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1
Flat Luneburg Lens via Transformation Optics
for Directive Antenna Applications
Carolina Mateo-Segura, Member, IEEE, Amy Dyke, Hazel Dyke, Sajad Haq and Yang Hao, Fellow,
IEEE

Abstract— The great flexibility offered by transformation optics
for controlling electromagnetic radiation by virtually re-shaping
the electromagnetic space has inspired a myriad of dreamtailored electromagnetic devices. Here we show a 3D-transformed
microwave Luneburg lens antenna which demonstrates high
directivity, low side-lobe level, broadband response and steerable
capabilities. A conventional Luneburg lens is redesigned
accounting for dielectric materials that implement a coordinate
transformation, modifying the lens geometry to accommodate its
size and shape for easy integration with planar microwave
antenna applications. An all dielectric lens is manufactured
following a thorough holistic analysis of ceramic materials with
different volume fractions of bi-modal distributed titanate fillers.
Fabrication and measurements of a 3-D flat Luneburg lens
antenna validate the design and confirm a high-directivity
performance. A maximum directivity of 17.96dBi, low side-lobe
levels for both main planes ~ -26dB, bandwidth limited only by
the source employed and beam-steering up to 34° were achieved.
Index Terms — Lens
Transformation Optics.
Antennas,
Directive
limited by a narrow operational bandwidth along with high SL
levels due to both the substrates employed and phase
differences. Solutions to address the bandwidth limitation [8]
and reduce the SL levels [9] have been presented; however,
both approaches come at a cost of increasing the overall
thickness of the antenna and a reduced gain, respectively.
Parabolic reflector antennas are popular candidates due to
their high-directivity, narrow beam-width and broadband
response [1-4]. Additionally, steerable beams can be obtained
by using a cluster of horns at their focal plane as a feeder;
conversely, such configurations produce bulky off-set
systems, due to the constraint on their physical dimensions,
and cause a high level of side lobes (SL) which in turn makes
them more vulnerable to interference [2-3].
antennas,
I. INTRODUCTION
A
NTENNAS convert the energy of free propagating
radiation to localized energy, and vice versa. Continuous
demand to enhance the effectiveness of wireless and satellite
communication and radar systems have driven antenna
designers to develop microwave antennas with highlydirective beams. Traditionally, three main configurations have
been put forth namely reflectors [1-4], arrays of radiating
elements [5] or cavities such as Fabry-Perot (FP) [6-9] and
lenses [10-29]. Space arrays of radiators can produce directive
beams, albeit complex feeding structures are required [5]. FP
cavity antennas are low-cost and compact, but nevertheless
Manuscript received 30th May, 2013. Dr. C. Mateo-Segura and Prof. Yang
Hao acknowledge the support by the Office of Naval Research Global
(ONRG) under Naval International Cooperative Opportunities (NICOP) for
the funding support with Grant No. N00014-09- 1-1013, and the additional
support by the Advanced Technology Centre, BAE Systems in Bristol.
C. Mateo-Segura is with the Institute of Sensors, Signals and Systems,
School of Engineering and Physical Science, Heriot Watt University,
Edinburgh EH14 4AS, UK (e-mail: [email protected]).
Amy Dyke, Hazel Dyke and Sajad Haq are with the Advanced Technology
Centre, BAE Systems, Functional Materials Department, Sowerby Building,
Filton, Bristol, BS34 7QW, UK (e-mail: [email protected]).
Yang Hao is with the Antennas & Electromagnetics Group, School of
Electronic Engineering and Computer Science, Queen Mary College,
University
of
London,
London
E1
4NS,
UK.
(e-mail:
[email protected]).
a)
c)
b)
d)
Fig. 1 Ray tracing for a) original and b) transformed Luneburg lens. Shadow
areas show the variation of the index of refraction, both systems with
aplanicity. 2-D c) original and d) transformed LL’s refractive index
distribution for δ = 5 and R=λ (frequency 10GHz).
Similarly to parabolic reflectors, lens antennas can be
designed to transform spherical wave-fronts emitted from a
feed at their focus into plane wave-fronts. The lenses
employed range from homogeneous dielectric lenses [10], to
gradient-index lenses (GRIN) [11-22]. An example of the
latter is the Luneburg lens (LL) [11]. This lens is particularly
attractive due to its focusing properties (i.e. free from optical
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aberrations), its symmetry, which allows multiple-beam
scanning, and its intrinsic broadband response, usually limited
by the bandwidth of the feed source. The gradient index
requirement imposes certain limitations in the manufacturing
process. Traditionally, Luneburg lenses have been
manufactured by a series of concentric inhomogeneous
dielectric shells, whose electric permittivity is varied
discretely [14]. Although several authors have presented
simpler configurations with only two layer shells [15-16], the
accuracy requirement of the material properties and shell
dimensions causes a constraint. Moreover, the large spherical
size poses a challenge for non-planar microwave antenna
applications together with the spherical focal locus of the lens
which complicates the spatial control of an antenna source for
beam steering. Recent progress towards flattening the focal
plane of the lens has been achieved for 2-D [17-18] and 3-D
[19-22] lenses using transformation optics (TO) [23-27].
Almost all current transformed lenses have been manufactured
by using metamaterials, i.e. man-made materials consisting of
many sub-wavelength resonators; however, metamaterials are
inherently narrowband, lossy and dispersive [28-29] and
fabrication
is
a
cumbersome
process.
Artificial
electromagnetic structures printed on circuit boards have also
been demonstrated to achieve the refractive index of the LL
[30-33]. Such lenses are generally compact and easily
integrable 2-D structures, but only able to collimate in the
same plane of the lens.
In this paper, we report a compact 3-D Flat LL antenna
design using TO to demonstrate high-directivity, low side-lobe
levels, broadband performance and steerable capabilities, with
similar properties to the original spherical lens. The flat LL is
designed and manufactured using a suite of all-dielectric
materials, thus avoiding the use of metamaterials which would
limit the bandwidth performance. The materials were designed
following a holistic analysis of their properties with varying
volume fractions of nano-sized particles. Resin composites
loaded with ceramic powders are used to produce the required
range of permittivity values. The fabricated materials proved
to have a stable dielectric constant across the X-band and
withstand relatively high powers making the antenna a good
candidate for high power applications. The manufacturing
technique is low-cost and easy to reproduce. Thus, we
anticipate that the proposed approach will open up
possibilities of novel material research and development with
an aim of achieving true industrial and societal impact. The
antenna design exhibits a high-directivity bandwidth
performance across the X-band with low SL level. The
prototype demonstrated a steering angle of up to 34 degrees.
The high-performance of the 3-D flat LL antenna together
with a novel manufacturing process shows the capability of
this method to develop a plethora of TO-based structures.
2
shape, or equivalently the space inside the lens is distorted.
The deformed coordinates could be described by a coordinate
transformation with the set of equations in (1), where  takes
any real values and represents the compression of the original
LL and R its radius, Fig. 1 a) and b).

z' 
(1)
z
y'  y
R  y2
It has long been known that a coordinate transformation is
equivalent to renormalizing the constitutive parameters, fields
and sources owing to Maxwell’s equations remaining invariant
[23-27]. Therefore, the new values of the permittivity and
permeability tensors are related to the original ones by the
relationships in Equation (2).
2
A B

R 2  y '2 z '2 y ' 
2
2 
1
 '   2 

R

y
'

2

B




R




(2)
A B
2
2 
 '  R  y'  B 1 




where the coefficients A and B are as in (3),
2

A

 R 2  y '2   z '2 y '2
R
2
 y'

2 3
B
z' y'
R 2  y '2 R 2  y '2 
(3)
The transformation does not change the nature of the
original LL performance and only reshapes it. The new
designed media performs the coordinate transformation
described by (1) allowing electromagnetic waves to take up
the distorted virtual electromagnetic space. The nonconformality nature of the proposed transformation leads to
the permittivity tensor being non-diagonal and thus the need of
including inhomogeneous anisotropic materials. However,
anisotropic materials are very difficult to manipulate for
practical applications and thus are desirable to avoid [34]. On
this basis and owing to the symmetry of the permittivity map
in (2), we apply a basic principle of linear algebra which
allows us to rewrite (2) as a diagonal tensor by means of
calculating its eigenvalues,
l1, 2 
C  C 2  4 2 z 3
2 2
(4)
where C    2     z ' 2 y ' 2 and   R 2  y ' 2 .
II. TRANSFORMED LUNEBURG LENS - DESIGN METHOD
To understand how transformation optics is applied to our
work, let us consider the original LL whose refractive index is
a function of the spatial coordinates y and z, Fig. 1c.
Subsequently, the LL is ‘squeezed’ into a slim cylindrical
Fig. 2 Peak value of the required permittivity as a fuction of the transformed
LL’s profile, δ.
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As can be seen from Eq.(1)-(4), the compression coefficient,
δ, has a significant impact in the permittivity map of the
transformed LL. Figure 2 shows the peak value of the required
permittivity as a function of δ, assuming the radius of the
original LL, R=λ for a frequency of operation 10GHz. In
particular, for thinner profiles (i.e. lower values of δ) the peak
value of the required permittivity increases. A value of δ=5
which provides a sufficiently thin planar lens profile (λ/6) and
a peak value of εr~12 achievable over a relatively broad
bandwidth is considered here. On the other hand, the lateral
compression of the lens (along z-axis) is accompanied by a
longitudinal extension (along the y-axis) as the lens strives to
maintain constant the atomic spacing and bulk volume, in
accordance with the Poisson effect [35]. The diagonal
permittivity map with the eigenvalues in (4) is shown in Fig.
1d. Here, ε’zz is less than 1 within all the space inside the
transformed lens; also, ε’yy was less than 1 in some regions.
Although this permittivity map can be realised using
metamaterials, fields are focused in the high refractive index
regions and avoid the low index ones (except for the highest
angles of incidence [17-18]). Thus, the approximation ε’zz
equal to the unitary matrix I and ε’yy equal to 1 for those
components whose value is lower than unity can be taken
without significantly affecting the performance of the
transformed lens [18]. This approximation obviates the use of
metamaterial periodic cells. In addition, we take the
approximation ’yy = ’zz = 1, and therefore ’~1 which
likewise does not detrimentally affect the lens behaviour.
a)
3
transformed lens is presented in Fig. 3 for a point source
located at different positions along the focal plane. The
position of the transformed lens is indicated with a white dash
line. As expected, the cylindrical wave front emanating from
the point source is transformed into a plane wave front by
means of a phase delay inside the transformation lens at any
position of the source (i.e. similarly as with the original LL). It
is to be noted that unlike the case of the LL the point source
(i.e. focus) is not located along the surface of the lens; this is
due to the change in the transformed lens’ gradient refractive
index distribution and the size of the lens (to account for
diffraction) which causes the focus to be located at a different
position along the optical axis, z-axis for a given frequency. In
particular, the focal plane is at z’=±/2 and y’= 0. The steering
capability of the proposed lens is also confirmed in Fig. 3
where the radiating source is placed at (/2, 0), (/2, /2), (/2, 0) and (-/2, -/2). The broadband nature of the
transformed LL is also studied. The lens response to a point
source located at z’=/2 and y’= /4 is observed for a broad
frequency band, particularly at 7, 10 and 14 GHz, Fig. 4. The
material response is assumed broadband and the effect of the
lens does not qualitatively change within the frequency band.
Similarly, as in the original LL the incoming wave is steered
and transformed into a plane wave for each frequency.
b)
Fig. 4 Normalized magnetic field intensity images obtained with FDTD
code where a point source is radiating a cylindrical wave at frequencies
f1=7GHz, f2=10GHz and f3=14 GHz from left to right.
c)
d)
Fig. 3 Normalized magnetic field intensity images obtained with FDTD
code where regions of ε’<1 have been approximated to unity. The cylindrical
wave radiated by a point source at a) (/2,0), b) (-/2,0), c) (/2, /2) and d) (/2,- /2), passes the transformed lens (dash white line) and is collimated to a
plane wave (frequency 10GHz) and permittivity map as in Fig. 1d.
A. FDTD Analysis of the Flat Luneburg Lens
The performance of the transformed lens is initially
observed in a 2-D scenario using an in-house FDTD code [35].
In the local coordinate, the required permittivity response of
the materials is in the range of 1 to 12, as shown in Fig.1d.
The normalized magnetic field, Hz propagating through the
B. Discretization of the Flat Luneburg Lens and Antenna
System
The 2-D design is extrapolated to a real 3-D scenario by
rotation of the 2-D permittivity map in Fig. 1d around the zaxis [19]. Thus, any cross-section of the 3-D lens will consist
of the 2-D permittivity map in Fig. 1d. By virtue of this
symmetry, a purely isotropic material will be required and the
lens response will not depend on the polarization of the wave
emanating from the source. Gradient permittivity materials as
in Eq. (3) are very difficult to manufacture and as a
consequence, the permittivity map should be discretized [1416]; in the discretization process a thorough study of the
number of dielectric layers and their dimensions is required
due to the significant effect on the focusing properties and
hence the transformation capabilities of the lens. An initial
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study of the depth of focus of the flat lens, not shown here for
the sake of brevity, was carried out. The electric field
amplitude along the z-axis was monitored for a different
number of dielectric layers. Our results proved that the depth
of focus increases with the number of dielectric layers. The
minimum number of layers ensuring adequate lens
performance and ease of the fabrication process is 6. The
dimensions of each layer were optimised by full-wave
electromagnetic simulations using CST Microwave Studio to
account for maximum directivity and also the non-ideal
spherical wave emanating from the real source employed. In
this case, and without loss of generality, an X-band pyramidal
horn is considered. The directivity and SL level are monitored
for different positions of the horn along the z-axis, h (Fig. 4),
in order to find the focal point (i.e maximum directivity) of the
lens after discretization. In order to compare the performance
of the proposed flat LL, a spherical LL with the same number
of shells as the flat lens is also examined. Fig. 4 illustrates the
3-D antenna system consisting of the discretized lenses, whose
dimensions are shown in Table I for each cylinder and sphere
respectively. Simulations of the complete antenna systems are
shown henceforth.
Cyl.
1
2
3
4
5
6
Hz
(mm)
3.32
6.62
8.8
11
12.4
13.8
Hy
(mm)
31.2
53.2
69.7
76.5
89.1
95.4
εr
Sph.
12
10
8
6
4
2
1
2
3
4
5
6
Hy
(mm)
31.4
53.5
69.8
76.9
89.3
95.9
4
owing to the phase delay introduced by the dielectric
materials. In particular the maximum directivity emerges at
10GHz with a value of ~18.2dBi and 18.3dBi respectively.
The SL ratio (i.e. difference between main lobe and SL level)
remain lower than -10dB across the band for H- and E-plane.
The figure further validates that the flat LL preserves similar
directivity performance to the original LL with the same
physical size along Y.
Fig. 5 Directivity within the frequency band of interest of the original LL
and transformed LL, both fed by an X-band pyramidal horn placed at their
focal points and not shifted along Y (d=0). The reflection coefficient and
directivity of the X-band pyramidal horn is also depicted.
εr
2
1.8
1.6
1.4
1.2
1.08
Fig. 4 Cross-section of the discretized flat LL and spherical LL after
considering 6 shells of dielectric material with the optimised dimensions as in
Table I; An X-band pyramidal horn located at the focal point is used as real
source of the antenna system.
In Fig. 5 the return loss, directivity of the horn and
directivity of the antenna including either the flat LL or the
spherical LL are depicted. The horn antenna is located at the
focal point, which corresponds to h and h/8 respectively.
In both cases the lens is not shifted along the planar/spherical
focal plane, d=0, thus the directive beam appears at boresight.
Both antennas are well matched and in both cases the lenses
contribute to increase the directivity of the horn within the Xband. This can be explained by observation of Fig. 6a-b where
the x-polarised wave radiated by the pyramidal horn is
transformed into a plane wave on the other side of the lens.
The generated far-field pattern shows a main beam that is
diffraction limited by the aperture of the discretized lens,
a)
b)
c)
d)
Fig. 6 Simulated a-b) E-field images of a) a common 6-shell LL and b) the
transformed LL, both fed by an X-band pyramidal horn located at their focal
points and c-d) 3-D far-field directivity patterns at 10 GHz for different
positions of the horn, c) at d (0, 10, 20 and 30mm) along the YZ plane and d)
at an arc length of d (0, 8.32, 16.6 and 24.9) along the boundary of the LL.
The steering capability of the antenna has been proven for
both cases. Fig.6c-d depicts the 3-D far field pattern when the
horn is shifted along the planar/spherical focal plane, d. In
particular, four positions of the patch antenna are examined, at
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the centre (y=d=0 and z=0) and at d=10, 20 and 30mm in the
case of the flat LL and at an arc length of d=8.32, 16.6 and
24.9mm in the case of the LL. As the beam steers by an
angle, the directivity of the flat LL decreases up to -1.1dBi
whilst the SL level increases by a maximum of 5dB. The
inability of the transformed LL to keep the same performance
as the original LL for different positions of the patch antenna
along the y-axis can be explained by consideration of the
previous approximations. For the flat LL with longitudinal
size ~1.4, the beam steers ~±30o mainly limited by the
physical aperture of the flat LL. Higher steering angles can be
obtained by increasing the longitudinal size of the lens (y- and
x-dimension). The matching at the interface between media
will slightly affect the performance of the lens; however, for
the flat LL antenna presented internal reflections are
negligible. The front-to-back ratio is calculated and the
radiated field in the backward direction, shown in Fig. 7,
remains lower than -13dB across the band. To further diminish
it extra matching layers may be added [37].
Fig. 7 Radiated Field of the flat LL in the backward direction within the Xband (8 to 12 GHz).
Cylinder
ε
μ
Density of
composite
(g/cm3)
1
2
3
4
5
6
12.35
10.50
7.92
5.46
4.54
2.10
1.01
0.99
1.01
1.00
1.01
1.00
2.50
2.30
2.00
1.90
1.54
1.1
Estimated
Breakdown
Strength
(kV/mm)
32
35
38
44
50
>100
5
c)
d)
Fig. 8 Powders illustrating alternative size distributions a) micron sized and b)
nanosized material). c-d) Fabrication process issues. c) Picture of voids
present in composite structure, d) Composite sample showing phase separated
region.
III. ANTENNA FABRICATION PROCESS AND MEASUREMENTS
The discretized transformed LL antenna was manufactured
following a thorough holistic analysis of the materials and
fabrication processes. Each cylinder within the lens was
tailored according to the values presented in Fig. 4 with a
maximum tolerance of 13% in the permittivity values. The
corresponding values of the fabricated materials are shown in
Table II. The loss tangent varied between 0.0004 and 0.006 for
the highest permittivity material. The morphology and
electrical properties of the materials through the entire
manufacturing lifecycle of the transformed LL were controlled
through novel techniques developed in our laboratory.
The key issues addressed in the manufacturing process
concerned the selection and modification of the filler material,
its compatibility with the matrix resin, the tailoring of the
dispersion of the fillers, particle size distribution, and
consideration of the particulate shape. All of these factors
have to be considered holistically to obtain the process control
and the viscosity requirements to ensure successful device
fabrication whilst achieving the requisite dielectric properties.
The interdependency of these matters requires a deep
understanding of the factors which affect them.
Table II. Relative permittivity, permeability, density values and estimated
breakdown strength of the composites used to prepare the cylinders forming
the lens in Fig. 4.
Fig. 9 Bimodal size distributions used for the different cylinders in Fig. 4.
a)
b)
A. Manufacturing Lifecycle of the Flat Luneburg Lens
The manufacturing lifecycle consisted of three-stages
namely, particulate filler preparation, composite (filler-resin)
production and multi-cast sequential fabrication of the lens.
The fabrication approach was a combination of casting and
machining of resin composites loaded with ceramic powders.
The challenge was to prepare casting mixtures that would
deliver the required permittivity values whilst controlling the
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rheology of the mixtures. This was achieved by inspection of
the volume fraction of the filler within the resin system, the
particle size and size distribution in relation to both the
permittivity and homogeneity of cast samples. It was
necessary to ensure that the dispersed powder did not settle
out of suspension during the cure cycle as otherwise it may
have led to an inhomogeneity in the electromagnetic
properties of the composite structures. Proprietary titanate
based ceramic materials were processed using an in-house
milling regime to produce the powders needed.
A
combination of particle size reduction methods were used to
attain the necessary bimodal particle size distribution in the
composite. Ball milling was used to generate powders in the
micron size ranges and bead milling to produce the nano-sized
powders, Fig. 8 (a-b). Bimodal size distributions allowed
higher volume fractions of filler to be incorporated within the
composite structures to achieve the higher values of dielectric
constant, and relate to better packing of the particles.
Alternative distributions are used for cylinders with lower
permittivity values. In particular, the bimodal size
distributions used for the different cylinders are presented in
Fig. 9. Distribution ‘a’ was used for the highest permittivity
zone, with the others (e.g. ‘b’) for successively reduced
permittivity values. The use of nanosized fillers in
combination with the larger particles sizes allows achievement
of higher volume fractions and is illustrated by the increase in
volume of smaller particle sizes, as the particle distribution is
tailored towards a more strongly bimodal distribution.
An analytical study to eliminate major sources of voids was
also performed. This entailed optimisation of the viscosity of
the resin mix, methods of mixing the particulate into the resin
system (to ensure homogeneous dispersions) and control of
the reaction exotherms. These processes are intimately linked
to the volume fraction of the mix and the duration of the
curing process. Therefore, in order to ensure that the required
dielectric properties with minimal voids and uniform
dispersion were achieved, a process design space for each
zone was established through a semi-analytical approach. The
filler material and the resin systems were pre-processed
separately in a vacuum system prior to their mixing.
Subsequently, the filler was added and stirred into the resin
system under vacuum to reduce the void content. Fig. 8 c)
shows a section of a composite with a high void content.
Fig.10 Series of cylinders made using the composite materials, casting and
assembling process of the manufactured lens.
Another difficulty in controlling the dielectric properties of
the structure concerns the phase separation of the mix during
the cure process. Phase separation results in in-homogeneities
6
and consequently a variation of the dielectric properties
around interfaces between particle and the resin matrix. This is
illustrated in Fig. 8 d) which shows a micrograph of a sample
cast using a composite resin mix, where the phase separation
of the filler-resin has resulted in resin rich areas. The final
density of the prepared composites in g/cm3 is presented in
Table II. Next, the permittivity value of the prepared materials
was measured between 8GHz and 12GHz, and in all cases
exhibit a steady dielectric constant and loss tangent over the
frequency band. The dielectric strength of the composite
materials was also measured to evaluate the maximum electric
field strength that the composites can intrinsically withstand
without breaking down. The values are presented in Table II in
kV/mm proving a reasonably high-power handling of the
transformed LL.
Upon preparation of the required dielectric materials a
sequential casting process coupled with machining was
employed. The structure was assembled in two halves, as
shown in Fig. 10, and formed together to obtain the
transformed flat LL.
a)
b)
c)
d)
Fig. 11 Reconstructed far-field of the antenna system formed by the 3-D flat
LL and the X-band horn antenna when the horn antenna is located at the focal
plane and different positions along Y a) (0, 0, ), b) (0, /3, ), c) (0, /1.5,
), d) (0, , ) for a frequency of operation at 10GHz.
B. Experimental Set-up and Measurements
The measurements of the flat LL antenna were performed
with a near field scanner NSI-200V-3x3. The fabricated flat
LL was embedded in a microwave transparent substrate
(ε=1.005) for mechanical support, and a typical pyramidal
horn working in the X-band was used as a feeder located at a
distance of ~=30mm (for an operating frequency of 10GHz)
from the lens. The near field at the horn aperture is similar to
the TE11 mode of a cylindrical waveguide thus producing a
spherical-like wave which will be converted into a plane
wave, resulting in highly-directive radiation in the far-field
region, by the transformed lens. The near field scanner was
employed to measure the field in the vicinity of the antenna
which was later on transformed into the far-field radiation
pattern. Fig. 11 a) to d) depicts the normalized far-field
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radiation pattern of the antenna as a function of the vertical (θ)
and horizontal (φ) angles when the horn is shifted along the Hplane. Four positions are shown d= 0, 10, 20 and 30 mm. The
series of figures demonstrate the correct performance of the
lens with a high directive beam, low SL level and steering
capability (likewise steering in the E-plane can be achieved if
the feeding source is shifted along the x-axis). The
asymmetries in the measurement were mainly due to the
asymmetric configuration in the chamber, the cables and the
metallic structure of the scanner. The cross-polarization was
also measured resulting in a maximum value of -15dB which
is sufficient for most practical applications. In Table III, the
directivity, SL ratio and steering angle achieved are gathered.
A maximum directivity of 17.96dBi, SL ratio of ~ -26dB and a
maximum steering of 34o is attained.
d
0
5
10
15
20
25
30
35
Directivity
(dBi)
17.96
17.82
17.13
16.96
16.52
16.02
16.09
15.95
SL
(dB)
28.3
27.62
26.43
23.76
23.52
20.22
19.14
18.1
Steer angle
(°)
0
4.5
9
12.6
19.8
29.7
32.4
34.1
ACKNOWLEDGMENTS
This work was sponsored by the Office of Naval Research
Global (ONRG) under Naval International Cooperative
Opportunities (NICOP) for the funding support with Grant No.
N00014-09- 1-1013. The authors wish to acknowledge the
support of Dr. O. Quevedo-Teruel and Dr. L. Zhang at Queen
Mary University, on the antenna measurements and Dr. Mike
Dunleavy at the Advance Technology Centre, BAE Systems
for their support on the transformed lens fabrication.
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Table III. Directivity, SL ratio and steer angle for different positions of the
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IV. CONCLUSIONS
We have successfully used a space compression
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