The dispersion characteristic of the left-handed coplanar waveguide were calculated ..... work on optical Bloch waves in (singly and doubly) periodic planar waveguides. .... ters of a three-dimensional lossless lattice of dielectric particles with complex ..... Chiral media provide another example of metamaterials with promising ...
Czech Technical University in Prague Faculty of Electrical Engineering
Doctoral Thesis
October 2007
Ing. Martin Hudliˇ cka
Czech Technical University in Prague Faculty of Electrical Engineering Department of Electromagnetic Field
PROPAGATION OF ELECTROMAGNETIC WAVES IN PERIODIC STRUCTURES Doctoral Thesis
Ing. Martin Hudliˇ cka
Prague, October 2007
Ph.D. Programme: Electrical Engineering and Information Technology Branch of Study: Radioelectronics
Supervisor: Supervisor-Specialist:
Doc. Ing. Jan Mach´ aˇ c, DrSc. Prof. Ing. J´ an Zehentner, DrSc.
Contents Abstract
iii
Acknowledgement
v
List of symbols
vi
List of abbreviations
viii
1 Introduction
1
1.1
Properties of metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
First references to metamaterials . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.2
Revival of interest in recent years . . . . . . . . . . . . . . . . . . . . . . .
5
Main research directions in metamaterials . . . . . . . . . . . . . . . . . . . . . .
6
1.3.1
Basic theory and general issues . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.2
Negative permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.3
Negative permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3.4
Perfect lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.5
Transmission-line implementations . . . . . . . . . . . . . . . . . . . . . .
13
1.3.6
Chiral and bianisotropic materials . . . . . . . . . . . . . . . . . . . . . .
15
1.3.7
THz and optical applications . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.3.8
Tunable metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3.9
Practical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Aims of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3
1.4
2 Left-handed coplanar waveguide
19
2.1
Transmission line model of a metamaterial . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Left-handed coplanar waveguide
. . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.1
Concept of a left-handed coplanar waveguide . . . . . . . . . . . . . . . .
24
2.2.2
Equivalent circuit of the LHCPW . . . . . . . . . . . . . . . . . . . . . . .
27
2.2.3
Equivalent circuit elements . . . . . . . . . . . . . . . . . . . . . . . . . .
31
i
CONTENTS 2.2.4
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3 Triple wire medium
35
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Effective permittivity dyadic, dispersion equation . . . . . . . . . . . . . . . . . .
36
3.3
Dispersion characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.4
Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.5
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4 Uniaxial Ω-medium
56
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.2
Antenna model for wire-and-loop elements . . . . . . . . . . . . . . . . . . . . . .
60
4.2.1
A simple LC model for electrically small particles . . . . . . . . . . . . . .
60
4.2.2
Antenna model of the particles . . . . . . . . . . . . . . . . . . . . . . . .
62
A perfectly matched backward-wave layer . . . . . . . . . . . . . . . . . . . . . .
64
4.3.1
Use of a simple LC model for electrically small particles . . . . . . . . . .
68
4.3.2
Antenna model of the particles . . . . . . . . . . . . . . . . . . . . . . . .
72
4.4
Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.5
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.3
5 Conclusion
90
A Derivations
93
A.1 Conventional coplanar waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
A.2 Coplanar waveguide with finite extent ground planes . . . . . . . . . . . . . . . .
94
Bibliography
95
ii
Abstract Propagation of electromagnetic waves in periodic structures Keywords - metamaterial, periodic structure, left-handed coplanar waveguide, triple wire medium, bi-anisotropic particle, omega particle, perfectly matched layer
The doctoral thesis summarizes the study of the propagation of electromagnetic waves in periodic structures, namely in metamaterials, whereas three forms of a metamaterial are investigated. The introductory part gives a brief historical overview of metamaterial research, and the main research directions in this area are outlined. The first part of the thesis summarizes an investigation of the properties of a novel metamaterial in the form of a left-handed coplanar waveguide and its equivalent circuit. The left-handed coplanar waveguide was designed, fabricated and measured. The dispersion characteristic of the left-handed coplanar waveguide were calculated analytically and then numerically in a full-wave electromagnetic field simulator. The second part deals with a volumetric form of a metamaterial in the form of a triple wire medium, with the aim to apply it as a negative permittivity isotropic metamaterial. The triple wire medium is formed by a three dimensional lattice of mutually perpendicular metallic connected wires. It was shown that the triple wire medium behaves as an isotropic material near the plasma frequency for the wave propagating parallel with any coordinate plane. The value of the effective permittivity of the wire lattice was numerically estimated. For a practical implementation, the planar form of the triple wire medium was proposed. The transmission coefficient and distribution of the electric field within the lattice were measured. In the third part of the thesis, a perfectly matched backward-wave slab was theoretically studied. The bianisotropic approach utilizing Ω-shaped metallic particles was used. The known implementations of backward-wave media imply negative permeability and permittivity. The novelty of the approach in this work is that negative permeability is not needed and moreover for matching the slab with free space, the condition ²r = µr is not necessary. First, the Ω-particles were studied analytically and basic mixing formulas were used for the design of the slab. From the derivation of the backward-wave condition it follows that a negative permittivity background material is needed for the existence of a backward-wave. It was proposed to use a single wire medium for the realization of a negative permittivity hosting material. The composite material consisting of Ω-particles and wires was studied. A form utilizing planar technology was proposed for a practical implementation of the slab. The slab was fabricated and measured and the results are in good agreement with the theoretical predictions and full-wave simulations.
iii
Anotace ˇ ıˇ S´ ren´ı elektromagnetick´ ych vln v periodick´ ych struktur´ ach Kl´ıˇ cov´ a slova - metamateri´al, periodick´a struktura, koplan´ arn´ı veden´ı se zpˇetnou vlnou, tˇr´ırozmˇern´e dr´atov´e prostˇred´ı, bianisotropn´ı ˇc´ astice, omega ˇc´ astice, ide´alnˇe pˇrizp˚ usoben´a vrstva
Disertaˇcn´ı pr´ace shrnuje v´ ysledky v´ yzkumu ˇs´ıˇren´ı elektromagnetick´ ych vln v periodick´ ych struktur´ach, jmenovitˇe tzv. metamateri´alech. V pr´aci jsou zkoum´ any celkem tˇri r˚ uzn´e formy metama´ teri´alu. Uvodn´ ı ˇc´ast pr´ace pod´av´a pˇrehled souˇcasn´eho stavu v´ yzkumu metamateri´al˚ u ve svˇetˇe. V prvn´ı ˇc´asti pr´ace je navrˇzena p˚ uvodn´ı forma plan´arn´ıho metamateri´alu zaloˇzen´ a na pouˇzit´ı koplan´arn´ıho veden´ı se zpˇetnou vlnou. Je odvozen n´ahradn´ı obvod takov´eto struktury a jsou nalezeny hodnoty prvk˚ u n´ahradn´ıho obvodu. Z n´ahradn´ıho obvodov´eho modelu koplan´ arn´ıho veden´ı se zpˇetnou vlnou jsou odvozeny disperzn´ı charakteristiky, kter´e jsou pot´e porovn´ any s v´ ypoˇctem simul´atoru elektromagnetick´eho pole. Struktura byla pot´e zkonstruov´ ana a byly zmˇeˇreny jej´ı parametry, kter´e jsou porovn´ any se simulac´ı. Druh´a ˇc´ ast pr´ace se zab´ yv´ a isotropn´ım prostˇred´ım se z´apornou efektivn´ı permitivitou ve formˇe tˇr´ırozmˇern´eho prostˇred´ı sloˇzen´eho ze syst´emu tˇr´ı navz´ajem kolm´ ych spojen´ ych vodiv´ ych dr´atk˚ u. Tˇr´ırozmˇern´e dr´atov´e prostˇred´ı se obecnˇe vyznaˇcuje prostorovou disperz´ı, kter´a se ovˇsem jev´ı jako zanedbateln´a v okol´ı plasmov´eho kmitoˇctu mˇr´ıˇze a prostˇred´ı tak lze pouˇz´ıt jako isotropn´ı materi´al se z´apornou efektivn´ı permitivitou v bl´ızkosti plasmov´eho kmitoˇctu. Pro dr´atkov´e prostˇred´ı byly nalezeny disperzn´ı charakteristiky ve vˇsech v´ yznaˇcn´ ych smˇerech ˇs´ıˇren´ı vlny v z´akladn´ı buˇ nce a porovn´ any se simul´ atorem elektromagnetick´eho pole. Dr´atkov´e prostˇred´ı bylo zhotoveno v p˚ uvodn´ı podobˇe vyuˇz´ıvaj´ıc´ı plan´arn´ı technologie a namˇeˇren´ y koeficient pˇrenosu a intenzita elektrick´eho pole uvnitˇr prostˇred´ı byly porovn´any se simulac´ı. Namˇeˇren´e v´ ysledky potvrzuj´ı teoretick´e pˇredpoklady. Ve tˇret´ı ˇc´asti pr´ace je zkoum´ana ide´alnˇe impedanˇcnˇe pˇrizp˚ usoben´a vrstva skl´adaj´ıc´ı se z tzv. jednoos´eho omega prostˇred´ı. Omega ˇc´astice patˇr´ı do obecn´e tˇr´ıdy bianisotropn´ıch ˇc´ astic a pˇri interakci s dopadaj´ıc´ı elektromagnetickou vlnou se vyznaˇcuj´ı nˇekter´ ymi unik´atn´ımi vlastnostmi. Je analyticky uk´az´ ano, ˇze pro souˇcasn´e splnˇen´ı podm´ınek ide´aln´ıho impedanˇcn´ıho pˇrizp˚ usoben´ı a existence zpˇetn´e vlny v prostˇred´ı mus´ı b´ yt omega ˇc´astice vloˇzeny do prostˇred´ı se z´apornou efektivn´ı permitivitou. Je uk´az´ano, ˇze konstrukce pˇrizp˚ usoben´e vrstvy se zpˇetnou vlnou pomoc´ı omega ˇc´ astic m´a nˇekter´e v´ yhody oproti konvenˇcn´ım ˇreˇsen´ım. Vrstva byla n´aslednˇe realizov´ana plan´arn´ı technologi´ı a byly zmˇeˇreny jej´ı pˇrenosov´e vlastnosti. Namˇeˇren´e v´ ysledky se dobˇre shoduj´ı s teoretick´ ymi pˇredpoklady.
iv
Acknowledgement I would like to express thanks to my supervisor, Doc. Ing. Jan Mach´ aˇc, DrSc. from the Czech Technical University in Prague, for his many suggestions and constant support during this research. I also would like to thank the following people for many fruitful discussions which have helped to improve this thesis: Prof. Ing. J´an Zehentner, DrSc. and Ing. Luk´aˇs Jel´ınek, Ph.D. from the Czech Technical University in Prague, Prof. Sergei A. Tretyakov from Helsinki University of Technology in Finland, Prof. Constantin R. Simovski from the State University of Information Technologies, Mechanics and Optics in Russia and Prof. Igor S. Nefedov from the Institute of Radio Engineering and Electronics of Russian Academy of Sciences. I also wish to thank Prof. Ekmel Ozbay and his research team from the Bilkent University in Ankara in Turkey for the cooperation with gaining the experimental results presented in Chapter 4 of this thesis. The research work has been sponsored by the Czech Ministry of Education, Youth and Sports in the framework of the project MSM 102/03/H086 “Research in the Area of Prospective Information and Navigation Technologies” and by the Grant Agency of the Czech Republic under project 102/06/1106 “Metamaterials, nanostructures and their applications”.
Prague, October 2007
Ing. Martin Hudliˇcka
v
List of symbols Symbol
Definition
a
lattice period (wire medium); loop radius (Ω-medium)
B c
magnetic induction vector √ speed of light, c = 1/ ²0 µ0
D
displacement vector
E
electric field intensity vector
f
frequency
fp
plasma frequency
H
magnetic field intensity vector
I
unit dyadic, I = x0 x0 + y0 y0 + z0 z0
j
imaginary unit
J
current density
J0
Bessel function of the first kind and order zero
J1
Bessel function of the first kind and first order
k
wave vector
k0
√ free space wave number, k0 = ω ²0 µ0
kp
plasma wavenumber
K
magnetoelectric coupling
Kn
normalized magnetoelectric coupling parameter
l
loop stem length
m
magnetic dipole moment
r0
wire radius
n
refraction index
N
inclusion concentration in unit volume
p
electric dipole moment
R
reflection coefficient
T
transmission coefficient
vg
group velocity
vi
vp
phase velocity
x0
unit vector along Cartesian axis x
y0
unit vector along Cartesian axis y
z0
unit vector along Cartesian axis z
Z0
characteristic impedance of a transmission line
α
attenuation constant
β
phase constant
²
permittivity dyadic
²0
permittivity of vacuum
ε = εr
(relative) dielectric permittivity, ε = ²/²0
εeff
effective permittivity
²t
transversal component of permittivity
η
wave impedance of an isotropic medium, η =
p
η0
µ/² p wave impedance of plane waves in vacuum, η0 = µ0 /²0
γ
propagation constant of a transmission line
κ
chirality parameter
λ
wavelength
µ
permeability dyadic
µ0
permeability of vacuum
µeff
effective permeability
µt
transversal component of permeability
ρ
charge density
χ
nonreciprocity parameter
ω
angular frequency
vii
List of abbreviations Abbreviation
Definition
BC-SRR
broadside coupled SRR
BW
backward wave
CPW
coplanar waveguide
CRLH
composite right/left-handed
DNG
double negative
ENG
ε-negative material, material with negative permittivity
LH
left-handed
LHM
left handed material
LHCPW
left handed coplanar waveguide
LHM
left-handed medium
MNG
µ-negative material, material with negative permeability
MWO
Microwave Office (linear circuit simulator by Applied Wave Research, Inc., USA)
MWS
Microwave Studio (full-wave electromagnetic simulator by CST, Germany)
NRI
negative refraction index
RH
right-handed
SD
spatial dispersion
SNG
single negative material
SRR
split-ring resonator
TL
transmission line
WM
wire medium
viii
Chapter 1
Introduction 1.1
Properties of metamaterials
In the last few years, the term metamaterials can be found frequently in the literature. How can they be defined? The prefix meta comes from Greek and means beyond and also of a higher kind. One can find various definitions of metamaterials and they are still being refined as new phenomena appear. According to a general definition, metamaterials are artificial structures which can be characterized by some unique properties that cannot be found in nature1 . These new properties emerge due to specific interactions with electromagnetic fields or due to external electrical control. The specific interactions are caused by inclusions (usually metallic), which are periodically (rarely randomly or irregularly) inserted into the hosting material. The existence of homogeneous substances with negative permittivity ε and/or permeability µ in nature was first discussed in 1967 by Veselago in [2]. Theoretically, four kinds of substances according to the values of ε and µ were considered: • µ > 0 and ε > 0 • µ > 0 and ε < 0 • µ < 0 and ε > 0 • µ < 0 and ε < 0 If we consider only isotropic µ and ε, then the first case involves most isotropic dielectrics. In the second case there will be both gaseous plasma and solid-state plasma. The product of µ and ε is negative, which leads to an imaginary refraction index n and to the propagation of evanescent waves. This fact is well confirmed by experiment, for example, in the ionosphere. 1
This definition implies philosophical questions about what is “naturally formed”. Every engineered material, e.g., teflon or steel, is non-natural and one could classify them as metamaterials too. Moreover, some researchers believe that such materials can exist in nature, since our universe is very extensive and we still do not know everything about it. More about these approaches can be found in [1].
1
1.1. PROPERTIES OF METAMATERIALS The third and fourth case are not possible in nature for isotropic substances, and we denote them with the term metamaterials. It can be derived [2] that waves with a phase that moves in the direction opposite to that of the energy flow (backward waves) can exist in metamaterials, ˇ and also that the Doppler and the Vavilov-Cerenkov effects are reversed. In metamaterials, the wavelength increases and the phase constant decreases with growing frequency. If we consider anisotropic substances, then the quantities µ and ε are generally tensors. Gyrotropic substances are especially interesting in this respect. A well known example of a gyrotropic substance is a plasma in a magnetic field, where ε is a tensor and two waves with opposite polarizations can propagate. Other examples of gyrotropic substances are various magnetic materials in which, in contrast to plasma, it is µ and not ε that is a tensor. Examples of substances where both µ and ε are tensors are pure ferromagnetic metals and semiconductors. Various kinds of passive artificial media are under consideration, and these media can be divided into two large classes. In one of the classes, the spatial period of the inclusions in the hosting material is small compared to the wavelength. This means that the spatial dispersion effects are weak. In the other case we have structures in which those characteristic sizes are comparable with the wavelength. This is a situation of strong spatial dispersion, and the usual material description in terms of its permittivity and permeability loses its sense. It is also worth mentioning that the term metamaterial should be used for three-dimensional volumetric or bulk structures, whereas two-dimensional forms of periodic structures, for instance transmission lines or impedance surfaces, should be called metasurfaces (the terminology is still not unified in the scientific community). In [2] Veselago introduced the term left-handed materials for metamaterials. This term is widely used today, but can have different meanings from different points of view [1,3,4]. Several other names for these materials are also in use: • Veselago media • media with simultaneously negative permittivity and permeability • double-negative (DNG) media • negative index of refraction (NIR or NRI) media • negative phase-velocity media • backward-wave (BW) media • left-handed (LH) media Media with only one negative parameter are sometimes called single-negative (SNG), specifically either ε-negative (ENG) or µ-negative (MNG) media.
2
1.2. HISTORICAL OVERVIEW Many authors use the term left-handed media. It should be noted that when we say something is left- or right-handed, this term can have three different denotations in electromagnetism: • right- or left-handed in structure of materials • right- or left-handed in polarization of the propagating wave • right- or left-handed basis of the three vectors characterizing the wave The first meaning comes from the basic sense: for example, the left hand cannot be superimposed on the right hand by only making rotations and translations. They are mirror images of each other. The Greek word, cheir, for hand, gives us the term chiral for objects that are right- or left-handed (e.g., screws). Chiral materials can be formed from chiral objects: when one mixes left-handed objects into a neutral host medium the result is a left-handed material. The well-known effect of chirality on wave propagation through such a medium is that the polarization plane rotates along the propagation path [5]. Another meaning of right- or left-handed orientation is related to the polarization of the electromagnetic wave, which is determined by the behavior of the electric field vector. Because there are two possibilities in the direction of the rotation, we come to two choices of polarization: left-handed and right-handed. The orientation of the polarization is dependent on the direction of observation. The third use of right- or left-handed orientation is more abstract and is connected with the vector system associated with wave propagation. In an “ordinary” medium with µ > 0 and ε > 0, the system (E, H, k) forms a right-handed triplet, whereas if µ and ε are negative, the triplet is left-handed. For such media, the propagation constant k is opposite to the power flow. From these examples it is obvious that the widely used term left-handed is not very suitable for metamaterials. The term gives the impression that the right- or left-handed orientation is a characteristic of the material itself, which generally is not true. Moreover, it is only the triplet (E, H, k) that is left-handed; taking instead (E, B, k) or (D, H, k) we can derive that these are both right-handed, independently of the signs of µ and ε. The right- or left-handed orientation of the field and the wave vector only makes sense for linearly polarized waves. The terms lefthanded (LH) media and right-handed (RH, i.e., conventional) media, however, are widely used, particularly in metamaterials based on transmission line theory. Other issues concerning the terminology and philosophical aspects of metamaterials can be found, e.g., in [3, 1, 6].
1.2
Historical overview
This section provides a brief historical overview of interest in metamaterials and other general periodic structures. Only the main research directions will be outlined, without aiming to make a comprehensive and complete list of published papers.
3
1.2. HISTORICAL OVERVIEW
1.2.1
First references to metamaterials
In general, structures in which the backward wave can propagate may be found in many fields of physics. Backward waves were observed in mechanical systems by Lamb [7] in 1904. Seemingly, the first person who discussed backward waves in electromagnetism was Schuster [8]. He briefly noted Lamb’s work and gave a speculative discussion of its implications for optical refraction, when a material with such properties were to be found. Around the same time, Pocklington [9] showed that in a specific backward-wave medium a suddenly activated source produces a wave whose group velocity is directed away from the source, while its phase velocity moves toward the source. In electromagnetism, metamaterials are characterized by negative permittivity and permeability. One of the first references to substances with negative material parameters comes from Mandelshtam [10], who proposed the idea of the reversed Snell’s law. The idea of a transmission line with the structure of transversal capacitors and shunt inductors comes from Malyuzhinets [11]. The possibility of negative phase velocity of an electromagnetic wave was predicted by Sivukhin [12]. He observed that media with negative parameters are backward-wave media, but had to state that “media with µ < 0 and ε < 0 were not known. The question of the possibility of their existence was not clarified.” The first more extensive paper on hypothetical materials with negative permittivity and permeability was introduced by Veselago [2]. For centuries people had believed that the refraction index can only be positive, but in this paper it was suggested that it could also be negative. The term left-handed material was used for the first time there. In the case of µ > 0 and ε > 0, the vectors E, H and k form a right-handed triplet, and if µ < 0 and ε < 0, they form a left-handed set. Since vector k is in the direction of the phase velocity, it is clear that left-handed substances are substances with negative group velocity, which occurs in particular in anisotropic substances or when there is spatial dispersion. In [2], several examples are discussed, including the reversed ˇ Doppler effect, the reversed Vavilov-Cerenkov effect, and the refraction of a ray at the boundary between left-handed and conventional media. Veselago introduced the term rightness, which can be positive or negative, to divide materials into two categories: right-handed and left-handed. The question of the proper use of the term left-handed was discussed above. There exist, however, several other works related to metamaterials, which are not usually mentioned. In early 1950s, several artificial dielectrics composed of periodically spaced lattices of metallic rods, wire mesh, or rodded structures were studied. The refraction index of these structures may be less than unity due to their plasma-like behavior [13, 14]. In 1972, Silin published a review [15] which focuses on artificial dielectrics formed as various periodic arrangements of conductive and dielectric elements. Propagation properties are explained in terms of equi- or iso-frequency surfaces. It is argued that flat parallel slabs of artificial dielectrics may convert a divergent beam into a convergent one, total reflection may appear at small incidence angles and partial reflections at larger incident angles. Increasing the incidence angle may re-
4
1.2. HISTORICAL OVERVIEW duce the refraction angle. Depending on the incidence angle, birefringence may occur. The use of a negative refractive medium as a plane-parallel lens was investigated by Silin in [16]. One of the first references to photonic crystals comes from Zengerle and Ulrich [17]. They presented work on optical Bloch waves in (singly and doubly) periodic planar waveguides. As possible applications, the paper discusses interference, transitions to nonperiodic guides, beam steering, negative refraction, and focusing. The authors presented both theoretical simulations and an experimental verification of their predictions. The work [17] was later revised and extended in [18]. Here Zengerle clearly demonstrates (both theoretically and experimentally) negative refraction in the visible spectra and focusing properties of the doubly periodic planar waveguide between unmodulated regions with parallel straight boundaries.
1.2.2
Revival of interest in recent years
Since the end of the 1990s, there has clearly been renewed interest in structures with negative material parameters. To the best knowledge of the author of this thesis, there are now eight monographs [5,19,20,21,22,23,24,25], one journal focused on metamaterials (the Metamaterials journal, published by Elsevier, first appeared at the beginning of 2007), and between the early 1990s and mid 2007 over 800 journal and conference papers referring to metamaterials have appeared (it is worth mentioning that only few of these papers have described a new research direction or breakthrough, whereas the others usually just improve or refine the results achieved earlier). The interest is really enormous nowadays and one could say that metamaterials have formed a new research direction. The possible practical applications of metamaterials are still challenging. The paper published by Veselago in 1967 provoked considerable interest at the time. Surprisingly, however, real metamaterial research did not start until about three decades later, when several seminal papers were published. The idea of having negative effective material parameters and thus a negative refractive index was stunning, but it lacked practical applications. In 1999, Pendry et al. [26] showed that microstructures built from nonmagnetic conducting sheets exhibit an effective magnetic permeability µeff , which can be tuned to values not accessible in naturally occurring materials, including large imaginary components of µeff . The microstructure is on a scale much smaller than the wavelength of radiation, and is not resolved by incident microwave radiation. An example of such a structure is an array of nonmagnetic thin sheets of metal, planar metallic split-rings or so called Swiss rolls [26]. Using Swiss rolls, a very strong magnetic response can be achieved. The idea of a material with a negative refractive index was worked out by Smith in 2000 in the paper [27]. There he drew inspiration from Veselago’s paper and proposed an artificial material which can produce both negative effective permittivity and negative effective permeability. Veselago had been looking for such a material for many years, and had expended huge amounts of time and resources, but unfortunately he did not find the
5
1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS proper way. He had intended to build the metamaterial as a mixture of natural materials with electric and/or magnetic anisotropy. The problem was always with high losses and the resonance of magnetically and electrically anisotropic materials at different frequencies [28]. The idea of Smith et al. was different, since it utilized a composite material, not a homogeneous one. One of the revolutionary metamaterial applications was proposed in 2000 by Pendry in [29]. It is well known that with a conventional optical lens the sharpness of the image is always limited by the wavelength of light used for illuminating the object. An unconventional alternative to a lens, a slab of negative refractive index material, has the power to focus all Fourier components of a two-dimensional image, even those that do not propagate (evanescent waves). The waves decay in amplitude, not in phase, as the distance from the object plane grows. Therefore to focus them we need to amplify them rather than to correct their phase. Pendry showed that such “superlenses” can be implemented in the microwave band with current technology. With this new lens both propagating and evanescent waves contribute to the resolution of the image [29]. Therefore there is no physical obstacle to perfect reconstruction of the image beyond the practical limitations of apertures and perfection of the lens surface. In 2000 Smith et al. [27] demonstrated a composite medium, based on a periodic array of interspaced conducting nonmagnetic splitring resonators (SRRs) and straight wires, which exhibits a frequency region in the microwave regime with simultaneously negative values of effective permeability µeff and permittivity εeff . This array forms an anisotropic medium. A split ring as a magnetic particle had been shown many years before in the well-known antenna textbook by Schelkunoff and Friis [30]. Several works with theoretical derivation of the negative refraction index in metamaterials composed of SRRs and thin wires have been presented [31, 32]. One of the first fabricated and measured structures was introduced by Shelby in 2001 [33]. The material consists of a two-dimensional array of repeated unit cells of copper strips and SRRs on interlocking strips of a standard circuit board material. This structure can be referred to as a bulk form, and was later studied by many authors in various modifications. By measuring the scattering angle of the transmitted beam through a prism fabricated from this material, the negative effective refractive index n was observed.
1.3
Main research directions in metamaterials
After publication of the seminal papers [27,29,33], a great number of research groups started to work in this novel and promising direction. Many different approaches and areas of particular interest appeared, but only the main approaches will be listed below.
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS
1.3.1
Basic theory and general issues
In this category we could include various general approaches and abstract medium considerations regardless of the particular implementation, such as backward-wave media (BWmedia) [34], media with indefinite ε and µ tensors [35] or negative-definite media [36]. Enhancement of radiated and scattered power by a metamaterial dipole coating was proposed in [37], and general radiation enhancement and radiation suppression was proposed in [38, 39]. It is worth mentioning that one must be careful with the conclusions of some papers that claim that the antenna radiates more power with the metamaterial covering than without it. This follows from the causality limitations on the material parameters of any passive media with negligible losses, as shown in [40], where the influence of the quality factor of the covering on the radiation enhancement was studied. One of the possible applications of LHM is to guide waves in an unusual way. Guidance of electromagnetic waves in a dielectric slab is based on total internal reflection. In a material with negative constitutive parameter(s), the supported guided waves cannot be explained only by total internal reflection. An example is guidance with a slab of electric plasma in which the guided waves have an imaginary transverse wavenumber. The implementation of such a medium by wires and the corresponding guided waves were discussed in [41]. Besides guiding with a slab, it is also possible to support such waves when there is only a single interface, instead of two, between regular medium and metamaterials, as described in [42,43]. In these papers, the surface polaritons on the boundary of ordinary media and double-negative materials are described. Waves in a slab of negative refractive medium with simultaneous negative permittivity and permeability were first discussed in [34, 44]. In [45, 46, 47], the guidance conditions of a slab waveguide with imaginary transverse wavenumbers was first shown to exist in addition to the guided modes with real transverse wavenumbers. These modes can be experimentally measured, and can serve as a means to identify an isotropic material that simultaneously possesses negative values of µ and ε. Wave propagation along a rectangular metallic waveguide longitudinally loaded with a metamaterial slab was studied in [48]. Guided modes in a waveguide filled with a pair of single-negative, double-negative, and/or double-positive layers were observed in [49]. Structures with index of refraction equal to zero (at the frequency of interest), sometimes referred to as “chiral nihility”, are introduced in [50, 51]. For a particular class of problems, it is analytically proven that such matched zero-index metamaterials may help to improve the transmission through a waveguide bend. Propagation of waves within a general multilayered metamaterial structure was studied in [52]. A detailed analysis and explanation of how metamaterial layers may be employed to enhance the wave transmission through a single subwavelength aperture in a perfectly conducting flat screen, when used as cover for the screen, is given in [53]. Bounds for the values of permittivity, permeability and refractive index and also the im-
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS portant question of losses are discussed in [54, 55]. This can help with an interpretation of the correctness of the extracted effective material parameters (both calculated and measured), since some authors report results that are evidently physically incorrect. The electromagnetic field energy stored in artificial microwave materials with strong dispersion and losses discussed in [56] is important for further investigation of possible new devices utilizing metamaterials. Composite materials with active inclusions to obey the medium passivity limitation were first proposed in [57]. These active inclusions forming negative capacitance and negative inductance can be realized as impedance inverters with ideal operational amplifiers. Another comprehensive research area is metamaterial homogenization. It is used mainly to validate the agreement between theoretically and experimentally obtained results. Metamaterial structures are usually intrinsically inhomogeneous, however, because they contain very small inclusions compared to the operating wavelength, they exhibit bulk properties, and can be characterized by macroscopic constitutive parameters. There have been many homogenization methods, and the pioneering work is linked with names such as Ewald and Lorentz, from the 1920s. Generally different approaches than those for ordinary materials are needed for homogenization of metamaterials, and there is still certain confusion in this area in the scientific community. Both numerical and experimental techniques for characterization of metamaterials have been developed. Smith et al. calculated the effective parameters by averaging the field in the structure [32], but this is difficult to carry out in an experiment. More suitable for experiment are methods using scattering parameters to calculate the effective permittivity and permeability [58], also extended for bianisotropic materials [59]. Other simple waveguide methods for retrieving effective parameters of single-negative materials are described in [60, 61]. A method for characterizing the band structure of general three-dimensional metallic crystals was proposed in [62]. A homogenization technique for computing the quasi-static effective parameters of a three-dimensional lossless lattice of dielectric particles with complex shape was proposed in [63]. A two-step approach which homogenizes composite materials, composed of, e.g., SRRs and wires, where the effective medium model is built layer by layer as an anisotropic single negative slab, is proposed in [64]. In the homogenization of metamaterials, additional boundary conditions have to be introduced [65]. One must be careful when extracting the parameters, since these parameters must be physical and satisfy the Kramer-Kronig relations. In many cases, an insufficient number of structural elements are used in simulations and experiments, and the conclusions become physically meaningless since they do not satisfy the conditions for passivity, causality, and absence of radiation losses2 . 2
It is theoretically possible to extract effective material parameters of the slab with few or so far just one inclusion across the layer as a slab of a continuous medium. In a layer with only one particle across it there is no uniform refraction index, no uniform wave-vector, and one particle cannot form the medium. However, a layer with single crystal plane can really be presented as a layer of a bulk medium if we introduce two thin transition layers with special properties at its interfaces (see, e.g., [66]). It is necessary to consider this fact in the process of retrieving effective parameters.
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS Several attempts to build an isotropic double-negative medium have been reported. A study of a composite medium consisting of insulating magnetodielectric spherical particles embedded in a background matrix in order to obtain a double negative medium was reported in [67]. In [68], a fully symmetric multigap single-ring split-ring resonator design crossing continuous wires was proposed. This idea is, as the authors report, not very suitable for an experimental realization and remains as only a theoretical approach. A way to realize an isotropic doublenegative material up to optical frequencies using a homogeneous structure consisting of dielectric spheres embedded randomly in a negative permittivity host medium was proposed in [69]. A structure made of periodic arrays of pairs of H-shaped metallic wires that offer a potentially simpler approach in building negative-index materials is proposed in [70]. A hybrid structure composed of S-shaped and Ω-shaped inclusions is designed, fabricated and measured in [71]. Generally, structures with properties similar to those of electromagnetic metamaterials may be found even in other parts of physics. Let us give an example from acoustics: the concept of an acoustic superlens opening opportunities to design acoustic metamaterials for ultrasonic imaging is introduced in [72]. The authors report that a negative effective-mass density is the necessary condition for the existence of surface states on acoustic metamaterials. Another interesting practical application of metamaterials is a cloaking device. Let us imagine an object covered by a hypothetical material which is able to bend the electromagnetic waves passing through so that they go around the object and then return undisturbed to their original trajectories. Then for an observer outside it seems that the object is “invisible”. The idea was first suggested by Pendry et al. in [73] and then elaborated in [74] where spherical and cylindrical cloaks were worked out. The authors described a method in which the transformation properties of Maxwell equations and the constitutive relations can yield material descriptions that allow such manipulations with the beam trajectory. Leonhardt described a similar method, where the two-dimensional Helmholtz equation is transformed to produce similar effects in the geometric limit [75]. Probably the first practical realization of a two-dimensional cloaking device was demonstrated under certain approximations in [76]. A copper cylinder was “hidden” inside a cloak constructed with the use of artificially structured metamaterials, designed for operation over a band of microwave frequencies. The cloak decreases the scattering from the hidden object while at the same time reduces its shadow, so that the cloak and object combined begin to resemble empty space. No real three-dimensional cloaking device has been fabricated so far.
1.3.2
Negative permittivity
The medium formed by a lattice of ideally conducting parallel thin wires (wire medium) has been known for a long time [13,77,78] 3 . When the wavelength of the incident radiation is much 3
A lattice of parallel wires in one direction is referred to as a (single) WM, while a three dimensional lattice formed of two mutually perpendicular single WM arrays is referred to as a double WM. A three dimensional lattice formed by three mutually perpendicular single WM arrays is referred to as a triple WM.
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS longer than the intrinsic length-scales of the structure, it is very helpful to consider this medium as a homogeneous material with averaged constitutive material parameters (effective medium theory). In the framework of effective medium theory, wire media (WM) are described by the plasma model, where the corresponding component of the permittivity dyadic is expressed by the Drude formula. Pendry et al. [79, 80] and Sievenpiper [81] independently demonstrated that metallic wire-mesh structures have a low frequency stop band from zero frequency to the cutoff frequency. They attributed this to the motion of the electrons in the metal wires and resonances of surface plasmons4 . To describe the existence and properties of surface plasmons, one can choose from various models. The simplest way to approach the problem is to treat each material as a homogeneous continuum, described by a dielectric constant. The condition for excitation of electronic surface plasmons is that the real part of the dielectric constant of the metal must be negative and its magnitude must be greater than that of the dielectric. This condition is met in the IR-visible wavelength region for air/metal and water/metal interfaces (where the real dielectric constant of a metal is negative and that of air or water is positive). The low frequency stop band caused by the plasmon excitation can be attributed to effective negative dielectric permittivity, and, when the wire lattice dimensions are chosen properly, negative permittivity can be obtained even at microwave frequencies [79]. The plasma model has been corrected by introducing spatial dispersion (SD) into the Drude formula [82]. The propagating waves “feel” the structure of the medium (the length scale or the lattice period is comparable to the wavelength) and complex diffraction and scattering phenomena take place, which have to be analyzed locally, on the level of individual elements. We speak about strong spatial dispersion in structures, where the spatial period is close to the half-wavelength of the wave propagating in the medium (photonic bandgaps in the optical band or electrical filters in the microwave region). When the characteristic size of the structure (lattice period or inclusion size) is much smaller than the wavelength but still cannot be neglected, then we speak about weak spatial dispersion. Spatial dispersion can also be defined as a nonlocal dispersive behavior of the material, i.e., the constitutive permittivity and permeability tensors depend not only on the frequency, but also on the spatial derivatives of the electric and magnetic field vectors or, for plane electromagnetic waves, on the wave-vector components determining the direction of propagation. Initially, a strong spatial dispersion effect in WM was predicted only in the shortwavelength limit [82]. In recent years, modes in both connected and nonconnected double wire media have been studied [83,84,85,86] and spatial dispersion effects have been discussed not only at low frequencies, but even above plasma resonance frequencies. By comparing the effective medium approach with the full-wave 3D Green’s function method, it was shown in [85] that 4
Surface plasmons, also known as surface plasmon polaritons, are surface electromagnetic waves that propagate parallel along a metal/dielectric (or metal/vacuum) interface. Since the wave is on the boundary of the metal and the external medium (air or water for example), these oscillations are very sensitive to any change of this boundary, such as the adsorption of molecules to the metal surface.
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS effective medium theory is applicable above plasma resonance if the wires are thin compared to the lattice period. A simple model of cold plasma was used by Shapiro et al. [87], who observed the surface waves at the interface between the wire mesh and the free space, and studied the influence of spatial dispersion on these waves. The band-structure and plasma frequency of a two-dimensional array of generally-shaped metallic rods was presented in [88] up to THz frequencies. Plasmon modes of a two-dimensional lattice of long conducting circular wires are investigated in [89]. An exact calculation of plasmon modes in a general periodic wire medium is still an unsolved task. A triple wire medium composed of thin connected metallic wires will be the subject of Chapter 3 of this thesis. Both calculated and measured results will be presented. It will be shown that the triple wire medium can be considered as an isotropic negative permittivity medium in the vicinity of its plasma frequency. There exist as well different approaches for achieving a negative effective permittivity. An alternative to the wire medium was presented in [90]. An inductive-capacitive resonator with a strong electric response is described which can be fabricated using a planar technology. This idea was used in [91] for a proposal of an isotropic negative permittivity medium. The effective permittivity of the material, however, is extracted only from a very small number of unit cells and thus its physical correctness is questionable. The single-negative material described in [90] was recently extended to a double-negative material by placing electrical resonators and magnetic SRRs together [92].
1.3.3
Negative permeability
The basic unit of most of the negative permeability materials is a split-ring resonator (SRR). It had been known for many years [30] and was adapted by Pendry in 1999 [26]. A modified version of the SRR that does not present magnetoelectric coupling, the so-called modified or broadside coupled SRR (BC-SRR), was introduced in [93]. In [94], it was shown that electromagnetic transmission in a hollow metallic waveguide filled with SRRs is feasible within a certain frequency band even if the transverse dimensions of the waveguide are much smaller than the associated free-space wavelength. This effect can be explained by metamaterial theory and further experimental validation can be provided. The magnetic resonances in SRR are studied in detail numerically and experimentally in [95]. A planar form of wires and SRRs introduced in [33] was proposed in [96]. Several compact band pass filters based on planar structures with three metal levels were proposed. The central layer consists of a coplanar waveguide with periodic wire connections between the central strip and the ground planes. In the upper and lower metal levels, SRRs are etched and aligned with the slots. The wires cause the structure to behave as a microwave plasma with negative effective permittivity covering a wide frequency range. Split-ring resonators, which are magnetically coupled to the CPW, provide negative magnetic
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS permeability in a narrow frequency range above their resonant frequency. The equivalent circuit of this structure was presented in [97]. The role of the magnetoelectric coupling in SRRs can be minimized by the proper choice of geometrical dimensions or by applying symmetries in the design, but some residual coupling still remains. A method for calculation and measurement of the magnetoelectric coupling (bianisotropy) of SRRs is proposed in [98]. The possibility of designing isotropic 3D magnetic resonators by properly arranging modified SRRs was studied in [99]. A new possible way to design artificial magnetic materials for microwave frequencies was introduced in [100]. A novel magnetic particle (metasolenoid) formed by a stack of many parallel and very closely spaced single broken loops was proposed and studied analytically, numerically, and experimentally. This structure was subsequently used enhancing antenna properties [101, 102]
1.3.4
Perfect lens
From optics it is well known that no lens can focus light on an area smaller than the square wavelength, thus it is not able to provide an image of objects smaller than the wavelength of the light illuminating the object. The hypothetical perfect lens overcomes this rule. Unlike conventional optical components, it will focus both the propagating spectra and the evanescent waves, and thus be capable of achieving diffraction-free imaging. In 2000, Pendry [29] predicted an intriguing property of such a LH lens (later refined in [103,104], a solution of a spherical perfect lens was given in [105]). Such a lens is formed by a planar slab of double-negative material. Currently, samples of double-negative materials and are created mainly in the microwave region [33, 106, 107, 108, 109] due to the difficulty of achieving a strong magnetic response at higher frequencies. Planar lenses, however, can operate only when the source is close to the lens (the distance is related with the slab thickness). For practical applications, such as telescopes and microwave communications, distant radiation needs to be focused. In order to focus far field radiation, the negative refractive index (NRI) lens with a concave surface is generally used. The monochromatic imaging quality of a conventional optical lens can be characterized by the five Seidel aberrations: spherical, coma, astigmatism, field curvature, and distortion. In [110], aberrations in a general NRI concave lens have been studied by Smith and Schurig. They found that this lens allows reduction or elimination of more aberrations than can be achieved with only positive refractive index media. The 3D superresolution in metamaterial slabs was experimentally shown in [111], and the theory was later generalized to an arbitrary subdiffraction imaging system [112]. The sensitivity of NRI lens resolution to the material imperfection is studied in [113]. The creation of left-handed materials at THz frequencies and in the optical range meets with problems in getting the required magnetic properties [114, 115], which have to be created
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS artificially. In the absence of magnetic properties, the lenses formed by materials with negative permittivity only (for example, silver at optical frequencies [29, 116] or an array of metallic nanorods [117]) are still able to create images with subwavelength resolution, but the operation is restricted to p-polarization only and the lens has to be thin as compared to the wavelength. This idea was confirmed by experimental results [118], which demonstrated the reality of subwavelength imaging using silver slabs in the optical frequency range. The resolution of such lenses is restricted by losses in the silver. At the present time there is no recipe for increasing the thickness of such lenses other than the introduction of artificial magnetism.
1.3.5
Transmission-line implementations
In 2002, Eleftheriades [119] introduced a metamaterial based on the transmission line approach. The dual transmission line model was used, which is composed of transversal capacitors and shunt inductors instead of transversal inductors and shunt capacitors as in a conventional transmission line. An antenna was fabricated and the existence of a backward wave was experimentally verified. In [120], a two-dimensional periodic version of the dual transmission line network was used to demonstrate negative refraction and focusing. The fabricated structure, full-wave field simulations illustrating negative refraction and focusing, and the first experimental verification of focusing using such an implementation were presented. This structure was later improved in [121]. A quasi-lumped element microstrip implementation of a left-handed coupler was proposed [122] by Caloz, Itoh et al. It was demonstrated that this coupler requires a shorter coupling length than do conventional right-handed coupled-line forward couplers, and this makes the device much more compact. In [123] various compact one-dimensional phase shifters were proposed using alternating sections of negative metamaterials and printed transmission lines in coplanar waveguide (CPW) technology. The metamaterial sections consist of lumped element capacitors and inductors, arranged in a dual transmission line (high-pass) configuration. Many microwave elements using metamaterial properties were introduced by Itoh, Caloz et al. In [122] the zeroth-order resonator is presented. The resonant frequency of the zeroth-order resonator is independent from the physical length, so that the resonator can be arbitrarily small under the condition that sufficient reactance is squeezed in a small length. This type of resonator is characterized by a theoretically infinite-wavelength wave. A planar antenna based on this resonator was introduced in [124]. In [125] a multi-layered left-handed transmission line composed of vertically stacked unit cells was presented. The possibility of exploiting the left-handed phase advance of this transmission line for dual-band microwave components was suggested. This structure was later improved in [126], where a 1 GHz/2 GHz diplexer using this multilayered transmission line was demonstrated. Arbitrary dual-band microstrip components using composite right/left-handed transmission lines (CRLH TL) are presented in [127]. The dual-band λ/4 open/short-circuit stub, dual-band λ/4 CRLH TL, dual-band branch-line
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS coupler, dual-band rat-race coupler are also demonstrated. In these devices, the frequency ratio of the two operating frequencies can be a noninteger, in contrast to conventionally designed devices. The use of metamaterials in highly-selective bandpass filters and diplexers is presented in [128]. A novel CRLH TL is composed of capacitively coupled dielectric resonators. The narrow band pass response can be obtained with the help of the dielectric resonator resonance to ruin the balanced condition of CRLH TL. A novel N-port series divider based on a CRLH TL supporting a wave with an infinite wavelength is presented in [129]. This novel divider evenly divides power in phase to an infinite number of ports at its infinite wavelength frequency. In addition, it is shown that the performance of the novel series divider is not dependent on the location of its output ports. The operating principle is based on the fact that a CRLH TL is able to support a wave with a theoretically infinite wavelength. As a result, all points along the CRLH TL have the same magnitude and phase. 3-port and 5-port series dividers consisting of an 8 unit cell and a 13 unit-cell CRLH TL were shown. A left-handed transmission line without the need for lumped elements or vias was demonstrated in [130]. The line consists of a multilayered dielectric substrate and a two-layer microstrip line. Applying the electromagnetic field appropriately, one can achieve negative permittivity and permeability. Two-layer SRRs consist of two concentric square rings, whose splits are oriented opposite to each other. To demonstrate the line properties, a microstrip combline antenna array was demonstrated. Most of the results of Caloz, Itoh et al. were summarized in the monograph [22] and the results of Eleftheriades et al. in the monograph [23]. The homogenization of transmission line based metamaterials (rather metasurfaces) is discussed in [131], where full-wave simulations and experiments with CRLH mushroom structures are performed. A concept of a dual composite right/left-handed (D-CRLH) transmission line metamaterial is introduced in [132]. The D-CRLH is the “dual” of the conventional CRLH in the sense that it has a series parallel (instead of series) LC tank and a shunt series (instead of parallel) LC tank. This topological duality results in dual properties. The LC equivalent structure is not necessarily needed for the design of a TRL metamaterial as shown in [133]. The TRL metamaterial, having shunt and series inductances electromagnetically coupled together, is introduced. Most of the transmission line metamaterials are based on a microstrip line and need via holes connecting the two metallized sides of the substrate, or lumped elements. These vias add parasitic unwanted elements to the equivalent circuit. A fully uniplanar structure which does not contain any lumped elements or vias, the left-handed coplanar waveguide (LHCPW), will be introduced in Chapter 2 of this thesis. Its layout is a modification of the line described in [134], but is more compact. The line presented in Chapter 2 is applied in microwave circuits, whereas the structure in [134] was aimed for a leaky wave antenna design. Almost at the same time as
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS the LHCPW structure was being studied by the author of this thesis, a very similar LHCPW was presented by Mao et al. in [135], where also the equivalent circuit of the unit cell was determined. The equivalent circuit elements were extracted simultaneously from the S-parameters based on effective medium theory and from the equivalent circuit [130]. The circuit elements, however, are valid only for one specific fabricated line, and no dependence on the geometrical dimensions of the structure is described. Unfortunately, the study of the LHCPW presented in Chapter 2 of this thesis has not led to any practical application. However, a power splitter and balun [136], bandpass and dual-passband CPW filters [137] and a 3 dB directional coupler [138] based on the line from [135] were recently fabricated and measured by Mao et al.
1.3.6
Chiral and bianisotropic materials
Chiral media provide another example of metamaterials with promising properties [5]. In chiral media, the spatial geometrical character of the internal structure (antisymmetry or nonsymmetry with respect to the mirror reflection) rotates the polarization of the propagating plane wave. This rotation is due to the magnetoelectric coupling caused by chiral elements. A further generalization is bianisotropy, i.e., the medium shows both anisotropy and magnetoelectric coupling [19]. The bianisotropy of SRRs was studied in [93]. A way to obtain negative refraction with chiral materials was proposed by Pendry in [139] and later extended by several research groups, see, e.g., [50, 140, 141, 142, 143, 144, 145]. The advantage of these structures is that they can provide both negative effective permittivity and negative effective permeability with just one particle instead of the two (e.g., SRRs and wires) needed in conventional metamaterials. There are also some possible ways to excite backward waves in anisotropic media [34, 146, 147, 148] and in anisotropic waveguides [149], which do not necessarily require magnetic properties of the materials, but are restricted to strongly anisotropic structures. In addition, difficulties to fabricate backward-wave samples matched to free space inhibit potential applications both in the microwave and in the optical regions. The possibility of achieving a negative refractive index in gyrotropically magnetoelectric materials was studied in [150] and [151] (see also the Comment [152] followed by the Reply to Comment [153]). A proposal for bianisotropic implementation of a perfectly-matched slab will be shown in Chapter 4 of this thesis. The design is based on a pseudochiral, so called Ω-particle, proposed by Saadoun and Engheta in [154]. Ω-media were studied by Tretyakov, Simovski et al. in [155, 156, 157] and the electrodynamics of bianisotropic media was summarized in [19]. Other bianisotropic particles, namely S-ring resonators, have been introduced recently in [158].
1.3.7
THz and optical applications
The idea of metamaterials has been quickly adopted even to THz and optical frequencies, since nanofabrication and subwavelength imaging have developed rapidly in recent years. When
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS the light interacts with a conventional material, then from the electric and magnetic component of the light only the electric component probes the atoms of a material effectively whereas the interaction with the magnetic component is usually weak. However, metamaterials allow both components of the light to be coupled, when they are designed properly. This can lead to fascinating applications such as superlenses, optical nanolithography, “invisibility cloaks”, etc. The metamaterials known from microwave frequencies, however, are difficult to fabricate at optical frequencies. For materials at optical frequencies, the dielectric permittivity ε is very different from that in a vacuum, whereas the magnetic permeability µ (for natural materials) is close to that in a vacuum. This is because of the weak interaction with the magnetic component of the light, mentioned above [159]. Conventional SRRs operating at GHz frequencies were scaled down up to 1 THz frequencies [114], and modified SRRs up to tens of THz [160, 161]. The design principle of SRRs for realizing a negative magnetic metamaterial is proposed through a theoretical investigation of the magnetic properties of SRRs from THz to the visible light region in [162]. A substrate-free three-layer terahertz metamaterial structure, based on Pendry’s nested split rings and discontinuous bars, resonant at ≈ 4.92 THz, is proposed in [163]. The first experimental demonstrations of a negative refractive index in the optical range were accomplished, nearly at the same time, for pairs of metal rods [164] and for the inverted system of pairs of dielectric voids in metal [165]. Electromagnetic wave scattering by many small particles of arbitrary shapes is analytically studied in [166]. So far, the largest figure of merit for negative refractive index materials in the optical range was achieved in [167].
1.3.8
Tunable metamaterials
Most of the practical metamaterial implementations so far have been narrow-band resonant structures. No wonder that researchers would like to achieve more broadband and tunable devices. Metamaterial applications can be divided into two big groups: the first one usually uses active devices to tune the capacitance or inductance in order to achieve left-handed behavior in various frequency bands (varactor-loaded split-ring resonators [168, 95] coupled to microstrip lines can lead to metamaterial transmission lines with tuning capability [169], a tunable impedance surface [170], varactors using reverse biased Schottky diodes [171], etc.), whereas the other group utilizes generally anisotropic natural materials with properties that are dependent on external electric or magnetic field intensity. Such examples are nonlinear dielectric substrates [172], a combination of yttrium iron garnet and a periodic array of copper wires [173, 174] or tunable phase shifters based on ferroelectric films [175] which have properties dependent on external fields. Gorkunov and Lapine studied ways of tuning the metamaterial band gap by applying an external magnetic field to change the properties of the resistance insertions or the capacitance insertions [176]. Utilization of micro-electro-mechanical structures (MEMS) as tunable capacitors is shown in [177]. An interesting way to make a tunable meta-
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1.3. MAIN RESEARCH DIRECTIONS IN METAMATERIALS material, a left-handed transmission line based on the periodic structure of an array of unbiased Josephson Junctions, was proposed in [178]. The effect of substrate thickness and losses and the resulting tunability of the refractive index in metamaterials is studied in [179]. A general issue of a two-dimensional wire medium loaded by a tunable capacity was proposed in [180]. This material would serve as a metasurface with tunable impedance and thus with tunable reflection properties. Another extensive class of tunable metamaterials is formed by liquid crystals. The properties of liquid crystals can be changed easily by applying an external electric field. Hence, they are adaptable for tuning. Bush and John [181] predicted the tunability of the band structure in photonic crystals utilizing liquid crystals. Takeda and Yoshino [182] showed tunable refraction effects in photonic crystal structures infiltrated by liquid crystals. Liu and Chen [183] demonstrated that the photonic band gap of a two-dimensional photonic crystal can be modulated by a nematic liquid crystal. Electrically controlled negative refraction in a nematic liquid crystal can be found in [184, 185, 186]. Generally, there are two kinds of negative refraction phenomena in photonic crystals [187]. The first one is the left-handed behavior, as described by Veselago. The other is realized without a negative refraction index or left-handed behavior, but by higher order Bragg scattering [188] or anisotropy [189].
1.3.9
Practical applications
As indicated above, metamaterials and metasurfaces are characterized by many unusual properties which can be utilized in practical applications. Most of the fabricated samples enhance the properties of known devices (mainly microwave passive components), but some of them introduce novel devices utilizing the unusual properties of backward-wave materials. Only some of them will be outlined in this section. A broadband microstrip balun with the use of a CRLH metamaterial was presented in [190]. The reported properties of the balun are significantly better than those of a conventional balun. An electronically scanned periodic microstrip leaky-wave antenna based on the concept of CRLH metamaterials is presented in [191]. This antenna includes varactors modulating the capacitive loading of the unit cell and therefore the propagation constant of the structure, which results in voltage scanning of the radiated beam. This antenna was later improved in [192]. A way to implement compact phase shifter circuits with the principle of periodically L-C loaded lines with left-handed wave propagation is demonstrated in this paper. A phase shifting circuit with varactor diodes is developed as a demonstrator exhibiting reasonable accordance with simulation results. Enhancing the directivity of patch antennas using a metamaterial superstrate is discussed in [193]. An all-dimensional subwavelength cavity breaking the size restrictions is introduced in [194]. High selectivity low-pass filters using negative-ε metamaterial resonators are introduced in [195]. Electrically small antenna elements using negative permittivity resonators
17
1.4. AIMS OF THE THESIS are demonstrated in [196], and a more comprehensive theoretical background for the improvement of antenna properties is given in [197]. Artificial magnetic materials are used for enhancing the properties of patch antennas, both as substrates (magnetic [198, 102] or electric [199]) and also as superstrates [200, 201]. A 5-bit time delay module with an LH and conventional transmission line is designed and fabricated in [202]. The delay line is more compact and shows lower losses in comparison to conventional solutions. Very wideband and compact filters are designed in [203] with the use of a microstrip line loaded with complementary SRRs, series gaps and grounded stubs. A metamaterial insulator able to suppress mutual coupling between densely packed array elements, i.e., in antenna arrays, is proposed in [204]. The most promising applications of metamaterials, however, are expected in the range of THz and even visible light frequencies. The most important applications so far are namely the superlens, photolithography or near-field microscopy, and their successful practical implementation is still challenging.
1.4
Aims of the thesis
The general aim of this work is to study the propagation of electromagnetic waves in periodic structures, namely in metamaterials. When research in this field began, metamaterial structures in the form of transmission lines were widely studied by many research teams all over the world for their simplicity and relatively easy practical implementation. The first aim of this doctoral thesis is to study a uniplanar transmission line metamaterial utilizing a coplanar waveguide. The achieved results will be presented in Chapter 2. Another challenging metamaterial research direction is the construction of isotropic materials with either negative effective permittivity or negative effective permeability. An isotropic negative permittivity material in the form of a triple wire medium will be introduced in Chapter 3. At first there were even attempts to build an isotropic negative permittivity material with small metallic chiral or bianisotropic inclusions in the shape of the Greek letter Ω. Later it was proved, however, that these structures are not suitable for constructing such materials, and a different promising application utilizing these inclusions was found. A perfectly matched bianisotropic slab supporting a backward wave will be shown in Chapter 4. Such a slab has several advantages over conventional metamaterial slab designs.
18
Chapter 2
Left-handed coplanar waveguide This section will provide a brief description of the general properties of planar metamaterials based on the transmission line approach. An extensive theoretical background and practical applications of transmission-line metamaterials can be found, e.g., in [22], along with numerous references. In particular, the left-handed coplanar waveguide will be introduced1 .
2.1
Transmission line model of a metamaterial
A medium transmitting an electromagnetic wave can be modeled by an equivalent homogeneous transmission line. A circuit model of this line consists of elementary (unit) cells. The equivalent circuit of the unit cell is shown in Fig. 2.1a. It consists of a series impedance Zs0 and a parallel admittance Yp0 taken per unit length d, which must be infinitesimally short. The characteristic impedance Z0 and the propagation constant γ of this transmission line are [205] s Z0 =
Zs0 , Yp0
q γ = ± Zs0 Yp0 ,
(2.1)
(2.2)
where γ = α + jβ, α is the attenuation constant and β is the phase constant2 . A standard transmission line has the inductive series impedance representing the stored energy of the magnetic field and the capacitive parallel admittance representing the stored 0 , energy of the electric field, Fig. 2.1b. Thus for a lossless line we have Zs0 = jωL0R , Yp0 = jωCR 1 The term left-handed is not very suitable for the general description of metamaterials, as discussed in the introductory section. In the theory of metamaterials based on transmission lines, however, this term is widely used and it will also be adopted in this chapter. 2 In the theory of transmission lines, the propagation constant is usually denoted as γ = α + jβ, where α stands for the attenuation factor and β for the phase constant. In other parts of physics, the widely accepted general notation of the propagation constant of the form k = β − jα is usually adopted, with the same meaning of symbols α and β. The relation γ = jk holds between these two notations. The difference between these two notations will be stressed in the text in cases where confusion may occur.
19
2.1. TRANSMISSION LINE MODEL OF A METAMATERIAL
Z's
L'R Y'p
C'R
d (a)
(b)
C'L
C'L
L'L
L'R
C'R
(c)
L'L
(d)
Figure 2.1: The equivalent circuit of a general transmission line (a), a standard L–C line (b), a C–L left-handed line (c), and a real LH transmission line (d). All circuit elements are taken per unit length d.
and the well known formulas follow from (2.1), (2.2) are now s Z0 =
L0R 0 , CR
(2.3)
q γ = jβ = jω
0 . L0R CR
(2.4)
This propagation constant γ (2.4) corresponds to the propagation of a standard forward wave along the line known as a right-handed (RH) wave. From (2.4) we get the phase velocity equal to the group velocity ω 1 =p 0 0 , β LR CR
(2.5)
1 1 =p 0 0 . ∂β/ LR CR ∂ω
(2.6)
vp =
vg =
Now let us consider the dual case with exchanged positions of the capacitor and the inductor, Fig. 2.1c. In this way we have changed the original L–C low-pass structure into a C–L high-pass structure. For a lossless line now we have3 Zs0 = 1/(jωCL0 ) and Yp0 = 1/(jωL0L ). Inserting them into (2.1), (2.2) we get
s Z0 =
L0L , CL0
1 β=− p 0 0 . ω L L CL
(2.7)
(2.8)
3 The impedance Zs and admittance Yp taken per unit length have the physical dimension of [Ω/m] and [S/m], respectively. In the case of the circuit from Fig. 2.1b, the series inductance is L0R = LR /d and the shunt capacitance 0 is CR = CR /d. By contrast, the circuit elements from Fig. 2.1c are given by L0L = LL d and CL0 = CL d.
20
2.1. TRANSMISSION LINE MODEL OF A METAMATERIAL From the phase constant (2.8) we get the phase and group velocities q ω = −ω 2 L0L CL0 , β
vp =
q 1 2 vg = = ω L0L CL0 . ∂β/ ∂ω
(2.9)
(2.10)
There are different signs in (2.9) and (2.10). The group velocity has the opposite direction to the phase velocity. This is a feature of a backward wave sometimes referred to as a lefthanded wave. The phase propagates in the direction opposite to the flow of the transmitted power. The magnitude of the phase constant (2.8) decreases with frequency, which means that the wavelength increases with frequency. An LH transmission line with the equivalent circuit from Fig. 2.1c cannot be fabricated as simply as it was introduced here. The inductors and capacitors are inserted into a real hosting environment, which has the character shown in Fig. 2.1b. Therefore we have the cell equivalent circuit as shown in Fig. 2.1d. The series impedance, the parallel admittance and the phase constant are Zs0 = jωL0R +
1 , jωCL0
(2.11)
0 Yp0 = jωCR +
1 , jωL0L
(2.12)
s β=
0 ω 2 L0R CR
1 + 2 0 0 − ω LL CL
µ
0 ¶ L0R CR + 0 . L0L CL
(2.13)
This general transmission line consists of a combination of a conventional right-handed (RH) transmission line (series branch, low-pass) and its dual, LH transmission line (parallel branch, high-pass), forming a composite right/left handed (CRLH) transmission line. At low frequencies, p the line is dominantly LH with the hyperbolic β = −1/(ω L0L CL0 ), while at high frequencies it p 0 . One can introduce new variables [22] is dominantly RH, with the linear β = ω L0R CR 1 0 L0R CR
(rad · m)/s,
(2.14a)
1 ωL0 = p 0 0 L L CL
rad/(m · s),
(2.14b)
0 ωR =p
0 ξ = L0R CL0 + L0L CR
21
(s/rad)2 ,
(2.14c)
2.1. TRANSMISSION LINE MODEL OF A METAMATERIAL and the shunt and series resonance frequencies 1 ωsh = p 0 0 LL CR
rad/s,
(2.15a)
1 L0R CL0
rad/s,
(2.15b)
ωse = p
and subsequently an explicit expression for the complex propagation constant can be obtained by inserting equations (2.11) and (2.12) into (2.2) with the use of eq. (2.14) sµ γ = α + jβ = js (ω)
ω 0 ωR
¶2
µ +
ωL0 ω
¶2 0
− ξωL2 ,
(2.16)
where s(ω) is following sign function −1 if s (ω) = +1 if
ω < min (ωse , ωsh )
LH range,
ω > max (ωse , ωsh )
RH range.
(2.17)
The propagation constant in (2.16) is not necessarily purely imaginary γ = jβ, which corresponds to a pass band; it can be purely real γ = α in some frequency ranges, which corresponds to a stop band4 . The dispersion/attenuation relation for the CRLH (2.16) is plotted in Fig. 2.2 together with curves corresponding to pure right- and left-handed transmission lines. The gap in the dispersion diagram (stop band) is due to the different series and shunt resonance frequencies ωse , ωsh . Then we speak about a balanced (ωse = ωsh ) or unbalanced (ωse 6= ωsh ) CRLH transmission line. The frequency ω0 corresponds to the frequency of maximum attenuation and can be derived as
q ω0 =
1 0 ω0 = p ωR . L 0 0 L0 C 0 4 LR CR L L
(2.18)
More information about the important case of balanced and unbalanced CRLH transmission lines and effects on the design of practical applications can be found in [22]. The previous theory was valid only for the case that the unit cell length is infinitesimally short. In the case of cell length comparable to the wavelength, formula (2.13) is not valid and the magnitude of the error increases with increasing cell length. In this case, the line must be treated as a periodic structure with a finite unit cell length. To derive the dispersion characteristic of a periodic structure, the transmission matrix of the cell is calculated according to [206], whereas the unit cell equivalent circuit must be modified according to Fig. 2.3. It is the n-th unit cell with length d of the periodic structure, or of the transmission line, which is periodically loaded with a T-network. The relationships between the input variables un , in and the output variables un+1 , 4
Note that the line is lossless and the purely real propagation constant γ = α does not have a physical meaning of losses, since the wave is evanescent in this band.
22
2.1. TRANSMISSION LINE MODEL OF A METAMATERIAL w bRH
bPLH
max(wse , wsh ) w0 min(wse , wsh )
bPRH
a
bLH b, a Figure 2.2: Dispersion/attenuation diagram according to (2.16) for a CRLH transmission line for energy propagation along the +z direction (vg > 0). The meaning of the symbols is as follows: βRH = phase constant of the right-handed section of the CRLH line, βLH = phase constant of the left-handed section of the CRLH line, βP LH = phase constant of a purely left-handed line, βP RH = phase constant of a purely right-handed line.
in
un
Z/2
Z/2
Z0 k0 d/2
Z0 k0 d/2
Y
i n+1
un+1
Figure 2.3: The unit cell of the equivalent circuit modified for derivation of the dispersion characteristic.
in+1 are readily found using the ABCD matrix [206]. The un and in are the total voltage and current amplitudes, respectively. The background hosting medium is represented5 by a section of a transmission line with characteristic impedance Z0 and free space propagation constant k0 = ω/c0 , where c0 is the velocity of light in a vacuum. The relationship between the input and output variables is of the form
un in
=
A B C
D
un+1 in+1
,
(2.19)
where the ABCD matrix is given by the product of the matrices of three elements connected in a cascade
¡ d¢ ¡ d¢ A B cos k0 2 j sin k0 2 1 + Y2Z = ¡ d¢ ¡ d¢ Y j sin k0 2 C D cos k0 2 Y0
4Z+Z 2 Y 4Z0 1 + Y2Z
¡ d¢ ¡ d¢ cos k0 2 j sin k0 2 ¡ d¢ ¡ ¢ j sin k0 2 cos k0 d2 (2.20)
and where Y0 = 1/Z0 is the characteristic admittance of the hosting medium. The input and output variables are bound according to Floquet’s theorem with the propagation constant k in 5 The background hosting medium with Z0 and k0 is chosen only for simplicity of the derivations, however it can be formed by a general medium with parameters not equal to those of free space.
23
2.2. LEFT-HANDED COPLANAR WAVEGUIDE the medium un+1 = un e−jkd ,
(2.21)
in+1 = in e−jkd .
(2.22)
Substituting these relations in (2.19) we get
A − ejkd
B
C
D − ejkd
un+1 in+1
=
0 0
.
(2.23)
The dispersion equation is obtained from the condition for existence of a non-trivial solution of this equation system. The determinant of the matrix term in (2.23) must be zero AD + ej2kd − (A + D) ejkd − BC = 0.
(2.24)
The cascade matrix in (2.19) describes a symmetric network, thus AD−BC = 1 and for reciprocal circuits we get cos(kd) =
A+D . 2
(2.25)
The final dispersion equation for the equivalent circuit in Fig. 2.3 is obtained by multiplying the matrices in (2.20). In the following text, the propagation constant k of the waves in the metamaterial will be denoted as q, whereas it still holds q = β − jα, and we get µ ¶ µ ¶ YZ j Y Z Z 2Y cos(qd) = cos(k0 d) 1 + + sin(k0 d) + + . 2 2 Y0 Z0 4Z0
2.2 2.2.1
(2.26)
Left-handed coplanar waveguide Concept of a left-handed coplanar waveguide
The concept of a left-handed coplanar waveguide (LHCPW) comes from the ideal left-handed transmission line with an equivalent circuit containing serial capacitors and parallel inductors. The layout of the unit cell of this structure is shown in Fig. 2.4. An interdigital capacitor represents the serial capacitor and the shortcut CPW stubs connected between the CPW central strip conductor and the ground represent the parallel inductor. These planar elements are, however, frequency dependent and have the character of resonant L–C circuits. A simple equivalent circuit of this unit cell will be described in the following sections. By periodically translating the unit cell shown in Fig. 2.4 we get the layout of the LHCPW shown in Fig. 2.5. The line was fabricated and measured using the ROGERS RO4003C substrate with permittivity εr = 3.38 and substrate height h = 0.813 mm. The central conductor width is s = 2.4 mm and the CPW slot width is w = 0.4 mm (Fig. 2.4). These dimensions were
24
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
metal
w s
slot
Figure 2.4: Unit cell of the left-handed coplanar waveguide.
chosen in order to achieve a 50 Ω CPW line. Closed form expressions for the coplanar waveguide characteristic impedance were used, see Appendix A.1, with the following dimensions: the length of the short-circuited CPW stub is 7 mm, the CPW stub central conductor width is 0.125 mm and the CPW stub slot width is 0.275 mm. The layout dimensions were optimized in the CST Microwave Studio (MWS) [207] in order to achieve the maximum of the transmission coefficient at a frequency of about 5 GHz and minimal reflection coefficient.
Figure 2.5: Layout of the LHCPW.
In the previous section, the dispersion characteristic of a general LH transmission line was shown analytically. Dispersion characteristics of the proposed LHCPW were found first numerically and then compared with the analytical approach. The dispersion characteristic of an arbitrary transmission line can be calculated by MWS in the following way [32]: the elementary cell of the line is terminated at the input and output faces by periodical boundaries with varying mutual phase shift ϕ. The resonant frequencies of this cell, which now represents a resonator, are calculated by the MWS eigenmode solver in dependence on ϕ. This phase shift determines the phase constant β=
25
ϕ . d
(2.27)
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
4000 1st
2nd
3rd
4th
(1/m)
3000 2000 1000 0
5
10
15
20
f (GHz)
Figure 2.6: Dispersion characteristics of the four lowest modes on the LHCPW.
The calculated resonant frequency represents the second variable of the dispersion relation. The dispersion diagram of the proposed LHCPW is plotted in Fig. 2.6. The first left-handed mode can propagate from 5.1 to 6.4 GHz. The second mode propagates from 7.13 to 7.20 GHz and is right-handed. The third mode, which is left-handed, can propagate between 16.2 and 18.8 GHz. The fourth mode propagates from 21.2 to 21.5 GHz and is right-handed. The first and third modes transmit a backward wave, since the propagation constant derivative is negative in this frequency band. The corresponding distribution of the longitudinal component of the electric field over the unit cell is depicted in Fig. 2.7. Only the first mode is of practical importance. The transmission coefficient calculated by MWS is shown in Fig. 2.8. The pass-bands shown in Fig. 2.6 correspond to those shown in Fig. 2.8. Theoretically, the first left-handed mode can propagate from 5.1 to 6.4 GHz. The practical transmission band of this left-handed mode is however narrower, from about 5.4 up to 6.2 GHz, i.e., about 13 % of the middle frequency. The peak of the frequency characteristic, in Fig. 2.8 at 7.2 GHz, corresponds to the propagation of the second mode. Some of the results achieved in this section were published by the author of this thesis and his colleagues in [208], [209] and [210]. Normalized E-field [V/m]
1st mode, f = 5.39 GHz
2nd mode, f = 7.15 GHz
3rd mode, f = 17.00 GHz
4th mode, f = 21.30 GHz
Figure 2.7: Distribution of the longitudinal component of the electric field over the unit cell of the LHCPW from Fig. 2.4.
26
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
0
-20 -30
|S
21
(dB)|
-10
-40 -50 -60 -70 0
5
10
15
20
25
f (GHz)
Figure 2.8: Calculated modulus of S21 of the manufactured LHCPW.
2.2.2
Equivalent circuit of the LHCPW
In the design of applications based on the LHCPW, it is very helpful to characterize the metamaterial structure with its equivalent circuit and to calculate the projected microwave circuit properties using a linear circuit simulator instead of a full-wave simulation. The general equivalent circuit, however, is quite complicated to find, and thus the equivalent circuit is usually valid only for a limited class of metamaterial inclusion shapes or is valid only in a narrow frequency band. An applicable equivalent circuit should approximate well the transmission and reflection properties of the metamaterial and also its dispersion characteristic. Periodic structures can be considered as homogeneous materials when their unit cells are much smaller in size than the wavelength. Each unit cell of the periodic medium consists of a hosting material (a section of the transmission line with equivalent circuit elements LR , CR in the case of a planar structure), into which appropriate defects have been inserted. These defects must have the character of a series capacitor and a parallel inductor [128]. One half of the unit cell of the LHCPW together with its equivalent circuit is presented in Fig. 2.9.
s1 w1 D
Z in L1
CR
C2
C1
L2
LR
Figure 2.9: One half of the unit cell of the LHCPW and its equivalent circuit.
27
2.2. LEFT-HANDED COPLANAR WAVEGUIDE In order to calculate the ABCD matrix and thus to derive the dispersion characteristic of the circuit in Fig. 2.9, it is convenient to modify it to the equivalent circuit shown in Fig. 2.10, 0 are replaced by a section of the transmission line with characteristic impedance where L0R , CR
Z0 , phase constant k and length d0 . The equivalent circuit consists of 4 unknown elements C1 , C2 , L1 , L2 , frequency dependent impedance Zin and the hosting transmission line represented 0 . Values of L0 , C 0 may be obtained from the known characteristic by the elements L0R , CR R R
impedance Z0 and the effective permittivity εeff of the hosting medium s Z0 =
L0R 0 , CR
k=
q ω√ 0 , εeff = ω L0R CR c
(2.28)
where (’) denotes the value taken per unit length d0 . The impedance Zin represents the input impedance of a short-circuited CPW with finite-extent ground planes of length D, Fig. 2.9, which is given by the relation Zin = jZ0e tan(β0 D)/2,
(2.29)
where Z0e is the characteristic impedance of the even mode of the CPW with finite-extent ground planes. The characteristic impedance of this line is given in Appendix A.2. The geometrical dimensions of the line are a = 0.0625 mm, b = 0.3375 mm, c = 0.4625 mm (see Fig. A.2 in the Appendix A.2), the substrate height h = 0.813 mm, the stub length is D = 7 mm, Z0e = 148.6 Ω and Zin shows the inductive behavior. The effect of coupling between neighboring cells is not taken into account. The equivalent circuit is aimed to model the dispersion characteristic of the first LH mode. Z0
Z s1
Z s2
d'
Yp
d Figure 2.10: Equivalent circuit from Fig. 2.9 modified for derivation of the dispersion characteristic.
Z in L1
Z in
CR
Z 01
C2 C1
L2
CR
LR
Z 03
Z 02
L2
Figure 2.11: Rearrangement of the equivalent circuit from Fig. 2.9.
28
LR
2.2. LEFT-HANDED COPLANAR WAVEGUIDE Taking over the transmission matrix of the circuit shown in Fig. 2.10 we get
A B C
D
=
cos(kd0 ) jY0
jZ0
sin(kd0 )
sin(kd0 )
cos(kd0 )
1 Zs1 0
1
1
0
Yp 1
1 Zs2 0
1
,
(2.30)
where the circuit elements Zs1 , Zs2 and Yp were obtained by the star-delta transformation of elements L1 , C1 , C2 and by some additional circuit rearrangements, see Fig. 2.11. Finally, they are given by the relations Zs1 = Z03 =
Zs2 = Z02 + jωL2 = µ Yp = (Z01 + Zin )
−1
=
L1 /C1 , jωL1 + 1/(jωC1 ) + 1/(jωC2 )
(2.31)
−1/(ω 2 C1 C2 ) + jωL2 , jωL1 + 1/(jωC1 ) + 1/(jωC2 )
(2.32)
¶−1 L1 /C2 + jZ0e tan (β0 D) . jωL1 + 1/(jωC1 ) + 1/(jωC2 )
(2.33)
The dispersion equation is obtained from the condition for existence of a non-trivial solution of equation system (2.23) with the elements A, B, C, D from eq. (2.30) ¡ ¢ ¡ ¢ A = (1 + Yp Zs1 ) cos kd0 + jYp Z0 sin kd0 ,
(2.34a)
¡ ¢ ¡ ¢ B = (Zs1 + Zs2 + Yp Zs1 Zs2 ) cos kd0 + jZ0 (1 + Yp Zs2 ) sin kd0 ,
(2.34b)
¡ ¢ (1 + Yp Zs1 ) sin (kd0 ) C = Yp cos kd0 + j , Z0
(2.34c)
¡ ¢ (Zs1 + Zs2 + Yp Zs1 Zs2 ) sin (kd0 ) D = (1 + Yp Zs2 ) cos kd0 + j . Z0
(2.34d)
It can be easily proved that AD − BC = sin2 (kd0 ) + cos2 (kd0 ) = 1 (reciprocal network). Applying Floquet’s theorem (2.21), (2.22), the frequency dependence of the phase constant of the wave on the transmission line, i.e., the dispersion characteristic, was derived in the form (a lossless medium is assumed, α = 0) cos (qd) =
¡ ¢ ¡ ¢¢ ¡ ¢ 1 j ¡ (2 + Yp (Zs1 + Zs2 )) cos kd0 + Zs1 + Zs2 + Yp Z02 + Zs1 Zs2 sin kd0 . 2 2Z0 (2.35)
A comparison of the dispersion characteristics computed using (2.35) and by MWS is shown in Fig. 2.12, where only the first LH mode is depicted. It can be seen that the curve calculated using (2.35) approximates the dispersion characteristic of the first LH mode very accurately
29
2.2. LEFT-HANDED COPLANAR WAVEGUIDE for a specific combination of L1 , L2 , C1 and C2 , and thus the equivalent circuit in Fig. 2.9 is able to describe the dispersion properties of the fabricated structure well. Changing particular dimensions of the LHCPW layout we change the frequency band of the first LH mode dispersion characteristic. This band is defined in Fig. 2.12 by lower frequency fL and by upper frequency fH . With a change in the structure geometry, or with a significant change in the dimensions of
(1/m)
the structure geometry, the equivalent circuit setup would have to be modified.
4000 Equation (2.35) 3000
Microwave Studio
f
L
2000 f
H
1000
0 5.0
5.5
6.0
6.5 f (GHz)
Figure 2.12: Comparison of dispersion characteristics computed by the Microwave Studio and by equation (2.35). The propagation constant q is equal to the phase constant β when assuming α = 0.
A comparison of the modulus of the transmission coefficient S21 calculated by MWS and using the equivalent circuit, Fig. 2.9, is shown in Fig. 2.13. For the transmission coefficient computation, a cascade of 20 unit cells was used (the same number as in the fabricated line). The transmission coefficient of the equivalent circuit was determined using the linear circuit simulator Microwave Office (MWO) [211], and the equivalent circuit elements correspond to the geometry of the fabricated line, L1 = 99 nH, C1 = 0.091 pF, L2 = 7.25 nH and C2 = 0.34 pF. It can be seen that the equivalent circuit is able to determine quite correctly the frequency band in which the first LH mode propagates. The amplitude of the transmission coefficient does not agree well with the MWS simulation, partly due to the coupling effect between the neighboring cells, which was neglected, and mainly due to the principle of the computation. MWS uses a full-wave simulation and the structure is fed from two sections of homogeneous CPW (see Fig. 2.5) terminated by waveguide ports, whereas MWO computes the response of a linear circuit with lumped parameters and is terminated by 50 Ω ports. The impedance step between the feeding line and the LH structure is different in both cases and thus leads to different impedance matching of the two structures. Results achieved in this section were published by the author of this thesis and his colleagues in papers [212] and [213].
30
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
0
Microwave Studio
-10
S
21
(dB)
Equivalent circuit
-20
-30
-40
-50 5.0
5.5
6.0
6.5
7.0
f (GHz)
Figure 2.13: Modulus of the transmission coefficient S21 computed by the Microwave Studio and using the equivalent circuit, Fig. 2.9.
2.2.3
Equivalent circuit elements
In order to determine the equivalent circuit element values, several structures with the same shape as in Fig. 2.9 were simulated by MWS. Stub length D, stub central conductor width s1 and stub slot width w1 , Fig. 2.9, were varied. The dispersion characteristics computed using (2.35) were fitted to the characteristics computed by MWS and the resulting element values are summarized in Fig. 2.14. The simulations were performed in the range D = 3 ÷ 8 mm, s1 = 0.125 ÷ 0.4 mm and w1 = 0.1 ÷ 0.5 mm for several combinations of these dimensions. Each combination of the geometrical dimensions determines a different frequency band in which the 1st LH mode propagates. It can be seen from Fig. 2.14 that the value of C1 is roughly constant, the value of C2 decreases slowly with growing D and the value of L1 varies ±0.5 % of its maximal value for each combination of geometrical dimensions. Element L2 shows a linear dependence on stub length D, which determines the frequency band of the 1st LH mode on the line. The dependencies of the lower frequency fL and the upper frequency fH of the LHCPW dispersion characteristic on stub length D are depicted in Fig. 2.15. It can be seen that the frequency band shifts down with increasing stub length D. The relative frequency band of the 1st LH mode on the line is shown in Fig. 2.16.
2.2.4
Experimental results
The fabrication and measurement of the proposed LHCPW was enabled by using the ROGERS RO4003C substrate with permittivity εr = 3.38, thickness of the substrate h = 0.813 mm and thickness of the copper metallization t = 0.035 mm. The line consists of 20 unit cells6 . A comparison of the modulus of S21 computed by MWS and measured is shown in Fig. 2.17. The 6
For a simple demonstration of the left-handed behavior of the line, much fewer unit cells are needed. Such a large number of unit cells was used in order to get a line long enough to measure the wavelength in the longitudinal direction.
31
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
100.1
0.20 1 2
0.18 100.0
3
(pF)
4 5
0.14
1
99.9
L
C
1
(nH)
0.16
0.12
99.8
0.10 99.7 0.08 99.6 3
4
5
6
7
0.06 3
8
4
5
D (mm)
6
7
8
7
8
D (mm)
14 0.60 12
0.55
(pF)
0.50 0.45
C
2
8
L
2
(nH)
10
6
0.40
4
0.35
2
0.30
3
4
5
6
7
3
8
4
5
6 D (mm)
D (mm)
Figure 2.14: Dependence of the equivalent circuit element values on the physical dimensions of LHCPW. 1 s1 = 0.125 mm, w1 = 0.5 mm, ° 2 s1 = 0.125 mm, w1 = 0.1 mm, ° 3 s1 = 0.25 mm, w1 = 0.25 mm, ° 4 s1 = 0.4 mm, w1 = 0.1 mm, ° 5 s1 = 0.125 mm, w1 = 0.275 mm. °
unwrapped phase of S21 is plotted in Fig. 2.18. The measured records fit the calculated patterns well. Fig. 2.19 shows the measured dispersion characteristic in the first LH pass band. It agrees well with the calculated characteristics. The phase constant was obtained from the measured wavelength of the standing wave created by the open-circuited termination of the line. The work presented in this chapter was in part published by the author of this thesis and his colleagues in references [214], [215] and [216].
2.2.5
Summary
This chapter summarizes the investigation of the properties of a metamaterial in the form of a left-handed coplanar waveguide and its equivalent circuit. General properties of the lefthanded transmission line were given and the dispersion characteristic of a simple left-handed transmission line was derived. A novel left-handed coplanar waveguide was designed, fabricated and measured. The dispersion characteristic computed by the CST Microwave Studio predicts the pass-bands of the left-handed and right-handed modes. The lowest pass band of the lefthanded wave verified experimentally is about 0.8 GHz in width with the center frequency 5.8 GHz. The simple equivalent circuit of a left-handed coplanar waveguide was introduced together with the values of its elements. The dispersion characteristic of the first left-handed mode
32
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
9
1
8
3
7
5
2
4
f
L
(GHz)
10
6 5 4
3
4
5
6
7
8
D (mm)
(a)
f
H
(GHz)
11
1
10
2
9
4
3
5
8 7 6 3
4
5
6
7
8
D (mm)
(b) Figure 2.15: The lower frequency (a) and the upper frequency (b) of the LHCPW dispersion characteristic 1 s1 = 0.125 mm, w1 = 0.5 mm, ° 2 s1 = 0.125 mm, w1 = 0.1 mm, calculated for different D, s1 , and w1 . ° 3 s1 = 0.25 mm, w1 = 0.25 mm, ° 4 s1 = 0.4 mm, w1 = 0.1 mm, ° 5 s1 = 0.125 mm, w1 = 0.275 mm. °
28
rel. BW (%)
26 24 22
1
2
3
4
5
20 18 16 3
4
5
6
7
8
D (mm)
Figure 2.16: Relative bandwidth of the 1st LH mode of the LHCPW. The legend is the same as in Fig. 2.15.
computed using of this circuit was compared to the curve computed by the CST Microwave Studio. The aim of the equivalent circuit design was to reduce the time required to design
33
2.2. LEFT-HANDED COPLANAR WAVEGUIDE
0 -10
(dB)|
-20
|S
21
-30 -40 -50
measured
-60
Microwave Studio
-70
4
5
6
7
8
f (GHz)
Figure 2.17: Measured and calculated modulus of S21 of the manufactured LHCPW.
0
) (deg)
-500
arg(S
21
-1000 -1500 measured
-2000
Microwave Studio
-2500 4
5
6
7
8
f (GHz)
Figure 2.18: Measured and calculated phase of S21 of the manufactured LHCPW.
(1/m)
4000 Microwave Studio equivalent circuit (2.35)
3000
experiment
2000
1000
0 5.0
5.5
6.0
6.5 f (GHz)
Figure 2.19: Calculated and measured dispersion characteristic of the LHCPW from Fig. 2.5.
practical applications using the proposed line. The dependence of the equivalent circuit elements on the geometrical dimensions of the line was presented.
34
Chapter 3
Triple wire medium In this chapter, an example of a bulk (volume) version of a metamaterial will be presented. First, the triple wire medium will be discussed as a candidate for construction of isotropic three dimensional metamaterials and then practical implementations and measurement results will be presented.
3.1
Introduction
The wire medium as a negative permittivity medium was introduced in Section 1.3, together with a brief historical overview of wire medium research. In 2005, Silveirinha and Fernandes [83] described the behavior of both connected and nonconnected triple wire lattices. In their paper, a permittivity tensor was derived, which corresponds to a magnetized plasma model describing the wire lattice [217]. For the construction of an isotropic negative permittivity material, a triple connected wire mesh system seems to be promising [83], as it exhibits weak spatial dispersion.
a
2rw
Figure 3.1: Triple wire mesh structure formed by a lattice of infinitely long connected wires.
The analysis of the medium consisting of the 3D mesh of connected wires presented here uses the effective medium approach proposed in [83]. Assuming propagation of a plane wave, Maxwell equations with the applied tensor of an effective permittivity, derived in [83], give an eigenvalue system of equations for possible eigenmodes propagating in the medium. The
35
3.2. EFFECTIVE PERMITTIVITY DYADIC, DISPERSION EQUATION resulting dispersion equation takes into account the spatial dispersion. The geometry of the unit cell is shown in Fig. 3.1. The wire mesh has a period a, and the wires have radius rw . The structure is assumed to be lossless.
3.2
Effective permittivity dyadic, dispersion equation
In this work, the relative effective permittivity dyadic describing the homogenized medium of 3D connected wires obtained in [83] has been adopted. This effective permittivity dyadic corresponds to the (relative) permittivity of a nonmagnetized plasma, where the pressure forces are considered [217], and reads kp2 ε=I− 2 k0
µ I−
kk 2 k − l0 k02
¶ ,
(3.1)
where kp is the plasma wavenumber, k0 is the free-space wavenumber, I = x0 x0 + y0 y0 + z0 z0 is the unit dyadic, k = kx x0 +ky y0 +kz z0 (does not have the meaning of the free space propagation constant k0 ) and l0 =
3 , 1 + 2kp2 /β12
(3.2)
³ a ´2 X £J ¡ 2πlrw ¢¤ 1 0 a = , 2 2π l2 β1
(3.3)
l6=0
where a is the lattice period, rw is the wire radius and J0 stands for the Bessel function of the first kind and order zero. In [20] there are two formulas for the plasma wavenumber of such a wire array (kp a)2 ≈
(kp a)2 ≈
³ ln ³ ln
a 2πrw
2π ´ + 0.5275
2π a2 4rw (a−rw )
´
(rw < a/100) ,
(3.4)
(a/20 < rw < a/5) .
(3.5)
The effective permittivity dyadic depends explicitly on the wave vector, and thus the medium suffers in general from spatial dispersion. The dispersion equation can be derived from Maxwell’s equations of the form ∇ × E = −jωµ0 H,
(3.6)
∇ × H = jω²0 εE,
(3.7)
where the electric and magnetic field intensity are expected to be in the form of plane waves E (r) = E0 exp [−j (k · r)] = E0 exp [−j (kx x + ky y + kz z)] ,
36
(3.8)
3.3. DISPERSION CHARACTERISTICS
H (r) = H0 exp [−j (k · r)] = H0 exp [−j (kx x + ky y + kz z)] .
(3.9)
By solving the set of equations (3.6), (3.7) and assuming k02 = ω 2 ²0 µ0 and k 2 = kx2 + ky2 + kz2 and the electromagnetic field in the form (3.8) and (3.9), we come to the eigenvalue equation
k02 εxx − kz2 − ky2
kx ky + k02 εxy
kx ky + k02 εyx
k02 εyy − kx2 − kz2
kx kz + k02 εzx
ky kz + k02 εzy
kx kz + k02 εxz
Ex
Ey = 0. 2 2 2 k0 εzz − ky − kx Ez ky kz + k02 εyz
(3.10)
Solving (3.10) means finding the eigenvalues kx , ky , kz for eigenwaves Ex , Ey , Ez . Rewriting the determinant of (3.10) into dyadic form we obtain the dispersion equation as in [83] ³ ´ det kk − k 2 I + k02 ε = 0.
(3.11)
The dispersion characteristics of the particular modes are determined by solving this dispersion equation.
3.3
Dispersion characteristics
In this section it is considered a = 10 mm, rw = 0.5 mm (see Fig. 4.1), and thus the plasma wavenumber is kp = 194.509 m−1 , β1 = 394.303 m−1 and the constant l0 = 2.018. The corresponding plasma frequency calculated using (3.5) is fp ≈ 9.28 GHz. First, it is assumed the case of kz = 0 and kx 6= ky , thus we can obtain isofrequencies kx as a function of ky for the wave propagating in the x–y plane. In this simplified case, the determinant in (3.10) has the form of a polynomial of the variable ky of the sixth order, and thus generally has 6 roots, which correspond to 3 eigen waves, i.e., 3 sets of solutions with positive and negative sign. The determinant of (3.10) in this case reads ¡ 2 ¢£ ¡ ¢¡ ¢¤ k0 − kp2 − kx2 − ky2 k06 l0 − 4kp2 kx2 ky2 + k02 kp2 + kx2 + ky2 kx2 + ky2 + kp2 l0 − kx2 + ky2 − k02 l0 ¢ ¤ ¡ £ k04 2kp2 l0 + kx2 + ky2 (l0 + 1) − = 0. kx2 + ky2 − k02 l0
(3.12)
The dependence of ky on kx can be expressed for the first solution as follows 2 ky1 = k02 − kx2 − kp2 ,
(3.13)
and the other solutions read 2 ky2 =
1 (K1 − K2 ) , 2k02
37
(3.14)
3.3. DISPERSION CHARACTERISTICS
2 ky3 =
where
1 (K1 + K2 ) , 2k02
(3.15)
£ ¤ K1 = k04 (l0 + 1) + 4kx2 kp2 − k02 2kx2 + kp2 (l0 + 1) ,
K2 =
(3.16)
q¡ q ¢ k02 − kp2 k06 − 16kp2 kx4 − k04 kp2 (l0 − 1)2 + 8k02 kp2 kx2 (l0 + 1).
(3.17)
The propagation constant (3.13) is shown in Fig. 3.2 for two different ratios k0 /kp . The plots in Figs. 3.2a and 3.2b represent circles, which means that the corresponding wave propagates in the x-y plane with the same propagation constant in all directions. The solutions (3.14) and (3.15) are considered as nonphysical and they are shown in Figs. 3.3 and 3.4. Equation (3.13) corresponds to equation (7) in [82], setting kz = 0. The frequency dependence of the propagation constant modulus from (3.13) is shown in Fig. 3.5 both below and above the plasma frequency fp . The curve below the plasma frequency corresponds to the case when both kx and ky are pure imaginary. Thus the size of k represents the size of the attenuation constant. In the case above the plasma frequency, kx and ky are considered to be real and the size of k represents the size of the phase constant. The point where both the real and the imaginary parts drop to zero corresponds to the plasma frequency fp . By changing the frequency, we only change the size of k and the dependence of ky on kx is always a circle for any frequency.
0.05
k /k
0.00
y
y
k /k
0
0
0.05
real
-0.05 0.00
imag
-0.05
0.05
0.00
0.05
Re(k /k )
Im(k /k ) x
real
-0.05
imag
-0.05
0.00
x
0
(a)
0
(b)
Figure 3.2: Dependence ky1 = f (kx ) (3.13) for the ratio k0 /kp = 0.999 below the plasma frequency (a), and k0 /kp = 1.001 above the plasma frequency (b). The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part of the propagation constant ky1 .
The other specific case is ki 6= 0 (i = x, y, z) and the other two components are equal to zero, thus we observe only the wave propagation along one axis. This corresponds to the Γ–X
38
3.3. DISPERSION CHARACTERISTICS
0.05
k /k
k /k
0
0
0.05
y
y
0.00
real
-0.05 -0.10
0.00
0.05
real
-0.05
imag
-0.05
0.00
-0.10
0.10
imag
-0.05
0.05
0.10
Re(k /k )
Im(k /k ) x
0.00 x
0
(a)
0
(b)
Figure 3.3: Dependence ky2 = f (kx ) (3.14) for the ratio k0 /kp = 0.999 below the plasma frequency (a), and k0 /kp = 1.001 above the plasma frequency (b). The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part of the propagation constant ky2 .
0.2
0.2
real
real
imag
imag
0.1
k /k
0.0
y
y
k /k
0
0
0.1
-0.1
-0.1
-0.2 -0.10
0.0
-0.05
0.00
0.05
-0.2 -0.10
0.10
-0.05
x
0.00
0.05
0.10
Re(k /k )
Im(k /k )
x
0
(a)
0
(b)
Figure 3.4: Dependence ky3 = f (kx ) (3.15) for the ratio k0 /kp = 0.999 below the plasma frequency (a), and k0 /kp = 1.001 above the plasma frequency (b). The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part of the propagation constant ky3 .
direction in the first Brillouin zone1 . The determinant of (3.10) in this case reads ¡ ¢2 £ ¡ 2 ¢ ¤ k02 ki2 − k02 + kp2 l0 k0 − kp2 − ki2 = 0, k02 l0 − ki2
i = x, y, z.
(3.18)
The frequency dependence of component ki now reads 2 2 ki1 = ki2 = k02 − kp2 , 1
i = x, y, z,
(3.19)
In the further text, the propagation of the wave along one coordinate axis will be denoted Γ–X (propagation along the x-coordinate), even if the wave can generally propagate along all the coordinate axes x, y, z.
39
3.3. DISPERSION CHARACTERISTICS
k a /
1.5
1.0
0.5 real imag 0.0 0
1
2 k /k 0
3
p
q Figure 3.5: Frequency dependence of the size of the propagation constant k =
kx2 + ky2 both below
(dashed line, imaginary part) and above (solid line, real part) the plasma frequency, for the case kz = 0 and kx 6= ky 6= 0.
¡ ¢ 2 ki3 = k02 − kp2 l0 ,
i = x, y, z.
(3.20)
It is obvious that (3.19) is a simplification of (3.13), and the plot in Fig. 3.5 can be adopted q for it, changing kx2 + ky2 on its vertical axis to just ki , since the propagation of the wave along one axis is equivalent to the propagation along the unit cell face, i.e., in the x-y plane. Solution (3.19) was identified as a TEM mode and solution (3.20) as a longitudinal mode in [83]. A study of the excitation of the longitudinal mode would need a more detailed analysis, and one would need to know the boundary conditions for the electromagnetic field at the interface with the surrounding medium. Such boundary conditions were proposed for instance in [218]. The full-wave simulations in MWS show, however, that the longitudinal mode is excited very weakly. Solution (3.20) is identical with (3.19), except for the value of the propagation constant, and has the same frequency dependence. The case of kx = ky and kz = 0 (or in general two of the components are equal and the remaining is equal to zero) corresponds to the propagation of a wave along the unit cell face diagonal, the Γ–M direction in the first Brillouin zone. The determinant of (3.10) is now ¢¤ ¢£ ¡ ¡ 2 ¢¡ k0 − kp2 k02 − kp2 − 2ki2 4ki2 + k04 l0 − k02 kp2 l0 + 2ki2 (l0 + 1) = 0, k02 l0 − 2ki2
i = x, y, z,
(3.21)
and the frequency dependence of the propagation constant component ki now reads 2 ki1 =
¢ 1¡ 2 k0 − kp2 , 2
40
i = x, y, z,
(3.22)
3.3. DISPERSION CHARACTERISTICS · ¸ q 1 2 k0 (l0 + 1) − k04 (l0 − 1)2 + 4k02 kp2 l0 , 4
i = x, y, z,
(3.23)
· ¸ q 1 2 2 4 2 2 = k (l0 + 1) + k0 (l0 − 1) + 4k0 kp l0 , 4 0
i = x, y, z,
(3.24)
2 ki2 =
2 ki3
whereas solution (3.22) corresponds to a physical wave and is identical with (3.19), except for the value, and has the same frequency dependence. Solutions (3.23) and (3.24) are considered as nonphysical and the dependencies of the propagation constant on the frequency are shown in Figs. 3.6a and 3.6b.
1.5
1.0 real
1.0
imag
real
0.5
imag
k a /
0.0
0.0
i
i
k a /
0.5
-0.5 -0.5
-1.0
-1.0 0
1
k /k 0
2
-1.5 0
3
1
k /k 0
p
(a)
2
3
p
(b)
Figure 3.6: Frequency dependence of the propagation constant ki2 (i = x, y, z) (3.23) (a) and ki3 (i = x, y, z) (3.24) (b) both below and above the plasma frequency for the case that two of the propagation constant components are equal and the remaining component is equal to zero. The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part, of the propagation constant.
In the case of kx = ky = kz , the wave propagates along the unit cell diagonal, corresponding to the Γ–R direction in the first Brillouin zone. The determinant of (3.10) is in this case £ 2¡ 2 ¢ ¤£ ¡ ¢ ¡ ¢¤2 k0 3ki + kp2 l0 − 4kp2 ki2 − k04 l0 ki2 kp2 + 9ki2 + k04 l0 − k02 kp2 l0 + 3ki2 (l0 + 1) = 0, ¡ 2 ¢3 3ki − k02 l0 i = x, y, z. (3.25) The physical solutions of the frequency dependence of the component kx = ky = kz are now 2 2 ki1 = ki2 =
¤ 1 £ 2 3k0 (l0 + 1) − kp2 − K3 , 18
i = x, y, z,
(3.26)
and the other two solutions (considered as nonphysical) read 2 ki3
¢ ¡ k02 l0 kp2 − k02 , = 4kp2 − 3k02 41
i = x, y, z,
(3.27)
3.3. DISPERSION CHARACTERISTICS
2 2 ki4 = ki5 =
where
¤ 1 £ 2 3k0 (l0 + 1) − kp2 + K3 , 18
i = x, y, z,
(3.28)
q
9k04 (l0 − 1)2 + 6k02 kp2 (5l0 − 1) + kp4 .
K3 =
(3.29)
The real and imaginary parts of (3.26) are plotted in Fig. 3.7 and the real and imaginary parts of (3.27) and (3.28) are plotted in Figs. 3.8a and 3.8b, respectively.
0.5
k
i1,2
a /
1.0
real imag 0.0 0
1
2 k /k 0
3
p
Figure 3.7: Frequency dependence of the propagation constant ki1,2 (i = x, y, z) (3.26) both below and above the plasma frequency for the case that all of the propagation constant components are equal. The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part, of the propagation constant.
4
1.5 1.0
2
a / i4,5
0
0.0
k
k
i3
a /
0.5
-0.5 -2
-1.0
real imag
-4 0
real imag
-1.5 1
2 k /k 0
3
4
0
1
k /k 0
p
(a)
2
3
p
(b)
Figure 3.8: Frequency dependence of the propagation constant ki (i = x, y, z) (3.27) (a) and (3.28) (b) both below and above the plasma frequency for the case that two of the propagation constant components are equal and the remaining is equal to zero. The solid line corresponds to the real part, and the dashed line corresponds to the imaginary part, of the propagation constant.
The results presented above were obtained by solving Maxwell equations with a dielectric
42
3.3. DISPERSION CHARACTERISTICS
1.0
0.6
) Im(k a /
0.6
0.4
i
Re(k a /
)
0.8
0.4
theory
MWS
0.2
-X 0.2
0.0
-M
-X and
-R
-R
1.0
1.5 k
0
/ k
0.0 0.0
2.0
-M
0.5
1.0 k /k 0
p
(a)
1.5
p
(b)
q Figure 3.9: A comparison of the real part of the propagation constant k = kx2 + ky2 + kz2 calculated by MWS and predicted by the presented theory for directions Γ–X, Γ–M and Γ–R (a) and the imaginary part of the propagation constant predicted by the theory (b).
permittivity tensor in the form (3.1). It was shown, however, that the propagation constant solutions only suffer from weak spatial dispersion in the vicinity of the plasma frequency, moreover they are the same for the Γ–X and Γ–M directions and only the direction Γ–R differs. Thus we can assume that the medium is almost isotropic near the plasma frequency (the term kk in (3.1) is negligible for small negative ε and becomes significant for ε ¿ 0). The limitation of the isotropy to the vicinity of the plasma frequency, however, is still useful, since the triple WM is supposed to be practically used for small negative values of ε and not for large negative values. By substituting the simple isotropic model of permittivity in the form à ²=
kp2 1− 2 k0
! I
(3.30)
into the eigenvalue equation (3.10) we naturally obtain identical solutions for all of the important lattice directions, and they are equal to (3.19). The comparison between the propagation constant predicted by the presented theory and the curves calculated by MWS is shown in Fig. 3.9a. The dispersion characteristics, i.e., the frequency dependence of the propagation constant, was calculated by MWS as described in Section 2.2.1. The propagation constant k 2 = kx2 + ky2 + kz2 now has one (Γ–X), two (Γ–M) or three (Γ–R) components and can be calculated as k 2 = nki2 ,
i = x, y, z,
n = 1, 2, 3,
(3.31)
where n = 1 and ki is calculated by (3.19) for the direction Γ–X, n = 2 and ki is calculated by (3.22) for the direction Γ–M and n = 3 and ki is calculated by (3.26) for the direction
43
3.4. NUMERICAL ANALYSIS Γ–R. MWS calculates the dispersion characteristics of the wave propagating in the periodical medium [32]. Consequently, the curve calculated by MWS shows an upper stop band where no wave propagates. The boundary of this stop band is described by the condition of Bragg reflection ka = π. The presented theory uses a homogenized medium (for ka ¿ π), therefore no such stop band can be seen and the propagation constant grows to infinity above the plasma frequency. Moreover, the homogenization of the medium for ka ≈ π loses its sense. The imaginary part of the propagation constant predicted by the theory is shown in Fig. 3.9b. The comparison with MWS is not possible in this case, since MWS is able to calculate only the real eigenvalues of the wire system. Fig. 3.9a shows quite good agreement between the propagation constant predicted by the presented theory and the curves calculated by MWS in the close vicinity above the the plasma frequency. A difference can also be seen between the simulated curves for the Γ–X and Γ–M direction, which are the same, and the curve for the Γ–R direction. The propagation constant below the plasma frequency cannot be directly computed using MWS, but, the value of the phase constant calculated by the presented theory shows good agreement in the close vicinity below the plasma frequency, see Figs. 3.5 and 3.7. The theoretical calculations presented in this section were published by author of this thesis and his colleagues in reference [219].
3.4
Numerical analysis
An original numerical experiment is proposed to verify the theory. The triple wire medium is modeled by MWS as a sphere filled with a system of mutually perpendicular connected wires, see Fig. 3.10. This system is excited by a dipole located at the center of the sphere. The electric field distribution is calculated. This field is naturally perturbed by the wires, and is not averaged as the field used to determine the effective permittivity tensor [83]. Nevertheless, it can be compared with the field radiated into an empty space, except for attenuation due to the negative permittivity. This simulates the omnidirectional propagation of a wave in the triple WM both below and above the plasma frequency. The results confirm the suitability of the triple wire medium to be applied as an isotropic negative permittivity metamaterial near the plasma frequency. The dispersion characteristics calculated by MWS are shown in Figs. 3.9a and 3.9b. To verify the behavior of the wave in the lower stop band under the plasma frequency, the structure shown in Fig. 3.10 was analyzed. The structure forms one octant of the hollow sphere of triple wire material with the space period a = 10 mm. The inner sphere radius is 25 mm, and the outer radius is 105 mm. The wires are rectangular2 in cross section, with sides 0.85 mm in length. This shape simplifies the numerical simulation and at the same time 2 The theoretical formula (3.5) holds for wires with circular cross-section. However, a rectangular cross-section is much more suitable for the numerical experiment and the error in determining the plasma frequency is acceptably small.
44
3.4. NUMERICAL ANALYSIS shows the same plasma frequency fp ≈ 9.28 GHz as the wire medium composed of cylinders. The structure is fed by a dipole located at its center. Taking into account the excitation field symmetries, the octant is terminated at the x-y plane by a perfectly conducting wall and in the x-z and y-z planes by perfectly magnetic walls. The field radiated by the dipole at frequencies below the plasma frequency is spread in the wire medium as evanescent waves with the distance dependence modified by the 1/x function representing a spherical wave [220] |E| ≈ E0
e−αr , r
(3.32)
where α is the attenuation constant represented by the imaginary part of the propagation constant, and r is radial distance. z
x dipole arm
y Figure 3.10: Analyzed 3D wire system.
3.0 -X
2.5 |E| (arb. units)
-M -R
2.0
approx (3.32) 1.5
1.0
0.5
0.0 30
40
50
60
70
80
r (mm)
Figure 3.11: Distribution of the electric field in the medium from Fig. 3.10 at particular directions at frequency 7 GHz (k0 /kp ≈ 0.75). The line ”approx” represents (3.32) with the value of α estimated as 120 m−1 .
The electric field distribution calculated by MWS in the medium shown in Fig. 3.10 is plotted
45
3.4. NUMERICAL ANALYSIS
Figure 3.12: Cubic wire mesh formed by a lattice of 8 × 8 × 8 connected wires.
in Fig. 3.11 at the frequency 7 GHz (k0 /kp ≈ 0.75). The calculated field represents the field in the microstructure, which is perturbed by the presence of the wires, as distinct from the averaged field, which defines the effective permittivity (3.1) [83]. However, the calculated field can be approximated by (3.32), as shown in Fig. 3.11. Similar plots have been obtained in the frequency band from 5 to 9 GHz. It may be noted from Fig. 3.11 that the field is attenuated at approximately the same rate in all shown characteristic directions. This verifies the isotropy of the medium below the plasma frequency. Attenuation constant α can be estimated from the calculated field distributions, but does not fit well the data read from Fig. 3.9b since the electric field distribution is quickly changing for small radial distances r. The values read from Fig. 3.9b for k0 /kp = 0.75 are Im(ka/π) = 0.411 (directions Γ–X and Γ–M) and Im(ka/π) = 0.340 (direction Γ–R), respectively, which corresponds to the attenuation constant α = 129.4 m−1 and α = 106.8 m−1 , respectively. Another numerical experiment was performed with use of a plane wave excitation. The simulated structure is shown in Fig. 3.12. It is formed by a lattice of 8 × 8 × 8 rectangular wires with a cross-section of 1 × 1 mm with the lattice period a = 20 mm. The mesh is illuminated sequentially by a plane wave in all the important directions, i.e., the Γ–X, Γ– M and Γ–R directions. The plasma frequency of the mesh calculated using (3.5) is approx. fp ≈ 3.9 GHz. The distribution of the electric field intensity in the plane crossing WM along all the important directions at frequency 2.5 GHz is shown in Figs. 3.13, 3.14 and 3.15, and the orientation of vectors E, H and k is also indicated. It is evident from Figs. 3.13, 3.14 and 3.15 that the wave is an evanescent wave within WM
46
3.4. NUMERICAL ANALYSIS with a purely imaginary propagation constant √ k = ω ²0 εeff µ ≡ −jα.
(3.33)
The attenuation constant can be determined from the behavior of the plane wave. The intensity of the electric field can be approximated [205] E (x) ≈ E0 e−αx ,
(3.34)
where x is the coordinate in the propagation direction and E0 is the amplitude of the electric field intensity at the plane wave source. As an example, the electric field intensity inside the wire mesh for direction of propagation Γ–X is depicted in Fig. 3.16 for several frequencies below the lattice plasma frequency. Eq. (3.34), however, is valid only for a large homogeneous medium illuminated by a plane wave without the presence of reflections (in our case, the electric field inside the cube is partly influenced by multiple reflections). From the calculated distribution of the electric field, the attenuation constant α and consequently the effective permittivity εeff of the lattice can be determined by fitting the electric field distribution inside the structure to eq. (3.34). The calculated effective permittivity is depicted in Fig. 3.17. According to (3.30), the effective permittivity is equal in all the important propagation directions. The differences observed in Fig. 3.17 are caused mainly by the error in determining the permittivity from the electric field distribution3 . It can be seen from Fig. 3.17 that the extracted permittivity agrees well with that simulated for small negative values of permittivity (in the vicinity of the plasma frequency), and differs for larger negative values, where the spatial dispersion is not negligible and model (3.1) instead of (3.30) is valid. The numerical simulations presented in this section were published by author of this thesis and his colleagues in the paper [221]. 3
For the Γ–M and Γ–R directions, the electric field distribution is influenced by the presence of metallic wires, and the intensity of the electric field varies rapidly in their vicinity.
E
z
k
x H
y
Figure 3.13: Distribution of the electric field intensity component Ey in the x-z plane crossing the wire mesh from Fig. 3.12 at the half height between two wire layers. The plane wave illuminating WM propagates in the Γ–X direction. The mesh boundary is marked with the dashed line.
47
3.4. NUMERICAL ANALYSIS
z x k
E
y
H Figure 3.14: Distribution of the electric field intensity component Ey in the x-z plane crossing the wire mesh from Fig. 3.12 at the half height between two wire layers. The plane wave illuminating WM propagates in the Γ–M direction. The mesh boundary is marked with the dashed line.
z
x
y
Figure 3.15: The distribution of the transversal component of the electric field of the plane wave illuminating the structure in the Γ–R direction. The mesh boundary is marked with the dashed line, and the plane where the cube is cut is indicated by the bold line.
1.8 f = 3.5 GHz
1.6
f = 3.0 GHz f = 2.5 GHz
1.2
f = 2.0 GHz
1.0
f = 1.5 GHz f = 1.0 GHz
0.8
f = 0.5 GHz
E
y
(arb. units)
1.4
0.6 0.4 0.2 0.0 30
40
50
60
70
80
90
100
x (mm)
Figure 3.16: Amplitude of the Ey component in the wire mesh from Fig. 3.12 (x-z plane, the lattice is crossed at half of its height). The plane wave source is situated at the coordinate x = 0 mm and the face of the simulated cube is situated at the coordinate x = 30 mm.
48
3.5. EXPERIMENTAL RESULTS
0
-10
eff
-20 theory (3.30)
-30
-40
-50
1
2
3
MWS (
-X)
MWS (
-M)
MWS (
-R)
4
5
f (GHz)
Figure 3.17: Calculated effective permittivity of the wire mesh from Fig. 3.12.
3.5
Experimental results
The idea of a triple WM as an isotropic ε-negative material is quite simple, but practical fabrication is cumbersome. To the author’s best knowledge, nobody has yet experimentally verified the theoretical results. An idea for fabricating an analogy to the wire lattice shown in Fig. 3.12 is proposed using planar technology. One unit cell of the proposed mesh is shown in Fig. 3.18. A photograph of the fabricated structure is shown in Fig. 3.19, where also one unit cell of the lattice is shown as a detail. It consists of a set of parallel dielectric sheets. The 2D planar lattice is etched in the x-y plane on each substrate layer, the third orthogonal set of planar wires is etched in the y-z planes, and the set of substrate layers without metallization in the x-z planes is inserted to preserve the 3D structure symmetry. The galvanic connection of the planar wires in the y-z plane is performed by via holes and miniature wires soldered in the x-y plane substrate layers. The lattice period is now 11.71 mm, and the strip width is w = 2 mm (twice the equivalent circular wire diameter, see, e.g., [222]). For the fabrication, FR4 substrate with permittivity εr = 4.3 and thickness 0.71 mm was used. The theoretical plasma frequency of a system with circular wires is now fp ≈ 7.6 GHz. The plasma frequency of the fabricated structure containing thin metallic strips, however, does not differ dramatically according to the MWS simulation. Since the wire lattice is not a resonant structure (in the long wavelength limit), the presence of the supporting dielectric substrates does not significantly affect the plasma frequency of the lattice. The lattice now consists of only 4 unit cells in order to make the cube acceptably small and to measure its transmission spectrum in a waveguide. The implementation of the lattice with a planar technology results in a structure that is not fully isotropic. However, the propagation of the electromagnetic wave in all the important directions does not differ dramatically from the ideal lattice. The effective permittivity was determined from the simulated electric field distribution inside the lattice fabricated by the planar technology in the same way as for the ideal wire lattice, and is shown in Fig. 3.20. The
49
3.5. EXPERIMENTAL RESULTS
Figure 3.18: Proposal for manufacturing the wire mesh shown in Fig. 3.12 with a planar technology.
metal traces
dielectric substrate
Figure 3.19: Fabricated wire mesh.
0 -10
eff
-20 -30
theory (3.30)
-40 -50 -60
1
2
3
4
5
MWS (
-X)
MWS (
-M)
MWS (
-R)
6
7
8
f (GHz)
Figure 3.20: The calculated effective permittivity of the wire mesh with unit cell from Fig. 3.18.
error of determination is comparable with that shown in shown in Fig. 3.17 (now in a different frequency band). Measurement of the transmission spectrum of the lattice was performed in a waveguide structure with a cross-section of 120×60 mm. This waveguide was fed from two R48 waveguides through two horn transitions, and the transmission coefficient was measured using the HP 8510B vector network analyzer, see Fig. 3.21. A comparison between the simulated and measured transmission coefficient normalized to its maximal value is shown in Fig. 3.22. First, the face of the metamaterial cube from Fig. 3.19 was placed perpendicularly to the propagation direction (transmission in the Γ–X direction), and then the cube was rotated by 45 degrees
50
3.5. EXPERIMENTAL RESULTS (transmission in the Γ-M direction). The remaining space in the waveguide cross-section was filled by an absorbing material. The difference between the measured transmission in the Γ– X and Γ–M direction is caused mainly by the different length of the propagation path. The transmission in the Γ–R was not measured due to the limited dimensions of the waveguide used in the experiment. The dip in the measured transmission coefficient in the vicinity of frequency 9 GHz is caused by higher modes propagating in the waveguide. HP8510B VECTOR NETWORK ANALYZER
MEASURED SAMPLE R48 TO COAXIAL
R48 TO COAXIAL HORN TRANSITION
HORN TRANSITION
MICROWAVE ABSORBER
Figure 3.21: Outline of the system for measuring of the transmission coefficient.
sim.
0
-X
meas.
-M
-20 -30
s
21
(dB)
-10
-X
meas.
-40 -50 -60
3
4
5
6
7
8
9
10
f (GHz)
Figure 3.22: Transmission of the energy through the lattice from Fig. 3.19, simulated and measured for directions Γ-X and Γ-M.
The measurement of the electric field intensity inside the lattice was enabled by drilling a hole into the cube and inserting an electric field probe (monopole antenna) inside the lattice, see Fig. 3.23. A horn antenna was used to illuminate the lattice in all the important directions and the transmission coefficient proportional to the electric field intensity inside the cube was measured by the vector network analyzer. The lattice was surrounded by a microwave absorbing material in all directions except the face in the direction of the incident electromagnetic wave. The simulated and measured intensity of the electric field inside the lattice normalized to its maximal value is shown in Figs. 3.24, 3.25 and 3.26. Both measured and simulated results suffer
51
3.6. SUMMARY from an error caused by the small number of unit cells, i.e., the cube cannot be considered as a fully homogeneous material. The experimental results were published by author of this thesis and his colleagues in reference [223].
HP8510B VECTOR NETWORK ANALYZER
MEASURED SAMPLE approx. 0.5 m ELECTRIC FIELD PROBE
HORN ANTENNA
MICROWAVE ABSORBER
Figure 3.23: Outline of the system for measuring the electric field intensity within the lattice.
0 simulated measured
-X -X
|E| (dB)
-10
-20
-30
-40
2
4
6
8
10
f (GHz)
Figure 3.24: Electric field intensity inside the lattice from Fig. 3.19, simulated and measured for the Γ-X direction.
3.6
Summary
In this section, the behavior of the triple wire medium was studied with the aim to apply it as an isotropic negative permittivity metamaterial. The derivation of the dispersion equation of modes in the 3D lattice is quite straightforward, and the result is in accord with [83]. The numerical examples show that there exist eigen modes both below and above the plasma frequency in all important lattice directions Γ–X, Γ–M and Γ–R, and two different models for the relative permittivity of the medium were used. Using the simple isotropic model of permittivity (3.30), only physical solutions were found, and with the use of a more accurate model (3.1) both physical and nonphysical solutions were found. The physical modes comply with physical
52
3.6. SUMMARY
0
|E| (dB)
-10
-20
-30 simulated measured
-40
2
4
6
-M -M
8
10
f (GHz)
Figure 3.25: Electric field intensity inside the lattice from Fig. 3.19, simulated and measured for the Γ-M direction. 0
|E| (dB)
-10
-20
-30 simulated measured
-40
2
4
6
8
-R -R
10
f (GHz)
Figure 3.26: Electric field intensity inside the lattice from Fig. 3.19, simulated and measured for the Γ-R direction.
intuition, i.e., they are evanescent below the plasma frequency, and become propagating above the plasma frequency. In the author’s opinion, the other modes (Figs. 3.6 and 3.8) do not correspond to physical waves. In the direction Γ–M, the attenuation and phase constants have the same amplitude as in the case of the direction Γ–X, since the propagation of the wave along one axis is equivalent to the propagation along the unit cell face. From Fig. 3.2, it is obvious that the curves below (imaginary part) and above (real part) the plasma frequency form a circle, i.e., equation (3.19) is an analytical expression of a circle. This implies isotropic propagation of the plane wave along one face of the triple wire cube. Since the structure is periodic, pass bands and stop bands should appear in the dispersion diagram. The curves calculated by MWS are in accord with this hypothesis, as it analyzes the structure as periodic, whereas the curves representing the results of the presented theory are not. The reason is that a homogenized model of the triple wire medium was used. The triple wire medium behaves as an isotropic material for the wave propagating parallel with any coordinate plane. Taking into consideration the wave propagating in a general direction, the isotropy is now in
53
3.6. SUMMARY general removed. However, a comparison of the dispersion characteristics calculated in various directions shows that in the first approximation the triple wire medium can be considered as an isotropic material in the close vicinity of the plasma frequency. The modeling by the Microwave Studio verifies these theoretical results. Two numerical experiments were performed. The first one shows the distribution of the electric field excited by the dipole located in the center of the sphere of the 3D wire material, and the other shows the distribution of the electric field inside a wire cube with 8×8×8 unit cells. The wave spreads in this system in the same way, except for the attenuation due to the negative permittivity, as in the empty space. The value of the attenuation constant and consequently the permittivity can be estimated from the resulting electric field distributions. The effective permittivity extracted from the electric field distribution is in accord with the theoretical predictions. The wave amplitude decreases at the same rate in all directions. This validates the isotropy of the medium even below the plasma frequency. For a practical implementation, a form of the triple wire medium utilizing the planar technology was proposed. It consists of three sets of mutually orthogonal dielectric substrate layers, which support cross connected etched planar wires. The fabricated specimen forms a cube with 4×4×4 unit cells. It is shown that the properties of the proposed planar lattice are very similar to those of the ideal wire lattice. The transmission coefficient of the lattice was measured in a waveguide setup in the direction along the unit cell axis and along the unit cell face diagonal. The intensity of the electric field was measured using an electric field probe inside the lattice for all three important propagation directions, i.e., along the unit cell axis, along the unit cell face diagonal and along the unit cell diagonal. The measured and simulated results show good agreement (except for some systematic measurement errors), which validates the isotropy of the triple wire medium below the plasma frequency. The proposed form of the 3D structure fabricated by the planar technology is suitable for future combination with an isotropic negative permeability material using modified split ring resonators (see, e.g., [99]) in order to design a double-negative isotropic metamaterial. The split ring resonators would be etched on dielectric substrates as well as the wires. The top view of the proposed junction structure is shown in Fig. 3.27. The split-ring resonators must fulfil certain spatial symmetry conditions and their orientation at the unit cell cube faces is not arbitrary [224]. Since effective permittivity of the triple WM is negative in a wide band, it would be possible to achieve both εeff (triple WM) and µeff (SRRs) at a certain frequency. Such a combination, however, would require much more detailed analysis, and some issues are still open.
54
3.6. SUMMARY
dielectric substrate split-ring resonator metal strips
Figure 3.27: Proposed junction of the triple wire lattice fabricated using planar technology and split ring resonators (top view).
55
Chapter 4
Uniaxial Ω-medium In this chapter, the design of a perfectly matched backward-wave layer will be presented. The layer utilizes some properties of bianisotropic materials, namely Ω-particles. It will be shown that the design of such a slab has some advantages over conventional ways of implementing and matching metamaterial slabs.
4.1
Introduction
Most researchers have focused on the design of magneto-dielectrics, where the backwardwave regime is realized when both the permittivity ² and permeability µ have negative real parts. Backward waves, however, can exist in more general linear media, namely in bianisotropic media (e.g., in chiral media [50, 139, 142]). It was demonstrated that in chiral media it is possible to improve the characteristics of the backward-wave regime [142, 150]. Moreover, it is possible to benefit from the greater design freedom offered by additional material parameters. Following paper [93], bianisotropy is usually considered as a factor that should be avoided in the design of backward-wave materials, and efforts are often concentrated on designing symmetrical variants of split rings to minimize the magnetoelectric coupling [68]. It will be shown that it is possible to design a bianisotropic material in such a way that it supports linearly-polarized backward waves, and a slab made of this material is perfectly matched to free space for the normal direction of propagation. This section deals with the construction of a perfectly matched backward-wave layer. The effect of material bianisotropy1 will be utilized here. bianisotropic materials are characterized by various electromagnetic properties not usual in conventional isotropic materials. In the past, the main area of research with these materials was in the field of microwave absorbers and non-reflecting coatings (see, e.g., [225]). The material parameters ε, µ were considered as positive in the frequency range of interest. The electrodynamics of bianisotropic materials is 1 A bianisotropic medium is the most general linear complex medium that is anisotropic in both the electric and the magnetic component.
56
4.1. INTRODUCTION comprehensively given in [19] and only basic properties will be summarized in this section. The well known Maxwell equations in the differential form µ−1 0 ∇ × B = ²0
∂E + J, ∂t
∇×E=−
∂B , ∂t
²0 ∇ · E = ρ,
∇ · B = 0,
(4.1)
allow us to solve microscopic electromagnetic fields within the structure. In macroscopic electromagnetics we usually deal with waves whose wavelengths are much larger than atomic sizes and distances between atoms and equations (4.1) cannot be practically used due to the high number of particles. We speak about spatial dispersion2 , and more complex quantities than just scalar permittivity and permeability are needed to fully describe the medium. Averaging the fields over a period of the unit cell, we lose very essential information about the spatial structure, which determines the main electromagnetic phenomena. However, such averaging is possible if the characteristic scale (the period of the structure or the inclusion size) is still considerably smaller than the wavelength. The dimensions d of the averaging volume should satisfy the inequalities a ¿ d ¿ λ, where a is the inclusion scale and λ is the characteristic wavelength. The process of homogenization might be described as a transition from the microscopic Maxwell equations (4.1) to the macroscopic Maxwell equations (see, e.g., [205]) ∇×H=
∂D + Jext , ∂t
∇×E=−
∂B , ∂t
∇ · D = ρext ,
∇ · B = 0.
(4.2)
These equations must be supplemented by a pair of constitutive relations. General bianisotropic constitutive material relations can be written as D = ² · E + a · H,
(4.3)
B = µ · H + b · E.
(4.4)
Note that such a medium comprises generally 36 different scalar constitutive parameters (dyadics ², µ, a, b are matrices 3 × 3) and we can distinguish several classes of materials according to these parameters. From the Lorentz reciprocity theorem (see, e.g., [19]) the following relations result T
²=² ,
T
µ=µ ,
T
b = −a .
(4.5)
The first two conditions for permittivity and permeability dyadics represent the symmetry, and the last condition represents the reciprocity of the material. A more convenient form that 2
The effect of the spatial dispersion was discussed in Section 1.3.
57
4.1. INTRODUCTION separates parameters describing spatial dispersion and non-reciprocity is the following D=²·E+
¡ ¢ √ ²0 µ0 χ − jκ · H,
(4.6)
B=µ·H+
¡ ¢T √ ²0 µ0 χ + jκ · E.
(4.7)
Here κ corresponds to the chirality parameter3 and the parameter χ models the non-reciprocity √ of the medium. The coefficient ²0 µ0 is introduced to make the chirality parameter κ dimensionless. Bianisotropic materials can be classified by many criteria, whereas the basic division is between reciprocal and non-reciprocal magnetoelectric media. The reciprocal case means that the reciprocity dyadics is zero χ = 0 and also the material parameters ² and µ have to be symmetric. Now the magnetoelectric coupling is represented by the chirality parameter κ. We can rewrite the coupling dyadics as κ = κI + M ,
(4.8)
© ª where I = x0 x0 + y0 y0 + z0 z0 is the unit dyadics and trM = 0. The parameter κ = 13 tr κ is the pseudoscalar complex chirality parameter4 . Media with κ 6= 0 are called chiral media and media with κ = 0 represent bianisotropic media with a mirror-symmetrical microstructure. For a better classification of reciprocal bianisotropic media, we can perform a decomposition of M into symmetric and antisymmetric parts [19] M = N + J, T
(4.9) T
where N = (M + M )/2 is a symmetric dyadic and J = (M − M )/2 is an antisymmetric P dyadic. The symmetric part can be diagonalized, i.e., expressed as N = κi ai ai , where κi are P complex numbers and ai are real vectors. For the coefficients holds κi = 0 [19]. Generally, the magnetoelectric coupling provided by the parameter N can be described as a coupling of virtual chiral inclusions oriented along axes ai . At least two sets of inclusions with opposite handedness are needed, because trN = 0. Then there can be media with not necessarily chiral inclusions which have behaviour similar to that of chiral media. Saadoun and Engheta proposed a term pseudochiral for these media in [154] for Ω-type media. In pseudochiral media κ = 0 but N 6= 0. The antisymmetric dyadic in (4.9) can always be presented as b × I, where b is a 3 The term chiral corresponds to media with mirror-symmetric geometry, i.e., composites with mirror-symmetric inclusions. An inversion transformation of spatial coordinates will not cause any change in the properties of the medium. 4 The operation trA, where A is an arbitrary dyadics, means the trace of dyadics A and is defined using the colon operator (:) as follows: let us suppose a, b, c, d to be arbitrary vectors. Then (ab) : (cd) = (a · c) (b · d) and for the trace of dyadics A holds trA = A : I (see, e.g., [226]).
58
4.1. INTRODUCTION complex vector. After splitting into real and imaginary part we have [19] J = K1 b1 × I + jK2 b2 × I,
(4.10)
where b1,2 are real unit vectors and K1,2 are real numbers. If both the real and imaginary parts of b point along the same direction one can normalize that vector and write J = Kb0 × I, where K is a scalar complex material parameter and b0 is the unit vector along b. In this case, K is the field coupling parameter of an effective uniaxial omega medium [225]. The uniaxial omega medium will be the principal subject of this chapter. Materials with nonzero coefficients K1,2 can be called omega media although, of course, they do not necessary contain Ω-shaped inclusions. In the general case we can model “antisymmetric” coupling effects by two sets of uniaxial omega elements (“hats”) oriented along vectors b1 and b2 . Based on a decomposition of the coupling dyadic κ, a classification for reciprocal bianisotropic materials is summarized in Table 4.1. The classification and properties of nonreciprocal bianisotropic media are not a subject of this section, and a detailed description can be found in [19]. Coupling κ 6= 0, N κ 6= 0, N κ = 0, N κ = 0, N κ 6= 0, N κ = 0, N κ 6= 0, N
parameters = 0, J = 0 6= 0, J = 0 6= 0, J = 0 = 0, J 6= 0 = 0, J 6= 0 6= 0, J 6= 0 6= 0, J 6= 0
Class Isotropic chiral medium Anisotropic chiral medium Pseudochiral medium Omega medium Chiral omega medium Pseudochiral omega medium General reciprocal bianisotropic medium
Table 4.1: Classification of reciprocal bianisotropic media.
The material can be realized as a composite with Ω-shaped metal inclusions, and can be called an omega-medium [154]. Since reliable analytical models of Ω-particles and Ω-media have been developed and checked numerically and experimentally for the microwave range [19, 155, 156], these models (valid below 70 − 100 GHz) will be shown for a demonstration of the concept. Reciprocal uniaxial Ω-media obey the following constitutive relations [19] √ D = ² · E + j ²0 µ0 KJ · H,
√ B = µ · H + j ²0 µ0 KJ · E.
(4.11)
Denoting the unit vector along the optical axis as x0 we can write the permittivity and permeability dyadics in the form ³ ´ ² = ²0 εt I t + εx x0 x0 ,
³ ´ µ = µ0 µt I t + µx x0 x0 ,
59
(4.12)
4.2. ANTENNA MODEL FOR WIRE-AND-LOOP ELEMENTS where I t is the two-dimensional unit dyadic defined in the plane orthogonal to x0 : I t = y0 y0 + z0 z0 (transversal plane). The magnetoelectric dyadic is antisymmetric: J = x0 × I t . Complex dimensionless parameter K measures the magnetoelectric coupling effect (its form for the special case of uniaxial Ω-medium will be given later). Eigenwaves in such media are linearly polarized plane waves, similarly to simple magneto-dielectrics.
4.2
Antenna model for wire-and-loop elements
4.2.1
A simple LC model for electrically small particles
Omega particles are composed of a wire loop connected to two straight wire elements (Fig. 4.1b). The electromagnetic analysis is based on replacing the particles by two connected antennas, representing the wire and loop portions. As will be shown later, the analysis of an omega particle is based on the analysis of a chiral particle (Fig. 4.1a). For electrically small particles, a lumped-element equivalent circuit can be constructed, and the polarizabilities can be expressed in terms of equivalent circuit parameters. Small particles of complex shapes in the bianisotropic approximation can be characterized by dyadic electric and magnetic polarizabilities, which define the bianisotropic relations between induced electric and magnetic dipole moments p, m and external electric and magnetic fields E, H [19] p = αee · E + αem · H,
(4.13)
m = αme · E + αmm · H.
(4.14)
Due to the reciprocity principle, the cross-coupling coefficients αem and αme are related to each other as T
αem = −αme
(4.15)
Also, the dyadics αee and αmm are symmetrical T
αee = αee ,
T
αmm = αmm .
(4.16)
A very simple particle model is assumed, where the input admittance of the loop antenna is modeled by an inductance and Ohmic losses, and the input impedance of the wire antenna is modeled by a capacitance, see Fig. 4.2. It is also assumed that ² = ²0 εm , where εm is the host material permittivity and µ = µ0 in following equations [19] Yl = 1/(jωL0 + Rl ),
60
(4.17)
4.2. ANTENNA MODEL FOR WIRE-AND-LOOP ELEMENTS
z y
a
a 2l
x
2r0
l
l
(a)
(b)
Figure 4.1: Geometry of a left-handed chiral (a) and omega (b) particle. The loop of the chiral particle lies in the x − y plane, whereas the loop of the Ω particle lies in the x − z plane.
wire antenna
loop antenna L0
C0 Rl
Figure 4.2: A simple equivalent circuit of the Ω-particle composed from a small loop antenna and short wire antenna.
Zw = 1/jωC0 ,
C0 =
πl²0 εm , ln (2l/r0 )
· µ ¶ ¸ 8a −2 , L0 = µ0 a ln r0 r Rl =
ωµ a , 2σ r0
(4.18)
(4.19)
(4.20)
(4.21)
where σ is the metal conductivity and dimensions r0 , a and l are defined in Fig. 4.1. For the simplicity, only the normal incidence of the wave is assumed. The dyadic character of the polarizabilities vanishes and then the polarizabilities of small chiral particles can be calculated in the following way [19] zz αee =
l2 , jω (Zl + Zw )
(4.22)
jωS 2 , Zl + Zw
(4.23)
zz αmm = −µ20
61
4.2. ANTENNA MODEL FOR WIRE-AND-LOOP ELEMENTS
zz αme = µ0
Sl , Zl + Zw
(4.24)
where S is the loop area. By substituting eqs. (4.17) – (4.21) into (4.22) – (4.24) the following notation can be obtained zz αee =
ω02
zz αmm =
zz αme =
A , − ω 2 + jωΓ
A = l2 /L0 ,
(4.25)
Bω 2 , ω02 − ω 2 + jωΓ
B=
µ2 S 2 , L0
(4.26)
jωD , − ω 2 + jωΓ
D=
µSl , L0
(4.27)
ω02
√ where ω0 = 1/ L0 C0 and Γ = Rl /L0 . It can be proved that the following condition is satisfied zz zz zz zz zz 2 αee αmm = αem αme = − (αme ) .
(4.28)
Polarizabilities of omega particles are related to polarizabilities of chiral particles in the following way [19] zz αee |omega
particle
zz yy yz = (αee + αee + 2αee )|chiral
xx αee |omega
particle
xx = αee |chiral
yy αmm |omega
particle
zz = αmm |chiral
yz αme |omega
particle
particle
particle
particle
zz zy = (αme + αme )|chiral
particle
(4.29)
(4.30)
(4.31)
(4.32)
Since a very simple omega particle model was introduced, the omega particle polarizabilities yy yz are reduced to chiral ones, i.e., terms αee and αee are assumed to be negligible compared with zz and the terms αxx and αzy are neglected. αee me ee
4.2.2
Antenna model of the particles
There exists a more accurate LC (but still analytically relatively simple) model of the wire and loop [19] which holds for electrically small antennas. The most accurate wire and loop model, however, is based on exact antenna models for small wire and loop antennas [155]. Moreover,
62
4.2. ANTENNA MODEL FOR WIRE-AND-LOOP ELEMENTS the antenna model takes into account the radiation losses5 . For the geometrical dimensions of the particles it is assumed that the electrical sizes are moderate, so that |k|l < 0.3 and |k|a < 1, √ where k = ω ²µ is the wave number in the background medium. For such particles analytical expressions for the antenna parameters can be used [19]. The input admittance of the linear wire antenna can be expressed as · ¸ kl 1 2 2F 3 3 Yw = 2πj 1+k l − jk l , ηΨdr 3 3 (Ω − 3) where F =1+ and η =
1.08 , Ω−3
µ Ω = 2 ln
2l r0
¶
µ ,
Ψdr = 2 ln
l r0
(4.33) ¶ ,
(4.34)
p
µ/² is wave impedance of the surrounding medium. Circular-loop antennas are
analyzed with the use of the Fourier series expansion of the current distribution function by keeping only the most important terms in the expansion −j Yl = πη
µ
1 2 2 + + A0 A1 A2
¶ (4.35)
with the Fourier coefficients A0 =
· µ ¶ ¸ ¤ £ ¤ ka 8a 1£ ln −2 + 0.667(ka)3 − 0.267(ka)5 − j 0.167(ka)4 − 0.033(ka)6 , (4.36) π r0 π ¶ · µ ¶ ¸ µ ¤ 8a 1£ 1 1 ln −2 + −0.667(ka)3 + 0.207(ka)5 A1 = ka − ka π r0 π −j[0.333(ka)2 − 0.133(ka)4 + 0.026(ka)6 ],
(4.37)
µ ¶ · µ ¶ ¸ ¤ 4 1 8a 1£ A2 = ka − ln − 2.667 + −0.4ka + 0.21(ka)3 − 0.086(ka)5 ka π r0 π −j[0.05(ka)4 − 0.012(ka)6 ].
(4.38)
The co- and cross-polarizabilities of chiral particles can be expressed as follows [19] zz αee =
2 leff 1 , jω (Zl + Zw )
(4.39)
5 The term describing radiation losses can actually be introduced only in material samples of a limited size or for lossy materials. For any regular periodic lattice of particles, radiation losses are not present in case that the background material is lossless. The energy radiated from one particle is absorbed in neighboring particles. At the same time, the energy radiated from this one particle is compensated by the energy absorbed from the neighboring (radiating) particles. Thus the energy balance is zero and no energy is lost due to radiation within the material. In a lossy background material, each particle radiates, but the other particles are not affected by this radiation, because all the energy is lost in the background material. The energy radiated by the particle is lost and is not compensated by the energy received from the other radiating particles - radiation losses.
63
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER
Yw Zl √ leff a2 √ leff a2 zz αme = ∓j ²µ = ∓j ²µ , A0 (Yl + Yw ) A0 (Zl + Zw ) yz αee =∓
zz αmm
4aleff J10 (ka) Yw , ωη A1 (Yl + Yw )
3 J1 (ka)
= −2µπa
A0
zz αem = ±j
j 1 Yl + Yw πηA0
¶ ,
(4.42)
(4.43)
4a2 J1 (ka) 1 , ωη A0 A1 Yl + Yw
(4.44)
4a2 J10 (ka) , ωηA1
(4.45)
xx αee =−
yy αee
(4.41)
2aleff J1 (ka) Zl ω A0 Zl + Zw
yz αem =−
zy αme =
µ 1+
(4.40)
1 4²a3 J10 (ka) , A0 A1 Yl + Yw
4πa2 J10 (ka) =− ωηA1
µ 1+
2j 1 πηA1 Yl + Yw
(4.46) ¶ ,
(4.47)
where leff is the effective dipole length and for short dipole antennas (l < λ/4) satisfies leff ≈ l, while the total geometrical length of the wire antenna is 2l, J1 (ka) is the Bessel function of the first order, J10 (ka) is the derivative of the Bessel function of the first order. The co- and cross polarizabilities of Ω-particles can be calculated from the relations above, using eqs. (4.29) to (4.32).
4.3
A perfectly matched backward-wave layer
The aim of this part of the work is to design a perfectly matched backward-wave slab, for simplicity for the normal incidence of the wave. The slab consists of small Ω-shaped conductive particles which are distributed in a background material in two directions. The entire slab is situated in the free space (or in a general hosting medium, as will be shown later). Fig. 4.3 shows the slab, and the other set of Ω-particles perpendicular to the first one (plane x − y) is not marked for better readability of the figure. A detail of one set of Ω-particles is shown in Fig. 4.4. The other perpendicular set, however, does not interact with the eigenwave in the case that the electric field is polarized along the z axis. All the derivations in the further text are valid for the general case of the electric field vector polarized in the y − z plane (Fig. 4.4), and the geometry
64
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER shown in Fig. 4.3 is considered only for a simplicity. The slab is infinite in directions y and z and has thickness d in direction x. The slab is illuminated by a plane wave coming in the positive x direction. In previous works dealing with perfectly-matched layers, the reflection and transmission coefficients of the slab were derived only for positive material parameters of the background. In this work, the aim is to achieve simultaneously a perfect impedance matching to the free space and propagation of a backward-wave within the medium at a certain frequency6 .
E z x H
y d
Figure 4.3: Geometry of the Ω-particle slab. The slab is infinite in the y and z directions and has thickness d. The slab is illuminated by a plane wave with the electric field vector parallel to the z axis. The uniaxial structure has a second identical set of particles lying in the x − y plane (not shown).
z y
x
Figure 4.4: Detailed appearance of one set of particles of the uniaxial Ω-medium. It consists of two Ω-particles with bodies perpendicular to each other and connected together (side and free view).
In the analysis of complex structures possessing uniaxial symmetry it is common to split the fields into normal and transverse parts with respect to the geometrical axis x (by analogy to the waveguide theory, where there also exists one distinguished direction). As stressed above, the eigenwaves in a uniaxial Ω-medium are linearly polarized waves. Their propagation factors read [19] βT2 M =
¢ εt ¡ 2 k0 εx µt − kt2 − k02 K 2 , εx
(4.48)
βT2 E =
¢ µt ¡ 2 k0 εt µx − kt2 − k02 K 2 , µx
(4.49)
where the TM and TE indices correspond to linearly polarized transversal magnetic and transversal electric waves, respectively, and the subscripts x ant t mean the longitudinal and transversal 6 The backward wave, i.e., the wave whose phase and group velocities have opposite signs, is usually observed in materials with both material parameters ², µ negative. In bianisotropic mixtures, the backward-wave can propagate even if only one of the material parameters is negative, as will be shown later.
65
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER component, respectively. The characteristic impedances and admittances of the medium can be expressed as follows [19] r TM Z±
r TE Z±
=
=
µ0 µt ²0 εt
Ãs
k2 1 − 2 t − Kn2 ± jKn k0 εx µt Ãs
µ0 µt 1 ²0 εt 1 − 2kt2
k0 εt µx
r Y±T M =
1 ²0 εt µ0 µt 1 − 2kt2 r
Y±T E =
²0 εt µ0 µt
Ãs
,
k2 1 − 2 t − Kn2 ± jKn k0 εt µx
Ãs
k0 εx µt
! (4.50) !
k2 1 − 2 t − Kn2 ∓ jKn k0 εx µt
k2 1 − 2 t − Kn2 ∓ jKn k0 εt µx
,
(4.51)
,
(4.52)
!
! .
(4.53)
As can be seen from (4.50) to (4.53), the impedances and admittances are different for the waves traveling in opposite x-directions7 . This is very unusual in passive reciprocal media, and this is physically possible because the position of uniaxial omega inclusions (hats) defines a preferred direction along the symmetry axis [19]. For media with symmetric coupling dyadics (K = 0) the impedances are symmetric. A general analysis of such materials can be made using the theory of nonreciprocal uniform transmission lines [227]. Similar properties can be observed for waves in magnetized ferrites or plasmas, which are nonreciprocal due to the external bias field. In the case of a slab with the optical axis orthogonal to the slab surface and the normalincidence excitation, both eigenwaves in the slab (4.48), (4.49) have the same propagation constant [19] β = k0
p
p √ εt µt − K 2 = k0 εt µt 1 − Kn2 ,
(4.54)
√ where k0 is the free-space wavenumber and Kn = K/ εt µt is the normalized magnetoelectric coupling parameter. Below, the matching problem for a wave traveling along axis x will be solved, however the condition of backward waves will be generally considered for waves propagating in the plane x − y. For the characteristic admittances which are different for the waves traveling in the positive and negative directions of axis x (denoted below as Y+ and Y− , respectively), the following relation holds [19] r Y± = Y0
´ εt ³p 1 − Kn2 ∓ jKn , µt
7
(4.55)
This leads to many unusual electromagnetic applications, such as a “trap” for electromagnetic waves: let us imagine a bianisotropic material slab with admittance Y±T E surrounded by free space with admittance Y0 . Now the admittance Y±T E is different for the two propagation directions. The interface between free space and the slab is matched in one direction, but not in the other, i.e., the wave propagates through the slab in one direction, but no wave is reflected back. The wave, which is partially reflected inside the slab, will be attenuated by multiple reflections in the slab.
66
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER where Y0 =
p
²0 /µ0 is the free-space wave admittance and from now on, the polarization is
considered as TE in the text. The reflection and transmission coefficients for an Ω−medium slab in free space can be expressed as ¡ ¢ (1 − y+ ) (1 + y− ) 1 − e−2jβd R= , (1 + y− ) (1 + y+ ) − e−2jβd (1 − y− ) (1 − y+ )
(4.56)
2 (y− + y+ ) e−jβd , (1 + y− ) (1 + y+ ) − e−2jβd (1 − y− ) (1 − y+ )
(4.57)
T =
where y+ = Y+ /Y0 , y− = Y− /Y0 are normalized admittances and d is layer thickness. In [225] it was shown that omega composites can be used for the design of absorbing layers which are matched to free space. To obtain the reflection coefficient R = 0 in the +x direction, the wave impedance (4.55) must be equal to that of free space Y0 . Generally for the normal incidence of the wave, the Y± = Y0 condition is satisfied for both the TE and TM waves when the coupling parameter is equal to [19] K=
j (µt − εt ) . 2
(4.58)
Then the reflection coefficient from one side of the slab of arbitrary thickness d equals R = 0 and the transmission coefficient reads T = exp(−jβd). Note that the matching condition can be satisfied for arbitrary permittivity and permeability values, provided one can control the coupling coefficient K. From the constitutive relations for uniaxial Ω-media (4.11) and (4.12), there are in general five complex parameters which can be chosen to make the interface between the slab and free space ideally matched). Since only the normal wave incidence considered, the number reduces to three. The perfectly matched layer (see, e.g., [228, 229]) is defined by εt = µt ,
εx = 1/εt ,
µx = 1/µt ,
K = 0,
(4.59)
where the permittivities and permeabilities are considered in a form εt,x = ε0t,x − jε00t,x , and µt,x = µ0t,x − jµ00t,x . The trivial absorbing condition εt = εx = µt = µx = 1 is of no interest and ε00t , µ00t > 0 (the imaginary part of εt , µt must be negative). To satisfy the Kramers-Kronig relations, such a medium is always active [230], because ε00x , µ00x < 0 in this case (the imaginary part of εx , µx is positive). Though the matching condition for any dielectric layer illuminated by a normally incident wave can be obtained at the thickness resonance, the mismatch in this case will be very strong for the oblique incidence (even for small angles) since the phase shift of the refracted wave across the layer varies sharply versus the refraction angle. Relation (4.58) provides the matching for a normally incident wave at the two interfaces separately (εx = µx = 1) and does not depend on the layer thickness. In this case the mismatch for small angles of incidence will be acceptable.
67
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER
4.3.1
Use of a simple LC model for electrically small particles
Now we have the zero-reflection condition (4.58) that we want to satisfy and we have a very rough electromagnetic model of the Ω-particle described in Section 4.2. This model is valid for electrically small particles. Due to the simplicity of the problem, condition (4.58) can be used in an analytical approach. The derived qualitative conclusions will be used later for a more sophisticated Ω-particle model. For the (relative) transversal permittivity and permeability and the coupling coefficient the simple Maxwell-Garnet mixing model was used. The form of the coupling coefficient was chosen in accordance with [225], and the polarizabilities are defined in (4.22) to (4.24) zz N αee NA ¢, = εr + ¡ 2 ²0 ²0 ω0 − ω 2 + jωΓ
(4.60)
zz N αmm N Bω 2 ¡ 2 ¢, = µr + µ0 µ0 ω0 − ω 2 + jωΓ
(4.61)
ωD αzz ¢√ , K = −N j √ me = N ¡ 2 2 ²0 µ0 ω0 − ω + jωΓ ²0 µ0
(4.62)
εt = εr +
µt = µr +
where εr and µr are the relative permittivity and permeability of the background material, respectively, and N denotes the number of particles in a unit volume (particle concentration). In formulas (4.60) to (4.62) a different notation for the background material is used than in (4.19) because in our structure εr 6= εm . The reason will be given later. An example of the shape of εt and µt calculated for certain set of particle geometrical dimensions and concentration is shown in Fig. 4.5 for illustration.
15 10
real imaginary
5
t
,
t
0 -5 -10 -15 -20 -25 2.6
2.8
3.0
3.2
f (GHz)
Figure 4.5: An example of the shape of εt and µt calculated using (4.60), (4.61) for a certain set of particle geometrical dimensions and concentration. The quantities are considered in the form εt = ε0t − jε00t , µt = µ0t − jµ00t .
68
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER Rewriting condition (4.58) by substituting (4.60), (4.61) and (4.62) we obtain αzz j −N j √ me = ²0 µ0 2
µ ¶ zz zz N αmm N αee µr − εr + − , µ0 ²0
(4.63)
" # jωD N Bω 2 1 NA ¡ ¢= ¡ ¢− ¡ 2 ¢ , −N √ 1 − εr + 2 ²0 µ0 ω02 − ω 2 + jωΓ µ0 ω02 − ω 2 + jωΓ ²0 ω0 − ω 2 + jωΓ (4.64) · ¸ ¡ 2 ¢ N Bω 2 N A −N jωD 1 2 = (1 − εr ) ω0 − ω + jωΓ + − . √ ²0 µ0 2 µ0 ²0
(4.65)
To satisfy condition (4.58), the real and imaginary part on both sides of (4.65) has to be equal. They read 2D (1 − εr ) = Γ, √ ²0 µ0 N (1 − εr ) 2 (ω0 − ω 2 ) = N
µ
A Bω 2 − ²0 µ0
(4.66) ¶ .
(4.67)
It can be naturally assumed that condition (4.58) is also satisfied for the normalized coupling coefficient Kn , i.e., the reflection coefficient is equal to zero and the transmission coefficient is only an exponentially decaying function. The normalized zero-reflection condition then reads K j Kn = √ = √ (µt − εt ) . εt µ t 2 εt µt
(4.68)
By substituting the normalized zero-reflection condition (4.68) into equations (4.54) to (4.57) one obtains r y1,2 =
s r 2 ³ ´ p εt εt (µt − εt ) µt − εt 1 − Kn2 ± jKn = 1+ ∓ √ µt µt 4εt µt 2 εt µt ¶ r µ εt εt + µt µ t − εt = ∓ √ , √ µt 2 εt µt 2 εt µt
y1 = εt /µt ,
R = 0,
T =
(4.69)
y2 = 1,
(4.70)
2 (1 + εt /µt ) e−jβd = e−jβd , 2 (1 + εt /µt )
(4.71)
p εt + µt k0 √ √ β = k0 εt µt 1 − Kn2 = k0 εt µt √ = (εt + µt ) . 2 εt µt 2
(4.72)
To satisfy the backward-wave condition at the working frequency, the real part of the propagation
69
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER constant β has to be negative. Note that now it is enough to satisfy Re (εt + µt ) < 0 instead of εt < 0 and µt < 0 simultaneously, which is required in metamaterials without magnetoelectric coupling. In the following text we will derive the conditions, under which this is possible. Substituting (4.60), (4.61) into (4.72), assuming µr = 1 and applying basic rules for inequalities8 one obtains " # √ ω ²0 µ0 k0 µ0 N A + ²0 N Bω 2 ¡ ¢ β= (εt + µt ) = εr + µr + 2 2 ²0 µ0 ω02 − ω 2 + jωΓ
(4.73)
¡ ¢¡ 2 ¢ √ 2 2 µ N A + ²0 N Bω ω0 − ω ω ²0 µ0 εr + µr + 0 h¡ i , Re{β} = ¢ 2 2 2 2 2 2 ²0 µ0 ω0 − ω +ω Γ
(4.74)
¡ ¢¡ ¢ µ0 N A + ²0 N Bω 2 ω02 − ω 2 h¡ i < 0, 1 + εr + ¢2 ²0 µ0 ω02 − ω 2 + ω 2 Γ2
(4.75)
Re{β} < 0
⇒
µ0 N A + ²0 N Bω 2
− (1 + εr ) ²0 µ0
h¡ i ¢2 ω02 − ω 2 + ω 2 Γ2
ω02 − ω 2 − (1 + εr ) ²0 µ0
h¡ i ¢2 ω02 − ω 2 + ω 2 Γ2
ω02 − ω 2
for ω0 > ω,
(4.76)
for ω0 < ω.
(4.77)
The parameters A, B are assumed to be positive to achieve a physically realizable geometrical dimension of the particle, see (4.25), (4.26). It is also assumed that parameters A, B are related by eq. (4.67), which can be rewritten as ¡ ¢ µ0 N A − ²0 N Bω 2 = −²0 µ0 (1 − εr ) ω02 − ω 2 .
(4.78)
First the case ω0 > ω will be considered. Then adding identity (4.78) to inequality (4.76) we obtain
0 < 2µ0 N A < −²0 µ0 (1 + εr )
h¡ i ¢2 ω02 − ω 2 + ω 2 Γ2 (ω02 − ω 2 )
+ ²0 µ0 (1 − εr ) (ω02 − ω 2 ).
(4.79)
The right-hand side of this inequality is positive, and from (4.79) it follows that (1 − εr ) > a(1 + εr ), 8
(4.80)
The allowed operations are: 1). adding/subtracting the same number on both sides, 2). switching sides and changing the orientation of the inequality sign, 3). multiplying/dividing by the same positive number on both sides, 4). multiplying/dividing by the same negative number on both sides and changing the orientation of the inequality sign.
70
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER where it is denoted a=1+
ω 2 Γ2 − ω 2 )2
(4.81)
(ω02
and near the resonance a À 1. Only εr < 0 (practically εr < −1 or εr ≈ −1) allows to satisfy (4.80). If εr < −1 it makes the right-hand side of (4.80) negative and this is sufficient for (4.80). It is easy to show that in order to satisfy the more restrictive inequality (4.79) containing parameter A > 0 the absolute value of this negative permittivity must be larger than in order to satisfy (4.80). The case ω0 < ω can be considered similarly. One can subtract identity (4.78) from inequality (4.77) and, using the fact that 2²0 N Bω 2 > 0, one comes to the same condition (4.80). The form (4.77) was chosen, since the working frequency ω is required to be greater than ω0 (above the particle resonant frequency). In the derivations above, the permeability µr was assumed as positive. Assuming a general positive or negative value of the permeability, the solutions would become much more complicated and one would need to distinguish among many possible solutions of the problem. Moreover, one must be careful in the limiting cases ω ≈ ω0 , since it is important whether ω0 is approaching ω from the left or from the right side (the same holds for εr ≈ −1) and more detailed analysis is required. Let us pick some reasonable values of parameters Γ, εm , a, l, r0 (so that a > l and a, l À r0 ) and calculate ω0 . Then from (4.66) and (4.67) the term (1 − εr )/N and ω can be found. At frequency ω the medium under design is perfectly matched. Then varying N the condition εr < −1 can be satisfied. Finally the required metal conductivity can be expressed through Γ ωµ0 σ= 2
µ
a ΓL0 r0
¶2 .
(4.82)
An important conclusion of the above mentioned derivation is that for satisfying both zeroreflection and the backward-wave condition, a hosting material with εr < −1 surrounding the Ω-particles is needed. This may sound unusual and this requirement was not expected at the beginning of the work. With a positive-permittivity surrounding material, one would always satisfy only one of the conditions (either the impedance matching, or the existence of a backward wave in the slab), but never both of them. This was the case of chiral and bianisotropic absorbers studied in the past [225]. In a practical implementation of the slab, negative permittivity of the host medium can be accomplished by a wire medium surrounding the Ω-particles (as will be shown later). The idea of the slab was published by the author of this thesis and his colleagues in references [231, 232], and possible practical applications were proposed in [233]. The derivations and design proposals of the slab were published in [234].
71
4.3. A PERFECTLY MATCHED BACKWARD-WAVE LAYER
4.3.2
Antenna model of the particles
If the antenna model of Ω-particles is used, the analytical solution of the zero-reflection condition becomes almost impossible, since the loop and wire admittances are rather complicated functions. However, it is possible to numerically tune the geometrical dimensions of the particle in order to make the reflection coefficient as small as possible. The conclusion from the simple LC model (the background medium effective permittivity has to be negative) was used in the design. The proper geometrical dimensions of Ω-particles were found by looking for a global minimum of the function ¡ ¢ j Λ r0 , l, a, N, ε0m , ε00m = K − (µt − εt ) 2
(4.83)
for a chosen angular frequency ω, where αzz K = −N j √ me , ²0 µ0
εt = εr +
µt = µr +
zz αee
4πa2 J10 (ka) l2 − = jω (Zl + Zw ) ωηA1
µ 1+
zz αme = µ0
zz N αee , ²0
(4.85)
yy N αmm , µ0
(4.86)
2j 1 πηA1 Yl + Yw
yy αmm = −µ20
(4.84)
¶ +
4aleff J10 (ka) Yw , ωη A1 (Yl + Yw )
jωS 2 , Zl + Zw
4²a3 J10 (ka) 1 Sl + . Zl + Zw A0 A1 Yl + Yw
(4.87)
(4.88)
(4.89)
Note that the Ω-particle polarizabilities can be calculated from the polarizabilities of a chiral particle, see eqs. (4.29) to (4.32). The bounds for variables in Λ were chosen to be physically realizable. The optimization criteria were the following: the reflection coefficient equal to zero and the transmission coefficient as high as possible at the working frequency, the sum of the real parts of transversal permittivity and permeability negative in the vicinity of the working frequency, and the Ω-particle loop diameter and sum of the stem lengths smaller than the unit cell period. It is possible to use the dimensions resulting from the very simple LC model as a starting point for the optimization and consequently the dimensions from the accurate antenna model as
72
4.4. NUMERICAL EXAMPLES a starting point for a full-wave electromagnetic simulation. Neither the simple LC model nor the accurate antenna model is not able to predict accurately the behavior of a real implementation of the Ω-slab, since the real system is finite in dimensions9 and the higher order multipoles influence the resonant frequencies of the system. The theoretical approach was taken mainly because it offered a quick prediction of the slab background properties and particle dimensions before performing the time-consuming full-wave simulation.
4.4
Numerical examples
In this section, the properties of a slab consisting of small omega particles surrounded by a negative permittivity background will be numerically studied. The very simple LC model of the Ω particles, the antenna model and then the full-wave simulations performed in MWS will be presented. First, the permittivity was chosen to be constant and negative in the whole frequency band of interest. Figs. 4.6 and 4.7 show the composite effective permittivity and permeability and the slab transmission and reflection coefficients together with the longitudinal propagation constant. It can be seen that the reflection coefficient is zero at the working frequency f = 3 GHz and also both effective material parameters εt , µt are negative at this frequency. The particle resonance frequency was chosen to be f0 = 2.55 GHz. The transmission coefficient is dependent on the variable Γ, which is related with the Ohmic losses in the wires. Thus the unity transmission coefficient can be obtained at the working frequency by choosing the losses properly. The thickness of the slab was chosen in order to obtain a slab-thickness resonance to enhance the low-reflection frequency band. The dependence of the slab parameters on the loss coefficient Γ is demonstrated in Figs. 4.8 and 4.9. It can be seen that increasing the loss parameter leads to a worse transmission coefficient of the slab and the effective material parameters do not contain any sharp peaks within the resonance band. A more realistic background material model than the constant negative permittivity medium can be used. Let us assume that the composite slab is inserted into an array of long thin metal wires of radius rw and a lattice period of aw (single wire medium). This array behaves as a low frequency plasma for the electric field oriented along the wires, as discussed in Chapter 3. The effective permittivity and the plasma angular frequency are given by à εr (ω) = εm
ωp2 1− 2 ω
! ,
(4.90)
9 The above mentioned derivations assumed the slab to be infinite in the y and z directions and of thickness d, see Fig. 4.3.
73
4.4. NUMERICAL EXAMPLES
100
0.0 -0.2
0.000
-0.4
-0.004
0
real
-0.008
imaginary
-0.8
t
t
-0.6
50
-50
-1.396
-1.0
-1.400
-1.2
-100 -150
-1.4 2.0
2.5
3.0
real imaginary
-1.404
3.5
2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.6: Effective medium parameters for a very simple LC model of the omega particle. Permittivity of the surrounding matrix εr = -1.4, concentration N = 106 , slab thickness 33 mm, loss coefficient Γ = 6 · 107 , working frequency f = 3 GHz, particle resonance frequency f0 = 2.55 GHz (the permittivity resonance is low and is shown in detailed windows).
0
0
-10
k
x
/ k
0
|R|, |T| (dB)
-4
-8
-20 -30 -40
real imaginary
-12
2.0
2.5
3.0
3.5
|R|
-50
|T|
-60 2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.7: Normalized longitudinal propagation constant and the reflection and transmission coefficients of the slab with parameters from Fig. 4.6.
40
-1.38
20
-1.40
Real(
Real(
t
t
)
)
-1.36
7
= 6*10
-1.42
0 7
= 6*10
-20
8
8
= 3*10 = 6*10
-1.44 2.0
= 3*10
8
2.5
3.0
3.5
8
-40 2.0
4.0
= 6*10
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.8: Dependence of the effective medium parameters for a very simple LC model of the omega particle on parameter Γ. The other slab parameters are the same as in Fig. 4.6.
74
0
0
-10
-20
-20
-40
|T| (dB)
|R| (dB)
4.4. NUMERICAL EXAMPLES
-30 -40
-60 7
-80
7
= 6*10
= 6*10
8
= 3*10
8
= 3*10
-50
-100
8
8
= 6*10
= 6*10
-60 2.0
2.5
3.0
3.5
-120 2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.9: Dependence of the reflection and transmission coefficient for a very simple LC model of the omega particle on parameter Γ. The other slab parameters are the same as in Fig. 4.6.
ωp2 =
2π 2 a2w ε0 µ0 ln 4rw (aaww−rw )
,
(4.91)
where εm is the permittivity of a medium surrounding the wires, rw is the wire radius and aw is the lattice period. The same notation εm for the matrix permittivity was used in (4.19) because in our structure εr 6= εm . The background permittivity is, of course, affected by the presence of metal wires. However, the capacitance C0 (4.19) is determined by the quasi-stationary electric field in the small spatial domain of the particle. The quasi-static electric field produced by the wire lattice vanishes at the centers of the Ω-particles. This is why the matrix permittivity εm and not εr enters C0 . Figs. 4.10 and 4.11 show an example of the case when the slab is inserted into the thin wires array. The background relative permittivity is now negative and slowly increasing in the frequency band of interest.
0.0
10 -0.5
0
-1.5
t
t
-1.0
-10
-2.0
-20
real
-2.5 -3.0 2.0
real
imaginary
2.5
3.0
3.5
imaginary
-30 2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.10: Effective medium parameters for a very simple LC model of the omega particle. The particles are inserted into a wire medium with wire radius r0 = 0.2 mm and lattice period l0 = 10 mm. Concentration N = 106 , slab thickness 30 mm, loss coefficient Γ = 6 · 107 , working frequency f = 3 GHz, particle resonance frequency f0 = 2.55 GHz.
75
4.4. NUMERICAL EXAMPLES
0
0
-1
|T| (dB)
k
x
/ k
0
-20 -2
-3
|R|
real
-4
-60
imaginary
-5 2.0
-40
2.5
3.0
3.5
2.0
4.0
|T|
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.11: Longitudinal propagation constant and the reflection and transmission coefficients of the slab with parameters from Fig. 4.10.
In the numerical examples in Figs. 4.6 to 4.11, there are plotted curves for arbitrarily chosen parameters f , f0 , Γ, N and εr . The question is, whether they correspond to some physically realistic geometrical dimensions of the particle. The procedure for finding the realistic geometrical dimensions was described in Section 4.3.1. This procedure is quite time-consuming and sensitive to numerical errors, however it always results in a certain (more or less physically realistic) set of geometrical dimensions. The required geometrical dimensions can be restricted arbitrarily, for instance one may want the loop radius to be smaller than λ/5, etc. Now the initial values of the geometrical dimensions can be used for a more sophisticated antenna model of the Ω-particles and subsequent optimization. First, the homogeneous dielectrics with negative permittivity was used and the Ω-particle dimensions were tuned in order to achieve R = 0 and Re (εt + µt ) < 0 at the working frequency f = 3 GHz. The parameters of such a slab are shown in Figs. 4.12 and 4.13.
0.0
5
-0.2
0
-0.6
t
t
-0.4
-0.8
-5 -1.0
imaginary
imaginary
-1.2 2.0
real
real
2.5
3.0
3.5
-10 2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.12: Effective medium parameters for the antenna model of the omega particle. Permittivity of the surrounding matrix εr = −1.25, concentration N = 2.838 · 106 , slab thickness 44 mm, loss coefficient Γ = 5.8 · 107 , working frequency f = 3 GHz, r0 = 0.2 mm, a = 5.5 mm, l = 7.2 mm (see Fig. 4.1).
In the practical design, it was proposed to use the uniaxial wire medium as a negative effective
76
4.4. NUMERICAL EXAMPLES
0
0
|R|, |T| (dB)
-10
k
x
/ k
0
-1
-2
-20
-30
-40
real
|R| |T|
imaginary
-3 2.0
2.5
3.0
3.5
-50 2.0
4.0
2.5
3.0
3.5
4.0
f (GHz)
f (GHz)
(a)
(b)
Figure 4.13: Longitudinal propagation constant and the reflection and transmission coefficients of the slab with parameters from Fig. 4.12.
permittivity medium. Such wires can be naturally embedded in a homogeneous dielectrics with εr > 1 to obtain the desired effective permittivity, see (4.90). The analytical model is valid for a slab infinite in the z and y directions (Fig. 4.3), whereas the practical implementation has only limited dimensions. Several full-wave simulations were performed in MWS to confirm the validity of the theoretical model and to compare results. The general slab from Fig. 4.3 now contains extra wires to realize the negative effective permittivity. The top and side view of the slab is shown in Figs. 4.14 and 4.15, respectively. Only a limited number of particles are shown for illustration (the slab is infinite in the y and z directions). The wires are located at the x-coordinate of the centers of the bodies of the Ω-particles to minimize the mutual coupling, see Fig. 4.15. y x z
aw
aw Figure 4.14: Geometry of the slab composed of Ω-particles and wires (top view). The slab is infinite in the y and z directions, the wire lattice period is aw .
For a better understanding of the behavior of the whole slab, first the wire medium, then the Ω-particles slab and finally the composite of Ω-particles and wires will be studied. As an example
77
4.4. NUMERICAL EXAMPLES z aw x
y
aw
Figure 4.15: Geometry of the slab composed of Ω-particles and wires (side view). The slab is infinite in the y and z directions, the wire lattice period is aw .
of the wire medium, a lattice with wire radius rw = 0.5 mm and lattice period aw = 20 mm was chosen. The slab consists of 20 wires in the x-direction and is infinite in the y and z directions and embedded in the free space (see Fig. 4.14, now without the Ω-particles). The transmission and reflection coefficients calculated by the theoretical model and by the full-wave simulation in MWS are compared in Fig. 4.16. The theoretical reflection and transmission coefficients are calculated by (4.56) and (4.57) with β given by (4.54), whereas now K = 0, εt is equal to (4.90) and µt = µr = 1. Notice the difference between the theoretical approach and the full-wave simulation: the theoretical model calculates the reflection and transmission of a homogenized slab with effective material parameters εt , µt and thus is not able to represent the higher pass bands. By contrast, from the full-wave simulation in MWS the higher pass bands (given by Bragg reflections) can be seen. The plasma frequency, however, is in good agreement for both approaches. With increasing number of wires along the propagation direction, the ripples on the curves are less significant and the curves are smooth for an infinite number of wires.
1.0
0.8
0.8
0.6
0.6
|T|
|R|
1.0
0.4
0.2
0.0 0
0.4
MWS
0.2
theory
2
4
6
8
10
0.0 0
12
MWS
theory
2
4
6
8
10
12
f (GHz)
f (GHz)
(a)
(b)
Figure 4.16: Reflection and transmission coefficients of a wire medium slab consisting of 20 wires in the direction of the plane wave propagation and infinite in the other two coordinates.
As a next step, the slab consisting of only Ω-particles is considered. Such slabs were studied in the past for microwave absorber applications (see, e.g., [225, 235]). Table 4.2 summarizes several examples of simulated structures. The Ω-particle dimensions are indicated in Fig. 4.1b. Parameter n corresponds to the number of Ω-particles in a row (i.e., in the x-direction) in the full-wave simulation, d is the layer thickness in the theoretical model, and aw is the unit cell
78
4.4. NUMERICAL EXAMPLES length. The concentration of particles in the theoretical model is N = 1/a3w in all the examples. a [mm] 3.5 3.5 2 2
l [mm] 5.8 5.8 3.5 3.5
r0 [mm] 0.66 0.66 0.4 0.4
εm [−] 1 3 1 3
aw [mm] 15 15 12 12
n [−] 6 6 6 6
d [mm] 90 90 72 72
aopt [mm] 2.66 2.66 1.58 1.58
dopt [mm] 140 84 77.6 77.6
Figure Fig. Fig. Fig. Fig.
4.17 4.18 4.19 4.20
Table 4.2: Summary of studied geometrical dimensions, slab consists of Ω-particles only.
The absorbing character of the slab can be seen from Figs. 4.17 to 4.20. First, the slab was designed using the analytical approach (“theory”). Then a full-wave simulation in MWS was performed (curves “MWS”) with the geometry depicted in Fig. 4.14. On the x − z planes, a perfect magnetic conductors and, on the x − y planes, perfect electric conductors are applied to make the slab infinite in the y and z directions, and the slab is excited by a plane wave propagating in the positive x direction with the E vector parallel to the z axis (in this simplified case, the second set of particles perpendicular to the first set is not needed). The full-wave simulation always shows a frequency shift of the reflection minimum. Various positions of Ω-particles and wires in the slab were examined by MWS, but this frequency shift still remains. Therefore the dimensions in the theoretical model were changed slightly to obtain preferably good agreement with the full-wave simulation, and this change can be assumed as a known systematical error of the model (curves “theory opt.”). This correction is almost of the same order for all the dimensions examples. Tab. 4.2 shows that only a and d were varied (corresponding aopt and dopt ). The examples in Figs. 4.17 to 4.20 only show an absorbing layer with various geometrical parameters. Note that the position of the resonance fits relatively well after changing the Ωparticle radius from a to aopt . The properties of the slab change most significantly with varying the loop diameter a. a [mm] 3.5 3.5 3 3
l [mm] 6.5 6.5 5.2 4.67
r0 [mm] 0.66 0.66 1.65 1.15
rw [mm] 0.485 0.485 2.26 0.96
εm [−] 1 1 1 2.145
aw [mm] 15 15 20 15
n [−] 6 3 4 4
d [mm] 90 45 87 61
aopt [mm] 3.14 3.29 2.49 2.7
dopt [mm] 90 45 87 61
Figure Fig. Fig. Fig. Fig.
4.21 4.22 4.23 4.24
Table 4.3: Summary of studied geometrical dimensions, slab consists of a junction of Ω-particles and wires.
The geometrical dimensions of several simulated layers composed of both Ω-particles and wires are summarized in Tab. 4.3. Here aw and rw mean the lattice period and wire radius of the wire medium surrounding the Ω-particles. The reflection and transmission coefficients of
79
4.4. NUMERICAL EXAMPLES the slabs are shown in Figs. 4.21 to 4.24. The theoretical, optimized theoretical and full-wave simulation curves are compared in the same way as for the Ω-particles slab. In addition, the theoretical sum of the effective permittivity and permeability is shown. The slab was designed in order to obtain the backward-wave condition Re(εt + µt ) < 0 at the working frequency. The change of the loop radius from a to aopt also changes the Re(εt + µt ) and, moreover, it is questionable, where the resonant frequency from the MWS simulation is (the resonant band is
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
quite wide and has several reflection minims, see Figs. 4.21 to 4.24).
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60
|R| MWS
-60
|T| theory opt
2
3
4
5
6
7
|T| MWS
2
8
3
4
5
6
7
8
f (GHz)
f (GHz)
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
Figure 4.17: Comparison of the theoretical and full-wave analysis, Ω-particles only, geometrical dimensions: a = 3.5 mm, l = 5.8 mm, r0 = 0.66 mm, εm = 1, aw = 15 mm, n = 6, d = 90 mm, aopt = 2.66 mm, dopt = 140 mm (see Tab. 4.2).
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60 0
|R| MWS
|T| theory opt
1
2
3
4
5
6
7
-60 0
8
|T| MWS
1
2
3
4
5
6
7
8
f (GHz)
f (GHz)
Figure 4.18: Comparison of the theoretical and full-wave analysis, Ω-particles only, geometrical dimensions: a = 3.5 mm, l = 5.8 mm, r0 = 0.66 mm, εm = 3, aw = 15 mm, n = 6, d = 90 mm, aopt = 2.66 mm, dopt = 84 mm (see Tab. 4.2).
The only example where the sum Re(εt + µt ) is negative even after loop radius optimization is shown in Fig. 4.24. Theoretically, this slab should support the backward-wave. This effect can really be seen in the MWS simulation when the phase of the electric field inside the slab is animated. To verify the backward-wave existence assumption, the structure was simulated separately: first a slab composed of just wires, then a slab composed of just Ω-particles, and finally their junction. The result is shown in Fig. 4.25. It can be seen that the wire medium
80
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
4.4. NUMERICAL EXAMPLES
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60
|R| MWS
-60
|T| theory opt
6
8
10
|T| MWS
6
12
8
10
12
f (GHz)
f (GHz)
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
Figure 4.19: Comparison of the theoretical and full-wave analysis, Ω-particles only, geometrical dimensions: a = 2 mm, l = 3.5 mm, r0 = 0.4 mm, εm = 1, aw = 12 mm, n = 6, d = 72 mm, aopt = 1.58 mm, dopt = 77.6 mm (see Tab. 4.2).
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60 2
|R| MWS
-60
|T| theory opt
3
4
5
6
7
2
8
|T| MWS
3
4
5
6
7
8
f (GHz)
f (GHz)
Figure 4.20: Comparison of the theoretical and full-wave analysis, Ω-particles only, geometrical dimensions: a = 2 mm, l = 3.5 mm, r0 = 0.4 mm, εm = 3, aw = 12 mm, n = 6, d = 72 mm, aopt = 1.58 mm, dopt = 77.6 mm (see Tab. 4.2).
has some plasma frequency and the wave is not transmitted below it. The omega particles show a frequency stop band. The junction of omega particles and wires shows a pass band in the frequency region, where no wave should be transmitted. The existence of a backward-wave is, however, disputable. The authors of [236] observed a similar effect in a slab composed of π-shaped particles. They simulated a wedge structure to investigate the negative refractive index of their π-structure, but they did not find any negatively refracted beam. Then from the extracted effective permittivity and permeability, they claim that both parameters should be negative to achieve LHM. In this work it was shown that the omega medium together with wires supports a backward wave, but the refractive index is also positive in our case (µ is positive and ε is negative at the frequency of interest). It could be problematic to simulate a wedge consisting of the combination of Ω-particles and wires. The backward wave and matching conditions derived in this work hold only for the normal incidence of the wave. The conditions for oblique incidence were not derived, and it would require a detailed analysis (and probably also another set of Omegas and wires perpendicular to the first one - uniaxial medium, see Fig. 4.4). The
81
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
4.4. NUMERICAL EXAMPLES
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60 2
|R| MWS
-60
|T| theory opt
3
4
5
6
7
|T| MWS
2
8
3
4
5
6
7
8
f (GHz)
f (GHz)
4
0
Re(
t
+
t
)
2
-2
-4
theory theory opt
-6 2
3
4
5
6
f (GHz)
Figure 4.21: Comparison of the theoretical and full-wave analysis, composite of Ω-particles + wires, geometrical dimensions: a = 3.5 mm, l = 6.5 mm, r0 = 0.66 mm, rw = 0.485, εm = 1, aw = 15 mm, n = 6, d = 90 mm, aopt = 3.14 mm, dopt = 90 mm (see Tab. 4.3).
experimental verification of the backward-wave in bianisotropic composites is still open. From the examples shown above it can be seen that: • For the Ω-particles slab (Figs. 4.17 to 4.20), the position of the resonance simulated in MWS agrees relatively well with the theoretical model after a slight change of the Ω-particle loop radius; the amplitudes do not fit very well. • For the slab composed of Ω-particles and wires (Figs. 4.21 to 4.24), the position of the particle resonance agrees with the theoretical model after a slight change of the loop radius, the position of the plasma frequency does not agree well (in the simulation of a pure wire medium, however, the theoretical and MWS curves are in good agreement); the amplitudes do not agree very well. • The examples in Figs. 4.21, 4.22, 4.23 and 4.24 should support a backward wave, since Re (εt + µt ) < 0 at the operational frequency. The practical implementation becomes cumbersome for Ω-particles and wires formed by a circular wire. A structure utilizing the planar technology was proposed. Both the Ω-particles
82
0
0
-10
-10 |R|, |T| (dB)
|R|, |T| (dB)
4.4. NUMERICAL EXAMPLES
-20
-30 |R| theory |R| theory opt
-40
-20
-30 |R| theory opt
-40
|T| theory opt
|T| theory
|R| MWS
|T| theory opt
-50 2
3
4
5
6
7
|T| MWS
-50 2
8
3
4
5
6
7
8
f (GHz)
f (GHz)
4
0
Re(
t
+
t
)
2
-2
-4
theory theory opt
-6 2
3
4
5
6
f (GHz)
Figure 4.22: Comparison of the theoretical and full-wave analysis, composite of Ω-particles + wires, geometrical dimensions: a = 3.5 mm, l = 6.5 mm, r0 = 0.66 mm, rw = 0.485, εm = 1, aw = 15 mm, n = 3, d = 45 mm, aopt = 3.29 mm, dopt = 45 mm (see Tab. 4.3).
and wires are placed on dielectric substrates, see Fig. 4.26. The geometrical dimensions of one Ω-particle are shown in Fig. 4.27. The slab is illuminated by a plane wave in the same way as the wire-structure from Fig. 4.3. Now both Ω-particles and wires are not surrounded by a homogeneous dielectric, but are placed on dielectric substrates with certain permittivity and surrounded by air. Thus the theoretically predicted resonant frequency of the Ω-particles is changed. The agreement with the theory is acceptably good up to substrate permittivity εr ≈ 2 and then the frequency shift becomes larger and it is also more difficult to find the proper dimensions of the Ω-particles in order to achieve a backward-wave propagation in the slab. As an example, two slabs with commercially available substrates were simulated in MWS. The following geometrical dimensions were used (see Fig. 4.27): a = 3.2 mm, l1 = 4.16 mm, strip width of the Ω-particle10 w = 1.52 mm, strip width of the wire medium ww = 1.96 mm, lattice period and also the wire medium spacing aw = 15 mm. Parameters of microwave dielectric substrates Rogers RT/Duroid 5880 (εr = 2.2, substrate thickness h = 0.508 mm, dissipation factor tanδ = 0.0009, metal thickness 30 µm) and Rogers RO3003 (εr = 3.0, substrate thickness h = 0.762 mm, dissipation factor tanδ = 0.0013, metal thickness 30 µm) were used in the 10
The effective width of a planar strip corresponding to a circular wire with radius r0 is w = 4r0 [222].
83
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
4.5. EXPERIMENTAL RESULTS
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt |R| MWS
|T| theory
-60 0
-60
|T| theory opt
2
4
6
8
|T| MWS
0
10
2
4
6
8
10
f (GHz)
f (GHz)
20
0
Re(
t
+
t
)
10
-10
-20
theory theory opt
-30
2
3
4
5
f (GHz)
Figure 4.23: Comparison of the theoretical and full-wave analysis, composite of Ω-particles + wires, geometrical dimensions: a = 3 mm, l = 5.2 mm, r0 = 1.65 mm, rw = 2.26, εm = 1, aw = 20 mm, n = 4, d = 87 mm, aopt = 2.49 mm, dopt = 87 mm (see Tab. 4.3).
simulations. The transmission and reflection coefficients of the simulated slabs are shown in Fig. 4.28.
4.5
Experimental results
The experiments were accomplished in cooperation with the Department of Electrical and Electronics Engineering, Bilkent University in Ankara, Turkey, on the basis of the theoretical results obtained in the previous section. All the measured results given in this chapter are reproduced with a permission of Prof. Ekmel Ozbay and his research team. For the construction of a backward-wave perfectly matched slab, a set of Ω-particles embedded in a wire medium is needed, as shown in [234]. The planar Ω-particles and wires were deposited on a microwave substrate and their dimensions were designed in order to work in the frequency band about 10 GHz. For the experiments, an FR4 printed circuit board with permittivity εr = 4.3, thickness h = 1.6 mm and copper cladding thickness 30 µm was used. Two monopole antennae were used to transmit and detect the electromagnetic waves through the single Ω unit cell and consequently through the slab. The antennae were connected to the
84
0
0
-10
-10
-20
-20
|R|, |T| (dB)
|R|, |T| (dB)
4.5. EXPERIMENTAL RESULTS
-30 -40 |R| theory
-50
-30 -40 |R| theory opt
-50
|R| theory opt
|T| theory opt
|T| theory
-60 2
|R| MWS
-60
|T| theory opt
3
4
5
6
7
|T| MWS
2
8
3
4
5
6
7
8
f (GHz)
f (GHz)
-10
Re(
t
+
t
)
0
theory theory opt
-20 2.0
2.5
3.0
3.5
4.0
f (GHz)
Figure 4.24: Comparison of the theoretical and full-wave analysis, composite of Ω-particles + wires, geometrical dimensions: a = 3 mm, l = 4.67 mm, r0 = 1.15 mm, rw = 0.96, εm = 2.145, aw = 15 mm, n = 4, d = 61 mm, aopt = 2.7 mm, dopt = 61 mm (see Tab. 4.3).
0
|T| (dB)
-20
-40
wires
-60
-particles -particles + wires
-80 0
1
2
3
4
5
f (GHz)
Figure 4.25: Comparison of the transmission coefficient of a slab composed of only wires, only Ω-particles and finally their junction. The geometrical dimensions are: a = 3 mm, l = 4.67 mm, r0 = 1.15 mm, rw = 0.96, εm = 2.145, aw = 15 mm, n = 4, d = 61 mm, aopt = 2.7 mm, dopt = 61 mm.
HP-8510C network analyzer to measure the transmission coefficients. First, a transmission spectrum of single Ω-particles with two orientations with respect to the incident electric field was investigated. The loop diameter a = 1.19 mm, strip width w = 0.45 mm, effective tail length l2 = 1.8 mm (see Fig. 4.27). The incident field propagates along
85
4.5. EXPERIMENTAL RESULTS
Figure 4.26: Proposal for the practical realization of the Ω-slab, both Ω-particles and wires are placed on dielectric substrates.
ww
w z y
z
w x
y
l1 l2
a
x
aw
aw
aw/2 (a)
(b)
Figure 4.27: Unit cell dimensions together with the geometry of the planar form of the Ω-particle (a) and the wire medium (b).
the x direction with the E field vector along the z direction. A dip in the transmission spectrum was observed at frequency 10.75 GHz and attributed to the resonant nature of these structures. After rotating the particle by 90◦ around the y axis, the resonance frequency was observed at frequency 11.1 GHz. After closing the gap in the Ω particle, no resonance was observed in the frequency band about 10 GHz. Several MWS simulations were performed and the field monitor facility was used to show the electric field within the Ω unit cell. It was found that the electric field localized in the gap between the wires is enhanced by factor ≈ 40. The enhancement of the field inside SRR reported in [237] was ≈ 150. This result is to be expected since SRR is formed from several more capacitive elements. A further step in composing the backward-wave slab is a study of the transmission of a slab composed of only Ω particles. The Ω particles are arranged periodically with 5, 40 and 30 unit cells in the x, y and z directions, respectively. The lattice period in all the directions is
86
4.5. EXPERIMENTAL RESULTS
0
|R|, |T| (dB)
-10
-20
-30
-40
-50
|R|, substrate 1
|R|, substrate 2
|T|, substrate 1
|T|, substrate 2
3
4
5
f (GHz)
Figure 4.28: Comparison of the reflection and transmission coefficients of two slabs designed utilizing the planar technology. The Rogers RT/Duroid 5880 with εr = 2.2, substrate thickness t = 0.508 mm and dissipation factor tanδ = 0.0009 is referred to as “substrate 1” and the Rogers RO3003 substrate with εr = 3.0, substrate thickness t = 0.762 mm and dissipation factor tanδ = 0.0013 is referred to as “substrate 2”.
aw = 5 mm (see Fig. 4.27a). Three different geometrical configurations were measured. The curves labeled “Normal Ω” correspond to a = 1.19 mm, w = 0.45 mm, l2 = 1.8 mm, the curves labeled “Long Ω” correspond to a = 1.19 mm, w = 0.45 mm, l2 = 2.2 mm, and the curves labeled “Large Ω” correspond to a = 1.43 mm, w = 0.54 mm, l2 = 2.16 mm. The measurement of the transmission spectrum was performed in free space and two horn antennae connected to the HP8510C network analyzer were used as transmitters and receivers. Several geometrical configurations were studied and the transmission spectra with a single bandgap in the vicinity of frequency 10 GHz were observed (for further details, see [238]). The frequency bandgap of the slab was wider than that observed in a single Ω particle due to the electromagnetic coupling between neighboring resonators, and both measured and simulated data are shown in Fig. 4.29. The results simulated in MWS are in good agreement with the measured data.
0
0 (a)
-10 -20
-20 |T| (dB)
|T| (dB)
(b)
-10
-30 -40
Normal
-30 -40
Normal
Long
-50
Long
-50
Large
-60 7
Large
-60 8
9
10
11
12
13
14
15
7
8
9
10
11
12
13
14
15
f (GHz)
f (GHz)
Figure 4.29: Transmission coefficient of the periodic arrangement of three different Ω-structures, both measured (a) and simulated (b) results. The measured curves are adopted with permission from [238].
Finally, the transmission spectrum of a material composed of periodic omega and thin wire
87
4.6. SUMMARY
0
|T| (dB)
-10
Comp. 1 Wire
-20
0
-10
(b)
Long
-30
-30
-30
-40
-40
-40
-50
-50
-50
-60
-60
-60
0
0 Normal Comp. 1 Wire
-20
-10
Long Comp. 2
-30
-30
-30
-40
-40
-40
-50
-50
-50
8
9
-60 10 11 12 13 14 15 7 f (GHz)
(f)
-10 -20
7
Comp. 3
0 (e)
-20
-60
Large
Comp. 2
-20
(d)
(c)
-10
-20
-10 |T| (dB)
0 Normal
(a)
Large Comp. 3
-60 8
9
10 11 12 13 14 15
7
f (GHz)
8
9
10 11 12 13 14 15 f (GHz)
Figure 4.30: Measured (a), (b), (c) and simulated (d), (e), (f) results of the transmission coefficient of three different composite materials (red curves) corresponding to three different geometrical dimensions of the Ω-particles (blue lines). The black line in (a) and (d) correspond to the transmission coefficient of only a thin wire medium slab. The measured curves are adopted with permission from [238].
media was studied. The measurement setup was identical to the case of the slab composed of only Ω particles. The unit cell of a continuous wire is shown in Fig. 4.27b. Three different geometrical configurations of composite material consisting of Ω-particles and wires labeled as “Comp. 1”, “Comp. 2” and “Comp. 3” were simulated and measured. The geometrical dimensions of Ωparticles used in the above mentioned composites correspond to “Normal Ω”, “Long Ω” and “Large Ω”, respectively. The wire strip width is w = 1.44 mm and height (equal to the unit cell period) is aw = 5 mm. The results simulated in MWS and the measured results are shown in Fig. 4.30. The difference between the simulated and measured transmission peak is caused mainly by the losses in the substrate. The FR4 printed circuit board has rather poor dielectric losses tan δ ≈ 0.05 at frequencies about 10 GHz. Use of a substrate with lower dielectric losses would improve the transmission coefficient of the slab. The measured and simulated curves in Fig. 4.30 confirm the theoretically predicted assumption of the existence of a backward wave below the WM plasma frequency, where no wave should propagate.
4.6
Summary
In this chapter, the perfectly matched backward-wave slab was studied. The bianisotropic approach utilizing Ω-shaped particles was used. The goal of the chapter was to propose a new route to realization and matching of backward-wave slabs. The known realizations of backwardwave media imply negative permeability and permittivity. The novelty of the approach in this chapter is that negative permeability is not needed. The known realizations imply that for
88
4.6. SUMMARY matching of the slab with free space the condition εr = µr is necessary. The novelty of the approach in this work lies in the fact that a slab can be matched even without satisfying this condition. First, the Ω-particles were studied analytically and basic mixing formulas were used for the design of the slab. A very simple LC model of the Ω-particles was used to derive the zeroreflection and backward-wave conditions for such an effective medium and it was found that a negative permittivity background material is needed for the existence of a backward-wave. It was proposed to use a single wire medium for the implementation of a negative permittivity hosting material and the composite consisting of Ω-particles and wires was studied, first analytically and then in the CST Microwave Studio full-wave simulator. The analytical approach takes into account the homogenized medium, whereas the full-wave simulation calculates the real geometrical structure. The results of the two approaches are in relatively good accord. A form utilizing planar technology was proposed for a practical implementation of the slab. The slab was fabricated and measured, and the results are in good agreement with the theoretical predictions and full-wave simulations.
89
Chapter 5
Conclusion The aim of this doctoral thesis has been to study the propagation of electromagnetic waves in periodic structures, namely in metamaterials. These structures are studied nowadays by many research teams all over the world, due to their intriguing properties and promising future practical applications. One could say that metamaterials have formed a new research direction in recent years. The thesis deals with three main topics related to the propagation of electromagnetic waves in metamaterials. The introductory part gives a brief historical overview of metamaterial research. The main research directions in this area are outlined, together with a list of crucial papers published in each subject field in recent years. Chapter 2 summarizes an investigation of the properties of a metamaterial in the form of a left-handed coplanar waveguide and its equivalent circuit. The properties of a general left-handed transmission line were given. A novel left-handed coplanar waveguide was designed, fabricated and measured. The coplanar waveguide was chosen due to some advantageous properties over other transmission lines used for metamaterials. The dispersion characteristic of the left-handed coplanar waveguide calculated by the CST Microwave Studio predicts the pass-bands of the left-handed and right-handed modes. The lowest pass band of the left-handed wave was verified experimentally. The simple equivalent circuit of the line was introduced together with the values of its elements. The dispersion characteristic of the first left-handed mode computed using this circuit was compared to the curve computed by the CST Microwave Studio, and good agreement was observed. The aim of the equivalent circuit design was to reduce the time needed to design practical applications using the proposed line. The left-handed coplanar waveguide is a structure suitable for constructing coplanar waveguide based microwave devices with improved properties in comparison with conventional solutions. This was recently proved by other research teams, which have fabricated and measured several devices. These devices, however, can be further improved and novel coplanar waveguide based devices, for instance passive microwave circuits used in monolithic microwave integrated circuits utilizing the properties of metamaterials, can be studied as a continuation of this work.
90
5. CONCLUSION Chapter 3 investigates a volumetric form of a metamaterial in the form of a triple wire medium, with the aim to apply it as a negative permittivity isotropic metamaterial. The triple wire medium is formed by a three dimensional lattice of mutually perpendicular metallic connected wires. The electromagnetic modes propagating in this medium were studied first analytically and then numerically in a full-wave simulator. The numerical examples show that there exist eigen modes both below and above the plasma frequency in all important directions Γ–X, Γ–M and Γ–R of the first Brillouin zone. Two different models for the relative permittivity of the medium were used. Using only a simple isotropic model of permittivity (3.30), only physical solutions were found, while with the use of a more sophisticated model (3.1) both physical and nonphysical solutions were found. The physical modes comply with physical intuition, i.e., they are evanescent below the plasma frequency, and become propagating above the plasma frequency. In the direction Γ–M, the attenuation and phase constants have the same amplitude as in the case of the direction Γ–X, since the propagation of the wave along one axis is equivalent to the propagation along the unit cell face. The triple wire medium behaves as an isotropic material near the plasma frequency for the wave propagating parallel with any coordinate plane. Two numerical experiments were performed. The first shows the distribution of the electric field excited by the dipole located in the center of the sphere of the 3D wire material, while the other shows the distribution of the electric field inside a wire cube with 8×8×8 unit cells. The value of the attenuation constant and consequently the effective permittivity of the wire lattice can be estimated from the resulting electric field distributions. The effective permittivity extracted from the electric field distribution is in accord with the theoretical predictions. For a practical implementation, the planar form of the triple wire medium was proposed. It is shown that the properties of the proposed planar lattice are very similar to those of the ideal wire lattice. The transmission coefficient of the lattice was measured in a waveguide setup in the direction along the unit cell axis and along the unit cell face diagonal. The intensity of the electric field was measured using an electric field probe inside the lattice for all three important propagation directions. The measured and simulated results show relatively good agreement, which validates the isotropy of the triple wire medium below the plasma frequency. The proposed form of the 3D structure fabricated by the planar technology is suitable for future combination with an isotropic negative permeability material using modified split ring resonators in order to design a double-negative isotropic metamaterial. The split ring resonators would be etched on dielectric substrates as well as the wires. Since the effective permittivity of the triple WM is negative in a wide band, it would be possible to achieve both εeff (triple WM) and µeff (SRRs) at a certain frequency. Such a combination, however, would require much more detailed analysis, and some issues are still open. The construction of a fully isotropic three-dimensional double-negative material remains still challenging. In Chapter 4, the perfectly matched backward-wave slab was theoretically studied. The
91
5. CONCLUSION bianisotropic approach utilizing Ω-shaped metallic particles was used. The goal of the chapter was to propose a new route to realization and matching of backward-wave slabs. The known implementations of backward-wave media imply negative permeability and permittivity. The novelty of the approach in this work is that negative permeability is not needed. The known realizations imply that for matching the slab with free space the condition εr = µr is necessary. The novelty of the approach in this work lies in the fact that we can match a slab without satisfying this condition. First, the Ω-particles were studied analytically and basic mixing formulas were used for the design of the slab. A very simple LC model of the Ω-particles was used to derive the zero-reflection and backward-wave conditions for such an effective medium, and it was found that a negative permittivity background material is needed for the existence of a backward-wave. It was proposed to use a single wire medium for the realization of a negative permittivity hosting material and the composite consisting of Ω-particles and wires was studied, first analytically and then in the CST Microwave Studio full-wave simulator. The results of the two approaches are in relatively good accord. A form utilizing planar technology was proposed for a practical implementation of the slab. The slab was fabricated and measured and the results are in good agreement with the theoretical predictions and full-wave simulations. A frequency band was observed where the backward wave exists. As a continuation of the work presented in Chapter 4, one could fabricate a more general structure, namely the uniaxial Ω-medium. Then the properties of the slab for a normal wave incidence would be independent of the wave polarizations. The fabrication, however, seems to be cumbersome. In this work, the bianisotropic effects in omega type metamaterials were investigated for the parallel propagation of electromagnetic waves. However, bianisotropic metamaterials also present interesting properties for waves with normal incidence. In the future, these structures are necessary to investigate more in detail both numerically and experimentally. Bianisotropic materials form alternative metamaterials with prospecting physical properties.
92
Appendix A
Derivations A.1
Conventional coplanar waveguide
The conventional coplanar waveguide cross-section is depicted in Fig. A.1. The characteristic impedance Z0 and effective permittivity εeff are given by the following relations [239] 30π K(k00 ) Z0 = √ , εeff K(k0 ) where εeff = 1 + s , k0 = s + 2w
(A.1)
(εr − 1) K(k1 ) K(k00 ) , 2 K(k10 ) K(k0 )
¡ πs ¢ sinh 4h ´, ³ k1 = sinh π(s+2w) 4h
k10 =
(A.2) p 1 − k1 ,
and K (·) denotes the complete elliptic integral of the first kind [240].
s/2
h
w
er
Figure A.1: Cross-section of a conventional coplanar waveguide.
93
(A.3)
APPENDIX A
A.2
Coplanar waveguide with finite extent ground planes
A cross-section of the line with dimensions is shown in Fig. A.2 and its characteristic impedance and effective permittivity are given by [241] 30π K(k20 ) Z0e = √ , εeff2 K(k2 ) where εeff2 = 1 + q (εr − 1) , s a k2 = b
1 − b2 /c2 , 1 − a2 /c2 k20 =
k3 =
sinh
q=
¡ πa ¢ s
¡ 2h ¢ πb sinh 2h
p 1 − k2 ,
k30 =
(A.4)
1 K(k20 ) K(k3 ) , 2 K(k2 ) K(k30 ) 1 − sinh2 1−
¡ πb ¢
(A.5) ¡
¢
2 πc 2h ¢ / sinh ¡ 2h ¢ ¡ 2 πc , sinh2 πa / sinh 2h 2h
p 1 − k3 ,
and β0 is the phase constant of this line.
c b a
h
er
Figure A.2: Cross-section of a coplanar waveguide with finite extent ground planes.
94
(A.6)
(A.7)
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