# Method for Extracting the Equivalent Admittance from Time-Varying
Metasurfaces and Its Application to Self-Tuned Spatiotemporal Wave
Manipulation

Ashif Aminulloh Fathnan<EMAIL_ADDRESS>Department of
Engineering, Graduate School of Engineering,
Nagoya Institute of Technology, Nagoya, Aichi, 466-8555, Japan  Haruki Homma
Department of Engineering, Graduate School of Engineering,
Nagoya Institute of Technology, Nagoya, Aichi, 466-8555, Japan  Shinya Sugiura
Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505,
Japan  Hiroki Wakatsuchi  Department of Engineering, Graduate School of
Engineering,
Nagoya Institute of Technology, Nagoya, Aichi, 466-8555, Japan

###### Abstract

With their self-tuned time-varying responses, waveform-selective metasurfaces
embedded with nonlinear electronics have shown fascinating applications,
including distinguishing different electromagnetic waves depending on the
pulse width. However, thus far they have only been realized with a spatially
homogeneous scattering profile. Here, by modeling a metasurface as time-
varying admittance sheets, we provide an analytical calculation method to
predict the metasurface time-domain responses. This allows derivation of
design specifications in the form of equivalent sheet admittance, which is
useful in synthesizing a metasurface with spatiotemporal control, such as to
realize a metasurface with prescribed time-dependent diffraction
characteristics. As an example, based on the proposed equivalent admittance
sheet modeling, we synthesize a waveform-selective Fresnel zone plate with
variable focal length depending on the incoming pulse width. The proposed
synthesis method of pulse-width-dependent metasurfaces may be extended to
designing metasurfaces with more complex spatiotemporal wave manipulation,
benefiting applications such as sensing, wireless communications and signal
processing.

††preprint: APS/123-QED

##  I  Introduction

Metamaterials and their two-dimensional versions, metasurfaces, have shown
unprecedented methods for wave manipulation, marking a new milestone in the
development of functional artificial structures. They are characterized by
subwavelength engineered elements, called meta-atoms, and have distinct
properties such as a near-zero index [1], balanced electric-magnetic responses
[2], or maximal cross-polarization [3], intended to feature novel
functionalities in the macrostructures [4, 5]. While many initial works on
metasurfaces were on achieving control of a static wave, which does not vary
with time, recent developments show dynamic wave shaping via time-varying
meta-atoms [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. These active-type
metasurfaces have been realized across a wide-range of the electromagnetic
spectrum, achieving various interesting spatiotemporal applications such as
second harmonic generation, nonlinear beam shaping and non-reciprocity [19,
20, 21, 22]. However, the realization of these spatiotemporal metasurfaces
often requires complicated external excitation subsystems, which could
substantially limit the applicability. For example, in controlling individual
meta-atom states, complex electrical biasing is required in microwave
metasurfaces, which provides advanced control but in return becomes a
formidable challenge for a large metasurface area and at higher operating
frequencies [23, 11, 9].

To achieve dynamic wave control without a complex chain of external
excitation, a resonant metasurface may be combined with nonlinear media. The
introduction of nonlinear media transforms the metasurface into a self-tunable
device that is sensitive to impinging wave properties such as the power level
or polarization state, removing the need for complicated electrical biasing
[24, 25, 26, 27, 28]. In resonant metamaterials, one example of introducing
nonlinearity is by inserting nonlinear Kerr nanoparticles within resonant
nanorod arrays, which results in the metamaterial having flux-dependent
transmission and absorption [29]. At microwave frequencies, a method of
embedding nonlinear lumped components such as diodes and varactors into
resonant metallic meta-atoms has realized various self-tunable power-dependent
metamaterials [30, 31, 32, 33, 34]. One of the representative works in Ref.
[30] demonstrated microwave beam steering and focusing depending on the
impinging monochromatic light intensity. In recent works, more complex
nonlinear power-dependent wave shaping has also been realized, such as
obtaining controllable spatial dispersion in a metasurface absorber [35] or
steering surface wave propagation in a mushroom-type metasurface [36]. All
these works utilized either diodes, varactors, or transistors in realizing
nonlinearity within the meta-atom.

Figure 1: (a) Metasurface with a temporal response enabling waveform
selectivity, and (b) its time-varying equivalent admittance. (c, d)
Spatiotemporal wave manipulation using the proposed metasurface where the
focal length (FL) of the metasurface changes with pulse width (PW) or time.

To exploit the temporal scattering of metasurfaces, more recent works have
utilized transient responses of reactive circuits under equivalent direct
current (DC) input [37, 38, 39, 40, 41]. These metasurfaces consist of
homogeneous patch structures embedded with a diode bridge full-wave rectifier
connected to a combination of a resistor, an inductor, and a capacitor ($R,L,$
and $C$). When an incident microwave beam illuminates the metasurface, the
output currents (or voltages) of the diode bridge do not immediately change
with time due to the existence of the reactive circuits, producing the effect
of time-varying transmission, reflection, and absorption. Unlike other
temporal metasurfaces that utilize periodic modulation of meta-atoms to
generate sidebands, in these transient metasurfaces, the meta-atoms are
designed to transform the incident energy into zero frequency. Their time
constant is much longer than the incident beam period; hence, they can perform
waveform selectivity or time-domain filtering, i.e., differentiate incident
beams depending on how long they propagate through the metasurface even at a
constant frequency [42]. An exemplary model of such waveform-selective
metasurfaces is shown in Fig. 1(a), which allows propagation of an incoming
pulse but absorbs or reflects a continuous wave. These metasurfaces can enable
various important applications such as designing waveform-selective cloaked
antennas [43] and realizing reconfigurable surfaces for Wi-Fi signals [40] as
well as for microwave imaging [10]. Despite the fascinating properties of
these temporal metasurfaces, they have thus far been realized strictly in a
spatially homogeneous condition. Moreover, very limited works on analytical
modeling of these metasurfaces have been reported, such as in Ref. [44], which
were focused only on analyzing the transient circuits. Thus, equivalent
admittance modeling, commonly used in other thin metasurfaces [45], remains
unexplored for these transient metasurfaces.

In this study, we extract the equivalent admittance of a self-tunable temporal
metasurface to facilitate design of spatiotemporal wave manipulation. We
consider a meta-atom consisting of a subwavelength metallic slit connected to
a full-wave rectifier and a combination of $R,L,$ and $C$ components. Such a
metasurface can be represented as a self-tuned time-varying electric
admittance and a magnetic impedance, in which the time constant corresponds to
the circuit parameters being implemented. As seen in Fig. 1(b), an equivalent
circuit model can be used to accurately predict the admittance $Y_{se}$ and
the time-varying scattering parameters at the resonant frequency
($\omega_{0}$). This analytical modeling provides a convenient design
methodology to realize a spatiotemporal metasurface, through which a Fresnel
zone plate (FZP) [46, 47] has been designed that exhibits a time-varying focal
point. In this FZP, the focal point changes depending on the pulse width of
the incoming wave (Fig. 1(c) and 1(d)). This work represents an important
finding on equivalent admittance sheet modeling of temporal metasurfaces and
paves the way for subsequent research on self-tuned spatiotemporal
metasurfaces.

##  II Metasurface with a Time-Varying Equivalent Admittance

###  II.1 Metasurface Configuration and Admittance Sheet Modeling

The metasurface considered within this study consists of a thin metallic slit
on top of a dielectric layer, as shown in Fig. 2(a). The same metasurface has
also been reported in Ref. [39]. Within the metallic slit, an embedded circuit
is connected. This embedded circuit consists of a diode bridge rectifier
loaded with a transient circuit (see Fig. 2(b)). The transient circuit can be
any combination of $R,L,$ and $C$ components, but for simplification, here, we
consider four combinations, i.e., $RL$ series, $RC$ parallel, $RLC$ series and
$RLC$ parallel (see Fig. 2(c)-(f). Note that in an ideal case, this
metasurface unit cell extends into an infinite two-dimensional homogeneous
array.

When a transverse electric (TE) wave impinges on the metasurface, it generates
a potential difference across the slit. Since the slit is connected to the
embedded circuit, when the potential difference across the slit exceeds the
diode turn-on voltage, a current starts to flow into the diode bridge and is
rectified. With the rectified input voltage or current from the diode bridge,
the capacitor or inductor prevents instantaneous changes in both parameters, a
response that is commonly found in a DC-driven transient circuit. Therefore,
the embedded circuit introduced within the slit is responsible for the
transient response observed in the scattering parameters of the metasurface.
The temporal scattering profile, e.g., transmission increasing or decreasing
with time or increasing then decreasing, depends on the implemented circuit
parameters, where the four types of transient circuits in Fig. 2(c)-(f) give
distinctive results.

As this metasurface has previously been extensively studied in numerical
simulations and experiments [37, 42, 39, 40], here, we focus on modeling this
metasurface as equivalent surface parameters to further assist in the design
methodology. In modeling such a transmissive metasurface, since the thickness
of the metasurface is much smaller than the wavelength, an impedance boundary
condition can be applied, in which the metasurface is equivalent to a
distribution of an electric sheet admittance ($Y_{se}$) and a magnetic sheet
impedance ($Z_{sm}$). The tangential components of the electric and magnetic
field can be equivalent to the voltage and current in circuit theory; hence,
the scattering problem of a normally incident plane wave can be modeled by
three shunt electric admittances separated by transmission lines, as seen in
Fig. 2(g). Such modeling has been widely used to describe subwavelength
resonant metasurfaces [48, 49, 2, 50, 51], where, in the case of no
magnetoelectric coupling considered, the resonant structures are called
Huygens’ metasurfaces [49].

Figure 2: (a) Physical configuration of the self-tuned time-varying
metasurface considered within this study. (b) Embedded circuit within the
slit: a diode bridge full-wave rectifier loaded with a transient circuit (TC).
(c)-(f) Transient circuits with four different transient responses: (c) $RL$
series, (d) $RL$ parallel, (e) $RLC$ series, and (f) $RLC$ parallel circuits.
$R_{d}$ represents the resistive component of two diodes and is connected to
all the circuits in series. (g) Ideal model for the metasurface where the
scattering properties can be physically represented as three shunt
admittances. (h) Metasurface model considered here with only a single-layer
shunt admittance $Y_{se}$ due to limited influence of $Z_{sm}$. (i) Equivalent
transmission line model of the metasurface with certain dielectric constant
$\epsilon_{s}$ and thickness $h$.

Here, since the metasurface only consists of a single-layer metallic
structure, the magnetic response is weak. Therefore, in Subsec. II.2, we will
show that the modeling of such a metasurface can be realized by considering
only a single electric shunt admittance. In this simplified model, the
subwavelength slit structure resembles a resonant meta-atom equivalent to a
transmission line loaded with a single electric sheet admittance, as shown in
Fig. 2(h)-(i).

###  II.2 Extraction of the Time-Varying Electric Admittance and Magnetic
Impedance

Similar to other thin metasurfaces that introduce field discontinuities into
the incoming wave, the metasurface considered here can also be treated as a
distribution of electric and magnetic sheet impedances [49, 2, 50, 51]. With
the transient response, however, the equivalent admittance and impedance of
the metasurface should also have a time dependence. The time-varying sheet
impedance and admittance can be obtained by using a calculation with complex
transmission and reflection, both having time and frequency dependence, as
follows [45]:

$\displaystyle Y_{se}(t,f)=\frac{2(1-T(t,f)-R(t,f))}{\eta(1+T(t,f)+R(t,f))},$
(1) $\displaystyle
Z_{sm}(t,f)=\frac{2(1-T(t,f)+R(t,f))}{\eta(1+T(t,f)-R(t,f))}.$ (2)

Here, $T$ is the complex transmission coefficient ($S_{21}$), $R$ is the
complex reflection coefficient ($S_{11}$), $f$ is the frequency, $t$ is time,
and $\eta=\sqrt{\mu/\epsilon}$ is the free-space wave impedance. It is noted
that essentially time and frequency are interchangeable and interdependent.
The time-frequency representation of Eqs. (1)-(2) is therefore bounded by a
resolution limit where Fourier transform cannot localize simultaneously both
the time domain and frequency domain [52]. With this limit, a trade-off
relation exists in which optimizing frequency resolution leads to poor
temporal resolution and vice versa. However, if a system has a slow variation
in the time domain, reducing the temporal resolution is acceptable, which in
turn would relax the time-frequency resolution bounds. This condition is
consistent with the metasurface considered here as it has a much larger time
constant $t_{c}$ than the time period of the waveform $t_{w}$. Hence,
$t_{c}\gg t_{w}$ guarantees the validity of Eqs. (1)-(2).

Figure 3: Extraction of equivalent surface parameters of the metasurface. (a)
Spectrogram of the metasurface transmission intensity. (b) Spectrogram of the
metasurface transmission phase. (c)-(d) $Y_{se}$ and (e)-(f) $Z_{sm}$
(normalized to the wave impedance ($\eta$), both real and imaginary parts)
over frequency and time, calculated using Eqs. (1) and (2). (g) Comparison of
transmission intensity over frequency at t=200 ns between the numerical
simulation and analytical calculation using the single shunt admittance model.
(h) The same comparison as in (g) over time at 4.2 GHz. The metasurface here
is with $RL$-series circuit where $R_{L}$=10 $\Omega$ and $L$=0.1 mH.

In extracting the impedance and admittance, we use a co-simulation method
based on ANSYS Electronics Desktop 2020 R2 [36, 37, 38], as explained in
Appendix A. This simulation method integrates a full-wave numerical result
within a circuit analysis. The meta-atom, as seen in Fig. 2(a), is first
analyzed using a full-wave numerical solver, where the meta-atom period is
$p=$18 mm, having slit dimensions of $s_{l}=$16 mm and $s_{h}=$5 mm, a
dielectric thickness of $h=$1.5 mm and permittivity of $\epsilon_{s}$=3. In
the circuit simulation, the electronic components seen in Fig. 2(c)-(f) are
embedded, and a transient solver is used in which the results of the circuit
simulation are voltages corresponding to the scattering parameters of the
meta-atom, i.e., $v_{T}$ for transmission voltage and $v_{R}$ for reflection
voltage. These voltages are sine waves with certain frequencies but
continuously time-varying amplitudes. To be able to calculate Eqs. (1) and
(2), complex representation of the transmission and reflection voltages are
needed. Therefore, a postprocessing procedure is implemented for these
voltages using the short time Fourier transform (STFT) [53]. In the STFT, the
voltages are divided into small segments of time before applying the Fourier
transform, which can be formulated as

${V}_{T,R,I}(\tau_{w},f)=\int_{-\infty}^{\infty}{v}_{T,R,I}(t)w(t-\tau_{w})e^{-j2\pi
ft}\mathrm{d}t.$ (3)

Here, $V_{T}$, $V_{R}$, and $V_{I}$ are Fourier transform outputs from
windowed voltages (transmission, reflection, and input voltages,
respectively), and $w(t)$ is a window function of length $M$ (4 ns in this
study). The window function is centered about time $\tau_{w}=nP$ where $P$ is
the interval time between successive Fourier transforms (set to 2 ns) and $n$
is an integer. To obtain consistent phase differences, the transmission phase
($\phi_{T}=\angle{T}$) and reflection phases ($\phi_{R}=\angle{R}$) are
calculated by subtracting the phase of a reference input voltage $V_{I}$ from
the phases of transmission and reflection voltages, e.g.,
$\phi_{T}(\tau_{w},f)=\angle{{V}_{T}-\angle{V}_{I}}.$ Similarly, the
amplitudes of transmission ($A_{T}=|T|$) and reflection ($A_{R}=|R|$) are
obtained by dividing the amplitudes of the transmission and reflection
voltages by that of the reference input voltage $V_{I}$, e.g.,
$A_{T}(\tau_{w},f)=|V_{T}|/|V_{I}|$.

Having both time- and frequency-dependent amplitude and phase, we can plot a
spectrogram of the metasurface transmission, as seen in Fig. 3(a) and (b).
Here, we consider a metasurface with an $RL$ series transient circuit (Fig.
2(c)), where the transmission is decreasing with time. The inductor here is
$L=0.1\,$mH, and the resistor is $R_{L}=10\,\Omega$. We can see in the
transmission amplitude spectrogram of Fig. 3(a) that the metasurface is
transparent around the resonant frequency of 4.2 GHz at the initial time,
while beyond 1000 ns, the transmission amplitude is drastically reduced. While
this variation in transmission is rapid, the phase variation over time is
insignificant, as depicted in Fig. 3(b). Using this transmission amplitude and
this transmission phase, the complex scattering parameters are then calculated
as

$\displaystyle T(\tau_{w},f)=A_{T}(\cos{\phi_{T}}+j\sin{\phi_{T}}),$ (4)
$\displaystyle R(\tau_{w},f)=A_{R}(\cos{\phi_{R}}+j\sin{\phi_{R}}).$

Based on the complex $T$ and $R$, we can calculate the electric sheet
admittance and magnetic sheet impedance of the metasurface as in Eqs. (1) and
(2), and the result is plotted in Fig. 3(c)-(f). Fig. 3(c) and (e) shows the
electric admittance ($Y_{se}$) and magnetic impedance ($Z_{sm}$) varying with
frequency at 200 ns, as indicated by the horizontal dashed line in Fig. 3(a).
Fig. 3(d) and (f) shows the electric admittance ($Y_{se}$) and magnetic
impedance ($Z_{sm}$) varying with time at 4.2 GHz, as indicated by the
vertical dashed line in Fig. 3(a).

We see in Fig. 3(e) and (f) a modest variation in the magnetic impedance
$Z_{sm}$ with time and frequency. This is in contrast to the electric
admittance $Y_{se}$ in Fig. 3(c) and (d), where the variation with time and
frequency is significant. Therefore, the magnetic contribution to the overall
scattering properties is expected to be minimal. This corresponds to the fact
that the metasurface here consists of only a single metallic structure
supported by a thin dielectric. Although the entire physical representation of
such a transmissive metasurface contains three shunt electric admittances
separated by dielectric layers, as seen in Fig. 2(g), here, we simplify the
physical model into a single shunt electric admittance, as seen in Fig.
2(h)-(i). In this simplified model, we only consider the electric sheet
admittance of the meta-atom $Y_{se}$ as the main contributor to the scattering
properties. In the two-port network representation, based on the $ABCD$
matrix, we can analytically calculate the scattering profile of the
metasurface (see Appendix B). The result is depicted by the dashed lines in
Fig. 3(g)-(h), where we see that the calculated scattering parameters agree
well with the numerical simulation, especially near the resonant frequency of
4.2 GHz. At lower and higher frequency regions, relatively large discrepancies
appear as magnetic impedance sheets ($Z_{sm}$) are omitted for the sake of
simplicity. However, this result ensures that the single equivalent admittance
is adequate for modeling the temporal response of the metasurface at the
resonant frequency (Fig. 3(h)).

###  II.3 Equivalent Circuit Model

To calculate the temporal response of the metasurface and assist in the design
methodology, we use an equivalent circuit model for the shunt admittance
$Y_{se}$. This equivalent circuit model is important to enable the design of
spatiotemporal metasurfaces as it provides a direct relation between a meta-
atom temporal response and its embedded circuit parameters.

As the metasurface here exhibits slow time variations relative to the time
period of the waveform, the time-frequency resolution limit is relaxed (as
discussed after Eqs. (1)- (2)). Therefore, two independent physical
contributions can be used to describe the scattering properties: first, a
shunt admittance that varies with frequency $Y_{freq}(f)$, which originates
from the dispersion characteristics of the metallic slit, and second, a shunt
admittance that varies with time $Y_{time}(t)$, which originates from the
embedded circuit transient responses. Note that $Y_{freq}(f)$ may contain
extra parameters that relate to the frequency dependence, for instance, the
parasitic capacitance of diodes. Also, with the relaxed time-frequency
resolution limit, both $Y_{freq}(f)$ and $Y_{time}(t)$ appear independent from
each other. Therefore, we can take their summation as the total shunt
admittance, as follows:

$Y_{se}(t,f)=Y_{freq}(f)+Y_{time}(t).$ (5)

For a normally incident plane wave, the metallic slit structure resembles a
resonant element that has dispersion characteristics similar to a parallel
$LC$ circuit [54]. Hence, the frequency-dependent admittance $Y_{freq}(f)$
from Eq. (5) can be attributed to a simple $LC$ admittance as follows:

$Y_{freq}(f)=j\bigg{(}\omega C_{m}-\frac{1}{\omega L_{m}}\bigg{)}+\dot{a},$
(6) $\dot{a}=a^{\prime}+ja^{\prime\prime},$ (7)

where $L_{m}$ and $C_{m}$ are the circuit parameters for the metallic slit,
which can be extracted from a full-wave simulation [55, 54], and $\omega$ is
the angular frequency. In the above equation, a complex adjustment parameter
$\dot{a}$ is added to account for a resonance shift that may be observed in a
numerical simulation or experiment.

Figure 4: Analytically calculated complex sheet admittances and corresponding
transmission intensity (dashed lines) compared to the numerical simulation
(solid lines) (a) for an $RL$ series metasurface with varied $L$ and fixed
$R_{L}$=10 $\Omega$, (b) for an $RC$ parallel metasurface with varied $C$ and
fixed $R_{C}$=10 k$\Omega$, (c) for an $RLC$ series with varied $C$ and fixed
$L$=0.1 mH, $R_{L}$=10 $\Omega$, and $R_{C}$=10 k$\Omega$, and (d) for an RLC
parallel with varied $L$ and fixed $C$=0.1 nF, $R_{L}$=10 $\Omega$, and
$R_{C}$=10 k$\Omega$. In these analytical calculations, complex adjustment
parameters $\dot{a}$ and $\dot{b}$ presented in Table 3 are used.

The second part of the admittance in Eq. (5), which is a time-dependent
admittance $Y_{time}(t)$, emerges mainly due to the transient response of the
embedded circuit. Hence, it can be quantified based on the time evolution of
currents or voltages within the embedded circuit. In this equivalent circuit
modeling, we assume that the incident wave is fully rectified and that the
circulating current is DC. In such a DC-driven circuit, we have a transient
current $i(t)$ and an initial applied potential $E_{0}$; therefore, we can
have $i(t)/E_{0}$ as a DC admittance. Since this DC admittance is directly
related to the current induced by the incident wave, it should also depend on
the coupling between the incident wave and the metasurface or the embedded
circuit system. Therefore, to adjust the significance of the coupling, we
assign another complex parameter $\dot{b}$, yielding

$Y_{time}(t)=\dot{b}\frac{i(t)}{E_{0}},$ (8)
$\dot{b}=b^{\prime}+jb^{\prime\prime}.$ (9)

As the DC admittance $i(t)/E_{0}$ in Eq. (8) is dependent on the $RLC$ circuit
configuration, it is predictable by an equivalent circuit model. In Appendix
C, we present a derivation of the DC admittance for the four circuit types
seen in Fig. 2(c)-(f), as has also been partially reported in Ref. [44].
Particularly, analytical expressions of the DC admittances for the four
circuit types are formulated in Eqs. (19), (22), (26), and (30), and have been
validated by a circuit simulation as detailed in Appendix D. Throughout this
equivalent circuit modeling, the diode bridge is modeled as resistor $R_{d}$
connected in series with the transient circuit (see Fig. 2(c)). Note that
since in the rectification process electric charges pass two diodes in series
connection, therefore $R_{d}$ is double the effective resistance of a single
diode $R_{0}$, i.e., $R_{d}=2R_{0}(=680\,\Omega$ in this study) [44]. The same
series $R_{d}$ connection should also be present for the other transient
circuits in Fig. 2(d)-(f), although we only draw it in Fig. 2(c). In
calculating the time-varying admittance, both $\dot{a}$ and $\dot{b}$ in Eqs.
(6) and (8) are chosen by fitting to a numerical or experimental result (see
the last paragraph of Appendix C for details). Analytically calculated time
varying admittances from this circuit modeling are shown by dashed lines in
Fig. 4 for both the real part (right column) and imaginary part (middle
column). The corresponding calculated transmission is obtained by the
transmission line model (Appendix B). Four different transient circuits, (a)
$RL$ series, (b) $RC$ parallel, (c) $RLC$ series, and (d) $RLC$ parallel are
considered. In these four cases, the inductance and capacitance are varied to
observe the practicality of the circuit modeling. For a comparison, we
numerically simulate the four metasurface types using the co-simulation method
and extract the complex admittances using Eq. (1). As depicted in Fig. 4, the
numerical simulations (solid lines) and analytical calculations (dashed lines)
agree with each other. This result indicates that the circuit modeling
efficiently represents the meta-atom responses and is useful in designing a
metasurface with variation in temporal scattering properties.

###  II.4 Full-wave Simulation of the Time-Varying Admittance

As the temporal admittance of the metasurface can be extracted and
analytically predicted, a full-wave simulation based on the admittance sheet
model may be conducted to obtain the scattered field distribution from the
metasurface at a particular time. In this admittance model, we use a
dielectric slab and one impedance boundary condition. The simulation setup
using ANSYS Electronics Desktop 2020 R2 is shown in Fig. 5(a) and (b),
comparing the physically realistic model based on the co-simulation method (a)
with the simplified admittance sheet model (b). In the admittance sheet model,
a separate simulation is performed for different time samples. For example,
here, we simulate the admittance profile in six samples at 10 ns, 34 ns, 70
ns, 100 ns, 200 ns, and 1000 ns. In each simulation, the scattering parameters
and the scattered fields can be directly obtained without a post-processing
procedure.

Figure 5: Comparison between the metasurface simulation using the realistic
structure and the admittance sheet model at 4.2 GHz over time. (a) Simulation
setup as shown in ANSYS Electronics Desktop 2020 R2 using the realistic
structure (co-simulation) and (b) using admittance sheet model. (c)
Transmission intensity for the metasurface with the $RL$ series transient
circuit having $L$=0.1 mH and $R_{L}$=10 $\Omega$. (d) Transmission intensity
for the metasurface with the $RC$ parallel transient circuit having $C$=0.1 nF
and $R_{C}$=10 k$\Omega$.

Simulation results for the metasurfaces with $RL$ series and $RC$ parallel
transient circuits are shown in Fig. 5(c) and (d), respectively. For the $RL$
series case, the inductor is $L$=0.1 nH, and the resistor is $R_{L}$=10
$\Omega$, while for the $RC$ parallel case, the capacitor is $C$=0.1 nF, and
the resistor is $R_{C}$=10 k$\Omega$. The admittance sheet model (circles) and
the physically realistic simulation model (blue line) results agree with each
other. A slight discrepancy can be observed due to the admittance sheet model
not taking into account the magnetic response $Z_{sm}$, which, as discussed in
Sec. II.1, has very little influence on the overall scattering parameters.

Figure 6: (a)-(b) Design of a binary amplitude self-tuned FZP with the
distribution of the meta-atom transmission profile varying in space and time.
The meta-atom size is 18 mm, and the designed focal distance is $F$=500, 300,
and 200 mm at three different times. The diameter of the FZP is $d$=756 mm.
(c) Simulated temporal transmission intensity profile of five meta-atoms used
for the time-varying FZP.

##  III Application to Self-Tuned Spatiotemporal Wave Manipulation

###  III.1 Design of a Fresnel Zone Plate with Time-Varying Focal Lengths

In the previous subsection, we have modeled the metasurface as a time-varying
admittance and confirmed that a full-wave simulation of the admittance sheet
model agrees with the physically realistic metasurface simulation. Now, we use
the method to synthesize a metasurface with a more complex spatiotemporal
configuration. Since the metasurface discussed here has temporal variation of
the transmission amplitude while exhibiting a relatively constant phase, we
can utilize it to realize prescribed time-varying diffraction characteristics.
As an example, here, we synthesize a Fresnel Zone Plate (FZP) with a variable
focal point depending on the incoming pulse width, as illustrated in Fig.
1(c)-(d).

In FZPs, opaque and transparent regions within concentric rings alternate over
a two-dimensional surface to obtain constructive interference at a focal
distance $F$. Only the transmission amplitude varies in these regions while
the transmission phase should be the same. As seen in Fig. 6(a), the ring
radii have a binary transmission amplitude (dark blue for opaque shields and
light blue for transparent windows), and the radius $r_{n}$ changes by zone
number $n$. The key to realizing a time-varying FZP is to engineer the
distribution of the binary transmission amplitude over time such that the
focal point is prescribed at different distances for different times, or
equivalently pulse widths. Using Fresnel zone theory [46], the radii for the
time-varying FZP can be formulated as

$r_{n}(t)=\sqrt{n\lambda F(t)+\frac{1}{4}n^{2}\lambda^{2}}.$ (10)

Here, $\lambda$ is the operating wavelength. Note that the focal distance $F$
has a prescribed variation with time $t$.

To realize this time-varying focal point FZP, we consider the meta-atom in
Sec. II.1 with four different transient circuits. Using these meta-atoms as
building block of the FZP, we may obtain a time-varying distribution of opaque
and transparent regions, while maintaining almost the same phase. As the FZP
requires a binary amplitude profile, using the proposed metasurfaces, in
accordance with Eq. (10), three different focal points can be designed for
three time segments. Fig. 6(b) depicts the distribution of opaque and
transparent meta-atoms over space and time in half of the FZP diameter
($d/2$). Note that to simplify the design process, here we consider only one-
dimensional (1D) profile of the FZP along the horizontal axis. Similar to the
metasurface described in Sec. II.2, the size of each meta-atom is 18 mm, and
there are 21 cells arranged in half of the diameter ($d=756$ mm) of the FZP,
as indicated by cell numbers 1-21. The operating frequency is 4.2 GHz
($\lambda$=71 mm). The dashed square in Fig. 6(a) indicates the particular
zoomed-in space in Fig. 6(b). The FZP is designed to have a decreasing focal
point with time, where at an initial time range $t<t_{1}=$100 ns, the focal
distance is $F=500\,$mm ($F/d=0.66$), at an intermediate time range
$t_{1}<t<t_{2}=$2000 ns, the focal distance is $F=300$ mm ($F/d=0.4$), and in
the steady-state condition at $t>t_{2}$, the focal distance is $F=200\,$mm
($F/d=0.27$). As shown in Fig. 6(b), if we consider the time domain and that
each cell represents a meta-atom with a certain time-varying transmission,
then we obtain five different meta-atom types (MA1-MA5). From cells 1 to 6,
the transmission is always the maximum; hence, they can be left blank, i.e.,
only a dielectric without metal patterns or circuits. For cells 7-21, the
transmission profile changes over time and can be represented by different
meta-atom designs. For example, in cells 7 and 8, the transmission is always
the maximum before $t_{2}$. This time-varying transmission profile is
equivalent to a meta-atom with the $RL$ series transient circuit. Similarly,
for cells 14, 15 and 16, the transmission is the maximum only in the
intermediate time of $t_{1}<t<t_{2}$, which is equivalent to the transmission
of a meta-atom with the $RLC$ parallel transient circuit.

With the required spatiotemporal distribution of opaque and transparent
regions for the FZP, we design meta-atoms suitable for the realization.
Accordingly, in each meta-atom, we choose circuit parameters to satisfy the
binary amplitude profile in Fig. 6(b). As the transmission of the metasurface
is directly related to the exponential terms of the corresponding DC
admittance, we can reduce the problem of finding suitable circuit parameters
by only considering the exponential decay constants (see exponential terms in
Eqs. (19), (22), (26), and (30) in Appendix C). First, we assume that the
boundary between transparent and opaque regions is 50% transmission (e.g.,
transparent if $|T|>$0.5). Therefore, we can use the exponential decay terms
to find the suitable $L$ or $C$ to realize the transmission profile at a
particular designed time. For the metasurface with an $RL$ series circuit, if
we take the exponential term in Eq. (19) and set the exponential decay to 50%,
i.e., $e^{\left(-\frac{R_{L}+R_{d}}{L}\right)t}=0.5$, then we obtain

$L=-t_{b}\frac{R_{L}R_{d}}{\mathrm{ln}0.5}.$ (11)

Similarly, for the metasurface with an $RC$ parallel circuit, using the
exponential term in Eq. (22), by setting
$e^{-\frac{RC+R_{d}}{CR_{C}R_{d}}t}=0.5$, we obtain

$C=-t_{b}\frac{R_{C}+R_{d}}{(\mathrm{ln}0.5)R_{C}R_{d}}.$ (12)

Here, $t_{b}$ is the time in which the exponential decay reaches 50% of its
initial value. Using these equations, we calculate the required $L$ and $C$
for MA1 to MA5 by setting $t_{1}$=100 ns and $t_{2}$=2000 ns. For example, in
MA1, as seen in Fig. 6(b), the boundary between opaque and transparent regions
occurs at $t_{2}$. Thus, by substituting $t_{b}=2000$ ns and $R_{L}$=10
$\Omega$ into Eq. (11), we obtain $L$=0.1 mH. For MA2, using the same method,
we obtain $L$=2 mH. For the $RLC$ series and parallel cases, the corresponding
$L$ and $C$ within the circuit can be calculated using the same equations, as
these cases are a combination of the $RL$ series and the $RC$ parallel cases.
In the $RLC$-series case, Eq. (11) is used to obtain $L$ when the transmission
decreases from the maximum to 50% at $t_{1}$, while Eq. (12) is used to obtain
$C$ when the transmission increases from the minimum to 50% at $t_{2}$. A
similar calculation is performed for the $RLC$ parallel case, but with an
inverse temporal transmission profile. In Fig. 6(c), the transmissions of
designed meta-atoms MA1 to MA5 are plotted. Details of the meta-atom types
used and the circuit parameters are presented in Table 1.

Figure 7: Binary amplitude FZP with a pulse-width-dependent focal point. Simulated electric field intensity (normalized ${|E_{y}|}^{2}$) from (a) ideal static binary FZPs based on PEC layers with different focal point designs and (b) FZPs based on equivalent admittance sheets that represent three fixed time points. (c) The electric field intensity at $x$=0 mm (normalized ${|E_{y}|}^{2}$) for the FZPs based on the PEC layers and the equivalent admittance sheets (red and blue curves, left $y$-axes) compared to the probe voltage (normalized $[v_{pr}^{2}]_{avg}$) for full FZP based on the rigorous waveform-selective metasurface (diamonds, right $y$-axes). Table 1: Meta-atom circuit parameters for the binary amplitude self-tuned FZP  | Circuit type | $R_{d}$ | $R_{L}$ | $R_{C}$ | L | C
---|---|---|---|---|---|---
MA1 |  RL Series | 680 $\Omega$ | 10 $\Omega$ | - | 2 mH | -
MA2 |  RL Series | 680 $\Omega$ | 10 $\Omega$ | - | 0.1 mH | -
MA3 |  RLC Series | 680 $\Omega$ | 10 $\Omega$ | 10 k$\Omega$ | 0.1 mH | 1 nF
MA4 |  RC Parallel | 680 $\Omega$ | - | 10 k$\Omega$ | - | 0.1 nF
MA5 |  RLC Parallel | 680 $\Omega$ | 10 $\Omega$ | 10 k$\Omega$ | 2 mH | 0.1 nF

###  III.2 Numerical Validation of the Spatiotemporal Response

To confirm the designed FZP with a time-varying focal point, we first simulate
an ideal static FZP case. In this simulation, the same transparent and opaque
region as illustrated in Fig. 6(b) is used where only perfect electric
conductor (PEC) layers on top of the dielectric substrate are simulated (see
Appendix E for details). Three simulation instances are conducted, and the
resulting $y$-polarized electric field intensity are shown in Fig. 7(a), where
three different focal points can be seen located at approximately 500, 300,
and 200 mm. Note that since the FZPs are discretized with a cell size of 18
mm, the ideal Fresnel zone radius $r_{n}$ may not be accurately realized for
all the instances. As a result, a slight shift in the focal distance $F$ from
the designed location may occur as seen from Fig. 7(a). After simulating this
ideal static FZP case, we simulate the time-varying vocal point of the FZP;
first using the admittance sheet model and then using a rigorous waveform-
selective metasurface based on the co-simulation method. In the admittance
sheet model, the method explained in Subsec. II.4 is used, in which the meta-
atoms are simulated at three specific times, i.e., $t$=10 ns, 340 ns, and 5000
ns. At each time, the admittance distribution corresponding to MA1 to MA5 is
utilized to obtain the $y$-polarized electric fields scattered in front of the
FZP. The result is presented in Fig. 7(b), where we can see that focal point
changes at the three different time instances, corresponding to the self-tuned
admittance of the designed metasurface.

To further validate the time-varying focal points, we simulate the full FZP
including the rigorous waveform-selective metasurface by using the co-
simulation method inside ANSYS Electronics Desktop 2020 R2. Since it is not
possible to obtain the transient scattered field in the co-simulation method,
a monopole antenna is used as a probe to capture the time-varying voltages
within the scattering area (see Appendix E for details). In contrast to the
previous simulation (the FZP based on admittance sheets) which gives scattered
field intensity profiles for specific discretized instants, this simulation
enables us to produce voltages varying in the time domain at the location of
the probe. Fig. 7(c) depicts one-dimensional field profiles of the FZP based
on the rigorous waveform-selective metasurface (diamonds) as well as those
based on the ideal PEC layers (blue curve) and the equivalent admittance
sheets (red curve). We see that despite a slight shift in the maximum level of
voltages in the steady state condition ($t$=5000 ns in Fig. 7(c)), the time-
varying focal point is also confirmed by the full FZP using the rigorous
waveform-selective metasurface. This result supports that the metasurface-
based FZP exhibits the desired spatiotemporal response, i.e., the time-varying
focal point, and that the design methodology proposed here is valid in
realizing such a spatiotemporal metasurface. The focusing efficiencies
calculated as a ratio between the field intensities in the focal plane with
and without the presence of the metasurface are 43.59 %, 41.74 %, and 39.58 %,
for t = 10 ns, 340 ns and 5000 ns, respectively. These efficiencies are lower
than other impedance-matched metasurfaces [48] due in part to the losses
considered in the admittance model as well as reflected waves in the opaque
zones of the FZP.

##  IV Discussion

It should be noted that the circuit modeling presented here accurately
predicts the temporal response of the metasurface only near the resonant
frequency. In a lower/higher frequency range, the meta-atom model should be
modified to include not only the electric responses $Y_{se}$ but also the
magnetic responses $Z_{sm}$. In addition, the presented equivalent circuit
model requires prior information of a numerical or experimental result for an
accurate prediction of the time-varying admittance. Therefore, future studies
may be aimed in obtaining analytical formulations for both the coupling of the
metasurface to the incident wave and frequency shifts due to parasitic
elements. While the extraction of time-varying admittance is applied to only a
transmissive meta-atom structure here, the method can also be implemented to a
reflective meta-atom structure [39]. The time-varying admittance concept
discussed in Subsec. II.2 is applicable to various frequency bands, although
the corresponding analytical model in Subsec. II.4 may require adjustments in
the optical bands due to higher ohmic losses involved in most metallic
nanostructures.

Despite the fact that our study only shows an FZP with a predesigned time-
varying focal point, various other spatiotemporal metasurfaces could be
envisaged based on the proposed design and simulation method. For example, the
binary FZP has been widely used as a beam-steering device when a spherical
source is placed at its focal point with variable lateral shift [47]. Similar
time-space variation for such an FZP with a spherical source may also be
conducted in which the time-varying admittance mimics the lateral shift of the
spherical source. Such a metasurface can perform beam-steering based on pulse
width of the incoming wave. Moreover, in the present work, we only consider a
metallic slit structure embedded with nonlinear circuits, which gives weak
magnetic responses. Therefore, almost no time-varying phases was observed from
the scattering properties. Further improvements can be achieved by designing a
metasurface with both time-varying electric and magnetic sheet impedances,
which could lead to time-varying phase responses.

With such innovative engineering and tunability, the proposed metasurface may
find a variety of applications in communications and signal processing. For
example, recently, metasurface-based intelligent reflecting surfaces (IRSs)
have been extensively investigated for the next-generation wireless standard
[56, 57, 58]. Most existing studies assume that each unit cell of IRSs is
connected with a control line, and the input voltage is optimized to form a
reflected beam toward a targeted direction [58, 59, 60]. However, such a
configuration with multiple control lines increases the hardware cost,
especially when the number of IRS elements is huge. Furthermore, it is
typically assumed that the IRSs are precisely synchronized with a base station
in a symbol or frame level, which also escalates the complexity of the system.
By contrast, the IRS based on waveform-selective metasurfaces is capable of
combating the above-mentioned limitations as the explicit benefits of our
pulse width-dependent metasurface tuning. More specifically, the beam of the
proposed metasurface is controlled by optimizing the pulse width, which does
not need any control lines connected to the unit cells or precise
synchronization between the base station and the IRSs.

##  V Conclusion

Analytical and numerical studies of a self-tuned time-varying metasurface have
been conducted, considering a metasurface with a metallic slit structure
embedded with nonlinear and transient electronic circuits. With the
metasurface having a low profile and deeply subwavelength dimensions, it can
be modeled as a single electric shunt admittance connected to a transmission
line. Since the time constant of the transient variation is much larger than
the time period of the waveform, we decompose the admittance into two
independent parameters to account for both the metallic slit frequency
dispersion and the embedded circuit time dependence. We show that such time-
and frequency-dependent sheet admittances can reconstruct the scattering
profile of the metasurface, where the analytical calculation results agree
well with the numerical simulation results. Four different transient circuits
are used in this study, each corresponding to a distinct temporal scattering
profile.

We then confirm that such a time-varying admittance can be simulated using an
impedance boundary condition layer within a commercially available full-wave
simulator (ANSYS Electronics Desktop HFSS). The scattering parameters of the
metasurface at a particular time simulated by the admittance sheet model are
consistent with the results obtained using the physically realistic structure.
Therefore, the full-wave simulation method based on the admittance sheet can
be used to predict the temporal response from a more complicated structure,
such as from an inhomogeneous metasurface. This can complement conventional
co-simulation methods, which do not provide time-dependent scattered fields
for such time-varying surfaces. As an exemplary model, here, we use the
proposed concept to design an FZP where the focal point changes depending on
time or the pulse width of the incoming wave. The presented metasurface holds
great potential to achieve other self-tunable time-varying wave control, and
the admittance sheet modeling offers a versatile method to design such
spatiotemporal metasurfaces.

###### Acknowledgements.

This work was supported in part by the Japanese Ministry of Internal Affairs
and Communications (MIC) under the Strategic Information and Communications
R&D Promotion Program (SCOPE) No. 192106007, the Japan Science and Technology
Agency (JST) under the Precursory Research for Embryonic Science and
Technology (PRESTO) No. JPMJPR1933 and JPMJPR193A, and the Japan Society for
the Promotion of Science (JSPS) KAKENHI No. 17KK0114 and 21H01324.

##  Appendix A Metasurface Co-simulation Method

ANSYS Electronics Desktop 2020 R2 is used to simulate the metasurface in Fig.
2(a), based on a co-simulation method that integrates a full-wave numerical
simulation with a circuit simulation. In such a simulation method, the
electromagnetic field distribution cannot be obtained since the final results
are produced by circuit simulations. A 3D full-wave simulation of a single
meta-atom is first conducted using periodic boundaries and Floquet port
excitation, as depicted in Fig. 8(a). The distances between the metasurface
and the Floquet ports are $z_{w}=100$ mm. Within the metallic slit, a 5 mm
$\times$ 5 mm lumped port is connected to represent a connection of the
embedded circuit. Next, we import a model based on the full-wave simulation
result of the meta-atom into a circuit simulation, as depicted in Fig. 8(b).
In the circuit simulation, the embedded circuit consisting of a diode bridge
rectifier and $RLC$ components (see Fig. 2 (a)-(f)) are connected. Other
necessary components are also placed within the circuit simulation to obtain
time-varying currents and voltages at the corresponding transmission and
reflection ports. For the diodes, we use the SPICE model based on Avago’s HSMS
286x Schottky diodes series as detailed in Table 2. The input power in the
circuit simulation is set to 0 dBm and a transient solver is used within a
time span of 10 $\mu$s and over a frequency variation range from 2 to 6 GHz
with a 0.1 GHz step. Using this simulation setup, we can monitor the voltages
in the Floquet ports (both at the transmission and reflection sides), which
has both time and frequency dependency.

Figure 8: Co-simulation method integrating electromagnetic simulator with circuit simulator. (a) Schematic of the 3D full-wave simulation of the metasurface using periodic boundaries and Floquet ports. (b) Schematic of the circuit simulation. The metasurface full-wave simulation result is imported into the circuit simulator as an embedded circuit component, along with other necessary components connected to it. Table 2: SPICE parameters for the diodes  SPICE parameter | Value | SPICE parameter | Value
---|---|---|---
$B_{V}$ | 0.7 V | N | 1.08
$C_{J0}$ | 0.18 pF | $R_{S}$ | 6.0 $\Omega$
EG | 0.69 eV | $P_{B}(VJ)$ | 0.65 V
$I_{BV}$ | 1$\times$10-5 A | $P_{T}(XTI)$ | 2
$I_{S}$ | 5$\times$10-8 A | M | 0.5

##  Appendix B Two-Port Network Representation

To analytically obtain the scattering parameters of the metasurface, we use a
two-port network representation. In this method, $ABCD$ transfer matrices of
the sheet admittance Mse and the transmission line Mtl are utilized to obtain
an $ABCD$ transfer matrix of the metasurface Mms, formulated as

$\bm{M_{ms}}=\bm{M_{se}}\cdot\bm{M_{tl}},$ (13)

with

$\displaystyle\bm{M_{se}}=\begin{bmatrix}1&0\\\ Y_{se}&1\end{bmatrix},$ (14)

$\displaystyle\bm{M_{tl}}=\begin{bmatrix}\cos(\beta_{s}h)&jZ_{s}\sin(\beta_{s}h)\\\
jY_{s}\sin(\beta_{s}h)&\cos(\beta_{s}h)\end{bmatrix},$ (15)

where $Z_{s}=1/Y_{s}=\sqrt{\mu_{s}/\epsilon_{s}}$ is the substrate impedance,
and $\beta_{s}=\omega\sqrt{\mu_{s}\epsilon_{s}}$, where $\epsilon_{s}$ is the
dielectric permittivity and $\mu_{s}$ is the permeability. Transformation of
the $ABCD$ transfer matrix Mms into other network representations such as the
scattering matrix Sms can be performed, through which we obtain $T=S_{21}$ as
the metasurface transmission and $R=S_{11}$ as the metasurface reflection,
written as [61]

$\displaystyle T$ $\displaystyle=\frac{2}{(A_{ms}+(B_{ms}/\eta)+(\eta
C_{ms})+D_{ms})}$ (16) $\displaystyle R$
$\displaystyle=\frac{(A_{ms}+(B_{ms}/\eta)-(\eta
C_{ms})-D_{ms})}{A_{ms}+(B_{ms}/\eta)+(\eta C_{ms})+D_{ms}},$ (17)

where $A_{ms},B_{ms},C_{ms}$, and $D_{ms}$ are elements of matrix Mms in Eq.
(13).

##  Appendix C Calculation of the DC Admittance

To derive the DC admittance as presented in Eq. (8), we use transient analysis
for the corresponding DC circuits. First, let us consider an $RL$ series
metasurface, in which the embedded circuit is equivalent to a series
configuration of an $RL$ circuit and $R_{d}$ that represents the resistive
components of two diodes at their turn-on voltage [44] (see Fig. 2(c)). In
this circuit, using the Kirchhoff voltage law (KVL), we find that initial
potential $E_{0}$ is equivalent to the sum of all transient voltages within
the resistors and inductor, i.e.,

$E_{0}=(R_{L}+R_{d})i(t)+L\frac{di(t)}{dt}.$ (18)

Hence, solving for circuit current $i(t)$ using the first-order differential
equation and rearranging it, we obtain the DC admittance as [44]

$\frac{i(t)}{E_{0}}=\frac{1}{R_{L}+R_{d}}\left(1-e^{\left(-\frac{R_{L}+R_{d}}{L}\right)t}\right).$
(19)

Second, we consider an $RC$ parallel circuit, as seen in Fig. 2(d). In this
circuit, we can use the Kirchhoff current law (KCL) to obtain the transient
current as follows:

$i(t)=i_{RC}(t)+i_{C}(t),$ (20)

Where $i_{RC}$ and $i_{C}$ represent the currents of $R$ and $C$,
respectively. Using the first-order differential equation, we can derive
$i_{C}$ and $i_{RC}$ as follows [44]:

$\displaystyle
i_{C}(t)=\frac{E_{0}}{R_{d}}e^{-\frac{R_{C}+R_{d}}{CR_{C}R_{d}}t},$ (21)
$\displaystyle
i_{RC}(t)=\frac{E_{0}\left(1-e^{-\frac{R_{C}+R_{d}}{CR_{C}R_{d}}t}\right)}{R_{d}+R_{C}}.$

Hence, the DC admittance is

$\frac{i(t)}{E_{0}}=\frac{i_{RC}(t)+i_{C}(t)}{E_{0}}.$ (22)

Next, we consider the series and parallel $RLC$ circuits seen in Fig. 2(e) and
(f). In contrast to our previous study [44], in these circuits, the transient
currents can be readily calculated by removing the $R_{C}$ from Fig. 2(d) and
$R_{L}$ from Fig. 2(e), assuming that both $R_{C}$ and $R_{L}$ have small
contributions to the overall transient response (see Appendix D). In the
series $RLC$, we use the KVL to obtain the following relationship:

$E_{0}=(R_{d}+R_{L})i(t)+L\frac{di(t)}{dt}+\frac{1}{C}\int i(t)dt.$ (23)

Note that here, we do not have $R_{C}$ connected in parallel to $C$. In
differential equation form, using $s$ to represent the roots, the above
formula can be written as

$\left(s^{2}+\frac{R_{d}+R_{L}}{L}s+\frac{1}{LC}\right)i=0.$ (24)

The solution of this second-order, linear, homogenous differential equation is
then

$i(t)=K_{1}e^{s_{1}t}+K_{2}e^{s_{2}t},$ (25)

where $s_{1}$ and $s_{2}$ are the roots of the differential equation in Eq.
(24), and $K_{1}$ and $K_{2}$ are two constants that are to be selected to
satisfy the initial conditions. First, at $t=0$, the initial condition is
$e^{s_{1}t}=e^{s_{2}t}=1$; hence, $0=K_{1}+K_{2}$. Second, as the current in
the inductor cannot change instantaneously, the initial condition of the
voltage at $t=0^{+}$ across the inductor is the same as $E_{0}$, i.e.,
$L\frac{di}{dt}=E_{0}$. Hence, we obtain $L(s_{1}K_{1}+s_{2}K_{2})=E_{0}$.
Substituting these two equations, we obtain
$K_{2}=\frac{E_{0}}{L(s_{2}-s_{1})}$ and
$K_{1}=-\frac{E_{0}}{L(s_{2}-s_{1})}$, and therefore, the DC admittance is
derived from

$\frac{i(t)}{E_{0}}=\frac{1}{L(s_{2}-s_{1})}\big{(}e^{s_{1}t}-e^{s_{2}t}\big{)}.$
(26)

For the parallel RLC circuit, a similar derivation as for the series RLC
circuit can be conducted. Here, application of the KCL to the parallel $RLC$
circuit yields

$\frac{E_{0}-v_{p}}{R_{d}}=\frac{v_{p}(t)}{R_{C}}+C\frac{dV_{p}(t)}{dt}+\frac{1}{L}\int
v_{p}(t)dt,$ (27)

where $v_{p}$ is the voltage applied to the entire circuit except for $R_{d}$
(see Fig. 2(f)). In differential equation form, using $p$ to represent the
roots, the above formula can be written as

$\left(p^{2}+\frac{R_{d}+R_{C}}{CR_{C}R_{d}}p+\frac{1}{LC}\right)v_{p}=0.$
(28)

The solution of this second-order, linear, homogenous differential equation is

$v_{p}(t)=\kappa_{1}e^{p_{1}t}+\kappa_{2}e^{p_{2}t},$ (29)

where $p_{1}$ and $p_{2}$ are the roots of the above differential equation,
and $\kappa_{1}$ and $\kappa_{2}$ are two constants that are to be selected to
satisfy the initial conditions. First, at $t=0$, the initial condition is
$e^{p_{1}t}=e^{p_{2}t}=1$; hence, $0=\kappa_{1}+\kappa_{2}$. Second, as the
voltage in the capacitor cannot change instantaneously, the initial condition
of the current at $t=0^{+}$ across the capacitor is same as initial current
$i(t)=\frac{E_{0}-v_{p}}{R_{d}}$, i.e.,
$C\frac{dv_{p}}{dt}=\frac{E_{0}}{R_{d}}$. Hence, we obtain
$R_{d}C(p_{1}\kappa_{1}+p_{2}\kappa_{2})=E_{0}$. Substituting these two
equations, we obtain $\kappa_{1}=\frac{E_{0}}{R_{d}C(p_{2}-p_{1})}$ and
$\kappa_{2}=-\frac{E_{0}}{R_{d}C(p_{2}-p_{1})}$, and hence, the DC admittance
is

$\displaystyle\frac{i(t)}{E_{0}}=\frac{1}{R_{d}}-\frac{1}{R_{d}^{2}C(p_{2}-p_{1})}\big{(}e^{p_{1}t}-e^{p_{2}t}\big{)}.$
(30)

To obtain the analytically calculated complex admittance in Eq. (5), we use
the following method. First, we derive the equivalent circuit of the metallic
slit based on the procedure described in [55, 54]. This extraction requires
full-wave simulation of the slit structure without the embedded circuit. From
the extraction, we obtain the equivalent $LC$ circuit for the metallic slit,
which, in this case, consists of $C_{m}$=0.12 pF and $L_{m}$=3.15 nH (see Eq.
(6)). Note that here, the metallic slit geometries are identical to those used
in Fig. 2(a). Through Eq. (6), we can obtain the frequency-dependent
admittance of the slit structure utilizing $C_{m}$ and $L_{m}$. We then take
the admittance only at the considered frequency (4.2 GHz), which is to be
added to the time-varying admittance in Eq. (8). Here, after obtaining the DC
admittance (either from Eqs. (19), (22), (26), or (30)), the adjustment
factors $\dot{a}$ and $\dot{b}$ are chosen to fit the corresponding numerical
simulation results. This fitting is conducted only _once_ for a particular
transient circuit type, and therefore is valid to be used in predicting the
admittance of other circuit components within the same circuit type. For
example, in the $RL$ series circuit, we adjust $\dot{a}$ and $\dot{b}$ to fit
the simulation result of the metasurface with inductor $L$=0.01 mH. Once
obtained, further alteration of the adjustment factors is not required, even
when the inductor changes to $L=$0.1 mH and $L=$1 mH. The elements of
$\dot{a}$ and $\dot{b}$ for the four different circuits considered in this
simulation are presented in Table 3.

Table 3: Adjustment factors for $\dot{a}$ and $\dot{b}$ of Eqs. (6) and (8) Circuit Type | $a^{\prime}$ | $a^{\prime\prime}$ | $b^{\prime}$ | $b^{\prime\prime}$
---|---|---|---|---
$RL$ Series and $RLC$ Series | 0.1 | 2 | 2.7 | 2.7
$RC$ Parallel and $RLC$ Parallel | 0.3 | 2.15 | 2.7 | 1.8

##  Appendix D DC Circuit Simulation Validating the Analytical Calculation

Figure 9: Comparison between the circuit simulation and analytically
calculated DC admittance as explained in Appendix C for (a) $RL$ series, (b)
$RC$ parallel, (c) $RLC$ series, and (d) $RLC$ parallel. Comparison with the
analytical expression of DC admittances from [44] for (e) $RLC$ series and (f)
$RLC$ parallel. In the above simulations and analytical calculations,
$R_{L}=10\,\Omega$ and $R_{C}=10\,$k$\Omega$.

After obtaining an analytical expression of the DC admittances for the four
different circuit types, i.e., Eqs. (19), (22), (26), and (30), we use the
circuit simulator inside ANSYS Electronics Desktop 2020 R2 to compare the
results. As can be seen from Fig. 9(a)-(d), both the analytically calculated
DC admittances and numerical results from the circuit simulator agree with
each other. In Fig. 9(c)-(d), a discrepancy can be found starting around 500
ns due to the simplification used in the $RLC$ series and parallel circuits.
As discussed in Appendix C, $R_{L}$ and $R_{C}$ are removed from the $RLC$
series case and the $RLC$ parallel case, respectively, to propose the
simplified yet realistic calculation method of the DC admittances, assuming
that both resistances only slightly affect the overall admittance. This
assumption remains valid for $R_{L}$ approaching zero, indicating a closed
connection, and $R_{C}$ approaching infinity, indicating an open connection.
With $R_{L}=10\,\Omega$ and $R_{C}=10\,$k$\Omega$ used in this study, both the
$RLC$ series and parallel admittances remain largely consistent. In [44], the
DC admittance was derived considering both $R_{L}$ and $R_{C}$ within the
circuit, which gives results more consistent with those of the circuit
simulator, as can be seen from Fig. 9(e) and (f). In this calculation, we
obtain $v_{p}$ as the voltage across the circuit of the $RLC$ parallel case,
except for the diodes, as follows [44]:

$v_{p}\ (t)=R_{p}E_{0}\bigg{[}R_{l}\bigg{(}{1+\frac{\beta e^{\alpha t}-\alpha
e^{\beta t}}{\alpha-\beta}}\bigg{)}+L\frac{\alpha\beta(e^{\alpha t}-e^{\beta
t})}{\alpha-\beta}\ \bigg{]},$ (31)

where $R_{p}=\frac{R_{c}}{R_{l}R_{d}+R_{l}R_{c}+R_{d}R_{c}}$, and $\alpha$ and
$\beta$ are roots of the differential equations derived in [44], which are
rewritten here as

$\alpha=-\frac{L(R_{d}+R_{c})+CR_{l}R_{d}R_{c}}{2LCR_{d}R_{c}}\\\
+\sqrt{\bigg{(}{\frac{L(R_{d}+R_{c})+CR_{l}R_{d}R_{c}}{2LCR_{d}R_{c}}\bigg{)}}^{2}-\frac{1}{LCR_{d}R_{p}}},$
(32) $\beta=-\frac{L(R_{d}+R_{c})+CR_{l}R_{d}R_{c}}{2LCR_{d}R_{c}}\\\
-\sqrt{\bigg{(}{\frac{L(R_{d}+R_{c})+CR_{l}R_{d}R_{c}}{2LCR_{d}R_{c}}\bigg{)}}^{2}-\frac{1}{LCR_{d}R_{p}}}.$
(33)

Figure 10: Simulation method for the FZPs based on (a) ideal PEC layers or
equivalent admittance sheets and (b) rigorous waveform-selective metasurface.
In these simulations, only 1D horizontal profiles of the FZPs are considered
by applying PEC boundaries to both positive and negative $y$-axis directions
(red arrows in (a) and (b)). For the FZP based on the admittance sheets, the
metasurface is approximated by impedance boundaries on top of a dielectric
slab. For the full FZP based on the rigorous waveform-selective metasurface, a
monopole probe is used to capture the time-varying voltage. (c) The monopole
placement within the metasurface scattering area. (d) Schematic of the circuit
simulation of the full FZP. (e) Normalized squared and averaged time-varying
voltages ($[v_{pr}^{2}]_{avg}$) for different probe locations ($z_{pr}$).

In the $RLC$ series case, considering that $R_{C}$ is connected in parallel to
the capacitor $C$ (see Fig. 2(e)), we obtain $v_{s}$ as the voltage within the
circuit, except for the diodes, as follows [44]:

$v_{s}(t)=\frac{E_{0}}{R_{d}+R_{l}+R_{c}}\bigg{[}(R_{c}+R_{l}){1-e^{ht}(\cos
kt-\frac{h}{k}\sin kt)}\\\
+\frac{h^{2}+k^{2}}{k}e^{ht}{(CR_{c}R_{l}+L(1+CR_{c}h))\sin kt+LCR_{c}k\cos
kt}\bigg{]},$ (34)

in which the real and imaginary roots $h$ and $k$ are

$h=-\frac{L+CR_{c}(R_{d}+R_{l})}{2LCR_{c}},$ (35)
$k=\sqrt{\frac{R_{d}+R_{l}+R_{c}}{LCR_{c}}-{\frac{L+CR_{c}(R_{d}+R_{l})}{2LCR_{c}}}^{2}}.$
(36)

More information is provided in Ref. [44] for $v_{p}$ and $v_{s}$ which have
two formulations depending on the combination of $RLC$ parameters. For
simplicity here we only show one version of $v_{p}$ and $v_{s}$. We confirmed
that both Eqs. (31) and (34) can accurately calculate the admittance within
the considered $RLC$ parameters as seen in Fig. 9(e) and (f).

##  Appendix E FZP Simulations

The simulation environment for the FZPs based on ideal PEC layers or
equivalent admittance sheets is depicted in Fig. 10(a), while that for the FZP
based on rigorous waveform-selective metasurface is shown in Fig. 10(b). The
simulations are performed only at the operating frequency of 4.2 GHz. In these
simulations, we only consider 1D horizontal profiles of the FZPs along the
x-axis by applying PEC boundaries to the y-axis directions. Waveguide port
excitation is assigned with one of the z-axis directions with a minimum
distance of $z_{w1}=200$ mm (${\sim}3\lambda$) from the metasurface. All the
other boundaries are set to be radiation boundaries. The simulation area in
front of the metasurface (positive $z-$axis) is set to a distance $z_{w2}=800$
mm (${\sim}11\lambda$).

In the admittance sheet model, impedance boundary conditions are used to
approximate the time-varying admittance only for three discretized instants.
This simulation is similar to the single meta-atom simulation as explained in
Subsec. II.4. However, here the metasurface is modeled to have inhomogeneous
scattering profiles over the horizontal x-axis. Using this full-wave
simulation of the admittance sheet model, we obtain scattered field profiles
where a focusing effect can be observed (see Fig. 7(a) and (b)).

In the full FZP based on the rigorous waveform-selective metasurface, the co-
simulation method explained in Appendix A is used. The meta-atoms here are
realistic metallic (PEC) slits with lumped ports. Note that in the full-wave
simulation there is no variation in either the lumped ports or the slit
dimensions. The variation to realize the inhomogeneous temporal profile is
made inside the circuit simulation by implementing the corresponding circuit
configuration, i.e., MA1 to MA5, where the detailed circuit parameters are
presented in Table 1. Although the full FZP consists of 42 cells in total, the
center 12 cells have no temporal variation and thus are left blank. To observe
the time-varying scattered field, a monopole probe is used with a variable
$z$-axis coordinate at the center of the metasurface since the co-simulation
method adopted cannot produce the transient field profile. As illustrated in
Fig. 10(c), the monopole is a metallic (copper) cylinder with a length of 16
mm and a diameter of 1 mm. At one of its corner, a lumped port (1 mm $\times$
1 mm) is connected. The circuit simulation of the full FZP is depicted in Fig.
10(d). The electromagnetic simulation result is imported to the circuit
simulator of ANSYS Electronics Desktop (2020 R2). After the necessary
components are connected via the lumped ports, we run a transient simulation
over a 5-$\mu$s time span. Note that the input power here is set to 30 dBm to
ensure that the incident fields generate a sufficiently large potential across
diodes so that the diodes are turned on. The voltage received by the monopole
probe ($v_{pr}$) is calculated for three distances corresponding to three
simulations with $z_{pr}=$500, 300, and 100 mm. The voltages are squared and
averaged ($[v_{pr}^{2}]_{avg}$) over 20 times wave period ($20\times 0.24$ ns)
in Fig. 10(e) and are normalized to be compared with the results in Fig. 7(c).

## References

  * Alu  _et al._ [2007] A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern, Phys. Rev. B 75, 155410 (2007).
  * Pfeiffer and Grbic [2013] C. Pfeiffer and A. Grbic, Metamaterial huygens’ surfaces: tailoring wave fronts with reflectionless sheets, Phys. Rev. Lett. 110, 197401 (2013).
  * Ding  _et al._ [2015] X. Ding, F. Monticone, K. Zhang, L. Zhang, D. Gao, S. N. Burokur, A. de Lustrac, Q. Wu, C.-W. Qiu, and A. Alu, Ultrathin pancharatnam–berry metasurface with maximal cross-polarization efficiency, Adv. Mater. 27, 1195 (2015).
  * Yuan  _et al._ [2020] Y. Yuan, S. Sun, Y. Chen, K. Zhang, X. Ding, B. Ratni, Q. Wu, S. N. Burokur, and C.-W. Qiu, A fully phase-modulated metasurface as an energy-controllable circular polarization router, Advanced Science 7, 2001437 (2020).
  * Zhang  _et al._ [2021] K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, Generating dual-polarized vortex beam by detour phase: From phase gradient metasurfaces to metagratings, IEEE Transactions on Microwave Theory and Techniques 70, 200 (2021).
  * Ramaccia  _et al._ [2019] D. Ramaccia, D. L. Sounas, A. Alu, A. Toscano, and F. Bilotti, Phase-induced frequency conversion and doppler effect with time-modulated metasurfaces, IEEE Trans. Antennas Propag. 68, 1607 (2019).
  * Hadad  _et al._ [2015] Y. Hadad, D. L. Sounas, and A. Alu, Space-time gradient metasurfaces, Phys. Rev. B 92, 100304 (2015).
  * Taravati and Eleftheriades [2019] S. Taravati and G. V. Eleftheriades, Generalized space-time-periodic diffraction gratings: Theory and applications, Phys. Rev. App. 12, 024026 (2019).
  * Zhang  _et al._ [2019] L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, Breaking reciprocity with space-time-coding digital metasurfaces, Adv. Mater. 31, 1904069 (2019).
  * F. Imani and Smith [2020] M. F. Imani and D. R. Smith, Temporal microwave ghost imaging using a reconfigurable disordered cavity, Appl. Phy. Lett. 116, 054102 (2020).
  * Zhao  _et al._ [2019] J. Zhao, X. Yang, J. Y. Dai, Q. Cheng, X. Li, N. H. Qi, J. C. Ke, G. D. Bai, S. Liu, S. Jin, _et al._ , Programmable time-domain digital-coding metasurface for non-linear harmonic manipulation and new wireless communication systems, Natl. Sci. Rev. 6, 231 (2019).
  * Hall  _et al._ [2021] L. A. Hall, M. Yessenov, S. A. Ponomarenko, and A. F. Abouraddy, The space–time talbot effect, APL Photonics 6, 056105 (2021).
  * Liu  _et al._ [2018] M. Liu, D. A. Powell, Y. Zarate, and I. V. Shadrivov, Huygens’ metadevices for parametric waves, Phys. Rev. X 8, 031077 (2018).
  * Shaltout  _et al._ [2019] A. M. Shaltout, K. G. Lagoudakis, J. van de Groep, S. J. Kim, J. Vučković, V. M. Shalaev, and M. L. Brongersma, Spatiotemporal light control with frequency-gradient metasurfaces, Science 365, 374 (2019).
  * Salary  _et al._ [2019] M. M. Salary, S. Farazi, and H. Mosallaei, A dynamically modulated all-dielectric metasurface doublet for directional harmonic generation and manipulation in transmission, Adv. Opt. Mater. 7, 1900843 (2019).
  * Sedeh  _et al._ [2020] H. B. Sedeh, M. M. Salary, and H. Mosallaei, Time-varying optical vortices enabled by time-modulated metasurfaces, Nanophotonics 9, 2957 (2020).
  * Zhang  _et al._ [2022] N. Zhang, K. Chen, Q. Hu, J. Zhao, J. Zhao, T. Jiang, and Y. Feng, Spatiotemporal metasurface to control electromagnetic wave scattering, Phys. Rev. App. 17, 054001 (2022).
  * Hu  _et al._ [2022] Q. Hu, J. Zhao, K. Chen, K. Qu, W. Yang, J. Zhao, T. Jiang, and Y. Feng, An intelligent programmable omni-metasurface, Laser & Photonics Reviews 16, 2100718 (2022).
  * Caloz and Deck-Leger [2019] C. Caloz and Z.-L. Deck-Leger, Spacetime metamaterials—part ii: theory and applications, IEEE Trans. Antennas Propag. 68, 1583 (2019).
  * Caloz and Deck-Léger [2019] C. Caloz and Z.-L. Deck-Léger, Spacetime metamaterials—part i: general concepts, IEEE Trans. Antennas Propag. 68, 1569 (2019).
  * Taravati and Eleftheriades [2022] S. Taravati and G. V. Eleftheriades, Microwave space-time-modulated metasurfaces, ACS Photonics 9, 305 (2022).
  * Tang  _et al._ [2019] W. Tang, J. Y. Dai, M. Chen, X. Li, Q. Cheng, S. Jin, K.-K. Wong, and T. J. Cui, Programmable metasurface-based rf chain-free 8psk wireless transmitter, Electron. Lett. 55, 417 (2019).
  * Li  _et al._ [2017a] L. Li, T. Jun Cui, W. Ji, S. Liu, J. Ding, X. Wan, Y. Bo Li, M. Jiang, C.-W. Qiu, and S. Zhang, Electromagnetic reprogrammable coding-metasurface holograms, Nat. Commun. 8, 1 (2017a).
  * Denz  _et al._ [2010] C. Denz, S. Flach, Y. S. Kivshar, _et al._ ,  _Nonlinearities in periodic structures and metamaterials_ , Vol. 150 (Springer Berlin, Heidelberg, 2010).
  * Wakatsuchi  _et al._ [2013a] H. Wakatsuchi, S. Kim, J. J. Rushton, and D. F. Sievenpiper, Circuit-based nonlinear metasurface absorbers for high power surface currents, App. Phys. Lett. 102, 214103 (2013a).
  * Kiani  _et al._ [2020a] M. Kiani, A. Momeni, M. Tayarani, and C. Ding, Spatial wave control using a self-biased nonlinear metasurface at microwave frequencies, Opt. Express 28, 35128 (2020a).
  * Kiani  _et al._ [2020b] M. Kiani, M. Tayarani, A. Momeni, H. Rajabalipanah, and A. Abdolali, Self-biased tri-state power-multiplexed digital metasurface operating at microwave frequencies, Opt. Express 28, 5410 (2020b).
  * Quevedo-Teruel  _et al._ [2019] O. Quevedo-Teruel, H. Chen, A. Díaz-Rubio, G. Gok, A. Grbic, G. Minatti, E. Martini, S. Maci, G. V. Eleftheriades, M. Chen, _et al._ , Roadmap on metasurfaces, J. Opt. 21, 073002 (2019).
  * Chen and Alù [2010] P.-Y. Chen and A. Alù, Optical nanoantenna arrays loaded with nonlinear materials, Phys. Rev. B 82, 235405 (2010).
  * Shadrivov  _et al._ [2012] I. V. Shadrivov, P. V. Kapitanova, S. I. Maslovski, and Y. S. Kivshar, Metamaterials controlled with light, Phys. Rev. Lett. 109, 083902 (2012).
  * Luo  _et al._ [2015] Z. Luo, X. Chen, J. Long, R. Quarfoth, and D. Sievenpiper, Self-focusing of electromagnetic surface waves on a nonlinear impedance surface, App. Phys. Lett. 106, 211102 (2015).
  * Luo  _et al._ [2019] Z. Luo, Q. Wang, X. G. Zhang, _et al._ , Intensity-dependent metasurface with digitally reconfigurable distribution of nonlinearity, Adv. Opt. Mater. 7, 1900792 (2019).
  * Kim  _et al._ [2020] S. Kim, A. Li, J. Lee, and D. F. Sievenpiper, Active self-tuning metasurface with enhanced absorbing frequency range for suppression of high-power surface currents, IEEE Trans. Antennas Propag. 69, 2759 (2020).
  * Li  _et al._ [2017b] A. Li, S. Kim, Y. Luo, Y. Li, J. Long, and D. F. Sievenpiper, High-power transistor-based tunable and switchable metasurface absorber, IEEE Trans. Microw. Theory Tech. 65, 2810 (2017b).
  * Luo  _et al._ [2022] Z. Luo, X. Ren, L. Zhou, Y. Chen, Q. Cheng, H. F. Ma, and T. J. Cui, A high-performance nonlinear metasurface for spatial-wave absorption, Adv. Funct. Mater. 32, 2109544 (2022).
  * Homma  _et al._ [2022] H. Homma, M. R. Akram, A. A. Fathnan, J. Lee, C. Christopoulos, and H. Wakatsuchi, Anisotropic impedance surfaces activated by incident waveform, Nanophotonics 11, 1989 (2022).
  * Wakatsuchi  _et al._ [2015] H. Wakatsuchi, D. Anzai, J. J. Rushton, F. Gao, S. Kim, and D. F. Sievenpiper, Waveform selectivity at the same frequency, Sci. Rep. 5, 1 (2015).
  * Wakatsuchi  _et al._ [2013b] H. Wakatsuchi, S. Kim, J. J. Rushton, and D. F. Sievenpiper, Waveform-dependent absorbing metasurfaces, Phy. Rev. Lett. 111, 245501 (2013b).
  * Wakatsuchi  _et al._ [2019] H. Wakatsuchi, J. Long, and D. F. Sievenpiper, Waveform selective surfaces, Adv. Funct. Mater. 29, 1806386 (2019).
  * Ushikoshi  _et al._ [2020] D. Ushikoshi, M. Tanikawa, K. Asano, K. Sanji, M. Ikeda, D. Anzai, and H. Wakatsuchi, Experimental demonstration of waveform-selective metasurface varying wireless communication characteristics at the same frequency band of 2.4 ghz, Electron. Lett. 56, 160 (2020).
  * Vellucci  _et al._ [2019a] S. Vellucci, A. Monti, M. Barbuto, A. Toscano, and F. Bilotti, Waveform-selective mantle cloaks for intelligent antennas, IEEE Trans. Antennas Propag. 68, 1717 (2019a).
  * Wakatsuchi [2015] H. Wakatsuchi, Time-domain filtering of metasurfaces, Sci. Rep. 5, 1 (2015).
  * Vellucci  _et al._ [2019b] S. Vellucci, A. Monti, M. Barbuto, A. Toscano, and F. Bilotti, Waveform-selective mantle cloaks for intelligent antennas, IEEE Trans. Antennas Propag. 68, 1717 (2019b).
  * Asano  _et al._ [2020] K. Asano, T. Nakasha, and H. Wakatsuchi, Simplified equivalent circuit approach for designing time-domain responses of waveform-selective metasurfaces, App. Phys. Lett. 116, 171603 (2020).
  * Holloway  _et al._ [2005] C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles, IEEE Trans. Electromagn. Compat. 47, 853 (2005).
  * Fowles [1989] G. R. Fowles,  _Introduction to modern optics_ (Dover Publications, New York, 1989).
  * Shams  _et al._ [2017] M. I. B. Shams, Z. Jiang, S. M. Rahman, L.-J. Cheng, J. L. Hesler, P. Fay, and L. Liu, A 740-ghz dynamic two-dimensional beam-steering and forming antenna based on photo-induced reconfigurable fresnel zone plates, IEEE Trans. Terahertz Sci. Technol. 7, 310 (2017).
  * Fathnan  _et al._ [2020a] A. A. Fathnan, M. Liu, and D. A. Powell, Achromatic huygens’ metalenses with deeply subwavelength thickness, Adv. Opt. Mater. 8, 2000754 (2020a).
  * Epstein and Eleftheriades [2016] A. Epstein and G. V. Eleftheriades, Huygens’ metasurfaces via the equivalence principle: design and applications, JOSA B 33, A31 (2016).
  * Monticone  _et al._ [2013] F. Monticone, N. M. Estakhri, and A. Alu, Full control of nanoscale optical transmission with a composite metascreen, Phys. Rev. Lett. 110, 203903 (2013).
  * Decker  _et al._ [2015] M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, High-efficiency dielectric huygens’ surfaces, Adv. Opt. Mater. 3, 813 (2015).
  * Boashash [2015] B. Boashash,  _Time-frequency signal analysis and processing: a comprehensive reference_ (Academic press, London, 2015).
  * Sejdić  _et al._ [2009] E. Sejdić, I. Djurović, and J. Jiang, Time–frequency feature representation using energy concentration: An overview of recent advances, Digit. Signal Process. 19, 153 (2009).
  * Fathnan  _et al._ [2020b] A. A. Fathnan, A. E. Olk, and D. A. Powell, Bandwidth limit and synthesis approach for single resonance ultrathin metasurfaces, J. Phys. D: Appl. Phys. 53, 495304 (2020b).
  * Olk and Powell [2019] A. E. Olk and D. A. Powell, Accurate metasurface synthesis incorporating near-field coupling effects, Phys. Rev. App. 11, 064007 (2019).
  * Wu and Zhang [2019] Q. Wu and R. Zhang, Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming, IEEE Trans. Wirel. Commun. 18, 5394 (2019).
  * Özdogan  _et al._ [2019] Ö. Özdogan, E. Björnson, and E. G. Larsson, Intelligent reflecting surfaces: Physics, propagation, and pathloss modeling, IEEE Wirel. Commun. 9, 581 (2019).
  * Sugiura  _et al._ [2021] S. Sugiura, Y. Kawai, T. Matsui, T. Lee, and H. Iizuka, Joint beam and polarization forming of intelligent reflecting surfaces for wireless communications, IEEE Trans. Veh. Technol. 70, 1648 (2021).
  * Dai  _et al._ [2020] L. Dai, B. Wang, M. Wang, X. Yang, J. Tan, S. Bi, S. Xu, F. Yang, Z. Chen, M. Di Renzo, _et al._ , Reconfigurable intelligent surface-based wireless communications: Antenna design, prototyping, and experimental results, IEEE Access 8, 45913 (2020).
  * Zhang  _et al._ [2018] L. Zhang, X. Q. Chen, S. Liu, Q. Zhang, J. Zhao, J. Y. Dai, G. D. Bai, X. Wan, Q. Cheng, G. Castaldi, _et al._ , Space-time-coding digital metasurfaces, Nat. Commun. 9, 1 (2018).
  * Pozar [1988] D. M. Pozar,  _Microwave engineering_ (Wiley, New York, 1988).