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Optimizing radiation patterns of mechanically reconfigurable phased arrays using flexible meta-gaps

Authors:

Abstract

In order to take on arbitrary geometries, shape-changing arrays must introduce gaps between their elements. To enhance performance, this unused area can be filled with meta-material inspired switched passive networks on flexible sheets in order to compensate for the effects of increased spacing. These flexible meta-gaps can easily fold and deploy when the array changes shape. This work investigates the promise of meta-gaps through the measurement of a 5-by-5 λ -spaced array with 40 meta-gap sheets and 960 switches. The optimization and measurement problems associated with such a high-dimensional phased array are discussed. Simulated and in-situ optimization experiments are conducted to examine the differential performance of metaheuristic algorithms and characterize the underlying optimization problem. Measurement results demonstrate that in our implementation meta-gaps increase the average main beam power within the field of view (FoV) by 0.46 dB, suppress the average side lobe level within the FoV by 2 dB, and enhance the field-of-view by 23.5 ∘ compared to a ground-plane backed array.
International Journal of
Microwave and Wireless
Technologies
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Research Paper
Cite this article: Williamstyer DE, Hajimiri A
(2024) Optimizing radiation patterns of
mechanically reconfigurable phased arrays
using flexible meta-gaps. International
Journal of Microwave and Wireless
Technologies 16(5), 838–851. https://doi.org/
10.1017/S1759078723001526
Received: 23 June 2023
Revised: 24 November 2023
Accepted: 29 November 2023
Keywords:
adaptive arrays; flexible electronics; genetic
algorithms; kirigami antennas; measurement
techniques; metaheuristics; metamaterials;
optimization; origami antennas; particle
swarm optimization; phased arrays;
simulated annealing
Corresponding author:
D. Elliott Williamstyer;
Email: d.e.williamstyer@hofstra.edu
© The Author(s), 2024. Published by
Cambridge University Press in association
with the European Microwave Association.
This is an Open Access article, distributed
under the terms of the Creative Commons
Attribution licence (http://creativecommons.
org/licenses/by/4.0), which permits
unrestricted re-use, distribution and
reproduction, provided the original article is
properly cited.
Optimizing radiation patterns of mechanically
reconfigurable phased arrays using flexible
meta-gaps
D. Elliott Williamstyer1,2and Ali Hajimiri1
1Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, USA and 2Department
of Engineering, Hofstra University, Hempstead, NY, USA
Abstract
In order to take on arbitrary geometries, shape-changing arrays must introduce gaps between
their elements. To enhance performance, this unused area can be lled with meta-material
inspired switched passive networks on exible sheets in order to compensate for the eects
of increased spacing. ese exible meta-gaps can easily fold and deploy when the array
changes shape. is work investigates the promise of meta-gaps through the measurement
of a 5-by-5 𝜆-spaced array with 40 meta-gap sheets and 960 switches. e optimization and
measurement problems associated with such a high-dimensional phased array are discussed.
Simulated and in-situ optimization experiments are conducted to examine the dierential per-
formance of metaheuristic algorithms and characterize the underlying optimization problem.
Measurement results demonstrate that in our implementation meta-gaps increase the average
main beam power within the eld of view (FoV) by 0.46 dB, suppress the average side lobe level
within the FoV by 2 dB, and enhance the eld-of-view by 23.5compared to a ground-plane
backed array.
Introduction
e recent development of mechanically shape-changing phased arrays shows the promise of
using geometric reconguration to optimize for a given array behavior [1]. However, Gauss’s
eorema Egregium requires that the surface area of an array must stretch or contract as it
changes between shapes with dierent Gauss curvature [2]. If the number of radiators in an
array is xed, the spacing between them must change as the array shis between planar and
spherical shapes. us, arrays of rigid tiles can only change shape if gaps are introduced between
the tiles or tiles are removed from the surface [3].
e required increased element spacing alters array performance by reducing the ll fac-
tor, introducing grating lobes, and changing the antenna coupling. But these gaps also present
an opportunity in the form of extra unused area on the radiation surface that can be lled
with switch-controlled exible passive structures capable of distorting the near-eld environ-
ment. Such meta-surface inspired structures, henceforth referred to as meta-gaps, can be used
to mitigate the eects of increased element spacing.
Figure 1 illustrates the advantages of exible meta-gaps. In a planar shape there are no gaps,
so the meta-gaps fold behind ground-plane backed antennas where they will minimally interact
with the radiating surface. When the array changes into a cylinder, the meta-gaps are deployed
between the antennas and are congured to mitigate the eects of increased antenna spacing or
otherwise enhance the array performance.
In-situ meta-gap optimization concept
Meta-gaps introduce parasitic metal structures in the gaps of an array in order to distort
the near-eld environment, thus altering the far-eld behavior, antenna coupling, and port
impedances. e challenge is identifying which metal patterns benet the array performance.
Unfortunately, electromagnetic elds are nonlinear with respect to their boundary conditions
and so adding conductors can drastically change their fundamental behavior. e diculty of
this design problem is exacerbated in the context of a phased array because the near-eld is sub-
ject to dierent eld excitations, each of which will induce dierent currents on the conductor
surfaces and therefore alter their impact of the array behavior. us the behavior of the structure
must be considered under every desired excitation.
Inspired by the work of Lavaei and Babakhani [4], we present a switch-controlled meta-gap
structure capable of dynamic reconguration. e structure consists of a grid of metal squares
connected to each other by RF switches. ese switches can be turned on and o to create
conductive pathways that control how and where currents are excited. With a ne enough grid,
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 839
Figure 1. Illustration of how meta-gaps can be deployed to fill the gaps in a
shape-changing array. (a) In planar configuration, the sheets fold behind the tiles.
(b) In cylindrical configuration they expand to fill the gaps. Reprinted with
permission from the copyright holder, EuMA.
the switches can essentially program the shape of conductors on the
surface into an arbitrary conguration. is concept is similar to
the pixel-based automated design of waveguide couplers presented
in [5].
By using programmable switches, the meta-gap conductor pat-
terns can be dynamically recongured in-situ, allowing optimiza-
tion methods to be used on a physical array to explore a large
number of candidate patterns to identify solutions that enhance
array performance. is in-situ optimization avoids the accuracy
and model complexity issues associated with simulating a large
array, providing an excellent platform to explore meta-gap per-
formance. In-situ optimization establishes a minimum bound of
performance for optimal realized meta-gap structures and allows
the nature of the design space and properties of optimal solu-
tions to be inferred from the relative performance of dierent
optimization algorithms.
Prior art
e use of switches and meta-materials to alter antenna and array
performance is well explored. PIN diodes have been used to alter
the operation frequency [6], polarization [7], and radiation pat-
terns of both antennas and arrays [8,9]. Techniques include using
switches to recongure the conductor geometry [10,11], alter res-
onator modes [12], change the array feed [13], or select dierent
elements [1416]. Switches have also been used to modify parasitic
conductors and adjust the near-eld environment [17,18].
Embedding meta-materials in radiating structures is another
well-traveled research avenue. Meta-materials have been used in
arrays to reduce antenna coupling [1921] and reduce the size of
a linear array [22]. ey have also been used in antenna designs to
reduce the antenna size [23], increase its bandwidth [24], and alter
its polarization [25]. Switch based meta-materials have been used
to change the selectivity of a frequency-selective surface [26] and
modify the radiation pattern of a dipole antenna [27].
Switched RF structures and meta-materials are useful for a
broad range of applications. Switched parasitic structures can be
used to modulate the radiation pattern to enable physically secure
wireless communication [28]. Pattern reconguration enables
blind optimization of transceiver patterns to mitigate interference
and multi-path eects using a constant modulus algorithm [29].
Recongurable antennas are also useful for in-orbit satellite adap-
tation, enhanced MIMO systems, and cognitive radio techniques
[30]. Surfaces of programmable meta-materials have enabled pro-
grammable reector arrays [31,32], passive relays [33], and holog-
raphy at THz frequencies [34].
Outline
An earlier version of this paper was presented at the 2022 European
Microwave Conference and was published in its Proceedings [35].
is paper expands on that work with additional experiments that
relax the identical sheet restriction, simulated experiments to iden-
tify the statistical properties of the optimization algorithms, and
more in-depth discussions, explanations, and analyses.
Section “Phased-Array Optimization discusses the complexi-
ties of phased array characterization and the meta-gap optimiza-
tion problem. Section “Simulated Statistical Analysis” presents
statistical results of optimizations preformed on a simulated
model. Section “Demonstration Array” describes the hardware
and measurement approach used to perform in-situ optimiza-
tion. ree rounds of in-situ optimization experiments are pre-
sented in In-Situ Optimization Experiments” section. Finally sec-
tion “Discussion takes a holistic view of the experimental and
simulated results before section “Conclusions concludes.
Phased-array optimization
Completely characterizing the performance of a phased array
requires numerous measurements. A full 3D radiation pattern
must be measured for every beam angle within the steering range
in order to get a complete picture of the power radiated in both
the desired (main beam) and undesired (side lobe) directions.
Managing this measurement complexity is critical for characteriz-
ing meta-gaps because each switch conguration eectively creates
a new array that requires complete characterization.
Array figures of merit
In order to reduce the number of measurements used for charac-
terization, we evaluate the performance of the array using three
criteria: the power radiated in the main beam direction (MBP),
the side lobe level (SLL), and the eld of view (FoV) in the E-
and H-plane cuts. SLL is here dened as the relative strength of
the peak side lobe1with respect to the MBP and FoV is dened as
the range over which the SLL is negative. ese measures provide
insight into the array’s maximum eective isotropic radiated power
(EIRP), 3-dB steering range, and gain, without requiring a full 3D
scan.
In order to characterize the performance of the array with a sin-
gle number for optimization purposes, the average MBP and SLL
over the FoV, and the average FoV over the dierent 𝜙cuts, are used
as the optimization criteria as shown in Figure 2. Using an average
ensures that the array performance is over its entire steering range
and not just in one direction.
Beams are only steered within the FoV when characterizing the
MBP and SLL because, due to spatial aliasing, the phase settings
required to steer a beam outside of the FoV are the same as those
that steer a beam at a dierent angle inside of the FoV. Since the
1Note that by this denition, grating lobes are considered side lobes and thus the SLL
can be positive.
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
840 D. Elliott Williamstyer and Ali Hajimiri
-90
-75
-60
-45
-30
-15
0
15
30
45
60
75
90
Theta [deg]
-10
-5
0
5
10
15
20
25
Measured Power [dBm]
Average Main Beam Power
Main Beam
Peak Side Lobe
Average Side Lobe Level
-90
-75
-60
-45
-30
-15
0
15
30
45
60
75
90
Theta [deg]
-10
-5
0
5
10
15
20
25
Measured Power [dBm]
Main Beam
Peak Side Lobe
(a) (b) (c)
Figure 2. Visualization of optimization criteria. (a) The main beam power is integrated within the theoretical field of view, e.g. ±60. (b) The dierence between the main
beam power and peak side lobe power, the side lobe level, is integrated within the theoretical field of view. (c) The field of view optimization maximizes the angular
dierence between the first crossings of the main beam power and the peak side lobe power.
beam strength is higher within the FoV, the beam on the outside
is not truly the main beam, but a grating lobe of the beam on the
inside. e FoV of an ideal 𝜆-spaced array is ±30, greatly reducing
the number of measurements.
Array characterization time
An ecient measurement approach is necessary to perform in-
situ optimization. In a radiation pattern measurement, the average
time to move the antenna under test between measurement points,
Tmove, is on the order of seconds, while the measurement time,
Tmeas, can theoretically be less than a microsecond. Assuming
Tmeas Tmove, the time to characterize a phased array is:
Tchar Nb(Nm+1)Tmove,(1)
where Nmis the number of measurement points per beam and Nb
is the number of beams used to characterize the array.
Equation 1 is the xed time required to move between all the
measurement points. If Tmove is one second, then it takes 90 min-
utes to characterize a single meta-gap conguration along two cuts
with 5precision. If Mstates are measured sequentially than the
total measurement time is MTchar.
Instead, multiple congurations can be measured in parallel by
iterating through them at each measurement point. us a large
number of congurations can be characterized while only mov-
ing through measurement points once. In this case, the time to
characterize a batch of Mcongurations is:
Tbatch (Nb+Nm)Tmove
+MNb[Nm+(Ntiles 1)D]Tmeas,(2)
where Ntiles is the number of tiles and Dis the number of measure-
ments required to optimize the beam phase settings.2
For a suciently large batch size the total characterization
time is decreased by orders of magnitude. However, results are
not available until the entire batch is processed. us algorithms
must operate on batches of measurements, as no decision can be
2Note thatt hereis an error in Equation (2) of the original paper [35] that underestimates
the speed increase.
made until the completion of each batch. e measurement time
of this method of sequential batches is simply Equation 2 times the
number of batches.
Characterizing the meta-gap optimization problem
Meta-gaps present a high-dimensional conguration space with
a far smaller subspace of useful solutions; a meta-gap structure
with kswitches has 2kpossible states. Finding an optimal state for
this kind of switching network has been proven to be NP-Hard in
general and there is no known convex heuristic that consistently
identies solutions close to the optimal one [4].
us, non-convex heuristic algorithms must be employed in
order to identify preferred, if not optimal congurations. While
problem-specic heuristics will outperform abstract metaheuris-
tic algorithms [36], it is not obvious how a given set of switches
will alter the array behavior due to the strongly nonlinear behav-
ior of switched networks and the complex interactions of parasitic
elements in response to dierent electromagnetic excitations for
dierent steering angles.
Four dierent metaheuristic algorithms and a simple random
search are employed to both explore meta-gap performance and
to provide insight into the general structure of the optimiza-
tion problem and its solutions. e selected algorithms—Genetic
Optimization (GEN) [37], variable neighborhood search (VNS)
[38], Particle Swarm (PS) [39], and Simulated Annealing (SA)
[40]—ake advantage of dierent properties of the optimization
problem [41]. Genetic optimization excels at problems that con-
tain underlying “features” that can be exploited. VNS takes a local
to global approach, performing well on problems with clustered
optima. SA has the opposite approach, performing a wide area
search before settling into a local optima. Like SA, PS takes a global
to local approach but incorporates several agents that investigate
local optima. Finally, the random search serves as a baseline com-
parison. us, properties of the problem structure can be identied
by comparing the relative performance of these methods.
Optimization framework
e same optimization framework is used in both the simulated
statistical experiments discussed in Section “Simulated Statistical
Analysis” and the in-situ optimization experiments discussed in
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 841
Figure 3. Finite element simulation model. (a) The five element 𝜆-spaced array
with embedded meta-gaps. (b) Closeup of antenna model with excitation port. (c)
Closeup of meta-gap model with control ports. (d) Closeup of dipole sense antenna.
(e) Simulation volume with array and sense antennas in the far-field.
In-Situ Optimization Experiments” section. e metaheuristic
algorithm selects switch states to be measured, the measure-
ment range (either real or simulated) measures the desired array
characteristic for the selected states, and the results are pro-
vided back to the algorithm to select the next states to be mea-
sured. Aer a predened number of iterations, the algorithm is
terminated.
As discussed in “Demonstration Array” section, the demonstra-
tion array has 2960 dierent possible meta-gap states. In the simu-
lated experiment and the initial in-situ experiment, each meta-gap
sheet is constrained to have the same switch settings. e rational
for this relatively arbitrary restriction is twofold, rst, it restricts
the total search space to 224 possible switch settings and, second,
the dierence in eld perturbations between neighboring states is
larger and thus easier to measure. ese restrictions are relaxed in
the later in-situ experiments.
Due to the complexity of phased array characterization, states
are measured in batches in order to reduce the total measurement
time. e algorithms operate in batches of 24 states so that search
based approaches, like VNS and SA, can evaluate every neighbor-
ing state in a single batch during the identical sheet experiments.
e algorithms are iterated for 30 batches, thus exploring a total of
720 states.
To ensure consistency, the same tuning parameters, run lengths,
and batch sizes are used in the dierent experiments. is con-
sistency allows direct comparison of the relative performance of
the dierent algorithms both in dierent experiments and when
optimizing dierent criteria.
Simulated statistical analysis
Due to the stochastic nature of the employed metaheuristics, they
are best characterized by their performance over a large num-
ber of trials. Unfortunately, as discussed in Section “Phased-Array
Optimization, the time required to measure a physical array pre-
vents a large number of trials. Instead, the Lavaei–Babakhani
method [4] can be used to perform a large number of simulated
trials. While deviations in a simulated model might give inaccu-
rate characterization results, they do not fundamentally change
the structure of the optimization problem. us, metaheuristic
algorithms should perform similarly in both simulationand in-situ.
e dierent optimization algorithms are run 1000 times with
random initial conditions. Optimization of MBP, SLL, and FoV are
treated separately with their own set of runs. e results of the
runs are used to calculate the average and standard deviation of the
optimal state identied over time. ese statistics provide insight
into the convergence time of the algorithms, their relative expected
performance, and the variance in runs.
Simulation setup
Figure 3 shows the simulation model used for statistical analysis. It
is a ve element 𝜆-spaced linear array of patch antennas separated
by four meta-gap sheets. Both the meta-gap sheets and patch anten-
nas have the same metal patterns and dimensions as their physical
equivalent. e meta-gap sheets also model the 30 nH choke induc-
tors in parallel with switches. Figure 3(e) shows the array and the
37 short dipole sensor antennas located in the far-eld every 5
in the steering-plane. is model contains 138 ports—5 excitation
ports, 37 sensor ports, and 96 control ports—that are represented
by teal rectangles. e total simulation volume including the array
and sensor ports is 605𝜋𝜆3.
e array was simulated using HFSSs nite element solver over
15 h using 23 cores and 218 GB of RAM. e 138 port s-parameter
model derived from the FEM simulation enables the behavior of
dierent meta-gap settings to be quickly calculated. is calcula-
tion method is encapsulated into a simulated range” enabling the
same optimization algorithms to be used for both simulated and
measured optimization experiments.
It is important to note that the simulated problem is only a one
dimensional array instead of the full two dimensional demonstra-
tion array used for the in-situ experiments. A simulation of the full
2D array was not possible due to the signicantly larger simula-
tion volume required (closer to 7100𝜋𝜆3) and the larger number
of ports (1058).
Algorithmic performance
e variable performance of the stochastic algorithms between
runs is best described by a probability distribution. Figure 4 com-
pares the average and standard deviations of the algorithms per-
formance over time for all three optimization criteria. A general
summary of the simulated algorithm performance is shown in
Table 1.
As can be seen, the algorithms’ behavior when maximizing
the MBP are similar to their behavior when minimizing the SLL.
For both criteria, genetic optimization performs the best on aver-
age, but also has the largest standard deviation for performance
between runs. VNS converges rapidly on a local optima, but the
standard deviation of the performance of that optima is high. e
performance of PS and SA is very similar as they both broadly
explore the space before converging on a local optima.
e statistical behavior of the algorithms when optimizing the
FoV is quite dierent. For these criteria, PS and SA both perform
the best. e most striking result is that both VNS and genetic
optimization perform worse than the random search. However, the
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
842 D. Elliott Williamstyer and Ali Hajimiri
Table 1. Summary of algorithm performance in simulated experiments.
Algorithm Random Genetic VNS PS SA
MBP/SLL performance Low High Medium Medium Medium
FoV performance Low Low Very low High High
Variability Low High Medium Medium Medium
Convergence Fast Slow Fast Slow Slow
Dependence on initial conditions None High Very high Low None
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Avg MBP within FoV vs. Baseline [dB]
Average Performance: Maximize Main Beam Power
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(a)
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
0
0.05
0.1
0.15
0.2
0.25
Avg MBP within FoV vs. Baseline [dB]
Performance STD: Maximize Main Beam Power
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(b)
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Avg SLL within FoV vs. Baseline [dB]
Average Performance: Minimize Side Lobe Levels
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(c)
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Avg SLL within FoV vs. Baseline [dB]
Performance STD: Minimize Side Lobe Levels
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(d)
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
Average Field of View [deg]
Average Performance: Maximize Field of View
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(e)
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
696
720
Number of States Measured
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
Average Field of View [deg]
Performance STD: Maximize Field of View
Random
Genetic
VNS
Particle Swarm
Simulated Annealing
(f)
Figure 4. Comparison of the statistical performance of dierent algorithms over
1000 trials when (a)–(b) maximizing main beam power, (c)–(d) minimizing side lobe
level, and (e)–(f) maximizing the field of view. (a), (c), and (e) show the average
performance of the best identified state over time. (b), (d), and (f) show the
standard deviation of performance of the best identified state over time.
dierence between the average performances is less than 0.5and
so it is possible that the statistical power of the experiment is not
large enough for the dierence to be statistically signicant. Finally,
the algorithms have similar standard deviations in performance
when optimizing the FoV.
Distribution of states
Because they are randomly selected, the states measured by the ran-
dom search algorithm provide a statistical sample of the underlying
distribution of states. For each of the optimization criteria, the ran-
dom algorithm explored approximately 705,000 unique states out
of 16.8 million over the 1000 runs.
Figure 5 contains histograms of state performance for the dif-
ferent optimization criteria and thus illuminates the underlying
distribution of states. e distributions for MBP and SLL are
closely related, with a long tail of states that perform worse than
baseline and a sharp drop-o in states that perform better. For
MBP optimization, the sample mean and standard deviation are
−0.35 and 0.23 dB, respectively, while they are 0.92 and 0.50 dB
for SLL optimization. e distribution of states indicates that most
meta-gap settings are detrimental to performance, but there a small
number of states that oer performance improvements. is aligns
with the intuition that random parasitic metal patterns in close
proximity to an antenna will generally degrade its pattern and
matching.
e histogram for the FoV optimization shown in Figure 5(c)
is radically dierent. e distribution is trimodal, with a sharp
thin collection of states close to 60, the ideal FoV for a 𝜆-spaced
array, and two wider loci around 67and 45. e high concen-
tration of states indicates that most switch settings do not really
aect the FoV. However, there exist sub-spaces that either enhance
or suppress the grating lobes to some degree. e sample mean
and standard deviation for FoV optimization are 59.7and 7.8.
is standard deviation is misleading, however, as the variation is
mostly attributed to the distinct modes with the center and right
modes exhibiting far lower variability.
Demonstration array
Array design
To experimentally test the performance of exible meta-gaps,
they are incorporated into a planar 𝜆-spaced 2D demonstration
array. e well-established grating lobe behavior of this struc-
ture makes it an ideal test case for examining the behavior of
meta-gaps.
e 2.5 GHz array, shown in Figure 6(a), consists of 25 radi-
ating tiles separated by 40 meta-gap sheets. Both the tiles and the
sheets are 6 cm squares—𝜆
2at the operating frequency—and are
mounted through corner holes to nylon standos connected to a
rigid wooden backboard. As each sheet contains 24 switches, the
total array contains 960 independently programmable switches. In
total, the array has 2960 dierent possible meta-gap congurations.
Each tile in the array is a 2.5 GHz radiator with programmable
phase and amplitude. e tiles’ RF outputs are synchronized by
locking to a 78.125 MHz reference. e signal is radiated by a patch
antenna on the opposite side of the tile. e patch ground-plane
isolates the radiator from the electronics. Figures 6(b) and 6(d)
show both sides of the tile and Figure 6(c) shows the measured
and FEM-simulated radiation patterns. e simulated antenna
eciency is 84%.
Each tile also serves as a controller for two neighboring meta-
gaps using programmable headers. Power, programming, and ref-
erence signals are distributed through the array using a single
cable. To reduce the total current carried by the cables, power
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 843
Sampled State Distribution: Maximize Main Beam Power
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
Average MBP within Steering Range vs. Baseline [dB]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Number of States [%]
(a)
Sampled State Distribution: Minimize Side Lobe Levels
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
Average SLL within Steering Range vs. Baseline [dB]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Number of States [%]
(b)
Sampled State Distribution: Maximize Field of View
30 35 40 45 50 55 60 65 70 75 80 85 90
Average Field of View [deg]
0
5
10
15
20
25
30
35
Number of States [%]
(c)
Figure 5. Distribution of the performance of randomly sampled states in the simulation experiment. (a) Main beam power. (b) Side lobe level. (c) Field of view.
Figure 6. (a) Demonstration array with embedded meta-gaps. (b) Tile antenna. (c)
Single tile element pattern. (d) Tile electronics. (e) Measured beam patterns of array
without meta-gaps.
is distributed at 30 V and down-converted to 3.3 V by a local
DC-to-DC converter on each tile.
Switched meta-gap design
e 6-cm meta-gap sheet shown in Figure 7(a) is comprised of
a 0.5-oz copper layer between two 1-mil layers of exible poly-
imide. e metal forms a 4-by-4 grid of 9.7mm squares with
5.3 mm gaps. Neighboring squares are connected by an RF switch
as shown in Figure 7(b). e metal grid is designed to achieve max-
imum variation in reectivity and transparency when all switches
are simultaneously turned on and o, respectively. When all the
switches are on, the surface ideally behaves like a ground-plane,
maximizing the reection of incident waves. When all the switches
are o, the surface ideally is transparent to incident waves. An
FEM analysis characterizing the relationship between metal spac-
ing and the surface’s ability to reect incident plane waves is shown
in Figure 7(c). A grid spacing of 5.3 mm is selected to minimize the
power reected when the switches are o while reecting at least
90% of the power when the switches are on. Less than 5.5% of the
incident wave power is dissipated by the structure in both the on
and o states.
e RF switch, shown in Figure 7(d), consists of a pair of back-
to-back PIN diodes biased with a 30 nH RF choke inductor. A com-
mon DC ground is established by using 30 nH inductors to connect
adjacent squares. PIN diodes are used to maximize the change in
switch impedance at RF frequencies and the back-to-back cong-
uration allows an independent DC bias to be established for each
switch. e measured switching behavior of the structure is shown
in Figure 7(e). At 2.5GHz, the diode insertion loss is −0.67 dB
when on, and the isolation is 15.3 dB when o. Each switch is
controlled with a 10 mA current via a header containing 24 con-
trol wires and 2 ground wires. e 10 mA bias current optimizes
the tradeo between insertion loss and power consumption. e
lightweight 32 AWG wires are connected to the sheet from the back
side. e thin substrate and wire conguration maximize sheet
exibility and minimize unwanted near-eld interactions.
In-situ optimization experiments
Using an automated test setup, the metahueristic algorithms are
used to perform in-situ optimization of the demonstration array
in a series of experiments. e results of these experiments pro-
vide insight into both the realized capabilities of meta-gaps and the
underlying optimization problem.
In the initial experiment, each meta-gap sheet is constrained to
have the same switch settings in order to restrict the search space.
en, the identical sheet restriction is replaced by dierent patterns
of enforced symmetry in order to explore an increase in the size of
the search space and the associated trade-o between the quality of
the optimal state and the ability to nd it. e best performing sym-
metry identied in this symmetry experiment, referred to as the
E&H near-eld mapping, is then used to repeat the evaluation of
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
844 D. Elliott Williamstyer and Ali Hajimiri
Figure 7. (a) Meta-gap sheet. (b) Close-up of switching network. (c) Simulation of
reflectivity and transparency versus gap size. (d) RF switch schematic. (e) Measured
switch S21.
the optimization criteria and optimization algorithms in the third
experiment.
Measurement setup
Figure 8 is a diagram of the automated range measurement system
used to perform the in-situ optimization. e demonstration array
is mounted on a far-eld scanner and rotated about the 𝜃and 𝜙
axes. A frequency generator provides a constant 2.5 GHz signal that
is converted to a 78.125MHz reference using a pair of frequency
dividers (total division ratio of 32). is reference is buered by
the array motherboard and distributed through the array under test
(AUT). Each tile up-converts the reference back to 2.5 GHz, adjusts
its phase, and radiates a xed power. e combined radiated eld
is sensed by a horn antenna 2.95m from the array and is measured
by a vector network analyzer set to measure S21 at 2.5 GHz. is
measured S21 is thus a measure of the radiated power relative to a
xed reference.
e host computer runs the metahueristic algorithms and man-
ages the optimization and measurements by programming the
array phase and meta-gap settings, controlling the scanner posi-
tion, and measuring the power from the VNA. e host computer
also monitors the array current, radiated spectrum, and tempera-
ture throughout the experiment to ensure reliability.
For MBP and SLL optimization, the array is measured by steer-
ing beams every 5from 30to 30in the E- and H-plane cuts,
while beams are steered every 5from 90to 90for the FoV opti-
mization. Beam steering is achieved by optimizing the tile phases
to maximize power in the beam direction. For all criteria, each
beam pattern is measured every 5from 90to 90in the E- and
H-plane cuts. MBP and SLL optimizations take approximately 23 h
while FoV optimizations take 60.
Figure 8. Diagram of measurement setup. Blue triangles indicate RF absorbers and
the boundaries of the range. Green components indicate the RF signal path. Grey
squares are other critical measurement devices. Blue shapes are auxiliary
equipment that monitor the measurement environment.
Figure 9. Baseline measurements of (a), (c) main beam power and (b), (d) side
lobe level in E- and H-planes for “all o” state, “all on” state, and ground-plane
backed array.
To compensate for low frequency noise and thermal dri over
the long measurement periods, each batch measures a reference
state, in which every switch is turned o, and uses it to normal-
ize the other patterns. Due to the interwoven nature of the batched
measurements, dierent states in the batch happen in quick suc-
cession at a given measurement point; therefore, the dierence
between these measurements are mostly immune to low-frequency
noise and temperature uctuations. By normalizing each measured
pattern to the same reference state, the relative performance of
states in dierent batches can be accurately compared.
Baseline measurements
While useful for measurement purposes, the all o state used
as a baseline is not indicative of the array performance without
the meta-gap sheets. A more appropriate comparison is the rel-
ative performance of the meta-gap sheet compared to a simpler
approach expected to improve array performance. A reasonable
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 845
Figure 10. (a)–(c) Optimization curves and (d)–(f) E-plane and (g)–(i) H-plane cuts of the optimal array characteristics for the random search, genetic optimization, VNS, PS,
and SA algorithms’ optimizations of main beam power, side lobe levels, and FoV under the identical sheet restriction. Optimization plots include the optimal state in
addition to the value of the explored states averaged over two batches.
solution is to ll the gaps in the array with a ground-plane, eec-
tively creating a uniform ground-plane commonly found behind
elements in an array. However, while a exible ground-plane would
make a superior baseline for comparison, it cannot be rapidly
switched to electronically during an experiment.
To enable comparisons between meta-gap performance and
a exible ground-plane, the gaps in the array are lled with a
ground-plane comprised of copper tape. By measuring the absolute
power of the ground-plane backed array and the “all o” meta-gap
state, the relative performance can be directly compared. With the
relative performance of the all o baseline established, the per-
formance of any meta-gap state measured relative to this baseline
can be compared to the ground-plane backed array.
Figure 9 shows the MBP and SLL for the ground-plane backed
array and meta-gap array in the all on” and all o states nor-
malized by the peak broadside power of the all o baseline. e
“all on state is included because it is the meta-gap state that cor-
responds with a solid ground-plane, even though the “all on state
performs worse than the copper tape ground-plane due to lossy
switches and parasitic metals. e average MBP of the “all on state
is 1 dB lower than the ground-plane backed array. In general the
ground-plane backed array has higher MBP and lower SLL than
the meta-gap array. e only exception is the E-plane SLL which is
lower when the meta-gap array is in the all o state.
Identical sheet experiments
In the rst set of experiments, each of the ve algorithms are
used to perform in-situ optimization of the MBP, the SLL, and the
FoV under the restriction that each meta-gap sheet has the same
switch settings. Note that unlike the statistical results in Section
“Simulated Statistical Analysis, the presented optimization results
are for a single run of the experiment. erefore, caution is war-
ranted when drawing conclusions about the relative performance
of the algorithms.
Experimental results are presented in multiple formats. Figures
10(a)10(c) show both the algorithms’ performance over time and
a moving average of the performance of states explored over two
batches. is average provides another useful view of algorithm
behavior. In addition, Figures 10(d)10(i) shows the optimal array
characteristic identied by each algorithm and the all o char-
acteristic for each of the three optimization criteria. Finally, an
illustration of the best performing meta-gap switch settings for
each of the three criteria is shown in Figure 11.
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
846 D. Elliott Williamstyer and Ali Hajimiri
For both MBP and SLL, genetic optimization, PS, and SA have
similar nal results and outperform both VNS and random search.
In both, genetic optimization identies the best performing state,
mirroring the results of the statistical experiments. When opti-
mizing the FoV, all of the algorithms perform similarly, with PS
identifying the best state.
e average value of the explored states provides addi-
tional insight. While the nal MBP and SLL results of genetic
optimization, PS, and SA are similar, the genetic optimizations
average state value increases much faster initially before leveling
o. During this period, the algorithm quickly prunes lower per-
forming genes. SA and PS however converge slower as they explore
the full search space. e average performance of states explored
by VNS oscillates as the algorithm switches between a focused
local search and an unfocused perturbation to escape local optima.
In the FoV optimization, however, the performance of the aver-
age state explored is similar for each of the algorithms and slowly
increases over the course of the run. In all optimization criteria,
the stability of the average state explored by the random search
indicates long term measurement stability.
Under the identical sheet restriction, meta-gaps are able to
increase the average MBP within the FoV by 1.13 dB and decrease
the average SLL within the FoV by 1.7 dB when compared to the all
o baseline. Comparing these improvements to the ground-plane
measurements in Section “Baseline Measurements” indicates a
0.23 dB improvement in average MBP and a 2 dB average reduction
in SLL compared to the ground-plane backed array. e perfor-
mance improvement is even higher broadside; meta-gaps increase
the MBP by 0.9 dB and a suppress the SLL by 4.6 dB when the beam
is steered broadside.
Interestingly, even the random search is capable of suppress-
ing the average SLL 1.3 dB more than the ground-plane backed
array and improving the average MBP by at least 0.89 dB compared
to the baseline, only 0.01 dB lower than the ground-plane backed
antenna. erefore it is relatively straightforward to identify meta-
gap states that perform at least as well as the ground-plane backed
array.
Meta-gaps are also able to increase the FoV to 83.5, 23.5higher
than the theoretical value of a 𝜆spaced array. However, on exam-
ining the optimal E-plane SLL characteristic in Figure 10(f), it is
clear that the FoV is improved by increasing SLL. e SLL is just
barely negative for most of the expanded FoV, indicating the main
lobe is only slightly higher than the peak side lobe. While this tech-
nically meets the denition of FoV, it does not follow the spirit
of the concept and thus renders the results relatively impractical.3
Nonetheless, the fact that FoV extension is even possible suggests
that meta-gaps can change the aliasing behavior of the array.
Figure 11 shows the optimal meta-gap sheet switch settings for
the three optimization criteria. Interestingly, the optimal MBP and
SLL states are nearly identical. Both shapes suggest a split “H” shape
with vertical symmetry. e optimal FoV state, however, exhibits
limited symmetry with no recognizable pattern.
Symmetry experiments
e optimal states identied under the identical sheet restric-
tion are within the larger 960 degree of freedom space and thus
3is behavior is not particularly surprising as exploiting the technicalities in the
framing of a problem is the modus operandi of optimization algorithms.
(a) (b) (c)
Figure 11. Visualization of optimal solution states for (a) main beam power, (b) side
lobe level, and (c) field of view optimizations under the identical sheet restriction.
The meta-gap sheet is represented using a four-by-four grid of light copper-colored
squares separated by either black, or darker copper-colored lines. These lines
indicate the status of each switch in the state; black lines indicate the switch is o
while copper colored lines indicate it is on. Thus the shape of the formed conductor
pattern can be visualized while still identifying the location of switches.
Figure 12. Diagram of mappings used to reduce degrees of freedom. The dark
purple squares with arrows indicate the location and polarization of the array
antennas. Black squares are the empty gaps between meta-gap sheets. Dashed
lines indicate lines of enforced symmetry with mirrored switched settings across the
entire array. Sheets with the same color have identical or mirrored switch settings.
Each mapping is labeled by the name of the mapping and the number of degrees of
freedom.
represent a minimum bound on the unrestricted optimal array per-
formance. erefore, it is possible to identify further improvements
in performance by relaxing this restriction. However, this potential
to increase performance comes at the cost of exponentially increas-
ing the search space making it less likely that the algorithms can
nd the optimal state. us, there is a trade-o between the per-
formance of the optimal state and the likelihood of identifying the
state.
By considering properties that the optimal solutions are likely to
have, it is possible to restrict the search space without eliminating
the best performing states. It is likely that the optimal state will be
symmetric due to the array symmetry. In addition, it is possible that
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 847
Figure 13. Main beam power optimization curves for (a) random search, (b)
genetic optimization, and (c) simulated annealing under dierent switch mappings.
(d) Main beam power optimization curve for variable neighborhood search under
dierent switch mappings when initialized with the best performing state identified
in the identical sheet experiments. Plots include the optimal state in addition to the
value of the explored states averaged over two batches.
regularity within the E- and H-planes is desirable to create a con-
sistent near-eld environment for each antenna. However, it is not
clear which combination of enforced symmetries and restrictions
must be true for the optimal state.
To explore the impact of dierent symmetries and restrictions,
the MBP is optimized under seven switch mappings with dif-
ferent enforced symmetries and degrees of freedom. Figure 12
includes diagrams of the seven mappings; each diagram is a sim-
plied representation of the array using a grid of squares. e
degrees of freedom in each mapping ranges from 24 for the pre-
viously explored identical sheet mapping shown in Figure 12(g),
to 960 for the completely unrestricted independent switch map-
ping shown in Figure 12(a). e remaining mappings enforce some
combination of horizontal and vertical symmetry, interior sheet vs.
edge sheet symmetry, or near-eld symmetry. In should be noted
that, except for the identical sheet mapping, the more restrictive
mappings are sub-spaces of the less restrictive mappings.
Array performance under each mapping is optimized using the
SA, genetic optimization, and random search optimization algo-
rithms. Random search is selected to establish a baseline, genetic
optimization is selected due to its high performance in the iden-
tical sheet experiments, and SA is selected because it has lower
variance than genetic optimization, yet still performs well. As
another test, the optimal MBP states identied in the identical sheet
experiments are used as initial seeds for VNS on the dierent map-
pings. By comparing the results of these dierent experiments, the
optimal mapping is identied.
Figure 13 shows the optimization results of these experiments.
For the random search, the mappings with more degrees of free-
dom perform worse than those that are more restricted. is is
expected as the performance of the random search increases with
the variability of the search space, and restricting the search space
increases the electromagnetic variation between randomly selected
states.
In the genetic optimization experiment, the identical sheet
restriction does not give the best results. Rather, slightly increas-
ing the degrees of freedom with the interior vs. edge mapping
and the E&H near-eld mapping allows a better state to be iden-
tied. In both cases, the average state explored did not converge
before the end of the experiment, indicating that a longer experi-
ment might improve performance further. However, larger degrees
of freedom degrade performance, with the independent switch
mapping performing the worst.
Unlike genetic optimization, none of the mappings used in the
SA experiment resulted in a better performing state than the iden-
tical sheet restriction. However the general order of the mappings’
performance is the same,4indicating that the interior vs. edge
and E&H near-eld mappings are indeed able to identify better
solutions than the others.
e initialized VNS experiments demonstrated the same gen-
eral trend; adding degrees of freedom improves performance ini-
tially before it starts to degrade. Due to the clarity of this trend,
the E-plane symmetry and independent switch mappings were
not explored. As can clearly be seen, the E&H near-eld mapping
resulted in signicantly better performance than any other map-
ping with a 1.36 dB improvement over the baseline, or 0.46dB
improvement over a ground-plane backed array. e remarkable
performance of this mapping suggests that it could be an outlier
and that if the optimization was run many more times, on aver-
age the performance of this mapping would not be quite as high.
However, it is unlikely that the average performance would be
substantially worse than this reported result.
An important point of comparison in Figure 13(d) is the per-
formance of the identical sheet mapping. While all of the map-
pings were initialized using the results of the previous identical
sheet experiments, this mapping was also subject to the identical
sheet restriction. is optimization thus serves eectively as an
extension of the run-time of the previous identical sheet exper-
iment. erefore, the greater performances demonstrated by the
other mappings over this second identical sheet optimization are
likely due to the change in mapping, and not due to the extended
run-time of the VNS algorithm.
In totality, the experiments indicate that while additional
degrees of freedom can improve performance, at some point the
search space is too large to nd a high performing state in a reason-
able amount of time. Both the interior vs. edge mapping and the
E&H near-eld mapping strike a balance and result in increased
performance. e initialized VNS experiment indicates that the
E&H near-eld mapping is likely the better performing of the two.
E&H near-field experiments
In the nal set of experiments, the E&H near-eld mapping is thor-
oughly explored by optimizing the MBP, the SLL, and the FoV
using each of the ve optimization algorithms. As with the iden-
tical sheet experiments, the algorithms are not initialized in order
to emulate the practical scenario where little is known about the
array in advance.
Figure 14 shows the algorithm’s performance over time in addi-
tion to the optimal array characteristic identied by each algo-
rithm. e similarities and dierences between the Identical Sheet
experiments and the E&H near-eld experiments are illuminating.
4e only exception being a particularly poor performance by the E-plane symmetry
mapping.
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
848 D. Elliott Williamstyer and Ali Hajimiri
Figure 14. (a)–(c) Optimization curves and (d)–(f) E-plane and (g)–(i) H-plane cuts of the optimal array characteristics for the random search, genetic optimization, variable
neighborhood search, particle swarm, and simulated annealing algorithms’ optimizations of main beam power, side lobe levels, and field of view under the E&H near-field
mapping. Optimization plots include the optimal state in addition to the value of the explored states averaged over two batches.
While genetic optimization still performs the best when optimizing
the MBP and SLL, VNS outperforms SA and PS. For FoV optimiza-
tion, only the VNS is able to make substantial improvements over
random search; the other algorithms performed similar to, or even
worse than the random search. For all of the optimization criteria,
the global to local approaches are less successful as the search space
increases.
e magnitude of the optimization results indicate that only
modest improvements are gained by relaxing the identical sheet
restriction. e state that maximizes the average MBP within the
FoV increased it by 1.2 dB compared to the “all o” baseline, a
0.07 dB improvement over the identical sheet restriction. is cor-
responds to a 0.3 dB enhancement compared to the ground-plane
backed array. e state that minimizes the average SLL within
the FoV suppresses it by 1.7 dB compared to the baseline, the
same suppression achieved by the optimal state under the identical
sheet restriction. e optimal FoV identied in the E&H near-eld
mapping, 76.8, is actually worse that the optimal FoV identi-
ed by random search in the identical sheet experiment. us,
increasing the degrees of freedom strictly resulted in worse per-
formance for FoV optimization. e array characteristics of the
optimal state resemble those of the optimal states under the iden-
tical sheet restriction, which is not surprising given their similar
performance.
e increased diculty of the search is demonstrated by the
slower convergence rates of both genetic optimization and VNS
when optimizing the MBP and SLL. e average values of explored
states also converge slower than in the identical sheet experiments.
Indeed VNS does not begin to oscillate until over halfway through
the run, indicating that it takes a long time to identify a local max-
ima. ese convergence rates and average state performance when
optimizing the FoV indicate that each of the algorithms, except
VNS, essentially behave like a random search. On the other hand,
VNS is able to improve performance as it quickly optimizes a local
maxima if it randomly encounters one.
Discussion
Meta-gap performance
e in-situ optimization experiments demonstrated that meta-gaps
are able to increase the average MBP within the FoV by 0.46 dB
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
International Journal of Microwave and Wireless Technologies 849
and decrease the average SLL within the FoV by 2 dB, compared
to a ground-plane backed array. e FoV can be increased by 23.5
compared to the theoretical 60FoV for a 𝜆−spaced array. ese
measured performance improvements represent a minimum bound
on the possible achievable improvements, as it is unlikely any of
the optimal states identied are the true global optima. Given that
only around 43,000 of the 2960 dierent states were characterized,
it is likely that even greater performance is possible.
However, another conclusion is that nding optimal solutions
is dicult due to the vastness of the search space. At some point,
additional degrees of freedom reduce the performance of the state
that can be found within a limited time-frame. Indeed, the identical
sheet restriction produced states with similar performance to those
under the E&H near-eld mapping despite it being an arbitrary and
sub-optimal restriction.
Structure of the optimization problem
e relative performance of the dierent algorithms within the
dierent experiments gives insight into the underlying optimiza-
tion problem structure. All of the experiments indicate that the
MBP and SLL optimization problems are tightly linked and dif-
ferent than FoV optimization. us, it is likely that algorithms or
approaches that work well on one of the problems will work well
on the other.
In addition, the identical sheet experiments suggest that the
MBP/SLL problem has many sparsely-spaced local minima in a
relatively gradual global basin. Because VNS quickly identies a
decent solution and then plateaus, it is likely that local minima are
relatively shallow and far apart. As the global to local approach
of SA and PS are able to make consistent improvements, it is
likely that there is some sort of global basin. e existence of
this gradual basin is supported by the fact that a random search
is able to quickly identify a solution with performance on par
with a ground-plane backed array, despite random switch settings
degrading performance on average.
However, the most critical observation is that genetic optimiza-
tion out performs the other algorithms under both the identical
sheet restriction and the E&H near-eld mapping, suggesting that
the MBP/SLL problem solutions are comprised of features with
intrinsic value. is matches intuition about the electromagnetic
problem as certain patterns of conductors, like closed loops or
lines of a specic length, will have a pronounced eect on the
electromagnetic elds. e large variance of genetic optimization
demonstrated in the simulated experiments is likely caused by
which features, if any, are present in the initial batch of elements,
as the algorithms do not run long enough for mutation to play a
substantial role. e success of genetic optimization suggests that
there likely exists a smaller set of basis solutions that could greatly
reduce the search space.
Another conclusion is that the FoV optimization problem is
quite dierent; the optimization space is much atter with most
states having little eect and only pockets of minor improve-
ment. is is demonstrated by the very gradual average-state
enhancement under the identical sheet restriction and the almost
non-existent average-state enhancement in the E&H near-eld
mapping. In fact, the problem under the E&H near-eld mapping
does not contain enough global variation for the algorithms to take
advantage of, as their convergence behaviors resemble that of ran-
dom search. Only VNS is able to make improvement because it
ascends to a local maxima. In addition, genetic optimization does
not oer substantial improvement over the othermetho ds, suggest-
ing that the FoV optimization problem likely does not contain a set
of basis solutions.
Possibility of enhanced algorithmic performance
e possible existence of a set of basis states for the MBP/SLL
optimization problem shows great promise for reducing the opti-
mization time, and thus enhancing the meta-gap performance.
While the binary switching optimization problem is NP-Hard in
general, particular constraints can reduce the problem complex-
ity, potentially making it convex. If a set basis can be identied,
then genetic optimization can become signicantly more ecient,
as it can be initialized with only the features that are required to
produce high-quality solutions. Depending on the properties of
the basis set, it could be possible to design the meta-gaps to only
contain basis states, reducing the number of switches and their par-
asitic eect on the radiated elds. Finally, just the existence of some
sort of basis set suggests that machine learning could be used to
eciently optimize meta-gaps without specically identifying the
exact basis states.
Alternatively, the similarity in the algorithmic behavior in both
the simulated and measured experiments suggests that simulation
and modeling errors do not drastically change the optimization
problem. erefore it is possible that optimal states could be iden-
tied in simulation and then used to initialize the in-situ opti-
mization. In this combined approach, a rapid simulated global
search would identify regions of good performance and the in-situ
optimization would nd the local optima.
ese enhanced optimization techniques could greatly improve
optimization time and allow more of the search space to be covered.
is in turn would allow higher performing states to be identied,
further increasing SLL suppression and MBP enhancement.
Arrays with dierent element spacings
is work exclusively studies the optimization problem and per-
formance of meta-gaps in a 𝜆-spaced array. Yet, it is important
to consider how the optimization problem and performance will
change for an array with dierent element spacing.
While the meta-gap concept is broadly applicable to any ele-
ment spacing, their impact on the near-eld environment depends
on the specic array and element design. erefore, both the opti-
mization problem and the performance of the optimal state will be
dierent.
Metahueristic algorithms will work on any meta-gap array
because they are agnostic of the underlying optimization prob-
lem. However, it is also likely that the basis states for the MBP/SLL
optimization problem are fundamental to the switched meta-gap
structure, and not on the particular properties of a 𝜆-spaced array.
erefore, enhanced algorithmic techniques that perform well on
a𝜆-spaced array will likely also perform well on arrays of dierent
spacing.
Further research into the performance of the meta-gaps and
their underlying optimization problem in dierent array designs
will be crucial once high-performing meta-gap optimization tech-
niques are developed.
Conclusions
is paper presents the design of a 𝜆-spaced 2D array with pro-
grammable meta-gaps between radiators. e platform is used
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
850 D. Elliott Williamstyer and Ali Hajimiri
to explore the ability of meta-gaps to alter the array characteris-
tics. Measurement results demonstrate that meta-gaps can enhance
the main beam power, SLL, and FoV of a sparse array. e com-
parative performance of algorithms in both in-situ optimization
experiments and simulated statistical experiments suggest that
switched passive networks are good candidates for feature based
optimization, such as genetic optimization and machine learning.
Experimental results also indicate that intelligently restricting the
size of the search space is critical for identifying high performing
states.
Acknowledgements. e authors thank A. Fikes, C. Ives, O. Mizrahi, and S.
Nooshabadi for their help.
Funding statement. is work was supported in part by the MURI Grant
FA9550-16-1-0566 via AFOSR.
Competing interests. None declared.
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D. Elliott Williamstyer received the B.S. and
M.Eng. degrees in Electrical Engineering and
Computer Science from the Massachusetts
Institute of Technology (MIT), Cambridge,
MA, USA, in 2015 and 2016 respectively, and a
Ph.D. degree in electrical engineering from the
California Institute of Technology, Pasadena,
CA, USA, in 2022. He joined the faculty of
Hofstra University in 2022. His research focuses
on developing adaptive electromagnetic systems
using novel degrees of freedom. His vision is to create devices with enhanced
control of electromagnetic elds that can dynamically adapt to changing
environments and needs. Dr. Williamstyer was a recipient of the Analog
Devices Outstanding Student Designer Award in 2017, won the 2022 EuMC
Young Engineer Prize for his work on exible meta-gaps, and was awarded a
2023 Innovation Teaching Fellow grant at Hofstra University.
Ali Hajimiri is Bren Professor of Electrical
Engineering at California Institute of Technology
(Caltech), where he is Director of the
Microelectronics Laboratory and Co-Director of
Space Solar Power Project. Before joining Caltech
in 1998, he worked at Philips Semiconductor,
Sun Microsystems, and Bell Laboratories. He has
authored and coauthored 2 books, several book
chapters, and more than 250 refereed journal
and conference technical articles. He has been granted more than 160 U.S.
patents with many more pending applications. His research interests include
high-speed and high-frequency integrated circuits for applications in sensors,
photonics, wireless energy transfer, biomedical devices, and communication
systems. Prof. Hajimiri is a Fellow of National Academy of Inventors and
IEEE. He was selected to the TR35 top innovator’s list. He was a Distinguished
Lecturer of the IEEE Solid-State and Microwave Societies. He was the recipient
of the Microwave Prize, Feynman Prize for Excellence in Teaching, and several
other teaching prizes. He has also won several best paper awards in various
conferences. He co-founded Axiom Microdevices Inc., whose fully-integrated
CMOS PA has shipped around 400,000,000 units, and was acquired by
Skyworks Inc. He is a Co-founder of GuRu Wireless Inc.
https://doi.org/10.1017/S1759078723001526 Published online by Cambridge University Press
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