I
11'
Directive Antenna Using Metamaterial Substrates
by
Weijen Wang
B. S. in Electrical Engineering and Computer Science
Massachusetts Institute of Technology, Cambridge, June 2002
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2004
@ Massachusetts Institute of Technology 2004. All rights reserved.
Author .............
Depadment of Electrical En geering and Computer Science
May 20, 2004
Certified by...........
Jin Au Kong
Thesis Supervisor
Accepted by .......
................
Arthur C. Smith
Chairman, Department Committee on Graduate Students
ASSACHUSETTS INST
OF TECHNOLOGY
E.
EJUL 2 0 200
BARKER
LIBRARIES
A
2
Directive Antenna Using Metamaterial Substrates
by
Weijen Wang
Submitted to the Department of Electrical Engineering and Computer Science
on May 20, 2004, in partial fulfillment of the
requirements for the Degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Using a commercially available software(CST Microwave Studio®), two kinds of simulations have been carried out on different metamaterials in the microwave regime.
One is transmission and reflection of a unit cell in a waveguide, and the other is
parallel plate slab farfield radiation. The S-parameters are obtained from the waveguide simulation and are used to retrieve the effective permittivity and permeability
with which we can estimate the farfield radiation using analytic method. Thus, by
comparing the farfield radiation from two different methods, analytic and slab simulation, we find that the analytic method is able to indicate many major features of
the slab simulation's farfield results, implying that within a certain frequency range,
we can treat the metamaterial as being homogeneous. After comparing the radiation
performance of different metamaterial as antenna substrates, a structure is chosen
to be optimized in such a way that it improves in radiation power, beamwidth, and
bandwidth.
Thesis Supervisor: Jin Au Kong
Title: Professor, Department of Electrical Engineering and Computer Science
3
4
Acknowledgments
First, I would like to thank Professor Jin Au Kong for giving me the opportunity to do
research in electromagnetics. I will always remember how his excellent teaching has
inspired me, and helped me in understanding one of the most difficult subjects in the
electrical engineering area. I am also thankful for his kindness and encouragement. I
am indebted to Dr. George A. Kocur, Professor Steven R. Lerman, Dr. V. Judson
Harward, and Professor Ruaidhri M. O'Connor, with whom I have gained valuable
experience as a teaching assistant.
I would also like to thank the CETA group members, namely Benjamin E. Barrowes, Jianbing Chen, Xudong Chen, Tomasz M. Grzegorczyk, May Lai, Jie Lu,
Christopher Moss, Madhusudhan Nikku, Joe Pacheco, Zachary M. Thomas, and BaeIan Wu, for their valuable advice and discussions on various topics. I am grateful for
the friendship and comradeship from my former and current fellow teaching assistants: Sehyun Ahn, Jeffrey M. Bartelma, Liou Cao, Christopher A. Cassa, Curtis R.
Eubanks, Peilei Fan, Abdallah W. Jabbour, Bharath K. Krishnan, Hariharan Lakshmanan, Jedidiah B. Northridge, Fernando Perez, and Anamika Prasad.
It has been almost six years since I first joined the great minds at the Massachusetts Institute of Technology. As what the rumors have spread, it has not been
easy surviving in this institution. I still remember being doubtful about my college
choice after meeting several international Math, Physics, Chemistry, and Informatics
Olympiad's medalists during my first day of international orientation. However, the
friendships that I have established at MIT have helped me tremendously throughout
my undergraduate and graduate years. To all my friends, I am thankful for the company when I stayed up at night, the encouragement when I felt down, the counsel
when I was lost, and mostly, the friendship that they have given me. Special thanks
to Senkodan Thevendran and Felicia Cox, who have accepted me regardless of good
or bad, for their love and patience, and making me a better person. Lastly, I would
like to acknowledge Bae-Ian Wu, who has been both a mentor and a friend, for his
guidance and support without which this thesis would not have been possible.
5
Finally, I would like to thank my parents for always trying to provide me with the
best education opportunities. Without them, I will not be where I am today. They
have never stopped believing in me and giving me the reason and the energy to thrive
at wherever I am. To my brother and sister, thank you for always being there for me
and bringing joy to my life.
I dedicate this thesis to my family and friends.
6
To my family and friends
7
8
Contents
1
Introduction
17
2
Methodology
21
3
2.1
Prior art . . . . . . . . . . . ..
2.2
Simulation ...............
22
. . .
23
2.2.1
Radiation setup . . . . . . . .
24
2.2.2
Waveguide setup . . . . . . .
26
2.3
Analytic method for farfield radiation
28
2.4
Radiation results and normalization .
30
Comparative Study of Different Metamaterial Substrate
37
3.1
2-D Smith structure
. . . . . . . . . . . . . . . . . . . . .
38
3.2
1-D Smith structure
. . . . . . . . . . . . . . . . . . . . .
44
3.3
Pendry structure
. . . . . . . . . . . . . . . . . . . . . . .
49
3.4
Omega structure
. . . . . . . . . . . . . . . . . . . . . . .
55
3.5
S structure................
3.6
Sum m ary
.....
....
......
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Optimized Metamaterial Structure
60
66
67
4.1
Optimized unit cell's geometry . . . . . . . . . . . . .
68
4.2
Optimization of cell orientation and antenna position
70
4.3
Comparison with analytic method . . . . . . . . . . .
77
4.4
Summary
. . . . . . . . . . . . . . . . . . . . . . . .
79
9
5 Conclusion
81
Bibliography
83
10
List of Figures
1-1
Dipole emission in a substrate with n = 0 . . . . . . . . . . . . . . . .
19
2-1
M ethodology chart . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2-2
Full size rod structure
. . . . . . . . . . . . . . . . . . . . . . . . . .
24
2-3
Mesh for full size rod structure
. . . . . . . . . . . . . . . . . . . . .
25
2-4
Slab of metamaterial(rod medium)
. . . . . . . . . . . . . . . . . . .
26
2-5
Unit cell rod structure in a waveguide . . . . . . . . . . . . . . . . . .
27
2-6
Retrieval results for rod medium . . . . . . . . . . . . . . . . . . . . .
27
2-7
Radiation configuration of a linesource in an infinite isotropic slab of
+
.
28
2-8
Radiation plane of interest . . . . . . . . . . . . . . . . . . . . . . . .
30
2-9
Radiated power and normalized radiation from analytic method for
thickness d 1
d2
..
. . . . . . . ..
rod structure ........
. . ..
. . . . . . . . . ..
...............................
31
2-10 Radiated power and normalized radiation from simulation for rod full
size and slab structure . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2-11 Beamwidth for full size rod medium . . . . . . . . . . . . . . . . . . .
33
2-12 Beamwidth for a slab of rod medium . . . . . . . . . . . . . . . . . .
34
3-1
Unit cell 2-D Smith structure in a waveguide . . . . . . . . . . . . . .
38
3-2
Retrieval results for 2-D Smith structure . . . . . . . . . . . . . . . .
39
3-3
Slab of 2-D Smith structure
. . . . . . . . . . . . . . . . . . . . . . .
40
3-4
Mesh for 2-D Smith structure . . . . . . . . . . . . . . . . . . . . . .
40
3-5
Radiated power and normalized radiation from analytic method and
slab simulation for 2-D Smith structure . . . . . . . . . . . . . . . . .
11
42
3-6
Beamwidth for 2-D Smith structure slab simulation . . . . . . . . . .
43
3-7
Unit cell 1-D Smith structure in a waveguide . . . . . . . . . . . . . .
44
3-8
Slab of 1-D Smith structure . . . . . . . . . . . . . . . . . . . . . . .
45
3-9
Retrieval results for 1-D Smith structure . . . . . . . . . . . . . . . .
46
3-10 Mesh for 1-D Smith structure
. . . . . . . . . . . . . . . . . . . . . .
46
3-11 Radiated power and normalized radiation from analytic method and
slab simulation for 1-D Smith structure . . . . . . . . . . . . . . . . .
47
3-12 Beamwidth for 1-D Smith structure slab simulation . . . . . . . . . .
48
3-13 Unit cell Pendry structure in a waveguide . . . . . . . . . . . . . . . .
49
3-14 Retrieval results for Pendry structure . . . . . . . . . . . . . . . . . .
50
3-15 Slab of Pendry structure . . . . . . . . . . . . . . . . . . . . . . . . .
51
3-16 Mesh for Pendry structure . . . . . . . . . . . . . . . . . . . . . . . .
52
3-17 Radiated power and normalized radiation from analytic method and
slab simulation for Pendry structure . . . . . . . . . . . . . . . . . . .
53
3-18 Beamwidth for Pendry structure slab simulation . . . . . . . . . . . .
54
3-19 Unit cell Omega structure in a waveguide . . . . . . . . . . . . . . . .
55
3-20 Retrieval results for Omega structure . . . . . . . . . . . . . . . . . .
56
3-21 Slab of Omega structure . . . . . . . . . . . . . . . . . . . . . . . . .
57
3-22 Mesh for Omega structure . . . . . . . . . . . . . . . . . . . . . . . .
57
3-23 Radiated power and normalized radiation from analytic method and
slab simulation for Omega structure . . . . . . . . . . . . . . . . . . .
58
3-24 Beamwidth for Omega structure slab simulation . . . . . . . . . . . .
59
3-25 Unit cell S structure in a waveguide . . . . . . . . . . . . . . . . . . .
60
3-26 Retrieval results for S structure . . . . . . . . . . . . . . . . . . . . .
61
3-27 Slab of S structure
.
62
. . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3-28 Mesh for S structure
. . . . ..
.........
. . . . . . ..
. ...
3-29 Radiated power and normalized radiation from analytic method and
slab simulation for S structure . . . . . . . . . . . . . . . . . . . . . .
64
3-30 Beamwidth for S structure slab simulation . . . . . . . . . . . . . . .
65
12
4-1
Unit cell optimized structure in a waveguide . . . . . . . . . . . . . .
68
4-2
Retrieval results for the optimized structure
. . . . . . . . . . . . . .
69
4-3
Two different 1-D orientations of unit cells arrangement . . . . . . . .
70
4-4
Mesh for the optimized structure
. . . . . . . . . . . . . . . . . . . .
71
4-5
Radiated power and normalized radiation from slab simulation for
Pendry version x and Pendry version y structure . . . . . . . . . . . .
72
4-6
Beamwidth for Pendry version x structure slab simulation
. . . . . .
73
4-7
Beamwidth for Pendry version y structure slab simulation
. . . . . .
74
4-8
Two different antenna positions . . . . . . . . . . . . . . . . . . . . .
75
4-9 Radiated power and normalized radiation from slab simulation for different antenna positions
. . . . . . . . . . . . . . . . . . . . . . . . .
76
4-10 Radiation configuration of a linesource in an infinite anisotropic slab
of thickness d 1
+
d 2 . . . . . . . . . . ..
. ...
.
.
. .
.. .
. .
.
77
4-11 Radiated power and normalized radiation from analytic method for
Pendry version x and Pendry version y structure . . . . . . . . . . . .
13
78
14
List of Tables
3.1
Comparison among different metamaterial substrates
15
. . . . . . . . .
66
16
Chapter 1
Introduction
Recently, there has been growing interest in the study of metamaterials both theoretically and experimentally. Metamaterials are artificial materials synthesized by
embedding specific inclusions, for example, periodic structures, in the host media [1][17]. Some of these materials exhibit either negative permittivity or negative permeability [2]-[16]. If both permittivity and permeability of such materials are negative at
the same time, then the composite possesses an effective negative index of refraction
[18] and is referred to as a left-handed metamaterial. The name is used because the
electric field, the magnetic field, and the wave vector form a left-handed system [18].
These metamaterials are typically realized artificially as composite structures that are
composed of periodic metallic patterns printed on dielectric substrates. These inclusions affect the macroscopic properties of the bulk composite medium which exhibits
a negative effective permittivity and/or permeability for a certain frequency band.
One of the first theoretical studies was done by Veselago in the 1960's [18]. He
examined the propagation of plane waves in a hypothetical substance with simultaneous negative permittivity and permeability. He found that the Poynting vector of
the plane wave is antiparallel to the direction of the phase velocity, which is contrary
to the conventional case of plane wave propagation in natural media. It has been
shown by Pendry et al. that a medium constructed with periodic metallic thin wires
behaves as a homogeneous material with a corresponding plasma frequency when the
lattice constant of the structure and the diameter of the wire are small in compar17
ison with the wavelength of interest [2].
Pendry et al. also showed that split ring
resonators can result in an effective negative permeability over a particular frequency
region [3]. Only a couple of years ago, Smith, Schultz, and Shelby from the University of California-San Diego constructed the first left-handed metamaterial in the
microwave regime, and demonstrated experimentally the negative index of refraction
[5].
Many properties and potential applications of metamaterials have been explored
and analyzed theoretically. Pendry proposed that left-handed metamaterials could
be used to build a perfect lens with sub-wavelength resolution [19]. Studies have been
done on backward waves propagation [20, 21], waveguides [22, 23], Cerenkov radiation
[24], resonators [25], and growing evanescent waves [26], etc.
We can view metamaterials as a class of materials broader than left-handed metamaterials. It is a class of materials that enable us to manipulate the bulk permittivity
and permeability. To this date, such technology in left-handed metamaterials is best
suited for our purpose.
Little research has been done on applications of metamaterials in antenna systems.
Emission in metamaterials using an antenna has been recently presented in 2002 by
Enoch et al. [27].
Two features are of interest regarding the control of emission:
direction and power of emission. Enoch et al. have demonstrated the feasibility of
using a rod medium to direct the emission of an embedded source towards the normal
of the substrate, thus confining the radiated energy to a small solid angle.
The metamaterial they used was a metallic mesh of thin wires. Such medium
can be characterized by a plasma frequency [2]. The effective permittivity can be
expressed as
EP =
W-
/w2
where w, is the plasma frequency and w is the frequency of the propagating electromagnetic wave. From this equation, the effective permittivity is negative when
the frequency is below the plasma frequency. When operating at the plasma frequency, the effective permittivity is zero, and hence yields a zero index of refraction
18
air n = 1
n=0
Figure 1-1: Dipole emission in a substrate with n = 0
(n =
jyreC).
From Snell's law
sin Ot /sin Oi = ni/nt
(1.2)
where i denotes the incident medium and t denotes the transmitted medium, for
ni ~ 0, we obtain a Ot of zero regardless of what 9i is. As shown in Figure 1-1, if we
place a dipole in a substrate with index of refraction n = 0, the exiting ray from the
substrate will be normal to the surface. Therefore, the closer the operating frequency
is to the plasma frequency, the better the directivity. The permittivity just above
the plasma frequency can be positive but still less than one. This will correspond to
an index of refraction of less than one and close to zero. Then for any incident ray
from inside such a medium to free space, the angle of refraction will be close to zero
and the refracted rays will be close to normal. This property can be used to control
the direction of emission. More specifically, We can expect directive radiation when
the absolute value of the index of refraction is less than one and its imaginary part is
small. Inspired by Enoch, we proposed to use left-handed like metamaterial, where
ceff
and peff will be zero for certain frequencies.
Through the manipulation of a
structure's geometry, where Eeff = 0 and peff = 0 are can be tuned to the desired
frequencies to produce directional emission in a wider band.
By using a commercially available software called CST Microwave Studio® [28],
studies will be done on a dipole embedded in different metamaterial substrates. In
19
Chapter 2, methodology for studying antenna radiation in metamaterial substrates
will be presented. In Chapter 3, different metamaterials are analyzed for antenna
substrates application.
Simulation results are compared with specific focuses on
beamwidth, bandwidth, and power. In Chapter 4, we will show our optimized results and how other parameters like position of antenna can effect radiation results.
Lastly, Chapter 5 is conclusion.
20
Chapter 2
Methodology
As discussed in the previous chapter, the objective of this thesis is to develop a
methodology to analyze and design metamaterial substrates for directive antenna. In
this chapter, we will present the methodology in detail. The flow chart in Figure 2-1
shows the basic elements in our methodology and the process of analysis. We will
use the rod medium as an example to illustrate how our methodology works and
how it can assist us to optimize a metamaterial structure as a substrate for directive
antenna.
one cell
> S parameters
PEC/PMC
wavegu ide
Metamaterials
Analytic formula
radiation
Microwave Studio
parallel plate waveguide
radiation
Farfield
results
Figure 2-1: Methodology chart
As seen in the methodology chart, the metamaterial is the starting point of the
analysis and is usually composed of periodic structures of metal and dielectric for the
microwave region. We can build the structure and perform experiments to determine
21
what its performance is as an antenna substrate, or we can do numerical simulations,
or we can do theoretical studies assuming the substrate is homogeneous. Since experiment is the most expensive and time consuming, we will use numerical simulation
and theoretical studies first to study the properties of different metamaterials, and
later design a metamaterial structure that is good for antenna substrate. Simulating
the real size structure requires a lot of memory and simulation time, therefore we will
use a slice of metamaterial which is placed in a parallel plate waveguide to approximate the radiation effect. We can also simulate a unit cell of the metamaterial in a
waveguide and extract the S-parameters in order to find the effective E and p for all
frequencies [29]. With the effective permittivity and permeability, we can calculate the
corresponding S-parameters and compare with the ones we obtained from simulation.
Furthermore, we can use analytic formula to obtain farfield results with peff and Eeff.
This theoretical farfield results is faster to acquire than the ones from simulating in a
parallel plate waveguide. Therefore, for optimization, we will use theoretical method
to find a structure with the best theoretical farfield radiation first, then simulate the
metamaterial in the parallel plate waveguide.
2.1
Prior art
Enoch et al. has used metamaterial as antenna substrate[27].
The metamaterial
they used were layers of copper grids separated by foam. The copper grids has a
square lattice with a period of 5.8 mm; each layer has a separation of 6.3 mm. This
metamaterial possesses a microwave plasma frequency at about 14.5 GHz. The source
of excitation used is a monopole antenna fed by a coaxial cable. The emitting part of
the monopole is approximately located at the center of the metamaterial substrate. In
the experiment, a ground plane is added to the metamaterial substrate. At 14.65 GHz,
it was shown to have the best directivity.
The block of metamaterial substrate can be treated as a homogenous material.
Since it has a plasma frequency of _14.5 GHz, the permittivity is closed to zero at
this frequency, which means that the index of refraction is closed to zero as well.
22
From ray theory, the exiting ray from the substrate will be very closed to the normal
of the substrate(as shown in Chapter 1).
Changes in the copper grid metamaterial can only change where ecf
= 0 is.
However, for the index of refraction n = 0, we do not necessarily have to make
permittivity zero; we can also have peff = 0. The known technology to manipulate both permittivity and permeability of a structure from the study of left handed
metamaterials can be applied. In the next chapter, we will study some known left
handed metamaterials as antenna substrates. However, in this chapter we will use rod
medium as an example, since it is the simplest metamaterial and we have previous
literature to compare our simulation results with. We will use similar aperture size
for all our metamaterial substrates, and keep our region of interest in the microwave
region. Ground planes will not be used in our study; the main effect will be that the
beamwidth will be wider without the ground plane. In the next subsection, we will
show how to simulate metamaterials using CST Microwave Studio®.
2.2
Simulation
For simulation, we use CST Microwave Studio®. It uses Finite Integration Technique (FIT)[30] for general purpose electromagnetic simulations.
FIT applied to
Cartesian grids in the time domain is computationally equivalent with the standard
Finite Difference Time Domain(FDTD) method. For high frequency electromagnetic
applications, time domain simulations methods are highly desirable, especially when
broadband results are needed. FIT therefore shares FDTD's advantageous properties like low memory requirements and efficient time stepping algorithm. However,
standard FDTD has poor modeling quality for arbitrarily shaped geometries since
it uses staircase approximations. FIT combined with Perfect Boundary Approximation(PBA) can maintain the convenient structured Cartesian grids and permit an
accurate modeling of curved structures[31].
The solver that we used for all our simulations is a transient solver. For a wide
frequency range, it uses only one computational run for the simulation of a structure's
23
behaviors[31]. The version of CST Microwave Studio@ that we are using is v4.2. We
will be using it for radiation simulation and waveguide simulation.
2.2.1
Radiation setup
Here we will show our simulation setup for getting the farfield radiation. To illustrate
the methodology in detail, a simple rod medium will be used as demonstration, which
is very similar to what Enoch et al. has used[27]. The full size structure setup for
the rod medium is shown in Figure 2-2. Each rod is a cylindrical Perfect Electric
Conductor(PEC) structure that has a radius of 0.2 mm, and a length of 250 mm.
The period in the x direction is 5.8 mm and in the y direction is 6.3 mm. There
are 6 layers of rods in the y direction and 40 repetitions in the x direction. A 50 Q
y
6.3mm x 6
-
-
~-~~-~-'
'250mm
5.8mm x 40
Figure 2-2: Full size rod structure
S-parameter discrete port(dipole) of 1mm in length is placed at the center of the
structure for radiation. Mesh type of PBA(usually staircase type will be used if there
is no curvature in the structure to save simulation time) is used with mesh density set
at 10 lines per wavelength with refinement at PEC edges by 3. The resultant mesh in
the x - y plane is shown in Figure 2-3. How the mesh is setup is very important to the
accuracy of our simulation results. It needs to be fine enough to capture the details
of the metamaterial structure. It is not as essential here for the simple rod medium
as for some other structure which we will encounter later. The open boundary is
24
Figure 2-3: Mesh for full size rod structure
modeled with Perfectly Matched Layer(PML) of 8 layers and a reflection coefficient
of 0.0001.
The automatic minimum distance to structure(when using "open(add
space)" as the boundary condition) is one wavelength. Farfield monitors are set up
for frequencies from 11 GHz to 17 GHz. Radiation power are calculated at a distance
10 meters away from the excitation source(dipole in all our cases). These are the
typical parameters we use in all our simulations, except for the mesh parameters and
farfield monitor frequency range. Farfield results are shown in Section 2.4. Simulating
the full size structure takes a lot of memory and as most metamaterial structures are
more complicated than periodic rods, more time and even more memory are needed
for simulation as well. Therefore, we use a slab of metamaterial in a PEC parallel
plate waveguide to approximate the full size structure to save simulation time and
memory. The slab setup is as shown in Figure 2-4. The dipole is again placed at the
center of the structure. Mesh parameters are the same as before with the additional
option of "Merge fixpoints on thin PEC and lossy metal sheet" chosen. All other
parameters stay the same.
25
5.8mm x 40
6.3mm x 6
-3.33mm
Figure 2-4: Slab of metamaterial(rod medium)
2.2.2
Waveguide setup
In order to study the metamaterial properties in a waveguide, a unit cell is identified
from the full size structure and placed in a waveguide to collect the S-parameters.
The unit cell for the rod medium is shown in Figure 2-5. The rod here is again
modeled with PEC material, and the background as air. The top and bottom surface has PEC boundary condition, whereas the left and right has perfect magnetic
conductor(PMC) boundary condition, and front and back with open boundary condition. These boundary conditions will be the same for all structures' unit cells that
we presented in this thesis. A waveguide port is placed at the open boundaries. Mesh
density is 10 lines per wavelength; the options of refinement at PEC edges by factor 4
and inside dielectric materials are chosen. With the S-parameter data obtained from
the waveguide simulation, we can retrieve the effective p and E for all frequency[29].
An electric plasma frequency of 13.5 GHz is observed from retrieval(see Figure 26). Therefore, for farfield radiation, we will be interested in the frequencies around
13.5 GHz, as the index of refraction will be close to zero in that region, and thus will
possibly have beam sharpening effect. Using a rod medium, we can only have eeff
to be zero at a certain frequency by changing the periodicity or the radius of the
rods and thus making the index of refraction n = 0 at the corresponding frequency.
Ultimately, our goal is to find a metamaterial structure where we can also get
[eff
to be zero at a frequency that is close to where eeff = 0 is, such that the region
26
I
3.33mm
F.3mm
Figure 2-5: Unit cell rod structure in a waveguide
5
4
3
2
1
0
-1
-2
-3
-4
-5
5
43-
Real(z)
Imag(z)
-
-- Real(n)
-Imag n)
g (-
......
..
21
-.
...-.
0
-1
-2
-3
-4
-5
----- ------
2
6
4
8
0
10 12 14 16 18 20
f/GHz
-
.
2
4
6
8
10 12 14 16 18 2C
f/GHz
5
4
3
2
1
-
Real(p)
-
Imag(p)
433 . . .
.
Real(e)
-...
-..Imag(e)
2-1
U.
-1
-2
-1
-
-3
-4 .
-5
0
-
-
-
-.
-...-2
-3
. ..
- 4 .. .. .. . ..
.
-.-.-.
2
4
6
8
0
10 12 14 16 18 20
f/GHz
2
4
6
.. . I... .. .
8
10 12
f/GH2
Figure 2-6: Retrieval results for rod medium
27
14 16 18 20
where n ~ 0 can be broadened. Consequently we can potentially have a wider band
where we can expect beam sharpening. Retrieval results (peff and eff) enable us
to relate metamaterials to a homogeneous material, and further assist us to estimate
the radiation characteristic as we will demonstrate in the next section.
2.3
Analytic method for farfield radiation
y
Region 0
0 , /t0
Region 1
E 1,1A
0
z
Region 2
- d2
60, P0
X
1
Figure 2-7: Radiation configuration of a linesource in an infinite isotropic slab of
thickness d, + d 2
Consider a slab with permittivity c, and permeability p1 as shown in Figure 2-7.
Given an embedded linesource oriented in the z direction at the origin, its electric
field can be expressed as the following[1])
Eiz =
J
dkxe(kxxiklY)Ein
(2.1)
where
E
=
-in
I(2.2)
is the spectrum which corresponds to the source at x = 0, y = 0 with magnitude I.
The electric fields in the different regions then can be expressed as follows:
28
In region 0,
Eoz
In region 1,
Elz =
J
dkxEfin(Tieikoy)eikxx
dkx Efin(e±ik1
In region 2,
J
E2Z
+ Aeiklyy +
Be-iky
(2.3)
(2.4)
)eikxx
(2.5)
dkxEin(T 2 e-ikyy)e ikxx
By matching the boundary conditions for the tangential electric and magnetic
fields at y
=
di and y = -d 2 , we can find the coefficients T1 , T2 , A, and B.
2 ei(kiy-koy)dipo1 (1 + ei2k1yd2(_
2 -(1
ei2kly (d +d2)(~ 1+ poi)
e
ei2klY (d+d2)(-1 + P01) 2
2 ei(klykoy)d2po1
T2
+Pi 1) +i2 P()
+ pol)
(26
(1 +| P01)2
-
+ ei 2 klydi
+ POO + P01)
(2.7)
P01) (i + ei2kiyd1(_1
+ P01) +2 P01)
2
e i2kiy(d1+d2)(1 + P01) _ (1 + Pa)
(28)
A
=
-
e ei2kid2(- ±
B
=
-
ei2kiyd2(_1 +pi)
(i
ei2kiy(d1+d2)(-1
+
ei2kiyd2(
poi)
2
(2.9)
+ pi0) ±oi)
(1 + P01) 2
_
where
pi
=
k
(2.10)
pu 1 Aikoy
Using farfield approximation, the electric field in region 0 can be simplified as
follows:
Eo= EnkoyT1]
f-00
dkx I eikoyeik;X
koy
=
EeinkoyT1
iko
eikor
(211
Normalizing this electric field to a free space case, we get
Er=
(2.12)
Poi
where Er denotes the relative electric field with respect to a dipole in free space.
29
Radiation results and normalization
2.4
In this section, we will show all our different radiation results from different methods
and how we compare them. Radiation results are plotted for the plane of interest,
the x - y plane, as shown in Figure 2-8 using a parallel plate radiation as an example.
Angle
#
is the angle from the +x axis in the plane of interest. In the frequency band
Yx
Figure 2-8: Radiation plane of interest
where index of refraction n ~~0, the main beam(most power) is expected to occur at
0 = 900 or
#
= 270' or both. We have two different radiation results: simulation and
analytic. These two methods plot different aspects of farfield radiation, and we want
to normalize them in some way such that we can compare our results.
Analytic method shown in Section 2.3 calculates radiation using electric field from
an embedded linesource. The resultant farfield calculation is the ratio of the electric
field in metamaterial to the electric field in free space. It can then be squared to
show the relative power in dB. Simulated farfield radiation calculates the electric
fields or power by using farfield approximation. In order to compare radiation results
from these different methods, we need to find a way to normalize these data first.
We noticed that in all these results, power is involved. In the analytic method, for
each direction, the power of each frequency is divided by the power corresponding
to the free space case. Therefore, for the same frequency, different radiated powers
at different angles will be divided by the same number. However, if the frequency is
different, the powers will be divided with a different number. This tells us that there
is a different scaling factor for different frequency. Hence, our method for normalizing
30
all radiation figures is that for each frequency, we calculate the average power in that
frequency and normalize(divide) all power data in that frequency with this average.
The 3 dB beamwidth calculation will not be affected by this normalization.
Figure 2-9 shows the radiation results obtained from analytic method.
From
retrieval as discussed in Section 2.2.2, we expect to see directive radiation around the
electric plasma frequency of 13.5 GHz. The most directive and high power radiation
is seen to center at 13.6 GHz and 13.9 GHz respectively. Shortly after 13.9 GHz, the
main beam starts to divert away from q
=
90', and form a "U" shape radiation
pattern.
Normalized Radiation
Radiated Power
17
15
16
10
-5
M15
5
O 14
-10
C14
0
13
-15
213
-5
12
-20
12
-10
-25
11
17
5
16
0
N
M15
11
0
30
60 90 120
Angle(degree)
150 180
N
0
30
60 90 120
Angle(degree)
150
180
-15
Figure 2-9: Radiated power and normalized radiation from analytic method for rod
structure
Next, lets look at the radiation results acquired from simulation for both the
full size and the parallel plate slab cases in Figure 2-10. For the full size structure
simulation, the most directive beam is centered at around 13.7GHz and the high
power beam is centered at around 14.4GHz. Similarly like what the analytic results
have predicted, the highest power beam takes place after the most directive beam, but
with both these frequencies shifted to a slightly higher frequencies. The parallel plate
slab case exhibits a directive and high power beam at around 13.8 GHz and 13.9 GHz
respectively. The parallel plate slab radiation has higher sidelobes which might be
caused by the addition of the two PEC parallel plates. The two simulations are both
a little bit different from what is predicted from the analytic method, however, they
31
do demonstrate similar behavior of a "U" shape radiation pattern as the analytic
case.
Normalized Radiation(Fullsize)
Radiated Power(Fullsize)
N
M
17
-45
17
16
-50
16
15
-55
N
M
20
*L"\-WIAU
15
15
10
c 14
-60
14
5
13
-65
13
0
12
-70
12
-5
-75
-75
11
S1
11 0
30
60 90 120 150 180
Angle(degree)
-I
0
30
60
90
120
150
180
Angle(degree)
Normalized Radiation(Slab)
Radiated Power(Slab)
17
-45
17
20
16
-50
16,
15
M15
-55
'15
c 14
-60
S14
5
13
-65
13
0
12
-70
12
-5
-75
11
0
U-
i1
0
30
60
90
120
150
180
a a
NV
10
-10
30
60
90 12U
1bU
180
Angle(degree)
Angle(degree)
Figure 2-10: Radiated power and normalized radiation from simulation for rod full
size and slab structure
Besides power and directivity, we are also interested in the 3 dB beamwidth of
the antenna system.
The only beamwidth we are interested in are in the region
where index of refraction n ~ 0, and where the mainlobe of the radiation in one
particular frequency at
#=
90 (normal to the substrate). Due to possible numerical
simulation errors/approximations, we allow the main beam to be slightly away from
the normal, given that the beam power at the normal direction is still within the
3 dB range of the main beam. We also allow the beam power to oscillates up and
down as long as it is all within a 3 dB window. The bandwidth is decided by having
the side lobes to be 10 dB lower than the main beam, given that the beam in the
32
normal direction is within 3 dB of the main beam. Figure 2-11 shows the maximum
power angle at different frequencies and the corresponding 3 dB beamwidths for the
full size rod structure; interested bandwidths are the regions colored in yellow. The
smallest beamwidth of 200 occurs at 12.8 GHz, and the bandwidth is from 12.8 GHz
to 15 GHz. Figure 2-12 shows the figures for the parallel plate slab case, where the
smallest beamwidth of 6' happens at 13.7 GHz, and the bandwidth is from 13.7 GHz
to 14GHz.
3dB Beamwidth
bu
50
M40
(D
4)
"a
30
-. .. . .-.. ..-. . ..-.
..
..
......
. ..-
- ...........
-. . . . . .. . . . .
-
-
20
10
11
12
13
14
Frequency(GHz)
15
16
17
Max Power Angle
100
a)
LD
90 -
--
80 - -
-
70 -
- -
-
-
-- --
--
-.-
.
-.-.-
-
- ---
- -
- -
-.-
60
11
12
13
14
Frequency(GHz)
15
16
17
Figure 2-11: Beamwidth for full size rod medium
In Enoch's experiment[27], the best directivity was observed at 14.65 GHz, and
the beamwidth at this frequency is 8.9 0 (Note: the experimental results is based on
an antenna substrate backed with a ground plane, which we did not put in our
analytic method or simulation).
Our simulation and analytic results yield similar
results regarding where the antenna system has high power and directivity. The
exact frequencies differ with each other and with Enoch's experimental results can
33
3dB Beamwidth
100
(D
. . .. . . .
60
..................
-
. -
-
80
. . . . . . . . . . . . . . . . . . . ...
................ .
0
'a 40
C
.. ... .
.. ... .. ... .. .. ... ..
20
0
14
Frequency(GHz)
13
12
1
15
16
17
16
17
Max Power Angle
135
(D105
S
()
-
--
-
120
90
-
-
60
45
11
-
-
75 -
.. ...
-
--
- -
--
3.
1
1.
-
-
- - -
12
13
14
Frequency(GHz)
15
Figure 2-12: Beamwidth for a slab of rod medium
34
be contributed from the difference in exact structure setup and excitation source,
simulation's numerical errors/approximations, and experimental noise. We conclude
that with this methodology, we can study how a metamaterial structure perform as
a directive antenna substrate and further make improvement on the metamaterial
structure to yield better results. In the next chapter, we will show how different
metamaterials perform as a substrate, and compare their performance.
35
36
Chapter 3
Comparative Study of Different
Metamaterial Substrate
In this chapter, we will examine different published metamaterial structures as antenna substrate. For each structure, the dimensions of a unit cell will be illustrated.
Next, the effective permittivity and permeability will be obtained from the retrieval
results based on the S-parameter scattering simulation in a waveguide. For parallel plate slab radiation simulation, dimension of the slab and the mesh used will be
shown. Lastly, analytic farfield radiation based on retrieved results and simulated
farfield radiation will be presented and compared.
These structures are all left handed metamaterials, there will be a region where
n < 0 and the index of refraction n = 0 would occur at the frequencies where either
Eeff = 0 or peff = 0. We are going to evaluate these different structures and see which
one would work best for our purposes. Ideally, we would like to find a structure which
is easy for tuning and manufacturing; then, we would optimize the structure such that
it will exhibit higher power, better beamwidth, and wider bandwidth.
At the end of this chapter, a summary of results and comparisons of these different
structures will be presented.
37
2-D Smith structure
3.1
The first classic metamaterial structure is Smith's structure, so we'll first examine
how it works as an antenna substrate. A unit cell 2-D Smith structure can be seen in
Figure 3-1 and the 1-D dimensions is as shown in Figure 3-7. The dimension of this
metamaterial structure is taken from a paper published by Shelby et al. in 2001[12].
The unit cell is then placed in a waveguide to collect the S-parameters scattering
data. Since the waveguide has PEC and PMC boundaries, we need to choose a unit
cell that is as symmetrical as possible.
The metal part of the structure is modeled with PEC. The dielectric is lossless with
a relative permittivity of 4. Background is free space. These parameters will be used
for all the structures we investigate in this chapter and beyond. Figure 3-2 shows the
0.125mm
0.12
3.33mm
5mm
Figure 3-1: Unit cell 2-D Smith structure in a waveguide
retrieval results from the unit cell. It is not a very clean results compared to the 1-D
unit cell one as shown in Figure 3-9. From both the 1-D and 2-D retrieval results, we
38
can estimate that we will see some directive radiation in the normal direction(# = 0)
from around 10 to 11 GHz. The radiation results from analytic method using the 2-D
retrieval data is shown in Figure 3-5. A slab of this 2-D Smith metamaterial substrate
10
10
Real(z)
-Imag(z)
5
5
.-.
. .-..
....
..... ....
-.
5.......
0
0
-5
--
-5
-10,
- Real(n)
-- lmag(n)
..
- -..
-- .
. -.. . . . .
0
5
10
15
-10
20
-.-
--
0
5
10
10
20
10
- Real(p)
-
5
--
Imag(g)
Real(e)
Imag(E)
-5
0.... ...
0
-5
-5-
-101
0
15
f/GHz
f/GHz
5
10
f/GHz
15
-10,
20
0
5
10
f/GHz
15
20
Figure 3-2: Retrieval results for 2-D Smith structure
will look like the one shown in Figure 3-3. The period in both the x and y direction is
5 mm. There are a total of 35 layers in the x direction and 5 layers in the y direction.
The excitation dipole is again placed in the center of the substrate. Farfield monitors
are set up for frequencies from 8 GHz to 22 GHz with 0.1 GHz intervals. In order
to get accurate simulation results, the mesh setup is important. Before starting the
radiation simulation, we need to make sure our mesh looks fine enough to capture
the details of the structure. In Figure 3-4, the mesh used for parallel plate radiation
is shown. Mesh line density is 15 lines per wavelength. The mesh type is staircase
mesh. No refinement at PEC edges or inside dielectric. Fixed points are merged for
thin PEC and lossy metal sheets.
39
Figure 3-3: Slab of 2-D Smith structure
Figure 3-4: Mesh for 2-D Smith structure
40
Lastly, the radiation results from both analytic and simulation are shown in Figure 3-5, and the 3 dB beamwidth from the simulation results are shown in Figure 3-6.
As we mentioned in Chapter 1, we are likely to see directive radiation when Inj < 1
with imaginary part of n being small(small loss). According to the 2-D retrieval
results, we should expect to see directive radiation from about 11.5 GHz to 12GHz,
since in this region, both the real part and the imaginary part of n is small. Therefore,
we see directive radiation in such range in the analytic radiation, but it is obviously
not very sharp beam, as the strength is almost evenly distributed among all the different
#
angles for one single frequency. This is mainly due to the uncleaned 2-D
retrieval results. In the slab farfield simulation, the directivity is not very strong until higher frequencies are reached, which should correspond to the plasma frequencies
of the rods(vertical metal strips on one side of the dielectric). There are somewhat
directive radiation between 12GHz and 17GHz, however not very strong, which is
probably due to the fact that the loss is still relatively high in that region. The best
directivity starts around 19 GHz, and the best beamwidth is about 290. To conclude,
this structure is relatively isotropic in two directions, but harder to retrieve and hence
harder to predict from analytic method what the radiation figure would look like. As
for its performance as an antenna substrate, it did not improve on what rods alone
could offer us.
41
Normalized Radiation(Analytic)
Radiated Power(Analytic)
22
0
22
15
20
-5
20
10
'Ri18
-18
M
-10
5
. 16
S16
-15
$Cr
2 14
Cr
-20
u- 12
S14
8
0
30
60 90 120 150 18C
Angle(degree)
-5
u- 12
-25
10
0
C
-10
10
-30
81
0
60 90 120
Angle(degree)
150 18(
-110
22
20
-115
20
-N18
-120
N
-
-
-
15
10
18
5
'16
-125
8 14
2
0
14
-5
-130
u. 12
-135
10
8
0
30
60 90 120
Angle(degree)
150 180
-15
Normalized Radiation(Simulation)
Radiated Power(Simulation)
22
316
30
-10
10
-140
8
0
30
60 90 120
Angle(degree)
150
18
-15
Figure 3-5: Radiated power and normalized radiation from analytic method and slab
simulation for 2-D Smith structure
42
3dB Beamwidth
120
ji0
0)
0
0)
C
8 0 --
-
-.-.-.-.-.-.-
60 -
-
40 20
-
-.-.-
100 -
-
-
-
-
-
-
-
10
8
18
14
16
Frequency(GHz)
12
22
20
Max Power Angle
I00
-
--
90 0)
0
0)
0
V
0
0)
C
80 -
-
-
70 60 50
8
.
- ..
10
.. .-..
12
-.-.-
- -..
16
14
Frequency(GHz)
.
. .. . . -..
. .. . . . .-..
.-..
18
20
Figure 3-6: Beamwidth for 2-D Smith structure slab simulation
43
22
3.2
1-D Smith structure
2-D metamaterials are nice in the sense that it provides us with a relatively isotropic
material property in the x - y plane. However, construction of a rigid 2-D structure is
hard. Looking at the waveguide transmission/reflection results for 1-D Smith structure, the S21 has a wide band where its value is close to 0 dB. It suggests that 1-D
Smith might give better radiation power. 1-D structures are easier to fabricate and
construct. The unit cell dimension is as shown in Figure 3-7. The 1-D structure we
investigate here has the dielectric strips aligned in the y direction for slab simulation
and the setup is shown in Figure 3-8. There are 6 unit cells in the y direction and
36 unit cells in the x direction; periodicity in both directions is 5 mm. The materials
used are the same as the ones used for the 2-D structure.
The retrieval results are
2.375mm
5mm
Figure 3-7: Unit cell 1-D Smith structure in a waveguide
shown in Figure 3-9 and the corresponding analytic radiation pattern is shown in
44
Figure 3-8: Slab of 1-D Smith structure
Figure 3-11. The radiation mesh setup for this structure is very similar to 2-D Smith;
the main difference is the mesh in the x - z plane(see right half of Figure 3-10). Most
of the mesh parameters are kept the same as the 2-D case, where the main change
is the mesh line density to 18 lines per wavelength from 15. The radiation results
are shown in Figure 3-11. The analytic and simulation results are similar but with
some slight frequency shift. This implies that the retrieval of effective permittivity
and permeability are more accurate for the 1-D case, and hence gave a better analytic radiation pattern. The radiation power has improved compared to the 2-D case,
however, not the directivity, nor the bandwidth. The radiation beamwidth resulted
from simulation can be seen in Figure 3-12.
45
10
Real(n)
- mag(n)
-Real(z)
-Imag(z)
...
...
.. -.. ....
5
0
0
--
-5 ---
-5
15
10
f/GHz
05
-10
2(0
20
15
10
f/GHz
5
0
10
10
-
- Real(g)
-
Real(E)
Imag(A)
-
Imag(E)
5-
5
0
0
-5
-
~4n
0
5
10
f/GHz
15
5
0
20
15
10
f/GHz
Figure 3-9: Retrieval results for 1-D Smith structure
I
-
t
Figure 3-10: Mesh for 1-D Smith structure
46
-
20
S
Normalized Radiation(Analytic)
Radiated Power(Analytic)
22
5
22
15
20
0
20
10
-18
-N
-5
5
516
16
-10
-15
LL 12
0
30
60 90 120
Angle(degree)
150
180
-10
10
8
0
-25
N
22
-30
22
20
-35
20
16
-40
150
180
-15
15
218
I
-45
S14
LL
60 90 120
Angle(degree)
16
0
C
30
Normalized Radiation(Simulation)
Radiated Power(Simulation)
M
-5
u 12
-20
10
0
S14
S14
8
18
Cr
-50
12
18%
IC
-55
IC
6
0
-5
S2
0*
30
60 90 120
Angle(degree)
150 180
-60
-10
0
30
60 90 120
Angle(degree)
150
180
-15
Figure 3-11: Radiated power and normalized radiation from analytic method and
slab simulation for 1-D Smith structure
47
3dB Beamwidth
60
50 --
-
0
0
40 -
-
0
0
~30
-
-.-.-.-
-
-
a
a
C
~
---
- --
--
20 -
.--
-
10
01
8
10
16
14
Frequency(GHz)
12
18
20
22
18
20
22
Max Power Angle
150 0
0)
0
V
0
-..
12 0 - . . .
-.-.-
..
-.-.
-.-.-.--
-
90
6
0)
C
3 0 - .-.-..-.-.--.-.-.
n
8
10
12
16
14
Frequency(GHz)
Figure 3-12: Beamwidth for 1-D Smith structure slab simulation
48
3.3
Pendry structure
Another classic example of metamaterials is the Pendry structure[32]. The unit cell
we used and its dimensions are as illustrated in Figure 3-13. We mainly scaled the
original Pendry structure to work in the microwave regime. With the S-parameters
0.5mm
1.17mm
6mm
5.04mm
Figure 3-13: Unit cell Pendry structure in a waveguide
obtained from the PEC-PMC waveguide simulation, retrieval is done and shown in
Figure 3-14. There is a resonant frequency at about 8 GHz. From 8.5 to 12 GHz, index
of refraction n ~ 0, but the imaginary part of n is slightly large. Therefore, instead of
seeing directive radiation for all frequencies between this range, we might only have it
around the two end frequencies. As what we expect, the analytic results as shown in
Figure 3-17 show that there are stronger directivity at around 8.5 GHz and 12 GHz,
but the region in between have a much lower directivity. Pendry structure is onedimensional, so we will use the similar radiation simulation setup as the 1-D Smith
49
-H I
IV
-
Real(z)
--
---- Real(n)
Imag n)
Imag(z)
5
5
. ....
.-..
.....
-.
0
0
-5
-5
-10
0
15
10
f/GHz
5
10
20
0
5
10
f/GHz
15
20
in
10
-
Real(E)
Imag(s)
..
-.
.
..... -..
-
....
.....-
5
Real(g)
--
Imag(E)
5
--
0
0
-
. .
. --..
-5-
-5
_1 "
0
5
10
f/GHz
15
0
20
5
10
f/GHz
Figure 3-14: Retrieval results for Pendry structure
50
15
20
structure. We have 6 unit cell repeated in the y direction and 64 unit cell repeated in
the x direction. The periodicity is 5.04 mm in the y direction and 2.84 mm in the x
direction. The mesh setup for this radiation simulation is as seen in Figure 3-16 and
Figure 3-15: Slab of Pendry structure
the radiation results from analytic method and simulation is presented in Figure 317. The corresponding beamwidth results for simulation is shown in Figure 3-18.
Pendry structure has its maximum beam power mainly directed along the normal
direction(<$ = 900) which is good for our directive antenna purposes. The beamwidth
however is somewhat a bit too large to be ideal. The simulation results is not exactly
what we have predicted from retrieval and analytic method; we do not see two strong
directivity areas. It might be that the loss is too large in the area of interest (8.5 GHz
to 12 GHz), which we can somewhat see in our simulation results.
51
Figure 3-16: Mesh for Pendry structure
52
Normalized Radiation(Analytic)
Radiated Power(Analytic)
N
22
10
22
15
20
5
20
10
18
0
18
5
16
N
16
-5
0
8 14
-5
-10
L 12
12
-15
10
-10
10
8
22
-100
22
15
20
-105
20
10
0
30
60 90 120 150 180
Angle(degree)
0
18
16
LL 12
-125
0
30
60 90 120 150 180
Angle(degree)
0
-5
-120
8
5
D 14
S14
10
18C
C 16
-115
LL 12
60 90 120 150
Angle(degree)
-18
-110
C
30
Normalized Radiation(Simulation)
Radiated Power(Simulation)
N
-15
-20
8
-10
10
8
0
-130
30
60 90 120 150
Angle(degree)
18(
-15
Figure 3-17: Radiated power and normalized radiation from analytic method and
slab simulation for Pendry structure
53
3dB Beamwidth
120
100
80
C
*560
40
20
12
10
8
14
Frequency(GHz)
16
18
20
Max Power Angle
110
-
100
2
0)
a)
*0
a)
0)
C
80
70
12
-..
...
310
90
- ..............
-.
-.-.-.-.-.-.-.-.
8
10
12
..
.. .. .. .. ..- .. .. ..-..
..
-.-.-
- ...-14
161
.....
.......
14
Frequency(GHz)
16
.........
18
Figure 3-18: Beamwidth for Pendry structure slab simulation
54
20
3.4
Omega structure
Recently, there are some new metamaterial structures being explored. One of which
is an Omega-shaped structure[14].The unit cell and its dimensions are presented in
Figure 3-19. The retrieval results for this Omega structure is shown in Figure 3-20.
0.4mm
1.05mm
3.33mm
4mm
2.5mm
Figure 3-19: Unit cell Omega structure in a waveguide
We can see that the retrieval results are not very clean for frequencies below 11 GHz.
This might be caused by the increased complexity of the structure, since the rod
are coupled with the rings, which means that the permittivity and permeability are
coupled. However, we see that at around 11.4 GHz and 16.8 GHz, n ~ 0 and the
imaginary part of n is closed to zero, so we should be able to have regions where we
observe directive radiation. The analytic radiation, which is plotted in Figure 3-23,
only has one directive radiation region centered around 17 GHz. We also see that
there are a lot of "ringing" effects from 12 to 14 GHz, which we would not be able to
55
tell from the retrieved data. The slab of metamaterial used for parallel plate radiation
20
-Real(z)
...
-.
10
----
IMag(Z)
.
.
-..
-.. .-.-.
.
Real(n)
Imag(n)
10
. .-..
.-.
0
0
-
V~
-...- ....
-10
-10
-20
10
f/GHz
5
15
20
0
5
5
10
10
15
20
20 I
-
-_Real(p)
10 1
. .. -..
-
- .....
-
Imag(g)
0-
-1 0
-10[
-
0A
0
10
f/GHz
5
Real(E)
Imag(E)
1I0
- ... . -.
0
20
f/GHz
15
.. . ..-
20
0
20
5
. . . -..
-
10
f/GHz
15
20
Figure 3-20: Retrieval results for Omega structure
is shown in Figure 3-21. There are a total of 6 by 72 unit cells with periodicity of 4 mm
and 2.5 mm respectively. Figure 3-22 shows the mesh used for radiation. Due to the
more complicated structure, limited memory, and a big aperture size, the mesh would
have been better if it is even finer, but it is not permitted with the available resource.
However, it should be able to tell us the approximate behavior of such structure as an
antenna substrate. The radiation results from both analytic method and simulation
are shown in Figure 3-23. The corresponding beamwidth from the simulation result
is shown in Figure 3-24. We see some "ringing" effect in our simulation results similar
to the analytic results and there are some relatively high radiation directivity from 16
to 17 GHz and around 14 GHz. However, the beamwidth at these frequencies are too
wide. Therefore, the Omega structure is not a good structure to use as a substrate
for directive antenna.
56
Figure 3-21: Slab of Omega structure
Figure 3-22: Mesh for Omega structure
57
Normalized Radiation(Analytic)
Radiated Power(Analytic)
22
5
15
20
0
10
18
-5
5
16
16
C
1N18
-10
S14
0
C1
a)
-15
u- 12
-20
10
-5
LiL 12
-10
10
-15
-25
8
22
-50
22
15
20
-55
20
10
8
0
30
60 90 120 150 180
Angle(degree)
30
60 90 120
Angle(degree)
150 180
Normalized Radiation(Simulation)
Radiated Power(Simulation)
18
'N' 18
-60
5
316
16
-65
C
$ 14
a'
a)
-70
LL 12
U-
-75
10
30
60 90 120 150
Angle(degree)
180
0
2 14
Cr
8
0
0
-5
12.
-10
10
-80
8
0
30
60 90 120
Angle(degree)
150 180
-15
Figure 3-23: Radiated power and normalized radiation from analytic method and
slab simulation for Omega structure
58
3dB Beamwidth
80
4
0
-
. . . -. . . .
-. .-. . .-. -.
-.
C
C
4
0
8
22
20
18
16
14
Frequency(GHz)
12
10
120Max Power
Angle
10
1 50 - 0
C)
0)
C)
V
ii-
- . . . . . -. .. .. ..
-... -... . -... . .
. ..
1 20
-.. .
.. .
.. .
. .. . .. . ..
...
.. .
90
0)
60
3
-
0
-...
-..-.
-.-..
8
10
12
14
16
Frequency(GHz)
18
20
Figure 3-24: Beamwidth for Omega structure slab simulation
59
22
3.5
S structure
0.5mm
1mm
5.4mm
0.
4mm
2.5mm
Figure 3-25: Unit cell S structure in a waveguide
Another novel metamaterial is a coupled "S" shaped structure[15]. There are no
obvious ring or rod parts any more, but it still has the properties of having an electric
plasma frequency and an magnetic resonant frequency. The unit cell is shown in Figure 3-25 and the corresponding retrieval results in Figure 3-26. Since the S structure
does not have any curvature, the retrieval results are relatively clean. Similarly, S also
have two frequencies where loss is low and n ~ 0. They are around 12.2 and 20 GHz,
and index of refraction is approximately zero in between these two frequencies but
the loss is large. Therefore, in the analytic method, we see that in this region there
are relatively high directivity, however, the radiation power is relatively low.
The slab of S structure is as seen in Figure 3-27. Identical to the Omega structure,
the slab is consist of 6 by 72 unit cells with the corresponding periodicity of 4mm and
60
10
10
Real(z)
-
Imag(z)
5
-.. . .. -..
.- . .
.
5
. . . -..
0
-5
-5
15
10
f/GHz
5
20
5
0
10
Imag(n)
-..
....
0
0
Real(n)
15
10
f/GHz
20
10
--
Real(p)
-
Imag(p)
-
Real(E)
Imag(E,)
-10
-10
0
-0
5
-0
-1
- -
-5
10
f/GHz
15
-10
20
- 5
10
f/GHz
Figure 3-26: Retrieval results for S structure
61
15
20
2.5 mm. The mesh setup for slab simulation is shown in Figure 3-28.
In Figure 3-
Figure 3-27: Slab of S structure
29, we show the analytic and simulation results and they display similar radiation
patterns. They both show some "ringing" effects. Furthermore, as predicted in the
analytic method, there are some directivity from about 12 GHz to 20 GHz, but with
low power. The only very high directivity only takes place at higher frequency of
20.2 GHz for the analytic case and 21.3 GHz for the simulation case. The beamwidth
for the slab simulation is shown in Figure 3-30. The best is 180 at 21 GHz.
62
-
Figure 3-28: Mesh for S structure
63
-~
--I
Normalized Radiation (Analytic)
Radiated Power(Analytic)
24
10
24
15
22
5
22
10
,20
N
N
0
20
0 18
0
16
-5
S16
5
I
(D 18
-5
-10
L- 12
L 12
-10
-15
10
-20
8
24
-50
24
22
-55
10
8
0
30
60 90 120
Angle(degree)
150 180
Radiated Power(Simulation)
-20
-5
12
-10
10
60 90 120 150 180
Angle(degree)
15
-0
16
10-75
30
15
I5
L- 12
0
180
920
-70
8
60 90 120 150
Angle(degree)
10
-65
C 16
30
Normalized Radiation(Simulation)
-60
1M
0
8
-80
0
30
--
60 90 120
Angle(degree)
150 180
15
Figure 3-29: Radiated power and normalized radiation from analytic method and
slab simulation for S structure
64
3dB Beamwidth
I
IUU
00
8
0
-... . . .
-..
-.
60
0)
R.
. -.-.-.
-
40
20
12
10
8
14
18
16
Frequency(GHz)
20
22
24
22
24
Max Power Angle
17n
...
.
- ..
150
.-
130
. . . .-. . .. . . .
12 110
0
0)
90
S.
-..
- - . .-.
-..
-.......
-.
. . .. ..
0
0
- -
.
50
.
.
-..
. ..
. .- .-..
-
-
-
-
-
-
.
30
10
8
10
12
14
16
Frequency(GHz)
18
20
Figure 3-30: Beamwidth for S structure slab simulation
65
. . . . ..
.
3.6
Summary
From studying and analyzing all these different metamaterial structures, we can conclude that a 2-D structure is harder to construct than a 1-D one. Additionally a
1-D structure yield higher power than a corresponding 2-D one. Permittivity and
permeability are coupled in the Omega structure and the S structure, so these two
structures would be harder to tune. The Pendry structure and the Smith structure
differ mainly in the implementation of their rings. The Pendry structure shows better
directional beam and is easier to tune its permeability since its rings are symmetrical.
For example, we can change the Pendry ring dimensions while keeping the same gap
while the Smith ring consists of one bigger ring and one smaller ring, so if we make
the bigger ring smaller then the gap would become smaller too unless we make the
small ring smaller as well. Hence, more parameters are involved in the Smith ring
and therefore harder to tune.
We are interested in the 8 GHz to 18 GHz frequency range, and in Table 3.1, we
summarize how the different metamaterial structure perform as directive antenna
substrate. Beamwidth and power shown will be the best within the bandwidth.
Table 3.1: Comparison among different metamaterial substrates
Structure
Aperture size
(mm x mm)
Beamwidth
2-D Smith
175 x 3.33
1-D Smith
180 x 3.33
Pendry
181.76 x 6
Omega
180 x 3.33
S
180 x 5.4
29
28
42
36, 56
28
Bandwidth
(GHz)
Power (dB)
10.2-18
12-12.3
16.1-18
10.7-10.8
-115
~ -35
~ -108
Ease of
construction
Retrieval
Tunability
hard
easy
easy
13.7-14.1
15.6-17.2
-55
~-65
easy
unclean
medium
clean
medium
clean
easy
unclean
hard
unclean
hard
(0)
66
~ -75
easy
Chapter 4
Optimized Metamaterial Structure
From the previous chapter, the Pendry structure has the main beam directed mostly
along the normal direction, which is where we want for a directive antenna. Therefore, we change some parameters in order to yield a wider band, higher power, and
smaller beamwidth. From experience, if we observe from retrieval that there are two
frequencies which have index of refraction n = 0 and are far from each other, then
the imaginary part of n is usually too large(loss is large) for the frequencies in between to maintain high power. This is what we see in the retrieval results of the
Pendry structure(Figure 3-14). Changes will be made on top of this Pendry structure
to make the frequencies where E = 0 and p = 0 closer to each other. Therefore,
instead of having two small regions where we see high directivity and high power, we
hope to have one larger region with high directivity and high power. We carry out
the one cell PEC-PMC waveguide simulation, and use the obtained S-parameters to
retrieve the effective permittivity, permeability, and index of refraction, with which
we can calculate the analytic farfield radiation. If the analytic radiation results are
not satisfactory, then the geometries of the unit cell will be further modified until the
analytic radiation exhibits high power in a wider band with high directivity.
67
4.1
Optimized unit cell's geometry
The optimized structure is shown in Figure 4-1. Material properties are the same as
the Pendry structure presented in the previous chapter.
0.5mm
2.m
m
.24mm
5mm
5mm
Figure 4-1: Unit cell optimized structure in a waveguide
Retrieval results are shown in Figure 4-2. The two frequencies where n
=
0 are
very closed to each other. The imaginary part of n is small for all frequencies in
between. Essentially, we have lowered the electric plasma frequency, and increased
the magnetic resonance frequency. The electric plasma frequency is related to the
size of the rod, and its periodicity. The smaller the period, the higher the electric
plasma frequency. Therefore, by increasing the size of the unit cell, we also increase
the spacing between the rods, and hence lower the electric plasma frequency. As for
magnetic resonance frequency, it is related to the dimensions of the ring. Decreasing
the perimeter of the ring will cause an increase in the magnetic resonance frequency.
68
10
10
-Real(z)
-Imag(z)
Real(n)
--
5
5
0
0
-5
-5
0
5
10
f/GHz
15
20
5
0
-Real(g)
-Imag(g)
. . ..
--
20
-5
-5
10
-
1IA
5
10
f/GHz
-
Real()
-
Image
..
-.
0
0
15
10
f/GHz
1
10
5
Imag(n)
15
20
. ..
-.
-.
0
5
10
f/GHz
.. . ..
15
Figure 4-2: Retrieval results for the optimized structure
69
20
Here we assume that the material property do not change: metal is still modeled with
PEC, and the dielectric is lossless with a relative permittivity of 4. Changes in the
material properties would cause changes in the effective permittivity and permeability
as well.
4.2
Optimization of cell orientation and antenna
position
After we have decided what the optimized unit cell should be, we need to figure
out how to put these unit cells together(orientation) and where to place the dipole
antenna within the bulk of the metamaterial. After all, the metamaterial substrate
is not homogeneous and radiation results could possibly change depending on how
everything is put together.
Since we are interested in making a 1-D structure, there are two possible arrangements: the dielectric strips align with the x direction or the y direction as shown in
Figure 4-3. We will refer to the one that has the dielectric strips aligned in the x
direction as Pendry version x, and the other as Pendry version y. The mesh setup for
Figure 4-3: Two different 1-D orientations of unit cells arrangement
farfield radiation for both orientations is shown in Figure 4-4.The radiation results
for these two different orientations are presented in Figure 4-5, and the corresponding
beamwidth results are shown in Figure 4-7 and Figure 4-6.
70
o
I
Figure 4-4: Mesh for the optimized structure
Comparing to the other substrates we presented in the previous chapter, both of
these metamaterial substrates have shown to have a better combination of radiation
power, beamwidth, and bandwidth. From the radiation figures alone, we can see
that there is an obvious improvement in directivity compared to the original Pendry
structure(region 8 GHz to 13 GHz in Figure 3-17), hence we can expect the beamwidth
to improved as well. In addition, radiation power has substantial improvement of
approximately 20 dB. Between Pendry version x and Pendry version y, it is seen
that Pendry version x has slightly higher power and directivity for a wider frequency
range than Pendry version y. Overall, Pendry version x has a better performance
than Pendry version y as a substrate for directive antenna.
71
Radiated Power(Pendry version x)
18
Normalized Radiation(Pendry version x)
18
1
-80
10
-85
16
N
N
-90
r
5
M
14
Cr
15
0
12
.2
-95
C
0
-100
a)
02
a)
-5
U-
10
8
-10
-105
0
30
60 90 120 150 18
Angle(degree)
-110
8
0
-80
-85
16
N
60
90
120
150
180
Angle(degree)
Radiated Power(Pendry version y)
18
-1A
30
Normalized Radiation (Pend ry version y)
18
15
16
10
N
7:
-90
14
0
5
14
0
-95
12
12
-100
10
80
0
30
60 90 120 150 180
Angle(degree)
-5
LL
-105
10
-110
8
-10
0
30
60 90 120
Angle(degree)
150
180
-15
Figure 4-5: Radiated power and normalized radiation from slab simulation for Pendry
version x and Pendry version y structure
72
80
3dB Beamwidth
- ....
-. ..........
70
.-.
. .... -
.. . .. . .. ..
...
.. ...
|60
-..
-. . . .-.--.
-.
-. . .-.. ...
-..-.-.-.-
,0)
-50
-......
..
.....
. ....
.....-
16
17
-
40
-
30
20
8
9
11
10
12
14
13
Frequency(GHz)
15
18
Max Power Angle
120
. _ .-.-.
-.-.
.-.-
110
.
-
-
-.-.-....-.-.-.-.-.-.-.-..-.-.-.-.-
100
SO 0)
-.
80
-.
......
.-..
70
0
....
8
9
10
11
-.
... .-..-. .-..-......
12
-..
..
-....
14
13
Frequency(GHz)
-..
..........
15
16
17
-
18
Figure 4-6: Beamwidth for Pendry version x structure slab simulation
73
3dB Beamwidth
100
-
90
g>
0)
. . ...-.
70
... .... ..
......
. ...
- -
40
-
. . .. . .
30
. -.. - .
i
14
13
Frequency(GHz)
12
11
10
. ...
.. .
.--..
p
9
-..
. .. . ..-. .
-.
-..
..
- .
20
. .. .-
. ...
-..
.
. . . .I . . . . .
.
-.
60
50
. ..-.
. .. .
.-. -.
. . ..
-.
80a)
. . .-.
.
. . . . ..-.
18
17
16
15
Max Power Angkc
40
1)
-
3 0 -
-
2120 1)
1)
1)
.
- - -.-.-
-
-
.-
-.-.-.-.-
- -
1 0-
-..
-..-...-----..
0090
-
-
-
j0
-
80 --8 .0
. ..
. ..
. ..
. _.. .
. _.
. _.
. ..
. . . . .
.
.
..
.
.
..
.
..
..
.
.
..
.
..
.
.
0
8
9
10
11
12
14
13
Frequency(GHz)
15
16
17
18
Figure 4-7: Beamwidth for Pendry version y structure slab simulation
74
Figure 4-8: Two different antenna positions
Is the position of the antenna going to affect our radiation results? This is something that analytic method cannot tell us. Therefore, we can only run simulations
to find out. For symmetry reason, there are two different antenna positions that we
can explore; they are shown in Figure 4-8. In Figure 4-9, comparison of the farfield
radiation from the two different positions of antenna is shown. One is always consistently 2 dB higher in radiation power than the other(varies between 2.06 dB to
2.37 dB). Therefore, for the metamaterial substrate to output higher radiation power,
it is better to align the antenna with the boundary between two adjacent unit cells,
and not with the rod of an unit cell.
75
Normalized Radiation(Aligned)
Radiated Power(Aligned)
18
16
-80
18
-85
16
10
N
-90
5
14
14
0
-95
0
cm12
12
-100
10
-5
-10
-105
81
0
30
60 90 120 150
Angle(degree)
180
Radiated Power(Not Aligned)
18
16
-110
0
30
60 90 120 150
Angle(degree)
1I
Normalized Radiation(Not Aligned)
-80
18
-85
16
N
15
5
214
14
-95
12
-15
10
-90
2.)
15
0
a)
.12
-5
-100
LL
10
81
0
10
-105
30
60 90 120
Angle(degree)
150 180
-110
-10
0
30
60 90 120 150
Angle(degree)
18 0
-15
Figure 4-9: Radiated power and normalized radiation from slab simulation for different antenna positions
76
4.3
Comparison with analytic method
y
Region 0
60, P0
di
Region 1
51,771
z
Region 2
-d2
Co, Ao
Figure 4-10: Radiation configuration of a linesource in an infinite anisotropic slab of
thickness d, + d 2
We have been using an isotropic analytic method up to this point; we can modify
the formulas to include the anisotropic case. Now the figure for anisotropic case(shown
in Figure 4-10) will be slightly different from the one for the isotropic case(Figure 27.) The substrate's permittivity and permeability will be expressed by using tensors
instead of scalar variables, which assumes isotropy. Now the permittivity and permeability take the form
1 0 0
0 Ey 0
1=
(4.1)
0 0 1
0 0
0 1 0
1=
(4.2)
0 0 p1 t
All the equations remain the same as shown in Section 2.3, except the expressions
for Eejn and poi.
E
-
47rklu
77
(4.3)
Poi = pok(44)
p1xkoy
Using these anisotropic analytic equations, the radiation patterns for Pendry version x and Pendry version y are different as shown in Figure 4-11. Pendry version x
structure yields better radiation power and directivity than Pendry version y structure
over a broader band. The same conclusion were drawn from the simulation results.
If anisotropy is taken into consideration, the analytic method has the capability to
demonstrate the difference between the two different setup of the unit cells.
Normalized Radiation(Pendry versio n x)
Radiation Power(Pendry version x)
10
18
16
5
16
N
(9 14
10
5
0
I
C.)
C
0)
0~
0)
U-
15
18
14
0
-5
12
.12
10
8
0)
-10
10
-15
0
30
60
90
120
Angle(degree)
150
180
-20
1V
0
60 90 120
Angle(degree)
150 180
Normalized Radiation(Pendry version y)
18
5
16
0
5
0
C
()
.12
-5
10
-15
0
30
60 90 120 150 180
Angle(degree)
15
10
-10
10
_1U
014
-5
8
30
10
14
= 12
-10
O0
Radiation Power(Pendry version y)
1V
-5
LL
-20
E
0
30
-I 150
60 90 120
Angle(degree)
-10
-15
180
Figure 4-11: Radiated power and normalized radiation from analytic method for
Pendry version x and Pendry version y structure
78
4.4
Summary
Through the manipulation of a metamaterial's geometry, we can tune its effective
permittivity and permeability to our desired specifications. We can potentially incorporate anisotropic properties of the metamaterial into our analytic radiation calculation. Analytic method has its own constraint, for example, it is unable to tell us if the
difference in antenna positions would affect radiation power. Hence, these metamaterials are only homogeneous to a certain extent. To summarize, the best structure is
using Pendry version x with the dipole in the "not aligned" position(aligned with the
boundary of two adjacent unit cells). This optimized structure is shown to improve
in radiation power, beamwidth, and bandwidth.
79
80
Chapter 5
Conclusion
After employing our methodology on different metamaterials, we can conclude that it
works reasonably well. Our analytic method is able to help us predict what we might
obtain from the numerical farfield simulation. Furthermore, we can model a metamaterial as a homogeneous medium for both the waveguide reflection/transmission study
and the farfield radiation study. This has helped us to simplify our design procedure,
because we can use our analytic method to arrive at a good structure first before
proceeding to the time and memory consuming farfield simulation. Additionally, the
analytic method is not limited to using isotropic metamaterial models, but also the
anisotropic ones. However, analytic method always assumes homogeneity, which cannot incorporate certain details of the antenna system. For example, the position of
the dipole antenna, which does affect the farfield radiation results. If the bulk of the
metamaterial substrate is indeed homogeneous, our radiation pattern should always
be symmetric, which is not the case as seen in Figure 3-5. Therefore, if we look at
the radiation in the
#
= 180' to
#
= 3600 range, we would not always expect to see
the same radiation pattern as what we will obtain from the
#=
0' to
#
= 180' range.
Our optimized structure uses both ring and rod elements. Additionally, we are able
to make the electric plasma frequency and the magnetic plasma frequency to be very
closed to each other. This is an improvement over the rod only medium.
81
82
Bibliography
[1] J. A. Kong, "Electromagnetic wave interaction with stratified negative isotropic
media," PIER, vol. 35, pp. 1-52, 2002.
[2] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency
plasmons in thin-wire structures,"
J. Physics-Condensed Matter, vol. 10, pp.
4785-4809, 1998.
[3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart,
"Magnetism
from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave
Theory Tech., vol. 47, no. 11, pp. 2075-2084, Nov. 1999.
[4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz,
"Composite medium with simutaneously negative permeability and permittivity," Phys. Rev. Lett., vol. 84, no. 18, pp. 4184-4187, May 2000.
[5] R. A. Shelby, Smith D. R., and S. Schultz,
"Experimental verification of a
negative index of refraction," Science, vol. 292, pp. 77-79, Apr. 2001.
[6] T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R.
Smith, D. C. Vier, and S. Schultz, "Ab initio numerical simulation of left-handed
metamaterials: Comparison of calculations and experiments,"
J. Appl. Phys.,
vol. 90, no. 10, pp. 5419-5424, 2001.
[7] C. G Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian,
"Experimental verification and simulation of negative index of refraction using
snell's law," Phys. Rev. Lett., vol. 90, pp. 107401, 2003.
83
[8] R. B. Greegor, C. G Parazzoli, K. Li, and M. Tanielian, "Origin of dissipative
losses in negative index of refraction materials," Appl. Phys. Lett., vol. 82, no.
14, pp. 2356-2358, 2003.
[9] K. Li, J. McLean, R. B. Greegor, C. G Parazzoli, and M. Tanielian, "Free-space
focused-beam characterization of left-handed materials," Appl. Phys. Lett., vol.
82, no. 15, pp. 2535-2537, 2003.
[10] D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design
and measurement of anisotropic metamaterials that exhibit negative refraction,"
JEICE Trans. Electron., vol. E87-C, no. 3, pp. 359-370, 2004.
[11] J. Pacheco, Theory and Application of Left-Handed Metamaterials, Ph.D. thesis,
Massachusetts Institute of Technology, Cambridge, 2004.
[12] R. A. Shelby, Smith D. R., S. C. Nemat-Nasser, and S. Schultz,
"Microwave
transmission through a two-dimensional, isotropic, left-handed metamaterial,"
Appl. Phys. Lett., vol. 78, no. 4, pp. 489-491, Jan. 2001.
[13] D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett., vol.
77, no. 14, pp. 2246-2248, Oct. 2000.
[14] J. Huangfu, L. Ran, H. Chen, X. Zhang, K. Chen, T.M. Grzegorczyk, and J.A.
Kong, "Experimental confirmation of negative refractive index of a metamaterial
composed of Q - like metallic patterns," Appl. Phys. Lett., vol. 84, no. 9, pp.
1537-1539, Mar. 2004.
[15] H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T.M. Grzegorczyk, and J.A.
Kong, "Left-handed metamaterials composed of only S-shaped resonators," To
Be Published.
[16] P. Marko and C.M. Soukoulis,
"Absorption losses in periodic arrays of thin
metallic wires," Optics Letters, vol. 28, pp. 846, May 2003.
84
[17] D. Sievenpiper, High-Impedance Electromagnetic Surfaces, Ph.D. thesis, University of California, Los Angeles, 1999.
[18] V. G. Veselago, "The electrodynamics of substances with simultaneously negative
values of E and p," Soviet Physics USPEKHI, vol. 10, no. 4, pp. 509-514, Jan.Feb. 1968.
[19] J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett., vol.
85, no. 18, pp. 3966-3969, Oct. 2000.
[20] I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW-Media
with negative parameters, capable of supporting backward waves,"
Microwave
Opt. Technol. Lett., vol. 31, no. 2, pp. 129-133, Oct. 2001.
[21] A. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave
radiation from a negative refractive index metamaterial,"
J. Appl. Phys., vol.
92, no. 10, pp. 5930-5935, 2002.
[22] B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with
imaginary transverse wavenumber in a slab waveguide with negative permittivity
and permeability," J. Appl. Phys., June 2003.
[23] C. Caloz, C.-C. Chang, and T. Itoh, "Full-wave verification of the fundamental
properties of left-handed materials in waveguide configurations," J. Appl. Phys.,
vol. 90, no. 11, pp. 5483-5486, 2001.
[24] J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco, B.-I. Wu, J. A. Kong, and
M. Chen, "Cerenkov radiation in materials with negative permittivity and permeability," Optics Express, vol. 11, no. 7, pp. 723-734, Apr. 2003.
[25] N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas and Wireless
PropagationLetters, vol. 1, no. 1, pp. 10-13, 2002.
85
[26] A. Grbic and G. V. Eleftheriades,
"Growing evanescent waves in negative-
refractive-index transmission-line media,"
Appl. Phys. Lett., vol. 82, no. 12,
pp. 1815-17, 2003.
[27] S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial
for directive emission," Phys. Rev. Lett., vol. 89, no. 21, pp. 213902, Nov. 2002.
[28] Microwave Studio is a registered trademark of CST GmbH, Darmstadt, Germany.
[29] X. Chen, T.M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr., and J.A. Kong,
"An
improved method to retrieve the constitutive effective parameters of metamaterials," To Be Published.
[30] T. Weiland,
"Time domain electromagnetic field computation with finite dif-
ference methods,"
International Journal of Numerical Modelling, vol. 9, pp.
259-319, 1996.
[31] CST-Computer Simulation Technology, CST Microwave Studio® Advanced Topics Version 4, 2002.
[32] S. O'Brien and J. B. Pendry, "Magnetic activity at infrared frequencies in structured metallic photonic crystals," J. Physics-Condensed Matter, vol. 14, no. 25,
pp. 6383-6394, July 2002.
[33] J. A. Kong, Electromagnetic Wave Theory, EMW, Cambridge, MA, 2000.
[34] S. Enoch, G. Tayeb, and D. Maystre,
"Dispersion diagrams of bloch modes
applied to the design of directive sources," PIER, vol. 41, pp. 61-81, 2003.
86