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Broadband microwave reflectarray using stacked circular apertures

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B r o a d b a n d M ic r o w a v e R e f l e c t a r r a y u s in g S t a c k e d C ir c u l a r
A pertu res
by
Manuel Felipe Romero Paz
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
Copyright © 2006 by Manuel Felipe Romero Paz
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A b stract
Broadband Microwave Reflectarray using Stacked Circular Apertures
Manuel Felipe Romero Paz
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2006
A multilayered, polarization independent beam-redirecting surface (BRS) at X-band
is presented with the goal of suppressing specular angle reflection to reduce radar cross
section. Beam-redirection is achieved by creating a linear reflection phase variation along
one dimension of the surface, effectively emulating a tilted metallic sheet. The beamredirecting surface was designed to exhibit a linear reflection phase over a 10% bandwidth
around 10 GHz.
The beam-redirecting surface unit cell consists of stacked circular
apertures separated by dielectric layers. A comprehensive equivalent circuit of the BRS
unit cell is presented th at aids in gaining physical understanding and facilitates the
design process. Each BRS unit cell can achieve a phase variation greater than 360°.
Furthermore, a high 10 dB specular angle suppression was achieved over a wide range of
incident angles at 10 G H z and greater than 9 dB between 8.5 G H z and 10.5 GHz.
ii
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A cknow ledgem ents
I would like to thank my supervisor, Professor George Eleftheriades for all the advice,
support and knowledge he provided me throughout this project. I would also like to
acknowledge my wife for her constant and unrelenting support throughout my graduate
career, without her I would be lost. I am grateful for my family’s love and support, you
have all helped me achieve my goals. Special thanks to all the electromagnetics graduate
students for all the discussions and fun times; you’ve all made my graduate career that
much more enjoyable. I would also like to thank Gerald Dubois for his help in the antenna
chamber and for all our interesting discussions.
Financial support from the Department of National Defence in Ottawa (DRCD) and
from the Natural Sciences and Engineering Research Council of Canada (NSERC) are
gratefully acknowledged.
iii
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C ontents
1
Introdu ction
1
1.1 M otivation...............................................................................................................
1
1.1.1 Stealth Technologies and the Need forB eam -redirection...................
1
1.1.2 Requirements for a Beam-redirecting S u rface.......................................
5
1.1.3 Reflectarrays as Beam-redirecting S u rfa c e s ..........................................
6
1.1.4 Multilayered R eflectarray s......................................................................
11
1.2 Proposed Design
1.3
2
..................................................................................................
12
Overview..................................................................................................................
13
B ackground
14
2.1 Antenna Array T h e o r y .........................................................................................
14
2.1.1 Array Factor B asics..................................................................................
15
2.1.2 Linear Antenna A r r a y s ............................................................................
16
2.1.3 Planar Antenna A rra y s............................................................................
18
2.2 Beam R ed irectio n..................................................................................................
19
2.2.1 T h e o ry .........................................................................................................
20
2.2.2 Linearity and B an d w id th .........................................................................
24
2 .2 .3
P r o o f o f C o n c e p t S i m u l a t i o n s .........................................................................
26
2.2.4 Specular Reflection S u p p ressio n ............................................................
31
2.3 Periodic S tr u c tu r e s ...............................................................................................
32
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3
4
2.3.1
Floquet T h e o re m ....................................................................................
33
2.3.2
Bloch Im p e d a n c e ....................................................................................
35
2.3.3
Terminated Periodic S tr u c tu r e s ..........................................................
36
A nalysis of th e P rop osed B eam -redirecting U n it C ell
38
3.1 Proposed BRS Unit C e l l .....................................................................................
38
3.2 Circular Aperture Equivalent C i r c u i t ..............................................................
40
3.3 Elementary Beam-redirecting Surface (EBRS) Unit C e ll..............................
47
3.3.1
Surface Impedance M atching.........................................................
48
3.3.2
Multiple Stacked EBRS C e l l s .....................................................
54
3.4 Reflection Phase Characteristics of Proposed BRS Unit C e l l ....................
57
D esign and Fabrication o f a B eam -redirecting Surface
62
4.1 D e s ig n .....................................................................................................................
62
4.1.1
Inter-cell Phase Difference and Number of BRS UnitCells . . . .
63
4.1.2 BRS Unit Cells Radii and LinearReflection Phase Gradient . . .
65
4.1.3
4.2 Fabrication
5
.......................
68
............................................................................................................
75
Semi-infinite Beam-redirecting Surface Simulations
4.2.1
Beam-redirecting Surface Size Considerations
.................................
75
4.2.2
Fabrication Method and Material S election.......................................
77
4.2.3
Sources of Fabrication Errors
82
..............................................................
E xp erim en tal R esu lts
83
5.1 Experimental C o n fig u ra tio n ...............................................................................
83
5.1.1
Far-Field C o n s tr a in t..............................................................................
87
5.1.2
Incident Polarization for Chosen Experimental Configuration . . .
89
5.1.3
A rra y F a c to r A p p ro a c h for M e a su re m e n t C o m p a r i s o n .......................
91
5.2 M easurem ents........................................................................................................
96
5.2.1
Normal In cid e n ce.....................................................................................
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96
6
7
5.2.2
Off-normal In c id e n c e ..............................................................................
99
5.2.3
Frequency V aria tio n s..............................................................................
103
5.2.4
Specular Angle S u p p re s sio n .................................................................
104
C onclusions and Future D irection s
106
6.1 C o n clu sio n s............................................................................................................
106
6.2 Future D ire c tio n s..................................................................................................
108
A p p en d ix
112
7.1 Reciprocity of A B C D M atrix used forZB D e riv a tio n ..................................
112
7.2 Coordinate Transformation to Facilitate AF P l o t t i n g ..................................
113
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List of Tables
1.1 Typical RCS values representative at X-band [ 1 ] ...........................................
3
3.1 Aperture radii configurations (in mm ) for Fig. 3 . 1 8 .....................................
57
4.1 Reflection phase and aperture radii configurations derived using BRS equiv­
alent circuit at 10 G H z .......................................................................................
65
4.2 Reflection phase and aperture radii configurations obtained via HFSS sim­
ulations at 10 G H z ..............................................................................................
66
4.3 Beam-redirecting surface reflection phase p r o f ile ...........................................
77
4.4 Final reflection phase and aperture radii configurations at 10 G H z . . . .
79
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List o f Figures
1.1 Monostatic radar system [ 2 ] ...............................................................................
2
1.2 F —117 Nighthawk stealth b o m b e r ..................................................................
2
1.3 Bistatic radar system [ 2 ] .....................................................................................
4
1.4 Reflectarray concept [3]........................................................................................
6
1.5 Reflection phase vs. frequency for patches of different size
.......................
8
1.6 Stub loaded patches [4 ]........................................................................................
8
1.7 Electronically tunable reflectarray [ 5 ] ..............................................................
9
1.8 Electronically tunable impedance surface [ 6 ] ..................................................
10
1.9 Three-layer reflectarray using patches of variable size [7]
11
1.10 Proposed beam-redirecting su rface.....................................................................
12
2.1 Linear antenna array in receiving mode [ 8 ] ......................................................
15
2.2 Linear antenna array [ 8 ] .....................................................................................
16
2.3 Planar array with equal grid spacing [8]...........................................................
19
2.4 Conceptual beam-redirecting surface
...............................................................
20
2.5 Beam-redirection via varying index of re fra c tio n ...........................................
21
2.6 Equivalence between beam-redirecting surface and tilted metal sheet.
24
. .
2.7 Correlation between reflection phase vs. frequency and operation bandwidth 25
2.8
H F S S p ro o f o f c o n c e p t sim u la tio n s e t u p .....................................................................
27
2.9 HFSS proof of concept simulated phase g ra d ie n t.............................................
28
2.10 Proof of concept simulation: Varying phase g ra d ie n t....................................
29
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2.11 Proof of concept simulation: Varying number of cells N ............................
29
2.12 Equivalent linear array of structure in Fig. 2 . 8 ............................................
30
2.13 AF approach: Varying phase g ra d ie n t............................................................
31
2.14 Specular reflection of metal sheet and beam-redirecting su rfa c e ...............
32
2.15 Periodic structure: Bloch impedance [9].........................................................
35
2.16 Termination of periodic structure [9]
37
3.1
Proposed BRS unit cell and its equivalent circuit
......................................
39
3.2
Inductively loaded rectangular waveguide and equivalent circuit [10] . . .
40
3.3
Inductive mesh and equivalent circuit [11]
41
3.4
Single circular a p e rtu re .......................................................................................
42
3.5
T-model for reciprocal two-port network [9]
43
3.6
Circular aperture extracted inductance vs. aperture r a d i u s ......................
44
3.7
Circular aperture extracted capacitance vs. aperture ra d iu s ......................
45
3.8
Two-port S-parameters for circular aperture of Fig. 3 . 4 ............................
46
3.9
Elementary beam-redirecting surface (EBRS) unit c e ll................................
47
3.10 Modified EBRS equivalent circuit for Bloch impedance derivation . . . .
48
3.11 Bloch impedance of EBRS unit c e l l ................................................................
50
3.12 Two-port S-parameter Simulation Results for the EBRS unit c e ll............
50
3.13 Grounded EBRS unit cell
52
................................................................................
3.14 Analogy between one-port grounded EBRS cell and two-port system of
two stacked EBRS c e lls ........................................................................................
52
3.15 Reflection phase properties of grounded EBRS unit c e l l ............................
53
3.16 Normalized Bloch impedance vs. aperture radius
......................................
55
T w o s ta c k e d E B R S cells a n d reflec tio n p h a s e l i n e a r i t y ....................................
55
3 .1 7
3.18 Reflection phase vs. frequency characteristics for proposed EBRS unit cell
and the radii configuration of Table 3 . 1 ..........................................................
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58
3.19 Reflection phase vs. frequency: Proposed BRS unit cell equivalent circuit
v a lid a tio n ..............................................................................................................
59
3.20 Reflection phase vs. aperture radius for proposed BRS unit cell of Fig. 3.1(a) 59
4.1
Proposed beam redirecting su rface...................................................................
63
4.2
Desired reflection angle for normal in c id e n c e ................................................
63
4.3
Reflection phase vs. frequency characteristics for N y = 12 BRS unit cells
66
4.4
Reflection phase gradient along y -d ire c tio n ....................................................
67
4.5
HFSS simulation setup for 12 BRS cell beam-redirecting s u r f a c e .............
68
4.6
Reflection phase gradient along y-direction at 10 G H z
70
4.7
Normalized reflected power patterns
for normalincidence (0,= 0°) . . . . 71
4.8
Normalized reflected power patterns
for off-normalin c id e n c e ..............
72
4.9
Reflection angle vs. incidence a n g l e .................................................................
73
4.10 Normalized reflection power at the specular a n g l e .......................................
73
4.11 Simulated specular angle su p p re ssio n ..............................................................
74
4.12 Chosen Experimental Configuration [ 6 ] ..........................................................
76
4.13 Beam redirecting surface fabrication p ro ced u re.............................................
78
4.14 Final reflection phase characteristics for the twelve distinct BRS unit cells
80
4.15 Reflection coefficient magnitude vs. frequency for N y = 12 BRS unit cells
81
4.16 Normalized reflection coefficient magnitude vs. BRS unit cell position y .
81
5.1
Bistatic measurement experimental configuration
.......................................
84
5.2
Chosen experimental configuration [6 ]..............................................................
85
5.3
3D sketch of chosen experimental configuration
..........................................
86
5.4
Maximum aperture dimension for a horn a n t e n n a .......................................
87
5.5
Cross-talk between RX and TX horn a n te n n a s ..............................................
88
5.6
Cross-talk m easurem ents....................................................................................
89
5.7
Projection of incidence plane on to the x = 0 p l a n e ....................................
90
.............................
x
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5.8
Antenna array equivalent of proposed beam-redirecting s u rfa c e ................
91
5.9
Amplitude taper along x-axis due to free space l o s s ....................................
93
5.10 Amplitude taper along x-oriented BRS unit cells due to free space loss . .
93
5.11 Coordinate transformation to facilitate AF p l o t t i n g ....................................
95
5.12 Measured normalized reflected power patterns for normal incidence at 10
GHz .
.................................................................................................................
97
5.13 Measured normalized power pattern for tilted metal s h e e t..........................
98
5.14 Measured normalized cross-polarization for a TM polarized incident wave
at 10 G H z ..............................................................................................................
99
5.15 Measured normalized reflected power patterns for off-normal incidence at
10 G H z .................................................................................................................
100
5.16 Reflection Angle vs. Incidence Angle at 10 G H z ..........................................
101
5.17 AF Argument for high sidelobes away from b ro a d s id e ................................
102
5.18 Frequency variations of reflected power pattern at 10 G H z for normal
incidence.................................................................................................................
103
5.19 Measured normalized specular reflected power vs. incidence angle . . . .
104
5.20 Measured specular angle suppression vs. incidence angle
..........................
105
.......................
108
6.1
Surface with random reflection phase along its dimensions
6.2
Inductive loading of circular apertures
...........................................................
110
6.3
Inductance control via multiple s w itc h e s .......................................................
Ill
7.1
Coordinate transformation to facilitate AF p l o t t i n g ....................................
114
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C hapter 1
Introduction
1.1
1.1.1
M otivation
S te a lth T ech n ologies an d th e N e e d for B e a m -r e d ir e c tio n
Stealth technologies encompass all materials and applications th a t aid in preventing the
recognition or categorization of an object. These technologies are of paramount value for
the military sector of many nations worldwide; they assist in keeping their fighter jets
and battle ships hidden from unwanted persons. This branch of research will continue
to enjoy much development for decades to come as nations develop evermore powerful
weapons which must be concealed.
A particular branch of stealth technologies deals with the concealment of objects from
radar signals. The term radar stands for radio detection and ranging, and refers to a
system which transm its an electromagnetic (EM) pulse, then “listens” for reflections,
or echoes, from a target [2]. A radar system which utilizes one antenna as both the
transm itter and receiver is referred to as a monostatic radar and is shown in Fig. 1.1.
A n e x a m p le o f a s te a lth te c h n o lo g y w h ich a im s to co n c ea l a n o b je c t fro m a m o n o s ta tic
radar includes materials th at absorb and dissipate incident EM power. In general, an
absorber permits an incident EM wave to penetrate its volume while at the same time
1
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C h a p t e r 1.
2
In t r o d u c t io n
Target crn
''
x
#
..-i,
J'\
5
V s"
Transmitter and receiver
Figure 1.1: Monostatic radar system [2]
Figure 1.2: F —117 Nighthawk stealth bomber
attenuating its magnitude. By covering an object with such an absorber, an incident pulse
sent by a radar system would be highly attenuated and only small amplitude reflections
would reach the radar antennas. If the backscattered power (defined in Fig. 1.1) is lower
than the noise floor of the receiving radar system, the object is effectively undetectable.
However, since the power transm itted by radar systems has increased over time, it has
become increasingly difficult to absorb all incident power. Also since some of the absorbed
e n e rg y is d is s ip a te d a s h e a t, a b s o rp tio n h a s b e e n a s s o c ia te d w ith d e te c ta b le in fra re d (IR )
emissions. In order to overcome this obstacle, the reflection properties of the object to
be concealed can be designed to provide minimum backscatter.
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C h a p t e r 1.
3
In t r o d u c t io n
Table 1.1: Typical RCS values representative at X-band [1]
Object
RCS (m2)
Pickup truck
200
Jumbo jet airliner
100
Cabin cruiser boat
10
Adult male
1
Bird
1 x 10~2
Insect
1 x 10~5
Advanced tactical fighter
1 x 10~6
The reflection properties of an object can be manipulated so as to minimize the
backscattered power reaching the radar antennas. For example, the surface of the US
F — 117 Nighthawk stealth bomber, Fig. 1.2, consists of various facets th a t are tilted
at different angles from one another. The variations in tilt angle of the facets are de­
signed such that an incident radar pulse is reflected at various directions except in the
backscatter direction, thus minimizing the EM power reaching the radar system. The
aforementioned facets are a method by which the radar cross section (RCS) of the bomber
was manipulated so as to achieve minimal backscattering. The RCS, a, of a structure is
defined as “the area intercepting th a t amount of power which, when scattered isotropically, produces at the receiver a [power] density which is equal to th a t scattered by the
actual target” [1]. In other words, the radar cross section is the effective area th at the
radar system detects. Radar cross section manipulation is not limited to the military
sector. Commercial jet liners have a relatively high radar cross section so th at radars at
various control towers, and airplanes, can maintain their current location and velocity.
F o r example, the tail of a B o e in g 737 has the vertical and horizontal stabilizers perpendic­
ular to each other so as to emulate a corner reflector. Since corner reflectors are effective
retro-reflectors, most of the incident power striking the tail will be reflected back to its
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C h a p t e r 1.
4
In t r o d u c t io n
D ire ct alqno
Receiver
o -------
I
— >4
Figure 1.3: Bistatic radar system [2]
source, thus facilitating the tracking of the aircraft. For completeness, some typical RCS
values are summarized in Table 1.1. It is worth noting th a t in a corner reflector case the
absorbers would outperform any reflecting surface. However, in designing objects that
must be invisible to radars, designers stay away from any type of corner reflector.
Current stealth technology has succeeded in concealing objects from monostatic radar
systems. However, if the radar system contains separate transm itter and receiver, placed
far from each other, the object then becomes harder to conceal. This radar configuration
is termed bistatic and is illustrated in Fig. 1.3. The location and velocity of the target
can be calculated from the following [2]:
• The transit time of the scattered signal (time required for an EM pulse to travel a
distance of D t + D r),
• The angle of arrival at the receiver ipe,
• The frequency of both incident and scattered signals (determines a velocity com­
ponent of the target).
T h e fre q u e n c y c o m p o n e n ts o f th e in c id e n t p u lse are obtained from the direct path signal
between the transm itter and receiver. Of course, the receiver must also have a knowledge
of the time origin of the transm itted pulse in order to determine the transit time. If the
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C h a p t e r 1.
In t r o d u c t io n
surface of the target reflects the incident energy away from the specular direction, the
receiver is less likely to detect any reflected energy. Hence, by designing a surface that
can redirect incident energy away from the specular angle, and thus minimize the bistatic
RCS, an enhanced stealth mode might be achievable. Furthermore, a beam-redirecting
surface may also be used to change the radar signature of an object; this is possible
because the reflection properties of such a surface can be changed arbitrarily.
Given the numerous applications for a structure th a t controls the bistatic radar cross
section, it was desired to design a beam-redirecting surface to control the direction of the
reflected power. In particular, it was desired to minimize the bistatic RCS by redirecting
incident power away from the specular angle.
1 .1.2
R e q u ir e m e n ts for a B e a m -r e d ir e c tin g Su rface
To minimize the bistatic RCS, it was desired to create a surface capable of redirecting
power away from the specular angle. In other words, it was desired to minimize the
reflected power at the specular angle by redirecting the incident EM wave.
The beam-redirecting surface to be designed was to operate at the centre of the X
frequency band, which includes the range between 8 —12 G H z and has a centre frequency
of 10 GHz.
This frequency range corresponds to the operational frequency of many
radar systems. To increase the usability of this beam-redirecting surface, it was desired
to maximize its operational bandwidth around its centre frequency. Specifically, the
reflections at the specular angle were to be minimized over at least a 10% bandwidth. To
further increase the versatility of this beam-redirecting surface, it had to accommodate
for any incident linear polarization. In other words, the reflection properties of the desired
surface should remain relatively constant for any incident linearly polarized EM wave.
F in a lly , th e su rfa c e sh o u ld re s p o n d to in c id e n t p la n e w aves fro m all directions.
As a
result, it was desired to maximize the angular range of operation of the surface so as
increase its functionality.
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C h a p t e r 1.
6
In t r o d u c t io n
\
feed
reflectarray
Figure 1.4: Reflectarray concept [3]
Other general requirements include the minimization of cost and weight. These are
im portant criterion which must be considered when designing a surface th a t is hoped to
be used for defense purposes. Furthermore, minimal weight allows the beam-redirecting
surface to be easily mounted on any vehicle/object.
1 .1 .3
R eflecta rra y s as B e a m -r ed ire ctin g Su rfaces
By analyzing the requirements for a beam-redirecting structure, one is lead into the field
of reflectarrays.
Reflectarrays, a marriage between antenna arrays and reflector antennas, have existed
for many years [12] and have found many applications in radar, communication and
medical imaging systems. They have attracted much attention due to their low cost,
ease of fabrication and ability to be readily mounted on flat surfaces. These benefits
make reflectarrays a potential means to achieve beam-redirection and thus to reduce
s p e c u la r reflectio n s.
A re fle c ta rra y is co m p o sed o f v ario u s u n it cells, w h o se re flec tio n
phase can be independently controlled. Via antenna array theory, the reflection phase of
each cell comprising the array can be selected so as to obtain reflection at a particular
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C h a p t e r 1.
In t r o d u c t io n
7
angle, as shown in Fig. 1.4. This ability to redirect incident waves can be exploited to
reduce specular reflections. Note th a t an array-element whose reflection phase can be
controlled and varied over a large range (at least 360°), is in general required to direct
the incident wave at any angle. Furthermore, it should be noted th a t in general, the term
“reflectarray” refers to surfaces which are utilized to mimic the properties of parabolic
reflector antennas and not to surfaces th a t simply redirect incident waves.
There has been an extensive amount of work in the field of reflectarrays. In general,
each publication proposes a new or improved reflectarray cell, then proceeds to design
and build a reflectarray based on this unit cell. Some examples of array unit cells include
dipoles [13], [14], patches (both circular and rectangular), and circular metallic loops [15].
However, the microstrip patch has become a particularly popular reflectarray element due
to its ease of fabrication and analysis.
Microstrip patch reflectarrays use microstrip patch antennas as their main radiating
elements.
These reflectarrays benefit from low cost, low loss and low profile.
Most
reflectarrays are fabricated via chemical etching procedures th a t are easily reproducible,
which in turn lowers the overall cost. The loss in a reflect array is minimized by employing
substrates with small loss tangents. Furthermore, as outlined in [3], thick dielectrics also
lead to lower losses; this is a result of the fields below the patch not being concentrated
over a small volume. However, thicker dielectrics tend to reduce the reflection phase
variation available to each patch reflectarray element. Furthermore, by employing flexible
substrates, a reflectarray can conform to any desired surface thus achieving a low profile.
The reflection phase of a microstrip patch antenna can be controlled by varying the
size [3], rotation angle [16] or loading of the patches [17], [4]. In this manner, a desired
phase distribution is obtained and thus also a desired reflection angle. In the case of a
v a ria b le -p a tc h -s iz e re fle c ta rra y , v a ry in g th e p a tc h le n g th L s h ifts th e resonance frequency
of the patch, resulting in a similar shift in its reflection phase characteristic as shown in
Fig. 1.5; this shift can then be exploited to vary the reflection phase across a structure.
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C h a p t e r 1.
In t r o d u c t io n
350
- - L+AL
300
- - L -A L
•g 250
•2 150
50
1.5
Normalized Frequency
Figure 1.5: Reflection phase vs. frequency for patches of different size
M■
■
■F
■V
■
p
p_
■
■
■■
■m
h
■
TT
(a) Single linear polarization configuration
-
(bj Dual linear polarization configuration
F ig u re 1.6: S tu b lo a d e d p a tc h e s [4]
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C h a p t e r 1.
9
In t r o d u c t io n
BIAS ^
network
Figure 1.7: Electronically tunable reflectarray [5]
The reflection phase of the reflectarray unit cell can also be controlled by loading a patch
with open circuited transmission-line stubs, as shown in Fig. 1.6. Specifically, if the
transmission-line stub is impedance matched to the radiation resistance of the patch, then
the reflection phase contributed by the stub is simply twice its electrical length [4]. This
relationship between length and reflection phase results from induced currents traveling
along the stub, reflecting from the open end back towards the patch and re-radiating.
Figure 1.6(a) depicts an array of stub-loaded patches so as to respond to only one linear
polarization, while Fig. 1.6(b) shows the configuration for the dual linearly polarized
version. Stub-loaded patches have been reported to provide poor polarization purity,
since the stubs are usually bent to fit in a fixed area and thus generate some unwanted
cross-polarized fields [18]. However, the length of the stub can be arbitrarily selected to
provide large phase variations within the limits of the reflectarray configuration.
Active beam steering surfaces, although not the focus of this thesis, have also been
of much interest as of late. Specifically, [5] uses a patch antenna th at is actively loaded
with a varactor diode as seen in Fig. 1.7. By varying the voltage across the diode, the
effective d io d e c a p a c ita n c e v aries, effectively changing the resonance frequency of the
patch; this frequency shift, similar to the one shown in Fig. 1.5, can then be used to
design a reflectarray. Furthermore, [6] uses a tunable impedance surface in order to
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C h a p t e r 1.
10
In t r o d u c t io n
Varactor Diodes
Bias Voltages
Figure 1.8: Electronically tunable impedance surface [6]
control the reflection phase of each element as seen in Fig. 1.8. The sheet capacitance
and inductance form a resonance which results in a high impedance surface. At this
resonance, the reflection phase varies quite sharply from 180° to —180° having a value of
0° at the resonant frequency. By varying the sheet capacitance via the varactor diodes,
the resonant frequency can be adjusted and a reflection phase variation is obtained.
It is worth mentioning th a t most reflectarrays based on the patch antenna element
suffer from small bandwidths since the patch is inherently narrowband; typical patch
bandwidths are on the order of 3% [1]. Bandwidth extension mechanisms have been
studied, among these, using thicker substrates and stacking multiple patches are the
most popular. Although thicker substrates in reflectarrays can lead to higher bandwidth,
up to 15% [16], they also reduce the reflection phase variation available to each patch,
th u s lim itin g th e ac h ie v a b le re fle ctio n an g le s [3]. It w as th e re fo re , desirable to consider a
multilayered design in order to overcome the tradeoff between bandwidth and available
phase variation.
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C h a p t e r 1.
In t r o d u c t io n
11
Figure 1.9: Three-layer reflectarray using patches of variable size [7]
1 .1 .4
M u ltila y ered R eflecta rra y s
In order to alleviate the bandwidth problem of single patch layer reflectarrays, multilay­
ered structures have also been proposed and analyzed. For example, [18] and [7] used
multiple stacked layers of patch antennas and multiple resonances to achieve larger band­
widths. In [7], by simultaneously varying the size of all three stacked patches, a large
phase variation in excess of 600° was reported while achieving 0.5 dB gain variations
within a 10% bandwidth around 12 G H z ; a schematic of this reflectarray is shown in
Fig 1.9. Other multilayered reflectarray structures include proximity coupled patches
[19] and stacked circular loops [15]. Note th at for the stacked circular loops, only the
reflection phase characteristics have been presented and no reflectarray based on this unit
cell has been reported as of this time.
The operating bandwidth of a reflectarray can be maximized by utilizing a multi­
layered structure since more degrees of freedom are available to change the reflection
phase. Of course, multi-layered structures introduce additional weight and cost. However,
by d e v e lo p in g a b e a m -re d ire c tin g re fle c ta rra y (su rface ) unit cell which uses an arbitrary
number of layers, the flexibility in designing for cost/weight or bandwidth is passed down
to the designer. Thus, this work presents a new beam-redirecting surface unit cell that
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C h a p t e r 1.
12
In t r o d u c t io n
-r— Metal 1 -
er
- — Metal 2 Metal 3 Proposed BRS Unit C ell'
(a) 3D view
(b) Front view
Figure 1.10: Proposed beam-redirecting surface
can be utilized for any number of layers, which can be used to design a beam-redirecting
surface to manipulate the radar cross section via minimization of the specular reflections.
1.2
P rop osed D esign
It was required to design a beam-redirecting surface to suppress specular reflections. For
this purpose, a beam-redirecting surface (BRS) is presented using a new architecture of
stacked circular apertures, as shown in Fig. 1.10, designed to exhibit a linear reflection
phase profile along one axis. As shown, the proposed surface unit cell consists of three
metallic layers. Two of these layers have printed circular aperture patterns while the
third layer is a ground plane. The metallic layers are separated by dielectric layers. By
individually varying the radii of the apertures within the BRS unit cell, it is possible to
vary the impedance of each unit cell and thus the overall reflection phase. The proposed
beam-redirecting surface also exploits microstrip technology which benefits from low
fabrication cost and high fidelity.
It was decided to establish a linear reflection phase along only one dimension in
o rd e r to d e m o n s tr a te th e b e a m -re d ire c tin g c a p a b ilitie s o f th e su rfac e; a linear reflection
phase gradient results in beam-redirection along one plane. Furthermore, the circular
aperture was chosen due to its inherent symmetry, resulting in a polarization independent
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C h a p t e r 1.
In t r o d u c t io n
13
structure.
It was desired to extract a comprehensive equivalent circuit of the BRS unit cell in
order to gain physical understanding and facilitate the design process. Furthermore, this
equivalent circuit should accurately characterize the proposed beam-redirecting unit cell
over the frequency band of interest, i.e. 8 — 12 GHz. This would permit the use of
the equivalent circuit in the design of the full beam-redirecting surface and thus would
minimize the number of full-wave simulations required.
The proposed BRS unit cell should be designed in such a way as to provide maximum
flexibility to the reflectarray designer. In other words, the proposed cell should allow
for opposite designs, i.e. optimizing for bandwidth or cost, to be achieved via only
one elementary unit cell. Finally, it was also desired to minimize the reflection phase
sensitivity to manufacturing errors.
1.3
O verview
This thesis is divided into six chapters. Chapter 2 reviews some necessary theoretical
background required for the design of a beam-redirecting surface. Chapter 3 outlines the
analysis of the proposed beam-redirecting surface (BRS) unit cell and its components.
Chapter 4 describes the design process of a beam-redirecting surface using the proposed
BRS unit cell; the fabrication methodology and associated considerations are also intro­
duced. Chapter 5 discusses the chosen experimental configuration along with measured
reflected power patterns. Finally, Chapter 6 summarizes the key points of the thesis and
suggests future directions for this work.
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Chapter 2
Background
This chapter provides the necessary background required for the analysis and design of
a beam-redirecting surface. A summary of array theory will be presented followed by
the principles of beam redirection. This chapter will conclude with a concise analysis of
periodic structures.
2.1
A n ten n a A rray T heory
Antenna arrays refer to a spatial arrangement of antennas in which the antennas are
excited in such a manner so as to create a directional radiation pattern. In what follows,
it will be assumed th a t all the antennas in a particular array are of the same type,
differing only in excitation amplitude and phase. Most of the following material will
fo cu s o n th e d e riv a tio n of th e a rr a y fa c to r (AF) as it will become useful for comparison
with reflected power patterns from the designed beam-redirecting surface. A large part
of what is summarized here is based on the theory presented in [8].
14
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C h a p t e r 2.
B ackground
15
Rays
Phase = t/o
"Reference
wavefront
E F ( Q ) d % Z / E ¥ ( e W \ Z / EF(0)ei5n
Phase (/„)
Radiation Pattern= EF(0)AF(0)
Figure 2.1: Linear antenna array in receiving mode [8 ]
2 .1 .1
A rray F actor B a sics
Consider the linear configuration of antennas as shown in Fig. 2.1. This array can be
analyzed in the transm itting mode or the receiving mode, whichever is most convenient,
since antennas usually satisfy the conditions for reciprocity. For now, let us consider the
receiving mode. One of the main advantages of using antenna arrays is their ability to
steer the maximum radiation beam in space. This is achieved by individually varying
the amplitude and excitation phase of each element. In the figure, the phase shifters
and attenuators attached to each antenna can be individually controlled, allowing the
amplitude and phase of the signal received by each element in the array to be selected;
thus allowing the array to receive a signal from any angle 9.
T h e o v e ra ll r a d ia tio n p a tt e r n a ch iev ed by th e array is th e product of the element
factor and the array factor. The element factor (EF) mathematically describes the spatial
variation of the radiation pattern of the individual antenna, while the AF describes the
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16
B ackground
dcos fi,
Figure 2.2: Linear antenna array [8]
effects of superposition on the radiation pattern when all array elements are considered.
To obtain the AF for a particular array, each antenna in the array is replaced with
an isotropic radiator, while conserving the position and excitation of each antenna. An
isotropic radiator is a hypothetical antenna th a t is lossless, occupies only a point in space
and radiates uniformly in all directions, i.e. EF(d,(f>) = 1. By replacing all antennas in
Fig. 2.1 with isotropic point radiators, one can obtain a general equation for the AF by
summing all the received signals of all antennas in the array.
JV -l
A F = £ / ne*»,
(2 -1)
n =0
where £„ corresponds to a free space phase delay from the reference wavefront to the
nth element in the array, In corresponds to the amplitude and phase adjusted induced
current on the nth element.
2 .1 .2
L inear A n te n n a A rrays
The general formulation for the AF given in (2.1) can be further developed for a linear
antenna array with equally spaced elements as shown in Fig. 2.2. Considering the array
to b e in th e re c e iv in g m o d e , it is e v id e n t t h a t a p la n e w ave, in c id e n t a t an angle 9 from
the axis of the array, will reach the array elements at different times. Specifically, element
1 will detect the incident plane wave before element 0. This time delay can be expressed
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C h a p t e r 2.
B
17
ackground
as a phase difference between these two receiving antennas. Since the path to element
0 is longer by d cos(6) and if element 0 is taken as a phase reference, i.e £o = 0 , then
element 1 leads element 0 in phase by k0d cos(0 ), where k0 is the free space wave number.
By letting /„ correspond to the weights applied to each antenna upon receiving, the AF
for the linear array is given by
iV -l
A F = I q + Iie^kod cos^ + / 2ej2fcod cos^ + . .. =
I ne’nkod cos(e\
(2-2)
n—0
If the array is considered to be in the transm itting mode, then the term I n can be
considered to be the excitation current on antenna element n. The excitation currents
can be expressed as
In = A ne ^ \
(2.3)
where A n is the excitation current amplitude feeding antenna n while the constant n A a is
the excitation current phase. This particular form of the excitation currents correspond
to a linear array with a constant progressive phase differencebetween
elements. By
exciting all elements in the array with currents th a t havea constant progressive phase
difference, Act, it is possible to steer the radiation maximum to any desired direction.
By substituting (2.3) into (2.2) one obtains
N —l
A F = Y 1 A nejn{kod cosW+Aa),
(2.4)
71=0
If
ip — k0d cos(9) + A a,
(2.5)
then
N- 1
A F = Y j A nejn^.
(2.6)
71=0
This form of the array factor, which is now a function of ip is useful in calculations but
o n e g e n e ra lly p lo ts th e A F a s a fu n c tio n o f 6. N o te t h a t (2.6) is a F o u rie r series.
As mentioned previously, the progressive phase difference between elements is utilized
to perform beam steering. The maximum of the array factor occurs when ip = 0. This
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C h a pt e r 2.
18
B ackground
results in
0o = c° s - 1(^ ) ,
(2-7)
where 60 corresponds to the angle at which the AF is maximum (refer to Fig. 2.2 for
definition of 0). Note th a t by varying the progressive phase difference, A a, between
elements, the beam maximum can be steered. This equation will be further discussed in
Section 2.2.1 when beam redirection is considered.
2 .1 .3
P la n a r A n te n n a A rrays
Linear arrays, although widely used, have numerous limitations. First, the radiation
maximum can only be steered along the plane th a t passes through the centre of all
elements in the array. Secondly, the beamwidth in the direction perpendicular to the
scanning plane is set by the element factor of the antenna and is generally wider than what
is desired. Some applications require beam steering in a 3D environment and control of
the beamwidth, for these applications planar arrays have been employed. In this section,
a planar array with a rectangular perimeter and equal grid spacing in both x and y axis
will be considered. A detailed study of various other planar array configurations can be
found in [8 ].
Consider the planar array show in Fig. 2.3. The m n th element in the array can be
referenced by a position vector r'mn given by
r'm n =
X'mn*
+
V'mnf
+
(2-8)
Z'mn?-
The array pattern is now given by
N
M
AF(6,4>) —
(2.9)
n —l m
=1
where I mn is the excitation current, both magnitude and phase, of the m n th element and
r is the unit vector in the radial direction. This general form of the array factor
directly applied to the configuration shown in Fig. 2.3.However, the
can be
final AF can be
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C h a pt e r 2.
19
B ackground
/ Planar
surface
Rectangular
grid
Rectangular
perimeter
Figure 2.3: Planar array with equal grid spacing [8 ]
simplified by assuming th a t the amplitude of the currents on the x-axis are proportional
to those on the y-axis [1]. This assumption allows the excitation current term, Imn in
(2.9) to be separated such th a t Imn = IxmI yn, which leads to
M
A F ( 6 , 0) =
N
sin (0 ) cos(</>))
Pj(k°y'n sin(0) sinW )
Y . I X™ej{k°X'mSi
771=1
71=1
(2 .10)
Note th a t the above AF is simply the product of linear array factors along the x and y
axes.
2.2
B eam R ed irection
Most works on reflectarrays use array theory in order to design for a particular reflection
angle. In these cases, the incident wave is considered the excitation and the relative
reflection phase of each element contributes to the reflected pattern as outlined in the
p re v io u s se c tio n . H ow ever, th e go al o f m o st re fle c ta rra y d esig n s is to mimic parabolic
reflectors. The work presented in this thesis deals with beam redirection in general, with
the goal of manipulating the radar cross section via suppression of reflections at the
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20
B ackground
S id e V ie w
Nd
Nd
(a)
Conceptual
beam-redirecting
surface (b) Required phase gradient for reflection at 0o,
(BRS), cell size d, N cells in both x and y
ki is incident wave vector, kr is reflected wave
direction.
vector and ky is the x-component of kr.
Figure 2.4: Conceptual beam-redirecting surface
specular angle. Although, array theory can be used to design a beam-redirecting surface,
an alternate approach is presented in this section. This alternate method provides further
insight into the reflection properties of the beam-redirecting surface.
2 .2.1
T h eo r y
In order to achieve beam redirection, a surface must produce a reflection phase gradient
in a particular direction. This can be best understood by considering Fig. 2.4. In order
to have a reflected beam at an angle 0o for normal incidence, the y-component of the
reflected wave vector must have a magnitude given by ky = kr sin( 0 o), where kr = |kr|.
Note th a t throughout this text, bolded variables will refer to vectors, while unbolded
variables with the same name as bolded ones refer to the magnitude of the vector. The
desired phase gradient vector, ky, is achieved by varying the reflection phase across the
structure such th at (f)r(y) = —kvy, where 4>r(y) is the reflection phase at position y. This
re fle c tio n p h a s e is e q u iv a le n t to w h a t is p ro d u c e d by a m e ta l sh e e t tilte d a t 90/ 2 w ith
respect to the horizontal y-axis as depicted in Fig. 2.6; this equality will be discussed
further ahead. If the phase gradient is kept constant with frequency, the reflection angle
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C h a p t e r 2.
21
B ackground
Reflected
wave-front
h
T
Figure 2.5: Beam-redirection via varying index of refraction
will vary as
d<t>r {y)
dy
= —kr sin( 0o) => 60 = sin - l
d.</>r(y) '
dy
(2 .11)
This derivation was performed for normal incidence, but the concept can be extended to
off-normal incidence as well. Notice th a t the reflection angle, 60, can be controlled by
varying the phase gradient along the structure. Furthermore, this equation is quite similar
to (2.7), which was introduced in the array theory section. By letting d<t>^
= — in
d<j>r(y) \
( ^ ). The only differences between the two equations are their
derivation procedures and the definition of the angle 6. Specifically, in this derivation,
the reflection angle was determined by analyzing the reflected wave vector.
There are many different ways to achieve a phase gradient across a surface. For exam­
ple, a dielectric with a linearly varying index of refraction can lead to beam redirection.
Consider Fig. 2.5 for a normally incident plane wave. The reflection phase is given by
M y ) = —2krhn(y), where h is the height of the substrate and n(y) is the index of
refraction at position y. This formula holds when there are no surface reflections; this
requires the relative permittivity, er , and permeability, /ir , to be equal so th a t the wave
impedance, rj, of the substrate is equal to th a t of free space, rj0. Mathematically,
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22
B ackground
v =
\fW *
V Mo/
=
r)0 = 1207T.
The variation of both perm ittivity and permeability as a function of position in a structure
th a t responds to plane waves is difficult to implement. Recently, there has been extensive
amount of work in the field of isotropic 3D metamaterials, [20], [21], [22], [23] and
[24]. In theory, these materials support plane waves of any polarization, incident from
any angle, while perm itting independent control of the perm ittivity and permeability of
the medium, both of which can be made negative. These volumetric metamaterials, or
negative refractive index materials, may possibly lead to new beam redirection technology
in the near future. However, the current state of this technology does not permit its
application to beam-redirection as the fabrication process is complicated and costly.
When these materials become practical, the reflection angle for a normally incident plane
wave will be given by
(2 .12)
The reflection angle could then be controlled by varying the gradient of the index of
refraction.
As mentioned in Chapter 1 , reflectarrays have the ability to shape the reflected wave
to have a maximum in a particular direction. This direction is controlled by the reflection
phase difference between array elements, as discussed in Section 2 .1 . Most reflectarrays
in the literature mimic the characteristics of parabolic reflector antennas through ap­
propriate phasing of its array elements. However, this same concept can be used for
b e a m -re d ire c tio n . B e a m -re d ire c tio n c a n b e ach iev ed b y p h a s in g e a c h a rr a y e le m e n t such
that the overall reflection phase as a function of position is equal to 4>r{y) as given by
(2.11). For example, for the discrete surface shown in Fig. 2.4, the reflection phase func­
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C h a p t e r 2.
23
B ackground
tion can be estimated as (j>r{Yn) = ~ k yYn, where Yn = n A y for n € [1,2, ...N] and N
denotes the number of unit cells in the structure along the y-direction. A discrete phase
function results in an approximation of the phase gradient given by
( 2 .13)
dy
Ay
where A<f>ry is the phase difference between adjacent cells in the y-direction and A y is
the size of the unit cell in the y-direction, which will be referred to as d from now on.
To design a structure which yields a reflection angle of 90 for normal incidence, the
following procedure is applied:
= —kr sin(0o),
• Determine the required phase gradient
• Obtain the required progressive phase between cells,
A *, -
(2.14)
• Determine the number of cells required to span a
N
range in reflection phase,
= —
a <j>-
The first N y cells will have a unique reflection phase ranging from 0 —2-7Twith a cell-to-cell
phase difference of A (f>ry. The reflection phase of the first N y elements is given by
(j)rn = A 4>ry{n - 1) + 4>o, for n € [1,2,..., N y],
(2.15)
where (j)0 is a constant. The cells can then be arranged along the y-axis so as to achieve
a reflection phase gradient; this is achieved by setting
< !> r(Y n)
=
, for
<!>rn
71
€ [1, 2, ...,N y].
(2.16)
N o te t h a t th e to ta l n u m b e r o f cells, N , in th e s tr u c tu r e c a n b e ch o se n a rb itra rily . If TV <
N y then all the cells along the y axis will be unique in terms of reflection phase. However,
if N > N y, then additional cells will repeat the reflection phase pattern set by the first N y
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C h a pt e r 2.
24
B ackground
(a) Beam-redirecting surface.
(b) Equivalent tilted metal sheet.
Figure 2.6: Equivalence between beam-redirecting surface and tilted metal sheet.
cells in such a way as to preserve the linear phase gradient along the y-axis, i.e. the N y
cells will constitute one period. In other words, the pattern <j>r{Yn) = <pri,<pr2 , ••■,<PrNv is
repeated periodically to obtain the desired size along the y-axis. It should be pointed out
that one is not limited to a linear phase gradient along one direction; this methodology
can be extended to both x and y directions to obtain any phase profile.
As mentioned previously, a surface with a linear reflection phase along one direction is
equivalent to a tilted metal sheet at the design frequency f a. Note from Fig. 2.6 th a t both
the tilted metal sheet and the beam-redirecting surface have the same reflected ky vector.
Assuming they are both infinite in length, equivalent ky vectors render them equivalent
in terms of reflection angle at the design frequency. For the metallic sheet, the angle
of reflection remains constant for all frequencies. However, if the phase gradient,
is relatively constant, then there will exist some frequency dependence of the reflection
angle as implied by (2.11). This frequency variation is actually desirable for the intended
application of radar cross section manipulation; a frequency varying reflection angle leads
to a frequency varying RCS resulting in a structure which is less resemblant of a regular
reflecting surface.
2 .2 .2
L in ea rity and B a n d w id th
T h e lin e a rity of th e re fle c tio n p h a s e vs. fre q u e n c y c h a ra c te ris tic for a surface u n it cell is a
measure of the operational bandwidth for the overall beam-redirecting surface. Consider
the curves shown in Fig. 2.7(a). Each curve represents the reflection phase vs. frequency
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C h a p t e r 2.
25
B ackground
cell at NAy
O
cell at 2Ay
cell at Ay
fo
Frequency
Cell Position
Ay 2Ay
NAy
(a) Reflection phase vs. frequency characteristics
(b) Corresponding reflection phase vs. position,
for different surface unit cells along the y-axis.
4>r{y)> obtained by sampling traces from plot (a)
at frequency f a
Figure 2.7: Correlation between reflection phase vs. frequency and operation bandwidth
characteristic of each unit cell on the y-axis. The bottom most characteristic corresponds
to the cell located at y = Ay, the one immediately above to the cell at y = 2 A y, etc.
By sampling these phase characteristics at a particular frequency, f 0 in the figure, the
phase as a function of position, <pr(y), is obtained and is shown in Fig. 2.7(b).
As
mentioned previously, a tilted metal sheet is effectively emulated by keeping the slope of
4>r(y) constant along the y-axis at one particular frequency. Notice th a t if all reflection
phase characteristics in Fig. 2.7(a) are parallel and separated by a constant A <pry, then
4>r(y) will retain its slope regardless of the operational frequency, f Q. Thus, by having
a linear reflection phase vs. frequency characteristic, one can maintain linearity in the
4>r{y) function over a large frequency band, allowing the redirection of power away from
the specular angle.
It should be noted th a t instead of plotting the phase vs. frequency characteristic,
most published works in the area of reflectarrays plot the reflection phase vs. a control
parameter, the param eter th at is varied to achieve the required phase. For example, if the
re fle c ta rra y e le m e n t is a m ic ro s trip p a tc h , o n e m ig h t p lo t th e reflec tio n p h a s e vs. p a tc h
length. This type of plot provides a convenient means to assess the operation bandwidth
of the corresponding reflectarray. Specifically, high slope regions are associated with high
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C h a pt e r 2.
B ackground
26
phase errors since a small change in the control parameter, caused by manufacturing
errors, results in a large phase change; these phase errors limit the usable bandwidth.
On the other hand, a shallow slope will result in less phase error when the electrical size
of the control parameter varies with frequency, thus resulting in a higher bandwidth. A
more thorough discussion of this m atter is presented in [3].
To keep in line with previous published works, both the phase vs. frequency plot and
the phase vs. control parameter plot, where the control param eter is the aperture radius,
will be discussed.
2 .2 .3
P r o o f o f C o n ce p t S im u la tio n s
The equations developed thus far will be verified by considering full-wave simulations us­
ing Ansoft’s commercial software package, High Frequency Structure Simulator (HFSS).
HFSS is a finite element method simulator which solves directly for the field configura­
tion in the computational domain defined by the boundary conditions. It is an iterative
simulator which halts when the calculated error of the computed fields reaches a user
defined threshold. This error term is can be approximately given by
Errorn(x0, ya, z0) oc Fieldsn(xot yot zQ) - Fieldsn^ ( x 0, y0, z0),
(2.17)
where n represents the iteration number and (x 0,y 0, z 0) is a point at which the error is
calculated. The field solution converges when the change in Fields from one iteration to
the next is lower than the user specified threshold.
In order to perform the proof of concept simulations, a discrete surface, as shown in
Fig. 2.8, was drawn within HFSS. The structure is of finite length, L, along the y-axis,
while E-walls were placed along the x — 0, A x planes, to simulate periodicity along the
x -ax is; A x w as s e t to 5 m m for a ll sim u la tio n s. Since it was d e sire d to keep th e h e ig h t
of all cells in the structure constant for each simulation, the phasing across the structure
was achieved by emulating a substrate with a varying index of refraction as discussed in
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C h a pt e r 2.
27
B ackground
H
*----- ® E
AL
t
n = l n=2
h
n=N
dielectric
m etal-
Figure 2.8: HFSS proof of concept simulation setup
Section 2.2.1. The simulated structure in the figure had a linearly increasing index of
refraction given by
rii = i, , for i= [l,2 ,.. .,N],
(2.18)
where n.t is the index of refraction of cell i, and
G< = (J>n = rii , for i= [l,2 ,.. .,N],
so th a t
(2.19)
the surface ofeach cell is impedance matched to free-space, i.e. \J / i / e
= r]0,
and the wave can incur the proper phase. To set the angle of reflection, onefollows the
procedure outlined in Section 2.2.1, using a cell size of d = A L . Once the progressive
phase difference, A<j)ry, was determined from the desired reflection angle, the height of
the simulated structure of Fig. 2.8 was set to
A<f>ry = - 2 kah ^ h = 1 ~ ^
2fC0
•
(2.20)
This value of h, guaranteed th at for the value of n* given in (2.18), the desired linear
phase gradient is preserved.
The simulation configuration discussed thus far, resulted in the reflection phase vs.
cell position (y ) shown in Fig. 2.9. The characteristic in this figure comes about due to
the lin e a r progression in the index of refraction along the y-axis. The cell at y = AL, has
a height h, given by (2.20), such th at its reflection phase is A 4>ry < 0. Since the index of
refraction of the cell at y = A L is
= 2 , the reflection phase is twice th at of the first
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C h a pt e r 2.
B
28
ackground
AL
AL
2AL
NAL
Figure 2.9: HFSS proof of concept simulated phase gradient
cell. This phase progression continues until the last cell of the structure is reached. Note
that the gradient of the characteristic shown in Fig. 2.9 can be adjusted by varying the
inter-cell phase difference, A (f>ry, by changing the height of the simulated structure, or
by varying the cell dimension A L .
In order to change the reflection angle, the progressive phase difference, A<firy, was
varied. The simulated results of these variations are shown in Fig. 2.10. Notice th a t as
the total phase across the structure, (f>T — NA<j>rv, is varied, while keeping the number
of cells and the total length constant, the angle of reflection is also varied. This behavior
is expected, since variation in
< \> t
results in a changing phase gradient. From (2.13), the
gradient is approximately
d(pr(y)
dy
0T
L ’
(2 .21)
since A y — L / N and A<fiTy — <pT/N . It is then possible to use (2.11) and (2.21) to obtain
the theoretical reflection angle, which is also plotted in Fig. 2.10(b). Good agreement
between simulation and theory was obtained.
B y in c re a sin g th e n u m b e r o f cells, N , in th e s tr u c tu r e a n d k e ep in g th e o v erall le n g th
and phase gradient constant, Fig. 2.11 was obtained. Note th a t as long as the phase
gradient was fixed, the angle of reflection does not change, as seen in Fig. 2.11(b); this
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C h a p t e r 2.
29
B ackground
-•-S im u late d
-■•Theoretical
-3 0
r-so
20
-9 0
Total phase acro ss sheet, <t>T (rad/rc)
(a) Simulated normalized reflected power pat-
(b) Simulated reflection angle vs.
terns for varying phase gradient, tpr — NAtpT ,
across structure (j>r
total phase
L = 4A„, N = 20
Figure 2.10: Proof of concept simulation: Varying phase gradient
-3 0
-6 0
°
27
0.2
-9 0
Number of cells
(a) Simulated normalized reflected power pat-
(b) Simulated reflection angle vs. number of cells
terns for <j>{y) =
for the case in (a)
y, where L = 2A0
Figure 2.11: Proof of concept simulation: Varying number of cells N
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C h a p t e r 2.
30
B ackground
1
2
AL —H«- AL —H
r2
N- 1
•
►
" V -----------• —
-y
1*N-1
Figure 2.12: Equivalent linear array of structure in Fig. 2.8
agrees with the theory of ( 2 . 11 ).
The equivalent array factor of the structure shown in Fig. 2.8 can be used to compare
with HFSS simulations. To obtain the AF of the system simulated in HFSS, each cell in
Fig. 2.8 is considered to be an independent isotropic radiator, as shown in Fig. 2.12. The
reflection phase gradient of the simulated surface is achieved by setting the progressive
phase difference between array elements to A(j>ry. Note th a t all elements in the array
are excited with equal amplitude for a normally incident wave. In Chapter 5, it will
be shown th a t the variations in magnitude between array elements should be considered
when calculating the AF for the off-normal incidence case. For the configuration shown in
Fig. 2.12, it is straightforward to derive the array factor (AF) of the equivalent system as
discussed in Section 2.1. By using (2.10) and plotting only the AFy term with y'n — n A L
and Iyn
yn = e
, the AF can be expressed as
1
1 N
iY-~X
AFy{9, (P) = — ^ 2 e M k o A L
N 71=0
sin(0) s i n ( 0 ) - A ^ ) ;
where the excitation amplitude terms were set to 1, i.e. A n — 1. The 1 /N term normal­
ized the AF so its maximum is one. Since the pattern will only be plotted in the x = 0
plane, the angle <fi was set to 90°.
Now th a t the AF for the HFSS simulated structure has been determined, it is possible
to c o m p a re th e H F S S s im u la te d re s u lts w ith th e A F a p p ro a c h . I t is w o rth n o tin g t h a t in
deriving the AF it was implicitly assumed th a t zero mutual coupling was present between
the cells in Fig. 2.8. Furthermore, the element factor of each cell has been assumed to
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C h a pt e r 2.
31
B ackground
70
-3 0
r -50
40
- • - A F Approach
•■■HFSS Simulation
-9 0
Total phase acro ss sheet, 4>T (radIn)
(a) Normalized AF power patterns for y'n = nAL,
(b) Reflection angle vs. total phase across struc-
I Vn = e~inA,l>rv and 0 = 90°
ture 0 t
Figure 2.13: AF approach: Varying phase gradient
be isotropic, an assumption which is difficult to validate by any means. Keeping the
implicit assumptions in mind, it is worth comparing the reflected patterns predicted by
the AF. The reflected power pattern and reflection angle for the AF approach are shown
in Fig. 2.13. By comparing Fig. 2.10 and Fig. 2.13 it can be seen th a t the AF approach
matches with the HFSS simulation results. Note th at because the AF assumes that
the array elements are isotropic radiators, the reflected field in Fig. 2.13(a) does not
approach zero at endfire for the case when 4>t = 77t. It has been shown th at the AF
approach provides an alternative method to predict the reflected power patterns for a
beam-redirecting surface.
2 .2 .4
S p ecu lar R e fle ctio n S u p p ressio n
To assess the suppression of specular reflections, it is necessary to compare the reflected
powers of a reference m etal sheet with that of the beam-redirecting surface at the specular
angle. A m etal sheet will always reflect maximum power at the specular angle 9S. Figure
2.14(a) sh o w s the definition of the specular angle; note that in this diagram, and for the
rest of this work, 6 is measured from broadside and has positive values in the clockwise
direction. For an incident angle of 0*, the m aximum reflected power for a m etal sheet will
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C h a p t e r 2.
32
B ackground
■
(a) Metal sheet: 9S = —9i = 9r
(b) Beam-redirecting surface: 9a ^ 9r
Figure 2.14: Specular reflection of metal sheet and beam-redirecting surface
occur at 6S = —0j. The reflection phase gradient on the proposed beam-redirecting surface
reflects the incident plane wave away from the specular angle as shown in Fig. 2.14(b).
In order to quantify this suppression, a specular angle suppression (SAS) figure of merit
is hereby introduced to describe how well the beam-redirecting surface diverts energy
away from the specular angle and is given by
S A S = 101ogIO ( ^ a s f c g ) .
(2.23)
In other words, SAS is the ratio of the total reflected power off a reference metal plate,
Pmtotai >f° the total reflected power off the beam-redirecting surface, PStotal, at the specular
angle. For a useful comparison, it is required for the beam-redirecting surface and the
metal sheet to be of equal surface area. Note th a t a positive value of the SAS means
that, at the specular angle, the beam-redirecting surface reflects less power than the
reference metal sheet; the beam-redirecting surface is then said to suppress the specular
angle reflection.
2.3
P eriod ic Structures
Before considering the design of a beam-redirecting surface, it is of importance to review
the general theory of periodic structures since this theory will be employed in deriving
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C h a p t e r 2.
B ackground
33
the impedance of the proposed beam-redirecting unit cell. In this section, the Floquet
theorem will be discussed; this will be followed by a concise overview of Bloch impedance
and the termination of periodic structures. The major sources for this section include
[25], [26], [27] and [9].
2 .3 .1
F lo q u e t T h eo r em
The Floquet theorem will be demonstrated by considering wave propagation in a ID
structure th a t is periodic along the x-axis. Simply put, the theorem states th a t the field
configuration, ip(x), in a periodic structure with period p, has the following property
ip(x + p) =
(2.24)
where 7 is in general a complex number. In other words, the field configurations separated
by the structure’s period, p, are related by a complex multiplicative constant. This
relationship will become apparent in the following example.
Consider a structure th at is periodic in the x direction, in which propagation along
the axis of periodicity will be analyzed. In this medium, the field configuration 'ip(x)
must satisfy the wave equation
d2
ip(x) = 0 .
Note th at the propagation constant along the direction of periodicity, kx, must be a pe­
riodic function. If for example, the perm ittivity varied periodically along the x direction,
then kx would reflect the periodicity in permittivity.
The approach can be made more general by considering the following form of the
wave equation,
^ 2 + p (x )
where P ( x ) is a periodic function in x with period p. Now, since
d2ip(x + p) _ d2il>{x+p)
d{x + p)2
dx2
’
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(2.25)
C h a pt er 2.
34
B ackground
then
dx2
+ P{x + p) ip(x + p)
= 0, but P ( x + p) = P(x),
'jL + P(x) ip(x + p) — 0 .
dx 2
(2.26)
So if ip(x) is a solution to (2.25), then from (2.26) it can be seen th a t ip(x + p) is also
a solution. However, note th at in general ^ ( x ) ^ iJj {x + p) since 'ip(x) is not necessarily
periodic with period p.
The second order equation of (2.25) has only two linearly independent solutions, let
ipi(x) and ip2 (x ) represent these solutions. All other solutions are a linear combination
of the two independent solutions. Therefore,
ipi(x + p) = Aipi(x) + Bip2 (x),
ip2(:T + p)
(2.27)
= C'tpiix) + Dip2(x).
Note th a t for the case when 'ipi(x) is periodic then A = D = 1 and B = C = 0. Similarly,
the general solution to (2.25) can be written as
(2.28)
\l/(x) = U ^ x ) + Vtp2{x)-
Since 'F (t) represents a wave propagating along the x direction, the constants U and V
must be chosen such th a t
^ ( x + p ) = e~™y(x).
(2.29)
where 7 is in general a complex number. The constants U and V can be solved for by
using (2.27), (2.28), (2.29), and the fact th a t ip\(x) and ^ ( x ) are linearly independent.
The details of this solution can be found in [25]. The linear independence condition
between ipi(x) and ^ { x ) , results is a set of linear equations given by
A — e- i p
\ tv\
C
=
B
A ■
0.
-iv
\ v
1
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(2.30)
C h a pt e r 2.
35
B ackground
ln+l
—
Unit
cell
Unit
cell
►
n+1
Unit
cell
Figure 2.15: Periodic structure: Bloch impedance [9]
A non-trivial solution for the constants U and V requires the determinant of (2.30) to
be zero; this leads to
e- , p = (A + D ) ± ^ / ( A + D y - i ( A D - B C j
(2 ^
This solution can be utilized to find the amplitude ratio U/ V, which can then be used
to find the desired solution \h(a;). By finding a solution for T/ (a?) with the constraint of
(2.29) Floquet’s theorem has been illustrated.
2 .3 .2
B lo ch Im p ed a n ce
Transmission-lines are defined by their propagation constant and their characteristic
impedance. When discussing periodic structures, the characteristic impedance is referred
to as the Bloch impedance. Consider Fig. 2.15, the current and voltage at node n are
related to those at node n + 1 by the ABCDs-parameters of the unit cell such that
^
K r.
1
V
A
A
B
a
(2.32)
C D
\
*n+l
From the Floquet theorem, Section 2.3.1, it is known th a t the voltages at the two
nodes of interest are related by a complex propagation constant. Assuming propagation
in the positive x direction, this relationship is given by
( Vn+1 ^
e-7P
0
I,n + 1
0
e- 7 P
/
v„
Vn
\ !n J
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(2.33)
C h a p t e r 2.
36
B ackground
It is now possible to introduce the equation for the Bloch impedance, it is given by
ry
^n+ 1
Ab = 7 ---- •
■Ml+1
From (2.32) and (2.33) one can obtain [9]
(A - e~™)Vn+i + B I n+i = 0.
This leads to
A - e-T'P
The term e~lv can be solved for by using (2.32) and (2.33) and by assuming th at the
system in Fig. 2.15 is reciprocal (AD —BC=1), which by no means is a tight constraint.
The details of the derivation
are found in [9].Only
the final result is of relevance here
and bysolving for e~lv oneobtains
~ A - D = f y / ( A + D y - 4'
(2'34^
This result can be further simplified if the system of Fig. 2.15 is assumed to be symmet­
rical, for which A — D. W ith this assumption, (2.34) becomes
A
=
(2-35)
Note th a t Z g and Zg refer to positively and negatively traveling waves respectively.
2 .3 .3
T erm in a ted P er io d ic S tr u c tu r es
All real structures are of finite extent. For this reason, it is im portant to consider how
to properly term inate a finite periodic structure so as to prevent any reflections from
the load impedance. Consider Fig. 2.16, where a periodic structure of period p has been
term inated in a load impedance Z l ■ The voltage waves at node n are given by
K+ =
V0+e~
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C h a p t e r 2.
37
B ackground
In
lN
— o—
-------- o -------
Unit
cell
+
Unit
cell
vn
-------- o--------
-----
—o—
• • •
—
—
0
o-----
Unit
cell
+
VN
ZL
Figure 2.16: Termination of periodic structure [9]
where the + ( —) corresponds to a wave traveling in the positive(negative) x direction.
Now, the total voltage and current at node n can be expresses as
V„ =
T
—
ln
—
v+ + v - ,
V+
T+ 4- T~ — —2 - 4 -
Xn
'
n
ry+
“B
'
Vn
ry-
’
“B
However, at the load where n = N , the following must also hold
VN =
+
VZ
=
Zl In = ZL ( ^ r + ^ : ) .
Z+
B ' ZB
So, the reflection coefficient, V is given by
r = va
This expression can be simplified if the system in Fig. 2.16 is assumed to be symmetrical,
for which Z B = Z B = Z b - This assumption results in
Zi, - Z r
ZL + ZB ’
(2.36)
which is the familiar transmission line equation for the reflection coefficient. It can be
seen th a t by setting Z B = Z B) the reflection coefficient can be set to zero, effectively
emulating an infinite periodic structure.
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C hapter 3
A nalysis of th e P roposed
B eam -redirecting U nit Cell
The required background for designing a beam-redirecting surface was presented in Chap­
ter 2. This chapter describes in detail the proposed beam-redirecting surface (BRS) unit
cell th a t was used to design a beam-redirecting surface. The full BRS unit cell will be pre­
sented first, followed by the analysis of its components. Specifically, the circular aperture
employed in the BRS unit cell will be characterized; this is followed by the development
of an elementary beam-redirecting surface (EBRS) unit cell. The EBRS unit cells are
then used to create the proposed beam-redirecting unit cell. Finally, the reflection phase
properties of the proposed cell will be examined.
3.1
P rop osed B R S U n it C ell
As mentioned in Section 2 .2 .1 , a unit cell th a t can achieve a reflection phase variation,
or phase agility, of 360° is required in order to design a beam-redirecting surface. This
re q u ire m e n t allow s th e su rfa c e to b e d e sig n e d fo r any re fle ctio n an g le.
The proposed
beam-redirecting surface unit cell is shown in Fig. 3.1(a). As shown, the proposed sur­
face unit cell consists of three metallic layers. Two of these layers have printed circular
38
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C h a pt e r 3.
A n a l y s is
of the
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C ell
39
Incident W ave
&
PiLh, Z TL
2pT L h, Z tl
pTLh , Z -TL
(a) Proposed BRS unit cell: d = 5
(b) Equivalent circuit: (3t
m m , er = 10, h = 75 mil.
Z t l = \/?-
l
=
Figure 3.1: Proposed BRS unit cell and its equivalent circuit
aperture patterns while the third layer is a ground plane. The metallic layers are sepa­
rated by dielectric layers. The top most dielectric layer is necessary to eliminate surface
reflections and allows the incident wave to penetrate into the structure and thus incur
a phase change within the cell. By individually varying the radii of the apertures, it is
possible to vary the impedance of each layer and thus the overall reflection phase.
The equivalent circuit of the proposed unit cell is shown in Fig. 3.1(b). This equivalent
circuit has two sets of parallel L-C tanks th a t model the top and bottom circular apertures
of Fig. 3.1(a). The transmission-line (TL) sections separating the parallel L-C tanks
model the dielectric slabs th at separate the apertures. The characteristic impedance, Z t l ,
and propagation constant, (3t
l ,
of the TL sections are determined from the constitutive
parameters of the dielectric slabs and are given by
Z tl
(3.1)
Ptl
(3.2)
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C h a p te r 3.
A n a ly s is o f t h e P r o p o s e d B e a m - r e d ir e c t in g U n it C e l l
z„
40
z„
O------------- 1-------------o
I-
Zo
*—a —*•l
- d
-I
Cross sectional view
5L
Z„
O------------- '------------- o
Ton \
(a) Inductively loaded rectangular waveg-
(b) Equivalent circuit of (a)
uide
Figure 3.2: Inductively loaded rectangular waveguide and equivalent circuit [10]
3.2 Circular
A pertu re E quivalent C ircuit
In order to accurately analyze the proposed BRS unit cell
shownin Fig. 3.1(a), it is
instructive to first consider a single inductive sheet of circular apertures.
Inductive loading in waveguides was studied extensively by Markcuvitz and docu­
mented in his Waveguide Handbook [10]. In this work, he noticed th a t the equivalent
circuit of a zero-thickness aperture, within a rectangular waveguide, th a t “cuts” the
magnetic field, as shown in Fig. 3.2(a), resulted in an inductive shunt impedance as il­
lustrated in Fig. 3.2(b). The inductive nature of the aperture can be better understood
by considering what happens on the surface of the metal when an electromagnetic wave
impinges on it. The boundary condition governing the magnetic field as it impinges on
the metallic aperture is given by [28]
h x (Hj - H m) = J s,
(3.3)
where n is a normal unit vector perpendicular to the surface of the aperture, H; is the
incidentmagnetic field vector,H m is the magnetic field
vector just inside the
comprising the aperture and J s is the induced current density vector
metal
on the aperture
surface. If the conductor comprising the aperture is assumed to be perfectly conducting,
then H m = 0 and (3.3) becomes
n x Hj = J s.
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(3.4)
C h a p t e r 3.
A n a l y s is
of the
P
r o po sed
B
e a m - r e d ir e c t in g
U
n it
C ell
41
Input
A.
k® l l
(a) Inductive Mesh
(b) Equivalent circuit of (a)
Figure 3.3: Inductive mesh and equivalent circuit [11]
This equation implies th a t the tangential component of the incident magnetic field, Hi,
will induce a surface current density on the surface of the aperture. The flow of current on
the aperture surface generates its own magnetic field, Hg. The magnetic flux generated
by Hg is then linked by the current on the aperture, thus resulting in the shunt inductance
as shown in Fig. 3.2(b). A similar analysis can be performed when the aperture “cuts”
the electric field instead, this results in a capacitive shunt impedance.
The work of Marcuvitz was further developed by Ulrich in his study of inductive
meshes. The inductive mesh considered by Ulrich in [11] is shown in Fig. 3.3(a). Ulrich
was able to determine the equivalent circuit of a single inductive unit cell in an infinite
inductive sheet. This equivalent circuit consisted of a parallel L-C tank as shown in
Fig. 3.3(b). The capacitance is a product of the incident electric field being “cut” as it
crosses the mesh. The term “inductive mesh” comes about from the equivalent impedance
of the L-C tank at low frequencies. The equivalent shunt impedance of the parallel L-C
tank is given by
7
_
eq
i t, >T.
3“>L
1 -uPLC'
(3.5)
By letting the radian frequency, a;, approach zero, it can be seen th a t the equivalent
impedance is given by
%eq ~ jwL,
(3.6)
and is inductive in nature. The equivalent impedance remains inductive until the tank
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C h a p te r 3.
A n a ly s is
of
t h e P r o p o s e d B e a m -r e d ir e c tin g U n it C e l l
42
Port 1
er
-'port
X
Surface
/ S a
?S,
d ■
Jport
Port 2
OS,
(a) Single circular aperture
(b) Equivalent circuit of (a)
Figure 3.4: Single circular aperture
reaches its resonance at u = ^ = , after which the impedance becomes capacitive. Note
th a t in the equivalent circuit of Fig. 3.3(b), the resistor th a t models the losses in the
mesh has been left out. The ultimate goal of the proposed beam-redirecting surface is to
reduce specular reflections and thus the amount of loss in the mesh is not of paramount
importance in this work.
Ulrich provided a starting point in determining the equivalent circuit of a single
circular apertures in an infinite array of apertures. However, in order to extract values of
inductance and capacitance, it was required to conduct two-port S-parameter simulations
of the circular aperture in HFSS. Figure 3.4(a) shows the single circular aperture that
was used as a unit cell to model an infinite inductive sheet. The equivalent circuit for
the single circular aperture, in an infinite inductive sheet, is shown in Fig. 3.4(b). In the
simulations, E walls were placed on the x = 0, d planes and H walls on the y = 0,d planes
to simulate an infinite periodic array of apertures for normal incidence, thus effectively
leading to an equivalent circuit extraction for normal incidence. Upon simulation, the
S-parameters were de-embedded by a length of Lport towards the metallic surface, Sa,
in o rd e r to c h a ra c te riz e th e a p e r tu r e itself. H ence, th e e q u iv a le n t c irc u it is v a lid a t th e
aperture surface Sa. From the two-port S-parameter simulations, the Z-parameters were
extracted and the equivalent T-model as shown in Fig. 3.5 was employed to determine
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C h a p t e r 3.
A
n a l y s is o f t h e
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C ell
43
Z „-Z :
Figure 3.5: T-model for reciprocal two-port network [9]
the values of inductance and capacitance as a function of aperture radius. Specifically,
the inductive unit cell of Fig. 3.4(a) was simulated at a low frequency of 0.5 G H z in
vacuum. From the extracted Z 2 1 , the shunt inductance can be extracted by equating
Z eq = Z 21 in (3.6) and solving for L to obtain
L =
( 3 .7 )
The use of (3.7) is justified since at 0.5 G H z the effect of the shunt capacitance can be
ignored.
Similarly, the capacitance was extracted at a high frequency of 15 G H z with the
inductive sheet immersed in a dielectric of er . The inclusion of the dielectric constant
was necessary since the shunt capacitance varies with the dielectric perm ittivity whereas
the inductance does not. This is also the reason behind the extraction procedure for the
inductance; by simulating the aperture in vacuum, er = 1, the capacitance is minimized
and the accuracy of the extracted inductance value increases. Capacitance extraction at
a high frequency aids in increasing the value of the shunt capacitance and thus results
in a more accurate extraction. In order to extract the value of capacitance, one equates
Z eq = Z 21 in (3.5) and solve for C to obtain
^
u 2L
uImag(Z2i)
N o te t h a t th e in d u c ta n c e v alu e, L , m u s t b e e x tr a c te d p rio r to o b ta in in g a value for the
aperture capacitance. Unless otherwise stated, the extracted value for L at er — 1 and
0.5 G H z will be used to obtain the value of C.
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C h apter 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
44
1.5
c 1.5
< 0.5
£ 0.5
1.5
A perture Radius (mm)
2.5
1.5
2
A perture R adius (mm)
2.5
(a) Extracted inductance for varying permittivity (b) Extracted inductance for varying cell dimen­
sion
Figure 3.6: Circular aperture extracted inductance vs. aperture radius
The extracted inductance as a function of aperture radius is shown in Fig. 3.6. Note
from Fig. 3.6(a) th a t as the permittivity varies, i.e. varying index of refraction, the
aperture inductance remains relatively constant as expected. Furthermore, it can be
seen from Fig. 3.6(b) th a t by increasing the size of the cell dimension, d, the maximum
aperture inductance is also increased. This is expected since the inductance of a line is
proportional to its length; in a similar manner, the larger cell dimension requires currents
to travel a longer distance, thus resulting in higher inductance per unit cell value. Also,
for all values of aperture radius, R, the extracted inductance for the d — 6 m m cell is
lower than those obtained for the d = 5 m m case. This result is intuitive since inductance
is inversely proportional to conductor width; narrow strips result in higher inductance.
For the circular aperture case, larger radius values lead to current having to flow along
narrower metallic strips.
Extracted values of capacitance are shown in Fig. 3.7. Note th at capacitance increases
as aperture radius decreases. This behavior was expected; it is known th a t the capaci­
ta n c e o f a sim p le p a ra lle l p la te c a p a c ito r is in v ersely p ro p o rtio n a l to th e d is ta n c e b e tw e e n
the plates. It was also observed th a t an increased value in the perm ittivity resulted in
higher extracted aperture capacitance as seen in the first two traces of Fig. 3.7(a) (as
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C h a p t e r 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
45
- » - £ rs 10,ds5nram
- ■ -e r*4,d“ 5mm
2.5
5/2x(eMtdg5mm curve)
1.5
2 0.5
1.5
A perture Radius (mm)
2.5
2.5
A perture Radius (mm)
(a) Extracted capacitance for varying permittivity (b) Extracted capacitance for varying cell dimen­
sion
Figure 3.7: Circular aperture extracted capacitance vs. aperture radius
per the legend). The relationship between extracted capacitance and refractive index is
given by [29]
f t - ( £ ) ’ c..
(3.9)
where the refractive index n* corresponds to the extracted capacitance C\. Note th at by
taking the extracted capacitance for the er = 4 case from Fig. 3.7(a), and multiplying it
by
= § one obtains the third curve in Fig. 3.7(a). Notice th at for R > 1.3 mm,
the adjusted er = 4 curve closely followed the er = 10 capacitance curve; thus confirming
(3.9) for the large aperture radii. The deviation between the adjusted er = 4 curve, via
(3.9), and the er = 10 curve at low aperture radii can be attributed to the small values
of extracted inductance; any magnitude change in inductance at these low values would
result in a large chance in aperture capacitance due to their inverse relationship. Figure
3.7(b) shows th a t the capacitance increases for increasing cell dimension. This comes
about due to the fact th a t the extracted capacitance value is inversely proportional to
the extracted inductance value as seen in (3.8), and because the aperture inductance for
la rg e r cell d im e n sio n s is low er.
The equivalent circuit of Fig. 3.4(b) was used to determine the resonance frequency of
the aperture. Figure 3.8 shows the S-parameters for a two-port simulation of the periodic
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C h a p t e r 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
46
-10
Region of Interest
S ., FW
-50
-60
er=10
d=5mm
R=2.45mm
-70.
Frequency (GHz)
Figure 3.8: Two-port S-parameters for circular aperture of Fig. 3.4
inductive sheet of Fig. 3.4(a) and the corresponding plot for the equivalent circuit for
R = 2.45 m m. Note th at the resonance of the aperture lies away from our region of
interest, which is within the range of 8 —12 GHz. This is predicted by both the equivalent
circuit and the full-wave simulations. The discrepancy in the resonance frequency is due
to the fact the equivalent circuit was derived for operation within the frequency band of
interest. Furthermore, in the region of interest the aperture has an inductive reactance
and the equivalent circuit closely matches the full-wave simulation. By operating away
from the resonance, the inductive reactance can be used over a wide frequency band,
while possibly reducing losses due to conduction currents. From this simulation it was
verified th a t the equivalent circuit of Fig. 3.4(b), along with the param eter extractions,
a c c u ra te ly m o d e ls th e c irc u la r a p e r tu r e in th e fre q u e n c y b a n d o f in te re s t. F u rth e rm o re ,
the equivalent circuit would help in designing a BRS unit cell with non-resonant circular
apertures.
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C h a p t e r 3.
A
n a l y s is o f t h e
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C ell
47
; PtlI1, Z tl
(a) EBRS unit cell
(b) Equivalent circuit:
Z t l
(3t
l
=
wyTJt,
= -AF
Figure 3.9: Elementary beam-redirecting surface (EBRS) unit cell
3.3
E lem entary B eam -red irecting Surface (E B R S) U n it
C ell
A linear reflection phase vs. frequency characteristic can be achieved by adding a dielec­
tric layer of finite height h above and below the circular aperture unit cell of Fig. 3.4(a),
effectively creating an elementary beam-redirecting surface (EBRS) unit cell as shown in
Fig. 3.9(a) with an equivalent circuit as show in Fig. 3.9(b). The extra dielectric layers
help match the surface of the structure to free space, allowing a normally incident wave
to incur phase inside the EBRS unit cell. By matching the surface of the two-port EBRS
unit cell, a bandpass filter is effectively created since the matching condition can only be
maintained over a finite bandwidth, ft will be shown th a t this bandpass filter exhibits a
linear transmission phase as a function of frequency. This linear phase will be exploited
by short-circuiting one end of the two-port EBRS unit cell, and converting the linear
transmission phase to a linear reflection phase. In this manner, a linear reflection phase
c a n b e u se d in th e d e sig n of a b e a m -re d ire c tin g su rfa ce . T h e n b y v a ry in g the aperture
radius, and therefore the inductance and capacitance of the aperture, the reflection phase
can be controlled. This variation in aperture radius will, in general, result in a surface
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C h a p t e r 3.
A
n a l y s is o f t h e
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C ell
48
TL
• • •
Y(R)
n+l
• • •
\/
EBRS Unit Cell
Figure 3.10: Modified EBRS equivalent circuit for Bloch impedance derivation
impedance mismatch leading to a non-linear reflection phase. However, this non-linearity
can be avoided by utilizing multiple EBRS unit cells and short-circuiting one end of the
resulting structure. By stacking multiple EBRS on top of one another, larger reflection
phase variation can be achieved. Furthermore, by varying the radii of certain cells, the
linear reflection phase can be preserved.
3 .3 .1
Su rface Im p ed a n c e M a tc h in g
To achieve a large reflection phase variation, a requirement in the design a beamredirecting surface, it is necessary to stack multiple EBRS cells. Therefore, in order
to determine the height, h, and the dielectric constant, er , of the extra dielectric slabs,
such th a t the surface is impedance matched to free space, the Bloch impedance of the
EBRS unit cell equivalent circuit must be derived. In this derivation, an infinite periodic
structure, with the unit cell shown in Fig. 3.10, is analyzed. This unit cell is a modified
version of the EBRS cell equivalent circuit of Fig. 3.9(b). The capacitor and inductor
have been grouped into the shunt adm ittance term Y(R) , which is a function of aperture
ra d iu s R a n d is g iv en b y
vim
nR) -
- 1)
1
z^ r
) - —
z m
—
'
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(3 0)
C h a pt e r 3.
A n a l y s is
of the
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C e ll
49
where L ( R ) and C(R) are the aperture inductance and capacitance respectively; both
of which are a function of the aperture radius. As discussed in Chapter 2, the voltage
and current at node n and n + 1 are related by the ABCD parameters of the periodic
cell, i.e. the EBRS cell, as in (2.32). By solving for the ABCD parameters, the Bloch
impedance can be determined using (2.35). The ABCD parameters for the modified
equivalent circuit is given by
C D
V
where
cos(0)
J Z tl sin(0)
I
0
j Y TL sin(0)
cos{6)
Y(R)
1
Q =
Pt l
=
Z tl
=
V
cos(8)
j Z Ti sin(0)
j Y TL sin(0)
cos(6)
PrLh,
(3.11)
(3.12)
1
YnT L
(3.13)
Carrying out the m atrix multiplication results in
A
= D = cos2(0) - Z t l Ytl sin2(0) + j Y (R ) Z Tl sin(0) cos(0),
B
= 2j Z tl sin(0) cos(0) —Z ^ LY {R ) sin2(0),
C = 2j Y TL sin(0) cos(0) + Y ( R ) cos2(6).
The above can be simplified by substituting the following:
Z tlY tl
=
1,
sin(20)
=
2sin(0) cos(0),
cos(20) =
cos2(0) —sin2(0).
Upon substitution, the ABCD param eter of the EBRS unit cell are simplified to
A
= D = cos(20) + ^-Y(R)Z tl sin(20),
z
(3.14)
B
= j Z TL sin(20) —Z t LY ( R ) sin2(0),
(3.15)
C
= JY tl sin(20) + Y ( R ) cos2(0).
(3.16)
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C h a pt e r 3.
A
n a l y s is o f t h e
P
B
r o po sed
— Theoretical
-- -F u ll-w a v e (FW)
o Equivalent Circuit (EC)
e a m - r e d ir e c t in g
U n it C ell
50
Theoretical
Full-w ave (FW)
Equivalent Circuit (EC)
3.5
2.5
o
£
0.5
F requency (GHz)
F requency (GHz)
(b) Imaginary part of the Bloch impedance
(a) Real part of the Bloch impedance
Figure 3.11: Bloch impedance of EBRS unit cell
400
300
-10
200
3
a. -100
2*-30
-40
100
S11
- - -S21
-* -S 1 1
- ♦ -S21
FW
FW
EC
EC
S11 FW
- - -S21 FW
-300 — S11 EC
- * -S21 EC
-400,
-200
F requency (GHz)
Frequency (GHz)
(a) Two-port S-parameter magnitude
(b) Two-port S-parameter phase
Figure 3.12: Two-port S-parameter Simulation Results for the EBRS unit cell
Note th a t since A = D the EBRS equivalent circuit of Fig. 3.10 is symmetric. Further
since A D — B C = 1, the details of which are in the Appendix, the equivalent circuit
is also reciprocal. By satisfying these conditions, the Bloch impedance is calculated via
(2.35) and is given by
Z B± =
± (JZ
tl
sin(20) - Z*LY ( R ) sin2(0))
(3.17)
\ J (cos(20) + §Y { R ) Z tl sin(20))2 - l ’
where again the + ( —) corresponds to a positive (negative) traveling wave.
Figure 3.11 shows the free space normalized Bloch impedance, i.e.
Vo
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Note that
C h a p t e r 3.
A
n a l y s is o p t h e
P
ro po sed
B
e a m - r e d ir e c t in g
U
n it
C ell
51
in the X-band, which corresponds to the range between 8 — 12 G H z , the real part of
the Bloch impedance is relatively close to one, indicating a good surface impedance
matching. Furthermore, since the imaginary part of the Bloch impedance in this region
is zero, it is known th a t the EBRS unit cell is operating in a passband. It can also
be seen from Fig. 3.11, th at the theoretical Bloch impedance given by (3.17) closely
matches the extracted Bloch impedance of a two-port S-parameter HFSS simulation of
one EBRS unit cell. The close correlation between full-wave simulations and theory
implies th a t it is possible to use (3.17) directly for design purposes, even for only one
EBRS cell th at is term inated in free space. In order to validate the equivalent circuit
of Fig. 3.9(b) and the derived Bloch impedance of (3.17), the two-port S-parameters of
the EBRS unit cell were simulated in HFSS while the equivalent circuit was simulated in
Agilent’s Advanced Design System (ADS). The two-port simulation results are shown in
Fig. 3.12. Note firstly th a t the full-wave (FW) simulation results closely match those of
the equivalent circuit (EC); this effectively validates the EBRS equivalent circuit shown in
Fig. 3.9(b). Secondly, the passband predicted via an S-parameter simulation is equivalent
to th a t predicted by the Bloch impedance; thus validating the Bloch impedance equation
of (3.17). Finally, note th a t the transmission phase is linear; this fact will be exploited by
adding a ground plate at one end of the EBRS unit cell of Fig. 3.9(a). The addition of a
ground plate effectively creates a short-circuit, allowing the two-port linear transmission
phase to be converted to a one-port linear reflection phase.
The exploitation of the linear transmission phase via the addition of a ground plate
can be better appreciated by considering the reflection phase of the grounded EBRS unit
cell shown in Fig. 3.13. In order to prevent reflections within a finite periodic structure, its
load impedance must equal its Bloch impedance. It was shown th a t a linear transmission
p h a s e w as o b ta in e d b y v a ry in g th e s u b s tr a te p a r a m e te rs su c h t h a t th e Bloch impedance
of the EBRS cell is equal to th at of free space. By adding a ground plate at one end of the
EBRS cell, the cell is no longer term inated in its Bloch impedance and one might expect
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C h a p t e r 3.
A n a l y s is
of the
P
r o po sed
B
e a m - r e d ir e c t in g
U
n it
C ell
52
P t iX Z T L
PiLh, Z'T L
(a) Grounded EBRS unit cell
(b) Equivalent circuit of (a)
Figure 3.13: Grounded EBRS unit cell
Free Space (r|0)
Free Space (r|0)
/
/
/
£r
£r
Metal
/
/
/
Plane o f
Symmetry
£r
/
Analogous
to
/
/
y>
£r
£'
.•T T T -^
)
’f
___ 1
i
/
k l ______________K
/ Image
/
Free Space (r|0)
Figure 3.14: Analogy between one-port grounded EBRS cell and two-port system of two
stacked EBRS cells
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C h a pter 3.
A n a l y s is
of the
P
B
r o po sed
e a m - r e d ir e c t in g
U
n it
C ell
53
200
d=5mm
h»1.905mm
3
100
50
'■§ -50 - - -R=0.9mm FW
R=2.04mm FW
& -100 - - R=2.45mm FW
-■-R «0.9m m EC
-®- R*2.04mm EC
R=2.45mm EC
- 200,
Frequency (GHz)
(a) Reflection phase vs. frequency
«
ttv
-5 0
-100
-150
f*9GHz FW
MOGHz FW
M 1G H z FW
M G H z EC
MOGHz EC
M 1 G H z EC
1.5
2
A perture Radius (mm)
2.5
(b) Reflection phase vs. aperture radius
Figure 3.15: Reflection phase properties of grounded EBRS unit cell
surface reflections to dominate. However, via the image theory in [28], termination in
a metallic ground plate results in an analogy between the one-port grounded EBRS cell
and the two-port structure consisting of two stacked EBRS cells as shown in Fig. 3.14;
the metal ground plate effectively acts as a “mirror” , thus resulting in two stacked EBRS
cells. Note th at because Z B = rj0, as was previously designed, and because the image
is also term inated in the free space impedance, r]a, an incident wave impinging on the
grounded EBRS cell effectively sees a two-cell periodic structure th a t is term inated in
Z l — Z B = r)0. Therefore, an incident wave impinging on a single grounded EBRS unit
cell does not reflect at the surface, instead it fully penetrates the cell, reflects from the
ground plate and reaches free space without any reflections occurring at the air-dielectric
interface.
The above argument is valid for an EBRS cell which has its Bloch impedance equal
to the free space impedance. This can only occur at some values of aperture radii. For
all other values of aperture radii, some surface reflection will be present. This surface
im p e d a n c e m is m a tc h r e s u lts in a d e g r a d a tio n o f th e reflec tio n p h a s e vs. fre q u e n c y c h a ra c ­
teristic as shown in Fig. 3.15(a). Note th a t the most linear reflection phase vs. frequency
is obtained for a radius R = 2.04 m m , which corresponds to the case when Z B = rj0.
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C h a pt e r 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U n it C e ll
54
As the aperture radius diverges from this value, the linear region of the reflection phase
reduces, thus resulting in a loss in bandwidth. It can also be seen from this plot that
the reflection phase obtained via the equivalent circuit closely matched th a t obtained via
HFSS; note th at the inductance and capacitance of the equivalent circuit were extracted
from HFSS simulations of the circular aperture as mentioned in Section 3.2. From the
reflection phase vs. aperture radius characteristic shown in Fig. 3.15(b), it can be seen
th a t a maximum phase variation of only 180° is obtained at 10 GHz. This small phase
variation is in general insufficient in the design of a beam-redirecting surface. In order
to obtain the desired 360° phase variation, stacking of EBRS unit cells was considered.
3 .3 .2
M u ltip le S tack ed E B R S C ells
By stacking multiple EBRS unit cells, a larger phase variation can be achieved since the
radii of more than one EBRS cell can be varied. However, it was previously mentioned
that the variation in aperture radius results in a change in Bloch impedance th at leads
to a non-linear reflection phase vs. frequency characteristic. Nevertheless, it is possible
to maintain a linear reflection phase vs. frequency characteristic by varying the radii of
the bottom most EBRS cells, while keeping the top EBRS cell radius fixed such th a t its
surface impedance is matched to free space.
Consider Fig. 3.16, which shows the variation of the normalized Bloch impedance as
a function of aperture radius for an EBRS unit cell. This figure was obtained via the
Bloch impedance equation given in (3.17); Y ( R ) was obtained from the full-wave HFSS
simulations of the circular aperture as presented in Section 3.2. The resulting figure is
reliable since the extracted aperture admittance, Y (R ) , has already been validated. Note
that approximately at R = 2.04 m m and for 10 GHz, the normalized Bloch impedance
is e q u a l to the free sp a c e w ave im p e d a n c e ; this v alu e of aperture radius results in no
surface reflections and a linear reflection phase. Furthermore, note th a t as the radius
tends to zero, the EBRS unit cell enters a bandstop or cutoff region, since the imaginary
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C h apter 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U
C ell
n it
55
— f"9GHz
— MOGHz
- ■ M 1G H z
---f* 9 G H z
— M OGHz
M 1G H z
o
tt
at
n
E
d=5mm
h*1.905mm
e=10
d=5mm
h=1. 905mm
E=10
1.5
A perture Radius (mm)
2.5
(a) Real part of Z b
1.5
A perture R adius (mm)
2.5
(b) Imaginary part of Z b
Figure 3.16: Normalized Bloch impedance vs. aperture radius
Free Space (r\0)
Free Space (t]0)
/
EBRS
Cell 1*
Er
£r
/
/
EBRS
Cell 2<
/
/
/
er
/
Cr
Surface Acts
■like Metal
Sheet
_
/
f
/
/
_________
(ZB > 0 as Rj —^ 0)
Sr
/
/
F ig u re 3.17: T w o sta c k e d E B R S cells a n d re flec tio n p h a s e lin e a rity
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
/
C h a pt e r 3.
A
n a l y s is o f t h e
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
56
part of the Bloch impedance is non-zero; in this region the real part of Z b vanishes,
much like a metallic sheet, and all incident power is reflected. This behavior is expected
since as the aperture radius tends to zero, the aperture inductance tends to zero, Y ( R )
approaches infinity and because the Bloch impedance equation of (3.17) has a Y 2(R) term
in the denominator, the Bloch impedance also tends to zero. As the Bloch impedance
approaches zero, the interface at which it was defined begins to act as a short, i.e. a
metallic sheet. If two EBRS cells are stacked, as shown in Fig. 3.17, and the aperture
of the bottom most cell is gradually “closed” , then via the image theory, one obtains a
similar analogy as was depicted in Fig. 3.14. As a result, from the arguments in Section
3.3.1 and by varying the aperture of the bottom most EBRS cell, while keeping the
top cell matched to free space, the linear reflection phase as a function of frequency is
preserved. This approach of preserving the reflection phase linearity, while achieving the
desired reflection phase variation, can be generalized for N b b r s stacked EBRS unit cells:
• Begin with the top EBRS cell surface matched to free space
• The bottom
N
e b r s
~
1 are set to their maximum aperture value (Note from Fig. 3.16
th a t at maximum aperture radius, R = 2.45, Z B is still relatively matched)
• Starting from the bottom most EBRS cell, gradually close the aperture until further
reductions in radius result in minute phase variations
• Continue “closing” the bottom most EBRS cells th at remain open until the required
phase variation is achieved
As mentioned previously, a phase variation of only 180° at 10 G H z can be obtained
for a single grounded EBRS cell. This phase variation is in general insufficient for the
d e sig n o f a b e a m -re d ire c tin g su rfac e, sin ce it is d e sire d to have a reflection phase variation
of 360° as discussed in Section 3.1. It was just concluded th at by stacking an extra EBRS
cells, it is possible to preserve the reflection phase linearity. However, this comes at the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a p t e r
3.
A n a l y s is
o f
t h e
P r o p o s e d
B e a m - r e d ir e c t i n g
U n it
C e l l
57
Table 3.1: Aperture radii configurations (in m m ) for Fig. 3.18
Cell Number
1
2
3
4
5
6
Ri
2.04
2.04
2.04
1.79
1.38
0.97
i?2
2.45
2.09
1.55
0.97
0.97
0.97
expense of extra materials which increases both the cost and weight of the structure. As
mentioned in Chapter 1, it is desired to keep both the cost and weight at a minimum.
Nevertheless, it is the goal of this project to design and build a beam-redirecting surface,
so the required phase variation must be achieved. This is possible by stacking only
two EBRS cells together and varying both radii to achieve the desired reflection phase;
variation of the top radius results in some non-linearities in the reflection phase. Note
also th a t as stated in [3], greater linearity, and therefore wider bandwidth, becomes
possible by using thicker substrates. However, thicker substrates would also inevitably
result in increased weight and cost. In order to minimize the cost and obtain the required
reflection phase, our proposed beam-redirecting surface (BRS) unit cell consists of two
stacked EBRS cells.
3.4
R eflection P h ase C haracteristics o f P rop osed B R S
U n it C ell
The reflection phase properties of the proposed BRS unit cell, which is composed of two
stacked EBRS cells with a ground at one end, are shown in Fig. 3.18. In this figure, the
radii are varied as summarized in Table 3.1. Note th a t by varying both radii, a total phase
variation of
= 355° was obtained. Furthermore, as expected for the cases when the
to p E B R S cell ra d iu s , R i , is c o n s ta n t, th e reflec tio n p h a se re m a in s lin e a rly proportional to
frequency over the desired bandwidth. As seen in Fig. 3.18, variations of the top aperture
radius introduce some non-linearities in the reflection phase vs. frequency characteristic,
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a pt e r 3.
A
n a l y s is o f t h e
P
r o po sed
B
e a m - r e d ir e c t in g
U n it C ell
58
100
f -200
s -300
£ -400
-500
Cell 1
- - -C e ll 2
Cell 3
- - Cell 4
Cell 5
- - Cell 6
d=5mm
h=1,905mm
e=10
-600.
Frequency (GHz)
Figure 3.18: Reflection phase vs. frequency characteristics for proposed EBRS unit cell
and the radii configuration of Table 3.1
with the advantage of achieving the higher phase variation required to design a beamredirecting surface. These non-linearities are expected since, as mentioned previously,
the surface of the EBRS unit cell is no longer impedance matched. However, since it
was the goal of this project to develop a beam-redirecting surface with minimal cost and
weight, some linearity, and therefore bandwidth, had to be sacrificed.
The reflection phase response of the BRS unit cell equivalent of Fig. 3.1(b) was also
considered. Figure 3.19 shows the reflection phase for both the equivalent circuit and
the full-wave simulation. It can be seen th at the equivalent circuit exactly predicts the
full-wave simulated reflection phase, thus validating the proposed equivalent circuit. This
close correlation was evident for all radii configurations of Table 3.1, but only one example
is shown for brevity.
It is also worthwhile to study the reflection phase vs. aperture radius characteristic
for the p ro p o s e d BRS u n it cell o f Fig. 3 .1 (a ). The v a ria tio n in the reflection phase as a
function of the top aperture radius is shown in Fig. 3.20(a). From Fig. 3.20(a), it was
observed th a t variations of the top aperture radius resulted in a high slope region between
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a pt e r 3.
A n a l y s is
of the
P
r o po sed
B
e a m - r e d ir e c t in g
U
n it
C ell
59
200
Full-wave
- • - Equivalent Circuit
100
-100
-200
-5 -300
£ -400
-500
d=5mm
h=1.905mm
er =10
R =R =2.04mm
-600.
Frequency (GHz)
Figure 3.19: Reflection phase vs. frequency: Proposed BRS unit cell equivalent circuit
validation
150
d»5mm
h*1.905mm
er»10
R *2.04mm
100
•3
50
4) 0
W
<0
£
c
o
-50
* -100
£ -150
-200
-250
- - -f*9.5GHz FW
M OGHz FW
- - M 0.5GHZ FW
-•-f*9.5G H z EC
-•-M O G H z EC
*«- M 0.5GHz EC
1
1.5
2
Top A perture Radius, R1 (mm)
- -f=9.5GHz FW
— M OGHz FW
■ • f«10.5GHz FW
• -M .5G H Z EC
■♦-MOGHz EC
♦ - M 0.5GHz EC
2.5
1
1.5
2
Bottom A perture Radius, R2 (mm)
2.5
(a) Reflection phase vs. top radius, R i, for R-z = (b) Reflection phase vs. bottom radius, Rz- for
0.97 m m .
R i = 2.04 rnm.
Figure 3.20: Reflection phase vs. aperture radius for proposed BRS unit cell of Fig. 3.1(a)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h apter 3.
A n a l y s is
of the
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
60
1.5mm < R < 2mm; this is associated with high phase errors and a narrow bandwidth as
discussed in Section 2.2.2. However, in Fig. 3.20(b) it can be seen th at by keeping the top
radius constant, such th at no surface reflections are present, the reflection phase remains
linear for a larger range of aperture radii while its slope is reduced. This shallower slope
assists in enhancing the bandwidth by reducing phase sensitivity. Finally, note the close
correlation between the equivalent circuit of Fig. 3.1(b) and the full-wave simulations.
If cost and weight are not an issue, extra EBRS cells can be stacked to achieve either
better reflection phase linearity or smaller cell dimension d. The former was previously
discussed. Since the stacking of an extra EBRS unit cell results in an increased reflection
phase variation, one can tradeoff some of the available phase variation for a smaller cell
dimension d. The advantage of a smaller cell size d is evident by carefully analyzing the
design procedure described in Section 2.2.1. By making d small, one can accommodate
for larger phase gradients, which are necessary to deflect the beam at large angles away
from broadside for normally incident plane waves. Specifically, it can be seen from (2.14)
that the inter-cell phase difference, A<firy, is equal to the product of the phase gradient,
d^ y V>, and the cell dimension, d. If the desired phase gradient results in a large and
impractical A<j)ry, then the cell dimension can be decreased to achieve a smaller inter-cell
phase difference; thus allowing a high-gradient beam-redirecting surface to be designed.
From Section 3.2 note th a t if d is reduced, the aperture radius is reduced, and the range
of achievable inductances is reduced; thus resulting in a lower reflection phase variation.
Of course, by reducing the phase variations available to each EBRS cell, one is forced to
again vary the top most layer to achieve the necessary phase.
A variation in reflection phase greater than 360°, a non-resonant aperture and a centre
frequency of 10 G H z were achieved by proper selection of substrate height, number of
sta c k e d EBRS u n it cells a n d u n it cell d im e n sio n d. T h ro u g h th e equations developed in
this chapter and intuition, the final beam-redirecting surface parameters were obtained
as shown in Fig. 3.1(a). Specifically, er = 10, d = 5 m m, h = 1.905 m m and Ri = R 2 =
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a pt e r 3.
A n a l y s is
of the
P
r o po sed
B e a m - r e d ir e c t in g U
n it
C ell
61
2.04 mm; the value of R corresponds to the aperture radius of an impedance matched
EBRS surface at 10 GHz. Furthermore, it was shown in this chapter th a t the developed
equivalent circuit of Fig. 3.1(b) was able to properly predict the reflection phase of the
proposed BRS unit cell; thus allowing its use in the design of a beam-redirecting surface.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C hapter 4
D esign and Fabrication o f a
B eam -redirecting Surface
The previous chapter discussed the proposed beam-redirecting surface (BRS) unit cell
in detail, from its composition to its reflection phase properties. This chapter will focus
on the design of a beam-redirecting surface using the proposed BRS unit cell developed
in Chapter 3 and the theory outlined in Chapter 2; this surface was designed to have a
reflection angle of —30° off broadside for a normally incident plane wave. This is followed
by a detailed description of the fabrication considerations, procedure and sources of error.
4.1
D esign
In order to achieve a high operational bandwidth, the reflection phase vs. aperture radius
linearity was maximized for each of the BRS unit cells comprising the beam-redirecting
surface. This requires minimizing variations of the top layer radii as discussed in Chapter
3. It was mentioned in Section 3.3.2 th a t a linear phase is maintained by varying the
radius of the bottom aperture while keeping the top one fixed. Therefore, in designing
the surface it was desired to vary the bottom aperture radii as much as possible, while
keeping the top radii constant. This approach ensured th at no surface reflections were
62
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h apter 4.
D e s ig n
and
F a b r ic a t io n
of a
B
e a m - r e d ir e c t in g
Surface
63
Nd
Nd
Figure 4.1: Proposed beam redirecting surface
I
0+
e0=-3o°!,
z
k X ^ l ki
t
1
1
1
.... _. 1
,.
1
1
1
. 1
,
1
1
1
!1
er
£r
£r
I*- ti­
Phase Gradient (pr(y)=(7t/30)y
Figure 4.2: Desired reflection angle for normal incidence
present and resulted in a linear reflection phase vs. frequency characteristic for a large
range of bottom layer radii. The aperture radii of the top layer were only varied to
achieve the required phase variation. Using this methodology, the steps from Section
2.2.1 and the BRS unit cell equivalent circuit of Fig. 3.1(b), a beam-redirecting surface
with a reflection angle of —30° off broadside, for a normally incident plane wave, was
designed at 10 GHz. A sketch of the proposed beam-redirecting surface is shown in
Fig. 4.1.
4 .1 .1
In ter -ce ll P h a se D ifferen ce a n d N u m b e r o f B R S U n it C ells
The equations presented in Section 2.2.1 can be directly used to determine the required
reflection phase difference between adjacent BRS unit cells. It was chosen to design a
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h apter 4.
D e s ig n
and
F a b r ic a t io n
of a
B
e a m - r e d ir e c t in g
S urface
64
beam-redirecting surface having a reflection angle of 60 = —30° for a normally incident
plane wave as shown in Fig. 4.2; note th a t as in Chapter 2, the angle 6 has been defined
to be positive in the clockwise direction from broadside. A —30° reflection angle was
sufficiently far from broadside to demonstrate beam-redirection and facilitated the design
procedure with a BRS unit cell dimension of d = 5 mm.
In order to achieve the desired reflection angle, a linear reflection phase gradient
was implemented along the y-direction as shown in Fig. 4.2; the phase gradient was
designed at the centre frequency of 10 GHz. From (2.11), the required phase gradient
was calculated as
=
dy
d4r{y)
dy
- k r sin(60),
-27t
sin(—30°),
A10G H z
^
=
^ ra d /m m .
30
dy
(4.1)
v '
The term AioGHz corresponds to the free space wavelength at the frequency of 10 GHz.
Using (2.14) and the above value for the phase gradient, the inter-cell phase difference
was calculated to be
d<f)r ( y )
&<Pry
=
—2 y
M rv =
d,
^ rad = 30°,
(4.2)
where d = 5 m m was used. The number of cells with unique reflection phases, ranging
from 0 —27r, is given by
N>
=
i t r 12-
(4 3 )
It was now possible to design a beam redirecting surface of arbitrary size via equations
(2.15) and (2.16). H ow ever, it w as first necessary to determine the B R S unit cell radii
values for each of the twelve distinct cells such th a t the inter-cell phase difference equalled
M r y = 30°.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a pt e r 4.
D
e s ig n a n d
F a b r ic a t io n
B
of a
e a m - r e d ir e c t in g
Surface
65
Table 4.1: Reflection phase and aperture radii configurations derived using BRS equiva­
lent circuit at 10 G H z
4 .1 .2
Eqv. cct. Reflection
Phase Error
Cell
Target Reflection
Ri
R2
No.
Phase 4>target (deg)
(mm)
(mm.)
1
-270
2.05
2.45
-267.6
-2.4
2
-240
2.05
2.3
-243.9
3.9
3
-210
2.05
2.1
-216.1
6.1
4
-180
2.05
1.85
-182.7
2.7
5
-150
2.05
1.55
-154.2
4.2
6
-120
2.05
0.95
-127.5
7.5
7
-90
1.95
0.95
-97.9
7.9
8
-60
1.85
0.95
-61.6
1.6
9
-30
1.8
0.95
-39.9
9.9
10
0
1.7
0.95
-4.2
4.2
11
30
1.6
0.95
23.3
6.7
12
60
1.4
0.95
56.7
3.3
Phase
(deg)
fic c t
~ ^target
fy c c t,
(deg)
B R S U n it C ells R a d ii an d L inear R e fle c tio n P h a se G ra­
d ien t
The equivalent circuit of the proposed BRS unit cell, shown in Fig. 3.1(b), along with the
extracted aperture inductance and capacitance, were utilized to approximate the aperture
radii, for each of the N y = 12 BRS unit cells, such th a t the inter-cell phase difference was
A<pry = 30°. The first step in this approximation was to determine the upper and lower
reflection phase limits of the proposed BRS unit cell. From the simulations performed
in Chapter 3, it was determined th at the lower reflection phase limit at 10 G H z was
—270° (for Ri = 2.04 m m and i?2 = 2.45 mm).
This lower limit was taken as the
re fe re n ce p h a s e v a lu e fro m w h ich th e reflec tio n p h a se s of th e re m a in in g 11 BRS unit
cells were determined. Note th at the chosen reference radii configurations resulted in a
linear reflection phase vs. frequency characteristic, since the top radius of Ri = 2.04 m m
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h apter 4.
D e s ig n
and
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
66
Table 4.2: Reflection phase and aperture radii configurations obtained via HFSS simula­
tions at 10 G H z
Cell
Target Reflection
Ri
R'2
HFSS Reflection
No.
Phase 4>targe.t (deg)
(mm)
(mm)
Phase <j>hfaH (deg)
1
-270
2.04
2.45
-270.4
0.4
2
-240
2.04
2.27
-239.4
-0.6
3
-210
2.04
2.09
-210.6
0.6
4
-180
2.04
1.85
-179.1
-0.9
5
-150
2.04
1.55
-149.7
-0.3
6
-120
2.04
0.97
-119.2
-0.8
7
-90
1.95
0.965
-90.1
0.1
8
-60
1.861
0.965
-59.0
-1.0
9
-30
1.79
0.965
-31.5
1.5
10
0
1.7
0.965
0.0
0.0
11
30
1.58
0.965
31.0
-1.0
12
60
1.38
0.965
59.5
0.5
8
9
10
Frequency (GHz)
(a) Reflection phase vs.
simulations
11
12
8
Phase Error
~ ^target
9
$hfss (deg)
10
Frequency (GHz)
11
12
frequency for full-wave (b) Reflection phase vs. frequency for equivalent
circuit simulations
Figure 4.3: Reflection phase vs. frequency characteristics for N y — 12 BRS unit cells
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h apter 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B
e a m - r e d ir e c t in g
Surface
67
100
-50
Jl -100
-150
9.5GHz FW
- * - 10GHz FW
- 4 - 10.5GHz FW
-•-9 .5 G H z EC
-■ -10G H z EC
-♦-10.5G H z EC
« -200
-250
-300
-350,
4d
6d
8c
Cell position along y-axis
10d
Figure 4.4: Reflection phase gradient along y-direction
results in no surface reflections at the air-dielectric boundary. The design procedure set
out in Section 3.3.2 was then utilized to design the reflection phase of the remaining BRS
unit cells. From the reference radii configuration, the bottom radius, R 2 , was gradually
closed to achieve the required inter-cell phase difference. After the 6th cell, reductions in
i ?2 did not provide sufficient phase variation to obtain the required reflection phase of
the 7th BRS unit cell. As a result, it was required to vary the top radius, i?i, in order to
complete the design for the remaining cells.
Approximate aperture radii values were derived via the equivalent circuit of Fig. 3.1(b).
Full-wave HFSS simulations were then used to fine tune the reflection phase to achieve
an error of less than ±2°. The summary of the radii configurations for the equivalent
circuit and full-wave simulations, along with the target reflection phase are summarized
in Tables 4.1 and 4.2; R i and _R2 correspond to the aperture radii of the top and bottom
layer respectively. The reflection phase vs. frequency characteristics of all twelve BRS
unit cells in Tables 4.1 and 4.2 are plotted in Fig. 4.3(b) and 4.3(a) respectively. Note
the close correlation between the full-wave simulated reflection phase and th a t obtained
via the equivalent circuit. It can be seen from both figures th a t the phase vs. frequency
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a p t e r 4.
D e s ig n
and
Fa b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
TE Polarization
H
E
68
TM Polarization
H
z
71
Cell k------------------------------------------------------------------------------------------------ 1
number
1
2
3
4
5
6
2
------------------------- * - Lbrs = Nyd = 2 A.ioghz = 6 cm
8
9
10
11
12
--------------------------------
■*
Figure 4.5: HFSS simulation setup for 12 BRS cell beam-redirecting surface
traces for cells 1 —6 are the most linear and correspond to the case when the top surface
is matched. These twelve BRS unit cells are then arranged, utilizing (2.16), so as to pro­
duce a linear phase profile along the y-axis. Specifically, cell number 1 is located between
0 < y < d, cell number 2 between d < y < 2d, cell number 3 between 2d < y < 3d, etc.
After the proper arrangement of cells, the reflection phase as a function of position was
obtained by sampling all twelve curves in Fig. 4.3(a) and 4.3(b) at 9.5 G H z, 10 G H z and
10.5 GHz-, the resulting phase profile is shown in Fig. 4.4. As mentioned in the previous
chapter, F W corresponds to the full-wave simulated results, while E C corresponds to
the equivalent circuit results. Note th a t even across a 10% bandwidth around 10 G H z,
a fairly linear reflection phase characteristic was obtained. Furthermore, as outlined in
Section 2.2.1, the slope of the lines in Fig. 4.4, along with the frequency of operation,
determine the angle of reflection.
4 .1 .3
S em i-in fin ite B e a m -r e d ir e c tin g Su rface S im u la tio n s
Some pre-fabrication simulations were required to assess the performance of the proposed
beam-redirecting surface. Thus, as shown in Fig. 4.5, HFSS was used to simulate a beam
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a pt e r 4.
D e s ig n
and
F a b r ic a t io n
op a
B
e a m - r e d ir e c t in g
Surface
69
redirecting surface with the twelve unique BRS unit cells placed along the y-direction so
as to create the desired linear phase progression. Since a linear phase gradient only along
the y-direction is desired, the inter-cell phase difference along the x-direction is set to zero.
Therefore, the reflection phase profile along the y-direction was maintained for all values
of x by placing E-walls(H-walls) at the x = 0, d planes for an incident TE(TM ) linearly
polarized plane wave, effectively simulating an infinite periodic structure along the x-axis.
A TE(TM ) polarized wave is defined as having its electric (magnetic) field parallel to the
reflecting surface upon incidence, as shown in Fig. 4.5. All simulated incident waves were
TEM in nature with wave vectors lying in the x = 0 plane; however, it was necessary to
distinguish between the incident polarizations in order to demonstrate the polarization
independence of the proposed structure. Note th a t the simulated structure is finite along
the y-direction, having a length equal to L b r s — N yd — 2Aioghz = 6 cm. Radiation
boundaries were setup around the simulated structure to absorb the reflected wave; in
HFSS, the radiation boundaries are implemented via absorbing boundary conditions.
The reflected wave, which was absorbed by the radiation boundaries, was used to plot
reflected power patterns as a function of the angle 6. Since the reflected wave vector lies
on the x = 0 plane, the reflected power patterns in the x = 0(0 = 90°) plane were then
plotted for further analysis.
In the previous section, the reflection phase gradient was designed by individually
simulating each of the twelve BRS unit cells in an infinite array(due to the boundary
conditions). However, this provides a somewhat idealized picture of the reflection gradi­
ent along the surface, since it ignores mutual coupling between dissimilar adjacent BRS
unit cells. Figure 4.6 plots the reflection phase of the semi-infinite structure for normal
incidence and th at obtained via individual simulations of each BRS unit cell at 10 GHz.
T h e re fle c tio n p h a s e of th e se m i-in fin ite b e a m -re d ire c tin g surface was obtained by sam­
pling the phase of the reflected electric field along a line placed above the surface. Note
that mutual coupling between the cells does result in some non-linearities in the reflec-
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
C h a p t e r 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B
e a m - r e d ir e c t in g
S urface
70
100
-Individual BRS cell Sim
-All 12 BRS ce lls Sim
o>
Q>
3
-5 0
</>
J2 -1 0 0
Q.
| -1 5 0
■J
0 -2 0 0
0
DC
-2 5 0
**-
-300'
-3 5 0
2d
4d
6d
8d
Cell position along y -a x is
10d
Figure 4.6: Reflection phase gradient along y-direction at 10 G H z
tion phase. However, the overall linearity of the curve is maintained and close correlation
between the two is observed.
Both TE and TM incident polarizations were simulated at various angles of incidence
in order to show the polarization independence of the proposed beam-redirecting surface.
Figure 4.7 shows the normalized reflected power patterns for a broadside incident plane
wave. The pattern maximum corresponds to the reflection angle. Note that, for both
incident polarizations, the simulated reflection angle is close to the theoretical value
of 0o = —30° predicted by (2.11). Discrepancies in the simulated reflection angle are
expected since the developed theory assumes an infinite beam-redirecting surface, while
simulations dealt with only a semi-infinite structure. Another possible cause for the
discrepancies in the reflected patterns is the non-linear reflection phase of the semi­
infinite structure as seen in Fig. 4.6. Notice th a t both incident polarizations result in
similar reflected patterns; thus validating the polarization independence of the proposed
beam-redirecting structure. Furthermore, it can be seen th at the reflected power pattern
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C h apter 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
71
-3 0
0.8
0.6
-6 0
0.4
0.2
-9 0
- - - IE
-----TM
—- A F
Figure 4.7: Normalized reflected power patterns for normal i n c i d e n c e = 0°)
predicted via the AF closely follows the other two simulated patterns. The reflected
power pattern for the AF approach was obtained by plotting \AF(9)\2, where AF{9) was
derived from the AFy(9, <j>) term in (2.10) by setting y'n = (n — l)d and <fi = 90°,
N
AF(9) =
I nej{n~l)kod BtaW+* » .
(4.4)
n= 1
The excitation term I n was set to the HFSS simulated reflection coefficient, both magni­
tude and phase, of each individual BRS unit cell comprising the beam-redirecting surface
in question. The extra phase term, cf>in, corresponds to the implicit phasing of the BRS
unit cells due to the angle of incidence, 9i, and is given by
4>in = ( n - 1)k0d sin(0j),
(4.5)
w h e re 0, is m e a s u re d fro m b ro a d sid e a n d is p o sitiv e in th e clockw ise direction. Note th at
for normal incidence, 9 is set to zero. Considering the lossless case, it is known th at the
magnitude of the BRS unit cell reflection coefficient is equal to one. The excitation term
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C h a p t e r 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
72
-3 0
-30
0.8
-60
0.4
-9 0
-90
(a) Incident TE polarization
(b) Incident TM polarization
Figure 4.8: Normalized reflected power patterns for off-normal incidence
is then given by
In = ei<t>hfss„(/))
(4.6)
where <j>hfssn{f) represents the HFSS simulated reflection phase, see Fig. 4.3(a), for a BRS
unit cell number n. Note th at the reflection phase of the BRS unit cells is a function of
frequency.
Off-normal incidence was also considered, the simulation results are shown in Fig. 4.8.
Again notice th at both polarization result in similar reflected patterns. The reflection
angles of these patterns are summarized in Fig. 4.9. From this figure, the close correlation
between both incident polarizations and the AF predicted reflection angle curves is readily
seen; effectively validating the theory of Chapter 2.
The potential of the proposed beam-redirecting to perform specular angle suppression
(SAS) was also considered (see Section 2.2.4). The definition of the SAS value, given by
(2.23), is the ratio of the total reflected power from a metal reference sheet to the total
reflected power from the test surface. As a result, a reference metal sheet having equal
su rfa c e a r e a to th e s im u la te d b e a m -re d ire c tin g su rfac e o f F ig . 4.5 w as also s im u la te d
in HFSS. The normalized reflected power, at the specular angle, for both the reference
metal sheet and the proposed beam-redirecting surface are shown in Fig. 4.10. The
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C h apter 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
-20
o>
-3 0
-4 0
-5 0
-•-T E
— -TM
^ AF
-1 0
Incidence A ngle (deg)
Figure 4.9: Reflection angle vs. incidence angle
Metal
-■ -B R S 9 G H z
-«- BRS 10GHz
— BRS 11GHz
§ -20
-1 0
Specular A ngle 6s (deg)
Figure 4.10: Normalized reflection power at the specular angle
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C h a p t e r 4.
D
e s ig n a n d
F a b r ic a t io n
op a
B e a m - r e d ir e c t in g S u r f a c e
74
< 20
- • - B R S 9GHz
-■ -B R S 10GHz
-«- BRS 11GHz
w 15
-1 0
S pecular A ngle 0s (deg)
Figure 4.11: Simulated specular angle suppression
plotted values were normalized to the maximum reflected specular power; this value
corresponded to the power reflected by the metallic sheet at normal incidence. Note
th a t only one frequency curve (10 G H z ) is plotted for the metal sheet; this is the case
because all frequencies resulted in the same normalized curve. Furthermore, the drop in
specularly reflected power for the metal sheet away from broadside is due to the variation
in the effective area “seen” by the incident plane wave; it is in other words the cos(6)
loss. Note also th a t the normalized reflected power for the beam-redirecting surface at
the centre frequency of 10 G H z was the lowest. From Fig. 4.10, the SAS was computed
and is plotted in Fig. 4.11. From this figure it can be seen th a t over a 20% bandwidth
around 10 G Hz, the simulated SAS is higher than 5 dB] at 10 G H z the suppression is
higher than 15 dB. A maximum specular reflection suppression of 22.2 dB occurs at the
d e sig n fre q u e n c y at w h ic h most of the reflected power is away from specular angle, as
desired. Maximum specular angle suppression occurs at 6S — 0°. This maximum value
is due to a frequency independent null in the reflected pattern of the beam-redirecting
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C h apter 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
75
surface at 6 = 0°. It is worth noting th a t the SAS value is dependant on the angular
location of the reflected sidelobes and nulls, as well as their absolute power values.
From the aforementioned simulations, it was concluded th a t the proposed BRS unit
cell could be used in the design of a fully functional beam-redirecting surface.
4.2
4 .2 .1
Fabrication
B e a m -r ed ire ctin g Su rface S ize C o n sid era tio n s
The twelve BRS unit cells described in the previous section can be used to make a
beam-redirecting surface of arbitrary size. The goal of the experimental aspect of this
project was primarily to show th at a beam-redirecting surface, based on stacked circular
apertures, could be realized. In determining the total number of BRS unit cells, and
therefore the total size of the final beam-redirecting surface, it was required to consider
the fabrication and experimental procedures.
The size of the final beam-redirecting surface would be limited by the fabrication
procedure chosen. The in-house precision milling machine perm itted the fabrication of
a larger surface when compared to the chemical etching facilities. The major advantage
of using the milling machine, over chemical etching, was the ability to accurately align
the aperture patterns on opposite sides of a substrate. It was therefore decided to fab­
ricate the proposed beam-redirecting surface via the in-house precision milling machine.
The alignment ability of the milling machine was exploited in the fabrication procedure
discussed in the following section. Note th a t although the milling machine permits the
fabrication of a larger surface, too large a structure would result in hours worth of milling,
requiring frequent replacement of aged milling bits; this increases the fabrication error
sin ce n o tw o m illin g b its h a v e th e sa m e tr a c e o u tlin e . T h is so u rc e o f error w as considered
when determining the total size of the beam-redirecting surface.
The experimental configuration, shown in Fig. 4.12, also helped determine the final
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C h apter 4.
D e s ig n
and
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
76
lR X
Rx
T X
Tx
Figure 4.12: Chosen Experimental Configuration [6]
beam-redirecting surface dimensions. The inner workings of this configuration will be
discussed in the following chapter. For now, only the configuration is of interest. In par­
ticular, the distance between the transm itter(Tx) horn and the beam-redirecting surface
had to be long enough so th at the beam-redirecting surface was in the far-field region of
the transm itter. Furthermore, the receiving(Rx) horn had to be in the far-field region
of the beam-redirecting surface. The far-field placement constraint guaranteed th at the
incident and measured waves were plane-wave in nature. One final constraint on the size
of the surface to be fabricated, was the dimensions of the in house anechoic chamber.
The testing horns and the beam-redirecting surface had to all fit in the chamber while
remaining in the far-field of each other.
Considering all the aforementioned constraints, it was decided to design a beamredirecting surface consisting of N = 17 BRS unit cells in both the x and y directions.
An N x N beam-redirecting surface, with N = 17 and d = 5 m m results in a total surface
area of 8.5 x 8.5 cm2.
Only twelve BRS unit cells with a unique reflection phase are required to design
a su rfa c e w ith a re fle c tio n an g le o f —30° for a n o rm a lly in c id e n t p la n e w ave.
I t w as
determined th a t N = 17 BRS unit cells would be used in both dimensions. This means
th a t in order to keep the linear reflection phase along the y-axis, the last five cells along
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C h a p t e r 4.
D e s ig n
and
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
77
Table 4.3: Beam-redirecting surface reflection phase profile
0 < y < d d < y < 2d
0<x <d
(f)ri
07'2
d < x < 2d
<f>ri
0T2
16d < x < 17d
(f>ri
07-2
11d < y < 12d
12d < y < I'M
16d < y < 17d
^7-12
4>ri
firs
(f)Tl
$T\2
4*r\
the y-direction will repeat the pattern set by the first five y-directed cells. Since it was
desired to create a linear phase gradient only along the y-direction, the inter-cell phase
difference along the x-direction was set to zero. If we define <t>rn to be the reflection phase
for cell number n, as outlined in Tables 4.1, 4.2 and 4.4, then the phase profile of the
beam redirecting surface can be summarized as shown in Table 4.3.
4 .2 .2
F a b rica tio n M e th o d and M a te ria l S e le c tio n
It was decided in Chapter 3 th at the dielectric perm ittivity of the substrate should be 10.
This value allows the proposed BRS unit cell to have a centre frequency of 10 G H z while
having a small cell size of d = 5 m m . It is possible to reduce the required permittivity
value at the expense of a larger cell dimension. By reducing the permittivity, the effective
aperture capacitance is reduced, thus increasing the operational frequency of the BRS
unit cell. This can be offset by increasing the cell dimension, since as seen in Section 3.2,
the aperture capacitance tends to increase with increasing d.
The desired substrate thickness of h = 1.905 m m (75 m il), is in general considered
quite thick and therefore is hard to come by, especially at high dielectric permittivity
values. Note th at the middle dielectric section of the proposed BRS unit cell shown in
Fig. 3.1(a), has a thickness of 2h = 150 mil. Since high dielectric substrates with this
la rg e v a lu e o f th ic k n e s s a re n o t re a d ily av a ila b le, it w as decided to glu e to g e th e r two
sections of 75 m il substrate to create the 150 m il thick middle dielectric section.
The metallization thickness on the substrate would ideally be as thin as possible;
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C h a p t e r 4.
D e s ig n
and
Fa b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
Front View
A
75 mil
A.
4F
r v f y? Y fg n fr
B
C
78
I
_ _L
150 mil
&
D
~ r
75 mil
I
D
Figure 4.13: Beam redirecting surface fabrication procedure
this guarantees maximum inductance values. However, from full-wave HFSS simulations
it was concluded th a t the extracted inductance for a zero thickness metal layer, a half
ounce(17.5 fim ) metallization layer and a one ounce(35 fim) metallization layer differed
only slightly. It was therefore determined th at the metallization thickness was not of
major concern. From the above considerations, a 75 mil Taconic CER-10 substrate with
one ounce copper was used in the fabrication process. This substrate is low loss, having
a loss tangent of only 0.0035 at 10 GHz.
The fabrication procedure utilized in realizing the beam-redirecting surface can be
broken down into six steps (see accompanying Fig. 4.13):
1. Generate a mask file using Agilent ADS th at contains the desired printed circular
aperture patterns for the top and bottom layers.
2. Cut the 75 m il substrate into four (Nd)x(Nd) sections (A,B,C,D), where N is the
total number of BRS unit cells comprising the surface.
3. Remove one full side of copper laminate on sections B, Cand D and both sides of
copper laminate on section A via chemical etching.
4. Glue sections B and C together with the copper layers facing outwards to create a
150 m il substrate with copper cladding on both sides.
5. Use the precision milling machine to create the pattern from the generated mask
on both sides of 150 mil substrate.
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C h a pter 4.
D e s ig n
and
Fa b r ic a t io n
of a
B
e a m - r e d ir e c t in g
Surface
79
Table 4.4: Final reflection phase and aperture radii configurations at 10 G H z
Phase Error
Cell
Target Reflection
Ri
R2
Simulated Reflection
No.
Phase (ptarget (deg)
(mm)
(mm.)
Phase (f>r (deg)
1
-290
2.04
2.45
-291.2
1.2
2
-260
2.04
2.3
-259.6
-0.4
3
-230
2.04
2.15
-230.0
0.0
4
-200
2.04
1.985
-199.8
-0.2
5
-170
2.04
1.77
-170.0
0.0
6
-140
2.04
1.42
-139.5
-0.5
7
-110
1.99
0.97
-109.9
-0.1
8
-80
1.895
0.97
-80.6
0.6
9
-50
1.803
0.97
-50.8
0.8
10
-20
1.705
0.97
-19.1
-0.9
11
10
1.583
0.97
10.8
-0.8
12
40
1.4
0.97
39.7
0.3
target
4*r
(deg)
6. Glue section A on the top of the 150 m il section and section D on the bottom.
In order to glue the different sections as outlined in the above procedure, it was required
to find a high permittivity, er = 10, glue with a thin finish. However, since a high per­
m ittivity glue was not available for purchase, a low perm ittivity prepreg(a thin sheet
of glue) and a high pressure, high tem perature compressor were used to bond the di­
electrics together. The selected prepreg was the Rogers RO4450B, which has a dielectric
perm ittivity of 3.54 at 10 G H z and 23° C and dries to a nominal thickness of 100 urn.
The inclusion of the low permittivity dielectric resulted in some reflection phase errors.
It was therefore required to re-simulate the twelve BRS unit cells, in order to minimize
the phase errors. In this final simulation, all non-idealities, including dielectric loss and
metallization thickness and loss were included. The final aperture radii configurations
a n d r e s u lta n t full-w ave s im u la te d re fle c tio n p h a s e for th e tw elv e u n iq u e BRS u n it cells
are summarized in Table 4.4. The reflection phase vs. frequency characteristics and the
reflection phase vs. position are shown in Fig 4.14. Note th a t even with the inclusion of
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C h apter 4.
D
e s ig n a n d
F a b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
80
-50
o -100
a. -200
Cell 1
Cell 2
Cell 3
Cell 4
♦ -Cell 5
Cell
Cell 7
Cell
Cell 9
Cell 10
Cell 11
Cell 12
-150
£ -200
-250
-300
10
Frequency (GHz)
(a) Reflection phase vs. frequency
0
2d
4d
6d
8d
Cell position along y-ax is
10d
(b) Reflection phase vs. position
Figure 4.14: Final reflection phase characteristics for the twelve distinct BRS unit cells
the prepreg, the reflection phase vs. frequency characteristics remain quite linear. The
reflection phase vs. position curve still enjoys a 10% linearity bandwidth around 10 G Hz.
The losses in each of the twelve proposed B R S unit cells differs from cell to cell
since the aperture radii also vary from cell to cell. This cell to cell loss variation results
in a varying reflection coefficient magnitude for the twelve B R S unit cells, effectively
creating an amplitude taper along the direction of the phase gradient. The magnitude
of the reflection coefficient vs. frequency is plotted in Fig. 4.15. Note th at within a 10%
bandwidth around 10 G H z , the maximum loss (—S n {dB)) is around 0.5 dB. Figure 4.16
shows the magnitude of the reflection coefficient as a function of position y\ recall that
each cell position corresponds to a B R S unit cell with a unique radii configuration. Note
that the B R S cell with the highest reflection magnitude corresponds to the B R S unit cell
with the largest excitation amplitude. Furthermore, notice th a t the losses create a nonu n ifo rm a m p litu d e ta p e r a lo n g th e y -d ire c tio n . As o u tlin e d in S e c tio n 2.1, th is amplitude
taper contributes to the shape of the overall reflected pattern. The loss analysis performed
thus far was utilized to derive a reflected pattern via an AF in order to compare with
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C h apter 4.
D e s ig n
and
Fa b r ic a t io n
of a
B e a m - r e d ir e c t in g S u r f a c e
81
-0 .5
-1
m
■a
V)
-1 .5
-2
-2.5.
- - Cell
-• -C e ll
- a -C ell
-n- Cell
—• -C e ll
-■ -C e ll
-o- Cell
-— Cell
-• -C e ll
• -* -Cell
3
4
5
6
7
8
9
10
11
12
10
Frequency (GHz)
in e
12
11
Figure 4.15: Reflection coefficient magnitude vs. frequency for N y = 12 BRS unit cells
00
"O
-
0.1
-
0.2
-0.
-0 .4
cO
-0.5.
-
0.6
9GHz
10GHz
11GHz
-0 .7
-
0.8
4d
Cell Position y
10d
Figure 4.16: Normalized reflection coefficient magnitude vs. BRS unit cell position y
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C h a p t e r 4.
D e s ig n
and
F a b r ic a t io n
of a
B
e a m - r e d ir e c t in g
Surface
82
measured patterns as outlined in the next chapter.
4 .2 .3
S ou rces o f F ab rication Errors
It is im portant to recognize th at every fabrication process has some implicit sources of
error. It is worthwhile to point out a few.
The radii of the top and bottom aperture layers control the reflection phase gradient
along the surface, which in turn controls the effective reflection angle. The milling ma­
chine used in creating the inductive aperture sheet had some associated error. A sample
board, consisting of three rows of the N y = 12 BRS unit cells apertures was milled.
The diameter of each of the 3 x N y apertures on this test sample were measured via
a microscope. The measured aperture diameter values resulted in a maximum error of
±25fim. These random errors in radii contribute to phase errors which in the end affect
the overall shape of the reflected pattern produced by the beam-redirecting surface.
In designing the radii of the loss-corrected reflection phase response of the twelve BRS
unit cells (Table 4.4), it was assumed in the full-wave simulations th a t the prepreg had
a constant thickness of 100 fim after adhesion. However, by cutting a glued sample in
half and examining it under the microscope, it was discovered th a t the thickness varied
considerably. Upon various measured test samples, a minimum thickness of 86 fim was
measured, whereas the maximum measured thickness was 122 gm; these correspond to a
variation of at least ±14% of the expected nominal value. The variation in the prepreg
thickness may have been caused by the inability to reproduce the required temperaturepressure vs. time curves with the in-house compressor, or a non-uniform pressure by the
compressor plates.
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C hapter 5
Experim ental R esults
The previous chapter elaborated on the design and fabrication of the proposed beamredirecting surface. Specifically, a surface consisting of N = 17 BRS unit cells along
both x and y dimensions was fabricated using the in-house precision milling machine.
This chapter describes the selected experimental configuration used to test the proposed
surface. The discussion then focuses on the measured reflection power patterns and how
they compare to the theory developed in Chapter 2.
5.1
E xp erim en tal C onfiguration
It was desired to measure the reflected power pattern for various angles of incidence and
for both TE and TM incident polarizations. In order to achieve this, it was required
to use two separate horns; one designated as the transm itter (TX) and the other as the
receiver (RX). Since the power pattern for fixed incidence angles was desired, the receiver
would ideally move around the test surface in order to capture the reflected pattern. In
effect, the required experimental setup w as th a t of a bistatic radar configuration. A
g e n e ra l b is ta tic e x p e rim e n ta l c o n fig u ra tio n is sh o w n in F ig . 5.1. In th is configuration,
the transm itting horn is fixed, while the reflecting test surface is mounted on a rotating
platform; rotation of this platform sets up the incident angle
. After setting the angle
83
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C h a p t e r 5.
84
E x p e r im e n t a l R e su l t s
Rx
\
Tx -(<-4(fixed) |
Phase gradient
H
Surface Under Test
Figure 5.1: Bistatic measurement experimental configuration
of incidence, the platform remains static while the receiver captures the reflected pattern.
The receiving horn then measures the reflected power pattern as a function of angle 6 as
it rotates about the semicircular path shown in Fig. 5.1. The bistatic setup allows both
the incident and reflected wave vectors to lie in the same plane, in this case it is the x = 0
plane. This property of the bistatic measurement technique permits the measurements to
be carried out in the two-dimensional space, i.e. the x = 0 plane. Although the bistatic
measurement setup of Fig. 5.1 is the ideal experimental configuration for testing the
proposed beam-redirecting surface, an alternative configuration had to be implemented.
The lack of an actively controlled dynamic receiving horn and the space limitations
of the in-house anechoic chamber deemed the bistatic measurement configuration im­
practical. It was therefore required to devise an alternative testing configuration, which
would emulate the bistatic setup while utilizing the in-house anechoic chamber. To this
e x te n t, th e ch o sen e x p e rim e n ta l c o n fig u ra tio n sh o w n in F ig . 5.2 was u tiliz e d ; n o te t h a t
this setup is similar to the one used by Sievenpiper in [6]. In the chosen configuration,
the transm itter and receiver horns are orthogonal to each other and at 45° to the beam-
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C h a pt e r 5.
85
E x p e r im e n t a l R e su l t s
Rotation to obtain full
scattered field =s>
lr x
Beam-redirecting surface. ^
Rx
Rotation about x
to adjust the
incidence angle
T X
Tx
Figure 5.2: Chosen experimental configuration [6]
redirecting surface test sample; both horns lie in the y = 0 plane. The reflection phase of
the beam-redirecting surface under test varies along the y-axis. Due to specular reflection
in the y = 0 plane, the incident and reflected wave vectors lie as shown in Fig. 5.2; the
orientation of the reflected wave vector was exploited, and pattern measurements were
performed in the x = z plane by rotating the sample and the transm itter along their
centre (vertical) axis as shown. In order to change the angle of incidence, the surface
under test is rotated along its secondary central axis (x-axis) as shown in Fig. 5.2.
The experimental configuration can be better visualized by considering a 3D sketch
from the reference point of the surface under test as shown in Fig. 5.3. Both the incidence
and reflection planes are shown in the diagram and are both 45° to the surface under
test; the incident (reflection) plane contains the incident (reflected) wave vector kj (kr).
To the test surface, both the transm itter and receiver appear to be moving along the
hemispherical paths outlined in Fig. 5.3. The incidence angle, 0i, and the observation
angle, & are both defined as shown in the figure; the observation angle dictates the angular
locations at which the reflected field is measured. As was done in the previous chapters,
incidence at
= 0° will be referred to as normal or broadside incidence. However, notice
t h a t for o u r ch o sen e x p e rim e n ta l s e tu p a n d for 0, = 0°, th e in c id e n c e w ave v e c to r lies in
the incidence plane, which is slanted at 45° to the surface under test. The term “normal”
incidence is still applied for the chosen configuration, since for
= 0°, the incident wave
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JH M W T A L R B S ® «
C H A M ® 5 - EXI>E
Incidence
Plane
phase Gradient
-O. 3 D sketch of chosen
Figure 5.o.
of th e co p y n 9 ht ° 'wner*
R eproduced
with pedH|SS'on
C h a p t e r 5.
87
E x p e r im e n t a l R e su l t s
a
b
Figure 5.4: Maximum aperture dimension for a horn antenna
vectors lies in the normal line (dashed line in the figure) th a t bisects the incidence plane
into two equal areas.
5 .1.1
F ar-F ield C o n stra in t
In determining the distances between the test sample and the receiver and transm itter
horns, (Ir x and dTx respectively as shown in Fig. 5.2, the far-field constraint discussed
in Section 4.2.1 had to be enforced. This constraint required the reflecting test surface
to be in the far field of the transm itter horn and the receiver horn to be in the far field
of the reflecting test surface. In order to satisfy this constraint, the minimum distance
from the horn’s aperture to its far-field region was calculated via [1]
(5.1)
{m i n
where \ mm corresponds to the maximum measurement frequency, in this case f max —
11.5G H z, and D is the maximum dimension of the horn’s aperture, as shown in Fig. 5.4.
The D R H —118 horn antenna from Trimillennium Corporation, operates between 1 —18
G H z and is generally used for most measurements in the in-house anechoic chamber
belo w 7 G H z . H ow ever, at 11.5 G H z , the far field region of this horn starts 6.2 m
from the horn’s aperture. This distance was too large to accommodate in the in-house
anechoic chamber. As a result, a smaller pair of X-band horn antennas was purchased
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C h a pt e r 5.
E x p e r im e n t a l R e su l t s
88
Without Absorber
With Absorber
Rx
Rx
Cross-talk
eliminated
Cross-talk
Tx
Tx
Absorber
Figure 5.5: Cross-talk between RX and TX horn antennas
to allow the implementation of the proposed experimental setup in the in-house anechoic
chamber. Specifically, ATM’s (Advanced Technical Materials) Inc. 90 —441 —06 model
horn antenna was selected to conduct the measurements. This smaller horn has a gain of
15 dB at 10 G H z and has its far-field region start at a distance of 54 cm form the horn’s
aperture. As a result, the beam-redirecting surface was placed at d xx = 60 cm above the
transm itter. The surface under test was held in place via a special styrofoam structure
which is transparent to electromagnetic waves in the X-band.
The receiver was set on
the opposite end of the anechoic chamber, dux ~ 4 m. This large distance resulted in
some unwanted cross-talk between the transm itter and receiver horns as shown in Fig
5.5. Via superposition, the desired reflected power pattern would be superimposed over
the crosstalk pattern, resulting in erroneous measurements. As a result, an absorber was
placed in the line-of-sight path between the transm itter and receiver to eliminate most of
the interaction between the two horns as shown in Fig. 5.5. Note th a t the absorber was
placed such th at the line-of-sight between the reflecting surface and the receiver would
remain unobstructed. Furthermore, it was placed in the a far field of both antennas
so as to minimize interference. The selected absorber was the A N —74 from Emerson
a n d C u m m in g M icrow ave P r o d u c ts , sin ce it r e p o r te d a n a b s o rb tio n o f more th a n 20 dB
between 8 —12 GHz.
In order to test the effectiveness of the absorber, simple cross-talk, or noise, measure-
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C h a pt e r 5.
89
E x p e r im e n t a l R e su l t s
-4 0
-6 0
2
-7 0
® -8 0
-9 0
-100
JO
-1 1 0
Without A bsorber
With A bsorber
-5 0
100
Azimuth A ngle 6’ (deg)
Figure 5.6: Cross-talk measurements
ments were conducted. For these measurements, the testing configuration of Fig. 5.2 was
established within the anechoic chamber and the surface under test was removed from
its pedestal so as to characterize only the crosstalk between the transm itter and receiver.
The resulting measurements are shown in Fig. 5.6. Note the significant improvement
achieved by the inclusion of the absorber. As a result, the absorber was kept in place
for all measured data to follow. Furthermore, the cross-talk was frequently measured to
ensure the integrity of the data collected.
5 .1 .2
In cid en t P o la r iz a tio n for C h o sen E x p e r im e n ta l C on figu ­
ration
D u e to th e n a tu r e o f th e ch o sen e x p e rim e n ta l s e tu p , a n in c id e n t T E o r T M p o la riz e d
plane wave can only be achieved at normal incidence (#, = 0°). For this case, the electric
(magnetic) field vector lies in the incidence plane and is parallel to the beam-redirecting
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C h a pt er 5.
E x p e r im e n t a l R
90
esu lt s
E
l
Incidence
Plane
Beam-redirecting Surface
Phase Gradient
Figure 5.7: Projection of incidence plane on to the x = 0 plane
surface, thus resulting in a TE (TM) incident plane wave polarization. If off-normal
incidence is considered, then the incident polarization cannot be categorized as either
for TE or TM, but rather as a linear combination of the two. This behavior can be
better visualized by considering Fig. 5.3. As the angle of incidence diverges from 0°,
the E field, although still in the plane of incidence, is no longer parallel to the beamredirecting surface and the incident polarization cannot be categorized as either T E or
TM. A better visualization of this phenomena can be obtain by considering the projection
of the incidence plane on to the x = 0 plane as defined by the coordinate system in
Fig. 5.3. The resulting projection leads to Fig. 5.7. From this figure it is readily seen
th a t a non-zero value for the incidence angle results in a non-zero normal component of
the electric field (Ex)- A non-zero E± implies th a t not all the electric field is parallel to
the beam-redirecting surface and thus the incident polarization is a linear combination
of both TE and TM.
The inability to distinguish between incident polarizations for off-normal incidence is
in no way a limiting factor. In fact, by demonstrating th at the proposed beam-redirecting
surface responds to a linear combination of incident polarizations, it can be concluded
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C h a p t e r 5.
91
E x p e r im e n t a l R e su l t s
z“
y=-8d •
•
•••
•
>y
Figure 5.8: Antenna array equivalent of proposed beam-redirecting surface
that the proposed surface is polarization independent. Furthermore, by measuring the
reflected patterns on a 45° inclined reflection plane, the flexibility and angular dynamic
range available to the proposed surface will be demonstrated.
Finally note th a t for the bistatic measurement configuration, incident TE and TM
polarized plane waves are easily implemented for any incidence angle. This is possible
since the incidence and reflected wave vectors lie in the same plane, effectively resulting
in a 2D measurement environment. On the other hand, the chosen experimental configu­
ration of Fig. 5.2, relies on two separate planes for incidence and reflection, thus resulting
in a more complex 3D measurement environment. However, as mentioned previously, the
chosen experimental setup, perm itted the use of the in-house anechoic chamber.
5 .1 .3
A rray F actor A p p roach for M ea su re m en t C om p arison
It was desired to derive the equivalent array factor (AF) of the proposed beam-redirecting
surface, when placed in the experimental configuration of Fig. 5.2. The derived AF power
patterns could then be used to compare with the measured reflected power patterns. The
AF approach was selected for comparison with measured data because HFSS simulations
w ere im p ra c tic a l to c a r r y o u t since th e s tr u c tu r e is e le c tric a lly la rg e (3A0 x 3A0) at 10
GHz.
In order to derive the AF for the proposed surface while considering the experimental
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C h a p t e r 5.
92
E x p e r im e n t a l R e su l t s
configuration, each BRS unit cell comprising the beam-redirecting surface was modeled
as an omnidirectional point source radiator as discussed in Section 2.1. This assumption
leads to the array representation of the proposed beam-redirecting surface as shown in
Fig 5.8.
From this figure, the overall array factor can be determined by separately
considering the array factor along the x-axis, AFx(d,(j>), and the array factor along the
y-axis, AFy(9,<j>). From Section 2.1, the array factors are given by
AF(9,</>) =
=
^
I xmej{koX- sinWcosW)- J 2 l y ne-,j(k 0y'n
m=—8
AFX(9, <f>) ■AFy(9, <f>),
am(B) sin(<6))
(5.2)
where the location of the point sources along the x-axis are give by x'm = m d for m =
[—8, —7, ...,8] and those along the y are given by y'n = nd for n = [—8, —7,..., 8]. The
array factor of the entire beam-redirecting surface can then be simplified to
AF(6,<t>)
=
=
^ 2
pj n k Qd sin(0) sin(^)
I xmejmkodsin{e)cos{,p)- J 2 l y n e
r a = —8
=
n = —8
AFX(9,4>) ■AFy(9, <j>).
(5.3)
The excitation term, Iyn, was derived from the magnitude and phase of the HFSS sim­
ulated reflection coefficient, S u n — |<S'ii„hJasW <t>nhSsai °f each y-directed BRS unit cell n
and the inherent phasing due to the incidence angle 0*. Mathematically,
I Vn = S llne’nk°d,*nM .
(5.4)
To determine the excitation term along the x-axis, I xn, it was required to consider
the free space loss (FSL) inherent in a propagating EM wave. This “loss” is given by
Pr = F S L - P , =
where Pr
(A5ypl.
(5.5)
and Ptcorrespond to received and transm itted power respectively,A is the
wavelength ofoperation,and D is the distance between receiverand transm itter.
Note
th a t the FSL, is not actually a loss as the word suggests; it basically describes the spherical
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C h a p t e r 5.
93
E x p e r im e n t a l R e su l t s
x=8d
Amplitude
Taper V
Incident
.Wavefront
x=-8d
Tx
Figure 5.9: Amplitude taper along x-axis due to free space loss
< -0 .3
■0.4
"0.5
H - 0 .6
t -0 .7
- 0 .:
-4 dI
0
4d
BRS Unit Cell Position along x -a x is
Figure 5.10: Amplitude taper along x-oriented BRS unit cells due to free space loss
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C h a pt e r 5.
E x p e r im e n t a l R
94
esu lts
spreading of power. This equation is only valid in the far-field of the transm it antenna,
where the plane wave approximation holds.
Now consider Fig. 5.9 which shows the
incident wave fronts and the beam-redirecting surface. Note th at as discussed previously,
the distance D was chosen such th at the beam-redirecting surface is in the far-field of the
horn; therefore, the excitation amplitudes can be calculated via (5.5). From this figure,
it can readily be seen th a t the bottom left BRS unit cells, located at x = —8d, will
receive the most incident power, while the cells at x = 8d will receive the least. The FSL
effectively creates an amplitude taper along only the x-oriented BRS unit cells. Since the
AF multiplies a field quantity instead of a power quantity, and if the reference excitation
amplitude for the BRS unit cell at x = —8d is one, then
\Ixn\
= ^
| 1ndfsi j
for n = [0 ,1 ,.., 16],
(5.6)
where d fsi is the vertical distance between BRS unit cells as shown in Fig. 5.9. The
amplitude taper along the x-axis is plotted in Fig. 5.10 for reference. Note th a t because
dfSi is much smaller than D, the variation in excitation amplitude along the x-direction
is quite small. However, for larger reflectarrays this taper may become of importance
in deriving the overall array factor. The x-directed taper analysis is included here for
completeness. The phase of
I xn
is computed by considering the inherent phasing of the
x-directed BRS unit cells do to the inclination of the incidence plane, as shown in Fig. 5.9.
From this figure, the BRS phasing along the x-axis is given as
^Ixn
— —m k 0d sin(45°) = —m k0d
^ ^
v2
The above derivation resulted in an AF whose angles are referenced to the coordinate
system shown in Fig. 5.11(a). However, since the reflected pattern is measured along
the x = z plane, plotting the AF in this plane using the spherical coordinate system
b e c o m e s d ifficu lt.
I t w as th e re fo re d e c id e d to p e rfo rm a c o o rd in a te tra n s fo rm a tio n , a
counterclockwise rotation about the y-axis, as shown in Fig. 5.11(b) in order to simplify
the plotting of the array factor along the measurement plane, i.e. the x = z plane.
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C h a p t e r 5.
95
E x p e r im e n t a l R e su l t s
Rx
T x\l
(a) Original coordinate system (x , y, z)
(b) Transformed coordinate system (x1, y',z')
Figure 5.11: Coordinate transformation to facilitate AF plotting
The original coordinate system is shown in Fig. 5.11(a), along with the relative location
of the transm itter and receiver.
The transformation was carried out with a general
counterclockwise rotation angle, 9CCW, the details of which can be found in the Appendix.
Only the final results are relevant here. The angles 9 and <j>, as functions of the primed
coordinate angles 9' and (f>r are given by
9(6') - cos - l
cos(0')"
V2
</>(9') = tan 1[V^tan(^')],
(5.8)
(5.9)
where 9CCW= 45° was used. Note th a t since the measurements are done over the x' — 0
plane, then the angle 4>is set to 90° and therefore the angles 9 and <p are only a function
of 9'.
In order to measure the reflected pattern with the experimental configuration of
Fig. 5.2, both the surface under test and the transm itter are rotated about the x'-axis
as shown in Fig. 5.11; the two rotate in unison to maintain a fixed incidence angle and
incident polarization. In this measurement technique, the receiving horn effectively ob­
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C h a pt e r 5.
E x p e r im e n t a l R e su l t s
96
serves the reflected pattern at an angle O' on the x' = 0 plane. This measured pattern can
now be compared to the AF-predicted reflected pattern by plotting \AF[0(0'), 4>{0’)]\2 for
O' € [—90°, 90°] while performing the coordinate transformation via (5.8) and (5.9). The
variation in O' is equivalent to moving the receiver horn along the hemispherical path
shown in Fig. 5.3.
The derived AF will be utilized in the next section to compare with measured data.
5.2
M easurem ents
The experimental configuration shown in Fig. 5.2, was utilized to derive the following re­
flected patterns. Measurements were conducted for TE and TM incident polarizations at
normal incidence. Off normal incidence was also considered, but as previously discussed
these incident polarization are a linear combination of both TE and TM polarizations.
Frequency variations of the reflected patterns were also analyzed in a range from 8.5
G H z to 10.5 G H z. Finally, the specular angle suppression was also analyzed between
8.5 G H z to 10.5 G H z.
5 .2 .1
N o r m a l In cid en ce
The measured normalized reflected patterns for broadside incident TE and TM polarized
waves at 10 G H z are shown in Fig. 5.12. Note th a t both TE and TM incident waves
result in a reflection angle close the designed 0o = —30°. Furthermore, both polarizations
result in similar patterns demonstrating the polarization independence of the proposed
structure.
Note th a t the reflected power pattern predicted by the AF, Fig. 5.12(c), does not
c o m p le te ly m a tc h th e m e a s u re d re s u lts . Specifically, n o te t h a t th e measured sidelobe at
broadside is not predicted by the AF. The reflected power pattern of a reference metal
sheet of equal area, which was tilted such th a t its reflection angle was —30°, was also
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C h a p t e r 5.
97
E x p e r im e n t a l R e su l t s
0.8
0.6
0.4
90
■90
(a) TE Polarization
0.8
0.6
0.4
0.2
90
-90
(b) TM Polarization
0.8
0.6
0.4
-90
90
(c) AF Prediction
Figure 5.12: Measured normalized reflected power patterns for normal incidence at 10
GHz
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C h a pt e r 5.
98
E x p e r im e n t a l R e su l t s
-30
0.8
0.6
-60
0.4
-90
Figure 5.13: Measured normalized power pattern for tilted metal sheet
measured and is shown in Fig. 5.13. Note th a t the pattern of the tilted metal sheet does
not have sidelobes at broadside. It is possible to rule out edge diffraction as a possible
cause for the measured sidelobes of the beam-redirecting surface, since the metal sheet
would equally experience any edge diffraction. The high sidelobe can be due the mutual
coupling between BRS unit cells. The induced currents are shared between adjacent cells,
thus possibly increasing mutual coupling and leading to an increased specular sidelobe.
Another possible cause for the discrepancy may be the fabrication errors in aperture radii,
which were not modeled in the AF due to the random nature of the error. However, this
lobe is not of too great of a concern since most of the incident power is still diverted
away from the specular angle.
In order to assess the polarization purity of the proposed beam-redirecting surface, the
cross-polarized pattern was measured for both TE and TM incident polarizations. For an
incident TM (TE) polarized wave, the cross-polarized reflected field is TE(TM ) polarized.
Figure 5.14 shows only the TM cross-polarization for brevity. From this figure, it is seen
th at the cross-polarization levels are lower than —12.5 dB. For reflectarray systems,
this is on the high side. However, since the surface was not meant to operate as an
antenna, this cross-polarization level is acceptable. Furthermore, note th a t even in the
cross-polarized reflection pattern, most of the reflected energy is concentrated away from
the specular angle. Similar results were obtained for the TE incident polarization.
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C h a pt e r 5.
99
E x p e r im e n t a l R e su l t s
C o-p ol
C ro ss-p o l
ST - 5
p
c -1 0
0)
S
S. -15
U
O
5 -20
o
Q_
■g -2 5
N
"
E -3 0
o
z
-3 5
100
-5 0
Azimuth A ngle O’ (deg)
Figure 5.14: Measured normalized cross-polarization for a TM polarized incident wave
at 10 G H z
5 .2 .2
O ff-norm al In cid en ce
Off-broadside incidence was also considered and the results are shown in Fig. 5.15 at 10
G H z. As mentioned previously, for off-normal incidence the wave is neither TE nor TM
polarized but a linear combination of the two. Note th at as the angle of incidence varies
from —10° to 30°, the reflected main beam always points away from the specular angle.
Furthermore, the AF-predicted pattern follows closely the measured pattern except for
the 0i = 10° case. This discrepancy can be due to the nature of the experimental setup
as will be explained ahead and possibly due to the fact th at the array factor does not
model the coupling between cells nor the element factor of the BRS unit cell. The element
factor of the individual BRS unit cell is difficult to determine by any means since the cells
a re m e a n t to o p e r a te in c o n c e rt with adjacent cells. Furthermore, note th a t the cos(0)
loss (see Section 4.1.3) will lower the peak reflected power at angles far from broadside
and since the plots are normalized, this will effectively lead to higher relative sidelobes.
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C h a pt e r 5.
E x p e r im e n t a l R e su l t s
100
-30
-30
0.8
0.8
0.6
0.6
-60
-60
0.4
-90
90
90
— Measured
AF-predlcted
- Measured
-AF-predicted
(a) 0i = 10°
(b) 0i
-10
o
-30
-30
0.8
0.8
0.6
0.6
-60
-60
0.4
■90
90
90
— Measured
- -AF-predicted
(c) &i
-
20 °
Measured
AF-predlcted
(d)
,
6
-30°
Figure 5.15: Measured normalized reflected power patterns for off-normal incidence at
10 G H z
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C h a p t e r 5.
E x p e r im e n t a l R
101
esu lts
-•-M e a s u r e d
-■ -A F Predicted
o>
2 .-1 0
JQ)
O)
C
< -20
<u
c
o
o
*-<
0)
ic -3 0
V
o'
-4 0
-20
-1 0
Incidence A ngle (deg)
Figure 5.16: Reflection Angle vs. Incidence Angle at 10 G H z
Figure 5.16 displays the measured reflection angle vs. incidence angle at 10 G H z along
with the reflection angle predicted via the AF. From this figure it can be seen th a t the
measurements closely agree with the theoretically predicted reflection angle.
The range of incidence angles was limited by the chosen experimental configuration,
which, for large angles of incidence, resulted in high sidelobes th a t dominated the reflected
power pattern. To understand why the chosen experimental configuration of Fig. 5.2 can
lead to high sidelobes as $i increases, the array factor derived in the previous section
was utilized. Using the transformed coordinate system introduced previously, the array
factors along the x and y axes (n o t along the x' and y' axes), AFX[6(6'), cf>(6')} = AFX and
AFy[9(0'), <j>(6')\ = AFy correspondingly, together with the overall AF, AF[9{9'),4>(6')] =
AFxAFy, are plotted in Fig. 5.17 for the broadside incidence case at 10 G H z. As expected,
the peak of AFy is at the designed reflection angle of —30°, since the linear reflection phase
gradient was designed along the y-axis. Note th a t AFX is not constant for all O' angles.
When the angle of incidence increases beyond zero, the peak of AFy, shown in Fig. 5.17,
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C h a p t e r 5.
102
E x p e r im e n t a l R e su l t s
---A F
AF
0.8
AF=AF AF
,
5 0.6
■i 0.4
0.2
-?00
-5 0
-3 0
0
Azimuth A ngle 6’ (deg)
100
Figure 5.17: AF Argument for high sidelobes away from broadside
occurs at more negative values of O'. As a result, the peak of the overall AF also shifts to
the left. Since the magnitude of AFX in this experimental configuration decreases away
from O' = 0, the peak of the overall AF is reduced and thus results in higher relative
sidelobes. The non-constant A F X is a result of the separate incidence and reflection
planes, both of which are 45° to the surface under test as shown in Fig. 5.3. The measured
sidelobes would be considerably lower if the bistatic measurement configuration, shown
in Fig. 5.1, was used instead. This is due to the fact th a t the incidence and reflection
planes for the bistatic setup are one and the same, in this case the x = 0 plane. Thus for
the bistatic case, AFX is constant for all scanning angles 6 (angle defined in Fig. 5.1), thus
resulting in lower sidelobes when compared with the chosen experimental configuration.
However, it is worth noting th at the chosen experimental setup allowed the demonstration
of b e a m -re d ire c tio n w ith in th e lim ita tio n s of o u r in-house anechoic chamber. A more
thorough experimental study of the sidelobes requires a larger chamber with bistatic
measurement capabilities similar to th a t shown in Fig. 5.1.
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C h a p t e r 5.
103
E x p e r im e n t a l R e su l t s
-26
-30
I
!
!
-90
-•-M e a s u re d
-"•T h e o re tic al
8.5GHz
10GHz
- - - 11GHz
10.5
Frequency (GHz)
(a) Reflected power pattern variations
(b) Reflection angle vs. frequency
Figure 5.18: Frequency variations of reflected power pattern at 10 G H z for normal
incidence
5 .2 .3
F req u en cy V ariation s
The frequency variations of the reflected pattern for a normally incident TM polarized
wave is shown in Fig. 5.18(a). Note th at the reflected patterns retain their shape across
the indicated frequency range. Furthermore, note th at the maximum reflected power
points away from the specular angle as desired. The variation in reflection angle as a
function of frequency is shown in Fig. 5.18(b). From this figure, note th a t the measured
data closely follows the theory of (2.11) from 8.5 G H z to 10.5 G H z. In Section 2.2.1, it
was mentioned th a t a surface with a linear reflection phase gradient along one direction is
effectively equivalent to a tilted metal sheet at the surface’s design frequency. It is in this
plot where the difference between a tilted metal sheet and the beam-redirecting surface
is evident, since the angle of reflection of the former is constant with frequency. Note
th at this frequency variation of the reflection angle would assist in further concealing a
ta r g e t fro m a b is ta tic r a d a r , sin ce d iffe re n t in c id e n t freq u e n c ies w o u ld re s u lt in d ifferen t
reflection angles. Similar frequency variations were obtained for the TE polarization but
are excluded for conciseness.
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C h a p t e r 5.
104
E x p e r im e n t a l R e su l t s
u -1 0
-1 ^
- • - Metal S h eet
- • - B R S at 8.5GHz
-■ -B R S at 10GHz
-■ -B R S at 10.5GHz
« -3 0
-20
-1 0
Incidence A ngle 6. (deg)
Figure 5.19: Measured normalized specular reflected power vs. incidence angle
5 .2 .4
S p ecu la r A n g le S u p p ressio n
To assess the suppression of specular reflections, it was necessary to compare the reflected
powers of a reference metal sheet with th a t of the beam-redirecting surface at the specular
angle for various angles of incidence. As a result, the reflected power patterns for an
8.5 x 8.5 cm 2 metal sheet were measured for 6^ e [—30°, —20°, —10°, 0°, 10°].
As mentioned in Section 2.2.4, a metal sheet will always reflect maximum power at the
specular angle 6S. In other words, for an incident angle of
the maximum reflected field
will occur at 9S = —0*. On the other hand, the reflection phase gradient of the proposed
beam-redirecting surface reflects the incident power away from the specular angle. Figure
5.19 shows the normalized measured specular reflected power vs. incidence angle for both
a reference metal sheet and the beam-redirecting surface at 8.5 G H z , 10 G H z and 10.5
G H z. The n o rm a liz a tio n v alu e w as th e s p e c u la r reflected power off the reference metal
sheet at normal incidence. Only the specular reflected power at 10 G H z is shown for
the metal sheet since all frequencies resulted in the same normalized curve. Note that
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C h a pt e r 5.
105
E x p e r im e n t a l R e su l t s
- • - S A S at 8.5GHz
— - S A S at 10GHz
-■ -S A S at 10.5GHz
■o
-20
-10
Incidence A ngle 0. (deg)
Figure 5.20: Measured specular angle suppression vs. incidence angle
as expected, the specular reflection of our structure is much lower than th a t of a metal
sheet of equal area. Furthermore, since for off-normal incidence the incident polarization
is a linear combination of both TE and TM polarizations, Fig. 5.19 encompasses to both
incident polarizations.
The normalized absolute reflected power of Fig. 5.19 was used to calculate the spec­
ular angle suppression via (2.23). The measured specular angle suppression (SAS) vs.
incidence angle is plotted Fig. 5.20. Note th at more than 10 dB suppression over a wide
range of incident angles at 10 G H z was achieved. Furthermore, between 8.5 G H z and
10.5 G H z the SAS is greater than 9 dB. The high suppression at 6i = —20° occurs due
to a null in the reflected pattern of the beam-redirecting surface th a t is fairly constant
with frequency. Higher SAS values would be measured with a full bistatic measurement
setup, since this configuration will result in lower sidelobe levels, which in turn lower the
SAS. However, these measured values are representative of a worst case scenario.
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Chapter 6
C onclusions and Future D irections
6.1
C onclusions
A broadband, multilayered beam-redirecting surface consisting of circular apertures has
been presented. This surface is polarization independent and exhibits a linear reflection
phase grating over a 10% bandwidth along one axis, effectively emulating a tilted metal
sheet. One is not limited to a linear phase gradient along a single direction; the outlined
methodology can be extended to achieve any phase profile along both the x and y axes.
Using full-wave S-parameter simulations, an equivalent circuit for the circular aperture
was extracted and used to ensure th a t the apertures were operated away from their
resonances, thus enhancing the operating bandwidth. This circular aperture equivalent
circuit was then extended with transmission-line sections to obtain the equivalent circuit
of the full BRS unit cell. Pull wave simulations of the BRS unit cell reflection phase were
compared to those of the BRS equivalent circuit and close agreement between the two
was observed. The BRS unit cell was then used to design a 17 x 17 cell (8.5cmx8.5cm)
b e a m -re d ire c tin g su rfa c e w ith a re fle ctio n an g le o f 0o = —30° fo r a n o rm a lly incident
plane wave. W ith a small sample size of 8.5cm x 8.5cm, it was possible to experimentally
demonstrate beam redirection at a designed angle of —30° for broadside incidence and
106
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C h a p t e r 6.
C o n c l u s io n s
and
F
uture
D
ir e c t io n s
107
for both TE and TM incident polarizations. Beam-redirection was also observable at
various angles of incidence, specifically for 0* 6 [10°,—30°]. Finally, it was possible to
achieve high speculax angle suppression greater than 10 d B over a wide range of incident
angles at 10 G H z and greater than 9 dB between 8.5 G H z and 10.5 G H z.
A comprehensive procedure to reduce phase sensitivity a for multilayered beamredirecting surface based on stacked circular apertures was presented. Specifically by
sequentially closing the bottom most radii while keeping the top one fixed, such th at the
surface is impedance matched to free space, a linear reflection phase vs. aperture radius
is maintained. This procedure can be extended to any multilayered reflectarray based on
any stacked elementary unit cell in order to reduce reflection phase sensitivity.
The proposed BRS unit cell and the corresponding beam-redirecting surface provide
an alternative to the current reflectarrays presented in literature. However, one major
constraint is the weight of the overall structure, which was dominated by the substrate
and limits deployment. When lighter materials with high perm ittivity values and low loss
become available, the proposed surface will become even more relevant. Note th a t the
weight can be reduced by utilizing a larger unit cell dimension. A larger value of d would
result in larger shunt inductance values which would eliminate the need for a high per­
m ittivity substrate. Furthermore, the larger shunt impedance could be readily matched
to free space via thinner dielectric slabs, thus leading to a lighter structure. However, as
mentioned previously a larger cell dimension will limit the achievable reflection angles.
If a large reflection angle (from broadside) for an normally incident wave is not a design
requirement, the proposed surface could be readily utilized to lower the specular reflec­
tio n o f a n y d e sire d o b je c t. F u rth e rm o re , flexible s u b s tr a te s c o u ld b e u tiliz e d to further
lower the profile of the proposed beam-redirecting surface while allowing it to conform
to any shape.
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C h a pt e r 6.
C o n c l u s io n s
and
F
uture
D
ir e c t io n s
108
ki
Figure 6.1: Surface with random reflection phase along its dimensions
6.2
Future D irection s
Given the short duration of the project, it was not possible to perform a comparative
study of the available fabrication methodologies. It would be of great value, to perform
careful comparison between the milling machine fabrication process and the chemical
etching process, including fabrication sources of errors and reproducibility. Such a study
would lower reflection phase errors due to fabrication errors and, as mentioned in Section
2.2.2, would in turn enhance the bandwidth and possibly the reflected patterns.
The proposed beam-redirecting surface is only one of many possible designs.
In
this work, only a linear phase progression along one axis was implemented in order to
demonstrate beam-redirection. However, by having phase gradients along both surface
axes, the redirected beam can be steered in virtually any direction in space. The theory
developed in this work can be directly extended to design a surface with linear reflection
phase profiles along two dimensions. Furthermore, a non-linear reflection phase gradient
along both directions can also be created with the proposed BRS unit cell. For this
configuration, the beam-redirecting surface would effectively mimic a parabolic reflector
antenna, a topic which has been explored via reflect arrays. Furthermore, random phase
variations across the surface could be implemented in order to design a surface which
scatters in all angular directions as shown in Fig. 6.1. Such a surface m a y be of use in
military radar applications.
It was desired to further reduce the weight of the EBRS unit cell in order to facilitate
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C h a pt e r 6.
C o n c l u s io n s
and
F
uture
D
ir e c t io n s
109
its fabrication and deployment. The substrate utilized in realizing the proposed beamredirecting surface was rather heavy. If lighter high perm ittivity substrates were available,
this would reduce the weight of the surface, thus making it more viable. Furthermore,
it is possible to lower the weight of the structure by sacrificing the small cell dimension.
By increasing d, the aperture inductance increases and thus the frequency of operation
decreases. By reducing the perm ittivity of the substrate, it may be possible to utilize
lighter and commercially available substrates. However, by increasing the cell dimension,
d, fine control of the reflection angle is lost as discussed in Section 3.3.2. Also, if some
reflection phase linearity is sacrificed, which results in a narrowing of the operational
bandwidth, it is possible to utilize only one EBRS unit cell to comprise the BRS unit
cell while lowering the substrate thickness. As mentioned in [3], thicker dielectrics result
in additional operating bandwidth at the expense of reflection phase variation. Hence,
by reducing the dielectric thickness, the reflection phase variation available to a single
EBRS unit cell is increased, thus eliminating the need for two stacked EBRS unit cells.
Of course, this approach leads to lower reflection phase linearity and therefore a smaller
bandwidth.
Another possible extension to this work could be creating a dynamic EBRS unit cell
and thus a dynamic beam-redirecting surface. The proposed beam-redirecting surface
is static, i.e its reflection properties are fixed, since the aperture radii of each EBRS
unit cell on the structure were fixed. Since by varying the aperture radii, and thus the
shunt aperture impedance of the EBRS unit cell, results in a reflection phase variation,
a dynamic EBRS unit cell could be implemented by adding a shunt variable inductor
as shown in Fig. 6.2(a). The load shunt inductor L var could then be varied to achieve
the desired reflection phase and thus eliminating the need to mechanically change the
a p e r tu r e ra d ii of th e E B R S u n it cell.
N o te t h a t a C M O S variable in d u c to r has been
proposed in [30] and could be used in the fabrication of a dynamic beam-redirecting
surface. For a polarization independent EBRS unit cell, two inductors placed orthogonal
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C o n c l u s io n s
C h a p t e r 6.
and
F uture D
110
ir e c t io n s
'var
d
'var
'var
(a) Single polarization
(b) Dual-polarization
Figure 6.2: Inductive loading of circular apertures
to each other, as shown in Fig. 6.2(b), would allow the surface to respond to both an
incident TE or TM polarized wave. Note th a t the addition of surface mount components
will inevitably change the reflection phase properties of the EBRS unit cell. Thus, time
must be invested in order to properly place the inductors so as to minimize the unwanted
and unpredictable variations in the reflection phase.
An alternative to above dynamic beam-redirecting surface is presented in Fig. 6.3.
In this configuration, the switches around the aperture’s perimeter are individually con­
trolled. By opening a switch, the current induced by the incident magnetic field would
be forced to travel around the gap, resulting in an increased inductance. Therefore, by
controlling the number of switches to close, it is possible to control aperture’s inductance
and therefore the reflection phase.
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C h a pt e r 6.
C o n c l u s io n s
and
F uture D
ir e c t io n s
switch
Figure 6.3: Inductance control via multiple switches
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C hapter
7
A ppendix
This appendix includes the proof th a t the A B C D matrix in Section 3.3.1 used to derive
the Bloch impedance of the EBRS unit cell is reciprocal and details on the coordinate
transformation discussed in Section 5.1.3.
7.1
R ecip rocity o f
ABCD M atrix used for Zb D eriva­
tion
The A B C D matrix relating the currents and voltages at terminals n and n + 1 of the
EBRS unit cell of Fig. 3.10 was given in Section 3.3.1 as
A
B
^ cos(20) + { Y { R )Z tl sin(20) j Z TL sin(28) - Z%LY (R ) sin2(0) ^
yi C
^ D
^ J,
y jY TL sin(28) + Y (R ) cos2(8)
cos(20) + {Y { R )Z tl sin(20)
j
To show th a t the EBRS unit cell is a reciprocal network, it is sufficient to prove th at
A D — B C = 1, therefore
AD
=
(cos(28) + -JY ( R ) Z T L sm(28))2
z
-
cos2(20) + jY { R ) Z TL sin(28) cos(28) - ^ Z $ LY 2{R) sin2(28),
112
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(7.1)
C h a p t e r 7.
A
113
p p e n d ix
and
BC
=
(jZ TL sin(20) - Z 2LY (R ) sin2(8))(jYTL sin(26) + Y (R ) cos2(0))
= jZ T iX { R ) sin(26)[cos2(9) —sin2(0)] —sin2(20) —Z ^ LY 2(R) sin2(0) cos2(0)
-
j Z tl Y {R )
sin(20) cos(20) - sin2(20) - ^ Z $ LY 2(R) sin2(29).
(7.2)
Subtracting (7.1) and (7.2) leads to
A D -B C
= cos2(20) + jY ( R ) Z TL sin(20) cos(20) - ^ Z 2LY 2(R ) sin2(20)
—(j Z t l Y { R ) sin(20) cos(20) —sin2(20) — ^ Z ^ LY 2(R) sin2(20))
=
cos2(20) + sin2(20) = 1.
(7.3)
Therefore the EBRS unit cell of Fig. 3.10 is reciprocal.
7.2
C oordinate T ransform ation to F acilitate A F P lo t­
tin g
In Section 5.1.3, a coordinate rotation about the y-axis was desired. The original coordi­
nate system, (x , y, z ), the desired coordinate system, (x1, y', z 1), and the relative locations
of the beam-redirecting surface and horn antennas are shown in Fig 7.1.
From Fig. 7.1(b), it is readily seen th a t a rotation about the y-axis results in y' = y,
^rVicT- two axes
ovqo are
cm related by
thea other
f x '\
\z' )
( cos(8ccw) - sin (0CCW) \
(x\
(
R‘CCW
\
sin(0cclt))
cos(0CCUJ)
\ z 1
(7.4)
\ ZI
However, it was desired to obtain x and 2: as a function of x f and z l. Therefore,
/
\
( x >\
/
COs($ccu;)
= R.'CCW
\ zJ
\ z>)
V
sin(0CCUJ) cos(^cciy) y
( ^ \
(7.5)
^ z y
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C h a p t e r 7.
A p p e n d ix
114
y =y
Tx
i
i
(a) Original coordinate system (x, y , z )
(b) Transformed coordinate system (x',y',z')
Figure 7.1: Coordinate transformation to facilitate AF plotting
Having determined the relationship between the primed and unprimed coordinates, it
was then required to obtain the angles of the spherical coordinate system as a functions
of the primed coordinates, i.e. 6(9', cf>') and 4>(9', cp'). It must first be noted th a t a rotation
of a coordinate system about one of its axis, does not perturb the origin of the coordinate
system; in other words, the point in space described by (x , y , z ) = (0,0,0) is the same
as ( x ',y ',z f) — (0,0,0). From this observation, it can be concluded th a t the distance
from the origin remains the same after the coordinate transformation; in the spherical
coordinate system this implied th at r’ = r. The angle 9(6', <j>') can be obtained by using
(7.5) and the fact th at z = r cos(6); from these two equations one obtains
cos[#(#', <j)')\ =
_
- sin(9ccw)x' + cos(9ccw)z'
rt
— cos($cciu) i
sin($CCU;) -
= cos(9CCW) cos(9') - sin(0CCU)) sin(0;) cos(</>'),
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(7.6)
C h a p t e r 7.
115
A p p e n d ix
where x — r' sm(d') cos(ft) and z' = r' cos{&) were used. This expression can be simpli­
fied by considering th a t the actual measurement plane in Fig. 7.1. The reflected wave
vector will now lie on the x' = 0 plane(i.e. f t = 90°), and the pattern measured by the
receiving horn is effectively a function of 8'. Therefore, (7.6) was simplified by setting
8ccw = 45° and f t = 90 to obtain
8{9') — cos 1
"cos(0')'
(7.7)
y/2
Similarly, the angle
ft) was obtained via (7.5) and the fact th a t tan<f>=
resulting
m
tan[<^(0', ft)]
=
—
y’
cos (9ccw)x' + sin (9ccw)z'
sin(0') sin(<^/)
cos{8CCW) sin(6,/) cos(ft) + sin(0ccu,) cos(0') ’
(7.8)
where y' = r' sin(#') sin(0') was used. To observe the same measurement plane as the
receiving horn of Fig. 7.1, ^ = 90° and (7.8) simplifies to
4>{8') = ta n -1 [\/2 tan(0')],
where 6CCW= 45° was used.
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(7.9)
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