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[10] B. T. Perry, E. J. Rothwell, and L. L. Nagy, “Analysis of switch failures
in a self-structuring antenna system,” IEEE Antennas Wireless Propag.
Lett., vol. 4, pp. 68–70, 2005.
Integration of Circular Polarized Array and LNA in LTCC
as a 60-GHz Active Receiving Antenna
Mei Sun, Ya-Qiong Zhang, Yong-Xin Guo,
Muhammad Faeyz Karim, Ong Ling Chuen, and Mook Seng Leong
Fig. 6. Radiation pattern of the steerable HLWA at 7.5 GHz optimized to steer
the beam to: A) 0 ; B) 40 ; C) 60 .
0
0
IV. CONCLUSION
Simulations show that it is possible to steer the main beam of a halfwidth leaky-wave antenna through nearly 180 by capacitively loading
the free edge, and selectively switching the capacitors to ground. This
may be accomplished at any frequency within the operating band of the
antenna by using a binary search algorithm to locate switch states that
provide an acceptable performance.
It is anticipated that the antenna will be operated in one of two
modes. In the first mode, appropriate switch configurations are determined prior to the operation of the antenna and stored in memory.
These switch states are then recalled to place the antenna main beam
along a desired angle at a certain frequency. By this technique the
beam may either be scanned at a fixed frequency or kept fixed along a
certain angle as the frequency is swept. In a second mode the switch
states may be found in real time to respond to dynamically changing
conditions. This requires a feedback mechanism (such as a measure
of signal strength) and an efficient search algorithm to quickly sort
through potential states. In this mode the antenna acts similar to the
self-structuring antennas reported in [8]–[10].
REFERENCES
[1] A. A. Oliner and K. S. Lee, “The nature of leakage from higher modes
on microstrip line,” in IEEE MTT-S Int. Microw. Symp. Dig., 1986, pp.
57–60.
[2] W. Menzel, “A new traveling wave antenna in microstrip,” in Proc. 8th
Eur. Microw. Conf., 1978, pp. 302–306.
[3] D. Sievenpiper, J. Schaffner, J. J. Lee, and S. Livingston, “A steerable
leaky-wave antenna using a tunable impedance ground plane,” IEEE
Antennas Wireless Propag. Lett., vol. 1, pp. 179–182, 2002.
[4] D. Sievenpiper, “Forward and backward leaky wave radiation with large
effective aperture from an electronically tunable textured surface,” IEEE
Trans. Antennas Propag., vol. 53, no. 1, pp. 236–247, 2005.
[5] C. Luxey and J. M. Laheurte, “Effect of reactive loading in microstrip
leaky wave antennas,” IEEE Electron. Lett., vol. 36, no. 15, pp.
1259–1260, 2000.
[6] M. Archbold, E. J. Rothwell, L. C. Kempel, and S. W. Schneider,
“Beam steering of a half-width microstrip leaky-wave antenna using
edge loading,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp.
203–206, 2010.
[7] G. M. Zelinski, G. A. Thiele, M. L. Hastriter, M. J. Havrilla, and A.
J. Terzouli, “Half width leaky wave antennas,” IET Microw., Antennas
Propag., vol. 1, no. 2, pp. 341–348, 2007.
[8] C. M. Coleman, E. J. Rothwell, J. E. Ross, and L. L. Nagy, “Self-structuring antennas,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp.
11–22, 2002.
[9] C. M. Coleman, E. J. Rothwell, and J. E. Ross, “Investigation of simulated annealing, ant-colony optimization, and genetic algorithms for
self-structuring antennas,” IEEE Trans. Antennas Propag., vol. 52, no.
4, pp. 1007–1014, 2004.
Abstract—A compact 60-GHz active receiving antenna array is designed
and fabricated using low temperature co-fired ceramic (LTCC) technology.
It integrates a 4 4 circularly polarized (CP) patch array and a 21-dB
low noise amplifier (LNA) into the same package substrate. By applying a
stripline sequential rotation feeding scheme the CP array exhibits a wide
2) and 3-dB axial ratio bandwidth, both
impedance bandwidth (SWR
over 8 GHz, as well as a beam-shaped pattern with a 3-dB beam width of 20
and a peak gain of 16.8 dBi. By applying a bond wire compensation scheme
and low-loss transition optimization, the LNA is successfully integrated into
the package. The final fabricated prototype measures only 13 20 1.4
mm , demonstrates the good integration performance as well as the CP polarization characteristics in measurement, and is estimated to have a peak
overall gain of at least 35 dBi.
Index Terms—Aperture-coupled patch antenna array, millimeter wave,
sequential rotation, 60-GHz radio, LTCC, wireless personal area network,
receiver front end.
I. INTRODUCTION
Designs towards low-cost highly-integrated 60-GHz radios have
been carried out using multi-layer low temperature co-fired ceramic (LTCC) based system-in-package (SiP) technology [1]–[4].
Antenna designs are shifting from conventional discrete designs to
antenna-in-package (AiP) solutions [5]–[9]. They have advantages
over antenna-on-chip (AoC) solutions by providing higher gain and
better package solutions [10]. The current AiP has developed from
a single element to an array to achieve higher gain [11]. In addition,
in view of wireless access applications the circularly-polarized (CP)
property is very desirable for 60-GHz antennas. The commonly used
linearly-polarized (LP) antenna necessitates rotating the transmitting
and receiving antenna properly for polarization matching, particularly
in the case of the line-of-sight (LOS) radio links. Using the CP antenna
this problem can be mitigated while also allowing for reduction in
interference from multi-path reflections. Thus many researches have
been pursued on 60-GHz CP array antennas [12]–[16]. However, there
is no report on the integration of the CP antenna with active circuits
at 60 GHz in the package. Based on our previous preliminary results
Manuscript received November 11, 2009; revised November 16, 2010; accepted November 19, 2010. Date of publication June 07, 2011; date of current
version August 03, 2011. The work was supported in part by the National University of Singapore Young Investigator Award 2009 and in part by the Agency
for Science, Technology and Research (A*STAR), Singapore.
M. Sun, M. F. Karim, and L. C. Ong are with the Institute for Infocomm
Research, Singapore 138632, Singapore.
Y. Q. Zhang, Y. X. Guo and M. S. Leong are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore
117576, Singapore (e-mail: eleguoyx@nus.edu.sg).
Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAP.2011.2158781
0018-926X/$26.00 © 2011 IEEE
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presented in [17], [18], in this communication, we report the details
of the integration of the CP antenna with a low-noise amplifier (LNA)
that boosts the receiving power at 60 GHz.
Design of an integrated CP antenna is challenging at 60 GHz to
achieve a high gain as well as wide axial ratio (AR) and impedance
bandwidths. The high gain is achieved by a patch array of 4 2 4 elements. The wide AR and impedance bandwidths both are achieved
by a unique stripline (SL) sequential rotation feeding scheme [17].
This feeding scheme is a well known technique to enhance AR and
impedance bandwidth [19]. The antenna also features aperture-coupled topology and SL feeding structures [17]. All these make the antenna performance less sensitive to the surrounding dielectric and metal
layers. They also decouple the design of the antenna from the exact
physical properties of the package, simplifying simulation and modeling complexity in the antenna integration with active devices.
Integration of the LNA into the package is challenging with wire
bonding technology at 60 GHz. A number of studies on the electrical
performance of wire-bonding interconnection have been reported for
microstrip and coplanar configurations, indicating that the insertion
loss for a bonding wire will be drastically increased with frequency as
the bond wire acts as a series inductor [20]–[22]. The flip-chip bonding
technology uses metallic bumps for device connections, which are kept
small (less than 100 m) compared to the length of the bond wire.
This results in better impedance matching, and reduces interconnection
losses and parasitic effects of transition discontinuities in mm-wave
systems [23]. The advanced ribbon or double bond wire techniques are
also possible to use to decrease the bond wire inductance, where the
ribbon is much more commonly used to bond mm-wave devices than
double bond wires due to their complexity [20], [24]. Nonetheless, the
wire-bonding technique, well established in consumer electronics, remains a very attractive solution since it is robust and inexpensive. In addition, it has the advantage of being tolerant to chip thermal expansion,
an important requirement for many applications. In order to improve
the high-frequency performance of a bond-wire interconnect, efforts
have usually focused on reducing the length of the bond wire and also
reducing the chip-to-package spacing. However, limitations in manufacturing require longer bond-wire lengths and wider chip-to-package
spacing to improve the yields of mm-wave chip-package assemblies.
In this communication we use a T-network bond wire compensation
scheme [25] to enable a bond wire with a length of 500 m and a diameter of 50 m, to be used. This is almost a tripled length of the shortest
bond wire supported by the current technology and would thus greatly
improve the yield of assembly of the chip with the package. This makes
our integration solution suitable for mass production. From chip-antenna interconnect aspect, low loss transitions have to be developed. In
this communication, we optimize these transitions to achieve the lowest
interconnection loss.
In Section II we present our active circular polarized array antenna
in a Ferro-A6M LTCC package with a relative dielectric constant of
"r = 5:9 and a loss tangent of tan = 0:002. The design of the antenna array and the study of its integration will be done in the HFSS
from Ansoft. The key integration challenges are solved by designing
the bond wire compensation scheme and developing low loss transitions. A proof-of-concept prototype is finally designed. It is then fabricated and finally tested to obtain integration performance as well as
polarization characteristics in Section III. Finally, Section IV concludes
the communication.
II. INTEGRATION OF ANTENNA AND LNA
Fig. 1 shows the geometry of the 60-GHz active receiving antenna
in LTCC package. It integrates the 4 2 4 antenna array presented
in [17] with a LNA. As shown the package consists of five cofired
Fig. 1. Geometry of the integrated array antenna with LNA (size:
13 19.85 1.4 mm ): (a) 3D top view, (b) 3D explored view, and (c) zoom
in view of the transitions with a 50-
dummy microstrip line on chip.
2
2
laminated ceramic layers with their thickness denoted respectively in
Fig. 1(b). There are also five metallic layers. The layer M1 provides the
metallization for the patch array antenna. The layer M2 provides the
metallization for the antenna ground with aperture and the grounded
co-planar waveguide (GCPW) for chip interconnection as well as
chip DC biasing layout. The layer M3 provides the metallization
for the antenna strip line feeding traces. The layer M4 provides the
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3085
TABLE I
BOND WIRE EFFECT ON jS j PERFORMANCE OF TRANSITIONS
BOND WIRE EFFECT ON jS
TABLE II
= S j PERFORMANCE OF TRANSITIONS
j j
metallization for the GCPW ground, and the layer M5 the metallization for the system ground plane. The size of the whole package is
13 2 19.85 2 1.4 mm3 . Note that the active antenna features standard
wire bonding to chip. A T-junction structure as shown in Fig. 1(c)
is used to compensate the bond wire larger inductance at 60 GHz.
This enables the longer bond wire of 500 m to be used. This also
makes our integration solution suitable for mass production. The bond
wire compensation will be presented in the following Section II.A. In
addition, the active antenna features low loss transitions as denoted in
Fig. 1(c). Their design and performance optimization will be presented
in the following Section II.B. Based on the bond wire compensation
and transition optimization, by using a 50-
dummy microstrip line
on chip as shown in Fig. 1(c) the whole receiving antenna is simulated.
This will give us an idea of the effects of transitions on the integration
performance.
A. Bonding-Wire Compensation
As shown in Fig. 1(c) we use a T-network [25] to compensate for the
series inductance introduced by the bond wire. The bond wire length
is fixed at 500 m to facilitate the compensation structure design. The
transition 1 as shown in Fig. 1(c) is simulated, where a 50-
GCPW
is cascaded by a T-junction bond wire compensation line (by layers of
M2 and M4) to connect to the chip 50-
dummy line using bond wires.
The effect of the bond wire compensation network on the electrical performance is shown in Fig. 2, where the 2-mil bond wire is used. As we
can see, the matching is optimized at 60 GHz, leading to a better return
loss from 9 dB to 18 dB and a reduction in insertion loss from 1.18 to
0.6 dB at this frequency. In simulation, it is also found that the bond
wire length variation of 625 m will have insignificant effect on performance. Tables I and II also show the variation of the return loss and
insertion loss with the bond wire diameter. As expected, with thinner
bond wires the transition 1 has the worse performance, indicated by an
increased insertion loss value. However, if the wire bonding condition
in fabrication is limited to 1-mil, the performance is still acceptable
with less stringent requirement.
To study the bond wire compensation experimentally, one 1-mil
bonding wire and two 1-mil bonding wires for back-to-back cases are
employed as shown in Fig. 3(c). Figs. 3(a)–(b) show the measured
results. It is demonstrated that the wire bondings with compensation
show the better performance at 60-GHz band than their un-compensated counterparts. Also as expected the 2-wire bondings with
compensation achieve the best performance in terms of matching and
insertion loss.
Fig. 2. Simulated results for transition 1 compared with the results without
compensation (500-m long 2-mil bond wire is used): (a) jS j & jS j and
(b) jS j.
B. Low Loss Transitions
In the integration design, attenuation caused by radiations at the discontinuities and impedance mismatch along the transmission lines and
integrated devices should be minimized for power efficiency and noise
performance of the mm-wave system. By properly designing the structure and placing grounding vias around the transitions, the attenuation
can be remedied. In our transition 2 design, the GCPW is cascaded by
a signal via and finally fed to the strip line of the antenna. Through
this transition the strip line fed antenna is transformed to GCPW-fed
format making it convenient for probe-touching test. Fig. 4 shows the
performance of this transition. It is seen that the matching is optimized
at 60 GHz, leading to a good return loss of 20 dB and insertion loss of
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Fig. 4. Simulated results for transitions 2–4: (a) jS
j
& jS
j
and (b) jS
j
.
Fig. 3. Measured results for bond wire compensation study (1-mil bond wires
are used): (a) jS j, (b) jS j, and (c) photo of bondings.
0.28 dB. In the transition 3 design, the bond wire compensation structure is cascaded by the transition 2. This increases the insertion loss
to 0.57 dB at 60 GHz with input and output matching still better than
20 dB. In the transition 4 design, the transition 1 and transition 3 combined to lead to a further increased insertion loss of 0.7 dB at 60 GHz
while input and output matching still better than 20 dB. It should be
noted that 0.7 dB attenuation is quite satisfactory for this complex transition with bond wires. Tables I and II also show the variation of the
return loss and insertion loss with the bond wire diameter. As expected,
with thinner bond wires the transitions 3 and 4 have the worse performance, indicated by an increased insertion loss value. It should be
noted that the whole transition performance in some bond wire cases is
better than the individual performance of its parts, owing to the global
optimization.
C. Integration Simulation
With a dummy 50-
microstrip line on chip, we evaluate the performance of the whole antenna as shown in Fig. 1. It is found from Fig. 5
that transitions degrade the impedance bandwidth. It is reduced to 16.5
GHz (53–69.5 GHz) from 18.5 GHz (51.5–70 GHz) for VSWR < 2.
In addition, the peak gain is reduced from 16.8 dBi to 14.9 dBi by
1.9 dB. The 3-dB gain bandwidth remains at 10 GHz but shifts from
56.8–66.8 GHz to the lower frequency band of 54.5–64.5 GHz. The
Fig. 5. Simulated performance of the integrated array antenna compared with
un-integration counterpart in [17]: (a) jS j, (b) gain and axial ratio.
3-dB axial ratio bandwidth broadens from 8 GHz (55.5–63.5 GHz) to
9 GHz (54.25–63.25 GHz) due to the AR performance improvement at
the lower frequency.
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3087
Fig. 6. Photograph of the fabricated samples for test: (a) the referenced array
antenna without amplifier and (b) the active array antenna with amplifier.
Fig. 7. Antenna test set up: (a) pattern measurement and (b) polarization study.
III. ANTENNA TESTING
The active antenna is tested by comparison with a counterpart sample
without LNA. The CP array without LNA was fabricated as shown in
Fig. 6(a). Note that the GCPW-SL transition is used in this sample to facilitate testing. In simulation, it is found that GCPW-SL transition does
not obviously degrade the antenna performance except for a 0.6-dB
peak gain penalty (from 16.8 dBi to 16.2 dBi). The antennas were tested
by Agilent E8361A vector network analyzer (VNA) up to a frequency
of 67 GHz. A GSG RF probe with a pitch of 250 m was touched on the
GCPW line of the antenna for testing. The antenna patterns were measured with the set up at the National University of Singapore as shown
in Fig. 7(a). With proper probe touching, the tested antenna senses the
radiation in the boresight direction from a WR-15 standard pyramidal
horn antenna. The horn antenna has an aperture size of 13 2 10 mm2
and a gain of 15 dBi. The far field region limit of the horn antenna can
be calculated as 2 2 0 10 8 cm, where is the largest dimension
of the horn and 0 is the free space wavelength. The distance between
the horn aperture and the antenna under test is set around 15 cm.
Fig. 8 shows the simulated and measurement results for the antenna
without LNA. It is found from Fig. 8(a) that the antenna has a wide
impedance matching bandwidth larger than 8 GHz, which is 11.5 GHz
2. If there was no measured hump
from 50.5–62 GHz for SWR
around 63.5 GHz, the bandwidth would be even larger. The occurrence
of the hump enlightens us to improve our basic array matching performance to eliminate this hump by optimizing the antenna dimensions as
shown in Fig. 5(a). Fig. 8(b) shows a measured peak gain 16 dBi over
60 GHz band. Fig. 8(c) also confirms the good 3-dB AR bandwidth performance in measurement. Figs. 8(d)–(f) present the measured patterns
in the upper half-plane.
A 21-dB Low noise amplifier (LNA), HMC-ALH382 [26] is used
in the finally fabricated active antenna. The amplifier die has a size
of 1.55 2 0.73 2 0.1 mm3 with pads laid out as illustrated in Fig. 1
to facilitate the wire bonding. Fig. 6(b) shows the photograph of the
final fabricated active antenna for measurement. It measures only
13 2 20 2 1.4 mm3 . The active antenna was also measured with a
D =
:
D
<
>
Fig. 8. Measured and simulated performance for the antenna without LNA:
(a) jS j, (b) peak gain, (c) axial ratio at the main radiation direction, and xz-plane
& yz-plane radiation patterns at (d) 57 GHz, (e) 60 GHz, and (f) 64 GHz.
VNA analyzer. Another DC probe was touched to provide a biasing
drain voltage of +2 5 V and a gate voltage of 00 3 V. A total 64 mA
:
:
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Fig. 9. Measured jS
j
for the antenna with LNA.
Fig. 10. Measured jS j for the antenna with and without LNA (0 position:
E field of horn in the y -direction, and 90 position: E field of horn is in the xdirection).
drain current was thus obtained in the measurement. Fig. 9 shows a
measured bandwidth of 4.3 GHz (60.7–65 GHz) for SWR < 2.
Finally, the set up as shown in Fig. 7(b) is used to study the polarization characteristics of the antenna. The horn antenna is placed on
the top of the AUT antenna. The apertures of the two antennas are both
paralleled to the xy-plane. The tests are then conducted with horn at 0
position (E field of horn is in the y - direction) and 90 position (E field
of horn is in the x- direction), respectively. Fig. 10 shows the measured
transmission performance. The circularly polarized characteristics are
observed with both transmission coefficients at 0 and 90 positions
having the similar amplitude. In addition, compared with the transmission coefficients measured in the case of without amplifier, those measured with amplifier show an average increase of approximately 19 dB
from 60.7–65 GHz, which is very close to the amplifier gain value of 21
dB deducted by a 1.9-dB loss. Based on the above, the final estimated
peak gain of the active antenna is at least 35 dBi.
IV. CONCLUSION
A wideband circular polarized patch array of 4 2 4 elements was integrated with a 21-dB LNA to form a compact 60-GHz active receiving
antenna, where the low loss transition structures were optimized and the
bond wire compensations were used to improve the performance of the
60-GHz chip-package interconnection. The fabricated active antenna
prototype measures only 13 2 20 2 1.4 mm3 . The antenna test was finally conducted to study antenna performance. The wide impedance
bandwidth was confirmed by measurement. The circularly polarized
characteristic was also observed. The final estimated peak gain of the
active antenna is at least 35 dBi. The designed active receiving antenna
will find application in the 60-GHz wireless personal area network.
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Fig. 1. Geometry and notations of the problem. The curve line indicates the
branch cut in real space due to CSP oriented to radiate at '
.
= 180
Two-Shell Radially Symmetric Dielectric Lenses as
Low-Cost Analogs of the Luneburg Lens
where R is the lens radius, or its discrete analog for the uniformlylayered LL
A. V. Boriskin, A. Vorobyov, and R. Sauleau
"n = 2
Abstract—This communication provides guidelines for the design of twoshell radially symmetric dielectric lenses with collimating capabilities compatible with those of the classical Luneburg lens. Unlike earlier publications, it is demonstrated that such a lens can be designed using any standard
low permittivity dielectric materials, provided the optimal shell thickness
is selected. The lens characteristics are studied in 2-D formulation using
exact series representation. A detailed description is given for lenses with
and core made of Rexolite. The design recommendations
radius of 10
and cores made of
are then generalized for lenses with radii of 5 to 15
Teflon, Fused Silica and Quartz. Finally, a chart defining the optimal shell
thickness for lenses made of two arbitrary dielectric materials is provided.
Validity of the recommendations for design of 2-D and 3-D radially symmetric lenses is proven by comparison with optimal designs reported by
other authors.
Index Terms—Dielectric lens, lens antennas, luneburg lens.
I. INTRODUCTION
A radially symmetric dielectric lens capable of collecting rays in a
focus on its rear surface is an attractive solution for many applications
from microwave to optical ranges [1]–[10]. Such a lens, known as the
Luneburg lens (LL) [11], can be designed as a radially inhomogeneous
or multi-shell sphere, e.g., [2], [5], [12]. A critical aspect in the design
of multi-shell LLs is the limited number of available low-loss dielectric
materials whose permittivity belongs to the range of 1–2, as suggested
by the ray-optics focusing rule
"(r ) = 2
0
r 2
;
R
r
2 [0; R]
(1)
Manuscript received September 20, 2010; revised November 14, 2010; accepted December 13, 2010. Date of publication June 07, 2011; date of current
version August 03, 2011. This work was supported in part by the Université Européenne de Bretagne, France, by the ESF in the framework of the RNP-NEWFOCUS, by the Fondation Michel Métivier, and in part by the North Atlantic
Treaty Organization under Grant RIG983313.
A.V. Boriskin is with the Institute of Radiophysics and Electronics NASU,
Kharkov, Ukraine, and also with the Institute of Electronics and Telecommunications of Rennes (IETR), UMR CNRS 6164, University of Rennes 1, Rennes,
France (e-mail: artem.boriskin@ieee.org).
A. Vorobyov and R. Sauleau are with the Institute of Electronics and Telecommunications of Rennes (IETR), UMR CNRS 6164, University of Rennes 1,
Rennes, France.
Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAP.2011.2158793
0
(n
0 0:5)
N
2
;
n = 1; . . . ; N
(2)
where "n is permittivity of the n-th layer and N is the total number of
layers (n = 1 corresponds to the most inner layer).
There are well known ways to fabricate artificial materials with desired properties via milling holes or adding compounding materials,
e.g., [13]–[15]. Nevertheless each of them increases complexity and
cost of the technology especially for multi-shell lenses. Therefore a favorable solution for low-cost radially symmetric lenses is a single- or
double-shell design with layers made of the available low-loss dielectric materials, e.g., [1], [3], [4], [6], [12].
As demonstrated in [12], the minimum number of layers needed to
provide the collimating capabilities compatible to those of a classical
LL is two. The optimal design for the two-shell lens, found in [12]
using global optimization technique, is a dense quartz-like core covered with a quarter-wavelength matching layer. Such a design exhibits
critical drawbacks, namely: increased overall weight, difficulties in fabrication and further exploitation of a fragile matching layer, and finally
involvement of the whispering gallery (WG) modes whose impact on
the antenna characteristics grows rapidly with increase of the dielectric
contrast on the shell boundaries [16].
In this communication we demonstrate that a two-shell lens with desired collimating capabilities can be designed using any standard low
permittivity dielectric materials. We also present a chart with tabulated
data generated from many simulation cycles that enables one to determine the optimal parameters of a two-shell lens made of two arbitrary dielectric materials without solving the corresponding diffraction
problem.
The communication is built as follows. After a brief outline of
the solution given in Section II, a Rexolite-core lens illuminated by
a wave beam is studied in detail in Section III-A. Then the impacts
of the lens size, feed location and core material are investigated in
Sections III-B–III-D, respectively. Finally, conclusions are outlined in
Section IV.
II. OUTLINE OF THE SOLUTION
We consider the problem in two-dimensional (2-D) formulation and
model the lens as a multi-shell circular dielectric cylinder (Fig. 1). To
study its collimating capabilities, the lens is illuminated by a wave
beam radiated by a complex-source point (CSP) feed, i.e., a line current
located in a point with a complex coordinate. In real space such a feed
radiates a beam whose waist is controlled by the value of the imaginary
part of its coordinate, b [17].
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