxial version of first LHM is
realized by Smith in 2000 [1]. It is composed of periodic arrangement of thin wires and
split-ring resonators proposed by Pendry. It is shown in Figure 2.3.
7
Figure 2.3. LHM geometry
In this design, the periodic wires (wire medium) exhibits negative permittivity and the
periodic split-ring resonators exhibits negative permeability. However, LH structures
presented originally were less practical for microwave applications because of their lossy
and narrow bandwidth characteristics. Therefore, a transmission line (TL) approach of LH
materials based on nonresonant components is introduced to realize an artificial LH-TL with
low losses and broad bandwidth [4,5]. Several microwave components with improved
performances have been realized with this design method [4-6]. In Chapter 5, resonant type
LHM materials are explained in detail.
8
3. Negative Permeability Metamaterials
3.1 Introduction
This chapter explains the main principles of artificial magnetism. The artificial
diamagnetism is addressed at first with the analytical calculations on infinitely long metallic
cylinders. This analytical study is important to understand how to obtain controllable
nonresonant magnetism in 2D. Then two concentric cylindrical sheets with opposite splits
are introduced as artificial magnetic inclusion. This cell geometry is important to obtain
resonant magnetic permeability with negative permeability region. Split-ring resonator
(SRR), which is proposed by Pendry, is addressed next to design 3D artificial materials. The
numerical calculations are performed for a specific SRR model. The negative permeability
phenomenon is confirmed with retrieved material parameters. The effect of electromagnetic
coupling among the cells on the resonance frequency is investigated. 1D dispersion relation
is studied. At the end of the chapter, a unit cell based on fractal spiral resonator is proposed
to design more homogenous negative permeability materials.
3.2 Theoretical Analysis
The response of natural materials to the magnetic field is determined by the permeability.
The bulk permeability is a macroscopic description of how readily the material experiences
the magnetization when an external magnetic field is applied. Magnetization is the measure
of total magnetic moment resulting from the parallel or anti-parallel alignment of orbital and
spin motion of moving charges to the magnetic field. The lack of magnetic charge leads the
permeability of natural materials to be positive. This makes them not to be useful in LHM
design. However, the effective electromagnetic parameters of any material can be
engineered in a controlled manner by embedding electrically small inclusions into the host
material. This is the main design strategy to be used in the design of negative permeability
materials. On the other hand, there must be some restrictions on the cell size. If macroscopic
electromagnetic parameters are to be assigned as in homogenous materials, the operation
wavelength has to be much larger than the cell size. This results the internal composite
nature of the medium not be identified by the electromagnetic wave. In other words, long
wavelength radiation is too myopic to detect the internal structure. In this limit, to assign an
effective permittivity and permeability is a valid concept even for the inhomogeneous
material. If this condition were not obeyed, there would be the possibility that internal
structure could diffract as well as scatter the radiation. This invalidates the effective material
definition [3-5],[7]. As a rule of thumb, the composite materials, made up of metallic
inclusions with period p can be treated as effective homogeneous materials under an
effective homogeneity condition. It is stated as [5]
4
g
p
(3.1)
where
g
is the guided wavelength.
9
3.2.1 Periodic Array of Cylindrical Metallic Sheets
In 1999, Pendry et al. introduced several configurations of conducting scattering elements
as the unit cells of artificial magnetic materials. In this section, the first structure in Pendry's
work is studied. It is shown in Figure 3.1.
0
H
Figure 3.1 Periodic arrangement of infinitely long cylindrical sheets excited by axial magnetic field
0
H
[7]
This bulk material is composed of periodic infinitely long metallic cylinders in 2D. The
period and radius are a and r, respectively. In terms of effective medium approach, these
sheets can be attributed as electrically small constituents of the host material. The sheet
thickness is comparably smaller than the skin depth at the frequency of interest. The
magnetic response to the axial magnetic field can be calculated by the quasi-static analysis
in the following manner. The external time-varying magnetic field induces current in the
cylinder. The induced current generates in turn magnetic field inside the cylinder opposing
to the time variation of external magnetic field. The φ-directed current leads the magnetic
field to change by a factor of S. S can be regarded as a magnetization factor. The induced
voltage on the cylinder surface is calculated by Faraday's induction law as
0
..
. . .
CF
o
CF
E dl j B dF
E dl j SH dF
(3.2)
Because cylinder thickness, Δr is much smaller than cylinder radius and operation
wavelength, the current can be approximated as a surface current. The surface current,
F
J
,
is calculated from the magnetic field difference between the inner and outer cylinder regions
as
0
( 1)
F
J S H e
. (3.3)
(3.2) can be alternatively formulated in terms of
F
J
as
10
2
00
2
F
Jr j SH r
r
(3.4)
where κ is the electric conductivity.
S can then be calculated by the substitution of (3.3) into (3.4) as
2
2
0
1
12
s
s
j
Sj r r j r r
(3.5)
where
s
is the skin-depth.
The magnetization vector,
M
can then be formulated in terms of
0
H
and effective
magnetic susceptibility,
m
as
22
00
2 2 2
F z m
s
r r r r
M J e H H
a a j r r
. (3.6)
The effective permeability is calculated in the form of
0
2
022
(1 )
(1 )
eff m
eff
s
r r r
a j r r
. (3.7a)
In (3.7a), for infinite conductivity or in the high frequency limit,
eff
is reduced by the ratio
of the cylinder surface to the cell surface. This surface ratio is an important factor in the
permeability calculation of further alternative forms of this model. An important remark is
that
eff
can never be less than zero or greater than
0
. It has a nonresonant diamagnetic
frequency dependence.
The same formulation can be alternatively derived for the finite cylinders of length, lcyr. and
inductance, L as an approximation to the present model (see Appendix A)
22
0
02
cyr.
()
(1 )
2 l
eff
j
r
r
ajL
r
. (3.7b)
In (3.7b), if L is loaded with a series capacitance, C in the form of longitudinal slots along
the cylinder axis, this array exhibits resonant magnetic response with a negative
permeability region in a certain frequency band. This point can be better understood after
the substitution of
2
0
22
1
(1 ) (1 )LL
LC
into L in (3.7b). The effective permeability in
(3.7b) is then in the form of
11
2 2 2 2
00
00
2
22
0
cyr. cyr.
2
( ) ( )
(1 ) (1 )
21 2
l (1 ) l
'
eff
jj
rr
rr
aa
jL jL
r j C r
(3.8)
2
022
0
1
eff
F
j
where C´ is the series capacitance per unit length, C/lcyr..,
0
is the magnetic resonance
frequency, F is the filling factor and
is the loss factor.
These parameters can be defined as
2
0
2
0
21
, ,
r
Fa r r LC
. (3.9)
The generic form of relative permeability is shown in Figure 3.2 for F= 1/3,
0
f
= 2GHz and
=20 MHz.
Figure 3.2 Real (blue) and imaginary (red) part of effective relative permeability
In Figure 3.2, the effective relative permeability exhibits Lorentzian type frequency
dependence with the resonance frequency,
0
f
. In addition, the resonance phenomenon
results the artificial material to be highly magnetized and demagnetized just below and
above
0
f
. There is a negative permeability region between
0
f
and
p
f
.
p
f
is defined as the
magnetic plasma frequency at which the effective permeability is zero analogous to the
electric plasma frequency of metals, ionic molecular and atomic gases.
p
f
is calculated as
0
1
1
p
ff F
(3.10)
12
3.2.2 Periodic Array of Concentric Cylindrical Metallic Sheets with Splits
In order to design more homogeneous medium, the resonance frequency has to be reduced
with the larger capacitive loading in the same cell. One cell of this type is proposed by
Pendry. It is composed of two concentric metallic cylinders with the splits at the opposite
ends as shown in Figure 3.3.
Figure .3.3. One unit cell of square array of concentric metallic
sheets divided into a “split ring” structure and separated
from each other by a distance d [7]
In Figure 3.3, the large capacitance between two rings enables the current to flow from one
ring to the other ring in the form of displacement current. The ring inductance is then loaded
by the gap capacitance between the outer and inner rings with a result of resonant magnetic
response. Therefore, due to the current interrupted by the splits, in the equivalent circuit
modeling, the first and second half of the total gap capacitance without splits are connected
in series manner [3],[40,41]. The equivalent circuit model is shown in Figure 3.4 where
0
C
is the capacitance of two concentric cylinders without splits. The effective permeability can
be calculated by substituting the equivalent capacitance per unit length into the formulation
in (3.8). The capacitance per unit length, C´ and inductance, L are approximately (see
Appendix A)
Figure 3.4. Equivalent circuit model of concentric cylindrical sheets with
opposite splits [40]
00
cyr.
2
0
cyr.
'4l 2ln( )
l
C
Crd
r
r
L
(3.11)
Therefore, the effective permeability has the form of
2 2 2
0
00
22
2
02
00
00
( ) 1
(1 ) (1 )
2ln( ) 2ln( )
2
21
eff
j
rr
r d r d
aa
j
rrr
jr
r j r r r
. (3.12)
13
E
k
H
It can be expressed as in (3.8) with the respective parameters,
2
02
2
000
2ln( )
2
, ,
rd
rr
Fa r r r
. (3.13)
In order to verify the analytical formulation in (3.12), the electromagnetic response of the
cell model in Figure 3.3 is numerically calculated under plane wave excitation. The perfect
electric (PEC) and magnetic (PMC) boundary conditions are imposed at two y-planes and z-
planes, respectively. They are also important to satisfy the transversal periodicity with one
unit cell. The numerical calculations are done with FEM based full-wave EM simulator
HFSS. The numerical model is shown in Figure 3.5. The geometrical parameters are
tabulated in Table 3.1. The transmission and reflection parameters are also calculated
analytically to compare the analytical and numerical calculations. They are shown in Figure
3.6. and in reasonable agreement.
Table 3.1. Geometrical parameters of infinitely extended concentric cylinders with opposite splits
Figure 3.5. Numerical model of infinitely extended concentric cylindrical sheets with opposite splits and boundary
conditions
Geometrical Parameters
(mm)
radius of inner cylinder (rin)
2.9
radius of outer cylinder (rout)
3.3
gap distance (d)
0.3
split width (ws)
0.2
metal width (w)
0.1
lattice period (a)
10
14
Figure 3.6.a: Numerically calculated transmission (blue) and reflection (red) parameters of infinitely extended cylinders
with opposite splits
Figure 3.6.b: Analytically calculated transmission (blue) and reflection (red) parameters of infinitely extended cylinders
with opposite splits
In the analytical calculations, due to the finite cylinder thickness, resonance frequency,
filling factor and loss parameter in (3.13) are modified as
2
02
2
000
2ln( )
2
, ,
in
out in
out s out
r w d
rrw
Far r
. (3.14)
In the numerical calculation, the resonance frequency is 1.88 GHz. In the analytical
calculation, it is 1.95 GHz. The resonance frequency formulated by Pendry for the same
structure [7] is 2.92 GHz. (3.12) estimates the resonance frequency better than the original
one with 3.72 % error. The cell size is 10 mm, which is 1/14.5 of the free space wavelength
at the resonance frequency.
15
Therefore, this one cell thick periodic array can be attributed as a resonant magnetic
material. The wave impedance is also analytically calculated and shown in Figure 3.7. It is
increasing monotonically from low frequencies up to the resonance frequency. At the lower
edge of the resonance frequency, it is high and then capacitive in the negative permeability
band between the magnetic resonance and plasma frequencies.
Figure 3.7 Analytically calculated real (blue) and imaginary (red) part of effective wave impedance of infinitely
extended cylinders with opposite splits
3.2.3 Periodic Array of Split- Ring Resonators
As stated in the previous section, the periodic concentric cylinders with the opposite splits
exhibit resonant magnetic response. However, the magnetic resonance is only valid for the
exciting magnetic field aligned along the cylinder axis. In addition, they exhibit nonresonant
electric response to the axially polarized wave with low transmission. This polarization
dependency of the material parameters makes 2D magnetic materials not to be useful in
some applications. Therefore, Pendry proposed one unit cell geometry in the form of split-
ring resonator (SRR) to design 3D isotropic magnetic materials. The material isotropy can
be potentially restored by printing one SRR cell on each of six surfaces of a cubic magnetic
material as in [7].
Figure 3.8a. Split-ring resonator geometry as negative permeability material
16
E
k
H
The resonance frequency can be sufficiently good estimated in the same manner as in
Section 3.2.2 (see Appendix A). The resonance frequency, filling and loss factors can be
calculated as
2
00
22
0
22
, , ,
2
ln(1 ) (1 )
3
= + +
22
+2 +
out z
out s out av r
av
out
ra
Fc
w
ar rr
d
wd
rr
r r w d
(3.15)
where
r
is the substrate relative permittivity and az is the period in the axial direction.
A 2D SRR array with the geometrical parameters in Table 3.2 is numerically calculated. It is
shown in Figure 3.8b.
Table 3.2. Geometrical parameters of SRR
Geometrical Parameters
mm
inner ring radius (Ri)
2.4
outer ring radius (Ro)
3.1
width of each ring (w)
0.6
spacing between ring edges (d)
0.1
split width
0.2
ax
10
ay
10
az
1
r
1
Figure 3.8b Split-ring resonator model with boundary conditions
17
In the numerical calculation, the resonance frequency is 2.58 GHz. In the analytical
calculation, it is 2.48 GHz. The percentage error is 3.87%. In the work of Pendry, it is highly
overestimated as 8 GHz.
The resonance frequency can be better estimated from alternative analytical formulations. In
[42, 43], the resonance frequency is calculated from the solution of a set of differential
equations, which are formulated in terms of ring currents. In [44], there is one improved
analysis, which is based on the treatment of doubly-split double rings as two coupled split
rings for resonance frequency calculation. In [41,45], the equivalent circuit models are
derived from the quasistatic calculations on the modified forms of SRRs. In these circuit
models, the average inductance is numerically calculated from the variational formulation
and the capacitance per unit length is calculated from the closed form expressions in [46].
Modified geometries of SRRs are also studied [47-49]. They are important to understand
how SRRs with multiple splits can be effectively used to tune the resonance frequency
towards higher frequencies. Alternative cell geometries are also proposed to squeeze the
resonant electrical size for the design of homogeneous magnetic materials [50-54]. An
intuitive method to reduce the electrical size without increasing the resonator area is to use
multiple split-ring resonators [3,55]. This cell design is extensively studied and equivalent
circuit model is derived in [55].
18
3.3 Numerical Simulations
3.3.1 Resonance Frequency of SRR Periodic Array
In this section, the electromagnetic response of one cell thick SRR array is numerically
calculated under plane wave excitation. The unit cell is shown along with the boundary
conditions in Figure 3.9. The geometrical parameters are tabulated in Table 3.3.
(a) (b)
Figure 3.9 (a) One unit cell of SRR geometry (b) boundary conditions imposed on the edges of one unit cell
Table 3.3 Geometrical parameters of one unit cell of SRR array
Geometrical Parameters
mm
inner ring radius (Ri)
0.7
outer ring radius (Ro)
1
width of each ring (w)
0.2
spacing between ring edges (d)
0.1
split width
0.1
substrate thickness
0.5
ax
2.8
ay
2.8
az
2
PEC is imposed on two y-planes at the edges of the unit cell as shown in Figure 3.9b. It
couples the periodic cells along y-direction electrically. In the same manner, PMC is
imposed on two z-planes at the edges to couple the periodic cells along z-direction
magnetically. Under these boundary conditions, one cell thick SRR array can be excited by
y-polarized in x-direction propagating plane wave. TEM wave is the fundamental mode of
the surrounding PEC-PMC guiding structure. The substrate is Rogers/RT duroid 5870. The
relative permittivity and tan(δ) are 2.33 and 0.0012, respectively. The transmission and
reflection parameters are numerically calculated with FEM based software HFSS. They are
shown in Figure 3.10.
19
Figure 3.10 Reflection (red) and transmission (blue) parameters of one unit cell thick SRR sample
The magnetic resonance frequency is 9.61 GHz. The operation wavelength is 31.21 mm. It
is approximately 11 times larger than the cell size. Thus, one cell thick SRR array can be
considered as an effectively homogeneous material. At the resonance frequency, the wave
transmission is highly degraded as explained in Section 3.2.2. The transmission between the
magnetic resonance and plasma frequency is also low as a result of wave attenuation and
impedance mismatch due to negative permeability. However, it is high at the resonance
frequency of 8.7GHz due to increasing effective permeability near the magnetic resonance
frequency as in Figure 3.2 and resulting impedance match with the port impedance. These
points are confirmed from the effective material parameters and wave impedance. The
magnetic field, electric field and surface current distribution at the magnetic resonance
frequency are shown in Figure 3.11.
Figure 3.11a Surface current distribution of one unit cell at the magnetic resonance frequency
20
Figure 3.11b Magnitude of electric field distribution of one unit cell at the magnetic resonance frequency
Figure 3.11c Magnitude of magnetic field distribution of one unit cell at the magnetic resonance frequency
The electric and magnetic field distributions are unsymmetrical despite geometrical
symmetry of unit cell. This is mainly because of finite metallic loss inside the cell and
resulting nonzero transmission at the magnetic resonance frequency. In addition to the
resonant field distributions, the effect of transversal periods on the resonance frequency has
to be investigated. It is important in order to figure out the role of electric and magnetic
coupling among the neighboring cells on the material parameters. Therefore, the
transmission parameter of one cell thick sample is numerically calculated for different
periods in the transversal directions. The effect of electric coupling among y-direction
oriented cells is shown in Figure 3.12a. The lower electric coupling resulting from the larger
period in y-direction, ay increases the resonance frequency. The transversal periodicity in y-
direction has however minor effect. It is because the electric field is mainly concentrated in
the split region of the rings and the gap region between the split rings as shown in Figure
3.11b. The resonant magnetic polarizability of SRR cell results the neighboring cells to
couple magnetically in more effective manner. This can be implied from the effect of
magnetic coupling among z-direction oriented cells on the resonance frequency. It is shown
in Figure 3.12b for the different periods in z-direction, az. Because the magnetic field is
mainly axially directed, the larger az reduces SRR inductance due to lower magnetic
coupling between axially oriented cells. This is the reason why the resonance frequency is
lower for the larger az.
21
Figure 3.12a Transmission parameter with different transversal periodicity in y- direction
Figure 3.12b Transmission parameter with different transversal periodicity in z- direction
3.3.2 Dispersion Relation of SRR Periodic Array
In this section, 1D Brillion diagram is numerically studied to investigate the passband and
stopband of SRR array. In the numerical model, eigenfrequencies are calculated with the
periodic boundary conditions in the propagation direction with different phase shifts. The
same PEC and PMC on the transversal y- and z-planes in the port mode simulation are
imposed. The dispersion diagram of two lowest bands is shown in Figure 3.13. In Figure
3.13, SRR array can be regarded as a transmission medium. The lower frequency band is
from 0 to 9.05 GHz. The higher frequency band is from 11.1 GHz to 22.41 GHz. However,
it can not be attributed as an effective material for the higher frequencies of second band. It
can only be regarded as an effective material in the first band. This is because the unit cell
size is comparable with the free space wavelength at the higher frequencies in the second
band. On the other hand, there is no wave propagation between the higher end of first band,
9.05 GHz and lower end of second band, 11.1 GHz. In this bandgap, it exhibits magnetic
resonance and negative permeability behavior. This is verified from the effective parameter
retrieval procedure in the next section.
22
Figure 3.13 Dispersion diagram of SRR with periodic boundary conditions of different phase shifts in propagation
direction
3.3.3 Effective Material Parameters of SRR Periodic Array
In this section, the effective material parameters are retrieved to confirm the negative
permeability. There are a number of proposed analytical methods for the material parameter
extraction in the literature [56-60]. The common point of these approaches is to calculate the
transmission and reflection parameters of one cell thick sample under plane wave excitation.
The wave impedance and propagation constant are then calculated from transmission and
reflection data. Effective permittivity and permeability are then retrieved from the wave
impedance and propagation constant. In these retrieval procedures, one cell thick sample in
the propagation direction is assumed to be sufficient for effective parameter assignment due
to electrically small cell size [56-60]. Instead of using conventional retrieval procedures, an
alternative extraction method is introduced. In this method, the longitudinal cell periodicity
in the propagation direction is included in the parameter extraction by using Bloch Theorem
unlike in conventional procedures. The common point is that the transmission and reflection
parameters have to calculated only for the fundamental mode in the extraction procedure as
a first step. These parameters have to be then deembedded upto the left and right surface of
the cell. As a second step, S parameters have to be transformed into Z parameters. ABCD
parameters of one unit cell are then calculated from Z parameters in order to use Bloch
Theorem in the following manner,
2
11 11 22 21 22
21 21 21 21
1
, , , =
Z Z Z Z Z
A B C D
Z Z Z Z
. (3.16)
The complex propagation constant, γ and Bloch impedance, ZB are calculated from ABCD
parameters by Bloch Theorem as
arccosh( )
2, ,
1
d
Bd
x
AD
Be
Zj
a Ae
(3.17)
23
where
and
are the phase and attenuation constants, respectively.
As a last step, effective relative permeability and permittivity are calculated from γ and ZB
in the form of
, , y
B line
reff reff line o
line o B o z
a
j Z j Z ZZ
Z k Z k a
(3.18)
where ko is the free space wave number and Zline is the line impedance.
The effective material parameters, complex propagation constant and characteristic
impedance of SRR array are retrieved by the above procedure. They are shown in Figure
3.14.
Figure 3.14a. Real (red) and imaginary (blue) part of effective relative permeability of SRR periodic array
Figure 3.14b. Real (red) and imaginary (blue) part of effective relative permittivity of SRR periodic array
24
In Figure 3.14a, SRR array exhibits magnetic resonance at 9.22 GHz. The relative
permeability has negative real part between magnetic resonance and plasma frequency,
11.36 GHz. The imaginary part is negative and minimum near the resonance frequency.
Small amount of power is transmitted through SRR array due to high field attenuation. In
addition, SRR cell has largest current distribution at this frequency due to high field
concentration. This increases ohmic loss in the metallic parts. It is the natural consequence
of field enhancement with accompanying high loss at the resonance frequency. The negative
permeability frequency band is in the bandgap region of the dispersion diagram in Figure
3.13. On the other hand, relative permittivity is positive as expected. However, the
imaginary part is positive between 9.2 GHz and 9.7 GHz. This is a nonphysical artifact for
the passive materials. There are typically three main reasons for this unphysical effect. The
first reason is the possible neglection of the effect of higher order modes excited by SRR.
The higher order modes are exponentially decaying evanescent fields. Even though no real
power is transported by the evanescent waves, the total reactive power in the near field has
effect on the reactive part of Z parameters. The second reason is the neglection of reactive
intercoupling of each cell with the adjacent cell by higher order modes even though the
longitudinal cell periodicity is taken into account. This is the common point, which is
lacking in alternative retrieval procedures [57-59]. The third reason is although the cell size
is smaller than the operation wavelength, the wavelength to cell size ratio is not sufficiently
large to regard SRR array as an ideal homogeneous material.
Figure 3.14c. Attenuation (blue) and phase (red) constant of SRR periodic array
25
Figure 3.14d. Real (red) and imaginary (blue) part of wave impedance of SRR periodic array
In Figure 3.14c, the attenuation constant is positive between 9.22 GHz and 11.36 GHz. It
corresponds well with the bandgap region of dispersion diagram in Figure 3.13. In addition,
this frequency band coincides with the negative permeability band. The wave attenuation is
maximum at 9.7GHz, which is the frequency of minimum negative permeability in Figure
3.14a. The wave impedance is capacitive in the bandgap region. Therefore, the field
transmission is highly degraded due to wave attenuation and high reflection resulting from
the impedance mismatch. On the other hand, SRR array responds to the exciting field as a
high impedance surface at 9.2 GHz as shown in Figure 3.14d. This feature has a potential
application in the design of high efficient directive antennas [4-6]. The reactive impedance
in the negative permeability band can be implied from the wave impedance formulation of
homogeneous materials
= = -j
eff eff
eff
wave
eff eff eff
Z
. (3.19)
As a result, analytical and numerical results prove the effective negative permeability of
SRR array. However, effective material parameters can only be assigned to the composite
materials if the unit cell size is much smaller than the wavelength. Therefore, it is important
to investigate alternative designs for the miniaturization of unit cells for more homogeneous
materials [3-4],[50-55]. In the next section, one electrically small negative permeability cell
is proposed for the design of more homogeneous materials.
26
3.4 Fractal Spiral Resonator as Magnetic Metamaterial
As a further step in more homogeneous material realization, an artificial magnetic unit cell
based on spiral fractal geometry is proposed [51]. The cell geometry is explained first in
Section 3.4.1. The magnetic resonance is illustrated from the numerically calculated field
pattern in Section 3.4.2. The effective permeability is analytically calculated from the
numerical data. The dispersion diagram and Bloch impedance are illustrated in Section
3.4.3.
3.4.1 Structural Description
The unit cell is shown in Figure 3.15. The resonator is basically formed by connecting two
fractal ring resonators in a spiral form. Upper half of the inner and outer rings is the mirror
image of the lower half. It is in the form of first order Hilbert fractal. These two concentric
rings are then connected at one end to obtain the spiral form. The marked inner section is
the extension of the inner curve. It is important in order to decrease the resonance frequency
due to inductive and capacitive coupling among the different sections. The substrate is 0.5
mm thick FR4 with dielectric constant 4.4 and tan(δ) 0.02. The metallization is copper. The
copper line width and minimum distance between any two lines are 0.2 mm. The other
geometrical parameters are L1= 2.2mm, L2= 0.8mm and L3= lmm. The unit cell size is ax=
5mm, ay = 2mm, az = 5mm. Only one side of the substrate is structured with the prescribed
fractal geometry.
Figure 3.15 Geometry of a fractal spiral resonator
3.4.2 Simulation Results
In order to obtain the magnetic resonance, the fractal spiral resonator array has to be excited
with out-of-plane directed magnetic field of the plane wave. Therefore, PEC at two x-planes
and PMC at two y-planes are imposed at the edges of the unit cell. These boundary
conditions are necessary to excite the array with x-direction polarized and z-direction
propagating plane wave.
27
The numerically calculated S-parameters are shown in Figure 3.16. The resonance
frequency is 1.52 GHz. The unit cell size is 1/40 of free space wavelength. Therefore, the
composite material made up of fractal spiral resonators can be regarded as a more
homogeneous material than SRRs-based materials. The magnetic field and surface current
distribution at the resonance frequency are shown in Figure 3.17a and Figure 3.17b.
Figure 3.16 Transmission (red) and reflection (blue) parameters of fractal spiral resonator
Figure 3.17a Magnetic field distribution of fractal spiral resonator at the resonance frequency
Figure 3.17b Surface current distribution of fractal spiral resonator at the resonance frequency
28
Due to the spiral form of surface current and resulting out-of-plane directed magnetic field,
the magnetic resonance is confirmed. Therefore, this electrically small cell can be
considered as a resonant magnetic dipole. The low transmission in Figure 3.16 is the result
of demagnetization effect of resonant inclusion to the exciting magnetic field in the negative
permeability band. This is the reason why this unit cell can be used in the realization of
negative permeability materials.
3.4.3 Effective Parameters
As a next step, the effective material parameters are retrieved to confirm the negative
permeability and justify the above-mentioned remarks. The extraction method explained in
Section 3.3.3 is exploited to calculate the propagation constant, Bloch impedance and
effective relative permeability. They are shown in Figure 3.18.
Figure 3.18a Phase (red) and attenuation (blue) constant of fractal spiral resonator array
Figure 3.18b Real (red) and imaginary (blue) part of Bloch impedance of fractal spiral resonator array
29
Figure 3.18c Real (red) and imaginary (blue) part of effective relative permeability of fractal spiral resonator array
In the frequency band of high field attenuation in Figure 3.18a, the wave impedance is
capacitive as shown in Figure 3.18b. This bandgap region is between 1.52GHz and
1.95GHz. The wave impedance has its highest value at the lower edge of magnetic
resonance frequency, 1.52GHz. The wave attenuation is therefore maximum at this
frequency, which is the reason of lowest transmission in Figure 3.16. The reason of low
transmission between 1.4 GHz and 1.52 GHz is the impedance mismatch resulting from
increasing wave impedance rather than negative permeability. The magnetic resonance is
also implied from Figure 3.18c. In Figure 3.18c, the fractal spiral resonator exhibits
negative permeability between the resonance and plasma frequency, 1.96GHz. As a result,
numerical simulations confirm the magnetic resonance and negative permeability.
3.5 Chapter Conclusion
In this chapter, the artificial magnetism obtained from periodic metallic cylinders and SRRs
is explained to introduce the design approach of negative permeability materials. The
effective permeability is analytically formulated for the plane wave excitation with axially
directed magnetic field. The transmission and reflection parameters are calculated. The
numerical calculations are additionally done to verify the analytical calculations. They are in
good agreement. The effective permeability formulations are compared with the original
formulations of Pendry. The analytical formulations in this chapter estimate the resonance
frequency better than the analytical formulations of Pendry.
A typical SRR model is numerically analyzed. The transmission and reflection parameters
are calculated. The surface current, electric and magnetic field distributions are presented.
The main reason in the demagnetization of incoming magnetic field is the excitation of
electrically small magnetic dipoles in the cells. They are formed from circular form of
surface currents on the metallic rings. 1D Brillion diagram is numerically calculated. The
bandgap region is determined. In the bandgap, the composite material has negative
permeability and capacitive wave impedance. An alternative parameter retrieval method is
proposed. It is based on the calculation of the effective material parameters of 3D array as a
result of periodic continuation of one cell in the propagation direction by Bloch Theorem.
The effective material parameters, Bloch impedance and dispersion diagram are calculated.
30
At the end of the chapter, a unit cell based on fractal spiral resonator is proposed. It is
numerically investigated in terms of effective permeability, Bloch impedance and complex
propagation constant. The numerical calculations confirm the effectiveness of the resonant
inclusion in the design of more homogeneous composite materials.
31
4. Negative Permittivity Metamaterials
4.1 Introduction
This chapter explains the basic principles of negative permittivity materials starting with the
periodic wire array. This was originally proposed by Pendry [8,9]. The plasma-like
permittivity response obtained from the periodic long wires is analytically calculated. This
calculation is important to understand the fundamental principle of nonresonant electric
depolarization in 2D. The effect of wire loading on effective permeability is also addressed.
Alternative designs are also pointed out for more homogeneous realizations. The numerical
calculations are conducted for a wire strip model structured on a low loss substrate as a
microwave realization of negative permittivity materials. The effective material parameters
are retrieved. 1D dispersion relation is studied to determine the plasma frequency and
negative permittivity band. At the end of the chapter, a unit cell based on fractal antispiral
resonator is proposed for the design of more homogeneous negative permittivity materials.
4.2 Theoretical Analysis
The response of natural materials to the electric field is determined by the permittivity. The
electric permittivity quantifies the polarisability strength of molecules and atoms in the
microscopic scale with the resulting dipole moment. The relative permittivity of natural
dielectrics is positive and larger than 1. On the other hand, the relative permittivity of metals
and dilute gases, plasmas can be negative. The electromagnetic response of metals is
determined by the dissipation at low frequencies below infrared. The metals respond to the
electromagnetic fields with the negative permittivity at the frequencies smaller than the
plasma frequency [6],[8,9], which is in the visible and UV frequency regions. To achieve
negative permittivity at lower frequencies in the microwave regime, periodic metallic wires
are proposed by Pendry. It is shown in Figure 4.1 [8,9]. Pendry examined how the magnetic
field of wire electrons increase the electron momentum in the wire array under plane wave
excitation [8,9]. The larger electron momentum can be described by the larger effective
mass of wire electrons. This larger effective mass reduces the plasma frequency from the
range of UV to the microwave regime. However, in Pendry’s formulation, the effective
permittivity is calculated under the assumption of Drude type permittivity response without
any high frequency field calculations. The generic form of Drude type permittivity is
expressed in (4.1) with the plasma frequency,
p
and dissipation factor,
2
2
( ) 1 .
p
rj
(4.1)
32
Figure 4.1. One unit cell of periodic structure composed of infinitely long wires in a simple cubic lattice [9]
In (4.1), the real part of relative permittivity is negative at the frequencies smaller than
p
for small
. For larger
, the frequency at which the real part is zero, is calculated as
22
new p
. (4.2)
In this section, the effective permittivity of periodic infinitely long wires of width w is
calculated under plane wave excitation. This formulation is more generalized than Pendry’s
formulation [8,9]. The equivalent model to be used in the analytical calculation is shown in
Figure 4.2.
(a) (b)
Figure 4.2. Excitation and boundary conditions of equivalent model of infinitely long wire array (a) on x-y plane (b) on
y-z plane
In this model, PEC boundary conditions are imposed at two y-planes to extend the wire
length to infinity. Because of infinite extension in y-direction and y-independency of the
incoming wave, the scattering parameters, Bloch impedance, complex propagation constant
and effective parameters are independent of the periodicity in y-direction, ay. In a similar
manner, PMC boundary conditions are imposed at two x-planes to excite the plane wave
mode of PEC-PMC guiding medium. This boundary condition extends the wire strip
periodically in x-direction. Thus, these boundary conditions model 1D wire array with one
unit cell in the propagation direction. This array has to be excited by y-polarized z-direction
propagating plane wave.
33
However, due to the material inhomogenity in x-direction, TE waves are also excited in
addition to TEM wave. Therefore, TE wave components, which are y-independent, have to
be calculated, as well. They are formulated as,
TE wave field components:
22
cos( )
cos( )
sin( )
z
z
z
jk z
TE TE
yx
jk z
TE TE z
xx
zjk z
TE TE
zx
x
E A k x e
k
H A k x e
kk
H jA k x e
k
(4.3)
The resulting field components can be described as the superposition of field components of
incoming plane wave and all excited modes of TE wave. They are formulated for z ≤ 0 as,
0
0
0
0,1,2,3...
0
0,1,2,3...
22
0,1,2,3...
cos( )
cos( )
sin( )
zm
zm
zm
jk z jk z
TE TE
ym
mx
jk z jk z
TE TE zm
xm
mxo
o zm jk z
TE TE
zm
mx
x
m
E A x e E e
a
k m E
H A x e e
aZ
kk m
H jA x e
ma
a
(4.4)
with kx =
x
m
a
, kzm =
2
2
o
x
m
ka
and ko =
0
c
, where ax is the cell size in x-direction.
In (4.4), the first terms are the field components excited by the current on the wire at z=0.
The field components for m=0 are the plane wave components of the reflected wave. The
second terms are the field components of incoming plane wave. To calculate the reflection
coefficient, unknown modal constants,
TE
m
A
have to be determined.
TE
m
A
s can be calculated
by setting the total electric field on the wire surface to zero. This boundary condition can be
easily satisfied at one point on the wire instead of whole wire surface from xo-w/2 to xo+w/2
if w is much smaller than free space wavelength. Therefore, to take one point on the wire
surface as the field calculation point is a sufficiently good approximation with negligible
deviations in the calculated
TE
m
A
s. The relation between
TE
m
A
’s and Eo can then be obtained
as
00
1,2,3...
w
cos ( )
2
TE TE
mo
mx
m
A A x E
a
. (4.5)
The reflection coefficient, R is defined as
0
0
TE
A
RE
. (4.6)
34
An additional relation between
TE
o
A
and
0
E
has to be derived to calculate R. This relation
can be obtained if the wire current, I wire is expressed in terms of
TE
m
A
. I wire can be calculated
from the tangential magnetic field at x=x0 and z=0 as
0,1,2,3...
2 cos( ) ( )
TE z
m wire o
mx
km
A x I x x
a
. (4.7a)
In the above expression, the current distribution is assumed to be concentrated at x=x0.
TE
m
A
’s can be calculated from (4.7a) by the infinite series expansion of the current
distribution in terms of modal magnetic field as
0
0
cos( ) m=1,2,3...
.
2
TE
m wire o
x x zm
TE wire
x
m
A I x
a a k
I
Aak
(4.7b)
By the substitution of
TE
m
A
’s in (4.7b) into (4.5),
wire
I
can be calculated in terms of E0 in
addition to R as
0
2
1,2,3...
02
w
cos cos ( )
2
1
2
x
wire
oo
xx
m
x
Ea
Imm
xx
aa
km
ka
(4.8a)
0
2
1,2,3... 2
0
1.
w
2 cos cos ( )
2
1
oo
xx
m
x
Rmm
k x x
aa
m
ka
(4.8b)
After the calculation of R, an equivalent circuit model can be built for the wire loaded PEC-
PMC guiding medium. As implied from Figure 4.2, the equivalent circuit can be
conveniently modeled as two transmission line sections of length,
2
z
a
loaded with a shunt
inductance, Lwire. It is shown in Figure 4.3.
35
(a) (b)
Figure 4.3. (a) Equivalent circuit and (b) impedance model of one cell thick long wire array
ko and Z0 are the free space propagation constant and wave impedance, respectively. The
wire inductance per unit length is calculated from the equivalent impedance in Figure 4.3b
as
1
2
o
wire
y
R
Z
Lj a R
. (4.9)
In order to validate the equivalent circuit model, the transmission and reflection parameters
of wire array are numerically calculated. The model is shown in Figure 4.4 with the
boundary conditions. The important issue in the analytical modeling is that two successive
wire elements do not have to be equally spaced in x-direction. However, in the original
work of Pendry, the wires are equally spaced in 2D array [8,9]. In addition, the above
analytical modeling allows two wires to be located inside the same cell in a more
generalized form.
k
E
H
Figure 4.4. Numerical model of infinitely extended wire array with PEC and PMC boundary conditions
36
In the model, the wire radius is 0.05 mm. The wire is located at a separation distance of 2
mm from the left PMC wall. The separation distance of two PMC walls is 5 mm. The wire
length between two PEC walls is 2.8 mm. The cell size in the propagation direction is 5mm.
In the modeling, the cutoff frequency of first propagating TE mode has to be larger than the
largest frequency in the interested band. This condition imposes the cell size to be smaller
than the operation wavelength. This is an important requirement to fulfill the effective
homogeneity condition. The numerically calculated scattering parameters are compared
with the analytical calculations. The results are shown in Figure 4.5. The agreement is very
good. The wire inductance per unit length is also calculated. It is shown in Figure 4.6.
Figure 4.5a. Magnitude of analytically (red) and numerically(blue) calculated transmission parameters
Figure 4.5b. Magnitude of analytically (red) and numerically(blue) calculated reflection parameters
37
Figure 4.6. Analytically calculated per-unit-length inductance of wire array
In Figure 4.6, the inductance is approximately constant as expected with a maximum change
smaller than 1%. This result confirms the validity of the wire modeling with the equivalent
circuit in Figure 4.3. However, there is no periodicity in the propagation direction.
Therefore, this 1D wire array has to be extended periodically in the propagation direction
for 2D wire array modeling. To retrieve the effective material parameters of 2D wire array,
the equivalent ABCD matrix of the unit cell in Figure 4.3 has to be calculated. ABCD
matrix of transmission line section of length az/2 is expressed as
0
_
0
cos( ) sin( )
22
sin( )
2cos( )
2
zz
oo
ABCD trline z
oz
o
aa
k jZ k
Ma
ka
jk
Z
. (4.10a)
ABCD matrix of Lwire is expressed as
_
10
11
ABCD shuntL
MjL
. (4.10b)
Therefore, the ABCD matrix of one wire cell is calculated as in (4.10c) with the matrix
elements in (4.10d)
38
00
_
00
_
cos( ) sin( ) cos( ) sin( )
10
2 2 2 2
1
sin( ) sin( )
1
22
cos( ) cos( )
22
z z z z
o o o o
ABCD cell zz
oo
zz
oo
ABCD cell
a a a a
k jZ k k jZ k
Maa
kk
j
aa
L
j k j k
ZZ
AB
MCD
(4.10c)
22
2
2
0
2
cos ( /2) sin( ) sin ( /2)
2
sin( ) sin ( )
2
11
sin( ) cos ( )
2
o
o z o z o z
z
o o z o
z
o z o
o
Z
A k a k a k a
L
Za
B jZ k a j k
L
a
C j k a j k
ZL
DA
(4.10d)
The complex propagation constant,
per j
follows from Bloch theorem together with
(4.10c) as
cos( ) sin( ) cosh( )
2
o
o z o z per z
Z
k a k a a
L
. (4.11a)
where
and
are the attenuation and phase constants, respectively. az is the cell period in
the propagation direction.
The Bloch impedance is then calculated in the form of
2
2
0
1
sin( ) sin ( )
2
cos( ) sin( ) 1
2
per z
per z
per z
per z
a
Bloch a
a
z
o o z o
Bloch a
o
o z o z
Be
ZAe
Za
jZ k a j k e
L
ZZ
k a k a e
L
. (4.11b)
The effective relative permittivity and permeability are calculated from Bloch impedance
and complex propagation constant as in Section 3.3.3. They are shown in Figure 4.7 along
with Bloch impedance and complex propagation constant.
39
Figure 4.7a Effective relative permittivity of analytically calculated periodic wire array
Figure 4.7b Effective relative permeability of analytically calculated periodic wire array
Figure 4.7c Attenuation (blue) and phase constant (red) of analytically calculated periodic wire array
40
Figure 4.7d Real (red) and imaginary (blue) part of Bloch impedance of analytically calculated periodic wire array
In Figure 4.7a, the wire array exhibits Drude type permittivity response as in dilute plasmas
and metals. The plasma frequency is 10.11 GHz. The part of the dielectric loss due to
metallic loss is neglected in the analytical modeling. On the other hand, the relative
permeability is positive as expected and approximately 1.106. It is larger than 1 due to the
inductive loading of shunt wires between two PEC surfaces. In Figure 4.7c, the attenuation
constant is positive below the plasma frequency as expected. This is the band of high field
attenuation with low transmission. The incoming wave is also reflected due to the inductive
wave impedance shown in Figure 4.7d. For the higher frequencies than the plasma
frequency, the wire array is a frequency dispersive transmission medium as shown in Figure
4.7c. This can also be implied from the nonlinear dependence of
per
on the frequency in
(4.11a). However, it tends to be less dispersive at higher frequencies due to the reducing
effect of inductive loading with increasing frequency. There is one important issue in the
wire array modeling. The pure inductive modeling of wire loading in Figure 4.3 is an
approximation with a maximum change smaller than 1%. Due to frequency dependency of
inductance, the shunt impedance has to be modeled with the insertion of a capacitance in
parallel with the inductance to take the frequency dependence as in Fig. 4.6 into account
[4-6].
For the realization of negative permittivity materials, the wire array has to be long enough to
have similar electromagnetic response of infinitely long array. However, Lorentzian type
permittivity response is obtained instead of Drude type response due to finite wire length. In
this case, the wire length is then comparable with the wavelength. Because the effective
homogeneity condition is invalidated, electromagnetic properties of such artificial materials
change according to the block size in the transversal and longitudinal directions. This causes
important dispersion and application problems in the composite materials made up of these
unit cells. Therefore, some more homogenous alternatives are also introduced such as
meander line or its modified forms and fractal based designs. They are valid design
strategies to obtain more homogeneous designs [4-6],[61,62]. An intuitive approach in the
cell miniaturization is the use of complementary designs of negative permeability materials
by invoking the concept of duality and complementarity [3-6]. The complementary SRR
(CSRR) is the negative image of SRR.
41
It is structured as the slotted form of SRR by etching the slots in the ground plane or in the
conductor strip of the transmission line [3]. The magnetic dipole of negative permeability
materials is replaced by the electric dipole of negative permittivity materials [3]. Multiple
slotted circular or rectangular rings and fractal resonators are well-known complementary
modifications of SRRs to design negative permittivity materials [51-53],[63].
42
4.3 Numerical Simulations
4.3.1 Frequency Response of a 1D Wire Array
In this section, the scattering parameters of one cell thick sample are numerically calculated
under plane wave excitation. The unit cell and imposed boundary conditions are shown in
Figure 4.8. The geometrical parameters are tabulated in Table 4.1.
(a) (b)
Figure 4.8. (a) One unit cell of wire strip geometry (b) boundary conditions imposed on the edges of one unit cell
Table 4.1. Geometrical parameters of one unit cell of periodic wire array
Geometrical Parameters
mm
Wire width
0.2
Wire length (ay)
2.8
ax
2.8
az
2
PEC and PMC are imposed on two y- and z-planes, respectively. The wire array can then be
excited by y-polarized x-direction propagating plane wave under these boundary conditions.
The substrate is 0.5mm thick Rogers/RT duroid 5870 with the relative permittivity of 2.33
and tan(δ) 0.0012. The transmission and reflection parameters are shown in Figure 4.9.
Figure 4.9 Reflection (red) and transmission (blue) parameters of one unit cell thick wire array
43
They have quite similar frequency dependence as the theoretical results in Figure 4.5. As
shown in Figure 4.10a, there is no resonant current distribution. Therefore, PEC boundary
condition extends the unit cell infinitely in y-direction to model the wires with no y-
dependence. This 1D array can only have higher transmission in the same frequency band if
it is extended in the propagation direction as 2D array. For this case, the transmission peaks
are resulting from the periodic arrangement in the propagation direction. The resonance
wavelength is therefore comparable with the cell period, which makes the effective
homogeneity condition invalid. This causes the material parameters to be dependent on the
number of cells in the propagation direction.
As stated in Section 4.2, the wire strip loads the guiding medium inductively. This results
accordingly the effective permittivity to be negative. These are the main reasons why there
is low transmission through the wire medium as shown in Figure 4.9. The magnetic field,
electric field and surface current distribution at 9.6 GHz are shown in Figure 4.10.
Figure 4.10a. Surface current distribution of one unit cell at 9.6 GHz
Figure 4.10b. Magnitude of electric field distribution of one unit cell at 9.6 GHz
44
Figure 4.10c. Magnitude of magnetic field distribution of one unit cell at 9.6 GHz
The field distributions are calculated at 9.6 GHz. It is because 9.6 GHz is the frequency at
which minimum negative permeability of SRR array is analytically retrieved in Section
3.3.3. In the next chapter, it is confirmed that 9.6 GHz is in the LH band where negative
permittivity of wire and negative permeability of SRR array are simultaneously obtained.
As in Chapter 3, the effect of electric and magnetic coupling between the neighboring cells
on the plasma frequency has to be figured out. This study is important in order to engineer
the effective material parameters with different coupling levels. The shift of plasma
frequency to lower frequencies can be deduced from how effectively the plane wave is
transmitted. It is because the wave has a higher attenuation for a higher plasma frequency.
Therefore, the transmission parameter of one cell thick sample with different periodicities in
the transversal directions is numerically calculated. There is no effect of electric coupling
among the cells in y-direction as expected. The magnetic coupling among the cells in z-
direction has an important effect due to the dependence of plasma frequency on cell
periodicity. As in the analytical formulations [8,9], the plasma frequency decreases with
increasing wire separation in z-direction.
4.3.2 Dispersion Relation of Periodic Wire Array
As a next step, the eigenfrequencies of one cell with periodic boundary conditions of
different phase shifts in the propagation direction are numerically calculated. The PEC and
PMC on the transversal y- and z-planes are imposed as in Figure 4.8. 1D dispersion diagram
of first two bands is shown in Figure 4.11. In Figure 4.11, there is no transmission through
wire medium in the frequency band of 7-12 GHz. The incoming field is attenuated due to
the negative permittivity and reflected due to the resulting inductive wave impedance. This
remark will be confirmed from the effective material parameters to be retrieved in the next
section. On the other hand, there is one important issue to be explained about the eigenmode
calculation. In the eigenmode calculation, all propagating modes are taken into account to
satisfy the phase shift along the propagation direction. Therefore, also higher TE and TM
modes are present in the high frequency band. This is the reason why Brillouin diagram
tends to be smoother for the phase shifts larger than 140° in the first band. These are
frequencies at which higher order modes with y-dependence are propagating, which is not
physically true for infinitely long wires.
45
However, the calculated eigenfrequencies and modes are correct. In other words, this is a
modeling artifact due to the finite cell size in y-direction for the frequencies higher than the
lowest cutoff frequency of PEC-PMC medium. This problem can be solved by decreasing
the cell size in y-direction or taking the frequencies in the second band as the extension of
the first band. In addition, the wire medium exhibits left-handed (LH) behavior for the phase
shifts larger than 150° in the first band. This is because of the capacitive coupling of
successive cells in the propagation direction under the excitation of higher order modes.
Figure 4.11 Dispersion diagram of wire strip with periodic boundary conditions of different phase shifts in propagation
direction
4.3.3 Effective Parameters of Periodic Wire Array
In this section, the effective relative permittivity and permeability are retrieved to confirm
the negative permittivity and positive permeability. The retrieval procedure in Section 3.3.3
is implemented. The effective material parameters, complex propagation constant and
characteristic impedance are shown in Figure 4.12.
Figure 4.12a. Real (red) and imaginary (blue) part of effective relative permittivity of periodic wire array
46
Figure 4.12b. Real (red) and imaginary (blue) part of effective relative permeability of periodic wire array
In Figure 4.12a, the wire array exhibits Drude type permittivity response. The relative
permittivity has negative real part in the frequency band of 7-12 GHz. The imaginary part is
negligibly small in comparison to the real part. It is in the range of 0.05. The plasma
frequency can be estimated with an approximate calculation. The relative permittivity
values in Figure 4.12a can be fit analytically into Drude type permittivity formulation in
(4.1). This analytic formulation estimates the plasma frequency as 28.56GHz. The lowest
frequency of the first band in Figure 4.11 corresponds quite well with this estimated
frequency. On the other hand, relative permeability is positive. The magnetic loss is in the
range of 8.10-4. The real part is greater than 1 and approximately 1.3. It is because of the
inductive loading of the wire strip as explained in Section 4.2.
Figure 4.12c. Attenuation (blue) and phase (red) constant of periodic wire array
47
Figure 4.12d. Real (red) and imaginary (blue) part of wave impedance of periodic wire array
The wave impedance is inductive with a negligible resistive part in the range of 0.25Ω.
Thus, the field transmission is highly degraded due to the wave attenuation and reflection
resulting from the impedance mismatch at the port. As a result, the numerical and analytical
results prove effective negative permittivity of wire array.
In the next section, a special negative permittivity cell is proposed. This design illustrates
how to decrease the effect of material inhomogenity with electrically small cell sizes. To
investigate such alternative designs for miniaturized cells is an important task in artificial
material design.
4.4 Fractal Anti-Spiral Resonator as Dielectric Metamaterial
In this section, a unit cell based on fractal spiral resonator is proposed for the design of
negative permittivity materials [62]. The cell geometry is explained at first. The electric
resonance is illustrated from the numerically calculated field pattern. The effective
permittivity is calculated from the numerical data in addition to the complex propagation
constant and Bloch impedance.
4.4.1 Structural Description
The geometry of one unit cell is shown in Figure 4.13. The red marked section is a second
order fractal Hilbert curve with modified side length ratio. It is surrounded closely with
another Hilbert curve in order to decrease the resonance frequency by capacitive coupling
inbetween. The design is similar to the magnetic metamaterial in Section 3.4. The important
difference is how first half of Hilbert curve is connected with the second half. The second
half is the mirror image of the first half. The substrate material is 0.5 mm thick FR4 with
dielectric constant 4.4 and tan(δ) 0.02. The metallization is copper. The copper line width
and minimum distance between any two lines are 0.2 mm. The other geometrical parameters
are L1 = 0.6 mm, L2 = 2 mm and L3 = 2 mm. The unit cell size is ax = 2 mm, ay = 5 mm, az
= 5 mm. Only one side of the substrate is structured with the prescribed fractal geometry.
48
Figure 4.13. Geometry of fractal anti-spiral resonator as artificial dielectric material
4.4.2 Simulation Results
In order to obtain resonant negative permittivity, this structure is excited by a y-direction
polarized, z-direction propagating plane wave. The blue marked overlapping sections in
Figure 4.13 show the enhanced capacitively coupled sections to reduce the resonance
frequency. In the numerical model, PEC at two y-planes and PMC at two x-planes are
imposed on one unit cell. The simulated S parameters are shown in Figure 4.14.
Figure 4.14. Transmission (red) and reflection (blue) parameters of fractal anti-spiral resonator
It is important to note that to determine the right polarization and propagation direction for
the cell excitation, additional S parameter calculations are done for both x- and z-polarized
plane waves with the corresponding boundary conditions and port locations. Electric
resonance was obtained at the lowest frequency and therefore, the above mentioned y-
direction polarized, z-direction propagating plane wave corresponds to the right excitation.
In Figure 4.14, the resonance frequency is 2.74 GHz. In order to understand why the electric
resonance is more effectively excited, the current distribution is investigated along with the
electric field distribution at the resonance frequency.
49
In Figure 4.15a, the current distributions on the upper and lower half are nearly anti-
symmetrical. The induced magnetic fields are thus oppositely directed. This results into a
small net out-of-plane magnetic dipole moment.
Figure 4.15a. Surface current distribution of fractal anti-spiral resonator at the resonance frequency
However, in Figure 4.15b, due to the electrically coupled sections in the middle part, the y-
directed electric field has a similar field distribution as a resonant y-directed electric dipole.
The transmission minimum is due to the depolarization effect of the excited dipole on the
incoming electric field. It can be noticed from the weaker field distribution on the right-
hand side than that on the left-hand side as in Figure 4.15b. Therefore, this artificial material
is regarded as a negative permittivity material in a certain frequency band. On the other
hand, the excited electric dipole is an in-plane dipole rather than out-off plane dipole. This
is an important difference between the proposed design and conventional CSRR-based
designs. In CSRR-based designs, slots etched in the ground plane excite out-off plane
directed dipoles. Thus, this design method has the advantage of exciting the structure
effectively with an in-plane feeding line on the same substrate layer [3-5], which is not
preferred for CSRR-based designs.
Figure 4.15b. E-field distribution of fractal anti-spiral resonator at the resonance frequency
50
4.4.3 Effective Parameters
As a next step, the effective permittivity is retrieved. The complex propagation constant and
Bloch impedance are also illustrated. They are shown in Figure 4.16.
Figure 4.16a. Real (red) and imaginary (blue) part of effective relative permittivity of fractal anti-spiral resonator array
Figure 4.16b. Phase (red) and attenuation (blue) constant of fractal anti-spiral resonator array
51
Figure 4.16c. Real (red) and imaginary (blue) part of Bloch impedance of fractal anti-spiral resonator array
In Figure 4.16a, the effective permittivity has Lorentzian type dispersion characteristics.
The plasma frequency is 3.13 GHz. The negative permittivity band extends from 2.67 GHz
to 3.13 GHz with the bandwidth of 460 MHz. The transmission minimum is at the
frequency of lowest negative permittivity, 2.74 GHz. It is 70 MHZ shifted from the electric
resonance frequency due to the metallic and substrate losses. In the negative permittivity
band, the attenuation constant is high and the wave impedance is inductive.
As a result, the numerical simulations confirm the electric resonance and negative
permittivity of fractal anti-spiral resonator. The unit cell size is 1/22 of free space
wavelength. It can be used to design more homogeneous dielectric metamaterials.
4.5 Chapter Conclusion
In this chapter, artificial dielectrics based on periodic metallic wires are first addressed. The
fundamental approach in the design of negative permittivity materials is introduced. The
effective permittivity is analytically formulated under plane wave excitation. The electric
field has to be directed parallel to the wire axis to obtain Drude type permittivity response.
The transmission and reflection parameters are analytically and numerically calculated.
They have quite good agreement. The equivalent circuit model is derived. The effective
material parameters are then analytically calculated from the equivalent circuit by Bloch’s
Theorem. The effective permittivity formulation has Drude type frequency dispersion as
expected from the original formulations of Pendry. The analytical calculations in this
chapter do not use any assumptions on the frequency dependency of wire array unlike in the
analytical formulation of Pendry. A wire model structured on low loss substrate is
numerically analyzed. The transmission and reflection parameters are calculated. Surface
current, electric and magnetic field distributions are presented. They are important to
understand the main driving principle of negative permittivity. The reason why the incoming
electric field is depolarized, is the excitation of electrically small electric dipoles, formed
from the wire currents. Thus, the wire array has an inductive impedance in the negative
permittivity region as in the case of short-circuited transmission line. 1D Brillion diagram is
numerically calculated. The bandgap region and plasma frequency are determined.
52
The effective material parameters, Bloch impedance and complex propagation constant are
calculated. At the end of the chapter, a unit cell based on fractal anti-spiral resonator is
proposed and numerically investigated. The effective permittivity, Bloch impedance and
complex propagation constant are retrieved. The numerical calculations confirm the
effectiveness of this electrically small cell in the design of more homogeneous negative
permittivity materials.
53
5. Left-Handed Metamaterials
5.1 Introduction
In this chapter, a third class of artificial materials, ‘Left-handed Metamaterials’ (LHM) is
explained. LHMs are realized by embedding negative permittivity and permeability unit
cells into same host medium to obtain artificial magnetism and dielectricity in the same
frequency band. The eigenmode equation of periodically loaded negative permittivity and
permeability materials is therefore first analytically calculated. This calculation is also
important to confirm subwavelength resonance and negative phase velocity in LHMs. The
numerical calculations are additionally done for a specific LHM cell geometry. This unit
cell is composed of same SRR geometry in Chapter 3 and wire geometry in Chapter 4. The
main reason to use same cell geometries is to confirm LH propagation in the frequency band
of both negative permeability and permittivity. The effective parameters are retrieved.
Dispersion relation is studied. The frequency band of negative refractive index is
determined. At the end of the chapter, a unit cell based on thin wire loaded spiral resonator
is studied to realize more homogeneous LH materials.
5.2 Theoretical Analysis
The theoretical research on LHM was initiated by Veselago in 1968 [2]. He considered
electromagnetic wave propagation through a homogenous isotropic material in which both
permittivity and permeability were negative. Because the direction of Poynting vector is
opposite to that of phase velocity in this material, he referred to this medium as a left-
handed medium. Due to the natural inexistence of such exotic materials, his ideas have
received little attention in scientific community. However, the work of Pendry on
electromagnetic engineering of permeability and permittivity of a host material has made
Veselago’s ideas realisable [7-9]. In 2000, Smith constructed a composite LH medium with
periodic metallic wires and SRRs in the microwave regime [1]. He demonstrated negative
refraction at the interface of LH medium and air. There have now been several theoretical
and experimental studies confirming negative refractive index [56-59], [64-69]. The
negative refractive index in LHMs can be simply understood from a similar procedure in
Section 3.3 and Section 4.3 by calculating wave impedances of LHM filled waveguide. H-
and E-wave impedances are calculated as
0
LHM
Hmn
zmn zmn
zmn zmn
Emn
LHM
zmn
eff
Zkk
kk
Z
k
nk
(5.1)
where
zmn
k
is phase constant at the operation frequency,
.
and
are negative
permeability and permittivity of LHM, respectively.
54
Because the wave impedance is always positive,
zmn
k
has to be negative in the impedance
formulation. This makes LHM to have negative refractive index. This is an alternative
explanation to understand why the refractive index is negative in LHMs. The effective
electromagnetic parameters can also be retrieved experimentally and numerically [56-59].
There are also engineering applications derived from this concept in addition to negative
refraction. Some of them are electrically small resonators [13], sub-wavelength waveguides
[15],[70-72], enhanced focusing [17,18], backward wave antennas [19], enhanced
electrically small antennas [20], compact filters [14], [73] and improved microwave
components [3,4],[6].
In this section, eigenmode equation of H-waves in a rectangular waveguide, which is
periodically loaded with negative permittivity and permeability materials is analytically
calculated [73]. The eigenmode equation of E-waves can be similarly calculated by
replacing H-wave impedance with E-wave impedance in ABCD matrix formulation. The
analytic model is shown in Figure 5.1.
Figure 5.1. Analytical model of periodically loaded negative permeability and permittivity materials
ABCD matrix of negative permittivity material of length az1 is formulated with H-wave
impedance, ZH1 and phase constant, kz1 as
1 1 1 1 1
_11
11
1
cos( ) sin( )
sin( ) cos( )
z z H z z
ABCD negeps zz
zz
H
k a jZ k a
Mka
j k a
Z
(5.2a)
2 2 2
1 1 1 ( ) ( )
z
mn
kab
1
1
1
H
z
Zk
where a and b are the waveguide side lengths in x- and y-directions, respectively.
In the same manner, ABCD matrix of negative permeability material of length az2 is
formulated with H-wave impedance, ZH2 and phase constant, kz2 as
2 2 2 2 2
_22
22
2
cos( ) sin( )
sin( ) cos( )
z z H z z
ABCD negmu zz
zz
H
k a jZ k a
Mka
j k a
Z
2 2 2
2 2 2 ( ) ( )
z
mn
kab
(5.2b)
55
2
2
2
H
z
Zk
Thus, ABCD matrix of one cell of length, az1+az2 is calculated by cascading ABCD matrices
as
2 2 2 2
2 2 2 2 2 2
1 1 1 1 1
_22
11
22
11
22
1
22
22
cos( ) sin( ) cos( ) sin( )
cos( ) sin( )
2 2 2 2
sin( )
sin( ) sin( )
cos( )
22
cos( ) cos( )
22
z z z z
z H z z H z
z z H z z
ABCD cell zz
zz
zz
zz
zz
H
zz
HH
a a a a
k jZ k k jZ k
k a jZ k a
Maa
ka
kk
j k a
aa
Z
j k j k
ZZ
_ABCD cell
AB
MCD
(5.2c)
where the matrix elements are calculated in (5.2.d).
21
1 1 2 2 1 1 2 2
12
2 2 2 2
11
2 1 1 2 2 1 2 1 2 2 2
1
2
11
1 1 2 2 1
2
2 1 2
1
cos( )cos( ) sin( )sin( )
2
sin( )
cos( )sin( ) cos( )
2
1 sin( )
cos( )sin( ) 2
HH
z z z z z z z z
HH
zz
H z z z z H H H H z z
H
zz
z z z z H
H H H
ZZ
A k a k a k a k a
ZZ
ka
B jZ k a k a j Z Z Z Z k a
Z
ka
C j k a k a j Z Z
Z Z Z
2 2 2
2 2 1 2 2
21
1 1 2 2 1 1 2 2
12
cos( )
1
cos( )cos( ) sin( )sin( )
2
H H H z z
HH
z z z z z z z z
HH
Z Z k a
ZZ
D k a k a k a k a
ZZ
(5.2d)
The dispersion relation can then be calculated from ABCD parameters to determine
complex propagation constant,
eff
as
12
1 2 1 1 2 2 1 1 2 2
21
1
cosh( ( )) cos( )cos( ) sin( )sin( )
2
HH
eff z z z z z z z z z z
HH
ZZ
a a k a k a k a k a
ZZ
(5.2e)
It can be formulated in an alternative form for lossless negative material parameters as
12
1 2 1 1 2 2 1 1 2 2
21
1
cosh( ( )) cosh( )cosh( ) sinh( )sinh( )
2
zeff z z z z z z z z z z
ZZ
a a a a a a
ZZ
(5.3.a)
where Z1,2 are the wave reactances and
1,2z
are the attenuation constants of negative
permittivity and permeability medium, respectively. The wave reactances and attenuation
constants are calculated from (5.2a) and (5.2b) as
56
1
11
2 2 2
11
2
22
2 2 2
22
( ) ( )
( ) ( )
H
H
Z jZ
mn
jab
Z jZ
mn
jab
(5.3b)
2 2 2
1 1 1
2 2 2
2 2 2
( ) ( )
( ) ( )
z
z
mn
ab
mn
ab
. (5.3c)
There are important issues to be discussed about the eigenmode equation. In (5.3b), the
wave impedances of negative permittivity and permeability materials are inductive and
capacitive, respectively. Thus, the wave propagation can be in principle obtained at any
frequency by engineering the material parameters and adjusting the slab thicknesses. In
addition, in (5.3a), the eigenmode equation is dependent on the monotonically increasing
hyperbolic functions. Therefore, due to lack of periodic functions in (5.3a) unlike in the
eigenmode equation of RHM and LHM loaded waveguides, one resonance frequency may
be calculated at a certain phase shift per cell for each mode. The same conclusion is also
valid for the parallel plate waveguides which are loaded with negative permittivity and
permeability materials in the transverse plane [15]. Another important issue is the design
possibility of compact LHM and RHM loaded cavity resonators with the resonance
frequencies dependent on the ratio of each slab thickness [13]. The main reason is the phase
compensation feature of LHM in RHM-based transmission lines and resulting zeroth order
resonance unlike in the conventional resonators [5],[73,74]. Thus, compact transmission
media and resonators can be realised by pairing any slab thickness of LHM with RHM
[13],[75]. The design possibility of subwavelength guided wave structures with lateral
dimensions below diffraction limits is analysed in [15,16].
As a case study, one ε-negative material with Drude type permittivity response is
periodically loaded with another µ-negative material with Lorentzian type permeability
response inside a rectangular waveguide. The eigenfrequencies of H10 mode are calculated.
The slab lengths of negative permittivity and permeability materials in the propagation
direction are az1= 1mm and az2=2mm, respectively. The side lengths of waveguide are
a=10mm and b=5mm. The magnetic resonance, magnetic plasma frequency and loss
parameter are 2.10 GHz, 2.21 GHz and 100 Hz. The electric plasma frequency and loss
parameter are 10 GHz and 100 Hz. The dispersion diagram and Bloch impedance are
analytically calculated. They are shown in Figure 5.2 between the magnetic resonance and
magnetic plasma frequencies along with numerically calculated eigenfrequencies at certain
phase shifts. The analytical and numerical results are in good agreement with maximum
error smaller than 1%.
57
Figure 5.2a. (*) Numerically and (solid )analytically calculated 1D dispersion diagram of periodically arranged
negative permeability and permittivity materials
Figure 5.2b. Resistance (red) and reactance (blue) of periodically arranged negative permeability and permittivity
materials
In Figure 5.2a, the propagation constant is negative between 2.106 GHz and 2.146 GHz
with 40 MHz bandwidth. However, in addition to LH band, RH band is also obtained
between 2.173 GHz and 2.21GHz with 37 MHz bandwidth. In both LH and RH bands, the
transmission loss is negligibly small due to low dissipation factors assumed in the
permeability and permittivity formulations. There is 27 MHz bandgap between LH and RH
bands. It extends from 2.146 GHz to 2.173 GHz. In Figure 5.2b, in the bandgap, Bloch
impedance is low ohmic with capacitive reactive part. However, there is no reactive part as
in the case of resonant circuits at the frequencies of band edges.
58
Thus, the composite material can be modeled as a combination of parallel and series
resonant circuits with the resonance frequencies of 2.146 GHz and 2.173 GHz, respectively.
The equivalent circuit models of negative permittivity and permeability materials are
illustrated in Figure 5.3 to confirm the emergence of RH band at the higher frequencies [4-
6].
(a) (b)
Figure 5.3. Equivalent circuit model of one unit cell of (a) negative permittivity and (b) negative permeability
materials
The periodic loading of negative permittivity and permeability materials can be modeled
with a unit cell shown in Figure 5.4.
Figure 5.4. Equivalent circuit model of one unit cell of periodically loaded negative permittivity and permeability
materials
In Figure 5.4, for the frequencies higher than the series resonance frequency, the impedance
of series branches is inductive. The impedance of parallel branches is capacitive for the
frequencies higher than the parallel resonance frequency. Thus, this unit cell can be modeled
as a series inductor loaded with a shunt capacitor for the frequencies larger than the parallel
and series resonance frequencies. This is the equivalent circuit model of conventional RH
transmission lines. It is the main reason why this material has an additional RH band at the
frequencies higher than LH band. The equivalent circuit model of LH materials can be
similarly derived from same circuit model for the frequencies lower than the parallel and
series resonance frequencies. For this case, the circuit model has an equivalent form of a
series capacitor loaded with a shunt inductor. It is the dual of RH circuit model. One
important remark is therefore the negative permittivity and permeability materials can be
alternatively designed without relying on high lossy resonance phenomenon. In other words,
low loss LHMs can also be realized by periodic loading of conventional microstrip
transmission lines with series capacitors and shunt inductors in planar microwave
technology [4,5]. Many microwave circuits have been implemented by using this
methodology such as compact broadband couplers [4], broadband phase shifters [5],
compact wideband filters [3],[14], compact resonant antennas [21-23],[76].
59
5.3 Numerical Simulations
5.3.1 Resonance Frequency of LHM Periodic Array
In this section, transmission and reflection parameters of one cell thick LHM array is
numerically calculated under plane wave excitation. The unit cell is shown in Figure 5.5
along with the boundary conditions. The geometrical parameters of SRR and wire are the
same as the ones of SRR in Section 3.3 and of wire in Section 4.3.
(a) (b) (c)
Figure 5.5. (a) Top and (b) bottom view of one unit cell of LHM geometry (c)PEC and PMC boundary conditions
In Figure 5.5, PEC is imposed on two y-planes at the edges to excite the wire by y-polarized
plane wave. This boundary condition is important to obtain Drude type permittivity
response. In the same manner, PMC is imposed on two z-planes to excite SRR with z-
directed magnetic field. It is important to obtain Lorentzian type permeability response. In
addition, these boundary conditions result the cells in y- and z-direction to be coupled
electrically and magnetically, respectively. The unit cell can then be excited by y-polarized
x-direction propagating plane wave. The substrate is 0.5mm thick Rogers/RT Duroid 5870.
The relative permittivity and tan(δ) are 2.33 and 0.0012, respectively. The transmission and
reflection parameters of LHM model are numerically calculated with HFSS. They are
shown in Figure 5.6.
Figure 5.6. Reflection (red) and transmission (blue) parameters of one unit cell thick LHM sample
60
The resonance frequency is 9.58GHz. The resonance wavelength is 31.48 mm. It is
approximately 11 times larger than the cell size. The cell size is sufficiently small to
attribute the composite material as a homogeneous material. The resonance frequency
agrees well with the magnetic resonance frequency of SRR array in Section 3.3.1 with only
30 MHz frequency shift. As explained in Section 3.3 and 4.3, there is low amount of power
transport through SRR and wire array due to negative permeability and permittivity,
respectively. However, the wave can propagate through LHM slab at the overlapping
frequency band of negative permeability and permittivity. This is because the capacitive
impedance of SRR array is compensated with the inductive impedance of wire array at
LHM band. This issue is confirmed from the extracted material parameters in the next
section. The magnetic field, electric field and surface current distribution at the resonance
frequency are shown in Figure 5.7.
Figure 5.7a Surface current distribution of one unit cell at the resonance frequency
Figure 5.7b Magnitude of electric field distribution of one unit cell at the resonance frequency
61
Figure 5.7c Magnitude of magnetic field distribution of one unit cell at the resonance frequency
The coupling between SRR and wire is important to determine the optimum wire position
for maximum LH transmission. The transmission and reflection parameters for different
wire positions with respect to SRR symmetry axis are numerically calculated. They are
shown in Figure 5.8. In Figure 5.8, the highest transmission is obtained for the wire position
at the symmetry axis of SRR. The resonance frequency is quite near to the magnetic
resonance frequency of SRR array with 30MHz frequency shift. The first reason of this
frequency overlap is ineffective magnetic coupling between SRR and wire because of no net
magnetic flux penetrating through SRR at this position. The second reason is the negligible
electrical coupling between SRR and wire since there is no y-directed electric field along
SRR symmetry axis. Therefore, maximum transmission is quite near to the magnetic
resonance frequency. In other words, the wire and SRR are excited almost independently
from each other with negligible field coupling inbetween.
Figure 5.8a Reflection parameter of one unit cell thick LHM medium for different wire positions
62
Figure 5.8b Transmission parameter of one unit cell thick LHM medium for different wire positions
In Figure 5.8a, LH resonance is not so noticeable for increasing wire displacement from
SRR symmetry axis. The electrical coupling between SRR and wire at increasing distances
reduces the magnetic flux penetrating through SRR. This degrades the demagnetization
effect of SRR in negative permeability band with the result of low LH transmission. The
effect of electric and magnetic coupling among the cells is also investigated in Chapter 3
and Chapter 4. The lower electric coupling along y-direction increases the resonance
frequency slightly as in Section 3.3.1. The lower magnetic coupling along z-direction
increases the magnetic resonance frequency and decreases the electric resonance frequency
as in Section 3.3.1 and Section 4.3.1.
5.3.2 Dispersion Relation of LHM Periodic Array
In this section, 1D Brillouin diagram of LHM cell is numerically studied. In the numerical
model, eigenfrequencies of one unit cell with periodic boundary conditions in the
propagation direction is numerically calculated for different phase shifts. PEC and PMC on
the transversal planes are imposed as in the case of port mode calculation. The dispersion
diagram of two lowest bands is shown in Figure 5.9. Because the phase velocity and group
velocity are oppositely directed in the first band, this band is LH band. LH band coincides
quite well with the stop band of only SRR array. LH resonance at 9.53 GHz in Figure 5.6 is
the result of pure real Bloch impedance of LHM array, which matches with the line
impedance of PEC-PMC medium. These periodic cells can thus be regarded as LH and RH
transmission medium in the first and second band, respectively. The lower band is from 9.2
GHz to 11.16 GHz, whereas the higher band is from 14.42 GHz to 22.43 GHz. There is no
transmission between the higher end of first band and lower end of second band. The
bandgap is 3.26 GHz.
63
Figure 5.9. Dispersion diagram of LHM with periodic boundary conditions of different phase shifts in propagation
direction
The effective refractive indices are calculated from the dispersion diagram as
eff
n2
o
c
pf
(5.4)
where
is phase shift per unit cell,
p
is cell period,
o
c
is speed of light in free space and
f
is eigenfrequency. They are shown in Figure 5.10 and are negative in LH band and
positive in RH band. They are decreasing in magnitude with increasing frequency for LH
band. This results the composite LH medium to be more homogeneous for the higher
frequencies because of smaller phase shift per unit cell unlike in RH medium.
(a) (b)
Figure 5.10. Effective refractive index of one unit cell thick LHM medium in (a) LH band and (b) RH band
64
In addition to the refractive indices, Q factors are numerically calculated at each frequency
in LH and RH band. They are shown in Figure 5.11. In Figure 5.11a, the larger metallic
losses result Q factor in LH band to decrease with increasing frequency. Q factor in RH
band is larger than Q factor in LH band upto 18.8 GHz. However, for the larger frequencies
than 18.8 GHz, the metallic losses are more effective than the total stored energies inside the
cell. Therefore, Q factor tends to decrease rapidly.
(a) )
(a) (b)
Figure 5.11. Q factor of one unit cell thick LHM array in (a) LH and (b) RH band
5.3.3 Effective Parameters of LHM Periodic Array
In this section, the effective permeability and permittivity are retrieved with the procedure
of Section 3.3.3. They are shown in Figure 5.12 along with Bloch impedance, complex
propagation constant and effective refractive index.
Figure 5.12a. Real (red) and imaginary (blue) part of effective relative permeability of LHM periodic array
65
Figure 5.12b. Real (red) and imaginary (blue) part of effective relative permittivity of LHM periodic array
In Figure 5.12a, LHM array exhibits resonant permeability response. The real part of
effective permeability is negative between the magnetic resonance, 9.38GHz and plasma
frequency, 11.4 GHz. The imaginary part is negative. The highest magnetic loss is obtained
near the magnetic resonance frequency. The magnetic resonance frequency of LHM array is
quite near the magnetic resonance frequency of only SRR array with 160 MHz frequency
shift. This 1.7 % frequency shift points out the negligible magnetic coupling between the
wire and SRR. Therefore, the material parameters of LHM array can be controlled
independently due to the electromagnetic decoupling of two metallic inclusions.
On the other hand, relative permittivity has Drude type response between 8.4 GHz and 9.28
GHz as expected. However, the imaginary part is positive between 8.47 GHz and 9.28 GHz.
This is a nonphysical artifact for the passive materials with the possible reasons as explained
in Section 3.3.3. This phenomenon has also been observed and pointed out in [57],[77].
Figure 5.12c. Attenuation (blue) and phase (red) constant of LHM periodic array
66
Figure 5.12d. Effective refractive index of LHM periodic array
Figure 5.12e. Real (red) and imaginary (blue) part of wave impedance of LHM periodic array
In Figure 5.12c, the attenuation constant is positive at the frequencies lower than 9.3 GHz
and higher than 11.4 GHz. The retrieved LH cutoff frequencies are higher than LH cutoff
frequencies in the dispersion diagram in Figure 5.9. This is possibly due to the neglection of
cell intercoupling in the retrieval method whereas cell intercoupling is not neglected in the
eigenmode calculation. The phase constant is real and negative as expected from 1D
dispersion relation. The refractive index is shown in Figure 5.12d. It has quite similar form
with the one calculated from the dispersion diagram in Figure 5.10.
In Figure 5.12e, the wave impedance is inductive for the frequencies, which are not in LH
band. Because SRR is not excited as a magnetic dipole in this bandgap, the wire loading
yields an inductive wave impedance. The wave impedance is real between 9.3 GHz and 11.4
GHz, which are the lowest and highest LH frequencies, respectively. LH material has high
impedance at 9.3 GHz and low impedance at 11.4 GHz, which is typical for LH materials.
67
As a result, analytical and numerical calculations prove effective negative permeability and
permittivity of periodic LHM arrays. In the next section, an LHM unit cell is proposed for
the design of more homogeneous negative index materials.
5.4 Wire loaded Spiral Resonator as LHM
In this section, an alternative LHM cell is proposed [76]. The design principle is based on
the wire loading of spiral resonator (SR) on both sides of the unit cell. Because the
resonance frequency of SR is one half of SRR resonance frequency, LHM cell design with
the spiral resonator is an alternative miniaturization strategy to the fractal based designs in
Chapter 3 and Chapter 4. The cell geometry is introduced first to explain how to obtain LH
behavior. The reflection and transmission parameters of one cell thick LHM array are
numerically calculated with 1D dispersion diagram. The effective parameters are
analytically retrieved from the numerical data as a last step to illustrate the negative material
parameters.
5.4.1 Structural Description
The negative material parameters are synthesized by the simultaneous excitation of electric
and magnetic dipoles in LHM unit cell. The nonresonant electric and resonant magnetic
dipoles are obtained by the excitation of wires and SRRs, respectively. As an alternative
model to the conventional designs, more homogeneous LHM cell can be designed by the
same methodology as shown in Figure 5.13. The geometrical parameters are tabulated in
Table 5.1.
(a) (b) (c)
Figure 5.13. (a) Front and (b)back side of one LHM unit cell with indicated geometrical parameters and boundary
conditions
The substrate is FR4-Epoxy with the relative permittivity of 4.4 and loss tangent of 0.02. In
Figure 5.13a, the inner ring of SR is connected to the outer one with a 0.2 mm wide line.
The line width is selected so as not to cause unwanted overlapping of the corner points of
the inner and outer rings because of poor resolution in the fabrication process. SR is used as
an alternative magnetic material to SRR because SR resonance frequency is one half of SRR
resonance frequency [40]. The wire strips and SR are directly connected with each other on
both sides of the substrate. This design method enhances electromagnetic coupling between
wire strips and SRs on each substrate side by increasing the electrical length effectively in
small cell size.
68
Table 5.1. Geometrical parameters of LHM unit cell (mm)
Length of outer SR ring (Lo)
2.6
Length of inner SR ring (Li)
1.5
Width of SR and wire strip
0.3
Gap between inner and outer ring of SR
0.25
Split length of inner/outer ring of SR (Ws)
0.4
Unit cell size (x)
3
Unit cell size (y)
3.5
Unit cell size (z)
2
Wire strip – SR coupling width (Wc)
0.4
Wire strip – SR coupling length (Lc)
1.7
Distance from substrate edge (x)
0.2
Distance from substrate edge (y)
0.1
The geometrical parameters of the front and back side cells are same except 0.6 mm shorter
wire length on the front side. Different wire lengths decrease the resonance frequency and
increase the bandwidth, as discussed in Section 5.4.3. In order to prove this issue, in
addition to the proposed model, two different alternative designs are also investigated to
determine the optimum LHM geometry. The first and second alternative models are LHM
cell designs with the identical resonator geometry in Figure 5.13a and Figure 5.13b on both
front and back sides, respectively. The opposite direction of group and phase velocity in
LHMs is verified for each of three models from the eigenmode calculations. However, the
validity of proposed LHM model in Figure 5.13 is confirmed by retrieving the effective
material parameters from S parameters in addition to eigenmode calculations.
5.4.2 Numerical Simulations
5.4.2.1 Transmission and Reflection Parameters of Wire loaded Spiral Resonator
In order to determine the resonance frequency, scattering parameters of one cell thick LHM
array are numerically calculated under plane wave excitation. PEC and PMC are imposed at
two x- and z-planes, respectively. These boundary conditions result the cell to be excited by
x-polarized y-direction propagating plane wave. The boundary conditions are shown along
with the port settings in Figure 5.13. The reflection and transmission parameters are shown
in Figure 5.14.
Figure 5.14. Reflection (red) and transmission (blue) parameters of one unit cell thick wire loaded spiral resonator
69
The resonance frequency is 1.96GHz. The resonance wavelength is 153.06 mm. It is
approximately 43.73 times larger than the cell size. The transmission and reflection at the
resonance frequency are approximately -20 dB and -9.5dB, which are worse than the
reflection and transmission parameters of LHM model in the previous section. This is the
natural consequence of the resonance phenomenon. In this cell model, the high current and
field concentration at the resonance frequency increase the metallic and dielectric losses
with the result of low transmission. The transmission and reflection parameters can be
enhanced by the cell design with one sided metallization or low loss substrate as in Section
5.3.1. As a result, the wire loading of spiral resonator on both substrate sides reduces the
resonance frequency. However, it increases the transmission loss and decreases Q at the
resonance frequency. The numerical results point out this trade off between lower resonance
frequency and higher losses. As a next step, 1D dispersion relation is studied to confirm LH
behavior.
5.4.2.2 1D Brillouin Diagram of Wire loaded Spiral Resonator
1D Brillouin diagram is studied in this section. In order to obtain the dispersion relation,
LHM cells are excited with the magnetic field perpendicular to the SR plane (z-direction),
and the electric field in the direction of wires(x-direction) as in Figure 5.13b. Unlike in the
port mode calculation, periodic boundary conditions are imposed in the propagation
direction with different phase shifts. The dispersion diagram is shown in Figure 5.15.
Oppositely directed phase and group velocities are observed in the first two bands between
2.15–2.56 GHz with 410 MHz bandwidth and 5.10-5.48GHz with 380MHz bandwidth.
Figure 5.15 Dispersion diagram of LHM with periodic boundary conditions of different phase shifts in propagation
direction
The resonance frequencies in the first LH band can be estimated from the total length of
spiral resonator and wire strip. In the first band, the open-circuited end of spiral resonator is
transformed to the short-circuited end of wire strip as in λ/4 transmission lines.
70
Similarly, the resonance frequencies in the second LH band are resulting from λ/2 resonance
between spiral resonator and wire ends. The current direction at λ/2 resonance is same as the
current direction at λ/4 resonance. This is the main reason why LH feature is obtained in
first two bands and not in the third band. The eigenfrequencies of alternative LH geometries
mentioned in Section 5.4.1 are also numerically calculated. LH band of first alternative
model is between 3.21-3.25GHz with 40MHz bandwidth. LH band of the second model is
between 2.72–3.03 GHz with 310MHz bandwidth. The second model is better than the first
model. However, the proposed model has lower eigenfrequencies and broader bandwidth
than those of the alternative designs. These additional calculations point out why the
proposed cell is selected in LHM design instead of two alternative forms.
5.4.3 Effective Parameters
Even though the left-handedness is proved with the opposite phase and group velocities in
Figure 5.15, the values and sign of the effective parameters have to be determined.
Therefore, effective material parameters are retrieved from the scattering parameters of one
cell thick LHM array. The effective relative permittivity, permeability, complex propagation
constant and wave impedance are shown in Figure 5.16. There is one important remark on
the exploited extraction method. Even though there are alternative retrieval methods, one
reason why to exploit this method is the cell asymmetry. Because the effective parameters
are retrieved from the transmission and reflection data, there is one inherent problem in the
parameter extraction of asymmetric cells [56-59]. In these cells, the reflection parameter at
the first port is different from the one at the second port. In order to solve this problem, the
geometrical mean of two reflection parameters is used as an effective reflection parameter in
the conventional retrieval methods [57]. On the other hand, this is not necessary in the
proposed retrieval procedure. This is an important advantage.
Figure 5.16a. Real (red) and imaginary (blue) part of effective relative permeability of wire loaded SR
71
Figure 5.16b. Real (red) and imaginary (blue) part of effective relative permittivity of wire loaded SR
Figure 5.16c. Phase (red) and attenuation (blue) constant of wire loaded SR
72
Figure 5.16d. Real (red) and imaginary (blue) part of wave impedance of wire loaded SR
There are important issues to be discussed about the frequency dependence of retrieved
parameters. The retrieval procedure leads in general to the satisfying results—an expected
Lorentzian type magnetic response for the effective permeability with the magnetic
resonance frequency at 1.95GHz. The negative permeability region extends from 1.93GHz
to 2.55GHz. Drude type permittivity response is identified upto the excitation of spiral
resonator through the wire. One important remark is on the possible reason of unphysical
artifacts such as positive imaginary part of permittivity and permeability at certain
frequencies. One reason is that magnetic field induced polarization and electric field
induced magnetization effects are not taken into account in the analytic formulation of
extraction method. However, these effects are important to characterize the material. For
instance, the wire can be excited with z-directed magnetic field through the spiral resonator
in addition to x-directed electric field. Thus, x-directed electric field excites the spiral
resonator with the resonant field distribution through the wire. In addition, the splits in the
inner and outer rings and the gap region between the inner and outer rings are the other
electric coupling locations for the magnetic response. In other words, both magnetic and
electric fields can excite each inclusion simultaneously due to the direct connection of spiral
resonator with the wire. Therefore, negative permittivity and permeability can be
conveniently obtained simultaneously in a certain band in this design. The negative phase
constant and negative refractive index, which is deduced from (5.4) because of negative
phase shift along the propagation direction, are clearly identified in Figure 5.16c. The
refractive index is therefore negative between 2.24-2.56GHz with the bandwidth of
320MHz. This LH band corresponds quite well with LH band in 1D dispersion diagram in
Figure 5.15. In Figure 5.16d, the resonance in Bloch impedance is identified as in the case
of wire loaded SRR in the previous section. The resonance frequency is 1.96GHz. It
coincides well with the magnetic resonance frequency as expected. Bloch impedance is
capacitive between the resonance frequency and higher edge of LH band, 2.56GHz.
However, it is less capacitive as expected in LH band to allow plane wave transmission
inside LHM array.
73
As a result, numerical calculations confirm negative refractive index of wire loaded spiral
resonator. Because the unit cell size is approximately 1/43 of the wavelength at 2 GHz, this
model can be used to design more homogeneous LH materials.
5.5 Chapter Conclusion
In this chapter, LHM design is explained. The basic design principle is based on the periodic
arrangement of negative permittivity and permeability cells in the same host medium.
Therefore, the eigenmode equation of periodically arranged negative permittivity and
permeability materials in a rectangular waveguide is analytically calculated. The analytical
calculations are confirmed by the numerical calculations with maximum error smaller than
1%. Drude type permittivity and Lorentzian type permeability responses with low electric
and magnetic losses are assumed. The LH and RH bands are obtained. The possible reason
of emerging RH band at the higher frequency is explained with the equivalent circuit
models of negative permittivity and permeability materials. They introduce an alternative
realization method of LHMs in planar microwave technology. The possible engineering
applications of LHMs with both realization strategies are additionally referenced.
An LHM cell model is numerically analyzed. It is composed of same SRR and thin wire
models with the geometrical dimensions indicated in Chapter 3 and Chapter 4. The main
reason in selecting the same cell geometries and dimensions is to indicate that LH
transmission band is the overlapping band of negative permittivity and permeability. The
transmission and reflection parameters are calculated. The resonance frequency is
determined. The surface current, electric and magnetic field distributions are presented. The
LH band is the resulting effect of inductive loading of capacitive impedance of SRR array
with the wire array. 1D Brillouin diagram is numerically calculated. LH passband is
determined. The effective material parameters, Bloch impedance and complex propagation
constant are additionally retrieved. At the end of the chapter, an electrically small inclusion
based on the wire loading of spiral resonator is proposed as an LHM unit cell. It is
numerically investigated in terms of effective permeability, permittivity, Bloch impedance
and complex propagation constant. The numerical calculations confirm the effectiveness of
LH cell in the design of more homogeneous composite materials with negative material
parameters.
74
6. Metamaterial-based Antenna Design
6.1 Introduction
The fundamental theories of negative permittivity, negative permeability and left-handed
metamaterials are illustrated in Chapter 3, Chapter 4 and Chapter 5. As one potential
application of artificial materials in microwave communication systems, metamaterial-based
antenna design is explained in this chapter. Basic concepts in electrically small antennas are
introduced first. The fundamental performance limitations are explained. The minimum Q
and maximum gain are the main parameters to be optimized. Therefore, these parameters
are investigated to point out the effect of electromagnetic material parameters and physical
dimensions on the antenna performance. The concept of artificial ground plane is
additionally included as an alternative approach for directive and high gain antennas. As
meta-antenna design examples, the design of broadband LHM-loaded dipole antenna is
explained first. How to exploit the unit cells in the antenna design is figured out. As a
second design, an alternative higher profile meta-antenna is designed. The same cell
geometry in the first design is used to design a slot antenna with higher gain.
6.2 Fundamental Limits of Small Antennas
Electrically small antenna design has long been the current trend as one significant and
interesting research topic in microwave community. The large dimensions of conventional
λ/2 or λ/4 antennas in short-, or long-wave radio communication has further triggered to
miniaturize the antenna dimensions. Especially, nowadays the demand on multifunctional
complex systems imposes compact mobile terminals. The challenging issue to be overcome
in the design is not only to miniaturize their physical dimensions but also not to degrade the
radiation performance. The small antennas have an additional advantage especially in
wireless/contactless measurement systems. They allow more reliable measurements to be
done due to insignificant influence on the measured field. Hence, to design compact, high
efficient/gain, broadband antennas is the main design target in any high performance
wireless system.
An electrically small antenna is commonly defined as an antenna occupying a small fraction
of one radiansphere, which circumscribes the maximum antenna dimension [78,79]. The
radiansphere is by definition the spherical volume having the radius of λ/ 2π. This defines
logically the maximum dimension to be smaller than λ/π (0.318λ) where λ is the free space
wavelength. A conventional upper limit for the greatest dimension is less than one-quarter
wavelength (including any image resulting from the ground plane). This results the
electrically small antenna to have maximum ka value of 0.785. k is the free space wave
number and a is the radius of the imaginary enclosing sphere. The main reason to have the
radiansphere as a logical reference is that this radius is also the distance at which the
reactive part of wave impedance is equal to the resistive part for infinitesimal electric and
magnetic dipoles [80,81]. Because the radiansphere volume is larger than the near field
volume for electrically small antennas, they can be modelled as a capacitive electric or an
inductive magnetic dipole.
75
At larger distances than the radian distance, the real part of complex power dominates the
imaginary part. This leads the radiating field to have more plane wave form like in the
transition from the Fresnel to the Fraunhofer regime in electrically large antennas.
The operation bandwidth of any antenna for each mode can be calculated from the
impedance formulation with reference to the modal equivalent circuits on the sphere of
radius, a [82]. The modal impedance has the form of Z(ω) = R(ω) + j X(ω) at the reference
terminal for the excited mode current, In . It can be derived from Poynting formulation in
the form of
22
* * 2
1 1 1 1 1
. 2 ( )
2 4 4 2 2 n
S V V
ExH dA j H E dV E J dV I Z
(6.1)
The real part consists of both the radiation and losses. The imaginary part is proportional to
the difference of time averaged stored magnetic and electric energy outside the spherical
volume.
The impedance bandwidth is often characterized by Q factor at the resonance frequency.
The antenna Q is inversely proportional to the fractional bandwidth. It has a theoretical
lower bound. Therefore, no electrically small antenna can exhibit a Q value less than its
theoretical lower bound or bandwidth larger than a specific upper bound. The antenna Q is
generally defined as in (6.2). We and Wm are the time average stored electric and magnetic
energy. Ploss is the sum of total radiated and metallic loss power at the resonance frequency,
ω0 [82,83].
0
loss
0
loss
W
2 ,W W
P
W
2 ,W W
P
eem
mme
Q
(6.2)
This formulation is actually the same formulation, which can be derived from the Poynting
formulation in (6.1). It is the ratio of imaginary and real part of complex power for high Q
antennas. In the lower bound calculation of Q, an arbitrary current distribution is assumed
on a wire antenna. The near field distribution is then calculated in terms of a series
expansion of all higher order spherical modes excited by In. The modal Qn can then be
calculated in the terms of modal tangential field quantities on the spherical surface of radius
a as [82],
2
n
n n n n
n
2
''
n n n n n n
2ωW 1d
Qρh (ρ) ρ X (ρ) X (ρ)
P 2 dρ
Xρj (ρ) ρj (ρ) ρn (ρ) ρn (ρ) ρh (ρ)
(6.3)
where hn is the nth order spherical Hankel function of second kind. It is defined as hn=jn- j.nn.
jn and nn are the spherical Bessel functions of the first and second kind.
76
ρ is ka. The above relation is plotted in Figure 6.1. In Figure 6.1, for a specific antenna size
and excited mode, with increasing operational frequency the stored energy decreases and
radiation resistance increases. This is consistent with the complex power calculations on
elementary electric dipole and can be derived [80].
Figure 6.1. Modal Qn of vertically polarized omnidirectional antenna [82]
On the other hand, higher order modes result the antenna to have higher Q. However, to
excite higher order modes is in practice quite challenging. It is because of the difficulty in
matching high reactive impedance. Therefore, the main conclusion in minimum Q
calculation is that Q can only be minimized under one mode excitation without multimode
operation. Because Q of lowest order mode is minimum, the antenna in the form of an
infinitesimally small dipole has potentially the broadest bandwidth [82]. Therefore, the
optimum method to obtain broad bandwidth is to operate the antenna with the lowest of all
possible modes. The theoretical lower bound for small directional and omnidirectional
antennas is expressed as in [83]
min
small 3
11
2
Qka
ka
(6.4)
Q can be alternatively calculated from the input impedance of a tuned antenna exhibiting a
single resonance frequency as [84]
2
20
''
0
0 0 0
00
(ω)
ω
( ) (ω ) (ω )
2(ω ) ω
X
Q R X
R
(6.5)
77
where
'0
(ω)R
and
'0
(ω)X
are the frequency derivatives of real and imaginary parts of
feed point impedance at the resonance frequency,
0
ω
. As implied from (6.5), one method to
minimize Q can be to increase the radiation resistance by using self-resonant radiators. This
has an additional advantage of no matching network requirement. This reduces not only the
antenna size but also the matching losses. It is an important issue. The loss resistance within
the matching network often exceeds the radiation resistance in electrically small antennas.
This degrades the overall efficiency [85]. Thus, one strategy could be to design self
resonating structures to increase the radiation resistance and enhance the operation
bandwidth.
In gain optimization, it is well known that there is no mathematical limit in the maximum
gain [82],[86]. However, a small antenna with extremely high gain produces high field
intensity in the vicinity of the antenna. The antenna gain is therefore limited with an upper
bound in practice. The maximum gain obtainable is calculated as Gmax= N(N+2) with the
maximum excited mode order N [86]. As N increases, the maximum gain increases with a
disadvantage of practical difficulty in the design of matching network. In other words, the
excitation of higher order modes to increase the radiation resistance results total stored
energy to increase in the antenna near field as in Figure 6.1. Due to this trade-off between
gain and Q, there is one additional parameter to be maximized for the optimum antenna
design. This parameter is the ratio of gain (G) to Q. The maximum G/Q ratio for the small
antennas is expressed as [83],
3
2
3
2
6
max 21
3
max 21
small
dir
small
omni
ka
G
Qka
ka
G
Qka
(6.6)
Antenna gain can be alternatively expressed with an additional parameter [81],[87]. It is
called radiation power factor (PF). It is an alternative description relating the radiating
energy with the stored reactive energy inside the antenna near field. It is a valid parameter
for either kind of electrically small electric or magnetic antennas. It is equal to 1/Q. It can be
expressed in terms of input impedance as the ratio of the input resistance to the reactance at
the resonance frequency. For any shape and type of electrically small antenna, the radiation
PF at one frequency is proportional to its physical volume. Moreover, the radiation PFs of
electric and magnetic dipoles are approximately same for similar physical volume.
The main difference of radiation PF from the radiation parameters introduced previously is
the definition of the calculation volume. The near field distribution of any small electric or
magnetic antenna can be assumed to be confined in an effective spherical volume of V’ with
radius a’. The radiation PF can then be calculated in terms of V' and volume of radiansphere,
Vs as [81],[87]
3
2 V' 2 2πa'
PF = =
9 Vs 9 λ
(6.7)
78
Thus, one method to enhance the radiation PF is to increase the effective volume of electric
and magnetic dipoles. This can be achieved by decreasing the substrate permittivity or
increasing the core permeability, respectively. The logical approach in this method is to
eliminate or minimize the avoidable stored energy inside the core or substrate by leaving
unavoidable amount of stored energy outside the core or substrate. The enhancement of
radiation resistance with low permittivity and high permeability is also consistent with the
radiation resistance formulations of Hertzian electric and magnetic dipoles.
In addition, the design of high permeability materials has an additional potential application.
They can be used in the realization of artificial ground planes to increase the gain and
directivity. [6]. In general, the performance of low profile wire antennas is degraded by their
ground plane backings. It is because of out-off phase image current when the antenna is
horizontally oriented in close proximity to the ground. If the separation distance between the
antenna and ground is λ/4, the ground plane reflects the antenna radiation in phase with an
approximately 3 dB increase in the antenna gain. The radiation degradation occurs if the
ground plane-antenna separation distance is smaller than λ/4. It can not provide gain
increase. The reflected back-radiation interferes destructively with the antenna forward-
radiation. Therefore, the antenna can be attributed to be partially “short circuited”. A second
problem especially in the microstrip antenna design is the generation of surface waves. The
field distribution on the feeding line and antenna near field excite the propagating surface
wave modes of the ground-substrate-air system. This results into the radiation efficiency
degradation. The guided waves can deteriorate the antenna radiation pattern by reflecting
from and diffracting at the substrate edges and other metallic parts. To solve these problems,
a Perfect Magnetic Conductor (PMC) would be an ideal solution for low profile antennas.
The antenna radiation is reflected from PMC without a phase-shift due to high surface
impedance as in negative permeability materials at the resonance frequency. They can be
designed by introducing artificial magnetic materials on the substrate surface. These
surfaces are called alternatively as Electromagnetic Bandgap (EBG) surfaces or Artificial
Magnetic Conductors (AMC) [6],[88]. There are two regions in such structures. The first
one is caused as a result of array resonance and array periodicity. This is the region where
the surface waves are suppressed and reflected within an electromagnetic bandgap. The
second region is obtained due to the cavity resonance formed by the ground plane and
structures on the substrate. This is the region where the radiating waves are reflected with
no phase shift as in the case of high impedance surfaces. The most commonly known EBG
surface is the mushroom EBG [89]. It consists of an array of metal patches. Each of them is
connected with a via to the ground through the substrate. The capacitively coupled metal
patches and inductive vias create a grid of LC resonators. A planar EBG can also be
designed, which does not have any vias and acts as a periodic frequency selective surface
(FSS). A widely used EBG surface is the Jerusalem-cross [90,91]. It consists of metal pads
connected with narrow lines to create an LC network. On the other hand, the negative
permeability metamaterials have also been frequently used in AMC design [92].
After the explanation of small antenna fundamentals, two metamaterial-based compact
antenna designs are proposed and numerically investigated in the next sections.
79
6.3 LHM-based Broadband Dipole Antenna
In this section, the design of a metamaterial-based microstrip antenna is explained [76]. The
antenna is composed of six LHM unit cells, which are directly connected to the dipole.
These cells have the same geometry in Section 5.4. The input impedance is matched to 50Ω
with the stepped impedance transformer and rectangular slot in the truncated ground. The
configuration and operation principle of the dipole antenna are explained in Section 6.3.1.
The simulated and measured return losses, radiation pattern and numerically computed
radiation parameters are illustrated in Section 6.3.2.
6.3.1 Antenna Design
The antenna model is shown in Figure 6.2a and Figure 6.2b. In the first step, how to
interconnect LHM cells has to be figured out to design broadband load for the dipole. The
front and back side resonators in Figure 5.13 are connected symmetrically with the adjacent
cells in the x-direction. They are then connected in a periodic form with the ones in the y-
direction. This interconnection form is implied from the boundary conditions in the
eigenmode calculation in Section 5.4. Six cells are used and arranged in a 2x3 array. The
front sides are directly connected to the dipole in order to increase the field coupling. This
results additionally the impedance of LH load to be transformed by the dipole. The ground
plane is a truncated ground plane with a rectangular slot. This results the dipole to radiate
more effectively due to lack of out-off phase image current. In addition, tuning the slot size
makes the antenna to be better impedance matched. The rectangular slot can be modeled by
a shunt element as a parallel LC resonator in series with the capacitance [93]. The width of
the slot is appreciably smaller than half of the wavelength in the substrate. The fabricated
prototype is shown in Figure 6.2c. The geometrical parameters are tabulated in Table 6.1.
(a)
(b)
80
(c)
Figure 6.2. (a) Top (b) bottom geometry and (c) prototype of LHM loaded dipole antenna
Table 6.1 Geometrical Parameters of LHM loaded Dipole Antenna (mm)
Length of substrate
55
Width of substrate
14
Length of feeding line (Lfeed)
12
Width of feeding line
1
Length of dipole line (Ldipole)
30
Width of dipole line
0.4
Dipole-LHM coupling length (LLHM)
12
Length of ground plane (Lground)
24
LHM-ground separation length (L sep)
19.3
Slot width
5
Slot length
10
The overall size of the dipole antenna is 55x14 mm. The size of the main radiating section is
30x14 mm. The operation principle is based on the radiation of the dipole and excitation of
LHM cells with the dipole field. The cell excitation in their modal currents couples the
electric and magnetic dipoles in the same way as in the eigenmode calculation in Section
5.4. These dipoles are also the radiation sources in addition to the exciting dipole antenna.
The magnetic and electric dipole moments are expressed by the surface current density,
e
J
as [94].
0
m
e
p = ( ') '
2
p = ( ') '
e
S
e
S
rxJ r ds
jJ r r ds
(6.8)
81
where
r
is the displacement vector directed from the surface current element to the
observational point.
'r
is the current element position.
'ds
is the current carrying surface
element.
Therefore, the electric and magnetic dipoles are in principle simultaneously excited at each
cell. However, the magnetic dipoles are more effectively excited than the electric dipoles.
The first reason is that magnetic dipole fields do not cancel in the far field because of the
electric coupling among the cells. In addition, the current on the back side wire is partially
oppositely directed to the current on the front side wire. This is the second reason why the
electric dipoles can not be excited as effectively as the magnetic dipoles. As a last reason,
the surface current on the back side cell spirals in the same direction as the surface current
on the front side cell. In that respect, the front and back side cells are mainly magnetically
coupled. However, as stated in the design principle, the antenna radiates mainly in the
dipole mode. This is the reason why it is called as an LHM-loaded dipole antenna.
6.3.2 Experimental and Numerical Results
The return loss is measured with the vector network analyzer HP 8722C. It is shown in
Figure 6.3 together with the numerical result. The operation band extends from
approximately 1.3 GHz to 2.5 GHz with the center frequency of 1.9 GHz. The impedance
bandwidth is 63.16 % of the center frequency. Three cell resonances can be clearly
identified from the numerical calculation. However, there are resonance frequency shifts
between the measured and simulated return losses. The uncertainty in the substrate
permittivity and loss tangent is one reason of these frequency shifts. The deviations of the
actual values from the model values result the input impedance not to be 50 Ohm. This
degrades the impedance matching at the antenna feeding point. The low frequency ripples
can be attributed to the impedance mismatch and inaccurate modeling of the coax-
microstrip line transition. In summary, the measured and simulated return losses are in good
agreement.
Figure 6.3. Measured (blue) and simulated (red) reflection coefficients of LHM-loaded dipole antenna
82
There are nevertheless some other issues to be discussed about the measured and simulated
results. First of all, in the numerical result, there are lower resonance frequencies than the
LH resonance frequencies in Section 5.4. These lower frequencies are due to the direct
coupling between the dipole and LHM cells. They are not emerging from LH resonances.
The LH resonances can be clearly identified from the foot-point impedance shown in Figure
6.6. In order to prove this reasoning, the surface current distributions at 1.72 GHz and 2.07
GHz are shown in Figure 6.4 and Figure 6.5.
(a)
(b)
Figure 6.4. Surface current distribution on the (a) top and (b) bottom side of LHM-loaded dipole antenna at 1.72GHz
(a)
83
(b)
Figure 6.5. Surface current distribution on the (a) top and (b) bottom side of LHM-loaded dipole antenna at 2.07GHz
Figure 6.6. Real (blue) and imaginary (red) parts of the foot-point impedance of LHM-loaded dipole antenna
In Figure 6.4, the dipole line is excited more effectively than LH cells at 1.72GHz. It is an
expected result because the lowest LH frequency is higher than 1.72GHz. In Figure 6.5, LH
load is effectively excited with the dipole at 2.07GHz. In other words, the resonance
frequency, 1.72 GHz is the result of impedance transformation of LH load through the
dipole with the stepped impedance and rectangular slot in the ground. It is not an LH
resonance. However, LH load is important for broadband operation. The simultaneous
excitation of different LHM and dipole sections in the operation band is the reason why the
antenna has broad bandwidth. The LH resonances at the lower frequencies are closer than
the ones at the higher frequencies. This unique property is one reason of broadband feature
at the lower frequencies. This is not the case in RHM-based antennas [5]. The same
reasoning can also be confirmed from the dispersion diagram in Figure 5.15. Therefore, the
coupled resonance feature of LHM cells results the antenna input impedance to have
smooth frequency dependence. The third important issue is the radiation of LHM cells. It
could be verified not only from the current distribution and return loss but also from E- and
H-plane radiation patterns. Therefore, the normalized radiation patterns in y-z and x-z
planes at 1.72 GHz and 2.07 GHz are shown in Figure 6.7. They are mainly dipole-like co-
polarization radiation patterns. This is the reason why to call the microstrip antenna as an
LHM-loaded dipole. The radiation of LHM cells is verified from the radiation pattern at
2.07 GHz.
84
The radiation patterns of cross-polarization in y-z and x-z planes are similar to the co-
polarization radiation patterns of the magnetic and electric dipoles with shifted phase
centers, respectively. The radiation intensity of cross-polarization in y-z plane at 2.07GHz is
4.77 dB more than that at 1.72 GHz due to the excitation of electric and magnetic dipoles.
The gain is unfortunately small. The maximum gain and directivity are -1 dBi and 3 dB. The
radiation efficiency is 40% at 2.5 GHz. The directivity, gain and radiation efficiency in
whole operation band are shown in Figure 6.8. In addition, the gain is higher than that of the
different kinds of miniaturized and narrow band antennas in the literature [95-97].
(a) (b)
(a) (b)
(a) (b)
(c) (d)
Figure 6.7. Normalized cross-polarization (blue ) and co-polarization (red) radiation patterns at 1.72 GHz in (a) y-z and
(c) x-z plane, and at 2.07 GHz in (b) y-z and (d) x-z plane
85
(a)
(b)
(c)
Figure 6.8. Directivity, radiation efficiency and gain of LHM-loaded dipole antenna
86
On the other hand, there are alternative design techniques instead of loading a narrowband
dipole antenna with LHM cells for broadband operation. Some of these techniques are to
increase the substrate thickness, use different shaped slots or radiating patches [98], stack
and load different radiating elements with the antenna laterally or vertically [99,100], utilize
magnetodielectric substrates [101] and engineer the ground plane as in the case of EBG
structures [6]. These techniques result additionally into the gain improvement of microstrip
antennas. The main reasons of low gain in the current design are the substrate/copper loss
and the horizontal orientation of the radiating section over the ground. It is like in the case
of gain reduction of the horizontal dipole with a small separation distance with the ground.
Another reason is that the increasing phase difference per cell with decreasing frequency
results LHM cells not to radiate effectively. This is the reason why there is a gain and
efficiency improvement at the increasing frequencies in Figure 6.8.b and Figure 6.8.c.
In the next section, a metamaterial-inspired slot antenna is introduced to illustrate how to
design a high gain antenna with LHM cells.
6.4 Metamaterial-Inspired Slot Antenna
In this section, a high profile metamaterial-inspired antenna is explained [102]. It consists of
LHM-based slot radiator and a T-formed feeding line. The slot radiator is composed of four
slotted cells of the same form as in the previous antenna. In Section 6.4.1, the geometrical
model of the slot radiator is introduced. In Section 6.4.2, the antenna geometry is explained
along with the design and operation principles. In Section 6.4.3, the numerically calculated
return loss and radiation patterns are illustrated.
6.4.1 Metamaterial Slot Radiator Design
The radiating section of the antenna is a slotted metal plate as shown in Figure 6.9. It is
designed by perforating four resonators in the same form of LH cells as in Section 6.3. The
electrical length of each resonator is increased by the direct connection with the adjacent
resonator. It is in principle in the form of two coupled λ/2 slot resonators. This perforated
structure is located horizontally with an optimum separation distance from the ground.
Therefore, the feeding line has to be located vertically to the ground plane for optimum field
coupling to the radiator. The overall size of the slot radiator is 14 mm x 6 mm. The
separation distance between each pair of the resonators is 0.4 mm.
Figure 6.9 Metamaterial slot radiator geometry
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6.4.2 Antenna Design
The metamaterial-inspired slot antenna is shown in Figure 6.10. In the model, the inner
conductor of SMA is connected with a T-formed feeding line. It is shown in Figure 6.10a
and 6.10b. The length of the extended inner conductor is 2 mm. The substrate is FR4 with
the relative permittivity of 4.4 and loss tangent 0.02. The width, length and thickness of the
substrate are 14 mm, 8 mm and 0.5 mm, respectively. The ground plane length, Lgrn, is 6
mm. There is a small coupling gap, wgap between the slot radiator and feeding line. It
improves the impedance matching and capacitive coupling from the feeding line to the slot
resonators. The width and length of coupling gap are 0.2 mm and 14 mm, respectively. The
arm width, Wmat and length, Lmat of T-formed feeding line are 6.5 mm and 6 mm,
respectively. The feeding line is located exactly at the middle position of the resonators as
shown in Figure 6.10c. Therefore, two pairs of slot resonators are always coupled
magnetically with each other. This feeding method has two main advantages. The first
advantage is that the substrate of T-formed feeding line is a supporting material on which
the slot radiator is located as shown in Figure 6.10a. The second advantage is that it allows
the matching network to be adequately designed without increasing the antenna size as
shown in Figure 6.10b.
(a) (b)
(c)
Figure 6.10 (a-b) Side and (c)top views of metamaterial-inspired slot antenna
88
The design principle is based on the simultaneous excitation of slotted spiral resonators and
slotted wires in LH cells at the resonance frequency. In this design, slotted spiral resonators
can be regarded as vertical electric dipoles due to spiralling magnetic current, whereas
straight slots can be regarded as horizontal magnetic dipoles. Because vertical electric and
horizontal magnetic dipoles have image dipoles in the same direction due to underlying
ground plane, the proposed antenna can radiate effectively in the far field.
6.4.3 Simulation Results
The return loss is numerically calculated and shown in Figure 6.11. The resonance
frequency is 5.25 GHz.
Figure 6.11. Return loss of the metamaterial-inspired slot antenna
The truncated ground plane of feeding line increases the field coupling from the
transmission line to the slot resonators. Therefore, the antenna is better impedance matched.
Another advantage of the ground plane is the effective coupling of the incoming field from
SMA connector to the slot radiators without any leakage to the larger ground. The current
distribution at the resonance frequency is shown in Figure 6.12. In Figure 6.12a, the slot
resonators are excited with eigenmode field distributions through the feeding line.
The normalized radiation patterns on horizontal and vertical planes at 5.25 GHz are shown
in Figure 6.13. In Figure 6.13, the co-polarization radiation pattern on horizontal plane is
similar to horizontal plane radiation pattern of elementary electric dipole. It is because of the
excitation of the slotted form of spiral resonator in the form of virtual magnetic current.
Spiralling magnetic current results the near field distribution to be in the form of the electric
dipole due to Babinet's principle and duality. However, the radiation pattern on vertical
plane is not similar to the radiation pattern of electric dipole. Two possible reasons of
enhanced radiation on the dipole axis are the superposition of the dipole fields in four
element array and the shift of phase center. The third reason can be the excitation of another
radiator in addition to the slotted wires connecting the spiral resonators. The small gap
region between the feeding line and slot resonators is another excitation source in the form
of magnetic dipole due to the electric field. The cross-polarization levels are better than -
90dB on vertical and -80dB on horizontal plane.
89
Figure 6.12. Resonant current distribution of (a) slot resonators, (b) ground plane of the feeding line and (c) T-formed
transmission line of metamaterial-inspired slot antenna
Figure 6.13 Co-polarization (red) and cross-polarization (blue) radiation patterns on (a) vertical and (b) horizontal
planes at the resonance frequency
90
The antenna gain is 5.5 dBi. The overall efficiency is more than 90%. There are two main
reasons why it is a quite efficient radiator. The first reason is the simultaneous excitation of
vertical electric and horizontal magnetic dipoles over the ground plane. The second reason
is the reduced level of field leakage through the feeding line from SMA to the slot
resonators. The overall antenna size is 0.24λ x 0.1λ x 0.14λ with ka value of 1.16. It is
slightly larger than the radiansphere at the resonance frequency. Some alternative compact
antenna designs in the literature are [102-107].
6.5 Chapter Conclusion
In this chapter, the fundamental radiation parameters of small antennas are introduced. The
basic performance limitations are explained. They are quite important parameters to have
realizable radiation parameters in a limited volume. The formulations of minimum Q and
maximum gain are presented. The effect of electromagnetic material parameters and
physical dimensions on the antenna radiation is pointed out. It is concluded that the
minimum radiation Q can only obtained with the excitation of the lowest order mode
without multimode operation. In contrary to the minimum Q limit, there is no mathematical
limit in the maximum gain. However, the excitation of higher order modes in a small
antenna results high field intensity in the vicinity of the antenna. In practice, this increases
the stored electric and magnetic energy with an additional increase in the losses and
degradation in the radiation efficiency. Therefore, maximum gain is limited by the gain of
lowest order mode for possibly broad bandwidth. In the next sections, two meta-antenna
design examples are illustrated. The first design is the broadband LHM-loaded dipole
antenna. The antenna design is based on the broadband loading of narrowband dipole with
LHM cells in Section 5.4. The reflection coefficient is numerically calculated and
experimentally verified. The surface current distribution and radiation patterns are
illustrated. The radiation of LHM cells is confirmed. The maximum gain and directivity are
-1 dBi and 3 dB. The radiation efficiency is 40% at 2.5 GHz. The directivity, gain and
radiation efficiency are illustrated. It has the overall dimension of 0.34λ x 0.08λ with the
main radiating section of 0.19λ x 0.08λ at the center frequency. The possible reasons of low
radiation efficiency are highlighted. An alternative metamaterial-inspired slot antenna is
introduced as a second meta-antenna. It consists of four slotted forms of the same cell as the
main radiator. The slot resonators are excited capacitively with T-formed feeding line. The
antenna is narrowband. The gain is 5.5dBi with an overall efficiency of better than 90%.
The effective excitation of the vertical electric and horizontal magnetic dipoles formed in
the slot resonators makes the antenna to radiate quite efficiently. The overall antenna size is
0.24λ x 0.1λ x 0.14λ with ka value of 1.16 at the resonance frequency.
91
7. Metamaterial-based Filter Design
7.1 Introduction
In Chapter 6, as a potential application of artificial materials, two different compact
antennas are proposed. In this chapter, metamaterial-based filter design is investigated. It is
an another important microwave application of artificial materials. The fundamental design
approaches are explained first with representative filter designs in the literature. Compact,
narrowband/wideband, band-stop/band-pass filters are presented. Important geometrical
parameters are discussed in these illustrative examples. These parameters are important to
tune the frequency band and improve the filter selectivity. As a next step, the design of a
compact band-stop filter (BSF) is numerically and experimentally studied. The design
approach how to excite the compact resonators in an effective manner is discussed. As a
second design example, a compact band-pass filter (BPF) is designed with the same cell
geometry as in the first design. The important design parameters are indicated. The main
reason to use the same cell geometry is to point out its effectiveness in different filter
designs.
7.2 Fundamental Principles of Metamaterial-based Filter Design
The metamaterial cells are electrically small and frequency selective as illustrated in Section
3.4, Section 4.4 and Section 5.4. The self-resonant nature allows them to be conveniently
used in the design of compact filters. These filters are in principle designed by loading of a
host transmission line with different combinations of electrically small resonators as LHM
with negative permittivity and permeability materials [3-5]. In which filter type they can be
utilized is deduced from 1D dispersion diagram. The unit cells of negative permittivity,
negative permeability and LH materials can be used in BPFs at the bandpass frequencies.
They can be alternatively used in BSFs at the bandgap frequencies. The high potential of
metamaterial cells in compact filter design can also be deduced from the circuit models. In
Section 5.2, LH cells are modeled as series capacitance and shunt inductance loaded with
series inductance and shunt capacitance. This property points out an additional potential of
LHM cells. They can be used in the design of compact broadband filters, which operate both
in RH and LH bands.
In this section, representative filter examples in the literature are presented. They are mainly
SRR and CSRR based designs. However, alternative designs composed of different
resonator geometries are also referenced. Most microwave filters in this section are
composed of small number of cells. In that respect, taking the effective parameters of these
structures into account for the filter design has no meaning [14]. Therefore, to attribute these
filters as “meta-filters” is the right approach. Sub-wavelength resonance features allow also
to model the filters as series or parallel resonance circuits with discrete elements. In meta-
filters, the filter parameters can be tuned by the resonator geometry, size and relative
location in the host medium. This controllable manipulation of electrical characteristics
makes them to be quite promising alternatives to the conventional designs [3-5].
The first design approach is based on how to improve the filter parameters of conventional
designs with the metamaterial inclusions. In that respect, as a meta-filter example, SRRs are
inserted into a conventional low pass filter for out-of-band rejection improvement
[108,109].
92
They can be alternatively located close to the coupled line sections of a conventional
coupled-line filter without entailing a considerable area increment. If they are properly
geometrically tuned, the spurious bands can be eliminated. The out-of-band rejection level
is additionally improved. The detailed design considerations are in [110].
In the second design approach, the design of narrowband LH BPF is investigated. It is
shown in Figure 7.1. SRRs and metallic parts of CPW are in black and gray, respectively.
Figure 7.1. SRR based LH transmission line [111]
In Figure 7.1, four pairs of SRRs are located in a periodic manner at the back side of a
short-circuited CPW. The short-circuited sections are shown in red. These sections excite
SRRs effectively at the resonance frequency through the magnetic field. Therefore,
optimum SRR positioning with respect to the shorting wires is an important design
parameter. The wave transmission is based on the loading of capacitive SRRs with inductive
shorting wires. Therefore, the transmission medium has narrow LH passband [3],[14],[111].
In analogy to bulk LH realizations, the short-circuiting lines are planar realizations of
infinitely long wires in artificial plasma. This can be deduced from the wave impedance of
short-circuited CPW, which is inductive as the wave impedance of periodic wire array under
plane wave excitation derived in Section 4.2. The measured and numerically calculated
filter parameters are extensively investigated with an equivalent circuit model in [111]. This
design approach is applied in other filter realizations [112-115]. BPFs can be alternatively
realized with different cell geometries [116-118]. Two cell geometries to be conveniently
used in microstrip technology are shown in Figure 7.2. The ground plane and signal line are
in gray and black, respectively.
(a) (b)
Figure 7.2. (a) Purely resonant and (b) hybrid CSRR-based LHM cell geometry [116],[119]
93
These cells are basically composed of CSRR etched in the ground plane and capacitive gaps
etched in the microstrip line. The main difference inbetween is that hybrid cell is contacted
to the ground with the vias at the left and rights sides of the signal line as indicated in red in
Figure 7.4. In CSRR-based hybrid cell, the signal transmission can be obtained at a lower
band due to the additional shunt inductance loading. Therefore, it is an improved version of
the purely resonant type [14]. They have one important design advantage.The bandgap
between RH and LH bands can be compensated with the appropriate geometrical parameters
[14],[120]. They can be therefore used conveniently in the broadband filter design [3].
The third design approach is based on broadband filter realizations with these LHM cells.
As a design example, broadband BPF with purely resonant cell is shown in Figure 7.3.
Figure 7.3. Broadband LHM BPF geometry [120]
In Figure 7.3, CSRRs are etched in the ground plane. Because they can be modeled as shunt
parallel resonance circuits, the shunt inductive loading is obtained for the frequencies lower
than the resonance frequency [3]. The series capacitive loading for LH transmission is
realized by the interdigital capacitances in the gap region. Thus, the operation principle
relies on the series loading of the interdigital capacitances with inductive CSRRs.
Measurement and simulation results are included in [120]. There are similar filter
realizations with the same design approach in [3],[121-123].
The fourth design approach is a frequently addressed methodology in narrowband filter
realizations [124-126]. It is based on the alternating arrangement of RH and LH cells along
the wave propagation direction. This alternating loading enhances the filter selectivity by
LH and RH cells at the lower and higher edge of the passband, respectively. It does not
degrade the insertion loss significantly. One exemplar BPF is shown in Figure 7.4.
Figure 7.4. Narrowband two LH and one RH alternating BPF geometry [124]
94
In Figure 7.4, SRRs and metallic parts of CPW are in gray and black, respectively. LH and
RH cells are indicated in red. RH lines are formed by smaller SRRs loaded with the
capacitive gaps etched in CPW signal line. LH lines are formed by larger SRRs loaded with
the short-circuited CPW line as in Figure 7.1. To use smaller SRRs in RH and larger SRRs
in LH is important. The frequencies lower than the resonance frequency of SRRs in LH line
are suppressed with high rejection level. Additionally, the frequencies higher than the
resonance frequency of SRRs in RH line are suppressed with high selectivity. Therefore, the
resulting filter is a narrow band, high selective, composite RH/LH filter [124]. If it is
compared with a conventional coupled line filter of the similar filter characteristics, the total
filter length is roughly three times shorter than the conventional filter [14]. The same design
approach can be conveniently implemented in microstrip technology. For this case, RH lines
are designed by CSRRs etched in the ground plane loaded with the shunt inductances. LH
lines are formed by CSRRs etched in the ground plane loaded with the series capacitances
[3].
In the fifth design approach, the realization of ultra-wide pass-band filters (UWBPF) is
explained. This approach allows the upper frequency limit to be controlled and the spurious
bands to be rejected at the specific frequencies. One UWBPF is shown in Figure 7.5.
Figure 7.5. Ultra-wide band-pass filter geometry [127]
In Figure 7.5, the signal line and ground plane of microstrip line are in black and gray,
respectively. This filter is the ultra-wideband version of the broadband filter in Figure 7.3 In
this design, there is no control on the upper limit of RH passband without smaller CSRRs.
Therefore, the upper frequency limit is controlled with the additional CSRRs of the smaller
radius inside the CSRRs of the larger radius. The higher resonance frequency of smaller
CSRRs limits the upper frequency with a high rejection level near the resonance frequency.
The geometrical dimensions can be accordingly adjusted for the desired resonance
frequency. Therefore, the upper frequency limit can be conveniently tuned. The upper
frequency and frequency selectivity can be optimized in a controlled manner with the
relative placement to the larger CSRRs and interdigital capacitances. The design and
operation principles are studied more extensively in [127]. This design principle can be
conveniently modified for improved filter realizations. Additional resonators can be
included to reject unwanted interfering signals. Two filter realizations with controlled
frequency notches in the passband are shown in Figure 7.6a and Figure 7.6b.
95
Figure 7.6a. UWB filter geometry with CSRR and complementary spiral resonators [127]
Figure 7.6b. UWB filter geometry with SRR and spiral resonators [128]
The filter in Figure 7.6a is a modified version of the filter in Figure 7.5. In Figure 7.6a, the
complementary spiral resonators (CSR) are etched in the ground plane under the feeding
line. They are located at the beginning and ending parts of the filter to have attenuation
poles in the passband [127]. They can be tuned precisely to eliminate the spurious signals at
the desired frequencies. The filter in Figure 7.6b is a modified version of the filter in Figure
7.3. In Figure 7.6b, all the additional resonators inserted into the host line are metallic. They
are located on the top layer of the substrate. Four SRRs located near the filter extremes are
designed to control the upper frequency limit with high selectivity. Two SRs are located in
the central part to have desired attenuation poles in the passband [128]. As in the previous
filters, the notch frequency and upper frequency limit can be adjusted conveniently by the
geometrical dimensions of SR and SRR. The relative displacements of each resonator from
the other resonators and LH/RH host line are also optimization parameters [3], [124-128].
In these typical design approaches, the design of pure resonant CSRRs is quite important for
the optimum filter performance. Hybrid cells can be alternatively applied in the filter
design. The design flexibility of hybrid cells leads not only UWBPFs to be implemented,
but also standard filters to be designed. A complete methodology for the design of planar
BPFs based on CSRRs was introduced in [129]. These filters can be modeled as in Figure
7.7a. In this model, the admittance inverters with normalized admittance of
J
=1 are
cascaded with alternating shunt resonators. These resonators can be designed as parallel LC
resonant tanks shown in Figure 7.7b. As long as the admittances of the resonant elements
match with those of LC tanks at least around the resonance frequency, the targeted
approximation (Chebyshev and Butterworth) can be applied in the filter design. The
admittance inverters can be implemented by cascading the hybrid cells. The filter
realizations based on this design methodology are illustrated in [130-133].
96
(a)
(b)
Figure 7.7 (a) BPF model consisting of impedance inverters and shunt resonators (b) BPF model
with LC resonant tanks as shunt resonators [129]
7.3 Metamaterial-based Band-Stop Filter
In this section, a compact, high selective BSF is explained [134]. The filter is composed of
four LHM cells as shown in Figure 7.8. One cell in each pair is anti-symmetrically
connected with its mirror image. Two identical resonator pairs are located anti-
symmetrically along the propagation direction. This filter topology is selected for symmetric
return losses at both ports. The compact resonators are the same cells introduced in Section
5.4. This cell geometry is utilized to point out the use of same cell topology in the design of
versatile microwave components. In Section 7.3.1, the geometrical model is introduced at
first. The important parameters to be optimized are pointed out. The design principle is
explained. In Section 7.3.2, the numerical and experimental results are presented. The
operation principle is described. The numerical calculations on the geometrical parameters
are illustrated.
7.3.1 Band-Stop Filter Design
LHM-based BSF is shown in Figure 7.8. In the filter model, the separation distance between
each of two x-direction oriented cells is 0.4 mm. Each resonator pair is arranged in y-
direction with the gap distance of 0.2 mm. The filter width (Wf) and length (Lf) are 6.5 mm
and 7.6 mm, respectively. The width of each metallic line is 0.3 mm except the width of
feeding line. The feeding line width is 0.9 mm to have 50 Ω line impedance. The length of
transmission line sections from the feeding ports is 6.2mm. The substrate is FR4 with the
thickness of 0.5mm. The relative permittivity and loss tangent are set as 4.4 and 0.02,
respectively. The total width and length of the filter are 7.7 mm and 20 mm, respectively.
In the filter design, two cells are connected anti-symmetrically on the either side of the
transmission line. This results the current distribution in each pair to be symmetrical at all
frequencies.
97
The resulting confinement of field lines in each pair reduces the possible radiation losses.
Each resonator pair is directly connected with the feeding line. The direct connection
enhances the field coupling from the feeding line to the resonators for high field rejection at
the resonance frequency. It results also the resonant current distribution to have similar form
with the current distribution in eigenmode calculation in Section 5.4.2.2. The cells in the
proposed filter are therefore electrically (red circle) and magnetically (blue circle) coupled
with the feeding line through the wires and spiral resonators. One resonator pair is arranged
as the mirror image of the other resonator pair in the propagation direction. Therefore, the
same return losses at both ports are obtained. In conventional BSFs, resonator shaped slots
are structured in the ground plane to have high rejection at the resonance frequency.
However, it requires an additional concern how to suppress the resulting back radiation of
the slotted ground. Therefore, this type of feeding method is a good alternative to the
conventional feeding method.
Figure. 7.8 LHM based Band-Stop Filter
However, there are important geometrical parameters to be optimised in the filter design.
These parameters are
separation distance of one resonator pair from the other resonator pair along y-direction
separation distance of both resonator pairs from the feeding line along x-direction
separation distance of both resonator pairs from the feeding line along y-direction
In Section 6.3, the uncertainty of the substrate permittivity is indicated as a possible reason
for the discrepancy between the measured and simulated resonance frequency of the meta-
antenna. It is therefore important to investigate the effect of substrate permittivity to have
better arguments for the possible deviations between the numerical and experimental results.
7.3.2 Experimental and Numerical Results of Band-Stop Filter
The performance of BSF is numerically calculated by using FEM based commercial
software HFSS. The transmission and reflection parameters are also measured by the vector
network analyser 8722C. The simulated and measured transmission parameters are shown
along with the reflection parameters in Figure 7.9.
98
Figure 7.9a. Measured (red) and calculated (blue) transmission parameters of band-stop filter
In Figure 7.9a, two coupled stop-band frequencies can be identified at 3.26 and 3.44 GHz in
the numerical calculation. It is an expected result of near field coupling from one resonator
pair to the other resonator pair and feeding line. The measured stopband frequencies are
3.67GHz and 4.12 GHz. They are shifted by 410 MHz and 680 MHz from the numerically
calculated lower and upper stop-band frequencies. There are two main reasons of frequency
shifts. The first reason is the geometrical deviations between the numerical model and
fabricated prototype. There are some metallic lines, which are not well-resolved in the
fabrication process. This can be verified from the discrepancy in the measured return losses
at both ports. The reflection parameter at port 1 must be same as the reflection parameter at
port 2. However, they are not the same as shown in Figure 7.9b. Therefore, measured
reflection parameter is different from calculated reflection parameter in Figure 7.9c. There
are some other effects of cables and even calibration errors, which can be implied from low
frequency ripples.
Figure 7.9b. Measured S11 (red) and S22 (blue) of band-stop filter
99
Figure 7.9c. Calculated reflection parameter of band-stop filter
The second reason is the uncertainty in the substrate permittivity. As pointed out in Section
6.3, this was a possible reason of resonance frequency shift in the experimental results. In
order to identify the effect of substrate permittivity, the transmission parameter is
numerically calculated for different permittivity values. It is shown in Figure 7.10.
Figure 7.10. Transmission parameter of band-stop filter for different substrate permittivity
In Figure 7.10, the stopband frequencies shift to the higher frequencies for lower
permittivity. The higher coupling among both resonator pairs and enhanced magnetic
coupling with the feeding line are the main reasons. In addition, this uncertainty changes the
input impedance from the designed value. Therefore, unwanted resonances can be observed
as in Figure 7.9a.
100
As a result, the numerically calculated return loss is larger than 10 dB in two frequency
bands of 2.41- 2.65 GHz and 4.26-5.72 GHz as shown in Figure 7.9c. The calculated
insertion loss is larger than 30 dB in the frequency band of 3.16-3.54 GHz. However, the
experimental results indicate signal suppression level better than 15dB between 3.6 GHz
and 4.34 GHz with 740MHz stopband. The surface current distribution at the higher
resonance frequency, 3.44 GHz is shown in Figure 7.11.
Figure 7.11. Surface current distribution of band-stop filter at 3.44 GHz
The current distribution on each of two resonators on the left side indicates λ/4 resonance at
this frequency. High impedance at the open end of these resonators is transformed into the
low impedance at this frequency. Thus, the incoming field is coupled electrically to the wire
and magnetically to the spiral resonator to excite coupled λ/4 resonance as in the case of
eigenmode calculation in Section 5.4.2.2. The magnetic coupling among the resonators can
be deduced from the symmetric current distribution. The magnetic coupling between the
feeding line and resonator pairs affects both upper and lower resonance frequencies.
However, the frequency shift at the lower stopband frequency is higher than that at the
higher stopband frequency. The transmission parameters are shown in Figure 7.12 for
different separation distances in x-direction between the feeding line and resonator pairs.
Figure 7.12. Transmission parameter of band-stop filter for different magnetic coupling distances
101
Why the lower resonance frequency is more dependent on the magnetic coupling is
explained in the following manner. The higher stop-band frequency is the resonance
frequency of two magnetically coupled resonators. This can be implied from the additional
numerical calculation without second resonator pair. The resonance frequency without
second resonator pair is negligibly different from the higher resonance frequency. However,
the lower resonance frequency is due to the magnetic coupling between the first and second
resonator pairs along the propagation direction. The higher magnetic coupling between the
feeding line and excited resonator pair reduces the magnetic field from the first resonator
pair to the second one. This is the reason why stop-bandwidth is smaller with the smaller
separation distance between the feeding line and resonator pairs.
In this section, a design approach for compact BSF with LHM cells is explained. The
dimensions of main filtering section are λo/12.57 x λo/10.75 at the lower stopband
frequency. The filter selectivity is approximately 40 dB/GHz at the higher stopband
frequency.
7.4 Metamaterial-based Band-Pass Filter Design
In this section, the design of a compact, LHM-based BPF is explained [73]. BPF is
composed of two LHM cells, which have the same geometry of thin wire loaded spiral
resonators as in Section 5.4. The design principle is based on direct connection of each
resonator to the feeding line and a certain coupling distance between two resonators. This
separation distance affects the inductive and capacitive coupling of each resonator with the
other resonator and feeding line. The field coupling among the resonators at the resonance
frequency results them to be effectively excited through the feeding line. In Section 7.4.1,
the geometrical model is introduced. The design parameters to be optimized are discussed.
The design principle is explained. In Section 7.4.2, the numerical results are presented.
Surface current density and effects of geometrical parameters are illustrated. The operation
principle is explained.
7.4.1 Band-Pass Filter Design
The BPF is shown in Figure 7.13. Two LHM resonators are located antisymmetrically
along the x-axis. They are directly connected with the feeding line to excite each resonator.
The excited resonator has to be coupled to the other resonator in an effective manner for low
insertion loss. The separation distance of two x-direction oriented cells is 0.2 mm. The
separation distance between one cell and feeding line section in y-direction, Lsep is 0.2 mm.
One resonator is shifted from the other one along y-direction with a distance of 0.2 mm.
This is an important design parameter to be investigated. Instead of using high lossy FR4,
low loss Rogers 5880 material is used. The relative permittivity and loss tangent are 2.2 and
0.0009. Low-loss substrate is advantageous to reduce the insertion loss by enhancing the
field coupling from one resonator to the other resonator. The lengths of extended feeding
line in x- (Lextx ) and y-directions (Lexty ) are 2.5mm and 4.3mm, respectively. The width
and length (Lfeed) of feeding line are 1.5mm and 7.5mm, respectively. The direct connection
length of the feeding line, (Lextfeed) in the extended sections is 1.9mm. The substrate width
and length are 30 mm and 18.8mm, respectively. The total filter size is 6.4mm x 4.8 mm.
102
Figure 7.13. Thin wire loaded spiral resonator based BPF
The design principle is to feed one resonator directly through the feeding line. The highly
concentrated field results the directly excited resonator to couple to the other resonator. The
excited field in the first resonator is also enhanced by the magnetic field of the extended
section of the feeding line. The magnetic and electric intercoupling as in the case of 0 and л
resonant modes among the resonators result the incoming field at the first port to transmit
with different phase shifts to the second port. How effectively these resonators are excited
and coupled is quite important. This affects the filter selectivity significantly. Therefore,
some geometrical parameters have to be investigated to figure out the field coupling
strength.
These parameters are mainly
1-) separation distance of one resonator from the other resonator along x-direction
2-) shift of one resonator from the other resonator along y-direction
3-) separation distance of both resonators from the feeding line sections
This feeding method results the resonators to couple with the feeding line in an optimal
manner. As a next step, the return and insertion losses are numerically calculated. The
resonant current distribution is illustrated. The effects of geometrical parameters on filter
performance are discussed.
7.4.2 Numerical Results of Band-Pass Filter
To validate the filter design principle, the reflection and transmission parameters are
numerically calculated. They are shown in Figure 7.14. The return loss is larger than 10 dB
with the insertion loss smaller than 1 dB between 4.58GHz and 5.24GHz. The lowest
insertion loss is 0.4 dB at 4.7GHz. The filter selectivities are approximately 124.25 dB/GHz
and 36.71 dB/GHz at the lower and higher edge of the passband, respectively.
103
Figure 7.14 S21(blue) and S11 (red) parameters of band-pass filter
In the design process, the geometrical parameters are optimized for low insertion loss. How
the resonators have to be arranged along y-direction is an important design parameter. In
Figure 7.15, the transmission parameters for different shift of these resonators along y-
direction are shown.
Figure 7.15 Transmission parameter of band-pass filter for different resonator shift along y-direction
In Figure 7.15, higher field coupling among the resonators with smaller shift results the
bandwidth to be broader. The insertion loss is lower. The filter selectivity at the lower and
higher edges of the passband are degraded. In addition, the field coupling from one
resonator to the another resonator is enhanced through the magnetic coupling from the
feeding line. Therefore, another design parameter is the separation distance between
extended feeding line in y-direction and resonators along x-direction. The transmission
parameters are shown in Figure 7.16 for different separation distances between each
resonator and feeding line.
104
Figure 7.16. Transmission parameter of band-pass filter for different resonator separation distances from the feeding
line
In Figure 7.16, the higher field coupling with small separation distance increases the
bandwidth. The lower stopband frequency decreases and higher stopband frequency
increases. These frequencies are the resonance frequencies of two coupled resonators. They
can be modelled as parallel or series resonators at these frequencies. In analogy to the band
diagram, these are the lower and higher edge of the passband in the first Brillioun zone. The
surface current distributions at 4.2GHz and 6.36GHz in the optimum filter design illustrate
the above mentioned remarks in Figure 7.17. In Figure 7.17, the first resonator is directly
excited through the feeding line on the left side. The second resonator is excited through the
field coupling from the first resonator and feeding line. In Figure 7.17a and Figure 7.17b,
the field distributions of the excited resonator and feeding line couple the resonators
electrically in 0 mode at 4.2GHz and magnetically in л mode at 6.36 GHz. However, in the
passband, the incoming field is transmitted at different phase shifts by the excitation of
second resonator through the electric and magnetic coupling from the first resonator.
(a) (b)
Figure 7.17: Surface current distribution of band-pass-filter at (a) 4.2GHz and (b) 6.36GHz
105
In this section, the design of a compact BPF composed of two LHM cells is explained. The
filter size without the feeding line sections is λo/10.23 x λo/13.64 at the lower passband
frequency, 4.58GHz. It is quite compact in comparison to the conventional stepped
impedance or coupled line filter designs [3],[14],[135]. One important advantage is no need
of an additional matching network. This reduces the filter size significantly. The proposed
filter has satisfactory insertion loss. It is smaller than 1 dB in the frequency band between
4.58 GHz and 5.24 GHz with the bandwidth of 660 MHz. The filter selectivities with
reference to 3dB insertion loss are 124.25 dB/GHz and 36.71 dB/GHz at the lower and
higher edge of the passband, respectively.
7.5 Chapter Conclusion
In this chapter, basic design approaches of metamaterial-based filters are highlighted.
Representative filter designs in the literature are illustrated. Compact broadband/
narrowband BPF and UWBPF realizations are explained because of their design challenges
in wireless communication systems. The bandgap separating LH band from RH band can be
conveniently eliminated by purely resonant and hybrid CSRR cells. This is very important
in broadband filter realizations. SRR and CSRR-based filter designs have inherently narrow
stop-band characteristics with high selectivity. Therefore, BSFs can be realized in a
straightforward manner by cascading SRR/CSRR cells. This property can be used in the
design of high selective narrowband filters. How to suppress the unwanted spurious
frequencies in UWBPFs is explained with design examples.
A compact LHM-based BSF is experimentally and numerically studied. The design and
operation principles are explained in detail. The experimental results indicate signal
suppression level better than 15dB between 3.6 GHz and 4.34 GHz with 740MHz
stopband. The possible reasons of the deviation between the measured and calculated
transmission parameters are discussed. The filter selectivity is approximately 40 dB/GHz at
the higher stopband frequency. The dimensions of main filtering section are λo/12.57 x
λo/10.75 at the lower stopband frequency. As a second meta-filter, a compact BPF
composed of two LHM cells is designed. The insertion loss is smaller than 1 dB in the
frequency band between 4.58 GHz and 5.24 GHz with the bandwidth of 660 MHz. The
filter selectivities are numerically calculated with reference to 3dB insertion loss. They are
124.25 dB/GHz and 36.71 dB/GHz at the lower and higher edge of the passband,
respectively. The size of main filtering section is λo/10.23 x λo/13.64 at the lower passband
frequency. It is quite compact in comparison to the conventional stepped impedance or
coupled line filter designs. The important advantage of both filter designs is no need of
additional matching network. This reduces the filter physical size significantly.
106
8. Conclusion
The main goal of this thesis is to understand and investigate the design and operational
principles of metamaterial-based antennas and filters with the focus on compact microwave
realizations. Therefore, unit cells of negative permeability, permittivity and Left-Handed
materials are extensively studied with numerical and analytical calculations. These
calculations are important in order to understand the underlying physical mechanism in the
design of artificial materials. For each of these metamaterial classes, one novel cell design
with improved homogeneity is proposed. In Chapter 2, a short overview is given to
understand the design principles.
In Chapter 3, artificial magnetism resulting from periodic metallic cylinders with/without
splits is studied analytically and numerically. Analytical calculations for transmission and
reflection parameters of concentric cylinders with splits are confirmed by numerical
simulations and are in good agreement. The effective permeability formulation is compared
with the original formulation of Pendry and approximates the resonance frequency better. A
conventional negative permeability cell, Split Ring Resonator, is numerically analyzed in
order to understand the principle of resonant negative permeability. The source for the
demagnetization of the incoming magnetic field is the excitation of electrically small
magnetic dipoles. Therefore, the Split Ring Resonator array has a capacitive impedance in
the negative permeability region between the magnetic resonance and the plasma frequency.
A 1D Brillouin diagram is numerically calculated to verify the bandgap region with
negative permeability. An alternative parameter retrieval method, based on Bloch’s
Theorem for infinitely periodic structures is proposed. At the end of this chapter, a novel
geometry based on fractal spiral resonator is proposed. The cell has dimensions, λ0/40 x
λ0/100 x λ0/40 at the magnetic resonance frequency and shows a more homogeneous
behaviour of the composite material.
In Chapter 4, artificial dielectrics based on periodic metallic wires are analyzed first to study
non-resonant negative permittivity materials. The effective permittivity is analytically
calculated under plane wave excitation and agrees well with numerical values. An
equivalent circuit model based on the reflection parameter is derived and a Drude type
permittivity function is confirmed by numerical calculations. The analytical calculations of
the permittivity of a wire array make no assumptions on the frequency dispersion unlike the
one of Pendry. A wire strip model in planar technology on a low loss substrate is
numerically analyzed. The transmission and reflection parameters are calculated in order to
understand the principle of negative permittivity. The incoming electric field is depolarized
due to excitation of electrically small electric dipoles, formed from the wire currents. Thus,
the wire array has inductive impedance in the negative permittivity region as in the case of a
short-circuited transmission line. At the end of the chapter, a novel geometry based on
fractal anti-spiral resonators is proposed. The cell dimensions are λ0/55 x λ0/22 x λ0/22 at the
electric resonance frequency. More homogeneous composite materials with negative
permittivity could be therefore designed.
In Chapter 5, the design of Left Handed Materials, based on the periodic arrangement of
negative permittivity and permeability cells in the same host medium, is studied. The
eigenmode equation of a loaded rectangular waveguide is analytically and numerically
calculated. They are in good aggrement. The negative refractive index and subwavelength
resonance phenomenon are confirmed. Drude type electric and Lorentzian type magnetic
parameters with low losses are assumed to observe LH and RH bands.
107
The reason of a high frequency RH band is explained by equivalent circuit models of
negative permittivity and permeability materials. An LHM unit cell is numerically analyzed
and confirms that the LH transmission band is the overlapping region of negative
permittivity and permeability. It is due to inductive loading of the capacitive impedance of
SRRs by thin wires. Thus, the stored electric energy in the near field of SRR is compensated
with the stored magnetic energy in the near field of thin wire. At the end of the chapter, a
novel geometry based on wire loading of spiral resonators is proposed as LH cell. The cell
dimensions are λ0/51 x λ0/44 x λ0/77 at the LH resonance frequency. It can be used in the
design of more homogeneous LH materials.
In Chapter 6, the design principle of metamaterial-based compact antennas is studied. The
first design is a broadband dipole antenna loaded with LHM cells. The reflection coefficient
is numerically calculated and experimentally confirmed. The radiation of electrically small
LHM cells is confirmed by the surface current distribution and radiation patterns. The
maximum gain and directivity are -1 dBi and 3 dB with 40% radiation efficiency at 2.5 GHz.
Possible reasons for the low antenna gain are the inherent substrate/copper loss, the
horizontal orientation of the radiating section over the ground plane and the increasing
phase difference per unit cell with the decreasing frequency. Higher phase differences cause
the successive LH cells to radiate destructively in the far field due to out-of-phase
excitation. This is the reason why there is a certain gain and efficiency improvement with
increasing frequencies. The overall dimensions of the antenna are 0.34λ x 0.08λ with the
main radiating section of 0.19λ x 0.08λ at the center frequency. The second design is a
metamaterial-based slot antenna. This antenna consists of four slotted cells as the main
radiator. The slot resonators are excited capacitively with T-shaped electrically small
monopoles. The antenna is a narrowband antenna with a gain of 5.5dBi and overall
efficiency better than 90%. The effective excitation of vertical electric and horizontal
magnetic dipoles formed by the slot resonators is the reason for the high efficieny. The
overall antenna size is 0.24λ x 0.1λ x 0.14λ with ka value of 1.16 at the resonance frequency.
Finally, in Chapter 7, the design principles of metamaterial-based compact filters are studied
with two compact filter designs. The first filter is a Band Stop Filter, which is studied
experimentally as well as numerically. The experimental results indicate a signal rejection
level better than 15dB between 3.6 GHz and 4.34 GHz with a 740MHz stopband. Possible
reasons for the differences between the measured and calculated transmission parameters
are discussed. The filter selectivity is 40 dB/GHz at the higher stopband frequency. The
dimensions are λo/12.57 x λo/10.75 at the lower stopband frequency. The second design is a
Band Pass Filter with two unit cells of same LHM geometry as for the Band Stop Filter. The
numerically calculated insertion loss is smaller than 1 dB in the frequency band between
4.58 GHz and 5.24 GHz with the bandwidth of 660 MHz. The filter selectivities at 3dB
insertion loss are 124.25 dB/GHz and 36.71 dB/GHz for the lower and higher edge of the
passband, respectively. The physical size without the feeding line is λo/10.23 x λo/13.64 at
the lower passband frequency, 4.58GHz. It is quite compact as compared to conventional
stepped impedance or coupled line filter designs. Both filter designs do not require any
matching network, which reduces the physical size significantly.
108
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116
APPENDIX A
Quasi-Static Analysis of Negative Permeability Cells
The effective permeability of periodic long cylinders can also be derived with an equivalent
inductance, L and resistance, R of the cylinder in one unit cell. The total induced current, I
is calculated from the magnetic induction law in the form of
0
2
00
. . (A1.1)
( )
CF
E dl j µ H dF
R j L I j µ H r
2
00
(A1.2)
j µ H r
IR j L (A1.3)
The magnetization vector ,
M
, can then be calculated from
I
in terms of
o
H
and
m
as
2
2
0
2
0
22
cyr. cyr.
0
11
l l
z
m
j µ r
M I r e H
a a R j L
MH (A2.1)
where lcyr. is the cylinder length.
Therefore, the effective permeability has the form of
2
2
0
02
cyr.
11 (A2.2)
l
eff
j µ r
µµ
a R j L
In (A2.2), R can be expressed as
cyr.
Length
1 1 2
(A2.3)
Cross Section Area l
r
Rr
L is formulated by calculating the magnetic field from Amper’s Law in Figure A1
0
cyr.
2
0
cyr.
.
l
.
(A2.4)
l
C
z
mF
H dl I
I
B µ e
B dF r
Lµ
II
117
where
m
is the magnetic flux
Figure A1 Metallic cylinder cell with the radius, r and length,
cyr.
l
After the substitution of L and R into (A2.2), the effective permeability is formulated as
2
2
0
02
cyr.
2
02
0
11 (A2.5)
12 l
= 1 2
eff
j µ r
µµ r
ajL
r
r r r
µaj r r
µ
(A2.6)
In this formulation, the cell size in z-direction has to be sufficiently long to reduce the edge
effects at the cylinder ends. (A2.5) and (A2.6) are the same formulations of (3.7b) and
(3.7a) after the substitution of skin depth into (3.7a).
Effective permeability formulations in terms of equivalent circuit elements in (A2.2) makes
the resonant permeability of concentric slotted cylinder array to be conveniently obtained.
The equivalent circuit is then in the form of series loading of L with an equivalent
capacitance of slotted cylinder, C in Figure 3.3. The effect of C can be taken into account by
the modification of L in the following form.
total
total eq
eq 2
1
Z = R+j L+ jC
Z = R+j L
1
L L(1- ) (A3.1)
CL
Thus, the most important point in the resonant permeability formulations is to calculate the
capacitance of one cell with longitudinal slots, C in terms of quasi-static analysis. C can be
calculated from the capacitance of two concentric cylinders without longitudinal slots, Co in
Figure A2. Co is formulated by calculating the electrical field and resulting potential
difference,
as in the following form,
118
Figure A2 Two concentric metallic cylinders with inner radius r and outer radius r+d and length
cyr.
l
0
21
0
l cyr.
00
0 cyr.
0
2
1
=- - 2
q l
= ln( ) =
2
2 l
=
ln( )
l
r d r d
l
rr
l
q
Ee
E d q d
qrd
rC
Crd
r
(A3.2)
where
l
q
is the total charge density per unit length, r and d are the radius of the inner
cylinder and the separation distance between two concentric cylinders, respectively.
In the formulation of SRR resonance frequency, L is assumed to be approximately same as
L in (A2.4). It is due to the effect of high magnetic coupling between the cells in the axial
direction. However, the equivalent capacitance is no more the same as the one calculated in
(A3.2) due to axial dependence of field lines. It can be calculated approximately from the
capacitance of two infinitely long rectangular lines of width
w
and separation distance d in
Figure A3.
(a) (b)
Figure A3 (a) Two parallel rectangular lines with width
w
and separation distance d (b) approximate electric
field lines
119
The rotation symmetrical electric field can be formulated from the potential difference,
o
in the form of
y>0
y<0
o
o
Ee
Ee (A4.1)
The total charge per unit length,
'total
Q
on the upper and lower sides of one rectangular plate
and total capacitance per unit length,
'SRR
C
are calculated as
22
0
22
2
0 s 0 s
2
0s
' . .
2
' ln( )
2
' = ln( )
dd
ww
dd
total y s y
d
woo
d
total
SRR
Q E e dx E e dx
wd
Q dx
xd
wd
Cd (A4.2)
where
s
is the substrate permittivity. Thus, total CSRR is in the form of
0s
2
= 2 ln( ) (A4.3)
SRR avg
wd
Cr d
with an average radius,
avg
r
.
In this formulation, d has to be sufficiently small in comparison to
w
to reduce the edge
effects for rotational symmetrical field distribution.
120
Appendix B
List of Author’s Publications
Journal/ Conference Papers
1. M. Palandöken, M. Aksoy und M. Tümay," A fuzzy-controlled single-phase active
power filter operating with fixed switching frequency for reactive power and current
harmonics compensation", Electrical Engineering, Volume 86, Number 1, 2003
2. Merih Palandöken, Murat Aksoy and Mehmet Tümay,"Application of fuzzy logic
controller to active power filters", Electrical Engineering Vol: 86 No:4, (2004)
3. Merih Palandöken, Mehmet Tümay and Murat Aksoy, " A novel approach to active
power filter control", Electrical Engineering Vol: 87 No: 1 , (2005)
4. M. Palandoken, H. Henke, "Fractal Spiral Resonator as Magnetic Metamaterial", IEEE
Applied Electromagnetics Conference (AEMC), 2009
5. Merih Palandoken, Andre Grede, and Heino Henke, "Broadband Microstrip Antenna
With Left-Handed Metamaterials", IEEE Transactions on Antennas and Propagation,
Vol. 57, No. 2, 2009
6. M. Palandoken, H. Henke, "Fractal Negative-Epsilon Metamaterial", IEEE International
Workshop on Antenna Technology (iWAT), 2010
7. Merih Palandöken, Heino Henke, "Compact LHM-based Band-Stop Filter", IEEE
Mediterranean Microwave Symposium (MMS), 2010
8. Merih Palandöken, Heino Henke, "Miniaturized Self-resonant Metamaterial-based
Antenna",International Journal of Microwave and Optical Technology, Vol.6, No., 2011
9. Merih Palandöken, "Compact LHM-inspired Microstrip Antenna", International Journal
of Microwave and Optical Technology, Vol.6, No.6, 2011
10. Merih Palandöken, "Dual-Band Epsilon-Negative Material Inspired Fractal Antenna",
International Journal on Science and Technology, Volume 2, Issue 1, 2011
11. B. Bouhlal, S. Lutzmann, M. Palandöken, V. Rymanov, A. Stöhr, T. Tekin, “Integration
platform for 72 GHz photodiode-based wireless transmitter”, SPIE Photonics West
2012, San Francisco, Jan. 21-26, Proc. SPIE 8259, 82590H, 2012
12. M. Palandöken, B. Bouhlal, S. Lutzmann, V. Rymanov, A. Stöhr, T. Tekin, “Integration
platform for 1.55 µm waveguide-photodiode based 71-76 GHz wireless transmitter”,
International Forum on Terahertz Spectroscopy and Imaging, 5th Workshop on Terahertz
Technology, Kaiserslautern, March 6-7, Paper Identity No. 47, 2012
121
13. V. Rymanov, S. Babiel, A. Stöhr, S. Lutzmann, M. Palandöken, B. Bouhlal, T. Tekin,
“ Integrated E-Band Photoreceiver Module for Wideband (71-76 GHz) Wireless
Transmission”, European Microwave Week 2012, European Microwave Conference,
EUMC 2012, Amsterdam,The Netherlands, 29 October - 1 November, paper no. 1751,
2012, (accepted)
14. V. Rymanov, S. Lutzmann, M. Palandöken, T. Tekin, A. Stöhr, “Wideband 1.55 µm
Waveguide Photodiodes Employing Planar Resonant Circuits for E-band (60-90 GHz)
Operation” , Progress In Electromagnetics Research Symposium, PIERS 2012, Moscow,
Russia, 19-23 August, 2012, (accepted)
15. V. Rymanov, M. Palandöken, S. Lutzmann, B. Bouhlal, T. Tekin, A. Stöhr, “Integrated
Photonic 71-76 GHz Transmitter Module Employing High Linearity Double
Mushroom-Type 1.55 µm Waveguide Photodiodes”, IEEE International Topical
Meeting on Microwave Photonics, MWP 2012, Noordwijk, The Netherlands, 11-14
September, paper no. 2594588, 2012, (accepted)
16. Merih Palandöken, Sascha Lutzmann, Vitaly Rymanov, Andreas Stöhr, and Tolga Tekin,
"Grounded CPW-WR12 Transition Design for 1.55 µm Photodiode Based E-band
Transmitter", PIERS 2012 , (accepted)
17. Merih Palandöken, Vitaly Rymanov, Andreas Stöhr, and Tolga Tekin, "Compact
Metamaterial-based Bias Tee Design for 1.55 µm Waveguide-photodiode Based 71-76
GHz Wireless Transmitter", PIERS 2012 , (accepted)
Book Chapters
18. Merih Palandöken, "Artificial Materials based Microstrip Antenna Design", in
Microstrip Antennas, ISBN 978-953-307-247-0, InTech, 2011
19. Merih Palandöken, "Metamaterial-Based Compact Filter Design", in Metamaterial,
ISBN 978-953-51-0591-6, InTech, 2012
