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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 18, NO. 8, AUGUST 2019
An Axial Compression Planar Lens Antenna Based
on Discrete Transformation Electromagnetics Using
High-Impedance Surface Cells
Yonghong Zhou , Xin Duan , Lin Zhou , Xing Chen , Senior Member, IEEE, and Pan Feng
Abstract—A two-dimensional (2-D) convex lens is reshaped and
compressed into a 2-D flat lens with a compression ratio of 52.6%
by discrete transformation electromagnetics. The single-layer highimpedance surface cells are used to fill with the compressed zone
to form a planar flat lens. A parallel-plate waveguide and a 1/4 λ
reflector are used to ensure that the electromagnetic energy generated by the monopole flows to the lens. The planar lens antenna,
4.35 λ × 4 λ on the lens plane and 0.375 λ tall, is fabricated and measured in TM mode at 7.5 GHz. The measured data indicate that the
proposed planar lens antenna achieves an 11.3 dBi gain, 16.7° halfpower beamwidth on H-plane, and almost 260 MHz impedance
bandwidth for S11 . The joint design of discrete transform electromagnetics and high-impedance surface cells is a new method,
which has strong reference value for the miniaturization of other
lens antennas.
Index Terms—Discrete transform electromagnetics, highimpedance surface cells, planar lens antenna, two-dimensional
(2-D) flat lens.
I. INTRODUCTION
LANAR metasurface lenses are low-profile and simpler to
manufacture compared with standard dielectric lenses, thus
offering advantages for new communication antennas and sensor
applications [1]. In recent years, the research on planar lens antenna is mainly focused on Luneburg lens [1]–[3], half Maxwell
fish-eye (HMFE) lens [4], [5], and gradient index flat lens [6],
[7]. A lot of studies have been conducted on new unit structures [1], [2], [4], profile height reduction [3], and multi-beams
and beam steering [5]–[7]. However, little attention is paid to
the compression of the axial size of the lens. Luneberg turned
a Luneberg lens into an HMFE lens in 1944 [8], but the compression method of other lenses was not involved. Furthermore,
a circular profile of the HMFE makes it difficult to be applied in
some scenarios. The gradient index flat lens has no issues on the
shape profile, but its axial dimension cannot be reduced easily.
P
Manuscript received May 31, 2010; accepted June 20, 2019. Date of publication June 26, 2019; date of current version August 2, 2019. This work was
supported in part by the Major Project for Natural Science of the Education Department of Sichuan Province under Grant 17AZ0384, in part by the Meritocracy
Research Funds of China West Normal University under Grant 17YC057, and
in part by the Initial Doctor Research Funds of China West Normal University
in 2018 under Grant 18Q072. (Corresponding author: Yonghong Zhou.)
Y. Zhou and P. Feng are with the School of Electronic and Information Engineering, China West Normal University, Nanchong 637002, China (e-mail:
scnczyh@163.com; panpanff@outlook.com).
X. Duan, L. Zhou, and X. Chen are with the College of Electronics and
Information Engineering, Sichuan University, Chengdu 610065, China (e-mail:
xinduan@outlook.com; linzhouworking@126.com; xingc@live.cn).
Digital Object Identifier 10.1109/LAWP.2019.2925074
Shrinking the axial dimensions and reshaping the profile of the
lens poses a great challenge.
Transformation electromagnetics provides a potential solution that reshapes the profiles of any lens while compressing
its dimension [9]–[11]. Discrete transform electromagnetics is
a discretization technology of transformation electromagnetics
proposed by Tang [12]. Since the discretization avoids magneticdependent tensor media that transformation electromagnetics
could lead to [12], [13], it allows the metasurface, in particular
the high-impedance surface, to be used. The most popular
way to realize high-impedance surfaces is placing periodic
metallization on stratified dielectrics, without or with shorting
vias; the latter is the case of the popular mushroom surface [5],
[14]–[19]. Its equivalent relative permittivity can be designed as
needed, and its efficiency is higher. The non-resonant microstrip
lines could also be used to realize the flat lens with a broader
bandwidth, but the efficiency is relatively low. In [20], a spatial
multi-layers structure was utilized to improve the efficiency,
and the directivity of the flat lens antenna was only 5 dBi.
A new design method of the axial compressed planar lens
antenna with a single-layer structure lens and high gain is presented in this letter. The arc profile of the two-dimensional (2D) convex lens is reshaped into a straight line, its axial length
is compressed by 47.4% using the discrete transformation electromagnetics method, and the high-impedance surface cells are
used to fill the compressed zone to achieve high efficiency. The
lens is a real planar flat lens because only one layer of cells is
used to fill the zone and the outline is a straight line. Finally, a
high-gain planar lens antenna is achieved by putting the planar
lens into a parallel planar waveguide. This design paves a feasible path toward the axial miniaturization of other planar lens
antennas. The contents of this letter are arranged as follows.
In Section II, 2-D convex lens reshaping and compression, as
well as the impedance value calculation, are presented in detail.
The designs of the high-impedance surface cells and its phase
adjustment performance are discussed in Section III. The simulation and test results of the proposed 2-D axial compression
planar lens antenna are given in Section IV. The conclusion and
prospect are presented in Section V.
II. 2-D CONVEX LENS COMPRESSION AND IMPEDANCE
VALUE CALCULATION
A. Compression Formula
n2 εr (x, y)μr (x, y)
ΔxΔy
Δx Δy (1)
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ZHOU et al.: AXIAL COMPRESSION PLANAR LENS ANTENNA BASED ON DISCRETE TRANSFORMATION ELECTROMAGNETICS
Fig. 3.
Fig. 1. (a) 2-D thick convex lens’ structure parameters and meshes.
(b) Reshaped and compressed 2-D flat lens’ structure parameters and meshes.
Fig. 2.
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FWHM index is about 2° by the meshes in Fig. 1(a).
TABLE I
EFFECTIVE IMPEDANCE VALUES OF THE GRIDS (UNIT: jΩ)
Corresponding relation of variables.
where εr (x, y) and μr (x, y) are the relative permittivity and
relative permeability of the distorted coordinate system, respectively. Δx and Δy also belong to the distorted coordinate systems, and they should be near-orthogonal to each other. Δx’ and
Δy’ belong to the orthogonal one [12], as shown in Fig. 1.
B. Lens Reshaping and Compression and Mesh Generation
A 2-D thick convex and the reshaped and compressed 2-D
flat lens with all the structure parameters are shown in Fig. 1(a)
and (b). To meet the requirement of formula (1), every mesh cell
should be near-rectangle. A software named as Pointwise 17.0
is used to mesh the grids, and the result is also shown in Fig. 1(a)
and (b).
C. Mesh Quality Evaluation
After meshing the 2-D convex lens by Pointwise software,
one can export the coordinates of the grid nodes. The full-width
at half-maximum (FWHM) index is used to evaluate the mesh
quality [12]. To calculate the FWHM index, Δx, Δy and their
included angle A of each grid should be obtained.
Fig. 2 shows the corresponding relationship of the variables.
The start grid M(1, 1) and the start node N(1, 1) are shown in
Fig. 1(a). Let us assume that the grid of the ith row, jth column is
M(i, j), the coordinate for node N(i, j) is (xi,j , yi,j ), N(i, j+1) is
(xi,j+1 , yi,j+1 ), and so on. According to Fig. 2, the following
formulas can be obtained by the cosine law:
Δxi,j = sqrt((xi,j − xi,j+1 )2 + (yi,j − yi,j+1 )2 )
(2)
Δyi,j = sqrt((xi,j − xi+1,j )2 + (yi,j − yi+1,j )2 )
(3)
Ai,j = cos−1
2
Δx2i,j +Δyi,j
−((xi+1,j − xi,j+1 )2 +(yi+1,j − yi,j+1 )2 )
×
2Δxi,j Δyi,j
(4)
Ai,j+1 = cos−1
2
Δx2i,j +Δyi,j+1
−((xi,j −xi+1,j+1 )2 +(yi,j −yi+1,j+1 )2 )
×
.
2Δxi,j Δyi,j+1
(5)
Using (2)–(4), the FWHM index is about 2°, as shown in
Fig. 3. This indicates that the orthogonality of the grids in
Fig. 1(a) meets the requirement of formula (1) [12].
D. Impedance Values Calculation
The refraction index of grids in the orthogonal coordinate system can be obtained by formula (1); then the effective impedance
of grid can be obtained by
Zi,j = jη0 n2 − 1
(6)
where η0 is impedance of free space. Table I shows the effective
impedance values corresponding to the grids of the enclosed
area in the blue dashed line in Fig. 1(b). The other impedance
values may be derived from symmetrical relation.
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 18, NO. 8, AUGUST 2019
Fig. 5. 2-D flat lens simulation models and results (phase snapshot of Ez );
the PEC blocks at both ends of the lens are to prevent the leakage wave [12].
Fig. 4. Structure of the impedance surface cell. (a) Top view of non-slotted
square cell. (b) Side view of non-slotted square cell. (c) Top view of slotted
square cell. (d) Side view of slotted square cell.
TABLE II
EFFECTIVE IMPEDANCE VALUES OF THE PROPOSED CELLS NEAR 7.5 GHZ
III. IMPEDANCE SURFACE CELLS DESIGN AND
PERFORMANCE VERIFICATION
A. Impedance Surface Cells Design
By using the eigenmode solver of HFSS 15.0 [15]–[16], the
final structure of the impedance surface cell and its effective
impedance values are shown in Fig. 4 and in Table II, respectively. Because the high-impedance surface cells are dispersive,
the impedance values shown in Table II are obtained at the operating frequency of 7.5 GHz.
B. Phase Adjustment Performance Verification
There are 40 different impedance values in Table I, but only
seven different ones in Table II, which leads to a low-resolution
2-D flat lens. Therefore, the phase adjustment performance of
impedance values provided by Tables I and II must be evaluated. For comparison, the simulation result of the original 2-D
convex lens is given. Fig. 5(a) is the original convex lens. The
phase of the vertical electric filed (Ez ) in the black box on the
right presents straight lines, but concentric circles in the black
box on the left. Fig. 5(b) is the 2-D flat lens with high resolution.
The wavefront on the right is approximate straight lines, which
indicates that the meshing and impedance value calculation are
correct. Fig. 5(c) is the 2-D flat lens with low resolution. Its performance is not much different from that of the high-resolution
one, which indicates that the high-impedance surface cells in
Fig. 4 can be used to fill with the compressed 2-D flat lens.
Fig. 6. Planar flat lens antenna. (a) Simulation model. (b) Manufactured sample of the planar flat lens. (c) Proposed antenna without cap. (d) With cap.
IV. PLANAR LENS ANTENNA DESIGN AND RESULTS ANALYSIS
To prevent the leakage of the electromagnetic energy, a metal
cover and enclosure are used on top and bottom, shown in
Fig. 6(a). In total, 960 cells are used in the design, of which
the type of C1 has 96 cells, C2 has 96 cells, C3 has 96 cells,
C4 has 96 cells, C5 has 192 cells, C6 has 192 cells, and C7 has
192 cells, shown in Fig. 6(b) and Table II. The unit cell structure and parameters of C1–C7 are shown in Fig. 4 and Table II,
respectively.
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ZHOU et al.: AXIAL COMPRESSION PLANAR LENS ANTENNA BASED ON DISCRETE TRANSFORMATION ELECTROMAGNETICS
Fig. 9.
Fig. 7.
one.
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Gain of the proposed antenna on H-plane.
Phase distribution of Ez . (a) With the planar flat lens. (b) Without the
Fig. 10.
Fig. 8. Directivity of the antenna on H-plane with the planar flat lens and the
one without the planar flat lens.
Although the proposed antenna in Fig. 6 looks like a horn
antenna with a metasurface cover [21]–[25], the antenna we
designed is a planar metasurface lens antenna. Because it is
the same way that planar metasurface lens antennas and the
proposed one adjust the phase of electromagnetic wave. The
TM polarization surface-wave excited by the probe is adjusted
to in-phase radiation by the planar flat lens to achieve beam
convergence and gain enhancement [1], [5].
The phase distribution of Ez of the proposed antenna is shown
in Fig. 7(a). For comparison, we also show the phase distribution without the planar flat lens in Fig. 7(b). It is evident that
the cylindrical wave which the monopole antenna produced is
modulated into a plane wave.
The directivity of the antenna on H-plane is about 11.5 dBi,
while the one without lens is about 9 dBi, which are plotted in
red and black lines in Fig. 8, respectively. From Fig. 8 we can
also see that the sidelobe with the planar flat lens is larger, which
is mainly due to the impedance mismatch between the planar flat
lens and free space [6], [7]. Since the planar flat lens can only
Simulated and measured S11 of the proposed antenna.
achieve in-phase manipulation rather than equal-amplitude and
in-phase manipulation, the directivity improvement is limited,
as for this case, the improvement is about 2.5 dB. The measured
gain on the H-plane is 11.3 dBi, it is in good agreement with
the directivity, which indicates the losses of the antenna is only
0.2 dB, shown in Fig. 9. The planar flat lens also has a narrowing effect on the beam width. After adding the planar flat
lens, the half-power beam width is compressed from about 36°
to 19°. The reflection coefficient S11 of the antenna is shown
in Fig. 10. The simulated and measured data agree well. The
impedance bandwidth is about 260 MHz. Compared with similar lens antennas composed of non-resonant metamaterials [20],
the antenna we proposed has a narrower bandwidth but higher
gain.
V. CONCLUSION
A 2-D convex lens is reshaped and compressed into a 2-D flat
lens with a compression ratio of 52.6% by discrete transformation electromagnetics, and the high-impedance surface cells are
used to fill with the compressed zone. Therefore, an axial compressed planar lens antenna with a single-layer structure lens
and high gain is presented. Thanks to the design method, the
lens’s axial length is reduced by 47.4% compared with the original convex lens, and a measured gain of the proposed antenna
11.3 dB is achieved. In addition, the single-layer printed circuit
board (PCB) configuration makes the lens easy to fabricate at
low cost. The proposed design method may be an inspiration to
the axial compression of other lens antennas.
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 18, NO. 8, AUGUST 2019
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