Two Dimensional Control of Metamaterial Parameters for Radiation Directivity by
Eleanor R. Foltz
B. S. in Electrical Engineering
Massachusetts Institute of Technology, Cambridge, June 2005
Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering at the
Massachusetts Institute of Technology
February, 2006
© 2006 Eleanor R. Foltz. All rights reserved.
The author hereby grants to M.I.T. permission to reproduce and distribute publicly paper and electronic copies of this thesis and to grant others the right to do so.
Author
Certified by
Certified by
-
Department of Electrical Engin ring and Computer Science
February 3, 2006
'- .
Jin Au Kong
Thesis Supervisor j3ae-lan Wu
Accepted by
Arthur C. Smith
Chairman, Department Committee on Graduate Theses
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Eleanor R. Foltz
Submitted to the Department of Electrical Engineering and Computer Science on February 3, 2006, in partial fulfillment of the requirements for the Degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
This work examines the feasibility of using metamaterials to direct radiation. The limits of required index of refraction and the required material depth are explored using
MATLAB simulations. A wedge of connected S-shape metamaterial is chosen and simulated in CST Microwave Studio®. The incident radiation is Transverse Magnetic
(TM) and negative deflection is achieved. The S-shape wedge is adjusted in small ways, and a specific wedge is chosen for further study. The S-shape metamaterial wedge is then adjusted by adding lumped elements of capacitance throughout the structure. A beam through this adjustable material is deflected -76o to +580 by adding OpF to 6pF additional capacitance. The deflection is not monotonic, but most pronounced between
0.1pF and 0.8pF. The deflection is discussed, as well as the regions of strongest signal power.
Thesis Supervisor: Jin Au Kong
Title: Professor of Electrical Engineering
Director, Center for Electromagnetic Theory and Applications,
Research Laboratory for Electronics, MIT
Thesis Supervisor: Dr. Bae-Ian Wu
Title: Research Scientist, Center for Electromagnetic Theory and Applications,
Research Laboratory for Electronics, MIT

I would like to thank Professor Jin Au Kong for allowing me to work with his group on this thesis. Great thanks also to Dr. Bae-Ian Wu for his regular admonishments to push on and help with every aspect of this thesis. He was incredibly helpful in identifying the crucial elements in whatever I was working on and steering away from immaterial fragments. Professor Erich Ippen has been a most reliable resource and advisor over my many years haunting of the Institute halls.
The friendship and support of numerous members of Women's Independent Living
Group (WILG), United Christian Fellowship (UCF), Crossproducts and Graduate
Christian Fellowship (GCF) has helped me in immeasurable ways. My parents' willingness to finance the very best in education allowed me to both apply to and attend this school. My parents and sister can always be depended on for counsel, encouragement, and laughter in any situation.

Abstract .............................................................................................
2
Acknowledgement ..............................................................................
3
Table of Contents ........................................ .................................. 4
List of Figures ................................................................................... 6
List of Tables ......................................................................................
9
Chapter 1 Introduction .............................................................. 10
Chapter 2 Theoretical Background ........................................... 13
2.1 Theoretical Possibility of LHM ................................ ... 13
2.2 Negative Permittivity ...... .......................................... .................... 14
2.3 Negative Permeability .................................. 16
2.4 Practical Uses: Directed Radiation.... ............................... 17
Chapter 3 Methodology and Approach .................................... 22
3.1 Ray Analysis .................
23
3.1.1 Method 1: Phase Comparison ...................................... ........... 24
3.1.2 Method 2: Refracted Rays ................................................ ...... .. 26
3.2 M ATLAB Simulations .............................................................. 31
3.2.1 MATLAB E-field Calculations............................. ............... 31
3.2.2 MATLAB Material Depth Estimates............................................. 34
3.3 CST Microwave Studio Simulations ........................ .... 41
3.3.1 Homogeneous Wedge ......................................... ................ 41
3.3.2 S-shape, TM incidence................................................................. 46

Chapter 4 Deflection due to Added Capacitance ..................... 61
4.2 Signal Power compared to Air Transmission........................ 73
Chapter 5 Conclusions and Suggestions ................................... 75
Bibliography ....................... .. ........... ...... ..................... ................. 7

Figure 1-1: Forward Calculation of Gain Pattern Block Diagram............................ . 10
Figure 1-2: Inverse Calculation of Gain Pattern Block Diagram ..................................... 11
Figure 2-1: Example Permittivity vs. Frequency.................... ............. 15
Figure 2-2: Example M etamaterials....................................................................... 19
Figure 2-3: S-Structure from Hongsheng Chen et al [35].................................. ..... 20
Figure 3-1: Position of Wedges for Analytic Setup ................................................. 23
Figure 3-2: Method 1 Phase Comparison Illustration........................... ......... 25
Figure 3-3: Method 2 - Refraction Arrangement 1, nmax .................................. 26
Figure 3-4: Method 2 - Refraction Arrangement 1, nmin ........................................ 27
Figure 3-5: Method 2 - Refraction Arrangement 2, nmax............................................... 29
Figure 3-6: Demonstration of correct E-field calculation.......................... ........ 32
Figure 3-7: Sensitivity of E-field calculation to dipole density ..................................... 33
Figure 3-8: E-field Achievable with d = 1.7 ........................................ ............ 35
Figure 3-9: Multiple Wedges Arrangement ........................................ ............ 35
Figure 3-10: Changes in E-field due to # of wedge pairs ........................................... . 36
Figure 3-11: E-field from fixed wedge shape, various n ........................................ 36
Figure 3-12: E-field w ith variable n
2
............................................................
.............
....... 37
Figure 3-13: Minimum depth for reflection, various n........................................ 38
Figure 3-14: Deflection Angles Possible from 3k wide wedges.............................. . 39
Figure 3-15: Deflection Angles Possible from Fixed Total Width................................ 40
Figure 3-16: Positive Wedge Simulation Setup................................... ........... 42
Figure 3-17: Homogeneous Positive Wedge Refraction ........................................ 42
Figure 3-18: Drude Model Parameter Simulation ...................................... ........ 43
Figure 3-19: Drude Model Wedge Simulation Setup ........................................ ...... 43
Figure 3-20: Homogeneous Wedge Refraction for n=0. ........................................ 44
Figure 3-21: Homogeneous Negative Wedge Refraction ....................................... 45
Figure 3-22: Two Views of Basic S-structure ....................................... ........... 47
Figure 3-23: Mesh Lines from X-direction........................................................... 48

Figure 3-24: Mesh Lines from Y-direction........................................................... 48
Figure 3-25: Mesh Lines from Z-direction onto single S .......................................
Figure 3-26: View of Unattached S-pair Setup...................................
49
............ 49
Figure 3-27: View of 5 rows of Unattached S-pairs ........................................ ...... 50
Figure 3-28: Farfield Results of 2 rows of unattached S-pairs ..................................... 50
Figure 3-29: Farfield Results of 5 rows of unattached S-pairs ..................................... 51
Figure 3-30: Joint S-structure, internal vacuum block shown ..................................... 52
Figure 3-31: Mesh Lines from Z-direction, Linked S-Structure ................................... 53
Figure 3-32: Two Views of Adjusted 3 S-pair Wedge Setup .................................... . 54
Figure 3-33: S
1
,
1
Results from Adjusted 3 S-pair wedge .......................................
Figure 3-34: Deflection from 3 S-pair wedge....................................
54
............. 55
Figure 3-35: Farfield Results from Adjusted 3 S-pair Wedge at 10.25 GHz ................. 56
Figure 3-36: Two Views of 4 S-pair Wedge Setup.................................. ......... 57
Figure 3-37: S
1
,
1
Results from 4 S-pair wedge ....................................... .......... 57
Figure 3-38: Negative Refraction due to 4 S-pairs ...................................... ....... 58
Figure 3-39: Deflection vs. Frequency for 3 and 4 S-pair Wedges ............................... 59
Figure 3-40: Deflection vs. Frequency, Narrow Frequency Range ............................... 59
Figure 4-1: S-shape with Extra Capacitance locations shown.................................... 62
Figure 4-2: CST rendering of S-shape with Capacitance locations shown ................... 62
Figure 4-3: Deflection vs. Frequency with minimal Capacitance ................................. 63
Figure 4-4: Deflection vs. Frequency with substantial Capacitance.......................... 64
Figure 4-5: Initial Results: Deflection vs. Frequency for Various Capacitances ............. 65
Figure 4-6: Deflection vs. Capacitance for various Frequencies .................................. 65
Figure 4-7: Deflection vs. Capacitance for various Frequencies on log scale.............. 66
Figure 4-8: Deflection vs. Frequency for various Capacitances, Complete Results ........ 66
Figure 4-9: Deflection vs. Capacitance for Various Frequencies on log scale ................. 67
Figure 4-10: Radiation from 0.0003pF and 0.03pF Added Capacitance ....................... 68
Figure 4-11: Radiation from 0. lpF and 0.2pF Added Capacitance .............................. 68
Figure 4-12: Radiation from 0.25pF and 0.26pF Added Capacitance .............................. 69
Figure 4-13: Radiation from 0.28pF and 0.3pF Added Capacitance ................................ 69
Figure 4-14: Radiation from 0.32pF and 0.34pF Added Capacitance ........................... 69

Figure 4-15: Radiation from 0.36pF and 0.38pF Added Capacitance .......................... 69
Figure 4-16: Radiation from 0.4pF and 0.42pF Added Capacitance......................... 70
Figure 4-17: Radiation from 0.44pF and 0.45pF Added Capacitance ........................... 70
Figure 4-18: Radiation from 0.5pF and 0.56pF Added Capacitance ............................. 70
Figure 4-19: Radiation from 0.6pF and 0.7pF Added Capacitance.......................... 70
Figure 4-20: Radiation from 0.8pF and 1.5pF Added Capacitance .............................. 71
Figure 4-21: Radiation from 3pF and 4.5pF Added Capacitance........................... 71
Figure 4-22: Radiation from 6pF Added Capacitance ........................................... 71
Figure 4-23: Power as a function of Capacitance, Various Frequencies ....................... 73
Figure 5-1: Two geometries for future investigation .............................................. 76

Table 1: Method 1 Phase Comparison Results ..................................... ......... 25
Table 2: Method 2 Refraction Results for Arrangement 1 .................................... 28
Table 3: Method 2 Refraction for Arrangement 2 Results ..................................... 30
Table 4: Analytic Results Summary ..................................................................... 30
Table 5: Angle of Deflection due to applied Capacitance, Initial Results.................... 64

This project examines the possibility of controlling the direction of radiation by controlling the permittivity and permeability of a metamaterial. The introduction to the field of metamaterials and the possibility of radiating through them is included in Chapter
2. This chapter outlines the thesis and the problem to be discussed. This thesis considers the results of two methods of calculating the E-field that would result, as seen in Figure
1-1. First, the details of the structure are converted into an effective media that varies in two dimensions. This media is then broken up into sections, which are treated as if each
Figure 1-1: Forward Calculation of Gain Pattern Block Diagram one creates a dipole source. The antenna gain pattern is then calculated from a sum of the fields from dipoles whose phases and amplitudes are functions of the index of refraction.
The MATLAB® methods used for the dipole model and gain pattern calculation are

described in Section 3.2. The second method uses the simulation tool CST Microwave
Studio* [42] to calculate the emitted field for comparison.
This analysis examines the possibility of creating precise control over the exiting wave.
This is the inverse problem of that described in the previous paragraph, seen in Figure
1-2. One method would be to determine a two dimensional set of dipoles that most closely create a three dimensional desired gain pattern, leading to a two dimensional local effective E's and pt's of the material, and finally to the 3D state of that material.
Figure 1-2: Inverse Calculation of Gain Pattern Block Diagram
A decision was made early on that two dimensional control over the metamaterial would be greatly simplified if (1) a wedge shape with essentially one set of effective parameters was used to control the beam in one dimension with the assumption that a second wedge would control the second dimension and (2) the incident beam was transverse magnetic.

A transverse magnetic beam allows the metamaterial inclusions to be attached, as described in Section 3.3.2. Attached inclusions can be adjusted with fewer points of control. This simplifies the control system, but reduces the flexibility of the created gain pattern. After simulating a TM incident beam deflected by a metamaterial, the remainder of the thesis considers the effect of adjusting the parameters of a metamaterial by adding capacitance to metamaterial geometry. This was accomplished in CST Microwave
Studio@ and is discussed in Chapter 4.

2.1 Theoretical Possibility of LHM
From Maxwell's Equations, the dispersion equation for isotropic medium can be directly found. The four well known Maxwell Equations are:
V xH = 8/6t(D) + J
V xE = -8/6t(B)
V D=p
V B=O
(1)
(2)
(3)
(4)
For a source-less environment and homogeneous vacuum, EoE replaces D and poH replaces B. Taking the curl of Equation 2,
Vx (V xE) = p0 6/8t(V xH ) (5)
Substitute Equation I for the curl of H and use the following vector identity on the left side of Equation 5: Ax(BxC) = (A-C)B (A-B)C
V (V .E) - (V *V)E
= o 6/6t( o 8/t(E)) to arrive at the Wave Equation:
( V
2 L 60 62/8t
2 )E = 0
A simple case is E = Eo cos (kz ot) 9. Then equation 7 simplifies to
(-k 2 + 8
0
0 2 62/
2 )
E o cos (kz - cot) = 0
This yields the Dispersion Relation:
(6)
(7)
(8)

or where k 2 = (2 [o E0 k 2 = ( 2 /C
2
) n
2 n2 =
(9)
(10)
(11)
Victor Veselago observed in [13] that the index of refraction, n, is not theoretically constrained to be positive. In his paper published in 1968, Veselago explored how materials that have n<O might behave. A negative index of refraction implied both negative permittivity e and permeability p. He called this hypothetical material a "lefthanded" substance because of the direction of the k-vector. In most substances
(including a vacuum), the k-vector, E-field vector, and H-field vector form a right-hand set. Power flow, as shown by the Poynting vector, is in the same direction as the kvector.
S=ExH (12)
In left-handed media (LHM), n<0 implies k<0 or k anti-parallel to S. The k-vector, E- field vector, and H-field vector form a left-hand set. Photonic Crystals demonstrate some of the properties similar to left-handed-materials (LHM) but photonic crystals exhibit negative refraction with respect to power flow, but not with respect to the direction of the k-vector [2]. This thesis examines left-handed media with an effective negative refractive index.
2.2 Negative Permittivity
Plasmas with negative permittivity were known, but a new composite was introduced which created negative permittivity similar to a plasma out of a dielectric and thin wires
[30]. The equations which govern permittivity in plasmas are [21]:

E =1 1- (Op
2
/)
2
)
COp2 ne
2
/(F me)
(13) where plasma frequency (14) and where n is the density of electrons.
Figure 2-1 shows how the permittivity varies with frequency for two example ~ps. It shows that the frequency region for negative permittivity extends into higher frequency regions if op is larger.
/
0 2 4
GHz
•wp le10
- -- -.
6 8 10
Figure 2-1: Example Permittivity vs. Frequency
The new composite media forced electrons to move solely on thin wires within the structure, which reduced the electron density because the electrons were confined to the wires. A wire of radius r in a unit cell of size a
3 lowers the electron density [21]: neff= n r
2
/a
2
(15)
The mass of the electrons is also affected, by an additional momentum component eA.
where poH = VxA
A = (gon r
2 nve/2n) ln(a/R)
(16)
(17) yielding an effective electron mass: meff = (poe2 tr
2 n/2n) In(a/r) (18)
The wires also have a finite resistance; including loss leads to the Drude model [21]: emetal = 1 - cp
2
/(02-io(y) (19)

The final result for a composite media with thin wires is Equation 19 using cop:
Op = neff e 2
/ (60 meff) = 2at c 2 / (a
2 ln(a/r)) (20) where c is the speed of light in a free space. This permittivity then has regions over which it is effectively negative.
Materials with negative permeability were unknown until 1999, when Pendry et al [32] suggested that since an effective permittivity could be formed by creating structures smaller than the wavelength of radiation, the same principle could be applied to create structures with an effective permeability. Specifically, electrically conducting sheets were placed in a variety of configurations yielding, in principle, a "magnetic plasma" frequency. Analysis of metal cylinders showed that Leff could not be negative, but the inclusion of capacitance enabled enough control to allow negative permeability. In particular, Swiss rolls (coils) of sheet metal as well as split-ring double cylinders were introduced. By 2000, D. R. Smith et al had simulated composite materials using thinwires and slices of the double cylinders, as well as experimentally observing transmission
(which implied both E and g negative) through a single split ring [3]. Shelby collaborated to confirm the results of simultaneously negative E and negative [i with a 2D array of split ring resonators (SRRs) in 2001 [1].

Negative permeability material uses magnetic resonators to create a magnetic response to an incident field [3]. An equation for negative permeability comes from [32]:
Peffl 7 r
2 /a 2
1 + i (2a/gocor) - 3 / (n 2 o0 2 Cr 3
) (21) where r is the outer radius of the SRR, a is the length of a unit cell, o is the conductivity of the metal used to form the SRR, and C= Eo/d is the capacitance (per unit area) between the two layers.
Fabricating such media requires specifically shaped conducting and non-conducting materials at sizes smaller than the wavelength of the energy under consideration. The composite nature of the media has led to the label metamaterials [5].
Many researchers are investigating possible uses for these materials [6, 8, 13, 16, 19, 28,
40, 41]. A perfect lens was proposed by Pendry in 2000 [8] as a surprising consequence of the unusual qualities of this medium. There are limits to the effective medium assumption created by the periodicity of the structure [9]. In fact, Ziolkowski and
Heyman suggest that a perfect lens is not realizable [10]. Similarly, K. Webb et al suggest that loss in the metamaterial would prevent a perfect lens realization [ 11]. The concerns these researchers have raised on the ability of a metamaterial to create a perfect lens might apply similar limits on the ability to arbitrarily direct the energy emission through a slab.

This thesis specifically examines the possibility of using an LHM to redirect a beam of energy. The question under consideration is whether this possibility is feasible and if it might be a less expensive alternative to traditional phased array style radars. This work examines the possibility of creating a media which allows precise dynamic control over the direction of the wave exiting the media.
The first examination of emission direction control was by Enoch et al [6]. They considered a slab of very small index of refraction with an internally embedded point source. Snell's law governs the interaction at a boundary from this slab to air, causing the radiation to exit the slab nearly normal to the surface.
sin 0t / sin Oi = ni / nt (22)
Since the media can be created in such a way that both , and ýt are very small, ni is close to zero, causing incoming waves at any angle to be transmitted at an angle normal to the surface. Previous work at MIT has examined the effectiveness of this normal directed beam by comparing results from a variety of these metamaterial structures [31].
Specifically, Wang investigated one and two dimensional versions of the broken square inside broken square structure created by Shelby and Smith [27], the broken square with mirror image proposed by Pendry [39], the Omega structure proposed by Huangfu et al
[12] and the S-shape proposed by H. Chen et al [35]. Her results showed the Pendry broken square with mirror image and rod had the best normal directivity. Some example metamaterials are shown in Figure 2-2; each structure requires two layers, usually mounted on opposite sides of a dielectric substrate.

(a) broken square in broken square
Front Layer:
(b) broken square with mirror image
(c)
Omega structure
Back Layer:
U
Figure 2-2: Example Metamaterials
In 2004, Hongsheng Chen et al proposed an S-shaped inclusion for metamaterial fabrication [35]. Previous work had noted that structures based on a mirror image had a desirable lower bianisotropy [14]. Chen proposed a two-layer geometry where one layer contained an S-shape structure, and the other layer contained its mirror image no extra rods were needed to create negative permittivity because the structure was specifically tuned to overlap the regions of negative permittivity and permeability. The proposed structure from their paper is recreated in Figure 2-3. As shown in the figure, the dimensions of the shape were a = 5.2mm, b = 2.8mm, h = 0.4mm, d = 0.5mm, and with a unit cell = 4x2.5x5.4mm
3 . The experimental verification of this structure used a unit cell

of 4x2.5x10mm
3 which meant the S-shapes overlap. His simulation results were LHM between 15 and 20GHz. Experimental results were at a slightly lower frequency of 10.9
to 13.5GHz due to the effect of the dielectric mounting for the S-structure.
A unit cell height
5.4mm
unit cell height
5.4mm
separation between layers d=0.5mm
repeat depth
2.5mm
unit cell width 4mm
Figure 2-3: S-Structure from Hongsheng Chen et al [35]
H. Chen has also investigated the use of additional capacitance between the two S-shaped layers as a way to tune the LHM frequency band of the material [45]. As seen in Figure
2-3, a single S-shape has 3 inherent capacitances: these are between the horizontal portions of the S, each having an approximate area bxh. Weiland et al suggest [7] that more tune-ability is achieved through adjusting the permittivity and permeability of the embedded medium (surrounding the wires and SRRs). But perhaps the application of voltage to the structure can change the capacitance, changing the effective parameters of the medium.

While the segmentation of the metamaterial into individually controlled strips holds some promise, the individual control of each strip adds a significant layer of complexity to the problem. In 1996, Rao et al suggested two methods to reduce the cost of traditional phased array radar [40]. One method was to use ferroelectric materials sandwiched between two conductors and vary the voltage between the conductors. This resulted in one dimensional beam-steering. The other method was to use a Radant© Lens
(demonstrated by Radant Technologies, Inc.). The E-field propagates between parallel metal plates. The dielectric layer between the plates contains diode strips. When a diode strip is turned on, the phase shift through the dielectric layer changes. Many layers of strips are included, and the amount of phase shift in each strip depends on the amount of metallization. The lens used diode switches to change the phase shift through the media.
By restricting the adjustment to one dimension, the necessary control circuitry was greatly reduced. The authors suggested cascading two Radant© lenses that each scan in one dimension only, which would also require a 900 polarization rotator. The disadvantage of the system, briefly mentioned in their paper, was the lack of sidelobe control with only linear phase shift gradients. Traditional phased array radars reduce the energy level in their sidelobes by implementing some control over especially the end elements in both phase and amplitude. This thesis accepts the notion of simplified control by separation of the two dimensions into two systems, and leaves the sidelobe reduction problem to future developers.

This project assumes normal incident waves at frequencies that allow the media to be treated as having effective parameters gi and e. The possibility of creating approximately normal incident waves was demonstrated by Enoch [6]. Further research on the best type of metamaterial was performed at MIT by Wang [31]. She examined the use of a number of different metamaterials with the effective parameters set close to zero in order to create radiation pattern with a narrow beamwidth at an angle normal to the metamaterial surface.
The intent of this thesis is to show that adjusting the capacitance within a TM incident metamaterial can change the effective index of refraction enough to redirect the energy, and therefore further work could be done to use a voltage to cause this change. At that point, a voltage could be applied to the embedded metal in the structure. The voltage would be adjusted after the material is fabricated, causing the material properties, i.e. the effective index of refraction, to change after fabrication. Previous work assumed that a block of material has the S-shape is perpendicular to the plane in which deflection occurs
[35]. To change the deflection, i.e. to have an adjustable material, then requires multiple points of contact between the control system and the material. This thesis uses a wedge and assumes a single point of contact with the control system. The shift in orientation means that the incident energy is TM or transverse magnetic.

This chapter begins with three analytical methods of determining the sizing requirements on such a wedge. MATLAB simulations are then performed to determine the accuracy of this analysis. Results are shown that verify the accuracy of the method, and then
MATLAB is used to generate predictions as well.
3.1 Ray Analysis
This section uses ray analysis to find the depth requirements on a pair of wedges. The wedges are chosen to be 4X
0 wide in one direction (infinite in the other) and of unknown depth and are positioned in one of two ways see Figure 3-1. The width was chosen so as to be sufficiently large that grating effects could be ignored. The volume outside of the two wedges is assumed to be air - vacuum.
\T!
Figure 3-1: Position of Wedges for Analytic Setup

Two basic analytic methods are compared for a single wedge in one of two positions.
The first method assumes no refraction. The direction of the outgoing radiation is determined by the phase difference of the exiting rays. The second method assumes refraction and neglects the phase difference of adjacent rays, by focusing only on the bending of an individual ray. Both methods assume uniform transmission, meaning all rays are transmitted without loss. It is suspected that the methods will yield different results since they treat the rays differently.
Both methods assume that one of the wedges has n = 1. The goal is to find the depth d of the wedges so that the resulting beam is at ±600, which was chosen as a good standard for a significant range of beam deflection. To get a general idea of how this system might behave, the example maximum indices of refraction were taken as n=2 and n=3.
3.1.1 Method 1: Phase Comparison
For the first method, we calculate the phase of any two rays separated by X
0 as seen in
Figure 3-2. They must add coherently at 8 = 60'. Let the phase entering the first wedge be 0O. For convenience in calculations, we will take one of the incoming rays at the tip of the adjustable material wedge. This puts the phase exiting the first wedge as nikod. The second ray has phase (3/4)nlkod entering the second wedge. At the output of the wedge, the phase of the
2 nd ray has advanced by (/
4
)k
2 d = (/
4
)n
2 kod. In order for these two rays to add coherently at 6 = 600, then nlkod + koXosin6
0° = (3/4)nikod + (/)n
2 kod (23)

Xv Xv
Figure 3-2: Method 1 Phase Comparison Illustration
Xosin60 0 = (¼)n
2 d -(/4)nld
4Xsin6
0
0
/ (n2 - nl) = d
Using ni = 1, consider two cases: nmax = 2 and nmax = 3.
nmax = 2 => d = (24 = 3.5 o
(24)
(25) nmax = 3 => d = (I3)o = 1.7 o
The next step is to calculate the nmin required in order to allow the beam to steer all the way from its position at 8 = +600 to a new position of 8 = -60'.
osin-60' = (/4)n
2 d -(¼)nld
-2No (43) / d = (n
2
- ni) nmax =
2 nmax = 3
=> -24 (13) / d = -1 = (n2 - n) => nmin = 0
=> -2 (3) / d = -2 = (n
2
n) => min = -1
(26)
(27)
Table 1: Method 1 Phase Comparison Results nmax
2
3 d a nmin
3.5 4 41.20 0
1.7 o 230 -1

If the wedge positions are reversed, i.e. wedge I is nf and wedge II is variable material n2, then the phase calculation method is the same. By reversing the direction (sign) of 6, the equations are also the same leading to identical solutions for the reversed order wedge problem.
Figure 3-3 illustrates the method that accommodates refraction for wedge arrangement 1.
A single ray is analyzed for its direction. When the ray enters wedge II, it is not refracted since it is a normal surface. The first refraction happens at the boundary between wedge
II and wedge I, governed by Snell's Law. Note that the diagram assumes n2 > nl for the angles to be positive.
6s
Figure 3-3: Method 2 - Refraction Arrangement 1, nmax

nlsin a = n2sin 1
4Xotan a = d a-P-y=0 n2 sin y = sin 8
Set 8 = +600 : sin y = ('3)/(2n
2
) nmax =
2 :7 = 25.660 nmax = 3 : = 16.780
The solution for d is found by graphing over P and finding the intersection of: nlsin a = nisin(p+ y) = nm,, * sin 01 nmax = 2 : P
=
21.50 a =
47.1 => d = 4.3
0 nmax=3: =
80 a = 24.8
0 => X
(32)
(28)
(29)
(30)
(31) incoming ngnt
Figure 3-4: Method 2 - Refraction Arrangement 1, nmin
To direct the beam in the other direction as shown in Figure 3-4, the first assumption was that n2 is positive but near zero. However, the beam can only be deflected a maximum of
90
0
-a at the first refraction point (between the wedges), and then is refracted towards the

normal at the second refraction point (leaving wedge I). Therefore, the beam must have negative n.
In this new arrangement, 0, 6, and y are defined differently. Equations 28 and 31 still hold, with n negative. Equation 29 remains the same. Equation 30 is modified:
(33)
Substitute and simplify:
(1/n
2
)sin 6 = sin (a+3) = sin(a + sin [ sin (a) *(nl/n
2
)] ) (34)
Given a , the solution for nmin is obtained graphically using MATLAB: for a = 47.1o => nmin = -2 for a = 24.80 => nmin = -3
Table 2: Method 2 Refraction Results for Arrangement 1 nmax
2
3
21.50
80 a d nmin
47.10 4.3
0
-2
24.80 1.85 X -3
Applying the second method with the placement of the variable wedge and nl wedge reversed yields different results. See the illustration in Figure 3-5. Again, no refraction occurs at the entrance to wedge II since the incoming ray is normal to the surface.
For generality, Equation 35 is included: nlsin y = sin 8 n
2 sin a = n
1 sin 1
(35)
(36)

However, since ni = 1, we are able to introduce the following: a - 13 =- Y = - 5 sin ' [(n2/nl)sin a] a = 8 = 600
(37)
(38)
Plotting a graph of Equation 38 shows that the maximum allowed 6 for nmax = 2 is 600. In fact, a = 300 will always be a solution if nmax is greater than n = 2.
nmax = 2 : a = 30
0 => d = 2.3 4 nmax = 3 : a = 19.1' or 30 0 => d = 1.4 X0 or 2.3 Xo
Care should be taken when choosing a = 300 since this means that a +600 to -600 swing will include an attempt to send energy along the edge of the wedge.
------------
0P
Figure 3-5: Method 2 - Refraction Arrangement 2, nmax
By inspection, n2 = nl will give 8 = 0, so n
2
< n
1 gives 6 negative.
nmin = sin (a 600) / sin a (39)

Given a, the solution for nmin is obtained graphically using MATLAB: for a = 300 => nmin = -1 for a = 19.10 => nmin = -
2
Table 3: Method 2 - Refraction for Arrangement 2 Results nmax
2
3 a
300
19.10 d
2.3 X
1.4 k nmin
-1
-2
The three calculation methods resulted in a variety of required ranges for the variable deep.
for nl =1
Method 1
Table 4: Analytic Results Summary nmax
2
3
Method 2, Arrangement 1 2
3 a
41.20
230
47.10
24.80 d
3.5 X
1.7 X
0
1 nmin
-1
0
4.3 Xo 21.50 -2
1.85h 80 -3
Method 2, Arrangement 2 2
3
300
19.10
2.3 X
0
1.4 X
-1
-2
In this section, several methods for calculating the minimum depth of material and the range of n needed have been demonstrated. As seen in Table 4, each method for determining the smallest n necessary for nmax = 2 found a different value. However, in each case, when nmax was increased to 3 then the nmin found was more negative. This shows the necessity of including negative values of n when creating a wedge that will steer a beam 60* in either direction.

The commercial software MATLAB®, available from The MathWorks, Inc., was used for a variety of calculations. Version 6.5 and 7.0.4 were used.
In performing MATLAB simulations, radiation through an aperture was assumed to be equivalent to radiation from a set of equally spaced dipoles within that aperture adjusted to the correct amplitude and phase, requiring that these dipoles be spaced relatively close together. A function was written to find the farfield magnitude of the E-field that resulted from a single row of dipoles. To find this E-field, the formula from [17] for an array of equally spaced, equally phase shifted dipoles is:
E0 = -iwot (Ileikr / 4ir ) sin 0 [1, exp{-in (kd cos \ a)} ] (40) where cos x = sin 0 cos c (41)
This formula is adjusted to accommodate arbitrary phase and amplitude of the equally spaced dipoles and to find only the normalized E-field where 0 = 90' by adjusting the amplitude and the phase for arbitrary spacing between dipoles:
E = E amplitudeindex * exp{-i2n/spacing (index-l) cos p + i * phaseindex} (42)

Once E has been calculated for each <p between 00 and 1800, each abs(E) is divided by max(abs(E)) so that the data is normalized to 1. Thus the E-field pattern is normalized.
Figure 3-6 shows that the E-field is calculated correctly by this method by examining the simple case of 2 or 3 dipoles separated by X or V/2. This figure is included to demonstrate that the MATLAB E-field calculator that was written performs as expected.
lase
lambda/2,
1
C
1 of
Figure 3-6: Demonstration of correct E-field calculation
Koschny et al have shown that composite materials in which all internal dimensions are significantly smaller than the wavelength of incident radiation can be treated as having an

effective permeability and permittivity [9]. Every simulation using MATLAB assumed a homogeneous medium.
270 linear scale
270 log scale
Figure 3-7: Sensitivity of E-field calculation to dipole density
Figure 3-7 shows a simulation in which the number of dipoles per wavelength is varied.
If only one dipole per wavelength is included, the results are inaccurate, but the continuous phase change used in this example shows minimal sensitivity to the density of representative dipoles once a minimum number is included. Both linear and logarithmic scales are shown: in the logarithmic view, it is easy to see how the pattern converges
(5 and 10 dipoles per wavelength almost overlap).

3.2.2 MATLAB Material Depth Estimates
Since the hand-calculation methods of relating the minimum and maximum index of refraction to the depth of a wedge resulted in somewhat similar estimates, the first method, essentially ray-tracing, which had an intermediate value of d (depth), was chosen as the basis for MATLAB estimates. Functions were created to estimate the possible deflection (from a homogeneous medium) given a maximum and minimum achievable index of refraction. Each function returns a vector of the E-field at 0= 900 for the specified n. (The method for calculating the E-field is described in Section 3.2.1.)
In the first function, a single variable slab is joined with a
2 nd slab of n=1 (air or vacuum).
Representative dipoles are spaced at intervals of X/10, where X = 3cm at 10GHz, but the function is not frequency dependent. Representative dipoles are assumed to have equal amplitude. Their phase is determined by the phase passing through the first variable wedge and then the wedge of air. The resultant E-field is returned for a maximum and minimum input index of refraction and a given depth and a given width.
Figure 3-8 shows the E-field calculation that results from this function by setting the depth of the material d = 1.7 X. This value is chosen because the middle estimate for the minimum depth d in Table 4 was the result from method 1. And method 1 is the basic method for the MATLAB simulations.

18( aperture=4, depth = 1.7, nmax = 3, nmin = -1
90 1 .
.
HI I ICA nmin
0
270
Figure 3-8: E-field Achievable with d = 1.7
This first function was quickly expanded with two additional features. The first feature was the ability to specify the index of refraction for the second wedge. The second feature was the ability to place multiple sets of these two wedges side by side. (Note: the treatment of grating effects is that they are wrapped into the simulation by assuming the field is generated by many dipoles.)
II: n2 11: n2
I: ni I: ni
Figure 3-9: Multiple Wedges Arrangement
The new functional code allowed these inputs: width of wedge, depth of wedge, number of wedge pairs (minimum 1), maximum and minimum index of refraction for one of the wedges, single index of refraction for the second wedge.

wedgewidth=3, depth=2, nmax=2, nmin = 0.1, n2 = 1
90
270 only 1 wedge
nmin
270
2 wedges wide
Figure 3-10: Changes in E-field due to # of wedge pairs
Figure 3-10 shows how the E-field changes due to an increase in the number of wedge pairs that are used. The result makes sense because a higher number of wedge pairs increases the overall size of the aperture, which narrows the beamwidth.
wedgewidth=3, 2 wedges, depth=1, n2=1
90
270 nmax=2, nmin = 0.1
-
-------- nmax nmin nmax=3, nmin = -1
Figure 3-11: E-field from fixed wedge shape, various n
Figure 3-11 shows how the E-field changes when a fixed geometry is given different homogeneous index of refraction values for the first wedge.

The index of refraction for the second wedge can also be varied, as seen in Figure 3-12.
wedgewidth=3, 2 wedges, depth=1, nmax=3, nmin=-1
90
270
_- nmax 270 n2 = 1 -- nmin n2 = 2
Figure 3-12: E-field with variable n
2
The next step was to predict what range of angles could be produced from homogeneous wedges of various refractive indices. A function was written that took as inputs the maximum and minimum index of refraction that the wedge might be capable of producing, as well as its geometry. If a single wedge pair is used, where both wedges can vary from nmin to nmax, then there is a minimum depth required to obtain any given angle of deflection. See in Figure 3-13 that when the range of n increases, then a specific deflection can be obtained at a smaller minimum depth.

single wedge pair when wedge width=3 width=3 single wedge pair when wedge
80
70
0
S60
D 50
E
S40
E
E 30
/
/
/
/
/
I
20
10
.7"
--. for n range [1,-1] t-- [1 -1.5] n
0
I
0.2
I
0.4
I
0.6
I
0.8
I
.--------
I II
1 1.2 1.4 1.6 1.8
depth of wedges in lambda
Figure 3-13: Minimum depth for reflection, various n
If wedges of material are available only in one given geometry, but the number of wedge pairs is flexible, then the number of wedges affects the beamwidth of the resulting field just as a larger antenna would. Figure 3-14 shows the results. However an increased number of wedges does not significantly decrease the depth of material needed to get a certain deflection.

for n range [-1,1] and wedge width=3
0 0.2 0.4 0.6 0.8 1 depth of wedges in lambda
1.2 1.4
Figure 3-14: Deflection Angles Possible from 3k wide wedges
A more realistic situation, however, is that the geometry of the wedges will be set by a single total width related to the installation use. Figure 3-15 shows the deflection angles possible with a set total width and variable individual wedge width. The range for the index of refraction is set at +1 to -1. It shows, as can be expected, that subdividing the total aperture results in a wider deflection range. This is because the angle of the wedge is greater when the same depth is applied to a wedge of narrower width.

max total aperture width of 10
,,
Iu
80
70
60
50
40
30
20
10 n
0 0.5 1 1.5 2 depth of wedges in lambda
2.5
3
Figure 3-15: Deflection Angles Possible from Fixed Total Width
3.5
This section has shown through MATLAB simulation that significant deflection can be achieved with a limited range for the index of refraction by ensuring that the geometry of the two wedges is sufficiently deep. It has also shown that sufficient angular range can be achieved if the material is capable of effective negative index of refraction.

CST Microwave Studio@ version 5.0 is used to simulate the deflection through a wedge of metamaterial similar to [35]. Finite Integration Technique (FIT) will be reasonably accurate because the metamaterial chosen has sharp edges and no curved surfaces.
Simulators use Perfect Boundary Approximation (PBATM), Thin Sheet Technique
(TSTTM) [44] and also the new Multilevel Subgridding Scheme (MSSTM) to help with meshing. The transient solver was used for each simulation.
The first step in using the simulation is to ensure that both positive and negative refraction can be obtained from a simple case of a homogeneous wedge of material. The positive wedge was constructed out of material with E = p. = 2 so that n = 2. The setup for the simulation was periodic background material at +/- 0 in the Z-directions and +/-10mm in the X- and Y-directions. The port was just under 4X wide (120mm for 10GHz).
Blocks made from Perfect Electrical Conductor (PEC) where placed at the ends of the ports and small blocks of air or vacuum (n=l) were placed just +/-Z of the port to force the E-field to be along the long portion of the port and the H-field to be along the short portion of the port. The wedge overlaps the port on both ends as seen in Figure 3-16.
The wedge was 140mm long(x), 9mm high (z), and 20mm at its deepest (y).
tan-' (20/140)= 8.130 (43)

The result was a deflected beam to the positive by 8' as seen in Figure 3-17.
Figure 3-16: Positive Wedge Simulation Setup
Farfield 'farfeld (f=10) i1]' deg.
90
0
100
Main lobe magnitude
= 1.02
Main lobe direction = 82.0 dcg
VA/m2
Angular width 13 dB) = 12.4 deg.
Side lobe level = -2.9 dB
300
270
Figure 3-17: Homogeneous Positive Wedge Refraction
`~("" i·~
Microwave Studio allows for the creation of materials where the parameters are created assuming the form:
F(o) = Eo [1 0p2/(02-i07)]
(44)
This form is used for permeability as well. Using the Drude model option for both material parameters, wedges of n
= 0 and of n = -2 were created (see Figure 3-19). To

create n = 0, the value of the parameters (permittivity and permeability) at infinity was set to 2, plasma frequency was set to 8.88e10 rad/s, and the collision frequency was 1e7 rad/s which resulted in E(o = 10GHz) = 0, g(o = 10GHz) = 0. As seen in Figure 3-18, the value of epsilon and mu (as coefficients of so and lo) are identical and pass through zero
Eps
Mue'
Mi "
Frequency / GHz
Figure 3-18: Drude Model Parameter Simulation at 10GHz. To create n = -2, the plasma frequency was increased to 1.2566el 1 rad/s which resulted in e(o = 10GHz) = -2, g(ao = 10GHz) = -2. The dimensions of the homogeneous wedge of adjustable index material (Drude model) were 140mm long(x) by
3mm high (z) by 20mm (y) at its deepest. The angle of the wedge is the same as above, it is just not as high (z).
i
Z
Figure 3-19: Drude Model Wedge Simulation Setup

When the index of refraction was set near zero at 10GHz, then the farfield radiation pattern was deflected 80 (Figure 3-20). By setting the index of refraction to -2, the wedge deflected as expected at 240 (Figure 3-21). The large port size in the x-direction
(118mm) of about 4X
0 helped create the narrow beam of less than 130 seen in both figures.
Farlield farfield (f=101 P-Fieldr=lm)_Abs(Phiu; Theta= 90.0
90
120 60
10
Frequency
Main lobe nmagnitude 0.420 VAIm2
Main lobe
Angular wid direction = 98.0 th (3 12.9 deg.
Side lobe l evel dB
300
Figure 3-20: Homogeneous Wedge Refraction for n=0.

180
150 r·
Farfield 'farfield (f1=10) Theta= 90.0 deg.
90
60
\\~.·-·--r--·-..
30
0
210 330
Frequency = 10
Main lobe magnitude = 0.226
Main lobe direction =114.0 de
Angular width (3 dB)= 12.7 deg. 300
Side lobe level = -4.8 dB
270
Figure 3-21: Homogeneous Negative Wedge Refraction
This first section of simulation has shown predictable deflection for homogeneous wedges made from positive, near-zero and negative index of refraction materials.
45

The S-shape discussed in [35, 46] was chosen as the metamaterial to be used in simulation because of its simplicity and tunability. The metal is 0.4mm wide, and the entire S-structure is 5.2mm in the x-direction and 2.8mm in the y-direction. From the zdirection, the two empty squares appear which are 2x2mm
2
(each S has two cut-outs measuring 2x2.4mm
2
). The empty squares appear because the metamaterial requires a loop (for inductance) so there are always two S-shapes often called here an S-pair. The two layers in an S-pair are separated by 0.5mm of vacuum. A single S-pair is shown in two views in Figure 3-22. The electric field must be along the S (the x-direction as shown in the figure) and the magnetic field must pass through the empty square of the Sshape pair (the z-direction in the figure). The setup is the same as Chen's except the incident field is TM. This means the fields are deflected in the plane of the E-field rather than in the plane of the H-field. Figure 3-22 shows the direction of the incident E and H fields as well as the k-vector. To force the simulation to create the desired orientation of the incident energy, the port is abutted at both ends by small blocks of PEC, which forces the E-field to be perpendicular to the blocks. Above and below the port (in the zdirection) are small blocks of vacuum which separate the port from the periodic boundary.

X
Figure 3-22: Two Views of Basic S-structure
In addition, an internal block of vacuum was added between the S-pairs, meaning between each S and its mirror, to make sure that the mesh lines were symmetric. Mesh lines should be positioned symmetrically to obtain the most accurate results from the simulator. The separation between the S-pairs was 0.5mm, so the block was 0.25mm in the z-direction to force the simulator to add a mesh line in a symmetric fashion between the S-pairs.
Mesh properties were set in this manner: mesh density control was set to 20 lines per wavelength, the lower mesh limit was 20 and mesh line ratio limit was 30. Mesh type used was PBA. The mesh lines are shown in Figure 3-23 to Figure 3-31. Fixpoints were set to fixpoints inside shapes with less than 1,000,000 faces only. Subgridding was to a

maximum depth of 5. The background material was open in the x- and y- directions and periodic in the z-direction. The distance to the background material was 5mm in the xand y- directions and Omm in the z-direction. This resulted in the periodic boundary being 0.95mm away from the S-shapes which had a metal (PEC) thickness of 0.05mm.
The Steady State accuracy limit during simulation was usually set to -40dB, but occasionally also set to -30dB since the reduced accuracy was not seen to impact results.
p . ..
............
Figure 3-23: Mesh Lines from X-direction
.. i.......
o _ .... ,
.... i
T
3-24: Mesh Lines from Y-direction

1l l:::::::'~f
1-
-A
-4
I i r
Figure 3-25: Mesh Lines from Z-direction onto single S
3.3.2.1 Unattached S-pairs
In these simulations, the S-structures are unattached to their nearest neighbor as shown in
Figure 3-26 and Figure 3-27. Two row and five row simulations were performed. The resulting power pattern is straight through (that is, concentrated at 900) and was cleaner when more rows were included, as can be seen in Figure 3-28 and Figure 3-29. This may be due to reduced grating effects and a closer approximation to an effective medium when more rows are included.
Figure 3-26: View of Unattached S-pair Setup

mmmmm
Mwm m M mm M Mm M M
Figure 3-27: View of 5 rows of Unattached S-pairs
Farfield 'farfield
(f=10
[1]' P-Field[r=lm)_AbsIPhi Theta= 90.0 deg.
90
120 60
I
I
150 ma:
'
,
/ : t .
t ;
/C
I '
%- ;
-:
.rrD
-
*1
30
300
330
210
Frequency =10,
Main lobe magnitude = 0.0285 VAm2
Main lobe direction = 90.0 deg.,
Angular width (3 dB) = 13.1 deg.
Side lobe level = -2.5 dB
---------
Figure 3-28: Farfield Results of 2 rows of unattached S-pairs
50

150
Farfield 'farlield f=10j 1]' P-Fieldlr=lmiAbs(Phii Theta= 90.0 deg.
90
120 60
30
0
210
Frequency =10
Main lobe magnitude 0.34 VAlm2
Main lobe direction = 90.0 deg.
Angular width (3 dB) = 12.7 deg."
Sde lOo0 level = -12.2 dBO
300
330
270
Figure 3-29: Farfield Results of 5 rows of unattached S-pairs
The lack of reflection in Figure 3-29 is notable; it demonstrates that transmission is possible of a TM incident signal. However, the ultimate goal of applying voltage(s) to the S-shapes to cause deflection would require many, many points of contact to control the material. Further simulations had S-pairs attached to each other to accommodate future control system plans.
3.3.2.2 S-shape Wedge, Negative Deflection (15.50)
Each S-structure is joined to its neighbor as shown in Figure 3-30. The S-structures must be linked to create single points of contact at the end of the long S-shape. (In this figure, it can be seen that the metal inclusions have a consistent width (0.4mm), i.e. the end of the S-shape is not thinner than its intermediate metal.) The Ss are replicated and then

joined using Microwave Studio's Boolean addition feature. Each individual S is 5.2mm
by 2.8mm (as in [35]) leaving empty squares of 2mm by 2mm. The three S-pair segment is then (5.2-0.4) x3 =14.4mm long plus an additional 0.4mm for the end of the S. The height of the metal in the simulation (i.e. the thickness of the metal inclusions) was
0.05mm. Note that the second layer of the S-structure (the other half of the pair) is not shown in Figure 3-30 because the large block of vacuum is shown that separates the upper S-structure from its pair. The long S-structure is partially repeated at 4mm intervals in the y-direction. The angle created by this wedge is tanl'(4/14.4) = 15.50.
z
Figure 3-30: Joint S-structure, internal vacuum block shown
The mesh lines for this arrangement are shown in Figure 3-31.

/'% t 4 F io
Figure 3-31: Mesh Lines from Z-direction, Linked S-Structure
The port for the wedge in this simulation is 80mm and the metamaterial overlaps the port by about two S-shapes on both ends. The block of PEC and the two blocks of vacuum surrounding both ends of the port are there to force the E-field to be in the x-direction, or to force TM incidence. The PEC block is 2.5mm high and the height of the port is
2.3mm.
Adjustment A: Longer, Thicker, and Ends Narrower
Adjustments to the structure from Section 3.3.2.2 can be made in a variety of ways. For example, in this simulation, a number of small changes are made. Each S-structure had a
"top" and "bottom" metal widths of only 0.2mm instead of the normal metal width of
0.4mm so that any two S-structures which are placed end-to-end create the usual total metal width of 0.4mm. (Note this leaves the very ends of the combined S-structure with a smaller width.) The height of the S (with reduced top and bottom metal widths) is
5.6mm which is slightly longer than that considered previously. The empty squares seen

when viewing the S-pairs from the z-direction are 2mm x 2.4mm. The three S-pair length in the x-direction is 5.6 x3 = 16.8mm. The long S-structure is partially repeated at only 3.8mm in the y-direction. The metal (PEC) is also thicker at 0.1mm. Figure 3-32 shows the wedge of metamaterial one S-pair high and made from sets of three S-pairs to form an angle of tan-' (3.8/16.8) = 12.70. Figure 3-33 shows Si,1 results and Figure 3-35 shows the three lobes that resulted (as well as a small reflected lobe).
11-I f T71--
Figure 3-32: Two Views of Adjusted 3 S-pair Wedge Setup
S-Pararmneter Magnitude
1, 1
0.4
0.2
0.8
0.6
0
Frequency / GH-lz
Figure 3-33: Sl1, Results from Adjusted 3 S-pair wedge
1

For the first three S-pair wedge, there was significant reflection at all frequencies, but some transmission as well. Figure 3-34 shows the deflection at 7.1GHz. The largest lobe is reflected, but the negatively deflected TM lobe is clearly visible. Results across a wider frequency band will be discussed at the end of this section.
Farfield if 07.1000 [11' P-Field(r=1 m)_Abs(Phi); Theta= 90.0 deg.
90
180
Freq
Main
Main
Angt
Side lobe level = -4.0 dB
270
Figure 3-34: Deflection from 3 S-pair wedge

150
Farfield 'farfield (f=10.25) [ll' P-Field(r=l m) Abs(Phi); Theta= 90.0 deg.
90
120 60
30
I Ir 0
210
Frequency =10.25
Main lobe nagnitude 0.0307 VArnm2
Main lobe direction = 87.0 deg"
Angular wid Ith 3 = 10.6
=
-1.1 dB
_ 300
330
Figure 3-35: Farfield Results from Adjusted 3 S-pair Wedge at 10.25 GHz
The results for the adjusted S-pair wedge at a higher frequency show that the metamaterial must be selected with the frequency range in mind. At the shown frequency of 10.25GHz, which is the frequency with the lowest S
1
,
1
, there are three transmitted beams which are each larger than the reflected beam.
Adjustment B: Four S-pair Wedge (11.8")
A similar setup to that shown in Section 3.3.2.2 was used to simulate a wedge made from four pairs of the same S-structure as shown in Figure 3-36. The length of the 4 S-pair segment is (5.2 0.4) x 4 = 19.2mm. Each row has four fewer of the S-pair structure, so that eight of the empty squares (as seen from the z-direction) do not have any inclusions

in front of them in the y-direction. The long S-structure is partially repeated at 4mm intervals in the y-direction. The angle created by this wedge is tan-'(4/19.2) = 11.8
0 .The
S1,1 results can be seen in Figure 3-37. A low reflection point is visible around 6.3 GHz, and the farfield results for 6.3GHz are shown in Figure 3-38.
J."o
0.95
1
Figure 336: Two Views of 4 S-pair Wedge Setup
S-Parameter Magrnitude
Si, i
0.9
0.85
0.8
Frequency / GHz
Figure 3-37: S
1
,
1
Results from 4 S-pair wedge

Farfield 'ff_06.3000 111' Theta= 90.0 deg.
90
180 0
Frequency = 6.3
Main lobe magnitude =
Main lobe direction = 271
Angular width (3 dB) = 16
Side lobe level = -1.9 dO
270
Figure 3-38: Negative Refraction due to 4 S-pairs
In Figure 3-38 we see that negative deflection of the beam is occurring for the four S-pair wedge as well as the three S-pair wedge. Figure 3-39 shows a graph of the locations of the main and secondary lobes for the three S-pair wedge and the four S-pair wedge over a wide frequency range. The four S-pair wedge has a band of negative deflection below
6.7GHz, but the three S-pair wedge has a negative deflection band just above 6.7GHz.

m m•
15
-30
--
-60 -
--
Uf
U
* Main Lobe
2ndary lobes
9 11
Frequency (GFtz)
13
15 ommmmma mm- ir0m
II F ,
Ur
-60
.If
a Main Lobe o 2ndary lobes
9 11
Frequency (GHz)
Figure 3-39: Deflection vs. Frequency for 3 and 4 S-pair Wedges
30
AngI (I leg) im
-30
-60
-90
-9
-
~_ .5~I m...
Main Lobe
0 2ndary lobes
Frequency (GHz)
Figure 3-40: Deflection vs. Frequency, Narrow Frequency Range

The three S-pair wedge was chosen as the setup for adding capacitance because it had a good deflection from 6.8 7.7GHz, a significantly wider range than the four pair wedge had. The frequency band of interest is reproduced clearly in Figure 3-40. The addition of capacitance is described in Section 4.1.
This section has shown the achievement of negative deflection for TM incident waves on a wedge of attached S-shape metamaterial. The S-shape can be adjusted by changing the dimensions of the initial unit S and by changing how many S-shapes are present in each row (both of which will affect the angle of the wedge). This thesis does not discuss the optimization of the structure, but instead focuses on the ability of the structure to deflect beams (Chapter 3) and the change in deflection by adjusting the parameters of the metamaterial with additional capacitance (Chapter 4).

From Chapter 3, we have a wedge shape with negative deflection in the neighborhood of
7.3GHz. The effective index of refraction for this material is a function of the inherent inductance due to the loops created by the overlapping Ss, and the inherent capacitance of the overlapping metal. The index of refraction can be adjusted by changing the amount of capacitance. Microwave Studio® allows for the addition of lumped elements. In this chapter, the addition of capacitance is studied to see its effect on the deflection of any transmitted energy.
4.1 S-shape Wedge with Added Capacitance
A macro was written that added lumped element capacitance at the locations that the two
S-shapes overlap. The same capacitance was added to each location since the ultimate goal is to use a single control circuit. Figure 4-1 shows the locations of the added capacitance in an individual S-shape. The inductance of the loops of each S and the capacitance of the overlapping metal create a resonant circuit. By adding capacitance to this circuit, the resonant frequency is altered. Figure 4-2 shows the Microwave Studio rendering of the addition of lumped element capacitors at those locations.

Extra
Capacitance added at location of arrows
Figure 4-1: S-shape with Extra Capacitance locations shown
RI7~
IN
Figure 4-2: CST rendering of S-shape with Capacitance locations shown

The dimensions are the same as the three S-pair wedge from Chapter 3. In review, a single S-shape is 5.2mm by 2.8mm by 0.05mm, with consistent metal width of 0.4mm, and is paired with an opposite S at a distance of 0.5mm. A three S-segment is (5.2-0.4) x3 = 14.4mm and is repeated in the y-direction at intervals of 4mm, creating an angle of tan-'(4/14.4) = 15.50. A good estimate of the inherent capacitance is
C = 6A/d
= 8.854 x 10
1
2 F/m x (0.4mm x 2.8mm) / 0.5mm
= 1.98 *10-1
4
F
(45)
When the added capacitance is small compared with the inherent capacitance, the results were almost identical to the results without capacitance; this is seen in Figure 4-3 which is almost identical to Figure 3-40.
30
Capacitor = le-20F c- U
-60 Main Lobe o 2ndary lobes
-90
6.5 7
Frequency (GHz)
7.5 8
Figure 4-3: Deflection vs. Frequency with minimal Capacitance
When the capacitance is increased too much, the transmitted energy is reduced dramatically; effectively removing the bands of negative deflection. Figure 4-4 shows the results over a wide frequency band: there is no region of negative deflection and many frequencies are absent from the graph since any transmitted lobes are too small.

30 on
01 i
I
I- I-
........
=15pF
-'
11
1I
13
C
-60
·- ,- -
~
--
-90 -
Frequency (GHz)
... .. ..
.... .... ............. . . . . . . .
Figure 4-4: Deflection vs. Frequency with substantial Capacitance
The initial simulation results of applying some small capacitance to the S-structure is shown in Table 5. The angle in degrees of the transmitted lobe varies by the size of the lumped elements added to the S-structure. This deflection angle for various capacitances, also shown in Figure 4-5, shows the extremely promising initial results across the chosen small frequency band.
Table 5: Angle of Deflection due to applied Capacitance, Initial Results
0.0003 0.03
Capacitance: (pF)
0.1 0.2 1.5 3 4.5
Freq:
7
7.1
7.2
7.3
7.4
-54
-50
-43
-38
-34
-52
-46
-40
-36
-32
-42
-38
-34
-31
-29
-32
-27
-24
-29
-24
2
2
3
5
8
2
2
2
2
5 3
8 22
13 16

0 a,
, -30
-- -A --
-*--
-A- 3pF
1.5pF
- - -- 0.2pF
--E--
0.1pF
-,-
-- 0.0003pF
-60
Frequency (GHz)
Figure 4-5: Initial Results: Deflection vs. Frequency for Various Capacitances
Figure 4-6 and Figure 4-7 show this initial data on both linear and logarithmic scales.
Lu
10
0
-10
-20
-30
-40
-50
-Afn
,,
0 1 2 3 4
Figure 4-6: Deflection vs. Capacitance for various Frequencies

Figure 4-7: Deflection vs. Capacitance for various Frequencies on log scale
It seemed that the next step would be to investigate the deflection caused by capacitances between 0.2pF and 1.5pF. Unfortunately, further investigation showed that there is not a strict correspondence of both increased frequency and increased capacitance causing increase in the angle of deflection. The more complete results are shown in Figure 4-8.
30
0
-30
-60
-- 56pF
0.5pF
---- 0.44pF
0.42pF
-
- 0.4pF
0.38pF
0.36pF
0.34pF
0.32pF
0.3pF
0.28pF
0.26pF
0.25pF
0.2pF
* 0.1pF
an
Frequency (GHz)
Figure 4-8: Deflection vs. Frequency for various Capacitances, Complete Results

Figure 4-8 contains a lot of information, so the single frequency results are broken apart and shown separately in Figure 4-9.
Deflection vs Capacitance for 7GHz Deflection vs Capacitance for 7.1GHz
30
-30
0
0.1
)1
-60
-90
0.1
7d~=L~,
Capacitance (pF)
Deflection vs Canacitance for 7.2GHz
Capacitance (pF)
Deflection vs Capacitance for 7.3GHz
Capacitance (pF)
Capacitance (pF)
90
Deflection vs Capacitance for 7.4GHz
60
30
0
0)1
-30
-60
0.1
-90
Capacitance (pF)
Figure 4-9: Deflection vs. Capacitance for Various Frequencies on log scale

As can be seen in the Figure 4-9, the addition of a small amount of capacitance turns a negatively deflected beam slightly less negative. However, as the additional capacitance rises above ten times the inherent capacitance (2x 10
-
1 4 x 10 0.2pF), the angle appears unstable. Once the capacitance reaches approximately 0.8pF, the deflection appears to stabilize at near zero to slightly positive deflection.
The following sequence of figures shows the farfield radiation pattern from simulation at a fixed frequency of 7.3GHz in order of increasing capacitance levels.
Ffai u?7.3 rPF..iý.LMAb."i T757- g.8 dj.
.d107.31 d-. its
110
I
218
..... '" 330
WWN
677JZ
ýISA
I4
Sid. obe N-1 = -1.4 f
-
Figure 4-10: Radiation from 0.0003pF and 0.03pF Added Capacitance
F-Aw 7 .. in
NA.
---- -T- ' ý g
7 .3ND 7. "A "I.
30
330
A.WWý13dq - I u~cnaM
M. 7
6nu.067.37
rclLld. 24
Figure 4-11: Radiation from 0.1pF and 0.2pF Added Capacitance

,, i
60.
Uafid '1973111 Or PNJe1FMI.403.33 q Thet3 98A kip, so
17
120
30
,:·
:I ·,.
II~ kRw4
.
fig
0~
10
·rk~w
1.... .
s 1 240.
73MM
330
20
14.4n.M.3..rbi =3311 2*77.
M.3.. direo.
-
270.0
3.
Bid. " I.-I -4.6 do Sid. be 1
211
Figure 4-12: Radiation from 0.25pF and 0.26pF Added Capacitance
~i10 h FýSII i7-.3
Oor u.3.3.
14-.3.y -?.3
.
-2..i
- -21do oft W
Figure 4-13: Radiation from 0.28pF and 0.3pF Added Capacitance
I'se BLU POr l .8g.
so
.qo.4y = .3
13.3 i4F3407.
F.3q.-7
M.M
3 .
I, I &..a
3.33
.1
3 0
ram mmm=
Figure 4-14: Radi ation from 0.32pF and 0.34pF Added Capacitance
3
.N.O 7.4.14 173.03 r
P3iWi.IoLL4P.Th.333 3.3. ,.
so a as
* w
La
4.40270
Figure 4-15: Radiation from 0.36pF and 0.38pF Added Capacitance
69

l* =ftCI.Ab.U'h0 NJ lg.F
Go
39 r li
330
I bb0.
.- 1.3Sol V0
41-H.M. 0S.UOVAA,
A"W.dllh 13 41.5d.l.
2pF Added
Figure 4-16: Radiation from 0.4pF and 0.42pF Added Capacitance
F.oW IlPr 11.11
ý0-1 I A Vl o so
120
I·
Ai~,Al
" -
336
Y,.,•,, ·
S -Lim
17
Figure 4-17: Radiation from 0.44pF and 0.45pF Added Capacitance
F.00.1 01.30 Oj
12
G
Ise
F.20.0111.3M III' N.0 d.5
SO
* 1.
1I
,i N
F..q...y 1l.1
MOM .. Idkw. 1 V02..
MW.w.~ -
1. M..dd'0
--220IB
279
Figure 4-18: Radiation from 0.5pF and 0.56pF Added Capacitance
90.9 .0
.
I
120
~Y··1UIMI -1.JII
Figure 4-19: Radiation from 0.6pF and 0.7pF Added Capacitance

u7.uu r ei .14Ab''. U. g.T.hUe%-r Tt u meI
La
·
270 276
Figure 4-20: Radiation from 0.8pF and 1.5pF Added Capacitance t.old t .•, U.S
Figure 4-21: Radiation from 3pF and 4.5pF Added Capacitance
7 IN
Figure 4-22: Radiation from 6pF Added Capacitance
At the lowest capacitance levels (inherent plus up to 0.2pF), the reflected beam is substantial, but the angle increases with increasing capacitance. However, the angle becomes unstable around 0.3pF as multiple lobes of different strengths form in various directions. The most positive deflection occurs at 0.32pF of +580, but as the capacitance

continues to increase in small increments, the deflected angle falls quickly as low as -78' at 0.38pF. A secondary lobe is positioned around 300 to 400 when the capacitance is
0.4pF to 0.6pF; secondary lobes may complicate the use of these structures in future applications if they continue, even after more development, to have such a large magnitude. (Although the 0.4pF simulation had little reflection, this secondary lobe was only 2.6dB down from the main lobe strength.) The reflection increases again as the capacitance rises, and is the primary lobe once 0.6pF or greater is applied. Impedance matching may be the cause for this reflection. At capacitance of 0.8pF and above, the deflection is positive, as high as 16' at 7.4GHz and 6pF.
This section has shown that adjusting the applied capacitance to an S-shaped metamaterial will adjust the effective parameters of the medium in a sufficient manner to change the direction of a beam that passes through it although the deflection is not monotonic with capacitance.

Transmitted power must be sufficiently large to be useful in some application. To create a benchmark, the PEC portion of the metamaterial was changed to air, and CST
Microwave Studio® simulation run on the same geometry made of air was completed.
Figure 4-23 shows the power as a function of capacitance. As expected, the transmitted power through the metamaterial is clearly smaller than the power resulting from the simulation of the same geometry made with air only. The highest transmitted power is about 3dB lower than the transmission of air and happens around 0.4pF which corresponds to the most negative deflection and is within the generally unstable deflection region. The transmitted signal is strongest when it is negatively deflected by the addition of between 0.34pF and 0.44pF.
) 1 0.2
I I
PWR (dBW/m2)
0.3
F
0.4
I I I 1 I i i -
0.I 0.6 0.7 0.8 0.9 I
--- 7 GRk
---- 7.1Gz
7.2GFz
7.3GHz
--- 7.4GHz
-+- air
__
1,
-
Capacitan e log scale
Figure 4-23: Power as a function of Capacitance, Various Frequencies

The presence of all of the metal inclusions may make large signal power in the positive angle direction difficult with only single wedge geometry.

The goal of this project was to determine if steering could be achieved by using a controlled metamaterial. A brief study of segments of metamaterial led to the conclusion that shaping the metamaterial into a wedge would allow a simpler control system. A previously explored geometry was chosen, and negative deflection of a TM incident wave was simulated. The three S-pair wedge geometry was chosen and used as the basis of a study of the effect of added capacitance on the beam direction for a single wedge.
Capacitance was seen to have an effect on beam steering, causing a range of -76o to +580 at 7.3 GHz. The deflection angle was stable when the additional lumped element capacitance was less than 0.2pF or greater than 0.8pF. The unstable deflection region had the greatest range of deflection, so a working theory of the sensitive small signal deflection within this region would be helpful. An important component of that step is to determine the precision of capacitance necessary, and therefore the precision of the control system, if a specified angle is desired.
Work could additionally be done to develop a working correlation between the desired index of refraction (including the corresponding permittivity and permeability, E and pt) and the necessary additional capacitance.
The transmitted signal was strongest in the area of the most negative deflection. This suggests that the use of two wedges of metamaterial face-to-face might assist in both

increased signal power and a more straightforward control over a wide range of transmitted angle. Figure 5-1 shows some possible wedge geometries for future exploration. They would require two single point control systems.
I
I
II
II I
Figure 5-1: Two geometries for future investigation
A variety of methods of adjusting the S-shape wedge structure were briefly mentioned in
Section 3.3.2.2. Optimization of this structure is suggested for higher power transmission, lower sidelobe levels or perhaps, but unlikely, a more stable, more nearly monotonic angular deflection.

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