Design of Broadband Reflectarray Antenna Using

Machine-Learning-Assisted Optimization Method

Yi Cao, Qi Wu, Haiming Wang, and Wei Hong
State Key Laboratory of Millimeter Waves, Southeast University
No. 9, Mozhou East Road, Jiangning District, Nanjing 211111, China
Email: {yicao, qiwu, hmwang, weihong}@seu.edu.cn

Abstract-A machine-learning-assisted optimization (MLAO)

method is presented to design a broadband reflectarray antenna.
First, a Gaussian process regression (GPR) surrogate model is con-
structed on-line to predict the phase responses of the unit cells with
different design parameters. Patch size and frequency are selected
as two additional features in the training set to improve the predic-
tion accuracy. Then, the phase range and slope, which are two vital
parameters for reflectarray antennas, are mathematically formu-
lated as two objectives and optimized using the evolutionary algo-
rithm (EA) based on the surrogate model. Finally, a reflectarray (a) (b)
antenna composed of 576 elements is designed using optimized unit
cells. Good radiation pattern is achieved with simulated 1-dB gain Fig. 1 Structure of the dual-resonance unit cell: (a) Top view; (b) Front view.
bandwidth of 45.5%.
II. DESIGN PROCEDURE
I. INTRODUCTION A. Unit Cell Analysis
Reflectarray antennas have been widely used in both terres- Multi-resonanace unit cells are usually used to increase the
trial and satellite communication systems due to their ad- bandwidth of reflectarray antennas [5]-[7]. A dual resonance
vantages including planarity, controllable beam direction and unit cell is shown in Fig. 1, which is composed of a square patch
high gain, etc. Every element in different position of the array and a square ring. Its structure is similar to the one in [5]. The
should offer a phase compensation to achieve a good radiation side length of the square unit cell is L  3 mm , about 0.470 at
pattern. For variable-size elements, phase reflection coefficient 47 GHz. We use w, h1, and h2 to repectively repesent the width
versus patch size curve is an important characteristic and the ba- of the square ring, the thickness of the substrate and the height
sis tool to design the array. The phase range should cover 360° of the substrate suspended over the ground. The side lengths of
and the slope should be stable with the change of the patch size the square and the ring are denoted by Ls and Lr, and their ratio
over the design frequency band. Thus, the elements can offer of is representd by a = Ls / Lr. These four parameters w, h1, h2,
stable phase compensations in the array to overcome the narrow and a are set as designable parameters in the MLAO method.
operating bandwidth of microstrip reflectarray. And the relative permittivity of the substrate is  r  0.22 .
Surrogate model based methods have been applied to antenna As mentioned above, to guarantee good performance of the 1-
design over the last decade [1]-[2]. In [3]-[4], the learning-by- dB gain bandwidth and the radiation pattern, the phase range
example strategy based on kriging or support vector machines is should cover 360° and the phase curves at different frequencies
introduced to evaluate the scattering response of elements with should be as linear as possible. These two objectives can be
designable parameters, which can be considered as a regression mathematically formulated as:
problem. It is more efficient and convenient than building scat- 1 N
tering matrix-versus-descriptors lookup tables for complex- O1  x    rmse pn  x  , pˆ n  x 
N n 1
(1)
shaped elements through full-wave simulations.
In this work, a machine-learning-assisted optimization f fail  x 
O2  x   (2)
(MLAO) method is used to design a broadband reflectarray an- N
tenna. An online Gaussian process regression (GPR) surrogate where O1 ( x ) is the average value of the root-mean-square error
model is constructed to predict the phase responses of a variable- (RMSE) between the phase response curve pn ( x) and its fitted
size reflectarray element based on a small sample training set straight line pˆ n ( x) at N frequency points within the designated
calculated by a full-wave electromagnetic (EM) solver. The bandwidth, and O2 ( x ) is the ratio of the frequency points at
phase range and the slope of the phase reflection coefficient ver- which the phase range fail to cover 360° f fail ( x ) to total number
sus patch size curve are selected as the objectives in optimiza- of frequency points N.
tion algorithm to find the optimal shape of the unit cells. A re-
flectarray antenna is finally constructed to verify the method.

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 pre-defined reference point helps to find the Pareto-optimal re-
gion closest to the reference points. The optimal individual ob-
tain by EA in this loop will be validated through EM simulation
and then used to update the database. Go back to Step 2.
III. EXPERIMENTAL RESULTS AND DISCUSSION
Table I shows the variation ranges of the designable parame-
ters, which are firstly obtained following the design principle
and then optimized by the proposed optimization method. A
576-element reflectarray is designed to test the performance of
the unit cells, which is shown in Fig. 3. The CST Microwave
Studio is used for the design of array [10]. The incidence angle
Fig. 2 Flow diagram of the MLAO method.
of the feed is designed as 20° and the unit cells should be steered
to scan the main beam 20° off the other side in the vertical plane.
B. Surrogate Model Based Optimization
Fig. 4 shows the phase response of the unit cell with parame-
The optimization process can be assumed as minimizing
ters set to the initial value over the frequency band from 38 GHz
y  F ( x) , x  X d , where d is the number of the designable pa-
rameters. Based on an on-line constructed GPR surrogate model, to 54 GHz and the gain curve of the reflectarray antenna. The
EA is adopted to search for the optimal solution x* . Fig. 2 shows values of the objectives is O1initial  x  = 15.37, O2initial  x  = 1. The
the flow diagram of the (MLAO) method. The key steps of the 1-dB gain bandwidth is about 26.3% (39.9–52.2 GHz), and peak
optimization method employed are as follows. gain is 28.1 dBi at 47 GHz.
 Step 1: Initialize database. TABLE I
The Latin hypercube sampling (LHS) method is used in this DESIGN PARAMETERS OF THE UNIT CELL
section to select N points from the design space [ X min , X max ]d . Parameter w (mm) a h1 (mm) h2 (mm)
Initial database is composed of the sample points and their re- Initial value 0.080 0.400 0.508 1.200
sponses calculated by EM simulation, ANSYS High-Frequency Lower bound 0.050 0.300 0.127 0.600
Structure Simulator (HFSS) [8] is used in this paper to design Upper bound 0.150 0.600 1.575 1.400
and analyze the unit cells. The ring size Lr and the frequency f Optimized value 0.139 0.530 0.318 1.004
are selected as two additional features in the database, thus more
information from accurate full-wave EM simulation can be uti-
lized when training.
 Step 2: Check criterion.
The loop terminates when stopping criterions including the
achievement of the pre-defined goals or the maximum number
of iterations are reached.
 Step 3: Construct surrogate model.
A GPR surrogate model is constructed using current database,
which is updated in every loop, so the surrogate model will be-
come more accurate with the addition of new individual.
Fig. 3 Reflectarray antenna with 576 elements.
 Step 4: Optimize unit cells using MALO.
The trained surrogate model is applied in EA to predict the
performance of new points instead of calculating them by full-
wave EM simulation. Reference point [ R1 ,..., RM ] [9] is utilized
to transform a multi-objective problem to a single-objective one,
which can be expressed as

 
F  x   max mM1 m  Om  x   Rm  (3)
(a) (b)
where M is the number of objectives and m is the m-th compo- Fig. 4 Performance of the initial unit cell: (a) Phase response; (b) Simulated
nent of a chosen weight vector to scalarize the objectives. The gain of 576-elements reflectarray.

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 The designed reflectarray antenna with optimized unit cells
have a wider 1-dB gain bandwidth with high gain and good ra-
diation pattern compared with the other wideband single layer
reflectarray antenna listed in Table II. The good performance of
the reflectarray antenna validates the effectiveness of the pro-
posed optimization method.
TABLE II
COMPARISONS OF THE OPTIMIZED REFLECTARRAY ANTENNA AND THOSE IN
REFERENCES
Max Gain 1-dB Gain Element Freq.
Ref.
(dBi) BW (%) Num. (GHz)
This work 28.4 45.5 24×24 47
Fig. 5 Convergence tendency. [5] 20.0 30.0 45 10
[6] 37.8 26.7 - 30
[7] 27.5 39.8 24×24 45

IV. CONCLUSION
The MLAO methed has been presented to design a broadband
reflectarray antenna. The variable-size reflectarray unit cells
have been firstly optimized. The GPR surrogate model with ad-
ditional features is constructed online to predict the phase re-
(a) (b) sponse. Phase range and the slope are mathematically formu-
Fig. 6 Performance of the optimized unit cell: (a) Phase response; (b) Simula- lated as two objectives in the EA. The structures of unit cells
tion gain of 576-elements reflectarray. have been successfully optimized within a few iterations. Fin-
nally, the optimized unit cells have been used to construct a re-
flectarray antenna, which help to broaden the 1-dB gain band-
width from 26.3% to 45.5%.
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