# Control of the Scattering Properties of Complex Systems By Means of Tunable
Metasurfaces

Jared Erb Corresponding author<EMAIL_ADDRESS>Quantum Materials Center,
Department of Physics, University of Maryland, College Park, Maryland
20742-4111, USA David Shrekenhamer  Johns Hopkins University Applied Physics
Laboratory, Laurel, MD, 20723, USA  Timothy Sleasman  Johns Hopkins University
Applied Physics Laboratory, Laurel, MD, 20723, USA  Thomas M. Antonsen
Department of Physics, University of Maryland, College Park, Maryland
20742-4111, USA  Department of Electrical and Computer Engineering, University
of Maryland, College Park, Maryland 20742-3285, USA  Steven M. Anlage  Quantum
Materials Center, Department of Physics, University of Maryland, College Park,
Maryland 20742-4111, USA Department of Electrical and Computer Engineering,
University of Maryland, College Park, Maryland 20742-3285, USA

###### Abstract

We demonstrate the ability to control the scattering properties of a two-
dimensional wave-chaotic microwave billiard through the use of tunable
metasurfaces located on the interior walls of the billiard. The complex
reflection coefficient of the metasurfaces can be varied by applying a DC
voltage bias to varactor diodes on mushroom-shaped resonant patches, and this
proves to be very effective at perturbing the eigenmodes of the cavity.
Placing multiple metasurfaces inside the cavity allows us to engineer desired
scattering conditions, such as coherent perfect absorption (CPA), by actively
manipulating the poles and zeros of the scattering matrix through the
application of multiple voltage biases. We demonstrate the ability to create
on-demand CPA conditions at a specific frequency, and document the near-null
of output power as a function of four independent parameters tuned through the
CPA point. A remarkably low output-to-input power ratio of
$\frac{P_{out}}{P_{in}}=3.71\times 10^{-8}$ is achieved near the CPA point at
8.54 GHz.

##  1 Introduction

We consider bounded complex scattering environments, coupled to the outside
world through a finite number of scattering channels. Examples include
enclosed three dimensional spaces such as rooms, cabins in a ship or aircraft,
or larger enclosed spaces such as warehouses. Other examples include two-
dimensional microwave billiards and one-dimensional cable graphs with multiple
propagation paths between any two points in the billiard or nodes of the
graph. The scattering channels can be coupled to the system through antennas,
probes, apertures on the walls of the enclosure, or any means by which wave
energy can leave the enclosure and propagate outside it. We assume that these
systems are reverberant in the sense that waves propagate across the length
and breadth of the system multiple times before significantly decaying in
amplitude. Such systems are characterized by a scattering ($S$) matrix that
relates the set of in-going wave excitations on the channels to the
corresponding set of out-going waves on the same channels. Because the
scattering environment is complicated and typically lossy, the $S$-matrix is
sub-unitary and has complex matrix elements that are rapid and irregular
functions of the frequency of the waves.

The question arises as to how to control, or tame, a complex scattering
environment, such that it can be harnessed to perform specific and useful
tasks. Example tasks include establishing and maintaining a robust
communication link between two points inside the enclosure, or transferring
wave energy to a specific object inside the enclosure with high efficiency and
minimal interference. It is our belief that active and tunable metasurfaces
can be used to alter the scattering properties of complex structures and thus
create new opportunities to manipulate complex waves. Active metasurfaces have
the property that they can alter the reflection coefficient of one portion of
the boundary of a scattering environment [1, 2]. This establishes a degree of
control of the walls of the enclosure, creating the opportunity to alter the
waves everywhere in the enclosure, due to its reverberant nature.

The majority of work on active metasurfaces concerns single-pass reflection or
transmission interactions between waves and the metasurface, or involve
metasurface antennas that launch waves but do not interact with them again
[3]. Here we are concerned with the more challenging situation in which the
same waves interact with the same active metasurface multiple times during
their propagation. Efforts to control the wave properties of enclosures by
means of active and tunable metasurfaces are relatively few in number. Gros,
et al., showed that three programmable metasurfaces on the walls of a regular
six-sided enclosure could be used to effectively stir the modes of the cavity
to create a set of uncorrelated cavity configurations [4]. Frazier, et al.,
placed a 240-element tunable binary metasurface inside a 1 $m^{3}$ reverberant
three-dimensional system, but taking up only 1.5 % of the surface area, and
demonstrated the ability to create ’cold spots’ (minima in transmission
$S_{21}$) between two arbitrary points inside the enclosure [5]. That work
showed that coherent perfect absorption (CPA) could also be achieved through
variation of the metasurface pixel states, as long as the system was already
near the CPA condition at baseline [5]. Earlier, one-port perfect absorption
was demonstrated in a reverberant environment using a programmable metasurface
[6]. Wavefront shaping with a large number of scattering channels, and
creation of CPA states has also been demonstrated in three-dimensional
enclosures [7, 8]. The 240-pixel metasurface experiment also utilized a
machine learning algorithm that could find the pixel settings on the
metasurface required to generate a desired transmission $S_{21}$ spectrum as a
function of frequency over a substantial bandwidth [9]. Most recently, an
elegant formalism has been created to model the effects of tunable
metasurfaces in reverberant environments [10].

The ultimate goal of our experiment is to demonstrate control over all of the
scattering parameters of a given microwave billiard system. Often, the
functional properties of a cavity are designed into the shape and structure of
the cavity, usually including symmetries to help meet the design goals. In our
experiments, we alternatively use a chaotic cavity, which represents the most
general wave scattering setting possible, containing no geometrical or hidden
symmetries. To gain control over the system we instead place active
metasurfaces inside the cavity, which effectively allow electronic
manipulation of the boundary conditions of the cavity. Scattering environments
encountered in the real world are usually very complex and lack any
symmetries. By demonstrating control over this complex and low-symmetry
system, we are moving one step closer to actively controlling the scattering
environments of arbitrary real-life systems.

##  2 Experimental Setup

A two-dimensional wave chaotic quarter bow-tie billiard is loaded with three
tunable nonlinear metasurfaces. The billiard has been used before to
demonstrate the crossover from Gaussian Orthogonal to Gaussian Unitary
ensemble statistics in both the level spacings [11] and eigenfunctions, [12,
13] and more recently has been loaded with microwave diodes to act as a
reservoir computer [14]. The metasurfaces used in this work were fabricated by
the Johns Hopkins University Applied Physics Laboratory [15] and were designed
to be used for reflection amplitude variation between 11-18 GHz and reflection
phase variation between 14-16 GHz. The metasurfaces are a linear array of 18
varactor-loaded mushroom-shaped resonant elements, where each element is sub-
wavelength in size [16]. The PCB material of the metasurfaces is Rogers 5880
and the diode part number is MACOM MAVR-011020-1411. Each metasurface is 1.8
mm thick, 7.9 mm high, and 185 mm long, and is flexible enough to conformally
attach to a curved interior wall. The diodes on a given metasurface can be
tuned simultaneously with a globally applied DC voltage bias to the
metasurface through thin insulated wires that exit the cavity underneath the
lid. As the applied voltage bias is increased from 0 to 15 V, the capacitance
of the varactor diodes varies from 0.24 to 0.03 pF, thus increasing the
resonant frequency of the patches. The metasurfaces are placed along three
different walls of the billiard and are connected to a Keithley 2230G-30-1
triple channel programmable DC power supply, as shown in Fig. 1(a). Each of
the three metasurfaces cover approximately 12% of the perimeter of the
billiard. The quality factor of the billiard with the 3 metasurfaces present
is approximately 250 at 9 GHz.

(a) * (b) * Figure 1: Quarter bow-tie billiard with three tunable metasurfaces
(MS1-3) along the interior walls of the cavity. The left image is a schematic
of the quarter bow-tie billiard with the three metasurfaces inside, and the
lid lifted off the base. Two antennas are connected to the billiard through
the lid and are attached to coaxial transmission lines, representing the
scattering channels, and these are connected to the VNA. The right image is a
perspective photograph of the interior of the base of the cavity, showing the
three metasurfaces.

[.45] [.45]

An Agilent Technologies model N5242A or Keysight model N5242B microwave vector
network analyzer (VNA) is connected to the two ports of the billiard through
coaxial cables that support a single mode of propagation, and the $2\times 2$
complex scattering matrix ($S$) is measured over a chosen frequency range.
Through the variation of the reflection coefficient of the metasurfaces, we
can manipulate the poles and zeros of the scattering matrix to create
conditions for coherent perfect absorption (CPA), among other things. CPA
occurs when a zero of the scattering matrix is brought to the real frequency
axis [17, 18]. When the system is excited with the $S$-matrix eigenvector that
corresponds to the zero eigenvalue, all of the injected energy is absorbed
inside the scattering system, and none is reflected or transmitted through any
of the scattering channels. The perfect absorption condition requires this
very specific type of coherent eigenvector excitation of the billiard, and any
deviation causes significant reflection and/or transmission.

To establish CPA experimentally, we first choose a frequency range of interest
and measure the scattering matrix with the VNA, and with each successive
measurement the applied voltage bias of a particular metasurface is increased,
usually with a step size of 0.01 V. During the set of measurements, the other
metasurfaces in the cavity are held at a fixed applied voltage bias. From this
set of measured $S$-matrices, the complex Wigner-Smith time delay [19] is
calculated as a function of frequency at various applied biases to the
metasurface. The conditions for CPA are found by identifying parametric
settings where the magnitude of the complex time delay diverges.

Once the conditions for a CPA state are found, the scattering matrix is
diagonalized to determine the $\lambda_{S}=0+i0$ eigenvalue and the
corresponding eigenvector $\ket{\psi_{CPA}}$. For our system, the scattering
matrix has two eigenvalues $\lambda_{S}$ and eigenvectors $\ket{\psi}$ whose
elements are two complex numbers representing the amplitudes of the waves
injected at the two ports, but we can determine which one is the CPA
excitation by finding the one that has an identically zero eigenvalue. From
the CPA eigenvector, the relative amplitude and phase of the required
excitation that must be simultaneously injected into the two ports is
determined. With this information, the 2-port dual source mode of the VNA is
used to inject the CPA eigenvector into the billiard to verify that the
conditions for coherent perfect absorption are achieved. For $S$-parameter
measurements, the VNA is calibrated up to the antennas on the billiard. But
when the 2-port dual source mode is activated, the calibration is no longer
valid. This causes the necessary parameters determining the CPA excitation to
be slightly different than from the calibrated $S$-parameter measurement.
During the CPA injection, we measure the ratio of outgoing power $P_{out,j}$
to incident power $P_{in,j}$ on each port $j$ as a function of several
parameters near the CPA condition. To do this, the receivers on each port of
the VNA are used to measure both the power that goes into the cavity and the
power that comes back out.

##  3 Complex Wigner-Smith Time Delay

The Wigner-Smith time delay $\tau_{W}$ is a rough measure of how long a wave
lingers in a scattering system before leaving. In unitary systems, the time
delay is a real quantity, but for subunitary systems it becomes complex valued
[20, 8, 19]. The Wigner-Smith time delay is defined as:

$\tau_{W}=-\frac{i}{M}\frac{\partial}{\partial f}log\>det\>S$ (1)

where $i$ is the imaginary number, $M$ is the number of ports connected to the
system, $f$ is frequency, and $S$ is the scattering matrix. Experimentally,
the frequency derivative in the Wigner-Smith time delay is calculated using a
central difference formula.

It is known that the determinant of $S$ can be written as a Weierstrass
factorization extending over the complex frequency plane as [21, 22, 23, 24,
25, 26, 27, 28, 29]

$detS(f)\propto\prod_{n=1}^{N}\frac{f+i\eta-z_{n}}{f+i\eta-\varepsilon_{n}}$
(2)

where $N$ is the total number of modes of the closed system, $\eta$ is the
uniform attenuation, and $z_{n}$ and $\varepsilon_{n}$ are the zeros and
poles, respectively, of the scattering matrix. We further define $z_{n}$ and
$\varepsilon_{n}$ as:

$z_{n}=Re[z_{n}]+iIm[z_{n}]$ (3) $\varepsilon_{n}=f_{n}-i\Gamma_{n},$ (4)

and adopt the convention that $\Gamma_{n}>0$ in passive lossy systems. It has
been shown [19] that the complex Wigner-Smith time delay associated with each
mode $n$ can be written as a sum of two terms for both the real and imaginary
parts, one involving the scattering poles and the other the scattering zeros,
as follows.

$\displaystyle\text{Re}\>\tau_{W}$
$\displaystyle=\frac{1}{M}\sum\limits_{n=1}^{N}\left[\frac{Im[z_{n}]-\eta}{(f-Re[z_{n}])^{2}+(Im[z_{n}]-\eta)^{2}}+\frac{\Gamma_{n}+\eta}{(f-f_{n})^{2}+(\Gamma_{n}+\eta)^{2}}\right]$
(5) $\displaystyle\text{Im}\>\tau_{W}$
$\displaystyle=-\frac{1}{M}\sum\limits_{n=1}^{N}\left[\frac{f-Re[z_{n}]}{(f-Re[z_{n}])^{2}+(Im[z_{n}]-\eta)^{2}}-\frac{f-f_{n}}{(f-f_{n})^{2}+(\Gamma_{n}+\eta)^{2}}\right]$
(6)

Using this formalism, we can see that the Wigner-Smith time delay is divergent
at $f=Re[z_{n}]$ when the imaginary part of the zero $Im[z_{n}]$ of the
scattering matrix is equal to the uniform attenuation of the system $\eta$.
The diverging time delay for a wave propagating inside a lossy system is an
indication that an $S$-matrix zero has crossed the real frequency axis, which
is the enabling condition for coherent perfect absorption [8, 19]. Further,
from the measured Wigner-Smith time delay, the poles and zeros can be
systematically extracted using Eqs. (5) and (6), and their evolution with
metasurface voltage bias can be visualized. For example, note that the
divergent term in the real part of $\tau_{W}$ has a sign that depends on the
sign of $Im[z_{n}]-\eta$, which changes as the imaginary part of the
$S$-matrix zero is varied. Using information such as this, we can understand
how a complex scattering system interacts with incoming waves, and use this
knowledge to engineer conditions for achieving CPA.

Figure 2: Real part of the Wigner-Smith time delay $Re[\tau_{W}]/2\pi$ vs.
frequency and metasurface 2 voltage for a particular cavity realization. We
use a nonlinear color scale to help distinguish the areas of small time delay
from the areas of very large time delay due to the divergence of the Wigner-
Smith time delay at one point in this parameter space. With this color scale,
the majority of the colors between blue and red (green-yellow) are
concentrated near the mean value of the time delay in this parameter space
($\sim$3 ns). The single largest value of the Wigner-Smith time delay measured
in this domain has a value of 696.5 $\mu s$, but is excluded from this plot
for clarity. The Heisenberg time ($\tau_{H}=\frac{2\pi}{\Delta}$, with mean
mode-spacing $\Delta=\frac{c^{2}}{2\pi fA}$ for this 2D billiard of area $A$)
at 9.66 GHz is $\tau_{H}\approx$ 0.49 $\mu$s. Note that $\tau_{W}$ is divided
by $2\pi$ to convert to seconds. Left Inset: Real Wigner-Smith time delay vs.
voltage at 9.66 GHz. Right Inset: Real part of Wigner-Smith time delay vs.
frequency at a fixed voltage of 6.87 V.

##  4 Experimental Results

For a specific setup of the cavity, we take measurements over a particular
frequency range and voltage bias applied to one of the metasurfaces. Then the
applied bias is swept until the Wigner-Smith time delay shows a near
divergence. A typical result is shown in Fig. 2 for the real part of the
Wigner-Smith time delay in a narrow range of frequency and voltage. For the
mode at 9.66 GHz, the time delay becomes very large in magnitude around 6.87
V. This indicates that a CPA state is possibly located at the point of
divergence. The left inset of Fig. 2 shows the real part of the time delay as
the voltage bias is increased at a fixed frequency of 9.66 GHz, through the
point of divergence. The dramatic increase can be explained by the imaginary
part of scattering zero ($Im[z_{n}]$) decreasing in magnitude toward the value
of the uniform attenuation ($\eta$) of the system, causing the Wigner-Smith
time delay to increase. At the divergence, $Im[z_{n}]$ is equal to $\eta$, and
past the point of divergence, $Im[z_{n}]$ has decreased below the value of
$\eta$, causing the sign of the Wigner-Smith time delay to switch from
positive to negative (see Eq. (5)). In the right inset of Fig. 2, the voltage
is kept fixed at 6.87 V while the frequency is increased through the point of
divergence. A similar increase of the real part of the time delay is observed,
except that the sign of the Wigner-Smith time delay remains positive through
the divergence. The interpretation is that at a fixed voltage bias of 6.87 V,
the values of $Im[z_{n}]$ and $\eta$ remain essentially fixed as the frequency
is varied, while the divergence arises from the frequency dependence indicated
in Eq. (5). The variation of $Re[\tau_{W}]$ from +4 $\mu s$ to -3.6 $\mu s$
visible in Fig. 2 occurs diagonally across frequency and voltage bias because
the bias voltage changes both the real and imaginary parts of the S-matrix
zero simultaneously.

Figure 3: Zeros and poles of the scattering matrix for a particular mode of
the bowtie billiard, extracted from measured complex time delay, plotted in
the complex frequency plane as a function of voltage bias applied to
metasurface 1. For each successive point from blue to red, the applied bias to
metasurface 1 is increased while holding all other parameters fixed. The top
curve shows evolution of the $S$-matrix zero, and the bottom curve shows
evolution of the pole.

Using the Wigner-Smith time delay expression as a sum of two terms for the
real and imaginary parts, the location of the poles and zeros of the
scattering matrix are extracted as functions of applied bias to metasurface 1
for another mode of the bowtie billiard, as shown in Fig. 3. This is
accomplished by fitting the Lorentzian expressions (Eqs. (5) and (6)) to the
real and imaginary parts of the experimental data simultaneously, and from the
best fitting we extract the pole and zero information for each voltage bias.
Note that the fit parameters are $Re[z_{n}]$ and $Im[z_{n}]-\eta$ for the
zero, and $f_{n}$ and $\Gamma_{n}+\eta$ for the pole, for each mode $n$.
Figure 3 shows that as the applied bias is increased, the zeros are observed
to move upward in the complex frequency plane toward the real frequency axis,
and CPA is enabled at the point where $Im[z_{n}]-\eta=0$. With this
information, we now know how to engineer the cavity to have a specific time
delay, degree of absorption, etc., for this frequency range.

Figure 4: Zeros of the scattering matrix ($Re[z_{n}]$,$Im[z_{n}]-\eta$) for a
particular mode of the bowtie billiard, plotted in the complex frequency
plane. Each successive point from left to right is a 0.01V step increase in
the voltage applied to metasurface 3 (MS3), and each color represents a
different fixed voltage of metasurface 1 (MS1). During these measurements,
metasurface 2 was held at a fixed voltage bias. The inset shows a zoomed in
view of the real frequency axis near the axis crossings.

Taking measurements of the cavity under different conditions, a particularly
interesting case is found where a single mode has a zero that crosses the real
axis twice as the voltage bias to metasurface 3 is varied (see Fig. 4). These
real frequency-axis crossings for the $S$-matrix zero correspond to two CPA
states in the voltage sweep of one metasurface. Then, using the other two
metasurfaces in the cavity, it is possible to raise and lower this parabolic
evolution of the zeros, as illustrated in the inset of Fig. 4. This result
demonstrates control of the scattering zeros of the system through the
additional degrees of freedom that the other metasurfaces offer.

Using the extracted pole and zero locations as a function of voltage, we can
use the Lorentzian expression (Eq. (5)) to examine in detail the real part of
the Wigner-Smith time delay for one of the curves from Fig. 4. In Fig. 5 we
see that there are two locations where the time delay is nearly divergent, and
that in between the divergences there is a significant peak value of time
delay, approximately 9 $\mu$s, and its location can be finely tuned in
frequency. This delay corresponds to 2.7 km of free space travel, and with the
characteristic length scale of the cavity being $\sqrt{A}$ (with cavity area
$A=0.115\ m^{2}$), this corresponds to approximately 8,000 bounces around the
cavity. It should be noted that at all bias values the time delay-bandwidth
product is on the order of unity. From Fig. 5, we can also see from the inset
that we can move the scattering matrix zeros and push apart the frequencies of
diverging time delay by utilizing the other metasurfaces inside the cavity.
This also causes the peak time delay between the two zero crossings to
decrease.

Figure 5: Real part of the Wigner-Smith time delay $Re[\tau_{W}]/2\pi$ vs.
frequency recovered from a particular curve from Fig. 4. The sign changes of
the extreme values of time delay bracket the two CPA points. The lower
frequency CPA state has an extreme time delay value around 60 $\mu s$, and the
upper frequency CPA state has an extreme time delay value near -500 $\mu s$,
not shown for presentation clarity. The gaps in between the peaks for each
color show the finite voltage resolution of the measurement. The inset shows
the real part of Wigner-Smith time delay vs. frequency for different fixed
values of the metasurface 1 and 2 voltage biases, showing the two CPA points
separated by a much larger frequency.

Once the conditions where the Wigner-Smith time delay diverges are found, we
see that the $S$-matrix has at least one eigenvalue nearly equal to
$\lambda_{S}=0+i0$. We can calculate the eigenvector of the $S$-matrix
corresponding to this eigenvalue. To test for the CPA condition, we must
inject this eigenvector excitation into the billiard. The CPA eigenvector can
be defined as: $\ket{\psi_{CPA}}=\begin{pmatrix}Xe^{i\theta}\\\
Ye^{i\psi}\end{pmatrix}$, defining the amplitudes and phases of the excitation
at the CPA frequency. Hence for the CPA excitation, the relative phase of the
signal between ports 1 and 2 is $\psi-\theta$, and the relative amplitude
between ports 1 and 2 is $20\log_{10}\left(\frac{X}{Y}\right)$, when $X$ and
$Y$ are measured in linear voltage and need to be converted to dB. The VNA is
used to inject this specific excitation into the system at the CPA frequency.
The expectation is that the output vector
$\ket{\psi_{out}}=S\ket{\psi_{in}}=0$ in this case, meaning that all of the
input energy is perfectly absorbed.

To test this experimentally, we measure both the injected and received powers
from the cavity, and see how those powers vary as the parameters of the
system, and the excitation, are changed. In Fig. 6, the four parameters we
have control over are systematically varied to verify that the CPA state is
correctly identified. The four parameters are the voltage applied to the
metasurface (which controls the location of the S-matrix zero in the complex
frequency plane), the relative amplitudes of the signals on the two input
channels, the frequency $f$, and the relative phase of the signals injected
into the two channels. For each parameter sweep, the other parameters are set
to their optimal values. We measure the total input power delivered to the
system through the two channels, $P_{in}=P_{in,1}+P_{in,2}$, and the total
power that emerges from the cavity, $P_{out}=P_{out,1}+P_{out,2}$, and form
the ratio $P_{out}/P_{in}$. This process is carried out in an iterative manner
and converges to a point very close to the CPA condition for the system, which
corresponds to $P_{out}/P_{in}=0$. For example, in the frequency sweep in Fig.
6(c), the ratio of output power to input power changes by approximately seven
orders of magnitude for small changes in frequency, with a minimum output to
input power ratio of $\frac{P_{out}}{P_{in}}=3.71\times 10^{-8}$. We observe
that the exact conditions of the CPA state slowly drift over time, likely due
to the temperature fluctuations of the laboratory. We believe this temperature
drift to be the limiting factor on how precisely we can measure the CPA state.
This illustrates how sensitive the CPA state is to small perturbations to the
system, making it attractive for use as a sensor.

Figure 6: Evidence of Coherent Perfect Absorption in the ray-chaotic quarter
bow-tie microwave billiard, for four independent parametric sweeps. We measure
the ratio of the total output power to the total input power, $P_{out}/P_{in}$
as a function of parameter variation. The swept voltage bias to metasurface 2
is shown in (a). Small parametric deviations from the CPA input eigenvector
$\ket{\psi_{CPA}}$ are made through the relative power
$20\log_{10}\left(\frac{X}{Y}\right)$ (b), frequency (c), and relative phase
$\psi-\theta$ (d) sweeps.

##  5 Discussion

Much of the prior work to controllably alter the scattering matrix of complex
systems has been done by mechanical means, mainly for the purpose of
perturbing the poles of the scattering matrix [30]. With the metasurfaces, the
poles and zeros of the scattering matrix can be smoothly varied in the complex
frequency plane and placed in determined locations, purely by electronic
means. By moving the zero of the scattering matrix to the real frequency axis,
we can enable CPA and this is correspondingly seen as a divergence in the real
part of the Wigner-Smith time delay. We can inject the zero-eigenvalue
eigenvector and observe nearly complete absorption, but this is a singular
condition which is only possible at a single point in the parameter space.
Suwunnarat et al. have recently shown the creation of a nonlinear CPA by using
exceptional point degeneracies of the zeros of the scattering matrix [31].
Work such as this shows the future possibilities of creating larger parametric
regions of coherent near-perfect absorption, which can be beneficial for a
variety of applications in communications and wireless power transfer.

##  6 Conclusions

In this paper we show that using tunable metasurfaces inside a two-dimensional
wave-chaotic cavity, we can control the locations of scattering poles and
zeros in the complex frequency plane. By perturbing the system with the
metasurfaces, we can drag the scattering zeros across the real frequency axis
to create a coherent perfect absorption condition. The demonstration of nearly
complete absorption is illustrated in Figure 6 by having approximately seven
orders of magnitude less power leaving the system compared to that injected in
the CPA eigenvector. We also demonstrated precise manipulation of the location
of the CPA state by applying voltage biases to the other metasurfaces in the
cavity. This is equivalent to controlling the scattering properties of the
system, and this control gives us the ability to engineer specific conditions
in the cavity, such as having coherent perfect absorption at a particular
frequency, tuning a desired time delay for a signal, having regions of high
absorption for unwanted signals, and low absorption for desired signals, etc.

## Acknowledgements

We acknowledge insightful discussions with Tsampikos Kottos, Dan Sievenpiper,
and Lei Chen. This work was supported by NSF/RINGS under grant No.
ECCS-2148318, ONR under grant N000142312507, ONR under grant N0002413D6400,
DARPA WARDEN under grant HR00112120021, ONR DURIP FY21 under grant
N000142112924, and ONR DURIP FY22 under grant N000142212263.

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